LMFDB - Embedded newform 4010.2.a.k.1.13

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:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 7 x^{14} - 7 x^{13} + 133 x^{12} - 99 x^{11} - 941 x^{10} + 1290 x^{9} + 3031 x^{8} + \cdots - 512 $ x^15 - 7x^14 - 7x^13 + 133x^12 - 99x^11 - 941x^10 + 1290x^9 + 3031x^8 - 5452x^7 - 4098x^6 + 9986x^5 + 850x^4 - 7216x^3 + 1688x^2 + 1441x - 512
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-1.05167$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.72412 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.72412 q^{6} -1.94532 q^{7} -1.00000 q^{8} -0.0274146 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.72412 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.72412 q^{6} -1.94532 q^{7} -1.00000 q^{8} -0.0274146 q^{9} +1.00000 q^{10} +1.83810 q^{11} +1.72412 q^{12} -2.05133 q^{13} +1.94532 q^{14} -1.72412 q^{15} +1.00000 q^{16} +1.66033 q^{17} +0.0274146 q^{18} +1.54006 q^{19} -1.00000 q^{20} -3.35396 q^{21} -1.83810 q^{22} +0.544628 q^{23} -1.72412 q^{24} +1.00000 q^{25} +2.05133 q^{26} -5.21962 q^{27} -1.94532 q^{28} +1.03602 q^{29} +1.72412 q^{30} +1.23453 q^{31} -1.00000 q^{32} +3.16911 q^{33} -1.66033 q^{34} +1.94532 q^{35} -0.0274146 q^{36} +8.44302 q^{37} -1.54006 q^{38} -3.53673 q^{39} +1.00000 q^{40} +1.18085 q^{41} +3.35396 q^{42} -10.8505 q^{43} +1.83810 q^{44} +0.0274146 q^{45} -0.544628 q^{46} -9.33896 q^{47} +1.72412 q^{48} -3.21574 q^{49} -1.00000 q^{50} +2.86260 q^{51} -2.05133 q^{52} +3.77155 q^{53} +5.21962 q^{54} -1.83810 q^{55} +1.94532 q^{56} +2.65524 q^{57} -1.03602 q^{58} -3.94882 q^{59} -1.72412 q^{60} +4.41148 q^{61} -1.23453 q^{62} +0.0533300 q^{63} +1.00000 q^{64} +2.05133 q^{65} -3.16911 q^{66} -8.02420 q^{67} +1.66033 q^{68} +0.939003 q^{69} -1.94532 q^{70} -10.8768 q^{71} +0.0274146 q^{72} -2.02924 q^{73} -8.44302 q^{74} +1.72412 q^{75} +1.54006 q^{76} -3.57570 q^{77} +3.53673 q^{78} +16.4487 q^{79} -1.00000 q^{80} -8.91700 q^{81} -1.18085 q^{82} -5.10696 q^{83} -3.35396 q^{84} -1.66033 q^{85} +10.8505 q^{86} +1.78623 q^{87} -1.83810 q^{88} +0.564286 q^{89} -0.0274146 q^{90} +3.99048 q^{91} +0.544628 q^{92} +2.12848 q^{93} +9.33896 q^{94} -1.54006 q^{95} -1.72412 q^{96} -5.43989 q^{97} +3.21574 q^{98} -0.0503908 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 15 q^{2} - 6 q^{3} + 15 q^{4} - 15 q^{5} + 6 q^{6} - 5 q^{7} - 15 q^{8} + 19 q^{9}+O(q^{10}) $ 15 * q - 15 * q^2 - 6 * q^3 + 15 * q^4 - 15 * q^5 + 6 * q^6 - 5 * q^7 - 15 * q^8 + 19 * q^9	\( 15 q - 15 q^{2} - 6 q^{3} + 15 q^{4} - 15 q^{5} + 6 q^{6} - 5 q^{7} - 15 q^{8} + 19 q^{9} + 15 q^{10} - 2 q^{11} - 6 q^{12} - 13 q^{13} + 5 q^{14} + 6 q^{15} + 15 q^{16} + 11 q^{17} - 19 q^{18} - 15 q^{19} - 15 q^{20} - 2 q^{21} + 2 q^{22} - 3 q^{23} + 6 q^{24} + 15 q^{25} + 13 q^{26} - 12 q^{27} - 5 q^{28} + 28 q^{29} - 6 q^{30} - 12 q^{31} - 15 q^{32} - 22 q^{33} - 11 q^{34} + 5 q^{35} + 19 q^{36} - 23 q^{37} + 15 q^{38} - 2 q^{39} + 15 q^{40} + 24 q^{41} + 2 q^{42} - 24 q^{43} - 2 q^{44} - 19 q^{45} + 3 q^{46} - 3 q^{47} - 6 q^{48} + 20 q^{49} - 15 q^{50} - 5 q^{51} - 13 q^{52} + 10 q^{53} + 12 q^{54} + 2 q^{55} + 5 q^{56} - 11 q^{57} - 28 q^{58} + 2 q^{59} + 6 q^{60} + 15 q^{61} + 12 q^{62} - 2 q^{63} + 15 q^{64} + 13 q^{65} + 22 q^{66} - 48 q^{67} + 11 q^{68} + 21 q^{69} - 5 q^{70} + 15 q^{71} - 19 q^{72} - 47 q^{73} + 23 q^{74} - 6 q^{75} - 15 q^{76} + 7 q^{77} + 2 q^{78} - 34 q^{79} - 15 q^{80} + 43 q^{81} - 24 q^{82} - 32 q^{83} - 2 q^{84} - 11 q^{85} + 24 q^{86} + 14 q^{87} + 2 q^{88} + 25 q^{89} + 19 q^{90} - 32 q^{91} - 3 q^{92} - 42 q^{93} + 3 q^{94} + 15 q^{95} + 6 q^{96} - 34 q^{97} - 20 q^{98} + 4 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 - 6 * q^3 + 15 * q^4 - 15 * q^5 + 6 * q^6 - 5 * q^7 - 15 * q^8 + 19 * q^9 + 15 * q^10 - 2 * q^11 - 6 * q^12 - 13 * q^13 + 5 * q^14 + 6 * q^15 + 15 * q^16 + 11 * q^17 - 19 * q^18 - 15 * q^19 - 15 * q^20 - 2 * q^21 + 2 * q^22 - 3 * q^23 + 6 * q^24 + 15 * q^25 + 13 * q^26 - 12 * q^27 - 5 * q^28 + 28 * q^29 - 6 * q^30 - 12 * q^31 - 15 * q^32 - 22 * q^33 - 11 * q^34 + 5 * q^35 + 19 * q^36 - 23 * q^37 + 15 * q^38 - 2 * q^39 + 15 * q^40 + 24 * q^41 + 2 * q^42 - 24 * q^43 - 2 * q^44 - 19 * q^45 + 3 * q^46 - 3 * q^47 - 6 * q^48 + 20 * q^49 - 15 * q^50 - 5 * q^51 - 13 * q^52 + 10 * q^53 + 12 * q^54 + 2 * q^55 + 5 * q^56 - 11 * q^57 - 28 * q^58 + 2 * q^59 + 6 * q^60 + 15 * q^61 + 12 * q^62 - 2 * q^63 + 15 * q^64 + 13 * q^65 + 22 * q^66 - 48 * q^67 + 11 * q^68 + 21 * q^69 - 5 * q^70 + 15 * q^71 - 19 * q^72 - 47 * q^73 + 23 * q^74 - 6 * q^75 - 15 * q^76 + 7 * q^77 + 2 * q^78 - 34 * q^79 - 15 * q^80 + 43 * q^81 - 24 * q^82 - 32 * q^83 - 2 * q^84 - 11 * q^85 + 24 * q^86 + 14 * q^87 + 2 * q^88 + 25 * q^89 + 19 * q^90 - 32 * q^91 - 3 * q^92 - 42 * q^93 + 3 * q^94 + 15 * q^95 + 6 * q^96 - 34 * q^97 - 20 * q^98 + 4 * 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
$3$	1.72412	0.995420	0.497710	−	0.867343i	$-0.334174\pi$
$3$	1.72412	0.995420	0.497710	+	0.867343i	$0.334174\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.72412	−0.703869
$7$	−1.94532	−0.735261	−0.367630	−	0.929972i	$-0.619831\pi$
$7$	−1.94532	−0.735261	−0.367630	+	0.929972i	$0.619831\pi$
$8$	−1.00000	−0.353553
$9$	−0.0274146	−0.00913819
$10$	1.00000	0.316228
$11$	1.83810	0.554209	0.277105	−	0.960840i	$-0.410625\pi$
$11$	1.83810	0.554209	0.277105	+	0.960840i	$0.410625\pi$
$12$	1.72412	0.497710
$13$	−2.05133	−0.568936	−0.284468	−	0.958686i	$-0.591817\pi$
$13$	−2.05133	−0.568936	−0.284468	+	0.958686i	$0.591817\pi$
$14$	1.94532	0.519908
$15$	−1.72412	−0.445166
$16$	1.00000	0.250000
$17$	1.66033	0.402688	0.201344	−	0.979521i	$-0.435469\pi$
$17$	1.66033	0.402688	0.201344	+	0.979521i	$0.435469\pi$
$18$	0.0274146	0.00646167
$19$	1.54006	0.353314	0.176657	−	0.984273i	$-0.443472\pi$
$19$	1.54006	0.353314	0.176657	+	0.984273i	$0.443472\pi$
$20$	−1.00000	−0.223607
$21$	−3.35396	−0.731893
$22$	−1.83810	−0.391885
$23$	0.544628	0.113563	0.0567814	−	0.998387i	$-0.481916\pi$
$23$	0.544628	0.113563	0.0567814	+	0.998387i	$0.481916\pi$
$24$	−1.72412	−0.351934
$25$	1.00000	0.200000
$26$	2.05133	0.402299
$27$	−5.21962	−1.00452
$28$	−1.94532	−0.367630
$29$	1.03602	0.192385	0.0961924	−	0.995363i	$-0.469334\pi$
$29$	1.03602	0.192385	0.0961924	+	0.995363i	$0.469334\pi$
$30$	1.72412	0.314780
$31$	1.23453	0.221729	0.110864	−	0.993836i	$-0.464638\pi$
$31$	1.23453	0.221729	0.110864	+	0.993836i	$0.464638\pi$
$32$	−1.00000	−0.176777
$33$	3.16911	0.551671
$34$	−1.66033	−0.284743
$35$	1.94532	0.328819
$36$	−0.0274146	−0.00456909
$37$	8.44302	1.38802	0.694012	−	0.719964i	$-0.255842\pi$
$37$	8.44302	1.38802	0.694012	+	0.719964i	$0.255842\pi$
$38$	−1.54006	−0.249830
$39$	−3.53673	−0.566331
$40$	1.00000	0.158114
$41$	1.18085	0.184417	0.0922087	−	0.995740i	$-0.470607\pi$
$41$	1.18085	0.184417	0.0922087	+	0.995740i	$0.470607\pi$
$42$	3.35396	0.517527
$43$	−10.8505	−1.65468	−0.827340	−	0.561701i	$-0.810147\pi$
$43$	−10.8505	−1.65468	−0.827340	+	0.561701i	$0.810147\pi$
$44$	1.83810	0.277105
$45$	0.0274146	0.00408672
$46$	−0.544628	−0.0803010
$47$	−9.33896	−1.36223	−0.681114	−	0.732178i	$-0.738504\pi$
$47$	−9.33896	−1.36223	−0.681114	+	0.732178i	$0.738504\pi$
$48$	1.72412	0.248855
$49$	−3.21574	−0.459392
$50$	−1.00000	−0.141421
$51$	2.86260	0.400844
$52$	−2.05133	−0.284468
$53$	3.77155	0.518062	0.259031	−	0.965869i	$-0.416597\pi$
$53$	3.77155	0.518062	0.259031	+	0.965869i	$0.416597\pi$
$54$	5.21962	0.710301
$55$	−1.83810	−0.247850
$56$	1.94532	0.259954
$57$	2.65524	0.351696
$58$	−1.03602	−0.136037
$59$	−3.94882	−0.514093	−0.257046	−	0.966399i	$-0.582749\pi$
$59$	−3.94882	−0.514093	−0.257046	+	0.966399i	$0.582749\pi$
$60$	−1.72412	−0.222583
$61$	4.41148	0.564832	0.282416	−	0.959292i	$-0.408864\pi$
$61$	4.41148	0.564832	0.282416	+	0.959292i	$0.408864\pi$
$62$	−1.23453	−0.156786
$63$	0.0533300	0.00671895
$64$	1.00000	0.125000
$65$	2.05133	0.254436
$66$	−3.16911	−0.390091
$67$	−8.02420	−0.980312	−0.490156	−	0.871635i	$-0.663060\pi$
$67$	−8.02420	−0.980312	−0.490156	+	0.871635i	$0.663060\pi$
$68$	1.66033	0.201344
$69$	0.939003	0.113043
$70$	−1.94532	−0.232510
$71$	−10.8768	−1.29083	−0.645417	−	0.763830i	$-0.723316\pi$
$71$	−10.8768	−1.29083	−0.645417	+	0.763830i	$0.723316\pi$
$72$	0.0274146	0.00323084
$73$	−2.02924	−0.237504	−0.118752	−	0.992924i	$-0.537889\pi$
$73$	−2.02924	−0.237504	−0.118752	+	0.992924i	$0.537889\pi$
$74$	−8.44302	−0.981481
$75$	1.72412	0.199084
$76$	1.54006	0.176657
$77$	−3.57570	−0.407488
$78$	3.53673	0.400456
$79$	16.4487	1.85063	0.925313	−	0.379203i	$-0.123802\pi$
$79$	16.4487	1.85063	0.925313	+	0.379203i	$0.123802\pi$
$80$	−1.00000	−0.111803
$81$	−8.91700	−0.990778
$82$	−1.18085	−0.130403
$83$	−5.10696	−0.560562	−0.280281	−	0.959918i	$-0.590428\pi$
$83$	−5.10696	−0.560562	−0.280281	+	0.959918i	$0.590428\pi$
$84$	−3.35396	−0.365947
$85$	−1.66033	−0.180088
$86$	10.8505	1.17004
$87$	1.78623	0.191504
$88$	−1.83810	−0.195943
$89$	0.564286	0.0598142	0.0299071	−	0.999553i	$-0.490479\pi$
$89$	0.564286	0.0598142	0.0299071	+	0.999553i	$0.490479\pi$
$90$	−0.0274146	−0.00288975
$91$	3.99048	0.418316
$92$	0.544628	0.0567814
$93$	2.12848	0.220713
$94$	9.33896	0.963240
$95$	−1.54006	−0.158007
$96$	−1.72412	−0.175967
$97$	−5.43989	−0.552337	−0.276169	−	0.961109i	$-0.589065\pi$
$97$	−5.43989	−0.552337	−0.276169	+	0.961109i	$0.589065\pi$
$98$	3.21574	0.324839
$99$	−0.0503908	−0.00506447
$100$	1.00000	0.100000
$101$	−16.5279	−1.64459	−0.822296	−	0.569060i	$-0.807307\pi$
$101$	−16.5279	−1.64459	−0.822296	+	0.569060i	$0.807307\pi$
$102$	−2.86260	−0.283439
$103$	11.1161	1.09530	0.547651	−	0.836707i	$-0.315522\pi$
$103$	11.1161	1.09530	0.547651	+	0.836707i	$0.315522\pi$
$104$	2.05133	0.201149
$105$	3.35396	0.327313
$106$	−3.77155	−0.366325
$107$	−5.71380	−0.552374	−0.276187	−	0.961104i	$-0.589071\pi$
$107$	−5.71380	−0.552374	−0.276187	+	0.961104i	$0.589071\pi$
$108$	−5.21962	−0.502258
$109$	−17.0725	−1.63525	−0.817625	−	0.575751i	$-0.804710\pi$
$109$	−17.0725	−1.63525	−0.817625	+	0.575751i	$0.804710\pi$
$110$	1.83810	0.175256
$111$	14.5568	1.38167
$112$	−1.94532	−0.183815
$113$	−4.04593	−0.380609	−0.190304	−	0.981725i	$-0.560948\pi$
$113$	−4.04593	−0.380609	−0.190304	+	0.981725i	$0.560948\pi$
$114$	−2.65524	−0.248686
$115$	−0.544628	−0.0507868
$116$	1.03602	0.0961924
$117$	0.0562363	0.00519904
$118$	3.94882	0.363519
$119$	−3.22986	−0.296081
$120$	1.72412	0.157390
$121$	−7.62137	−0.692852
$122$	−4.41148	−0.399396
$123$	2.03592	0.183573
$124$	1.23453	0.110864
$125$	−1.00000	−0.0894427
$126$	−0.0533300	−0.00475101
$127$	−17.2277	−1.52871	−0.764357	−	0.644794i	$-0.776943\pi$
$127$	−17.2277	−1.52871	−0.764357	+	0.644794i	$0.776943\pi$
$128$	−1.00000	−0.0883883
$129$	−18.7075	−1.64710
$130$	−2.05133	−0.179913
$131$	11.5742	1.01124	0.505620	−	0.862757i	$-0.331264\pi$
$131$	11.5742	1.01124	0.505620	+	0.862757i	$0.331264\pi$
$132$	3.16911	0.275836
$133$	−2.99590	−0.259778
$134$	8.02420	0.693185
$135$	5.21962	0.449234
$136$	−1.66033	−0.142372
$137$	−4.60948	−0.393814	−0.196907	−	0.980422i	$-0.563090\pi$
$137$	−4.60948	−0.393814	−0.196907	+	0.980422i	$0.563090\pi$
$138$	−0.939003	−0.0799333
$139$	−16.5125	−1.40057	−0.700287	−	0.713862i	$-0.746944\pi$
$139$	−16.5125	−1.40057	−0.700287	+	0.713862i	$0.746944\pi$
$140$	1.94532	0.164409
$141$	−16.1015	−1.35599
$142$	10.8768	0.912758
$143$	−3.77056	−0.315310
$144$	−0.0274146	−0.00228455
$145$	−1.03602	−0.0860371
$146$	2.02924	0.167941
$147$	−5.54432	−0.457288
$148$	8.44302	0.694012
$149$	6.16182	0.504796	0.252398	−	0.967623i	$-0.418781\pi$
$149$	6.16182	0.504796	0.252398	+	0.967623i	$0.418781\pi$
$150$	−1.72412	−0.140774
$151$	8.49220	0.691085	0.345543	−	0.938403i	$-0.387695\pi$
$151$	8.49220	0.691085	0.345543	+	0.938403i	$0.387695\pi$
$152$	−1.54006	−0.124915
$153$	−0.0455171	−0.00367984
$154$	3.57570	0.288138
$155$	−1.23453	−0.0991601
$156$	−3.53673	−0.283165
$157$	−15.8470	−1.26473	−0.632364	−	0.774672i	$-0.717915\pi$
$157$	−15.8470	−1.26473	−0.632364	+	0.774672i	$0.717915\pi$
$158$	−16.4487	−1.30859
$159$	6.50260	0.515690
$160$	1.00000	0.0790569
$161$	−1.05947	−0.0834982
$162$	8.91700	0.700586
$163$	−7.53318	−0.590044	−0.295022	−	0.955490i	$-0.595327\pi$
$163$	−7.53318	−0.590044	−0.295022	+	0.955490i	$0.595327\pi$
$164$	1.18085	0.0922087
$165$	−3.16911	−0.246715
$166$	5.10696	0.396377
$167$	−18.5865	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$167$	−18.5865	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$168$	3.35396	0.258763
$169$	−8.79205	−0.676312
$170$	1.66033	0.127341
$171$	−0.0422200	−0.00322865
$172$	−10.8505	−0.827340
$173$	9.98907	0.759455	0.379727	−	0.925098i	$-0.376018\pi$
$173$	9.98907	0.759455	0.379727	+	0.925098i	$0.376018\pi$
$174$	−1.78623	−0.135414
$175$	−1.94532	−0.147052
$176$	1.83810	0.138552
$177$	−6.80824	−0.511739
$178$	−0.564286	−0.0422950
$179$	17.4916	1.30738	0.653691	−	0.756761i	$-0.273219\pi$
$179$	17.4916	1.30738	0.653691	+	0.756761i	$0.273219\pi$
$180$	0.0274146	0.00204336
$181$	−9.51269	−0.707073	−0.353536	−	0.935421i	$-0.615021\pi$
$181$	−9.51269	−0.707073	−0.353536	+	0.935421i	$0.615021\pi$
$182$	−3.99048	−0.295794
$183$	7.60591	0.562245
$184$	−0.544628	−0.0401505
$185$	−8.44302	−0.620743
$186$	−2.12848	−0.156068
$187$	3.05185	0.223174
$188$	−9.33896	−0.681114
$189$	10.1538	0.738582
$190$	1.54006	0.111728
$191$	15.1580	1.09679	0.548396	−	0.836218i	$-0.315238\pi$
$191$	15.1580	1.09679	0.548396	+	0.836218i	$0.315238\pi$
$192$	1.72412	0.124428
$193$	12.2141	0.879189	0.439595	−	0.898196i	$-0.355122\pi$
$193$	12.2141	0.879189	0.439595	+	0.898196i	$0.355122\pi$
$194$	5.43989	0.390561
$195$	3.53673	0.253271
$196$	−3.21574	−0.229696
$197$	−26.5971	−1.89496	−0.947481	−	0.319811i	$-0.896381\pi$
$197$	−26.5971	−1.89496	−0.947481	+	0.319811i	$0.896381\pi$
$198$	0.0503908	0.00358112
$199$	−17.2394	−1.22207	−0.611035	−	0.791603i	$-0.709247\pi$
$199$	−17.2394	−1.22207	−0.611035	+	0.791603i	$0.709247\pi$
$200$	−1.00000	−0.0707107
$201$	−13.8347	−0.975822
$202$	16.5279	1.16290
$203$	−2.01540	−0.141453
$204$	2.86260	0.200422
$205$	−1.18085	−0.0824740
$206$	−11.1161	−0.774496
$207$	−0.0149307	−0.00103776
$208$	−2.05133	−0.142234
$209$	2.83079	0.195810
$210$	−3.35396	−0.231445
$211$	−18.4808	−1.27227	−0.636135	−	0.771578i	$-0.719468\pi$
$211$	−18.4808	−1.27227	−0.636135	+	0.771578i	$0.719468\pi$
$212$	3.77155	0.259031
$213$	−18.7528	−1.28492
$214$	5.71380	0.390588
$215$	10.8505	0.739996
$216$	5.21962	0.355150
$217$	−2.40156	−0.163028
$218$	17.0725	1.15630
$219$	−3.49865	−0.236417
$220$	−1.83810	−0.123925
$221$	−3.40587	−0.229104
$222$	−14.5568	−0.976986
$223$	−18.0052	−1.20572	−0.602859	−	0.797848i	$-0.705972\pi$
$223$	−18.0052	−1.20572	−0.602859	+	0.797848i	$0.705972\pi$
$224$	1.94532	0.129977
$225$	−0.0274146	−0.00182764
$226$	4.04593	0.269131
$227$	17.2948	1.14789	0.573947	−	0.818892i	$-0.305411\pi$
$227$	17.2948	1.14789	0.573947	+	0.818892i	$0.305411\pi$
$228$	2.65524	0.175848
$229$	21.6346	1.42966	0.714828	−	0.699300i	$-0.246505\pi$
$229$	21.6346	1.42966	0.714828	+	0.699300i	$0.246505\pi$
$230$	0.544628	0.0359117
$231$	−6.16492	−0.405622
$232$	−1.03602	−0.0680183
$233$	−13.4681	−0.882323	−0.441161	−	0.897428i	$-0.645433\pi$
$233$	−13.4681	−0.882323	−0.441161	+	0.897428i	$0.645433\pi$
$234$	−0.0562363	−0.00367628
$235$	9.33896	0.609207
$236$	−3.94882	−0.257046
$237$	28.3596	1.84215
$238$	3.22986	0.209361
$239$	11.0188	0.712744	0.356372	−	0.934344i	$-0.384014\pi$
$239$	11.0188	0.712744	0.356372	+	0.934344i	$0.384014\pi$
$240$	−1.72412	−0.111291
$241$	−0.143227	−0.00922606	−0.00461303	−	0.999989i	$-0.501468\pi$
$241$	−0.143227	−0.00922606	−0.00461303	+	0.999989i	$0.501468\pi$
$242$	7.62137	0.489920
$243$	0.284892	0.0182758
$244$	4.41148	0.282416
$245$	3.21574	0.205446
$246$	−2.03592	−0.129806
$247$	−3.15917	−0.201013
$248$	−1.23453	−0.0783929
$249$	−8.80501	−0.557995
$250$	1.00000	0.0632456
$251$	1.97385	0.124589	0.0622943	−	0.998058i	$-0.480158\pi$
$251$	1.97385	0.124589	0.0622943	+	0.998058i	$0.480158\pi$
$252$	0.0533300	0.00335947
$253$	1.00108	0.0629376
$254$	17.2277	1.08096
$255$	−2.86260	−0.179263
$256$	1.00000	0.0625000
$257$	−4.40506	−0.274780	−0.137390	−	0.990517i	$-0.543871\pi$
$257$	−4.40506	−0.274780	−0.137390	+	0.990517i	$0.543871\pi$
$258$	18.7075	1.16468
$259$	−16.4243	−1.02056
$260$	2.05133	0.127218
$261$	−0.0284022	−0.00175805
$262$	−11.5742	−0.715054
$263$	30.7542	1.89638	0.948192	−	0.317699i	$-0.102910\pi$
$263$	30.7542	1.89638	0.948192	+	0.317699i	$0.102910\pi$
$264$	−3.16911	−0.195045
$265$	−3.77155	−0.231685
$266$	2.99590	0.183691
$267$	0.972896	0.0595402
$268$	−8.02420	−0.490156
$269$	25.4603	1.55234	0.776171	−	0.630522i	$-0.217159\pi$
$269$	25.4603	1.55234	0.776171	+	0.630522i	$0.217159\pi$
$270$	−5.21962	−0.317656
$271$	−17.3366	−1.05312	−0.526562	−	0.850137i	$-0.676519\pi$
$271$	−17.3366	−1.05312	−0.526562	+	0.850137i	$0.676519\pi$
$272$	1.66033	0.100672
$273$	6.88007	0.416401
$274$	4.60948	0.278469
$275$	1.83810	0.110842
$276$	0.939003	0.0565213
$277$	29.1086	1.74896	0.874482	−	0.485057i	$-0.161201\pi$
$277$	29.1086	1.74896	0.874482	+	0.485057i	$0.161201\pi$
$278$	16.5125	0.990355
$279$	−0.0338442	−0.00202620
$280$	−1.94532	−0.116255
$281$	21.3534	1.27384	0.636919	−	0.770931i	$-0.280208\pi$
$281$	21.3534	1.27384	0.636919	+	0.770931i	$0.280208\pi$
$282$	16.1015	0.958829
$283$	25.7112	1.52837	0.764187	−	0.644995i	$-0.223140\pi$
$283$	25.7112	1.52837	0.764187	+	0.644995i	$0.223140\pi$
$284$	−10.8768	−0.645417
$285$	−2.65524	−0.157283
$286$	3.77056	0.222958
$287$	−2.29712	−0.135595
$288$	0.0274146	0.00161542
$289$	−14.2433	−0.837842
$290$	1.03602	0.0608374
$291$	−9.37902	−0.549808
$292$	−2.02924	−0.118752
$293$	18.1782	1.06198	0.530992	−	0.847377i	$-0.321819\pi$
$293$	18.1782	1.06198	0.530992	+	0.847377i	$0.321819\pi$
$294$	5.54432	0.323352
$295$	3.94882	0.229909
$296$	−8.44302	−0.490740
$297$	−9.59421	−0.556713
$298$	−6.16182	−0.356945
$299$	−1.11721	−0.0646099
$300$	1.72412	0.0995420
$301$	21.1076	1.21662
$302$	−8.49220	−0.488671
$303$	−28.4961	−1.63706
$304$	1.54006	0.0883284
$305$	−4.41148	−0.252601
$306$	0.0455171	0.00260204
$307$	−13.6166	−0.777141	−0.388570	−	0.921419i	$-0.627031\pi$
$307$	−13.6166	−0.777141	−0.388570	+	0.921419i	$0.627031\pi$
$308$	−3.57570	−0.203744
$309$	19.1655	1.09029
$310$	1.23453	0.0701168
$311$	−0.107193	−0.00607837	−0.00303919	−	0.999995i	$-0.500967\pi$
$311$	−0.107193	−0.00607837	−0.00303919	+	0.999995i	$0.500967\pi$
$312$	3.53673	0.200228
$313$	−26.1637	−1.47886	−0.739430	−	0.673233i	$-0.764905\pi$
$313$	−26.1637	−1.47886	−0.739430	+	0.673233i	$0.764905\pi$
$314$	15.8470	0.894297
$315$	−0.0533300	−0.00300481
$316$	16.4487	0.925313
$317$	−14.5197	−0.815508	−0.407754	−	0.913092i	$-0.633688\pi$
$317$	−14.5197	−0.815508	−0.407754	+	0.913092i	$0.633688\pi$
$318$	−6.50260	−0.364648
$319$	1.90432	0.106622
$320$	−1.00000	−0.0559017
$321$	−9.85128	−0.549845
$322$	1.05947	0.0590422
$323$	2.55700	0.142275
$324$	−8.91700	−0.495389
$325$	−2.05133	−0.113787
$326$	7.53318	0.417224
$327$	−29.4350	−1.62776
$328$	−1.18085	−0.0652014
$329$	18.1672	1.00159
$330$	3.16911	0.174454
$331$	−13.7827	−0.757567	−0.378783	−	0.925485i	$-0.623657\pi$
$331$	−13.7827	−0.757567	−0.378783	+	0.925485i	$0.623657\pi$
$332$	−5.10696	−0.280281
$333$	−0.231462	−0.0126840
$334$	18.5865	1.01701
$335$	8.02420	0.438409
$336$	−3.35396	−0.182973
$337$	−0.152670	−0.00831645	−0.00415822	−	0.999991i	$-0.501324\pi$
$337$	−0.152670	−0.00831645	−0.00415822	+	0.999991i	$0.501324\pi$
$338$	8.79205	0.478225
$339$	−6.97566	−0.378866
$340$	−1.66033	−0.0900438
$341$	2.26920	0.122884
$342$	0.0422200	0.00228300
$343$	19.8729	1.07303
$344$	10.8505	0.585018
$345$	−0.939003	−0.0505542
$346$	−9.98907	−0.537016
$347$	−31.9305	−1.71412	−0.857059	−	0.515218i	$-0.827711\pi$
$347$	−31.9305	−1.71412	−0.857059	+	0.515218i	$0.827711\pi$
$348$	1.78623	0.0957519
$349$	−0.490583	−0.0262603	−0.0131302	−	0.999914i	$-0.504180\pi$
$349$	−0.490583	−0.0262603	−0.0131302	+	0.999914i	$0.504180\pi$
$350$	1.94532	0.103982
$351$	10.7072	0.571506
$352$	−1.83810	−0.0979713
$353$	22.3431	1.18920	0.594602	−	0.804020i	$-0.297310\pi$
$353$	22.3431	1.18920	0.594602	+	0.804020i	$0.297310\pi$
$354$	6.80824	0.361854
$355$	10.8768	0.577279
$356$	0.564286	0.0299071
$357$	−5.56866	−0.294725
$358$	−17.4916	−0.924459
$359$	−29.0754	−1.53454	−0.767270	−	0.641324i	$-0.778385\pi$
$359$	−29.0754	−1.53454	−0.767270	+	0.641324i	$0.778385\pi$
$360$	−0.0274146	−0.00144487
$361$	−16.6282	−0.875169
$362$	9.51269	0.499976
$363$	−13.1401	−0.689679
$364$	3.99048	0.209158
$365$	2.02924	0.106215
$366$	−7.60591	−0.397567
$367$	9.83490	0.513378	0.256689	−	0.966494i	$-0.417368\pi$
$367$	9.83490	0.513378	0.256689	+	0.966494i	$0.417368\pi$
$368$	0.544628	0.0283907
$369$	−0.0323724	−0.00168524
$370$	8.44302	0.438932
$371$	−7.33686	−0.380911
$372$	2.12848	0.110357
$373$	16.0683	0.831985	0.415992	−	0.909368i	$-0.363434\pi$
$373$	16.0683	0.831985	0.415992	+	0.909368i	$0.363434\pi$
$374$	−3.05185	−0.157808
$375$	−1.72412	−0.0890331
$376$	9.33896	0.481620
$377$	−2.12523	−0.109455
$378$	−10.1538	−0.522256
$379$	−30.0765	−1.54493	−0.772463	−	0.635059i	$-0.780976\pi$
$379$	−30.0765	−1.54493	−0.772463	+	0.635059i	$0.780976\pi$
$380$	−1.54006	−0.0790033
$381$	−29.7026	−1.52171
$382$	−15.1580	−0.775550
$383$	19.0125	0.971495	0.485747	−	0.874099i	$-0.338547\pi$
$383$	19.0125	0.971495	0.485747	+	0.874099i	$0.338547\pi$
$384$	−1.72412	−0.0879836
$385$	3.57570	0.182234
$386$	−12.2141	−0.621681
$387$	0.297461	0.0151208
$388$	−5.43989	−0.276169
$389$	−10.8795	−0.551614	−0.275807	−	0.961213i	$-0.588945\pi$
$389$	−10.8795	−0.551614	−0.275807	+	0.961213i	$0.588945\pi$
$390$	−3.53673	−0.179089
$391$	0.904259	0.0457304
$392$	3.21574	0.162420
$393$	19.9552	1.00661
$394$	26.5971	1.33994
$395$	−16.4487	−0.827625
$396$	−0.0503908	−0.00253224
$397$	−13.3201	−0.668519	−0.334260	−	0.942481i	$-0.608486\pi$
$397$	−13.3201	−0.668519	−0.334260	+	0.942481i	$0.608486\pi$
$398$	17.2394	0.864134
$399$	−5.16529	−0.258588
$400$	1.00000	0.0500000
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	13.8347	0.690011
$403$	−2.53243	−0.126149
$404$	−16.5279	−0.822296
$405$	8.91700	0.443090
$406$	2.01540	0.100022
$407$	15.5191	0.769256
$408$	−2.86260	−0.141720
$409$	15.4727	0.765075	0.382538	−	0.923940i	$-0.375050\pi$
$409$	15.4727	0.765075	0.382538	+	0.923940i	$0.375050\pi$
$410$	1.18085	0.0583179
$411$	−7.94729	−0.392011
$412$	11.1161	0.547651
$413$	7.68171	0.377992
$414$	0.0149307	0.000733806 0
$415$	5.10696	0.250691
$416$	2.05133	0.100575
$417$	−28.4695	−1.39416
$418$	−2.83079	−0.138458
$419$	30.2832	1.47943	0.739716	−	0.672919i	$-0.234960\pi$
$419$	30.2832	1.47943	0.739716	+	0.672919i	$0.234960\pi$
$420$	3.35396	0.163656
$421$	29.6085	1.44303	0.721516	−	0.692398i	$-0.243446\pi$
$421$	29.6085	1.44303	0.721516	+	0.692398i	$0.243446\pi$
$422$	18.4808	0.899631
$423$	0.256024	0.0124483
$424$	−3.77155	−0.183163
$425$	1.66033	0.0805376
$426$	18.7528	0.908578
$427$	−8.58172	−0.415299
$428$	−5.71380	−0.276187
$429$	−6.50089	−0.313866
$430$	−10.8505	−0.523256
$431$	18.6154	0.896673	0.448336	−	0.893865i	$-0.352017\pi$
$431$	18.6154	0.896673	0.448336	+	0.893865i	$0.352017\pi$
$432$	−5.21962	−0.251129
$433$	−9.69199	−0.465767	−0.232884	−	0.972505i	$-0.574816\pi$
$433$	−9.69199	−0.465767	−0.232884	+	0.972505i	$0.574816\pi$
$434$	2.40156	0.115278
$435$	−1.78623	−0.0856431
$436$	−17.0725	−0.817625
$437$	0.838759	0.0401233
$438$	3.49865	0.167172
$439$	21.1167	1.00785	0.503923	−	0.863749i	$-0.331890\pi$
$439$	21.1167	1.00785	0.503923	+	0.863749i	$0.331890\pi$
$440$	1.83810	0.0876282
$441$	0.0881582	0.00419801
$442$	3.40587	0.162001
$443$	−13.2897	−0.631413	−0.315707	−	0.948857i	$-0.602242\pi$
$443$	−13.2897	−0.631413	−0.315707	+	0.948857i	$0.602242\pi$
$444$	14.5568	0.690833
$445$	−0.564286	−0.0267497
$446$	18.0052	0.852571
$447$	10.6237	0.502484
$448$	−1.94532	−0.0919076
$449$	26.0789	1.23074	0.615370	−	0.788239i	$-0.289007\pi$
$449$	26.0789	1.23074	0.615370	+	0.788239i	$0.289007\pi$
$450$	0.0274146	0.00129233
$451$	2.17052	0.102206
$452$	−4.04593	−0.190304
$453$	14.6416	0.687920
$454$	−17.2948	−0.811684
$455$	−3.99048	−0.187077
$456$	−2.65524	−0.124343
$457$	−25.0095	−1.16989	−0.584947	−	0.811072i	$-0.698885\pi$
$457$	−25.0095	−1.16989	−0.584947	+	0.811072i	$0.698885\pi$
$458$	−21.6346	−1.01092
$459$	−8.66627	−0.404507
$460$	−0.544628	−0.0253934
$461$	−24.9188	−1.16058	−0.580292	−	0.814408i	$-0.697062\pi$
$461$	−24.9188	−1.16058	−0.580292	+	0.814408i	$0.697062\pi$
$462$	6.16492	0.286818
$463$	36.0122	1.67363	0.836815	−	0.547486i	$-0.184415\pi$
$463$	36.0122	1.67363	0.836815	+	0.547486i	$0.184415\pi$
$464$	1.03602	0.0480962
$465$	−2.12848	−0.0987060
$466$	13.4681	0.623896
$467$	−35.9059	−1.66153	−0.830763	−	0.556626i	$-0.812096\pi$
$467$	−35.9059	−1.66153	−0.830763	+	0.556626i	$0.812096\pi$
$468$	0.0562363	0.00259952
$469$	15.6096	0.720785
$470$	−9.33896	−0.430774
$471$	−27.3221	−1.25894
$472$	3.94882	0.181759
$473$	−19.9443	−0.917040
$474$	−28.3596	−1.30260
$475$	1.54006	0.0706627
$476$	−3.22986	−0.148040
$477$	−0.103395	−0.00473415
$478$	−11.0188	−0.503986
$479$	13.6305	0.622791	0.311396	−	0.950280i	$-0.399204\pi$
$479$	13.6305	0.622791	0.311396	+	0.950280i	$0.399204\pi$
$480$	1.72412	0.0786949
$481$	−17.3194	−0.789697
$482$	0.143227	0.00652381
$483$	−1.82666	−0.0831158
$484$	−7.62137	−0.346426
$485$	5.43989	0.247013
$486$	−0.284892	−0.0129229
$487$	−36.5213	−1.65494	−0.827469	−	0.561512i	$-0.810220\pi$
$487$	−36.5213	−1.65494	−0.827469	+	0.561512i	$0.810220\pi$
$488$	−4.41148	−0.199698
$489$	−12.9881	−0.587342
$490$	−3.21574	−0.145272
$491$	0.632414	0.0285404	0.0142702	−	0.999898i	$-0.495457\pi$
$491$	0.632414	0.0285404	0.0142702	+	0.999898i	$0.495457\pi$
$492$	2.03592	0.0917865
$493$	1.72014	0.0774711
$494$	3.15917	0.142138
$495$	0.0503908	0.00226490
$496$	1.23453	0.0554322
$497$	21.1587	0.949100
$498$	8.80501	0.394562
$499$	27.1807	1.21677	0.608387	−	0.793641i	$-0.291817\pi$
$499$	27.1807	1.21677	0.608387	+	0.793641i	$0.291817\pi$
$500$	−1.00000	−0.0447214
$501$	−32.0453	−1.43168
$502$	−1.97385	−0.0880974
$503$	21.9270	0.977677	0.488839	−	0.872374i	$-0.337421\pi$
$503$	21.9270	0.977677	0.488839	+	0.872374i	$0.337421\pi$
$504$	−0.0533300	−0.00237551
$505$	16.5279	0.735484
$506$	−1.00108	−0.0445036
$507$	−15.1585	−0.673215
$508$	−17.2277	−0.764357
$509$	10.4997	0.465391	0.232696	−	0.972550i	$-0.425245\pi$
$509$	10.4997	0.465391	0.232696	+	0.972550i	$0.425245\pi$
$510$	2.86260	0.126758
$511$	3.94751	0.174628
$512$	−1.00000	−0.0441942
$513$	−8.03852	−0.354909
$514$	4.40506	0.194299
$515$	−11.1161	−0.489834
$516$	−18.7075	−0.823551
$517$	−17.1660	−0.754959
$518$	16.4243	0.721644
$519$	17.2223	0.755977
$520$	−2.05133	−0.0899567
$521$	2.03064	0.0889641	0.0444821	−	0.999010i	$-0.485836\pi$
$521$	2.03064	0.0889641	0.0444821	+	0.999010i	$0.485836\pi$
$522$	0.0284022	0.00124313
$523$	−0.611405	−0.0267349	−0.0133674	−	0.999911i	$-0.504255\pi$
$523$	−0.611405	−0.0267349	−0.0133674	+	0.999911i	$0.504255\pi$
$524$	11.5742	0.505620
$525$	−3.35396	−0.146379
$526$	−30.7542	−1.34095
$527$	2.04973	0.0892875
$528$	3.16911	0.137918
$529$	−22.7034	−0.987104
$530$	3.77155	0.163826
$531$	0.108255	0.00469788
$532$	−2.99590	−0.129889
$533$	−2.42231	−0.104922
$534$	−0.972896	−0.0421013
$535$	5.71380	0.247029
$536$	8.02420	0.346593
$537$	30.1576	1.30140
$538$	−25.4603	−1.09767
$539$	−5.91087	−0.254599
$540$	5.21962	0.224617
$541$	−7.42798	−0.319354	−0.159677	−	0.987169i	$-0.551045\pi$
$541$	−7.42798	−0.319354	−0.159677	+	0.987169i	$0.551045\pi$
$542$	17.3366	0.744671
$543$	−16.4010	−0.703835
$544$	−1.66033	−0.0711859
$545$	17.0725	0.731306
$546$	−6.88007	−0.294440
$547$	−34.7352	−1.48517	−0.742584	−	0.669753i	$-0.766400\pi$
$547$	−34.7352	−1.48517	−0.742584	+	0.669753i	$0.766400\pi$
$548$	−4.60948	−0.196907
$549$	−0.120939	−0.00516154
$550$	−1.83810	−0.0783770
$551$	1.59554	0.0679722
$552$	−0.939003	−0.0399666
$553$	−31.9980	−1.36069
$554$	−29.1086	−1.23670
$555$	−14.5568	−0.617900
$556$	−16.5125	−0.700287
$557$	4.11454	0.174339	0.0871693	−	0.996194i	$-0.472218\pi$
$557$	4.11454	0.174339	0.0871693	+	0.996194i	$0.472218\pi$
$558$	0.0338442	0.00143274
$559$	22.2579	0.941407
$560$	1.94532	0.0822046
$561$	5.26176	0.222151
$562$	−21.3534	−0.900740
$563$	33.1422	1.39678	0.698389	−	0.715719i	$-0.253901\pi$
$563$	33.1422	1.39678	0.698389	+	0.715719i	$0.253901\pi$
$564$	−16.1015	−0.677995
$565$	4.04593	0.170214
$566$	−25.7112	−1.08072
$567$	17.3464	0.728480
$568$	10.8768	0.456379
$569$	9.19828	0.385612	0.192806	−	0.981237i	$-0.438241\pi$
$569$	9.19828	0.385612	0.192806	+	0.981237i	$0.438241\pi$
$570$	2.65524	0.111216
$571$	−10.5379	−0.440998	−0.220499	−	0.975387i	$-0.570769\pi$
$571$	−10.5379	−0.440998	−0.220499	+	0.975387i	$0.570769\pi$
$572$	−3.77056	−0.157655
$573$	26.1342	1.09177
$574$	2.29712	0.0958801
$575$	0.544628	0.0227126
$576$	−0.0274146	−0.00114227
$577$	−30.0687	−1.25178	−0.625888	−	0.779913i	$-0.715263\pi$
$577$	−30.0687	−1.25178	−0.625888	+	0.779913i	$0.715263\pi$
$578$	14.2433	0.592444
$579$	21.0585	0.875163
$580$	−1.03602	−0.0430186
$581$	9.93465	0.412159
$582$	9.37902	0.388773
$583$	6.93251	0.287115
$584$	2.02924	0.0839705
$585$	−0.0562363	−0.00232508
$586$	−18.1782	−0.750937
$587$	−20.0376	−0.827042	−0.413521	−	0.910495i	$-0.635701\pi$
$587$	−20.0376	−0.827042	−0.413521	+	0.910495i	$0.635701\pi$
$588$	−5.54432	−0.228644
$589$	1.90125	0.0783398
$590$	−3.94882	−0.162570
$591$	−45.8565	−1.88628
$592$	8.44302	0.347006
$593$	−33.8460	−1.38989	−0.694945	−	0.719063i	$-0.744571\pi$
$593$	−33.8460	−1.38989	−0.694945	+	0.719063i	$0.744571\pi$
$594$	9.59421	0.393655
$595$	3.22986	0.132411
$596$	6.16182	0.252398
$597$	−29.7228	−1.21647
$598$	1.11721	0.0456861
$599$	−25.9143	−1.05883	−0.529416	−	0.848363i	$-0.677589\pi$
$599$	−25.9143	−1.05883	−0.529416	+	0.848363i	$0.677589\pi$
$600$	−1.72412	−0.0703869
$601$	2.59188	0.105725	0.0528624	−	0.998602i	$-0.483166\pi$
$601$	2.59188	0.105725	0.0528624	+	0.998602i	$0.483166\pi$
$602$	−21.1076	−0.860281
$603$	0.219980	0.00895827
$604$	8.49220	0.345543
$605$	7.62137	0.309853
$606$	28.4961	1.15758
$607$	−19.9746	−0.810744	−0.405372	−	0.914152i	$-0.632858\pi$
$607$	−19.9746	−0.810744	−0.405372	+	0.914152i	$0.632858\pi$
$608$	−1.54006	−0.0624576
$609$	−3.47478	−0.140805
$610$	4.41148	0.178616
$611$	19.1573	0.775020
$612$	−0.0455171	−0.00183992
$613$	−1.89876	−0.0766903	−0.0383451	−	0.999265i	$-0.512209\pi$
$613$	−1.89876	−0.0766903	−0.0383451	+	0.999265i	$0.512209\pi$
$614$	13.6166	0.549521
$615$	−2.03592	−0.0820963
$616$	3.57570	0.144069
$617$	18.2366	0.734178	0.367089	−	0.930186i	$-0.380355\pi$
$617$	18.2366	0.734178	0.367089	+	0.930186i	$0.380355\pi$
$618$	−19.1655	−0.770949
$619$	−9.23342	−0.371122	−0.185561	−	0.982633i	$-0.559410\pi$
$619$	−9.23342	−0.371122	−0.185561	+	0.982633i	$0.559410\pi$
$620$	−1.23453	−0.0495801
$621$	−2.84275	−0.114076
$622$	0.107193	0.00429806
$623$	−1.09771	−0.0439790
$624$	−3.53673	−0.141583
$625$	1.00000	0.0400000
$626$	26.1637	1.04571
$627$	4.88062	0.194913
$628$	−15.8470	−0.632364
$629$	14.0182	0.558940
$630$	0.0533300	0.00212472
$631$	22.8126	0.908154	0.454077	−	0.890963i	$-0.349969\pi$
$631$	22.8126	0.908154	0.454077	+	0.890963i	$0.349969\pi$
$632$	−16.4487	−0.654295
$633$	−31.8631	−1.26644
$634$	14.5197	0.576651
$635$	17.2277	0.683661
$636$	6.50260	0.257845
$637$	6.59655	0.261365
$638$	−1.90432	−0.0753928
$639$	0.298182	0.0117959
$640$	1.00000	0.0395285
$641$	34.8840	1.37784	0.688918	−	0.724839i	$-0.258086\pi$
$641$	34.8840	1.37784	0.688918	+	0.724839i	$0.258086\pi$
$642$	9.85128	0.388799
$643$	−48.9655	−1.93101	−0.965505	−	0.260385i	$-0.916151\pi$
$643$	−48.9655	−1.93101	−0.965505	+	0.260385i	$0.916151\pi$
$644$	−1.05947	−0.0417491
$645$	18.7075	0.736607
$646$	−2.55700	−0.100604
$647$	5.10060	0.200525	0.100263	−	0.994961i	$-0.468032\pi$
$647$	5.10060	0.200525	0.100263	+	0.994961i	$0.468032\pi$
$648$	8.91700	0.350293
$649$	−7.25835	−0.284915
$650$	2.05133	0.0804597
$651$	−4.14057	−0.162282
$652$	−7.53318	−0.295022
$653$	3.64131	0.142495	0.0712477	−	0.997459i	$-0.477302\pi$
$653$	3.64131	0.142495	0.0712477	+	0.997459i	$0.477302\pi$
$654$	29.4350	1.15100
$655$	−11.5742	−0.452240
$656$	1.18085	0.0461044
$657$	0.0556307	0.00217036
$658$	−18.1672	−0.708233
$659$	−8.27962	−0.322528	−0.161264	−	0.986911i	$-0.551557\pi$
$659$	−8.27962	−0.322528	−0.161264	+	0.986911i	$0.551557\pi$
$660$	−3.16911	−0.123357
$661$	−16.2956	−0.633826	−0.316913	−	0.948455i	$-0.602646\pi$
$661$	−16.2956	−0.633826	−0.316913	+	0.948455i	$0.602646\pi$
$662$	13.7827	0.535680
$663$	−5.87213	−0.228055
$664$	5.10696	0.198188
$665$	2.99590	0.116176
$666$	0.231462	0.00896896
$667$	0.564248	0.0218478
$668$	−18.5865	−0.719132
$669$	−31.0431	−1.20020
$670$	−8.02420	−0.310002
$671$	8.10876	0.313035
$672$	3.35396	0.129382
$673$	−24.2733	−0.935668	−0.467834	−	0.883816i	$-0.654965\pi$
$673$	−24.2733	−0.935668	−0.467834	+	0.883816i	$0.654965\pi$
$674$	0.152670	0.00588062
$675$	−5.21962	−0.200903
$676$	−8.79205	−0.338156
$677$	44.0596	1.69335	0.846674	−	0.532112i	$-0.178601\pi$
$677$	44.0596	1.69335	0.846674	+	0.532112i	$0.178601\pi$
$678$	6.97566	0.267899
$679$	10.5823	0.406112
$680$	1.66033	0.0636706
$681$	29.8183	1.14264
$682$	−2.26920	−0.0868922
$683$	−52.0297	−1.99086	−0.995431	−	0.0954884i	$-0.969559\pi$
$683$	−52.0297	−1.99086	−0.995431	+	0.0954884i	$0.969559\pi$
$684$	−0.0422200	−0.00161432
$685$	4.60948	0.176119
$686$	−19.8729	−0.758749
$687$	37.3006	1.42311
$688$	−10.8505	−0.413670
$689$	−7.73669	−0.294744
$690$	0.939003	0.0357472
$691$	46.5556	1.77106	0.885529	−	0.464585i	$-0.153796\pi$
$691$	46.5556	1.77106	0.885529	+	0.464585i	$0.153796\pi$
$692$	9.98907	0.379727
$693$	0.0980261	0.00372371
$694$	31.9305	1.21206
$695$	16.5125	0.626355
$696$	−1.78623	−0.0677068
$697$	1.96059	0.0742627
$698$	0.490583	0.0185689
$699$	−23.2206	−0.878282
$700$	−1.94532	−0.0735261
$701$	1.35967	0.0513541	0.0256771	−	0.999670i	$-0.491826\pi$
$701$	1.35967	0.0513541	0.0256771	+	0.999670i	$0.491826\pi$
$702$	−10.7072	−0.404116
$703$	13.0027	0.490408
$704$	1.83810	0.0692762
$705$	16.1015	0.606417
$706$	−22.3431	−0.840894
$707$	32.1521	1.20920
$708$	−6.80824	−0.255869
$709$	−22.9948	−0.863587	−0.431794	−	0.901972i	$-0.642119\pi$
$709$	−22.9948	−0.863587	−0.431794	+	0.901972i	$0.642119\pi$
$710$	−10.8768	−0.408198
$711$	−0.450935	−0.0169114
$712$	−0.564286	−0.0211475
$713$	0.672361	0.0251801
$714$	5.56866	0.208402
$715$	3.77056	0.141011
$716$	17.4916	0.653691
$717$	18.9977	0.709480
$718$	29.0754	1.08508
$719$	−6.99189	−0.260753	−0.130377	−	0.991465i	$-0.541619\pi$
$719$	−6.99189	−0.260753	−0.130377	+	0.991465i	$0.541619\pi$
$720$	0.0274146	0.00102168
$721$	−21.6244	−0.805333
$722$	16.6282	0.618838
$723$	−0.246940	−0.00918381
$724$	−9.51269	−0.353536
$725$	1.03602	0.0384770
$726$	13.1401	0.487677
$727$	−45.5778	−1.69039	−0.845194	−	0.534459i	$-0.820515\pi$
$727$	−45.5778	−1.69039	−0.845194	+	0.534459i	$0.820515\pi$
$728$	−3.99048	−0.147897
$729$	27.2422	1.00897
$730$	−2.02924	−0.0751055
$731$	−18.0153	−0.666320
$732$	7.60591	0.281123
$733$	−14.8586	−0.548814	−0.274407	−	0.961614i	$-0.588482\pi$
$733$	−14.8586	−0.548814	−0.274407	+	0.961614i	$0.588482\pi$
$734$	−9.83490	−0.363013
$735$	5.54432	0.204505
$736$	−0.544628	−0.0200752
$737$	−14.7493	−0.543298
$738$	0.0323724	0.00119165
$739$	8.85326	0.325672	0.162836	−	0.986653i	$-0.447936\pi$
$739$	8.85326	0.325672	0.162836	+	0.986653i	$0.447936\pi$
$740$	−8.44302	−0.310371
$741$	−5.44678	−0.200092
$742$	7.33686	0.269345
$743$	−33.0448	−1.21229	−0.606147	−	0.795352i	$-0.707286\pi$
$743$	−33.0448	−1.21229	−0.606147	+	0.795352i	$0.707286\pi$
$744$	−2.12848	−0.0780339
$745$	−6.16182	−0.225752
$746$	−16.0683	−0.588302
$747$	0.140005	0.00512252
$748$	3.05185	0.111587
$749$	11.1152	0.406139
$750$	1.72412	0.0629559
$751$	−1.70576	−0.0622442	−0.0311221	−	0.999516i	$-0.509908\pi$
$751$	−1.70576	−0.0622442	−0.0311221	+	0.999516i	$0.509908\pi$
$752$	−9.33896	−0.340557
$753$	3.40316	0.124018
$754$	2.12523	0.0773962
$755$	−8.49220	−0.309063
$756$	10.1538	0.369291
$757$	19.1434	0.695780	0.347890	−	0.937535i	$-0.386898\pi$
$757$	19.1434	0.695780	0.347890	+	0.937535i	$0.386898\pi$
$758$	30.0765	1.09243
$759$	1.72599	0.0626493
$760$	1.54006	0.0558638
$761$	−18.6183	−0.674913	−0.337456	−	0.941341i	$-0.609567\pi$
$761$	−18.6183	−0.674913	−0.337456	+	0.941341i	$0.609567\pi$
$762$	29.7026	1.07601
$763$	33.2114	1.20233
$764$	15.1580	0.548396
$765$	0.0455171	0.00164567
$766$	−19.0125	−0.686951
$767$	8.10033	0.292486
$768$	1.72412	0.0622138
$769$	−25.8580	−0.932462	−0.466231	−	0.884663i	$-0.654388\pi$
$769$	−25.8580	−0.932462	−0.466231	+	0.884663i	$0.654388\pi$
$770$	−3.57570	−0.128859
$771$	−7.59484	−0.273522
$772$	12.2141	0.439595
$773$	10.8182	0.389102	0.194551	−	0.980892i	$-0.437675\pi$
$773$	10.8182	0.389102	0.194551	+	0.980892i	$0.437675\pi$
$774$	−0.297461	−0.0106920
$775$	1.23453	0.0443457
$776$	5.43989	0.195281
$777$	−28.3175	−1.01589
$778$	10.8795	0.390050
$779$	1.81857	0.0651572
$780$	3.53673	0.126635
$781$	−19.9926	−0.715393
$782$	−0.904259	−0.0323363
$783$	−5.40765	−0.193254
$784$	−3.21574	−0.114848
$785$	15.8470	0.565603
$786$	−19.9552	−0.711779
$787$	−1.39603	−0.0497633	−0.0248816	−	0.999690i	$-0.507921\pi$
$787$	−1.39603	−0.0497633	−0.0248816	+	0.999690i	$0.507921\pi$
$788$	−26.5971	−0.947481
$789$	53.0238	1.88770
$790$	16.4487	0.585220
$791$	7.87061	0.279847
$792$	0.0503908	0.00179056
$793$	−9.04939	−0.321353
$794$	13.3201	0.472714
$795$	−6.50260	−0.230624
$796$	−17.2394	−0.611035
$797$	46.8998	1.66128	0.830638	−	0.556813i	$-0.187976\pi$
$797$	46.8998	1.66128	0.830638	+	0.556813i	$0.187976\pi$
$798$	5.16529	0.182849
$799$	−15.5057	−0.548553
$800$	−1.00000	−0.0353553
$801$	−0.0154696	−0.000546593 0
$802$	1.00000	0.0353112
$803$	−3.72995	−0.131627
$804$	−13.8347	−0.487911
$805$	1.05947	0.0373415
$806$	2.53243	0.0892011
$807$	43.8966	1.54523
$808$	16.5279	0.581451
$809$	47.2936	1.66275	0.831377	−	0.555709i	$-0.187553\pi$
$809$	47.2936	1.66275	0.831377	+	0.555709i	$0.187553\pi$
$810$	−8.91700	−0.313312
$811$	28.4006	0.997281	0.498640	−	0.866809i	$-0.333833\pi$
$811$	28.4006	0.997281	0.498640	+	0.866809i	$0.333833\pi$
$812$	−2.01540	−0.0707265
$813$	−29.8904	−1.04830
$814$	−15.5191	−0.543946
$815$	7.53318	0.263876
$816$	2.86260	0.100211
$817$	−16.7104	−0.584621
$818$	−15.4727	−0.540990
$819$	−0.109397	−0.00382265
$820$	−1.18085	−0.0412370
$821$	16.0937	0.561675	0.280837	−	0.959755i	$-0.409388\pi$
$821$	16.0937	0.561675	0.280837	+	0.959755i	$0.409388\pi$
$822$	7.94729	0.277193
$823$	−3.54993	−0.123743	−0.0618714	−	0.998084i	$-0.519707\pi$
$823$	−3.54993	−0.123743	−0.0618714	+	0.998084i	$0.519707\pi$
$824$	−11.1161	−0.387248
$825$	3.16911	0.110334
$826$	−7.68171	−0.267281
$827$	44.2461	1.53859	0.769294	−	0.638895i	$-0.220608\pi$
$827$	44.2461	1.53859	0.769294	+	0.638895i	$0.220608\pi$
$828$	−0.0149307	−0.000518879 0
$829$	27.9947	0.972295	0.486147	−	0.873877i	$-0.338402\pi$
$829$	27.9947	0.972295	0.486147	+	0.873877i	$0.338402\pi$
$830$	−5.10696	−0.177265
$831$	50.1866	1.74096
$832$	−2.05133	−0.0711170
$833$	−5.33918	−0.184992
$834$	28.4695	0.985819
$835$	18.5865	0.643211
$836$	2.83079	0.0979049
$837$	−6.44380	−0.222730
$838$	−30.2832	−1.04612
$839$	−24.2642	−0.837694	−0.418847	−	0.908057i	$-0.637566\pi$
$839$	−24.2642	−0.837694	−0.418847	+	0.908057i	$0.637566\pi$
$840$	−3.35396	−0.115723
$841$	−27.9267	−0.962988
$842$	−29.6085	−1.02038
$843$	36.8158	1.26800
$844$	−18.4808	−0.636135
$845$	8.79205	0.302456
$846$	−0.256024	−0.00880227
$847$	14.8260	0.509427
$848$	3.77155	0.129516
$849$	44.3292	1.52137
$850$	−1.66033	−0.0569487
$851$	4.59830	0.157628
$852$	−18.7528	−0.642461
$853$	5.66471	0.193956	0.0969780	−	0.995287i	$-0.469082\pi$
$853$	5.66471	0.193956	0.0969780	+	0.995287i	$0.469082\pi$
$854$	8.58172	0.293660
$855$	0.0422200	0.00144389
$856$	5.71380	0.195294
$857$	0.292142	0.00997939	0.00498970	−	0.999988i	$-0.498412\pi$
$857$	0.292142	0.00997939	0.00498970	+	0.999988i	$0.498412\pi$
$858$	6.50089	0.221937
$859$	19.6835	0.671592	0.335796	−	0.941935i	$-0.390995\pi$
$859$	19.6835	0.671592	0.335796	+	0.941935i	$0.390995\pi$
$860$	10.8505	0.369998
$861$	−3.96051	−0.134974
$862$	−18.6154	−0.634043
$863$	−37.5654	−1.27874	−0.639371	−	0.768898i	$-0.720805\pi$
$863$	−37.5654	−1.27874	−0.639371	+	0.768898i	$0.720805\pi$
$864$	5.21962	0.177575
$865$	−9.98907	−0.339638
$866$	9.69199	0.329347
$867$	−24.5572	−0.834005
$868$	−2.40156	−0.0815142
$869$	30.2345	1.02563
$870$	1.78623	0.0605588
$871$	16.4603	0.557735
$872$	17.0725	0.578148
$873$	0.149132	0.00504736
$874$	−0.838759	−0.0283714
$875$	1.94532	0.0657637
$876$	−3.49865	−0.118208
$877$	33.1322	1.11879	0.559397	−	0.828900i	$-0.311033\pi$
$877$	33.1322	1.11879	0.559397	+	0.828900i	$0.311033\pi$
$878$	−21.1167	−0.712655
$879$	31.3415	1.05712
$880$	−1.83810	−0.0619625
$881$	53.1407	1.79036	0.895178	−	0.445709i	$-0.147048\pi$
$881$	53.1407	1.79036	0.895178	+	0.445709i	$0.147048\pi$
$882$	−0.0881582	−0.00296844
$883$	7.90991	0.266190	0.133095	−	0.991103i	$-0.457509\pi$
$883$	7.90991	0.266190	0.133095	+	0.991103i	$0.457509\pi$
$884$	−3.40587	−0.114552
$885$	6.80824	0.228856
$886$	13.2897	0.446477
$887$	29.8751	1.00311	0.501554	−	0.865126i	$-0.332762\pi$
$887$	29.8751	1.00311	0.501554	+	0.865126i	$0.332762\pi$
$888$	−14.5568	−0.488493
$889$	33.5134	1.12400
$890$	0.564286	0.0189149
$891$	−16.3904	−0.549099
$892$	−18.0052	−0.602859
$893$	−14.3825	−0.481294
$894$	−10.6237	−0.355310
$895$	−17.4916	−0.584679
$896$	1.94532	0.0649885
$897$	−1.92620	−0.0643141
$898$	−26.0789	−0.870264
$899$	1.27901	0.0426573
$900$	−0.0274146	−0.000913819 0
$901$	6.26200	0.208618
$902$	−2.17052	−0.0722705
$903$	36.3920	1.21105
$904$	4.04593	0.134566
$905$	9.51269	0.316213
$906$	−14.6416	−0.486433
$907$	51.8225	1.72074	0.860369	−	0.509672i	$-0.170233\pi$
$907$	51.8225	1.72074	0.860369	+	0.509672i	$0.170233\pi$
$908$	17.2948	0.573947
$909$	0.453106	0.0150286
$910$	3.99048	0.132283
$911$	35.7953	1.18595	0.592976	−	0.805220i	$-0.297953\pi$
$911$	35.7953	1.18595	0.592976	+	0.805220i	$0.297953\pi$
$912$	2.65524	0.0879239
$913$	−9.38713	−0.310669
$914$	25.0095	0.827240
$915$	−7.60591	−0.251444
$916$	21.6346	0.714828
$917$	−22.5154	−0.743524
$918$	8.66627	0.286030
$919$	−6.60188	−0.217776	−0.108888	−	0.994054i	$-0.534729\pi$
$919$	−6.60188	−0.217776	−0.108888	+	0.994054i	$0.534729\pi$
$920$	0.544628	0.0179558
$921$	−23.4766	−0.773582
$922$	24.9188	0.820657
$923$	22.3118	0.734402
$924$	−6.16492	−0.202811
$925$	8.44302	0.277605
$926$	−36.0122	−1.18344
$927$	−0.304743	−0.0100091
$928$	−1.03602	−0.0340092
$929$	46.8771	1.53799	0.768994	−	0.639256i	$-0.220758\pi$
$929$	46.8771	1.53799	0.768994	+	0.639256i	$0.220758\pi$
$930$	2.12848	0.0697957
$931$	−4.95243	−0.162309
$932$	−13.4681	−0.441161
$933$	−0.184814	−0.00605053
$934$	35.9059	1.17488
$935$	−3.05185	−0.0998062
$936$	−0.0562363	−0.00183814
$937$	40.8058	1.33307	0.666533	−	0.745475i	$-0.267777\pi$
$937$	40.8058	1.33307	0.666533	+	0.745475i	$0.267777\pi$
$938$	−15.6096	−0.509672
$939$	−45.1093	−1.47209
$940$	9.33896	0.304603
$941$	10.7011	0.348846	0.174423	−	0.984671i	$-0.444194\pi$
$941$	10.7011	0.348846	0.174423	+	0.984671i	$0.444194\pi$
$942$	27.3221	0.890202
$943$	0.643123	0.0209430
$944$	−3.94882	−0.128523
$945$	−10.1538	−0.330304
$946$	19.9443	0.648445
$947$	−42.3714	−1.37688	−0.688442	−	0.725291i	$-0.741705\pi$
$947$	−42.3714	−1.37688	−0.688442	+	0.725291i	$0.741705\pi$
$948$	28.3596	0.921076
$949$	4.16264	0.135125
$950$	−1.54006	−0.0499661
$951$	−25.0337	−0.811773
$952$	3.22986	0.104680
$953$	−15.3842	−0.498343	−0.249172	−	0.968459i	$-0.580158\pi$
$953$	−15.3842	−0.498343	−0.249172	+	0.968459i	$0.580158\pi$
$954$	0.103395	0.00334755
$955$	−15.1580	−0.490501
$956$	11.0188	0.356372
$957$	3.28328	0.106133
$958$	−13.6305	−0.440380
$959$	8.96689	0.289556
$960$	−1.72412	−0.0556457
$961$	−29.4759	−0.950836
$962$	17.3194	0.558400
$963$	0.156641	0.00504770
$964$	−0.143227	−0.00461303
$965$	−12.2141	−0.393185
$966$	1.82666	0.0587718
$967$	44.4010	1.42784	0.713919	−	0.700228i	$-0.246918\pi$
$967$	44.4010	1.42784	0.713919	+	0.700228i	$0.246918\pi$
$968$	7.62137	0.244960
$969$	4.40857	0.141624
$970$	−5.43989	−0.174664
$971$	−36.1636	−1.16055	−0.580273	−	0.814422i	$-0.697054\pi$
$971$	−36.1636	−1.16055	−0.580273	+	0.814422i	$0.697054\pi$
$972$	0.284892	0.00913790
$973$	32.1221	1.02979
$974$	36.5213	1.17022
$975$	−3.53673	−0.113266
$976$	4.41148	0.141208
$977$	6.33344	0.202625	0.101312	−	0.994855i	$-0.467696\pi$
$977$	6.33344	0.202625	0.101312	+	0.994855i	$0.467696\pi$
$978$	12.9881	0.415314
$979$	1.03722	0.0331496
$980$	3.21574	0.102723
$981$	0.468036	0.0149432
$982$	−0.632414	−0.0201811
$983$	44.3715	1.41523	0.707615	−	0.706598i	$-0.249771\pi$
$983$	44.3715	1.41523	0.707615	+	0.706598i	$0.249771\pi$
$984$	−2.03592	−0.0649028
$985$	26.5971	0.847453
$986$	−1.72014	−0.0547803
$987$	31.3225	0.997005
$988$	−3.15917	−0.100506
$989$	−5.90947	−0.187910
$990$	−0.0503908	−0.00160153
$991$	45.9359	1.45920	0.729602	−	0.683872i	$-0.239705\pi$
$991$	45.9359	1.45920	0.729602	+	0.683872i	$0.239705\pi$
$992$	−1.23453	−0.0391965
$993$	−23.7630	−0.754097
$994$	−21.1587	−0.671115
$995$	17.2394	0.546527
$996$	−8.80501	−0.278997
$997$	−44.2650	−1.40189	−0.700943	−	0.713217i	$-0.747237\pi$
$997$	−44.2650	−1.40189	−0.700943	+	0.713217i	$0.747237\pi$
$998$	−27.1807	−0.860389
$999$	−44.0694	−1.39429

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4010.2.a.k.1.13	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4010.2.a.k.1.13	✓	15	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.72412 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.72412 q^{6} -1.94532 q^{7} -1.00000 q^{8} -0.0274146 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.72412 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.72412 q^{6} -1.94532 q^{7} -1.00000 q^{8} -0.0274146 q^{9} +1.00000 q^{10} +1.83810 q^{11} +1.72412 q^{12} -2.05133 q^{13} +1.94532 q^{14} -1.72412 q^{15} +1.00000 q^{16} +1.66033 q^{17} +0.0274146 q^{18} +1.54006 q^{19} -1.00000 q^{20} -3.35396 q^{21} -1.83810 q^{22} +0.544628 q^{23} -1.72412 q^{24} +1.00000 q^{25} +2.05133 q^{26} -5.21962 q^{27} -1.94532 q^{28} +1.03602 q^{29} +1.72412 q^{30} +1.23453 q^{31} -1.00000 q^{32} +3.16911 q^{33} -1.66033 q^{34} +1.94532 q^{35} -0.0274146 q^{36} +8.44302 q^{37} -1.54006 q^{38} -3.53673 q^{39} +1.00000 q^{40} +1.18085 q^{41} +3.35396 q^{42} -10.8505 q^{43} +1.83810 q^{44} +0.0274146 q^{45} -0.544628 q^{46} -9.33896 q^{47} +1.72412 q^{48} -3.21574 q^{49} -1.00000 q^{50} +2.86260 q^{51} -2.05133 q^{52} +3.77155 q^{53} +5.21962 q^{54} -1.83810 q^{55} +1.94532 q^{56} +2.65524 q^{57} -1.03602 q^{58} -3.94882 q^{59} -1.72412 q^{60} +4.41148 q^{61} -1.23453 q^{62} +0.0533300 q^{63} +1.00000 q^{64} +2.05133 q^{65} -3.16911 q^{66} -8.02420 q^{67} +1.66033 q^{68} +0.939003 q^{69} -1.94532 q^{70} -10.8768 q^{71} +0.0274146 q^{72} -2.02924 q^{73} -8.44302 q^{74} +1.72412 q^{75} +1.54006 q^{76} -3.57570 q^{77} +3.53673 q^{78} +16.4487 q^{79} -1.00000 q^{80} -8.91700 q^{81} -1.18085 q^{82} -5.10696 q^{83} -3.35396 q^{84} -1.66033 q^{85} +10.8505 q^{86} +1.78623 q^{87} -1.83810 q^{88} +0.564286 q^{89} -0.0274146 q^{90} +3.99048 q^{91} +0.544628 q^{92} +2.12848 q^{93} +9.33896 q^{94} -1.54006 q^{95} -1.72412 q^{96} -5.43989 q^{97} +3.21574 q^{98} -0.0503908 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 15 q^{2} - 6 q^{3} + 15 q^{4} - 15 q^{5} + 6 q^{6} - 5 q^{7} - 15 q^{8} + 19 q^{9}+O(q^{10}) \) 15 * q - 15 * q^2 - 6 * q^3 + 15 * q^4 - 15 * q^5 + 6 * q^6 - 5 * q^7 - 15 * q^8 + 19 * q^9	\( 15 q - 15 q^{2} - 6 q^{3} + 15 q^{4} - 15 q^{5} + 6 q^{6} - 5 q^{7} - 15 q^{8} + 19 q^{9} + 15 q^{10} - 2 q^{11} - 6 q^{12} - 13 q^{13} + 5 q^{14} + 6 q^{15} + 15 q^{16} + 11 q^{17} - 19 q^{18} - 15 q^{19} - 15 q^{20} - 2 q^{21} + 2 q^{22} - 3 q^{23} + 6 q^{24} + 15 q^{25} + 13 q^{26} - 12 q^{27} - 5 q^{28} + 28 q^{29} - 6 q^{30} - 12 q^{31} - 15 q^{32} - 22 q^{33} - 11 q^{34} + 5 q^{35} + 19 q^{36} - 23 q^{37} + 15 q^{38} - 2 q^{39} + 15 q^{40} + 24 q^{41} + 2 q^{42} - 24 q^{43} - 2 q^{44} - 19 q^{45} + 3 q^{46} - 3 q^{47} - 6 q^{48} + 20 q^{49} - 15 q^{50} - 5 q^{51} - 13 q^{52} + 10 q^{53} + 12 q^{54} + 2 q^{55} + 5 q^{56} - 11 q^{57} - 28 q^{58} + 2 q^{59} + 6 q^{60} + 15 q^{61} + 12 q^{62} - 2 q^{63} + 15 q^{64} + 13 q^{65} + 22 q^{66} - 48 q^{67} + 11 q^{68} + 21 q^{69} - 5 q^{70} + 15 q^{71} - 19 q^{72} - 47 q^{73} + 23 q^{74} - 6 q^{75} - 15 q^{76} + 7 q^{77} + 2 q^{78} - 34 q^{79} - 15 q^{80} + 43 q^{81} - 24 q^{82} - 32 q^{83} - 2 q^{84} - 11 q^{85} + 24 q^{86} + 14 q^{87} + 2 q^{88} + 25 q^{89} + 19 q^{90} - 32 q^{91} - 3 q^{92} - 42 q^{93} + 3 q^{94} + 15 q^{95} + 6 q^{96} - 34 q^{97} - 20 q^{98} + 4 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 - 6 * q^3 + 15 * q^4 - 15 * q^5 + 6 * q^6 - 5 * q^7 - 15 * q^8 + 19 * q^9 + 15 * q^10 - 2 * q^11 - 6 * q^12 - 13 * q^13 + 5 * q^14 + 6 * q^15 + 15 * q^16 + 11 * q^17 - 19 * q^18 - 15 * q^19 - 15 * q^20 - 2 * q^21 + 2 * q^22 - 3 * q^23 + 6 * q^24 + 15 * q^25 + 13 * q^26 - 12 * q^27 - 5 * q^28 + 28 * q^29 - 6 * q^30 - 12 * q^31 - 15 * q^32 - 22 * q^33 - 11 * q^34 + 5 * q^35 + 19 * q^36 - 23 * q^37 + 15 * q^38 - 2 * q^39 + 15 * q^40 + 24 * q^41 + 2 * q^42 - 24 * q^43 - 2 * q^44 - 19 * q^45 + 3 * q^46 - 3 * q^47 - 6 * q^48 + 20 * q^49 - 15 * q^50 - 5 * q^51 - 13 * q^52 + 10 * q^53 + 12 * q^54 + 2 * q^55 + 5 * q^56 - 11 * q^57 - 28 * q^58 + 2 * q^59 + 6 * q^60 + 15 * q^61 + 12 * q^62 - 2 * q^63 + 15 * q^64 + 13 * q^65 + 22 * q^66 - 48 * q^67 + 11 * q^68 + 21 * q^69 - 5 * q^70 + 15 * q^71 - 19 * q^72 - 47 * q^73 + 23 * q^74 - 6 * q^75 - 15 * q^76 + 7 * q^77 + 2 * q^78 - 34 * q^79 - 15 * q^80 + 43 * q^81 - 24 * q^82 - 32 * q^83 - 2 * q^84 - 11 * q^85 + 24 * q^86 + 14 * q^87 + 2 * q^88 + 25 * q^89 + 19 * q^90 - 32 * q^91 - 3 * q^92 - 42 * q^93 + 3 * q^94 + 15 * q^95 + 6 * q^96 - 34 * q^97 - 20 * q^98 + 4 * q^99

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