LMFDB - Embedded newform 4010.2.a.k.1.6

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			1.6
Root			$-1.71302$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.21181 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.21181 q^{6} +0.441519 q^{7} -1.00000 q^{8} -1.53151 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.21181 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.21181 q^{6} +0.441519 q^{7} -1.00000 q^{8} -1.53151 q^{9} +1.00000 q^{10} +0.781934 q^{11} -1.21181 q^{12} -4.67597 q^{13} -0.441519 q^{14} +1.21181 q^{15} +1.00000 q^{16} +0.347476 q^{17} +1.53151 q^{18} +0.947001 q^{19} -1.00000 q^{20} -0.535038 q^{21} -0.781934 q^{22} -2.26644 q^{23} +1.21181 q^{24} +1.00000 q^{25} +4.67597 q^{26} +5.49134 q^{27} +0.441519 q^{28} +8.38155 q^{29} -1.21181 q^{30} +7.83398 q^{31} -1.00000 q^{32} -0.947558 q^{33} -0.347476 q^{34} -0.441519 q^{35} -1.53151 q^{36} +7.03784 q^{37} -0.947001 q^{38} +5.66640 q^{39} +1.00000 q^{40} -4.97470 q^{41} +0.535038 q^{42} +1.53149 q^{43} +0.781934 q^{44} +1.53151 q^{45} +2.26644 q^{46} +2.09399 q^{47} -1.21181 q^{48} -6.80506 q^{49} -1.00000 q^{50} -0.421076 q^{51} -4.67597 q^{52} -9.21875 q^{53} -5.49134 q^{54} -0.781934 q^{55} -0.441519 q^{56} -1.14759 q^{57} -8.38155 q^{58} +7.20809 q^{59} +1.21181 q^{60} -13.0437 q^{61} -7.83398 q^{62} -0.676190 q^{63} +1.00000 q^{64} +4.67597 q^{65} +0.947558 q^{66} -11.2388 q^{67} +0.347476 q^{68} +2.74650 q^{69} +0.441519 q^{70} +9.64673 q^{71} +1.53151 q^{72} -5.47699 q^{73} -7.03784 q^{74} -1.21181 q^{75} +0.947001 q^{76} +0.345238 q^{77} -5.66640 q^{78} -1.92618 q^{79} -1.00000 q^{80} -2.05995 q^{81} +4.97470 q^{82} +8.36611 q^{83} -0.535038 q^{84} -0.347476 q^{85} -1.53149 q^{86} -10.1569 q^{87} -0.781934 q^{88} +7.27170 q^{89} -1.53151 q^{90} -2.06453 q^{91} -2.26644 q^{92} -9.49332 q^{93} -2.09399 q^{94} -0.947001 q^{95} +1.21181 q^{96} +11.6018 q^{97} +6.80506 q^{98} -1.19754 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.21181	−0.699641	−0.349820	−	0.936817i	$-0.613757\pi$
$3$	−1.21181	−0.699641	−0.349820	+	0.936817i	$0.613757\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.21181	0.494721
$7$	0.441519	0.166878	0.0834392	−	0.996513i	$-0.473410\pi$
$7$	0.441519	0.166878	0.0834392	+	0.996513i	$0.473410\pi$
$8$	−1.00000	−0.353553
$9$	−1.53151	−0.510503
$10$	1.00000	0.316228
$11$	0.781934	0.235762	0.117881	−	0.993028i	$-0.462390\pi$
$11$	0.781934	0.235762	0.117881	+	0.993028i	$0.462390\pi$
$12$	−1.21181	−0.349820
$13$	−4.67597	−1.29688	−0.648441	−	0.761265i	$-0.724579\pi$
$13$	−4.67597	−1.29688	−0.648441	+	0.761265i	$0.724579\pi$
$14$	−0.441519	−0.118001
$15$	1.21181	0.312889
$16$	1.00000	0.250000
$17$	0.347476	0.0842753	0.0421376	−	0.999112i	$-0.486583\pi$
$17$	0.347476	0.0842753	0.0421376	+	0.999112i	$0.486583\pi$
$18$	1.53151	0.360980
$19$	0.947001	0.217257	0.108628	−	0.994082i	$-0.465354\pi$
$19$	0.947001	0.217257	0.108628	+	0.994082i	$0.465354\pi$
$20$	−1.00000	−0.223607
$21$	−0.535038	−0.116755
$22$	−0.781934	−0.166709
$23$	−2.26644	−0.472585	−0.236292	−	0.971682i	$-0.575932\pi$
$23$	−2.26644	−0.472585	−0.236292	+	0.971682i	$0.575932\pi$
$24$	1.21181	0.247360
$25$	1.00000	0.200000
$26$	4.67597	0.917034
$27$	5.49134	1.05681
$28$	0.441519	0.0834392
$29$	8.38155	1.55642	0.778208	−	0.628007i	$-0.216129\pi$
$29$	8.38155	1.55642	0.778208	+	0.628007i	$0.216129\pi$
$30$	−1.21181	−0.221246
$31$	7.83398	1.40703	0.703513	−	0.710683i	$-0.251614\pi$
$31$	7.83398	1.40703	0.703513	+	0.710683i	$0.251614\pi$
$32$	−1.00000	−0.176777
$33$	−0.947558	−0.164949
$34$	−0.347476	−0.0595916
$35$	−0.441519	−0.0746303
$36$	−1.53151	−0.255252
$37$	7.03784	1.15701	0.578507	−	0.815678i	$-0.303636\pi$
$37$	7.03784	1.15701	0.578507	+	0.815678i	$0.303636\pi$
$38$	−0.947001	−0.153624
$39$	5.66640	0.907351
$40$	1.00000	0.158114
$41$	−4.97470	−0.776918	−0.388459	−	0.921466i	$-0.626993\pi$
$41$	−4.97470	−0.776918	−0.388459	+	0.921466i	$0.626993\pi$
$42$	0.535038	0.0825582
$43$	1.53149	0.233550	0.116775	−	0.993158i	$-0.462744\pi$
$43$	1.53149	0.233550	0.116775	+	0.993158i	$0.462744\pi$
$44$	0.781934	0.117881
$45$	1.53151	0.228304
$46$	2.26644	0.334168
$47$	2.09399	0.305440	0.152720	−	0.988270i	$-0.451197\pi$
$47$	2.09399	0.305440	0.152720	+	0.988270i	$0.451197\pi$
$48$	−1.21181	−0.174910
$49$	−6.80506	−0.972152
$50$	−1.00000	−0.141421
$51$	−0.421076	−0.0589624
$52$	−4.67597	−0.648441
$53$	−9.21875	−1.26629	−0.633147	−	0.774032i	$-0.718237\pi$
$53$	−9.21875	−1.26629	−0.633147	+	0.774032i	$0.718237\pi$
$54$	−5.49134	−0.747277
$55$	−0.781934	−0.105436
$56$	−0.441519	−0.0590004
$57$	−1.14759	−0.152002
$58$	−8.38155	−1.10055
$59$	7.20809	0.938414	0.469207	−	0.883088i	$-0.344540\pi$
$59$	7.20809	0.938414	0.469207	+	0.883088i	$0.344540\pi$
$60$	1.21181	0.156444
$61$	−13.0437	−1.67008	−0.835038	−	0.550193i	$-0.814554\pi$
$61$	−13.0437	−1.67008	−0.835038	+	0.550193i	$0.814554\pi$
$62$	−7.83398	−0.994917
$63$	−0.676190	−0.0851919
$64$	1.00000	0.125000
$65$	4.67597	0.579983
$66$	0.947558	0.116636
$67$	−11.2388	−1.37304	−0.686522	−	0.727109i	$-0.740863\pi$
$67$	−11.2388	−1.37304	−0.686522	+	0.727109i	$0.740863\pi$
$68$	0.347476	0.0421376
$69$	2.74650	0.330639
$70$	0.441519	0.0527716
$71$	9.64673	1.14486	0.572428	−	0.819955i	$-0.306001\pi$
$71$	9.64673	1.14486	0.572428	+	0.819955i	$0.306001\pi$
$72$	1.53151	0.180490
$73$	−5.47699	−0.641034	−0.320517	−	0.947243i	$-0.603857\pi$
$73$	−5.47699	−0.641034	−0.320517	+	0.947243i	$0.603857\pi$
$74$	−7.03784	−0.818132
$75$	−1.21181	−0.139928
$76$	0.947001	0.108628
$77$	0.345238	0.0393436
$78$	−5.66640	−0.641594
$79$	−1.92618	−0.216712	−0.108356	−	0.994112i	$-0.534559\pi$
$79$	−1.92618	−0.216712	−0.108356	+	0.994112i	$0.534559\pi$
$80$	−1.00000	−0.111803
$81$	−2.05995	−0.228883
$82$	4.97470	0.549364
$83$	8.36611	0.918300	0.459150	−	0.888359i	$-0.348154\pi$
$83$	8.36611	0.918300	0.459150	+	0.888359i	$0.348154\pi$
$84$	−0.535038	−0.0583774
$85$	−0.347476	−0.0376891
$86$	−1.53149	−0.165144
$87$	−10.1569	−1.08893
$88$	−0.781934	−0.0833544
$89$	7.27170	0.770798	0.385399	−	0.922750i	$-0.374064\pi$
$89$	7.27170	0.770798	0.385399	+	0.922750i	$0.374064\pi$
$90$	−1.53151	−0.161435
$91$	−2.06453	−0.216421
$92$	−2.26644	−0.236292
$93$	−9.49332	−0.984412
$94$	−2.09399	−0.215978
$95$	−0.947001	−0.0971602
$96$	1.21181	0.123680
$97$	11.6018	1.17798	0.588992	−	0.808139i	$-0.299525\pi$
$97$	11.6018	1.17798	0.588992	+	0.808139i	$0.299525\pi$
$98$	6.80506	0.687415
$99$	−1.19754	−0.120357
$100$	1.00000	0.100000
$101$	1.12415	0.111857	0.0559286	−	0.998435i	$-0.482188\pi$
$101$	1.12415	0.111857	0.0559286	+	0.998435i	$0.482188\pi$
$102$	0.421076	0.0416927
$103$	−4.02781	−0.396872	−0.198436	−	0.980114i	$-0.563586\pi$
$103$	−4.02781	−0.396872	−0.198436	+	0.980114i	$0.563586\pi$
$104$	4.67597	0.458517
$105$	0.535038	0.0522144
$106$	9.21875	0.895405
$107$	0.728772	0.0704530	0.0352265	−	0.999379i	$-0.488785\pi$
$107$	0.728772	0.0704530	0.0352265	+	0.999379i	$0.488785\pi$
$108$	5.49134	0.528405
$109$	9.42649	0.902894	0.451447	−	0.892298i	$-0.350908\pi$
$109$	9.42649	0.902894	0.451447	+	0.892298i	$0.350908\pi$
$110$	0.781934	0.0745545
$111$	−8.52855	−0.809494
$112$	0.441519	0.0417196
$113$	−3.90239	−0.367106	−0.183553	−	0.983010i	$-0.558760\pi$
$113$	−3.90239	−0.367106	−0.183553	+	0.983010i	$0.558760\pi$
$114$	1.14759	0.107481
$115$	2.26644	0.211346
$116$	8.38155	0.778208
$117$	7.16130	0.662062
$118$	−7.20809	−0.663559
$119$	0.153417	0.0140637
$120$	−1.21181	−0.110623
$121$	−10.3886	−0.944416
$122$	13.0437	1.18092
$123$	6.02841	0.543563
$124$	7.83398	0.703513
$125$	−1.00000	−0.0894427
$126$	0.676190	0.0602398
$127$	−17.5384	−1.55628	−0.778139	−	0.628092i	$-0.783836\pi$
$127$	−17.5384	−1.55628	−0.778139	+	0.628092i	$0.783836\pi$
$128$	−1.00000	−0.0883883
$129$	−1.85588	−0.163401
$130$	−4.67597	−0.410110
$131$	−2.83317	−0.247536	−0.123768	−	0.992311i	$-0.539498\pi$
$131$	−2.83317	−0.247536	−0.123768	+	0.992311i	$0.539498\pi$
$132$	−0.947558	−0.0824743
$133$	0.418118	0.0362555
$134$	11.2388	0.970888
$135$	−5.49134	−0.472619
$136$	−0.347476	−0.0297958
$137$	10.1243	0.864975	0.432488	−	0.901640i	$-0.357636\pi$
$137$	10.1243	0.864975	0.432488	+	0.901640i	$0.357636\pi$
$138$	−2.74650	−0.233797
$139$	6.29885	0.534262	0.267131	−	0.963660i	$-0.413924\pi$
$139$	6.29885	0.534262	0.267131	+	0.963660i	$0.413924\pi$
$140$	−0.441519	−0.0373151
$141$	−2.53752	−0.213698
$142$	−9.64673	−0.809536
$143$	−3.65630	−0.305755
$144$	−1.53151	−0.127626
$145$	−8.38155	−0.696050
$146$	5.47699	0.453279
$147$	8.24646	0.680157
$148$	7.03784	0.578507
$149$	3.54263	0.290223	0.145112	−	0.989415i	$-0.453646\pi$
$149$	3.54263	0.290223	0.145112	+	0.989415i	$0.453646\pi$
$150$	1.21181	0.0989441
$151$	6.03883	0.491433	0.245716	−	0.969342i	$-0.420977\pi$
$151$	6.03883	0.491433	0.245716	+	0.969342i	$0.420977\pi$
$152$	−0.947001	−0.0768119
$153$	−0.532163	−0.0430228
$154$	−0.345238	−0.0278201
$155$	−7.83398	−0.629241
$156$	5.66640	0.453675
$157$	−9.79188	−0.781477	−0.390738	−	0.920502i	$-0.627780\pi$
$157$	−9.79188	−0.781477	−0.390738	+	0.920502i	$0.627780\pi$
$158$	1.92618	0.153239
$159$	11.1714	0.885950
$160$	1.00000	0.0790569
$161$	−1.00067	−0.0788642
$162$	2.05995	0.161845
$163$	−15.1332	−1.18532	−0.592661	−	0.805452i	$-0.701923\pi$
$163$	−15.1332	−1.18532	−0.592661	+	0.805452i	$0.701923\pi$
$164$	−4.97470	−0.388459
$165$	0.947558	0.0737673
$166$	−8.36611	−0.649336
$167$	−19.9819	−1.54625	−0.773124	−	0.634255i	$-0.781307\pi$
$167$	−19.9819	−1.54625	−0.773124	+	0.634255i	$0.781307\pi$
$168$	0.535038	0.0412791
$169$	8.86473	0.681902
$170$	0.347476	0.0266502
$171$	−1.45034	−0.110910
$172$	1.53149	0.116775
$173$	−2.42866	−0.184648	−0.0923238	−	0.995729i	$-0.529430\pi$
$173$	−2.42866	−0.184648	−0.0923238	+	0.995729i	$0.529430\pi$
$174$	10.1569	0.769991
$175$	0.441519	0.0333757
$176$	0.781934	0.0589405
$177$	−8.73486	−0.656552
$178$	−7.27170	−0.545037
$179$	−24.5676	−1.83627	−0.918134	−	0.396271i	$-0.870304\pi$
$179$	−24.5676	−1.83627	−0.918134	+	0.396271i	$0.870304\pi$
$180$	1.53151	0.114152
$181$	−5.73582	−0.426340	−0.213170	−	0.977015i	$-0.568379\pi$
$181$	−5.73582	−0.426340	−0.213170	+	0.977015i	$0.568379\pi$
$182$	2.06453	0.153033
$183$	15.8065	1.16845
$184$	2.26644	0.167084
$185$	−7.03784	−0.517432
$186$	9.49332	0.696084
$187$	0.271703	0.0198689
$188$	2.09399	0.152720
$189$	2.42453	0.176359
$190$	0.947001	0.0687027
$191$	−11.5893	−0.838571	−0.419285	−	0.907854i	$-0.637719\pi$
$191$	−11.5893	−0.838571	−0.419285	+	0.907854i	$0.637719\pi$
$192$	−1.21181	−0.0874551
$193$	1.55342	0.111818	0.0559088	−	0.998436i	$-0.482194\pi$
$193$	1.55342	0.111818	0.0559088	+	0.998436i	$0.482194\pi$
$194$	−11.6018	−0.832960
$195$	−5.66640	−0.405780
$196$	−6.80506	−0.486076
$197$	−10.9813	−0.782383	−0.391191	−	0.920309i	$-0.627937\pi$
$197$	−10.9813	−0.782383	−0.391191	+	0.920309i	$0.627937\pi$
$198$	1.19754	0.0851054
$199$	13.9734	0.990546	0.495273	−	0.868737i	$-0.335068\pi$
$199$	13.9734	0.990546	0.495273	+	0.868737i	$0.335068\pi$
$200$	−1.00000	−0.0707107
$201$	13.6194	0.960637
$202$	−1.12415	−0.0790950
$203$	3.70061	0.259732
$204$	−0.421076	−0.0294812
$205$	4.97470	0.347448
$206$	4.02781	0.280631
$207$	3.47107	0.241256
$208$	−4.67597	−0.324220
$209$	0.740492	0.0512209
$210$	−0.535038	−0.0369211
$211$	2.91823	0.200899	0.100449	−	0.994942i	$-0.467972\pi$
$211$	2.91823	0.200899	0.100449	+	0.994942i	$0.467972\pi$
$212$	−9.21875	−0.633147
$213$	−11.6900	−0.800988
$214$	−0.728772	−0.0498178
$215$	−1.53149	−0.104447
$216$	−5.49134	−0.373638
$217$	3.45885	0.234802
$218$	−9.42649	−0.638442
$219$	6.63709	0.448493
$220$	−0.781934	−0.0527180
$221$	−1.62479	−0.109295
$222$	8.52855	0.572398
$223$	−17.8176	−1.19315	−0.596576	−	0.802557i	$-0.703473\pi$
$223$	−17.8176	−1.19315	−0.596576	+	0.802557i	$0.703473\pi$
$224$	−0.441519	−0.0295002
$225$	−1.53151	−0.102101
$226$	3.90239	0.259583
$227$	15.5741	1.03369	0.516845	−	0.856079i	$-0.327106\pi$
$227$	15.5741	1.03369	0.516845	+	0.856079i	$0.327106\pi$
$228$	−1.14759	−0.0760009
$229$	8.00310	0.528860	0.264430	−	0.964405i	$-0.414816\pi$
$229$	8.00310	0.528860	0.264430	+	0.964405i	$0.414816\pi$
$230$	−2.26644	−0.149444
$231$	−0.418364	−0.0275264
$232$	−8.38155	−0.550276
$233$	14.7387	0.965565	0.482782	−	0.875740i	$-0.339626\pi$
$233$	14.7387	0.965565	0.482782	+	0.875740i	$0.339626\pi$
$234$	−7.16130	−0.468149
$235$	−2.09399	−0.136597
$236$	7.20809	0.469207
$237$	2.33417	0.151621
$238$	−0.153417	−0.00994455
$239$	0.273552	0.0176946	0.00884732	−	0.999961i	$-0.497184\pi$
$239$	0.273552	0.0176946	0.00884732	+	0.999961i	$0.497184\pi$
$240$	1.21181	0.0782222
$241$	0.940060	0.0605546	0.0302773	−	0.999542i	$-0.490361\pi$
$241$	0.940060	0.0605546	0.0302773	+	0.999542i	$0.490361\pi$
$242$	10.3886	0.667803
$243$	−13.9778	−0.896673
$244$	−13.0437	−0.835038
$245$	6.80506	0.434759
$246$	−6.02841	−0.384357
$247$	−4.42815	−0.281756
$248$	−7.83398	−0.497459
$249$	−10.1382	−0.642480
$250$	1.00000	0.0632456
$251$	3.17592	0.200462	0.100231	−	0.994964i	$-0.468042\pi$
$251$	3.17592	0.200462	0.100231	+	0.994964i	$0.468042\pi$
$252$	−0.676190	−0.0425960
$253$	−1.77220	−0.111418
$254$	17.5384	1.10045
$255$	0.421076	0.0263688
$256$	1.00000	0.0625000
$257$	−18.2747	−1.13995	−0.569973	−	0.821663i	$-0.693046\pi$
$257$	−18.2747	−1.13995	−0.569973	+	0.821663i	$0.693046\pi$
$258$	1.85588	0.115542
$259$	3.10734	0.193081
$260$	4.67597	0.289992
$261$	−12.8364	−0.794555
$262$	2.83317	0.175034
$263$	22.1564	1.36622	0.683111	−	0.730315i	$-0.260627\pi$
$263$	22.1564	1.36622	0.683111	+	0.730315i	$0.260627\pi$
$264$	0.947558	0.0583181
$265$	9.21875	0.566304
$266$	−0.418118	−0.0256365
$267$	−8.81194	−0.539282
$268$	−11.2388	−0.686522
$269$	−30.9675	−1.88812	−0.944060	−	0.329774i	$-0.893027\pi$
$269$	−30.9675	−1.88812	−0.944060	+	0.329774i	$0.893027\pi$
$270$	5.49134	0.334192
$271$	−0.286023	−0.0173747	−0.00868734	−	0.999962i	$-0.502765\pi$
$271$	−0.286023	−0.0173747	−0.00868734	+	0.999962i	$0.502765\pi$
$272$	0.347476	0.0210688
$273$	2.50182	0.151417
$274$	−10.1243	−0.611630
$275$	0.781934	0.0471524
$276$	2.74650	0.165320
$277$	−16.2895	−0.978739	−0.489370	−	0.872076i	$-0.662773\pi$
$277$	−16.2895	−0.978739	−0.489370	+	0.872076i	$0.662773\pi$
$278$	−6.29885	−0.377780
$279$	−11.9978	−0.718291
$280$	0.441519	0.0263858
$281$	−6.11821	−0.364982	−0.182491	−	0.983208i	$-0.558416\pi$
$281$	−6.11821	−0.364982	−0.182491	+	0.983208i	$0.558416\pi$
$282$	2.53752	0.151107
$283$	−29.3709	−1.74592	−0.872959	−	0.487793i	$-0.837802\pi$
$283$	−29.3709	−1.74592	−0.872959	+	0.487793i	$0.837802\pi$
$284$	9.64673	0.572428
$285$	1.14759	0.0679772
$286$	3.65630	0.216202
$287$	−2.19642	−0.129651
$288$	1.53151	0.0902451
$289$	−16.8793	−0.992898
$290$	8.38155	0.492182
$291$	−14.0592	−0.824165
$292$	−5.47699	−0.320517
$293$	−1.39480	−0.0814854	−0.0407427	−	0.999170i	$-0.512972\pi$
$293$	−1.39480	−0.0814854	−0.0407427	+	0.999170i	$0.512972\pi$
$294$	−8.24646	−0.480943
$295$	−7.20809	−0.419671
$296$	−7.03784	−0.409066
$297$	4.29387	0.249155
$298$	−3.54263	−0.205219
$299$	10.5978	0.612887
$300$	−1.21181	−0.0699641
$301$	0.676180	0.0389744
$302$	−6.03883	−0.347495
$303$	−1.36226	−0.0782599
$304$	0.947001	0.0543142
$305$	13.0437	0.746880
$306$	0.532163	0.0304217
$307$	14.3901	0.821286	0.410643	−	0.911796i	$-0.365304\pi$
$307$	14.3901	0.821286	0.410643	+	0.911796i	$0.365304\pi$
$308$	0.345238	0.0196718
$309$	4.88096	0.277668
$310$	7.83398	0.444940
$311$	10.3774	0.588448	0.294224	−	0.955736i	$-0.404939\pi$
$311$	10.3774	0.588448	0.294224	+	0.955736i	$0.404939\pi$
$312$	−5.66640	−0.320797
$313$	−1.21333	−0.0685817	−0.0342908	−	0.999412i	$-0.510917\pi$
$313$	−1.21333	−0.0685817	−0.0342908	+	0.999412i	$0.510917\pi$
$314$	9.79188	0.552588
$315$	0.676190	0.0380990
$316$	−1.92618	−0.108356
$317$	−12.8494	−0.721694	−0.360847	−	0.932625i	$-0.617512\pi$
$317$	−12.8494	−0.721694	−0.360847	+	0.932625i	$0.617512\pi$
$318$	−11.1714	−0.626461
$319$	6.55382	0.366944
$320$	−1.00000	−0.0559017
$321$	−0.883135	−0.0492918
$322$	1.00067	0.0557654
$323$	0.329060	0.0183094
$324$	−2.05995	−0.114442
$325$	−4.67597	−0.259376
$326$	15.1332	0.838149
$327$	−11.4231	−0.631701
$328$	4.97470	0.274682
$329$	0.924535	0.0509712
$330$	−0.947558	−0.0521613
$331$	−8.01850	−0.440737	−0.220368	−	0.975417i	$-0.570726\pi$
$331$	−8.01850	−0.440737	−0.220368	+	0.975417i	$0.570726\pi$
$332$	8.36611	0.459150
$333$	−10.7785	−0.590659
$334$	19.9819	1.09336
$335$	11.2388	0.614044
$336$	−0.535038	−0.0291887
$337$	13.1570	0.716708	0.358354	−	0.933586i	$-0.383338\pi$
$337$	13.1570	0.716708	0.358354	+	0.933586i	$0.383338\pi$
$338$	−8.86473	−0.482178
$339$	4.72897	0.256842
$340$	−0.347476	−0.0188445
$341$	6.12566	0.331723
$342$	1.45034	0.0784254
$343$	−6.09519	−0.329109
$344$	−1.53149	−0.0825722
$345$	−2.74650	−0.147866
$346$	2.42866	0.130566
$347$	−15.7431	−0.845132	−0.422566	−	0.906332i	$-0.638870\pi$
$347$	−15.7431	−0.845132	−0.422566	+	0.906332i	$0.638870\pi$
$348$	−10.1569	−0.544466
$349$	−4.78445	−0.256106	−0.128053	−	0.991767i	$-0.540873\pi$
$349$	−4.78445	−0.256106	−0.128053	+	0.991767i	$0.540873\pi$
$350$	−0.441519	−0.0236002
$351$	−25.6774	−1.37056
$352$	−0.781934	−0.0416772
$353$	5.67180	0.301880	0.150940	−	0.988543i	$-0.451770\pi$
$353$	5.67180	0.301880	0.150940	+	0.988543i	$0.451770\pi$
$354$	8.73486	0.464252
$355$	−9.64673	−0.511995
$356$	7.27170	0.385399
$357$	−0.185913	−0.00983955
$358$	24.5676	1.29844
$359$	−3.45971	−0.182597	−0.0912983	−	0.995824i	$-0.529102\pi$
$359$	−3.45971	−0.182597	−0.0912983	+	0.995824i	$0.529102\pi$
$360$	−1.53151	−0.0807176
$361$	−18.1032	−0.952799
$362$	5.73582	0.301468
$363$	12.5890	0.660752
$364$	−2.06453	−0.108211
$365$	5.47699	0.286679
$366$	−15.8065	−0.826221
$367$	−19.2846	−1.00665	−0.503325	−	0.864097i	$-0.667890\pi$
$367$	−19.2846	−1.00665	−0.503325	+	0.864097i	$0.667890\pi$
$368$	−2.26644	−0.118146
$369$	7.61880	0.396619
$370$	7.03784	0.365880
$371$	−4.07025	−0.211317
$372$	−9.49332	−0.492206
$373$	−21.5390	−1.11525	−0.557623	−	0.830094i	$-0.688287\pi$
$373$	−21.5390	−1.11525	−0.557623	+	0.830094i	$0.688287\pi$
$374$	−0.271703	−0.0140494
$375$	1.21181	0.0625778
$376$	−2.09399	−0.107989
$377$	−39.1919	−2.01849
$378$	−2.42453	−0.124704
$379$	4.78597	0.245839	0.122920	−	0.992417i	$-0.460774\pi$
$379$	4.78597	0.245839	0.122920	+	0.992417i	$0.460774\pi$
$380$	−0.947001	−0.0485801
$381$	21.2532	1.08884
$382$	11.5893	0.592959
$383$	6.11975	0.312705	0.156352	−	0.987701i	$-0.450026\pi$
$383$	6.11975	0.312705	0.156352	+	0.987701i	$0.450026\pi$
$384$	1.21181	0.0618401
$385$	−0.345238	−0.0175950
$386$	−1.55342	−0.0790670
$387$	−2.34549	−0.119228
$388$	11.6018	0.588992
$389$	3.87667	0.196555	0.0982775	−	0.995159i	$-0.468667\pi$
$389$	3.87667	0.196555	0.0982775	+	0.995159i	$0.468667\pi$
$390$	5.66640	0.286930
$391$	−0.787532	−0.0398272
$392$	6.80506	0.343708
$393$	3.43328	0.173186
$394$	10.9813	0.553228
$395$	1.92618	0.0969167
$396$	−1.19754	−0.0601786
$397$	−28.6478	−1.43779	−0.718896	−	0.695118i	$-0.755352\pi$
$397$	−28.6478	−1.43779	−0.718896	+	0.695118i	$0.755352\pi$
$398$	−13.9734	−0.700422
$399$	−0.506681	−0.0253658
$400$	1.00000	0.0500000
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	−13.6194	−0.679273
$403$	−36.6315	−1.82475
$404$	1.12415	0.0559286
$405$	2.05995	0.102360
$406$	−3.70061	−0.183658
$407$	5.50313	0.272780
$408$	0.421076	0.0208464
$409$	−5.74096	−0.283872	−0.141936	−	0.989876i	$-0.545333\pi$
$409$	−5.74096	−0.283872	−0.141936	+	0.989876i	$0.545333\pi$
$410$	−4.97470	−0.245683
$411$	−12.2687	−0.605172
$412$	−4.02781	−0.198436
$413$	3.18251	0.156601
$414$	−3.47107	−0.170594
$415$	−8.36611	−0.410676
$416$	4.67597	0.229258
$417$	−7.63303	−0.373791
$418$	−0.740492	−0.0362187
$419$	−3.90764	−0.190901	−0.0954505	−	0.995434i	$-0.530429\pi$
$419$	−3.90764	−0.190901	−0.0954505	+	0.995434i	$0.530429\pi$
$420$	0.535038	0.0261072
$421$	18.4802	0.900668	0.450334	−	0.892860i	$-0.351305\pi$
$421$	18.4802	0.900668	0.450334	+	0.892860i	$0.351305\pi$
$422$	−2.91823	−0.142057
$423$	−3.20696	−0.155928
$424$	9.21875	0.447702
$425$	0.347476	0.0168551
$426$	11.6900	0.566384
$427$	−5.75904	−0.278699
$428$	0.728772	0.0352265
$429$	4.43075	0.213919
$430$	1.53149	0.0738548
$431$	−38.8156	−1.86968	−0.934841	−	0.355067i	$-0.884458\pi$
$431$	−38.8156	−1.86968	−0.934841	+	0.355067i	$0.884458\pi$
$432$	5.49134	0.264202
$433$	9.40605	0.452026	0.226013	−	0.974124i	$-0.427431\pi$
$433$	9.40605	0.452026	0.226013	+	0.974124i	$0.427431\pi$
$434$	−3.45885	−0.166030
$435$	10.1569	0.486985
$436$	9.42649	0.451447
$437$	−2.14632	−0.102672
$438$	−6.63709	−0.317132
$439$	−2.85365	−0.136197	−0.0680987	−	0.997679i	$-0.521693\pi$
$439$	−2.85365	−0.136197	−0.0680987	+	0.997679i	$0.521693\pi$
$440$	0.781934	0.0372772
$441$	10.4220	0.496286
$442$	1.62479	0.0772833
$443$	−18.8725	−0.896660	−0.448330	−	0.893868i	$-0.647981\pi$
$443$	−18.8725	−0.896660	−0.448330	+	0.893868i	$0.647981\pi$
$444$	−8.52855	−0.404747
$445$	−7.27170	−0.344711
$446$	17.8176	0.843686
$447$	−4.29300	−0.203052
$448$	0.441519	0.0208598
$449$	21.0421	0.993038	0.496519	−	0.868026i	$-0.334611\pi$
$449$	21.0421	0.993038	0.496519	+	0.868026i	$0.334611\pi$
$450$	1.53151	0.0721960
$451$	−3.88989	−0.183168
$452$	−3.90239	−0.183553
$453$	−7.31793	−0.343826
$454$	−15.5741	−0.730929
$455$	2.06453	0.0967866
$456$	1.14759	0.0537407
$457$	−8.24588	−0.385726	−0.192863	−	0.981226i	$-0.561777\pi$
$457$	−8.24588	−0.385726	−0.192863	+	0.981226i	$0.561777\pi$
$458$	−8.00310	−0.373960
$459$	1.90811	0.0890629
$460$	2.26644	0.105673
$461$	28.0292	1.30545	0.652724	−	0.757596i	$-0.273626\pi$
$461$	28.0292	1.30545	0.652724	+	0.757596i	$0.273626\pi$
$462$	0.418364	0.0194641
$463$	−28.1590	−1.30866	−0.654329	−	0.756210i	$-0.727049\pi$
$463$	−28.1590	−1.30866	−0.654329	+	0.756210i	$0.727049\pi$
$464$	8.38155	0.389104
$465$	9.49332	0.440242
$466$	−14.7387	−0.682758
$467$	−1.74732	−0.0808565	−0.0404282	−	0.999182i	$-0.512872\pi$
$467$	−1.74732	−0.0808565	−0.0404282	+	0.999182i	$0.512872\pi$
$468$	7.16130	0.331031
$469$	−4.96216	−0.229131
$470$	2.09399	0.0965885
$471$	11.8659	0.546753
$472$	−7.20809	−0.331779
$473$	1.19752	0.0550621
$474$	−2.33417	−0.107212
$475$	0.947001	0.0434514
$476$	0.153417	0.00703186
$477$	14.1186	0.646447
$478$	−0.273552	−0.0125120
$479$	26.2763	1.20059	0.600297	−	0.799777i	$-0.295049\pi$
$479$	26.2763	1.20059	0.600297	+	0.799777i	$0.295049\pi$
$480$	−1.21181	−0.0553114
$481$	−32.9087	−1.50051
$482$	−0.940060	−0.0428185
$483$	1.21263	0.0551766
$484$	−10.3886	−0.472208
$485$	−11.6018	−0.526810
$486$	13.9778	0.634044
$487$	−23.8805	−1.08213	−0.541065	−	0.840981i	$-0.681979\pi$
$487$	−23.8805	−1.08213	−0.541065	+	0.840981i	$0.681979\pi$
$488$	13.0437	0.590461
$489$	18.3386	0.829299
$490$	−6.80506	−0.307421
$491$	−8.48580	−0.382959	−0.191479	−	0.981497i	$-0.561328\pi$
$491$	−8.48580	−0.382959	−0.191479	+	0.981497i	$0.561328\pi$
$492$	6.02841	0.271782
$493$	2.91239	0.131167
$494$	4.42815	0.199232
$495$	1.19754	0.0538254
$496$	7.83398	0.351756
$497$	4.25921	0.191052
$498$	10.1382	0.454302
$499$	−15.8989	−0.711731	−0.355866	−	0.934537i	$-0.615814\pi$
$499$	−15.8989	−0.711731	−0.355866	+	0.934537i	$0.615814\pi$
$500$	−1.00000	−0.0447214
$501$	24.2144	1.08182
$502$	−3.17592	−0.141748
$503$	−1.89612	−0.0845440	−0.0422720	−	0.999106i	$-0.513460\pi$
$503$	−1.89612	−0.0845440	−0.0422720	+	0.999106i	$0.513460\pi$
$504$	0.676190	0.0301199
$505$	−1.12415	−0.0500241
$506$	1.77220	0.0787841
$507$	−10.7424	−0.477086
$508$	−17.5384	−0.778139
$509$	34.6656	1.53653	0.768263	−	0.640134i	$-0.221121\pi$
$509$	34.6656	1.53653	0.768263	+	0.640134i	$0.221121\pi$
$510$	−0.421076	−0.0186456
$511$	−2.41819	−0.106975
$512$	−1.00000	−0.0441942
$513$	5.20031	0.229599
$514$	18.2747	0.806064
$515$	4.02781	0.177487
$516$	−1.85588	−0.0817004
$517$	1.63736	0.0720110
$518$	−3.10734	−0.136529
$519$	2.94308	0.129187
$520$	−4.67597	−0.205055
$521$	23.1428	1.01391	0.506953	−	0.861974i	$-0.330772\pi$
$521$	23.1428	1.01391	0.506953	+	0.861974i	$0.330772\pi$
$522$	12.8364	0.561835
$523$	18.2417	0.797656	0.398828	−	0.917026i	$-0.369417\pi$
$523$	18.2417	0.797656	0.398828	+	0.917026i	$0.369417\pi$
$524$	−2.83317	−0.123768
$525$	−0.535038	−0.0233510
$526$	−22.1564	−0.966065
$527$	2.72212	0.118577
$528$	−0.947558	−0.0412372
$529$	−17.8633	−0.776664
$530$	−9.21875	−0.400437
$531$	−11.0393	−0.479063
$532$	0.418118	0.0181277
$533$	23.2616	1.00757
$534$	8.81194	0.381330
$535$	−0.728772	−0.0315075
$536$	11.2388	0.485444
$537$	29.7713	1.28473
$538$	30.9675	1.33510
$539$	−5.32111	−0.229196
$540$	−5.49134	−0.236310
$541$	4.08936	0.175815	0.0879077	−	0.996129i	$-0.471982\pi$
$541$	4.08936	0.175815	0.0879077	+	0.996129i	$0.471982\pi$
$542$	0.286023	0.0122858
$543$	6.95074	0.298285
$544$	−0.347476	−0.0148979
$545$	−9.42649	−0.403786
$546$	−2.50182	−0.107068
$547$	−0.598670	−0.0255973	−0.0127986	−	0.999918i	$-0.504074\pi$
$547$	−0.598670	−0.0255973	−0.0127986	+	0.999918i	$0.504074\pi$
$548$	10.1243	0.432488
$549$	19.9766	0.852579
$550$	−0.781934	−0.0333418
$551$	7.93734	0.338142
$552$	−2.74650	−0.116899
$553$	−0.850445	−0.0361646
$554$	16.2895	0.692073
$555$	8.52855	0.362017
$556$	6.29885	0.267131
$557$	2.59015	0.109748	0.0548740	−	0.998493i	$-0.482524\pi$
$557$	2.59015	0.109748	0.0548740	+	0.998493i	$0.482524\pi$
$558$	11.9978	0.507908
$559$	−7.16119	−0.302886
$560$	−0.441519	−0.0186576
$561$	−0.329254	−0.0139011
$562$	6.11821	0.258081
$563$	−27.6580	−1.16565	−0.582823	−	0.812599i	$-0.698052\pi$
$563$	−27.6580	−1.16565	−0.582823	+	0.812599i	$0.698052\pi$
$564$	−2.53752	−0.106849
$565$	3.90239	0.164175
$566$	29.3709	1.23455
$567$	−0.909507	−0.0381957
$568$	−9.64673	−0.404768
$569$	39.1870	1.64280	0.821401	−	0.570351i	$-0.193193\pi$
$569$	39.1870	1.64280	0.821401	+	0.570351i	$0.193193\pi$
$570$	−1.14759	−0.0480672
$571$	−41.8064	−1.74954	−0.874772	−	0.484536i	$-0.838989\pi$
$571$	−41.8064	−1.74954	−0.874772	+	0.484536i	$0.838989\pi$
$572$	−3.65630	−0.152878
$573$	14.0440	0.586698
$574$	2.19642	0.0916770
$575$	−2.26644	−0.0945170
$576$	−1.53151	−0.0638129
$577$	−37.4265	−1.55808	−0.779042	−	0.626971i	$-0.784294\pi$
$577$	−37.4265	−1.55808	−0.779042	+	0.626971i	$0.784294\pi$
$578$	16.8793	0.702085
$579$	−1.88245	−0.0782321
$580$	−8.38155	−0.348025
$581$	3.69379	0.153244
$582$	14.0592	0.582773
$583$	−7.20846	−0.298544
$584$	5.47699	0.226640
$585$	−7.16130	−0.296083
$586$	1.39480	0.0576188
$587$	18.2023	0.751289	0.375645	−	0.926764i	$-0.377421\pi$
$587$	18.2023	0.751289	0.375645	+	0.926764i	$0.377421\pi$
$588$	8.24646	0.340078
$589$	7.41879	0.305686
$590$	7.20809	0.296752
$591$	13.3072	0.547387
$592$	7.03784	0.289253
$593$	12.1539	0.499099	0.249550	−	0.968362i	$-0.419717\pi$
$593$	12.1539	0.499099	0.249550	+	0.968362i	$0.419717\pi$
$594$	−4.29387	−0.176179
$595$	−0.153417	−0.00628949
$596$	3.54263	0.145112
$597$	−16.9331	−0.693026
$598$	−10.5978	−0.433376
$599$	24.6183	1.00588	0.502938	−	0.864323i	$-0.332252\pi$
$599$	24.6183	1.00588	0.502938	+	0.864323i	$0.332252\pi$
$600$	1.21181	0.0494721
$601$	−5.83318	−0.237940	−0.118970	−	0.992898i	$-0.537959\pi$
$601$	−5.83318	−0.237940	−0.118970	+	0.992898i	$0.537959\pi$
$602$	−0.676180	−0.0275590
$603$	17.2124	0.700943
$604$	6.03883	0.245716
$605$	10.3886	0.422356
$606$	1.36226	0.0553381
$607$	5.32952	0.216319	0.108159	−	0.994134i	$-0.465504\pi$
$607$	5.32952	0.216319	0.108159	+	0.994134i	$0.465504\pi$
$608$	−0.947001	−0.0384060
$609$	−4.48445	−0.181719
$610$	−13.0437	−0.528124
$611$	−9.79143	−0.396119
$612$	−0.532163	−0.0215114
$613$	3.17924	0.128408	0.0642041	−	0.997937i	$-0.479549\pi$
$613$	3.17924	0.128408	0.0642041	+	0.997937i	$0.479549\pi$
$614$	−14.3901	−0.580737
$615$	−6.02841	−0.243089
$616$	−0.345238	−0.0139101
$617$	26.6755	1.07391	0.536957	−	0.843609i	$-0.319574\pi$
$617$	26.6755	1.07391	0.536957	+	0.843609i	$0.319574\pi$
$618$	−4.88096	−0.196341
$619$	−27.3184	−1.09802	−0.549010	−	0.835816i	$-0.684995\pi$
$619$	−27.3184	−1.09802	−0.549010	+	0.835816i	$0.684995\pi$
$620$	−7.83398	−0.314620
$621$	−12.4458	−0.499432
$622$	−10.3774	−0.416096
$623$	3.21059	0.128630
$624$	5.66640	0.226838
$625$	1.00000	0.0400000
$626$	1.21333	0.0484946
$627$	−0.897338	−0.0358362
$628$	−9.79188	−0.390738
$629$	2.44548	0.0975077
$630$	−0.676190	−0.0269400
$631$	12.0937	0.481444	0.240722	−	0.970594i	$-0.422616\pi$
$631$	12.0937	0.481444	0.240722	+	0.970594i	$0.422616\pi$
$632$	1.92618	0.0766194
$633$	−3.53634	−0.140557
$634$	12.8494	0.510315
$635$	17.5384	0.695989
$636$	11.1714	0.442975
$637$	31.8203	1.26077
$638$	−6.55382	−0.259468
$639$	−14.7741	−0.584453
$640$	1.00000	0.0395285
$641$	−20.5534	−0.811809	−0.405905	−	0.913915i	$-0.633044\pi$
$641$	−20.5534	−0.811809	−0.405905	+	0.913915i	$0.633044\pi$
$642$	0.883135	0.0348546
$643$	−29.4893	−1.16294	−0.581472	−	0.813567i	$-0.697523\pi$
$643$	−29.4893	−1.16294	−0.581472	+	0.813567i	$0.697523\pi$
$644$	−1.00067	−0.0394321
$645$	1.85588	0.0730750
$646$	−0.329060	−0.0129467
$647$	14.6520	0.576029	0.288014	−	0.957626i	$-0.407005\pi$
$647$	14.6520	0.576029	0.288014	+	0.957626i	$0.407005\pi$
$648$	2.05995	0.0809225
$649$	5.63625	0.221242
$650$	4.67597	0.183407
$651$	−4.19148	−0.164277
$652$	−15.1332	−0.592661
$653$	−4.47836	−0.175252	−0.0876260	−	0.996153i	$-0.527928\pi$
$653$	−4.47836	−0.175252	−0.0876260	+	0.996153i	$0.527928\pi$
$654$	11.4231	0.446680
$655$	2.83317	0.110701
$656$	−4.97470	−0.194230
$657$	8.38807	0.327250
$658$	−0.924535	−0.0360421
$659$	−7.32171	−0.285213	−0.142607	−	0.989779i	$-0.545548\pi$
$659$	−7.32171	−0.285213	−0.142607	+	0.989779i	$0.545548\pi$
$660$	0.947558	0.0368836
$661$	38.1544	1.48403	0.742017	−	0.670381i	$-0.233869\pi$
$661$	38.1544	1.48403	0.742017	+	0.670381i	$0.233869\pi$
$662$	8.01850	0.311648
$663$	1.96894	0.0764673
$664$	−8.36611	−0.324668
$665$	−0.418118	−0.0162139
$666$	10.7785	0.417659
$667$	−18.9963	−0.735538
$668$	−19.9819	−0.773124
$669$	21.5915	0.834777
$670$	−11.2388	−0.434195
$671$	−10.1993	−0.393740
$672$	0.535038	0.0206395
$673$	−0.557535	−0.0214914	−0.0107457	−	0.999942i	$-0.503421\pi$
$673$	−0.557535	−0.0214914	−0.0107457	+	0.999942i	$0.503421\pi$
$674$	−13.1570	−0.506789
$675$	5.49134	0.211362
$676$	8.86473	0.340951
$677$	−25.7037	−0.987872	−0.493936	−	0.869498i	$-0.664442\pi$
$677$	−25.7037	−0.987872	−0.493936	+	0.869498i	$0.664442\pi$
$678$	−4.72897	−0.181615
$679$	5.12241	0.196580
$680$	0.347476	0.0133251
$681$	−18.8729	−0.723212
$682$	−6.12566	−0.234564
$683$	−11.4197	−0.436961	−0.218480	−	0.975841i	$-0.570110\pi$
$683$	−11.4197	−0.436961	−0.218480	+	0.975841i	$0.570110\pi$
$684$	−1.45034	−0.0554552
$685$	−10.1243	−0.386829
$686$	6.09519	0.232715
$687$	−9.69826	−0.370012
$688$	1.53149	0.0583874
$689$	43.1066	1.64223
$690$	2.74650	0.104557
$691$	−28.2101	−1.07316	−0.536582	−	0.843848i	$-0.680285\pi$
$691$	−28.2101	−1.07316	−0.536582	+	0.843848i	$0.680285\pi$
$692$	−2.42866	−0.0923238
$693$	−0.528736	−0.0200850
$694$	15.7431	0.597598
$695$	−6.29885	−0.238929
$696$	10.1569	0.384995
$697$	−1.72859	−0.0654750
$698$	4.78445	0.181094
$699$	−17.8606	−0.675548
$700$	0.441519	0.0166878
$701$	−25.3329	−0.956812	−0.478406	−	0.878139i	$-0.658785\pi$
$701$	−25.3329	−0.956812	−0.478406	+	0.878139i	$0.658785\pi$
$702$	25.6774	0.969130
$703$	6.66484	0.251369
$704$	0.781934	0.0294702
$705$	2.53752	0.0955686
$706$	−5.67180	−0.213461
$707$	0.496334	0.0186666
$708$	−8.73486	−0.328276
$709$	−40.4961	−1.52086	−0.760432	−	0.649418i	$-0.775013\pi$
$709$	−40.4961	−1.52086	−0.760432	+	0.649418i	$0.775013\pi$
$710$	9.64673	0.362035
$711$	2.94997	0.110632
$712$	−7.27170	−0.272518
$713$	−17.7552	−0.664939
$714$	0.185913	0.00695761
$715$	3.65630	0.136738
$716$	−24.5676	−0.918134
$717$	−0.331494	−0.0123799
$718$	3.45971	0.129115
$719$	−22.3153	−0.832221	−0.416111	−	0.909314i	$-0.636607\pi$
$719$	−22.3153	−0.832221	−0.416111	+	0.909314i	$0.636607\pi$
$720$	1.53151	0.0570760
$721$	−1.77835	−0.0662294
$722$	18.1032	0.673731
$723$	−1.13918	−0.0423664
$724$	−5.73582	−0.213170
$725$	8.38155	0.311283
$726$	−12.5890	−0.467222
$727$	23.5195	0.872289	0.436145	−	0.899877i	$-0.356344\pi$
$727$	23.5195	0.872289	0.436145	+	0.899877i	$0.356344\pi$
$728$	2.06453	0.0765165
$729$	23.1183	0.856232
$730$	−5.47699	−0.202713
$731$	0.532155	0.0196825
$732$	15.8065	0.584226
$733$	47.5357	1.75577	0.877885	−	0.478872i	$-0.158954\pi$
$733$	47.5357	1.75577	0.877885	+	0.478872i	$0.158954\pi$
$734$	19.2846	0.711809
$735$	−8.24646	−0.304175
$736$	2.26644	0.0835420
$737$	−8.78804	−0.323711
$738$	−7.61880	−0.280452
$739$	9.07806	0.333942	0.166971	−	0.985962i	$-0.446601\pi$
$739$	9.07806	0.333942	0.166971	+	0.985962i	$0.446601\pi$
$740$	−7.03784	−0.258716
$741$	5.36609	0.197128
$742$	4.07025	0.149424
$743$	28.6060	1.04945	0.524725	−	0.851272i	$-0.324168\pi$
$743$	28.6060	1.04945	0.524725	+	0.851272i	$0.324168\pi$
$744$	9.49332	0.348042
$745$	−3.54263	−0.129792
$746$	21.5390	0.788598
$747$	−12.8128	−0.468795
$748$	0.271703	0.00993445
$749$	0.321766	0.0117571
$750$	−1.21181	−0.0442492
$751$	−22.6726	−0.827335	−0.413668	−	0.910428i	$-0.635752\pi$
$751$	−22.6726	−0.827335	−0.413668	+	0.910428i	$0.635752\pi$
$752$	2.09399	0.0763599
$753$	−3.84862	−0.140251
$754$	39.1919	1.42729
$755$	−6.03883	−0.219775
$756$	2.42453	0.0881793
$757$	22.3643	0.812843	0.406421	−	0.913686i	$-0.366777\pi$
$757$	22.3643	0.812843	0.406421	+	0.913686i	$0.366777\pi$
$758$	−4.78597	−0.173834
$759$	2.14758	0.0779522
$760$	0.947001	0.0343513
$761$	2.61429	0.0947680	0.0473840	−	0.998877i	$-0.484912\pi$
$761$	2.61429	0.0947680	0.0473840	+	0.998877i	$0.484912\pi$
$762$	−21.2532	−0.769923
$763$	4.16197	0.150673
$764$	−11.5893	−0.419285
$765$	0.532163	0.0192404
$766$	−6.11975	−0.221116
$767$	−33.7048	−1.21701
$768$	−1.21181	−0.0437275
$769$	11.3662	0.409875	0.204938	−	0.978775i	$-0.434301\pi$
$769$	11.3662	0.409875	0.204938	+	0.978775i	$0.434301\pi$
$770$	0.345238	0.0124415
$771$	22.1456	0.797553
$772$	1.55342	0.0559088
$773$	−17.5362	−0.630734	−0.315367	−	0.948970i	$-0.602128\pi$
$773$	−17.5362	−0.630734	−0.315367	+	0.948970i	$0.602128\pi$
$774$	2.34549	0.0843068
$775$	7.83398	0.281405
$776$	−11.6018	−0.416480
$777$	−3.76551	−0.135087
$778$	−3.87667	−0.138985
$779$	−4.71105	−0.168791
$780$	−5.66640	−0.202890
$781$	7.54311	0.269914
$782$	0.787532	0.0281621
$783$	46.0260	1.64483
$784$	−6.80506	−0.243038
$785$	9.79188	0.349487
$786$	−3.43328	−0.122461
$787$	7.48945	0.266970	0.133485	−	0.991051i	$-0.457383\pi$
$787$	7.48945	0.266970	0.133485	+	0.991051i	$0.457383\pi$
$788$	−10.9813	−0.391191
$789$	−26.8494	−0.955864
$790$	−1.92618	−0.0685305
$791$	−1.72298	−0.0612620
$792$	1.19754	0.0425527
$793$	60.9920	2.16589
$794$	28.6478	1.01667
$795$	−11.1714	−0.396209
$796$	13.9734	0.495273
$797$	0.761516	0.0269743	0.0134871	−	0.999909i	$-0.495707\pi$
$797$	0.761516	0.0269743	0.0134871	+	0.999909i	$0.495707\pi$
$798$	0.506681	0.0179363
$799$	0.727610	0.0257410
$800$	−1.00000	−0.0353553
$801$	−11.1367	−0.393495
$802$	1.00000	0.0353112
$803$	−4.28265	−0.151131
$804$	13.6194	0.480318
$805$	1.00067	0.0352691
$806$	36.6315	1.29029
$807$	37.5268	1.32101
$808$	−1.12415	−0.0395475
$809$	−11.2517	−0.395587	−0.197794	−	0.980244i	$-0.563378\pi$
$809$	−11.2517	−0.395587	−0.197794	+	0.980244i	$0.563378\pi$
$810$	−2.05995	−0.0723793
$811$	−46.7416	−1.64132	−0.820661	−	0.571416i	$-0.806394\pi$
$811$	−46.7416	−1.64132	−0.820661	+	0.571416i	$0.806394\pi$
$812$	3.70061	0.129866
$813$	0.346607	0.0121560
$814$	−5.50313	−0.192884
$815$	15.1332	0.530092
$816$	−0.421076	−0.0147406
$817$	1.45032	0.0507402
$818$	5.74096	0.200728
$819$	3.16185	0.110484
$820$	4.97470	0.173724
$821$	−20.4555	−0.713904	−0.356952	−	0.934123i	$-0.616184\pi$
$821$	−20.4555	−0.713904	−0.356952	+	0.934123i	$0.616184\pi$
$822$	12.2687	0.427921
$823$	−29.6757	−1.03443	−0.517214	−	0.855856i	$-0.673031\pi$
$823$	−29.6757	−1.03443	−0.517214	+	0.855856i	$0.673031\pi$
$824$	4.02781	0.140316
$825$	−0.947558	−0.0329897
$826$	−3.18251	−0.110734
$827$	1.06553	0.0370520	0.0185260	−	0.999828i	$-0.494103\pi$
$827$	1.06553	0.0370520	0.0185260	+	0.999828i	$0.494103\pi$
$828$	3.47107	0.120628
$829$	56.0850	1.94791	0.973957	−	0.226735i	$-0.0728050\pi$
$829$	56.0850	1.94791	0.973957	+	0.226735i	$0.0728050\pi$
$830$	8.36611	0.290392
$831$	19.7398	0.684766
$832$	−4.67597	−0.162110
$833$	−2.36460	−0.0819284
$834$	7.63303	0.264310
$835$	19.9819	0.691503
$836$	0.740492	0.0256105
$837$	43.0191	1.48696
$838$	3.90764	0.134987
$839$	36.8965	1.27381	0.636905	−	0.770942i	$-0.280214\pi$
$839$	36.8965	1.27381	0.636905	+	0.770942i	$0.280214\pi$
$840$	−0.535038	−0.0184606
$841$	41.2504	1.42243
$842$	−18.4802	−0.636868
$843$	7.41413	0.255356
$844$	2.91823	0.100449
$845$	−8.86473	−0.304956
$846$	3.20696	0.110258
$847$	−4.58675	−0.157603
$848$	−9.21875	−0.316573
$849$	35.5920	1.22152
$850$	−0.347476	−0.0119183
$851$	−15.9508	−0.546787
$852$	−11.6900	−0.400494
$853$	21.5303	0.737182	0.368591	−	0.929592i	$-0.379840\pi$
$853$	21.5303	0.737182	0.368591	+	0.929592i	$0.379840\pi$
$854$	5.75904	0.197070
$855$	1.45034	0.0496006
$856$	−0.728772	−0.0249089
$857$	4.60168	0.157191	0.0785953	−	0.996907i	$-0.474957\pi$
$857$	4.60168	0.157191	0.0785953	+	0.996907i	$0.474957\pi$
$858$	−4.43075	−0.151263
$859$	−42.9929	−1.46690	−0.733450	−	0.679744i	$-0.762091\pi$
$859$	−42.9929	−1.46690	−0.733450	+	0.679744i	$0.762091\pi$
$860$	−1.53149	−0.0522233
$861$	2.66166	0.0907090
$862$	38.8156	1.32206
$863$	19.4634	0.662543	0.331272	−	0.943535i	$-0.392522\pi$
$863$	19.4634	0.662543	0.331272	+	0.943535i	$0.392522\pi$
$864$	−5.49134	−0.186819
$865$	2.42866	0.0825770
$866$	−9.40605	−0.319631
$867$	20.4545	0.694671
$868$	3.45885	0.117401
$869$	−1.50615	−0.0510925
$870$	−10.1569	−0.344350
$871$	52.5525	1.78067
$872$	−9.42649	−0.319221
$873$	−17.7683	−0.601364
$874$	2.14632	0.0726003
$875$	−0.441519	−0.0149261
$876$	6.63709	0.224247
$877$	7.50383	0.253386	0.126693	−	0.991942i	$-0.459564\pi$
$877$	7.50383	0.253386	0.126693	+	0.991942i	$0.459564\pi$
$878$	2.85365	0.0963061
$879$	1.69024	0.0570105
$880$	−0.781934	−0.0263590
$881$	−30.2278	−1.01840	−0.509200	−	0.860648i	$-0.670059\pi$
$881$	−30.2278	−1.01840	−0.509200	+	0.860648i	$0.670059\pi$
$882$	−10.4220	−0.350928
$883$	−38.4664	−1.29450	−0.647249	−	0.762279i	$-0.724081\pi$
$883$	−38.4664	−1.29450	−0.647249	+	0.762279i	$0.724081\pi$
$884$	−1.62479	−0.0546475
$885$	8.73486	0.293619
$886$	18.8725	0.634035
$887$	−22.3512	−0.750479	−0.375239	−	0.926928i	$-0.622439\pi$
$887$	−22.3512	−0.750479	−0.375239	+	0.926928i	$0.622439\pi$
$888$	8.52855	0.286199
$889$	−7.74351	−0.259709
$890$	7.27170	0.243748
$891$	−1.61075	−0.0539620
$892$	−17.8176	−0.596576
$893$	1.98301	0.0663588
$894$	4.29300	0.143579
$895$	24.5676	0.821204
$896$	−0.441519	−0.0147501
$897$	−12.8425	−0.428800
$898$	−21.0421	−0.702184
$899$	65.6610	2.18992
$900$	−1.53151	−0.0510503
$901$	−3.20330	−0.106717
$902$	3.88989	0.129519
$903$	−0.819403	−0.0272680
$904$	3.90239	0.129792
$905$	5.73582	0.190665
$906$	7.31793	0.243122
$907$	15.0899	0.501051	0.250526	−	0.968110i	$-0.419397\pi$
$907$	15.0899	0.501051	0.250526	+	0.968110i	$0.419397\pi$
$908$	15.5741	0.516845
$909$	−1.72165	−0.0571035
$910$	−2.06453	−0.0684385
$911$	54.6511	1.81067	0.905336	−	0.424696i	$-0.139619\pi$
$911$	54.6511	1.81067	0.905336	+	0.424696i	$0.139619\pi$
$912$	−1.14759	−0.0380004
$913$	6.54175	0.216500
$914$	8.24588	0.272749
$915$	−15.8065	−0.522548
$916$	8.00310	0.264430
$917$	−1.25090	−0.0413083
$918$	−1.90811	−0.0629770
$919$	−1.10681	−0.0365104	−0.0182552	−	0.999833i	$-0.505811\pi$
$919$	−1.10681	−0.0365104	−0.0182552	+	0.999833i	$0.505811\pi$
$920$	−2.26644	−0.0747222
$921$	−17.4381	−0.574605
$922$	−28.0292	−0.923091
$923$	−45.1079	−1.48474
$924$	−0.418364	−0.0137632
$925$	7.03784	0.231403
$926$	28.1590	0.925361
$927$	6.16863	0.202605
$928$	−8.38155	−0.275138
$929$	29.9498	0.982620	0.491310	−	0.870985i	$-0.336518\pi$
$929$	29.9498	0.982620	0.491310	+	0.870985i	$0.336518\pi$
$930$	−9.49332	−0.311298
$931$	−6.44440	−0.211207
$932$	14.7387	0.482782
$933$	−12.5755	−0.411702
$934$	1.74732	0.0571742
$935$	−0.271703	−0.00888565
$936$	−7.16130	−0.234074
$937$	60.4497	1.97480	0.987402	−	0.158232i	$-0.0505793\pi$
$937$	60.4497	1.97480	0.987402	+	0.158232i	$0.0505793\pi$
$938$	4.96216	0.162020
$939$	1.47033	0.0479825
$940$	−2.09399	−0.0682984
$941$	19.9986	0.651937	0.325968	−	0.945381i	$-0.394310\pi$
$941$	19.9986	0.651937	0.325968	+	0.945381i	$0.394310\pi$
$942$	−11.8659	−0.386613
$943$	11.2749	0.367160
$944$	7.20809	0.234603
$945$	−2.42453	−0.0788700
$946$	−1.19752	−0.0389348
$947$	24.5867	0.798960	0.399480	−	0.916742i	$-0.369191\pi$
$947$	24.5867	0.798960	0.399480	+	0.916742i	$0.369191\pi$
$948$	2.33417	0.0758104
$949$	25.6103	0.831345
$950$	−0.947001	−0.0307248
$951$	15.5711	0.504927
$952$	−0.153417	−0.00497228
$953$	53.7109	1.73987	0.869934	−	0.493169i	$-0.164161\pi$
$953$	53.7109	1.73987	0.869934	+	0.493169i	$0.164161\pi$
$954$	−14.1186	−0.457107
$955$	11.5893	0.375020
$956$	0.273552	0.00884732
$957$	−7.94201	−0.256729
$958$	−26.2763	−0.848949
$959$	4.47006	0.144346
$960$	1.21181	0.0391111
$961$	30.3713	0.979720
$962$	32.9087	1.06102
$963$	−1.11612	−0.0359665
$964$	0.940060	0.0302773
$965$	−1.55342	−0.0500064
$966$	−1.21263	−0.0390157
$967$	−33.7710	−1.08600	−0.543000	−	0.839732i	$-0.682712\pi$
$967$	−33.7710	−1.08600	−0.543000	+	0.839732i	$0.682712\pi$
$968$	10.3886	0.333902
$969$	−0.398759	−0.0128100
$970$	11.6018	0.372511
$971$	7.98673	0.256306	0.128153	−	0.991754i	$-0.459095\pi$
$971$	7.98673	0.256306	0.128153	+	0.991754i	$0.459095\pi$
$972$	−13.9778	−0.448337
$973$	2.78106	0.0891567
$974$	23.8805	0.765181
$975$	5.66640	0.181470
$976$	−13.0437	−0.417519
$977$	−15.7017	−0.502341	−0.251171	−	0.967943i	$-0.580816\pi$
$977$	−15.7017	−0.502341	−0.251171	+	0.967943i	$0.580816\pi$
$978$	−18.3386	−0.586403
$979$	5.68599	0.181725
$980$	6.80506	0.217380
$981$	−14.4368	−0.460930
$982$	8.48580	0.270793
$983$	−0.551020	−0.0175748	−0.00878740	−	0.999961i	$-0.502797\pi$
$983$	−0.551020	−0.0175748	−0.00878740	+	0.999961i	$0.502797\pi$
$984$	−6.02841	−0.192179
$985$	10.9813	0.349892
$986$	−2.91239	−0.0927493
$987$	−1.12036	−0.0356616
$988$	−4.42815	−0.140878
$989$	−3.47102	−0.110372
$990$	−1.19754	−0.0380603
$991$	4.47988	0.142308	0.0711540	−	0.997465i	$-0.477332\pi$
$991$	4.47988	0.142308	0.0711540	+	0.997465i	$0.477332\pi$
$992$	−7.83398	−0.248729
$993$	9.71693	0.308357
$994$	−4.25921	−0.135094
$995$	−13.9734	−0.442986
$996$	−10.1382	−0.321240
$997$	14.7494	0.467119	0.233559	−	0.972343i	$-0.424963\pi$
$997$	14.7494	0.467119	0.233559	+	0.972343i	$0.424963\pi$
$998$	15.8989	0.503270
$999$	38.6472	1.22274

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4010.2.a.k.1.6	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4010.2.a.k.1.6	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.21181 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.21181 q^{6} +0.441519 q^{7} -1.00000 q^{8} -1.53151 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.21181 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.21181 q^{6} +0.441519 q^{7} -1.00000 q^{8} -1.53151 q^{9} +1.00000 q^{10} +0.781934 q^{11} -1.21181 q^{12} -4.67597 q^{13} -0.441519 q^{14} +1.21181 q^{15} +1.00000 q^{16} +0.347476 q^{17} +1.53151 q^{18} +0.947001 q^{19} -1.00000 q^{20} -0.535038 q^{21} -0.781934 q^{22} -2.26644 q^{23} +1.21181 q^{24} +1.00000 q^{25} +4.67597 q^{26} +5.49134 q^{27} +0.441519 q^{28} +8.38155 q^{29} -1.21181 q^{30} +7.83398 q^{31} -1.00000 q^{32} -0.947558 q^{33} -0.347476 q^{34} -0.441519 q^{35} -1.53151 q^{36} +7.03784 q^{37} -0.947001 q^{38} +5.66640 q^{39} +1.00000 q^{40} -4.97470 q^{41} +0.535038 q^{42} +1.53149 q^{43} +0.781934 q^{44} +1.53151 q^{45} +2.26644 q^{46} +2.09399 q^{47} -1.21181 q^{48} -6.80506 q^{49} -1.00000 q^{50} -0.421076 q^{51} -4.67597 q^{52} -9.21875 q^{53} -5.49134 q^{54} -0.781934 q^{55} -0.441519 q^{56} -1.14759 q^{57} -8.38155 q^{58} +7.20809 q^{59} +1.21181 q^{60} -13.0437 q^{61} -7.83398 q^{62} -0.676190 q^{63} +1.00000 q^{64} +4.67597 q^{65} +0.947558 q^{66} -11.2388 q^{67} +0.347476 q^{68} +2.74650 q^{69} +0.441519 q^{70} +9.64673 q^{71} +1.53151 q^{72} -5.47699 q^{73} -7.03784 q^{74} -1.21181 q^{75} +0.947001 q^{76} +0.345238 q^{77} -5.66640 q^{78} -1.92618 q^{79} -1.00000 q^{80} -2.05995 q^{81} +4.97470 q^{82} +8.36611 q^{83} -0.535038 q^{84} -0.347476 q^{85} -1.53149 q^{86} -10.1569 q^{87} -0.781934 q^{88} +7.27170 q^{89} -1.53151 q^{90} -2.06453 q^{91} -2.26644 q^{92} -9.49332 q^{93} -2.09399 q^{94} -0.947001 q^{95} +1.21181 q^{96} +11.6018 q^{97} +6.80506 q^{98} -1.19754 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 15 q^{2} - 6 q^{3} + 15 q^{4} - 15 q^{5} + 6 q^{6} - 5 q^{7} - 15 q^{8} + 19 q^{9}+O(q^{10}) \) 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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