LMFDB - Embedded newform 4010.2.a.m.1.1

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			1.1
Root			$-3.16561$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -3.16561 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.16561 q^{6} +0.575814 q^{7} -1.00000 q^{8} +7.02106 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.16561 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.16561 q^{6} +0.575814 q^{7} -1.00000 q^{8} +7.02106 q^{9} -1.00000 q^{10} -3.79969 q^{11} -3.16561 q^{12} -3.71251 q^{13} -0.575814 q^{14} -3.16561 q^{15} +1.00000 q^{16} -7.82120 q^{17} -7.02106 q^{18} +1.62587 q^{19} +1.00000 q^{20} -1.82280 q^{21} +3.79969 q^{22} -8.73426 q^{23} +3.16561 q^{24} +1.00000 q^{25} +3.71251 q^{26} -12.7291 q^{27} +0.575814 q^{28} -10.0388 q^{29} +3.16561 q^{30} +8.56630 q^{31} -1.00000 q^{32} +12.0283 q^{33} +7.82120 q^{34} +0.575814 q^{35} +7.02106 q^{36} +3.11071 q^{37} -1.62587 q^{38} +11.7524 q^{39} -1.00000 q^{40} -1.35375 q^{41} +1.82280 q^{42} +9.34902 q^{43} -3.79969 q^{44} +7.02106 q^{45} +8.73426 q^{46} +0.624291 q^{47} -3.16561 q^{48} -6.66844 q^{49} -1.00000 q^{50} +24.7588 q^{51} -3.71251 q^{52} -10.7646 q^{53} +12.7291 q^{54} -3.79969 q^{55} -0.575814 q^{56} -5.14686 q^{57} +10.0388 q^{58} +5.65902 q^{59} -3.16561 q^{60} +0.795276 q^{61} -8.56630 q^{62} +4.04283 q^{63} +1.00000 q^{64} -3.71251 q^{65} -12.0283 q^{66} +12.1601 q^{67} -7.82120 q^{68} +27.6492 q^{69} -0.575814 q^{70} -3.78586 q^{71} -7.02106 q^{72} -10.2551 q^{73} -3.11071 q^{74} -3.16561 q^{75} +1.62587 q^{76} -2.18792 q^{77} -11.7524 q^{78} -13.7803 q^{79} +1.00000 q^{80} +19.2321 q^{81} +1.35375 q^{82} +0.0361901 q^{83} -1.82280 q^{84} -7.82120 q^{85} -9.34902 q^{86} +31.7789 q^{87} +3.79969 q^{88} -13.4197 q^{89} -7.02106 q^{90} -2.13772 q^{91} -8.73426 q^{92} -27.1175 q^{93} -0.624291 q^{94} +1.62587 q^{95} +3.16561 q^{96} +5.62339 q^{97} +6.66844 q^{98} -26.6779 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.16561	−1.82766	−0.913832	−	0.406093i	$-0.866891\pi$
$3$	−3.16561	−1.82766	−0.913832	+	0.406093i	$0.866891\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	3.16561	1.29235
$7$	0.575814	0.217637	0.108819	−	0.994062i	$-0.465293\pi$
$7$	0.575814	0.217637	0.108819	+	0.994062i	$0.465293\pi$
$8$	−1.00000	−0.353553
$9$	7.02106	2.34035
$10$	−1.00000	−0.316228
$11$	−3.79969	−1.14565	−0.572825	−	0.819678i	$-0.694153\pi$
$11$	−3.79969	−1.14565	−0.572825	+	0.819678i	$0.694153\pi$
$12$	−3.16561	−0.913832
$13$	−3.71251	−1.02967	−0.514833	−	0.857290i	$-0.672146\pi$
$13$	−3.71251	−1.02967	−0.514833	+	0.857290i	$0.672146\pi$
$14$	−0.575814	−0.153893
$15$	−3.16561	−0.817356
$16$	1.00000	0.250000
$17$	−7.82120	−1.89692	−0.948459	−	0.316899i	$-0.897358\pi$
$17$	−7.82120	−1.89692	−0.948459	+	0.316899i	$0.897358\pi$
$18$	−7.02106	−1.65488
$19$	1.62587	0.373000	0.186500	−	0.982455i	$-0.440286\pi$
$19$	1.62587	0.373000	0.186500	+	0.982455i	$0.440286\pi$
$20$	1.00000	0.223607
$21$	−1.82280	−0.397768
$22$	3.79969	0.810097
$23$	−8.73426	−1.82122	−0.910609	−	0.413268i	$-0.864387\pi$
$23$	−8.73426	−1.82122	−0.910609	+	0.413268i	$0.864387\pi$
$24$	3.16561	0.646177
$25$	1.00000	0.200000
$26$	3.71251	0.728084
$27$	−12.7291	−2.44972
$28$	0.575814	0.108819
$29$	−10.0388	−1.86416	−0.932078	−	0.362257i	$-0.882006\pi$
$29$	−10.0388	−1.86416	−0.932078	+	0.362257i	$0.882006\pi$
$30$	3.16561	0.577958
$31$	8.56630	1.53855	0.769277	−	0.638916i	$-0.220617\pi$
$31$	8.56630	1.53855	0.769277	+	0.638916i	$0.220617\pi$
$32$	−1.00000	−0.176777
$33$	12.0283	2.09386
$34$	7.82120	1.34132
$35$	0.575814	0.0973303
$36$	7.02106	1.17018
$37$	3.11071	0.511398	0.255699	−	0.966756i	$-0.417694\pi$
$37$	3.11071	0.511398	0.255699	+	0.966756i	$0.417694\pi$
$38$	−1.62587	−0.263751
$39$	11.7524	1.88188
$40$	−1.00000	−0.158114
$41$	−1.35375	−0.211421	−0.105710	−	0.994397i	$-0.533712\pi$
$41$	−1.35375	−0.211421	−0.105710	+	0.994397i	$0.533712\pi$
$42$	1.82280	0.281264
$43$	9.34902	1.42571	0.712856	−	0.701311i	$-0.247401\pi$
$43$	9.34902	1.42571	0.712856	+	0.701311i	$0.247401\pi$
$44$	−3.79969	−0.572825
$45$	7.02106	1.04664
$46$	8.73426	1.28780
$47$	0.624291	0.0910622	0.0455311	−	0.998963i	$-0.485502\pi$
$47$	0.624291	0.0910622	0.0455311	+	0.998963i	$0.485502\pi$
$48$	−3.16561	−0.456916
$49$	−6.66844	−0.952634
$50$	−1.00000	−0.141421
$51$	24.7588	3.46693
$52$	−3.71251	−0.514833
$53$	−10.7646	−1.47864	−0.739318	−	0.673356i	$-0.764852\pi$
$53$	−10.7646	−1.47864	−0.739318	+	0.673356i	$0.764852\pi$
$54$	12.7291	1.73221
$55$	−3.79969	−0.512350
$56$	−0.575814	−0.0769464
$57$	−5.14686	−0.681718
$58$	10.0388	1.31816
$59$	5.65902	0.736741	0.368371	−	0.929679i	$-0.379916\pi$
$59$	5.65902	0.736741	0.368371	+	0.929679i	$0.379916\pi$
$60$	−3.16561	−0.408678
$61$	0.795276	0.101825	0.0509123	−	0.998703i	$-0.483787\pi$
$61$	0.795276	0.101825	0.0509123	+	0.998703i	$0.483787\pi$
$62$	−8.56630	−1.08792
$63$	4.04283	0.509348
$64$	1.00000	0.125000
$65$	−3.71251	−0.460481
$66$	−12.0283	−1.48058
$67$	12.1601	1.48560	0.742798	−	0.669516i	$-0.233498\pi$
$67$	12.1601	1.48560	0.742798	+	0.669516i	$0.233498\pi$
$68$	−7.82120	−0.948459
$69$	27.6492	3.32858
$70$	−0.575814	−0.0688229
$71$	−3.78586	−0.449298	−0.224649	−	0.974440i	$-0.572124\pi$
$71$	−3.78586	−0.449298	−0.224649	+	0.974440i	$0.572124\pi$
$72$	−7.02106	−0.827440
$73$	−10.2551	−1.20027	−0.600134	−	0.799899i	$-0.704886\pi$
$73$	−10.2551	−1.20027	−0.600134	+	0.799899i	$0.704886\pi$
$74$	−3.11071	−0.361613
$75$	−3.16561	−0.365533
$76$	1.62587	0.186500
$77$	−2.18792	−0.249336
$78$	−11.7524	−1.33069
$79$	−13.7803	−1.55041	−0.775204	−	0.631711i	$-0.782353\pi$
$79$	−13.7803	−1.55041	−0.775204	+	0.631711i	$0.782353\pi$
$80$	1.00000	0.111803
$81$	19.2321	2.13690
$82$	1.35375	0.149497
$83$	0.0361901	0.00397238	0.00198619	−	0.999998i	$-0.499368\pi$
$83$	0.0361901	0.00397238	0.00198619	+	0.999998i	$0.499368\pi$
$84$	−1.82280	−0.198884
$85$	−7.82120	−0.848328
$86$	−9.34902	−1.00813
$87$	31.7789	3.40705
$88$	3.79969	0.405049
$89$	−13.4197	−1.42249	−0.711244	−	0.702945i	$-0.751868\pi$
$89$	−13.4197	−1.42249	−0.711244	+	0.702945i	$0.751868\pi$
$90$	−7.02106	−0.740085
$91$	−2.13772	−0.224094
$92$	−8.73426	−0.910609
$93$	−27.1175	−2.81196
$94$	−0.624291	−0.0643907
$95$	1.62587	0.166810
$96$	3.16561	0.323088
$97$	5.62339	0.570969	0.285484	−	0.958383i	$-0.407846\pi$
$97$	5.62339	0.570969	0.285484	+	0.958383i	$0.407846\pi$
$98$	6.66844	0.673614
$99$	−26.6779	−2.68123
$100$	1.00000	0.100000
$101$	9.11648	0.907124	0.453562	−	0.891225i	$-0.350153\pi$
$101$	9.11648	0.907124	0.453562	+	0.891225i	$0.350153\pi$
$102$	−24.7588	−2.45149
$103$	−10.5156	−1.03613	−0.518067	−	0.855340i	$-0.673348\pi$
$103$	−10.5156	−1.03613	−0.518067	+	0.855340i	$0.673348\pi$
$104$	3.71251	0.364042
$105$	−1.82280	−0.177887
$106$	10.7646	1.04555
$107$	−7.57558	−0.732359	−0.366179	−	0.930544i	$-0.619334\pi$
$107$	−7.57558	−0.732359	−0.366179	+	0.930544i	$0.619334\pi$
$108$	−12.7291	−1.22486
$109$	−0.340290	−0.0325939	−0.0162969	−	0.999867i	$-0.505188\pi$
$109$	−0.340290	−0.0325939	−0.0162969	+	0.999867i	$0.505188\pi$
$110$	3.79969	0.362286
$111$	−9.84729	−0.934663
$112$	0.575814	0.0544093
$113$	15.6164	1.46907	0.734536	−	0.678570i	$-0.237400\pi$
$113$	15.6164	1.46907	0.734536	+	0.678570i	$0.237400\pi$
$114$	5.14686	0.482047
$115$	−8.73426	−0.814474
$116$	−10.0388	−0.932078
$117$	−26.0658	−2.40978
$118$	−5.65902	−0.520955
$119$	−4.50355	−0.412840
$120$	3.16561	0.288979
$121$	3.43766	0.312514
$122$	−0.795276	−0.0720009
$123$	4.28545	0.386406
$124$	8.56630	0.769277
$125$	1.00000	0.0894427
$126$	−4.04283	−0.360164
$127$	0.412278	0.0365838	0.0182919	−	0.999833i	$-0.494177\pi$
$127$	0.412278	0.0365838	0.0182919	+	0.999833i	$0.494177\pi$
$128$	−1.00000	−0.0883883
$129$	−29.5953	−2.60572
$130$	3.71251	0.325609
$131$	15.3345	1.33978	0.669889	−	0.742461i	$-0.266342\pi$
$131$	15.3345	1.33978	0.669889	+	0.742461i	$0.266342\pi$
$132$	12.0283	1.04693
$133$	0.936197	0.0811786
$134$	−12.1601	−1.05047
$135$	−12.7291	−1.09555
$136$	7.82120	0.670662
$137$	15.0203	1.28327	0.641634	−	0.767011i	$-0.278257\pi$
$137$	15.0203	1.28327	0.641634	+	0.767011i	$0.278257\pi$
$138$	−27.6492	−2.35366
$139$	12.1218	1.02816	0.514080	−	0.857742i	$-0.328134\pi$
$139$	12.1218	1.02816	0.514080	+	0.857742i	$0.328134\pi$
$140$	0.575814	0.0486652
$141$	−1.97626	−0.166431
$142$	3.78586	0.317702
$143$	14.1064	1.17964
$144$	7.02106	0.585089
$145$	−10.0388	−0.833676
$146$	10.2551	0.848718
$147$	21.1097	1.74109
$148$	3.11071	0.255699
$149$	−12.7572	−1.04511	−0.522554	−	0.852606i	$-0.675021\pi$
$149$	−12.7572	−1.04511	−0.522554	+	0.852606i	$0.675021\pi$
$150$	3.16561	0.258471
$151$	4.45210	0.362307	0.181153	−	0.983455i	$-0.442017\pi$
$151$	4.45210	0.362307	0.181153	+	0.983455i	$0.442017\pi$
$152$	−1.62587	−0.131875
$153$	−54.9131	−4.43946
$154$	2.18792	0.176307
$155$	8.56630	0.688062
$156$	11.7524	0.940942
$157$	−21.4930	−1.71533	−0.857664	−	0.514210i	$-0.828085\pi$
$157$	−21.4930	−1.71533	−0.857664	+	0.514210i	$0.828085\pi$
$158$	13.7803	1.09630
$159$	34.0766	2.70245
$160$	−1.00000	−0.0790569
$161$	−5.02931	−0.396365
$162$	−19.2321	−1.51102
$163$	1.99474	0.156240	0.0781199	−	0.996944i	$-0.475108\pi$
$163$	1.99474	0.156240	0.0781199	+	0.996944i	$0.475108\pi$
$164$	−1.35375	−0.105710
$165$	12.0283	0.936404
$166$	−0.0361901	−0.00280889
$167$	−7.89712	−0.611097	−0.305549	−	0.952176i	$-0.598840\pi$
$167$	−7.89712	−0.611097	−0.305549	+	0.952176i	$0.598840\pi$
$168$	1.82280	0.140632
$169$	0.782759	0.0602122
$170$	7.82120	0.599858
$171$	11.4153	0.872951
$172$	9.34902	0.712856
$173$	−2.87367	−0.218481	−0.109240	−	0.994015i	$-0.534842\pi$
$173$	−2.87367	−0.218481	−0.109240	+	0.994015i	$0.534842\pi$
$174$	−31.7789	−2.40915
$175$	0.575814	0.0435274
$176$	−3.79969	−0.286413
$177$	−17.9142	−1.34652
$178$	13.4197	1.00585
$179$	23.7032	1.77166	0.885829	−	0.464012i	$-0.153590\pi$
$179$	23.7032	1.77166	0.885829	+	0.464012i	$0.153590\pi$
$180$	7.02106	0.523319
$181$	9.82702	0.730437	0.365218	−	0.930922i	$-0.380994\pi$
$181$	9.82702	0.730437	0.365218	+	0.930922i	$0.380994\pi$
$182$	2.13772	0.158458
$183$	−2.51753	−0.186101
$184$	8.73426	0.643898
$185$	3.11071	0.228704
$186$	27.1175	1.98836
$187$	29.7181	2.17321
$188$	0.624291	0.0455311
$189$	−7.32960	−0.533150
$190$	−1.62587	−0.117953
$191$	−12.2624	−0.887273	−0.443637	−	0.896207i	$-0.646312\pi$
$191$	−12.2624	−0.887273	−0.443637	+	0.896207i	$0.646312\pi$
$192$	−3.16561	−0.228458
$193$	−26.3060	−1.89355	−0.946773	−	0.321901i	$-0.895678\pi$
$193$	−26.3060	−1.89355	−0.946773	+	0.321901i	$0.895678\pi$
$194$	−5.62339	−0.403736
$195$	11.7524	0.841604
$196$	−6.66844	−0.476317
$197$	19.9865	1.42398	0.711990	−	0.702189i	$-0.247794\pi$
$197$	19.9865	1.42398	0.711990	+	0.702189i	$0.247794\pi$
$198$	26.6779	1.89591
$199$	−12.2601	−0.869097	−0.434549	−	0.900648i	$-0.643092\pi$
$199$	−12.2601	−0.869097	−0.434549	+	0.900648i	$0.643092\pi$
$200$	−1.00000	−0.0707107
$201$	−38.4942	−2.71517
$202$	−9.11648	−0.641434
$203$	−5.78048	−0.405710
$204$	24.7588	1.73346
$205$	−1.35375	−0.0945502
$206$	10.5156	0.732657
$207$	−61.3238	−4.26230
$208$	−3.71251	−0.257417
$209$	−6.17780	−0.427327
$210$	1.82280	0.125785
$211$	7.57243	0.521307	0.260654	−	0.965432i	$-0.416062\pi$
$211$	7.57243	0.521307	0.260654	+	0.965432i	$0.416062\pi$
$212$	−10.7646	−0.739318
$213$	11.9845	0.821166
$214$	7.57558	0.517856
$215$	9.34902	0.637598
$216$	12.7291	0.866106
$217$	4.93260	0.334847
$218$	0.340290	0.0230474
$219$	32.4636	2.19369
$220$	−3.79969	−0.256175
$221$	29.0363	1.95319
$222$	9.84729	0.660907
$223$	−2.59667	−0.173886	−0.0869430	−	0.996213i	$-0.527710\pi$
$223$	−2.59667	−0.173886	−0.0869430	+	0.996213i	$0.527710\pi$
$224$	−0.575814	−0.0384732
$225$	7.02106	0.468071
$226$	−15.6164	−1.03879
$227$	4.59711	0.305121	0.152561	−	0.988294i	$-0.451248\pi$
$227$	4.59711	0.305121	0.152561	+	0.988294i	$0.451248\pi$
$228$	−5.14686	−0.340859
$229$	−13.2087	−0.872856	−0.436428	−	0.899739i	$-0.643757\pi$
$229$	−13.2087	−0.872856	−0.436428	+	0.899739i	$0.643757\pi$
$230$	8.73426	0.575920
$231$	6.92608	0.455703
$232$	10.0388	0.659079
$233$	−16.0115	−1.04895	−0.524475	−	0.851426i	$-0.675738\pi$
$233$	−16.0115	−1.04895	−0.524475	+	0.851426i	$0.675738\pi$
$234$	26.0658	1.70397
$235$	0.624291	0.0407243
$236$	5.65902	0.368371
$237$	43.6231	2.83363
$238$	4.50355	0.291922
$239$	16.8663	1.09099	0.545495	−	0.838114i	$-0.316342\pi$
$239$	16.8663	1.09099	0.545495	+	0.838114i	$0.316342\pi$
$240$	−3.16561	−0.204339
$241$	12.6282	0.813453	0.406727	−	0.913550i	$-0.366670\pi$
$241$	12.6282	0.813453	0.406727	+	0.913550i	$0.366670\pi$
$242$	−3.43766	−0.220981
$243$	−22.6941	−1.45583
$244$	0.795276	0.0509123
$245$	−6.66844	−0.426031
$246$	−4.28545	−0.273230
$247$	−6.03606	−0.384065
$248$	−8.56630	−0.543961
$249$	−0.114564	−0.00726017
$250$	−1.00000	−0.0632456
$251$	−11.5879	−0.731423	−0.365712	−	0.930728i	$-0.619174\pi$
$251$	−11.5879	−0.731423	−0.365712	+	0.930728i	$0.619174\pi$
$252$	4.04283	0.254674
$253$	33.1875	2.08648
$254$	−0.412278	−0.0258686
$255$	24.7588	1.55046
$256$	1.00000	0.0625000
$257$	11.3793	0.709821	0.354910	−	0.934900i	$-0.384511\pi$
$257$	11.3793	0.709821	0.354910	+	0.934900i	$0.384511\pi$
$258$	29.5953	1.84252
$259$	1.79119	0.111299
$260$	−3.71251	−0.230240
$261$	−70.4830	−4.36279
$262$	−15.3345	−0.947366
$263$	11.0480	0.681248	0.340624	−	0.940200i	$-0.389362\pi$
$263$	11.0480	0.681248	0.340624	+	0.940200i	$0.389362\pi$
$264$	−12.0283	−0.740292
$265$	−10.7646	−0.661266
$266$	−0.936197	−0.0574019
$267$	42.4816	2.59983
$268$	12.1601	0.742798
$269$	8.21626	0.500954	0.250477	−	0.968123i	$-0.419413\pi$
$269$	8.21626	0.500954	0.250477	+	0.968123i	$0.419413\pi$
$270$	12.7291	0.774669
$271$	−11.3276	−0.688103	−0.344051	−	0.938951i	$-0.611799\pi$
$271$	−11.3276	−0.688103	−0.344051	+	0.938951i	$0.611799\pi$
$272$	−7.82120	−0.474230
$273$	6.76717	0.409568
$274$	−15.0203	−0.907408
$275$	−3.79969	−0.229130
$276$	27.6492	1.66429
$277$	−15.5857	−0.936456	−0.468228	−	0.883608i	$-0.655107\pi$
$277$	−15.5857	−0.936456	−0.468228	+	0.883608i	$0.655107\pi$
$278$	−12.1218	−0.727019
$279$	60.1446	3.60076
$280$	−0.575814	−0.0344115
$281$	−2.82717	−0.168655	−0.0843275	−	0.996438i	$-0.526874\pi$
$281$	−2.82717	−0.168655	−0.0843275	+	0.996438i	$0.526874\pi$
$282$	1.97626	0.117685
$283$	4.04455	0.240424	0.120212	−	0.992748i	$-0.461643\pi$
$283$	4.04455	0.240424	0.120212	+	0.992748i	$0.461643\pi$
$284$	−3.78586	−0.224649
$285$	−5.14686	−0.304873
$286$	−14.1064	−0.834129
$287$	−0.779510	−0.0460130
$288$	−7.02106	−0.413720
$289$	44.1711	2.59830
$290$	10.0388	0.589498
$291$	−17.8014	−1.04354
$292$	−10.2551	−0.600134
$293$	11.3309	0.661957	0.330978	−	0.943638i	$-0.392621\pi$
$293$	11.3309	0.661957	0.330978	+	0.943638i	$0.392621\pi$
$294$	−21.1097	−1.23114
$295$	5.65902	0.329481
$296$	−3.11071	−0.180806
$297$	48.3667	2.80652
$298$	12.7572	0.739003
$299$	32.4261	1.87525
$300$	−3.16561	−0.182766
$301$	5.38329	0.310288
$302$	−4.45210	−0.256190
$303$	−28.8592	−1.65792
$304$	1.62587	0.0932499
$305$	0.795276	0.0455374
$306$	54.9131	3.13917
$307$	−0.609194	−0.0347686	−0.0173843	−	0.999849i	$-0.505534\pi$
$307$	−0.609194	−0.0347686	−0.0173843	+	0.999849i	$0.505534\pi$
$308$	−2.18792	−0.124668
$309$	33.2883	1.89370
$310$	−8.56630	−0.486533
$311$	−1.38322	−0.0784354	−0.0392177	−	0.999231i	$-0.512487\pi$
$311$	−1.38322	−0.0784354	−0.0392177	+	0.999231i	$0.512487\pi$
$312$	−11.7524	−0.665346
$313$	20.7335	1.17193	0.585965	−	0.810337i	$-0.300716\pi$
$313$	20.7335	1.17193	0.585965	+	0.810337i	$0.300716\pi$
$314$	21.4930	1.21292
$315$	4.04283	0.227787
$316$	−13.7803	−0.775204
$317$	16.6162	0.933260	0.466630	−	0.884453i	$-0.345468\pi$
$317$	16.6162	0.933260	0.466630	+	0.884453i	$0.345468\pi$
$318$	−34.0766	−1.91092
$319$	38.1443	2.13567
$320$	1.00000	0.0559017
$321$	23.9813	1.33851
$322$	5.02931	0.280272
$323$	−12.7162	−0.707550
$324$	19.2321	1.06845
$325$	−3.71251	−0.205933
$326$	−1.99474	−0.110478
$327$	1.07722	0.0595707
$328$	1.35375	0.0747485
$329$	0.359476	0.0198185
$330$	−12.0283	−0.662138
$331$	−8.12049	−0.446343	−0.223171	−	0.974779i	$-0.571641\pi$
$331$	−8.12049	−0.446343	−0.223171	+	0.974779i	$0.571641\pi$
$332$	0.0361901	0.00198619
$333$	21.8405	1.19685
$334$	7.89712	0.432111
$335$	12.1601	0.664379
$336$	−1.82280	−0.0994419
$337$	25.8271	1.40689	0.703447	−	0.710748i	$-0.251643\pi$
$337$	25.8271	1.40689	0.703447	+	0.710748i	$0.251643\pi$
$338$	−0.782759	−0.0425765
$339$	−49.4355	−2.68497
$340$	−7.82120	−0.424164
$341$	−32.5493	−1.76264
$342$	−11.4153	−0.617270
$343$	−7.87048	−0.424966
$344$	−9.34902	−0.504065
$345$	27.6492	1.48858
$346$	2.87367	0.154489
$347$	17.9318	0.962631	0.481316	−	0.876547i	$-0.340159\pi$
$347$	17.9318	0.962631	0.481316	+	0.876547i	$0.340159\pi$
$348$	31.7789	1.70353
$349$	10.5591	0.565217	0.282608	−	0.959235i	$-0.408800\pi$
$349$	10.5591	0.565217	0.282608	+	0.959235i	$0.408800\pi$
$350$	−0.575814	−0.0307785
$351$	47.2570	2.52239
$352$	3.79969	0.202524
$353$	34.3788	1.82980	0.914900	−	0.403680i	$-0.132269\pi$
$353$	34.3788	1.82980	0.914900	+	0.403680i	$0.132269\pi$
$354$	17.9142	0.952130
$355$	−3.78586	−0.200932
$356$	−13.4197	−0.711244
$357$	14.2565	0.754533
$358$	−23.7032	−1.25275
$359$	−11.1704	−0.589554	−0.294777	−	0.955566i	$-0.595245\pi$
$359$	−11.1704	−0.589554	−0.294777	+	0.955566i	$0.595245\pi$
$360$	−7.02106	−0.370043
$361$	−16.3566	−0.860871
$362$	−9.82702	−0.516497
$363$	−10.8823	−0.571171
$364$	−2.13772	−0.112047
$365$	−10.2551	−0.536776
$366$	2.51753	0.131593
$367$	−16.8448	−0.879291	−0.439645	−	0.898171i	$-0.644896\pi$
$367$	−16.8448	−0.879291	−0.439645	+	0.898171i	$0.644896\pi$
$368$	−8.73426	−0.455305
$369$	−9.50479	−0.494800
$370$	−3.11071	−0.161718
$371$	−6.19843	−0.321806
$372$	−27.1175	−1.40598
$373$	7.73691	0.400602	0.200301	−	0.979734i	$-0.435808\pi$
$373$	7.73691	0.400602	0.200301	+	0.979734i	$0.435808\pi$
$374$	−29.7181	−1.53669
$375$	−3.16561	−0.163471
$376$	−0.624291	−0.0321954
$377$	37.2691	1.91946
$378$	7.32960	0.376994
$379$	34.9935	1.79750	0.898749	−	0.438464i	$-0.144477\pi$
$379$	34.9935	1.79750	0.898749	+	0.438464i	$0.144477\pi$
$380$	1.62587	0.0834052
$381$	−1.30511	−0.0668629
$382$	12.2624	0.627397
$383$	−8.56130	−0.437462	−0.218731	−	0.975785i	$-0.570192\pi$
$383$	−8.56130	−0.437462	−0.218731	+	0.975785i	$0.570192\pi$
$384$	3.16561	0.161544
$385$	−2.18792	−0.111507
$386$	26.3060	1.33894
$387$	65.6400	3.33667
$388$	5.62339	0.285484
$389$	−17.0282	−0.863365	−0.431683	−	0.902026i	$-0.642080\pi$
$389$	−17.0282	−0.863365	−0.431683	+	0.902026i	$0.642080\pi$
$390$	−11.7524	−0.595104
$391$	68.3124	3.45470
$392$	6.66844	0.336807
$393$	−48.5429	−2.44866
$394$	−19.9865	−1.00691
$395$	−13.7803	−0.693364
$396$	−26.6779	−1.34061
$397$	3.32722	0.166988	0.0834941	−	0.996508i	$-0.473392\pi$
$397$	3.32722	0.166988	0.0834941	+	0.996508i	$0.473392\pi$
$398$	12.2601	0.614545
$399$	−2.96363	−0.148367
$400$	1.00000	0.0500000
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	38.4942	1.91991
$403$	−31.8025	−1.58420
$404$	9.11648	0.453562
$405$	19.2321	0.955653
$406$	5.78048	0.286880
$407$	−11.8197	−0.585883
$408$	−24.7588	−1.22574
$409$	−2.02868	−0.100312	−0.0501560	−	0.998741i	$-0.515972\pi$
$409$	−2.02868	−0.100312	−0.0501560	+	0.998741i	$0.515972\pi$
$410$	1.35375	0.0668571
$411$	−47.5483	−2.34538
$412$	−10.5156	−0.518067
$413$	3.25854	0.160342
$414$	61.3238	3.01390
$415$	0.0361901	0.00177650
$416$	3.71251	0.182021
$417$	−38.3729	−1.87913
$418$	6.17780	0.302166
$419$	−3.84679	−0.187928	−0.0939641	−	0.995576i	$-0.529954\pi$
$419$	−3.84679	−0.187928	−0.0939641	+	0.995576i	$0.529954\pi$
$420$	−1.82280	−0.0889435
$421$	5.00412	0.243886	0.121943	−	0.992537i	$-0.461088\pi$
$421$	5.00412	0.243886	0.121943	+	0.992537i	$0.461088\pi$
$422$	−7.57243	−0.368620
$423$	4.38319	0.213118
$424$	10.7646	0.522777
$425$	−7.82120	−0.379384
$426$	−11.9845	−0.580652
$427$	0.457931	0.0221608
$428$	−7.57558	−0.366179
$429$	−44.6553	−2.15598
$430$	−9.34902	−0.450850
$431$	−23.0241	−1.10903	−0.554515	−	0.832174i	$-0.687096\pi$
$431$	−23.0241	−1.10903	−0.554515	+	0.832174i	$0.687096\pi$
$432$	−12.7291	−0.612429
$433$	−34.9392	−1.67907	−0.839535	−	0.543306i	$-0.817172\pi$
$433$	−34.9392	−1.67907	−0.839535	+	0.543306i	$0.817172\pi$
$434$	−4.93260	−0.236772
$435$	31.7789	1.52368
$436$	−0.340290	−0.0162969
$437$	−14.2007	−0.679314
$438$	−32.4636	−1.55117
$439$	−32.9251	−1.57143	−0.785714	−	0.618590i	$-0.787704\pi$
$439$	−32.9251	−1.57143	−0.785714	+	0.618590i	$0.787704\pi$
$440$	3.79969	0.181143
$441$	−46.8195	−2.22950
$442$	−29.0363	−1.38112
$443$	22.2432	1.05681	0.528404	−	0.848993i	$-0.322791\pi$
$443$	22.2432	1.05681	0.528404	+	0.848993i	$0.322791\pi$
$444$	−9.84729	−0.467331
$445$	−13.4197	−0.636156
$446$	2.59667	0.122956
$447$	40.3842	1.91011
$448$	0.575814	0.0272047
$449$	32.8365	1.54965	0.774826	−	0.632174i	$-0.217837\pi$
$449$	32.8365	1.54965	0.774826	+	0.632174i	$0.217837\pi$
$450$	−7.02106	−0.330976
$451$	5.14385	0.242214
$452$	15.6164	0.734536
$453$	−14.0936	−0.662175
$454$	−4.59711	−0.215753
$455$	−2.13772	−0.100218
$456$	5.14686	0.241024
$457$	−7.78746	−0.364282	−0.182141	−	0.983272i	$-0.558303\pi$
$457$	−7.78746	−0.364282	−0.182141	+	0.983272i	$0.558303\pi$
$458$	13.2087	0.617202
$459$	99.5568	4.64691
$460$	−8.73426	−0.407237
$461$	7.99316	0.372279	0.186139	−	0.982523i	$-0.440402\pi$
$461$	7.99316	0.372279	0.186139	+	0.982523i	$0.440402\pi$
$462$	−6.92608	−0.322230
$463$	9.87698	0.459022	0.229511	−	0.973306i	$-0.426287\pi$
$463$	9.87698	0.459022	0.229511	+	0.973306i	$0.426287\pi$
$464$	−10.0388	−0.466039
$465$	−27.1175	−1.25755
$466$	16.0115	0.741719
$467$	−17.8845	−0.827595	−0.413798	−	0.910369i	$-0.635798\pi$
$467$	−17.8845	−0.827595	−0.413798	+	0.910369i	$0.635798\pi$
$468$	−26.0658	−1.20489
$469$	7.00197	0.323321
$470$	−0.624291	−0.0287964
$471$	68.0384	3.13504
$472$	−5.65902	−0.260477
$473$	−35.5234	−1.63337
$474$	−43.6231	−2.00368
$475$	1.62587	0.0745999
$476$	−4.50355	−0.206420
$477$	−75.5792	−3.46053
$478$	−16.8663	−0.771446
$479$	−15.8343	−0.723488	−0.361744	−	0.932277i	$-0.617819\pi$
$479$	−15.8343	−0.723488	−0.361744	+	0.932277i	$0.617819\pi$
$480$	3.16561	0.144490
$481$	−11.5486	−0.526569
$482$	−12.6282	−0.575198
$483$	15.9208	0.724422
$484$	3.43766	0.156257
$485$	5.62339	0.255345
$486$	22.6941	1.02942
$487$	6.23707	0.282629	0.141314	−	0.989965i	$-0.454867\pi$
$487$	6.23707	0.282629	0.141314	+	0.989965i	$0.454867\pi$
$488$	−0.795276	−0.0360004
$489$	−6.31455	−0.285554
$490$	6.66844	0.301249
$491$	38.6738	1.74533	0.872663	−	0.488324i	$-0.162392\pi$
$491$	38.6738	1.74533	0.872663	+	0.488324i	$0.162392\pi$
$492$	4.28545	0.193203
$493$	78.5154	3.53615
$494$	6.03606	0.271575
$495$	−26.6779	−1.19908
$496$	8.56630	0.384638
$497$	−2.17995	−0.0977841
$498$	0.114564	0.00513371
$499$	12.3353	0.552206	0.276103	−	0.961128i	$-0.410957\pi$
$499$	12.3353	0.552206	0.276103	+	0.961128i	$0.410957\pi$
$500$	1.00000	0.0447214
$501$	24.9992	1.11688
$502$	11.5879	0.517194
$503$	−0.833776	−0.0371762	−0.0185881	−	0.999827i	$-0.505917\pi$
$503$	−0.833776	−0.0371762	−0.0185881	+	0.999827i	$0.505917\pi$
$504$	−4.04283	−0.180082
$505$	9.11648	0.405678
$506$	−33.1875	−1.47536
$507$	−2.47791	−0.110048
$508$	0.412278	0.0182919
$509$	−25.3064	−1.12169	−0.560843	−	0.827922i	$-0.689523\pi$
$509$	−25.3064	−1.12169	−0.560843	+	0.827922i	$0.689523\pi$
$510$	−24.7588	−1.09634
$511$	−5.90503	−0.261223
$512$	−1.00000	−0.0441942
$513$	−20.6958	−0.913744
$514$	−11.3793	−0.501919
$515$	−10.5156	−0.463373
$516$	−29.5953	−1.30286
$517$	−2.37211	−0.104325
$518$	−1.79119	−0.0787004
$519$	9.09690	0.399310
$520$	3.71251	0.162804
$521$	26.7536	1.17210	0.586048	−	0.810276i	$-0.300683\pi$
$521$	26.7536	1.17210	0.586048	+	0.810276i	$0.300683\pi$
$522$	70.4830	3.08496
$523$	17.0707	0.746448	0.373224	−	0.927741i	$-0.378252\pi$
$523$	17.0707	0.746448	0.373224	+	0.927741i	$0.378252\pi$
$524$	15.3345	0.669889
$525$	−1.82280	−0.0795535
$526$	−11.0480	−0.481715
$527$	−66.9987	−2.91851
$528$	12.0283	0.523466
$529$	53.2873	2.31684
$530$	10.7646	0.467586
$531$	39.7323	1.72424
$532$	0.936197	0.0405893
$533$	5.02583	0.217693
$534$	−42.4816	−1.83836
$535$	−7.57558	−0.327521
$536$	−12.1601	−0.525237
$537$	−75.0349	−3.23800
$538$	−8.21626	−0.354228
$539$	25.3380	1.09139
$540$	−12.7291	−0.547773
$541$	24.1256	1.03724	0.518621	−	0.855004i	$-0.326446\pi$
$541$	24.1256	1.03724	0.518621	+	0.855004i	$0.326446\pi$
$542$	11.3276	0.486562
$543$	−31.1085	−1.33499
$544$	7.82120	0.335331
$545$	−0.340290	−0.0145764
$546$	−6.76717	−0.289608
$547$	15.0304	0.642652	0.321326	−	0.946969i	$-0.395872\pi$
$547$	15.0304	0.642652	0.321326	+	0.946969i	$0.395872\pi$
$548$	15.0203	0.641634
$549$	5.58368	0.238306
$550$	3.79969	0.162019
$551$	−16.3217	−0.695330
$552$	−27.6492	−1.17683
$553$	−7.93491	−0.337427
$554$	15.5857	0.662174
$555$	−9.84729	−0.417994
$556$	12.1218	0.514080
$557$	21.3083	0.902860	0.451430	−	0.892307i	$-0.350914\pi$
$557$	21.3083	0.902860	0.451430	+	0.892307i	$0.350914\pi$
$558$	−60.1446	−2.54612
$559$	−34.7084	−1.46801
$560$	0.575814	0.0243326
$561$	−94.0759	−3.97189
$562$	2.82717	0.119257
$563$	2.83828	0.119619	0.0598096	−	0.998210i	$-0.480951\pi$
$563$	2.83828	0.119619	0.0598096	+	0.998210i	$0.480951\pi$
$564$	−1.97626	−0.0832156
$565$	15.6164	0.656989
$566$	−4.04455	−0.170005
$567$	11.0741	0.465070
$568$	3.78586	0.158851
$569$	44.8858	1.88171	0.940856	−	0.338807i	$-0.110023\pi$
$569$	44.8858	1.88171	0.940856	+	0.338807i	$0.110023\pi$
$570$	5.14686	0.215578
$571$	−38.0101	−1.59067	−0.795337	−	0.606168i	$-0.792706\pi$
$571$	−38.0101	−1.59067	−0.795337	+	0.606168i	$0.792706\pi$
$572$	14.1064	0.589819
$573$	38.8178	1.62164
$574$	0.779510	0.0325361
$575$	−8.73426	−0.364244
$576$	7.02106	0.292544
$577$	32.0009	1.33221	0.666107	−	0.745856i	$-0.267959\pi$
$577$	32.0009	1.33221	0.666107	+	0.745856i	$0.267959\pi$
$578$	−44.1711	−1.83728
$579$	83.2744	3.46077
$580$	−10.0388	−0.416838
$581$	0.0208388	0.000864537 0
$582$	17.8014	0.737893
$583$	40.9023	1.69400
$584$	10.2551	0.424359
$585$	−26.0658	−1.07769
$586$	−11.3309	−0.468074
$587$	−33.4007	−1.37859	−0.689297	−	0.724479i	$-0.742080\pi$
$587$	−33.4007	−1.37859	−0.689297	+	0.724479i	$0.742080\pi$
$588$	21.1097	0.870547
$589$	13.9277	0.573880
$590$	−5.65902	−0.232978
$591$	−63.2695	−2.60256
$592$	3.11071	0.127849
$593$	13.8186	0.567462	0.283731	−	0.958904i	$-0.408428\pi$
$593$	13.8186	0.567462	0.283731	+	0.958904i	$0.408428\pi$
$594$	−48.3667	−1.98451
$595$	−4.50355	−0.184628
$596$	−12.7572	−0.522554
$597$	38.8107	1.58842
$598$	−32.4261	−1.32600
$599$	45.3766	1.85404	0.927019	−	0.375013i	$-0.122362\pi$
$599$	45.3766	1.85404	0.927019	+	0.375013i	$0.122362\pi$
$600$	3.16561	0.129235
$601$	−8.64703	−0.352720	−0.176360	−	0.984326i	$-0.556432\pi$
$601$	−8.64703	−0.352720	−0.176360	+	0.984326i	$0.556432\pi$
$602$	−5.38329	−0.219407
$603$	85.3770	3.47682
$604$	4.45210	0.181153
$605$	3.43766	0.139761
$606$	28.8592	1.17232
$607$	17.8230	0.723411	0.361706	−	0.932292i	$-0.382195\pi$
$607$	17.8230	0.723411	0.361706	+	0.932292i	$0.382195\pi$
$608$	−1.62587	−0.0659376
$609$	18.2987	0.741501
$610$	−0.795276	−0.0321998
$611$	−2.31769	−0.0937637
$612$	−54.9131	−2.21973
$613$	−16.3922	−0.662076	−0.331038	−	0.943617i	$-0.607399\pi$
$613$	−16.3922	−0.662076	−0.331038	+	0.943617i	$0.607399\pi$
$614$	0.609194	0.0245851
$615$	4.28545	0.172806
$616$	2.18792	0.0881536
$617$	−42.3480	−1.70487	−0.852434	−	0.522835i	$-0.824874\pi$
$617$	−42.3480	−1.70487	−0.852434	+	0.522835i	$0.824874\pi$
$618$	−33.2883	−1.33905
$619$	−33.5895	−1.35008	−0.675039	−	0.737782i	$-0.735873\pi$
$619$	−33.5895	−1.35008	−0.675039	+	0.737782i	$0.735873\pi$
$620$	8.56630	0.344031
$621$	111.179	4.46147
$622$	1.38322	0.0554622
$623$	−7.72726	−0.309586
$624$	11.7524	0.470471
$625$	1.00000	0.0400000
$626$	−20.7335	−0.828679
$627$	19.5565	0.781010
$628$	−21.4930	−0.857664
$629$	−24.3295	−0.970080
$630$	−4.04283	−0.161070
$631$	13.3485	0.531394	0.265697	−	0.964057i	$-0.414398\pi$
$631$	13.3485	0.531394	0.265697	+	0.964057i	$0.414398\pi$
$632$	13.7803	0.548152
$633$	−23.9713	−0.952774
$634$	−16.6162	−0.659915
$635$	0.412278	0.0163608
$636$	34.0766	1.35122
$637$	24.7567	0.980895
$638$	−38.1443	−1.51015
$639$	−26.5807	−1.05152
$640$	−1.00000	−0.0395285
$641$	3.68482	0.145542	0.0727708	−	0.997349i	$-0.476816\pi$
$641$	3.68482	0.145542	0.0727708	+	0.997349i	$0.476816\pi$
$642$	−23.9813	−0.946466
$643$	−22.6913	−0.894858	−0.447429	−	0.894320i	$-0.647660\pi$
$643$	−22.6913	−0.894858	−0.447429	+	0.894320i	$0.647660\pi$
$644$	−5.02931	−0.198182
$645$	−29.5953	−1.16531
$646$	12.7162	0.500313
$647$	6.97361	0.274161	0.137080	−	0.990560i	$-0.456228\pi$
$647$	6.97361	0.274161	0.137080	+	0.990560i	$0.456228\pi$
$648$	−19.2321	−0.755510
$649$	−21.5025	−0.844048
$650$	3.71251	0.145617
$651$	−15.6147	−0.611987
$652$	1.99474	0.0781199
$653$	−0.716400	−0.0280349	−0.0140175	−	0.999902i	$-0.504462\pi$
$653$	−0.716400	−0.0280349	−0.0140175	+	0.999902i	$0.504462\pi$
$654$	−1.07722	−0.0421228
$655$	15.3345	0.599167
$656$	−1.35375	−0.0528552
$657$	−72.0017	−2.80905
$658$	−0.359476	−0.0140138
$659$	14.9053	0.580628	0.290314	−	0.956931i	$-0.406240\pi$
$659$	14.9053	0.580628	0.290314	+	0.956931i	$0.406240\pi$
$660$	12.0283	0.468202
$661$	16.1899	0.629713	0.314856	−	0.949139i	$-0.398044\pi$
$661$	16.1899	0.629713	0.314856	+	0.949139i	$0.398044\pi$
$662$	8.12049	0.315612
$663$	−91.9175	−3.56978
$664$	−0.0361901	−0.00140445
$665$	0.936197	0.0363042
$666$	−21.8405	−0.846302
$667$	87.6814	3.39504
$668$	−7.89712	−0.305549
$669$	8.22004	0.317805
$670$	−12.1601	−0.469787
$671$	−3.02180	−0.116655
$672$	1.82280	0.0703160
$673$	−16.3493	−0.630220	−0.315110	−	0.949055i	$-0.602041\pi$
$673$	−16.3493	−0.630220	−0.315110	+	0.949055i	$0.602041\pi$
$674$	−25.8271	−0.994824
$675$	−12.7291	−0.489943
$676$	0.782759	0.0301061
$677$	−6.57568	−0.252724	−0.126362	−	0.991984i	$-0.540330\pi$
$677$	−6.57568	−0.252724	−0.126362	+	0.991984i	$0.540330\pi$
$678$	49.4355	1.89856
$679$	3.23803	0.124264
$680$	7.82120	0.299929
$681$	−14.5527	−0.557659
$682$	32.5493	1.24638
$683$	−20.8303	−0.797050	−0.398525	−	0.917157i	$-0.630478\pi$
$683$	−20.8303	−0.797050	−0.398525	+	0.917157i	$0.630478\pi$
$684$	11.4153	0.436476
$685$	15.0203	0.573895
$686$	7.87048	0.300496
$687$	41.8136	1.59529
$688$	9.34902	0.356428
$689$	39.9639	1.52250
$690$	−27.6492	−1.05259
$691$	−0.0609403	−0.00231828	−0.00115914	−	0.999999i	$-0.500369\pi$
$691$	−0.0609403	−0.00231828	−0.00115914	+	0.999999i	$0.500369\pi$
$692$	−2.87367	−0.109240
$693$	−15.3615	−0.583535
$694$	−17.9318	−0.680683
$695$	12.1218	0.459807
$696$	−31.7789	−1.20457
$697$	10.5880	0.401048
$698$	−10.5591	−0.399668
$699$	50.6862	1.91713
$700$	0.575814	0.0217637
$701$	−6.82498	−0.257776	−0.128888	−	0.991659i	$-0.541141\pi$
$701$	−6.82498	−0.257776	−0.128888	+	0.991659i	$0.541141\pi$
$702$	−47.2570	−1.78360
$703$	5.05760	0.190751
$704$	−3.79969	−0.143206
$705$	−1.97626	−0.0744303
$706$	−34.3788	−1.29386
$707$	5.24940	0.197424
$708$	−17.9142	−0.673258
$709$	−31.7854	−1.19372	−0.596862	−	0.802344i	$-0.703586\pi$
$709$	−31.7854	−1.19372	−0.596862	+	0.802344i	$0.703586\pi$
$710$	3.78586	0.142081
$711$	−96.7526	−3.62851
$712$	13.4197	0.502925
$713$	−74.8203	−2.80204
$714$	−14.2565	−0.533535
$715$	14.1064	0.527550
$716$	23.7032	0.885829
$717$	−53.3920	−1.99396
$718$	11.1704	0.416877
$719$	−43.1948	−1.61089	−0.805447	−	0.592667i	$-0.798075\pi$
$719$	−43.1948	−1.61089	−0.805447	+	0.592667i	$0.798075\pi$
$720$	7.02106	0.261660
$721$	−6.05503	−0.225501
$722$	16.3566	0.608728
$723$	−39.9759	−1.48672
$724$	9.82702	0.365218
$725$	−10.0388	−0.372831
$726$	10.8823	0.403879
$727$	4.47722	0.166051	0.0830254	−	0.996547i	$-0.473542\pi$
$727$	4.47722	0.166051	0.0830254	+	0.996547i	$0.473542\pi$
$728$	2.13772	0.0792291
$729$	14.1441	0.523855
$730$	10.2551	0.379558
$731$	−73.1205	−2.70446
$732$	−2.51753	−0.0930506
$733$	−48.6175	−1.79573	−0.897864	−	0.440273i	$-0.854882\pi$
$733$	−48.6175	−1.79573	−0.897864	+	0.440273i	$0.854882\pi$
$734$	16.8448	0.621753
$735$	21.1097	0.778641
$736$	8.73426	0.321949
$737$	−46.2047	−1.70197
$738$	9.50479	0.349876
$739$	34.5631	1.27143	0.635713	−	0.771926i	$-0.280706\pi$
$739$	34.5631	1.27143	0.635713	+	0.771926i	$0.280706\pi$
$740$	3.11071	0.114352
$741$	19.1078	0.701942
$742$	6.19843	0.227551
$743$	23.4699	0.861027	0.430513	−	0.902584i	$-0.358333\pi$
$743$	23.4699	0.861027	0.430513	+	0.902584i	$0.358333\pi$
$744$	27.1175	0.994178
$745$	−12.7572	−0.467387
$746$	−7.73691	−0.283268
$747$	0.254093	0.00929677
$748$	29.7181	1.08660
$749$	−4.36212	−0.159388
$750$	3.16561	0.115592
$751$	−17.4606	−0.637146	−0.318573	−	0.947898i	$-0.603204\pi$
$751$	−17.4606	−0.637146	−0.318573	+	0.947898i	$0.603204\pi$
$752$	0.624291	0.0227656
$753$	36.6828	1.33680
$754$	−37.2691	−1.35726
$755$	4.45210	0.162029
$756$	−7.32960	−0.266575
$757$	24.2626	0.881839	0.440919	−	0.897547i	$-0.354652\pi$
$757$	24.2626	0.881839	0.440919	+	0.897547i	$0.354652\pi$
$758$	−34.9935	−1.27102
$759$	−105.059	−3.81338
$760$	−1.62587	−0.0589764
$761$	4.37594	0.158628	0.0793138	−	0.996850i	$-0.474727\pi$
$761$	4.37594	0.158628	0.0793138	+	0.996850i	$0.474727\pi$
$762$	1.30511	0.0472792
$763$	−0.195944	−0.00709364
$764$	−12.2624	−0.443637
$765$	−54.9131	−1.98539
$766$	8.56130	0.309332
$767$	−21.0092	−0.758597
$768$	−3.16561	−0.114229
$769$	−43.0046	−1.55079	−0.775393	−	0.631479i	$-0.782448\pi$
$769$	−43.0046	−1.55079	−0.775393	+	0.631479i	$0.782448\pi$
$770$	2.18792	0.0788470
$771$	−36.0224	−1.29731
$772$	−26.3060	−0.946773
$773$	27.4623	0.987752	0.493876	−	0.869532i	$-0.335580\pi$
$773$	27.4623	0.987752	0.493876	+	0.869532i	$0.335580\pi$
$774$	−65.6400	−2.35938
$775$	8.56630	0.307711
$776$	−5.62339	−0.201868
$777$	−5.67020	−0.203417
$778$	17.0282	0.610491
$779$	−2.20102	−0.0788599
$780$	11.7524	0.420802
$781$	14.3851	0.514739
$782$	−68.3124	−2.44284
$783$	127.785	4.56666
$784$	−6.66844	−0.238159
$785$	−21.4930	−0.767118
$786$	48.5429	1.73147
$787$	45.6473	1.62715	0.813576	−	0.581459i	$-0.197518\pi$
$787$	45.6473	1.62715	0.813576	+	0.581459i	$0.197518\pi$
$788$	19.9865	0.711990
$789$	−34.9736	−1.24509
$790$	13.7803	0.490282
$791$	8.99217	0.319725
$792$	26.6779	0.947957
$793$	−2.95247	−0.104845
$794$	−3.32722	−0.118079
$795$	34.0766	1.20857
$796$	−12.2601	−0.434549
$797$	26.2342	0.929264	0.464632	−	0.885504i	$-0.346187\pi$
$797$	26.2342	0.929264	0.464632	+	0.885504i	$0.346187\pi$
$798$	2.96363	0.104911
$799$	−4.88270	−0.172738
$800$	−1.00000	−0.0353553
$801$	−94.2207	−3.32913
$802$	1.00000	0.0353112
$803$	38.9662	1.37509
$804$	−38.4942	−1.35758
$805$	−5.02931	−0.177260
$806$	31.8025	1.12020
$807$	−26.0094	−0.915576
$808$	−9.11648	−0.320717
$809$	10.4555	0.367597	0.183798	−	0.982964i	$-0.441161\pi$
$809$	10.4555	0.367597	0.183798	+	0.982964i	$0.441161\pi$
$810$	−19.2321	−0.675749
$811$	−40.1532	−1.40997	−0.704985	−	0.709222i	$-0.749046\pi$
$811$	−40.1532	−1.40997	−0.704985	+	0.709222i	$0.749046\pi$
$812$	−5.78048	−0.202855
$813$	35.8587	1.25762
$814$	11.8197	0.414282
$815$	1.99474	0.0698725
$816$	24.7588	0.866732
$817$	15.2003	0.531790
$818$	2.02868	0.0709312
$819$	−15.0090	−0.524459
$820$	−1.35375	−0.0472751
$821$	−46.9092	−1.63714	−0.818571	−	0.574406i	$-0.805233\pi$
$821$	−46.9092	−1.63714	−0.818571	+	0.574406i	$0.805233\pi$
$822$	47.5483	1.65844
$823$	38.3510	1.33683	0.668415	−	0.743788i	$-0.266973\pi$
$823$	38.3510	1.33683	0.668415	+	0.743788i	$0.266973\pi$
$824$	10.5156	0.366329
$825$	12.0283	0.418773
$826$	−3.25854	−0.113379
$827$	−3.65207	−0.126995	−0.0634975	−	0.997982i	$-0.520225\pi$
$827$	−3.65207	−0.126995	−0.0634975	+	0.997982i	$0.520225\pi$
$828$	−61.3238	−2.13115
$829$	−30.6029	−1.06288	−0.531442	−	0.847095i	$-0.678350\pi$
$829$	−30.6029	−1.06288	−0.531442	+	0.847095i	$0.678350\pi$
$830$	−0.0361901	−0.00125618
$831$	49.3383	1.71153
$832$	−3.71251	−0.128708
$833$	52.1552	1.80707
$834$	38.3729	1.32875
$835$	−7.89712	−0.273291
$836$	−6.17780	−0.213664
$837$	−109.041	−3.76902
$838$	3.84679	0.132885
$839$	44.1879	1.52554	0.762768	−	0.646672i	$-0.223840\pi$
$839$	44.1879	1.52554	0.762768	+	0.646672i	$0.223840\pi$
$840$	1.82280	0.0628926
$841$	71.7773	2.47508
$842$	−5.00412	−0.172453
$843$	8.94972	0.308245
$844$	7.57243	0.260654
$845$	0.782759	0.0269277
$846$	−4.38319	−0.150697
$847$	1.97945	0.0680148
$848$	−10.7646	−0.369659
$849$	−12.8035	−0.439413
$850$	7.82120	0.268265
$851$	−27.1698	−0.931367
$852$	11.9845	0.410583
$853$	−32.6798	−1.11894	−0.559468	−	0.828852i	$-0.688995\pi$
$853$	−32.6798	−1.11894	−0.559468	+	0.828852i	$0.688995\pi$
$854$	−0.457931	−0.0156701
$855$	11.4153	0.390396
$856$	7.57558	0.258928
$857$	−5.65860	−0.193294	−0.0966471	−	0.995319i	$-0.530812\pi$
$857$	−5.65860	−0.193294	−0.0966471	+	0.995319i	$0.530812\pi$
$858$	44.6553	1.52451
$859$	33.0047	1.12611	0.563053	−	0.826421i	$-0.309627\pi$
$859$	33.0047	1.12611	0.563053	+	0.826421i	$0.309627\pi$
$860$	9.34902	0.318799
$861$	2.46762	0.0840963
$862$	23.0241	0.784203
$863$	−38.2300	−1.30136	−0.650682	−	0.759351i	$-0.725517\pi$
$863$	−38.2300	−1.30136	−0.650682	+	0.759351i	$0.725517\pi$
$864$	12.7291	0.433053
$865$	−2.87367	−0.0977076
$866$	34.9392	1.18728
$867$	−139.828	−4.74882
$868$	4.93260	0.167423
$869$	52.3610	1.77623
$870$	−31.7789	−1.07740
$871$	−45.1446	−1.52967
$872$	0.340290	0.0115237
$873$	39.4822	1.33627
$874$	14.2007	0.480347
$875$	0.575814	0.0194661
$876$	32.4636	1.09684
$877$	25.6312	0.865504	0.432752	−	0.901513i	$-0.357543\pi$
$877$	25.6312	0.865504	0.432752	+	0.901513i	$0.357543\pi$
$878$	32.9251	1.11117
$879$	−35.8691	−1.20983
$880$	−3.79969	−0.128088
$881$	30.2701	1.01983	0.509913	−	0.860226i	$-0.329678\pi$
$881$	30.2701	1.01983	0.509913	+	0.860226i	$0.329678\pi$
$882$	46.8195	1.57650
$883$	49.5485	1.66744	0.833719	−	0.552188i	$-0.186207\pi$
$883$	49.5485	1.66744	0.833719	+	0.552188i	$0.186207\pi$
$884$	29.0363	0.976596
$885$	−17.9142	−0.602180
$886$	−22.2432	−0.747276
$887$	12.1843	0.409109	0.204554	−	0.978855i	$-0.434425\pi$
$887$	12.1843	0.409109	0.204554	+	0.978855i	$0.434425\pi$
$888$	9.84729	0.330453
$889$	0.237396	0.00796199
$890$	13.4197	0.449830
$891$	−73.0762	−2.44815
$892$	−2.59667	−0.0869430
$893$	1.01501	0.0339662
$894$	−40.3842	−1.35065
$895$	23.7032	0.792310
$896$	−0.575814	−0.0192366
$897$	−102.648	−3.42732
$898$	−32.8365	−1.09577
$899$	−85.9953	−2.86811
$900$	7.02106	0.234035
$901$	84.1923	2.80485
$902$	−5.14385	−0.171271
$903$	−17.0414	−0.567102
$904$	−15.6164	−0.519395
$905$	9.82702	0.326661
$906$	14.0936	0.468228
$907$	−25.3154	−0.840584	−0.420292	−	0.907389i	$-0.638072\pi$
$907$	−25.3154	−0.840584	−0.420292	+	0.907389i	$0.638072\pi$
$908$	4.59711	0.152561
$909$	64.0074	2.12299
$910$	2.13772	0.0708646
$911$	2.31262	0.0766206	0.0383103	−	0.999266i	$-0.487802\pi$
$911$	2.31262	0.0766206	0.0383103	+	0.999266i	$0.487802\pi$
$912$	−5.14686	−0.170429
$913$	−0.137511	−0.00455095
$914$	7.78746	0.257586
$915$	−2.51753	−0.0832270
$916$	−13.2087	−0.436428
$917$	8.82980	0.291586
$918$	−99.5568	−3.28586
$919$	1.45208	0.0478996	0.0239498	−	0.999713i	$-0.492376\pi$
$919$	1.45208	0.0478996	0.0239498	+	0.999713i	$0.492376\pi$
$920$	8.73426	0.287960
$921$	1.92847	0.0635452
$922$	−7.99316	−0.263241
$923$	14.0550	0.462627
$924$	6.92608	0.227851
$925$	3.11071	0.102280
$926$	−9.87698	−0.324578
$927$	−73.8307	−2.42492
$928$	10.0388	0.329539
$929$	−44.8786	−1.47242	−0.736210	−	0.676753i	$-0.763386\pi$
$929$	−44.8786	−1.47242	−0.736210	+	0.676753i	$0.763386\pi$
$930$	27.1175	0.889219
$931$	−10.8420	−0.355332
$932$	−16.0115	−0.524475
$933$	4.37874	0.143354
$934$	17.8845	0.585198
$935$	29.7181	0.971887
$936$	26.0658	0.851987
$937$	−2.72044	−0.0888730	−0.0444365	−	0.999012i	$-0.514149\pi$
$937$	−2.72044	−0.0888730	−0.0444365	+	0.999012i	$0.514149\pi$
$938$	−7.00197	−0.228622
$939$	−65.6342	−2.14189
$940$	0.624291	0.0203621
$941$	32.1771	1.04894	0.524472	−	0.851428i	$-0.324263\pi$
$941$	32.1771	1.04894	0.524472	+	0.851428i	$0.324263\pi$
$942$	−68.0384	−2.21681
$943$	11.8240	0.385043
$944$	5.65902	0.184185
$945$	−7.32960	−0.238432
$946$	35.5234	1.15496
$947$	−3.13607	−0.101909	−0.0509544	−	0.998701i	$-0.516226\pi$
$947$	−3.13607	−0.101909	−0.0509544	+	0.998701i	$0.516226\pi$
$948$	43.6231	1.41681
$949$	38.0722	1.23588
$950$	−1.62587	−0.0527501
$951$	−52.6004	−1.70569
$952$	4.50355	0.145961
$953$	−0.661885	−0.0214406	−0.0107203	−	0.999943i	$-0.503412\pi$
$953$	−0.661885	−0.0214406	−0.0107203	+	0.999943i	$0.503412\pi$
$954$	75.5792	2.44697
$955$	−12.2624	−0.396801
$956$	16.8663	0.545495
$957$	−120.750	−3.90329
$958$	15.8343	0.511583
$959$	8.64888	0.279287
$960$	−3.16561	−0.102170
$961$	42.3816	1.36715
$962$	11.5486	0.372340
$963$	−53.1886	−1.71398
$964$	12.6282	0.406727
$965$	−26.3060	−0.846820
$966$	−15.9208	−0.512244
$967$	25.1365	0.808335	0.404168	−	0.914685i	$-0.367561\pi$
$967$	25.1365	0.808335	0.404168	+	0.914685i	$0.367561\pi$
$968$	−3.43766	−0.110491
$969$	40.2546	1.29316
$970$	−5.62339	−0.180556
$971$	40.5846	1.30242	0.651210	−	0.758897i	$-0.274262\pi$
$971$	40.5846	1.30242	0.651210	+	0.758897i	$0.274262\pi$
$972$	−22.6941	−0.727913
$973$	6.97991	0.223766
$974$	−6.23707	−0.199849
$975$	11.7524	0.376377
$976$	0.795276	0.0254562
$977$	21.1759	0.677478	0.338739	−	0.940880i	$-0.390000\pi$
$977$	21.1759	0.677478	0.338739	+	0.940880i	$0.390000\pi$
$978$	6.31455	0.201917
$979$	50.9908	1.62967
$980$	−6.66844	−0.213015
$981$	−2.38920	−0.0762813
$982$	−38.6738	−1.23413
$983$	15.5459	0.495836	0.247918	−	0.968781i	$-0.420254\pi$
$983$	15.5459	0.495836	0.247918	+	0.968781i	$0.420254\pi$
$984$	−4.28545	−0.136615
$985$	19.9865	0.636823
$986$	−78.5154	−2.50044
$987$	−1.13796	−0.0362216
$988$	−6.03606	−0.192033
$989$	−81.6567	−2.59653
$990$	26.6779	0.847879
$991$	10.1651	0.322904	0.161452	−	0.986881i	$-0.448382\pi$
$991$	10.1651	0.322904	0.161452	+	0.986881i	$0.448382\pi$
$992$	−8.56630	−0.271980
$993$	25.7063	0.815764
$994$	2.17995	0.0691438
$995$	−12.2601	−0.388672
$996$	−0.114564	−0.00363008
$997$	−51.1789	−1.62085	−0.810426	−	0.585841i	$-0.800764\pi$
$997$	−51.1789	−1.62085	−0.810426	+	0.585841i	$0.800764\pi$
$998$	−12.3353	−0.390468
$999$	−39.5966	−1.25278

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4010.2.a.m.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.16561 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.16561 q^{6} +0.575814 q^{7} -1.00000 q^{8} +7.02106 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.16561 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.16561 q^{6} +0.575814 q^{7} -1.00000 q^{8} +7.02106 q^{9} -1.00000 q^{10} -3.79969 q^{11} -3.16561 q^{12} -3.71251 q^{13} -0.575814 q^{14} -3.16561 q^{15} +1.00000 q^{16} -7.82120 q^{17} -7.02106 q^{18} +1.62587 q^{19} +1.00000 q^{20} -1.82280 q^{21} +3.79969 q^{22} -8.73426 q^{23} +3.16561 q^{24} +1.00000 q^{25} +3.71251 q^{26} -12.7291 q^{27} +0.575814 q^{28} -10.0388 q^{29} +3.16561 q^{30} +8.56630 q^{31} -1.00000 q^{32} +12.0283 q^{33} +7.82120 q^{34} +0.575814 q^{35} +7.02106 q^{36} +3.11071 q^{37} -1.62587 q^{38} +11.7524 q^{39} -1.00000 q^{40} -1.35375 q^{41} +1.82280 q^{42} +9.34902 q^{43} -3.79969 q^{44} +7.02106 q^{45} +8.73426 q^{46} +0.624291 q^{47} -3.16561 q^{48} -6.66844 q^{49} -1.00000 q^{50} +24.7588 q^{51} -3.71251 q^{52} -10.7646 q^{53} +12.7291 q^{54} -3.79969 q^{55} -0.575814 q^{56} -5.14686 q^{57} +10.0388 q^{58} +5.65902 q^{59} -3.16561 q^{60} +0.795276 q^{61} -8.56630 q^{62} +4.04283 q^{63} +1.00000 q^{64} -3.71251 q^{65} -12.0283 q^{66} +12.1601 q^{67} -7.82120 q^{68} +27.6492 q^{69} -0.575814 q^{70} -3.78586 q^{71} -7.02106 q^{72} -10.2551 q^{73} -3.11071 q^{74} -3.16561 q^{75} +1.62587 q^{76} -2.18792 q^{77} -11.7524 q^{78} -13.7803 q^{79} +1.00000 q^{80} +19.2321 q^{81} +1.35375 q^{82} +0.0361901 q^{83} -1.82280 q^{84} -7.82120 q^{85} -9.34902 q^{86} +31.7789 q^{87} +3.79969 q^{88} -13.4197 q^{89} -7.02106 q^{90} -2.13772 q^{91} -8.73426 q^{92} -27.1175 q^{93} -0.624291 q^{94} +1.62587 q^{95} +3.16561 q^{96} +5.62339 q^{97} +6.66844 q^{98} -26.6779 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} + 20 q^{5} - 4 q^{6} + 11 q^{7} - 20 q^{8} + 26 q^{9}+O(q^{10}) \) 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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