LMFDB - Embedded newform 4002.2.a.ba.1.1

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			1.1
Root			$-1.87996$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.29417 q^{5} +1.00000 q^{6} +2.46575 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.29417 q^{5} +1.00000 q^{6} +2.46575 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.29417 q^{10} +5.48709 q^{11} -1.00000 q^{12} -0.169761 q^{13} -2.46575 q^{14} +3.29417 q^{15} +1.00000 q^{16} +0.362680 q^{17} -1.00000 q^{18} +4.85158 q^{19} -3.29417 q^{20} -2.46575 q^{21} -5.48709 q^{22} +1.00000 q^{23} +1.00000 q^{24} +5.85158 q^{25} +0.169761 q^{26} -1.00000 q^{27} +2.46575 q^{28} -1.00000 q^{29} -3.29417 q^{30} -3.48709 q^{31} -1.00000 q^{32} -5.48709 q^{33} -0.362680 q^{34} -8.12260 q^{35} +1.00000 q^{36} +10.8843 q^{37} -4.85158 q^{38} +0.169761 q^{39} +3.29417 q^{40} +7.95284 q^{41} +2.46575 q^{42} +10.1226 q^{43} +5.48709 q^{44} -3.29417 q^{45} -1.00000 q^{46} +1.53425 q^{47} -1.00000 q^{48} -0.920090 q^{49} -5.85158 q^{50} -0.362680 q^{51} -0.169761 q^{52} -5.21426 q^{53} +1.00000 q^{54} -18.0754 q^{55} -2.46575 q^{56} -4.85158 q^{57} +1.00000 q^{58} -12.1226 q^{59} +3.29417 q^{60} +9.27102 q^{61} +3.48709 q^{62} +2.46575 q^{63} +1.00000 q^{64} +0.559222 q^{65} +5.48709 q^{66} -6.34827 q^{67} +0.362680 q^{68} -1.00000 q^{69} +8.12260 q^{70} -6.07544 q^{71} -1.00000 q^{72} -3.86299 q^{73} -10.8843 q^{74} -5.85158 q^{75} +4.85158 q^{76} +13.5298 q^{77} -0.169761 q^{78} -3.38403 q^{79} -3.29417 q^{80} +1.00000 q^{81} -7.95284 q^{82} +5.55741 q^{83} -2.46575 q^{84} -1.19473 q^{85} -10.1226 q^{86} +1.00000 q^{87} -5.48709 q^{88} -10.2828 q^{89} +3.29417 q^{90} -0.418588 q^{91} +1.00000 q^{92} +3.48709 q^{93} -1.53425 q^{94} -15.9820 q^{95} +1.00000 q^{96} -16.1458 q^{97} +0.920090 q^{98} +5.48709 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.29417	−1.47320	−0.736600	−	0.676329i	$-0.763570\pi$
$5$	−3.29417	−1.47320	−0.736600	+	0.676329i	$0.763570\pi$
$6$	1.00000	0.408248
$7$	2.46575	0.931965	0.465982	−	0.884794i	$-0.345701\pi$
$7$	2.46575	0.931965	0.465982	+	0.884794i	$0.345701\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	3.29417	1.04171
$11$	5.48709	1.65442	0.827210	−	0.561892i	$-0.189926\pi$
$11$	5.48709	1.65442	0.827210	+	0.561892i	$0.189926\pi$
$12$	−1.00000	−0.288675
$13$	−0.169761	−0.0470832	−0.0235416	−	0.999723i	$-0.507494\pi$
$13$	−0.169761	−0.0470832	−0.0235416	+	0.999723i	$0.507494\pi$
$14$	−2.46575	−0.658999
$15$	3.29417	0.850552
$16$	1.00000	0.250000
$17$	0.362680	0.0879628	0.0439814	−	0.999032i	$-0.485996\pi$
$17$	0.362680	0.0879628	0.0439814	+	0.999032i	$0.485996\pi$
$18$	−1.00000	−0.235702
$19$	4.85158	1.11303	0.556515	−	0.830838i	$-0.312138\pi$
$19$	4.85158	1.11303	0.556515	+	0.830838i	$0.312138\pi$
$20$	−3.29417	−0.736600
$21$	−2.46575	−0.538070
$22$	−5.48709	−1.16985
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	5.85158	1.17032
$26$	0.169761	0.0332929
$27$	−1.00000	−0.192450
$28$	2.46575	0.465982
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−3.29417	−0.601431
$31$	−3.48709	−0.626300	−0.313150	−	0.949704i	$-0.601384\pi$
$31$	−3.48709	−0.626300	−0.313150	+	0.949704i	$0.601384\pi$
$32$	−1.00000	−0.176777
$33$	−5.48709	−0.955180
$34$	−0.362680	−0.0621991
$35$	−8.12260	−1.37297
$36$	1.00000	0.166667
$37$	10.8843	1.78937	0.894687	−	0.446694i	$-0.147399\pi$
$37$	10.8843	1.78937	0.894687	+	0.446694i	$0.147399\pi$
$38$	−4.85158	−0.787031
$39$	0.169761	0.0271835
$40$	3.29417	0.520855
$41$	7.95284	1.24203	0.621013	−	0.783801i	$-0.286722\pi$
$41$	7.95284	1.24203	0.621013	+	0.783801i	$0.286722\pi$
$42$	2.46575	0.380473
$43$	10.1226	1.54368	0.771841	−	0.635815i	$-0.219336\pi$
$43$	10.1226	1.54368	0.771841	+	0.635815i	$0.219336\pi$
$44$	5.48709	0.827210
$45$	−3.29417	−0.491067
$46$	−1.00000	−0.147442
$47$	1.53425	0.223794	0.111897	−	0.993720i	$-0.464307\pi$
$47$	1.53425	0.223794	0.111897	+	0.993720i	$0.464307\pi$
$48$	−1.00000	−0.144338
$49$	−0.920090	−0.131441
$50$	−5.85158	−0.827539
$51$	−0.362680	−0.0507853
$52$	−0.169761	−0.0235416
$53$	−5.21426	−0.716234	−0.358117	−	0.933677i	$-0.616581\pi$
$53$	−5.21426	−0.716234	−0.358117	+	0.933677i	$0.616581\pi$
$54$	1.00000	0.136083
$55$	−18.0754	−2.43729
$56$	−2.46575	−0.329499
$57$	−4.85158	−0.642608
$58$	1.00000	0.131306
$59$	−12.1226	−1.57823	−0.789114	−	0.614247i	$-0.789460\pi$
$59$	−12.1226	−1.57823	−0.789114	+	0.614247i	$0.789460\pi$
$60$	3.29417	0.425276
$61$	9.27102	1.18703	0.593516	−	0.804822i	$-0.297739\pi$
$61$	9.27102	1.18703	0.593516	+	0.804822i	$0.297739\pi$
$62$	3.48709	0.442861
$63$	2.46575	0.310655
$64$	1.00000	0.125000
$65$	0.559222	0.0693630
$66$	5.48709	0.675414
$67$	−6.34827	−0.775565	−0.387782	−	0.921751i	$-0.626759\pi$
$67$	−6.34827	−0.775565	−0.387782	+	0.921751i	$0.626759\pi$
$68$	0.362680	0.0439814
$69$	−1.00000	−0.120386
$70$	8.12260	0.970837
$71$	−6.07544	−0.721022	−0.360511	−	0.932755i	$-0.617398\pi$
$71$	−6.07544	−0.721022	−0.360511	+	0.932755i	$0.617398\pi$
$72$	−1.00000	−0.117851
$73$	−3.86299	−0.452129	−0.226064	−	0.974112i	$-0.572586\pi$
$73$	−3.86299	−0.452129	−0.226064	+	0.974112i	$0.572586\pi$
$74$	−10.8843	−1.26528
$75$	−5.85158	−0.675683
$76$	4.85158	0.556515
$77$	13.5298	1.54186
$78$	−0.169761	−0.0192217
$79$	−3.38403	−0.380733	−0.190366	−	0.981713i	$-0.560968\pi$
$79$	−3.38403	−0.380733	−0.190366	+	0.981713i	$0.560968\pi$
$80$	−3.29417	−0.368300
$81$	1.00000	0.111111
$82$	−7.95284	−0.878244
$83$	5.55741	0.610005	0.305003	−	0.952352i	$-0.401343\pi$
$83$	5.55741	0.610005	0.305003	+	0.952352i	$0.401343\pi$
$84$	−2.46575	−0.269035
$85$	−1.19473	−0.129587
$86$	−10.1226	−1.09155
$87$	1.00000	0.107211
$88$	−5.48709	−0.584926
$89$	−10.2828	−1.08997	−0.544986	−	0.838445i	$-0.683465\pi$
$89$	−10.2828	−1.08997	−0.544986	+	0.838445i	$0.683465\pi$
$90$	3.29417	0.347236
$91$	−0.418588	−0.0438799
$92$	1.00000	0.104257
$93$	3.48709	0.361595
$94$	−1.53425	−0.158246
$95$	−15.9820	−1.63972
$96$	1.00000	0.102062
$97$	−16.1458	−1.63935	−0.819677	−	0.572826i	$-0.805847\pi$
$97$	−16.1458	−1.63935	−0.819677	+	0.572826i	$0.805847\pi$
$98$	0.920090	0.0929432
$99$	5.48709	0.551474
$100$	5.85158	0.585158
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0.362680	0.0359107
$103$	12.8480	1.26595	0.632974	−	0.774173i	$-0.281834\pi$
$103$	12.8480	1.26595	0.632974	+	0.774173i	$0.281834\pi$
$104$	0.169761	0.0166464
$105$	8.12260	0.792685
$106$	5.21426	0.506454
$107$	−10.0986	−0.976268	−0.488134	−	0.872769i	$-0.662322\pi$
$107$	−10.0986	−0.976268	−0.488134	+	0.872769i	$0.662322\pi$
$108$	−1.00000	−0.0962250
$109$	0.442589	0.0423924	0.0211962	−	0.999775i	$-0.493253\pi$
$109$	0.442589	0.0423924	0.0211962	+	0.999775i	$0.493253\pi$
$110$	18.0754	1.72343
$111$	−10.8843	−1.03310
$112$	2.46575	0.232991
$113$	10.5652	0.993890	0.496945	−	0.867782i	$-0.334455\pi$
$113$	10.5652	0.993890	0.496945	+	0.867782i	$0.334455\pi$
$114$	4.85158	0.454393
$115$	−3.29417	−0.307183
$116$	−1.00000	−0.0928477
$117$	−0.169761	−0.0156944
$118$	12.1226	1.11598
$119$	0.894277	0.0819782
$120$	−3.29417	−0.300716
$121$	19.1082	1.73711
$122$	−9.27102	−0.839358
$123$	−7.95284	−0.717083
$124$	−3.48709	−0.313150
$125$	−2.80527	−0.250911
$126$	−2.46575	−0.219666
$127$	6.72898	0.597101	0.298550	−	0.954394i	$-0.403497\pi$
$127$	6.72898	0.597101	0.298550	+	0.954394i	$0.403497\pi$
$128$	−1.00000	−0.0883883
$129$	−10.1226	−0.891245
$130$	−0.559222	−0.0490470
$131$	−21.1903	−1.85140	−0.925701	−	0.378256i	$-0.876524\pi$
$131$	−21.1903	−1.85140	−0.925701	+	0.378256i	$0.876524\pi$
$132$	−5.48709	−0.477590
$133$	11.9628	1.03730
$134$	6.34827	0.548407
$135$	3.29417	0.283517
$136$	−0.362680	−0.0310995
$137$	19.6193	1.67619	0.838094	−	0.545525i	$-0.183670\pi$
$137$	19.6193	1.67619	0.838094	+	0.545525i	$0.183670\pi$
$138$	1.00000	0.0851257
$139$	−7.06851	−0.599543	−0.299771	−	0.954011i	$-0.596910\pi$
$139$	−7.06851	−0.599543	−0.299771	+	0.954011i	$0.596910\pi$
$140$	−8.12260	−0.686485
$141$	−1.53425	−0.129207
$142$	6.07544	0.509840
$143$	−0.931495	−0.0778955
$144$	1.00000	0.0833333
$145$	3.29417	0.273566
$146$	3.86299	0.319703
$147$	0.920090	0.0758878
$148$	10.8843	0.894687
$149$	13.9288	1.14109	0.570547	−	0.821265i	$-0.306731\pi$
$149$	13.9288	1.14109	0.570547	+	0.821265i	$0.306731\pi$
$150$	5.85158	0.477780
$151$	10.8380	0.881986	0.440993	−	0.897511i	$-0.354626\pi$
$151$	10.8380	0.881986	0.440993	+	0.897511i	$0.354626\pi$
$152$	−4.85158	−0.393516
$153$	0.362680	0.0293209
$154$	−13.5298	−1.09026
$155$	11.4871	0.922666
$156$	0.169761	0.0135918
$157$	13.9528	1.11356	0.556779	−	0.830661i	$-0.312037\pi$
$157$	13.9528	1.11356	0.556779	+	0.830661i	$0.312037\pi$
$158$	3.38403	0.269219
$159$	5.21426	0.413518
$160$	3.29417	0.260427
$161$	2.46575	0.194328
$162$	−1.00000	−0.0785674
$163$	15.2702	1.19605	0.598026	−	0.801477i	$-0.295952\pi$
$163$	15.2702	1.19605	0.598026	+	0.801477i	$0.295952\pi$
$164$	7.95284	0.621013
$165$	18.0754	1.40717
$166$	−5.55741	−0.431339
$167$	−15.5662	−1.20455	−0.602273	−	0.798290i	$-0.705738\pi$
$167$	−15.5662	−1.20455	−0.602273	+	0.798290i	$0.705738\pi$
$168$	2.46575	0.190237
$169$	−12.9712	−0.997783
$170$	1.19473	0.0916317
$171$	4.85158	0.371010
$172$	10.1226	0.771841
$173$	−4.29236	−0.326342	−0.163171	−	0.986598i	$-0.552172\pi$
$173$	−4.29236	−0.326342	−0.163171	+	0.986598i	$0.552172\pi$
$174$	−1.00000	−0.0758098
$175$	14.4285	1.09069
$176$	5.48709	0.413605
$177$	12.1226	0.911191
$178$	10.2828	0.770726
$179$	−23.3600	−1.74601	−0.873005	−	0.487711i	$-0.837832\pi$
$179$	−23.3600	−1.74601	−0.873005	+	0.487711i	$0.837832\pi$
$180$	−3.29417	−0.245533
$181$	3.04450	0.226296	0.113148	−	0.993578i	$-0.463907\pi$
$181$	3.04450	0.226296	0.113148	+	0.993578i	$0.463907\pi$
$182$	0.418588	0.0310278
$183$	−9.27102	−0.685333
$184$	−1.00000	−0.0737210
$185$	−35.8549	−2.63610
$186$	−3.48709	−0.255686
$187$	1.99006	0.145527
$188$	1.53425	0.111897
$189$	−2.46575	−0.179357
$190$	15.9820	1.15945
$191$	18.3011	1.32422	0.662111	−	0.749406i	$-0.269661\pi$
$191$	18.3011	1.32422	0.662111	+	0.749406i	$0.269661\pi$
$192$	−1.00000	−0.0721688
$193$	22.7909	1.64052	0.820261	−	0.571989i	$-0.193828\pi$
$193$	22.7909	1.64052	0.820261	+	0.571989i	$0.193828\pi$
$194$	16.1458	1.15920
$195$	−0.559222	−0.0400467
$196$	−0.920090	−0.0657207
$197$	15.0213	1.07023	0.535113	−	0.844780i	$-0.320269\pi$
$197$	15.0213	1.07023	0.535113	+	0.844780i	$0.320269\pi$
$198$	−5.48709	−0.389951
$199$	22.1082	1.56721	0.783604	−	0.621261i	$-0.213379\pi$
$199$	22.1082	1.56721	0.783604	+	0.621261i	$0.213379\pi$
$200$	−5.85158	−0.413770
$201$	6.34827	0.447772
$202$	10.0000	0.703598
$203$	−2.46575	−0.173062
$204$	−0.362680	−0.0253927
$205$	−26.1980	−1.82975
$206$	−12.8480	−0.895160
$207$	1.00000	0.0695048
$208$	−0.169761	−0.0117708
$209$	26.6211	1.84142
$210$	−8.12260	−0.560513
$211$	−6.49850	−0.447375	−0.223688	−	0.974661i	$-0.571810\pi$
$211$	−6.49850	−0.447375	−0.223688	+	0.974661i	$0.571810\pi$
$212$	−5.21426	−0.358117
$213$	6.07544	0.416282
$214$	10.0986	0.690326
$215$	−33.3456	−2.27415
$216$	1.00000	0.0680414
$217$	−8.59829	−0.583690
$218$	−0.442589	−0.0299759
$219$	3.86299	0.261037
$220$	−18.0754	−1.21865
$221$	−0.0615689	−0.00414157
$222$	10.8843	0.730509
$223$	−2.34315	−0.156909	−0.0784543	−	0.996918i	$-0.524998\pi$
$223$	−2.34315	−0.156909	−0.0784543	+	0.996918i	$0.524998\pi$
$224$	−2.46575	−0.164750
$225$	5.85158	0.390106
$226$	−10.5652	−0.702786
$227$	5.80708	0.385430	0.192715	−	0.981255i	$-0.438271\pi$
$227$	5.80708	0.385430	0.192715	+	0.981255i	$0.438271\pi$
$228$	−4.85158	−0.321304
$229$	−5.08685	−0.336148	−0.168074	−	0.985774i	$-0.553755\pi$
$229$	−5.08685	−0.336148	−0.168074	+	0.985774i	$0.553755\pi$
$230$	3.29417	0.217211
$231$	−13.5298	−0.890195
$232$	1.00000	0.0656532
$233$	3.95731	0.259252	0.129626	−	0.991563i	$-0.458622\pi$
$233$	3.95731	0.259252	0.129626	+	0.991563i	$0.458622\pi$
$234$	0.169761	0.0110976
$235$	−5.05410	−0.329693
$236$	−12.1226	−0.789114
$237$	3.38403	0.219816
$238$	−0.894277	−0.0579674
$239$	16.7254	1.08187	0.540937	−	0.841063i	$-0.318070\pi$
$239$	16.7254	1.08187	0.540937	+	0.841063i	$0.318070\pi$
$240$	3.29417	0.212638
$241$	7.56700	0.487434	0.243717	−	0.969846i	$-0.421633\pi$
$241$	7.56700	0.487434	0.243717	+	0.969846i	$0.421633\pi$
$242$	−19.1082	−1.22832
$243$	−1.00000	−0.0641500
$244$	9.27102	0.593516
$245$	3.03094	0.193640
$246$	7.95284	0.507055
$247$	−0.823610	−0.0524050
$248$	3.48709	0.221431
$249$	−5.55741	−0.352187
$250$	2.80527	0.177421
$251$	−6.79086	−0.428635	−0.214318	−	0.976764i	$-0.568753\pi$
$251$	−6.79086	−0.428635	−0.214318	+	0.976764i	$0.568753\pi$
$252$	2.46575	0.155327
$253$	5.48709	0.344971
$254$	−6.72898	−0.422214
$255$	1.19473	0.0748169
$256$	1.00000	0.0625000
$257$	24.3670	1.51997	0.759985	−	0.649941i	$-0.225206\pi$
$257$	24.3670	1.51997	0.759985	+	0.649941i	$0.225206\pi$
$258$	10.1226	0.630206
$259$	26.8380	1.66763
$260$	0.559222	0.0346815
$261$	−1.00000	−0.0618984
$262$	21.1903	1.30914
$263$	28.2975	1.74490	0.872449	−	0.488705i	$-0.162531\pi$
$263$	28.2975	1.74490	0.872449	+	0.488705i	$0.162531\pi$
$264$	5.48709	0.337707
$265$	17.1767	1.05516
$266$	−11.9628	−0.733485
$267$	10.2828	0.629295
$268$	−6.34827	−0.387782
$269$	14.8044	0.902642	0.451321	−	0.892362i	$-0.350953\pi$
$269$	14.8044	0.902642	0.451321	+	0.892362i	$0.350953\pi$
$270$	−3.29417	−0.200477
$271$	−7.03969	−0.427631	−0.213815	−	0.976874i	$-0.568589\pi$
$271$	−7.03969	−0.427631	−0.213815	+	0.976874i	$0.568589\pi$
$272$	0.362680	0.0219907
$273$	0.418588	0.0253341
$274$	−19.6193	−1.18524
$275$	32.1082	1.93620
$276$	−1.00000	−0.0601929
$277$	29.6843	1.78356	0.891778	−	0.452473i	$-0.149458\pi$
$277$	29.6843	1.78356	0.891778	+	0.452473i	$0.149458\pi$
$278$	7.06851	0.423941
$279$	−3.48709	−0.208767
$280$	8.12260	0.485418
$281$	−9.66048	−0.576296	−0.288148	−	0.957586i	$-0.593039\pi$
$281$	−9.66048	−0.576296	−0.288148	+	0.957586i	$0.593039\pi$
$282$	1.53425	0.0913634
$283$	6.25002	0.371525	0.185763	−	0.982595i	$-0.440524\pi$
$283$	6.25002	0.371525	0.185763	+	0.982595i	$0.440524\pi$
$284$	−6.07544	−0.360511
$285$	15.9820	0.946690
$286$	0.931495	0.0550804
$287$	19.6097	1.15752
$288$	−1.00000	−0.0589256
$289$	−16.8685	−0.992263
$290$	−3.29417	−0.193441
$291$	16.1458	0.946481
$292$	−3.86299	−0.226064
$293$	21.1304	1.23445	0.617225	−	0.786787i	$-0.288257\pi$
$293$	21.1304	1.23445	0.617225	+	0.786787i	$0.288257\pi$
$294$	−0.920090	−0.0536608
$295$	39.9340	2.32505
$296$	−10.8843	−0.632639
$297$	−5.48709	−0.318393
$298$	−13.9288	−0.806876
$299$	−0.169761	−0.00981753
$300$	−5.85158	−0.337841
$301$	24.9598	1.43866
$302$	−10.8380	−0.623658
$303$	10.0000	0.574485
$304$	4.85158	0.278257
$305$	−30.5403	−1.74874
$306$	−0.362680	−0.0207330
$307$	−13.1767	−0.752034	−0.376017	−	0.926613i	$-0.622707\pi$
$307$	−13.1767	−0.752034	−0.376017	+	0.926613i	$0.622707\pi$
$308$	13.5298	0.770931
$309$	−12.8480	−0.730895
$310$	−11.4871	−0.652423
$311$	20.6683	1.17199	0.585995	−	0.810315i	$-0.300704\pi$
$311$	20.6683	1.17199	0.585995	+	0.810315i	$0.300704\pi$
$312$	−0.169761	−0.00961083
$313$	10.8152	0.611312	0.305656	−	0.952142i	$-0.401124\pi$
$313$	10.8152	0.611312	0.305656	+	0.952142i	$0.401124\pi$
$314$	−13.9528	−0.787404
$315$	−8.12260	−0.457657
$316$	−3.38403	−0.190366
$317$	−32.4805	−1.82428	−0.912142	−	0.409874i	$-0.865573\pi$
$317$	−32.4805	−1.82428	−0.912142	+	0.409874i	$0.865573\pi$
$318$	−5.21426	−0.292401
$319$	−5.48709	−0.307218
$320$	−3.29417	−0.184150
$321$	10.0986	0.563649
$322$	−2.46575	−0.137411
$323$	1.75957	0.0979052
$324$	1.00000	0.0555556
$325$	−0.993371	−0.0551023
$326$	−15.2702	−0.845737
$327$	−0.442589	−0.0244753
$328$	−7.95284	−0.439122
$329$	3.78308	0.208568
$330$	−18.0754	−0.995020
$331$	−23.2629	−1.27865	−0.639323	−	0.768938i	$-0.720785\pi$
$331$	−23.2629	−1.27865	−0.639323	+	0.768938i	$0.720785\pi$
$332$	5.55741	0.305003
$333$	10.8843	0.596458
$334$	15.5662	0.851742
$335$	20.9123	1.14256
$336$	−2.46575	−0.134518
$337$	−31.1391	−1.69626	−0.848128	−	0.529791i	$-0.822270\pi$
$337$	−31.1391	−1.69626	−0.848128	+	0.529791i	$0.822270\pi$
$338$	12.9712	0.705539
$339$	−10.5652	−0.573822
$340$	−1.19473	−0.0647934
$341$	−19.1340	−1.03616
$342$	−4.85158	−0.262344
$343$	−19.5289	−1.05446
$344$	−10.1226	−0.545774
$345$	3.29417	0.177352
$346$	4.29236	0.230759
$347$	−33.7795	−1.81338	−0.906688	−	0.421802i	$-0.861398\pi$
$347$	−33.7795	−1.81338	−0.906688	+	0.421802i	$0.861398\pi$
$348$	1.00000	0.0536056
$349$	−4.08901	−0.218880	−0.109440	−	0.993993i	$-0.534906\pi$
$349$	−4.08901	−0.218880	−0.109440	+	0.993993i	$0.534906\pi$
$350$	−14.4285	−0.771237
$351$	0.169761	0.00906117
$352$	−5.48709	−0.292463
$353$	20.2025	1.07527	0.537636	−	0.843177i	$-0.319318\pi$
$353$	20.2025	1.07527	0.537636	+	0.843177i	$0.319318\pi$
$354$	−12.1226	−0.644309
$355$	20.0136	1.06221
$356$	−10.2828	−0.544986
$357$	−0.894277	−0.0473302
$358$	23.3600	1.23462
$359$	15.2806	0.806480	0.403240	−	0.915094i	$-0.367884\pi$
$359$	15.2806	0.806480	0.403240	+	0.915094i	$0.367884\pi$
$360$	3.29417	0.173618
$361$	4.53788	0.238836
$362$	−3.04450	−0.160016
$363$	−19.1082	−1.00292
$364$	−0.418588	−0.0219400
$365$	12.7254	0.666076
$366$	9.27102	0.484604
$367$	24.8324	1.29624	0.648119	−	0.761539i	$-0.275556\pi$
$367$	24.8324	1.29624	0.648119	+	0.761539i	$0.275556\pi$
$368$	1.00000	0.0521286
$369$	7.95284	0.414008
$370$	35.8549	1.86401
$371$	−12.8571	−0.667505
$372$	3.48709	0.180797
$373$	−28.1404	−1.45706	−0.728529	−	0.685015i	$-0.759795\pi$
$373$	−28.1404	−1.45706	−0.728529	+	0.685015i	$0.759795\pi$
$374$	−1.99006	−0.102903
$375$	2.80527	0.144863
$376$	−1.53425	−0.0791230
$377$	0.169761	0.00874314
$378$	2.46575	0.126824
$379$	−16.2025	−0.832267	−0.416134	−	0.909304i	$-0.636615\pi$
$379$	−16.2025	−0.832267	−0.416134	+	0.909304i	$0.636615\pi$
$380$	−15.9820	−0.819858
$381$	−6.72898	−0.344736
$382$	−18.3011	−0.936366
$383$	15.5063	0.792334	0.396167	−	0.918179i	$-0.370340\pi$
$383$	15.5063	0.792334	0.396167	+	0.918179i	$0.370340\pi$
$384$	1.00000	0.0510310
$385$	−44.5695	−2.27147
$386$	−22.7909	−1.16002
$387$	10.1226	0.514561
$388$	−16.1458	−0.819677
$389$	−14.2680	−0.723417	−0.361708	−	0.932291i	$-0.617806\pi$
$389$	−14.2680	−0.723417	−0.361708	+	0.932291i	$0.617806\pi$
$390$	0.559222	0.0283173
$391$	0.362680	0.0183415
$392$	0.920090	0.0464716
$393$	21.1903	1.06891
$394$	−15.0213	−0.756764
$395$	11.1476	0.560895
$396$	5.48709	0.275737
$397$	−28.8200	−1.44643	−0.723217	−	0.690621i	$-0.757337\pi$
$397$	−28.8200	−1.44643	−0.723217	+	0.690621i	$0.757337\pi$
$398$	−22.1082	−1.10818
$399$	−11.9628	−0.598888
$400$	5.85158	0.292579
$401$	26.9706	1.34685	0.673423	−	0.739258i	$-0.264823\pi$
$401$	26.9706	1.34685	0.673423	+	0.739258i	$0.264823\pi$
$402$	−6.34827	−0.316623
$403$	0.591973	0.0294883
$404$	−10.0000	−0.497519
$405$	−3.29417	−0.163689
$406$	2.46575	0.122373
$407$	59.7234	2.96038
$408$	0.362680	0.0179553
$409$	−37.7786	−1.86803	−0.934016	−	0.357231i	$-0.883721\pi$
$409$	−37.7786	−1.86803	−0.934016	+	0.357231i	$0.883721\pi$
$410$	26.1980	1.29383
$411$	−19.6193	−0.967748
$412$	12.8480	0.632974
$413$	−29.8913	−1.47085
$414$	−1.00000	−0.0491473
$415$	−18.3071	−0.898659
$416$	0.169761	0.00832322
$417$	7.06851	0.346146
$418$	−26.6211	−1.30208
$419$	−10.9838	−0.536593	−0.268296	−	0.963336i	$-0.586461\pi$
$419$	−10.9838	−0.536593	−0.268296	+	0.963336i	$0.586461\pi$
$420$	8.12260	0.396342
$421$	20.8762	1.01745	0.508723	−	0.860930i	$-0.330118\pi$
$421$	20.8762	1.01745	0.508723	+	0.860930i	$0.330118\pi$
$422$	6.49850	0.316342
$423$	1.53425	0.0745979
$424$	5.21426	0.253227
$425$	2.12225	0.102944
$426$	−6.07544	−0.294356
$427$	22.8600	1.10627
$428$	−10.0986	−0.488134
$429$	0.931495	0.0449730
$430$	33.3456	1.60807
$431$	10.1698	0.489860	0.244930	−	0.969541i	$-0.421235\pi$
$431$	10.1698	0.489860	0.244930	+	0.969541i	$0.421235\pi$
$432$	−1.00000	−0.0481125
$433$	−20.0842	−0.965185	−0.482592	−	0.875845i	$-0.660305\pi$
$433$	−20.0842	−0.965185	−0.482592	+	0.875845i	$0.660305\pi$
$434$	8.59829	0.412731
$435$	−3.29417	−0.157944
$436$	0.442589	0.0211962
$437$	4.85158	0.232083
$438$	−3.86299	−0.184581
$439$	35.6097	1.69956	0.849779	−	0.527139i	$-0.176735\pi$
$439$	35.6097	1.69956	0.849779	+	0.527139i	$0.176735\pi$
$440$	18.0754	0.861713
$441$	−0.920090	−0.0438138
$442$	0.0615689	0.00292853
$443$	2.56916	0.122065	0.0610323	−	0.998136i	$-0.480561\pi$
$443$	2.56916	0.122065	0.0610323	+	0.998136i	$0.480561\pi$
$444$	−10.8843	−0.516548
$445$	33.8732	1.60575
$446$	2.34315	0.110951
$447$	−13.9288	−0.658811
$448$	2.46575	0.116496
$449$	−15.5207	−0.732467	−0.366233	−	0.930523i	$-0.619353\pi$
$449$	−15.5207	−0.732467	−0.366233	+	0.930523i	$0.619353\pi$
$450$	−5.85158	−0.275846
$451$	43.6380	2.05483
$452$	10.5652	0.496945
$453$	−10.8380	−0.509215
$454$	−5.80708	−0.272540
$455$	1.37890	0.0646439
$456$	4.85158	0.227196
$457$	18.5220	0.866423	0.433211	−	0.901292i	$-0.357380\pi$
$457$	18.5220	0.866423	0.433211	+	0.901292i	$0.357380\pi$
$458$	5.08685	0.237693
$459$	−0.362680	−0.0169284
$460$	−3.29417	−0.153592
$461$	−8.76173	−0.408075	−0.204037	−	0.978963i	$-0.565406\pi$
$461$	−8.76173	−0.408075	−0.204037	+	0.978963i	$0.565406\pi$
$462$	13.5298	0.629463
$463$	−9.49704	−0.441365	−0.220682	−	0.975346i	$-0.570828\pi$
$463$	−9.49704	−0.441365	−0.220682	+	0.975346i	$0.570828\pi$
$464$	−1.00000	−0.0464238
$465$	−11.4871	−0.532701
$466$	−3.95731	−0.183319
$467$	−34.0718	−1.57666	−0.788328	−	0.615255i	$-0.789053\pi$
$467$	−34.0718	−1.57666	−0.788328	+	0.615255i	$0.789053\pi$
$468$	−0.169761	−0.00784721
$469$	−15.6532	−0.722799
$470$	5.05410	0.233128
$471$	−13.9528	−0.642913
$472$	12.1226	0.557988
$473$	55.5437	2.55390
$474$	−3.38403	−0.155433
$475$	28.3895	1.30260
$476$	0.894277	0.0409891
$477$	−5.21426	−0.238745
$478$	−16.7254	−0.765000
$479$	20.7341	0.947366	0.473683	−	0.880696i	$-0.342924\pi$
$479$	20.7341	0.947366	0.473683	+	0.880696i	$0.342924\pi$
$480$	−3.29417	−0.150358
$481$	−1.84774	−0.0842495
$482$	−7.56700	−0.344668
$483$	−2.46575	−0.112195
$484$	19.1082	0.868554
$485$	53.1869	2.41509
$486$	1.00000	0.0453609
$487$	42.0111	1.90370	0.951852	−	0.306557i	$-0.0991771\pi$
$487$	42.0111	1.90370	0.951852	+	0.306557i	$0.0991771\pi$
$488$	−9.27102	−0.419679
$489$	−15.2702	−0.690541
$490$	−3.03094	−0.136924
$491$	21.6896	0.978838	0.489419	−	0.872049i	$-0.337209\pi$
$491$	21.6896	0.978838	0.489419	+	0.872049i	$0.337209\pi$
$492$	−7.95284	−0.358542
$493$	−0.362680	−0.0163343
$494$	0.823610	0.0370560
$495$	−18.0754	−0.812431
$496$	−3.48709	−0.156575
$497$	−14.9805	−0.671967
$498$	5.55741	0.249034
$499$	16.2125	0.725769	0.362885	−	0.931834i	$-0.381792\pi$
$499$	16.2125	0.725769	0.362885	+	0.931834i	$0.381792\pi$
$500$	−2.80527	−0.125455
$501$	15.5662	0.695445
$502$	6.79086	0.303091
$503$	4.87028	0.217155	0.108577	−	0.994088i	$-0.465370\pi$
$503$	4.87028	0.217155	0.108577	+	0.994088i	$0.465370\pi$
$504$	−2.46575	−0.109833
$505$	32.9417	1.46589
$506$	−5.48709	−0.243931
$507$	12.9712	0.576070
$508$	6.72898	0.298550
$509$	−20.9498	−0.928585	−0.464293	−	0.885682i	$-0.653691\pi$
$509$	−20.9498	−0.928585	−0.464293	+	0.885682i	$0.653691\pi$
$510$	−1.19473	−0.0529036
$511$	−9.52515	−0.421368
$512$	−1.00000	−0.0441942
$513$	−4.85158	−0.214203
$514$	−24.3670	−1.07478
$515$	−42.3234	−1.86499
$516$	−10.1226	−0.445623
$517$	8.41859	0.370249
$518$	−26.8380	−1.17919
$519$	4.29236	0.188414
$520$	−0.559222	−0.0245235
$521$	33.6188	1.47287	0.736433	−	0.676510i	$-0.236509\pi$
$521$	33.6188	1.47287	0.736433	+	0.676510i	$0.236509\pi$
$522$	1.00000	0.0437688
$523$	−4.73048	−0.206850	−0.103425	−	0.994637i	$-0.532980\pi$
$523$	−4.73048	−0.206850	−0.103425	+	0.994637i	$0.532980\pi$
$524$	−21.1903	−0.925701
$525$	−14.4285	−0.629713
$526$	−28.2975	−1.23383
$527$	−1.26470	−0.0550911
$528$	−5.48709	−0.238795
$529$	1.00000	0.0434783
$530$	−17.1767	−0.746108
$531$	−12.1226	−0.526076
$532$	11.9628	0.518652
$533$	−1.35008	−0.0584786
$534$	−10.2828	−0.444979
$535$	33.2665	1.43824
$536$	6.34827	0.274204
$537$	23.3600	1.00806
$538$	−14.8044	−0.638264
$539$	−5.04862	−0.217460
$540$	3.29417	0.141759
$541$	−11.8450	−0.509254	−0.254627	−	0.967039i	$-0.581953\pi$
$541$	−11.8450	−0.509254	−0.254627	+	0.967039i	$0.581953\pi$
$542$	7.03969	0.302380
$543$	−3.04450	−0.130652
$544$	−0.362680	−0.0155498
$545$	−1.45797	−0.0624524
$546$	−0.418588	−0.0179139
$547$	−12.5920	−0.538394	−0.269197	−	0.963085i	$-0.586758\pi$
$547$	−12.5920	−0.538394	−0.269197	+	0.963085i	$0.586758\pi$
$548$	19.6193	0.838094
$549$	9.27102	0.395677
$550$	−32.1082	−1.36910
$551$	−4.85158	−0.206684
$552$	1.00000	0.0425628
$553$	−8.34415	−0.354829
$554$	−29.6843	−1.26116
$555$	35.8549	1.52196
$556$	−7.06851	−0.299771
$557$	−36.0761	−1.52859	−0.764297	−	0.644865i	$-0.776914\pi$
$557$	−36.0761	−1.52859	−0.764297	+	0.644865i	$0.776914\pi$
$558$	3.48709	0.147620
$559$	−1.71842	−0.0726816
$560$	−8.12260	−0.343243
$561$	−1.99006	−0.0840203
$562$	9.66048	0.407503
$563$	25.1803	1.06122	0.530612	−	0.847615i	$-0.321962\pi$
$563$	25.1803	1.06122	0.530612	+	0.847615i	$0.321962\pi$
$564$	−1.53425	−0.0646037
$565$	−34.8036	−1.46420
$566$	−6.25002	−0.262708
$567$	2.46575	0.103552
$568$	6.07544	0.254920
$569$	27.4487	1.15071	0.575354	−	0.817904i	$-0.304864\pi$
$569$	27.4487	1.15071	0.575354	+	0.817904i	$0.304864\pi$
$570$	−15.9820	−0.669411
$571$	−26.8132	−1.12210	−0.561048	−	0.827783i	$-0.689602\pi$
$571$	−26.8132	−1.12210	−0.561048	+	0.827783i	$0.689602\pi$
$572$	−0.931495	−0.0389477
$573$	−18.3011	−0.764540
$574$	−19.6097	−0.818493
$575$	5.85158	0.244028
$576$	1.00000	0.0416667
$577$	4.37590	0.182171	0.0910855	−	0.995843i	$-0.470966\pi$
$577$	4.37590	0.182171	0.0910855	+	0.995843i	$0.470966\pi$
$578$	16.8685	0.701636
$579$	−22.7909	−0.947156
$580$	3.29417	0.136783
$581$	13.7032	0.568503
$582$	−16.1458	−0.669263
$583$	−28.6112	−1.18495
$584$	3.86299	0.159852
$585$	0.559222	0.0231210
$586$	−21.1304	−0.872888
$587$	0.808893	0.0333866	0.0166933	−	0.999861i	$-0.494686\pi$
$587$	0.808893	0.0333866	0.0166933	+	0.999861i	$0.494686\pi$
$588$	0.920090	0.0379439
$589$	−16.9179	−0.697091
$590$	−39.9340	−1.64406
$591$	−15.0213	−0.617896
$592$	10.8843	0.447343
$593$	6.59197	0.270700	0.135350	−	0.990798i	$-0.456784\pi$
$593$	6.59197	0.270700	0.135350	+	0.990798i	$0.456784\pi$
$594$	5.48709	0.225138
$595$	−2.94590	−0.120770
$596$	13.9288	0.570547
$597$	−22.1082	−0.904828
$598$	0.169761	0.00694204
$599$	39.8504	1.62824	0.814122	−	0.580694i	$-0.197219\pi$
$599$	39.8504	1.62824	0.814122	+	0.580694i	$0.197219\pi$
$600$	5.85158	0.238890
$601$	14.6575	0.597891	0.298945	−	0.954270i	$-0.403365\pi$
$601$	14.6575	0.597891	0.298945	+	0.954270i	$0.403365\pi$
$602$	−24.9598	−1.01728
$603$	−6.34827	−0.258522
$604$	10.8380	0.440993
$605$	−62.9457	−2.55911
$606$	−10.0000	−0.406222
$607$	−35.7714	−1.45191	−0.725957	−	0.687740i	$-0.758603\pi$
$607$	−35.7714	−1.45191	−0.725957	+	0.687740i	$0.758603\pi$
$608$	−4.85158	−0.196758
$609$	2.46575	0.0999171
$610$	30.5403	1.23654
$611$	−0.260456	−0.0105369
$612$	0.362680	0.0146605
$613$	24.0087	0.969704	0.484852	−	0.874596i	$-0.338873\pi$
$613$	24.0087	0.969704	0.484852	+	0.874596i	$0.338873\pi$
$614$	13.1767	0.531768
$615$	26.1980	1.05641
$616$	−13.5298	−0.545131
$617$	2.81039	0.113142	0.0565711	−	0.998399i	$-0.481983\pi$
$617$	2.81039	0.113142	0.0565711	+	0.998399i	$0.481983\pi$
$618$	12.8480	0.516821
$619$	−15.1659	−0.609569	−0.304785	−	0.952421i	$-0.598585\pi$
$619$	−15.1659	−0.609569	−0.304785	+	0.952421i	$0.598585\pi$
$620$	11.4871	0.461333
$621$	−1.00000	−0.0401286
$622$	−20.6683	−0.828722
$623$	−25.3547	−1.01582
$624$	0.169761	0.00679588
$625$	−20.0169	−0.800675
$626$	−10.8152	−0.432263
$627$	−26.6211	−1.06314
$628$	13.9528	0.556779
$629$	3.94753	0.157398
$630$	8.12260	0.323612
$631$	14.1190	0.562068	0.281034	−	0.959698i	$-0.409323\pi$
$631$	14.1190	0.562068	0.281034	+	0.959698i	$0.409323\pi$
$632$	3.38403	0.134609
$633$	6.49850	0.258292
$634$	32.4805	1.28996
$635$	−22.1664	−0.879649
$636$	5.21426	0.206759
$637$	0.156195	0.00618869
$638$	5.48709	0.217236
$639$	−6.07544	−0.240341
$640$	3.29417	0.130214
$641$	−8.45700	−0.334031	−0.167016	−	0.985954i	$-0.553413\pi$
$641$	−8.45700	−0.334031	−0.167016	+	0.985954i	$0.553413\pi$
$642$	−10.0986	−0.398560
$643$	48.9238	1.92936	0.964682	−	0.263416i	$-0.0848493\pi$
$643$	48.9238	1.92936	0.964682	+	0.263416i	$0.0848493\pi$
$644$	2.46575	0.0971641
$645$	33.3456	1.31298
$646$	−1.75957	−0.0692294
$647$	20.5129	0.806446	0.403223	−	0.915102i	$-0.367890\pi$
$647$	20.5129	0.806446	0.403223	+	0.915102i	$0.367890\pi$
$648$	−1.00000	−0.0392837
$649$	−66.5178	−2.61105
$650$	0.993371	0.0389632
$651$	8.59829	0.336994
$652$	15.2702	0.598026
$653$	−12.5013	−0.489213	−0.244606	−	0.969622i	$-0.578659\pi$
$653$	−12.5013	−0.489213	−0.244606	+	0.969622i	$0.578659\pi$
$654$	0.442589	0.0173066
$655$	69.8044	2.72748
$656$	7.95284	0.310506
$657$	−3.86299	−0.150710
$658$	−3.78308	−0.147480
$659$	31.1479	1.21335	0.606675	−	0.794950i	$-0.292503\pi$
$659$	31.1479	1.21335	0.606675	+	0.794950i	$0.292503\pi$
$660$	18.0754	0.703586
$661$	9.98594	0.388408	0.194204	−	0.980961i	$-0.437788\pi$
$661$	9.98594	0.388408	0.194204	+	0.980961i	$0.437788\pi$
$662$	23.2629	0.904139
$663$	0.0615689	0.00239114
$664$	−5.55741	−0.215669
$665$	−39.4075	−1.52816
$666$	−10.8843	−0.421759
$667$	−1.00000	−0.0387202
$668$	−15.5662	−0.602273
$669$	2.34315	0.0905912
$670$	−20.9123	−0.807913
$671$	50.8709	1.96385
$672$	2.46575	0.0951183
$673$	−7.50782	−0.289405	−0.144703	−	0.989475i	$-0.546223\pi$
$673$	−7.50782	−0.289405	−0.144703	+	0.989475i	$0.546223\pi$
$674$	31.1391	1.19943
$675$	−5.85158	−0.225228
$676$	−12.9712	−0.498892
$677$	15.7650	0.605900	0.302950	−	0.953006i	$-0.402028\pi$
$677$	15.7650	0.605900	0.302950	+	0.953006i	$0.402028\pi$
$678$	10.5652	0.405754
$679$	−39.8114	−1.52782
$680$	1.19473	0.0458158
$681$	−5.80708	−0.222528
$682$	19.1340	0.732679
$683$	44.1792	1.69047	0.845234	−	0.534396i	$-0.179461\pi$
$683$	44.1792	1.69047	0.845234	+	0.534396i	$0.179461\pi$
$684$	4.85158	0.185505
$685$	−64.6294	−2.46936
$686$	19.5289	0.745618
$687$	5.08685	0.194075
$688$	10.1226	0.385921
$689$	0.885179	0.0337226
$690$	−3.29417	−0.125407
$691$	−1.47322	−0.0560439	−0.0280220	−	0.999607i	$-0.508921\pi$
$691$	−1.47322	−0.0560439	−0.0280220	+	0.999607i	$0.508921\pi$
$692$	−4.29236	−0.163171
$693$	13.5298	0.513954
$694$	33.7795	1.28225
$695$	23.2849	0.883246
$696$	−1.00000	−0.0379049
$697$	2.88434	0.109252
$698$	4.08901	0.154771
$699$	−3.95731	−0.149679
$700$	14.4285	0.545347
$701$	−9.42950	−0.356147	−0.178074	−	0.984017i	$-0.556987\pi$
$701$	−9.42950	−0.356147	−0.178074	+	0.984017i	$0.556987\pi$
$702$	−0.169761	−0.00640722
$703$	52.8063	1.99163
$704$	5.48709	0.206803
$705$	5.05410	0.190348
$706$	−20.2025	−0.760332
$707$	−24.6575	−0.927340
$708$	12.1226	0.455595
$709$	−23.6365	−0.887686	−0.443843	−	0.896104i	$-0.646385\pi$
$709$	−23.6365	−0.887686	−0.443843	+	0.896104i	$0.646385\pi$
$710$	−20.0136	−0.751096
$711$	−3.38403	−0.126911
$712$	10.2828	0.385363
$713$	−3.48709	−0.130593
$714$	0.894277	0.0334675
$715$	3.06851	0.114756
$716$	−23.3600	−0.873005
$717$	−16.7254	−0.624620
$718$	−15.2806	−0.570267
$719$	−3.44078	−0.128319	−0.0641597	−	0.997940i	$-0.520437\pi$
$719$	−3.44078	−0.128319	−0.0641597	+	0.997940i	$0.520437\pi$
$720$	−3.29417	−0.122767
$721$	31.6798	1.17982
$722$	−4.53788	−0.168882
$723$	−7.56700	−0.281420
$724$	3.04450	0.113148
$725$	−5.85158	−0.217322
$726$	19.1082	0.709172
$727$	−2.08026	−0.0771525	−0.0385763	−	0.999256i	$-0.512282\pi$
$727$	−2.08026	−0.0771525	−0.0385763	+	0.999256i	$0.512282\pi$
$728$	0.418588	0.0155139
$729$	1.00000	0.0370370
$730$	−12.7254	−0.470987
$731$	3.67126	0.135787
$732$	−9.27102	−0.342667
$733$	0.751792	0.0277681	0.0138840	−	0.999904i	$-0.495580\pi$
$733$	0.751792	0.0277681	0.0138840	+	0.999904i	$0.495580\pi$
$734$	−24.8324	−0.916579
$735$	−3.03094	−0.111798
$736$	−1.00000	−0.0368605
$737$	−34.8336	−1.28311
$738$	−7.95284	−0.292748
$739$	28.0583	1.03214	0.516070	−	0.856547i	$-0.327395\pi$
$739$	28.0583	1.03214	0.516070	+	0.856547i	$0.327395\pi$
$740$	−35.8549	−1.31805
$741$	0.823610	0.0302561
$742$	12.8571	0.471997
$743$	31.7693	1.16550	0.582752	−	0.812650i	$-0.301976\pi$
$743$	31.7693	1.16550	0.582752	+	0.812650i	$0.301976\pi$
$744$	−3.48709	−0.127843
$745$	−45.8840	−1.68106
$746$	28.1404	1.03029
$747$	5.55741	0.203335
$748$	1.99006	0.0727637
$749$	−24.9006	−0.909848
$750$	−2.80527	−0.102434
$751$	17.9197	0.653901	0.326950	−	0.945042i	$-0.393979\pi$
$751$	17.9197	0.653901	0.326950	+	0.945042i	$0.393979\pi$
$752$	1.53425	0.0559484
$753$	6.79086	0.247473
$754$	−0.169761	−0.00618233
$755$	−35.7023	−1.29934
$756$	−2.46575	−0.0896784
$757$	41.8784	1.52210	0.761048	−	0.648695i	$-0.224685\pi$
$757$	41.8784	1.52210	0.761048	+	0.648695i	$0.224685\pi$
$758$	16.2025	0.588502
$759$	−5.48709	−0.199169
$760$	15.9820	0.579727
$761$	−51.0016	−1.84881	−0.924404	−	0.381415i	$-0.875437\pi$
$761$	−51.0016	−1.84881	−0.924404	+	0.381415i	$0.875437\pi$
$762$	6.72898	0.243765
$763$	1.09131	0.0395082
$764$	18.3011	0.662111
$765$	−1.19473	−0.0431956
$766$	−15.5063	−0.560265
$767$	2.05795	0.0743081
$768$	−1.00000	−0.0360844
$769$	−39.7447	−1.43323	−0.716615	−	0.697469i	$-0.754309\pi$
$769$	−39.7447	−1.43323	−0.716615	+	0.697469i	$0.754309\pi$
$770$	44.5695	1.60617
$771$	−24.3670	−0.877555
$772$	22.7909	0.820261
$773$	51.7714	1.86209	0.931043	−	0.364909i	$-0.118900\pi$
$773$	51.7714	1.86209	0.931043	+	0.364909i	$0.118900\pi$
$774$	−10.1226	−0.363849
$775$	−20.4050	−0.732970
$776$	16.1458	0.579599
$777$	−26.8380	−0.962809
$778$	14.2680	0.511533
$779$	38.5839	1.38241
$780$	−0.559222	−0.0200234
$781$	−33.3365	−1.19287
$782$	−0.362680	−0.0129694
$783$	1.00000	0.0357371
$784$	−0.920090	−0.0328604
$785$	−45.9631	−1.64049
$786$	−21.1903	−0.755832
$787$	−17.7235	−0.631776	−0.315888	−	0.948796i	$-0.602302\pi$
$787$	−17.7235	−0.631776	−0.315888	+	0.948796i	$0.602302\pi$
$788$	15.0213	0.535113
$789$	−28.2975	−1.00742
$790$	−11.1476	−0.396613
$791$	26.0511	0.926270
$792$	−5.48709	−0.194975
$793$	−1.57386	−0.0558893
$794$	28.8200	1.02278
$795$	−17.1767	−0.609195
$796$	22.1082	0.783604
$797$	45.1079	1.59780	0.798902	−	0.601462i	$-0.205415\pi$
$797$	45.1079	1.59780	0.798902	+	0.601462i	$0.205415\pi$
$798$	11.9628	0.423478
$799$	0.556443	0.0196855
$800$	−5.85158	−0.206885
$801$	−10.2828	−0.363324
$802$	−26.9706	−0.952364
$803$	−21.1966	−0.748011
$804$	6.34827	0.223886
$805$	−8.12260	−0.286284
$806$	−0.591973	−0.0208513
$807$	−14.8044	−0.521140
$808$	10.0000	0.351799
$809$	17.2458	0.606331	0.303165	−	0.952938i	$-0.401957\pi$
$809$	17.2458	0.606331	0.303165	+	0.952938i	$0.401957\pi$
$810$	3.29417	0.115745
$811$	−26.0410	−0.914423	−0.457212	−	0.889358i	$-0.651152\pi$
$811$	−26.0410	−0.914423	−0.457212	+	0.889358i	$0.651152\pi$
$812$	−2.46575	−0.0865308
$813$	7.03969	0.246893
$814$	−59.7234	−2.09330
$815$	−50.3026	−1.76202
$816$	−0.362680	−0.0126963
$817$	49.1107	1.71816
$818$	37.7786	1.32090
$819$	−0.418588	−0.0146266
$820$	−26.1980	−0.914875
$821$	15.7224	0.548714	0.274357	−	0.961628i	$-0.411535\pi$
$821$	15.7224	0.548714	0.274357	+	0.961628i	$0.411535\pi$
$822$	19.6193	0.684301
$823$	−33.8213	−1.17894	−0.589468	−	0.807792i	$-0.700663\pi$
$823$	−33.8213	−1.17894	−0.589468	+	0.807792i	$0.700663\pi$
$824$	−12.8480	−0.447580
$825$	−32.1082	−1.11786
$826$	29.8913	1.04005
$827$	−48.8726	−1.69947	−0.849734	−	0.527212i	$-0.823237\pi$
$827$	−48.8726	−1.69947	−0.849734	+	0.527212i	$0.823237\pi$
$828$	1.00000	0.0347524
$829$	−43.6461	−1.51589	−0.757945	−	0.652318i	$-0.773797\pi$
$829$	−43.6461	−1.51589	−0.757945	+	0.652318i	$0.773797\pi$
$830$	18.3071	0.635448
$831$	−29.6843	−1.02974
$832$	−0.169761	−0.00588540
$833$	−0.333698	−0.0115620
$834$	−7.06851	−0.244762
$835$	51.2776	1.77454
$836$	26.6211	0.920710
$837$	3.48709	0.120532
$838$	10.9838	0.379428
$839$	−40.4844	−1.39768	−0.698839	−	0.715279i	$-0.746300\pi$
$839$	−40.4844	−1.39768	−0.698839	+	0.715279i	$0.746300\pi$
$840$	−8.12260	−0.280256
$841$	1.00000	0.0344828
$842$	−20.8762	−0.719443
$843$	9.66048	0.332725
$844$	−6.49850	−0.223688
$845$	42.7293	1.46993
$846$	−1.53425	−0.0527487
$847$	47.1160	1.61892
$848$	−5.21426	−0.179059
$849$	−6.25002	−0.214500
$850$	−2.12225	−0.0727926
$851$	10.8843	0.373110
$852$	6.07544	0.208141
$853$	0.878017	0.0300627	0.0150314	−	0.999887i	$-0.495215\pi$
$853$	0.878017	0.0300627	0.0150314	+	0.999887i	$0.495215\pi$
$854$	−22.8600	−0.782253
$855$	−15.9820	−0.546572
$856$	10.0986	0.345163
$857$	−41.3838	−1.41365	−0.706823	−	0.707391i	$-0.749872\pi$
$857$	−41.3838	−1.41365	−0.706823	+	0.707391i	$0.749872\pi$
$858$	−0.931495	−0.0318007
$859$	−5.59413	−0.190869	−0.0954347	−	0.995436i	$-0.530424\pi$
$859$	−5.59413	−0.190869	−0.0954347	+	0.995436i	$0.530424\pi$
$860$	−33.3456	−1.13708
$861$	−19.6097	−0.668297
$862$	−10.1698	−0.346383
$863$	−52.5142	−1.78760	−0.893802	−	0.448461i	$-0.851972\pi$
$863$	−52.5142	−1.78760	−0.893802	+	0.448461i	$0.851972\pi$
$864$	1.00000	0.0340207
$865$	14.1398	0.480767
$866$	20.0842	0.682489
$867$	16.8685	0.572883
$868$	−8.59829	−0.291845
$869$	−18.5685	−0.629892
$870$	3.29417	0.111683
$871$	1.07769	0.0365161
$872$	−0.442589	−0.0149880
$873$	−16.1458	−0.546451
$874$	−4.85158	−0.164107
$875$	−6.91709	−0.233840
$876$	3.86299	0.130518
$877$	−11.4156	−0.385477	−0.192738	−	0.981250i	$-0.561737\pi$
$877$	−11.4156	−0.385477	−0.192738	+	0.981250i	$0.561737\pi$
$878$	−35.6097	−1.20177
$879$	−21.1304	−0.712710
$880$	−18.0754	−0.609323
$881$	−8.39265	−0.282756	−0.141378	−	0.989956i	$-0.545153\pi$
$881$	−8.39265	−0.282756	−0.141378	+	0.989956i	$0.545153\pi$
$882$	0.920090	0.0309811
$883$	4.53279	0.152541	0.0762703	−	0.997087i	$-0.475699\pi$
$883$	4.53279	0.152541	0.0762703	+	0.997087i	$0.475699\pi$
$884$	−0.0615689	−0.00207079
$885$	−39.9340	−1.34237
$886$	−2.56916	−0.0863128
$887$	−47.6443	−1.59974	−0.799869	−	0.600174i	$-0.795098\pi$
$887$	−47.6443	−1.59974	−0.799869	+	0.600174i	$0.795098\pi$
$888$	10.8843	0.365254
$889$	16.5920	0.556477
$890$	−33.8732	−1.13543
$891$	5.48709	0.183825
$892$	−2.34315	−0.0784543
$893$	7.44356	0.249089
$894$	13.9288	0.465850
$895$	76.9520	2.57222
$896$	−2.46575	−0.0823748
$897$	0.169761	0.00566816
$898$	15.5207	0.517932
$899$	3.48709	0.116301
$900$	5.85158	0.195053
$901$	−1.89111	−0.0630020
$902$	−43.6380	−1.45299
$903$	−24.9598	−0.830609
$904$	−10.5652	−0.351393
$905$	−10.0291	−0.333379
$906$	10.8380	0.360069
$907$	−0.757802	−0.0251624	−0.0125812	−	0.999921i	$-0.504005\pi$
$907$	−0.757802	−0.0251624	−0.0125812	+	0.999921i	$0.504005\pi$
$908$	5.80708	0.192715
$909$	−10.0000	−0.331679
$910$	−1.37890	−0.0457101
$911$	−19.8417	−0.657384	−0.328692	−	0.944437i	$-0.606608\pi$
$911$	−19.8417	−0.657384	−0.328692	+	0.944437i	$0.606608\pi$
$912$	−4.85158	−0.160652
$913$	30.4940	1.00921
$914$	−18.5220	−0.612653
$915$	30.5403	1.00963
$916$	−5.08685	−0.168074
$917$	−52.2498	−1.72544
$918$	0.362680	0.0119702
$919$	4.53557	0.149615	0.0748073	−	0.997198i	$-0.476166\pi$
$919$	4.53557	0.149615	0.0748073	+	0.997198i	$0.476166\pi$
$920$	3.29417	0.108606
$921$	13.1767	0.434187
$922$	8.76173	0.288552
$923$	1.03137	0.0339481
$924$	−13.5298	−0.445097
$925$	63.6906	2.09413
$926$	9.49704	0.312092
$927$	12.8480	0.421982
$928$	1.00000	0.0328266
$929$	19.7597	0.648296	0.324148	−	0.946006i	$-0.394922\pi$
$929$	19.7597	0.648296	0.324148	+	0.946006i	$0.394922\pi$
$930$	11.4871	0.376677
$931$	−4.46390	−0.146298
$932$	3.95731	0.129626
$933$	−20.6683	−0.676648
$934$	34.0718	1.11486
$935$	−6.55560	−0.214391
$936$	0.169761	0.00554881
$937$	3.25630	0.106379	0.0531893	−	0.998584i	$-0.483061\pi$
$937$	3.25630	0.106379	0.0531893	+	0.998584i	$0.483061\pi$
$938$	15.6532	0.511096
$939$	−10.8152	−0.352941
$940$	−5.05410	−0.164846
$941$	28.1404	0.917352	0.458676	−	0.888603i	$-0.348324\pi$
$941$	28.1404	0.917352	0.458676	+	0.888603i	$0.348324\pi$
$942$	13.9528	0.454608
$943$	7.95284	0.258980
$944$	−12.1226	−0.394557
$945$	8.12260	0.264228
$946$	−55.5437	−1.80588
$947$	−17.5953	−0.571770	−0.285885	−	0.958264i	$-0.592288\pi$
$947$	−17.5953	−0.571770	−0.285885	+	0.958264i	$0.592288\pi$
$948$	3.38403	0.109908
$949$	0.655785	0.0212877
$950$	−28.3895	−0.921076
$951$	32.4805	1.05325
$952$	−0.894277	−0.0289837
$953$	11.7512	0.380658	0.190329	−	0.981720i	$-0.439045\pi$
$953$	11.7512	0.380658	0.190329	+	0.981720i	$0.439045\pi$
$954$	5.21426	0.168818
$955$	−60.2871	−1.95084
$956$	16.7254	0.540937
$957$	5.48709	0.177373
$958$	−20.7341	−0.669889
$959$	48.3762	1.56215
$960$	3.29417	0.106319
$961$	−18.8402	−0.607748
$962$	1.84774	0.0595734
$963$	−10.0986	−0.325423
$964$	7.56700	0.243717
$965$	−75.0771	−2.41682
$966$	2.46575	0.0793341
$967$	2.50928	0.0806931	0.0403466	−	0.999186i	$-0.487154\pi$
$967$	2.50928	0.0806931	0.0403466	+	0.999186i	$0.487154\pi$
$968$	−19.1082	−0.614161
$969$	−1.75957	−0.0565256
$970$	−53.1869	−1.70773
$971$	−20.9414	−0.672043	−0.336021	−	0.941854i	$-0.609081\pi$
$971$	−20.9414	−0.672043	−0.336021	+	0.941854i	$0.609081\pi$
$972$	−1.00000	−0.0320750
$973$	−17.4291	−0.558753
$974$	−42.0111	−1.34612
$975$	0.993371	0.0318133
$976$	9.27102	0.296758
$977$	−15.3501	−0.491093	−0.245546	−	0.969385i	$-0.578967\pi$
$977$	−15.3501	−0.491093	−0.245546	+	0.969385i	$0.578967\pi$
$978$	15.2702	0.488286
$979$	−56.4225	−1.80327
$980$	3.03094	0.0968198
$981$	0.442589	0.0141308
$982$	−21.6896	−0.692143
$983$	−29.2533	−0.933036	−0.466518	−	0.884512i	$-0.654492\pi$
$983$	−29.2533	−0.933036	−0.466518	+	0.884512i	$0.654492\pi$
$984$	7.95284	0.253527
$985$	−49.4829	−1.57666
$986$	0.362680	0.0115501
$987$	−3.78308	−0.120417
$988$	−0.823610	−0.0262025
$989$	10.1226	0.321880
$990$	18.0754	0.574475
$991$	35.3747	1.12372	0.561858	−	0.827234i	$-0.310087\pi$
$991$	35.3747	1.12372	0.561858	+	0.827234i	$0.310087\pi$
$992$	3.48709	0.110715
$993$	23.2629	0.738227
$994$	14.9805	0.475153
$995$	−72.8282	−2.30881
$996$	−5.55741	−0.176093
$997$	36.7573	1.16411	0.582057	−	0.813148i	$-0.302248\pi$
$997$	36.7573	1.16411	0.582057	+	0.813148i	$0.302248\pi$
$998$	−16.2125	−0.513196
$999$	−10.8843	−0.344365

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.ba.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.29417 q^{5} +1.00000 q^{6} +2.46575 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.29417 q^{5} +1.00000 q^{6} +2.46575 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.29417 q^{10} +5.48709 q^{11} -1.00000 q^{12} -0.169761 q^{13} -2.46575 q^{14} +3.29417 q^{15} +1.00000 q^{16} +0.362680 q^{17} -1.00000 q^{18} +4.85158 q^{19} -3.29417 q^{20} -2.46575 q^{21} -5.48709 q^{22} +1.00000 q^{23} +1.00000 q^{24} +5.85158 q^{25} +0.169761 q^{26} -1.00000 q^{27} +2.46575 q^{28} -1.00000 q^{29} -3.29417 q^{30} -3.48709 q^{31} -1.00000 q^{32} -5.48709 q^{33} -0.362680 q^{34} -8.12260 q^{35} +1.00000 q^{36} +10.8843 q^{37} -4.85158 q^{38} +0.169761 q^{39} +3.29417 q^{40} +7.95284 q^{41} +2.46575 q^{42} +10.1226 q^{43} +5.48709 q^{44} -3.29417 q^{45} -1.00000 q^{46} +1.53425 q^{47} -1.00000 q^{48} -0.920090 q^{49} -5.85158 q^{50} -0.362680 q^{51} -0.169761 q^{52} -5.21426 q^{53} +1.00000 q^{54} -18.0754 q^{55} -2.46575 q^{56} -4.85158 q^{57} +1.00000 q^{58} -12.1226 q^{59} +3.29417 q^{60} +9.27102 q^{61} +3.48709 q^{62} +2.46575 q^{63} +1.00000 q^{64} +0.559222 q^{65} +5.48709 q^{66} -6.34827 q^{67} +0.362680 q^{68} -1.00000 q^{69} +8.12260 q^{70} -6.07544 q^{71} -1.00000 q^{72} -3.86299 q^{73} -10.8843 q^{74} -5.85158 q^{75} +4.85158 q^{76} +13.5298 q^{77} -0.169761 q^{78} -3.38403 q^{79} -3.29417 q^{80} +1.00000 q^{81} -7.95284 q^{82} +5.55741 q^{83} -2.46575 q^{84} -1.19473 q^{85} -10.1226 q^{86} +1.00000 q^{87} -5.48709 q^{88} -10.2828 q^{89} +3.29417 q^{90} -0.418588 q^{91} +1.00000 q^{92} +3.48709 q^{93} -1.53425 q^{94} -15.9820 q^{95} +1.00000 q^{96} -16.1458 q^{97} +0.920090 q^{98} +5.48709 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 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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