LMFDB - Embedded newform 4031.2.a.e.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4031,2,Mod(1,4031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4031, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4031 = 29 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4031.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:	$32.1876970548$
Analytic rank:	$0$
Dimension:	$103$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	4031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.38730 q^{2} -1.73906 q^{3} +3.69919 q^{4} +0.845395 q^{5} +4.15166 q^{6} -4.68935 q^{7} -4.05648 q^{8} +0.0243354 q^{9} +O(q^{10})$	\(q-2.38730 q^{2} -1.73906 q^{3} +3.69919 q^{4} +0.845395 q^{5} +4.15166 q^{6} -4.68935 q^{7} -4.05648 q^{8} +0.0243354 q^{9} -2.01821 q^{10} +5.15643 q^{11} -6.43312 q^{12} -3.80619 q^{13} +11.1949 q^{14} -1.47019 q^{15} +2.28563 q^{16} +5.22593 q^{17} -0.0580960 q^{18} +6.90151 q^{19} +3.12728 q^{20} +8.15506 q^{21} -12.3099 q^{22} +5.75113 q^{23} +7.05446 q^{24} -4.28531 q^{25} +9.08651 q^{26} +5.17486 q^{27} -17.3468 q^{28} -1.00000 q^{29} +3.50979 q^{30} +2.29481 q^{31} +2.65646 q^{32} -8.96735 q^{33} -12.4758 q^{34} -3.96435 q^{35} +0.0900215 q^{36} -7.67932 q^{37} -16.4760 q^{38} +6.61920 q^{39} -3.42933 q^{40} +11.0726 q^{41} -19.4686 q^{42} -1.88987 q^{43} +19.0746 q^{44} +0.0205731 q^{45} -13.7297 q^{46} -0.635447 q^{47} -3.97486 q^{48} +14.9900 q^{49} +10.2303 q^{50} -9.08821 q^{51} -14.0798 q^{52} -7.03192 q^{53} -12.3539 q^{54} +4.35922 q^{55} +19.0222 q^{56} -12.0022 q^{57} +2.38730 q^{58} -2.10839 q^{59} -5.43853 q^{60} -11.6564 q^{61} -5.47839 q^{62} -0.114117 q^{63} -10.9130 q^{64} -3.21774 q^{65} +21.4077 q^{66} -5.42874 q^{67} +19.3317 q^{68} -10.0016 q^{69} +9.46409 q^{70} +8.10417 q^{71} -0.0987162 q^{72} +12.7437 q^{73} +18.3328 q^{74} +7.45241 q^{75} +25.5300 q^{76} -24.1803 q^{77} -15.8020 q^{78} -14.3662 q^{79} +1.93227 q^{80} -9.07241 q^{81} -26.4337 q^{82} -16.9151 q^{83} +30.1671 q^{84} +4.41797 q^{85} +4.51168 q^{86} +1.73906 q^{87} -20.9169 q^{88} +7.44257 q^{89} -0.0491141 q^{90} +17.8485 q^{91} +21.2745 q^{92} -3.99081 q^{93} +1.51700 q^{94} +5.83451 q^{95} -4.61975 q^{96} +1.20934 q^{97} -35.7855 q^{98} +0.125484 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 103 q + q^{2} + 2 q^{3} + 127 q^{4} + 9 q^{5} + 19 q^{6} + 18 q^{7} + 149 q^{9}+O(q^{10}) $ 103 * q + q^2 + 2 * q^3 + 127 * q^4 + 9 * q^5 + 19 * q^6 + 18 * q^7 + 149 * q^9	\( 103 q + q^{2} + 2 q^{3} + 127 q^{4} + 9 q^{5} + 19 q^{6} + 18 q^{7} + 149 q^{9} + 20 q^{10} + 9 q^{11} + 36 q^{13} - 10 q^{14} + 16 q^{15} + 179 q^{16} + 21 q^{17} + 7 q^{18} + 42 q^{19} + 24 q^{20} + 28 q^{21} + 32 q^{22} + 25 q^{23} + 68 q^{24} + 194 q^{25} - 5 q^{26} + 14 q^{27} + 59 q^{28} - 103 q^{29} + 84 q^{30} + 34 q^{31} + 11 q^{32} + 42 q^{33} + 54 q^{34} + 35 q^{35} + 214 q^{36} + 34 q^{37} + 9 q^{38} + 23 q^{39} + 46 q^{40} + 16 q^{41} + 13 q^{42} + 68 q^{43} - 6 q^{44} + 25 q^{45} + 60 q^{46} + 6 q^{47} + 5 q^{48} + 257 q^{49} - 51 q^{50} + 68 q^{51} + 37 q^{52} + 35 q^{53} + 30 q^{54} + 66 q^{55} - 54 q^{56} + 78 q^{57} - q^{58} + 10 q^{59} - 24 q^{60} + 70 q^{61} + 29 q^{62} + 26 q^{63} + 276 q^{64} + 95 q^{65} + 77 q^{66} + 71 q^{67} - 21 q^{68} - 20 q^{69} + 48 q^{70} + 32 q^{71} + 32 q^{72} + 94 q^{73} + 35 q^{74} + 7 q^{75} + 134 q^{76} + 17 q^{77} + 58 q^{78} + 110 q^{79} + 78 q^{80} + 267 q^{81} - 71 q^{82} + 35 q^{83} + 96 q^{84} + 71 q^{85} + 33 q^{86} - 2 q^{87} + 100 q^{88} + 22 q^{89} - 134 q^{90} + 108 q^{91} - 11 q^{92} + 78 q^{93} + 90 q^{94} + 12 q^{95} + 177 q^{96} + 44 q^{97} - 18 q^{98} + 83 q^{99}+O(q^{100}) \) 103 * q + q^2 + 2 * q^3 + 127 * q^4 + 9 * q^5 + 19 * q^6 + 18 * q^7 + 149 * q^9 + 20 * q^10 + 9 * q^11 + 36 * q^13 - 10 * q^14 + 16 * q^15 + 179 * q^16 + 21 * q^17 + 7 * q^18 + 42 * q^19 + 24 * q^20 + 28 * q^21 + 32 * q^22 + 25 * q^23 + 68 * q^24 + 194 * q^25 - 5 * q^26 + 14 * q^27 + 59 * q^28 - 103 * q^29 + 84 * q^30 + 34 * q^31 + 11 * q^32 + 42 * q^33 + 54 * q^34 + 35 * q^35 + 214 * q^36 + 34 * q^37 + 9 * q^38 + 23 * q^39 + 46 * q^40 + 16 * q^41 + 13 * q^42 + 68 * q^43 - 6 * q^44 + 25 * q^45 + 60 * q^46 + 6 * q^47 + 5 * q^48 + 257 * q^49 - 51 * q^50 + 68 * q^51 + 37 * q^52 + 35 * q^53 + 30 * q^54 + 66 * q^55 - 54 * q^56 + 78 * q^57 - q^58 + 10 * q^59 - 24 * q^60 + 70 * q^61 + 29 * q^62 + 26 * q^63 + 276 * q^64 + 95 * q^65 + 77 * q^66 + 71 * q^67 - 21 * q^68 - 20 * q^69 + 48 * q^70 + 32 * q^71 + 32 * q^72 + 94 * q^73 + 35 * q^74 + 7 * q^75 + 134 * q^76 + 17 * q^77 + 58 * q^78 + 110 * q^79 + 78 * q^80 + 267 * q^81 - 71 * q^82 + 35 * q^83 + 96 * q^84 + 71 * q^85 + 33 * q^86 - 2 * q^87 + 100 * q^88 + 22 * q^89 - 134 * q^90 + 108 * q^91 - 11 * q^92 + 78 * q^93 + 90 * q^94 + 12 * q^95 + 177 * q^96 + 44 * q^97 - 18 * q^98 + 83 * 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 $

$2$	−2.38730	−1.68807	−0.844037	−	0.536285i	$-0.819827\pi$
$2$	−2.38730	−1.68807	−0.844037	+	0.536285i	$0.819827\pi$
$3$	−1.73906	−1.00405	−0.502024	−	0.864854i	$-0.667411\pi$
$3$	−1.73906	−1.00405	−0.502024	+	0.864854i	$0.667411\pi$
$4$	3.69919	1.84960
$5$	0.845395	0.378072	0.189036	−	0.981970i	$-0.439464\pi$
$5$	0.845395	0.378072	0.189036	+	0.981970i	$0.439464\pi$
$6$	4.15166	1.69491
$7$	−4.68935	−1.77241	−0.886203	−	0.463297i	$-0.846666\pi$
$7$	−4.68935	−1.77241	−0.886203	+	0.463297i	$0.846666\pi$
$8$	−4.05648	−1.43418
$9$	0.0243354	0.00811182
$10$	−2.01821	−0.638214
$11$	5.15643	1.55472	0.777361	−	0.629055i	$-0.216558\pi$
$11$	5.15643	1.55472	0.777361	+	0.629055i	$0.216558\pi$
$12$	−6.43312	−1.85708
$13$	−3.80619	−1.05565	−0.527823	−	0.849354i	$-0.676992\pi$
$13$	−3.80619	−1.05565	−0.527823	+	0.849354i	$0.676992\pi$
$14$	11.1949	2.99195
$15$	−1.47019	−0.379603
$16$	2.28563	0.571409
$17$	5.22593	1.26747	0.633737	−	0.773549i	$-0.281520\pi$
$17$	5.22593	1.26747	0.633737	+	0.773549i	$0.281520\pi$
$18$	−0.0580960	−0.0136934
$19$	6.90151	1.58332	0.791658	−	0.610964i	$-0.209218\pi$
$19$	6.90151	1.58332	0.791658	+	0.610964i	$0.209218\pi$
$20$	3.12728	0.699281
$21$	8.15506	1.77958
$22$	−12.3099	−2.62449
$23$	5.75113	1.19919	0.599597	−	0.800302i	$-0.295328\pi$
$23$	5.75113	1.19919	0.599597	+	0.800302i	$0.295328\pi$
$24$	7.05446	1.43999
$25$	−4.28531	−0.857061
$26$	9.08651	1.78201
$27$	5.17486	0.995903
$28$	−17.3468	−3.27824
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	3.50979	0.640798
$31$	2.29481	0.412159	0.206080	−	0.978535i	$-0.433929\pi$
$31$	2.29481	0.412159	0.206080	+	0.978535i	$0.433929\pi$
$32$	2.65646	0.469601
$33$	−8.96735	−1.56101
$34$	−12.4758	−2.13959
$35$	−3.96435	−0.670098
$36$	0.0900215	0.0150036
$37$	−7.67932	−1.26247	−0.631236	−	0.775591i	$-0.717452\pi$
$37$	−7.67932	−1.26247	−0.631236	+	0.775591i	$0.717452\pi$
$38$	−16.4760	−2.67276
$39$	6.61920	1.05992
$40$	−3.42933	−0.542224
$41$	11.0726	1.72926	0.864628	−	0.502412i	$-0.167554\pi$
$41$	11.0726	1.72926	0.864628	+	0.502412i	$0.167554\pi$
$42$	−19.4686	−3.00406
$43$	−1.88987	−0.288202	−0.144101	−	0.989563i	$-0.546029\pi$
$43$	−1.88987	−0.288202	−0.144101	+	0.989563i	$0.546029\pi$
$44$	19.0746	2.87561
$45$	0.0205731	0.00306685
$46$	−13.7297	−2.02433
$47$	−0.635447	−0.0926894	−0.0463447	−	0.998926i	$-0.514757\pi$
$47$	−0.635447	−0.0926894	−0.0463447	+	0.998926i	$0.514757\pi$
$48$	−3.97486	−0.573722
$49$	14.9900	2.14142
$50$	10.2303	1.44678
$51$	−9.08821	−1.27260
$52$	−14.0798	−1.95252
$53$	−7.03192	−0.965909	−0.482954	−	0.875646i	$-0.660436\pi$
$53$	−7.03192	−0.965909	−0.482954	+	0.875646i	$0.660436\pi$
$54$	−12.3539	−1.68116
$55$	4.35922	0.587797
$56$	19.0222	2.54195
$57$	−12.0022	−1.58972
$58$	2.38730	0.313468
$59$	−2.10839	−0.274489	−0.137245	−	0.990537i	$-0.543825\pi$
$59$	−2.10839	−0.274489	−0.137245	+	0.990537i	$0.543825\pi$
$60$	−5.43853	−0.702112
$61$	−11.6564	−1.49244	−0.746221	−	0.665698i	$-0.768134\pi$
$61$	−11.6564	−1.49244	−0.746221	+	0.665698i	$0.768134\pi$
$62$	−5.47839	−0.695756
$63$	−0.114117	−0.0143774
$64$	−10.9130	−1.36413
$65$	−3.21774	−0.399111
$66$	21.4077	2.63511
$67$	−5.42874	−0.663226	−0.331613	−	0.943415i	$-0.607593\pi$
$67$	−5.42874	−0.663226	−0.331613	+	0.943415i	$0.607593\pi$
$68$	19.3317	2.34431
$69$	−10.0016	−1.20405
$70$	9.46409	1.13118
$71$	8.10417	0.961788	0.480894	−	0.876779i	$-0.340312\pi$
$71$	8.10417	0.961788	0.480894	+	0.876779i	$0.340312\pi$
$72$	−0.0987162	−0.0116338
$73$	12.7437	1.49154	0.745770	−	0.666204i	$-0.232082\pi$
$73$	12.7437	1.49154	0.745770	+	0.666204i	$0.232082\pi$
$74$	18.3328	2.13115
$75$	7.45241	0.860530
$76$	25.5300	2.92849
$77$	−24.1803	−2.75560
$78$	−15.8020	−1.78922
$79$	−14.3662	−1.61633	−0.808163	−	0.588959i	$-0.799538\pi$
$79$	−14.3662	−1.61633	−0.808163	+	0.588959i	$0.799538\pi$
$80$	1.93227	0.216034
$81$	−9.07241	−1.00805
$82$	−26.4337	−2.91911
$83$	−16.9151	−1.85668	−0.928338	−	0.371738i	$-0.878762\pi$
$83$	−16.9151	−1.85668	−0.928338	+	0.371738i	$0.878762\pi$
$84$	30.1671	3.29150
$85$	4.41797	0.479197
$86$	4.51168	0.486507
$87$	1.73906	0.186447
$88$	−20.9169	−2.22975
$89$	7.44257	0.788911	0.394456	−	0.918915i	$-0.370933\pi$
$89$	7.44257	0.788911	0.394456	+	0.918915i	$0.370933\pi$
$90$	−0.0491141	−0.00517708
$91$	17.8485	1.87104
$92$	21.2745	2.21802
$93$	−3.99081	−0.413828
$94$	1.51700	0.156467
$95$	5.83451	0.598608
$96$	−4.61975	−0.471501
$97$	1.20934	0.122789	0.0613947	−	0.998114i	$-0.480445\pi$
$97$	1.20934	0.122789	0.0613947	+	0.998114i	$0.480445\pi$
$98$	−35.7855	−3.61488
$99$	0.125484	0.0126116
$100$	−15.8522	−1.58522
$101$	14.5264	1.44543	0.722716	−	0.691145i	$-0.242893\pi$
$101$	14.5264	1.44543	0.722716	+	0.691145i	$0.242893\pi$
$102$	21.6963	2.14825
$103$	17.8971	1.76345	0.881725	−	0.471765i	$-0.156383\pi$
$103$	17.8971	1.76345	0.881725	+	0.471765i	$0.156383\pi$
$104$	15.4397	1.51399
$105$	6.89425	0.672810
$106$	16.7873	1.63053
$107$	2.49001	0.240718	0.120359	−	0.992730i	$-0.461595\pi$
$107$	2.49001	0.240718	0.120359	+	0.992730i	$0.461595\pi$
$108$	19.1428	1.84202
$109$	8.28793	0.793839	0.396920	−	0.917853i	$-0.370079\pi$
$109$	8.28793	0.793839	0.396920	+	0.917853i	$0.370079\pi$
$110$	−10.4068	−0.992246
$111$	13.3548	1.26758
$112$	−10.7181	−1.01277
$113$	7.93139	0.746122	0.373061	−	0.927807i	$-0.378308\pi$
$113$	7.93139	0.746122	0.373061	+	0.927807i	$0.378308\pi$
$114$	28.6527	2.68357
$115$	4.86198	0.453382
$116$	−3.69919	−0.343461
$117$	−0.0926253	−0.00856321
$118$	5.03336	0.463358
$119$	−24.5062	−2.24648
$120$	5.96381	0.544419
$121$	15.5887	1.41716
$122$	27.8272	2.51936
$123$	−19.2560	−1.73626
$124$	8.48893	0.762328
$125$	−7.84976	−0.702104
$126$	0.272432	0.0242702
$127$	−7.13946	−0.633524	−0.316762	−	0.948505i	$-0.602596\pi$
$127$	−7.13946	−0.633524	−0.316762	+	0.948505i	$0.602596\pi$
$128$	20.7397	1.83315
$129$	3.28660	0.289369
$130$	7.68169	0.673729
$131$	−12.1240	−1.05927	−0.529637	−	0.848224i	$-0.677672\pi$
$131$	−12.1240	−1.05927	−0.529637	+	0.848224i	$0.677672\pi$
$132$	−33.1719	−2.88725
$133$	−32.3636	−2.80628
$134$	12.9600	1.11958
$135$	4.37481	0.376523
$136$	−21.1988	−1.81779
$137$	9.14713	0.781492	0.390746	−	0.920498i	$-0.372217\pi$
$137$	9.14713	0.781492	0.390746	+	0.920498i	$0.372217\pi$
$138$	23.8767	2.03252
$139$	1.00000	0.0848189
$139$	1.00000	0.0848189
$140$	−14.6649	−1.23941
$141$	1.10508	0.0930646
$142$	−19.3471	−1.62357
$143$	−19.6263	−1.64124
$144$	0.0556219	0.00463516
$145$	−0.845395	−0.0702063
$146$	−30.4231	−2.51783
$147$	−26.0685	−2.15009
$148$	−28.4073	−2.33506
$149$	−17.7118	−1.45101	−0.725505	−	0.688217i	$-0.758394\pi$
$149$	−17.7118	−1.45101	−0.725505	+	0.688217i	$0.758394\pi$
$150$	−17.7911	−1.45264
$151$	2.77057	0.225466	0.112733	−	0.993625i	$-0.464040\pi$
$151$	2.77057	0.225466	0.112733	+	0.993625i	$0.464040\pi$
$152$	−27.9958	−2.27076
$153$	0.127175	0.0102815
$154$	57.7255	4.65166
$155$	1.94002	0.155826
$156$	24.4857	1.96042
$157$	10.2353	0.816867	0.408433	−	0.912788i	$-0.366075\pi$
$157$	10.2353	0.816867	0.408433	+	0.912788i	$0.366075\pi$
$158$	34.2964	2.72848
$159$	12.2289	0.969818
$160$	2.24576	0.177543
$161$	−26.9691	−2.12546
$162$	21.6586	1.70166
$163$	−17.2794	−1.35343	−0.676713	−	0.736247i	$-0.736596\pi$
$163$	−17.2794	−1.35343	−0.676713	+	0.736247i	$0.736596\pi$
$164$	40.9598	3.19843
$165$	−7.58095	−0.590176
$166$	40.3814	3.13421
$167$	−21.6280	−1.67362	−0.836811	−	0.547493i	$-0.815582\pi$
$167$	−21.6280	−1.67362	−0.836811	+	0.547493i	$0.815582\pi$
$168$	−33.0808	−2.55224
$169$	1.48707	0.114390
$170$	−10.5470	−0.808920
$171$	0.167951	0.0128436
$172$	−6.99099	−0.533058
$173$	1.39712	0.106221	0.0531107	−	0.998589i	$-0.483086\pi$
$173$	1.39712	0.106221	0.0531107	+	0.998589i	$0.483086\pi$
$174$	−4.15166	−0.314736
$175$	20.0953	1.51906
$176$	11.7857	0.888381
$177$	3.66662	0.275600
$178$	−17.7676	−1.33174
$179$	−8.79553	−0.657409	−0.328704	−	0.944433i	$-0.606612\pi$
$179$	−8.79553	−0.657409	−0.328704	+	0.944433i	$0.606612\pi$
$180$	0.0761038	0.00567244
$181$	−25.9716	−1.93045	−0.965225	−	0.261419i	$-0.915810\pi$
$181$	−25.9716	−1.93045	−0.965225	+	0.261419i	$0.915810\pi$
$182$	−42.6098	−3.15845
$183$	20.2711	1.49848
$184$	−23.3293	−1.71986
$185$	−6.49206	−0.477306
$186$	9.52725	0.698572
$187$	26.9471	1.97057
$188$	−2.35064	−0.171438
$189$	−24.2667	−1.76514
$190$	−13.9287	−1.01049
$191$	−8.01959	−0.580277	−0.290139	−	0.956985i	$-0.593701\pi$
$191$	−8.01959	−0.580277	−0.290139	+	0.956985i	$0.593701\pi$
$192$	18.9784	1.36965
$193$	−6.58521	−0.474013	−0.237007	−	0.971508i	$-0.576166\pi$
$193$	−6.58521	−0.474013	−0.237007	+	0.971508i	$0.576166\pi$
$194$	−2.88704	−0.207278
$195$	5.59584	0.400726
$196$	55.4508	3.96077
$197$	15.2775	1.08848	0.544239	−	0.838930i	$-0.316818\pi$
$197$	15.2775	1.08848	0.544239	+	0.838930i	$0.316818\pi$
$198$	−0.299568	−0.0212893
$199$	−9.94743	−0.705155	−0.352577	−	0.935783i	$-0.614695\pi$
$199$	−9.94743	−0.705155	−0.352577	+	0.935783i	$0.614695\pi$
$200$	17.3832	1.22918
$201$	9.44092	0.665911
$202$	−34.6789	−2.44000
$203$	4.68935	0.329128
$204$	−33.6190	−2.35380
$205$	9.36076	0.653784
$206$	−42.7256	−2.97683
$207$	0.139956	0.00972764
$208$	−8.69956	−0.603206
$209$	35.5872	2.46162
$210$	−16.4586	−1.13575
$211$	12.7028	0.874499	0.437250	−	0.899340i	$-0.355953\pi$
$211$	12.7028	0.874499	0.437250	+	0.899340i	$0.355953\pi$
$212$	−26.0124	−1.78654
$213$	−14.0936	−0.965681
$214$	−5.94439	−0.406350
$215$	−1.59769	−0.108961
$216$	−20.9917	−1.42831
$217$	−10.7611	−0.730514
$218$	−19.7858	−1.34006
$219$	−22.1621	−1.49758
$220$	16.1256	1.08719
$221$	−19.8909	−1.33800
$222$	−31.8819	−2.13977
$223$	2.82118	0.188920	0.0944601	−	0.995529i	$-0.469888\pi$
$223$	2.82118	0.188920	0.0944601	+	0.995529i	$0.469888\pi$
$224$	−12.4571	−0.832323
$225$	−0.104285	−0.00695232
$226$	−18.9346	−1.25951
$227$	−1.00542	−0.0667320	−0.0333660	−	0.999443i	$-0.510623\pi$
$227$	−1.00542	−0.0667320	−0.0333660	+	0.999443i	$0.510623\pi$
$228$	−44.3983	−2.94035
$229$	10.1453	0.670423	0.335212	−	0.942143i	$-0.391192\pi$
$229$	10.1453	0.670423	0.335212	+	0.942143i	$0.391192\pi$
$230$	−11.6070	−0.765343
$231$	42.0510	2.76675
$232$	4.05648	0.266321
$233$	17.7926	1.16563	0.582815	−	0.812605i	$-0.301951\pi$
$233$	17.7926	1.16563	0.582815	+	0.812605i	$0.301951\pi$
$234$	0.221124	0.0144553
$235$	−0.537204	−0.0350433
$236$	−7.79935	−0.507694
$237$	24.9837	1.62287
$238$	58.5035	3.79222
$239$	15.2096	0.983828	0.491914	−	0.870644i	$-0.336298\pi$
$239$	15.2096	0.983828	0.491914	+	0.870644i	$0.336298\pi$
$240$	−3.36033	−0.216908
$241$	1.34337	0.0865342	0.0432671	−	0.999064i	$-0.486223\pi$
$241$	1.34337	0.0865342	0.0432671	+	0.999064i	$0.486223\pi$
$242$	−37.2150	−2.39227
$243$	0.252895	0.0162232
$244$	−43.1191	−2.76042
$245$	12.6725	0.809613
$246$	45.9698	2.93093
$247$	−26.2685	−1.67142
$248$	−9.30883	−0.591111
$249$	29.4164	1.86419
$250$	18.7397	1.18520
$251$	−0.844141	−0.0532817	−0.0266408	−	0.999645i	$-0.508481\pi$
$251$	−0.844141	−0.0532817	−0.0266408	+	0.999645i	$0.508481\pi$
$252$	−0.422142	−0.0265924
$253$	29.6553	1.86441
$254$	17.0440	1.06944
$255$	−7.68313	−0.481136
$256$	−27.6859	−1.73037
$257$	17.1489	1.06972	0.534860	−	0.844941i	$-0.320364\pi$
$257$	17.1489	1.06972	0.534860	+	0.844941i	$0.320364\pi$
$258$	−7.84609	−0.488476
$259$	36.0110	2.23761
$260$	−11.9030	−0.738194
$261$	−0.0243354	−0.00150633
$262$	28.9435	1.78813
$263$	−2.22718	−0.137334	−0.0686669	−	0.997640i	$-0.521875\pi$
$263$	−2.22718	−0.137334	−0.0686669	+	0.997640i	$0.521875\pi$
$264$	36.3758	2.23878
$265$	−5.94475	−0.365183
$266$	77.2615	4.73721
$267$	−12.9431	−0.792105
$268$	−20.0820	−1.22670
$269$	10.9386	0.666937	0.333469	−	0.942761i	$-0.391781\pi$
$269$	10.9386	0.666937	0.333469	+	0.942761i	$0.391781\pi$
$270$	−10.4440	−0.635600
$271$	−10.7625	−0.653776	−0.326888	−	0.945063i	$-0.606000\pi$
$271$	−10.7625	−0.653776	−0.326888	+	0.945063i	$0.606000\pi$
$272$	11.9446	0.724245
$273$	−31.0397	−1.87861
$274$	−21.8369	−1.31922
$275$	−22.0969	−1.33249
$276$	−36.9977	−2.22700
$277$	24.0792	1.44678	0.723391	−	0.690439i	$-0.242583\pi$
$277$	24.0792	1.44678	0.723391	+	0.690439i	$0.242583\pi$
$278$	−2.38730	−0.143181
$279$	0.0558451	0.00334336
$280$	16.0813	0.961042
$281$	−2.44941	−0.146119	−0.0730597	−	0.997328i	$-0.523276\pi$
$281$	−2.44941	−0.146119	−0.0730597	+	0.997328i	$0.523276\pi$
$282$	−2.63816	−0.157100
$283$	21.6522	1.28709	0.643545	−	0.765408i	$-0.277463\pi$
$283$	21.6522	1.28709	0.643545	+	0.765408i	$0.277463\pi$
$284$	29.9789	1.77892
$285$	−10.1466	−0.601031
$286$	46.8539	2.77053
$287$	−51.9235	−3.06494
$288$	0.0646462	0.00380931
$289$	10.3103	0.606488
$290$	2.01821	0.118513
$291$	−2.10311	−0.123286
$292$	47.1415	2.75874
$293$	2.28735	0.133628	0.0668141	−	0.997765i	$-0.478717\pi$
$293$	2.28735	0.133628	0.0668141	+	0.997765i	$0.478717\pi$
$294$	62.2332	3.62952
$295$	−1.78243	−0.103777
$296$	31.1510	1.81061
$297$	26.6838	1.54835
$298$	42.2834	2.44941
$299$	−21.8899	−1.26593
$300$	27.5679	1.59163
$301$	8.86225	0.510811
$302$	−6.61418	−0.380604
$303$	−25.2623	−1.45128
$304$	15.7743	0.904720
$305$	−9.85423	−0.564251
$306$	−0.303605	−0.0173560
$307$	10.4936	0.598903	0.299451	−	0.954112i	$-0.403196\pi$
$307$	10.4936	0.598903	0.299451	+	0.954112i	$0.403196\pi$
$308$	−89.4475	−5.09674
$309$	−31.1241	−1.77059
$310$	−4.63140	−0.263046
$311$	28.7971	1.63293	0.816466	−	0.577393i	$-0.195930\pi$
$311$	28.7971	1.63293	0.816466	+	0.577393i	$0.195930\pi$
$312$	−26.8506	−1.52012
$313$	−1.19355	−0.0674633	−0.0337317	−	0.999431i	$-0.510739\pi$
$313$	−1.19355	−0.0674633	−0.0337317	+	0.999431i	$0.510739\pi$
$314$	−24.4347	−1.37893
$315$	−0.0964743	−0.00543571
$316$	−53.1434	−2.98955
$317$	15.9402	0.895290	0.447645	−	0.894211i	$-0.352263\pi$
$317$	15.9402	0.895290	0.447645	+	0.894211i	$0.352263\pi$
$318$	−29.1941	−1.63713
$319$	−5.15643	−0.288705
$320$	−9.22583	−0.515740
$321$	−4.33028	−0.241693
$322$	64.3832	3.58793
$323$	36.0668	2.00681
$324$	−33.5606	−1.86448
$325$	16.3107	0.904754
$326$	41.2511	2.28468
$327$	−14.4132	−0.797053
$328$	−44.9159	−2.48007
$329$	2.97983	0.164283
$330$	18.0980	0.996262
$331$	−0.0250694	−0.00137794	−0.000688970	−	1.00000i	$-0.500219\pi$
$331$	−0.0250694	−0.00137794	−0.000688970		1.00000i	$0.500219\pi$
$332$	−62.5723	−3.43410
$333$	−0.186880	−0.0102409
$334$	51.6324	2.82520
$335$	−4.58943	−0.250748
$336$	18.6395	1.01687
$337$	−15.7559	−0.858276	−0.429138	−	0.903239i	$-0.641183\pi$
$337$	−15.7559	−0.858276	−0.429138	+	0.903239i	$0.641183\pi$
$338$	−3.55009	−0.193099
$339$	−13.7932	−0.749142
$340$	16.3429	0.886320
$341$	11.8330	0.640793
$342$	−0.400950	−0.0216809
$343$	−37.4677	−2.02307
$344$	7.66621	0.413334
$345$	−8.45529	−0.455217
$346$	−3.33535	−0.179310
$347$	−12.5358	−0.672959	−0.336480	−	0.941691i	$-0.609236\pi$
$347$	−12.5358	−0.672959	−0.336480	+	0.941691i	$0.609236\pi$
$348$	6.43312	0.344852
$349$	7.66006	0.410034	0.205017	−	0.978758i	$-0.434275\pi$
$349$	7.66006	0.410034	0.205017	+	0.978758i	$0.434275\pi$
$350$	−47.9734	−2.56429
$351$	−19.6965	−1.05132
$352$	13.6979	0.730098
$353$	24.9842	1.32977	0.664887	−	0.746944i	$-0.268480\pi$
$353$	24.9842	1.32977	0.664887	+	0.746944i	$0.268480\pi$
$354$	−8.75333	−0.465234
$355$	6.85123	0.363625
$356$	27.5315	1.45917
$357$	42.6178	2.25557
$358$	20.9976	1.10976
$359$	19.3854	1.02312	0.511561	−	0.859247i	$-0.329067\pi$
$359$	19.3854	1.02312	0.511561	+	0.859247i	$0.329067\pi$
$360$	−0.0834542	−0.00439842
$361$	28.6309	1.50689
$362$	62.0019	3.25875
$363$	−27.1098	−1.42290
$364$	66.0252	3.46066
$365$	10.7735	0.563910
$366$	−48.3932	−2.52955
$367$	−1.26265	−0.0659099	−0.0329549	−	0.999457i	$-0.510492\pi$
$367$	−1.26265	−0.0659099	−0.0329549	+	0.999457i	$0.510492\pi$
$368$	13.1450	0.685230
$369$	0.269458	0.0140274
$370$	15.4985	0.805728
$371$	32.9751	1.71198
$372$	−14.7628	−0.765414
$373$	−31.4749	−1.62971	−0.814854	−	0.579666i	$-0.803183\pi$
$373$	−31.4749	−1.62971	−0.814854	+	0.579666i	$0.803183\pi$
$374$	−64.3308	−3.32647
$375$	13.6512	0.704945
$376$	2.57767	0.132933
$377$	3.80619	0.196029
$378$	57.9319	2.97970
$379$	2.26358	0.116272	0.0581360	−	0.998309i	$-0.481484\pi$
$379$	2.26358	0.116272	0.0581360	+	0.998309i	$0.481484\pi$
$380$	21.5830	1.10718
$381$	12.4160	0.636089
$382$	19.1452	0.979552
$383$	13.2728	0.678207	0.339103	−	0.940749i	$-0.389876\pi$
$383$	13.2728	0.678207	0.339103	+	0.940749i	$0.389876\pi$
$384$	−36.0677	−1.84057
$385$	−20.4419	−1.04182
$386$	15.7208	0.800170
$387$	−0.0459908	−0.00233784
$388$	4.47356	0.227111
$389$	−18.8803	−0.957271	−0.478636	−	0.878014i	$-0.658868\pi$
$389$	−18.8803	−0.957271	−0.478636	+	0.878014i	$0.658868\pi$
$390$	−13.3589	−0.676456
$391$	30.0550	1.51995
$392$	−60.8065	−3.07119
$393$	21.0843	1.06356
$394$	−36.4720	−1.83743
$395$	−12.1451	−0.611088
$396$	0.464189	0.0233264
$397$	20.4786	1.02779	0.513896	−	0.857852i	$-0.328202\pi$
$397$	20.4786	1.02779	0.513896	+	0.857852i	$0.328202\pi$
$398$	23.7475	1.19035
$399$	56.2823	2.81764
$400$	−9.79464	−0.489732
$401$	−9.11225	−0.455044	−0.227522	−	0.973773i	$-0.573062\pi$
$401$	−9.11225	−0.455044	−0.227522	+	0.973773i	$0.573062\pi$
$402$	−22.5383	−1.12411
$403$	−8.73447	−0.435095
$404$	53.7360	2.67347
$405$	−7.66978	−0.381114
$406$	−11.1949	−0.555592
$407$	−39.5979	−1.96279
$408$	36.8661	1.82514
$409$	22.7266	1.12376	0.561880	−	0.827219i	$-0.310078\pi$
$409$	22.7266	1.12376	0.561880	+	0.827219i	$0.310078\pi$
$410$	−22.3469	−1.10364
$411$	−15.9074	−0.784656
$412$	66.2046	3.26167
$413$	9.88698	0.486507
$414$	−0.334118	−0.0164210
$415$	−14.3000	−0.701958
$416$	−10.1110	−0.495732
$417$	−1.73906	−0.0851622
$418$	−84.9572	−4.15539
$419$	−16.3103	−0.796809	−0.398404	−	0.917210i	$-0.630436\pi$
$419$	−16.3103	−0.796809	−0.398404	+	0.917210i	$0.630436\pi$
$420$	25.5032	1.24443
$421$	26.4037	1.28684	0.643418	−	0.765515i	$-0.277516\pi$
$421$	26.4037	1.28684	0.643418	+	0.765515i	$0.277516\pi$
$422$	−30.3255	−1.47622
$423$	−0.0154639	−0.000751880 0
$424$	28.5248	1.38529
$425$	−22.3947	−1.08630
$426$	33.6457	1.63014
$427$	54.6607	2.64522
$428$	9.21102	0.445231
$429$	34.1314	1.64788
$430$	3.81415	0.183935
$431$	9.26783	0.446416	0.223208	−	0.974771i	$-0.428347\pi$
$431$	9.26783	0.446416	0.223208	+	0.974771i	$0.428347\pi$
$432$	11.8278	0.569068
$433$	−16.9547	−0.814790	−0.407395	−	0.913252i	$-0.633563\pi$
$433$	−16.9547	−0.814790	−0.407395	+	0.913252i	$0.633563\pi$
$434$	25.6901	1.23316
$435$	1.47019	0.0704904
$436$	30.6586	1.46828
$437$	39.6915	1.89870
$438$	52.9076	2.52802
$439$	−17.3066	−0.825997	−0.412998	−	0.910732i	$-0.635518\pi$
$439$	−17.3066	−0.825997	−0.412998	+	0.910732i	$0.635518\pi$
$440$	−17.6831	−0.843008
$441$	0.364788	0.0173708
$442$	47.4854	2.25865
$443$	8.97033	0.426193	0.213097	−	0.977031i	$-0.431645\pi$
$443$	8.97033	0.426193	0.213097	+	0.977031i	$0.431645\pi$
$444$	49.4020	2.34451
$445$	6.29192	0.298266
$446$	−6.73500	−0.318912
$447$	30.8020	1.45688
$448$	51.1750	2.41779
$449$	9.43955	0.445480	0.222740	−	0.974878i	$-0.428500\pi$
$449$	9.43955	0.445480	0.222740	+	0.974878i	$0.428500\pi$
$450$	0.248959	0.0117360
$451$	57.0953	2.68851
$452$	29.3397	1.38002
$453$	−4.81820	−0.226379
$454$	2.40024	0.112649
$455$	15.0891	0.707387
$456$	48.6865	2.27995
$457$	2.38636	0.111629	0.0558147	−	0.998441i	$-0.482224\pi$
$457$	2.38636	0.111629	0.0558147	+	0.998441i	$0.482224\pi$
$458$	−24.2200	−1.13172
$459$	27.0435	1.26228
$460$	17.9854	0.838574
$461$	28.3684	1.32125	0.660625	−	0.750716i	$-0.270291\pi$
$461$	28.3684	1.32125	0.660625	+	0.750716i	$0.270291\pi$
$462$	−100.388	−4.67048
$463$	−7.57761	−0.352161	−0.176081	−	0.984376i	$-0.556342\pi$
$463$	−7.57761	−0.352161	−0.176081	+	0.984376i	$0.556342\pi$
$464$	−2.28563	−0.106108
$465$	−3.37381	−0.156457
$466$	−42.4762	−1.96767
$467$	0.664723	0.0307597	0.0153799	−	0.999882i	$-0.495104\pi$
$467$	0.664723	0.0307597	0.0153799	+	0.999882i	$0.495104\pi$
$468$	−0.342639	−0.0158385
$469$	25.4572	1.17551
$470$	1.28247	0.0591557
$471$	−17.7998	−0.820173
$472$	8.55264	0.393667
$473$	−9.74497	−0.448074
$474$	−59.6436	−2.73952
$475$	−29.5751	−1.35700
$476$	−90.6530	−4.15508
$477$	−0.171125	−0.00783527
$478$	−36.3099	−1.66077
$479$	29.2555	1.33672	0.668360	−	0.743838i	$-0.266997\pi$
$479$	29.2555	1.33672	0.668360	+	0.743838i	$0.266997\pi$
$480$	−3.90552	−0.178262
$481$	29.2289	1.33272
$482$	−3.20703	−0.146076
$483$	46.9008	2.13406
$484$	57.6658	2.62117
$485$	1.02237	0.0464233
$486$	−0.603736	−0.0273860
$487$	−11.5266	−0.522320	−0.261160	−	0.965296i	$-0.584105\pi$
$487$	−11.5266	−0.522320	−0.261160	+	0.965296i	$0.584105\pi$
$488$	47.2837	2.14043
$489$	30.0499	1.35890
$490$	−30.2529	−1.36669
$491$	20.1191	0.907964	0.453982	−	0.891011i	$-0.350003\pi$
$491$	20.1191	0.907964	0.453982	+	0.891011i	$0.350003\pi$
$492$	−71.2317	−3.21137
$493$	−5.22593	−0.235364
$494$	62.7107	2.82149
$495$	0.106084	0.00476810
$496$	5.24509	0.235511
$497$	−38.0032	−1.70468
$498$	−70.2258	−3.14689
$499$	5.41955	0.242613	0.121306	−	0.992615i	$-0.461292\pi$
$499$	5.41955	0.242613	0.121306	+	0.992615i	$0.461292\pi$
$500$	−29.0378	−1.29861
$501$	37.6123	1.68040
$502$	2.01522	0.0899435
$503$	1.64202	0.0732141	0.0366070	−	0.999330i	$-0.488345\pi$
$503$	1.64202	0.0732141	0.0366070	+	0.999330i	$0.488345\pi$
$504$	0.462914	0.0206198
$505$	12.2806	0.546478
$506$	−70.7960	−3.14727
$507$	−2.58611	−0.114853
$508$	−26.4102	−1.17176
$509$	−42.8046	−1.89728	−0.948641	−	0.316356i	$-0.897541\pi$
$509$	−42.8046	−1.89728	−0.948641	+	0.316356i	$0.897541\pi$
$510$	18.3419	0.812194
$511$	−59.7597	−2.64361
$512$	24.6149	1.08784
$513$	35.7144	1.57683
$514$	−40.9396	−1.80577
$515$	15.1301	0.666711
$516$	12.1578	0.535215
$517$	−3.27664	−0.144106
$518$	−85.9689	−3.77726
$519$	−2.42968	−0.106651
$520$	13.0527	0.572397
$521$	3.97585	0.174185	0.0870925	−	0.996200i	$-0.472242\pi$
$521$	3.97585	0.174185	0.0870925	+	0.996200i	$0.472242\pi$
$522$	0.0580960	0.00254279
$523$	−8.44329	−0.369199	−0.184600	−	0.982814i	$-0.559099\pi$
$523$	−8.44329	−0.369199	−0.184600	+	0.982814i	$0.559099\pi$
$524$	−44.8488	−1.95923
$525$	−34.9469	−1.52521
$526$	5.31694	0.231830
$527$	11.9925	0.522401
$528$	−20.4961	−0.891977
$529$	10.0755	0.438066
$530$	14.1919	0.616457
$531$	−0.0513087	−0.00222661
$532$	−119.719	−5.19048
$533$	−42.1446	−1.82548
$534$	30.8990	1.33713
$535$	2.10504	0.0910089
$536$	22.0216	0.951187
$537$	15.2960	0.660070
$538$	−26.1137	−1.12584
$539$	77.2947	3.32932
$540$	16.1832	0.696416
$541$	13.6163	0.585410	0.292705	−	0.956203i	$-0.405445\pi$
$541$	13.6163	0.585410	0.292705	+	0.956203i	$0.405445\pi$
$542$	25.6933	1.10362
$543$	45.1661	1.93826
$544$	13.8825	0.595206
$545$	7.00658	0.300129
$546$	74.1010	3.17123
$547$	−5.31344	−0.227186	−0.113593	−	0.993527i	$-0.536236\pi$
$547$	−5.31344	−0.227186	−0.113593	+	0.993527i	$0.536236\pi$
$548$	33.8370	1.44545
$549$	−0.283663	−0.0121064
$550$	52.7518	2.24935
$551$	−6.90151	−0.294014
$552$	40.5711	1.72682
$553$	67.3682	2.86479
$554$	−57.4843	−2.44228
$555$	11.2901	0.479238
$556$	3.69919	0.156881
$557$	35.0878	1.48672	0.743359	−	0.668892i	$-0.233231\pi$
$557$	35.0878	1.48672	0.743359	+	0.668892i	$0.233231\pi$
$558$	−0.133319	−0.00564384
$559$	7.19320	0.304240
$560$	−9.06106	−0.382900
$561$	−46.8627	−1.97854
$562$	5.84747	0.246661
$563$	27.5631	1.16165	0.580823	−	0.814030i	$-0.302731\pi$
$563$	27.5631	1.16165	0.580823	+	0.814030i	$0.302731\pi$
$564$	4.08791	0.172132
$565$	6.70516	0.282088
$566$	−51.6903	−2.17271
$567$	42.5437	1.78667
$568$	−32.8744	−1.37938
$569$	−22.4042	−0.939231	−0.469616	−	0.882871i	$-0.655608\pi$
$569$	−22.4042	−0.939231	−0.469616	+	0.882871i	$0.655608\pi$
$570$	24.2229	1.01459
$571$	−21.7372	−0.909672	−0.454836	−	0.890575i	$-0.650302\pi$
$571$	−21.7372	−0.909672	−0.454836	+	0.890575i	$0.650302\pi$
$572$	−72.6016	−3.03562
$573$	13.9466	0.582626
$574$	123.957	5.17386
$575$	−24.6454	−1.02778
$576$	−0.265574	−0.0110656
$577$	−12.6660	−0.527290	−0.263645	−	0.964620i	$-0.584925\pi$
$577$	−12.6660	−0.527290	−0.263645	+	0.964620i	$0.584925\pi$
$578$	−24.6138	−1.02380
$579$	11.4521	0.475932
$580$	−3.12728	−0.129853
$581$	79.3208	3.29078
$582$	5.02075	0.208117
$583$	−36.2596	−1.50172
$584$	−51.6946	−2.13914
$585$	−0.0783050	−0.00323751
$586$	−5.46058	−0.225574
$587$	−43.7442	−1.80552	−0.902759	−	0.430146i	$-0.858462\pi$
$587$	−43.7442	−1.80552	−0.902759	+	0.430146i	$0.858462\pi$
$588$	−96.4323	−3.97680
$589$	15.8376	0.652579
$590$	4.25518	0.175183
$591$	−26.5685	−1.09288
$592$	−17.5521	−0.721388
$593$	−15.9730	−0.655932	−0.327966	−	0.944689i	$-0.606363\pi$
$593$	−15.9730	−0.655932	−0.327966	+	0.944689i	$0.606363\pi$
$594$	−63.7022	−2.61373
$595$	−20.7174	−0.849331
$596$	−65.5195	−2.68378
$597$	17.2992	0.708009
$598$	52.2577	2.13698
$599$	−34.1617	−1.39581	−0.697904	−	0.716191i	$-0.745884\pi$
$599$	−34.1617	−1.39581	−0.697904	+	0.716191i	$0.745884\pi$
$600$	−30.2305	−1.23416
$601$	39.5622	1.61378	0.806888	−	0.590704i	$-0.201150\pi$
$601$	39.5622	1.61378	0.806888	+	0.590704i	$0.201150\pi$
$602$	−21.1568	−0.862288
$603$	−0.132111	−0.00537997
$604$	10.2489	0.417021
$605$	13.1787	0.535789
$606$	60.3087	2.44987
$607$	30.8069	1.25041	0.625207	−	0.780459i	$-0.285015\pi$
$607$	30.8069	1.25041	0.625207	+	0.780459i	$0.285015\pi$
$608$	18.3336	0.743526
$609$	−8.15506	−0.330460
$610$	23.5250	0.952499
$611$	2.41863	0.0978473
$612$	0.470446	0.0190166
$613$	−41.0767	−1.65907	−0.829537	−	0.558452i	$-0.811396\pi$
$613$	−41.0767	−1.65907	−0.829537	+	0.558452i	$0.811396\pi$
$614$	−25.0514	−1.01099
$615$	−16.2789	−0.656430
$616$	98.0867	3.95203
$617$	30.0320	1.20904	0.604522	−	0.796589i	$-0.293364\pi$
$617$	30.0320	1.20904	0.604522	+	0.796589i	$0.293364\pi$
$618$	74.3024	2.98888
$619$	28.2036	1.13360	0.566798	−	0.823857i	$-0.308182\pi$
$619$	28.2036	1.13360	0.566798	+	0.823857i	$0.308182\pi$
$620$	7.17650	0.288215
$621$	29.7613	1.19428
$622$	−68.7472	−2.75651
$623$	−34.9008	−1.39827
$624$	15.1291	0.605647
$625$	14.7904	0.591615
$626$	2.84935	0.113883
$627$	−61.8883	−2.47158
$628$	37.8624	1.51087
$629$	−40.1315	−1.60015
$630$	0.230313	0.00917588
$631$	10.6501	0.423974	0.211987	−	0.977272i	$-0.432006\pi$
$631$	10.6501	0.423974	0.211987	+	0.977272i	$0.432006\pi$
$632$	58.2762	2.31810
$633$	−22.0910	−0.878039
$634$	−38.0540	−1.51132
$635$	−6.03566	−0.239518
$636$	45.2372	1.79377
$637$	−57.0547	−2.26059
$638$	12.3099	0.487355
$639$	0.197219	0.00780184
$640$	17.5333	0.693064
$641$	−5.02340	−0.198413	−0.0992063	−	0.995067i	$-0.531630\pi$
$641$	−5.02340	−0.198413	−0.0992063	+	0.995067i	$0.531630\pi$
$642$	10.3377	0.407995
$643$	−0.263155	−0.0103778	−0.00518891	−	0.999987i	$-0.501652\pi$
$643$	−0.263155	−0.0103778	−0.00518891	+	0.999987i	$0.501652\pi$
$644$	−99.7637	−3.93124
$645$	2.77848	0.109402
$646$	−86.1022	−3.38765
$647$	25.8787	1.01740	0.508699	−	0.860945i	$-0.330127\pi$
$647$	25.8787	1.01740	0.508699	+	0.860945i	$0.330127\pi$
$648$	36.8020	1.44572
$649$	−10.8718	−0.426754
$650$	−38.9385	−1.52729
$651$	18.7143	0.733471
$652$	−63.9198	−2.50329
$653$	7.75215	0.303365	0.151683	−	0.988429i	$-0.451531\pi$
$653$	7.75215	0.303365	0.151683	+	0.988429i	$0.451531\pi$
$654$	34.4086	1.34548
$655$	−10.2495	−0.400482
$656$	25.3080	0.988112
$657$	0.310124	0.0120991
$658$	−7.11374	−0.277323
$659$	−38.8802	−1.51456	−0.757279	−	0.653091i	$-0.773472\pi$
$659$	−38.8802	−1.51456	−0.757279	+	0.653091i	$0.773472\pi$
$660$	−28.0434	−1.09159
$661$	−8.39517	−0.326534	−0.163267	−	0.986582i	$-0.552203\pi$
$661$	−8.39517	−0.326534	−0.163267	+	0.986582i	$0.552203\pi$
$662$	0.0598482	0.00232606
$663$	34.5914	1.34342
$664$	68.6158	2.66281
$665$	−27.3600	−1.06098
$666$	0.446137	0.0172875
$667$	−5.75113	−0.222685
$668$	−80.0059	−3.09552
$669$	−4.90621	−0.189685
$670$	10.9563	0.423281
$671$	−60.1051	−2.32033
$672$	21.6636	0.835692
$673$	21.1170	0.814002	0.407001	−	0.913428i	$-0.366575\pi$
$673$	21.1170	0.814002	0.407001	+	0.913428i	$0.366575\pi$
$674$	37.6139	1.44883
$675$	−22.1759	−0.853550
$676$	5.50097	0.211576
$677$	−0.766745	−0.0294684	−0.0147342	−	0.999891i	$-0.504690\pi$
$677$	−0.766745	−0.0294684	−0.0147342	+	0.999891i	$0.504690\pi$
$678$	32.9284	1.26461
$679$	−5.67099	−0.217633
$680$	−17.9214	−0.687255
$681$	1.74849	0.0670021
$682$	−28.2489	−1.08171
$683$	42.6367	1.63145	0.815723	−	0.578442i	$-0.196339\pi$
$683$	42.6367	1.63145	0.815723	+	0.578442i	$0.196339\pi$
$684$	0.621285	0.0237554
$685$	7.73295	0.295461
$686$	89.4466	3.41509
$687$	−17.6434	−0.673137
$688$	−4.31955	−0.164681
$689$	26.7648	1.01966
$690$	20.1853	0.768441
$691$	18.0926	0.688275	0.344137	−	0.938919i	$-0.388171\pi$
$691$	18.0926	0.688275	0.344137	+	0.938919i	$0.388171\pi$
$692$	5.16823	0.196467
$693$	−0.588438	−0.0223529
$694$	29.9268	1.13601
$695$	0.845395	0.0320677
$696$	−7.05446	−0.267399
$697$	57.8648	2.19179
$698$	−18.2869	−0.692168
$699$	−30.9424	−1.17035
$700$	74.3363	2.80965
$701$	35.3252	1.33422	0.667108	−	0.744961i	$-0.267532\pi$
$701$	35.3252	1.33422	0.667108	+	0.744961i	$0.267532\pi$
$702$	47.0214	1.77471
$703$	−52.9989	−1.99889
$704$	−56.2723	−2.12084
$705$	0.934231	0.0351852
$706$	−59.6447	−2.24476
$707$	−68.1194	−2.56189
$708$	13.5635	0.509749
$709$	49.7073	1.86680	0.933399	−	0.358841i	$-0.116828\pi$
$709$	49.7073	1.86680	0.933399	+	0.358841i	$0.116828\pi$
$710$	−16.3559	−0.613827
$711$	−0.349608	−0.0131113
$712$	−30.1906	−1.13144
$713$	13.1977	0.494259
$714$	−101.741	−3.80757
$715$	−16.5920	−0.620506
$716$	−32.5364	−1.21594
$717$	−26.4504	−0.987810
$718$	−46.2787	−1.72711
$719$	22.7922	0.850006	0.425003	−	0.905192i	$-0.360273\pi$
$719$	22.7922	0.850006	0.425003	+	0.905192i	$0.360273\pi$
$720$	0.0470225	0.00175243
$721$	−83.9255	−3.12555
$722$	−68.3505	−2.54374
$723$	−2.33621	−0.0868845
$724$	−96.0738	−3.57055
$725$	4.28531	0.159152
$726$	64.7192	2.40195
$727$	−44.8282	−1.66259	−0.831293	−	0.555835i	$-0.812399\pi$
$727$	−44.8282	−1.66259	−0.831293	+	0.555835i	$0.812399\pi$
$728$	−72.4022	−2.68340
$729$	26.7774	0.991757
$730$	−25.7195	−0.951922
$731$	−9.87631	−0.365289
$732$	74.9867	2.77159
$733$	34.4519	1.27251	0.636254	−	0.771480i	$-0.280483\pi$
$733$	34.4519	1.27251	0.636254	+	0.771480i	$0.280483\pi$
$734$	3.01433	0.111261
$735$	−22.0382	−0.812890
$736$	15.2777	0.563142
$737$	−27.9929	−1.03113
$738$	−0.643276	−0.0236793
$739$	−19.0482	−0.700699	−0.350349	−	0.936619i	$-0.613937\pi$
$739$	−19.0482	−0.700699	−0.350349	+	0.936619i	$0.613937\pi$
$740$	−24.0154	−0.882823
$741$	45.6825	1.67819
$742$	−78.7214	−2.88995
$743$	28.1008	1.03092	0.515459	−	0.856914i	$-0.327621\pi$
$743$	28.1008	1.03092	0.515459	+	0.856914i	$0.327621\pi$
$744$	16.1886	0.593504
$745$	−14.9735	−0.548587
$746$	75.1399	2.75107
$747$	−0.411637	−0.0150610
$748$	99.6825	3.64475
$749$	−11.6765	−0.426650
$750$	−32.5895	−1.19000
$751$	−5.13637	−0.187429	−0.0937144	−	0.995599i	$-0.529874\pi$
$751$	−5.13637	−0.187429	−0.0937144	+	0.995599i	$0.529874\pi$
$752$	−1.45240	−0.0529635
$753$	1.46801	0.0534974
$754$	−9.08651	−0.330911
$755$	2.34223	0.0852425
$756$	−89.7673	−3.26480
$757$	−39.3249	−1.42929	−0.714644	−	0.699489i	$-0.753411\pi$
$757$	−39.3249	−1.42929	−0.714644	+	0.699489i	$0.753411\pi$
$758$	−5.40383	−0.196276
$759$	−51.5724	−1.87196
$760$	−23.6675	−0.858512
$761$	27.7788	1.00698	0.503490	−	0.864001i	$-0.332049\pi$
$761$	27.7788	1.00698	0.503490	+	0.864001i	$0.332049\pi$
$762$	−29.6406	−1.07376
$763$	−38.8650	−1.40701
$764$	−29.6660	−1.07328
$765$	0.107513	0.00388715
$766$	−31.6861	−1.14486
$767$	8.02494	0.289764
$768$	48.1474	1.73737
$769$	29.1830	1.05237	0.526183	−	0.850371i	$-0.323623\pi$
$769$	29.1830	1.05237	0.526183	+	0.850371i	$0.323623\pi$
$770$	48.8009	1.75866
$771$	−29.8230	−1.07405
$772$	−24.3599	−0.876733
$773$	−25.8716	−0.930538	−0.465269	−	0.885169i	$-0.654042\pi$
$773$	−25.8716	−0.930538	−0.465269	+	0.885169i	$0.654042\pi$
$774$	0.109794	0.00394645
$775$	−9.83395	−0.353246
$776$	−4.90564	−0.176102
$777$	−62.6253	−2.24667
$778$	45.0730	1.61595
$779$	76.4180	2.73796
$780$	20.7001	0.741182
$781$	41.7886	1.49531
$782$	−71.7502	−2.56578
$783$	−5.17486	−0.184935
$784$	34.2616	1.22363
$785$	8.65289	0.308835
$786$	−50.3345	−1.79537
$787$	37.7608	1.34603	0.673014	−	0.739630i	$-0.264999\pi$
$787$	37.7608	1.34603	0.673014	+	0.739630i	$0.264999\pi$
$788$	56.5145	2.01324
$789$	3.87320	0.137890
$790$	28.9941	1.03156
$791$	−37.1930	−1.32243
$792$	−0.509023	−0.0180873
$793$	44.3663	1.57549
$794$	−48.8885	−1.73499
$795$	10.3383	0.366662
$796$	−36.7975	−1.30425
$797$	12.7850	0.452868	0.226434	−	0.974026i	$-0.427293\pi$
$797$	12.7850	0.452868	0.226434	+	0.974026i	$0.427293\pi$
$798$	−134.363	−4.75638
$799$	−3.32080	−0.117481
$800$	−11.3838	−0.402476
$801$	0.181118	0.00639950
$802$	21.7536	0.768148
$803$	65.7121	2.31893
$804$	34.9238	1.23167
$805$	−22.7995	−0.803577
$806$	20.8518	0.734472
$807$	−19.0229	−0.669637
$808$	−58.9261	−2.07301
$809$	24.3669	0.856696	0.428348	−	0.903614i	$-0.359096\pi$
$809$	24.3669	0.856696	0.428348	+	0.903614i	$0.359096\pi$
$810$	18.3100	0.643349
$811$	−9.95894	−0.349706	−0.174853	−	0.984595i	$-0.555945\pi$
$811$	−9.95894	−0.349706	−0.174853	+	0.984595i	$0.555945\pi$
$812$	17.3468	0.608753
$813$	18.7167	0.656422
$814$	94.5319	3.31334
$815$	−14.6079	−0.511693
$816$	−20.7723	−0.727177
$817$	−13.0430	−0.456315
$818$	−54.2552	−1.89699
$819$	0.434352	0.0151775
$820$	34.6273	1.20924
$821$	−34.7608	−1.21316	−0.606581	−	0.795022i	$-0.707459\pi$
$821$	−34.7608	−1.21316	−0.606581	+	0.795022i	$0.707459\pi$
$822$	37.9758	1.32456
$823$	30.0994	1.04920	0.524599	−	0.851349i	$-0.324215\pi$
$823$	30.0994	1.04920	0.524599	+	0.851349i	$0.324215\pi$
$824$	−72.5990	−2.52910
$825$	38.4278	1.33789
$826$	−23.6032	−0.821259
$827$	−39.3703	−1.36904	−0.684519	−	0.728995i	$-0.739988\pi$
$827$	−39.3703	−1.36904	−0.684519	+	0.728995i	$0.739988\pi$
$828$	0.517725	0.0179922
$829$	30.6365	1.06405	0.532025	−	0.846729i	$-0.321431\pi$
$829$	30.6365	1.06405	0.532025	+	0.846729i	$0.321431\pi$
$830$	34.1383	1.18496
$831$	−41.8753	−1.45264
$832$	41.5371	1.44004
$833$	78.3365	2.71420
$834$	4.15166	0.143760
$835$	−18.2842	−0.632750
$836$	131.644	4.55299
$837$	11.8753	0.410471
$838$	38.9375	1.34507
$839$	−53.9308	−1.86190	−0.930948	−	0.365151i	$-0.881017\pi$
$839$	−53.9308	−1.86190	−0.930948	+	0.365151i	$0.881017\pi$
$840$	−27.9664	−0.964932
$841$	1.00000	0.0344828
$842$	−63.0334	−2.17227
$843$	4.25967	0.146711
$844$	46.9902	1.61747
$845$	1.25717	0.0432478
$846$	0.0369169	0.00126923
$847$	−73.1010	−2.51178
$848$	−16.0724	−0.551929
$849$	−37.6545	−1.29230
$850$	53.4628	1.83376
$851$	−44.1648	−1.51395
$852$	−52.1351	−1.78612
$853$	15.5981	0.534069	0.267035	−	0.963687i	$-0.413956\pi$
$853$	15.5981	0.534069	0.267035	+	0.963687i	$0.413956\pi$
$854$	−130.491	−4.46532
$855$	0.141985	0.00485580
$856$	−10.1007	−0.345233
$857$	−19.3888	−0.662310	−0.331155	−	0.943576i	$-0.607438\pi$
$857$	−19.3888	−0.662310	−0.331155	+	0.943576i	$0.607438\pi$
$858$	−81.4819	−2.78174
$859$	−23.7623	−0.810758	−0.405379	−	0.914149i	$-0.632861\pi$
$859$	−23.7623	−0.810758	−0.405379	+	0.914149i	$0.632861\pi$
$860$	−5.91015	−0.201534
$861$	90.2981	3.07735
$862$	−22.1251	−0.753583
$863$	17.6593	0.601129	0.300565	−	0.953761i	$-0.402825\pi$
$863$	17.6593	0.601129	0.300565	+	0.953761i	$0.402825\pi$
$864$	13.7468	0.467677
$865$	1.18112	0.0401594
$866$	40.4759	1.37543
$867$	−17.9302	−0.608943
$868$	−39.8075	−1.35116
$869$	−74.0784	−2.51294
$870$	−3.50979	−0.118993
$871$	20.6628	0.700133
$872$	−33.6198	−1.13851
$873$	0.0294297	0.000996045 0
$874$	−94.7555	−3.20515
$875$	36.8102	1.24441
$876$	−81.9819	−2.76991
$877$	−13.0563	−0.440879	−0.220439	−	0.975401i	$-0.570749\pi$
$877$	−13.0563	−0.440879	−0.220439	+	0.975401i	$0.570749\pi$
$878$	41.3159	1.39434
$879$	−3.97784	−0.134169
$880$	9.96359	0.335872
$881$	9.78938	0.329813	0.164906	−	0.986309i	$-0.447268\pi$
$881$	9.78938	0.329813	0.164906	+	0.986309i	$0.447268\pi$
$882$	−0.870857	−0.0293233
$883$	13.6974	0.460954	0.230477	−	0.973078i	$-0.425971\pi$
$883$	13.6974	0.460954	0.230477	+	0.973078i	$0.425971\pi$
$884$	−73.5801	−2.47477
$885$	3.09975	0.104197
$886$	−21.4149	−0.719446
$887$	36.1901	1.21515	0.607573	−	0.794264i	$-0.292143\pi$
$887$	36.1901	1.21515	0.607573	+	0.794264i	$0.292143\pi$
$888$	−54.1735	−1.81794
$889$	33.4794	1.12286
$890$	−15.0207	−0.503495
$891$	−46.7813	−1.56723
$892$	10.4361	0.349426
$893$	−4.38554	−0.146757
$894$	−73.5335	−2.45933
$895$	−7.43570	−0.248548
$896$	−97.2558	−3.24909
$897$	38.0679	1.27105
$898$	−22.5350	−0.752004
$899$	−2.29481	−0.0765361
$900$	−0.385770	−0.0128590
$901$	−36.7483	−1.22426
$902$	−136.303	−4.53841
$903$	−15.4120	−0.512879
$904$	−32.1735	−1.07007
$905$	−21.9562	−0.729850
$906$	11.5025	0.382144
$907$	−41.2646	−1.37017	−0.685085	−	0.728463i	$-0.740235\pi$
$907$	−41.2646	−1.37017	−0.685085	+	0.728463i	$0.740235\pi$
$908$	−3.71924	−0.123427
$909$	0.353507	0.0117251
$910$	−36.0221	−1.19412
$911$	57.6631	1.91046	0.955232	−	0.295858i	$-0.0956054\pi$
$911$	57.6631	1.91046	0.955232	+	0.295858i	$0.0956054\pi$
$912$	−27.4325	−0.908382
$913$	−87.2216	−2.88661
$914$	−5.69696	−0.188439
$915$	17.1371	0.566535
$916$	37.5296	1.24001
$917$	56.8534	1.87746
$918$	−64.5608	−2.13082
$919$	−4.49239	−0.148190	−0.0740951	−	0.997251i	$-0.523607\pi$
$919$	−4.49239	−0.148190	−0.0740951	+	0.997251i	$0.523607\pi$
$920$	−19.7225	−0.650232
$921$	−18.2491	−0.601327
$922$	−67.7239	−2.23037
$923$	−30.8460	−1.01531
$924$	155.555	5.11737
$925$	32.9082	1.08202
$926$	18.0900	0.594475
$927$	0.435533	0.0143048
$928$	−2.65646	−0.0872026
$929$	21.4903	0.705074	0.352537	−	0.935798i	$-0.385319\pi$
$929$	21.4903	0.705074	0.352537	+	0.935798i	$0.385319\pi$
$930$	8.05430	0.264111
$931$	103.453	3.39055
$932$	65.8182	2.15595
$933$	−50.0799	−1.63954
$934$	−1.58689	−0.0519247
$935$	22.7810	0.745017
$936$	0.375732	0.0122812
$937$	−12.0580	−0.393917	−0.196958	−	0.980412i	$-0.563106\pi$
$937$	−12.0580	−0.393917	−0.196958	+	0.980412i	$0.563106\pi$
$938$	−60.7740	−1.98434
$939$	2.07565	0.0677364
$940$	−1.98722	−0.0648160
$941$	8.91419	0.290594	0.145297	−	0.989388i	$-0.453586\pi$
$941$	8.91419	0.290594	0.145297	+	0.989388i	$0.453586\pi$
$942$	42.4935	1.38451
$943$	63.6802	2.07371
$944$	−4.81901	−0.156846
$945$	−20.5150	−0.667352
$946$	23.2642	0.756383
$947$	11.4805	0.373066	0.186533	−	0.982449i	$-0.440275\pi$
$947$	11.4805	0.373066	0.186533	+	0.982449i	$0.440275\pi$
$948$	92.4196	3.00165
$949$	−48.5050	−1.57454
$950$	70.6046	2.29072
$951$	−27.7210	−0.898914
$952$	99.4087	3.22186
$953$	54.8859	1.77793	0.888964	−	0.457977i	$-0.151426\pi$
$953$	54.8859	1.77793	0.888964	+	0.457977i	$0.151426\pi$
$954$	0.408526	0.0132265
$955$	−6.77973	−0.219387
$956$	56.2632	1.81968
$957$	8.96735	0.289873
$958$	−69.8417	−2.25648
$959$	−42.8941	−1.38512
$960$	16.0443	0.517827
$961$	−25.7339	−0.830125
$962$	−69.7782	−2.24974
$963$	0.0605955	0.00195266
$964$	4.96939	0.160053
$965$	−5.56710	−0.179211
$966$	−111.966	−3.60246
$967$	37.2346	1.19738	0.598692	−	0.800979i	$-0.295687\pi$
$967$	37.2346	1.19738	0.598692	+	0.800979i	$0.295687\pi$
$968$	−63.2354	−2.03246
$969$	−62.7224	−2.01493
$970$	−2.44069	−0.0783660
$971$	38.5552	1.23729	0.618647	−	0.785669i	$-0.287681\pi$
$971$	38.5552	1.23729	0.618647	+	0.785669i	$0.287681\pi$
$972$	0.935508	0.0300064
$973$	−4.68935	−0.150334
$974$	27.5174	0.881714
$975$	−28.3653	−0.908416
$976$	−26.6422	−0.852795
$977$	−36.9838	−1.18322	−0.591608	−	0.806226i	$-0.701507\pi$
$977$	−36.9838	−1.18322	−0.591608	+	0.806226i	$0.701507\pi$
$978$	−71.7381	−2.29393
$979$	38.3771	1.22654
$980$	46.8778	1.49746
$981$	0.201690	0.00643948
$982$	−48.0304	−1.53271
$983$	−44.8186	−1.42949	−0.714746	−	0.699384i	$-0.753458\pi$
$983$	−44.8186	−1.42949	−0.714746	+	0.699384i	$0.753458\pi$
$984$	78.1115	2.49011
$985$	12.9155	0.411523
$986$	12.4758	0.397312
$987$	−5.18211	−0.164948
$988$	−97.1721	−3.09146
$989$	−10.8689	−0.345610
$990$	−0.253253	−0.00804891
$991$	−13.8680	−0.440531	−0.220266	−	0.975440i	$-0.570692\pi$
$991$	−13.8680	−0.440531	−0.220266	+	0.975440i	$0.570692\pi$
$992$	6.09607	0.193550
$993$	0.0435973	0.00138352
$994$	90.7251	2.87762
$995$	−8.40952	−0.266600
$996$	108.817	3.44800
$997$	34.0464	1.07826	0.539131	−	0.842222i	$-0.318753\pi$
$997$	34.0464	1.07826	0.539131	+	0.842222i	$0.318753\pi$
$998$	−12.9381	−0.409548
$999$	−39.7394	−1.25730

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.e.1.12	✓	103

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.e.1.12	✓	103	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4031 = 29 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.38730 q^{2} -1.73906 q^{3} +3.69919 q^{4} +0.845395 q^{5} +4.15166 q^{6} -4.68935 q^{7} -4.05648 q^{8} +0.0243354 q^{9} +O(q^{10})\)	\(q-2.38730 q^{2} -1.73906 q^{3} +3.69919 q^{4} +0.845395 q^{5} +4.15166 q^{6} -4.68935 q^{7} -4.05648 q^{8} +0.0243354 q^{9} -2.01821 q^{10} +5.15643 q^{11} -6.43312 q^{12} -3.80619 q^{13} +11.1949 q^{14} -1.47019 q^{15} +2.28563 q^{16} +5.22593 q^{17} -0.0580960 q^{18} +6.90151 q^{19} +3.12728 q^{20} +8.15506 q^{21} -12.3099 q^{22} +5.75113 q^{23} +7.05446 q^{24} -4.28531 q^{25} +9.08651 q^{26} +5.17486 q^{27} -17.3468 q^{28} -1.00000 q^{29} +3.50979 q^{30} +2.29481 q^{31} +2.65646 q^{32} -8.96735 q^{33} -12.4758 q^{34} -3.96435 q^{35} +0.0900215 q^{36} -7.67932 q^{37} -16.4760 q^{38} +6.61920 q^{39} -3.42933 q^{40} +11.0726 q^{41} -19.4686 q^{42} -1.88987 q^{43} +19.0746 q^{44} +0.0205731 q^{45} -13.7297 q^{46} -0.635447 q^{47} -3.97486 q^{48} +14.9900 q^{49} +10.2303 q^{50} -9.08821 q^{51} -14.0798 q^{52} -7.03192 q^{53} -12.3539 q^{54} +4.35922 q^{55} +19.0222 q^{56} -12.0022 q^{57} +2.38730 q^{58} -2.10839 q^{59} -5.43853 q^{60} -11.6564 q^{61} -5.47839 q^{62} -0.114117 q^{63} -10.9130 q^{64} -3.21774 q^{65} +21.4077 q^{66} -5.42874 q^{67} +19.3317 q^{68} -10.0016 q^{69} +9.46409 q^{70} +8.10417 q^{71} -0.0987162 q^{72} +12.7437 q^{73} +18.3328 q^{74} +7.45241 q^{75} +25.5300 q^{76} -24.1803 q^{77} -15.8020 q^{78} -14.3662 q^{79} +1.93227 q^{80} -9.07241 q^{81} -26.4337 q^{82} -16.9151 q^{83} +30.1671 q^{84} +4.41797 q^{85} +4.51168 q^{86} +1.73906 q^{87} -20.9169 q^{88} +7.44257 q^{89} -0.0491141 q^{90} +17.8485 q^{91} +21.2745 q^{92} -3.99081 q^{93} +1.51700 q^{94} +5.83451 q^{95} -4.61975 q^{96} +1.20934 q^{97} -35.7855 q^{98} +0.125484 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 103 q + q^{2} + 2 q^{3} + 127 q^{4} + 9 q^{5} + 19 q^{6} + 18 q^{7} + 149 q^{9}+O(q^{10}) \) 103 * q + q^2 + 2 * q^3 + 127 * q^4 + 9 * q^5 + 19 * q^6 + 18 * q^7 + 149 * q^9	\( 103 q + q^{2} + 2 q^{3} + 127 q^{4} + 9 q^{5} + 19 q^{6} + 18 q^{7} + 149 q^{9} + 20 q^{10} + 9 q^{11} + 36 q^{13} - 10 q^{14} + 16 q^{15} + 179 q^{16} + 21 q^{17} + 7 q^{18} + 42 q^{19} + 24 q^{20} + 28 q^{21} + 32 q^{22} + 25 q^{23} + 68 q^{24} + 194 q^{25} - 5 q^{26} + 14 q^{27} + 59 q^{28} - 103 q^{29} + 84 q^{30} + 34 q^{31} + 11 q^{32} + 42 q^{33} + 54 q^{34} + 35 q^{35} + 214 q^{36} + 34 q^{37} + 9 q^{38} + 23 q^{39} + 46 q^{40} + 16 q^{41} + 13 q^{42} + 68 q^{43} - 6 q^{44} + 25 q^{45} + 60 q^{46} + 6 q^{47} + 5 q^{48} + 257 q^{49} - 51 q^{50} + 68 q^{51} + 37 q^{52} + 35 q^{53} + 30 q^{54} + 66 q^{55} - 54 q^{56} + 78 q^{57} - q^{58} + 10 q^{59} - 24 q^{60} + 70 q^{61} + 29 q^{62} + 26 q^{63} + 276 q^{64} + 95 q^{65} + 77 q^{66} + 71 q^{67} - 21 q^{68} - 20 q^{69} + 48 q^{70} + 32 q^{71} + 32 q^{72} + 94 q^{73} + 35 q^{74} + 7 q^{75} + 134 q^{76} + 17 q^{77} + 58 q^{78} + 110 q^{79} + 78 q^{80} + 267 q^{81} - 71 q^{82} + 35 q^{83} + 96 q^{84} + 71 q^{85} + 33 q^{86} - 2 q^{87} + 100 q^{88} + 22 q^{89} - 134 q^{90} + 108 q^{91} - 11 q^{92} + 78 q^{93} + 90 q^{94} + 12 q^{95} + 177 q^{96} + 44 q^{97} - 18 q^{98} + 83 q^{99}+O(q^{100}) \) 103 * q + q^2 + 2 * q^3 + 127 * q^4 + 9 * q^5 + 19 * q^6 + 18 * q^7 + 149 * q^9 + 20 * q^10 + 9 * q^11 + 36 * q^13 - 10 * q^14 + 16 * q^15 + 179 * q^16 + 21 * q^17 + 7 * q^18 + 42 * q^19 + 24 * q^20 + 28 * q^21 + 32 * q^22 + 25 * q^23 + 68 * q^24 + 194 * q^25 - 5 * q^26 + 14 * q^27 + 59 * q^28 - 103 * q^29 + 84 * q^30 + 34 * q^31 + 11 * q^32 + 42 * q^33 + 54 * q^34 + 35 * q^35 + 214 * q^36 + 34 * q^37 + 9 * q^38 + 23 * q^39 + 46 * q^40 + 16 * q^41 + 13 * q^42 + 68 * q^43 - 6 * q^44 + 25 * q^45 + 60 * q^46 + 6 * q^47 + 5 * q^48 + 257 * q^49 - 51 * q^50 + 68 * q^51 + 37 * q^52 + 35 * q^53 + 30 * q^54 + 66 * q^55 - 54 * q^56 + 78 * q^57 - q^58 + 10 * q^59 - 24 * q^60 + 70 * q^61 + 29 * q^62 + 26 * q^63 + 276 * q^64 + 95 * q^65 + 77 * q^66 + 71 * q^67 - 21 * q^68 - 20 * q^69 + 48 * q^70 + 32 * q^71 + 32 * q^72 + 94 * q^73 + 35 * q^74 + 7 * q^75 + 134 * q^76 + 17 * q^77 + 58 * q^78 + 110 * q^79 + 78 * q^80 + 267 * q^81 - 71 * q^82 + 35 * q^83 + 96 * q^84 + 71 * q^85 + 33 * q^86 - 2 * q^87 + 100 * q^88 + 22 * q^89 - 134 * q^90 + 108 * q^91 - 11 * q^92 + 78 * q^93 + 90 * q^94 + 12 * q^95 + 177 * q^96 + 44 * q^97 - 18 * q^98 + 83 * q^99

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