LMFDB - Embedded newform 4002.2.a.bg.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:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 13x^{5} + 16x^{4} + 19x^{3} - 8x^{2} - 10x - 2 $ x^7 - x^6 - 13x^5 + 16x^4 + 19x^3 - 8x^2 - 10*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.99277$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.83000 q^{5} +1.00000 q^{6} -4.91072 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.83000 q^{5} +1.00000 q^{6} -4.91072 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.83000 q^{10} -1.87490 q^{11} -1.00000 q^{12} -3.25549 q^{13} +4.91072 q^{14} +3.83000 q^{15} +1.00000 q^{16} +5.70490 q^{17} -1.00000 q^{18} +0.533915 q^{19} -3.83000 q^{20} +4.91072 q^{21} +1.87490 q^{22} +1.00000 q^{23} +1.00000 q^{24} +9.66892 q^{25} +3.25549 q^{26} -1.00000 q^{27} -4.91072 q^{28} +1.00000 q^{29} -3.83000 q^{30} -6.65632 q^{31} -1.00000 q^{32} +1.87490 q^{33} -5.70490 q^{34} +18.8081 q^{35} +1.00000 q^{36} +0.389510 q^{37} -0.533915 q^{38} +3.25549 q^{39} +3.83000 q^{40} +2.74088 q^{41} -4.91072 q^{42} +12.3587 q^{43} -1.87490 q^{44} -3.83000 q^{45} -1.00000 q^{46} -3.23624 q^{47} -1.00000 q^{48} +17.1152 q^{49} -9.66892 q^{50} -5.70490 q^{51} -3.25549 q^{52} -3.80480 q^{53} +1.00000 q^{54} +7.18087 q^{55} +4.91072 q^{56} -0.533915 q^{57} -1.00000 q^{58} +7.62559 q^{59} +3.83000 q^{60} +5.94654 q^{61} +6.65632 q^{62} -4.91072 q^{63} +1.00000 q^{64} +12.4685 q^{65} -1.87490 q^{66} -6.17079 q^{67} +5.70490 q^{68} -1.00000 q^{69} -18.8081 q^{70} +2.94885 q^{71} -1.00000 q^{72} -9.05747 q^{73} -0.389510 q^{74} -9.66892 q^{75} +0.533915 q^{76} +9.20710 q^{77} -3.25549 q^{78} -5.70853 q^{79} -3.83000 q^{80} +1.00000 q^{81} -2.74088 q^{82} +3.17099 q^{83} +4.91072 q^{84} -21.8498 q^{85} -12.3587 q^{86} -1.00000 q^{87} +1.87490 q^{88} +11.3685 q^{89} +3.83000 q^{90} +15.9868 q^{91} +1.00000 q^{92} +6.65632 q^{93} +3.23624 q^{94} -2.04490 q^{95} +1.00000 q^{96} +3.97037 q^{97} -17.1152 q^{98} -1.87490 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} + 2 q^{5} + 7 q^{6} - 5 q^{7} - 7 q^{8} + 7 q^{9}+O(q^{10}) $ 7 * q - 7 * q^2 - 7 * q^3 + 7 * q^4 + 2 * q^5 + 7 * q^6 - 5 * q^7 - 7 * q^8 + 7 * q^9	\( 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} + 2 q^{5} + 7 q^{6} - 5 q^{7} - 7 q^{8} + 7 q^{9} - 2 q^{10} - q^{11} - 7 q^{12} + q^{13} + 5 q^{14} - 2 q^{15} + 7 q^{16} - q^{17} - 7 q^{18} - 9 q^{19} + 2 q^{20} + 5 q^{21} + q^{22} + 7 q^{23} + 7 q^{24} + q^{25} - q^{26} - 7 q^{27} - 5 q^{28} + 7 q^{29} + 2 q^{30} - 19 q^{31} - 7 q^{32} + q^{33} + q^{34} + 11 q^{35} + 7 q^{36} - 18 q^{37} + 9 q^{38} - q^{39} - 2 q^{40} + 4 q^{41} - 5 q^{42} - 9 q^{43} - q^{44} + 2 q^{45} - 7 q^{46} - 11 q^{47} - 7 q^{48} + 12 q^{49} - q^{50} + q^{51} + q^{52} + 8 q^{53} + 7 q^{54} - 11 q^{55} + 5 q^{56} + 9 q^{57} - 7 q^{58} + 12 q^{59} - 2 q^{60} - 5 q^{61} + 19 q^{62} - 5 q^{63} + 7 q^{64} + 23 q^{65} - q^{66} - 13 q^{67} - q^{68} - 7 q^{69} - 11 q^{70} - q^{71} - 7 q^{72} - 26 q^{73} + 18 q^{74} - q^{75} - 9 q^{76} + 6 q^{77} + q^{78} - 38 q^{79} + 2 q^{80} + 7 q^{81} - 4 q^{82} - 6 q^{83} + 5 q^{84} - 25 q^{85} + 9 q^{86} - 7 q^{87} + q^{88} + 20 q^{89} - 2 q^{90} + 2 q^{91} + 7 q^{92} + 19 q^{93} + 11 q^{94} - 31 q^{95} + 7 q^{96} - 8 q^{97} - 12 q^{98} - q^{99}+O(q^{100}) \) 7 * q - 7 * q^2 - 7 * q^3 + 7 * q^4 + 2 * q^5 + 7 * q^6 - 5 * q^7 - 7 * q^8 + 7 * q^9 - 2 * q^10 - q^11 - 7 * q^12 + q^13 + 5 * q^14 - 2 * q^15 + 7 * q^16 - q^17 - 7 * q^18 - 9 * q^19 + 2 * q^20 + 5 * q^21 + q^22 + 7 * q^23 + 7 * q^24 + q^25 - q^26 - 7 * q^27 - 5 * q^28 + 7 * q^29 + 2 * q^30 - 19 * q^31 - 7 * q^32 + q^33 + q^34 + 11 * q^35 + 7 * q^36 - 18 * q^37 + 9 * q^38 - q^39 - 2 * q^40 + 4 * q^41 - 5 * q^42 - 9 * q^43 - q^44 + 2 * q^45 - 7 * q^46 - 11 * q^47 - 7 * q^48 + 12 * q^49 - q^50 + q^51 + q^52 + 8 * q^53 + 7 * q^54 - 11 * q^55 + 5 * q^56 + 9 * q^57 - 7 * q^58 + 12 * q^59 - 2 * q^60 - 5 * q^61 + 19 * q^62 - 5 * q^63 + 7 * q^64 + 23 * q^65 - q^66 - 13 * q^67 - q^68 - 7 * q^69 - 11 * q^70 - q^71 - 7 * q^72 - 26 * q^73 + 18 * q^74 - q^75 - 9 * q^76 + 6 * q^77 + q^78 - 38 * q^79 + 2 * q^80 + 7 * q^81 - 4 * q^82 - 6 * q^83 + 5 * q^84 - 25 * q^85 + 9 * q^86 - 7 * q^87 + q^88 + 20 * q^89 - 2 * q^90 + 2 * q^91 + 7 * q^92 + 19 * q^93 + 11 * q^94 - 31 * q^95 + 7 * q^96 - 8 * q^97 - 12 * q^98 - q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−3.83000	−1.71283	−0.856415	−	0.516289i	$-0.827313\pi$
$5$	−3.83000	−1.71283	−0.856415	+	0.516289i	$0.827313\pi$
$6$	1.00000	0.408248
$7$	−4.91072	−1.85608	−0.928039	−	0.372484i	$-0.878506\pi$
$7$	−4.91072	−1.85608	−0.928039	+	0.372484i	$0.878506\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	3.83000	1.21115
$11$	−1.87490	−0.565303	−0.282652	−	0.959223i	$-0.591214\pi$
$11$	−1.87490	−0.565303	−0.282652	+	0.959223i	$0.591214\pi$
$12$	−1.00000	−0.288675
$13$	−3.25549	−0.902912	−0.451456	−	0.892294i	$-0.649095\pi$
$13$	−3.25549	−0.902912	−0.451456	+	0.892294i	$0.649095\pi$
$14$	4.91072	1.31244
$15$	3.83000	0.988902
$16$	1.00000	0.250000
$17$	5.70490	1.38364	0.691821	−	0.722069i	$-0.256809\pi$
$17$	5.70490	1.38364	0.691821	+	0.722069i	$0.256809\pi$
$18$	−1.00000	−0.235702
$19$	0.533915	0.122489	0.0612443	−	0.998123i	$-0.480493\pi$
$19$	0.533915	0.122489	0.0612443	+	0.998123i	$0.480493\pi$
$20$	−3.83000	−0.856415
$21$	4.91072	1.07161
$22$	1.87490	0.399730
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	9.66892	1.93378
$26$	3.25549	0.638455
$27$	−1.00000	−0.192450
$28$	−4.91072	−0.928039
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−3.83000	−0.699260
$31$	−6.65632	−1.19551	−0.597755	−	0.801679i	$-0.703941\pi$
$31$	−6.65632	−1.19551	−0.597755	+	0.801679i	$0.703941\pi$
$32$	−1.00000	−0.176777
$33$	1.87490	0.326378
$34$	−5.70490	−0.978382
$35$	18.8081	3.17914
$36$	1.00000	0.166667
$37$	0.389510	0.0640350	0.0320175	−	0.999487i	$-0.489807\pi$
$37$	0.389510	0.0640350	0.0320175	+	0.999487i	$0.489807\pi$
$38$	−0.533915	−0.0866125
$39$	3.25549	0.521296
$40$	3.83000	0.605577
$41$	2.74088	0.428054	0.214027	−	0.976828i	$-0.431342\pi$
$41$	2.74088	0.428054	0.214027	+	0.976828i	$0.431342\pi$
$42$	−4.91072	−0.757740
$43$	12.3587	1.88469	0.942343	−	0.334649i	$-0.108618\pi$
$43$	12.3587	1.88469	0.942343	+	0.334649i	$0.108618\pi$
$44$	−1.87490	−0.282652
$45$	−3.83000	−0.570943
$46$	−1.00000	−0.147442
$47$	−3.23624	−0.472055	−0.236027	−	0.971746i	$-0.575845\pi$
$47$	−3.23624	−0.472055	−0.236027	+	0.971746i	$0.575845\pi$
$48$	−1.00000	−0.144338
$49$	17.1152	2.44502
$50$	−9.66892	−1.36739
$51$	−5.70490	−0.798846
$52$	−3.25549	−0.451456
$53$	−3.80480	−0.522629	−0.261314	−	0.965254i	$-0.584156\pi$
$53$	−3.80480	−0.522629	−0.261314	+	0.965254i	$0.584156\pi$
$54$	1.00000	0.136083
$55$	7.18087	0.968268
$56$	4.91072	0.656222
$57$	−0.533915	−0.0707188
$58$	−1.00000	−0.131306
$59$	7.62559	0.992767	0.496384	−	0.868103i	$-0.334661\pi$
$59$	7.62559	0.992767	0.496384	+	0.868103i	$0.334661\pi$
$60$	3.83000	0.494451
$61$	5.94654	0.761376	0.380688	−	0.924704i	$-0.375687\pi$
$61$	5.94654	0.761376	0.380688	+	0.924704i	$0.375687\pi$
$62$	6.65632	0.845354
$63$	−4.91072	−0.618692
$64$	1.00000	0.125000
$65$	12.4685	1.54653
$66$	−1.87490	−0.230784
$67$	−6.17079	−0.753882	−0.376941	−	0.926237i	$-0.623024\pi$
$67$	−6.17079	−0.753882	−0.376941	+	0.926237i	$0.623024\pi$
$68$	5.70490	0.691821
$69$	−1.00000	−0.120386
$70$	−18.8081	−2.24799
$71$	2.94885	0.349965	0.174982	−	0.984572i	$-0.444013\pi$
$71$	2.94885	0.349965	0.174982	+	0.984572i	$0.444013\pi$
$72$	−1.00000	−0.117851
$73$	−9.05747	−1.06010	−0.530049	−	0.847967i	$-0.677826\pi$
$73$	−9.05747	−1.06010	−0.530049	+	0.847967i	$0.677826\pi$
$74$	−0.389510	−0.0452796
$75$	−9.66892	−1.11647
$76$	0.533915	0.0612443
$77$	9.20710	1.04925
$78$	−3.25549	−0.368612
$79$	−5.70853	−0.642259	−0.321130	−	0.947035i	$-0.604063\pi$
$79$	−5.70853	−0.642259	−0.321130	+	0.947035i	$0.604063\pi$
$80$	−3.83000	−0.428207
$81$	1.00000	0.111111
$82$	−2.74088	−0.302680
$83$	3.17099	0.348061	0.174030	−	0.984740i	$-0.444321\pi$
$83$	3.17099	0.348061	0.174030	+	0.984740i	$0.444321\pi$
$84$	4.91072	0.535803
$85$	−21.8498	−2.36994
$86$	−12.3587	−1.33267
$87$	−1.00000	−0.107211
$88$	1.87490	0.199865
$89$	11.3685	1.20506	0.602531	−	0.798096i	$-0.294159\pi$
$89$	11.3685	1.20506	0.602531	+	0.798096i	$0.294159\pi$
$90$	3.83000	0.403718
$91$	15.9868	1.67587
$92$	1.00000	0.104257
$93$	6.65632	0.690228
$94$	3.23624	0.333793
$95$	−2.04490	−0.209802
$96$	1.00000	0.102062
$97$	3.97037	0.403130	0.201565	−	0.979475i	$-0.435397\pi$
$97$	3.97037	0.403130	0.201565	+	0.979475i	$0.435397\pi$
$98$	−17.1152	−1.72889
$99$	−1.87490	−0.188434
$100$	9.66892	0.966892
$101$	18.8391	1.87456	0.937281	−	0.348574i	$-0.113334\pi$
$101$	18.8391	1.87456	0.937281	+	0.348574i	$0.113334\pi$
$102$	5.70490	0.564869
$103$	−7.01135	−0.690848	−0.345424	−	0.938447i	$-0.612265\pi$
$103$	−7.01135	−0.690848	−0.345424	+	0.938447i	$0.612265\pi$
$104$	3.25549	0.319227
$105$	−18.8081	−1.83548
$106$	3.80480	0.369554
$107$	−8.37729	−0.809863	−0.404932	−	0.914347i	$-0.632705\pi$
$107$	−8.37729	−0.809863	−0.404932	+	0.914347i	$0.632705\pi$
$108$	−1.00000	−0.0962250
$109$	−3.47602	−0.332942	−0.166471	−	0.986046i	$-0.553237\pi$
$109$	−3.47602	−0.332942	−0.166471	+	0.986046i	$0.553237\pi$
$110$	−7.18087	−0.684669
$111$	−0.389510	−0.0369706
$112$	−4.91072	−0.464019
$113$	−0.833643	−0.0784226	−0.0392113	−	0.999231i	$-0.512485\pi$
$113$	−0.833643	−0.0784226	−0.0392113	+	0.999231i	$0.512485\pi$
$114$	0.533915	0.0500057
$115$	−3.83000	−0.357150
$116$	1.00000	0.0928477
$117$	−3.25549	−0.300971
$118$	−7.62559	−0.701993
$119$	−28.0152	−2.56815
$120$	−3.83000	−0.349630
$121$	−7.48476	−0.680432
$122$	−5.94654	−0.538374
$123$	−2.74088	−0.247137
$124$	−6.65632	−0.597755
$125$	−17.8820	−1.59941
$126$	4.91072	0.437482
$127$	−16.7703	−1.48812	−0.744061	−	0.668111i	$-0.767103\pi$
$127$	−16.7703	−1.48812	−0.744061	+	0.668111i	$0.767103\pi$
$128$	−1.00000	−0.0883883
$129$	−12.3587	−1.08812
$130$	−12.4685	−1.09356
$131$	16.7185	1.46070	0.730351	−	0.683072i	$-0.239356\pi$
$131$	16.7185	1.46070	0.730351	+	0.683072i	$0.239356\pi$
$132$	1.87490	0.163189
$133$	−2.62191	−0.227348
$134$	6.17079	0.533075
$135$	3.83000	0.329634
$136$	−5.70490	−0.489191
$137$	−0.560201	−0.0478612	−0.0239306	−	0.999714i	$-0.507618\pi$
$137$	−0.560201	−0.0478612	−0.0239306	+	0.999714i	$0.507618\pi$
$138$	1.00000	0.0851257
$139$	−21.8843	−1.85620	−0.928102	−	0.372326i	$-0.878560\pi$
$139$	−21.8843	−1.85620	−0.928102	+	0.372326i	$0.878560\pi$
$140$	18.8081	1.58957
$141$	3.23624	0.272541
$142$	−2.94885	−0.247462
$143$	6.10372	0.510419
$144$	1.00000	0.0833333
$145$	−3.83000	−0.318064
$146$	9.05747	0.749602
$147$	−17.1152	−1.41163
$148$	0.389510	0.0320175
$149$	−0.858167	−0.0703038	−0.0351519	−	0.999382i	$-0.511192\pi$
$149$	−0.858167	−0.0703038	−0.0351519	+	0.999382i	$0.511192\pi$
$150$	9.66892	0.789464
$151$	−13.5924	−1.10613	−0.553065	−	0.833138i	$-0.686542\pi$
$151$	−13.5924	−1.10613	−0.553065	+	0.833138i	$0.686542\pi$
$152$	−0.533915	−0.0433062
$153$	5.70490	0.461214
$154$	−9.20710	−0.741929
$155$	25.4937	2.04771
$156$	3.25549	0.260648
$157$	3.57130	0.285021	0.142511	−	0.989793i	$-0.454483\pi$
$157$	3.57130	0.285021	0.142511	+	0.989793i	$0.454483\pi$
$158$	5.70853	0.454146
$159$	3.80480	0.301740
$160$	3.83000	0.302788
$161$	−4.91072	−0.387019
$162$	−1.00000	−0.0785674
$163$	18.4147	1.44235	0.721174	−	0.692754i	$-0.243603\pi$
$163$	18.4147	1.44235	0.721174	+	0.692754i	$0.243603\pi$
$164$	2.74088	0.214027
$165$	−7.18087	−0.559030
$166$	−3.17099	−0.246116
$167$	5.92710	0.458653	0.229327	−	0.973350i	$-0.426348\pi$
$167$	5.92710	0.458653	0.229327	+	0.973350i	$0.426348\pi$
$168$	−4.91072	−0.378870
$169$	−2.40176	−0.184751
$170$	21.8498	1.67580
$171$	0.533915	0.0408295
$172$	12.3587	0.942343
$173$	2.39898	0.182391	0.0911954	−	0.995833i	$-0.470931\pi$
$173$	2.39898	0.182391	0.0911954	+	0.995833i	$0.470931\pi$
$174$	1.00000	0.0758098
$175$	−47.4813	−3.58925
$176$	−1.87490	−0.141326
$177$	−7.62559	−0.573174
$178$	−11.3685	−0.852107
$179$	−1.32347	−0.0989208	−0.0494604	−	0.998776i	$-0.515750\pi$
$179$	−1.32347	−0.0989208	−0.0494604	+	0.998776i	$0.515750\pi$
$180$	−3.83000	−0.285472
$181$	20.3239	1.51066	0.755331	−	0.655344i	$-0.227476\pi$
$181$	20.3239	1.51066	0.755331	+	0.655344i	$0.227476\pi$
$182$	−15.9868	−1.18502
$183$	−5.94654	−0.439581
$184$	−1.00000	−0.0737210
$185$	−1.49182	−0.109681
$186$	−6.65632	−0.488065
$187$	−10.6961	−0.782177
$188$	−3.23624	−0.236027
$189$	4.91072	0.357202
$190$	2.04490	0.148352
$191$	20.2770	1.46719	0.733596	−	0.679586i	$-0.237840\pi$
$191$	20.2770	1.46719	0.733596	+	0.679586i	$0.237840\pi$
$192$	−1.00000	−0.0721688
$193$	−17.5746	−1.26505	−0.632523	−	0.774541i	$-0.717981\pi$
$193$	−17.5746	−1.26505	−0.632523	+	0.774541i	$0.717981\pi$
$194$	−3.97037	−0.285056
$195$	−12.4685	−0.892891
$196$	17.1152	1.22251
$197$	−14.7121	−1.04819	−0.524096	−	0.851659i	$-0.675597\pi$
$197$	−14.7121	−1.04819	−0.524096	+	0.851659i	$0.675597\pi$
$198$	1.87490	0.133243
$199$	−3.40969	−0.241706	−0.120853	−	0.992670i	$-0.538563\pi$
$199$	−3.40969	−0.241706	−0.120853	+	0.992670i	$0.538563\pi$
$200$	−9.66892	−0.683696
$201$	6.17079	0.435254
$202$	−18.8391	−1.32552
$203$	−4.91072	−0.344665
$204$	−5.70490	−0.399423
$205$	−10.4976	−0.733183
$206$	7.01135	0.488504
$207$	1.00000	0.0695048
$208$	−3.25549	−0.225728
$209$	−1.00104	−0.0692431
$210$	18.8081	1.29788
$211$	−12.2627	−0.844197	−0.422099	−	0.906550i	$-0.638706\pi$
$211$	−12.2627	−0.844197	−0.422099	+	0.906550i	$0.638706\pi$
$212$	−3.80480	−0.261314
$213$	−2.94885	−0.202052
$214$	8.37729	0.572660
$215$	−47.3339	−3.22814
$216$	1.00000	0.0680414
$217$	32.6873	2.21896
$218$	3.47602	0.235426
$219$	9.05747	0.612047
$220$	7.18087	0.484134
$221$	−18.5723	−1.24931
$222$	0.389510	0.0261422
$223$	−14.0063	−0.937932	−0.468966	−	0.883216i	$-0.655373\pi$
$223$	−14.0063	−0.937932	−0.468966	+	0.883216i	$0.655373\pi$
$224$	4.91072	0.328111
$225$	9.66892	0.644595
$226$	0.833643	0.0554531
$227$	26.1269	1.73410	0.867052	−	0.498217i	$-0.166012\pi$
$227$	26.1269	1.73410	0.867052	+	0.498217i	$0.166012\pi$
$228$	−0.533915	−0.0353594
$229$	18.3486	1.21251	0.606255	−	0.795270i	$-0.292671\pi$
$229$	18.3486	1.21251	0.606255	+	0.795270i	$0.292671\pi$
$230$	3.83000	0.252543
$231$	−9.20710	−0.605783
$232$	−1.00000	−0.0656532
$233$	13.5059	0.884804	0.442402	−	0.896817i	$-0.354127\pi$
$233$	13.5059	0.884804	0.442402	+	0.896817i	$0.354127\pi$
$234$	3.25549	0.212818
$235$	12.3948	0.808549
$236$	7.62559	0.496384
$237$	5.70853	0.370809
$238$	28.0152	1.81595
$239$	−25.7537	−1.66587	−0.832934	−	0.553373i	$-0.813341\pi$
$239$	−25.7537	−1.66587	−0.832934	+	0.553373i	$0.813341\pi$
$240$	3.83000	0.247226
$241$	−3.86531	−0.248987	−0.124493	−	0.992220i	$-0.539731\pi$
$241$	−3.86531	−0.248987	−0.124493	+	0.992220i	$0.539731\pi$
$242$	7.48476	0.481138
$243$	−1.00000	−0.0641500
$244$	5.94654	0.380688
$245$	−65.5511	−4.18791
$246$	2.74088	0.174752
$247$	−1.73816	−0.110596
$248$	6.65632	0.422677
$249$	−3.17099	−0.200953
$250$	17.8820	1.13096
$251$	28.6662	1.80940	0.904698	−	0.426054i	$-0.140097\pi$
$251$	28.6662	1.80940	0.904698	+	0.426054i	$0.140097\pi$
$252$	−4.91072	−0.309346
$253$	−1.87490	−0.117874
$254$	16.7703	1.05226
$255$	21.8498	1.36829
$256$	1.00000	0.0625000
$257$	−28.2206	−1.76035	−0.880176	−	0.474648i	$-0.842575\pi$
$257$	−28.2206	−1.76035	−0.880176	+	0.474648i	$0.842575\pi$
$258$	12.3587	0.769420
$259$	−1.91277	−0.118854
$260$	12.4685	0.773267
$261$	1.00000	0.0618984
$262$	−16.7185	−1.03287
$263$	3.91934	0.241677	0.120838	−	0.992672i	$-0.461442\pi$
$263$	3.91934	0.241677	0.120838	+	0.992672i	$0.461442\pi$
$264$	−1.87490	−0.115392
$265$	14.5724	0.895174
$266$	2.62191	0.160759
$267$	−11.3685	−0.695743
$268$	−6.17079	−0.376941
$269$	8.44951	0.515176	0.257588	−	0.966255i	$-0.417072\pi$
$269$	8.44951	0.515176	0.257588	+	0.966255i	$0.417072\pi$
$270$	−3.83000	−0.233087
$271$	0.259889	0.0157871	0.00789357	−	0.999969i	$-0.497487\pi$
$271$	0.259889	0.0157871	0.00789357	+	0.999969i	$0.497487\pi$
$272$	5.70490	0.345910
$273$	−15.9868	−0.967566
$274$	0.560201	0.0338430
$275$	−18.1282	−1.09317
$276$	−1.00000	−0.0601929
$277$	18.2011	1.09360	0.546798	−	0.837265i	$-0.315847\pi$
$277$	18.2011	1.09360	0.546798	+	0.837265i	$0.315847\pi$
$278$	21.8843	1.31253
$279$	−6.65632	−0.398504
$280$	−18.8081	−1.12400
$281$	9.68103	0.577522	0.288761	−	0.957401i	$-0.406757\pi$
$281$	9.68103	0.577522	0.288761	+	0.957401i	$0.406757\pi$
$282$	−3.23624	−0.192716
$283$	−11.0363	−0.656040	−0.328020	−	0.944671i	$-0.606381\pi$
$283$	−11.0363	−0.656040	−0.328020	+	0.944671i	$0.606381\pi$
$284$	2.94885	0.174982
$285$	2.04490	0.121129
$286$	−6.10372	−0.360921
$287$	−13.4597	−0.794501
$288$	−1.00000	−0.0589256
$289$	15.5459	0.914465
$290$	3.83000	0.224905
$291$	−3.97037	−0.232747
$292$	−9.05747	−0.530049
$293$	4.04428	0.236269	0.118135	−	0.992998i	$-0.462309\pi$
$293$	4.04428	0.236269	0.118135	+	0.992998i	$0.462309\pi$
$294$	17.1152	0.998176
$295$	−29.2060	−1.70044
$296$	−0.389510	−0.0226398
$297$	1.87490	0.108793
$298$	0.858167	0.0497123
$299$	−3.25549	−0.188270
$300$	−9.66892	−0.558235
$301$	−60.6901	−3.49812
$302$	13.5924	0.782152
$303$	−18.8391	−1.08228
$304$	0.533915	0.0306221
$305$	−22.7753	−1.30411
$306$	−5.70490	−0.326127
$307$	29.2154	1.66741	0.833706	−	0.552209i	$-0.186215\pi$
$307$	29.2154	1.66741	0.833706	+	0.552209i	$0.186215\pi$
$308$	9.20710	0.524623
$309$	7.01135	0.398862
$310$	−25.4937	−1.44795
$311$	−6.51297	−0.369316	−0.184658	−	0.982803i	$-0.559118\pi$
$311$	−6.51297	−0.369316	−0.184658	+	0.982803i	$0.559118\pi$
$312$	−3.25549	−0.184306
$313$	−7.39447	−0.417960	−0.208980	−	0.977920i	$-0.567014\pi$
$313$	−7.39447	−0.417960	−0.208980	+	0.977920i	$0.567014\pi$
$314$	−3.57130	−0.201540
$315$	18.8081	1.05971
$316$	−5.70853	−0.321130
$317$	−9.04641	−0.508097	−0.254048	−	0.967192i	$-0.581762\pi$
$317$	−9.04641	−0.508097	−0.254048	+	0.967192i	$0.581762\pi$
$318$	−3.80480	−0.213362
$319$	−1.87490	−0.104974
$320$	−3.83000	−0.214104
$321$	8.37729	0.467575
$322$	4.91072	0.273664
$323$	3.04593	0.169480
$324$	1.00000	0.0555556
$325$	−31.4771	−1.74604
$326$	−18.4147	−1.01989
$327$	3.47602	0.192224
$328$	−2.74088	−0.151340
$329$	15.8923	0.876170
$330$	7.18087	0.395294
$331$	5.63149	0.309535	0.154767	−	0.987951i	$-0.450537\pi$
$331$	5.63149	0.309535	0.154767	+	0.987951i	$0.450537\pi$
$332$	3.17099	0.174030
$333$	0.389510	0.0213450
$334$	−5.92710	−0.324317
$335$	23.6341	1.29127
$336$	4.91072	0.267902
$337$	6.41842	0.349634	0.174817	−	0.984601i	$-0.444067\pi$
$337$	6.41842	0.349634	0.174817	+	0.984601i	$0.444067\pi$
$338$	2.40176	0.130639
$339$	0.833643	0.0452773
$340$	−21.8498	−1.18497
$341$	12.4799	0.675826
$342$	−0.533915	−0.0288708
$343$	−49.6727	−2.68207
$344$	−12.3587	−0.666337
$345$	3.83000	0.206200
$346$	−2.39898	−0.128970
$347$	−11.0884	−0.595255	−0.297627	−	0.954682i	$-0.596195\pi$
$347$	−11.0884	−0.595255	−0.297627	+	0.954682i	$0.596195\pi$
$348$	−1.00000	−0.0536056
$349$	−5.04948	−0.270293	−0.135146	−	0.990826i	$-0.543150\pi$
$349$	−5.04948	−0.270293	−0.135146	+	0.990826i	$0.543150\pi$
$350$	47.4813	2.53798
$351$	3.25549	0.173765
$352$	1.87490	0.0999324
$353$	−24.9910	−1.33014	−0.665069	−	0.746782i	$-0.731598\pi$
$353$	−24.9910	−1.33014	−0.665069	+	0.746782i	$0.731598\pi$
$354$	7.62559	0.405296
$355$	−11.2941	−0.599430
$356$	11.3685	0.602531
$357$	28.0152	1.48272
$358$	1.32347	0.0699476
$359$	30.9096	1.63134	0.815672	−	0.578514i	$-0.196367\pi$
$359$	30.9096	1.63134	0.815672	+	0.578514i	$0.196367\pi$
$360$	3.83000	0.201859
$361$	−18.7149	−0.984997
$362$	−20.3239	−1.06820
$363$	7.48476	0.392848
$364$	15.9868	0.837937
$365$	34.6901	1.81577
$366$	5.94654	0.310831
$367$	−23.4141	−1.22221	−0.611103	−	0.791551i	$-0.709274\pi$
$367$	−23.4141	−1.22221	−0.611103	+	0.791551i	$0.709274\pi$
$368$	1.00000	0.0521286
$369$	2.74088	0.142685
$370$	1.49182	0.0775562
$371$	18.6843	0.970040
$372$	6.65632	0.345114
$373$	11.7633	0.609083	0.304541	−	0.952499i	$-0.401497\pi$
$373$	11.7633	0.609083	0.304541	+	0.952499i	$0.401497\pi$
$374$	10.6961	0.553083
$375$	17.8820	0.923421
$376$	3.23624	0.166897
$377$	−3.25549	−0.167666
$378$	−4.91072	−0.252580
$379$	−6.32423	−0.324854	−0.162427	−	0.986721i	$-0.551932\pi$
$379$	−6.32423	−0.324854	−0.162427	+	0.986721i	$0.551932\pi$
$380$	−2.04490	−0.104901
$381$	16.7703	0.859168
$382$	−20.2770	−1.03746
$383$	33.7745	1.72579	0.862897	−	0.505380i	$-0.168648\pi$
$383$	33.7745	1.72579	0.862897	+	0.505380i	$0.168648\pi$
$384$	1.00000	0.0510310
$385$	−35.2632	−1.79718
$386$	17.5746	0.894523
$387$	12.3587	0.628229
$388$	3.97037	0.201565
$389$	−13.3750	−0.678141	−0.339071	−	0.940761i	$-0.610113\pi$
$389$	−13.3750	−0.678141	−0.339071	+	0.940761i	$0.610113\pi$
$390$	12.4685	0.631370
$391$	5.70490	0.288509
$392$	−17.1152	−0.864446
$393$	−16.7185	−0.843337
$394$	14.7121	0.741183
$395$	21.8637	1.10008
$396$	−1.87490	−0.0942172
$397$	−12.4687	−0.625784	−0.312892	−	0.949789i	$-0.601298\pi$
$397$	−12.4687	−0.625784	−0.312892	+	0.949789i	$0.601298\pi$
$398$	3.40969	0.170912
$399$	2.62191	0.131259
$400$	9.66892	0.483446
$401$	−32.3691	−1.61644	−0.808218	−	0.588884i	$-0.799568\pi$
$401$	−32.3691	−1.61644	−0.808218	+	0.588884i	$0.799568\pi$
$402$	−6.17079	−0.307771
$403$	21.6696	1.07944
$404$	18.8391	0.937281
$405$	−3.83000	−0.190314
$406$	4.91072	0.243715
$407$	−0.730292	−0.0361992
$408$	5.70490	0.282435
$409$	15.8537	0.783917	0.391959	−	0.919983i	$-0.371798\pi$
$409$	15.8537	0.783917	0.391959	+	0.919983i	$0.371798\pi$
$410$	10.4976	0.518439
$411$	0.560201	0.0276327
$412$	−7.01135	−0.345424
$413$	−37.4471	−1.84265
$414$	−1.00000	−0.0491473
$415$	−12.1449	−0.596169
$416$	3.25549	0.159614
$417$	21.8843	1.07168
$418$	1.00104	0.0489623
$419$	−8.75952	−0.427930	−0.213965	−	0.976841i	$-0.568638\pi$
$419$	−8.75952	−0.427930	−0.213965	+	0.976841i	$0.568638\pi$
$420$	−18.8081	−0.917740
$421$	36.0980	1.75931	0.879654	−	0.475614i	$-0.157774\pi$
$421$	36.0980	1.75931	0.879654	+	0.475614i	$0.157774\pi$
$422$	12.2627	0.596938
$423$	−3.23624	−0.157352
$424$	3.80480	0.184777
$425$	55.1602	2.67566
$426$	2.94885	0.142872
$427$	−29.2018	−1.41317
$428$	−8.37729	−0.404932
$429$	−6.10372	−0.294690
$430$	47.3339	2.28264
$431$	−23.8375	−1.14821	−0.574105	−	0.818781i	$-0.694650\pi$
$431$	−23.8375	−1.14821	−0.574105	+	0.818781i	$0.694650\pi$
$432$	−1.00000	−0.0481125
$433$	−24.9294	−1.19803	−0.599015	−	0.800738i	$-0.704441\pi$
$433$	−24.9294	−1.19803	−0.599015	+	0.800738i	$0.704441\pi$
$434$	−32.6873	−1.56904
$435$	3.83000	0.183635
$436$	−3.47602	−0.166471
$437$	0.533915	0.0255406
$438$	−9.05747	−0.432783
$439$	33.4631	1.59711	0.798554	−	0.601923i	$-0.205599\pi$
$439$	33.4631	1.59711	0.798554	+	0.601923i	$0.205599\pi$
$440$	−7.18087	−0.342334
$441$	17.1152	0.815007
$442$	18.5723	0.883393
$443$	−23.2755	−1.10585	−0.552927	−	0.833230i	$-0.686489\pi$
$443$	−23.2755	−1.10585	−0.552927	+	0.833230i	$0.686489\pi$
$444$	−0.389510	−0.0184853
$445$	−43.5415	−2.06407
$446$	14.0063	0.663218
$447$	0.858167	0.0405899
$448$	−4.91072	−0.232010
$449$	29.2949	1.38251	0.691255	−	0.722611i	$-0.257058\pi$
$449$	29.2949	1.38251	0.691255	+	0.722611i	$0.257058\pi$
$450$	−9.66892	−0.455797
$451$	−5.13888	−0.241980
$452$	−0.833643	−0.0392113
$453$	13.5924	0.638625
$454$	−26.1269	−1.22620
$455$	−61.2295	−2.87048
$456$	0.533915	0.0250029
$457$	27.2353	1.27401	0.637007	−	0.770858i	$-0.280172\pi$
$457$	27.2353	1.27401	0.637007	+	0.770858i	$0.280172\pi$
$458$	−18.3486	−0.857374
$459$	−5.70490	−0.266282
$460$	−3.83000	−0.178575
$461$	−3.22761	−0.150325	−0.0751625	−	0.997171i	$-0.523948\pi$
$461$	−3.22761	−0.150325	−0.0751625	+	0.997171i	$0.523948\pi$
$462$	9.20710	0.428353
$463$	30.1032	1.39901	0.699506	−	0.714627i	$-0.253403\pi$
$463$	30.1032	1.39901	0.699506	+	0.714627i	$0.253403\pi$
$464$	1.00000	0.0464238
$465$	−25.4937	−1.18224
$466$	−13.5059	−0.625651
$467$	6.87890	0.318318	0.159159	−	0.987253i	$-0.449122\pi$
$467$	6.87890	0.318318	0.159159	+	0.987253i	$0.449122\pi$
$468$	−3.25549	−0.150485
$469$	30.3030	1.39926
$470$	−12.3948	−0.571731
$471$	−3.57130	−0.164557
$472$	−7.62559	−0.350996
$473$	−23.1713	−1.06542
$474$	−5.70853	−0.262201
$475$	5.16238	0.236866
$476$	−28.0152	−1.28407
$477$	−3.80480	−0.174210
$478$	25.7537	1.17795
$479$	6.31331	0.288462	0.144231	−	0.989544i	$-0.453929\pi$
$479$	6.31331	0.288462	0.144231	+	0.989544i	$0.453929\pi$
$480$	−3.83000	−0.174815
$481$	−1.26805	−0.0578180
$482$	3.86531	0.176060
$483$	4.91072	0.223445
$484$	−7.48476	−0.340216
$485$	−15.2065	−0.690492
$486$	1.00000	0.0453609
$487$	27.4092	1.24203	0.621015	−	0.783799i	$-0.286721\pi$
$487$	27.4092	1.24203	0.621015	+	0.783799i	$0.286721\pi$
$488$	−5.94654	−0.269187
$489$	−18.4147	−0.832740
$490$	65.5511	2.96130
$491$	−42.0045	−1.89564	−0.947819	−	0.318810i	$-0.896717\pi$
$491$	−42.0045	−1.89564	−0.947819	+	0.318810i	$0.896717\pi$
$492$	−2.74088	−0.123569
$493$	5.70490	0.256936
$494$	1.73816	0.0782034
$495$	7.18087	0.322756
$496$	−6.65632	−0.298878
$497$	−14.4810	−0.649561
$498$	3.17099	0.142095
$499$	−35.3516	−1.58256	−0.791278	−	0.611457i	$-0.790584\pi$
$499$	−35.3516	−1.58256	−0.791278	+	0.611457i	$0.790584\pi$
$500$	−17.8820	−0.799706
$501$	−5.92710	−0.264803
$502$	−28.6662	−1.27944
$503$	−21.3739	−0.953014	−0.476507	−	0.879171i	$-0.658097\pi$
$503$	−21.3739	−0.953014	−0.476507	+	0.879171i	$0.658097\pi$
$504$	4.91072	0.218741
$505$	−72.1539	−3.21081
$506$	1.87490	0.0833494
$507$	2.40176	0.106666
$508$	−16.7703	−0.744061
$509$	12.7335	0.564402	0.282201	−	0.959355i	$-0.408935\pi$
$509$	12.7335	0.564402	0.282201	+	0.959355i	$0.408935\pi$
$510$	−21.8498	−0.967525
$511$	44.4787	1.96762
$512$	−1.00000	−0.0441942
$513$	−0.533915	−0.0235729
$514$	28.2206	1.24476
$515$	26.8535	1.18331
$516$	−12.3587	−0.544062
$517$	6.06763	0.266854
$518$	1.91277	0.0840424
$519$	−2.39898	−0.105303
$520$	−12.4685	−0.546782
$521$	8.59574	0.376586	0.188293	−	0.982113i	$-0.439705\pi$
$521$	8.59574	0.376586	0.188293	+	0.982113i	$0.439705\pi$
$522$	−1.00000	−0.0437688
$523$	−4.59103	−0.200752	−0.100376	−	0.994950i	$-0.532005\pi$
$523$	−4.59103	−0.200752	−0.100376	+	0.994950i	$0.532005\pi$
$524$	16.7185	0.730351
$525$	47.4813	2.07226
$526$	−3.91934	−0.170891
$527$	−37.9737	−1.65416
$528$	1.87490	0.0815945
$529$	1.00000	0.0434783
$530$	−14.5724	−0.632984
$531$	7.62559	0.330922
$532$	−2.62191	−0.113674
$533$	−8.92293	−0.386495
$534$	11.3685	0.491964
$535$	32.0851	1.38716
$536$	6.17079	0.266538
$537$	1.32347	0.0571120
$538$	−8.44951	−0.364284
$539$	−32.0892	−1.38218
$540$	3.83000	0.164817
$541$	21.9295	0.942824	0.471412	−	0.881913i	$-0.343745\pi$
$541$	21.9295	0.942824	0.471412	+	0.881913i	$0.343745\pi$
$542$	−0.259889	−0.0111632
$543$	−20.3239	−0.872181
$544$	−5.70490	−0.244596
$545$	13.3132	0.570273
$546$	15.9868	0.684172
$547$	−11.2261	−0.479993	−0.239997	−	0.970774i	$-0.577146\pi$
$547$	−11.2261	−0.479993	−0.239997	+	0.970774i	$0.577146\pi$
$548$	−0.560201	−0.0239306
$549$	5.94654	0.253792
$550$	18.1282	0.772991
$551$	0.533915	0.0227455
$552$	1.00000	0.0425628
$553$	28.0330	1.19208
$554$	−18.2011	−0.773289
$555$	1.49182	0.0633244
$556$	−21.8843	−0.928102
$557$	−34.8655	−1.47730	−0.738650	−	0.674089i	$-0.764536\pi$
$557$	−34.8655	−1.47730	−0.738650	+	0.674089i	$0.764536\pi$
$558$	6.65632	0.281785
$559$	−40.2337	−1.70170
$560$	18.8081	0.794786
$561$	10.6961	0.451590
$562$	−9.68103	−0.408370
$563$	−34.2819	−1.44481	−0.722405	−	0.691471i	$-0.756963\pi$
$563$	−34.2819	−1.44481	−0.722405	+	0.691471i	$0.756963\pi$
$564$	3.23624	0.136270
$565$	3.19286	0.134324
$566$	11.0363	0.463891
$567$	−4.91072	−0.206231
$568$	−2.94885	−0.123731
$569$	11.1828	0.468806	0.234403	−	0.972140i	$-0.424687\pi$
$569$	11.1828	0.468806	0.234403	+	0.972140i	$0.424687\pi$
$570$	−2.04490	−0.0856513
$571$	−13.2459	−0.554324	−0.277162	−	0.960823i	$-0.589394\pi$
$571$	−13.2459	−0.554324	−0.277162	+	0.960823i	$0.589394\pi$
$572$	6.10372	0.255209
$573$	−20.2770	−0.847084
$574$	13.4597	0.561797
$575$	9.66892	0.403222
$576$	1.00000	0.0416667
$577$	−28.3523	−1.18032	−0.590161	−	0.807286i	$-0.700936\pi$
$577$	−28.3523	−1.18032	−0.590161	+	0.807286i	$0.700936\pi$
$578$	−15.5459	−0.646624
$579$	17.5746	0.730375
$580$	−3.83000	−0.159032
$581$	−15.5718	−0.646028
$582$	3.97037	0.164577
$583$	7.13361	0.295444
$584$	9.05747	0.374801
$585$	12.4685	0.515511
$586$	−4.04428	−0.167068
$587$	−35.4129	−1.46165	−0.730823	−	0.682568i	$-0.760863\pi$
$587$	−35.4129	−1.46165	−0.730823	+	0.682568i	$0.760863\pi$
$588$	−17.1152	−0.705817
$589$	−3.55391	−0.146436
$590$	29.2060	1.20239
$591$	14.7121	0.605174
$592$	0.389510	0.0160088
$593$	41.6833	1.71173	0.855864	−	0.517200i	$-0.173026\pi$
$593$	41.6833	1.71173	0.855864	+	0.517200i	$0.173026\pi$
$594$	−1.87490	−0.0769280
$595$	107.298	4.39880
$596$	−0.858167	−0.0351519
$597$	3.40969	0.139549
$598$	3.25549	0.133127
$599$	−39.9957	−1.63418	−0.817091	−	0.576509i	$-0.804415\pi$
$599$	−39.9957	−1.63418	−0.817091	+	0.576509i	$0.804415\pi$
$600$	9.66892	0.394732
$601$	30.8301	1.25759	0.628793	−	0.777573i	$-0.283549\pi$
$601$	30.8301	1.25759	0.628793	+	0.777573i	$0.283549\pi$
$602$	60.6901	2.47355
$603$	−6.17079	−0.251294
$604$	−13.5924	−0.553065
$605$	28.6666	1.16546
$606$	18.8391	0.765287
$607$	33.3434	1.35337	0.676683	−	0.736275i	$-0.263417\pi$
$607$	33.3434	1.35337	0.676683	+	0.736275i	$0.263417\pi$
$608$	−0.533915	−0.0216531
$609$	4.91072	0.198992
$610$	22.7753	0.922143
$611$	10.5356	0.426224
$612$	5.70490	0.230607
$613$	−19.8609	−0.802175	−0.401087	−	0.916040i	$-0.631368\pi$
$613$	−19.8609	−0.802175	−0.401087	+	0.916040i	$0.631368\pi$
$614$	−29.2154	−1.17904
$615$	10.4976	0.423304
$616$	−9.20710	−0.370965
$617$	−47.0395	−1.89374	−0.946870	−	0.321618i	$-0.895773\pi$
$617$	−47.0395	−1.89374	−0.946870	+	0.321618i	$0.895773\pi$
$618$	−7.01135	−0.282038
$619$	11.6425	0.467950	0.233975	−	0.972243i	$-0.424827\pi$
$619$	11.6425	0.467950	0.233975	+	0.972243i	$0.424827\pi$
$620$	25.4937	1.02385
$621$	−1.00000	−0.0401286
$622$	6.51297	0.261146
$623$	−55.8276	−2.23669
$624$	3.25549	0.130324
$625$	20.1434	0.805737
$626$	7.39447	0.295543
$627$	1.00104	0.0399775
$628$	3.57130	0.142511
$629$	2.22212	0.0886016
$630$	−18.8081	−0.749331
$631$	−10.6489	−0.423926	−0.211963	−	0.977278i	$-0.567986\pi$
$631$	−10.6489	−0.423926	−0.211963	+	0.977278i	$0.567986\pi$
$632$	5.70853	0.227073
$633$	12.2627	0.487398
$634$	9.04641	0.359279
$635$	64.2303	2.54890
$636$	3.80480	0.150870
$637$	−55.7183	−2.20764
$638$	1.87490	0.0742279
$639$	2.94885	0.116655
$640$	3.83000	0.151394
$641$	−28.0047	−1.10612	−0.553060	−	0.833142i	$-0.686540\pi$
$641$	−28.0047	−1.10612	−0.553060	+	0.833142i	$0.686540\pi$
$642$	−8.37729	−0.330625
$643$	−14.4751	−0.570842	−0.285421	−	0.958402i	$-0.592133\pi$
$643$	−14.4751	−0.570842	−0.285421	+	0.958402i	$0.592133\pi$
$644$	−4.91072	−0.193509
$645$	47.3339	1.86377
$646$	−3.04593	−0.119841
$647$	41.3074	1.62396	0.811981	−	0.583684i	$-0.198389\pi$
$647$	41.3074	1.62396	0.811981	+	0.583684i	$0.198389\pi$
$648$	−1.00000	−0.0392837
$649$	−14.2972	−0.561215
$650$	31.4771	1.23463
$651$	−32.6873	−1.28112
$652$	18.4147	0.721174
$653$	−12.9580	−0.507085	−0.253542	−	0.967324i	$-0.581596\pi$
$653$	−12.9580	−0.507085	−0.253542	+	0.967324i	$0.581596\pi$
$654$	−3.47602	−0.135923
$655$	−64.0319	−2.50193
$656$	2.74088	0.107013
$657$	−9.05747	−0.353366
$658$	−15.8923	−0.619546
$659$	26.9082	1.04819	0.524097	−	0.851658i	$-0.324403\pi$
$659$	26.9082	1.04819	0.524097	+	0.851658i	$0.324403\pi$
$660$	−7.18087	−0.279515
$661$	35.8148	1.39304	0.696518	−	0.717539i	$-0.254732\pi$
$661$	35.8148	1.39304	0.696518	+	0.717539i	$0.254732\pi$
$662$	−5.63149	−0.218874
$663$	18.5723	0.721287
$664$	−3.17099	−0.123058
$665$	10.0419	0.389408
$666$	−0.389510	−0.0150932
$667$	1.00000	0.0387202
$668$	5.92710	0.229327
$669$	14.0063	0.541515
$670$	−23.6341	−0.913067
$671$	−11.1492	−0.430408
$672$	−4.91072	−0.189435
$673$	−5.14212	−0.198214	−0.0991071	−	0.995077i	$-0.531599\pi$
$673$	−5.14212	−0.198214	−0.0991071	+	0.995077i	$0.531599\pi$
$674$	−6.41842	−0.247228
$675$	−9.66892	−0.372157
$676$	−2.40176	−0.0923754
$677$	−5.44358	−0.209214	−0.104607	−	0.994514i	$-0.533358\pi$
$677$	−5.44358	−0.209214	−0.104607	+	0.994514i	$0.533358\pi$
$678$	−0.833643	−0.0320159
$679$	−19.4973	−0.748240
$680$	21.8498	0.837901
$681$	−26.1269	−1.00119
$682$	−12.4799	−0.477881
$683$	−3.89519	−0.149045	−0.0745226	−	0.997219i	$-0.523743\pi$
$683$	−3.89519	−0.149045	−0.0745226	+	0.997219i	$0.523743\pi$
$684$	0.533915	0.0204148
$685$	2.14557	0.0819781
$686$	49.6727	1.89651
$687$	−18.3486	−0.700043
$688$	12.3587	0.471171
$689$	12.3865	0.471888
$690$	−3.83000	−0.145806
$691$	−22.3650	−0.850806	−0.425403	−	0.905004i	$-0.639868\pi$
$691$	−22.3650	−0.850806	−0.425403	+	0.905004i	$0.639868\pi$
$692$	2.39898	0.0911954
$693$	9.20710	0.349749
$694$	11.0884	0.420909
$695$	83.8170	3.17936
$696$	1.00000	0.0379049
$697$	15.6365	0.592273
$698$	5.04948	0.191126
$699$	−13.5059	−0.510842
$700$	−47.4813	−1.79463
$701$	16.0604	0.606593	0.303297	−	0.952896i	$-0.401913\pi$
$701$	16.0604	0.606593	0.303297	+	0.952896i	$0.401913\pi$
$702$	−3.25549	−0.122871
$703$	0.207965	0.00784356
$704$	−1.87490	−0.0706629
$705$	−12.3948	−0.466816
$706$	24.9910	0.940549
$707$	−92.5136	−3.47933
$708$	−7.62559	−0.286587
$709$	24.6413	0.925424	0.462712	−	0.886509i	$-0.346876\pi$
$709$	24.6413	0.925424	0.462712	+	0.886509i	$0.346876\pi$
$710$	11.2941	0.423861
$711$	−5.70853	−0.214086
$712$	−11.3685	−0.426054
$713$	−6.65632	−0.249281
$714$	−28.0152	−1.04844
$715$	−23.3773	−0.874260
$716$	−1.32347	−0.0494604
$717$	25.7537	0.961789
$718$	−30.9096	−1.15353
$719$	−16.1644	−0.602829	−0.301415	−	0.953493i	$-0.597459\pi$
$719$	−16.1644	−0.602829	−0.301415	+	0.953493i	$0.597459\pi$
$720$	−3.83000	−0.142736
$721$	34.4307	1.28227
$722$	18.7149	0.696498
$723$	3.86531	0.143753
$724$	20.3239	0.755331
$725$	9.66892	0.359095
$726$	−7.48476	−0.277785
$727$	−42.9921	−1.59449	−0.797245	−	0.603656i	$-0.793710\pi$
$727$	−42.9921	−1.59449	−0.797245	+	0.603656i	$0.793710\pi$
$728$	−15.9868	−0.592511
$729$	1.00000	0.0370370
$730$	−34.6901	−1.28394
$731$	70.5052	2.60773
$732$	−5.94654	−0.219790
$733$	34.7544	1.28368	0.641841	−	0.766838i	$-0.278171\pi$
$733$	34.7544	1.28368	0.641841	+	0.766838i	$0.278171\pi$
$734$	23.4141	0.864231
$735$	65.5511	2.41789
$736$	−1.00000	−0.0368605
$737$	11.5696	0.426172
$738$	−2.74088	−0.100893
$739$	−27.3490	−1.00605	−0.503026	−	0.864272i	$-0.667780\pi$
$739$	−27.3490	−1.00605	−0.503026	+	0.864272i	$0.667780\pi$
$740$	−1.49182	−0.0548405
$741$	1.73816	0.0638528
$742$	−18.6843	−0.685922
$743$	−35.0890	−1.28729	−0.643644	−	0.765325i	$-0.722578\pi$
$743$	−35.0890	−1.28729	−0.643644	+	0.765325i	$0.722578\pi$
$744$	−6.65632	−0.244033
$745$	3.28678	0.120418
$746$	−11.7633	−0.430686
$747$	3.17099	0.116020
$748$	−10.6961	−0.391089
$749$	41.1385	1.50317
$750$	−17.8820	−0.652957
$751$	13.2572	0.483761	0.241880	−	0.970306i	$-0.422236\pi$
$751$	13.2572	0.483761	0.241880	+	0.970306i	$0.422236\pi$
$752$	−3.23624	−0.118014
$753$	−28.6662	−1.04465
$754$	3.25549	0.118558
$755$	52.0588	1.89461
$756$	4.91072	0.178601
$757$	11.0990	0.403400	0.201700	−	0.979447i	$-0.435353\pi$
$757$	11.0990	0.403400	0.201700	+	0.979447i	$0.435353\pi$
$758$	6.32423	0.229706
$759$	1.87490	0.0680545
$760$	2.04490	0.0741762
$761$	1.03759	0.0376126	0.0188063	−	0.999823i	$-0.494013\pi$
$761$	1.03759	0.0376126	0.0188063	+	0.999823i	$0.494013\pi$
$762$	−16.7703	−0.607524
$763$	17.0698	0.617967
$764$	20.2770	0.733596
$765$	−21.8498	−0.789981
$766$	−33.7745	−1.22032
$767$	−24.8251	−0.896381
$768$	−1.00000	−0.0360844
$769$	29.8280	1.07563	0.537813	−	0.843064i	$-0.319251\pi$
$769$	29.8280	1.07563	0.537813	+	0.843064i	$0.319251\pi$
$770$	35.2632	1.27080
$771$	28.2206	1.01634
$772$	−17.5746	−0.632523
$773$	21.0462	0.756981	0.378490	−	0.925605i	$-0.376443\pi$
$773$	21.0462	0.756981	0.378490	+	0.925605i	$0.376443\pi$
$774$	−12.3587	−0.444225
$775$	−64.3594	−2.31186
$776$	−3.97037	−0.142528
$777$	1.91277	0.0686204
$778$	13.3750	0.479518
$779$	1.46340	0.0524317
$780$	−12.4685	−0.446446
$781$	−5.52880	−0.197836
$782$	−5.70490	−0.204007
$783$	−1.00000	−0.0357371
$784$	17.1152	0.611256
$785$	−13.6781	−0.488192
$786$	16.7185	0.596329
$787$	−44.6701	−1.59232	−0.796158	−	0.605088i	$-0.793138\pi$
$787$	−44.6701	−1.59232	−0.796158	+	0.605088i	$0.793138\pi$
$788$	−14.7121	−0.524096
$789$	−3.91934	−0.139532
$790$	−21.8637	−0.777874
$791$	4.09379	0.145558
$792$	1.87490	0.0666216
$793$	−19.3589	−0.687455
$794$	12.4687	0.442496
$795$	−14.5724	−0.516829
$796$	−3.40969	−0.120853
$797$	0.127673	0.00452240	0.00226120	−	0.999997i	$-0.499280\pi$
$797$	0.127673	0.00452240	0.00226120	+	0.999997i	$0.499280\pi$
$798$	−2.62191	−0.0928145
$799$	−18.4625	−0.653155
$800$	−9.66892	−0.341848
$801$	11.3685	0.401687
$802$	32.3691	1.14299
$803$	16.9818	0.599276
$804$	6.17079	0.217627
$805$	18.8081	0.662897
$806$	−21.6696	−0.763279
$807$	−8.44951	−0.297437
$808$	−18.8391	−0.662758
$809$	13.9555	0.490650	0.245325	−	0.969441i	$-0.421105\pi$
$809$	13.9555	0.490650	0.245325	+	0.969441i	$0.421105\pi$
$810$	3.83000	0.134573
$811$	10.2788	0.360939	0.180469	−	0.983581i	$-0.442238\pi$
$811$	10.2788	0.360939	0.180469	+	0.983581i	$0.442238\pi$
$812$	−4.91072	−0.172332
$813$	−0.259889	−0.00911471
$814$	0.730292	0.0255967
$815$	−70.5282	−2.47050
$816$	−5.70490	−0.199711
$817$	6.59850	0.230852
$818$	−15.8537	−0.554313
$819$	15.9868	0.558624
$820$	−10.4976	−0.366592
$821$	−20.9582	−0.731447	−0.365723	−	0.930724i	$-0.619178\pi$
$821$	−20.9582	−0.731447	−0.365723	+	0.930724i	$0.619178\pi$
$822$	−0.560201	−0.0195393
$823$	−36.4060	−1.26903	−0.634517	−	0.772909i	$-0.718801\pi$
$823$	−36.4060	−1.26903	−0.634517	+	0.772909i	$0.718801\pi$
$824$	7.01135	0.244252
$825$	18.1282	0.631144
$826$	37.4471	1.30295
$827$	45.7934	1.59239	0.796196	−	0.605039i	$-0.206842\pi$
$827$	45.7934	1.59239	0.796196	+	0.605039i	$0.206842\pi$
$828$	1.00000	0.0347524
$829$	33.8949	1.17722	0.588610	−	0.808417i	$-0.299675\pi$
$829$	33.8949	1.17722	0.588610	+	0.808417i	$0.299675\pi$
$830$	12.1449	0.421555
$831$	−18.2011	−0.631388
$832$	−3.25549	−0.112864
$833$	97.6403	3.38303
$834$	−21.8843	−0.757792
$835$	−22.7008	−0.785594
$836$	−1.00104	−0.0346216
$837$	6.65632	0.230076
$838$	8.75952	0.302592
$839$	−31.1241	−1.07452	−0.537261	−	0.843416i	$-0.680541\pi$
$839$	−31.1241	−1.07452	−0.537261	+	0.843416i	$0.680541\pi$
$840$	18.8081	0.648940
$841$	1.00000	0.0344828
$842$	−36.0980	−1.24402
$843$	−9.68103	−0.333433
$844$	−12.2627	−0.422099
$845$	9.19875	0.316447
$846$	3.23624	0.111264
$847$	36.7555	1.26293
$848$	−3.80480	−0.130657
$849$	11.0363	0.378765
$850$	−55.1602	−1.89198
$851$	0.389510	0.0133522
$852$	−2.94885	−0.101026
$853$	−7.20128	−0.246567	−0.123284	−	0.992371i	$-0.539342\pi$
$853$	−7.20128	−0.246567	−0.123284	+	0.992371i	$0.539342\pi$
$854$	29.2018	0.999264
$855$	−2.04490	−0.0699340
$856$	8.37729	0.286330
$857$	19.4479	0.664328	0.332164	−	0.943222i	$-0.392221\pi$
$857$	19.4479	0.664328	0.332164	+	0.943222i	$0.392221\pi$
$858$	6.10372	0.208378
$859$	14.7197	0.502231	0.251116	−	0.967957i	$-0.419203\pi$
$859$	14.7197	0.502231	0.251116	+	0.967957i	$0.419203\pi$
$860$	−47.3339	−1.61407
$861$	13.4597	0.458705
$862$	23.8375	0.811907
$863$	12.5146	0.426001	0.213000	−	0.977052i	$-0.431676\pi$
$863$	12.5146	0.426001	0.213000	+	0.977052i	$0.431676\pi$
$864$	1.00000	0.0340207
$865$	−9.18809	−0.312404
$866$	24.9294	0.847136
$867$	−15.5459	−0.527966
$868$	32.6873	1.10948
$869$	10.7029	0.363071
$870$	−3.83000	−0.129849
$871$	20.0890	0.680689
$872$	3.47602	0.117713
$873$	3.97037	0.134377
$874$	−0.533915	−0.0180599
$875$	87.8133	2.96863
$876$	9.05747	0.306024
$877$	10.4260	0.352061	0.176031	−	0.984385i	$-0.443674\pi$
$877$	10.4260	0.352061	0.176031	+	0.984385i	$0.443674\pi$
$878$	−33.4631	−1.12933
$879$	−4.04428	−0.136410
$880$	7.18087	0.242067
$881$	−3.50332	−0.118030	−0.0590150	−	0.998257i	$-0.518796\pi$
$881$	−3.50332	−0.118030	−0.0590150	+	0.998257i	$0.518796\pi$
$882$	−17.1152	−0.576297
$883$	29.6173	0.996700	0.498350	−	0.866976i	$-0.333939\pi$
$883$	29.6173	0.996700	0.498350	+	0.866976i	$0.333939\pi$
$884$	−18.5723	−0.624653
$885$	29.2060	0.981750
$886$	23.2755	0.781957
$887$	−9.83410	−0.330197	−0.165098	−	0.986277i	$-0.552794\pi$
$887$	−9.83410	−0.330197	−0.165098	+	0.986277i	$0.552794\pi$
$888$	0.389510	0.0130711
$889$	82.3542	2.76207
$890$	43.5415	1.45951
$891$	−1.87490	−0.0628115
$892$	−14.0063	−0.468966
$893$	−1.72788	−0.0578213
$894$	−0.858167	−0.0287014
$895$	5.06890	0.169434
$896$	4.91072	0.164056
$897$	3.25549	0.108698
$898$	−29.2949	−0.977582
$899$	−6.65632	−0.222001
$900$	9.66892	0.322297
$901$	−21.7060	−0.723131
$902$	5.13888	0.171106
$903$	60.6901	2.01964
$904$	0.833643	0.0277266
$905$	−77.8405	−2.58750
$906$	−13.5924	−0.451576
$907$	38.6451	1.28319	0.641595	−	0.767043i	$-0.278273\pi$
$907$	38.6451	1.28319	0.641595	+	0.767043i	$0.278273\pi$
$908$	26.1269	0.867052
$909$	18.8391	0.624854
$910$	61.2295	2.02974
$911$	19.3657	0.641614	0.320807	−	0.947145i	$-0.396046\pi$
$911$	19.3657	0.641614	0.320807	+	0.947145i	$0.396046\pi$
$912$	−0.533915	−0.0176797
$913$	−5.94528	−0.196760
$914$	−27.2353	−0.900864
$915$	22.7753	0.752927
$916$	18.3486	0.606255
$917$	−82.0998	−2.71118
$918$	5.70490	0.188290
$919$	−8.54419	−0.281847	−0.140923	−	0.990020i	$-0.545007\pi$
$919$	−8.54419	−0.281847	−0.140923	+	0.990020i	$0.545007\pi$
$920$	3.83000	0.126271
$921$	−29.2154	−0.962680
$922$	3.22761	0.106296
$923$	−9.59998	−0.315987
$924$	−9.20710	−0.302891
$925$	3.76614	0.123830
$926$	−30.1032	−0.989251
$927$	−7.01135	−0.230283
$928$	−1.00000	−0.0328266
$929$	−12.7564	−0.418525	−0.209263	−	0.977859i	$-0.567106\pi$
$929$	−12.7564	−0.418525	−0.209263	+	0.977859i	$0.567106\pi$
$930$	25.4937	0.835972
$931$	9.13804	0.299487
$932$	13.5059	0.442402
$933$	6.51297	0.213225
$934$	−6.87890	−0.225085
$935$	40.9661	1.33974
$936$	3.25549	0.106409
$937$	−38.1445	−1.24613	−0.623063	−	0.782172i	$-0.714112\pi$
$937$	−38.1445	−1.24613	−0.623063	+	0.782172i	$0.714112\pi$
$938$	−30.3030	−0.989428
$939$	7.39447	0.241310
$940$	12.3948	0.404275
$941$	34.0705	1.11067	0.555334	−	0.831627i	$-0.312590\pi$
$941$	34.0705	1.11067	0.555334	+	0.831627i	$0.312590\pi$
$942$	3.57130	0.116359
$943$	2.74088	0.0892554
$944$	7.62559	0.248192
$945$	−18.8081	−0.611826
$946$	23.1713	0.753365
$947$	−8.96900	−0.291453	−0.145727	−	0.989325i	$-0.546552\pi$
$947$	−8.96900	−0.291453	−0.145727	+	0.989325i	$0.546552\pi$
$948$	5.70853	0.185404
$949$	29.4865	0.957174
$950$	−5.16238	−0.167490
$951$	9.04641	0.293350
$952$	28.0152	0.907977
$953$	22.1281	0.716801	0.358400	−	0.933568i	$-0.383322\pi$
$953$	22.1281	0.716801	0.358400	+	0.933568i	$0.383322\pi$
$954$	3.80480	0.123185
$955$	−77.6610	−2.51305
$956$	−25.7537	−0.832934
$957$	1.87490	0.0606069
$958$	−6.31331	−0.203974
$959$	2.75099	0.0888341
$960$	3.83000	0.123613
$961$	13.3066	0.429245
$962$	1.26805	0.0408835
$963$	−8.37729	−0.269954
$964$	−3.86531	−0.124493
$965$	67.3107	2.16681
$966$	−4.91072	−0.158000
$967$	−16.4454	−0.528848	−0.264424	−	0.964406i	$-0.585182\pi$
$967$	−16.4454	−0.528848	−0.264424	+	0.964406i	$0.585182\pi$
$968$	7.48476	0.240569
$969$	−3.04593	−0.0978495
$970$	15.2065	0.488252
$971$	50.2627	1.61301	0.806503	−	0.591229i	$-0.201357\pi$
$971$	50.2627	1.61301	0.806503	+	0.591229i	$0.201357\pi$
$972$	−1.00000	−0.0320750
$973$	107.468	3.44526
$974$	−27.4092	−0.878248
$975$	31.4771	1.00807
$976$	5.94654	0.190344
$977$	−51.2445	−1.63946	−0.819728	−	0.572753i	$-0.805875\pi$
$977$	−51.2445	−1.63946	−0.819728	+	0.572753i	$0.805875\pi$
$978$	18.4147	0.588836
$979$	−21.3148	−0.681225
$980$	−65.5511	−2.09395
$981$	−3.47602	−0.110981
$982$	42.0045	1.34042
$983$	41.0361	1.30885	0.654424	−	0.756128i	$-0.272911\pi$
$983$	41.0361	1.30885	0.654424	+	0.756128i	$0.272911\pi$
$984$	2.74088	0.0873761
$985$	56.3473	1.79537
$986$	−5.70490	−0.181681
$987$	−15.8923	−0.505857
$988$	−1.73816	−0.0552981
$989$	12.3587	0.392984
$990$	−7.18087	−0.228223
$991$	32.0693	1.01871	0.509357	−	0.860555i	$-0.329883\pi$
$991$	32.0693	1.01871	0.509357	+	0.860555i	$0.329883\pi$
$992$	6.65632	0.211338
$993$	−5.63149	−0.178710
$994$	14.4810	0.459309
$995$	13.0591	0.414002
$996$	−3.17099	−0.100477
$997$	−11.0606	−0.350293	−0.175147	−	0.984542i	$-0.556040\pi$
$997$	−11.0606	−0.350293	−0.175147	+	0.984542i	$0.556040\pi$
$998$	35.3516	1.11904
$999$	−0.389510	−0.0123235

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bg.1.1	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bg.1.1	✓	7	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.83000 q^{5} +1.00000 q^{6} -4.91072 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.83000 q^{5} +1.00000 q^{6} -4.91072 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.83000 q^{10} -1.87490 q^{11} -1.00000 q^{12} -3.25549 q^{13} +4.91072 q^{14} +3.83000 q^{15} +1.00000 q^{16} +5.70490 q^{17} -1.00000 q^{18} +0.533915 q^{19} -3.83000 q^{20} +4.91072 q^{21} +1.87490 q^{22} +1.00000 q^{23} +1.00000 q^{24} +9.66892 q^{25} +3.25549 q^{26} -1.00000 q^{27} -4.91072 q^{28} +1.00000 q^{29} -3.83000 q^{30} -6.65632 q^{31} -1.00000 q^{32} +1.87490 q^{33} -5.70490 q^{34} +18.8081 q^{35} +1.00000 q^{36} +0.389510 q^{37} -0.533915 q^{38} +3.25549 q^{39} +3.83000 q^{40} +2.74088 q^{41} -4.91072 q^{42} +12.3587 q^{43} -1.87490 q^{44} -3.83000 q^{45} -1.00000 q^{46} -3.23624 q^{47} -1.00000 q^{48} +17.1152 q^{49} -9.66892 q^{50} -5.70490 q^{51} -3.25549 q^{52} -3.80480 q^{53} +1.00000 q^{54} +7.18087 q^{55} +4.91072 q^{56} -0.533915 q^{57} -1.00000 q^{58} +7.62559 q^{59} +3.83000 q^{60} +5.94654 q^{61} +6.65632 q^{62} -4.91072 q^{63} +1.00000 q^{64} +12.4685 q^{65} -1.87490 q^{66} -6.17079 q^{67} +5.70490 q^{68} -1.00000 q^{69} -18.8081 q^{70} +2.94885 q^{71} -1.00000 q^{72} -9.05747 q^{73} -0.389510 q^{74} -9.66892 q^{75} +0.533915 q^{76} +9.20710 q^{77} -3.25549 q^{78} -5.70853 q^{79} -3.83000 q^{80} +1.00000 q^{81} -2.74088 q^{82} +3.17099 q^{83} +4.91072 q^{84} -21.8498 q^{85} -12.3587 q^{86} -1.00000 q^{87} +1.87490 q^{88} +11.3685 q^{89} +3.83000 q^{90} +15.9868 q^{91} +1.00000 q^{92} +6.65632 q^{93} +3.23624 q^{94} -2.04490 q^{95} +1.00000 q^{96} +3.97037 q^{97} -17.1152 q^{98} -1.87490 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} + 2 q^{5} + 7 q^{6} - 5 q^{7} - 7 q^{8} + 7 q^{9}+O(q^{10}) \) 7 * q - 7 * q^2 - 7 * q^3 + 7 * q^4 + 2 * q^5 + 7 * q^6 - 5 * q^7 - 7 * q^8 + 7 * q^9	\( 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} + 2 q^{5} + 7 q^{6} - 5 q^{7} - 7 q^{8} + 7 q^{9} - 2 q^{10} - q^{11} - 7 q^{12} + q^{13} + 5 q^{14} - 2 q^{15} + 7 q^{16} - q^{17} - 7 q^{18} - 9 q^{19} + 2 q^{20} + 5 q^{21} + q^{22} + 7 q^{23} + 7 q^{24} + q^{25} - q^{26} - 7 q^{27} - 5 q^{28} + 7 q^{29} + 2 q^{30} - 19 q^{31} - 7 q^{32} + q^{33} + q^{34} + 11 q^{35} + 7 q^{36} - 18 q^{37} + 9 q^{38} - q^{39} - 2 q^{40} + 4 q^{41} - 5 q^{42} - 9 q^{43} - q^{44} + 2 q^{45} - 7 q^{46} - 11 q^{47} - 7 q^{48} + 12 q^{49} - q^{50} + q^{51} + q^{52} + 8 q^{53} + 7 q^{54} - 11 q^{55} + 5 q^{56} + 9 q^{57} - 7 q^{58} + 12 q^{59} - 2 q^{60} - 5 q^{61} + 19 q^{62} - 5 q^{63} + 7 q^{64} + 23 q^{65} - q^{66} - 13 q^{67} - q^{68} - 7 q^{69} - 11 q^{70} - q^{71} - 7 q^{72} - 26 q^{73} + 18 q^{74} - q^{75} - 9 q^{76} + 6 q^{77} + q^{78} - 38 q^{79} + 2 q^{80} + 7 q^{81} - 4 q^{82} - 6 q^{83} + 5 q^{84} - 25 q^{85} + 9 q^{86} - 7 q^{87} + q^{88} + 20 q^{89} - 2 q^{90} + 2 q^{91} + 7 q^{92} + 19 q^{93} + 11 q^{94} - 31 q^{95} + 7 q^{96} - 8 q^{97} - 12 q^{98} - q^{99}+O(q^{100}) \) 7 * q - 7 * q^2 - 7 * q^3 + 7 * q^4 + 2 * q^5 + 7 * q^6 - 5 * q^7 - 7 * q^8 + 7 * q^9 - 2 * q^10 - q^11 - 7 * q^12 + q^13 + 5 * q^14 - 2 * q^15 + 7 * q^16 - q^17 - 7 * q^18 - 9 * q^19 + 2 * q^20 + 5 * q^21 + q^22 + 7 * q^23 + 7 * q^24 + q^25 - q^26 - 7 * q^27 - 5 * q^28 + 7 * q^29 + 2 * q^30 - 19 * q^31 - 7 * q^32 + q^33 + q^34 + 11 * q^35 + 7 * q^36 - 18 * q^37 + 9 * q^38 - q^39 - 2 * q^40 + 4 * q^41 - 5 * q^42 - 9 * q^43 - q^44 + 2 * q^45 - 7 * q^46 - 11 * q^47 - 7 * q^48 + 12 * q^49 - q^50 + q^51 + q^52 + 8 * q^53 + 7 * q^54 - 11 * q^55 + 5 * q^56 + 9 * q^57 - 7 * q^58 + 12 * q^59 - 2 * q^60 - 5 * q^61 + 19 * q^62 - 5 * q^63 + 7 * q^64 + 23 * q^65 - q^66 - 13 * q^67 - q^68 - 7 * q^69 - 11 * q^70 - q^71 - 7 * q^72 - 26 * q^73 + 18 * q^74 - q^75 - 9 * q^76 + 6 * q^77 + q^78 - 38 * q^79 + 2 * q^80 + 7 * q^81 - 4 * q^82 - 6 * q^83 + 5 * q^84 - 25 * q^85 + 9 * q^86 - 7 * q^87 + q^88 + 20 * q^89 - 2 * q^90 + 2 * q^91 + 7 * q^92 + 19 * q^93 + 11 * q^94 - 31 * q^95 + 7 * q^96 - 8 * q^97 - 12 * q^98 - q^99

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