LMFDB - Embedded newform 4010.2.a.k.1.4

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			1.4
Root			$4.22659$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.24378 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.24378 q^{6} -2.96737 q^{7} -1.00000 q^{8} +2.03457 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.24378 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.24378 q^{6} -2.96737 q^{7} -1.00000 q^{8} +2.03457 q^{9} +1.00000 q^{10} +3.78210 q^{11} -2.24378 q^{12} +1.59128 q^{13} +2.96737 q^{14} +2.24378 q^{15} +1.00000 q^{16} -1.08728 q^{17} -2.03457 q^{18} -1.19988 q^{19} -1.00000 q^{20} +6.65813 q^{21} -3.78210 q^{22} -9.20416 q^{23} +2.24378 q^{24} +1.00000 q^{25} -1.59128 q^{26} +2.16623 q^{27} -2.96737 q^{28} +4.24818 q^{29} -2.24378 q^{30} -7.57530 q^{31} -1.00000 q^{32} -8.48621 q^{33} +1.08728 q^{34} +2.96737 q^{35} +2.03457 q^{36} +3.06583 q^{37} +1.19988 q^{38} -3.57049 q^{39} +1.00000 q^{40} -0.377871 q^{41} -6.65813 q^{42} -0.242242 q^{43} +3.78210 q^{44} -2.03457 q^{45} +9.20416 q^{46} +2.92557 q^{47} -2.24378 q^{48} +1.80526 q^{49} -1.00000 q^{50} +2.43961 q^{51} +1.59128 q^{52} +3.63063 q^{53} -2.16623 q^{54} -3.78210 q^{55} +2.96737 q^{56} +2.69227 q^{57} -4.24818 q^{58} +10.1896 q^{59} +2.24378 q^{60} +14.0200 q^{61} +7.57530 q^{62} -6.03730 q^{63} +1.00000 q^{64} -1.59128 q^{65} +8.48621 q^{66} +9.43061 q^{67} -1.08728 q^{68} +20.6521 q^{69} -2.96737 q^{70} -6.13717 q^{71} -2.03457 q^{72} -0.754336 q^{73} -3.06583 q^{74} -2.24378 q^{75} -1.19988 q^{76} -11.2229 q^{77} +3.57049 q^{78} +14.7209 q^{79} -1.00000 q^{80} -10.9642 q^{81} +0.377871 q^{82} +3.77840 q^{83} +6.65813 q^{84} +1.08728 q^{85} +0.242242 q^{86} -9.53199 q^{87} -3.78210 q^{88} +5.67276 q^{89} +2.03457 q^{90} -4.72191 q^{91} -9.20416 q^{92} +16.9973 q^{93} -2.92557 q^{94} +1.19988 q^{95} +2.24378 q^{96} -12.0035 q^{97} -1.80526 q^{98} +7.69492 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.24378	−1.29545	−0.647725	−	0.761875i	$-0.724279\pi$
$3$	−2.24378	−1.29545	−0.647725	+	0.761875i	$0.724279\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.24378	0.916021
$7$	−2.96737	−1.12156	−0.560779	−	0.827965i	$-0.689498\pi$
$7$	−2.96737	−1.12156	−0.560779	+	0.827965i	$0.689498\pi$
$8$	−1.00000	−0.353553
$9$	2.03457	0.678188
$10$	1.00000	0.316228
$11$	3.78210	1.14035	0.570173	−	0.821525i	$-0.306876\pi$
$11$	3.78210	1.14035	0.570173	+	0.821525i	$0.306876\pi$
$12$	−2.24378	−0.647725
$13$	1.59128	0.441342	0.220671	−	0.975348i	$-0.429175\pi$
$13$	1.59128	0.441342	0.220671	+	0.975348i	$0.429175\pi$
$14$	2.96737	0.793062
$15$	2.24378	0.579342
$16$	1.00000	0.250000
$17$	−1.08728	−0.263703	−0.131852	−	0.991269i	$-0.542092\pi$
$17$	−1.08728	−0.263703	−0.131852	+	0.991269i	$0.542092\pi$
$18$	−2.03457	−0.479552
$19$	−1.19988	−0.275272	−0.137636	−	0.990483i	$-0.543950\pi$
$19$	−1.19988	−0.275272	−0.137636	+	0.990483i	$0.543950\pi$
$20$	−1.00000	−0.223607
$21$	6.65813	1.45292
$22$	−3.78210	−0.806346
$23$	−9.20416	−1.91920	−0.959600	−	0.281369i	$-0.909211\pi$
$23$	−9.20416	−1.91920	−0.959600	+	0.281369i	$0.909211\pi$
$24$	2.24378	0.458010
$25$	1.00000	0.200000
$26$	−1.59128	−0.312076
$27$	2.16623	0.416890
$28$	−2.96737	−0.560779
$29$	4.24818	0.788867	0.394433	−	0.918925i	$-0.370941\pi$
$29$	4.24818	0.788867	0.394433	+	0.918925i	$0.370941\pi$
$30$	−2.24378	−0.409657
$31$	−7.57530	−1.36056	−0.680282	−	0.732950i	$-0.738143\pi$
$31$	−7.57530	−1.36056	−0.680282	+	0.732950i	$0.738143\pi$
$32$	−1.00000	−0.176777
$33$	−8.48621	−1.47726
$34$	1.08728	0.186466
$35$	2.96737	0.501576
$36$	2.03457	0.339094
$37$	3.06583	0.504019	0.252009	−	0.967725i	$-0.418909\pi$
$37$	3.06583	0.504019	0.252009	+	0.967725i	$0.418909\pi$
$38$	1.19988	0.194646
$39$	−3.57049	−0.571736
$40$	1.00000	0.158114
$41$	−0.377871	−0.0590135	−0.0295068	−	0.999565i	$-0.509394\pi$
$41$	−0.377871	−0.0590135	−0.0295068	+	0.999565i	$0.509394\pi$
$42$	−6.65813	−1.02737
$43$	−0.242242	−0.0369416	−0.0184708	−	0.999829i	$-0.505880\pi$
$43$	−0.242242	−0.0369416	−0.0184708	+	0.999829i	$0.505880\pi$
$44$	3.78210	0.570173
$45$	−2.03457	−0.303295
$46$	9.20416	1.35708
$47$	2.92557	0.426738	0.213369	−	0.976972i	$-0.431556\pi$
$47$	2.92557	0.426738	0.213369	+	0.976972i	$0.431556\pi$
$48$	−2.24378	−0.323862
$49$	1.80526	0.257894
$50$	−1.00000	−0.141421
$51$	2.43961	0.341614
$52$	1.59128	0.220671
$53$	3.63063	0.498706	0.249353	−	0.968413i	$-0.419782\pi$
$53$	3.63063	0.498706	0.249353	+	0.968413i	$0.419782\pi$
$54$	−2.16623	−0.294786
$55$	−3.78210	−0.509978
$56$	2.96737	0.396531
$57$	2.69227	0.356600
$58$	−4.24818	−0.557813
$59$	10.1896	1.32657	0.663284	−	0.748368i	$-0.269162\pi$
$59$	10.1896	1.32657	0.663284	+	0.748368i	$0.269162\pi$
$60$	2.24378	0.289671
$61$	14.0200	1.79508	0.897539	−	0.440935i	$-0.145353\pi$
$61$	14.0200	1.79508	0.897539	+	0.440935i	$0.145353\pi$
$62$	7.57530	0.962065
$63$	−6.03730	−0.760628
$64$	1.00000	0.125000
$65$	−1.59128	−0.197374
$66$	8.48621	1.04458
$67$	9.43061	1.15213	0.576066	−	0.817403i	$-0.304587\pi$
$67$	9.43061	1.15213	0.576066	+	0.817403i	$0.304587\pi$
$68$	−1.08728	−0.131852
$69$	20.6521	2.48622
$70$	−2.96737	−0.354668
$71$	−6.13717	−0.728349	−0.364174	−	0.931331i	$-0.618649\pi$
$71$	−6.13717	−0.728349	−0.364174	+	0.931331i	$0.618649\pi$
$72$	−2.03457	−0.239776
$73$	−0.754336	−0.0882883	−0.0441441	−	0.999025i	$-0.514056\pi$
$73$	−0.754336	−0.0882883	−0.0441441	+	0.999025i	$0.514056\pi$
$74$	−3.06583	−0.356395
$75$	−2.24378	−0.259090
$76$	−1.19988	−0.137636
$77$	−11.2229	−1.27896
$78$	3.57049	0.404279
$79$	14.7209	1.65623	0.828114	−	0.560560i	$-0.189414\pi$
$79$	14.7209	1.65623	0.828114	+	0.560560i	$0.189414\pi$
$80$	−1.00000	−0.111803
$81$	−10.9642	−1.21825
$82$	0.377871	0.0417289
$83$	3.77840	0.414733	0.207366	−	0.978263i	$-0.433511\pi$
$83$	3.77840	0.414733	0.207366	+	0.978263i	$0.433511\pi$
$84$	6.65813	0.726461
$85$	1.08728	0.117932
$86$	0.242242	0.0261217
$87$	−9.53199	−1.02194
$88$	−3.78210	−0.403173
$89$	5.67276	0.601312	0.300656	−	0.953733i	$-0.402794\pi$
$89$	5.67276	0.601312	0.300656	+	0.953733i	$0.402794\pi$
$90$	2.03457	0.214462
$91$	−4.72191	−0.494991
$92$	−9.20416	−0.959600
$93$	16.9973	1.76254
$94$	−2.92557	−0.301749
$95$	1.19988	0.123105
$96$	2.24378	0.229005
$97$	−12.0035	−1.21877	−0.609383	−	0.792876i	$-0.708583\pi$
$97$	−12.0035	−1.21877	−0.609383	+	0.792876i	$0.708583\pi$
$98$	−1.80526	−0.182359
$99$	7.69492	0.773369
$100$	1.00000	0.100000
$101$	3.63273	0.361470	0.180735	−	0.983532i	$-0.442152\pi$
$101$	3.63273	0.361470	0.180735	+	0.983532i	$0.442152\pi$
$102$	−2.43961	−0.241558
$103$	−5.51088	−0.543003	−0.271501	−	0.962438i	$-0.587520\pi$
$103$	−5.51088	−0.543003	−0.271501	+	0.962438i	$0.587520\pi$
$104$	−1.59128	−0.156038
$105$	−6.65813	−0.649767
$106$	−3.63063	−0.352638
$107$	−15.1815	−1.46765	−0.733827	−	0.679336i	$-0.762268\pi$
$107$	−15.1815	−1.46765	−0.733827	+	0.679336i	$0.762268\pi$
$108$	2.16623	0.208445
$109$	5.50994	0.527756	0.263878	−	0.964556i	$-0.414998\pi$
$109$	5.50994	0.527756	0.263878	+	0.964556i	$0.414998\pi$
$110$	3.78210	0.360609
$111$	−6.87905	−0.652930
$112$	−2.96737	−0.280390
$113$	15.8093	1.48721	0.743605	−	0.668620i	$-0.233114\pi$
$113$	15.8093	1.48721	0.743605	+	0.668620i	$0.233114\pi$
$114$	−2.69227	−0.252155
$115$	9.20416	0.858292
$116$	4.24818	0.394433
$117$	3.23757	0.299313
$118$	−10.1896	−0.938025
$119$	3.22634	0.295759
$120$	−2.24378	−0.204828
$121$	3.30426	0.300387
$122$	−14.0200	−1.26931
$123$	0.847861	0.0764490
$124$	−7.57530	−0.680282
$125$	−1.00000	−0.0894427
$126$	6.03730	0.537845
$127$	−10.7690	−0.955592	−0.477796	−	0.878471i	$-0.658564\pi$
$127$	−10.7690	−0.955592	−0.477796	+	0.878471i	$0.658564\pi$
$128$	−1.00000	−0.0883883
$129$	0.543539	0.0478560
$130$	1.59128	0.139565
$131$	−0.543170	−0.0474569	−0.0237285	−	0.999718i	$-0.507554\pi$
$131$	−0.543170	−0.0474569	−0.0237285	+	0.999718i	$0.507554\pi$
$132$	−8.48621	−0.738630
$133$	3.56049	0.308733
$134$	−9.43061	−0.814680
$135$	−2.16623	−0.186439
$136$	1.08728	0.0932331
$137$	0.150966	0.0128979	0.00644894	−	0.999979i	$-0.497947\pi$
$137$	0.150966	0.0128979	0.00644894	+	0.999979i	$0.497947\pi$
$138$	−20.6521	−1.75803
$139$	−2.83205	−0.240212	−0.120106	−	0.992761i	$-0.538323\pi$
$139$	−2.83205	−0.240212	−0.120106	+	0.992761i	$0.538323\pi$
$140$	2.96737	0.250788
$141$	−6.56434	−0.552818
$142$	6.13717	0.515020
$143$	6.01838	0.503282
$144$	2.03457	0.169547
$145$	−4.24818	−0.352792
$146$	0.754336	0.0624293
$147$	−4.05061	−0.334089
$148$	3.06583	0.252009
$149$	−3.14602	−0.257732	−0.128866	−	0.991662i	$-0.541134\pi$
$149$	−3.14602	−0.257732	−0.128866	+	0.991662i	$0.541134\pi$
$150$	2.24378	0.183204
$151$	1.76751	0.143838	0.0719191	−	0.997410i	$-0.477088\pi$
$151$	1.76751	0.143838	0.0719191	+	0.997410i	$0.477088\pi$
$152$	1.19988	0.0973232
$153$	−2.21213	−0.178840
$154$	11.2229	0.904364
$155$	7.57530	0.608463
$156$	−3.57049	−0.285868
$157$	−2.40237	−0.191730	−0.0958650	−	0.995394i	$-0.530562\pi$
$157$	−2.40237	−0.191730	−0.0958650	+	0.995394i	$0.530562\pi$
$158$	−14.7209	−1.17113
$159$	−8.14636	−0.646048
$160$	1.00000	0.0790569
$161$	27.3121	2.15249
$162$	10.9642	0.861432
$163$	−4.92711	−0.385921	−0.192961	−	0.981207i	$-0.561809\pi$
$163$	−4.92711	−0.385921	−0.192961	+	0.981207i	$0.561809\pi$
$164$	−0.377871	−0.0295068
$165$	8.48621	0.660650
$166$	−3.77840	−0.293260
$167$	4.05202	0.313555	0.156778	−	0.987634i	$-0.449889\pi$
$167$	4.05202	0.313555	0.156778	+	0.987634i	$0.449889\pi$
$168$	−6.65813	−0.513686
$169$	−10.4678	−0.805217
$170$	−1.08728	−0.0833902
$171$	−2.44124	−0.186686
$172$	−0.242242	−0.0184708
$173$	−10.4642	−0.795581	−0.397791	−	0.917476i	$-0.630223\pi$
$173$	−10.4642	−0.795581	−0.397791	+	0.917476i	$0.630223\pi$
$174$	9.53199	0.722619
$175$	−2.96737	−0.224312
$176$	3.78210	0.285086
$177$	−22.8632	−1.71850
$178$	−5.67276	−0.425192
$179$	−11.9204	−0.890973	−0.445486	−	0.895289i	$-0.646969\pi$
$179$	−11.9204	−0.890973	−0.445486	+	0.895289i	$0.646969\pi$
$180$	−2.03457	−0.151648
$181$	9.17128	0.681696	0.340848	−	0.940118i	$-0.389286\pi$
$181$	9.17128	0.681696	0.340848	+	0.940118i	$0.389286\pi$
$182$	4.72191	0.350012
$183$	−31.4579	−2.32543
$184$	9.20416	0.678539
$185$	−3.06583	−0.225404
$186$	−16.9973	−1.24631
$187$	−4.11218	−0.300713
$188$	2.92557	0.213369
$189$	−6.42799	−0.467567
$190$	−1.19988	−0.0870486
$191$	−21.6269	−1.56487	−0.782433	−	0.622735i	$-0.786022\pi$
$191$	−21.6269	−1.56487	−0.782433	+	0.622735i	$0.786022\pi$
$192$	−2.24378	−0.161931
$193$	−26.4655	−1.90503	−0.952513	−	0.304497i	$-0.901512\pi$
$193$	−26.4655	−1.90503	−0.952513	+	0.304497i	$0.901512\pi$
$194$	12.0035	0.861798
$195$	3.57049	0.255688
$196$	1.80526	0.128947
$197$	18.2780	1.30225	0.651126	−	0.758969i	$-0.274297\pi$
$197$	18.2780	1.30225	0.651126	+	0.758969i	$0.274297\pi$
$198$	−7.69492	−0.546854
$199$	20.6980	1.46724	0.733622	−	0.679558i	$-0.237828\pi$
$199$	20.6980	1.46724	0.733622	+	0.679558i	$0.237828\pi$
$200$	−1.00000	−0.0707107
$201$	−21.1602	−1.49253
$202$	−3.63273	−0.255598
$203$	−12.6059	−0.884761
$204$	2.43961	0.170807
$205$	0.377871	0.0263917
$206$	5.51088	0.383961
$207$	−18.7265	−1.30158
$208$	1.59128	0.110336
$209$	−4.53807	−0.313905
$210$	6.65813	0.459454
$211$	−16.9222	−1.16497	−0.582486	−	0.812841i	$-0.697920\pi$
$211$	−16.9222	−1.16497	−0.582486	+	0.812841i	$0.697920\pi$
$212$	3.63063	0.249353
$213$	13.7705	0.943539
$214$	15.1815	1.03779
$215$	0.242242	0.0165208
$216$	−2.16623	−0.147393
$217$	22.4787	1.52595
$218$	−5.50994	−0.373180
$219$	1.69257	0.114373
$220$	−3.78210	−0.254989
$221$	−1.73016	−0.116383
$222$	6.87905	0.461692
$223$	21.3560	1.43011	0.715053	−	0.699071i	$-0.246403\pi$
$223$	21.3560	1.43011	0.715053	+	0.699071i	$0.246403\pi$
$224$	2.96737	0.198265
$225$	2.03457	0.135638
$226$	−15.8093	−1.05162
$227$	28.5483	1.89482	0.947408	−	0.320029i	$-0.103693\pi$
$227$	28.5483	1.89482	0.947408	+	0.320029i	$0.103693\pi$
$228$	2.69227	0.178300
$229$	0.261496	0.0172802	0.00864009	−	0.999963i	$-0.497250\pi$
$229$	0.261496	0.0172802	0.00864009	+	0.999963i	$0.497250\pi$
$230$	−9.20416	−0.606904
$231$	25.1817	1.65683
$232$	−4.24818	−0.278907
$233$	−2.81531	−0.184437	−0.0922187	−	0.995739i	$-0.529396\pi$
$233$	−2.81531	−0.184437	−0.0922187	+	0.995739i	$0.529396\pi$
$234$	−3.23757	−0.211646
$235$	−2.92557	−0.190843
$236$	10.1896	0.663284
$237$	−33.0304	−2.14556
$238$	−3.22634	−0.209133
$239$	−18.5317	−1.19871	−0.599357	−	0.800482i	$-0.704577\pi$
$239$	−18.5317	−1.19871	−0.599357	+	0.800482i	$0.704577\pi$
$240$	2.24378	0.144836
$241$	−15.2647	−0.983283	−0.491642	−	0.870798i	$-0.663603\pi$
$241$	−15.2647	−0.983283	−0.491642	+	0.870798i	$0.663603\pi$
$242$	−3.30426	−0.212406
$243$	18.1027	1.16129
$244$	14.0200	0.897539
$245$	−1.80526	−0.115334
$246$	−0.847861	−0.0540576
$247$	−1.90935	−0.121489
$248$	7.57530	0.481032
$249$	−8.47790	−0.537265
$250$	1.00000	0.0632456
$251$	−14.4528	−0.912253	−0.456127	−	0.889915i	$-0.650764\pi$
$251$	−14.4528	−0.912253	−0.456127	+	0.889915i	$0.650764\pi$
$252$	−6.03730	−0.380314
$253$	−34.8110	−2.18855
$254$	10.7690	0.675706
$255$	−2.43961	−0.152774
$256$	1.00000	0.0625000
$257$	6.55811	0.409083	0.204542	−	0.978858i	$-0.434430\pi$
$257$	6.55811	0.409083	0.204542	+	0.978858i	$0.434430\pi$
$258$	−0.543539	−0.0338393
$259$	−9.09742	−0.565286
$260$	−1.59128	−0.0986871
$261$	8.64320	0.535000
$262$	0.543170	0.0335571
$263$	−1.30528	−0.0804872	−0.0402436	−	0.999190i	$-0.512813\pi$
$263$	−1.30528	−0.0804872	−0.0402436	+	0.999190i	$0.512813\pi$
$264$	8.48621	0.522290
$265$	−3.63063	−0.223028
$266$	−3.56049	−0.218307
$267$	−12.7285	−0.778969
$268$	9.43061	0.576066
$269$	18.5690	1.13217	0.566087	−	0.824345i	$-0.308457\pi$
$269$	18.5690	1.13217	0.566087	+	0.824345i	$0.308457\pi$
$270$	2.16623	0.131832
$271$	−28.2537	−1.71629	−0.858146	−	0.513406i	$-0.828383\pi$
$271$	−28.2537	−1.71629	−0.858146	+	0.513406i	$0.828383\pi$
$272$	−1.08728	−0.0659258
$273$	10.5950	0.641236
$274$	−0.150966	−0.00912018
$275$	3.78210	0.228069
$276$	20.6521	1.24311
$277$	−8.92264	−0.536110	−0.268055	−	0.963404i	$-0.586381\pi$
$277$	−8.92264	−0.536110	−0.268055	+	0.963404i	$0.586381\pi$
$278$	2.83205	0.169855
$279$	−15.4125	−0.922719
$280$	−2.96737	−0.177334
$281$	7.82655	0.466893	0.233447	−	0.972370i	$-0.425000\pi$
$281$	7.82655	0.466893	0.233447	+	0.972370i	$0.425000\pi$
$282$	6.56434	0.390901
$283$	−13.7382	−0.816650	−0.408325	−	0.912837i	$-0.633887\pi$
$283$	−13.7382	−0.816650	−0.408325	+	0.912837i	$0.633887\pi$
$284$	−6.13717	−0.364174
$285$	−2.69227	−0.159477
$286$	−6.01838	−0.355874
$287$	1.12128	0.0661872
$288$	−2.03457	−0.119888
$289$	−15.8178	−0.930461
$290$	4.24818	0.249462
$291$	26.9332	1.57885
$292$	−0.754336	−0.0441441
$293$	18.6529	1.08971	0.544857	−	0.838529i	$-0.316584\pi$
$293$	18.6529	1.08971	0.544857	+	0.838529i	$0.316584\pi$
$294$	4.05061	0.236236
$295$	−10.1896	−0.593259
$296$	−3.06583	−0.178197
$297$	8.19288	0.475399
$298$	3.14602	0.182244
$299$	−14.6464	−0.847023
$300$	−2.24378	−0.129545
$301$	0.718822	0.0414322
$302$	−1.76751	−0.101709
$303$	−8.15105	−0.468266
$304$	−1.19988	−0.0688179
$305$	−14.0200	−0.802783
$306$	2.21213	0.126459
$307$	−6.80077	−0.388140	−0.194070	−	0.980988i	$-0.562169\pi$
$307$	−6.80077	−0.388140	−0.194070	+	0.980988i	$0.562169\pi$
$308$	−11.2229	−0.639482
$309$	12.3652	0.703432
$310$	−7.57530	−0.430248
$311$	3.50135	0.198543	0.0992717	−	0.995060i	$-0.468349\pi$
$311$	3.50135	0.198543	0.0992717	+	0.995060i	$0.468349\pi$
$312$	3.57049	0.202139
$313$	−15.8948	−0.898425	−0.449212	−	0.893425i	$-0.648295\pi$
$313$	−15.8948	−0.898425	−0.449212	+	0.893425i	$0.648295\pi$
$314$	2.40237	0.135574
$315$	6.03730	0.340163
$316$	14.7209	0.828114
$317$	21.6901	1.21824	0.609118	−	0.793080i	$-0.291524\pi$
$317$	21.6901	1.21824	0.609118	+	0.793080i	$0.291524\pi$
$318$	8.14636	0.456825
$319$	16.0670	0.899581
$320$	−1.00000	−0.0559017
$321$	34.0641	1.90127
$322$	−27.3121	−1.52204
$323$	1.30460	0.0725900
$324$	−10.9642	−0.609124
$325$	1.59128	0.0882684
$326$	4.92711	0.272887
$327$	−12.3631	−0.683681
$328$	0.377871	0.0208644
$329$	−8.68123	−0.478612
$330$	−8.48621	−0.467150
$331$	−28.5289	−1.56809	−0.784044	−	0.620705i	$-0.786847\pi$
$331$	−28.5289	−1.56809	−0.784044	+	0.620705i	$0.786847\pi$
$332$	3.77840	0.207366
$333$	6.23762	0.341820
$334$	−4.05202	−0.221717
$335$	−9.43061	−0.515249
$336$	6.65813	0.363231
$337$	−13.8872	−0.756482	−0.378241	−	0.925707i	$-0.623471\pi$
$337$	−13.8872	−0.756482	−0.378241	+	0.925707i	$0.623471\pi$
$338$	10.4678	0.569375
$339$	−35.4725	−1.92660
$340$	1.08728	0.0589658
$341$	−28.6505	−1.55151
$342$	2.44124	0.132007
$343$	15.4147	0.832315
$344$	0.242242	0.0130608
$345$	−20.6521	−1.11187
$346$	10.4642	0.562561
$347$	−10.1110	−0.542785	−0.271393	−	0.962469i	$-0.587484\pi$
$347$	−10.1110	−0.542785	−0.271393	+	0.962469i	$0.587484\pi$
$348$	−9.53199	−0.510968
$349$	7.91140	0.423488	0.211744	−	0.977325i	$-0.432086\pi$
$349$	7.91140	0.423488	0.211744	+	0.977325i	$0.432086\pi$
$350$	2.96737	0.158612
$351$	3.44708	0.183991
$352$	−3.78210	−0.201586
$353$	−18.0971	−0.963210	−0.481605	−	0.876389i	$-0.659946\pi$
$353$	−18.0971	−0.963210	−0.481605	+	0.876389i	$0.659946\pi$
$354$	22.8632	1.21516
$355$	6.13717	0.325727
$356$	5.67276	0.300656
$357$	−7.23922	−0.383140
$358$	11.9204	0.630013
$359$	−7.99822	−0.422130	−0.211065	−	0.977472i	$-0.567693\pi$
$359$	−7.99822	−0.422130	−0.211065	+	0.977472i	$0.567693\pi$
$360$	2.03457	0.107231
$361$	−17.5603	−0.924225
$362$	−9.17128	−0.482032
$363$	−7.41404	−0.389136
$364$	−4.72191	−0.247496
$365$	0.754336	0.0394837
$366$	31.4579	1.64433
$367$	17.2894	0.902498	0.451249	−	0.892398i	$-0.350979\pi$
$367$	17.2894	0.902498	0.451249	+	0.892398i	$0.350979\pi$
$368$	−9.20416	−0.479800
$369$	−0.768803	−0.0400223
$370$	3.06583	0.159385
$371$	−10.7734	−0.559328
$372$	16.9973	0.881271
$373$	−20.5070	−1.06181	−0.530906	−	0.847431i	$-0.678148\pi$
$373$	−20.5070	−1.06181	−0.530906	+	0.847431i	$0.678148\pi$
$374$	4.11218	0.212636
$375$	2.24378	0.115868
$376$	−2.92557	−0.150875
$377$	6.76005	0.348160
$378$	6.42799	0.330620
$379$	13.1084	0.673332	0.336666	−	0.941624i	$-0.390701\pi$
$379$	13.1084	0.673332	0.336666	+	0.941624i	$0.390701\pi$
$380$	1.19988	0.0615526
$381$	24.1633	1.23792
$382$	21.6269	1.10653
$383$	−27.3794	−1.39902	−0.699510	−	0.714623i	$-0.746598\pi$
$383$	−27.3794	−1.39902	−0.699510	+	0.714623i	$0.746598\pi$
$384$	2.24378	0.114503
$385$	11.2229	0.571970
$386$	26.4655	1.34706
$387$	−0.492858	−0.0250534
$388$	−12.0035	−0.609383
$389$	9.70549	0.492088	0.246044	−	0.969259i	$-0.420869\pi$
$389$	9.70549	0.492088	0.246044	+	0.969259i	$0.420869\pi$
$390$	−3.57049	−0.180799
$391$	10.0075	0.506099
$392$	−1.80526	−0.0911793
$393$	1.21875	0.0614781
$394$	−18.2780	−0.920832
$395$	−14.7209	−0.740687
$396$	7.69492	0.386684
$397$	−16.3681	−0.821491	−0.410745	−	0.911750i	$-0.634732\pi$
$397$	−16.3681	−0.821491	−0.410745	+	0.911750i	$0.634732\pi$
$398$	−20.6980	−1.03750
$399$	−7.98896	−0.399948
$400$	1.00000	0.0500000
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	21.1602	1.05538
$403$	−12.0544	−0.600475
$404$	3.63273	0.180735
$405$	10.9642	0.544817
$406$	12.6059	0.625620
$407$	11.5952	0.574755
$408$	−2.43961	−0.120779
$409$	−21.5280	−1.06449	−0.532246	−	0.846590i	$-0.678652\pi$
$409$	−21.5280	−1.06449	−0.532246	+	0.846590i	$0.678652\pi$
$410$	−0.377871	−0.0186617
$411$	−0.338735	−0.0167085
$412$	−5.51088	−0.271501
$413$	−30.2361	−1.48782
$414$	18.7265	0.920355
$415$	−3.77840	−0.185474
$416$	−1.59128	−0.0780190
$417$	6.35451	0.311182
$418$	4.53807	0.221964
$419$	−3.86254	−0.188697	−0.0943486	−	0.995539i	$-0.530077\pi$
$419$	−3.86254	−0.188697	−0.0943486	+	0.995539i	$0.530077\pi$
$420$	−6.65813	−0.324883
$421$	−16.9250	−0.824873	−0.412437	−	0.910986i	$-0.635322\pi$
$421$	−16.9250	−0.824873	−0.412437	+	0.910986i	$0.635322\pi$
$422$	16.9222	0.823760
$423$	5.95226	0.289409
$424$	−3.63063	−0.176319
$425$	−1.08728	−0.0527406
$426$	−13.7705	−0.667183
$427$	−41.6025	−2.01329
$428$	−15.1815	−0.733827
$429$	−13.5039	−0.651977
$430$	−0.242242	−0.0116820
$431$	−9.30456	−0.448185	−0.224093	−	0.974568i	$-0.571942\pi$
$431$	−9.30456	−0.448185	−0.224093	+	0.974568i	$0.571942\pi$
$432$	2.16623	0.104223
$433$	36.6921	1.76331	0.881655	−	0.471894i	$-0.156430\pi$
$433$	36.6921	1.76331	0.881655	+	0.471894i	$0.156430\pi$
$434$	−22.4787	−1.07901
$435$	9.53199	0.457024
$436$	5.50994	0.263878
$437$	11.0439	0.528301
$438$	−1.69257	−0.0808739
$439$	24.8310	1.18512	0.592560	−	0.805526i	$-0.298117\pi$
$439$	24.8310	1.18512	0.592560	+	0.805526i	$0.298117\pi$
$440$	3.78210	0.180304
$441$	3.67292	0.174901
$442$	1.73016	0.0822954
$443$	18.5611	0.881864	0.440932	−	0.897541i	$-0.354648\pi$
$443$	18.5611	0.881864	0.440932	+	0.897541i	$0.354648\pi$
$444$	−6.87905	−0.326465
$445$	−5.67276	−0.268915
$446$	−21.3560	−1.01124
$447$	7.05899	0.333879
$448$	−2.96737	−0.140195
$449$	22.2341	1.04929	0.524647	−	0.851320i	$-0.324197\pi$
$449$	22.2341	1.04929	0.524647	+	0.851320i	$0.324197\pi$
$450$	−2.03457	−0.0959103
$451$	−1.42915	−0.0672958
$452$	15.8093	0.743605
$453$	−3.96592	−0.186335
$454$	−28.5483	−1.33984
$455$	4.72191	0.221367
$456$	−2.69227	−0.126077
$457$	−15.5214	−0.726059	−0.363029	−	0.931778i	$-0.618258\pi$
$457$	−15.5214	−0.726059	−0.363029	+	0.931778i	$0.618258\pi$
$458$	−0.261496	−0.0122189
$459$	−2.35529	−0.109935
$460$	9.20416	0.429146
$461$	−11.1142	−0.517639	−0.258819	−	0.965926i	$-0.583333\pi$
$461$	−11.1142	−0.517639	−0.258819	+	0.965926i	$0.583333\pi$
$462$	−25.1817	−1.17156
$463$	−34.3965	−1.59854	−0.799270	−	0.600972i	$-0.794780\pi$
$463$	−34.3965	−1.59854	−0.799270	+	0.600972i	$0.794780\pi$
$464$	4.24818	0.197217
$465$	−16.9973	−0.788233
$466$	2.81531	0.130417
$467$	−33.5408	−1.55208	−0.776040	−	0.630683i	$-0.782775\pi$
$467$	−33.5408	−1.55208	−0.776040	+	0.630683i	$0.782775\pi$
$468$	3.23757	0.149657
$469$	−27.9841	−1.29218
$470$	2.92557	0.134946
$471$	5.39040	0.248376
$472$	−10.1896	−0.469013
$473$	−0.916184	−0.0421262
$474$	33.0304	1.51714
$475$	−1.19988	−0.0550543
$476$	3.22634	0.147879
$477$	7.38676	0.338217
$478$	18.5317	0.847619
$479$	−4.84612	−0.221425	−0.110712	−	0.993852i	$-0.535313\pi$
$479$	−4.84612	−0.221425	−0.110712	+	0.993852i	$0.535313\pi$
$480$	−2.24378	−0.102414
$481$	4.87859	0.222445
$482$	15.2647	0.695286
$483$	−61.2824	−2.78845
$484$	3.30426	0.150194
$485$	12.0035	0.545049
$486$	−18.1027	−0.821155
$487$	−8.90754	−0.403639	−0.201820	−	0.979423i	$-0.564685\pi$
$487$	−8.90754	−0.403639	−0.201820	+	0.979423i	$0.564685\pi$
$488$	−14.0200	−0.634656
$489$	11.0554	0.499941
$490$	1.80526	0.0815532
$491$	−12.6983	−0.573066	−0.286533	−	0.958070i	$-0.592503\pi$
$491$	−12.6983	−0.573066	−0.286533	+	0.958070i	$0.592503\pi$
$492$	0.847861	0.0382245
$493$	−4.61894	−0.208027
$494$	1.90935	0.0859057
$495$	−7.69492	−0.345861
$496$	−7.57530	−0.340141
$497$	18.2112	0.816886
$498$	8.47790	0.379904
$499$	−22.7707	−1.01935	−0.509677	−	0.860366i	$-0.670235\pi$
$499$	−22.7707	−1.01935	−0.509677	+	0.860366i	$0.670235\pi$
$500$	−1.00000	−0.0447214
$501$	−9.09186	−0.406195
$502$	14.4528	0.645061
$503$	−27.5330	−1.22764	−0.613818	−	0.789447i	$-0.710367\pi$
$503$	−27.5330	−1.22764	−0.613818	+	0.789447i	$0.710367\pi$
$504$	6.03730	0.268923
$505$	−3.63273	−0.161654
$506$	34.8110	1.54754
$507$	23.4875	1.04312
$508$	−10.7690	−0.477796
$509$	−25.8731	−1.14681	−0.573403	−	0.819273i	$-0.694377\pi$
$509$	−25.8731	−1.14681	−0.573403	+	0.819273i	$0.694377\pi$
$510$	2.43961	0.108028
$511$	2.23839	0.0990205
$512$	−1.00000	−0.0441942
$513$	−2.59921	−0.114758
$514$	−6.55811	−0.289266
$515$	5.51088	0.242838
$516$	0.543539	0.0239280
$517$	11.0648	0.486629
$518$	9.09742	0.399718
$519$	23.4795	1.03063
$520$	1.59128	0.0697823
$521$	5.97102	0.261595	0.130798	−	0.991409i	$-0.458246\pi$
$521$	5.97102	0.261595	0.130798	+	0.991409i	$0.458246\pi$
$522$	−8.64320	−0.378302
$523$	−10.7992	−0.472218	−0.236109	−	0.971727i	$-0.575872\pi$
$523$	−10.7992	−0.472218	−0.236109	+	0.971727i	$0.575872\pi$
$524$	−0.543170	−0.0237285
$525$	6.65813	0.290584
$526$	1.30528	0.0569131
$527$	8.23644	0.358785
$528$	−8.48621	−0.369315
$529$	61.7165	2.68333
$530$	3.63063	0.157705
$531$	20.7313	0.899663
$532$	3.56049	0.154367
$533$	−0.601299	−0.0260452
$534$	12.7285	0.550814
$535$	15.1815	0.656355
$536$	−9.43061	−0.407340
$537$	26.7468	1.15421
$538$	−18.5690	−0.800568
$539$	6.82766	0.294088
$540$	−2.16623	−0.0932195
$541$	16.3100	0.701220	0.350610	−	0.936522i	$-0.385974\pi$
$541$	16.3100	0.701220	0.350610	+	0.936522i	$0.385974\pi$
$542$	28.2537	1.21360
$543$	−20.5784	−0.883102
$544$	1.08728	0.0466166
$545$	−5.50994	−0.236020
$546$	−10.5950	−0.453422
$547$	−44.9434	−1.92164	−0.960821	−	0.277170i	$-0.910604\pi$
$547$	−44.9434	−1.92164	−0.960821	+	0.277170i	$0.910604\pi$
$548$	0.150966	0.00644894
$549$	28.5246	1.21740
$550$	−3.78210	−0.161269
$551$	−5.09731	−0.217153
$552$	−20.6521	−0.879013
$553$	−43.6822	−1.85756
$554$	8.92264	0.379087
$555$	6.87905	0.291999
$556$	−2.83205	−0.120106
$557$	24.3539	1.03191	0.515953	−	0.856617i	$-0.327438\pi$
$557$	24.3539	1.03191	0.515953	+	0.856617i	$0.327438\pi$
$558$	15.4125	0.652461
$559$	−0.385476	−0.0163039
$560$	2.96737	0.125394
$561$	9.22685	0.389558
$562$	−7.82655	−0.330143
$563$	−12.2132	−0.514725	−0.257362	−	0.966315i	$-0.582853\pi$
$563$	−12.2132	−0.514725	−0.257362	+	0.966315i	$0.582853\pi$
$564$	−6.56434	−0.276409
$565$	−15.8093	−0.665100
$566$	13.7382	0.577459
$567$	32.5349	1.36634
$568$	6.13717	0.257510
$569$	2.29264	0.0961123	0.0480562	−	0.998845i	$-0.484697\pi$
$569$	2.29264	0.0961123	0.0480562	+	0.998845i	$0.484697\pi$
$570$	2.69227	0.112767
$571$	−9.26401	−0.387687	−0.193843	−	0.981032i	$-0.562095\pi$
$571$	−9.26401	−0.387687	−0.193843	+	0.981032i	$0.562095\pi$
$572$	6.01838	0.251641
$573$	48.5260	2.02720
$574$	−1.12128	−0.0468014
$575$	−9.20416	−0.383840
$576$	2.03457	0.0847736
$577$	6.56066	0.273124	0.136562	−	0.990632i	$-0.456395\pi$
$577$	6.56066	0.273124	0.136562	+	0.990632i	$0.456395\pi$
$578$	15.8178	0.657935
$579$	59.3828	2.46787
$580$	−4.24818	−0.176396
$581$	−11.2119	−0.465147
$582$	−26.9332	−1.11642
$583$	13.7314	0.568697
$584$	0.754336	0.0312146
$585$	−3.23757	−0.133857
$586$	−18.6529	−0.770545
$587$	−19.9120	−0.821855	−0.410928	−	0.911668i	$-0.634795\pi$
$587$	−19.9120	−0.821855	−0.410928	+	0.911668i	$0.634795\pi$
$588$	−4.05061	−0.167044
$589$	9.08947	0.374525
$590$	10.1896	0.419498
$591$	−41.0119	−1.68700
$592$	3.06583	0.126005
$593$	−28.2049	−1.15824	−0.579119	−	0.815243i	$-0.696603\pi$
$593$	−28.2049	−1.15824	−0.579119	+	0.815243i	$0.696603\pi$
$594$	−8.19288	−0.336158
$595$	−3.22634	−0.132267
$596$	−3.14602	−0.128866
$597$	−46.4419	−1.90074
$598$	14.6464	0.598936
$599$	−1.19218	−0.0487113	−0.0243557	−	0.999703i	$-0.507753\pi$
$599$	−1.19218	−0.0487113	−0.0243557	+	0.999703i	$0.507753\pi$
$600$	2.24378	0.0916021
$601$	−7.53346	−0.307296	−0.153648	−	0.988126i	$-0.549102\pi$
$601$	−7.53346	−0.307296	−0.153648	+	0.988126i	$0.549102\pi$
$602$	−0.718822	−0.0292970
$603$	19.1872	0.781363
$604$	1.76751	0.0719191
$605$	−3.30426	−0.134337
$606$	8.15105	0.331114
$607$	17.0915	0.693722	0.346861	−	0.937917i	$-0.387248\pi$
$607$	17.0915	0.693722	0.346861	+	0.937917i	$0.387248\pi$
$608$	1.19988	0.0486616
$609$	28.2849	1.14616
$610$	14.0200	0.567653
$611$	4.65540	0.188337
$612$	−2.21213	−0.0894202
$613$	42.7942	1.72844	0.864221	−	0.503113i	$-0.167812\pi$
$613$	42.7942	1.72844	0.864221	+	0.503113i	$0.167812\pi$
$614$	6.80077	0.274457
$615$	−0.847861	−0.0341891
$616$	11.2229	0.452182
$617$	30.7182	1.23667	0.618333	−	0.785916i	$-0.287808\pi$
$617$	30.7182	1.23667	0.618333	+	0.785916i	$0.287808\pi$
$618$	−12.3652	−0.497402
$619$	16.8722	0.678151	0.339075	−	0.940759i	$-0.389886\pi$
$619$	16.8722	0.678151	0.339075	+	0.940759i	$0.389886\pi$
$620$	7.57530	0.304232
$621$	−19.9383	−0.800096
$622$	−3.50135	−0.140391
$623$	−16.8332	−0.674406
$624$	−3.57049	−0.142934
$625$	1.00000	0.0400000
$626$	15.8948	0.635282
$627$	10.1824	0.406648
$628$	−2.40237	−0.0958650
$629$	−3.33340	−0.132911
$630$	−6.03730	−0.240532
$631$	−23.1968	−0.923452	−0.461726	−	0.887023i	$-0.652770\pi$
$631$	−23.1968	−0.923452	−0.461726	+	0.887023i	$0.652770\pi$
$632$	−14.7209	−0.585565
$633$	37.9698	1.50916
$634$	−21.6901	−0.861423
$635$	10.7690	0.427354
$636$	−8.14636	−0.323024
$637$	2.87267	0.113819
$638$	−16.0670	−0.636100
$639$	−12.4865	−0.493958
$640$	1.00000	0.0395285
$641$	−15.1934	−0.600105	−0.300052	−	0.953923i	$-0.597004\pi$
$641$	−15.1934	−0.600105	−0.300052	+	0.953923i	$0.597004\pi$
$642$	−34.0641	−1.34440
$643$	9.43775	0.372189	0.186094	−	0.982532i	$-0.440417\pi$
$643$	9.43775	0.372189	0.186094	+	0.982532i	$0.440417\pi$
$644$	27.3121	1.07625
$645$	−0.543539	−0.0214018
$646$	−1.30460	−0.0513289
$647$	16.9549	0.666565	0.333283	−	0.942827i	$-0.391844\pi$
$647$	16.9549	0.666565	0.333283	+	0.942827i	$0.391844\pi$
$648$	10.9642	0.430716
$649$	38.5379	1.51275
$650$	−1.59128	−0.0624152
$651$	−50.4373	−1.97680
$652$	−4.92711	−0.192961
$653$	35.5957	1.39297	0.696484	−	0.717573i	$-0.254747\pi$
$653$	35.5957	1.39297	0.696484	+	0.717573i	$0.254747\pi$
$654$	12.3631	0.483436
$655$	0.543170	0.0212234
$656$	−0.377871	−0.0147534
$657$	−1.53475	−0.0598761
$658$	8.68123	0.338430
$659$	8.50494	0.331306	0.165653	−	0.986184i	$-0.447027\pi$
$659$	8.50494	0.331306	0.165653	+	0.986184i	$0.447027\pi$
$660$	8.48621	0.330325
$661$	10.5547	0.410532	0.205266	−	0.978706i	$-0.434194\pi$
$661$	10.5547	0.410532	0.205266	+	0.978706i	$0.434194\pi$
$662$	28.5289	1.10881
$663$	3.88211	0.150769
$664$	−3.77840	−0.146630
$665$	−3.56049	−0.138070
$666$	−6.23762	−0.241703
$667$	−39.1009	−1.51399
$668$	4.05202	0.156778
$669$	−47.9183	−1.85263
$670$	9.43061	0.364336
$671$	53.0250	2.04701
$672$	−6.65813	−0.256843
$673$	−39.5530	−1.52465	−0.762327	−	0.647192i	$-0.775943\pi$
$673$	−39.5530	−1.52465	−0.762327	+	0.647192i	$0.775943\pi$
$674$	13.8872	0.534914
$675$	2.16623	0.0833781
$676$	−10.4678	−0.402609
$677$	27.9532	1.07433	0.537164	−	0.843478i	$-0.319496\pi$
$677$	27.9532	1.07433	0.537164	+	0.843478i	$0.319496\pi$
$678$	35.4725	1.36231
$679$	35.6187	1.36692
$680$	−1.08728	−0.0416951
$681$	−64.0562	−2.45464
$682$	28.6505	1.09709
$683$	17.8758	0.683999	0.342000	−	0.939700i	$-0.388896\pi$
$683$	17.8758	0.683999	0.342000	+	0.939700i	$0.388896\pi$
$684$	−2.44124	−0.0933430
$685$	−0.150966	−0.00576811
$686$	−15.4147	−0.588536
$687$	−0.586741	−0.0223856
$688$	−0.242242	−0.00923540
$689$	5.77736	0.220100
$690$	20.6521	0.786213
$691$	−8.61554	−0.327751	−0.163875	−	0.986481i	$-0.552399\pi$
$691$	−8.61554	−0.327751	−0.163875	+	0.986481i	$0.552399\pi$
$692$	−10.4642	−0.397791
$693$	−22.8337	−0.867379
$694$	10.1110	0.383807
$695$	2.83205	0.107426
$696$	9.53199	0.361309
$697$	0.410850	0.0155621
$698$	−7.91140	−0.299451
$699$	6.31695	0.238929
$700$	−2.96737	−0.112156
$701$	24.2297	0.915143	0.457571	−	0.889173i	$-0.348719\pi$
$701$	24.2297	0.915143	0.457571	+	0.889173i	$0.348719\pi$
$702$	−3.44708	−0.130102
$703$	−3.67863	−0.138742
$704$	3.78210	0.142543
$705$	6.56434	0.247228
$706$	18.0971	0.681092
$707$	−10.7796	−0.405410
$708$	−22.8632	−0.859251
$709$	6.55294	0.246101	0.123050	−	0.992400i	$-0.460732\pi$
$709$	6.55294	0.246101	0.123050	+	0.992400i	$0.460732\pi$
$710$	−6.13717	−0.230324
$711$	29.9506	1.12323
$712$	−5.67276	−0.212596
$713$	69.7243	2.61120
$714$	7.23922	0.270921
$715$	−6.01838	−0.225075
$716$	−11.9204	−0.445486
$717$	41.5811	1.55287
$718$	7.99822	0.298491
$719$	−24.7982	−0.924818	−0.462409	−	0.886667i	$-0.653015\pi$
$719$	−24.7982	−0.924818	−0.462409	+	0.886667i	$0.653015\pi$
$720$	−2.03457	−0.0758238
$721$	16.3528	0.609009
$722$	17.5603	0.653526
$723$	34.2506	1.27379
$724$	9.17128	0.340848
$725$	4.24818	0.157773
$726$	7.41404	0.275161
$727$	−12.7303	−0.472142	−0.236071	−	0.971736i	$-0.575860\pi$
$727$	−12.7303	−0.472142	−0.236071	+	0.971736i	$0.575860\pi$
$728$	4.72191	0.175006
$729$	−7.72583	−0.286142
$730$	−0.754336	−0.0279192
$731$	0.263384	0.00974162
$732$	−31.4579	−1.16272
$733$	−12.6583	−0.467544	−0.233772	−	0.972291i	$-0.575107\pi$
$733$	−12.6583	−0.467544	−0.233772	+	0.972291i	$0.575107\pi$
$734$	−17.2894	−0.638162
$735$	4.05061	0.149409
$736$	9.20416	0.339270
$737$	35.6675	1.31383
$738$	0.768803	0.0283000
$739$	−22.9874	−0.845607	−0.422803	−	0.906221i	$-0.638954\pi$
$739$	−22.9874	−0.845607	−0.422803	+	0.906221i	$0.638954\pi$
$740$	−3.06583	−0.112702
$741$	4.28417	0.157383
$742$	10.7734	0.395505
$743$	5.79413	0.212566	0.106283	−	0.994336i	$-0.466105\pi$
$743$	5.79413	0.212566	0.106283	+	0.994336i	$0.466105\pi$
$744$	−16.9973	−0.623153
$745$	3.14602	0.115261
$746$	20.5070	0.750814
$747$	7.68739	0.281267
$748$	−4.11218	−0.150356
$749$	45.0492	1.64606
$750$	−2.24378	−0.0819314
$751$	−25.6154	−0.934719	−0.467359	−	0.884067i	$-0.654795\pi$
$751$	−25.6154	−0.934719	−0.467359	+	0.884067i	$0.654795\pi$
$752$	2.92557	0.106685
$753$	32.4290	1.18178
$754$	−6.76005	−0.246186
$755$	−1.76751	−0.0643264
$756$	−6.42799	−0.233784
$757$	5.83461	0.212063	0.106031	−	0.994363i	$-0.466186\pi$
$757$	5.83461	0.212063	0.106031	+	0.994363i	$0.466186\pi$
$758$	−13.1084	−0.476118
$759$	78.1084	2.83515
$760$	−1.19988	−0.0435243
$761$	31.0981	1.12730	0.563652	−	0.826012i	$-0.309396\pi$
$761$	31.0981	1.12730	0.563652	+	0.826012i	$0.309396\pi$
$762$	−24.1633	−0.875343
$763$	−16.3500	−0.591910
$764$	−21.6269	−0.782433
$765$	2.21213	0.0799799
$766$	27.3794	0.989257
$767$	16.2145	0.585470
$768$	−2.24378	−0.0809656
$769$	−26.6210	−0.959978	−0.479989	−	0.877274i	$-0.659359\pi$
$769$	−26.6210	−0.959978	−0.479989	+	0.877274i	$0.659359\pi$
$770$	−11.2229	−0.404444
$771$	−14.7150	−0.529947
$772$	−26.4655	−0.952513
$773$	35.6842	1.28347	0.641735	−	0.766926i	$-0.278215\pi$
$773$	35.6842	1.28347	0.641735	+	0.766926i	$0.278215\pi$
$774$	0.492858	0.0177154
$775$	−7.57530	−0.272113
$776$	12.0035	0.430899
$777$	20.4127	0.732300
$778$	−9.70549	−0.347959
$779$	0.453400	0.0162448
$780$	3.57049	0.127844
$781$	−23.2114	−0.830569
$782$	−10.0075	−0.357866
$783$	9.20252	0.328871
$784$	1.80526	0.0644735
$785$	2.40237	0.0857442
$786$	−1.21875	−0.0434716
$787$	−6.90653	−0.246191	−0.123096	−	0.992395i	$-0.539282\pi$
$787$	−6.90653	−0.246191	−0.123096	+	0.992395i	$0.539282\pi$
$788$	18.2780	0.651126
$789$	2.92877	0.104267
$790$	14.7209	0.523745
$791$	−46.9118	−1.66799
$792$	−7.69492	−0.273427
$793$	22.3098	0.792243
$794$	16.3681	0.580882
$795$	8.14636	0.288921
$796$	20.6980	0.733622
$797$	4.88962	0.173199	0.0865996	−	0.996243i	$-0.472400\pi$
$797$	4.88962	0.173199	0.0865996	+	0.996243i	$0.472400\pi$
$798$	7.98896	0.282806
$799$	−3.18090	−0.112532
$800$	−1.00000	−0.0353553
$801$	11.5416	0.407803
$802$	1.00000	0.0353112
$803$	−2.85297	−0.100679
$804$	−21.1602	−0.746264
$805$	−27.3121	−0.962625
$806$	12.0544	0.424600
$807$	−41.6649	−1.46667
$808$	−3.63273	−0.127799
$809$	23.3285	0.820188	0.410094	−	0.912043i	$-0.365496\pi$
$809$	23.3285	0.820188	0.410094	+	0.912043i	$0.365496\pi$
$810$	−10.9642	−0.385244
$811$	3.86206	0.135615	0.0678076	−	0.997698i	$-0.478400\pi$
$811$	3.86206	0.135615	0.0678076	+	0.997698i	$0.478400\pi$
$812$	−12.6059	−0.442380
$813$	63.3952	2.22337
$814$	−11.5952	−0.406413
$815$	4.92711	0.172589
$816$	2.43961	0.0854035
$817$	0.290662	0.0101690
$818$	21.5280	0.752709
$819$	−9.60704	−0.335697
$820$	0.377871	0.0131958
$821$	25.3816	0.885825	0.442912	−	0.896565i	$-0.353945\pi$
$821$	25.3816	0.885825	0.442912	+	0.896565i	$0.353945\pi$
$822$	0.338735	0.0118147
$823$	17.3917	0.606238	0.303119	−	0.952953i	$-0.401972\pi$
$823$	17.3917	0.606238	0.303119	+	0.952953i	$0.401972\pi$
$824$	5.51088	0.191980
$825$	−8.48621	−0.295452
$826$	30.2361	1.05205
$827$	−36.9925	−1.28636	−0.643178	−	0.765716i	$-0.722385\pi$
$827$	−36.9925	−1.28636	−0.643178	+	0.765716i	$0.722385\pi$
$828$	−18.7265	−0.650789
$829$	−35.9495	−1.24858	−0.624289	−	0.781194i	$-0.714611\pi$
$829$	−35.9495	−1.24858	−0.624289	+	0.781194i	$0.714611\pi$
$830$	3.77840	0.131150
$831$	20.0205	0.694503
$832$	1.59128	0.0551678
$833$	−1.96281	−0.0680074
$834$	−6.35451	−0.220039
$835$	−4.05202	−0.140226
$836$	−4.53807	−0.156952
$837$	−16.4098	−0.567207
$838$	3.86254	0.133429
$839$	50.3204	1.73725	0.868627	−	0.495467i	$-0.165003\pi$
$839$	50.3204	1.73725	0.868627	+	0.495467i	$0.165003\pi$
$840$	6.65813	0.229727
$841$	−10.9530	−0.377689
$842$	16.9250	0.583274
$843$	−17.5611	−0.604836
$844$	−16.9222	−0.582486
$845$	10.4678	0.360104
$846$	−5.95226	−0.204643
$847$	−9.80494	−0.336902
$848$	3.63063	0.124676
$849$	30.8255	1.05793
$850$	1.08728	0.0372932
$851$	−28.2183	−0.967312
$852$	13.7705	0.471769
$853$	−29.0949	−0.996189	−0.498095	−	0.867123i	$-0.665967\pi$
$853$	−29.0949	−0.996189	−0.498095	+	0.867123i	$0.665967\pi$
$854$	41.6025	1.42361
$855$	2.44124	0.0834886
$856$	15.1815	0.518894
$857$	53.6120	1.83135	0.915676	−	0.401918i	$-0.131656\pi$
$857$	53.6120	1.83135	0.915676	+	0.401918i	$0.131656\pi$
$858$	13.5039	0.461017
$859$	−17.8064	−0.607547	−0.303774	−	0.952744i	$-0.598247\pi$
$859$	−17.8064	−0.607547	−0.303774	+	0.952744i	$0.598247\pi$
$860$	0.242242	0.00826040
$861$	−2.51591	−0.0857421
$862$	9.30456	0.316915
$863$	−22.2473	−0.757306	−0.378653	−	0.925539i	$-0.623613\pi$
$863$	−22.2473	−0.757306	−0.378653	+	0.925539i	$0.623613\pi$
$864$	−2.16623	−0.0736965
$865$	10.4642	0.355795
$866$	−36.6921	−1.24685
$867$	35.4918	1.20536
$868$	22.4787	0.762977
$869$	55.6758	1.88867
$870$	−9.53199	−0.323165
$871$	15.0068	0.508484
$872$	−5.50994	−0.186590
$873$	−24.4218	−0.826554
$874$	−11.0439	−0.373565
$875$	2.96737	0.100315
$876$	1.69257	0.0571865
$877$	−0.116856	−0.00394594	−0.00197297	−	0.999998i	$-0.500628\pi$
$877$	−0.116856	−0.00394594	−0.00197297	+	0.999998i	$0.500628\pi$
$878$	−24.8310	−0.838006
$879$	−41.8531	−1.41167
$880$	−3.78210	−0.127494
$881$	43.9679	1.48131	0.740657	−	0.671883i	$-0.234514\pi$
$881$	43.9679	1.48131	0.740657	+	0.671883i	$0.234514\pi$
$882$	−3.67292	−0.123674
$883$	−8.04145	−0.270616	−0.135308	−	0.990804i	$-0.543202\pi$
$883$	−8.04145	−0.270616	−0.135308	+	0.990804i	$0.543202\pi$
$884$	−1.73016	−0.0581916
$885$	22.8632	0.768537
$886$	−18.5611	−0.623572
$887$	19.6128	0.658533	0.329267	−	0.944237i	$-0.393199\pi$
$887$	19.6128	0.658533	0.329267	+	0.944237i	$0.393199\pi$
$888$	6.87905	0.230846
$889$	31.9555	1.07175
$890$	5.67276	0.190151
$891$	−41.4678	−1.38922
$892$	21.3560	0.715053
$893$	−3.51034	−0.117469
$894$	−7.05899	−0.236088
$895$	11.9204	0.398455
$896$	2.96737	0.0991327
$897$	32.8634	1.09728
$898$	−22.2341	−0.741962
$899$	−32.1812	−1.07330
$900$	2.03457	0.0678188
$901$	−3.94750	−0.131510
$902$	1.42915	0.0475853
$903$	−1.61288	−0.0536733
$904$	−15.8093	−0.525808
$905$	−9.17128	−0.304864
$906$	3.96592	0.131759
$907$	17.0659	0.566662	0.283331	−	0.959022i	$-0.408560\pi$
$907$	17.0659	0.566662	0.283331	+	0.959022i	$0.408560\pi$
$908$	28.5483	0.947408
$909$	7.39102	0.245145
$910$	−4.72191	−0.156530
$911$	−3.20485	−0.106181	−0.0530907	−	0.998590i	$-0.516907\pi$
$911$	−3.20485	−0.106181	−0.0530907	+	0.998590i	$0.516907\pi$
$912$	2.69227	0.0891501
$913$	14.2903	0.472939
$914$	15.5214	0.513401
$915$	31.4579	1.03996
$916$	0.261496	0.00864009
$917$	1.61178	0.0532257
$918$	2.35529	0.0777360
$919$	34.9045	1.15139	0.575697	−	0.817663i	$-0.304731\pi$
$919$	34.9045	1.15139	0.575697	+	0.817663i	$0.304731\pi$
$920$	−9.20416	−0.303452
$921$	15.2595	0.502816
$922$	11.1142	0.366026
$923$	−9.76597	−0.321451
$924$	25.1817	0.828416
$925$	3.06583	0.100804
$926$	34.3965	1.13034
$927$	−11.2122	−0.368258
$928$	−4.24818	−0.139453
$929$	−55.6170	−1.82473	−0.912367	−	0.409373i	$-0.865747\pi$
$929$	−55.6170	−1.82473	−0.912367	+	0.409373i	$0.865747\pi$
$930$	16.9973	0.557365
$931$	−2.16610	−0.0709909
$932$	−2.81531	−0.0922187
$933$	−7.85627	−0.257203
$934$	33.5408	1.09749
$935$	4.11218	0.134483
$936$	−3.23757	−0.105823
$937$	−9.00724	−0.294254	−0.147127	−	0.989118i	$-0.547003\pi$
$937$	−9.00724	−0.294254	−0.147127	+	0.989118i	$0.547003\pi$
$938$	27.9841	0.913712
$939$	35.6644	1.16386
$940$	−2.92557	−0.0954215
$941$	−61.0321	−1.98959	−0.994795	−	0.101896i	$-0.967509\pi$
$941$	−61.0321	−1.98959	−0.994795	+	0.101896i	$0.967509\pi$
$942$	−5.39040	−0.175629
$943$	3.47798	0.113259
$944$	10.1896	0.331642
$945$	6.42799	0.209102
$946$	0.916184	0.0297877
$947$	27.6805	0.899496	0.449748	−	0.893156i	$-0.351514\pi$
$947$	27.6805	0.899496	0.449748	+	0.893156i	$0.351514\pi$
$948$	−33.0304	−1.07278
$949$	−1.20036	−0.0389653
$950$	1.19988	0.0389293
$951$	−48.6678	−1.57816
$952$	−3.22634	−0.104566
$953$	−43.8513	−1.42048	−0.710242	−	0.703958i	$-0.751414\pi$
$953$	−43.8513	−1.42048	−0.710242	+	0.703958i	$0.751414\pi$
$954$	−7.38676	−0.239155
$955$	21.6269	0.699829
$956$	−18.5317	−0.599357
$957$	−36.0509	−1.16536
$958$	4.84612	0.156571
$959$	−0.447971	−0.0144657
$960$	2.24378	0.0724178
$961$	26.3852	0.851137
$962$	−4.87859	−0.157292
$963$	−30.8878	−0.995346
$964$	−15.2647	−0.491642
$965$	26.4655	0.851954
$966$	61.2824	1.97173
$967$	−31.0514	−0.998546	−0.499273	−	0.866445i	$-0.666399\pi$
$967$	−31.0514	−0.998546	−0.499273	+	0.866445i	$0.666399\pi$
$968$	−3.30426	−0.106203
$969$	−2.92724	−0.0940366
$970$	−12.0035	−0.385408
$971$	37.2884	1.19664	0.598321	−	0.801257i	$-0.295835\pi$
$971$	37.2884	1.19664	0.598321	+	0.801257i	$0.295835\pi$
$972$	18.1027	0.580644
$973$	8.40374	0.269411
$974$	8.90754	0.285416
$975$	−3.57049	−0.114347
$976$	14.0200	0.448769
$977$	−5.67674	−0.181615	−0.0908075	−	0.995868i	$-0.528945\pi$
$977$	−5.67674	−0.181615	−0.0908075	+	0.995868i	$0.528945\pi$
$978$	−11.0554	−0.353512
$979$	21.4549	0.685703
$980$	−1.80526	−0.0576669
$981$	11.2103	0.357918
$982$	12.6983	0.405219
$983$	45.5812	1.45381	0.726907	−	0.686736i	$-0.240957\pi$
$983$	45.5812	1.45381	0.726907	+	0.686736i	$0.240957\pi$
$984$	−0.847861	−0.0270288
$985$	−18.2780	−0.582385
$986$	4.61894	0.147097
$987$	19.4788	0.620017
$988$	−1.90935	−0.0607445
$989$	2.22964	0.0708983
$990$	7.69492	0.244561
$991$	−5.13391	−0.163084	−0.0815420	−	0.996670i	$-0.525984\pi$
$991$	−5.13391	−0.163084	−0.0815420	+	0.996670i	$0.525984\pi$
$992$	7.57530	0.240516
$993$	64.0126	2.03138
$994$	−18.2112	−0.577625
$995$	−20.6980	−0.656171
$996$	−8.47790	−0.268633
$997$	60.1615	1.90533	0.952666	−	0.304017i	$-0.0983281\pi$
$997$	60.1615	1.90533	0.952666	+	0.304017i	$0.0983281\pi$
$998$	22.7707	0.720793
$999$	6.64127	0.210121

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4010.2.a.k.1.4	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4010.2.a.k.1.4	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.24378 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.24378 q^{6} -2.96737 q^{7} -1.00000 q^{8} +2.03457 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.24378 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.24378 q^{6} -2.96737 q^{7} -1.00000 q^{8} +2.03457 q^{9} +1.00000 q^{10} +3.78210 q^{11} -2.24378 q^{12} +1.59128 q^{13} +2.96737 q^{14} +2.24378 q^{15} +1.00000 q^{16} -1.08728 q^{17} -2.03457 q^{18} -1.19988 q^{19} -1.00000 q^{20} +6.65813 q^{21} -3.78210 q^{22} -9.20416 q^{23} +2.24378 q^{24} +1.00000 q^{25} -1.59128 q^{26} +2.16623 q^{27} -2.96737 q^{28} +4.24818 q^{29} -2.24378 q^{30} -7.57530 q^{31} -1.00000 q^{32} -8.48621 q^{33} +1.08728 q^{34} +2.96737 q^{35} +2.03457 q^{36} +3.06583 q^{37} +1.19988 q^{38} -3.57049 q^{39} +1.00000 q^{40} -0.377871 q^{41} -6.65813 q^{42} -0.242242 q^{43} +3.78210 q^{44} -2.03457 q^{45} +9.20416 q^{46} +2.92557 q^{47} -2.24378 q^{48} +1.80526 q^{49} -1.00000 q^{50} +2.43961 q^{51} +1.59128 q^{52} +3.63063 q^{53} -2.16623 q^{54} -3.78210 q^{55} +2.96737 q^{56} +2.69227 q^{57} -4.24818 q^{58} +10.1896 q^{59} +2.24378 q^{60} +14.0200 q^{61} +7.57530 q^{62} -6.03730 q^{63} +1.00000 q^{64} -1.59128 q^{65} +8.48621 q^{66} +9.43061 q^{67} -1.08728 q^{68} +20.6521 q^{69} -2.96737 q^{70} -6.13717 q^{71} -2.03457 q^{72} -0.754336 q^{73} -3.06583 q^{74} -2.24378 q^{75} -1.19988 q^{76} -11.2229 q^{77} +3.57049 q^{78} +14.7209 q^{79} -1.00000 q^{80} -10.9642 q^{81} +0.377871 q^{82} +3.77840 q^{83} +6.65813 q^{84} +1.08728 q^{85} +0.242242 q^{86} -9.53199 q^{87} -3.78210 q^{88} +5.67276 q^{89} +2.03457 q^{90} -4.72191 q^{91} -9.20416 q^{92} +16.9973 q^{93} -2.92557 q^{94} +1.19988 q^{95} +2.24378 q^{96} -12.0035 q^{97} -1.80526 q^{98} +7.69492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 15 q^{2} - 6 q^{3} + 15 q^{4} - 15 q^{5} + 6 q^{6} - 5 q^{7} - 15 q^{8} + 19 q^{9}+O(q^{10}) \) 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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