LMFDB - Embedded newform 1849.2.a.j.1.2

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:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ 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)

$f(q)$	$=$	$q-0.445042 q^{2} -1.80194 q^{3} -1.80194 q^{4} +0.801938 q^{5} +0.801938 q^{6} +3.04892 q^{7} +1.69202 q^{8} +0.246980 q^{9} +O(q^{10})$	\(q-0.445042 q^{2} -1.80194 q^{3} -1.80194 q^{4} +0.801938 q^{5} +0.801938 q^{6} +3.04892 q^{7} +1.69202 q^{8} +0.246980 q^{9} -0.356896 q^{10} +5.29590 q^{11} +3.24698 q^{12} +3.96077 q^{13} -1.35690 q^{14} -1.44504 q^{15} +2.85086 q^{16} -0.890084 q^{17} -0.109916 q^{18} +2.75302 q^{19} -1.44504 q^{20} -5.49396 q^{21} -2.35690 q^{22} -4.38404 q^{23} -3.04892 q^{24} -4.35690 q^{25} -1.76271 q^{26} +4.96077 q^{27} -5.49396 q^{28} +6.00000 q^{29} +0.643104 q^{30} -6.94869 q^{31} -4.65279 q^{32} -9.54288 q^{33} +0.396125 q^{34} +2.44504 q^{35} -0.445042 q^{36} +3.46681 q^{37} -1.22521 q^{38} -7.13706 q^{39} +1.35690 q^{40} +7.10992 q^{41} +2.44504 q^{42} -9.54288 q^{44} +0.198062 q^{45} +1.95108 q^{46} -8.91185 q^{47} -5.13706 q^{48} +2.29590 q^{49} +1.93900 q^{50} +1.60388 q^{51} -7.13706 q^{52} -2.07069 q^{53} -2.20775 q^{54} +4.24698 q^{55} +5.15883 q^{56} -4.96077 q^{57} -2.67025 q^{58} -0.862937 q^{59} +2.60388 q^{60} +1.38404 q^{61} +3.09246 q^{62} +0.753020 q^{63} -3.63102 q^{64} +3.17629 q^{65} +4.24698 q^{66} +6.71379 q^{67} +1.60388 q^{68} +7.89977 q^{69} -1.08815 q^{70} +16.5918 q^{71} +0.417895 q^{72} +12.4547 q^{73} -1.54288 q^{74} +7.85086 q^{75} -4.96077 q^{76} +16.1468 q^{77} +3.17629 q^{78} -7.85086 q^{79} +2.28621 q^{80} -9.67994 q^{81} -3.16421 q^{82} +7.36658 q^{83} +9.89977 q^{84} -0.713792 q^{85} -10.8116 q^{87} +8.96077 q^{88} -14.1075 q^{89} -0.0881460 q^{90} +12.0761 q^{91} +7.89977 q^{92} +12.5211 q^{93} +3.96615 q^{94} +2.20775 q^{95} +8.38404 q^{96} -9.37867 q^{97} -1.02177 q^{98} +1.30798 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} - q^{3} - q^{4} - 2 q^{5} - 2 q^{6} - 4 q^{9}+O(q^{10}) $ 3 * q - q^2 - q^3 - q^4 - 2 * q^5 - 2 * q^6 - 4 * q^9	$ 3 q - q^{2} - q^{3} - q^{4} - 2 q^{5} - 2 q^{6} - 4 q^{9} + 3 q^{10} + 2 q^{11} + 5 q^{12} - q^{13} - 4 q^{15} - 5 q^{16} - 2 q^{17} - q^{18} + 13 q^{19} - 4 q^{20} - 7 q^{21} - 3 q^{22} - 3 q^{23} - 9 q^{25} + 12 q^{26} + 2 q^{27} - 7 q^{28} + 18 q^{29} + 6 q^{30} + 11 q^{31} + 4 q^{32} - 10 q^{33} + 10 q^{34} + 7 q^{35} - q^{36} + 7 q^{37} - 2 q^{38} - 16 q^{39} + 22 q^{41} + 7 q^{42} - 10 q^{44} + 5 q^{45} + 15 q^{46} - 23 q^{47} - 10 q^{48} - 7 q^{49} - 4 q^{50} - 4 q^{51} - 16 q^{52} + 6 q^{53} + 11 q^{54} + 8 q^{55} + 7 q^{56} - 2 q^{57} - 6 q^{58} - 8 q^{59} - q^{60} - 6 q^{61} - 6 q^{62} + 7 q^{63} + 4 q^{64} + 17 q^{65} + 8 q^{66} + 12 q^{67} - 4 q^{68} + q^{69} - 7 q^{70} + 22 q^{71} + 7 q^{72} + 15 q^{73} + 14 q^{74} + 10 q^{75} - 2 q^{76} + 21 q^{77} + 17 q^{78} - 10 q^{79} + 15 q^{80} - 5 q^{81} + 2 q^{82} - 4 q^{83} + 7 q^{84} + 6 q^{85} - 6 q^{87} + 14 q^{88} - 2 q^{89} - 4 q^{90} + 21 q^{91} + q^{92} + 22 q^{93} - 4 q^{94} - 11 q^{95} + 15 q^{96} - 21 q^{97} + 9 q^{99}+O(q^{100}) $ 3 * q - q^2 - q^3 - q^4 - 2 * q^5 - 2 * q^6 - 4 * q^9 + 3 * q^10 + 2 * q^11 + 5 * q^12 - q^13 - 4 * q^15 - 5 * q^16 - 2 * q^17 - q^18 + 13 * q^19 - 4 * q^20 - 7 * q^21 - 3 * q^22 - 3 * q^23 - 9 * q^25 + 12 * q^26 + 2 * q^27 - 7 * q^28 + 18 * q^29 + 6 * q^30 + 11 * q^31 + 4 * q^32 - 10 * q^33 + 10 * q^34 + 7 * q^35 - q^36 + 7 * q^37 - 2 * q^38 - 16 * q^39 + 22 * q^41 + 7 * q^42 - 10 * q^44 + 5 * q^45 + 15 * q^46 - 23 * q^47 - 10 * q^48 - 7 * q^49 - 4 * q^50 - 4 * q^51 - 16 * q^52 + 6 * q^53 + 11 * q^54 + 8 * q^55 + 7 * q^56 - 2 * q^57 - 6 * q^58 - 8 * q^59 - q^60 - 6 * q^61 - 6 * q^62 + 7 * q^63 + 4 * q^64 + 17 * q^65 + 8 * q^66 + 12 * q^67 - 4 * q^68 + q^69 - 7 * q^70 + 22 * q^71 + 7 * q^72 + 15 * q^73 + 14 * q^74 + 10 * q^75 - 2 * q^76 + 21 * q^77 + 17 * q^78 - 10 * q^79 + 15 * q^80 - 5 * q^81 + 2 * q^82 - 4 * q^83 + 7 * q^84 + 6 * q^85 - 6 * q^87 + 14 * q^88 - 2 * q^89 - 4 * q^90 + 21 * q^91 + q^92 + 22 * q^93 - 4 * q^94 - 11 * q^95 + 15 * q^96 - 21 * q^97 + 9 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.445042	−0.314692	−0.157346	−	0.987544i	$-0.550294\pi$
$2$	−0.445042	−0.314692	−0.157346	+	0.987544i	$0.550294\pi$
$3$	−1.80194	−1.04035	−0.520175	−	0.854060i	$-0.674133\pi$
$3$	−1.80194	−1.04035	−0.520175	+	0.854060i	$0.674133\pi$
$4$	−1.80194	−0.900969
$5$	0.801938	0.358637	0.179319	−	0.983791i	$-0.442611\pi$
$5$	0.801938	0.358637	0.179319	+	0.983791i	$0.442611\pi$
$6$	0.801938	0.327390
$7$	3.04892	1.15238	0.576191	−	0.817315i	$-0.304538\pi$
$7$	3.04892	1.15238	0.576191	+	0.817315i	$0.304538\pi$
$8$	1.69202	0.598220
$9$	0.246980	0.0823265
$10$	−0.356896	−0.112860
$11$	5.29590	1.59677	0.798387	−	0.602145i	$-0.205687\pi$
$11$	5.29590	1.59677	0.798387	+	0.602145i	$0.205687\pi$
$12$	3.24698	0.937322
$13$	3.96077	1.09852	0.549260	−	0.835651i	$-0.314910\pi$
$13$	3.96077	1.09852	0.549260	+	0.835651i	$0.314910\pi$
$14$	−1.35690	−0.362646
$15$	−1.44504	−0.373108
$16$	2.85086	0.712714
$17$	−0.890084	−0.215877	−0.107939	−	0.994158i	$-0.534425\pi$
$17$	−0.890084	−0.215877	−0.107939	+	0.994158i	$0.534425\pi$
$18$	−0.109916	−0.0259075
$19$	2.75302	0.631586	0.315793	−	0.948828i	$-0.397729\pi$
$19$	2.75302	0.631586	0.315793	+	0.948828i	$0.397729\pi$
$20$	−1.44504	−0.323121
$21$	−5.49396	−1.19888
$22$	−2.35690	−0.502492
$23$	−4.38404	−0.914136	−0.457068	−	0.889432i	$-0.651100\pi$
$23$	−4.38404	−0.914136	−0.457068	+	0.889432i	$0.651100\pi$
$24$	−3.04892	−0.622358
$25$	−4.35690	−0.871379
$26$	−1.76271	−0.345696
$27$	4.96077	0.954701
$28$	−5.49396	−1.03826
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0.643104	0.117414
$31$	−6.94869	−1.24802	−0.624011	−	0.781416i	$-0.714498\pi$
$31$	−6.94869	−1.24802	−0.624011	+	0.781416i	$0.714498\pi$
$32$	−4.65279	−0.822505
$33$	−9.54288	−1.66120
$34$	0.396125	0.0679348
$35$	2.44504	0.413288
$36$	−0.445042	−0.0741736
$37$	3.46681	0.569940	0.284970	−	0.958536i	$-0.408016\pi$
$37$	3.46681	0.569940	0.284970	+	0.958536i	$0.408016\pi$
$38$	−1.22521	−0.198755
$39$	−7.13706	−1.14284
$40$	1.35690	0.214544
$41$	7.10992	1.11038	0.555191	−	0.831723i	$-0.312645\pi$
$41$	7.10992	1.11038	0.555191	+	0.831723i	$0.312645\pi$
$42$	2.44504	0.377278
$43$	0	0
$43$	0	0
$44$	−9.54288	−1.43864
$45$	0.198062	0.0295254
$46$	1.95108	0.287671
$47$	−8.91185	−1.29993	−0.649964	−	0.759965i	$-0.725216\pi$
$47$	−8.91185	−1.29993	−0.649964	+	0.759965i	$0.725216\pi$
$48$	−5.13706	−0.741471
$49$	2.29590	0.327985
$50$	1.93900	0.274216
$51$	1.60388	0.224587
$52$	−7.13706	−0.989733
$53$	−2.07069	−0.284431	−0.142215	−	0.989836i	$-0.545423\pi$
$53$	−2.07069	−0.284431	−0.142215	+	0.989836i	$0.545423\pi$
$54$	−2.20775	−0.300437
$55$	4.24698	0.572663
$56$	5.15883	0.689378
$57$	−4.96077	−0.657070
$58$	−2.67025	−0.350621
$59$	−0.862937	−0.112345	−0.0561724	−	0.998421i	$-0.517890\pi$
$59$	−0.862937	−0.112345	−0.0561724	+	0.998421i	$0.517890\pi$
$60$	2.60388	0.336159
$61$	1.38404	0.177209	0.0886043	−	0.996067i	$-0.471759\pi$
$61$	1.38404	0.177209	0.0886043	+	0.996067i	$0.471759\pi$
$62$	3.09246	0.392743
$63$	0.753020	0.0948717
$64$	−3.63102	−0.453878
$65$	3.17629	0.393971
$66$	4.24698	0.522767
$67$	6.71379	0.820220	0.410110	−	0.912036i	$-0.365490\pi$
$67$	6.71379	0.820220	0.410110	+	0.912036i	$0.365490\pi$
$68$	1.60388	0.194498
$69$	7.89977	0.951021
$70$	−1.08815	−0.130058
$71$	16.5918	1.96908	0.984542	−	0.175150i	$-0.0560409\pi$
$71$	16.5918	1.96908	0.984542	+	0.175150i	$0.0560409\pi$
$72$	0.417895	0.0492494
$73$	12.4547	1.45772	0.728858	−	0.684665i	$-0.240051\pi$
$73$	12.4547	1.45772	0.728858	+	0.684665i	$0.240051\pi$
$74$	−1.54288	−0.179356
$75$	7.85086	0.906539
$76$	−4.96077	−0.569039
$77$	16.1468	1.84009
$78$	3.17629	0.359644
$79$	−7.85086	−0.883290	−0.441645	−	0.897190i	$-0.645605\pi$
$79$	−7.85086	−0.883290	−0.441645	+	0.897190i	$0.645605\pi$
$80$	2.28621	0.255606
$81$	−9.67994	−1.07555
$82$	−3.16421	−0.349429
$83$	7.36658	0.808588	0.404294	−	0.914629i	$-0.367517\pi$
$83$	7.36658	0.808588	0.404294	+	0.914629i	$0.367517\pi$
$84$	9.89977	1.08015
$85$	−0.713792	−0.0774216
$86$	0	0
$87$	−10.8116	−1.15913
$88$	8.96077	0.955221
$89$	−14.1075	−1.49539	−0.747697	−	0.664040i	$-0.768841\pi$
$89$	−14.1075	−1.49539	−0.747697	+	0.664040i	$0.768841\pi$
$90$	−0.0881460	−0.00929140
$91$	12.0761	1.26592
$92$	7.89977	0.823608
$93$	12.5211	1.29838
$94$	3.96615	0.409077
$95$	2.20775	0.226510
$96$	8.38404	0.855693
$97$	−9.37867	−0.952259	−0.476130	−	0.879375i	$-0.657961\pi$
$97$	−9.37867	−0.952259	−0.476130	+	0.879375i	$0.657961\pi$
$98$	−1.02177	−0.103214
$99$	1.30798	0.131457
$100$	7.85086	0.785086
$101$	12.5211	1.24590	0.622948	−	0.782263i	$-0.285935\pi$
$101$	12.5211	1.24590	0.622948	+	0.782263i	$0.285935\pi$
$102$	−0.713792	−0.0706759
$103$	4.38404	0.431973	0.215986	−	0.976396i	$-0.430703\pi$
$103$	4.38404	0.431973	0.215986	+	0.976396i	$0.430703\pi$
$104$	6.70171	0.657157
$105$	−4.40581	−0.429963
$106$	0.921543	0.0895081
$107$	0.643104	0.0621712	0.0310856	−	0.999517i	$-0.490104\pi$
$107$	0.643104	0.0621712	0.0310856	+	0.999517i	$0.490104\pi$
$108$	−8.93900	−0.860156
$109$	−1.87800	−0.179880	−0.0899400	−	0.995947i	$-0.528668\pi$
$109$	−1.87800	−0.179880	−0.0899400	+	0.995947i	$0.528668\pi$
$110$	−1.89008	−0.180212
$111$	−6.24698	−0.592937
$112$	8.69202	0.821319
$113$	−12.6286	−1.18800	−0.594001	−	0.804464i	$-0.702452\pi$
$113$	−12.6286	−1.18800	−0.594001	+	0.804464i	$0.702452\pi$
$114$	2.20775	0.206775
$115$	−3.51573	−0.327843
$116$	−10.8116	−1.00383
$117$	0.978230	0.0904374
$118$	0.384043	0.0353540
$119$	−2.71379	−0.248773
$120$	−2.44504	−0.223201
$121$	17.0465	1.54968
$122$	−0.615957	−0.0557661
$123$	−12.8116	−1.15519
$124$	12.5211	1.12443
$125$	−7.50365	−0.671147
$126$	−0.335126	−0.0298554
$127$	7.21744	0.640444	0.320222	−	0.947342i	$-0.396242\pi$
$127$	7.21744	0.640444	0.320222	+	0.947342i	$0.396242\pi$
$128$	10.9215	0.965337
$129$	0	0
$130$	−1.41358	−0.123979
$131$	9.13169	0.797839	0.398920	−	0.916986i	$-0.369385\pi$
$131$	9.13169	0.797839	0.398920	+	0.916986i	$0.369385\pi$
$132$	17.1957	1.49669
$133$	8.39373	0.727829
$134$	−2.98792	−0.258117
$135$	3.97823	0.342392
$136$	−1.50604	−0.129142
$137$	6.09246	0.520514	0.260257	−	0.965539i	$-0.416193\pi$
$137$	6.09246	0.520514	0.260257	+	0.965539i	$0.416193\pi$
$138$	−3.51573	−0.299279
$139$	7.14675	0.606180	0.303090	−	0.952962i	$-0.401982\pi$
$139$	7.14675	0.606180	0.303090	+	0.952962i	$0.401982\pi$
$140$	−4.40581	−0.372359
$141$	16.0586	1.35238
$142$	−7.38404	−0.619655
$143$	20.9758	1.75409
$144$	0.704103	0.0586753
$145$	4.81163	0.399584
$146$	−5.54288	−0.458732
$147$	−4.13706	−0.341219
$148$	−6.24698	−0.513499
$149$	4.44504	0.364152	0.182076	−	0.983284i	$-0.441718\pi$
$149$	4.44504	0.364152	0.182076	+	0.983284i	$0.441718\pi$
$150$	−3.49396	−0.285281
$151$	−14.2010	−1.15566	−0.577832	−	0.816155i	$-0.696101\pi$
$151$	−14.2010	−1.15566	−0.577832	+	0.816155i	$0.696101\pi$
$152$	4.65817	0.377827
$153$	−0.219833	−0.0177724
$154$	−7.18598	−0.579063
$155$	−5.57242	−0.447587
$156$	12.8605	1.02967
$157$	7.44265	0.593988	0.296994	−	0.954879i	$-0.404016\pi$
$157$	7.44265	0.593988	0.296994	+	0.954879i	$0.404016\pi$
$158$	3.49396	0.277964
$159$	3.73125	0.295907
$160$	−3.73125	−0.294981
$161$	−13.3666	−1.05343
$162$	4.30798	0.338467
$163$	3.34721	0.262174	0.131087	−	0.991371i	$-0.458153\pi$
$163$	3.34721	0.262174	0.131087	+	0.991371i	$0.458153\pi$
$164$	−12.8116	−1.00042
$165$	−7.65279	−0.595769
$166$	−3.27844	−0.254456
$167$	−2.03684	−0.157615	−0.0788075	−	0.996890i	$-0.525111\pi$
$167$	−2.03684	−0.157615	−0.0788075	+	0.996890i	$0.525111\pi$
$168$	−9.29590	−0.717194
$169$	2.68771	0.206747
$170$	0.317667	0.0243640
$171$	0.679940	0.0519963
$172$	0	0
$173$	1.02715	0.0780925	0.0390463	−	0.999237i	$-0.487568\pi$
$173$	1.02715	0.0780925	0.0390463	+	0.999237i	$0.487568\pi$
$174$	4.81163	0.364768
$175$	−13.2838	−1.00416
$176$	15.0978	1.13804
$177$	1.55496	0.116878
$178$	6.27844	0.470589
$179$	−9.75600	−0.729198	−0.364599	−	0.931165i	$-0.618794\pi$
$179$	−9.75600	−0.729198	−0.364599	+	0.931165i	$0.618794\pi$
$180$	−0.356896	−0.0266014
$181$	8.39075	0.623679	0.311840	−	0.950135i	$-0.399055\pi$
$181$	8.39075	0.623679	0.311840	+	0.950135i	$0.399055\pi$
$182$	−5.37435	−0.398374
$183$	−2.49396	−0.184359
$184$	−7.41789	−0.546854
$185$	2.78017	0.204402
$186$	−5.57242	−0.408589
$187$	−4.71379	−0.344707
$188$	16.0586	1.17119
$189$	15.1250	1.10018
$190$	−0.982542	−0.0712811
$191$	13.6582	0.988270	0.494135	−	0.869385i	$-0.335485\pi$
$191$	13.6582	0.988270	0.494135	+	0.869385i	$0.335485\pi$
$192$	6.54288	0.472191
$193$	4.08815	0.294271	0.147136	−	0.989116i	$-0.452995\pi$
$193$	4.08815	0.294271	0.147136	+	0.989116i	$0.452995\pi$
$194$	4.17390	0.299668
$195$	−5.72348	−0.409867
$196$	−4.13706	−0.295505
$197$	8.58211	0.611450	0.305725	−	0.952120i	$-0.401101\pi$
$197$	8.58211	0.611450	0.305725	+	0.952120i	$0.401101\pi$
$198$	−0.582105	−0.0413684
$199$	5.70948	0.404734	0.202367	−	0.979310i	$-0.435137\pi$
$199$	5.70948	0.404734	0.202367	+	0.979310i	$0.435137\pi$
$200$	−7.37196	−0.521276
$201$	−12.0978	−0.853315
$202$	−5.57242	−0.392074
$203$	18.2935	1.28395
$204$	−2.89008	−0.202346
$205$	5.70171	0.398225
$206$	−1.95108	−0.135938
$207$	−1.08277	−0.0752577
$208$	11.2916	0.782931
$209$	14.5797	1.00850
$210$	1.96077	0.135306
$211$	19.1564	1.31878	0.659392	−	0.751799i	$-0.270814\pi$
$211$	19.1564	1.31878	0.659392	+	0.751799i	$0.270814\pi$
$212$	3.73125	0.256263
$213$	−29.8974	−2.04853
$214$	−0.286208	−0.0195648
$215$	0	0
$216$	8.39373	0.571121
$217$	−21.1860	−1.43820
$218$	0.835790	0.0566068
$219$	−22.4426	−1.51653
$220$	−7.65279	−0.515951
$221$	−3.52542	−0.237145
$222$	2.78017	0.186593
$223$	−0.103211	−0.00691155	−0.00345578	−	0.999994i	$-0.501100\pi$
$223$	−0.103211	−0.00691155	−0.00345578	+	0.999994i	$0.501100\pi$
$224$	−14.1860	−0.947841
$225$	−1.07606	−0.0717376
$226$	5.62027	0.373855
$227$	−13.3545	−0.886369	−0.443185	−	0.896430i	$-0.646151\pi$
$227$	−13.3545	−0.886369	−0.443185	+	0.896430i	$0.646151\pi$
$228$	8.93900	0.592000
$229$	−28.3870	−1.87587	−0.937934	−	0.346814i	$-0.887263\pi$
$229$	−28.3870	−1.87587	−0.937934	+	0.346814i	$0.887263\pi$
$230$	1.56465	0.103170
$231$	−29.0954	−1.91434
$232$	10.1521	0.666520
$233$	−10.9638	−0.718259	−0.359130	−	0.933288i	$-0.616926\pi$
$233$	−10.9638	−0.718259	−0.359130	+	0.933288i	$0.616926\pi$
$234$	−0.435353	−0.0284599
$235$	−7.14675	−0.466203
$236$	1.55496	0.101219
$237$	14.1468	0.918930
$238$	1.20775	0.0782869
$239$	19.4209	1.25623	0.628116	−	0.778120i	$-0.283826\pi$
$239$	19.4209	1.25623	0.628116	+	0.778120i	$0.283826\pi$
$240$	−4.11960	−0.265919
$241$	3.49396	0.225066	0.112533	−	0.993648i	$-0.464104\pi$
$241$	3.49396	0.225066	0.112533	+	0.993648i	$0.464104\pi$
$242$	−7.58642	−0.487673
$243$	2.56033	0.164246
$244$	−2.49396	−0.159659
$245$	1.84117	0.117628
$246$	5.70171	0.363528
$247$	10.9041	0.693810
$248$	−11.7573	−0.746591
$249$	−13.2741	−0.841214
$250$	3.33944	0.211205
$251$	1.25129	0.0789808	0.0394904	−	0.999220i	$-0.487427\pi$
$251$	1.25129	0.0789808	0.0394904	+	0.999220i	$0.487427\pi$
$252$	−1.35690	−0.0854764
$253$	−23.2174	−1.45967
$254$	−3.21206	−0.201543
$255$	1.28621	0.0805455
$256$	2.40150	0.150094
$257$	7.74094	0.482866	0.241433	−	0.970417i	$-0.422383\pi$
$257$	7.74094	0.482866	0.241433	+	0.970417i	$0.422383\pi$
$258$	0	0
$259$	10.5700	0.656789
$260$	−5.72348	−0.354955
$261$	1.48188	0.0917259
$262$	−4.06398	−0.251074
$263$	−1.27844	−0.0788319	−0.0394159	−	0.999223i	$-0.512550\pi$
$263$	−1.27844	−0.0788319	−0.0394159	+	0.999223i	$0.512550\pi$
$264$	−16.1468	−0.993764
$265$	−1.66056	−0.102008
$266$	−3.73556	−0.229042
$267$	25.4209	1.55573
$268$	−12.0978	−0.738993
$269$	25.3773	1.54728	0.773642	−	0.633623i	$-0.218433\pi$
$269$	25.3773	1.54728	0.773642	+	0.633623i	$0.218433\pi$
$270$	−1.77048	−0.107748
$271$	−8.95108	−0.543740	−0.271870	−	0.962334i	$-0.587642\pi$
$271$	−8.95108	−0.543740	−0.271870	+	0.962334i	$0.587642\pi$
$272$	−2.53750	−0.153859
$273$	−21.7603	−1.31699
$274$	−2.71140	−0.163802
$275$	−23.0737	−1.39139
$276$	−14.2349	−0.856840
$277$	6.69740	0.402408	0.201204	−	0.979549i	$-0.435515\pi$
$277$	6.69740	0.402408	0.201204	+	0.979549i	$0.435515\pi$
$278$	−3.18060	−0.190760
$279$	−1.71618	−0.102745
$280$	4.13706	0.247237
$281$	−22.3666	−1.33428	−0.667139	−	0.744933i	$-0.732481\pi$
$281$	−22.3666	−1.33428	−0.667139	+	0.744933i	$0.732481\pi$
$282$	−7.14675	−0.425583
$283$	29.2078	1.73622	0.868110	−	0.496371i	$-0.165335\pi$
$283$	29.2078	1.73622	0.868110	+	0.496371i	$0.165335\pi$
$284$	−29.8974	−1.77408
$285$	−3.97823	−0.235650
$286$	−9.33513	−0.551998
$287$	21.6775	1.27959
$288$	−1.14914	−0.0677140
$289$	−16.2078	−0.953397
$290$	−2.14138	−0.125746
$291$	16.8998	0.990682
$292$	−22.4426	−1.31336
$293$	−9.65817	−0.564236	−0.282118	−	0.959380i	$-0.591037\pi$
$293$	−9.65817	−0.564236	−0.282118	+	0.959380i	$0.591037\pi$
$294$	1.84117	0.107379
$295$	−0.692021	−0.0402910
$296$	5.86592	0.340950
$297$	26.2717	1.52444
$298$	−1.97823	−0.114596
$299$	−17.3642	−1.00420
$300$	−14.1468	−0.816763
$301$	0	0
$302$	6.32006	0.363679
$303$	−22.5623	−1.29617
$304$	7.84846	0.450140
$305$	1.10992	0.0635536
$306$	0.0978347	0.00559284
$307$	−12.9855	−0.741123	−0.370562	−	0.928808i	$-0.620835\pi$
$307$	−12.9855	−0.741123	−0.370562	+	0.928808i	$0.620835\pi$
$308$	−29.0954	−1.65787
$309$	−7.89977	−0.449402
$310$	2.47996	0.140852
$311$	−5.91723	−0.335535	−0.167768	−	0.985827i	$-0.553656\pi$
$311$	−5.91723	−0.335535	−0.167768	+	0.985827i	$0.553656\pi$
$312$	−12.0761	−0.683673
$313$	−20.5676	−1.16255	−0.581276	−	0.813707i	$-0.697446\pi$
$313$	−20.5676	−1.16255	−0.581276	+	0.813707i	$0.697446\pi$
$314$	−3.31229	−0.186923
$315$	0.603875	0.0340245
$316$	14.1468	0.795817
$317$	18.3623	1.03133	0.515664	−	0.856791i	$-0.327545\pi$
$317$	18.3623	1.03133	0.515664	+	0.856791i	$0.327545\pi$
$318$	−1.66056	−0.0931197
$319$	31.7754	1.77908
$320$	−2.91185	−0.162778
$321$	−1.15883	−0.0646798
$322$	5.94869	0.331508
$323$	−2.45042	−0.136345
$324$	17.4426	0.969036
$325$	−17.2567	−0.957228
$326$	−1.48965	−0.0825039
$327$	3.38404	0.187138
$328$	12.0301	0.664253
$329$	−27.1715	−1.49801
$330$	3.40581	0.187484
$331$	−1.64848	−0.0906087	−0.0453044	−	0.998973i	$-0.514426\pi$
$331$	−1.64848	−0.0906087	−0.0453044	+	0.998973i	$0.514426\pi$
$332$	−13.2741	−0.728512
$333$	0.856232	0.0469212
$334$	0.906477	0.0496002
$335$	5.38404	0.294162
$336$	−15.6625	−0.854458
$337$	29.1444	1.58759	0.793797	−	0.608183i	$-0.208101\pi$
$337$	29.1444	1.58759	0.793797	+	0.608183i	$0.208101\pi$
$338$	−1.19614	−0.0650616
$339$	22.7560	1.23594
$340$	1.28621	0.0697544
$341$	−36.7995	−1.99281
$342$	−0.302602	−0.0163628
$343$	−14.3424	−0.774418
$344$	0	0
$345$	6.33513	0.341072
$346$	−0.457123	−0.0245751
$347$	25.2107	1.35338	0.676692	−	0.736267i	$-0.263413\pi$
$347$	25.2107	1.35338	0.676692	+	0.736267i	$0.263413\pi$
$348$	19.4819	1.04434
$349$	−0.261454	−0.0139953	−0.00699766	−	0.999976i	$-0.502227\pi$
$349$	−0.261454	−0.0139953	−0.00699766	+	0.999976i	$0.502227\pi$
$350$	5.91185	0.316002
$351$	19.6485	1.04876
$352$	−24.6407	−1.31335
$353$	−14.2862	−0.760378	−0.380189	−	0.924909i	$-0.624141\pi$
$353$	−14.2862	−0.760378	−0.380189	+	0.924909i	$0.624141\pi$
$354$	−0.692021	−0.0367805
$355$	13.3056	0.706187
$356$	25.4209	1.34730
$357$	4.89008	0.258811
$358$	4.34183	0.229473
$359$	−0.957787	−0.0505501	−0.0252750	−	0.999681i	$-0.508046\pi$
$359$	−0.957787	−0.0505501	−0.0252750	+	0.999681i	$0.508046\pi$
$360$	0.335126	0.0176627
$361$	−11.4209	−0.601099
$362$	−3.73423	−0.196267
$363$	−30.7168	−1.61221
$364$	−21.7603	−1.14055
$365$	9.98792	0.522792
$366$	1.10992	0.0580163
$367$	12.3123	0.642696	0.321348	−	0.946961i	$-0.395864\pi$
$367$	12.3123	0.642696	0.321348	+	0.946961i	$0.395864\pi$
$368$	−12.4983	−0.651517
$369$	1.75600	0.0914139
$370$	−1.23729	−0.0643237
$371$	−6.31336	−0.327773
$372$	−22.5623	−1.16980
$373$	−10.9632	−0.567651	−0.283826	−	0.958876i	$-0.591604\pi$
$373$	−10.9632	−0.567651	−0.283826	+	0.958876i	$0.591604\pi$
$374$	2.09783	0.108476
$375$	13.5211	0.698227
$376$	−15.0790	−0.777643
$377$	23.7646	1.22394
$378$	−6.73125	−0.346218
$379$	−11.1086	−0.570610	−0.285305	−	0.958437i	$-0.592095\pi$
$379$	−11.1086	−0.570610	−0.285305	+	0.958437i	$0.592095\pi$
$380$	−3.97823	−0.204079
$381$	−13.0054	−0.666286
$382$	−6.07846	−0.311001
$383$	7.96376	0.406929	0.203464	−	0.979082i	$-0.434780\pi$
$383$	7.96376	0.406929	0.203464	+	0.979082i	$0.434780\pi$
$384$	−19.6799	−1.00429
$385$	12.9487	0.659926
$386$	−1.81940	−0.0926048
$387$	0	0
$388$	16.8998	0.857956
$389$	11.4045	0.578230	0.289115	−	0.957294i	$-0.406639\pi$
$389$	11.4045	0.578230	0.289115	+	0.957294i	$0.406639\pi$
$390$	2.54719	0.128982
$391$	3.90217	0.197341
$392$	3.88471	0.196207
$393$	−16.4547	−0.830031
$394$	−3.81940	−0.192418
$395$	−6.29590	−0.316781
$396$	−2.35690	−0.118438
$397$	−9.50365	−0.476974	−0.238487	−	0.971146i	$-0.576652\pi$
$397$	−9.50365	−0.476974	−0.238487	+	0.971146i	$0.576652\pi$
$398$	−2.54096	−0.127367
$399$	−15.1250	−0.757196
$400$	−12.4209	−0.621044
$401$	10.8659	0.542618	0.271309	−	0.962492i	$-0.412543\pi$
$401$	10.8659	0.542618	0.271309	+	0.962492i	$0.412543\pi$
$402$	5.38404	0.268532
$403$	−27.5222	−1.37098
$404$	−22.5623	−1.12251
$405$	−7.76271	−0.385732
$406$	−8.14138	−0.404050
$407$	18.3599	0.910065
$408$	2.71379	0.134353
$409$	21.1739	1.04698	0.523491	−	0.852031i	$-0.324629\pi$
$409$	21.1739	1.04698	0.523491	+	0.852031i	$0.324629\pi$
$410$	−2.53750	−0.125318
$411$	−10.9782	−0.541516
$412$	−7.89977	−0.389194
$413$	−2.63102	−0.129464
$414$	0.481878	0.0236830
$415$	5.90754	0.289990
$416$	−18.4286	−0.903539
$417$	−12.8780	−0.630638
$418$	−6.48858	−0.317367
$419$	32.1075	1.56856	0.784278	−	0.620410i	$-0.213034\pi$
$419$	32.1075	1.56856	0.784278	+	0.620410i	$0.213034\pi$
$420$	7.93900	0.387384
$421$	−15.2198	−0.741769	−0.370885	−	0.928679i	$-0.620945\pi$
$421$	−15.2198	−0.741769	−0.370885	+	0.928679i	$0.620945\pi$
$422$	−8.52542	−0.415011
$423$	−2.20105	−0.107019
$424$	−3.50365	−0.170152
$425$	3.87800	0.188111
$426$	13.3056	0.644658
$427$	4.21983	0.204212
$428$	−1.15883	−0.0560143
$429$	−37.7972	−1.82486
$430$	0	0
$431$	37.4819	1.80544	0.902719	−	0.430230i	$-0.141568\pi$
$431$	37.4819	1.80544	0.902719	+	0.430230i	$0.141568\pi$
$432$	14.1424	0.680428
$433$	−22.9705	−1.10389	−0.551945	−	0.833881i	$-0.686114\pi$
$433$	−22.9705	−1.10389	−0.551945	+	0.833881i	$0.686114\pi$
$434$	9.42865	0.452590
$435$	−8.67025	−0.415707
$436$	3.38404	0.162066
$437$	−12.0694	−0.577356
$438$	9.98792	0.477241
$439$	1.45712	0.0695447	0.0347724	−	0.999395i	$-0.488929\pi$
$439$	1.45712	0.0695447	0.0347724	+	0.999395i	$0.488929\pi$
$440$	7.18598	0.342578
$441$	0.567040	0.0270019
$442$	1.56896	0.0746278
$443$	−20.9530	−0.995507	−0.497754	−	0.867318i	$-0.665842\pi$
$443$	−20.9530	−0.995507	−0.497754	+	0.867318i	$0.665842\pi$
$444$	11.2567	0.534218
$445$	−11.3134	−0.536304
$446$	0.0459334	0.00217501
$447$	−8.00969	−0.378845
$448$	−11.0707	−0.523041
$449$	−8.73855	−0.412397	−0.206199	−	0.978510i	$-0.566109\pi$
$449$	−8.73855	−0.412397	−0.206199	+	0.978510i	$0.566109\pi$
$450$	0.478894	0.0225753
$451$	37.6534	1.77303
$452$	22.7560	1.07035
$453$	25.5894	1.20229
$454$	5.94331	0.278933
$455$	9.68425	0.454005
$456$	−8.39373	−0.393072
$457$	−31.6703	−1.48147	−0.740736	−	0.671796i	$-0.765523\pi$
$457$	−31.6703	−1.48147	−0.740736	+	0.671796i	$0.765523\pi$
$458$	12.6334	0.590321
$459$	−4.41550	−0.206098
$460$	6.33513	0.295377
$461$	33.9748	1.58236	0.791181	−	0.611581i	$-0.209466\pi$
$461$	33.9748	1.58236	0.791181	+	0.611581i	$0.209466\pi$
$462$	12.9487	0.602428
$463$	6.40044	0.297454	0.148727	−	0.988878i	$-0.452483\pi$
$463$	6.40044	0.297454	0.148727	+	0.988878i	$0.452483\pi$
$464$	17.1051	0.794086
$465$	10.0411	0.465647
$466$	4.87933	0.226031
$467$	40.9506	1.89497	0.947484	−	0.319803i	$-0.103617\pi$
$467$	40.9506	1.89497	0.947484	+	0.319803i	$0.103617\pi$
$468$	−1.76271	−0.0814813
$469$	20.4698	0.945207
$470$	3.18060	0.146710
$471$	−13.4112	−0.617955
$472$	−1.46011	−0.0672069
$473$	0	0
$474$	−6.29590	−0.289180
$475$	−11.9946	−0.550351
$476$	4.89008	0.224137
$477$	−0.511418	−0.0234162
$478$	−8.64310	−0.395326
$479$	−19.8495	−0.906948	−0.453474	−	0.891269i	$-0.649816\pi$
$479$	−19.8495	−0.906948	−0.453474	+	0.891269i	$0.649816\pi$
$480$	6.72348	0.306883
$481$	13.7313	0.626091
$482$	−1.55496	−0.0708264
$483$	24.0858	1.09594
$484$	−30.7168	−1.39622
$485$	−7.52111	−0.341516
$486$	−1.13946	−0.0516868
$487$	−14.3937	−0.652242	−0.326121	−	0.945328i	$-0.605742\pi$
$487$	−14.3937	−0.652242	−0.326121	+	0.945328i	$0.605742\pi$
$488$	2.34183	0.106010
$489$	−6.03146	−0.272752
$490$	−0.819396	−0.0370165
$491$	38.4295	1.73430	0.867150	−	0.498048i	$-0.165950\pi$
$491$	38.4295	1.73430	0.867150	+	0.498048i	$0.165950\pi$
$492$	23.0858	1.04079
$493$	−5.34050	−0.240524
$494$	−4.85277	−0.218337
$495$	1.04892	0.0471453
$496$	−19.8097	−0.889482
$497$	50.5870	2.26914
$498$	5.90754	0.264723
$499$	21.1183	0.945384	0.472692	−	0.881228i	$-0.343282\pi$
$499$	21.1183	0.945384	0.472692	+	0.881228i	$0.343282\pi$
$500$	13.5211	0.604682
$501$	3.67025	0.163975
$502$	−0.556877	−0.0248546
$503$	−7.06770	−0.315133	−0.157567	−	0.987508i	$-0.550365\pi$
$503$	−7.06770	−0.315133	−0.157567	+	0.987508i	$0.550365\pi$
$504$	1.27413	0.0567541
$505$	10.0411	0.446825
$506$	10.3327	0.459346
$507$	−4.84309	−0.215089
$508$	−13.0054	−0.577020
$509$	−27.6136	−1.22395	−0.611975	−	0.790877i	$-0.709625\pi$
$509$	−27.6136	−1.22395	−0.611975	+	0.790877i	$0.709625\pi$
$510$	−0.572417	−0.0253470
$511$	37.9734	1.67985
$512$	−22.9119	−1.01257
$513$	13.6571	0.602976
$514$	−3.44504	−0.151954
$515$	3.51573	0.154922
$516$	0	0
$517$	−47.1963	−2.07569
$518$	−4.70410	−0.206686
$519$	−1.85086	−0.0812435
$520$	5.37435	0.235681
$521$	43.1323	1.88966	0.944830	−	0.327562i	$-0.106227\pi$
$521$	43.1323	1.88966	0.944830	+	0.327562i	$0.106227\pi$
$522$	−0.659498	−0.0288654
$523$	−13.6420	−0.596525	−0.298262	−	0.954484i	$-0.596407\pi$
$523$	−13.6420	−0.596525	−0.298262	+	0.954484i	$0.596407\pi$
$524$	−16.4547	−0.718828
$525$	23.9366	1.04468
$526$	0.568959	0.0248078
$527$	6.18492	0.269419
$528$	−27.2054	−1.18396
$529$	−3.78017	−0.164355
$530$	0.739020	0.0321010
$531$	−0.213128	−0.00924896
$532$	−15.1250	−0.655751
$533$	28.1608	1.21978
$534$	−11.3134	−0.489577
$535$	0.515729	0.0222969
$536$	11.3599	0.490672
$537$	17.5797	0.758621
$538$	−11.2940	−0.486918
$539$	12.1588	0.523718
$540$	−7.16852	−0.308484
$541$	−24.5623	−1.05601	−0.528007	−	0.849240i	$-0.677060\pi$
$541$	−24.5623	−1.05601	−0.528007	+	0.849240i	$0.677060\pi$
$542$	3.98361	0.171111
$543$	−15.1196	−0.648844
$544$	4.14138	0.177560
$545$	−1.50604	−0.0645117
$546$	9.68425	0.414448
$547$	21.8189	0.932910	0.466455	−	0.884545i	$-0.345531\pi$
$547$	21.8189	0.932910	0.466455	+	0.884545i	$0.345531\pi$
$548$	−10.9782	−0.468967
$549$	0.341830	0.0145890
$550$	10.2687	0.437861
$551$	16.5181	0.703696
$552$	13.3666	0.568920
$553$	−23.9366	−1.01789
$554$	−2.98062	−0.126635
$555$	−5.00969	−0.212649
$556$	−12.8780	−0.546149
$557$	−40.8980	−1.73290	−0.866451	−	0.499262i	$-0.833605\pi$
$557$	−40.8980	−1.73290	−0.866451	+	0.499262i	$0.833605\pi$
$558$	0.763774	0.0323331
$559$	0	0
$560$	6.97046	0.294556
$561$	8.49396	0.358615
$562$	9.95407	0.419887
$563$	0.679940	0.0286560	0.0143280	−	0.999897i	$-0.495439\pi$
$563$	0.679940	0.0286560	0.0143280	+	0.999897i	$0.495439\pi$
$564$	−28.9366	−1.21845
$565$	−10.1274	−0.426062
$566$	−12.9987	−0.546375
$567$	−29.5133	−1.23944
$568$	28.0737	1.17795
$569$	−16.4359	−0.689031	−0.344515	−	0.938781i	$-0.611957\pi$
$569$	−16.4359	−0.689031	−0.344515	+	0.938781i	$0.611957\pi$
$570$	1.77048	0.0741572
$571$	−21.0237	−0.879814	−0.439907	−	0.898043i	$-0.644989\pi$
$571$	−21.0237	−0.879814	−0.439907	+	0.898043i	$0.644989\pi$
$572$	−37.7972	−1.58038
$573$	−24.6112	−1.02815
$574$	−9.64742	−0.402675
$575$	19.1008	0.796559
$576$	−0.896789	−0.0373662
$577$	−40.9506	−1.70480	−0.852398	−	0.522893i	$-0.824853\pi$
$577$	−40.9506	−1.70480	−0.852398	+	0.522893i	$0.824853\pi$
$578$	7.21313	0.300027
$579$	−7.36658	−0.306145
$580$	−8.67025	−0.360013
$581$	22.4601	0.931802
$582$	−7.52111	−0.311760
$583$	−10.9661	−0.454171
$584$	21.0737	0.872035
$585$	0.784479	0.0324342
$586$	4.29829	0.177561
$587$	8.21446	0.339047	0.169523	−	0.985526i	$-0.445777\pi$
$587$	8.21446	0.339047	0.169523	+	0.985526i	$0.445777\pi$
$588$	7.45473	0.307428
$589$	−19.1299	−0.788233
$590$	0.307979	0.0126793
$591$	−15.4644	−0.636121
$592$	9.88338	0.406204
$593$	17.8321	0.732275	0.366138	−	0.930561i	$-0.380680\pi$
$593$	17.8321	0.732275	0.366138	+	0.930561i	$0.380680\pi$
$594$	−11.6920	−0.479729
$595$	−2.17629	−0.0892193
$596$	−8.00969	−0.328090
$597$	−10.2881	−0.421065
$598$	7.72779	0.316013
$599$	35.6829	1.45796	0.728982	−	0.684532i	$-0.239994\pi$
$599$	35.6829	1.45796	0.728982	+	0.684532i	$0.239994\pi$
$600$	13.2838	0.542309
$601$	8.47219	0.345588	0.172794	−	0.984958i	$-0.444721\pi$
$601$	8.47219	0.345588	0.172794	+	0.984958i	$0.444721\pi$
$602$	0	0
$603$	1.65817	0.0675259
$604$	25.5894	1.04122
$605$	13.6703	0.555775
$606$	10.0411	0.407894
$607$	−33.8407	−1.37355	−0.686776	−	0.726869i	$-0.740975\pi$
$607$	−33.8407	−1.37355	−0.686776	+	0.726869i	$0.740975\pi$
$608$	−12.8092	−0.519483
$609$	−32.9638	−1.33576
$610$	−0.493959	−0.0199998
$611$	−35.2978	−1.42800
$612$	0.396125	0.0160124
$613$	−35.8092	−1.44632	−0.723161	−	0.690680i	$-0.757311\pi$
$613$	−35.8092	−1.44632	−0.723161	+	0.690680i	$0.757311\pi$
$614$	5.77910	0.233226
$615$	−10.2741	−0.414293
$616$	27.3207	1.10078
$617$	3.95838	0.159358	0.0796792	−	0.996821i	$-0.474610\pi$
$617$	3.95838	0.159358	0.0796792	+	0.996821i	$0.474610\pi$
$618$	3.51573	0.141423
$619$	22.4916	0.904012	0.452006	−	0.892015i	$-0.350708\pi$
$619$	22.4916	0.904012	0.452006	+	0.892015i	$0.350708\pi$
$620$	10.0411	0.403262
$621$	−21.7482	−0.872727
$622$	2.63342	0.105590
$623$	−43.0127	−1.72327
$624$	−20.3467	−0.814521
$625$	15.7670	0.630681
$626$	9.15346	0.365846
$627$	−26.2717	−1.04919
$628$	−13.4112	−0.535165
$629$	−3.08575	−0.123037
$630$	−0.268750	−0.0107073
$631$	−11.7192	−0.466533	−0.233266	−	0.972413i	$-0.574941\pi$
$631$	−11.7192	−0.466533	−0.233266	+	0.972413i	$0.574941\pi$
$632$	−13.2838	−0.528402
$633$	−34.5187	−1.37200
$634$	−8.17198	−0.324551
$635$	5.78794	0.229687
$636$	−6.72348	−0.266603
$637$	9.09352	0.360298
$638$	−14.1414	−0.559862
$639$	4.09783	0.162108
$640$	8.75840	0.346206
$641$	−41.9885	−1.65845	−0.829223	−	0.558918i	$-0.811217\pi$
$641$	−41.9885	−1.65845	−0.829223	+	0.558918i	$0.811217\pi$
$642$	0.515729	0.0203542
$643$	19.2379	0.758668	0.379334	−	0.925260i	$-0.376153\pi$
$643$	19.2379	0.758668	0.379334	+	0.925260i	$0.376153\pi$
$644$	24.0858	0.949112
$645$	0	0
$646$	1.09054	0.0429067
$647$	−15.6926	−0.616940	−0.308470	−	0.951234i	$-0.599817\pi$
$647$	−15.6926	−0.616940	−0.308470	+	0.951234i	$0.599817\pi$
$648$	−16.3787	−0.643415
$649$	−4.57002	−0.179389
$650$	7.67994	0.301232
$651$	38.1758	1.49623
$652$	−6.03146	−0.236210
$653$	46.2344	1.80929	0.904646	−	0.426163i	$-0.140135\pi$
$653$	46.2344	1.80929	0.904646	+	0.426163i	$0.140135\pi$
$654$	−1.50604	−0.0588909
$655$	7.32304	0.286135
$656$	20.2693	0.791385
$657$	3.07606	0.120009
$658$	12.0925	0.471413
$659$	7.91915	0.308486	0.154243	−	0.988033i	$-0.450706\pi$
$659$	7.91915	0.308486	0.154243	+	0.988033i	$0.450706\pi$
$660$	13.7899	0.536769
$661$	39.6993	1.54412	0.772062	−	0.635547i	$-0.219225\pi$
$661$	39.6993	1.54412	0.772062	+	0.635547i	$0.219225\pi$
$662$	0.733643	0.0285138
$663$	6.35258	0.246714
$664$	12.4644	0.483713
$665$	6.73125	0.261027
$666$	−0.381059	−0.0147657
$667$	−26.3043	−1.01850
$668$	3.67025	0.142006
$669$	0.185981	0.00719043
$670$	−2.39612	−0.0925704
$671$	7.32975	0.282962
$672$	25.5623	0.986085
$673$	−30.3787	−1.17101	−0.585506	−	0.810668i	$-0.699104\pi$
$673$	−30.3787	−1.17101	−0.585506	+	0.810668i	$0.699104\pi$
$674$	−12.9705	−0.499603
$675$	−21.6136	−0.831906
$676$	−4.84309	−0.186273
$677$	11.7855	0.452955	0.226478	−	0.974016i	$-0.427279\pi$
$677$	11.7855	0.452955	0.226478	+	0.974016i	$0.427279\pi$
$678$	−10.1274	−0.388939
$679$	−28.5948	−1.09737
$680$	−1.20775	−0.0463151
$681$	24.0640	0.922134
$682$	16.3773	0.627121
$683$	−16.5711	−0.634075	−0.317038	−	0.948413i	$-0.602688\pi$
$683$	−16.5711	−0.634075	−0.317038	+	0.948413i	$0.602688\pi$
$684$	−1.22521	−0.0468470
$685$	4.88577	0.186676
$686$	6.38298	0.243703
$687$	51.1517	1.95156
$688$	0	0
$689$	−8.20152	−0.312453
$690$	−2.81940	−0.107333
$691$	−18.8538	−0.717234	−0.358617	−	0.933485i	$-0.616752\pi$
$691$	−18.8538	−0.717234	−0.358617	+	0.933485i	$0.616752\pi$
$692$	−1.85086	−0.0703590
$693$	3.98792	0.151488
$694$	−11.2198	−0.425899
$695$	5.73125	0.217399
$696$	−18.2935	−0.693413
$697$	−6.32842	−0.239706
$698$	0.116358	0.00440422
$699$	19.7560	0.747241
$700$	23.9366	0.904719
$701$	30.2403	1.14216	0.571080	−	0.820895i	$-0.306525\pi$
$701$	30.2403	1.14216	0.571080	+	0.820895i	$0.306525\pi$
$702$	−8.74440	−0.330036
$703$	9.54420	0.359966
$704$	−19.2295	−0.724740
$705$	12.8780	0.485014
$706$	6.35796	0.239285
$707$	38.1758	1.43575
$708$	−2.80194	−0.105303
$709$	−0.0682947	−0.00256486	−0.00128243	−	0.999999i	$-0.500408\pi$
$709$	−0.0682947	−0.00256486	−0.00128243	+	0.999999i	$0.500408\pi$
$710$	−5.92154	−0.222232
$711$	−1.93900	−0.0727182
$712$	−23.8702	−0.894575
$713$	30.4634	1.14086
$714$	−2.17629	−0.0814457
$715$	16.8213	0.629082
$716$	17.5797	0.656985
$717$	−34.9952	−1.30692
$718$	0.426256	0.0159077
$719$	26.8442	1.00112	0.500559	−	0.865702i	$-0.333128\pi$
$719$	26.8442	1.00112	0.500559	+	0.865702i	$0.333128\pi$
$720$	0.564647	0.0210431
$721$	13.3666	0.497798
$722$	5.08277	0.189161
$723$	−6.29590	−0.234147
$724$	−15.1196	−0.561916
$725$	−26.1414	−0.970866
$726$	13.6703	0.507351
$727$	7.80971	0.289646	0.144823	−	0.989458i	$-0.453739\pi$
$727$	7.80971	0.289646	0.144823	+	0.989458i	$0.453739\pi$
$728$	20.4330	0.757296
$729$	24.4263	0.904676
$730$	−4.44504	−0.164518
$731$	0	0
$732$	4.49396	0.166102
$733$	12.2959	0.454159	0.227080	−	0.973876i	$-0.427082\pi$
$733$	12.2959	0.454159	0.227080	+	0.973876i	$0.427082\pi$
$734$	−5.47948	−0.202251
$735$	−3.31767	−0.122374
$736$	20.3980	0.751882
$737$	35.5555	1.30971
$738$	−0.781495	−0.0287672
$739$	3.92261	0.144295	0.0721477	−	0.997394i	$-0.477015\pi$
$739$	3.92261	0.144295	0.0721477	+	0.997394i	$0.477015\pi$
$740$	−5.00969	−0.184160
$741$	−19.6485	−0.721805
$742$	2.80971	0.103148
$743$	22.5235	0.826307	0.413154	−	0.910661i	$-0.364427\pi$
$743$	22.5235	0.826307	0.413154	+	0.910661i	$0.364427\pi$
$744$	21.1860	0.776716
$745$	3.56465	0.130599
$746$	4.87907	0.178635
$747$	1.81940	0.0665682
$748$	8.49396	0.310570
$749$	1.96077	0.0716450
$750$	−6.01746	−0.219727
$751$	−2.58881	−0.0944670	−0.0472335	−	0.998884i	$-0.515040\pi$
$751$	−2.58881	−0.0944670	−0.0472335	+	0.998884i	$0.515040\pi$
$752$	−25.4064	−0.926476
$753$	−2.25475	−0.0821676
$754$	−10.5763	−0.385164
$755$	−11.3884	−0.414465
$756$	−27.2543	−0.991228
$757$	−11.2107	−0.407461	−0.203731	−	0.979027i	$-0.565307\pi$
$757$	−11.2107	−0.407461	−0.203731	+	0.979027i	$0.565307\pi$
$758$	4.94379	0.179566
$759$	41.8364	1.51856
$760$	3.73556	0.135503
$761$	10.4614	0.379227	0.189613	−	0.981859i	$-0.439277\pi$
$761$	10.4614	0.379227	0.189613	+	0.981859i	$0.439277\pi$
$762$	5.78794	0.209675
$763$	−5.72587	−0.207291
$764$	−24.6112	−0.890401
$765$	−0.176292	−0.00637385
$766$	−3.54420	−0.128057
$767$	−3.41789	−0.123413
$768$	−4.32736	−0.156150
$769$	−8.96807	−0.323397	−0.161698	−	0.986840i	$-0.551697\pi$
$769$	−8.96807	−0.323397	−0.161698	+	0.986840i	$0.551697\pi$
$770$	−5.76271	−0.207674
$771$	−13.9487	−0.502350
$772$	−7.36658	−0.265129
$773$	−46.0995	−1.65808	−0.829042	−	0.559187i	$-0.811113\pi$
$773$	−46.0995	−1.65808	−0.829042	+	0.559187i	$0.811113\pi$
$774$	0	0
$775$	30.2747	1.08750
$776$	−15.8689	−0.569660
$777$	−19.0465	−0.683290
$778$	−5.07547	−0.181965
$779$	19.5737	0.701302
$780$	10.3134	0.369277
$781$	87.8684	3.14418
$782$	−1.73663	−0.0621017
$783$	29.7646	1.06370
$784$	6.54527	0.233760
$785$	5.96854	0.213026
$786$	7.32304	0.261204
$787$	29.3303	1.04551	0.522757	−	0.852482i	$-0.324904\pi$
$787$	29.3303	1.04551	0.522757	+	0.852482i	$0.324904\pi$
$788$	−15.4644	−0.550897
$789$	2.30367	0.0820127
$790$	2.80194	0.0996885
$791$	−38.5036	−1.36903
$792$	2.21313	0.0786401
$793$	5.48188	0.194667
$794$	4.22952	0.150100
$795$	2.99223	0.106123
$796$	−10.2881	−0.364653
$797$	−41.1511	−1.45765	−0.728823	−	0.684702i	$-0.759932\pi$
$797$	−41.1511	−1.45765	−0.728823	+	0.684702i	$0.759932\pi$
$798$	6.73125	0.238284
$799$	7.93230	0.280624
$800$	20.2717	0.716714
$801$	−3.48427	−0.123111
$802$	−4.83579	−0.170758
$803$	65.9590	2.32764
$804$	21.7995	0.768811
$805$	−10.7192	−0.377801
$806$	12.2485	0.431436
$807$	−45.7284	−1.60972
$808$	21.1860	0.745320
$809$	35.0549	1.23246	0.616232	−	0.787565i	$-0.288658\pi$
$809$	35.0549	1.23246	0.616232	+	0.787565i	$0.288658\pi$
$810$	3.45473	0.121387
$811$	−27.4476	−0.963814	−0.481907	−	0.876222i	$-0.660056\pi$
$811$	−27.4476	−0.963814	−0.481907	+	0.876222i	$0.660056\pi$
$812$	−32.9638	−1.15680
$813$	16.1293	0.565679
$814$	−8.17092	−0.286390
$815$	2.68425	0.0940252
$816$	4.57242	0.160067
$817$	0	0
$818$	−9.42327	−0.329477
$819$	2.98254	0.104218
$820$	−10.2741	−0.358788
$821$	−9.93661	−0.346790	−0.173395	−	0.984852i	$-0.555474\pi$
$821$	−9.93661	−0.346790	−0.173395	+	0.984852i	$0.555474\pi$
$822$	4.88577	0.170411
$823$	3.46489	0.120779	0.0603893	−	0.998175i	$-0.480766\pi$
$823$	3.46489	0.120779	0.0603893	+	0.998175i	$0.480766\pi$
$824$	7.41789	0.258415
$825$	41.5773	1.44754
$826$	1.17092	0.0407414
$827$	0.467403	0.0162532	0.00812660	−	0.999967i	$-0.497413\pi$
$827$	0.467403	0.0162532	0.00812660	+	0.999967i	$0.497413\pi$
$828$	1.95108	0.0678048
$829$	24.0374	0.834854	0.417427	−	0.908710i	$-0.362932\pi$
$829$	24.0374	0.834854	0.417427	+	0.908710i	$0.362932\pi$
$830$	−2.62910	−0.0912575
$831$	−12.0683	−0.418645
$832$	−14.3817	−0.498594
$833$	−2.04354	−0.0708045
$834$	5.73125	0.198457
$835$	−1.63342	−0.0565267
$836$	−26.2717	−0.908627
$837$	−34.4709	−1.19149
$838$	−14.2892	−0.493612
$839$	−11.0435	−0.381265	−0.190633	−	0.981661i	$-0.561054\pi$
$839$	−11.0435	−0.381265	−0.190633	+	0.981661i	$0.561054\pi$
$840$	−7.45473	−0.257213
$841$	7.00000	0.241379
$842$	6.77346	0.233429
$843$	40.3032	1.38812
$844$	−34.5187	−1.18818
$845$	2.15538	0.0741472
$846$	0.979558	0.0336779
$847$	51.9734	1.78583
$848$	−5.90323	−0.202718
$849$	−52.6305	−1.80628
$850$	−1.72587	−0.0591970
$851$	−15.1987	−0.521003
$852$	53.8732	1.84567
$853$	−47.2804	−1.61885	−0.809424	−	0.587224i	$-0.800221\pi$
$853$	−47.2804	−1.61885	−0.809424	+	0.587224i	$0.800221\pi$
$854$	−1.87800	−0.0642639
$855$	0.545269	0.0186478
$856$	1.08815	0.0371921
$857$	−0.906477	−0.0309647	−0.0154823	−	0.999880i	$-0.504928\pi$
$857$	−0.906477	−0.0309647	−0.0154823	+	0.999880i	$0.504928\pi$
$858$	16.8213	0.574270
$859$	35.4486	1.20949	0.604746	−	0.796419i	$-0.293275\pi$
$859$	35.4486	1.20949	0.604746	+	0.796419i	$0.293275\pi$
$860$	0	0
$861$	−39.0616	−1.33122
$862$	−16.6810	−0.568157
$863$	10.3722	0.353075	0.176537	−	0.984294i	$-0.443510\pi$
$863$	10.3722	0.353075	0.176537	+	0.984294i	$0.443510\pi$
$864$	−23.0814	−0.785247
$865$	0.823708	0.0280069
$866$	10.2228	0.347385
$867$	29.2054	0.991866
$868$	38.1758	1.29577
$869$	−41.5773	−1.41041
$870$	3.85862	0.130820
$871$	26.5918	0.901029
$872$	−3.17762	−0.107608
$873$	−2.31634	−0.0783962
$874$	5.37137	0.181689
$875$	−22.8780	−0.773418
$876$	40.4403	1.36635
$877$	−7.39612	−0.249749	−0.124875	−	0.992173i	$-0.539853\pi$
$877$	−7.39612	−0.249749	−0.124875	+	0.992173i	$0.539853\pi$
$878$	−0.648481	−0.0218852
$879$	17.4034	0.587003
$880$	12.1075	0.408145
$881$	−6.17496	−0.208040	−0.104020	−	0.994575i	$-0.533171\pi$
$881$	−6.17496	−0.208040	−0.104020	+	0.994575i	$0.533171\pi$
$882$	−0.252356	−0.00849728
$883$	−23.2194	−0.781394	−0.390697	−	0.920519i	$-0.627766\pi$
$883$	−23.2194	−0.781394	−0.390697	+	0.920519i	$0.627766\pi$
$884$	6.35258	0.213661
$885$	1.24698	0.0419168
$886$	9.32496	0.313278
$887$	14.1438	0.474901	0.237451	−	0.971400i	$-0.423688\pi$
$887$	14.1438	0.474901	0.237451	+	0.971400i	$0.423688\pi$
$888$	−10.5700	−0.354707
$889$	22.0054	0.738037
$890$	5.03492	0.168771
$891$	−51.2640	−1.71741
$892$	0.185981	0.00622709
$893$	−24.5345	−0.821016
$894$	3.56465	0.119220
$895$	−7.82371	−0.261518
$896$	33.2989	1.11244
$897$	31.2892	1.04472
$898$	3.88902	0.129778
$899$	−41.6921	−1.39051
$900$	1.93900	0.0646334
$901$	1.84309	0.0614021
$902$	−16.7573	−0.557958
$903$	0	0
$904$	−21.3679	−0.710686
$905$	6.72886	0.223675
$906$	−11.3884	−0.378353
$907$	−38.3110	−1.27209	−0.636047	−	0.771650i	$-0.719432\pi$
$907$	−38.3110	−1.27209	−0.636047	+	0.771650i	$0.719432\pi$
$908$	24.0640	0.798591
$909$	3.09246	0.102570
$910$	−4.30990	−0.142872
$911$	−5.49934	−0.182201	−0.0911006	−	0.995842i	$-0.529038\pi$
$911$	−5.49934	−0.182201	−0.0911006	+	0.995842i	$0.529038\pi$
$912$	−14.1424	−0.468303
$913$	39.0127	1.29113
$914$	14.0946	0.466208
$915$	−2.00000	−0.0661180
$916$	51.1517	1.69010
$917$	27.8418	0.919416
$918$	1.96508	0.0648574
$919$	−46.2140	−1.52446	−0.762229	−	0.647307i	$-0.775895\pi$
$919$	−46.2140	−1.52446	−0.762229	+	0.647307i	$0.775895\pi$
$920$	−5.94869	−0.196122
$921$	23.3991	0.771027
$922$	−15.1202	−0.497957
$923$	65.7163	2.16308
$924$	52.4282	1.72476
$925$	−15.1045	−0.496634
$926$	−2.84846	−0.0936063
$927$	1.08277	0.0355628
$928$	−27.9168	−0.916412
$929$	−60.0635	−1.97062	−0.985310	−	0.170776i	$-0.945373\pi$
$929$	−60.0635	−1.97062	−0.985310	+	0.170776i	$0.945373\pi$
$930$	−4.46873	−0.146535
$931$	6.32065	0.207151
$932$	19.7560	0.647129
$933$	10.6625	0.349074
$934$	−18.2247	−0.596332
$935$	−3.78017	−0.123625
$936$	1.65519	0.0541014
$937$	32.3347	1.05633	0.528164	−	0.849143i	$-0.322881\pi$
$937$	32.3347	1.05633	0.528164	+	0.849143i	$0.322881\pi$
$938$	−9.10992	−0.297449
$939$	37.0616	1.20946
$940$	12.8780	0.420034
$941$	−27.8351	−0.907397	−0.453698	−	0.891155i	$-0.649896\pi$
$941$	−27.8351	−0.907397	−0.453698	+	0.891155i	$0.649896\pi$
$942$	5.96854	0.194466
$943$	−31.1702	−1.01504
$944$	−2.46011	−0.0800697
$945$	12.1293	0.394566
$946$	0	0
$947$	−32.2707	−1.04866	−0.524328	−	0.851516i	$-0.675683\pi$
$947$	−32.2707	−1.04866	−0.524328	+	0.851516i	$0.675683\pi$
$948$	−25.4916	−0.827928
$949$	49.3303	1.60133
$950$	5.33811	0.173191
$951$	−33.0877	−1.07294
$952$	−4.59179	−0.148821
$953$	−15.0032	−0.486003	−0.243001	−	0.970026i	$-0.578132\pi$
$953$	−15.0032	−0.486003	−0.243001	+	0.970026i	$0.578132\pi$
$954$	0.227602	0.00736889
$955$	10.9530	0.354431
$956$	−34.9952	−1.13183
$957$	−57.2573	−1.85086
$958$	8.83387	0.285409
$959$	18.5754	0.599831
$960$	5.24698	0.169346
$961$	17.2843	0.557558
$962$	−6.11098	−0.197026
$963$	0.158834	0.00511834
$964$	−6.29590	−0.202777
$965$	3.27844	0.105537
$966$	−10.7192	−0.344884
$967$	−25.0683	−0.806142	−0.403071	−	0.915169i	$-0.632057\pi$
$967$	−25.0683	−0.806142	−0.403071	+	0.915169i	$0.632057\pi$
$968$	28.8431	0.927052
$969$	4.41550	0.141846
$970$	3.34721	0.107472
$971$	2.73795	0.0878652	0.0439326	−	0.999034i	$-0.486011\pi$
$971$	2.73795	0.0878652	0.0439326	+	0.999034i	$0.486011\pi$
$972$	−4.61356	−0.147980
$973$	21.7899	0.698551
$974$	6.40581	0.205255
$975$	31.0954	0.995851
$976$	3.94571	0.126299
$977$	−17.4359	−0.557825	−0.278913	−	0.960316i	$-0.589974\pi$
$977$	−17.4359	−0.557825	−0.278913	+	0.960316i	$0.589974\pi$
$978$	2.68425	0.0858329
$979$	−74.7120	−2.38781
$980$	−3.31767	−0.105979
$981$	−0.463828	−0.0148089
$982$	−17.1027	−0.545770
$983$	51.2747	1.63541	0.817705	−	0.575638i	$-0.195246\pi$
$983$	51.2747	1.63541	0.817705	+	0.575638i	$0.195246\pi$
$984$	−21.6775	−0.691055
$985$	6.88231	0.219289
$986$	2.37675	0.0756910
$987$	48.9614	1.55846
$988$	−19.6485	−0.625101
$989$	0	0
$990$	−0.466812	−0.0148363
$991$	45.4677	1.44433	0.722164	−	0.691722i	$-0.243148\pi$
$991$	45.4677	1.44433	0.722164	+	0.691722i	$0.243148\pi$
$992$	32.3308	1.02650
$993$	2.97046	0.0942647
$994$	−22.5133	−0.714080
$995$	4.57865	0.145153
$996$	23.9191	0.757907
$997$	−52.1879	−1.65281	−0.826404	−	0.563078i	$-0.809617\pi$
$997$	−52.1879	−1.65281	−0.826404	+	0.563078i	$0.809617\pi$
$998$	−9.39852	−0.297505
$999$	17.1981	0.544123

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.2.a.j.1.2		3
43.32	odd	14		43.2.e.a.35.1	yes	6
43.39	odd	14		43.2.e.a.16.1	✓	6
43.42	odd	2		1849.2.a.k.1.2		3
129.32	even	14		387.2.u.c.379.1		6
129.125	even	14		387.2.u.c.145.1		6
172.39	even	14		688.2.u.b.145.1		6
172.75	even	14		688.2.u.b.465.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.e.a.16.1	✓	6	43.39	odd	14
43.2.e.a.35.1	yes	6	43.32	odd	14
387.2.u.c.145.1		6	129.125	even	14
387.2.u.c.379.1		6	129.32	even	14
688.2.u.b.145.1		6	172.39	even	14
688.2.u.b.465.1		6	172.75	even	14
1849.2.a.j.1.2		3	1.1	even	1	trivial
1849.2.a.k.1.2		3	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)

\(f(q)\)	\(=\)	\(q-0.445042 q^{2} -1.80194 q^{3} -1.80194 q^{4} +0.801938 q^{5} +0.801938 q^{6} +3.04892 q^{7} +1.69202 q^{8} +0.246980 q^{9} +O(q^{10})\)	\(q-0.445042 q^{2} -1.80194 q^{3} -1.80194 q^{4} +0.801938 q^{5} +0.801938 q^{6} +3.04892 q^{7} +1.69202 q^{8} +0.246980 q^{9} -0.356896 q^{10} +5.29590 q^{11} +3.24698 q^{12} +3.96077 q^{13} -1.35690 q^{14} -1.44504 q^{15} +2.85086 q^{16} -0.890084 q^{17} -0.109916 q^{18} +2.75302 q^{19} -1.44504 q^{20} -5.49396 q^{21} -2.35690 q^{22} -4.38404 q^{23} -3.04892 q^{24} -4.35690 q^{25} -1.76271 q^{26} +4.96077 q^{27} -5.49396 q^{28} +6.00000 q^{29} +0.643104 q^{30} -6.94869 q^{31} -4.65279 q^{32} -9.54288 q^{33} +0.396125 q^{34} +2.44504 q^{35} -0.445042 q^{36} +3.46681 q^{37} -1.22521 q^{38} -7.13706 q^{39} +1.35690 q^{40} +7.10992 q^{41} +2.44504 q^{42} -9.54288 q^{44} +0.198062 q^{45} +1.95108 q^{46} -8.91185 q^{47} -5.13706 q^{48} +2.29590 q^{49} +1.93900 q^{50} +1.60388 q^{51} -7.13706 q^{52} -2.07069 q^{53} -2.20775 q^{54} +4.24698 q^{55} +5.15883 q^{56} -4.96077 q^{57} -2.67025 q^{58} -0.862937 q^{59} +2.60388 q^{60} +1.38404 q^{61} +3.09246 q^{62} +0.753020 q^{63} -3.63102 q^{64} +3.17629 q^{65} +4.24698 q^{66} +6.71379 q^{67} +1.60388 q^{68} +7.89977 q^{69} -1.08815 q^{70} +16.5918 q^{71} +0.417895 q^{72} +12.4547 q^{73} -1.54288 q^{74} +7.85086 q^{75} -4.96077 q^{76} +16.1468 q^{77} +3.17629 q^{78} -7.85086 q^{79} +2.28621 q^{80} -9.67994 q^{81} -3.16421 q^{82} +7.36658 q^{83} +9.89977 q^{84} -0.713792 q^{85} -10.8116 q^{87} +8.96077 q^{88} -14.1075 q^{89} -0.0881460 q^{90} +12.0761 q^{91} +7.89977 q^{92} +12.5211 q^{93} +3.96615 q^{94} +2.20775 q^{95} +8.38404 q^{96} -9.37867 q^{97} -1.02177 q^{98} +1.30798 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} - q^{3} - q^{4} - 2 q^{5} - 2 q^{6} - 4 q^{9}+O(q^{10}) \) 3 * q - q^2 - q^3 - q^4 - 2 * q^5 - 2 * q^6 - 4 * q^9	\( 3 q - q^{2} - q^{3} - q^{4} - 2 q^{5} - 2 q^{6} - 4 q^{9} + 3 q^{10} + 2 q^{11} + 5 q^{12} - q^{13} - 4 q^{15} - 5 q^{16} - 2 q^{17} - q^{18} + 13 q^{19} - 4 q^{20} - 7 q^{21} - 3 q^{22} - 3 q^{23} - 9 q^{25} + 12 q^{26} + 2 q^{27} - 7 q^{28} + 18 q^{29} + 6 q^{30} + 11 q^{31} + 4 q^{32} - 10 q^{33} + 10 q^{34} + 7 q^{35} - q^{36} + 7 q^{37} - 2 q^{38} - 16 q^{39} + 22 q^{41} + 7 q^{42} - 10 q^{44} + 5 q^{45} + 15 q^{46} - 23 q^{47} - 10 q^{48} - 7 q^{49} - 4 q^{50} - 4 q^{51} - 16 q^{52} + 6 q^{53} + 11 q^{54} + 8 q^{55} + 7 q^{56} - 2 q^{57} - 6 q^{58} - 8 q^{59} - q^{60} - 6 q^{61} - 6 q^{62} + 7 q^{63} + 4 q^{64} + 17 q^{65} + 8 q^{66} + 12 q^{67} - 4 q^{68} + q^{69} - 7 q^{70} + 22 q^{71} + 7 q^{72} + 15 q^{73} + 14 q^{74} + 10 q^{75} - 2 q^{76} + 21 q^{77} + 17 q^{78} - 10 q^{79} + 15 q^{80} - 5 q^{81} + 2 q^{82} - 4 q^{83} + 7 q^{84} + 6 q^{85} - 6 q^{87} + 14 q^{88} - 2 q^{89} - 4 q^{90} + 21 q^{91} + q^{92} + 22 q^{93} - 4 q^{94} - 11 q^{95} + 15 q^{96} - 21 q^{97} + 9 q^{99}+O(q^{100}) \) 3 * q - q^2 - q^3 - q^4 - 2 * q^5 - 2 * q^6 - 4 * q^9 + 3 * q^10 + 2 * q^11 + 5 * q^12 - q^13 - 4 * q^15 - 5 * q^16 - 2 * q^17 - q^18 + 13 * q^19 - 4 * q^20 - 7 * q^21 - 3 * q^22 - 3 * q^23 - 9 * q^25 + 12 * q^26 + 2 * q^27 - 7 * q^28 + 18 * q^29 + 6 * q^30 + 11 * q^31 + 4 * q^32 - 10 * q^33 + 10 * q^34 + 7 * q^35 - q^36 + 7 * q^37 - 2 * q^38 - 16 * q^39 + 22 * q^41 + 7 * q^42 - 10 * q^44 + 5 * q^45 + 15 * q^46 - 23 * q^47 - 10 * q^48 - 7 * q^49 - 4 * q^50 - 4 * q^51 - 16 * q^52 + 6 * q^53 + 11 * q^54 + 8 * q^55 + 7 * q^56 - 2 * q^57 - 6 * q^58 - 8 * q^59 - q^60 - 6 * q^61 - 6 * q^62 + 7 * q^63 + 4 * q^64 + 17 * q^65 + 8 * q^66 + 12 * q^67 - 4 * q^68 + q^69 - 7 * q^70 + 22 * q^71 + 7 * q^72 + 15 * q^73 + 14 * q^74 + 10 * q^75 - 2 * q^76 + 21 * q^77 + 17 * q^78 - 10 * q^79 + 15 * q^80 - 5 * q^81 + 2 * q^82 - 4 * q^83 + 7 * q^84 + 6 * q^85 - 6 * q^87 + 14 * q^88 - 2 * q^89 - 4 * q^90 + 21 * q^91 + q^92 + 22 * q^93 - 4 * q^94 - 11 * q^95 + 15 * q^96 - 21 * q^97 + 9 * q^99

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