LMFDB - Embedded newform 4002.2.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4002 = 2 \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4002.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:	$31.9561308889$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.16448.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 14 $ x^4 - 2x^3 - 7x^2 + 8*x + 14
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.87996$ of defining polynomial
Character	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.46575 q^{5} +1.00000 q^{6} -2.29417 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.46575 q^{5} +1.00000 q^{6} -2.29417 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.46575 q^{10} -1.24445 q^{11} -1.00000 q^{12} -6.90131 q^{13} +2.29417 q^{14} -1.46575 q^{15} +1.00000 q^{16} +5.12260 q^{17} -1.00000 q^{18} -3.85158 q^{19} +1.46575 q^{20} +2.29417 q^{21} +1.24445 q^{22} +1.00000 q^{23} +1.00000 q^{24} -2.85158 q^{25} +6.90131 q^{26} -1.00000 q^{27} -2.29417 q^{28} -1.00000 q^{29} +1.46575 q^{30} +3.24445 q^{31} -1.00000 q^{32} +1.24445 q^{33} -5.12260 q^{34} -3.36268 q^{35} +1.00000 q^{36} -10.1270 q^{37} +3.85158 q^{38} +6.90131 q^{39} -1.46575 q^{40} -3.53863 q^{41} -2.29417 q^{42} +5.36268 q^{43} -1.24445 q^{44} +1.46575 q^{45} -1.00000 q^{46} +6.29417 q^{47} -1.00000 q^{48} -1.73676 q^{49} +2.85158 q^{50} -5.12260 q^{51} -6.90131 q^{52} -1.27102 q^{53} +1.00000 q^{54} -1.82405 q^{55} +2.29417 q^{56} +3.85158 q^{57} +1.00000 q^{58} -7.36268 q^{59} -1.46575 q^{60} +13.2143 q^{61} -3.24445 q^{62} -2.29417 q^{63} +1.00000 q^{64} -10.1156 q^{65} -1.24445 q^{66} +12.6914 q^{67} +5.12260 q^{68} -1.00000 q^{69} +3.36268 q^{70} +10.1759 q^{71} -1.00000 q^{72} +15.1767 q^{73} +10.1270 q^{74} +2.85158 q^{75} -3.85158 q^{76} +2.85499 q^{77} -6.90131 q^{78} -6.17232 q^{79} +1.46575 q^{80} +1.00000 q^{81} +3.53863 q^{82} +1.61416 q^{83} +2.29417 q^{84} +7.50844 q^{85} -5.36268 q^{86} +1.00000 q^{87} +1.24445 q^{88} -15.8594 q^{89} -1.46575 q^{90} +15.8328 q^{91} +1.00000 q^{92} -3.24445 q^{93} -6.29417 q^{94} -5.64545 q^{95} +1.00000 q^{96} -2.68267 q^{97} +1.73676 q^{98} -1.24445 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 + 6 * q^7 - 4 * q^8 + 4 * q^9	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9} - 2 q^{10} - 4 q^{12} - 6 q^{14} - 2 q^{15} + 4 q^{16} - 6 q^{17} - 4 q^{18} + 2 q^{19} + 2 q^{20} - 6 q^{21} + 4 q^{23} + 4 q^{24} + 6 q^{25} - 4 q^{27} + 6 q^{28} - 4 q^{29} + 2 q^{30} + 8 q^{31} - 4 q^{32} + 6 q^{34} - 6 q^{35} + 4 q^{36} + 10 q^{37} - 2 q^{38} - 2 q^{40} + 6 q^{41} + 6 q^{42} + 14 q^{43} + 2 q^{45} - 4 q^{46} + 10 q^{47} - 4 q^{48} + 6 q^{49} - 6 q^{50} + 6 q^{51} + 4 q^{53} + 4 q^{54} - 20 q^{55} - 6 q^{56} - 2 q^{57} + 4 q^{58} - 22 q^{59} - 2 q^{60} + 28 q^{61} - 8 q^{62} + 6 q^{63} + 4 q^{64} + 12 q^{65} + 24 q^{67} - 6 q^{68} - 4 q^{69} + 6 q^{70} + 28 q^{71} - 4 q^{72} - 10 q^{74} - 6 q^{75} + 2 q^{76} - 4 q^{77} + 12 q^{79} + 2 q^{80} + 4 q^{81} - 6 q^{82} + 20 q^{83} - 6 q^{84} - 10 q^{85} - 14 q^{86} + 4 q^{87} - 24 q^{89} - 2 q^{90} + 28 q^{91} + 4 q^{92} - 8 q^{93} - 10 q^{94} + 2 q^{95} + 4 q^{96} - 32 q^{97} - 6 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 + 6 * q^7 - 4 * q^8 + 4 * q^9 - 2 * q^10 - 4 * q^12 - 6 * q^14 - 2 * q^15 + 4 * q^16 - 6 * q^17 - 4 * q^18 + 2 * q^19 + 2 * q^20 - 6 * q^21 + 4 * q^23 + 4 * q^24 + 6 * q^25 - 4 * q^27 + 6 * q^28 - 4 * q^29 + 2 * q^30 + 8 * q^31 - 4 * q^32 + 6 * q^34 - 6 * q^35 + 4 * q^36 + 10 * q^37 - 2 * q^38 - 2 * q^40 + 6 * q^41 + 6 * q^42 + 14 * q^43 + 2 * q^45 - 4 * q^46 + 10 * q^47 - 4 * q^48 + 6 * q^49 - 6 * q^50 + 6 * q^51 + 4 * q^53 + 4 * q^54 - 20 * q^55 - 6 * q^56 - 2 * q^57 + 4 * q^58 - 22 * q^59 - 2 * q^60 + 28 * q^61 - 8 * q^62 + 6 * q^63 + 4 * q^64 + 12 * q^65 + 24 * q^67 - 6 * q^68 - 4 * q^69 + 6 * q^70 + 28 * q^71 - 4 * q^72 - 10 * q^74 - 6 * q^75 + 2 * q^76 - 4 * q^77 + 12 * q^79 + 2 * q^80 + 4 * q^81 - 6 * q^82 + 20 * q^83 - 6 * q^84 - 10 * q^85 - 14 * q^86 + 4 * q^87 - 24 * q^89 - 2 * q^90 + 28 * q^91 + 4 * q^92 - 8 * q^93 - 10 * q^94 + 2 * q^95 + 4 * q^96 - 32 * q^97 - 6 * q^98

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.46575	0.655502	0.327751	−	0.944764i	$-0.393709\pi$
$5$	1.46575	0.655502	0.327751	+	0.944764i	$0.393709\pi$
$6$	1.00000	0.408248
$7$	−2.29417	−0.867116	−0.433558	−	0.901126i	$-0.642742\pi$
$7$	−2.29417	−0.867116	−0.433558	+	0.901126i	$0.642742\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.46575	−0.463510
$11$	−1.24445	−0.375217	−0.187608	−	0.982244i	$-0.560074\pi$
$11$	−1.24445	−0.375217	−0.187608	+	0.982244i	$0.560074\pi$
$12$	−1.00000	−0.288675
$13$	−6.90131	−1.91408	−0.957039	−	0.289959i	$-0.906358\pi$
$13$	−6.90131	−1.91408	−0.957039	+	0.289959i	$0.906358\pi$
$14$	2.29417	0.613144
$15$	−1.46575	−0.378454
$16$	1.00000	0.250000
$17$	5.12260	1.24241	0.621207	−	0.783647i	$-0.286643\pi$
$17$	5.12260	1.24241	0.621207	+	0.783647i	$0.286643\pi$
$18$	−1.00000	−0.235702
$19$	−3.85158	−0.883614	−0.441807	−	0.897110i	$-0.645662\pi$
$19$	−3.85158	−0.883614	−0.441807	+	0.897110i	$0.645662\pi$
$20$	1.46575	0.327751
$21$	2.29417	0.500630
$22$	1.24445	0.265318
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	−2.85158	−0.570317
$26$	6.90131	1.35346
$27$	−1.00000	−0.192450
$28$	−2.29417	−0.433558
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	1.46575	0.267608
$31$	3.24445	0.582721	0.291360	−	0.956613i	$-0.405892\pi$
$31$	3.24445	0.582721	0.291360	+	0.956613i	$0.405892\pi$
$32$	−1.00000	−0.176777
$33$	1.24445	0.216631
$34$	−5.12260	−0.878519
$35$	−3.36268	−0.568397
$36$	1.00000	0.166667
$37$	−10.1270	−1.66486	−0.832432	−	0.554127i	$-0.813052\pi$
$37$	−10.1270	−1.66486	−0.832432	+	0.554127i	$0.813052\pi$
$38$	3.85158	0.624810
$39$	6.90131	1.10509
$40$	−1.46575	−0.231755
$41$	−3.53863	−0.552641	−0.276320	−	0.961066i	$-0.589115\pi$
$41$	−3.53863	−0.552641	−0.276320	+	0.961066i	$0.589115\pi$
$42$	−2.29417	−0.353999
$43$	5.36268	0.817801	0.408901	−	0.912579i	$-0.365912\pi$
$43$	5.36268	0.817801	0.408901	+	0.912579i	$0.365912\pi$
$44$	−1.24445	−0.187608
$45$	1.46575	0.218501
$46$	−1.00000	−0.147442
$47$	6.29417	0.918100	0.459050	−	0.888411i	$-0.348190\pi$
$47$	6.29417	0.918100	0.459050	+	0.888411i	$0.348190\pi$
$48$	−1.00000	−0.144338
$49$	−1.73676	−0.248109
$50$	2.85158	0.403275
$51$	−5.12260	−0.717308
$52$	−6.90131	−0.957039
$53$	−1.27102	−0.174588	−0.0872938	−	0.996183i	$-0.527822\pi$
$53$	−1.27102	−0.174588	−0.0872938	+	0.996183i	$0.527822\pi$
$54$	1.00000	0.136083
$55$	−1.82405	−0.245955
$56$	2.29417	0.306572
$57$	3.85158	0.510155
$58$	1.00000	0.131306
$59$	−7.36268	−0.958539	−0.479270	−	0.877668i	$-0.659098\pi$
$59$	−7.36268	−0.958539	−0.479270	+	0.877668i	$0.659098\pi$
$60$	−1.46575	−0.189227
$61$	13.2143	1.69191	0.845957	−	0.533252i	$-0.179030\pi$
$61$	13.2143	1.69191	0.845957	+	0.533252i	$0.179030\pi$
$62$	−3.24445	−0.412046
$63$	−2.29417	−0.289039
$64$	1.00000	0.125000
$65$	−10.1156	−1.25468
$66$	−1.24445	−0.153182
$67$	12.6914	1.55050	0.775252	−	0.631653i	$-0.217623\pi$
$67$	12.6914	1.55050	0.775252	+	0.631653i	$0.217623\pi$
$68$	5.12260	0.621207
$69$	−1.00000	−0.120386
$70$	3.36268	0.401917
$71$	10.1759	1.20766	0.603831	−	0.797112i	$-0.293640\pi$
$71$	10.1759	1.20766	0.603831	+	0.797112i	$0.293640\pi$
$72$	−1.00000	−0.117851
$73$	15.1767	1.77630	0.888149	−	0.459556i	$-0.151991\pi$
$73$	15.1767	1.77630	0.888149	+	0.459556i	$0.151991\pi$
$74$	10.1270	1.17724
$75$	2.85158	0.329273
$76$	−3.85158	−0.441807
$77$	2.85499	0.325356
$78$	−6.90131	−0.781419
$79$	−6.17232	−0.694441	−0.347220	−	0.937784i	$-0.612874\pi$
$79$	−6.17232	−0.694441	−0.347220	+	0.937784i	$0.612874\pi$
$80$	1.46575	0.163876
$81$	1.00000	0.111111
$82$	3.53863	0.390776
$83$	1.61416	0.177177	0.0885887	−	0.996068i	$-0.471764\pi$
$83$	1.61416	0.177177	0.0885887	+	0.996068i	$0.471764\pi$
$84$	2.29417	0.250315
$85$	7.50844	0.814405
$86$	−5.36268	−0.578273
$87$	1.00000	0.107211
$88$	1.24445	0.132659
$89$	−15.8594	−1.68109	−0.840545	−	0.541742i	$-0.817765\pi$
$89$	−15.8594	−1.68109	−0.840545	+	0.541742i	$0.817765\pi$
$90$	−1.46575	−0.154503
$91$	15.8328	1.65973
$92$	1.00000	0.104257
$93$	−3.24445	−0.336434
$94$	−6.29417	−0.649195
$95$	−5.64545	−0.579211
$96$	1.00000	0.102062
$97$	−2.68267	−0.272384	−0.136192	−	0.990682i	$-0.543486\pi$
$97$	−2.68267	−0.272384	−0.136192	+	0.990682i	$0.543486\pi$
$98$	1.73676	0.175440
$99$	−1.24445	−0.125072
$100$	−2.85158	−0.285158
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	5.12260	0.507213
$103$	17.6079	1.73496	0.867478	−	0.497475i	$-0.165739\pi$
$103$	17.6079	1.73496	0.867478	+	0.497475i	$0.165739\pi$
$104$	6.90131	0.676729
$105$	3.36268	0.328164
$106$	1.27102	0.123452
$107$	14.8560	1.43618	0.718090	−	0.695950i	$-0.245017\pi$
$107$	14.8560	1.43618	0.718090	+	0.695950i	$0.245017\pi$
$108$	−1.00000	−0.0962250
$109$	4.38584	0.420087	0.210044	−	0.977692i	$-0.432639\pi$
$109$	4.38584	0.420087	0.210044	+	0.977692i	$0.432639\pi$
$110$	1.82405	0.173917
$111$	10.1270	0.961210
$112$	−2.29417	−0.216779
$113$	9.74852	0.917063	0.458532	−	0.888678i	$-0.348376\pi$
$113$	9.74852	0.917063	0.458532	+	0.888678i	$0.348376\pi$
$114$	−3.85158	−0.360734
$115$	1.46575	0.136682
$116$	−1.00000	−0.0928477
$117$	−6.90131	−0.638026
$118$	7.36268	0.677790
$119$	−11.7521	−1.07732
$120$	1.46575	0.133804
$121$	−9.45134	−0.859213
$122$	−13.2143	−1.19636
$123$	3.53863	0.319067
$124$	3.24445	0.291360
$125$	−11.5084	−1.02935
$126$	2.29417	0.204381
$127$	2.78574	0.247194	0.123597	−	0.992332i	$-0.460557\pi$
$127$	2.78574	0.247194	0.123597	+	0.992332i	$0.460557\pi$
$128$	−1.00000	−0.0883883
$129$	−5.36268	−0.472158
$130$	10.1156	0.887194
$131$	2.94762	0.257535	0.128767	−	0.991675i	$-0.458898\pi$
$131$	2.94762	0.257535	0.128767	+	0.991675i	$0.458898\pi$
$132$	1.24445	0.108316
$133$	8.83621	0.766196
$134$	−12.6914	−1.09637
$135$	−1.46575	−0.126151
$136$	−5.12260	−0.439259
$137$	4.52285	0.386413	0.193207	−	0.981158i	$-0.438111\pi$
$137$	4.52285	0.386413	0.193207	+	0.981158i	$0.438111\pi$
$138$	1.00000	0.0851257
$139$	−16.5883	−1.40701	−0.703503	−	0.710693i	$-0.748382\pi$
$139$	−16.5883	−1.40701	−0.703503	+	0.710693i	$0.748382\pi$
$140$	−3.36268	−0.284198
$141$	−6.29417	−0.530065
$142$	−10.1759	−0.853947
$143$	8.58835	0.718194
$144$	1.00000	0.0833333
$145$	−1.46575	−0.121724
$146$	−15.1767	−1.25603
$147$	1.73676	0.143246
$148$	−10.1270	−0.832432
$149$	−17.7573	−1.45473	−0.727366	−	0.686250i	$-0.759256\pi$
$149$	−17.7573	−1.45473	−0.727366	+	0.686250i	$0.759256\pi$
$150$	−2.85158	−0.232831
$151$	7.23305	0.588617	0.294309	−	0.955710i	$-0.404911\pi$
$151$	7.23305	0.588617	0.294309	+	0.955710i	$0.404911\pi$
$152$	3.85158	0.312405
$153$	5.12260	0.414138
$154$	−2.85499	−0.230062
$155$	4.75555	0.381975
$156$	6.90131	0.552547
$157$	2.46137	0.196439	0.0982195	−	0.995165i	$-0.468685\pi$
$157$	2.46137	0.196439	0.0982195	+	0.995165i	$0.468685\pi$
$158$	6.17232	0.491044
$159$	1.27102	0.100798
$160$	−1.46575	−0.115877
$161$	−2.29417	−0.180806
$162$	−1.00000	−0.0785674
$163$	−9.68439	−0.758540	−0.379270	−	0.925286i	$-0.623825\pi$
$163$	−9.68439	−0.758540	−0.379270	+	0.925286i	$0.623825\pi$
$164$	−3.53863	−0.276320
$165$	1.82405	0.142002
$166$	−1.61416	−0.125283
$167$	20.8799	1.61573	0.807866	−	0.589366i	$-0.200622\pi$
$167$	20.8799	1.61573	0.807866	+	0.589366i	$0.200622\pi$
$168$	−2.29417	−0.176999
$169$	34.6280	2.66370
$170$	−7.50844	−0.575871
$171$	−3.85158	−0.294538
$172$	5.36268	0.408901
$173$	−6.26399	−0.476242	−0.238121	−	0.971235i	$-0.576531\pi$
$173$	−6.26399	−0.476242	−0.238121	+	0.971235i	$0.576531\pi$
$174$	−1.00000	−0.0758098
$175$	6.54203	0.494531
$176$	−1.24445	−0.0938041
$177$	7.36268	0.553413
$178$	15.8594	1.18871
$179$	−5.95368	−0.444999	−0.222500	−	0.974933i	$-0.571422\pi$
$179$	−5.95368	−0.444999	−0.222500	+	0.974933i	$0.571422\pi$
$180$	1.46575	0.109250
$181$	−7.63029	−0.567155	−0.283578	−	0.958949i	$-0.591521\pi$
$181$	−7.63029	−0.567155	−0.283578	+	0.958949i	$0.591521\pi$
$182$	−15.8328	−1.17361
$183$	−13.2143	−0.976827
$184$	−1.00000	−0.0737210
$185$	−14.8436	−1.09132
$186$	3.24445	0.237895
$187$	−6.37483	−0.466174
$188$	6.29417	0.459050
$189$	2.29417	0.166877
$190$	5.64545	0.409564
$191$	−12.2300	−0.884935	−0.442467	−	0.896785i	$-0.645897\pi$
$191$	−12.2300	−0.884935	−0.442467	+	0.896785i	$0.645897\pi$
$192$	−1.00000	−0.0721688
$193$	7.69442	0.553857	0.276928	−	0.960891i	$-0.410684\pi$
$193$	7.69442	0.553857	0.276928	+	0.960891i	$0.410684\pi$
$194$	2.68267	0.192604
$195$	10.1156	0.724391
$196$	−1.73676	−0.124055
$197$	13.0497	0.929754	0.464877	−	0.885375i	$-0.346099\pi$
$197$	13.0497	0.929754	0.464877	+	0.885375i	$0.346099\pi$
$198$	1.24445	0.0884394
$199$	−6.45134	−0.457323	−0.228662	−	0.973506i	$-0.573435\pi$
$199$	−6.45134	−0.457323	−0.228662	+	0.973506i	$0.573435\pi$
$200$	2.85158	0.201638
$201$	−12.6914	−0.895183
$202$	10.0000	0.703598
$203$	2.29417	0.161019
$204$	−5.12260	−0.358654
$205$	−5.18673	−0.362257
$206$	−17.6079	−1.22680
$207$	1.00000	0.0695048
$208$	−6.90131	−0.478520
$209$	4.79311	0.331547
$210$	−3.36268	−0.232047
$211$	10.5696	0.727639	0.363819	−	0.931469i	$-0.381473\pi$
$211$	10.5696	0.727639	0.363819	+	0.931469i	$0.381473\pi$
$212$	−1.27102	−0.0872938
$213$	−10.1759	−0.697244
$214$	−14.8560	−1.01553
$215$	7.86033	0.536070
$216$	1.00000	0.0680414
$217$	−7.44334	−0.505287
$218$	−4.38584	−0.297046
$219$	−15.1767	−1.02555
$220$	−1.82405	−0.122978
$221$	−35.3526	−2.37808
$222$	−10.1270	−0.679678
$223$	−2.34315	−0.156909	−0.0784543	−	0.996918i	$-0.524998\pi$
$223$	−2.34315	−0.156909	−0.0784543	+	0.996918i	$0.524998\pi$
$224$	2.29417	0.153286
$225$	−2.85158	−0.190106
$226$	−9.74852	−0.648462
$227$	7.77871	0.516291	0.258145	−	0.966106i	$-0.416889\pi$
$227$	7.77871	0.516291	0.258145	+	0.966106i	$0.416889\pi$
$228$	3.85158	0.255077
$229$	21.5011	1.42083	0.710415	−	0.703783i	$-0.248507\pi$
$229$	21.5011	1.42083	0.710415	+	0.703783i	$0.248507\pi$
$230$	−1.46575	−0.0966485
$231$	−2.85499	−0.187845
$232$	1.00000	0.0656532
$233$	7.90056	0.517583	0.258791	−	0.965933i	$-0.416676\pi$
$233$	7.90056	0.517583	0.258791	+	0.965933i	$0.416676\pi$
$234$	6.90131	0.451153
$235$	9.22567	0.601816
$236$	−7.36268	−0.479270
$237$	6.17232	0.400935
$238$	11.7521	0.761778
$239$	26.2452	1.69766	0.848831	−	0.528665i	$-0.177307\pi$
$239$	26.2452	1.69766	0.848831	+	0.528665i	$0.177307\pi$
$240$	−1.46575	−0.0946136
$241$	0.0187836	0.00120996	0.000604979	−	1.00000i	$-0.499807\pi$
$241$	0.0187836	0.00120996	0.000604979		1.00000i	$0.499807\pi$
$242$	9.45134	0.607555
$243$	−1.00000	−0.0641500
$244$	13.2143	0.845957
$245$	−2.54566	−0.162636
$246$	−3.53863	−0.225615
$247$	26.5810	1.69131
$248$	−3.24445	−0.206023
$249$	−1.61416	−0.102293
$250$	11.5084	0.727858
$251$	8.30558	0.524243	0.262122	−	0.965035i	$-0.415578\pi$
$251$	8.30558	0.524243	0.262122	+	0.965035i	$0.415578\pi$
$252$	−2.29417	−0.144519
$253$	−1.24445	−0.0782381
$254$	−2.78574	−0.174793
$255$	−7.50844	−0.470197
$256$	1.00000	0.0625000
$257$	−18.8106	−1.17337	−0.586687	−	0.809814i	$-0.699568\pi$
$257$	−18.8106	−1.17337	−0.586687	+	0.809814i	$0.699568\pi$
$258$	5.36268	0.333866
$259$	23.2330	1.44363
$260$	−10.1156	−0.627341
$261$	−1.00000	−0.0618984
$262$	−2.94762	−0.182105
$263$	11.2294	0.692436	0.346218	−	0.938154i	$-0.387466\pi$
$263$	11.2294	0.692436	0.346218	+	0.938154i	$0.387466\pi$
$264$	−1.24445	−0.0765908
$265$	−1.86299	−0.114443
$266$	−8.83621	−0.541783
$267$	15.8594	0.970577
$268$	12.6914	0.775252
$269$	−5.39021	−0.328647	−0.164324	−	0.986407i	$-0.552544\pi$
$269$	−5.39021	−0.328647	−0.164324	+	0.986407i	$0.552544\pi$
$270$	1.46575	0.0892025
$271$	31.0397	1.88553	0.942763	−	0.333463i	$-0.108217\pi$
$271$	31.0397	1.88553	0.942763	+	0.333463i	$0.108217\pi$
$272$	5.12260	0.310603
$273$	−15.8328	−0.958245
$274$	−4.52285	−0.273235
$275$	3.54866	0.213992
$276$	−1.00000	−0.0601929
$277$	−26.9564	−1.61965	−0.809826	−	0.586671i	$-0.800438\pi$
$277$	−26.9564	−1.61965	−0.809826	+	0.586671i	$0.800438\pi$
$278$	16.5883	0.994903
$279$	3.24445	0.194240
$280$	3.36268	0.200959
$281$	3.80261	0.226845	0.113422	−	0.993547i	$-0.463819\pi$
$281$	3.80261	0.226845	0.113422	+	0.993547i	$0.463819\pi$
$282$	6.29417	0.374813
$283$	24.1348	1.43466	0.717331	−	0.696732i	$-0.245363\pi$
$283$	24.1348	1.43466	0.717331	+	0.696732i	$0.245363\pi$
$284$	10.1759	0.603831
$285$	5.64545	0.334408
$286$	−8.58835	−0.507840
$287$	8.11823	0.479204
$288$	−1.00000	−0.0589256
$289$	9.24105	0.543591
$290$	1.46575	0.0860716
$291$	2.68267	0.157261
$292$	15.1767	0.888149
$293$	19.4970	1.13903	0.569514	−	0.821981i	$-0.307131\pi$
$293$	19.4970	1.13903	0.569514	+	0.821981i	$0.307131\pi$
$294$	−1.73676	−0.101290
$295$	−10.7918	−0.628325
$296$	10.1270	0.588619
$297$	1.24445	0.0722105
$298$	17.7573	1.02865
$299$	−6.90131	−0.399113
$300$	2.85158	0.164636
$301$	−12.3029	−0.709129
$302$	−7.23305	−0.416215
$303$	10.0000	0.574485
$304$	−3.85158	−0.220904
$305$	19.3688	1.10905
$306$	−5.12260	−0.292840
$307$	5.86299	0.334618	0.167309	−	0.985904i	$-0.446492\pi$
$307$	5.86299	0.334618	0.167309	+	0.985904i	$0.446492\pi$
$308$	2.85499	0.162678
$309$	−17.6079	−1.00168
$310$	−4.75555	−0.270097
$311$	10.3317	0.585859	0.292930	−	0.956134i	$-0.405370\pi$
$311$	10.3317	0.585859	0.292930	+	0.956134i	$0.405370\pi$
$312$	−6.90131	−0.390710
$313$	27.8833	1.57606	0.788028	−	0.615640i	$-0.211102\pi$
$313$	27.8833	1.57606	0.788028	+	0.615640i	$0.211102\pi$
$314$	−2.46137	−0.138903
$315$	−3.36268	−0.189466
$316$	−6.17232	−0.347220
$317$	−5.07588	−0.285090	−0.142545	−	0.989788i	$-0.545529\pi$
$317$	−5.07588	−0.285090	−0.142545	+	0.989788i	$0.545529\pi$
$318$	−1.27102	−0.0712751
$319$	1.24445	0.0696760
$320$	1.46575	0.0819378
$321$	−14.8560	−0.829179
$322$	2.29417	0.127849
$323$	−19.7301	−1.09781
$324$	1.00000	0.0555556
$325$	19.6797	1.09163
$326$	9.68439	0.536369
$327$	−4.38584	−0.242537
$328$	3.53863	0.195388
$329$	−14.4399	−0.796099
$330$	−1.82405	−0.100411
$331$	−25.2345	−1.38702	−0.693508	−	0.720449i	$-0.743936\pi$
$331$	−25.2345	−1.38702	−0.693508	+	0.720449i	$0.743936\pi$
$332$	1.61416	0.0885887
$333$	−10.1270	−0.554955
$334$	−20.8799	−1.14250
$335$	18.6024	1.01636
$336$	2.29417	0.125157
$337$	2.99699	0.163257	0.0816284	−	0.996663i	$-0.473988\pi$
$337$	2.99699	0.163257	0.0816284	+	0.996663i	$0.473988\pi$
$338$	−34.6280	−1.88352
$339$	−9.74852	−0.529467
$340$	7.50844	0.407202
$341$	−4.03757	−0.218647
$342$	3.85158	0.208270
$343$	20.0437	1.08226
$344$	−5.36268	−0.289136
$345$	−1.46575	−0.0789132
$346$	6.26399	0.336754
$347$	−29.0195	−1.55785	−0.778925	−	0.627117i	$-0.784235\pi$
$347$	−29.0195	−1.55785	−0.778925	+	0.627117i	$0.784235\pi$
$348$	1.00000	0.0536056
$349$	17.2606	0.923938	0.461969	−	0.886896i	$-0.347143\pi$
$349$	17.2606	0.923938	0.461969	+	0.886896i	$0.347143\pi$
$350$	−6.54203	−0.349686
$351$	6.90131	0.368365
$352$	1.24445	0.0663295
$353$	14.6259	0.778459	0.389229	−	0.921141i	$-0.372741\pi$
$353$	14.6259	0.778459	0.389229	+	0.921141i	$0.372741\pi$
$354$	−7.36268	−0.391322
$355$	14.9154	0.791625
$356$	−15.8594	−0.840545
$357$	11.7521	0.621989
$358$	5.95368	0.314662
$359$	15.6189	0.824333	0.412167	−	0.911108i	$-0.364772\pi$
$359$	15.6189	0.824333	0.412167	+	0.911108i	$0.364772\pi$
$360$	−1.46575	−0.0772517
$361$	−4.16529	−0.219226
$362$	7.63029	0.401039
$363$	9.45134	0.496067
$364$	15.8328	0.829864
$365$	22.2452	1.16437
$366$	13.2143	0.690721
$367$	−35.0750	−1.83090	−0.915450	−	0.402432i	$-0.868165\pi$
$367$	−35.0750	−1.83090	−0.915450	+	0.402432i	$0.868165\pi$
$368$	1.00000	0.0521286
$369$	−3.53863	−0.184214
$370$	14.8436	0.771681
$371$	2.91593	0.151388
$372$	−3.24445	−0.168217
$373$	29.6552	1.53549	0.767743	−	0.640758i	$-0.221380\pi$
$373$	29.6552	1.53549	0.767743	+	0.640758i	$0.221380\pi$
$374$	6.37483	0.329635
$375$	11.5084	0.594293
$376$	−6.29417	−0.324597
$377$	6.90131	0.355435
$378$	−2.29417	−0.118000
$379$	−10.6259	−0.545817	−0.272908	−	0.962040i	$-0.587986\pi$
$379$	−10.6259	−0.545817	−0.272908	+	0.962040i	$0.587986\pi$
$380$	−5.64545	−0.289605
$381$	−2.78574	−0.142718
$382$	12.2300	0.625743
$383$	1.56479	0.0799570	0.0399785	−	0.999201i	$-0.487271\pi$
$383$	1.56479	0.0799570	0.0399785	+	0.999201i	$0.487271\pi$
$384$	1.00000	0.0510310
$385$	4.18470	0.213272
$386$	−7.69442	−0.391636
$387$	5.36268	0.272600
$388$	−2.68267	−0.136192
$389$	15.9249	0.807423	0.403711	−	0.914886i	$-0.367720\pi$
$389$	15.9249	0.807423	0.403711	+	0.914886i	$0.367720\pi$
$390$	−10.1156	−0.512222
$391$	5.12260	0.259061
$392$	1.73676	0.0877198
$393$	−2.94762	−0.148688
$394$	−13.0497	−0.657435
$395$	−9.04707	−0.455207
$396$	−1.24445	−0.0625361
$397$	−14.8785	−0.746730	−0.373365	−	0.927684i	$-0.621796\pi$
$397$	−14.8785	−0.746730	−0.373365	+	0.927684i	$0.621796\pi$
$398$	6.45134	0.323376
$399$	−8.83621	−0.442364
$400$	−2.85158	−0.142579
$401$	26.9706	1.34685	0.673423	−	0.739258i	$-0.264823\pi$
$401$	26.9706	1.34685	0.673423	+	0.739258i	$0.264823\pi$
$402$	12.6914	0.632990
$403$	−22.3910	−1.11537
$404$	−10.0000	−0.497519
$405$	1.46575	0.0728336
$406$	−2.29417	−0.113858
$407$	12.6025	0.624685
$408$	5.12260	0.253607
$409$	−4.12088	−0.203765	−0.101882	−	0.994796i	$-0.532487\pi$
$409$	−4.12088	−0.203765	−0.101882	+	0.994796i	$0.532487\pi$
$410$	5.18673	0.256154
$411$	−4.52285	−0.223096
$412$	17.6079	0.867478
$413$	16.8913	0.831165
$414$	−1.00000	−0.0491473
$415$	2.36595	0.116140
$416$	6.90131	0.338364
$417$	16.5883	0.812335
$418$	−4.79311	−0.234439
$419$	6.08428	0.297237	0.148618	−	0.988895i	$-0.452517\pi$
$419$	6.08428	0.297237	0.148618	+	0.988895i	$0.452517\pi$
$420$	3.36268	0.164082
$421$	−2.10669	−0.102674	−0.0513369	−	0.998681i	$-0.516348\pi$
$421$	−2.10669	−0.102674	−0.0513369	+	0.998681i	$0.516348\pi$
$422$	−10.5696	−0.514518
$423$	6.29417	0.306033
$424$	1.27102	0.0617260
$425$	−14.6075	−0.708569
$426$	10.1759	0.493026
$427$	−30.3158	−1.46709
$428$	14.8560	0.718090
$429$	−8.58835	−0.414649
$430$	−7.86033	−0.379059
$431$	16.9013	0.814107	0.407054	−	0.913404i	$-0.366556\pi$
$431$	16.9013	0.814107	0.407054	+	0.913404i	$0.366556\pi$
$432$	−1.00000	−0.0481125
$433$	28.6700	1.37779	0.688896	−	0.724861i	$-0.258096\pi$
$433$	28.6700	1.37779	0.688896	+	0.724861i	$0.258096\pi$
$434$	7.44334	0.357292
$435$	1.46575	0.0702772
$436$	4.38584	0.210044
$437$	−3.85158	−0.184246
$438$	15.1767	0.725171
$439$	24.1182	1.15110	0.575550	−	0.817767i	$-0.304788\pi$
$439$	24.1182	1.15110	0.575550	+	0.817767i	$0.304788\pi$
$440$	1.82405	0.0869583
$441$	−1.73676	−0.0827030
$442$	35.3526	1.68155
$443$	0.259263	0.0123180	0.00615898	−	0.999981i	$-0.498040\pi$
$443$	0.259263	0.0123180	0.00615898	+	0.999981i	$0.498040\pi$
$444$	10.1270	0.480605
$445$	−23.2458	−1.10196
$446$	2.34315	0.110951
$447$	17.7573	0.839890
$448$	−2.29417	−0.108390
$449$	−25.3788	−1.19770	−0.598850	−	0.800861i	$-0.704375\pi$
$449$	−25.3788	−1.19770	−0.598850	+	0.800861i	$0.704375\pi$
$450$	2.85158	0.134425
$451$	4.40365	0.207360
$452$	9.74852	0.458532
$453$	−7.23305	−0.339838
$454$	−7.77871	−0.365073
$455$	23.2069	1.08796
$456$	−3.85158	−0.180367
$457$	4.72064	0.220822	0.110411	−	0.993886i	$-0.464783\pi$
$457$	4.72064	0.220822	0.110411	+	0.993886i	$0.464783\pi$
$458$	−21.5011	−1.00468
$459$	−5.12260	−0.239103
$460$	1.46575	0.0683408
$461$	7.48966	0.348828	0.174414	−	0.984672i	$-0.444197\pi$
$461$	7.48966	0.348828	0.174414	+	0.984672i	$0.444197\pi$
$462$	2.85499	0.132826
$463$	−11.1304	−0.517273	−0.258636	−	0.965975i	$-0.583273\pi$
$463$	−11.1304	−0.517273	−0.258636	+	0.965975i	$0.583273\pi$
$464$	−1.00000	−0.0464238
$465$	−4.75555	−0.220533
$466$	−7.90056	−0.365986
$467$	−31.2835	−1.44763	−0.723814	−	0.689995i	$-0.757613\pi$
$467$	−31.2835	−1.44763	−0.723814	+	0.689995i	$0.757613\pi$
$468$	−6.90131	−0.319013
$469$	−29.1163	−1.34447
$470$	−9.22567	−0.425548
$471$	−2.46137	−0.113414
$472$	7.36268	0.338895
$473$	−6.67360	−0.306852
$474$	−6.17232	−0.283504
$475$	10.9831	0.503940
$476$	−11.7521	−0.538659
$477$	−1.27102	−0.0581959
$478$	−26.2452	−1.20043
$479$	−2.24883	−0.102752	−0.0513758	−	0.998679i	$-0.516361\pi$
$479$	−2.24883	−0.102752	−0.0513758	+	0.998679i	$0.516361\pi$
$480$	1.46575	0.0669019
$481$	69.8894	3.18668
$482$	−0.0187836	−0.000855570 0
$483$	2.29417	0.104389
$484$	−9.45134	−0.429606
$485$	−3.93211	−0.178548
$486$	1.00000	0.0453609
$487$	32.8295	1.48765	0.743824	−	0.668376i	$-0.233010\pi$
$487$	32.8295	1.48765	0.743824	+	0.668376i	$0.233010\pi$
$488$	−13.2143	−0.598182
$489$	9.68439	0.437943
$490$	2.54566	0.115001
$491$	9.38146	0.423380	0.211690	−	0.977337i	$-0.432103\pi$
$491$	9.38146	0.423380	0.211690	+	0.977337i	$0.432103\pi$
$492$	3.53863	0.159534
$493$	−5.12260	−0.230710
$494$	−26.5810	−1.19593
$495$	−1.82405	−0.0819851
$496$	3.24445	0.145680
$497$	−23.3454	−1.04718
$498$	1.61416	0.0723323
$499$	19.0008	0.850590	0.425295	−	0.905055i	$-0.360170\pi$
$499$	19.0008	0.850590	0.425295	+	0.905055i	$0.360170\pi$
$500$	−11.5084	−0.514673
$501$	−20.8799	−0.932844
$502$	−8.30558	−0.370696
$503$	−27.9708	−1.24716	−0.623578	−	0.781761i	$-0.714322\pi$
$503$	−27.9708	−1.24716	−0.623578	+	0.781761i	$0.714322\pi$
$504$	2.29417	0.102191
$505$	−14.6575	−0.652249
$506$	1.24445	0.0553227
$507$	−34.6280	−1.53789
$508$	2.78574	0.123597
$509$	24.6778	1.09382	0.546911	−	0.837191i	$-0.315804\pi$
$509$	24.6778	1.09382	0.546911	+	0.837191i	$0.315804\pi$
$510$	7.50844	0.332479
$511$	−34.8180	−1.54026
$512$	−1.00000	−0.0441942
$513$	3.85158	0.170052
$514$	18.8106	0.829701
$515$	25.8087	1.13727
$516$	−5.36268	−0.236079
$517$	−7.83280	−0.344486
$518$	−23.2330	−1.02080
$519$	6.26399	0.274958
$520$	10.1156	0.443597
$521$	1.59441	0.0698524	0.0349262	−	0.999390i	$-0.488880\pi$
$521$	1.59441	0.0698524	0.0349262	+	0.999390i	$0.488880\pi$
$522$	1.00000	0.0437688
$523$	4.78936	0.209424	0.104712	−	0.994503i	$-0.466608\pi$
$523$	4.78936	0.209424	0.104712	+	0.994503i	$0.466608\pi$
$524$	2.94762	0.128767
$525$	−6.54203	−0.285518
$526$	−11.2294	−0.489626
$527$	16.6200	0.723980
$528$	1.24445	0.0541578
$529$	1.00000	0.0434783
$530$	1.86299	0.0809231
$531$	−7.36268	−0.319513
$532$	8.83621	0.383098
$533$	24.4212	1.05780
$534$	−15.8594	−0.686302
$535$	21.7751	0.941419
$536$	−12.6914	−0.548186
$537$	5.95368	0.256920
$538$	5.39021	0.232389
$539$	2.16132	0.0930947
$540$	−1.46575	−0.0630757
$541$	17.5312	0.753727	0.376864	−	0.926269i	$-0.377003\pi$
$541$	17.5312	0.753727	0.376864	+	0.926269i	$0.377003\pi$
$542$	−31.0397	−1.33327
$543$	7.63029	0.327447
$544$	−5.12260	−0.219630
$545$	6.42853	0.275368
$546$	15.8328	0.677581
$547$	10.3910	0.444285	0.222143	−	0.975014i	$-0.428695\pi$
$547$	10.3910	0.444285	0.222143	+	0.975014i	$0.428695\pi$
$548$	4.52285	0.193207
$549$	13.2143	0.563971
$550$	−3.54866	−0.151315
$551$	3.85158	0.164083
$552$	1.00000	0.0425628
$553$	14.1604	0.602161
$554$	26.9564	1.14527
$555$	14.8436	0.630075
$556$	−16.5883	−0.703503
$557$	14.6497	0.620727	0.310364	−	0.950618i	$-0.399549\pi$
$557$	14.6497	0.620727	0.310364	+	0.950618i	$0.399549\pi$
$558$	−3.24445	−0.137349
$559$	−37.0095	−1.56534
$560$	−3.36268	−0.142099
$561$	6.37483	0.269146
$562$	−3.80261	−0.160404
$563$	−7.32246	−0.308605	−0.154302	−	0.988024i	$-0.549313\pi$
$563$	−7.32246	−0.308605	−0.154302	+	0.988024i	$0.549313\pi$
$564$	−6.29417	−0.265033
$565$	14.2889	0.601137
$566$	−24.1348	−1.01446
$567$	−2.29417	−0.0963463
$568$	−10.1759	−0.426973
$569$	−23.2771	−0.975827	−0.487914	−	0.872892i	$-0.662242\pi$
$569$	−23.2771	−0.975827	−0.487914	+	0.872892i	$0.662242\pi$
$570$	−5.64545	−0.236462
$571$	25.8842	1.08322	0.541611	−	0.840629i	$-0.317815\pi$
$571$	25.8842	1.08322	0.541611	+	0.840629i	$0.317815\pi$
$572$	8.58835	0.359097
$573$	12.2300	0.510917
$574$	−8.11823	−0.338848
$575$	−2.85158	−0.118919
$576$	1.00000	0.0416667
$577$	−7.93225	−0.330224	−0.165112	−	0.986275i	$-0.552799\pi$
$577$	−7.93225	−0.330224	−0.165112	+	0.986275i	$0.552799\pi$
$578$	−9.24105	−0.384377
$579$	−7.69442	−0.319769
$580$	−1.46575	−0.0608618
$581$	−3.70317	−0.153633
$582$	−2.68267	−0.111200
$583$	1.58172	0.0655081
$584$	−15.1767	−0.628016
$585$	−10.1156	−0.418227
$586$	−19.4970	−0.805415
$587$	−3.95103	−0.163076	−0.0815382	−	0.996670i	$-0.525983\pi$
$587$	−3.95103	−0.163076	−0.0815382	+	0.996670i	$0.525983\pi$
$588$	1.73676	0.0716229
$589$	−12.4963	−0.514900
$590$	10.7918	0.444293
$591$	−13.0497	−0.536794
$592$	−10.1270	−0.416216
$593$	−16.3910	−0.673096	−0.336548	−	0.941666i	$-0.609259\pi$
$593$	−16.3910	−0.673096	−0.336548	+	0.941666i	$0.609259\pi$
$594$	−1.24445	−0.0510605
$595$	−17.2257	−0.706184
$596$	−17.7573	−0.727366
$597$	6.45134	0.264036
$598$	6.90131	0.282215
$599$	3.40440	0.139100	0.0695501	−	0.997578i	$-0.477844\pi$
$599$	3.40440	0.139100	0.0695501	+	0.997578i	$0.477844\pi$
$600$	−2.85158	−0.116415
$601$	−32.9417	−1.34372	−0.671861	−	0.740677i	$-0.734505\pi$
$601$	−32.9417	−1.34372	−0.671861	+	0.740677i	$0.734505\pi$
$602$	12.3029	0.501430
$603$	12.6914	0.516834
$604$	7.23305	0.294309
$605$	−13.8533	−0.563216
$606$	−10.0000	−0.406222
$607$	−29.0398	−1.17869	−0.589345	−	0.807882i	$-0.700614\pi$
$607$	−29.0398	−1.17869	−0.589345	+	0.807882i	$0.700614\pi$
$608$	3.85158	0.156202
$609$	−2.29417	−0.0929646
$610$	−19.3688	−0.784219
$611$	−43.4380	−1.75731
$612$	5.12260	0.207069
$613$	−8.49403	−0.343071	−0.171535	−	0.985178i	$-0.554873\pi$
$613$	−8.49403	−0.343071	−0.171535	+	0.985178i	$0.554873\pi$
$614$	−5.86299	−0.236611
$615$	5.18673	0.209149
$616$	−2.85499	−0.115031
$617$	−7.52612	−0.302990	−0.151495	−	0.988458i	$-0.548409\pi$
$617$	−7.52612	−0.302990	−0.151495	+	0.988458i	$0.548409\pi$
$618$	17.6079	0.708293
$619$	41.1365	1.65341	0.826707	−	0.562632i	$-0.190211\pi$
$619$	41.1365	1.65341	0.826707	+	0.562632i	$0.190211\pi$
$620$	4.75555	0.190987
$621$	−1.00000	−0.0401286
$622$	−10.3317	−0.414265
$623$	36.3841	1.45770
$624$	6.90131	0.276273
$625$	−2.61054	−0.104422
$626$	−27.8833	−1.11444
$627$	−4.79311	−0.191419
$628$	2.46137	0.0982195
$629$	−51.8765	−2.06845
$630$	3.36268	0.133972
$631$	22.8221	0.908535	0.454268	−	0.890865i	$-0.349901\pi$
$631$	22.8221	0.908535	0.454268	+	0.890865i	$0.349901\pi$
$632$	6.17232	0.245522
$633$	−10.5696	−0.420103
$634$	5.07588	0.201589
$635$	4.08318	0.162036
$636$	1.27102	0.0503991
$637$	11.9859	0.474900
$638$	−1.24445	−0.0492683
$639$	10.1759	0.402554
$640$	−1.46575	−0.0579387
$641$	−36.1999	−1.42981	−0.714904	−	0.699222i	$-0.753530\pi$
$641$	−36.1999	−1.42981	−0.714904	+	0.699222i	$0.753530\pi$
$642$	14.8560	0.586318
$643$	49.4021	1.94823	0.974115	−	0.226052i	$-0.0725819\pi$
$643$	49.4021	1.94823	0.974115	+	0.226052i	$0.0725819\pi$
$644$	−2.29417	−0.0904031
$645$	−7.86033	−0.309500
$646$	19.7301	0.776272
$647$	27.2445	1.07109	0.535545	−	0.844507i	$-0.320106\pi$
$647$	27.2445	1.07109	0.535545	+	0.844507i	$0.320106\pi$
$648$	−1.00000	−0.0392837
$649$	9.16251	0.359660
$650$	−19.6797	−0.771900
$651$	7.44334	0.291727
$652$	−9.68439	−0.379270
$653$	46.9277	1.83642	0.918211	−	0.396091i	$-0.129634\pi$
$653$	46.9277	1.83642	0.918211	+	0.396091i	$0.129634\pi$
$654$	4.38584	0.171500
$655$	4.32047	0.168815
$656$	−3.53863	−0.138160
$657$	15.1767	0.592099
$658$	14.4399	0.562927
$659$	−35.4910	−1.38253	−0.691267	−	0.722599i	$-0.742947\pi$
$659$	−35.4910	−1.38253	−0.691267	+	0.722599i	$0.742947\pi$
$660$	1.82405	0.0710012
$661$	−1.84380	−0.0717157	−0.0358578	−	0.999357i	$-0.511416\pi$
$661$	−1.84380	−0.0717157	−0.0358578	+	0.999357i	$0.511416\pi$
$662$	25.2345	0.980769
$663$	35.3526	1.37298
$664$	−1.61416	−0.0626416
$665$	12.9516	0.502243
$666$	10.1270	0.392412
$667$	−1.00000	−0.0387202
$668$	20.8799	0.807866
$669$	2.34315	0.0905912
$670$	−18.6024	−0.718674
$671$	−16.4445	−0.634834
$672$	−2.29417	−0.0884997
$673$	−46.4039	−1.78874	−0.894369	−	0.447330i	$-0.852375\pi$
$673$	−46.4039	−1.78874	−0.894369	+	0.447330i	$0.852375\pi$
$674$	−2.99699	−0.115440
$675$	2.85158	0.109758
$676$	34.6280	1.33185
$677$	−12.7945	−0.491732	−0.245866	−	0.969304i	$-0.579072\pi$
$677$	−12.7945	−0.491732	−0.245866	+	0.969304i	$0.579072\pi$
$678$	9.74852	0.374390
$679$	6.15451	0.236188
$680$	−7.50844	−0.287935
$681$	−7.77871	−0.298081
$682$	4.03757	0.154606
$683$	−16.0665	−0.614766	−0.307383	−	0.951586i	$-0.599453\pi$
$683$	−16.0665	−0.614766	−0.307383	+	0.951586i	$0.599453\pi$
$684$	−3.85158	−0.147269
$685$	6.62935	0.253295
$686$	−20.0437	−0.765270
$687$	−21.5011	−0.820317
$688$	5.36268	0.204450
$689$	8.77168	0.334174
$690$	1.46575	0.0558000
$691$	−46.2841	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$691$	−46.2841	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$692$	−6.26399	−0.238121
$693$	2.85499	0.108452
$694$	29.0195	1.10157
$695$	−24.3143	−0.922295
$696$	−1.00000	−0.0379049
$697$	−18.1270	−0.686608
$698$	−17.2606	−0.653323
$699$	−7.90056	−0.298826
$700$	6.54203	0.247266
$701$	34.0864	1.28742	0.643712	−	0.765268i	$-0.277394\pi$
$701$	34.0864	1.28742	0.643712	+	0.765268i	$0.277394\pi$
$702$	−6.90131	−0.260473
$703$	39.0049	1.47110
$704$	−1.24445	−0.0469021
$705$	−9.22567	−0.347459
$706$	−14.6259	−0.550454
$707$	22.9417	0.862813
$708$	7.36268	0.276706
$709$	10.0213	0.376356	0.188178	−	0.982135i	$-0.439742\pi$
$709$	10.0213	0.376356	0.188178	+	0.982135i	$0.439742\pi$
$710$	−14.9154	−0.559764
$711$	−6.17232	−0.231480
$712$	15.8594	0.594355
$713$	3.24445	0.121506
$714$	−11.7521	−0.439813
$715$	12.5883	0.470778
$716$	−5.95368	−0.222500
$717$	−26.2452	−0.980145
$718$	−15.6189	−0.582892
$719$	−14.1156	−0.526422	−0.263211	−	0.964738i	$-0.584782\pi$
$719$	−14.1156	−0.526422	−0.263211	+	0.964738i	$0.584782\pi$
$720$	1.46575	0.0546252
$721$	−40.3956	−1.50441
$722$	4.16529	0.155016
$723$	−0.0187836	−0.000698570 0
$724$	−7.63029	−0.283578
$725$	2.85158	0.105905
$726$	−9.45134	−0.350772
$727$	−13.2334	−0.490801	−0.245401	−	0.969422i	$-0.578920\pi$
$727$	−13.2334	−0.490801	−0.245401	+	0.969422i	$0.578920\pi$
$728$	−15.8328	−0.586803
$729$	1.00000	0.0370370
$730$	−22.2452	−0.823332
$731$	27.4709	1.01605
$732$	−13.2143	−0.488413
$733$	−23.8645	−0.881455	−0.440728	−	0.897641i	$-0.645279\pi$
$733$	−23.8645	−0.881455	−0.440728	+	0.897641i	$0.645279\pi$
$734$	35.0750	1.29464
$735$	2.54566	0.0938980
$736$	−1.00000	−0.0368605
$737$	−15.7939	−0.581775
$738$	3.53863	0.130259
$739$	30.3682	1.11711	0.558555	−	0.829467i	$-0.311356\pi$
$739$	30.3682	1.11711	0.558555	+	0.829467i	$0.311356\pi$
$740$	−14.8436	−0.545661
$741$	−26.5810	−0.976476
$742$	−2.91593	−0.107047
$743$	−44.7277	−1.64090	−0.820450	−	0.571718i	$-0.806277\pi$
$743$	−44.7277	−1.64090	−0.820450	+	0.571718i	$0.806277\pi$
$744$	3.24445	0.118947
$745$	−26.0277	−0.953580
$746$	−29.6552	−1.08575
$747$	1.61416	0.0590591
$748$	−6.37483	−0.233087
$749$	−34.0822	−1.24534
$750$	−11.5084	−0.420229
$751$	6.76655	0.246915	0.123457	−	0.992350i	$-0.460602\pi$
$751$	6.76655	0.246915	0.123457	+	0.992350i	$0.460602\pi$
$752$	6.29417	0.229525
$753$	−8.30558	−0.302672
$754$	−6.90131	−0.251331
$755$	10.6018	0.385840
$756$	2.29417	0.0834383
$757$	24.1338	0.877157	0.438579	−	0.898693i	$-0.355482\pi$
$757$	24.1338	0.877157	0.438579	+	0.898693i	$0.355482\pi$
$758$	10.6259	0.385951
$759$	1.24445	0.0451708
$760$	5.64545	0.204782
$761$	19.1021	0.692452	0.346226	−	0.938151i	$-0.387463\pi$
$761$	19.1021	0.692452	0.346226	+	0.938151i	$0.387463\pi$
$762$	2.78574	0.100917
$763$	−10.0619	−0.364264
$764$	−12.2300	−0.442467
$765$	7.50844	0.271468
$766$	−1.56479	−0.0565382
$767$	50.8121	1.83472
$768$	−1.00000	−0.0360844
$769$	22.4726	0.810382	0.405191	−	0.914232i	$-0.367205\pi$
$769$	22.4726	0.810382	0.405191	+	0.914232i	$0.367205\pi$
$770$	−4.18470	−0.150806
$771$	18.8106	0.677448
$772$	7.69442	0.276928
$773$	45.0398	1.61997	0.809985	−	0.586451i	$-0.199475\pi$
$773$	45.0398	1.61997	0.809985	+	0.586451i	$0.199475\pi$
$774$	−5.36268	−0.192758
$775$	−9.25183	−0.332336
$776$	2.68267	0.0963022
$777$	−23.2330	−0.833481
$778$	−15.9249	−0.570934
$779$	13.6293	0.488321
$780$	10.1156	0.362196
$781$	−12.6635	−0.453135
$782$	−5.12260	−0.183184
$783$	1.00000	0.0357371
$784$	−1.73676	−0.0620273
$785$	3.60775	0.128766
$786$	2.94762	0.105138
$787$	−33.9749	−1.21108	−0.605538	−	0.795817i	$-0.707042\pi$
$787$	−33.9749	−1.21108	−0.605538	+	0.795817i	$0.707042\pi$
$788$	13.0497	0.464877
$789$	−11.2294	−0.399778
$790$	9.04707	0.321880
$791$	−22.3648	−0.795201
$792$	1.24445	0.0442197
$793$	−91.1957	−3.23845
$794$	14.8785	0.528018
$795$	1.86299	0.0660734
$796$	−6.45134	−0.228662
$797$	17.7033	0.627083	0.313542	−	0.949574i	$-0.398484\pi$
$797$	17.7033	0.627083	0.313542	+	0.949574i	$0.398484\pi$
$798$	8.83621	0.312798
$799$	32.2425	1.14066
$800$	2.85158	0.100819
$801$	−15.8594	−0.560363
$802$	−26.9706	−0.952364
$803$	−18.8867	−0.666496
$804$	−12.6914	−0.447592
$805$	−3.36268	−0.118519
$806$	22.3910	0.788688
$807$	5.39021	0.189744
$808$	10.0000	0.351799
$809$	−39.8732	−1.40187	−0.700934	−	0.713226i	$-0.747233\pi$
$809$	−39.8732	−1.40187	−0.700934	+	0.713226i	$0.747233\pi$
$810$	−1.46575	−0.0515011
$811$	35.6979	1.25352	0.626761	−	0.779212i	$-0.284380\pi$
$811$	35.6979	1.25352	0.626761	+	0.779212i	$0.284380\pi$
$812$	2.29417	0.0805097
$813$	−31.0397	−1.08861
$814$	−12.6025	−0.441719
$815$	−14.1949	−0.497224
$816$	−5.12260	−0.179327
$817$	−20.6548	−0.722621
$818$	4.12088	0.144083
$819$	15.8328	0.553243
$820$	−5.18673	−0.181129
$821$	−8.89393	−0.310400	−0.155200	−	0.987883i	$-0.549602\pi$
$821$	−8.89393	−0.310400	−0.155200	+	0.987883i	$0.549602\pi$
$822$	4.52285	0.157752
$823$	3.77967	0.131751	0.0658756	−	0.997828i	$-0.479016\pi$
$823$	3.77967	0.131751	0.0658756	+	0.997828i	$0.479016\pi$
$824$	−17.6079	−0.613400
$825$	−3.54866	−0.123549
$826$	−16.8913	−0.587723
$827$	−39.3528	−1.36843	−0.684215	−	0.729280i	$-0.739855\pi$
$827$	−39.3528	−1.36843	−0.684215	+	0.729280i	$0.739855\pi$
$828$	1.00000	0.0347524
$829$	−6.38337	−0.221704	−0.110852	−	0.993837i	$-0.535358\pi$
$829$	−6.38337	−0.221704	−0.110852	+	0.993837i	$0.535358\pi$
$830$	−2.36595	−0.0821235
$831$	26.9564	0.935106
$832$	−6.90131	−0.239260
$833$	−8.89675	−0.308254
$834$	−16.5883	−0.574408
$835$	30.6046	1.05912
$836$	4.79311	0.165773
$837$	−3.24445	−0.112145
$838$	−6.08428	−0.210178
$839$	−11.5866	−0.400015	−0.200007	−	0.979794i	$-0.564097\pi$
$839$	−11.5866	−0.400015	−0.200007	+	0.979794i	$0.564097\pi$
$840$	−3.36268	−0.116023
$841$	1.00000	0.0344828
$842$	2.10669	0.0726014
$843$	−3.80261	−0.130969
$844$	10.5696	0.363819
$845$	50.7559	1.74606
$846$	−6.29417	−0.216398
$847$	21.6830	0.745037
$848$	−1.27102	−0.0436469
$849$	−24.1348	−0.828303
$850$	14.6075	0.501034
$851$	−10.1270	−0.347148
$852$	−10.1759	−0.348622
$853$	−41.9613	−1.43673	−0.718363	−	0.695668i	$-0.755108\pi$
$853$	−41.9613	−1.43673	−0.718363	+	0.695668i	$0.755108\pi$
$854$	30.3158	1.03739
$855$	−5.64545	−0.193070
$856$	−14.8560	−0.507766
$857$	19.2001	0.655862	0.327931	−	0.944702i	$-0.393649\pi$
$857$	19.2001	0.655862	0.327931	+	0.944702i	$0.393649\pi$
$858$	8.58835	0.293201
$859$	12.1505	0.414569	0.207285	−	0.978281i	$-0.433537\pi$
$859$	12.1505	0.414569	0.207285	+	0.978281i	$0.433537\pi$
$860$	7.86033	0.268035
$861$	−8.11823	−0.276668
$862$	−16.9013	−0.575661
$863$	9.70304	0.330295	0.165148	−	0.986269i	$-0.447190\pi$
$863$	9.70304	0.330295	0.165148	+	0.986269i	$0.447190\pi$
$864$	1.00000	0.0340207
$865$	−9.18142	−0.312178
$866$	−28.6700	−0.974245
$867$	−9.24105	−0.313842
$868$	−7.44334	−0.252643
$869$	7.68116	0.260566
$870$	−1.46575	−0.0496935
$871$	−87.5874	−2.96778
$872$	−4.38584	−0.148523
$873$	−2.68267	−0.0907945
$874$	3.85158	0.130282
$875$	26.4024	0.892563
$876$	−15.1767	−0.512773
$877$	38.9719	1.31599	0.657994	−	0.753023i	$-0.271405\pi$
$877$	38.9719	1.31599	0.657994	+	0.753023i	$0.271405\pi$
$878$	−24.1182	−0.813951
$879$	−19.4970	−0.657619
$880$	−1.82405	−0.0614888
$881$	−43.2053	−1.45562	−0.727812	−	0.685776i	$-0.759463\pi$
$881$	−43.2053	−1.45562	−0.727812	+	0.685776i	$0.759463\pi$
$882$	1.73676	0.0584799
$883$	27.9941	0.942077	0.471039	−	0.882113i	$-0.343879\pi$
$883$	27.9941	0.942077	0.471039	+	0.882113i	$0.343879\pi$
$884$	−35.3526	−1.18904
$885$	10.7918	0.362763
$886$	−0.259263	−0.00871011
$887$	−30.2380	−1.01529	−0.507646	−	0.861566i	$-0.669484\pi$
$887$	−30.2380	−1.01529	−0.507646	+	0.861566i	$0.669484\pi$
$888$	−10.1270	−0.339839
$889$	−6.39096	−0.214346
$890$	23.2458	0.779202
$891$	−1.24445	−0.0416907
$892$	−2.34315	−0.0784543
$893$	−24.2425	−0.811246
$894$	−17.7573	−0.593892
$895$	−8.72660	−0.291698
$896$	2.29417	0.0766430
$897$	6.90131	0.230428
$898$	25.3788	0.846902
$899$	−3.24445	−0.108209
$900$	−2.85158	−0.0950528
$901$	−6.51091	−0.216910
$902$	−4.40365	−0.146626
$903$	12.3029	0.409416
$904$	−9.74852	−0.324231
$905$	−11.1841	−0.371771
$906$	7.23305	0.240302
$907$	−44.4138	−1.47474	−0.737368	−	0.675492i	$-0.763931\pi$
$907$	−44.4138	−1.47474	−0.737368	+	0.675492i	$0.763931\pi$
$908$	7.77871	0.258145
$909$	−10.0000	−0.331679
$910$	−23.2069	−0.769301
$911$	−15.8984	−0.526739	−0.263369	−	0.964695i	$-0.584834\pi$
$911$	−15.8984	−0.526739	−0.263369	+	0.964695i	$0.584834\pi$
$912$	3.85158	0.127539
$913$	−2.00875	−0.0664799
$914$	−4.72064	−0.156145
$915$	−19.3688	−0.640312
$916$	21.5011	0.710415
$917$	−6.76236	−0.223313
$918$	5.12260	0.169071
$919$	−14.3640	−0.473825	−0.236912	−	0.971531i	$-0.576135\pi$
$919$	−14.3640	−0.473825	−0.236912	+	0.971531i	$0.576135\pi$
$920$	−1.46575	−0.0483243
$921$	−5.86299	−0.193192
$922$	−7.48966	−0.246659
$923$	−70.2273	−2.31156
$924$	−2.85499	−0.0939223
$925$	28.8779	0.949501
$926$	11.1304	0.365767
$927$	17.6079	0.578319
$928$	1.00000	0.0328266
$929$	−53.1323	−1.74321	−0.871607	−	0.490205i	$-0.836922\pi$
$929$	−53.1323	−1.74321	−0.871607	+	0.490205i	$0.836922\pi$
$930$	4.75555	0.155941
$931$	6.68929	0.219233
$932$	7.90056	0.258791
$933$	−10.3317	−0.338246
$934$	31.2835	1.02363
$935$	−9.34390	−0.305578
$936$	6.90131	0.225576
$937$	29.8442	0.974968	0.487484	−	0.873132i	$-0.337915\pi$
$937$	29.8442	0.974968	0.487484	+	0.873132i	$0.337915\pi$
$938$	29.1163	0.950682
$939$	−27.8833	−0.909936
$940$	9.22567	0.300908
$941$	−29.6552	−0.966731	−0.483365	−	0.875419i	$-0.660586\pi$
$941$	−29.6552	−0.966731	−0.483365	+	0.875419i	$0.660586\pi$
$942$	2.46137	0.0801959
$943$	−3.53863	−0.115234
$944$	−7.36268	−0.239635
$945$	3.36268	0.109388
$946$	6.67360	0.216977
$947$	17.6958	0.575036	0.287518	−	0.957775i	$-0.407170\pi$
$947$	17.6958	0.575036	0.287518	+	0.957775i	$0.407170\pi$
$948$	6.17232	0.200468
$949$	−104.739	−3.39997
$950$	−10.9831	−0.356340
$951$	5.07588	0.164597
$952$	11.7521	0.380889
$953$	34.7341	1.12515	0.562574	−	0.826747i	$-0.309811\pi$
$953$	34.7341	1.12515	0.562574	+	0.826747i	$0.309811\pi$
$954$	1.27102	0.0411507
$955$	−17.9262	−0.580077
$956$	26.2452	0.848831
$957$	−1.24445	−0.0402274
$958$	2.24883	0.0726563
$959$	−10.3762	−0.335065
$960$	−1.46575	−0.0473068
$961$	−20.4735	−0.660436
$962$	−69.8894	−2.25332
$963$	14.8560	0.478727
$964$	0.0187836	0.000604979 0
$965$	11.2781	0.363054
$966$	−2.29417	−0.0738139
$967$	22.7039	0.730109	0.365054	−	0.930986i	$-0.381050\pi$
$967$	22.7039	0.730109	0.365054	+	0.930986i	$0.381050\pi$
$968$	9.45134	0.303778
$969$	19.7301	0.633823
$970$	3.93211	0.126253
$971$	−19.7865	−0.634979	−0.317489	−	0.948262i	$-0.602840\pi$
$971$	−19.7865	−0.634979	−0.317489	+	0.948262i	$0.602840\pi$
$972$	−1.00000	−0.0320750
$973$	38.0566	1.22004
$974$	−32.8295	−1.05193
$975$	−19.6797	−0.630254
$976$	13.2143	0.422978
$977$	10.4212	0.333402	0.166701	−	0.986007i	$-0.446689\pi$
$977$	10.4212	0.333402	0.166701	+	0.986007i	$0.446689\pi$
$978$	−9.68439	−0.309673
$979$	19.7362	0.630773
$980$	−2.54566	−0.0813180
$981$	4.38584	0.140029
$982$	−9.38146	−0.299375
$983$	−34.8299	−1.11090	−0.555451	−	0.831549i	$-0.687454\pi$
$983$	−34.8299	−1.11090	−0.555451	+	0.831549i	$0.687454\pi$
$984$	−3.53863	−0.112807
$985$	19.1276	0.609456
$986$	5.12260	0.163137
$987$	14.4399	0.459628
$988$	26.5810	0.845653
$989$	5.36268	0.170523
$990$	1.82405	0.0579722
$991$	−4.67626	−0.148546	−0.0742731	−	0.997238i	$-0.523664\pi$
$991$	−4.67626	−0.148546	−0.0742731	+	0.997238i	$0.523664\pi$
$992$	−3.24445	−0.103011
$993$	25.2345	0.800794
$994$	23.3454	0.740471
$995$	−9.45603	−0.299776
$996$	−1.61416	−0.0511467
$997$	5.07116	0.160605	0.0803026	−	0.996771i	$-0.474411\pi$
$997$	5.07116	0.160605	0.0803026	+	0.996771i	$0.474411\pi$
$998$	−19.0008	−0.601458
$999$	10.1270	0.320403

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.ba.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.ba.1.3	✓	4	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:	\( N \)	\(=\)	\( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.46575 q^{5} +1.00000 q^{6} -2.29417 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.46575 q^{5} +1.00000 q^{6} -2.29417 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.46575 q^{10} -1.24445 q^{11} -1.00000 q^{12} -6.90131 q^{13} +2.29417 q^{14} -1.46575 q^{15} +1.00000 q^{16} +5.12260 q^{17} -1.00000 q^{18} -3.85158 q^{19} +1.46575 q^{20} +2.29417 q^{21} +1.24445 q^{22} +1.00000 q^{23} +1.00000 q^{24} -2.85158 q^{25} +6.90131 q^{26} -1.00000 q^{27} -2.29417 q^{28} -1.00000 q^{29} +1.46575 q^{30} +3.24445 q^{31} -1.00000 q^{32} +1.24445 q^{33} -5.12260 q^{34} -3.36268 q^{35} +1.00000 q^{36} -10.1270 q^{37} +3.85158 q^{38} +6.90131 q^{39} -1.46575 q^{40} -3.53863 q^{41} -2.29417 q^{42} +5.36268 q^{43} -1.24445 q^{44} +1.46575 q^{45} -1.00000 q^{46} +6.29417 q^{47} -1.00000 q^{48} -1.73676 q^{49} +2.85158 q^{50} -5.12260 q^{51} -6.90131 q^{52} -1.27102 q^{53} +1.00000 q^{54} -1.82405 q^{55} +2.29417 q^{56} +3.85158 q^{57} +1.00000 q^{58} -7.36268 q^{59} -1.46575 q^{60} +13.2143 q^{61} -3.24445 q^{62} -2.29417 q^{63} +1.00000 q^{64} -10.1156 q^{65} -1.24445 q^{66} +12.6914 q^{67} +5.12260 q^{68} -1.00000 q^{69} +3.36268 q^{70} +10.1759 q^{71} -1.00000 q^{72} +15.1767 q^{73} +10.1270 q^{74} +2.85158 q^{75} -3.85158 q^{76} +2.85499 q^{77} -6.90131 q^{78} -6.17232 q^{79} +1.46575 q^{80} +1.00000 q^{81} +3.53863 q^{82} +1.61416 q^{83} +2.29417 q^{84} +7.50844 q^{85} -5.36268 q^{86} +1.00000 q^{87} +1.24445 q^{88} -15.8594 q^{89} -1.46575 q^{90} +15.8328 q^{91} +1.00000 q^{92} -3.24445 q^{93} -6.29417 q^{94} -5.64545 q^{95} +1.00000 q^{96} -2.68267 q^{97} +1.73676 q^{98} -1.24445 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 + 6 * q^7 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9} - 2 q^{10} - 4 q^{12} - 6 q^{14} - 2 q^{15} + 4 q^{16} - 6 q^{17} - 4 q^{18} + 2 q^{19} + 2 q^{20} - 6 q^{21} + 4 q^{23} + 4 q^{24} + 6 q^{25} - 4 q^{27} + 6 q^{28} - 4 q^{29} + 2 q^{30} + 8 q^{31} - 4 q^{32} + 6 q^{34} - 6 q^{35} + 4 q^{36} + 10 q^{37} - 2 q^{38} - 2 q^{40} + 6 q^{41} + 6 q^{42} + 14 q^{43} + 2 q^{45} - 4 q^{46} + 10 q^{47} - 4 q^{48} + 6 q^{49} - 6 q^{50} + 6 q^{51} + 4 q^{53} + 4 q^{54} - 20 q^{55} - 6 q^{56} - 2 q^{57} + 4 q^{58} - 22 q^{59} - 2 q^{60} + 28 q^{61} - 8 q^{62} + 6 q^{63} + 4 q^{64} + 12 q^{65} + 24 q^{67} - 6 q^{68} - 4 q^{69} + 6 q^{70} + 28 q^{71} - 4 q^{72} - 10 q^{74} - 6 q^{75} + 2 q^{76} - 4 q^{77} + 12 q^{79} + 2 q^{80} + 4 q^{81} - 6 q^{82} + 20 q^{83} - 6 q^{84} - 10 q^{85} - 14 q^{86} + 4 q^{87} - 24 q^{89} - 2 q^{90} + 28 q^{91} + 4 q^{92} - 8 q^{93} - 10 q^{94} + 2 q^{95} + 4 q^{96} - 32 q^{97} - 6 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 + 6 * q^7 - 4 * q^8 + 4 * q^9 - 2 * q^10 - 4 * q^12 - 6 * q^14 - 2 * q^15 + 4 * q^16 - 6 * q^17 - 4 * q^18 + 2 * q^19 + 2 * q^20 - 6 * q^21 + 4 * q^23 + 4 * q^24 + 6 * q^25 - 4 * q^27 + 6 * q^28 - 4 * q^29 + 2 * q^30 + 8 * q^31 - 4 * q^32 + 6 * q^34 - 6 * q^35 + 4 * q^36 + 10 * q^37 - 2 * q^38 - 2 * q^40 + 6 * q^41 + 6 * q^42 + 14 * q^43 + 2 * q^45 - 4 * q^46 + 10 * q^47 - 4 * q^48 + 6 * q^49 - 6 * q^50 + 6 * q^51 + 4 * q^53 + 4 * q^54 - 20 * q^55 - 6 * q^56 - 2 * q^57 + 4 * q^58 - 22 * q^59 - 2 * q^60 + 28 * q^61 - 8 * q^62 + 6 * q^63 + 4 * q^64 + 12 * q^65 + 24 * q^67 - 6 * q^68 - 4 * q^69 + 6 * q^70 + 28 * q^71 - 4 * q^72 - 10 * q^74 - 6 * q^75 + 2 * q^76 - 4 * q^77 + 12 * q^79 + 2 * q^80 + 4 * q^81 - 6 * q^82 + 20 * q^83 - 6 * q^84 - 10 * q^85 - 14 * q^86 + 4 * q^87 - 24 * q^89 - 2 * q^90 + 28 * q^91 + 4 * q^92 - 8 * q^93 - 10 * q^94 + 2 * q^95 + 4 * q^96 - 32 * q^97 - 6 * q^98

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