LMFDB - Embedded newform 1045.2.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.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:	$8.34436701122$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 28x^{4} - 17x^{3} - 28x^{2} + 6x + 8 $ x^8 - 2x^7 - 9x^6 + 12x^5 + 28x^4 - 17x^3 - 28x^2 + 6*x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.865980$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.86598 q^{2} +2.01288 q^{3} +1.48188 q^{4} +1.00000 q^{5} -3.75600 q^{6} -2.04136 q^{7} +0.966800 q^{8} +1.05170 q^{9} +O(q^{10})$	\(q-1.86598 q^{2} +2.01288 q^{3} +1.48188 q^{4} +1.00000 q^{5} -3.75600 q^{6} -2.04136 q^{7} +0.966800 q^{8} +1.05170 q^{9} -1.86598 q^{10} +1.00000 q^{11} +2.98285 q^{12} -4.91831 q^{13} +3.80913 q^{14} +2.01288 q^{15} -4.76779 q^{16} -6.79891 q^{17} -1.96245 q^{18} -1.00000 q^{19} +1.48188 q^{20} -4.10901 q^{21} -1.86598 q^{22} +1.43601 q^{23} +1.94606 q^{24} +1.00000 q^{25} +9.17747 q^{26} -3.92170 q^{27} -3.02505 q^{28} -0.704749 q^{29} -3.75600 q^{30} -4.21862 q^{31} +6.96300 q^{32} +2.01288 q^{33} +12.6866 q^{34} -2.04136 q^{35} +1.55849 q^{36} -1.40495 q^{37} +1.86598 q^{38} -9.89999 q^{39} +0.966800 q^{40} -9.38658 q^{41} +7.66733 q^{42} +3.80388 q^{43} +1.48188 q^{44} +1.05170 q^{45} -2.67956 q^{46} -7.13622 q^{47} -9.59701 q^{48} -2.83286 q^{49} -1.86598 q^{50} -13.6854 q^{51} -7.28835 q^{52} +2.65027 q^{53} +7.31782 q^{54} +1.00000 q^{55} -1.97358 q^{56} -2.01288 q^{57} +1.31505 q^{58} +13.3502 q^{59} +2.98285 q^{60} +7.58071 q^{61} +7.87185 q^{62} -2.14689 q^{63} -3.45724 q^{64} -4.91831 q^{65} -3.75600 q^{66} +10.5714 q^{67} -10.0752 q^{68} +2.89051 q^{69} +3.80913 q^{70} +5.75745 q^{71} +1.01678 q^{72} -5.49519 q^{73} +2.62161 q^{74} +2.01288 q^{75} -1.48188 q^{76} -2.04136 q^{77} +18.4732 q^{78} +7.12218 q^{79} -4.76779 q^{80} -11.0490 q^{81} +17.5152 q^{82} -10.8625 q^{83} -6.08907 q^{84} -6.79891 q^{85} -7.09797 q^{86} -1.41858 q^{87} +0.966800 q^{88} -3.26592 q^{89} -1.96245 q^{90} +10.0400 q^{91} +2.12799 q^{92} -8.49158 q^{93} +13.3160 q^{94} -1.00000 q^{95} +14.0157 q^{96} -1.64413 q^{97} +5.28607 q^{98} +1.05170 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9}+O(q^{10}) $ 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9	\( 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9} - 6 q^{10} + 8 q^{11} - 7 q^{12} - 17 q^{13} + 12 q^{14} - 7 q^{15} + 18 q^{16} - 9 q^{17} - 2 q^{18} - 8 q^{19} + 10 q^{20} + q^{21} - 6 q^{22} - 8 q^{23} + q^{24} + 8 q^{25} + 10 q^{26} - 34 q^{27} - 22 q^{28} - 3 q^{29} - q^{31} - 37 q^{32} - 7 q^{33} - 8 q^{34} - 11 q^{35} + 30 q^{36} - 17 q^{37} + 6 q^{38} + 14 q^{39} - 18 q^{40} - 5 q^{41} + 15 q^{42} - 21 q^{43} + 10 q^{44} + 11 q^{45} - 2 q^{46} - 8 q^{47} + 10 q^{48} + 19 q^{49} - 6 q^{50} - 16 q^{51} + 9 q^{52} - 19 q^{53} - 3 q^{54} + 8 q^{55} + 24 q^{56} + 7 q^{57} + 37 q^{58} - 33 q^{59} - 7 q^{60} - q^{61} - 42 q^{62} - 20 q^{63} + 48 q^{64} - 17 q^{65} - 18 q^{67} - 37 q^{68} + 16 q^{69} + 12 q^{70} - 18 q^{71} + 13 q^{72} - 18 q^{73} + 15 q^{74} - 7 q^{75} - 10 q^{76} - 11 q^{77} - 51 q^{78} - 5 q^{79} + 18 q^{80} + 32 q^{81} + 12 q^{82} - 33 q^{83} - 51 q^{84} - 9 q^{85} - 16 q^{86} - 26 q^{87} - 18 q^{88} - 20 q^{89} - 2 q^{90} + 6 q^{91} - 3 q^{92} + 18 q^{93} + 30 q^{94} - 8 q^{95} + 21 q^{96} - 69 q^{98} + 11 q^{99}+O(q^{100}) \) 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9 - 6 * q^10 + 8 * q^11 - 7 * q^12 - 17 * q^13 + 12 * q^14 - 7 * q^15 + 18 * q^16 - 9 * q^17 - 2 * q^18 - 8 * q^19 + 10 * q^20 + q^21 - 6 * q^22 - 8 * q^23 + q^24 + 8 * q^25 + 10 * q^26 - 34 * q^27 - 22 * q^28 - 3 * q^29 - q^31 - 37 * q^32 - 7 * q^33 - 8 * q^34 - 11 * q^35 + 30 * q^36 - 17 * q^37 + 6 * q^38 + 14 * q^39 - 18 * q^40 - 5 * q^41 + 15 * q^42 - 21 * q^43 + 10 * q^44 + 11 * q^45 - 2 * q^46 - 8 * q^47 + 10 * q^48 + 19 * q^49 - 6 * q^50 - 16 * q^51 + 9 * q^52 - 19 * q^53 - 3 * q^54 + 8 * q^55 + 24 * q^56 + 7 * q^57 + 37 * q^58 - 33 * q^59 - 7 * q^60 - q^61 - 42 * q^62 - 20 * q^63 + 48 * q^64 - 17 * q^65 - 18 * q^67 - 37 * q^68 + 16 * q^69 + 12 * q^70 - 18 * q^71 + 13 * q^72 - 18 * q^73 + 15 * q^74 - 7 * q^75 - 10 * q^76 - 11 * q^77 - 51 * q^78 - 5 * q^79 + 18 * q^80 + 32 * q^81 + 12 * q^82 - 33 * q^83 - 51 * q^84 - 9 * q^85 - 16 * q^86 - 26 * q^87 - 18 * q^88 - 20 * q^89 - 2 * q^90 + 6 * q^91 - 3 * q^92 + 18 * q^93 + 30 * q^94 - 8 * q^95 + 21 * q^96 - 69 * q^98 + 11 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.86598	−1.31945	−0.659723	−	0.751508i	$-0.729327\pi$
$2$	−1.86598	−1.31945	−0.659723	+	0.751508i	$0.729327\pi$
$3$	2.01288	1.16214	0.581069	−	0.813854i	$-0.302635\pi$
$3$	2.01288	1.16214	0.581069	+	0.813854i	$0.302635\pi$
$4$	1.48188	0.740940
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−3.75600	−1.53338
$7$	−2.04136	−0.771560	−0.385780	−	0.922591i	$-0.626068\pi$
$7$	−2.04136	−0.771560	−0.385780	+	0.922591i	$0.626068\pi$
$8$	0.966800	0.341815
$9$	1.05170	0.350566
$10$	−1.86598	−0.590075
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	2.98285	0.861075
$13$	−4.91831	−1.36409	−0.682047	−	0.731308i	$-0.738910\pi$
$13$	−4.91831	−1.36409	−0.682047	+	0.731308i	$0.738910\pi$
$14$	3.80913	1.01803
$15$	2.01288	0.519724
$16$	−4.76779	−1.19195
$17$	−6.79891	−1.64898	−0.824489	−	0.565878i	$-0.808537\pi$
$17$	−6.79891	−1.64898	−0.824489	+	0.565878i	$0.808537\pi$
$18$	−1.96245	−0.462554
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.48188	0.331359
$21$	−4.10901	−0.896660
$22$	−1.86598	−0.397828
$23$	1.43601	0.299428	0.149714	−	0.988729i	$-0.452165\pi$
$23$	1.43601	0.299428	0.149714	+	0.988729i	$0.452165\pi$
$24$	1.94606	0.397237
$25$	1.00000	0.200000
$26$	9.17747	1.79985
$27$	−3.92170	−0.754732
$28$	−3.02505	−0.571680
$29$	−0.704749	−0.130869	−0.0654343	−	0.997857i	$-0.520843\pi$
$29$	−0.704749	−0.130869	−0.0654343	+	0.997857i	$0.520843\pi$
$30$	−3.75600	−0.685749
$31$	−4.21862	−0.757686	−0.378843	−	0.925461i	$-0.623678\pi$
$31$	−4.21862	−0.757686	−0.378843	+	0.925461i	$0.623678\pi$
$32$	6.96300	1.23090
$33$	2.01288	0.350398
$34$	12.6866	2.17574
$35$	−2.04136	−0.345052
$36$	1.55849	0.259749
$37$	−1.40495	−0.230973	−0.115486	−	0.993309i	$-0.536843\pi$
$37$	−1.40495	−0.230973	−0.115486	+	0.993309i	$0.536843\pi$
$38$	1.86598	0.302702
$39$	−9.89999	−1.58527
$40$	0.966800	0.152865
$41$	−9.38658	−1.46594	−0.732969	−	0.680262i	$-0.761866\pi$
$41$	−9.38658	−1.46594	−0.732969	+	0.680262i	$0.761866\pi$
$42$	7.66733	1.18310
$43$	3.80388	0.580087	0.290043	−	0.957014i	$-0.406330\pi$
$43$	3.80388	0.580087	0.290043	+	0.957014i	$0.406330\pi$
$44$	1.48188	0.223402
$45$	1.05170	0.156778
$46$	−2.67956	−0.395079
$47$	−7.13622	−1.04093	−0.520463	−	0.853885i	$-0.674240\pi$
$47$	−7.13622	−1.04093	−0.520463	+	0.853885i	$0.674240\pi$
$48$	−9.59701	−1.38521
$49$	−2.83286	−0.404695
$50$	−1.86598	−0.263889
$51$	−13.6854	−1.91634
$52$	−7.28835	−1.01071
$53$	2.65027	0.364043	0.182021	−	0.983295i	$-0.441736\pi$
$53$	2.65027	0.364043	0.182021	+	0.983295i	$0.441736\pi$
$54$	7.31782	0.995829
$55$	1.00000	0.134840
$56$	−1.97358	−0.263731
$57$	−2.01288	−0.266613
$58$	1.31505	0.172674
$59$	13.3502	1.73805	0.869023	−	0.494772i	$-0.164748\pi$
$59$	13.3502	1.73805	0.869023	+	0.494772i	$0.164748\pi$
$60$	2.98285	0.385085
$61$	7.58071	0.970610	0.485305	−	0.874345i	$-0.338709\pi$
$61$	7.58071	0.970610	0.485305	+	0.874345i	$0.338709\pi$
$62$	7.87185	0.999726
$63$	−2.14689	−0.270483
$64$	−3.45724	−0.432155
$65$	−4.91831	−0.610042
$66$	−3.75600	−0.462332
$67$	10.5714	1.29150	0.645751	−	0.763548i	$-0.276545\pi$
$67$	10.5714	1.29150	0.645751	+	0.763548i	$0.276545\pi$
$68$	−10.0752	−1.22179
$69$	2.89051	0.347977
$70$	3.80913	0.455278
$71$	5.75745	0.683283	0.341642	−	0.939830i	$-0.389017\pi$
$71$	5.75745	0.683283	0.341642	+	0.939830i	$0.389017\pi$
$72$	1.01678	0.119829
$73$	−5.49519	−0.643163	−0.321582	−	0.946882i	$-0.604214\pi$
$73$	−5.49519	−0.643163	−0.321582	+	0.946882i	$0.604214\pi$
$74$	2.62161	0.304756
$75$	2.01288	0.232428
$76$	−1.48188	−0.169983
$77$	−2.04136	−0.232634
$78$	18.4732	2.09168
$79$	7.12218	0.801307	0.400654	−	0.916230i	$-0.368783\pi$
$79$	7.12218	0.801307	0.400654	+	0.916230i	$0.368783\pi$
$80$	−4.76779	−0.533055
$81$	−11.0490	−1.22767
$82$	17.5152	1.93423
$83$	−10.8625	−1.19231	−0.596157	−	0.802868i	$-0.703306\pi$
$83$	−10.8625	−1.19231	−0.596157	+	0.802868i	$0.703306\pi$
$84$	−6.08907	−0.664371
$85$	−6.79891	−0.737446
$86$	−7.09797	−0.765393
$87$	−1.41858	−0.152087
$88$	0.966800	0.103061
$89$	−3.26592	−0.346187	−0.173094	−	0.984905i	$-0.555376\pi$
$89$	−3.26592	−0.346187	−0.173094	+	0.984905i	$0.555376\pi$
$90$	−1.96245	−0.206860
$91$	10.0400	1.05248
$92$	2.12799	0.221858
$93$	−8.49158	−0.880536
$94$	13.3160	1.37345
$95$	−1.00000	−0.102598
$96$	14.0157	1.43047
$97$	−1.64413	−0.166936	−0.0834680	−	0.996510i	$-0.526600\pi$
$97$	−1.64413	−0.166936	−0.0834680	+	0.996510i	$0.526600\pi$
$98$	5.28607	0.533974
$99$	1.05170	0.105700
$100$	1.48188	0.148188
$101$	−2.89820	−0.288382	−0.144191	−	0.989550i	$-0.546058\pi$
$101$	−2.89820	−0.288382	−0.144191	+	0.989550i	$0.546058\pi$
$102$	25.5367	2.52851
$103$	−12.7793	−1.25918	−0.629590	−	0.776927i	$-0.716777\pi$
$103$	−12.7793	−1.25918	−0.629590	+	0.776927i	$0.716777\pi$
$104$	−4.75502	−0.466268
$105$	−4.10901	−0.400998
$106$	−4.94535	−0.480335
$107$	−17.2758	−1.67012	−0.835059	−	0.550160i	$-0.814567\pi$
$107$	−17.2758	−1.67012	−0.835059	+	0.550160i	$0.814567\pi$
$108$	−5.81150	−0.559211
$109$	6.56114	0.628444	0.314222	−	0.949350i	$-0.398256\pi$
$109$	6.56114	0.628444	0.314222	+	0.949350i	$0.398256\pi$
$110$	−1.86598	−0.177914
$111$	−2.82800	−0.268422
$112$	9.73276	0.919659
$113$	−1.66685	−0.156804	−0.0784018	−	0.996922i	$-0.524982\pi$
$113$	−1.66685	−0.156804	−0.0784018	+	0.996922i	$0.524982\pi$
$114$	3.75600	0.351782
$115$	1.43601	0.133908
$116$	−1.04435	−0.0969658
$117$	−5.17258	−0.478205
$118$	−24.9112	−2.29326
$119$	13.8790	1.27229
$120$	1.94606	0.177650
$121$	1.00000	0.0909091
$122$	−14.1455	−1.28067
$123$	−18.8941	−1.70362
$124$	−6.25149	−0.561400
$125$	1.00000	0.0894427
$126$	4.00606	0.356888
$127$	−5.24891	−0.465766	−0.232883	−	0.972505i	$-0.574816\pi$
$127$	−5.24891	−0.465766	−0.232883	+	0.972505i	$0.574816\pi$
$128$	−7.47487	−0.660691
$129$	7.65677	0.674141
$130$	9.17747	0.804917
$131$	−9.05742	−0.791350	−0.395675	−	0.918391i	$-0.629489\pi$
$131$	−9.05742	−0.791350	−0.395675	+	0.918391i	$0.629489\pi$
$132$	2.98285	0.259624
$133$	2.04136	0.177008
$134$	−19.7260	−1.70407
$135$	−3.92170	−0.337526
$136$	−6.57319	−0.563646
$137$	−15.5957	−1.33243	−0.666217	−	0.745758i	$-0.732088\pi$
$137$	−15.5957	−1.33243	−0.666217	+	0.745758i	$0.732088\pi$
$138$	−5.39364	−0.459137
$139$	9.26096	0.785505	0.392752	−	0.919644i	$-0.371523\pi$
$139$	9.26096	0.785505	0.392752	+	0.919644i	$0.371523\pi$
$140$	−3.02505	−0.255663
$141$	−14.3644	−1.20970
$142$	−10.7433	−0.901556
$143$	−4.91831	−0.411290
$144$	−5.01428	−0.417857
$145$	−0.704749	−0.0585262
$146$	10.2539	0.848620
$147$	−5.70223	−0.470312
$148$	−2.08197	−0.171137
$149$	5.22633	0.428158	0.214079	−	0.976816i	$-0.431325\pi$
$149$	5.22633	0.428158	0.214079	+	0.976816i	$0.431325\pi$
$150$	−3.75600	−0.306676
$151$	20.7869	1.69161	0.845805	−	0.533492i	$-0.179120\pi$
$151$	20.7869	1.69161	0.845805	+	0.533492i	$0.179120\pi$
$152$	−0.966800	−0.0784178
$153$	−7.15041	−0.578076
$154$	3.80913	0.306948
$155$	−4.21862	−0.338847
$156$	−14.6706	−1.17459
$157$	11.2308	0.896317	0.448158	−	0.893954i	$-0.352080\pi$
$157$	11.2308	0.896317	0.448158	+	0.893954i	$0.352080\pi$
$158$	−13.2898	−1.05728
$159$	5.33469	0.423068
$160$	6.96300	0.550474
$161$	−2.93140	−0.231027
$162$	20.6173	1.61984
$163$	4.43995	0.347763	0.173882	−	0.984767i	$-0.444369\pi$
$163$	4.43995	0.347763	0.173882	+	0.984767i	$0.444369\pi$
$164$	−13.9098	−1.08617
$165$	2.01288	0.156703
$166$	20.2692	1.57319
$167$	−14.1852	−1.09768	−0.548841	−	0.835927i	$-0.684931\pi$
$167$	−14.1852	−1.09768	−0.548841	+	0.835927i	$0.684931\pi$
$168$	−3.97259	−0.306492
$169$	11.1898	0.860753
$170$	12.6866	0.973020
$171$	−1.05170	−0.0804254
$172$	5.63690	0.429810
$173$	−5.44057	−0.413639	−0.206819	−	0.978379i	$-0.566311\pi$
$173$	−5.44057	−0.413639	−0.206819	+	0.978379i	$0.566311\pi$
$174$	2.64704	0.200671
$175$	−2.04136	−0.154312
$176$	−4.76779	−0.359386
$177$	26.8724	2.01985
$178$	6.09415	0.456776
$179$	−8.45415	−0.631893	−0.315946	−	0.948777i	$-0.602322\pi$
$179$	−8.45415	−0.631893	−0.315946	+	0.948777i	$0.602322\pi$
$180$	1.55849	0.116163
$181$	−13.6320	−1.01326	−0.506629	−	0.862164i	$-0.669109\pi$
$181$	−13.6320	−1.01326	−0.506629	+	0.862164i	$0.669109\pi$
$182$	−18.7345	−1.38869
$183$	15.2591	1.12798
$184$	1.38833	0.102349
$185$	−1.40495	−0.103294
$186$	15.8451	1.16182
$187$	−6.79891	−0.497186
$188$	−10.5750	−0.771263
$189$	8.00559	0.582321
$190$	1.86598	0.135372
$191$	−12.4905	−0.903778	−0.451889	−	0.892074i	$-0.649250\pi$
$191$	−12.4905	−0.903778	−0.451889	+	0.892074i	$0.649250\pi$
$192$	−6.95902	−0.502224
$193$	−9.94947	−0.716179	−0.358089	−	0.933687i	$-0.616572\pi$
$193$	−9.94947	−0.716179	−0.358089	+	0.933687i	$0.616572\pi$
$194$	3.06791	0.220263
$195$	−9.89999	−0.708953
$196$	−4.19797	−0.299855
$197$	24.3654	1.73596	0.867982	−	0.496595i	$-0.165417\pi$
$197$	24.3654	1.73596	0.867982	+	0.496595i	$0.165417\pi$
$198$	−1.96245	−0.139465
$199$	−11.5042	−0.815509	−0.407754	−	0.913092i	$-0.633688\pi$
$199$	−11.5042	−0.815509	−0.407754	+	0.913092i	$0.633688\pi$
$200$	0.966800	0.0683631
$201$	21.2790	1.50090
$202$	5.40798	0.380504
$203$	1.43864	0.100973
$204$	−20.2802	−1.41989
$205$	−9.38658	−0.655587
$206$	23.8459	1.66142
$207$	1.51025	0.104969
$208$	23.4495	1.62593
$209$	−1.00000	−0.0691714
$210$	7.66733	0.529096
$211$	10.8040	0.743778	0.371889	−	0.928277i	$-0.378710\pi$
$211$	10.8040	0.743778	0.371889	+	0.928277i	$0.378710\pi$
$212$	3.92739	0.269734
$213$	11.5891	0.794070
$214$	32.2364	2.20363
$215$	3.80388	0.259423
$216$	−3.79150	−0.257979
$217$	8.61170	0.584600
$218$	−12.2430	−0.829198
$219$	−11.0612	−0.747445
$220$	1.48188	0.0999084
$221$	33.4392	2.24936
$222$	5.27700	0.354169
$223$	−4.86983	−0.326108	−0.163054	−	0.986617i	$-0.552135\pi$
$223$	−4.86983	−0.326108	−0.163054	+	0.986617i	$0.552135\pi$
$224$	−14.2140	−0.949711
$225$	1.05170	0.0701132
$226$	3.11030	0.206894
$227$	−27.6885	−1.83775	−0.918876	−	0.394547i	$-0.870902\pi$
$227$	−27.6885	−1.83775	−0.918876	+	0.394547i	$0.870902\pi$
$228$	−2.98285	−0.197544
$229$	5.51690	0.364567	0.182284	−	0.983246i	$-0.441651\pi$
$229$	5.51690	0.364567	0.182284	+	0.983246i	$0.441651\pi$
$230$	−2.67956	−0.176685
$231$	−4.10901	−0.270353
$232$	−0.681351	−0.0447329
$233$	24.3012	1.59202	0.796012	−	0.605281i	$-0.206939\pi$
$233$	24.3012	1.59202	0.796012	+	0.605281i	$0.206939\pi$
$234$	9.65193	0.630967
$235$	−7.13622	−0.465516
$236$	19.7834	1.28779
$237$	14.3361	0.931230
$238$	−25.8979	−1.67871
$239$	−10.6660	−0.689928	−0.344964	−	0.938616i	$-0.612109\pi$
$239$	−10.6660	−0.689928	−0.344964	+	0.938616i	$0.612109\pi$
$240$	−9.59701	−0.619484
$241$	−4.78267	−0.308079	−0.154039	−	0.988065i	$-0.549228\pi$
$241$	−4.78267	−0.308079	−0.154039	+	0.988065i	$0.549228\pi$
$242$	−1.86598	−0.119950
$243$	−10.4753	−0.671990
$244$	11.2337	0.719164
$245$	−2.83286	−0.180985
$246$	35.2560	2.24784
$247$	4.91831	0.312945
$248$	−4.07856	−0.258989
$249$	−21.8649	−1.38563
$250$	−1.86598	−0.118015
$251$	29.0361	1.83274	0.916370	−	0.400333i	$-0.131105\pi$
$251$	29.0361	1.83274	0.916370	+	0.400333i	$0.131105\pi$
$252$	−3.18144	−0.200412
$253$	1.43601	0.0902809
$254$	9.79437	0.614553
$255$	−13.6854	−0.857014
$256$	20.8624	1.30390
$257$	−2.04952	−0.127845	−0.0639227	−	0.997955i	$-0.520361\pi$
$257$	−2.04952	−0.127845	−0.0639227	+	0.997955i	$0.520361\pi$
$258$	−14.2874	−0.889493
$259$	2.86801	0.178209
$260$	−7.28835	−0.452004
$261$	−0.741183	−0.0458781
$262$	16.9010	1.04414
$263$	27.0100	1.66551	0.832755	−	0.553642i	$-0.186762\pi$
$263$	27.0100	1.66551	0.832755	+	0.553642i	$0.186762\pi$
$264$	1.94606	0.119771
$265$	2.65027	0.162805
$266$	−3.80913	−0.233553
$267$	−6.57392	−0.402317
$268$	15.6655	0.956925
$269$	12.3963	0.755818	0.377909	−	0.925843i	$-0.376643\pi$
$269$	12.3963	0.755818	0.377909	+	0.925843i	$0.376643\pi$
$270$	7.31782	0.445348
$271$	32.8661	1.99647	0.998236	−	0.0593651i	$-0.0189076\pi$
$271$	32.8661	1.99647	0.998236	+	0.0593651i	$0.0189076\pi$
$272$	32.4158	1.96550
$273$	20.2094	1.22313
$274$	29.1013	1.75808
$275$	1.00000	0.0603023
$276$	4.28340	0.257830
$277$	14.8509	0.892304	0.446152	−	0.894957i	$-0.352794\pi$
$277$	14.8509	0.892304	0.446152	+	0.894957i	$0.352794\pi$
$278$	−17.2808	−1.03643
$279$	−4.43671	−0.265619
$280$	−1.97358	−0.117944
$281$	7.48442	0.446483	0.223242	−	0.974763i	$-0.428336\pi$
$281$	7.48442	0.446483	0.223242	+	0.974763i	$0.428336\pi$
$282$	26.8036	1.59613
$283$	−10.6009	−0.630155	−0.315078	−	0.949066i	$-0.602031\pi$
$283$	−10.6009	−0.630155	−0.315078	+	0.949066i	$0.602031\pi$
$284$	8.53185	0.506272
$285$	−2.01288	−0.119233
$286$	9.17747	0.542675
$287$	19.1614	1.13106
$288$	7.32298	0.431511
$289$	29.2252	1.71913
$290$	1.31505	0.0772222
$291$	−3.30944	−0.194003
$292$	−8.14321	−0.476546
$293$	−22.4854	−1.31361	−0.656805	−	0.754060i	$-0.728093\pi$
$293$	−22.4854	−1.31361	−0.656805	+	0.754060i	$0.728093\pi$
$294$	10.6402	0.620551
$295$	13.3502	0.777278
$296$	−1.35831	−0.0789500
$297$	−3.92170	−0.227560
$298$	−9.75223	−0.564932
$299$	−7.06273	−0.408448
$300$	2.98285	0.172215
$301$	−7.76508	−0.447572
$302$	−38.7879	−2.23199
$303$	−5.83374	−0.335140
$304$	4.76779	0.273452
$305$	7.58071	0.434070
$306$	13.3425	0.762741
$307$	13.3031	0.759250	0.379625	−	0.925140i	$-0.376053\pi$
$307$	13.3031	0.759250	0.379625	+	0.925140i	$0.376053\pi$
$308$	−3.02505	−0.172368
$309$	−25.7232	−1.46334
$310$	7.87185	0.447091
$311$	−8.03547	−0.455650	−0.227825	−	0.973702i	$-0.573161\pi$
$311$	−8.03547	−0.455650	−0.227825	+	0.973702i	$0.573161\pi$
$312$	−9.57131	−0.541869
$313$	17.1926	0.971785	0.485892	−	0.874019i	$-0.338495\pi$
$313$	17.1926	0.971785	0.485892	+	0.874019i	$0.338495\pi$
$314$	−20.9565	−1.18264
$315$	−2.14689	−0.120964
$316$	10.5542	0.593721
$317$	3.68497	0.206969	0.103484	−	0.994631i	$-0.467001\pi$
$317$	3.68497	0.206969	0.103484	+	0.994631i	$0.467001\pi$
$318$	−9.95442	−0.558216
$319$	−0.704749	−0.0394584
$320$	−3.45724	−0.193266
$321$	−34.7742	−1.94091
$322$	5.46993	0.304828
$323$	6.79891	0.378302
$324$	−16.3733	−0.909630
$325$	−4.91831	−0.272819
$326$	−8.28485	−0.458855
$327$	13.2068	0.730339
$328$	−9.07495	−0.501080
$329$	14.5676	0.803136
$330$	−3.75600	−0.206761
$331$	−16.1229	−0.886192	−0.443096	−	0.896474i	$-0.646120\pi$
$331$	−16.1229	−0.886192	−0.443096	+	0.896474i	$0.646120\pi$
$332$	−16.0969	−0.883433
$333$	−1.47759	−0.0809712
$334$	26.4693	1.44833
$335$	10.5714	0.577577
$336$	19.5909	1.06877
$337$	−9.25224	−0.504001	−0.252001	−	0.967727i	$-0.581089\pi$
$337$	−9.25224	−0.504001	−0.252001	+	0.967727i	$0.581089\pi$
$338$	−20.8799	−1.13572
$339$	−3.35516	−0.182228
$340$	−10.0752	−0.546403
$341$	−4.21862	−0.228451
$342$	1.96245	0.106117
$343$	20.0724	1.08381
$344$	3.67759	0.198283
$345$	2.89051	0.155620
$346$	10.1520	0.545775
$347$	1.38539	0.0743715	0.0371858	−	0.999308i	$-0.488161\pi$
$347$	1.38539	0.0743715	0.0371858	+	0.999308i	$0.488161\pi$
$348$	−2.10216	−0.112688
$349$	−31.0884	−1.66412	−0.832062	−	0.554683i	$-0.812840\pi$
$349$	−31.0884	−1.66412	−0.832062	+	0.554683i	$0.812840\pi$
$350$	3.80913	0.203607
$351$	19.2882	1.02953
$352$	6.96300	0.371129
$353$	−29.4747	−1.56878	−0.784389	−	0.620269i	$-0.787023\pi$
$353$	−29.4747	−1.56878	−0.784389	+	0.620269i	$0.787023\pi$
$354$	−50.1433	−2.66509
$355$	5.75745	0.305574
$356$	−4.83971	−0.256504
$357$	27.9368	1.47857
$358$	15.7753	0.833749
$359$	−9.74843	−0.514502	−0.257251	−	0.966345i	$-0.582817\pi$
$359$	−9.74843	−0.514502	−0.257251	+	0.966345i	$0.582817\pi$
$360$	1.01678	0.0535891
$361$	1.00000	0.0526316
$362$	25.4370	1.33694
$363$	2.01288	0.105649
$364$	14.8781	0.779826
$365$	−5.49519	−0.287631
$366$	−28.4731	−1.48831
$367$	−23.1973	−1.21089	−0.605444	−	0.795888i	$-0.707004\pi$
$367$	−23.1973	−1.21089	−0.605444	+	0.795888i	$0.707004\pi$
$368$	−6.84658	−0.356903
$369$	−9.87186	−0.513908
$370$	2.62161	0.136291
$371$	−5.41015	−0.280881
$372$	−12.5835	−0.652425
$373$	32.5197	1.68381	0.841903	−	0.539629i	$-0.181435\pi$
$373$	32.5197	1.68381	0.841903	+	0.539629i	$0.181435\pi$
$374$	12.6866	0.656010
$375$	2.01288	0.103945
$376$	−6.89930	−0.355804
$377$	3.46617	0.178517
$378$	−14.9383	−0.768342
$379$	18.8601	0.968780	0.484390	−	0.874852i	$-0.339042\pi$
$379$	18.8601	0.968780	0.484390	+	0.874852i	$0.339042\pi$
$380$	−1.48188	−0.0760189
$381$	−10.5654	−0.541284
$382$	23.3069	1.19249
$383$	−8.07858	−0.412796	−0.206398	−	0.978468i	$-0.566174\pi$
$383$	−8.07858	−0.412796	−0.206398	+	0.978468i	$0.566174\pi$
$384$	−15.0460	−0.767815
$385$	−2.04136	−0.104037
$386$	18.5655	0.944960
$387$	4.00054	0.203359
$388$	−2.43640	−0.123690
$389$	32.3244	1.63891	0.819456	−	0.573143i	$-0.194276\pi$
$389$	32.3244	1.63891	0.819456	+	0.573143i	$0.194276\pi$
$390$	18.4732	0.935426
$391$	−9.76328	−0.493750
$392$	−2.73881	−0.138331
$393$	−18.2315	−0.919659
$394$	−45.4654	−2.29051
$395$	7.12218	0.358356
$396$	1.55849	0.0783172
$397$	2.99126	0.150127	0.0750634	−	0.997179i	$-0.476084\pi$
$397$	2.99126	0.150127	0.0750634	+	0.997179i	$0.476084\pi$
$398$	21.4665	1.07602
$399$	4.10901	0.205708
$400$	−4.76779	−0.238390
$401$	3.52554	0.176057	0.0880286	−	0.996118i	$-0.471943\pi$
$401$	3.52554	0.176057	0.0880286	+	0.996118i	$0.471943\pi$
$402$	−39.7061	−1.98036
$403$	20.7485	1.03356
$404$	−4.29479	−0.213674
$405$	−11.0490	−0.549031
$406$	−2.68448	−0.133228
$407$	−1.40495	−0.0696409
$408$	−13.2311	−0.655035
$409$	−23.0387	−1.13919	−0.569595	−	0.821926i	$-0.692900\pi$
$409$	−23.0387	−1.13919	−0.569595	+	0.821926i	$0.692900\pi$
$410$	17.5152	0.865013
$411$	−31.3924	−1.54847
$412$	−18.9374	−0.932978
$413$	−27.2525	−1.34101
$414$	−2.81809	−0.138501
$415$	−10.8625	−0.533219
$416$	−34.2462	−1.67906
$417$	18.6412	0.912865
$418$	1.86598	0.0912681
$419$	−29.8644	−1.45897	−0.729486	−	0.683996i	$-0.760241\pi$
$419$	−29.8644	−1.45897	−0.729486	+	0.683996i	$0.760241\pi$
$420$	−6.08907	−0.297116
$421$	21.8866	1.06669	0.533345	−	0.845898i	$-0.320935\pi$
$421$	21.8866	1.06669	0.533345	+	0.845898i	$0.320935\pi$
$422$	−20.1600	−0.981375
$423$	−7.50516	−0.364913
$424$	2.56228	0.124435
$425$	−6.79891	−0.329796
$426$	−21.6250	−1.04773
$427$	−15.4749	−0.748884
$428$	−25.6007	−1.23746
$429$	−9.89999	−0.477976
$430$	−7.09797	−0.342294
$431$	−12.7937	−0.616250	−0.308125	−	0.951346i	$-0.599701\pi$
$431$	−12.7937	−0.616250	−0.308125	+	0.951346i	$0.599701\pi$
$432$	18.6979	0.899601
$433$	8.06755	0.387702	0.193851	−	0.981031i	$-0.437902\pi$
$433$	8.06755	0.387702	0.193851	+	0.981031i	$0.437902\pi$
$434$	−16.0693	−0.771349
$435$	−1.41858	−0.0680156
$436$	9.72283	0.465639
$437$	−1.43601	−0.0686935
$438$	20.6399	0.986214
$439$	1.25574	0.0599330	0.0299665	−	0.999551i	$-0.490460\pi$
$439$	1.25574	0.0599330	0.0299665	+	0.999551i	$0.490460\pi$
$440$	0.966800	0.0460904
$441$	−2.97932	−0.141872
$442$	−62.3968	−2.96791
$443$	30.4664	1.44750	0.723752	−	0.690060i	$-0.242416\pi$
$443$	30.4664	1.44750	0.723752	+	0.690060i	$0.242416\pi$
$444$	−4.19076	−0.198885
$445$	−3.26592	−0.154820
$446$	9.08701	0.430282
$447$	10.5200	0.497579
$448$	7.05746	0.333433
$449$	2.35876	0.111317	0.0556584	−	0.998450i	$-0.482274\pi$
$449$	2.35876	0.111317	0.0556584	+	0.998450i	$0.482274\pi$
$450$	−1.96245	−0.0925107
$451$	−9.38658	−0.441997
$452$	−2.47007	−0.116182
$453$	41.8415	1.96589
$454$	51.6662	2.42482
$455$	10.0400	0.470684
$456$	−1.94606	−0.0911324
$457$	−37.1810	−1.73926	−0.869628	−	0.493708i	$-0.835641\pi$
$457$	−37.1810	−1.73926	−0.869628	+	0.493708i	$0.835641\pi$
$458$	−10.2944	−0.481027
$459$	26.6633	1.24454
$460$	2.12799	0.0992181
$461$	10.4784	0.488030	0.244015	−	0.969771i	$-0.421535\pi$
$461$	10.4784	0.488030	0.244015	+	0.969771i	$0.421535\pi$
$462$	7.66733	0.356717
$463$	−24.2378	−1.12642	−0.563212	−	0.826312i	$-0.690435\pi$
$463$	−24.2378	−1.12642	−0.563212	+	0.826312i	$0.690435\pi$
$464$	3.36010	0.155989
$465$	−8.49158	−0.393788
$466$	−45.3455	−2.10059
$467$	21.5240	0.996012	0.498006	−	0.867174i	$-0.334066\pi$
$467$	21.5240	0.996012	0.498006	+	0.867174i	$0.334066\pi$
$468$	−7.66515	−0.354322
$469$	−21.5800	−0.996471
$470$	13.3160	0.614223
$471$	22.6063	1.04164
$472$	12.9070	0.594091
$473$	3.80388	0.174903
$474$	−26.7509	−1.22871
$475$	−1.00000	−0.0458831
$476$	20.5670	0.942688
$477$	2.78729	0.127621
$478$	19.9026	0.910323
$479$	−7.29664	−0.333392	−0.166696	−	0.986008i	$-0.553310\pi$
$479$	−7.29664	−0.333392	−0.166696	+	0.986008i	$0.553310\pi$
$480$	14.0157	0.639727
$481$	6.90999	0.315068
$482$	8.92436	0.406493
$483$	−5.90057	−0.268485
$484$	1.48188	0.0673582
$485$	−1.64413	−0.0746561
$486$	19.5467	0.886655
$487$	−30.8245	−1.39679	−0.698396	−	0.715712i	$-0.746102\pi$
$487$	−30.8245	−1.39679	−0.698396	+	0.715712i	$0.746102\pi$
$488$	7.32903	0.331770
$489$	8.93709	0.404149
$490$	5.28607	0.238800
$491$	−30.9271	−1.39572	−0.697860	−	0.716234i	$-0.745864\pi$
$491$	−30.9271	−1.39572	−0.697860	+	0.716234i	$0.745864\pi$
$492$	−27.9988	−1.26228
$493$	4.79153	0.215799
$494$	−9.17747	−0.412914
$495$	1.05170	0.0472703
$496$	20.1135	0.903122
$497$	−11.7530	−0.527194
$498$	40.7995	1.82827
$499$	38.2965	1.71439	0.857193	−	0.514995i	$-0.172206\pi$
$499$	38.2965	1.71439	0.857193	+	0.514995i	$0.172206\pi$
$500$	1.48188	0.0662717
$501$	−28.5531	−1.27566
$502$	−54.1807	−2.41820
$503$	38.2324	1.70470	0.852349	−	0.522973i	$-0.175177\pi$
$503$	38.2324	1.70470	0.852349	+	0.522973i	$0.175177\pi$
$504$	−2.07561	−0.0924552
$505$	−2.89820	−0.128968
$506$	−2.67956	−0.119121
$507$	22.5237	1.00031
$508$	−7.77826	−0.345105
$509$	13.8877	0.615561	0.307780	−	0.951457i	$-0.400414\pi$
$509$	13.8877	0.615561	0.307780	+	0.951457i	$0.400414\pi$
$510$	25.5367	1.13078
$511$	11.2176	0.496239
$512$	−23.9791	−1.05974
$513$	3.92170	0.173147
$514$	3.82436	0.168685
$515$	−12.7793	−0.563123
$516$	11.3464	0.499498
$517$	−7.13622	−0.313851
$518$	−5.35164	−0.235138
$519$	−10.9512	−0.480706
$520$	−4.75502	−0.208522
$521$	−30.9995	−1.35811	−0.679057	−	0.734085i	$-0.737611\pi$
$521$	−30.9995	−1.35811	−0.679057	+	0.734085i	$0.737611\pi$
$522$	1.38303	0.0605337
$523$	−14.4767	−0.633020	−0.316510	−	0.948589i	$-0.602511\pi$
$523$	−14.4767	−0.633020	−0.316510	+	0.948589i	$0.602511\pi$
$524$	−13.4220	−0.586343
$525$	−4.10901	−0.179332
$526$	−50.4002	−2.19755
$527$	28.6820	1.24941
$528$	−9.59701	−0.417656
$529$	−20.9379	−0.910343
$530$	−4.94535	−0.214812
$531$	14.0404	0.609300
$532$	3.02505	0.131152
$533$	46.1661	1.99968
$534$	12.2668	0.530837
$535$	−17.2758	−0.746899
$536$	10.2204	0.441455
$537$	−17.0172	−0.734347
$538$	−23.1313	−0.997261
$539$	−2.83286	−0.122020
$540$	−5.81150	−0.250087
$541$	11.0122	0.473453	0.236726	−	0.971576i	$-0.423926\pi$
$541$	11.0122	0.473453	0.236726	+	0.971576i	$0.423926\pi$
$542$	−61.3275	−2.63424
$543$	−27.4396	−1.17755
$544$	−47.3408	−2.02972
$545$	6.56114	0.281049
$546$	−37.7103	−1.61385
$547$	2.09356	0.0895142	0.0447571	−	0.998998i	$-0.485749\pi$
$547$	2.09356	0.0895142	0.0447571	+	0.998998i	$0.485749\pi$
$548$	−23.1110	−0.987255
$549$	7.97262	0.340263
$550$	−1.86598	−0.0795656
$551$	0.704749	0.0300233
$552$	2.79455	0.118944
$553$	−14.5389	−0.618257
$554$	−27.7115	−1.17735
$555$	−2.82800	−0.120042
$556$	13.7236	0.582012
$557$	27.0490	1.14610	0.573052	−	0.819519i	$-0.305759\pi$
$557$	27.0490	1.14610	0.573052	+	0.819519i	$0.305759\pi$
$558$	8.27882	0.350470
$559$	−18.7087	−0.791293
$560$	9.73276	0.411284
$561$	−13.6854	−0.577799
$562$	−13.9658	−0.589111
$563$	11.5944	0.488646	0.244323	−	0.969694i	$-0.421434\pi$
$563$	11.5944	0.488646	0.244323	+	0.969694i	$0.421434\pi$
$564$	−21.2863	−0.896315
$565$	−1.66685	−0.0701247
$566$	19.7810	0.831456
$567$	22.5550	0.947221
$568$	5.56630	0.233557
$569$	−26.0673	−1.09280	−0.546399	−	0.837525i	$-0.684002\pi$
$569$	−26.0673	−1.09280	−0.546399	+	0.837525i	$0.684002\pi$
$570$	3.75600	0.157322
$571$	43.1051	1.80389	0.901946	−	0.431849i	$-0.142139\pi$
$571$	43.1051	1.80389	0.901946	+	0.431849i	$0.142139\pi$
$572$	−7.28835	−0.304741
$573$	−25.1418	−1.05032
$574$	−35.7547	−1.49237
$575$	1.43601	0.0598856
$576$	−3.63597	−0.151499
$577$	−7.45466	−0.310341	−0.155171	−	0.987888i	$-0.549593\pi$
$577$	−7.45466	−0.310341	−0.155171	+	0.987888i	$0.549593\pi$
$578$	−54.5337	−2.26830
$579$	−20.0271	−0.832299
$580$	−1.04435	−0.0433644
$581$	22.1742	0.919941
$582$	6.17535	0.255976
$583$	2.65027	0.109763
$584$	−5.31275	−0.219843
$585$	−5.17258	−0.213860
$586$	41.9573	1.73324
$587$	−27.8753	−1.15054	−0.575269	−	0.817964i	$-0.695103\pi$
$587$	−27.8753	−1.15054	−0.575269	+	0.817964i	$0.695103\pi$
$588$	−8.45002	−0.348473
$589$	4.21862	0.173825
$590$	−24.9112	−1.02558
$591$	49.0447	2.01743
$592$	6.69852	0.275307
$593$	29.0611	1.19339	0.596697	−	0.802466i	$-0.296479\pi$
$593$	29.0611	1.19339	0.596697	+	0.802466i	$0.296479\pi$
$594$	7.31782	0.300254
$595$	13.8790	0.568984
$596$	7.74480	0.317240
$597$	−23.1565	−0.947734
$598$	13.1789	0.538926
$599$	−39.5579	−1.61629	−0.808146	−	0.588982i	$-0.799529\pi$
$599$	−39.5579	−1.61629	−0.808146	+	0.588982i	$0.799529\pi$
$600$	1.94606	0.0794474
$601$	−4.44168	−0.181180	−0.0905900	−	0.995888i	$-0.528875\pi$
$601$	−4.44168	−0.181180	−0.0905900	+	0.995888i	$0.528875\pi$
$602$	14.4895	0.590547
$603$	11.1179	0.452757
$604$	30.8036	1.25338
$605$	1.00000	0.0406558
$606$	10.8856	0.442199
$607$	−15.6237	−0.634145	−0.317072	−	0.948401i	$-0.602700\pi$
$607$	−15.6237	−0.634145	−0.317072	+	0.948401i	$0.602700\pi$
$608$	−6.96300	−0.282387
$609$	2.89582	0.117345
$610$	−14.1455	−0.572733
$611$	35.0982	1.41992
$612$	−10.5961	−0.428320
$613$	−29.5049	−1.19169	−0.595845	−	0.803099i	$-0.703183\pi$
$613$	−29.5049	−1.19169	−0.595845	+	0.803099i	$0.703183\pi$
$614$	−24.8234	−1.00179
$615$	−18.8941	−0.761883
$616$	−1.97358	−0.0795179
$617$	−27.5007	−1.10714	−0.553569	−	0.832804i	$-0.686734\pi$
$617$	−27.5007	−1.10714	−0.553569	+	0.832804i	$0.686734\pi$
$618$	47.9990	1.93080
$619$	−24.6833	−0.992107	−0.496054	−	0.868292i	$-0.665218\pi$
$619$	−24.6833	−0.992107	−0.496054	+	0.868292i	$0.665218\pi$
$620$	−6.25149	−0.251066
$621$	−5.63159	−0.225988
$622$	14.9940	0.601206
$623$	6.66691	0.267104
$624$	47.2011	1.88956
$625$	1.00000	0.0400000
$626$	−32.0811	−1.28222
$627$	−2.01288	−0.0803868
$628$	16.6427	0.664117
$629$	9.55214	0.380869
$630$	4.00606	0.159605
$631$	−7.77127	−0.309370	−0.154685	−	0.987964i	$-0.549436\pi$
$631$	−7.77127	−0.309370	−0.154685	+	0.987964i	$0.549436\pi$
$632$	6.88572	0.273899
$633$	21.7472	0.864373
$634$	−6.87608	−0.273084
$635$	−5.24891	−0.208297
$636$	7.90537	0.313468
$637$	13.9329	0.552042
$638$	1.31505	0.0520632
$639$	6.05510	0.239536
$640$	−7.47487	−0.295470
$641$	−36.9773	−1.46051	−0.730257	−	0.683172i	$-0.760600\pi$
$641$	−36.9773	−1.46051	−0.730257	+	0.683172i	$0.760600\pi$
$642$	64.8880	2.56093
$643$	−1.65306	−0.0651902	−0.0325951	−	0.999469i	$-0.510377\pi$
$643$	−1.65306	−0.0651902	−0.0325951	+	0.999469i	$0.510377\pi$
$644$	−4.34399	−0.171177
$645$	7.65677	0.301485
$646$	−12.6866	−0.499149
$647$	−5.12322	−0.201415	−0.100707	−	0.994916i	$-0.532111\pi$
$647$	−5.12322	−0.201415	−0.100707	+	0.994916i	$0.532111\pi$
$648$	−10.6822	−0.419636
$649$	13.3502	0.524041
$650$	9.17747	0.359970
$651$	17.3343	0.679387
$652$	6.57947	0.257672
$653$	41.0422	1.60611	0.803053	−	0.595908i	$-0.203208\pi$
$653$	41.0422	1.60611	0.803053	+	0.595908i	$0.203208\pi$
$654$	−24.6437	−0.963643
$655$	−9.05742	−0.353903
$656$	44.7533	1.74732
$657$	−5.77928	−0.225471
$658$	−27.1828	−1.05970
$659$	−39.9155	−1.55489	−0.777443	−	0.628953i	$-0.783484\pi$
$659$	−39.9155	−1.55489	−0.777443	+	0.628953i	$0.783484\pi$
$660$	2.98285	0.116107
$661$	−35.6144	−1.38524	−0.692619	−	0.721303i	$-0.743543\pi$
$661$	−35.6144	−1.38524	−0.692619	+	0.721303i	$0.743543\pi$
$662$	30.0849	1.16928
$663$	67.3092	2.61407
$664$	−10.5019	−0.407551
$665$	2.04136	0.0791604
$666$	2.75714	0.106837
$667$	−1.01202	−0.0391857
$668$	−21.0208	−0.813317
$669$	−9.80241	−0.378983
$670$	−19.7260	−0.762082
$671$	7.58071	0.292650
$672$	−28.6111	−1.10370
$673$	−20.6495	−0.795982	−0.397991	−	0.917389i	$-0.630292\pi$
$673$	−20.6495	−0.795982	−0.397991	+	0.917389i	$0.630292\pi$
$674$	17.2645	0.665003
$675$	−3.92170	−0.150946
$676$	16.5819	0.637767
$677$	8.49349	0.326431	0.163216	−	0.986590i	$-0.447813\pi$
$677$	8.49349	0.326431	0.163216	+	0.986590i	$0.447813\pi$
$678$	6.26067	0.240440
$679$	3.35625	0.128801
$680$	−6.57319	−0.252070
$681$	−55.7338	−2.13572
$682$	7.87185	0.301429
$683$	−34.3263	−1.31346	−0.656729	−	0.754126i	$-0.728061\pi$
$683$	−34.3263	−1.31346	−0.656729	+	0.754126i	$0.728061\pi$
$684$	−1.55849	−0.0595904
$685$	−15.5957	−0.595883
$686$	−37.4547	−1.43003
$687$	11.1049	0.423677
$688$	−18.1361	−0.691433
$689$	−13.0349	−0.496589
$690$	−5.39364	−0.205332
$691$	−20.0621	−0.763198	−0.381599	−	0.924328i	$-0.624626\pi$
$691$	−20.0621	−0.763198	−0.381599	+	0.924328i	$0.624626\pi$
$692$	−8.06228	−0.306482
$693$	−2.14689	−0.0815537
$694$	−2.58511	−0.0981293
$695$	9.26096	0.351288
$696$	−1.37148	−0.0519858
$697$	63.8186	2.41730
$698$	58.0103	2.19572
$699$	48.9155	1.85015
$700$	−3.02505	−0.114336
$701$	−0.877252	−0.0331334	−0.0165667	−	0.999863i	$-0.505274\pi$
$701$	−0.877252	−0.0331334	−0.0165667	+	0.999863i	$0.505274\pi$
$702$	−35.9913	−1.35840
$703$	1.40495	0.0529888
$704$	−3.45724	−0.130300
$705$	−14.3644	−0.540994
$706$	54.9991	2.06992
$707$	5.91626	0.222504
$708$	39.8216	1.49659
$709$	31.6482	1.18857	0.594286	−	0.804254i	$-0.297435\pi$
$709$	31.6482	1.18857	0.594286	+	0.804254i	$0.297435\pi$
$710$	−10.7433	−0.403188
$711$	7.49038	0.280911
$712$	−3.15749	−0.118332
$713$	−6.05796	−0.226872
$714$	−52.1295	−1.95090
$715$	−4.91831	−0.183934
$716$	−12.5280	−0.468195
$717$	−21.4695	−0.801791
$718$	18.1904	0.678859
$719$	−26.0542	−0.971656	−0.485828	−	0.874054i	$-0.661482\pi$
$719$	−26.0542	−0.971656	−0.485828	+	0.874054i	$0.661482\pi$
$720$	−5.01428	−0.186871
$721$	26.0871	0.971533
$722$	−1.86598	−0.0694446
$723$	−9.62695	−0.358030
$724$	−20.2010	−0.750763
$725$	−0.704749	−0.0261737
$726$	−3.75600	−0.139398
$727$	−36.4040	−1.35015	−0.675075	−	0.737749i	$-0.735889\pi$
$727$	−36.4040	−1.35015	−0.675075	+	0.737749i	$0.735889\pi$
$728$	9.70670	0.359754
$729$	12.0615	0.446724
$730$	10.2539	0.379514
$731$	−25.8623	−0.956550
$732$	22.6121	0.835769
$733$	28.3926	1.04870	0.524352	−	0.851502i	$-0.324308\pi$
$733$	28.3926	1.04870	0.524352	+	0.851502i	$0.324308\pi$
$734$	43.2856	1.59770
$735$	−5.70223	−0.210330
$736$	9.99891	0.368565
$737$	10.5714	0.389402
$738$	18.4207	0.678075
$739$	22.7331	0.836251	0.418126	−	0.908389i	$-0.362687\pi$
$739$	22.7331	0.836251	0.418126	+	0.908389i	$0.362687\pi$
$740$	−2.08197	−0.0765348
$741$	9.89999	0.363685
$742$	10.0952	0.370607
$743$	−11.9651	−0.438958	−0.219479	−	0.975617i	$-0.570436\pi$
$743$	−11.9651	−0.438958	−0.219479	+	0.975617i	$0.570436\pi$
$744$	−8.20966	−0.300981
$745$	5.22633	0.191478
$746$	−60.6811	−2.22169
$747$	−11.4241	−0.417985
$748$	−10.0752	−0.368385
$749$	35.2661	1.28860
$750$	−3.75600	−0.137150
$751$	33.1233	1.20869	0.604343	−	0.796724i	$-0.293436\pi$
$751$	33.1233	1.20869	0.604343	+	0.796724i	$0.293436\pi$
$752$	34.0240	1.24073
$753$	58.4462	2.12990
$754$	−6.46781	−0.235544
$755$	20.7869	0.756511
$756$	11.8633	0.431465
$757$	31.0693	1.12923	0.564617	−	0.825353i	$-0.309024\pi$
$757$	31.0693	1.12923	0.564617	+	0.825353i	$0.309024\pi$
$758$	−35.1926	−1.27825
$759$	2.89051	0.104919
$760$	−0.966800	−0.0350695
$761$	9.25808	0.335605	0.167803	−	0.985821i	$-0.446333\pi$
$761$	9.25808	0.335605	0.167803	+	0.985821i	$0.446333\pi$
$762$	19.7149	0.714196
$763$	−13.3936	−0.484882
$764$	−18.5094	−0.669646
$765$	−7.15041	−0.258524
$766$	15.0745	0.544662
$767$	−65.6604	−2.37086
$768$	41.9936	1.51531
$769$	−18.6181	−0.671388	−0.335694	−	0.941971i	$-0.608971\pi$
$769$	−18.6181	−0.671388	−0.335694	+	0.941971i	$0.608971\pi$
$770$	3.80913	0.137272
$771$	−4.12544	−0.148574
$772$	−14.7439	−0.530646
$773$	−24.0735	−0.865864	−0.432932	−	0.901427i	$-0.642521\pi$
$773$	−24.0735	−0.865864	−0.432932	+	0.901427i	$0.642521\pi$
$774$	−7.46492	−0.268321
$775$	−4.21862	−0.151537
$776$	−1.58954	−0.0570613
$777$	5.77296	0.207104
$778$	−60.3166	−2.16246
$779$	9.38658	0.336309
$780$	−14.6706	−0.525292
$781$	5.75745	0.206018
$782$	18.2181	0.651477
$783$	2.76382	0.0987707
$784$	13.5065	0.482375
$785$	11.2308	0.400845
$786$	34.0197	1.21344
$787$	14.1014	0.502660	0.251330	−	0.967901i	$-0.419132\pi$
$787$	14.1014	0.502660	0.251330	+	0.967901i	$0.419132\pi$
$788$	36.1066	1.28625
$789$	54.3680	1.93555
$790$	−13.2898	−0.472831
$791$	3.40262	0.120983
$792$	1.01678	0.0361298
$793$	−37.2843	−1.32400
$794$	−5.58163	−0.198084
$795$	5.33469	0.189202
$796$	−17.0478	−0.604243
$797$	−42.7967	−1.51594	−0.757969	−	0.652290i	$-0.773808\pi$
$797$	−42.7967	−1.51594	−0.757969	+	0.652290i	$0.773808\pi$
$798$	−7.66733	−0.271421
$799$	48.5186	1.71646
$800$	6.96300	0.246179
$801$	−3.43477	−0.121362
$802$	−6.57859	−0.232298
$803$	−5.49519	−0.193921
$804$	31.5329	1.11208
$805$	−2.93140	−0.103318
$806$	−38.7162	−1.36372
$807$	24.9524	0.878365
$808$	−2.80198	−0.0985733
$809$	28.5997	1.00551	0.502756	−	0.864428i	$-0.332319\pi$
$809$	28.5997	1.00551	0.502756	+	0.864428i	$0.332319\pi$
$810$	20.6173	0.724417
$811$	−35.3795	−1.24234	−0.621171	−	0.783675i	$-0.713343\pi$
$811$	−35.3795	−1.24234	−0.621171	+	0.783675i	$0.713343\pi$
$812$	2.13190	0.0748150
$813$	66.1556	2.32018
$814$	2.62161	0.0918874
$815$	4.43995	0.155525
$816$	65.2492	2.28418
$817$	−3.80388	−0.133081
$818$	42.9897	1.50310
$819$	10.5591	0.368964
$820$	−13.9098	−0.485751
$821$	24.0141	0.838097	0.419048	−	0.907964i	$-0.362364\pi$
$821$	24.0141	0.838097	0.419048	+	0.907964i	$0.362364\pi$
$822$	58.5776	2.04313
$823$	−20.7259	−0.722458	−0.361229	−	0.932477i	$-0.617643\pi$
$823$	−20.7259	−0.722458	−0.361229	+	0.932477i	$0.617643\pi$
$824$	−12.3550	−0.430407
$825$	2.01288	0.0700796
$826$	50.8526	1.76939
$827$	28.6655	0.996797	0.498398	−	0.866948i	$-0.333922\pi$
$827$	28.6655	0.996797	0.498398	+	0.866948i	$0.333922\pi$
$828$	2.23800	0.0777760
$829$	21.3397	0.741158	0.370579	−	0.928801i	$-0.379159\pi$
$829$	21.3397	0.741158	0.370579	+	0.928801i	$0.379159\pi$
$830$	20.2692	0.703554
$831$	29.8931	1.03698
$832$	17.0038	0.589500
$833$	19.2604	0.667333
$834$	−34.7842	−1.20448
$835$	−14.1852	−0.490899
$836$	−1.48188	−0.0512519
$837$	16.5442	0.571850
$838$	55.7264	1.92504
$839$	−40.3234	−1.39212	−0.696060	−	0.717984i	$-0.745065\pi$
$839$	−40.3234	−1.39212	−0.696060	+	0.717984i	$0.745065\pi$
$840$	−3.97259	−0.137067
$841$	−28.5033	−0.982873
$842$	−40.8400	−1.40744
$843$	15.0653	0.518875
$844$	16.0102	0.551095
$845$	11.1898	0.384941
$846$	14.0045	0.481484
$847$	−2.04136	−0.0701418
$848$	−12.6359	−0.433920
$849$	−21.3383	−0.732328
$850$	12.6866	0.435148
$851$	−2.01752	−0.0691597
$852$	17.1736	0.588359
$853$	46.6593	1.59758	0.798792	−	0.601607i	$-0.205473\pi$
$853$	46.6593	1.59758	0.798792	+	0.601607i	$0.205473\pi$
$854$	28.8759	0.988113
$855$	−1.05170	−0.0359673
$856$	−16.7023	−0.570872
$857$	52.7689	1.80255	0.901275	−	0.433248i	$-0.142632\pi$
$857$	52.7689	1.80255	0.901275	+	0.433248i	$0.142632\pi$
$858$	18.4732	0.630664
$859$	−3.98543	−0.135981	−0.0679905	−	0.997686i	$-0.521659\pi$
$859$	−3.98543	−0.135981	−0.0679905	+	0.997686i	$0.521659\pi$
$860$	5.63690	0.192217
$861$	38.5696	1.31445
$862$	23.8728	0.813109
$863$	14.6185	0.497621	0.248811	−	0.968552i	$-0.419960\pi$
$863$	14.6185	0.497621	0.248811	+	0.968552i	$0.419960\pi$
$864$	−27.3068	−0.928997
$865$	−5.44057	−0.184985
$866$	−15.0539	−0.511552
$867$	58.8269	1.99787
$868$	12.7615	0.433154
$869$	7.12218	0.241603
$870$	2.64704	0.0897429
$871$	−51.9934	−1.76173
$872$	6.34331	0.214812
$873$	−1.72913	−0.0585221
$874$	2.67956	0.0906374
$875$	−2.04136	−0.0690104
$876$	−16.3913	−0.553812
$877$	−24.7365	−0.835294	−0.417647	−	0.908609i	$-0.637145\pi$
$877$	−24.7365	−0.835294	−0.417647	+	0.908609i	$0.637145\pi$
$878$	−2.34318	−0.0790784
$879$	−45.2605	−1.52660
$880$	−4.76779	−0.160722
$881$	−30.9602	−1.04308	−0.521538	−	0.853228i	$-0.674642\pi$
$881$	−30.9602	−1.04308	−0.521538	+	0.853228i	$0.674642\pi$
$882$	5.55935	0.187193
$883$	17.8695	0.601358	0.300679	−	0.953725i	$-0.402787\pi$
$883$	17.8695	0.601358	0.300679	+	0.953725i	$0.402787\pi$
$884$	49.5529	1.66664
$885$	26.8724	0.903305
$886$	−56.8498	−1.90991
$887$	28.5051	0.957108	0.478554	−	0.878058i	$-0.341161\pi$
$887$	28.5051	0.957108	0.478554	+	0.878058i	$0.341161\pi$
$888$	−2.73411	−0.0917509
$889$	10.7149	0.359366
$890$	6.09415	0.204276
$891$	−11.0490	−0.370156
$892$	−7.21651	−0.241627
$893$	7.13622	0.238805
$894$	−19.6301	−0.656529
$895$	−8.45415	−0.282591
$896$	15.2589	0.509763
$897$	−14.2164	−0.474673
$898$	−4.40140	−0.146877
$899$	2.97307	0.0991573
$900$	1.55849	0.0519497
$901$	−18.0190	−0.600299
$902$	17.5152	0.583192
$903$	−15.6302	−0.520140
$904$	−1.61151	−0.0535979
$905$	−13.6320	−0.453143
$906$	−78.0754	−2.59388
$907$	−26.4797	−0.879245	−0.439622	−	0.898183i	$-0.644888\pi$
$907$	−26.4797	−0.879245	−0.439622	+	0.898183i	$0.644888\pi$
$908$	−41.0311	−1.36166
$909$	−3.04803	−0.101097
$910$	−18.7345	−0.621042
$911$	14.7377	0.488281	0.244140	−	0.969740i	$-0.421494\pi$
$911$	14.7377	0.488281	0.244140	+	0.969740i	$0.421494\pi$
$912$	9.59701	0.317789
$913$	−10.8625	−0.359496
$914$	69.3791	2.29486
$915$	15.2591	0.504450
$916$	8.17539	0.270122
$917$	18.4894	0.610574
$918$	−49.7532	−1.64210
$919$	27.1844	0.896729	0.448365	−	0.893851i	$-0.352007\pi$
$919$	27.1844	0.896729	0.448365	+	0.893851i	$0.352007\pi$
$920$	1.38833	0.0457719
$921$	26.7777	0.882354
$922$	−19.5526	−0.643929
$923$	−28.3169	−0.932063
$924$	−6.08907	−0.200316
$925$	−1.40495	−0.0461945
$926$	45.2272	1.48626
$927$	−13.4400	−0.441426
$928$	−4.90717	−0.161086
$929$	28.3245	0.929295	0.464648	−	0.885496i	$-0.346181\pi$
$929$	28.3245	0.929295	0.464648	+	0.885496i	$0.346181\pi$
$930$	15.8451	0.519582
$931$	2.83286	0.0928434
$932$	36.0115	1.17959
$933$	−16.1745	−0.529528
$934$	−40.1634	−1.31419
$935$	−6.79891	−0.222348
$936$	−5.00085	−0.163458
$937$	37.8767	1.23738	0.618689	−	0.785636i	$-0.287664\pi$
$937$	37.8767	1.23738	0.618689	+	0.785636i	$0.287664\pi$
$938$	40.2678	1.31479
$939$	34.6067	1.12935
$940$	−10.5750	−0.344919
$941$	−49.2677	−1.60608	−0.803040	−	0.595925i	$-0.796786\pi$
$941$	−49.2677	−1.60608	−0.803040	+	0.595925i	$0.796786\pi$
$942$	−42.1829	−1.37439
$943$	−13.4792	−0.438943
$944$	−63.6509	−2.07166
$945$	8.00559	0.260422
$946$	−7.09797	−0.230775
$947$	−31.2750	−1.01630	−0.508151	−	0.861268i	$-0.669671\pi$
$947$	−31.2750	−1.01630	−0.508151	+	0.861268i	$0.669671\pi$
$948$	21.2444	0.689986
$949$	27.0271	0.877335
$950$	1.86598	0.0605404
$951$	7.41741	0.240526
$952$	13.4182	0.434887
$953$	17.0312	0.551694	0.275847	−	0.961202i	$-0.411042\pi$
$953$	17.0312	0.551694	0.275847	+	0.961202i	$0.411042\pi$
$954$	−5.20102	−0.168389
$955$	−12.4905	−0.404182
$956$	−15.8058	−0.511195
$957$	−1.41858	−0.0458561
$958$	13.6154	0.439893
$959$	31.8365	1.02805
$960$	−6.95902	−0.224601
$961$	−13.2033	−0.425912
$962$	−12.8939	−0.415716
$963$	−18.1690	−0.585487
$964$	−7.08734	−0.228268
$965$	−9.94947	−0.320285
$966$	11.0103	0.354252
$967$	−46.9228	−1.50894	−0.754468	−	0.656337i	$-0.772105\pi$
$967$	−46.9228	−1.50894	−0.754468	+	0.656337i	$0.772105\pi$
$968$	0.966800	0.0310741
$969$	13.6854	0.439639
$970$	3.06791	0.0985047
$971$	−38.3346	−1.23022	−0.615108	−	0.788443i	$-0.710888\pi$
$971$	−38.3346	−1.23022	−0.615108	+	0.788443i	$0.710888\pi$
$972$	−15.5231	−0.497905
$973$	−18.9049	−0.606064
$974$	57.5179	1.84299
$975$	−9.89999	−0.317053
$976$	−36.1432	−1.15692
$977$	−45.8913	−1.46819	−0.734096	−	0.679045i	$-0.762394\pi$
$977$	−45.8913	−1.46819	−0.734096	+	0.679045i	$0.762394\pi$
$978$	−16.6764	−0.533254
$979$	−3.26592	−0.104379
$980$	−4.19797	−0.134099
$981$	6.90035	0.220311
$982$	57.7093	1.84158
$983$	−29.4166	−0.938244	−0.469122	−	0.883133i	$-0.655430\pi$
$983$	−29.4166	−0.938244	−0.469122	+	0.883133i	$0.655430\pi$
$984$	−18.2668	−0.582325
$985$	24.3654	0.776347
$986$	−8.94089	−0.284736
$987$	29.3228	0.933356
$988$	7.28835	0.231873
$989$	5.46240	0.173694
$990$	−1.96245	−0.0623707
$991$	1.56657	0.0497637	0.0248819	−	0.999690i	$-0.492079\pi$
$991$	1.56657	0.0497637	0.0248819	+	0.999690i	$0.492079\pi$
$992$	−29.3742	−0.932633
$993$	−32.4534	−1.02988
$994$	21.9309	0.695605
$995$	−11.5042	−0.364707
$996$	−32.4012	−1.02667
$997$	54.1570	1.71517	0.857585	−	0.514343i	$-0.171964\pi$
$997$	54.1570	1.71517	0.857585	+	0.514343i	$0.171964\pi$
$998$	−71.4605	−2.26204
$999$	5.50980	0.174322

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.a.i.1.3	✓	8
3.2	odd	2		9405.2.a.bf.1.6		8
5.4	even	2		5225.2.a.o.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.i.1.3	✓	8	1.1	even	1	trivial
5225.2.a.o.1.6		8	5.4	even	2
9405.2.a.bf.1.6		8	3.2	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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q-1.86598 q^{2} +2.01288 q^{3} +1.48188 q^{4} +1.00000 q^{5} -3.75600 q^{6} -2.04136 q^{7} +0.966800 q^{8} +1.05170 q^{9} +O(q^{10})\)	\(q-1.86598 q^{2} +2.01288 q^{3} +1.48188 q^{4} +1.00000 q^{5} -3.75600 q^{6} -2.04136 q^{7} +0.966800 q^{8} +1.05170 q^{9} -1.86598 q^{10} +1.00000 q^{11} +2.98285 q^{12} -4.91831 q^{13} +3.80913 q^{14} +2.01288 q^{15} -4.76779 q^{16} -6.79891 q^{17} -1.96245 q^{18} -1.00000 q^{19} +1.48188 q^{20} -4.10901 q^{21} -1.86598 q^{22} +1.43601 q^{23} +1.94606 q^{24} +1.00000 q^{25} +9.17747 q^{26} -3.92170 q^{27} -3.02505 q^{28} -0.704749 q^{29} -3.75600 q^{30} -4.21862 q^{31} +6.96300 q^{32} +2.01288 q^{33} +12.6866 q^{34} -2.04136 q^{35} +1.55849 q^{36} -1.40495 q^{37} +1.86598 q^{38} -9.89999 q^{39} +0.966800 q^{40} -9.38658 q^{41} +7.66733 q^{42} +3.80388 q^{43} +1.48188 q^{44} +1.05170 q^{45} -2.67956 q^{46} -7.13622 q^{47} -9.59701 q^{48} -2.83286 q^{49} -1.86598 q^{50} -13.6854 q^{51} -7.28835 q^{52} +2.65027 q^{53} +7.31782 q^{54} +1.00000 q^{55} -1.97358 q^{56} -2.01288 q^{57} +1.31505 q^{58} +13.3502 q^{59} +2.98285 q^{60} +7.58071 q^{61} +7.87185 q^{62} -2.14689 q^{63} -3.45724 q^{64} -4.91831 q^{65} -3.75600 q^{66} +10.5714 q^{67} -10.0752 q^{68} +2.89051 q^{69} +3.80913 q^{70} +5.75745 q^{71} +1.01678 q^{72} -5.49519 q^{73} +2.62161 q^{74} +2.01288 q^{75} -1.48188 q^{76} -2.04136 q^{77} +18.4732 q^{78} +7.12218 q^{79} -4.76779 q^{80} -11.0490 q^{81} +17.5152 q^{82} -10.8625 q^{83} -6.08907 q^{84} -6.79891 q^{85} -7.09797 q^{86} -1.41858 q^{87} +0.966800 q^{88} -3.26592 q^{89} -1.96245 q^{90} +10.0400 q^{91} +2.12799 q^{92} -8.49158 q^{93} +13.3160 q^{94} -1.00000 q^{95} +14.0157 q^{96} -1.64413 q^{97} +5.28607 q^{98} +1.05170 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9}+O(q^{10}) \) 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9	\( 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9} - 6 q^{10} + 8 q^{11} - 7 q^{12} - 17 q^{13} + 12 q^{14} - 7 q^{15} + 18 q^{16} - 9 q^{17} - 2 q^{18} - 8 q^{19} + 10 q^{20} + q^{21} - 6 q^{22} - 8 q^{23} + q^{24} + 8 q^{25} + 10 q^{26} - 34 q^{27} - 22 q^{28} - 3 q^{29} - q^{31} - 37 q^{32} - 7 q^{33} - 8 q^{34} - 11 q^{35} + 30 q^{36} - 17 q^{37} + 6 q^{38} + 14 q^{39} - 18 q^{40} - 5 q^{41} + 15 q^{42} - 21 q^{43} + 10 q^{44} + 11 q^{45} - 2 q^{46} - 8 q^{47} + 10 q^{48} + 19 q^{49} - 6 q^{50} - 16 q^{51} + 9 q^{52} - 19 q^{53} - 3 q^{54} + 8 q^{55} + 24 q^{56} + 7 q^{57} + 37 q^{58} - 33 q^{59} - 7 q^{60} - q^{61} - 42 q^{62} - 20 q^{63} + 48 q^{64} - 17 q^{65} - 18 q^{67} - 37 q^{68} + 16 q^{69} + 12 q^{70} - 18 q^{71} + 13 q^{72} - 18 q^{73} + 15 q^{74} - 7 q^{75} - 10 q^{76} - 11 q^{77} - 51 q^{78} - 5 q^{79} + 18 q^{80} + 32 q^{81} + 12 q^{82} - 33 q^{83} - 51 q^{84} - 9 q^{85} - 16 q^{86} - 26 q^{87} - 18 q^{88} - 20 q^{89} - 2 q^{90} + 6 q^{91} - 3 q^{92} + 18 q^{93} + 30 q^{94} - 8 q^{95} + 21 q^{96} - 69 q^{98} + 11 q^{99}+O(q^{100}) \) 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9 - 6 * q^10 + 8 * q^11 - 7 * q^12 - 17 * q^13 + 12 * q^14 - 7 * q^15 + 18 * q^16 - 9 * q^17 - 2 * q^18 - 8 * q^19 + 10 * q^20 + q^21 - 6 * q^22 - 8 * q^23 + q^24 + 8 * q^25 + 10 * q^26 - 34 * q^27 - 22 * q^28 - 3 * q^29 - q^31 - 37 * q^32 - 7 * q^33 - 8 * q^34 - 11 * q^35 + 30 * q^36 - 17 * q^37 + 6 * q^38 + 14 * q^39 - 18 * q^40 - 5 * q^41 + 15 * q^42 - 21 * q^43 + 10 * q^44 + 11 * q^45 - 2 * q^46 - 8 * q^47 + 10 * q^48 + 19 * q^49 - 6 * q^50 - 16 * q^51 + 9 * q^52 - 19 * q^53 - 3 * q^54 + 8 * q^55 + 24 * q^56 + 7 * q^57 + 37 * q^58 - 33 * q^59 - 7 * q^60 - q^61 - 42 * q^62 - 20 * q^63 + 48 * q^64 - 17 * q^65 - 18 * q^67 - 37 * q^68 + 16 * q^69 + 12 * q^70 - 18 * q^71 + 13 * q^72 - 18 * q^73 + 15 * q^74 - 7 * q^75 - 10 * q^76 - 11 * q^77 - 51 * q^78 - 5 * q^79 + 18 * q^80 + 32 * q^81 + 12 * q^82 - 33 * q^83 - 51 * q^84 - 9 * q^85 - 16 * q^86 - 26 * q^87 - 18 * q^88 - 20 * q^89 - 2 * q^90 + 6 * q^91 - 3 * q^92 + 18 * q^93 + 30 * q^94 - 8 * q^95 + 21 * q^96 - 69 * q^98 + 11 * q^99

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