LMFDB - Embedded newform 4010.2.a.m.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4010,2,Mod(1,4010)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4010, 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("4010.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4010 = 2 \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4010.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:	$32.0200112105$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 35 x^{18} + 154 x^{17} + 460 x^{16} - 2392 x^{15} - 2591 x^{14} + 19157 x^{13} + \cdots + 104 $ x^20 - 4x^19 - 35x^18 + 154x^17 + 460x^16 - 2392x^15 - 2591x^14 + 19157x^13 + 2776x^12 - 83577x^11 + 34362x^10 + 190617x^9 - 150697x^8 - 189098x^7 + 211360x^6 + 37365x^5 - 68876x^4 - 8984x^3 + 7920x^2 + 1972*x + 104
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.97232$ of defining polynomial
Character	$\chi$	$=$	4010.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.97232 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.97232 q^{6} +4.46416 q^{7} -1.00000 q^{8} +0.890065 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.97232 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.97232 q^{6} +4.46416 q^{7} -1.00000 q^{8} +0.890065 q^{9} -1.00000 q^{10} +3.57251 q^{11} -1.97232 q^{12} -5.35054 q^{13} -4.46416 q^{14} -1.97232 q^{15} +1.00000 q^{16} -1.14025 q^{17} -0.890065 q^{18} -8.48136 q^{19} +1.00000 q^{20} -8.80478 q^{21} -3.57251 q^{22} -5.12011 q^{23} +1.97232 q^{24} +1.00000 q^{25} +5.35054 q^{26} +4.16148 q^{27} +4.46416 q^{28} +7.40661 q^{29} +1.97232 q^{30} +2.31182 q^{31} -1.00000 q^{32} -7.04615 q^{33} +1.14025 q^{34} +4.46416 q^{35} +0.890065 q^{36} +2.72367 q^{37} +8.48136 q^{38} +10.5530 q^{39} -1.00000 q^{40} -0.831224 q^{41} +8.80478 q^{42} +1.83158 q^{43} +3.57251 q^{44} +0.890065 q^{45} +5.12011 q^{46} -4.02634 q^{47} -1.97232 q^{48} +12.9288 q^{49} -1.00000 q^{50} +2.24894 q^{51} -5.35054 q^{52} +5.53708 q^{53} -4.16148 q^{54} +3.57251 q^{55} -4.46416 q^{56} +16.7280 q^{57} -7.40661 q^{58} -2.27345 q^{59} -1.97232 q^{60} +12.8462 q^{61} -2.31182 q^{62} +3.97340 q^{63} +1.00000 q^{64} -5.35054 q^{65} +7.04615 q^{66} +8.80246 q^{67} -1.14025 q^{68} +10.0985 q^{69} -4.46416 q^{70} -1.20504 q^{71} -0.890065 q^{72} -12.9254 q^{73} -2.72367 q^{74} -1.97232 q^{75} -8.48136 q^{76} +15.9483 q^{77} -10.5530 q^{78} +7.33713 q^{79} +1.00000 q^{80} -10.8780 q^{81} +0.831224 q^{82} -6.55527 q^{83} -8.80478 q^{84} -1.14025 q^{85} -1.83158 q^{86} -14.6082 q^{87} -3.57251 q^{88} +14.7996 q^{89} -0.890065 q^{90} -23.8857 q^{91} -5.12011 q^{92} -4.55966 q^{93} +4.02634 q^{94} -8.48136 q^{95} +1.97232 q^{96} -0.565280 q^{97} -12.9288 q^{98} +3.17976 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} + 20 q^{5} - 4 q^{6} + 11 q^{7} - 20 q^{8} + 26 q^{9}+O(q^{10}) $ 20 * q - 20 * q^2 + 4 * q^3 + 20 * q^4 + 20 * q^5 - 4 * q^6 + 11 * q^7 - 20 * q^8 + 26 * q^9	\( 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} + 20 q^{5} - 4 q^{6} + 11 q^{7} - 20 q^{8} + 26 q^{9} - 20 q^{10} + 10 q^{11} + 4 q^{12} - 9 q^{13} - 11 q^{14} + 4 q^{15} + 20 q^{16} - 11 q^{17} - 26 q^{18} + 17 q^{19} + 20 q^{20} - 2 q^{21} - 10 q^{22} - 3 q^{23} - 4 q^{24} + 20 q^{25} + 9 q^{26} - 2 q^{27} + 11 q^{28} + 6 q^{29} - 4 q^{30} + 28 q^{31} - 20 q^{32} + 2 q^{33} + 11 q^{34} + 11 q^{35} + 26 q^{36} + 33 q^{37} - 17 q^{38} + 36 q^{39} - 20 q^{40} + 32 q^{41} + 2 q^{42} + 30 q^{43} + 10 q^{44} + 26 q^{45} + 3 q^{46} + 13 q^{47} + 4 q^{48} + 43 q^{49} - 20 q^{50} + 43 q^{51} - 9 q^{52} + 2 q^{54} + 10 q^{55} - 11 q^{56} + 19 q^{57} - 6 q^{58} + 52 q^{59} + 4 q^{60} + 25 q^{61} - 28 q^{62} + 16 q^{63} + 20 q^{64} - 9 q^{65} - 2 q^{66} + 40 q^{67} - 11 q^{68} + 39 q^{69} - 11 q^{70} + 25 q^{71} - 26 q^{72} - 5 q^{73} - 33 q^{74} + 4 q^{75} + 17 q^{76} - 9 q^{77} - 36 q^{78} + 40 q^{79} + 20 q^{80} + 48 q^{81} - 32 q^{82} + 10 q^{83} - 2 q^{84} - 11 q^{85} - 30 q^{86} + 10 q^{87} - 10 q^{88} + 17 q^{89} - 26 q^{90} + 88 q^{91} - 3 q^{92} - 4 q^{93} - 13 q^{94} + 17 q^{95} - 4 q^{96} - 2 q^{97} - 43 q^{98} + 50 q^{99}+O(q^{100}) \) 20 * q - 20 * q^2 + 4 * q^3 + 20 * q^4 + 20 * q^5 - 4 * q^6 + 11 * q^7 - 20 * q^8 + 26 * q^9 - 20 * q^10 + 10 * q^11 + 4 * q^12 - 9 * q^13 - 11 * q^14 + 4 * q^15 + 20 * q^16 - 11 * q^17 - 26 * q^18 + 17 * q^19 + 20 * q^20 - 2 * q^21 - 10 * q^22 - 3 * q^23 - 4 * q^24 + 20 * q^25 + 9 * q^26 - 2 * q^27 + 11 * q^28 + 6 * q^29 - 4 * q^30 + 28 * q^31 - 20 * q^32 + 2 * q^33 + 11 * q^34 + 11 * q^35 + 26 * q^36 + 33 * q^37 - 17 * q^38 + 36 * q^39 - 20 * q^40 + 32 * q^41 + 2 * q^42 + 30 * q^43 + 10 * q^44 + 26 * q^45 + 3 * q^46 + 13 * q^47 + 4 * q^48 + 43 * q^49 - 20 * q^50 + 43 * q^51 - 9 * q^52 + 2 * q^54 + 10 * q^55 - 11 * q^56 + 19 * q^57 - 6 * q^58 + 52 * q^59 + 4 * q^60 + 25 * q^61 - 28 * q^62 + 16 * q^63 + 20 * q^64 - 9 * q^65 - 2 * q^66 + 40 * q^67 - 11 * q^68 + 39 * q^69 - 11 * q^70 + 25 * q^71 - 26 * q^72 - 5 * q^73 - 33 * q^74 + 4 * q^75 + 17 * q^76 - 9 * q^77 - 36 * q^78 + 40 * q^79 + 20 * q^80 + 48 * q^81 - 32 * q^82 + 10 * q^83 - 2 * q^84 - 11 * q^85 - 30 * q^86 + 10 * q^87 - 10 * q^88 + 17 * q^89 - 26 * q^90 + 88 * q^91 - 3 * q^92 - 4 * q^93 - 13 * q^94 + 17 * q^95 - 4 * q^96 - 2 * q^97 - 43 * q^98 + 50 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.97232	−1.13872	−0.569361	−	0.822088i	$-0.692809\pi$
$3$	−1.97232	−1.13872	−0.569361	+	0.822088i	$0.692809\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.97232	0.805198
$7$	4.46416	1.68730	0.843648	−	0.536897i	$-0.180404\pi$
$7$	4.46416	1.68730	0.843648	+	0.536897i	$0.180404\pi$
$8$	−1.00000	−0.353553
$9$	0.890065	0.296688
$10$	−1.00000	−0.316228
$11$	3.57251	1.07715	0.538576	−	0.842577i	$-0.318963\pi$
$11$	3.57251	1.07715	0.538576	+	0.842577i	$0.318963\pi$
$12$	−1.97232	−0.569361
$13$	−5.35054	−1.48397	−0.741987	−	0.670415i	$-0.766116\pi$
$13$	−5.35054	−1.48397	−0.741987	+	0.670415i	$0.766116\pi$
$14$	−4.46416	−1.19310
$15$	−1.97232	−0.509252
$16$	1.00000	0.250000
$17$	−1.14025	−0.276551	−0.138276	−	0.990394i	$-0.544156\pi$
$17$	−1.14025	−0.276551	−0.138276	+	0.990394i	$0.544156\pi$
$18$	−0.890065	−0.209790
$19$	−8.48136	−1.94576	−0.972879	−	0.231314i	$-0.925698\pi$
$19$	−8.48136	−1.94576	−0.972879	+	0.231314i	$0.925698\pi$
$20$	1.00000	0.223607
$21$	−8.80478	−1.92136
$22$	−3.57251	−0.761661
$23$	−5.12011	−1.06762	−0.533808	−	0.845606i	$-0.679240\pi$
$23$	−5.12011	−1.06762	−0.533808	+	0.845606i	$0.679240\pi$
$24$	1.97232	0.402599
$25$	1.00000	0.200000
$26$	5.35054	1.04933
$27$	4.16148	0.800877
$28$	4.46416	0.843648
$29$	7.40661	1.37537	0.687686	−	0.726008i	$-0.258627\pi$
$29$	7.40661	1.37537	0.687686	+	0.726008i	$0.258627\pi$
$30$	1.97232	0.360096
$31$	2.31182	0.415215	0.207607	−	0.978212i	$-0.433432\pi$
$31$	2.31182	0.415215	0.207607	+	0.978212i	$0.433432\pi$
$32$	−1.00000	−0.176777
$33$	−7.04615	−1.22658
$34$	1.14025	0.195551
$35$	4.46416	0.754581
$36$	0.890065	0.148344
$37$	2.72367	0.447768	0.223884	−	0.974616i	$-0.428126\pi$
$37$	2.72367	0.447768	0.223884	+	0.974616i	$0.428126\pi$
$38$	8.48136	1.37586
$39$	10.5530	1.68983
$40$	−1.00000	−0.158114
$41$	−0.831224	−0.129815	−0.0649077	−	0.997891i	$-0.520675\pi$
$41$	−0.831224	−0.129815	−0.0649077	+	0.997891i	$0.520675\pi$
$42$	8.80478	1.35861
$43$	1.83158	0.279314	0.139657	−	0.990200i	$-0.455400\pi$
$43$	1.83158	0.279314	0.139657	+	0.990200i	$0.455400\pi$
$44$	3.57251	0.538576
$45$	0.890065	0.132683
$46$	5.12011	0.754919
$47$	−4.02634	−0.587302	−0.293651	−	0.955913i	$-0.594870\pi$
$47$	−4.02634	−0.587302	−0.293651	+	0.955913i	$0.594870\pi$
$48$	−1.97232	−0.284681
$49$	12.9288	1.84696
$50$	−1.00000	−0.141421
$51$	2.24894	0.314915
$52$	−5.35054	−0.741987
$53$	5.53708	0.760577	0.380288	−	0.924868i	$-0.375825\pi$
$53$	5.53708	0.760577	0.380288	+	0.924868i	$0.375825\pi$
$54$	−4.16148	−0.566305
$55$	3.57251	0.481717
$56$	−4.46416	−0.596549
$57$	16.7280	2.21568
$58$	−7.40661	−0.972535
$59$	−2.27345	−0.295978	−0.147989	−	0.988989i	$-0.547280\pi$
$59$	−2.27345	−0.295978	−0.147989	+	0.988989i	$0.547280\pi$
$60$	−1.97232	−0.254626
$61$	12.8462	1.64479	0.822396	−	0.568916i	$-0.192637\pi$
$61$	12.8462	1.64479	0.822396	+	0.568916i	$0.192637\pi$
$62$	−2.31182	−0.293601
$63$	3.97340	0.500601
$64$	1.00000	0.125000
$65$	−5.35054	−0.663653
$66$	7.04615	0.867321
$67$	8.80246	1.07539	0.537696	−	0.843139i	$-0.319295\pi$
$67$	8.80246	1.07539	0.537696	+	0.843139i	$0.319295\pi$
$68$	−1.14025	−0.138276
$69$	10.0985	1.21572
$70$	−4.46416	−0.533570
$71$	−1.20504	−0.143012	−0.0715061	−	0.997440i	$-0.522781\pi$
$71$	−1.20504	−0.143012	−0.0715061	+	0.997440i	$0.522781\pi$
$72$	−0.890065	−0.104895
$73$	−12.9254	−1.51281	−0.756403	−	0.654106i	$-0.773045\pi$
$73$	−12.9254	−1.51281	−0.756403	+	0.654106i	$0.773045\pi$
$74$	−2.72367	−0.316620
$75$	−1.97232	−0.227744
$76$	−8.48136	−0.972879
$77$	15.9483	1.81747
$78$	−10.5530	−1.19489
$79$	7.33713	0.825491	0.412746	−	0.910846i	$-0.364570\pi$
$79$	7.33713	0.825491	0.412746	+	0.910846i	$0.364570\pi$
$80$	1.00000	0.111803
$81$	−10.8780	−1.20866
$82$	0.831224	0.0917934
$83$	−6.55527	−0.719534	−0.359767	−	0.933042i	$-0.617144\pi$
$83$	−6.55527	−0.719534	−0.359767	+	0.933042i	$0.617144\pi$
$84$	−8.80478	−0.960680
$85$	−1.14025	−0.123677
$86$	−1.83158	−0.197505
$87$	−14.6082	−1.56617
$88$	−3.57251	−0.380831
$89$	14.7996	1.56875	0.784376	−	0.620286i	$-0.212983\pi$
$89$	14.7996	1.56875	0.784376	+	0.620286i	$0.212983\pi$
$90$	−0.890065	−0.0938211
$91$	−23.8857	−2.50390
$92$	−5.12011	−0.533808
$93$	−4.55966	−0.472814
$94$	4.02634	0.415285
$95$	−8.48136	−0.870170
$96$	1.97232	0.201300
$97$	−0.565280	−0.0573955	−0.0286978	−	0.999588i	$-0.509136\pi$
$97$	−0.565280	−0.0573955	−0.0286978	+	0.999588i	$0.509136\pi$
$98$	−12.9288	−1.30600
$99$	3.17976	0.319578
$100$	1.00000	0.100000
$101$	8.34629	0.830486	0.415243	−	0.909710i	$-0.363696\pi$
$101$	8.34629	0.830486	0.415243	+	0.909710i	$0.363696\pi$
$102$	−2.24894	−0.222678
$103$	19.5367	1.92501	0.962503	−	0.271272i	$-0.0874444\pi$
$103$	19.5367	1.92501	0.962503	+	0.271272i	$0.0874444\pi$
$104$	5.35054	0.524664
$105$	−8.80478	−0.859259
$106$	−5.53708	−0.537809
$107$	2.94265	0.284476	0.142238	−	0.989832i	$-0.454570\pi$
$107$	2.94265	0.284476	0.142238	+	0.989832i	$0.454570\pi$
$108$	4.16148	0.400438
$109$	−8.23634	−0.788898	−0.394449	−	0.918918i	$-0.629065\pi$
$109$	−8.23634	−0.788898	−0.394449	+	0.918918i	$0.629065\pi$
$110$	−3.57251	−0.340625
$111$	−5.37195	−0.509883
$112$	4.46416	0.421824
$113$	−12.0168	−1.13044	−0.565222	−	0.824939i	$-0.691209\pi$
$113$	−12.0168	−1.13044	−0.565222	+	0.824939i	$0.691209\pi$
$114$	−16.7280	−1.56672
$115$	−5.12011	−0.477452
$116$	7.40661	0.687686
$117$	−4.76233	−0.440278
$118$	2.27345	0.209288
$119$	−5.09026	−0.466623
$120$	1.97232	0.180048
$121$	1.76281	0.160256
$122$	−12.8462	−1.16304
$123$	1.63944	0.147824
$124$	2.31182	0.207607
$125$	1.00000	0.0894427
$126$	−3.97340	−0.353978
$127$	1.98808	0.176414	0.0882069	−	0.996102i	$-0.471886\pi$
$127$	1.98808	0.176414	0.0882069	+	0.996102i	$0.471886\pi$
$128$	−1.00000	−0.0883883
$129$	−3.61248	−0.318061
$130$	5.35054	0.469274
$131$	−5.00597	−0.437374	−0.218687	−	0.975795i	$-0.570177\pi$
$131$	−5.00597	−0.437374	−0.218687	+	0.975795i	$0.570177\pi$
$132$	−7.04615	−0.613288
$133$	−37.8622	−3.28307
$134$	−8.80246	−0.760416
$135$	4.16148	0.358163
$136$	1.14025	0.0977756
$137$	12.6618	1.08177	0.540886	−	0.841096i	$-0.318089\pi$
$137$	12.6618	1.08177	0.540886	+	0.841096i	$0.318089\pi$
$138$	−10.0985	−0.859643
$139$	−14.3273	−1.21523	−0.607613	−	0.794233i	$-0.707873\pi$
$139$	−14.3273	−1.21523	−0.607613	+	0.794233i	$0.707873\pi$
$140$	4.46416	0.377291
$141$	7.94125	0.668773
$142$	1.20504	0.101125
$143$	−19.1149	−1.59846
$144$	0.890065	0.0741721
$145$	7.40661	0.615085
$146$	12.9254	1.06972
$147$	−25.4997	−2.10318
$148$	2.72367	0.223884
$149$	4.53333	0.371385	0.185693	−	0.982608i	$-0.440547\pi$
$149$	4.53333	0.371385	0.185693	+	0.982608i	$0.440547\pi$
$150$	1.97232	0.161040
$151$	10.4460	0.850087	0.425044	−	0.905173i	$-0.360259\pi$
$151$	10.4460	0.850087	0.425044	+	0.905173i	$0.360259\pi$
$152$	8.48136	0.687930
$153$	−1.01490	−0.0820495
$154$	−15.9483	−1.28515
$155$	2.31182	0.185690
$156$	10.5530	0.844917
$157$	−2.96594	−0.236708	−0.118354	−	0.992971i	$-0.537762\pi$
$157$	−2.96594	−0.236708	−0.118354	+	0.992971i	$0.537762\pi$
$158$	−7.33713	−0.583711
$159$	−10.9209	−0.866086
$160$	−1.00000	−0.0790569
$161$	−22.8570	−1.80138
$162$	10.8780	0.854655
$163$	17.3398	1.35816	0.679078	−	0.734066i	$-0.262380\pi$
$163$	17.3398	1.35816	0.679078	+	0.734066i	$0.262380\pi$
$164$	−0.831224	−0.0649077
$165$	−7.04615	−0.548542
$166$	6.55527	0.508788
$167$	3.32690	0.257443	0.128722	−	0.991681i	$-0.458913\pi$
$167$	3.32690	0.257443	0.128722	+	0.991681i	$0.458913\pi$
$168$	8.80478	0.679303
$169$	15.6283	1.20218
$170$	1.14025	0.0874531
$171$	−7.54896	−0.577284
$172$	1.83158	0.139657
$173$	9.07488	0.689951	0.344975	−	0.938612i	$-0.387887\pi$
$173$	9.07488	0.689951	0.344975	+	0.938612i	$0.387887\pi$
$174$	14.6082	1.10745
$175$	4.46416	0.337459
$176$	3.57251	0.269288
$177$	4.48398	0.337037
$178$	−14.7996	−1.10928
$179$	21.6148	1.61556	0.807782	−	0.589481i	$-0.200668\pi$
$179$	21.6148	1.61556	0.807782	+	0.589481i	$0.200668\pi$
$180$	0.890065	0.0663415
$181$	16.2898	1.21081	0.605404	−	0.795918i	$-0.293011\pi$
$181$	16.2898	1.21081	0.605404	+	0.795918i	$0.293011\pi$
$182$	23.8857	1.77053
$183$	−25.3369	−1.87296
$184$	5.12011	0.377459
$185$	2.72367	0.200248
$186$	4.55966	0.334330
$187$	−4.07355	−0.297887
$188$	−4.02634	−0.293651
$189$	18.5775	1.35132
$190$	8.48136	0.615303
$191$	7.70953	0.557842	0.278921	−	0.960314i	$-0.410023\pi$
$191$	7.70953	0.557842	0.278921	+	0.960314i	$0.410023\pi$
$192$	−1.97232	−0.142340
$193$	−16.4253	−1.18232	−0.591158	−	0.806556i	$-0.701329\pi$
$193$	−16.4253	−1.18232	−0.591158	+	0.806556i	$0.701329\pi$
$194$	0.565280	0.0405847
$195$	10.5530	0.755716
$196$	12.9288	0.923482
$197$	−2.20129	−0.156835	−0.0784177	−	0.996921i	$-0.524987\pi$
$197$	−2.20129	−0.156835	−0.0784177	+	0.996921i	$0.524987\pi$
$198$	−3.17976	−0.225976
$199$	24.4277	1.73163	0.865817	−	0.500361i	$-0.166800\pi$
$199$	24.4277	1.73163	0.865817	+	0.500361i	$0.166800\pi$
$200$	−1.00000	−0.0707107
$201$	−17.3613	−1.22457
$202$	−8.34629	−0.587243
$203$	33.0643	2.32066
$204$	2.24894	0.157457
$205$	−0.831224	−0.0580552
$206$	−19.5367	−1.36118
$207$	−4.55723	−0.316749
$208$	−5.35054	−0.370993
$209$	−30.2997	−2.09588
$210$	8.80478	0.607588
$211$	25.1347	1.73034	0.865171	−	0.501477i	$-0.167210\pi$
$211$	25.1347	1.73034	0.865171	+	0.501477i	$0.167210\pi$
$212$	5.53708	0.380288
$213$	2.37674	0.162851
$214$	−2.94265	−0.201155
$215$	1.83158	0.124913
$216$	−4.16148	−0.283153
$217$	10.3203	0.700590
$218$	8.23634	0.557835
$219$	25.4931	1.72267
$220$	3.57251	0.240858
$221$	6.10095	0.410394
$222$	5.37195	0.360542
$223$	−19.9792	−1.33791	−0.668954	−	0.743304i	$-0.733258\pi$
$223$	−19.9792	−1.33791	−0.668954	+	0.743304i	$0.733258\pi$
$224$	−4.46416	−0.298274
$225$	0.890065	0.0593377
$226$	12.0168	0.799344
$227$	−3.63417	−0.241209	−0.120604	−	0.992701i	$-0.538483\pi$
$227$	−3.63417	−0.241209	−0.120604	+	0.992701i	$0.538483\pi$
$228$	16.7280	1.10784
$229$	14.3424	0.947770	0.473885	−	0.880587i	$-0.342851\pi$
$229$	14.3424	0.947770	0.473885	+	0.880587i	$0.342851\pi$
$230$	5.12011	0.337610
$231$	−31.4551	−2.06960
$232$	−7.40661	−0.486267
$233$	−8.19892	−0.537129	−0.268565	−	0.963262i	$-0.586549\pi$
$233$	−8.19892	−0.537129	−0.268565	+	0.963262i	$0.586549\pi$
$234$	4.76233	0.311323
$235$	−4.02634	−0.262649
$236$	−2.27345	−0.147989
$237$	−14.4712	−0.940005
$238$	5.09026	0.329952
$239$	−3.63359	−0.235038	−0.117519	−	0.993071i	$-0.537494\pi$
$239$	−3.63359	−0.235038	−0.117519	+	0.993071i	$0.537494\pi$
$240$	−1.97232	−0.127313
$241$	12.7878	0.823737	0.411868	−	0.911243i	$-0.364876\pi$
$241$	12.7878	0.823737	0.411868	+	0.911243i	$0.364876\pi$
$242$	−1.76281	−0.113318
$243$	8.97048	0.575456
$244$	12.8462	0.822396
$245$	12.9288	0.825988
$246$	−1.63944	−0.104527
$247$	45.3799	2.88745
$248$	−2.31182	−0.146801
$249$	12.9291	0.819350
$250$	−1.00000	−0.0632456
$251$	−18.9188	−1.19414	−0.597072	−	0.802187i	$-0.703669\pi$
$251$	−18.9188	−1.19414	−0.597072	+	0.802187i	$0.703669\pi$
$252$	3.97340	0.250300
$253$	−18.2916	−1.14998
$254$	−1.98808	−0.124743
$255$	2.24894	0.140834
$256$	1.00000	0.0625000
$257$	−25.9786	−1.62050	−0.810249	−	0.586085i	$-0.800668\pi$
$257$	−25.9786	−1.62050	−0.810249	+	0.586085i	$0.800668\pi$
$258$	3.61248	0.224903
$259$	12.1589	0.755517
$260$	−5.35054	−0.331827
$261$	6.59236	0.408057
$262$	5.00597	0.309270
$263$	−17.6021	−1.08539	−0.542695	−	0.839930i	$-0.682596\pi$
$263$	−17.6021	−1.08539	−0.542695	+	0.839930i	$0.682596\pi$
$264$	7.04615	0.433660
$265$	5.53708	0.340140
$266$	37.8622	2.32148
$267$	−29.1896	−1.78637
$268$	8.80246	0.537696
$269$	−14.8053	−0.902693	−0.451346	−	0.892349i	$-0.649056\pi$
$269$	−14.8053	−0.902693	−0.451346	+	0.892349i	$0.649056\pi$
$270$	−4.16148	−0.253259
$271$	20.3518	1.23628	0.618141	−	0.786067i	$-0.287886\pi$
$271$	20.3518	1.23628	0.618141	+	0.786067i	$0.287886\pi$
$272$	−1.14025	−0.0691378
$273$	47.1103	2.85125
$274$	−12.6618	−0.764929
$275$	3.57251	0.215430
$276$	10.0985	0.607859
$277$	−19.0366	−1.14380	−0.571900	−	0.820323i	$-0.693793\pi$
$277$	−19.0366	−1.14380	−0.571900	+	0.820323i	$0.693793\pi$
$278$	14.3273	0.859294
$279$	2.05767	0.123189
$280$	−4.46416	−0.266785
$281$	18.7645	1.11939	0.559697	−	0.828697i	$-0.310917\pi$
$281$	18.7645	1.11939	0.559697	+	0.828697i	$0.310917\pi$
$282$	−7.94125	−0.472894
$283$	18.7279	1.11326	0.556631	−	0.830760i	$-0.312094\pi$
$283$	18.7279	1.11326	0.556631	+	0.830760i	$0.312094\pi$
$284$	−1.20504	−0.0715061
$285$	16.7280	0.990882
$286$	19.1149	1.13028
$287$	−3.71072	−0.219037
$288$	−0.890065	−0.0524476
$289$	−15.6998	−0.923520
$290$	−7.40661	−0.434931
$291$	1.11492	0.0653575
$292$	−12.9254	−0.756403
$293$	−1.61768	−0.0945059	−0.0472529	−	0.998883i	$-0.515047\pi$
$293$	−1.61768	−0.0945059	−0.0472529	+	0.998883i	$0.515047\pi$
$294$	25.4997	1.48717
$295$	−2.27345	−0.132365
$296$	−2.72367	−0.158310
$297$	14.8669	0.862666
$298$	−4.53333	−0.262609
$299$	27.3953	1.58431
$300$	−1.97232	−0.113872
$301$	8.17649	0.471285
$302$	−10.4460	−0.601103
$303$	−16.4616	−0.945693
$304$	−8.48136	−0.486440
$305$	12.8462	0.735573
$306$	1.01490	0.0580177
$307$	14.2845	0.815261	0.407630	−	0.913147i	$-0.366355\pi$
$307$	14.2845	0.815261	0.407630	+	0.913147i	$0.366355\pi$
$308$	15.9483	0.908736
$309$	−38.5327	−2.19205
$310$	−2.31182	−0.131302
$311$	8.57485	0.486235	0.243117	−	0.969997i	$-0.421830\pi$
$311$	8.57485	0.486235	0.243117	+	0.969997i	$0.421830\pi$
$312$	−10.5530	−0.597446
$313$	2.55731	0.144548	0.0722740	−	0.997385i	$-0.476974\pi$
$313$	2.55731	0.144548	0.0722740	+	0.997385i	$0.476974\pi$
$314$	2.96594	0.167378
$315$	3.97340	0.223875
$316$	7.33713	0.412746
$317$	−20.0574	−1.12653	−0.563267	−	0.826275i	$-0.690456\pi$
$317$	−20.0574	−1.12653	−0.563267	+	0.826275i	$0.690456\pi$
$318$	10.9209	0.612415
$319$	26.4602	1.48148
$320$	1.00000	0.0559017
$321$	−5.80385	−0.323939
$322$	22.8570	1.27377
$323$	9.67087	0.538102
$324$	−10.8780	−0.604332
$325$	−5.35054	−0.296795
$326$	−17.3398	−0.960361
$327$	16.2447	0.898336
$328$	0.831224	0.0458967
$329$	−17.9742	−0.990951
$330$	7.04615	0.387878
$331$	9.36246	0.514607	0.257304	−	0.966331i	$-0.417166\pi$
$331$	9.36246	0.514607	0.257304	+	0.966331i	$0.417166\pi$
$332$	−6.55527	−0.359767
$333$	2.42424	0.132847
$334$	−3.32690	−0.182040
$335$	8.80246	0.480930
$336$	−8.80478	−0.480340
$337$	19.5062	1.06257	0.531286	−	0.847192i	$-0.321709\pi$
$337$	19.5062	1.06257	0.531286	+	0.847192i	$0.321709\pi$
$338$	−15.6283	−0.850067
$339$	23.7010	1.28726
$340$	−1.14025	−0.0618387
$341$	8.25899	0.447249
$342$	7.54896	0.408201
$343$	26.4669	1.42908
$344$	−1.83158	−0.0987524
$345$	10.0985	0.543686
$346$	−9.07488	−0.487869
$347$	−12.5977	−0.676282	−0.338141	−	0.941095i	$-0.609798\pi$
$347$	−12.5977	−0.676282	−0.338141	+	0.941095i	$0.609798\pi$
$348$	−14.6082	−0.783083
$349$	−20.5810	−1.10168	−0.550839	−	0.834612i	$-0.685692\pi$
$349$	−20.5810	−1.10168	−0.550839	+	0.834612i	$0.685692\pi$
$350$	−4.46416	−0.238620
$351$	−22.2662	−1.18848
$352$	−3.57251	−0.190415
$353$	−17.5284	−0.932944	−0.466472	−	0.884536i	$-0.654475\pi$
$353$	−17.5284	−0.932944	−0.466472	+	0.884536i	$0.654475\pi$
$354$	−4.48398	−0.238321
$355$	−1.20504	−0.0639570
$356$	14.7996	0.784376
$357$	10.0396	0.531354
$358$	−21.6148	−1.14238
$359$	−21.6013	−1.14008	−0.570038	−	0.821619i	$-0.693071\pi$
$359$	−21.6013	−1.14008	−0.570038	+	0.821619i	$0.693071\pi$
$360$	−0.890065	−0.0469105
$361$	52.9335	2.78598
$362$	−16.2898	−0.856171
$363$	−3.47684	−0.182487
$364$	−23.8857	−1.25195
$365$	−12.9254	−0.676548
$366$	25.3369	1.32438
$367$	−30.2971	−1.58150	−0.790748	−	0.612141i	$-0.790308\pi$
$367$	−30.2971	−1.58150	−0.790748	+	0.612141i	$0.790308\pi$
$368$	−5.12011	−0.266904
$369$	−0.739844	−0.0385147
$370$	−2.72367	−0.141597
$371$	24.7184	1.28332
$372$	−4.55966	−0.236407
$373$	31.1810	1.61449	0.807246	−	0.590215i	$-0.200957\pi$
$373$	31.1810	1.61449	0.807246	+	0.590215i	$0.200957\pi$
$374$	4.07355	0.210638
$375$	−1.97232	−0.101850
$376$	4.02634	0.207643
$377$	−39.6294	−2.04102
$378$	−18.5775	−0.955524
$379$	33.6983	1.73097	0.865483	−	0.500938i	$-0.167012\pi$
$379$	33.6983	1.73097	0.865483	+	0.500938i	$0.167012\pi$
$380$	−8.48136	−0.435085
$381$	−3.92114	−0.200886
$382$	−7.70953	−0.394454
$383$	38.8110	1.98315	0.991574	−	0.129540i	$-0.0413501\pi$
$383$	38.8110	1.98315	0.991574	+	0.129540i	$0.0413501\pi$
$384$	1.97232	0.100650
$385$	15.9483	0.812798
$386$	16.4253	0.836024
$387$	1.63023	0.0828691
$388$	−0.565280	−0.0286978
$389$	30.1932	1.53085	0.765427	−	0.643523i	$-0.222528\pi$
$389$	30.1932	1.53085	0.765427	+	0.643523i	$0.222528\pi$
$390$	−10.5530	−0.534372
$391$	5.83820	0.295250
$392$	−12.9288	−0.653001
$393$	9.87341	0.498048
$394$	2.20129	0.110899
$395$	7.33713	0.369171
$396$	3.17976	0.159789
$397$	15.8953	0.797762	0.398881	−	0.917003i	$-0.369399\pi$
$397$	15.8953	0.797762	0.398881	+	0.917003i	$0.369399\pi$
$398$	−24.4277	−1.22445
$399$	74.6765	3.73850
$400$	1.00000	0.0500000
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	17.3613	0.865903
$403$	−12.3695	−0.616168
$404$	8.34629	0.415243
$405$	−10.8780	−0.540531
$406$	−33.0643	−1.64095
$407$	9.73032	0.482314
$408$	−2.24894	−0.111339
$409$	−19.2788	−0.953277	−0.476639	−	0.879099i	$-0.658145\pi$
$409$	−19.2788	−0.953277	−0.476639	+	0.879099i	$0.658145\pi$
$410$	0.831224	0.0410512
$411$	−24.9732	−1.23184
$412$	19.5367	0.962503
$413$	−10.1490	−0.499402
$414$	4.55723	0.223976
$415$	−6.55527	−0.321786
$416$	5.35054	0.262332
$417$	28.2581	1.38380
$418$	30.2997	1.48201
$419$	−26.7443	−1.30654	−0.653271	−	0.757124i	$-0.726604\pi$
$419$	−26.7443	−1.30654	−0.653271	+	0.757124i	$0.726604\pi$
$420$	−8.80478	−0.429629
$421$	27.1888	1.32510	0.662551	−	0.749017i	$-0.269474\pi$
$421$	27.1888	1.32510	0.662551	+	0.749017i	$0.269474\pi$
$422$	−25.1347	−1.22354
$423$	−3.58370	−0.174246
$424$	−5.53708	−0.268905
$425$	−1.14025	−0.0553102
$426$	−2.37674	−0.115153
$427$	57.3477	2.77525
$428$	2.94265	0.142238
$429$	37.7007	1.82021
$430$	−1.83158	−0.0883268
$431$	−22.6120	−1.08918	−0.544590	−	0.838702i	$-0.683315\pi$
$431$	−22.6120	−1.08918	−0.544590	+	0.838702i	$0.683315\pi$
$432$	4.16148	0.200219
$433$	−14.8761	−0.714901	−0.357450	−	0.933932i	$-0.616354\pi$
$433$	−14.8761	−0.714901	−0.357450	+	0.933932i	$0.616354\pi$
$434$	−10.3203	−0.495392
$435$	−14.6082	−0.700411
$436$	−8.23634	−0.394449
$437$	43.4255	2.07732
$438$	−25.4931	−1.21811
$439$	9.90959	0.472959	0.236479	−	0.971636i	$-0.424006\pi$
$439$	9.90959	0.472959	0.236479	+	0.971636i	$0.424006\pi$
$440$	−3.57251	−0.170313
$441$	11.5074	0.547973
$442$	−6.10095	−0.290193
$443$	11.6999	0.555881	0.277941	−	0.960598i	$-0.410348\pi$
$443$	11.6999	0.555881	0.277941	+	0.960598i	$0.410348\pi$
$444$	−5.37195	−0.254942
$445$	14.7996	0.701567
$446$	19.9792	0.946044
$447$	−8.94121	−0.422905
$448$	4.46416	0.210912
$449$	−4.81032	−0.227013	−0.113507	−	0.993537i	$-0.536208\pi$
$449$	−4.81032	−0.227013	−0.113507	+	0.993537i	$0.536208\pi$
$450$	−0.890065	−0.0419581
$451$	−2.96956	−0.139831
$452$	−12.0168	−0.565222
$453$	−20.6030	−0.968013
$454$	3.63417	0.170560
$455$	−23.8857	−1.11978
$456$	−16.7280	−0.783361
$457$	18.2630	0.854305	0.427153	−	0.904180i	$-0.359517\pi$
$457$	18.2630	0.854305	0.427153	+	0.904180i	$0.359517\pi$
$458$	−14.3424	−0.670175
$459$	−4.74512	−0.221483
$460$	−5.12011	−0.238726
$461$	−32.4153	−1.50973	−0.754867	−	0.655878i	$-0.772298\pi$
$461$	−32.4153	−1.50973	−0.754867	+	0.655878i	$0.772298\pi$
$462$	31.4551	1.46343
$463$	30.9402	1.43791	0.718956	−	0.695056i	$-0.244620\pi$
$463$	30.9402	1.43791	0.718956	+	0.695056i	$0.244620\pi$
$464$	7.40661	0.343843
$465$	−4.55966	−0.211449
$466$	8.19892	0.379808
$467$	−13.3646	−0.618438	−0.309219	−	0.950991i	$-0.600068\pi$
$467$	−13.3646	−0.618438	−0.309219	+	0.950991i	$0.600068\pi$
$468$	−4.76233	−0.220139
$469$	39.2956	1.81450
$470$	4.02634	0.185721
$471$	5.84980	0.269544
$472$	2.27345	0.104644
$473$	6.54335	0.300863
$474$	14.4712	0.664684
$475$	−8.48136	−0.389152
$476$	−5.09026	−0.233312
$477$	4.92836	0.225654
$478$	3.63359	0.166197
$479$	13.1486	0.600774	0.300387	−	0.953817i	$-0.402884\pi$
$479$	13.1486	0.600774	0.300387	+	0.953817i	$0.402884\pi$
$480$	1.97232	0.0900239
$481$	−14.5731	−0.664476
$482$	−12.7878	−0.582470
$483$	45.0814	2.05128
$484$	1.76281	0.0801278
$485$	−0.565280	−0.0256680
$486$	−8.97048	−0.406909
$487$	−1.39094	−0.0630294	−0.0315147	−	0.999503i	$-0.510033\pi$
$487$	−1.39094	−0.0630294	−0.0315147	+	0.999503i	$0.510033\pi$
$488$	−12.8462	−0.581522
$489$	−34.1997	−1.54656
$490$	−12.9288	−0.584062
$491$	34.1511	1.54122	0.770608	−	0.637310i	$-0.219953\pi$
$491$	34.1511	1.54122	0.770608	+	0.637310i	$0.219953\pi$
$492$	1.63944	0.0739119
$493$	−8.44538	−0.380361
$494$	−45.3799	−2.04174
$495$	3.17976	0.142920
$496$	2.31182	0.103804
$497$	−5.37951	−0.241304
$498$	−12.9291	−0.579368
$499$	−3.58325	−0.160408	−0.0802042	−	0.996778i	$-0.525557\pi$
$499$	−3.58325	−0.160408	−0.0802042	+	0.996778i	$0.525557\pi$
$500$	1.00000	0.0447214
$501$	−6.56172	−0.293156
$502$	18.9188	0.844388
$503$	12.9401	0.576970	0.288485	−	0.957484i	$-0.406848\pi$
$503$	12.9401	0.576970	0.288485	+	0.957484i	$0.406848\pi$
$504$	−3.97340	−0.176989
$505$	8.34629	0.371405
$506$	18.2916	0.813162
$507$	−30.8241	−1.36895
$508$	1.98808	0.0882069
$509$	41.4779	1.83848	0.919238	−	0.393703i	$-0.128806\pi$
$509$	41.4779	1.83848	0.919238	+	0.393703i	$0.128806\pi$
$510$	−2.24894	−0.0995848
$511$	−57.7012	−2.55255
$512$	−1.00000	−0.0441942
$513$	−35.2950	−1.55831
$514$	25.9786	1.14587
$515$	19.5367	0.860889
$516$	−3.61248	−0.159030
$517$	−14.3841	−0.632613
$518$	−12.1589	−0.534231
$519$	−17.8986	−0.785662
$520$	5.35054	0.234637
$521$	−15.1941	−0.665666	−0.332833	−	0.942986i	$-0.608005\pi$
$521$	−15.1941	−0.665666	−0.332833	+	0.942986i	$0.608005\pi$
$522$	−6.59236	−0.288540
$523$	−10.8686	−0.475251	−0.237626	−	0.971357i	$-0.576369\pi$
$523$	−10.8686	−0.475251	−0.237626	+	0.971357i	$0.576369\pi$
$524$	−5.00597	−0.218687
$525$	−8.80478	−0.384272
$526$	17.6021	0.767487
$527$	−2.63605	−0.114828
$528$	−7.04615	−0.306644
$529$	3.21550	0.139804
$530$	−5.53708	−0.240516
$531$	−2.02352	−0.0878132
$532$	−37.8622	−1.64153
$533$	4.44750	0.192643
$534$	29.1896	1.26316
$535$	2.94265	0.127222
$536$	−8.80246	−0.380208
$537$	−42.6313	−1.83968
$538$	14.8053	0.638300
$539$	46.1881	1.98946
$540$	4.16148	0.179081
$541$	16.1402	0.693921	0.346961	−	0.937880i	$-0.387214\pi$
$541$	16.1402	0.693921	0.346961	+	0.937880i	$0.387214\pi$
$542$	−20.3518	−0.874183
$543$	−32.1287	−1.37877
$544$	1.14025	0.0488878
$545$	−8.23634	−0.352806
$546$	−47.1103	−2.01614
$547$	27.7708	1.18739	0.593697	−	0.804689i	$-0.297668\pi$
$547$	27.7708	1.18739	0.593697	+	0.804689i	$0.297668\pi$
$548$	12.6618	0.540886
$549$	11.4340	0.487991
$550$	−3.57251	−0.152332
$551$	−62.8181	−2.67614
$552$	−10.0985	−0.429821
$553$	32.7541	1.39285
$554$	19.0366	0.808789
$555$	−5.37195	−0.228027
$556$	−14.3273	−0.607613
$557$	38.0228	1.61108	0.805538	−	0.592544i	$-0.201876\pi$
$557$	38.0228	1.61108	0.805538	+	0.592544i	$0.201876\pi$
$558$	−2.05767	−0.0871081
$559$	−9.79996	−0.414494
$560$	4.46416	0.188645
$561$	8.03436	0.339211
$562$	−18.7645	−0.791531
$563$	3.51645	0.148201	0.0741004	−	0.997251i	$-0.476391\pi$
$563$	3.51645	0.148201	0.0741004	+	0.997251i	$0.476391\pi$
$564$	7.94125	0.334387
$565$	−12.0168	−0.505550
$566$	−18.7279	−0.787194
$567$	−48.5611	−2.03937
$568$	1.20504	0.0505625
$569$	−31.7340	−1.33036	−0.665179	−	0.746684i	$-0.731645\pi$
$569$	−31.7340	−1.33036	−0.665179	+	0.746684i	$0.731645\pi$
$570$	−16.7280	−0.700659
$571$	3.85389	0.161280	0.0806402	−	0.996743i	$-0.474304\pi$
$571$	3.85389	0.161280	0.0806402	+	0.996743i	$0.474304\pi$
$572$	−19.1149	−0.799232
$573$	−15.2057	−0.635227
$574$	3.71072	0.154882
$575$	−5.12011	−0.213523
$576$	0.890065	0.0370860
$577$	−31.3800	−1.30637	−0.653184	−	0.757199i	$-0.726567\pi$
$577$	−31.3800	−1.30637	−0.653184	+	0.757199i	$0.726567\pi$
$578$	15.6998	0.653027
$579$	32.3960	1.34633
$580$	7.40661	0.307543
$581$	−29.2638	−1.21407
$582$	−1.11492	−0.0462148
$583$	19.7813	0.819257
$584$	12.9254	0.534858
$585$	−4.76233	−0.196898
$586$	1.61768	0.0668257
$587$	0.452410	0.0186729	0.00933647	−	0.999956i	$-0.497028\pi$
$587$	0.452410	0.0186729	0.00933647	+	0.999956i	$0.497028\pi$
$588$	−25.4997	−1.05159
$589$	−19.6074	−0.807908
$590$	2.27345	0.0935964
$591$	4.34166	0.178592
$592$	2.72367	0.111942
$593$	−22.3476	−0.917706	−0.458853	−	0.888512i	$-0.651740\pi$
$593$	−22.3476	−0.917706	−0.458853	+	0.888512i	$0.651740\pi$
$594$	−14.8669	−0.609997
$595$	−5.09026	−0.208680
$596$	4.53333	0.185693
$597$	−48.1793	−1.97185
$598$	−27.3953	−1.12028
$599$	−23.4827	−0.959478	−0.479739	−	0.877411i	$-0.659269\pi$
$599$	−23.4827	−0.959478	−0.479739	+	0.877411i	$0.659269\pi$
$600$	1.97232	0.0805198
$601$	−13.2726	−0.541399	−0.270699	−	0.962664i	$-0.587255\pi$
$601$	−13.2726	−0.541399	−0.270699	+	0.962664i	$0.587255\pi$
$602$	−8.17649	−0.333249
$603$	7.83476	0.319056
$604$	10.4460	0.425044
$605$	1.76281	0.0716685
$606$	16.4616	0.668706
$607$	27.4876	1.11569	0.557843	−	0.829947i	$-0.311629\pi$
$607$	27.4876	1.11569	0.557843	+	0.829947i	$0.311629\pi$
$608$	8.48136	0.343965
$609$	−65.2135	−2.64259
$610$	−12.8462	−0.520129
$611$	21.5431	0.871540
$612$	−1.01490	−0.0410247
$613$	−38.2408	−1.54453	−0.772266	−	0.635299i	$-0.780877\pi$
$613$	−38.2408	−1.54453	−0.772266	+	0.635299i	$0.780877\pi$
$614$	−14.2845	−0.576476
$615$	1.63944	0.0661088
$616$	−15.9483	−0.642574
$617$	43.6364	1.75674	0.878368	−	0.477984i	$-0.158632\pi$
$617$	43.6364	1.75674	0.878368	+	0.477984i	$0.158632\pi$
$618$	38.5327	1.55001
$619$	4.33726	0.174329	0.0871646	−	0.996194i	$-0.472219\pi$
$619$	4.33726	0.174329	0.0871646	+	0.996194i	$0.472219\pi$
$620$	2.31182	0.0928449
$621$	−21.3072	−0.855029
$622$	−8.57485	−0.343820
$623$	66.0677	2.64695
$624$	10.5530	0.422458
$625$	1.00000	0.0400000
$626$	−2.55731	−0.102211
$627$	59.7609	2.38662
$628$	−2.96594	−0.118354
$629$	−3.10566	−0.123831
$630$	−3.97340	−0.158304
$631$	1.57583	0.0627327	0.0313663	−	0.999508i	$-0.490014\pi$
$631$	1.57583	0.0627327	0.0313663	+	0.999508i	$0.490014\pi$
$632$	−7.33713	−0.291855
$633$	−49.5737	−1.97038
$634$	20.0574	0.796579
$635$	1.98808	0.0788947
$636$	−10.9209	−0.433043
$637$	−69.1758	−2.74085
$638$	−26.4602	−1.04757
$639$	−1.07257	−0.0424301
$640$	−1.00000	−0.0395285
$641$	−25.1766	−0.994415	−0.497207	−	0.867632i	$-0.665641\pi$
$641$	−25.1766	−0.994415	−0.497207	+	0.867632i	$0.665641\pi$
$642$	5.80385	0.229060
$643$	32.3574	1.27605	0.638026	−	0.770015i	$-0.279751\pi$
$643$	32.3574	1.27605	0.638026	+	0.770015i	$0.279751\pi$
$644$	−22.8570	−0.900692
$645$	−3.61248	−0.142241
$646$	−9.67087	−0.380495
$647$	−2.53203	−0.0995442	−0.0497721	−	0.998761i	$-0.515849\pi$
$647$	−2.53203	−0.0995442	−0.0497721	+	0.998761i	$0.515849\pi$
$648$	10.8780	0.427327
$649$	−8.12192	−0.318813
$650$	5.35054	0.209866
$651$	−20.3551	−0.797778
$652$	17.3398	0.679078
$653$	34.1630	1.33690	0.668451	−	0.743756i	$-0.266958\pi$
$653$	34.1630	1.33690	0.668451	+	0.743756i	$0.266958\pi$
$654$	−16.2447	−0.635220
$655$	−5.00597	−0.195600
$656$	−0.831224	−0.0324539
$657$	−11.5045	−0.448832
$658$	17.9742	0.700708
$659$	21.3697	0.832445	0.416222	−	0.909263i	$-0.363354\pi$
$659$	21.3697	0.832445	0.416222	+	0.909263i	$0.363354\pi$
$660$	−7.04615	−0.274271
$661$	16.4409	0.639478	0.319739	−	0.947506i	$-0.396405\pi$
$661$	16.4409	0.639478	0.319739	+	0.947506i	$0.396405\pi$
$662$	−9.36246	−0.363882
$663$	−12.0331	−0.467325
$664$	6.55527	0.254394
$665$	−37.8622	−1.46823
$666$	−2.42424	−0.0939374
$667$	−37.9226	−1.46837
$668$	3.32690	0.128722
$669$	39.4055	1.52351
$670$	−8.80246	−0.340069
$671$	45.8933	1.77169
$672$	8.80478	0.339652
$673$	−9.80048	−0.377781	−0.188890	−	0.981998i	$-0.560489\pi$
$673$	−9.80048	−0.377781	−0.188890	+	0.981998i	$0.560489\pi$
$674$	−19.5062	−0.751352
$675$	4.16148	0.160175
$676$	15.6283	0.601088
$677$	16.7233	0.642728	0.321364	−	0.946956i	$-0.395859\pi$
$677$	16.7233	0.642728	0.321364	+	0.946956i	$0.395859\pi$
$678$	−23.7010	−0.910231
$679$	−2.52350	−0.0968431
$680$	1.14025	0.0437266
$681$	7.16777	0.274670
$682$	−8.25899	−0.316253
$683$	−0.944394	−0.0361362	−0.0180681	−	0.999837i	$-0.505752\pi$
$683$	−0.944394	−0.0361362	−0.0180681	+	0.999837i	$0.505752\pi$
$684$	−7.54896	−0.288642
$685$	12.6618	0.483784
$686$	−26.4669	−1.01051
$687$	−28.2878	−1.07925
$688$	1.83158	0.0698285
$689$	−29.6264	−1.12868
$690$	−10.0985	−0.384444
$691$	−17.4756	−0.664801	−0.332401	−	0.943138i	$-0.607859\pi$
$691$	−17.4756	−0.664801	−0.332401	+	0.943138i	$0.607859\pi$
$692$	9.07488	0.344975
$693$	14.1950	0.539223
$694$	12.5977	0.478204
$695$	−14.3273	−0.543466
$696$	14.6082	0.553724
$697$	0.947803	0.0359006
$698$	20.5810	0.779004
$699$	16.1709	0.611641
$700$	4.46416	0.168730
$701$	16.4302	0.620558	0.310279	−	0.950645i	$-0.399577\pi$
$701$	16.4302	0.620558	0.310279	+	0.950645i	$0.399577\pi$
$702$	22.2662	0.840382
$703$	−23.1004	−0.871248
$704$	3.57251	0.134644
$705$	7.94125	0.299085
$706$	17.5284	0.659691
$707$	37.2592	1.40128
$708$	4.48398	0.168518
$709$	44.0079	1.65275	0.826376	−	0.563119i	$-0.190399\pi$
$709$	44.0079	1.65275	0.826376	+	0.563119i	$0.190399\pi$
$710$	1.20504	0.0452245
$711$	6.53052	0.244914
$712$	−14.7996	−0.554638
$713$	−11.8368	−0.443290
$714$	−10.0396	−0.375724
$715$	−19.1149	−0.714855
$716$	21.6148	0.807782
$717$	7.16663	0.267643
$718$	21.6013	0.806155
$719$	21.5148	0.802366	0.401183	−	0.915998i	$-0.368599\pi$
$719$	21.5148	0.802366	0.401183	+	0.915998i	$0.368599\pi$
$720$	0.890065	0.0331708
$721$	87.2149	3.24805
$722$	−52.9335	−1.96998
$723$	−25.2218	−0.938007
$724$	16.2898	0.605404
$725$	7.40661	0.275074
$726$	3.47684	0.129037
$727$	35.1470	1.30353	0.651766	−	0.758420i	$-0.274028\pi$
$727$	35.1470	1.30353	0.651766	+	0.758420i	$0.274028\pi$
$728$	23.8857	0.885263
$729$	14.9412	0.553379
$730$	12.9254	0.478391
$731$	−2.08846	−0.0772445
$732$	−25.3369	−0.936481
$733$	−5.78078	−0.213518	−0.106759	−	0.994285i	$-0.534047\pi$
$733$	−5.78078	−0.213518	−0.106759	+	0.994285i	$0.534047\pi$
$734$	30.2971	1.11829
$735$	−25.4997	−0.940571
$736$	5.12011	0.188730
$737$	31.4468	1.15836
$738$	0.739844	0.0272340
$739$	2.85643	0.105076	0.0525378	−	0.998619i	$-0.483269\pi$
$739$	2.85643	0.105076	0.0525378	+	0.998619i	$0.483269\pi$
$740$	2.72367	0.100124
$741$	−89.5039	−3.28801
$742$	−24.7184	−0.907443
$743$	−25.7129	−0.943315	−0.471658	−	0.881782i	$-0.656344\pi$
$743$	−25.7129	−0.943315	−0.471658	+	0.881782i	$0.656344\pi$
$744$	4.55966	0.167165
$745$	4.53333	0.166089
$746$	−31.1810	−1.14162
$747$	−5.83462	−0.213477
$748$	−4.07355	−0.148944
$749$	13.1364	0.479995
$750$	1.97232	0.0720191
$751$	−3.37137	−0.123023	−0.0615115	−	0.998106i	$-0.519592\pi$
$751$	−3.37137	−0.123023	−0.0615115	+	0.998106i	$0.519592\pi$
$752$	−4.02634	−0.146825
$753$	37.3140	1.35980
$754$	39.6294	1.44322
$755$	10.4460	0.380171
$756$	18.5775	0.675658
$757$	−40.5597	−1.47417	−0.737084	−	0.675801i	$-0.763798\pi$
$757$	−40.5597	−1.47417	−0.737084	+	0.675801i	$0.763798\pi$
$758$	−33.6983	−1.22398
$759$	36.0770	1.30951
$760$	8.48136	0.307651
$761$	−37.0336	−1.34247	−0.671233	−	0.741247i	$-0.734235\pi$
$761$	−37.0336	−1.34247	−0.671233	+	0.741247i	$0.734235\pi$
$762$	3.92114	0.142048
$763$	−36.7684	−1.33110
$764$	7.70953	0.278921
$765$	−1.01490	−0.0366936
$766$	−38.8110	−1.40230
$767$	12.1642	0.439223
$768$	−1.97232	−0.0711701
$769$	2.07325	0.0747633	0.0373816	−	0.999301i	$-0.488098\pi$
$769$	2.07325	0.0747633	0.0373816	+	0.999301i	$0.488098\pi$
$770$	−15.9483	−0.574735
$771$	51.2382	1.84530
$772$	−16.4253	−0.591158
$773$	50.7601	1.82571	0.912857	−	0.408280i	$-0.133871\pi$
$773$	50.7601	1.82571	0.912857	+	0.408280i	$0.133871\pi$
$774$	−1.63023	−0.0585973
$775$	2.31182	0.0830430
$776$	0.565280	0.0202924
$777$	−23.9813	−0.860323
$778$	−30.1932	−1.08248
$779$	7.04992	0.252589
$780$	10.5530	0.377858
$781$	−4.30503	−0.154046
$782$	−5.83820	−0.208774
$783$	30.8224	1.10150
$784$	12.9288	0.461741
$785$	−2.96594	−0.105859
$786$	−9.87341	−0.352173
$787$	26.7694	0.954228	0.477114	−	0.878841i	$-0.341683\pi$
$787$	26.7694	0.954228	0.477114	+	0.878841i	$0.341683\pi$
$788$	−2.20129	−0.0784177
$789$	34.7170	1.23596
$790$	−7.33713	−0.261043
$791$	−53.6449	−1.90739
$792$	−3.17976	−0.112988
$793$	−68.7343	−2.44083
$794$	−15.8953	−0.564103
$795$	−10.9209	−0.387325
$796$	24.4277	0.865817
$797$	−27.9612	−0.990435	−0.495218	−	0.868769i	$-0.664912\pi$
$797$	−27.9612	−0.990435	−0.495218	+	0.868769i	$0.664912\pi$
$798$	−74.6765	−2.64352
$799$	4.59103	0.162419
$800$	−1.00000	−0.0353553
$801$	13.1726	0.465430
$802$	1.00000	0.0353112
$803$	−46.1762	−1.62952
$804$	−17.3613	−0.612286
$805$	−22.8570	−0.805603
$806$	12.3695	0.435696
$807$	29.2008	1.02792
$808$	−8.34629	−0.293621
$809$	−29.5117	−1.03758	−0.518789	−	0.854903i	$-0.673617\pi$
$809$	−29.5117	−1.03758	−0.518789	+	0.854903i	$0.673617\pi$
$810$	10.8780	0.382213
$811$	27.0695	0.950539	0.475269	−	0.879840i	$-0.342351\pi$
$811$	27.0695	0.950539	0.475269	+	0.879840i	$0.342351\pi$
$812$	33.0643	1.16033
$813$	−40.1403	−1.40778
$814$	−9.73032	−0.341047
$815$	17.3398	0.607386
$816$	2.24894	0.0787287
$817$	−15.5343	−0.543477
$818$	19.2788	0.674069
$819$	−21.2598	−0.742878
$820$	−0.831224	−0.0290276
$821$	−48.3557	−1.68763	−0.843814	−	0.536637i	$-0.819695\pi$
$821$	−48.3557	−1.68763	−0.843814	+	0.536637i	$0.819695\pi$
$822$	24.9732	0.871042
$823$	34.3389	1.19698	0.598490	−	0.801131i	$-0.295768\pi$
$823$	34.3389	1.19698	0.598490	+	0.801131i	$0.295768\pi$
$824$	−19.5367	−0.680592
$825$	−7.04615	−0.245315
$826$	10.1490	0.353131
$827$	2.99920	0.104292	0.0521462	−	0.998639i	$-0.483394\pi$
$827$	2.99920	0.104292	0.0521462	+	0.998639i	$0.483394\pi$
$828$	−4.55723	−0.158375
$829$	−30.4382	−1.05716	−0.528581	−	0.848883i	$-0.677276\pi$
$829$	−30.4382	−1.05716	−0.528581	+	0.848883i	$0.677276\pi$
$830$	6.55527	0.227537
$831$	37.5464	1.30247
$832$	−5.35054	−0.185497
$833$	−14.7420	−0.510780
$834$	−28.2581	−0.978498
$835$	3.32690	0.115132
$836$	−30.2997	−1.04794
$837$	9.62058	0.332536
$838$	26.7443	0.923865
$839$	−10.5117	−0.362905	−0.181453	−	0.983400i	$-0.558080\pi$
$839$	−10.5117	−0.362905	−0.181453	+	0.983400i	$0.558080\pi$
$840$	8.80478	0.303794
$841$	25.8578	0.891648
$842$	−27.1888	−0.936988
$843$	−37.0096	−1.27468
$844$	25.1347	0.865171
$845$	15.6283	0.537630
$846$	3.58370	0.123210
$847$	7.86948	0.270398
$848$	5.53708	0.190144
$849$	−36.9376	−1.26770
$850$	1.14025	0.0391102
$851$	−13.9455	−0.478044
$852$	2.37674	0.0814256
$853$	5.60021	0.191748	0.0958738	−	0.995393i	$-0.469435\pi$
$853$	5.60021	0.191748	0.0958738	+	0.995393i	$0.469435\pi$
$854$	−57.3477	−1.96240
$855$	−7.54896	−0.258169
$856$	−2.94265	−0.100578
$857$	−43.4657	−1.48476	−0.742381	−	0.669978i	$-0.766303\pi$
$857$	−43.4657	−1.48476	−0.742381	+	0.669978i	$0.766303\pi$
$858$	−37.7007	−1.28708
$859$	−6.11428	−0.208616	−0.104308	−	0.994545i	$-0.533263\pi$
$859$	−6.11428	−0.208616	−0.104308	+	0.994545i	$0.533263\pi$
$860$	1.83158	0.0624565
$861$	7.31875	0.249422
$862$	22.6120	0.770167
$863$	5.44962	0.185507	0.0927536	−	0.995689i	$-0.470433\pi$
$863$	5.44962	0.185507	0.0927536	+	0.995689i	$0.470433\pi$
$864$	−4.16148	−0.141576
$865$	9.07488	0.308555
$866$	14.8761	0.505511
$867$	30.9652	1.05163
$868$	10.3203	0.350295
$869$	26.2119	0.889179
$870$	14.6082	0.495265
$871$	−47.0979	−1.59585
$872$	8.23634	0.278918
$873$	−0.503136	−0.0170286
$874$	−43.4255	−1.46889
$875$	4.46416	0.150916
$876$	25.4931	0.861333
$877$	−28.1017	−0.948928	−0.474464	−	0.880275i	$-0.657358\pi$
$877$	−28.1017	−0.948928	−0.474464	+	0.880275i	$0.657358\pi$
$878$	−9.90959	−0.334432
$879$	3.19059	0.107616
$880$	3.57251	0.120429
$881$	−11.8078	−0.397815	−0.198907	−	0.980018i	$-0.563739\pi$
$881$	−11.8078	−0.397815	−0.198907	+	0.980018i	$0.563739\pi$
$882$	−11.5074	−0.387475
$883$	3.17123	0.106720	0.0533602	−	0.998575i	$-0.483007\pi$
$883$	3.17123	0.106720	0.0533602	+	0.998575i	$0.483007\pi$
$884$	6.10095	0.205197
$885$	4.48398	0.150727
$886$	−11.6999	−0.393067
$887$	53.9205	1.81047	0.905236	−	0.424909i	$-0.139694\pi$
$887$	53.9205	1.81047	0.905236	+	0.424909i	$0.139694\pi$
$888$	5.37195	0.180271
$889$	8.87513	0.297662
$890$	−14.7996	−0.496083
$891$	−38.8617	−1.30191
$892$	−19.9792	−0.668954
$893$	34.1488	1.14275
$894$	8.94121	0.299039
$895$	21.6148	0.722502
$896$	−4.46416	−0.149137
$897$	−54.0325	−1.80409
$898$	4.81032	0.160523
$899$	17.1227	0.571075
$900$	0.890065	0.0296688
$901$	−6.31365	−0.210338
$902$	2.96956	0.0988754
$903$	−16.1267	−0.536663
$904$	12.0168	0.399672
$905$	16.2898	0.541490
$906$	20.6030	0.684489
$907$	−47.2465	−1.56880	−0.784398	−	0.620258i	$-0.787028\pi$
$907$	−47.2465	−1.56880	−0.784398	+	0.620258i	$0.787028\pi$
$908$	−3.63417	−0.120604
$909$	7.42874	0.246396
$910$	23.8857	0.791803
$911$	−53.5931	−1.77562	−0.887809	−	0.460212i	$-0.847773\pi$
$911$	−53.5931	−1.77562	−0.887809	+	0.460212i	$0.847773\pi$
$912$	16.7280	0.553920
$913$	−23.4188	−0.775048
$914$	−18.2630	−0.604085
$915$	−25.3369	−0.837614
$916$	14.3424	0.473885
$917$	−22.3475	−0.737979
$918$	4.74512	0.156612
$919$	−13.1530	−0.433878	−0.216939	−	0.976185i	$-0.569607\pi$
$919$	−13.1530	−0.433878	−0.216939	+	0.976185i	$0.569607\pi$
$920$	5.12011	0.168805
$921$	−28.1737	−0.928355
$922$	32.4153	1.06754
$923$	6.44763	0.212226
$924$	−31.4551	−1.03480
$925$	2.72367	0.0895536
$926$	−30.9402	−1.01676
$927$	17.3889	0.571127
$928$	−7.40661	−0.243134
$929$	32.2632	1.05852	0.529261	−	0.848459i	$-0.322469\pi$
$929$	32.2632	1.05852	0.529261	+	0.848459i	$0.322469\pi$
$930$	4.55966	0.149517
$931$	−109.653	−3.59375
$932$	−8.19892	−0.268565
$933$	−16.9124	−0.553687
$934$	13.3646	0.437302
$935$	−4.07355	−0.133219
$936$	4.76233	0.155662
$937$	−48.8584	−1.59613	−0.798067	−	0.602568i	$-0.794144\pi$
$937$	−48.8584	−1.59613	−0.798067	+	0.602568i	$0.794144\pi$
$938$	−39.2956	−1.28305
$939$	−5.04385	−0.164600
$940$	−4.02634	−0.131325
$941$	−45.9637	−1.49837	−0.749186	−	0.662359i	$-0.769555\pi$
$941$	−45.9637	−1.49837	−0.749186	+	0.662359i	$0.769555\pi$
$942$	−5.84980	−0.190597
$943$	4.25596	0.138593
$944$	−2.27345	−0.0739945
$945$	18.5775	0.604327
$946$	−6.54335	−0.212743
$947$	−14.1427	−0.459576	−0.229788	−	0.973241i	$-0.573803\pi$
$947$	−14.1427	−0.459576	−0.229788	+	0.973241i	$0.573803\pi$
$948$	−14.4712	−0.470003
$949$	69.1580	2.24496
$950$	8.48136	0.275172
$951$	39.5596	1.28281
$952$	5.09026	0.164976
$953$	−59.8496	−1.93872	−0.969360	−	0.245646i	$-0.921000\pi$
$953$	−59.8496	−1.93872	−0.969360	+	0.245646i	$0.921000\pi$
$954$	−4.92836	−0.159562
$955$	7.70953	0.249475
$956$	−3.63359	−0.117519
$957$	−52.1880	−1.68700
$958$	−13.1486	−0.424811
$959$	56.5245	1.82527
$960$	−1.97232	−0.0636565
$961$	−25.6555	−0.827597
$962$	14.5731	0.469855
$963$	2.61915	0.0844008
$964$	12.7878	0.411868
$965$	−16.4253	−0.528748
$966$	−45.0814	−1.45047
$967$	−25.2112	−0.810738	−0.405369	−	0.914153i	$-0.632857\pi$
$967$	−25.2112	−0.810738	−0.405369	+	0.914153i	$0.632857\pi$
$968$	−1.76281	−0.0566589
$969$	−19.0741	−0.612748
$970$	0.565280	0.0181501
$971$	−19.4172	−0.623129	−0.311564	−	0.950225i	$-0.600853\pi$
$971$	−19.4172	−0.623129	−0.311564	+	0.950225i	$0.600853\pi$
$972$	8.97048	0.287728
$973$	−63.9594	−2.05044
$974$	1.39094	0.0445685
$975$	10.5530	0.337967
$976$	12.8462	0.411198
$977$	−17.7234	−0.567021	−0.283510	−	0.958969i	$-0.591499\pi$
$977$	−17.7234	−0.567021	−0.283510	+	0.958969i	$0.591499\pi$
$978$	34.1997	1.09358
$979$	52.8716	1.68978
$980$	12.9288	0.412994
$981$	−7.33088	−0.234057
$982$	−34.1511	−1.08980
$983$	45.1660	1.44057	0.720286	−	0.693677i	$-0.244010\pi$
$983$	45.1660	1.44057	0.720286	+	0.693677i	$0.244010\pi$
$984$	−1.63944	−0.0522636
$985$	−2.20129	−0.0701389
$986$	8.44538	0.268956
$987$	35.4510	1.12842
$988$	45.3799	1.44373
$989$	−9.37790	−0.298200
$990$	−3.17976	−0.101060
$991$	35.6917	1.13378	0.566892	−	0.823792i	$-0.308146\pi$
$991$	35.6917	1.13378	0.566892	+	0.823792i	$0.308146\pi$
$992$	−2.31182	−0.0734003
$993$	−18.4658	−0.585995
$994$	5.37951	0.170628
$995$	24.4277	0.774410
$996$	12.9291	0.409675
$997$	−24.8438	−0.786811	−0.393406	−	0.919365i	$-0.628703\pi$
$997$	−24.8438	−0.786811	−0.393406	+	0.919365i	$0.628703\pi$
$998$	3.58325	0.113426
$999$	11.3345	0.358607

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4010.2.a.m.1.5	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4010.2.a.m.1.5	✓	20	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:	\( N \)	\(=\)	\( 4010 = 2 \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4010.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.97232 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.97232 q^{6} +4.46416 q^{7} -1.00000 q^{8} +0.890065 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.97232 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.97232 q^{6} +4.46416 q^{7} -1.00000 q^{8} +0.890065 q^{9} -1.00000 q^{10} +3.57251 q^{11} -1.97232 q^{12} -5.35054 q^{13} -4.46416 q^{14} -1.97232 q^{15} +1.00000 q^{16} -1.14025 q^{17} -0.890065 q^{18} -8.48136 q^{19} +1.00000 q^{20} -8.80478 q^{21} -3.57251 q^{22} -5.12011 q^{23} +1.97232 q^{24} +1.00000 q^{25} +5.35054 q^{26} +4.16148 q^{27} +4.46416 q^{28} +7.40661 q^{29} +1.97232 q^{30} +2.31182 q^{31} -1.00000 q^{32} -7.04615 q^{33} +1.14025 q^{34} +4.46416 q^{35} +0.890065 q^{36} +2.72367 q^{37} +8.48136 q^{38} +10.5530 q^{39} -1.00000 q^{40} -0.831224 q^{41} +8.80478 q^{42} +1.83158 q^{43} +3.57251 q^{44} +0.890065 q^{45} +5.12011 q^{46} -4.02634 q^{47} -1.97232 q^{48} +12.9288 q^{49} -1.00000 q^{50} +2.24894 q^{51} -5.35054 q^{52} +5.53708 q^{53} -4.16148 q^{54} +3.57251 q^{55} -4.46416 q^{56} +16.7280 q^{57} -7.40661 q^{58} -2.27345 q^{59} -1.97232 q^{60} +12.8462 q^{61} -2.31182 q^{62} +3.97340 q^{63} +1.00000 q^{64} -5.35054 q^{65} +7.04615 q^{66} +8.80246 q^{67} -1.14025 q^{68} +10.0985 q^{69} -4.46416 q^{70} -1.20504 q^{71} -0.890065 q^{72} -12.9254 q^{73} -2.72367 q^{74} -1.97232 q^{75} -8.48136 q^{76} +15.9483 q^{77} -10.5530 q^{78} +7.33713 q^{79} +1.00000 q^{80} -10.8780 q^{81} +0.831224 q^{82} -6.55527 q^{83} -8.80478 q^{84} -1.14025 q^{85} -1.83158 q^{86} -14.6082 q^{87} -3.57251 q^{88} +14.7996 q^{89} -0.890065 q^{90} -23.8857 q^{91} -5.12011 q^{92} -4.55966 q^{93} +4.02634 q^{94} -8.48136 q^{95} +1.97232 q^{96} -0.565280 q^{97} -12.9288 q^{98} +3.17976 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} + 20 q^{5} - 4 q^{6} + 11 q^{7} - 20 q^{8} + 26 q^{9}+O(q^{10}) \) 20 * q - 20 * q^2 + 4 * q^3 + 20 * q^4 + 20 * q^5 - 4 * q^6 + 11 * q^7 - 20 * q^8 + 26 * q^9	\( 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} + 20 q^{5} - 4 q^{6} + 11 q^{7} - 20 q^{8} + 26 q^{9} - 20 q^{10} + 10 q^{11} + 4 q^{12} - 9 q^{13} - 11 q^{14} + 4 q^{15} + 20 q^{16} - 11 q^{17} - 26 q^{18} + 17 q^{19} + 20 q^{20} - 2 q^{21} - 10 q^{22} - 3 q^{23} - 4 q^{24} + 20 q^{25} + 9 q^{26} - 2 q^{27} + 11 q^{28} + 6 q^{29} - 4 q^{30} + 28 q^{31} - 20 q^{32} + 2 q^{33} + 11 q^{34} + 11 q^{35} + 26 q^{36} + 33 q^{37} - 17 q^{38} + 36 q^{39} - 20 q^{40} + 32 q^{41} + 2 q^{42} + 30 q^{43} + 10 q^{44} + 26 q^{45} + 3 q^{46} + 13 q^{47} + 4 q^{48} + 43 q^{49} - 20 q^{50} + 43 q^{51} - 9 q^{52} + 2 q^{54} + 10 q^{55} - 11 q^{56} + 19 q^{57} - 6 q^{58} + 52 q^{59} + 4 q^{60} + 25 q^{61} - 28 q^{62} + 16 q^{63} + 20 q^{64} - 9 q^{65} - 2 q^{66} + 40 q^{67} - 11 q^{68} + 39 q^{69} - 11 q^{70} + 25 q^{71} - 26 q^{72} - 5 q^{73} - 33 q^{74} + 4 q^{75} + 17 q^{76} - 9 q^{77} - 36 q^{78} + 40 q^{79} + 20 q^{80} + 48 q^{81} - 32 q^{82} + 10 q^{83} - 2 q^{84} - 11 q^{85} - 30 q^{86} + 10 q^{87} - 10 q^{88} + 17 q^{89} - 26 q^{90} + 88 q^{91} - 3 q^{92} - 4 q^{93} - 13 q^{94} + 17 q^{95} - 4 q^{96} - 2 q^{97} - 43 q^{98} + 50 q^{99}+O(q^{100}) \) 20 * q - 20 * q^2 + 4 * q^3 + 20 * q^4 + 20 * q^5 - 4 * q^6 + 11 * q^7 - 20 * q^8 + 26 * q^9 - 20 * q^10 + 10 * q^11 + 4 * q^12 - 9 * q^13 - 11 * q^14 + 4 * q^15 + 20 * q^16 - 11 * q^17 - 26 * q^18 + 17 * q^19 + 20 * q^20 - 2 * q^21 - 10 * q^22 - 3 * q^23 - 4 * q^24 + 20 * q^25 + 9 * q^26 - 2 * q^27 + 11 * q^28 + 6 * q^29 - 4 * q^30 + 28 * q^31 - 20 * q^32 + 2 * q^33 + 11 * q^34 + 11 * q^35 + 26 * q^36 + 33 * q^37 - 17 * q^38 + 36 * q^39 - 20 * q^40 + 32 * q^41 + 2 * q^42 + 30 * q^43 + 10 * q^44 + 26 * q^45 + 3 * q^46 + 13 * q^47 + 4 * q^48 + 43 * q^49 - 20 * q^50 + 43 * q^51 - 9 * q^52 + 2 * q^54 + 10 * q^55 - 11 * q^56 + 19 * q^57 - 6 * q^58 + 52 * q^59 + 4 * q^60 + 25 * q^61 - 28 * q^62 + 16 * q^63 + 20 * q^64 - 9 * q^65 - 2 * q^66 + 40 * q^67 - 11 * q^68 + 39 * q^69 - 11 * q^70 + 25 * q^71 - 26 * q^72 - 5 * q^73 - 33 * q^74 + 4 * q^75 + 17 * q^76 - 9 * q^77 - 36 * q^78 + 40 * q^79 + 20 * q^80 + 48 * q^81 - 32 * q^82 + 10 * q^83 - 2 * q^84 - 11 * q^85 - 30 * q^86 + 10 * q^87 - 10 * q^88 + 17 * q^89 - 26 * q^90 + 88 * q^91 - 3 * q^92 - 4 * q^93 - 13 * q^94 + 17 * q^95 - 4 * q^96 - 2 * q^97 - 43 * q^98 + 50 * q^99

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