LMFDB - Embedded newform 8007.2.a.f.1.23

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, 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("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.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:	$63.9362168984$
Analytic rank:	$1$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.23
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.206596 q^{2} -1.00000 q^{3} -1.95732 q^{4} -2.96369 q^{5} +0.206596 q^{6} -3.29821 q^{7} +0.817567 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.206596 q^{2} -1.00000 q^{3} -1.95732 q^{4} -2.96369 q^{5} +0.206596 q^{6} -3.29821 q^{7} +0.817567 q^{8} +1.00000 q^{9} +0.612286 q^{10} -2.02436 q^{11} +1.95732 q^{12} -4.52233 q^{13} +0.681397 q^{14} +2.96369 q^{15} +3.74573 q^{16} -1.00000 q^{17} -0.206596 q^{18} +2.84222 q^{19} +5.80088 q^{20} +3.29821 q^{21} +0.418225 q^{22} -6.15139 q^{23} -0.817567 q^{24} +3.78344 q^{25} +0.934296 q^{26} -1.00000 q^{27} +6.45564 q^{28} +3.35506 q^{29} -0.612286 q^{30} -3.97663 q^{31} -2.40899 q^{32} +2.02436 q^{33} +0.206596 q^{34} +9.77486 q^{35} -1.95732 q^{36} -6.02124 q^{37} -0.587192 q^{38} +4.52233 q^{39} -2.42301 q^{40} -5.50704 q^{41} -0.681397 q^{42} -0.805249 q^{43} +3.96231 q^{44} -2.96369 q^{45} +1.27085 q^{46} +2.71134 q^{47} -3.74573 q^{48} +3.87818 q^{49} -0.781645 q^{50} +1.00000 q^{51} +8.85163 q^{52} +7.81249 q^{53} +0.206596 q^{54} +5.99957 q^{55} -2.69651 q^{56} -2.84222 q^{57} -0.693142 q^{58} +0.000491989 q^{59} -5.80088 q^{60} +10.2945 q^{61} +0.821556 q^{62} -3.29821 q^{63} -6.99377 q^{64} +13.4028 q^{65} -0.418225 q^{66} +6.98424 q^{67} +1.95732 q^{68} +6.15139 q^{69} -2.01945 q^{70} +12.7098 q^{71} +0.817567 q^{72} -0.595084 q^{73} +1.24397 q^{74} -3.78344 q^{75} -5.56313 q^{76} +6.67676 q^{77} -0.934296 q^{78} +10.0119 q^{79} -11.1012 q^{80} +1.00000 q^{81} +1.13773 q^{82} -10.2303 q^{83} -6.45564 q^{84} +2.96369 q^{85} +0.166361 q^{86} -3.35506 q^{87} -1.65505 q^{88} +6.31716 q^{89} +0.612286 q^{90} +14.9156 q^{91} +12.0402 q^{92} +3.97663 q^{93} -0.560152 q^{94} -8.42345 q^{95} +2.40899 q^{96} +6.62727 q^{97} -0.801217 q^{98} -2.02436 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) $ 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * 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 $

$2$	−0.206596	−0.146086	−0.0730428	−	0.997329i	$-0.523271\pi$
$2$	−0.206596	−0.146086	−0.0730428	+	0.997329i	$0.523271\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.95732	−0.978659
$5$	−2.96369	−1.32540	−0.662701	−	0.748884i	$-0.730590\pi$
$5$	−2.96369	−1.32540	−0.662701	+	0.748884i	$0.730590\pi$
$6$	0.206596	0.0843425
$7$	−3.29821	−1.24661	−0.623303	−	0.781981i	$-0.714210\pi$
$7$	−3.29821	−1.24661	−0.623303	+	0.781981i	$0.714210\pi$
$8$	0.817567	0.289054
$9$	1.00000	0.333333
$10$	0.612286	0.193622
$11$	−2.02436	−0.610367	−0.305184	−	0.952294i	$-0.598718\pi$
$11$	−2.02436	−0.610367	−0.305184	+	0.952294i	$0.598718\pi$
$12$	1.95732	0.565029
$13$	−4.52233	−1.25427	−0.627134	−	0.778911i	$-0.715772\pi$
$13$	−4.52233	−1.25427	−0.627134	+	0.778911i	$0.715772\pi$
$14$	0.681397	0.182111
$15$	2.96369	0.765221
$16$	3.74573	0.936432
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−0.206596	−0.0486952
$19$	2.84222	0.652050	0.326025	−	0.945361i	$-0.394291\pi$
$19$	2.84222	0.652050	0.326025	+	0.945361i	$0.394291\pi$
$20$	5.80088	1.29712
$21$	3.29821	0.719728
$22$	0.418225	0.0891658
$23$	−6.15139	−1.28265	−0.641327	−	0.767268i	$-0.721616\pi$
$23$	−6.15139	−1.28265	−0.641327	+	0.767268i	$0.721616\pi$
$24$	−0.817567	−0.166885
$25$	3.78344	0.756688
$26$	0.934296	0.183230
$27$	−1.00000	−0.192450
$28$	6.45564	1.22000
$29$	3.35506	0.623018	0.311509	−	0.950243i	$-0.399166\pi$
$29$	3.35506	0.623018	0.311509	+	0.950243i	$0.399166\pi$
$30$	−0.612286	−0.111788
$31$	−3.97663	−0.714224	−0.357112	−	0.934062i	$-0.616238\pi$
$31$	−3.97663	−0.714224	−0.357112	+	0.934062i	$0.616238\pi$
$32$	−2.40899	−0.425853
$33$	2.02436	0.352396
$34$	0.206596	0.0354310
$35$	9.77486	1.65225
$36$	−1.95732	−0.326220
$37$	−6.02124	−0.989886	−0.494943	−	0.868926i	$-0.664811\pi$
$37$	−6.02124	−0.989886	−0.494943	+	0.868926i	$0.664811\pi$
$38$	−0.587192	−0.0952551
$39$	4.52233	0.724152
$40$	−2.42301	−0.383112
$41$	−5.50704	−0.860055	−0.430028	−	0.902816i	$-0.641496\pi$
$41$	−5.50704	−0.860055	−0.430028	+	0.902816i	$0.641496\pi$
$42$	−0.681397	−0.105142
$43$	−0.805249	−0.122799	−0.0613996	−	0.998113i	$-0.519556\pi$
$43$	−0.805249	−0.122799	−0.0613996	+	0.998113i	$0.519556\pi$
$44$	3.96231	0.597341
$45$	−2.96369	−0.441800
$46$	1.27085	0.187377
$47$	2.71134	0.395489	0.197745	−	0.980254i	$-0.436638\pi$
$47$	2.71134	0.395489	0.197745	+	0.980254i	$0.436638\pi$
$48$	−3.74573	−0.540650
$49$	3.87818	0.554026
$50$	−0.781645	−0.110541
$51$	1.00000	0.140028
$52$	8.85163	1.22750
$53$	7.81249	1.07313	0.536564	−	0.843860i	$-0.319722\pi$
$53$	7.81249	1.07313	0.536564	+	0.843860i	$0.319722\pi$
$54$	0.206596	0.0281142
$55$	5.99957	0.808981
$56$	−2.69651	−0.360336
$57$	−2.84222	−0.376461
$58$	−0.693142	−0.0910140
$59$	0.000491989 0	6.40516e−5 0	3.20258e−5	−	1.00000i	$-0.499990\pi$
$59$	0.000491989 0	6.40516e−5 0	3.20258e−5		1.00000i	$0.499990\pi$
$60$	−5.80088	−0.748890
$61$	10.2945	1.31808	0.659040	−	0.752108i	$-0.270963\pi$
$61$	10.2945	1.31808	0.659040	+	0.752108i	$0.270963\pi$
$62$	0.821556	0.104338
$63$	−3.29821	−0.415535
$64$	−6.99377	−0.874222
$65$	13.4028	1.66241
$66$	−0.418225	−0.0514799
$67$	6.98424	0.853261	0.426630	−	0.904426i	$-0.359701\pi$
$67$	6.98424	0.853261	0.426630	+	0.904426i	$0.359701\pi$
$68$	1.95732	0.237360
$69$	6.15139	0.740540
$70$	−2.01945	−0.241370
$71$	12.7098	1.50838	0.754189	−	0.656657i	$-0.228030\pi$
$71$	12.7098	1.50838	0.754189	+	0.656657i	$0.228030\pi$
$72$	0.817567	0.0963512
$73$	−0.595084	−0.0696493	−0.0348246	−	0.999393i	$-0.511087\pi$
$73$	−0.595084	−0.0696493	−0.0348246	+	0.999393i	$0.511087\pi$
$74$	1.24397	0.144608
$75$	−3.78344	−0.436874
$76$	−5.56313	−0.638134
$77$	6.67676	0.760887
$78$	−0.934296	−0.105788
$79$	10.0119	1.12643	0.563216	−	0.826310i	$-0.309564\pi$
$79$	10.0119	1.12643	0.563216	+	0.826310i	$0.309564\pi$
$80$	−11.1012	−1.24115
$81$	1.00000	0.111111
$82$	1.13773	0.125642
$83$	−10.2303	−1.12293	−0.561463	−	0.827502i	$-0.689761\pi$
$83$	−10.2303	−1.12293	−0.561463	+	0.827502i	$0.689761\pi$
$84$	−6.45564	−0.704368
$85$	2.96369	0.321457
$86$	0.166361	0.0179392
$87$	−3.35506	−0.359700
$88$	−1.65505	−0.176429
$89$	6.31716	0.669618	0.334809	−	0.942286i	$-0.391328\pi$
$89$	6.31716	0.669618	0.334809	+	0.942286i	$0.391328\pi$
$90$	0.612286	0.0645407
$91$	14.9156	1.56358
$92$	12.0402	1.25528
$93$	3.97663	0.412357
$94$	−0.560152	−0.0577752
$95$	−8.42345	−0.864228
$96$	2.40899	0.245866
$97$	6.62727	0.672898	0.336449	−	0.941702i	$-0.390774\pi$
$97$	6.62727	0.672898	0.336449	+	0.941702i	$0.390774\pi$
$98$	−0.801217	−0.0809352
$99$	−2.02436	−0.203456
$100$	−7.40540	−0.740540
$101$	4.37106	0.434937	0.217468	−	0.976067i	$-0.430220\pi$
$101$	4.37106	0.434937	0.217468	+	0.976067i	$0.430220\pi$
$102$	−0.206596	−0.0204561
$103$	10.8198	1.06610	0.533052	−	0.846082i	$-0.321045\pi$
$103$	10.8198	1.06610	0.533052	+	0.846082i	$0.321045\pi$
$104$	−3.69731	−0.362551
$105$	−9.77486	−0.953929
$106$	−1.61403	−0.156769
$107$	13.2838	1.28419	0.642097	−	0.766624i	$-0.278065\pi$
$107$	13.2838	1.28419	0.642097	+	0.766624i	$0.278065\pi$
$108$	1.95732	0.188343
$109$	7.51938	0.720226	0.360113	−	0.932909i	$-0.382738\pi$
$109$	7.51938	0.720226	0.360113	+	0.932909i	$0.382738\pi$
$110$	−1.23949	−0.118181
$111$	6.02124	0.571511
$112$	−12.3542	−1.16736
$113$	6.82917	0.642434	0.321217	−	0.947006i	$-0.395908\pi$
$113$	6.82917	0.642434	0.321217	+	0.947006i	$0.395908\pi$
$114$	0.587192	0.0549955
$115$	18.2308	1.70003
$116$	−6.56691	−0.609722
$117$	−4.52233	−0.418089
$118$	−0.000101643 0	−9.35701e−6 0
$119$	3.29821	0.302346
$120$	2.42301	0.221190
$121$	−6.90197	−0.627452
$122$	−2.12681	−0.192552
$123$	5.50704	0.496553
$124$	7.78353	0.698981
$125$	3.60550	0.322485
$126$	0.681397	0.0607037
$127$	−1.72390	−0.152971	−0.0764857	−	0.997071i	$-0.524370\pi$
$127$	−1.72390	−0.152971	−0.0764857	+	0.997071i	$0.524370\pi$
$128$	6.26286	0.553564
$129$	0.805249	0.0708982
$130$	−2.76896	−0.242854
$131$	12.7234	1.11165	0.555825	−	0.831300i	$-0.312403\pi$
$131$	12.7234	1.11165	0.555825	+	0.831300i	$0.312403\pi$
$132$	−3.96231	−0.344875
$133$	−9.37423	−0.812849
$134$	−1.44292	−0.124649
$135$	2.96369	0.255074
$136$	−0.817567	−0.0701058
$137$	−6.52930	−0.557836	−0.278918	−	0.960315i	$-0.589976\pi$
$137$	−6.52930	−0.557836	−0.278918	+	0.960315i	$0.589976\pi$
$138$	−1.27085	−0.108182
$139$	−16.1046	−1.36597	−0.682986	−	0.730431i	$-0.739319\pi$
$139$	−16.1046	−1.36597	−0.682986	+	0.730431i	$0.739319\pi$
$140$	−19.1325	−1.61699
$141$	−2.71134	−0.228336
$142$	−2.62580	−0.220352
$143$	9.15482	0.765564
$144$	3.74573	0.312144
$145$	−9.94333	−0.825749
$146$	0.122942	0.0101748
$147$	−3.87818	−0.319867
$148$	11.7855	0.968761
$149$	−11.1683	−0.914945	−0.457472	−	0.889224i	$-0.651245\pi$
$149$	−11.1683	−0.914945	−0.457472	+	0.889224i	$0.651245\pi$
$150$	0.781645	0.0638210
$151$	8.50804	0.692375	0.346187	−	0.938165i	$-0.387476\pi$
$151$	8.50804	0.692375	0.346187	+	0.938165i	$0.387476\pi$
$152$	2.32370	0.188477
$153$	−1.00000	−0.0808452
$154$	−1.37939	−0.111155
$155$	11.7855	0.946633
$156$	−8.85163	−0.708698
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	−2.06843	−0.164555
$159$	−7.81249	−0.619571
$160$	7.13948	0.564426
$161$	20.2886	1.59896
$162$	−0.206596	−0.0162317
$163$	−21.7533	−1.70385	−0.851925	−	0.523664i	$-0.824565\pi$
$163$	−21.7533	−1.70385	−0.851925	+	0.523664i	$0.824565\pi$
$164$	10.7790	0.841701
$165$	−5.99957	−0.467066
$166$	2.11355	0.164043
$167$	−2.07171	−0.160314	−0.0801569	−	0.996782i	$-0.525542\pi$
$167$	−2.07171	−0.160314	−0.0801569	+	0.996782i	$0.525542\pi$
$168$	2.69651	0.208040
$169$	7.45145	0.573189
$170$	−0.612286	−0.0469602
$171$	2.84222	0.217350
$172$	1.57613	0.120179
$173$	−6.97136	−0.530023	−0.265011	−	0.964245i	$-0.585376\pi$
$173$	−6.97136	−0.530023	−0.265011	+	0.964245i	$0.585376\pi$
$174$	0.693142	0.0525469
$175$	−12.4786	−0.943292
$176$	−7.58270	−0.571568
$177$	−0.000491989 0	−3.69802e−5 0
$178$	−1.30510	−0.0978215
$179$	−15.8036	−1.18122	−0.590609	−	0.806958i	$-0.701113\pi$
$179$	−15.8036	−1.18122	−0.590609	+	0.806958i	$0.701113\pi$
$180$	5.80088	0.432372
$181$	15.5742	1.15762	0.578812	−	0.815461i	$-0.303517\pi$
$181$	15.5742	1.15762	0.578812	+	0.815461i	$0.303517\pi$
$182$	−3.08150	−0.228416
$183$	−10.2945	−0.760994
$184$	−5.02917	−0.370755
$185$	17.8451	1.31200
$186$	−0.821556	−0.0602394
$187$	2.02436	0.148036
$188$	−5.30695	−0.387049
$189$	3.29821	0.239909
$190$	1.74025	0.126251
$191$	−9.33763	−0.675647	−0.337824	−	0.941209i	$-0.609691\pi$
$191$	−9.33763	−0.675647	−0.337824	+	0.941209i	$0.609691\pi$
$192$	6.99377	0.504732
$193$	−12.2457	−0.881468	−0.440734	−	0.897638i	$-0.645282\pi$
$193$	−12.2457	−0.881468	−0.440734	+	0.897638i	$0.645282\pi$
$194$	−1.36917	−0.0983006
$195$	−13.4028	−0.959792
$196$	−7.59083	−0.542202
$197$	−20.2355	−1.44172	−0.720861	−	0.693080i	$-0.756253\pi$
$197$	−20.2355	−1.44172	−0.720861	+	0.693080i	$0.756253\pi$
$198$	0.418225	0.0297219
$199$	0.253590	0.0179765	0.00898825	−	0.999960i	$-0.497139\pi$
$199$	0.253590	0.0179765	0.00898825	+	0.999960i	$0.497139\pi$
$200$	3.09322	0.218723
$201$	−6.98424	−0.492630
$202$	−0.903045	−0.0635380
$203$	−11.0657	−0.776658
$204$	−1.95732	−0.137040
$205$	16.3211	1.13992
$206$	−2.23532	−0.155742
$207$	−6.15139	−0.427551
$208$	−16.9394	−1.17454
$209$	−5.75367	−0.397990
$210$	2.01945	0.139355
$211$	−22.3520	−1.53877	−0.769386	−	0.638784i	$-0.779438\pi$
$211$	−22.3520	−1.53877	−0.769386	+	0.638784i	$0.779438\pi$
$212$	−15.2915	−1.05023
$213$	−12.7098	−0.870863
$214$	−2.74438	−0.187602
$215$	2.38651	0.162758
$216$	−0.817567	−0.0556284
$217$	13.1157	0.890355
$218$	−1.55347	−0.105215
$219$	0.595084	0.0402120
$220$	−11.7431	−0.791717
$221$	4.52233	0.304205
$222$	−1.24397	−0.0834895
$223$	9.37799	0.627996	0.313998	−	0.949424i	$-0.398331\pi$
$223$	9.37799	0.627996	0.313998	+	0.949424i	$0.398331\pi$
$224$	7.94534	0.530870
$225$	3.78344	0.252229
$226$	−1.41088	−0.0938503
$227$	−16.6181	−1.10298	−0.551490	−	0.834181i	$-0.685941\pi$
$227$	−16.6181	−1.10298	−0.551490	+	0.834181i	$0.685941\pi$
$228$	5.56313	0.368427
$229$	8.54414	0.564612	0.282306	−	0.959324i	$-0.408901\pi$
$229$	8.54414	0.564612	0.282306	+	0.959324i	$0.408901\pi$
$230$	−3.76641	−0.248350
$231$	−6.67676	−0.439298
$232$	2.74298	0.180086
$233$	−10.6121	−0.695224	−0.347612	−	0.937638i	$-0.613007\pi$
$233$	−10.6121	−0.695224	−0.347612	+	0.937638i	$0.613007\pi$
$234$	0.934296	0.0610768
$235$	−8.03555	−0.524182
$236$	−0.000962980 0	−6.26846e−5 0
$237$	−10.0119	−0.650346
$238$	−0.681397	−0.0441684
$239$	10.1801	0.658494	0.329247	−	0.944244i	$-0.393205\pi$
$239$	10.1801	0.658494	0.329247	+	0.944244i	$0.393205\pi$
$240$	11.1012	0.716578
$241$	−6.02177	−0.387896	−0.193948	−	0.981012i	$-0.562129\pi$
$241$	−6.02177	−0.387896	−0.193948	+	0.981012i	$0.562129\pi$
$242$	1.42592	0.0916617
$243$	−1.00000	−0.0641500
$244$	−20.1497	−1.28995
$245$	−11.4937	−0.734306
$246$	−1.13773	−0.0725392
$247$	−12.8534	−0.817845
$248$	−3.25116	−0.206449
$249$	10.2303	0.648322
$250$	−0.744882	−0.0471105
$251$	9.91638	0.625916	0.312958	−	0.949767i	$-0.398680\pi$
$251$	9.91638	0.625916	0.312958	+	0.949767i	$0.398680\pi$
$252$	6.45564	0.406667
$253$	12.4526	0.782890
$254$	0.356151	0.0223469
$255$	−2.96369	−0.185593
$256$	12.6937	0.793354
$257$	−14.5981	−0.910604	−0.455302	−	0.890337i	$-0.650469\pi$
$257$	−14.5981	−0.910604	−0.455302	+	0.890337i	$0.650469\pi$
$258$	−0.166361	−0.0103572
$259$	19.8593	1.23400
$260$	−26.2335	−1.62693
$261$	3.35506	0.207673
$262$	−2.62861	−0.162396
$263$	9.62897	0.593748	0.296874	−	0.954917i	$-0.404056\pi$
$263$	9.62897	0.593748	0.296874	+	0.954917i	$0.404056\pi$
$264$	1.65505	0.101861
$265$	−23.1538	−1.42233
$266$	1.93668	0.118746
$267$	−6.31716	−0.386604
$268$	−13.6704	−0.835051
$269$	−24.2163	−1.47650	−0.738248	−	0.674529i	$-0.764347\pi$
$269$	−24.2163	−1.47650	−0.738248	+	0.674529i	$0.764347\pi$
$270$	−0.612286	−0.0372626
$271$	0.936971	0.0569169	0.0284585	−	0.999595i	$-0.490940\pi$
$271$	0.936971	0.0569169	0.0284585	+	0.999595i	$0.490940\pi$
$272$	−3.74573	−0.227118
$273$	−14.9156	−0.902732
$274$	1.34893	0.0814917
$275$	−7.65904	−0.461858
$276$	−12.0402	−0.724736
$277$	−14.1399	−0.849582	−0.424791	−	0.905291i	$-0.639652\pi$
$277$	−14.1399	−0.849582	−0.424791	+	0.905291i	$0.639652\pi$
$278$	3.32714	0.199549
$279$	−3.97663	−0.238075
$280$	7.99160	0.477589
$281$	32.3428	1.92941	0.964704	−	0.263338i	$-0.0848236\pi$
$281$	32.3428	1.92941	0.964704	+	0.263338i	$0.0848236\pi$
$282$	0.560152	0.0333565
$283$	24.0543	1.42988	0.714940	−	0.699186i	$-0.246454\pi$
$283$	24.0543	1.42988	0.714940	+	0.699186i	$0.246454\pi$
$284$	−24.8772	−1.47619
$285$	8.42345	0.498962
$286$	−1.89135	−0.111838
$287$	18.1634	1.07215
$288$	−2.40899	−0.141951
$289$	1.00000	0.0588235
$290$	2.05425	0.120630
$291$	−6.62727	−0.388498
$292$	1.16477	0.0681629
$293$	27.9330	1.63186	0.815932	−	0.578148i	$-0.196224\pi$
$293$	27.9330	1.63186	0.815932	+	0.578148i	$0.196224\pi$
$294$	0.801217	0.0467279
$295$	−0.00145810	−8.48940e−5 0
$296$	−4.92277	−0.286130
$297$	2.02436	0.117465
$298$	2.30733	0.133660
$299$	27.8186	1.60879
$300$	7.40540	0.427551
$301$	2.65588	0.153082
$302$	−1.75773	−0.101146
$303$	−4.37106	−0.251111
$304$	10.6462	0.610601
$305$	−30.5098	−1.74698
$306$	0.206596	0.0118103
$307$	28.8610	1.64718	0.823592	−	0.567182i	$-0.191966\pi$
$307$	28.8610	1.64718	0.823592	+	0.567182i	$0.191966\pi$
$308$	−13.0685	−0.744649
$309$	−10.8198	−0.615515
$310$	−2.43484	−0.138289
$311$	−14.5043	−0.822465	−0.411232	−	0.911531i	$-0.634902\pi$
$311$	−14.5043	−0.822465	−0.411232	+	0.911531i	$0.634902\pi$
$312$	3.69731	0.209319
$313$	−4.22057	−0.238561	−0.119280	−	0.992861i	$-0.538059\pi$
$313$	−4.22057	−0.238561	−0.119280	+	0.992861i	$0.538059\pi$
$314$	0.206596	0.0116589
$315$	9.77486	0.550751
$316$	−19.5966	−1.10239
$317$	5.40649	0.303659	0.151829	−	0.988407i	$-0.451484\pi$
$317$	5.40649	0.303659	0.151829	+	0.988407i	$0.451484\pi$
$318$	1.61403	0.0905104
$319$	−6.79184	−0.380270
$320$	20.7274	1.15869
$321$	−13.2838	−0.741429
$322$	−4.19154	−0.233585
$323$	−2.84222	−0.158145
$324$	−1.95732	−0.108740
$325$	−17.1100	−0.949090
$326$	4.49415	0.248908
$327$	−7.51938	−0.415822
$328$	−4.50237	−0.248602
$329$	−8.94255	−0.493019
$330$	1.23949	0.0682315
$331$	23.6438	1.29958	0.649789	−	0.760114i	$-0.274857\pi$
$331$	23.6438	1.29958	0.649789	+	0.760114i	$0.274857\pi$
$332$	20.0240	1.09896
$333$	−6.02124	−0.329962
$334$	0.428008	0.0234195
$335$	−20.6991	−1.13091
$336$	12.3542	0.673977
$337$	−14.6867	−0.800035	−0.400017	−	0.916508i	$-0.630996\pi$
$337$	−14.6867	−0.800035	−0.400017	+	0.916508i	$0.630996\pi$
$338$	−1.53944	−0.0837346
$339$	−6.82917	−0.370909
$340$	−5.80088	−0.314597
$341$	8.05012	0.435939
$342$	−0.587192	−0.0317517
$343$	10.2964	0.555954
$344$	−0.658345	−0.0354956
$345$	−18.2308	−0.981513
$346$	1.44026	0.0774286
$347$	−0.0908107	−0.00487497	−0.00243749	−	0.999997i	$-0.500776\pi$
$347$	−0.0908107	−0.00487497	−0.00243749	+	0.999997i	$0.500776\pi$
$348$	6.56691	0.352023
$349$	−28.7455	−1.53871	−0.769356	−	0.638820i	$-0.779423\pi$
$349$	−28.7455	−1.53871	−0.769356	+	0.638820i	$0.779423\pi$
$350$	2.57803	0.137801
$351$	4.52233	0.241384
$352$	4.87665	0.259927
$353$	−3.49073	−0.185793	−0.0928963	−	0.995676i	$-0.529613\pi$
$353$	−3.49073	−0.185793	−0.0928963	+	0.995676i	$0.529613\pi$
$354$	0.000101643 0	5.40227e−6 0
$355$	−37.6680	−1.99921
$356$	−12.3647	−0.655327
$357$	−3.29821	−0.174560
$358$	3.26497	0.172559
$359$	−1.58297	−0.0835462	−0.0417731	−	0.999127i	$-0.513301\pi$
$359$	−1.58297	−0.0835462	−0.0417731	+	0.999127i	$0.513301\pi$
$360$	−2.42301	−0.127704
$361$	−10.9218	−0.574831
$362$	−3.21758	−0.169112
$363$	6.90197	0.362260
$364$	−29.1945	−1.53021
$365$	1.76364	0.0923133
$366$	2.12681	0.111170
$367$	7.06116	0.368590	0.184295	−	0.982871i	$-0.441000\pi$
$367$	7.06116	0.368590	0.184295	+	0.982871i	$0.441000\pi$
$368$	−23.0414	−1.20112
$369$	−5.50704	−0.286685
$370$	−3.68672	−0.191664
$371$	−25.7672	−1.33777
$372$	−7.78353	−0.403557
$373$	−1.51149	−0.0782621	−0.0391310	−	0.999234i	$-0.512459\pi$
$373$	−1.51149	−0.0782621	−0.0391310	+	0.999234i	$0.512459\pi$
$374$	−0.418225	−0.0216259
$375$	−3.60550	−0.186187
$376$	2.21670	0.114317
$377$	−15.1727	−0.781432
$378$	−0.681397	−0.0350473
$379$	−32.5267	−1.67079	−0.835393	−	0.549654i	$-0.814760\pi$
$379$	−32.5267	−1.67079	−0.835393	+	0.549654i	$0.814760\pi$
$380$	16.4874	0.845784
$381$	1.72390	0.0883180
$382$	1.92912	0.0987023
$383$	12.0659	0.616541	0.308271	−	0.951299i	$-0.400250\pi$
$383$	12.0659	0.616541	0.308271	+	0.951299i	$0.400250\pi$
$384$	−6.26286	−0.319600
$385$	−19.7878	−1.00848
$386$	2.52992	0.128770
$387$	−0.805249	−0.0409331
$388$	−12.9717	−0.658537
$389$	39.1118	1.98304	0.991522	−	0.129937i	$-0.0414774\pi$
$389$	39.1118	1.98304	0.991522	+	0.129937i	$0.0414774\pi$
$390$	2.76896	0.140212
$391$	6.15139	0.311089
$392$	3.17067	0.160143
$393$	−12.7234	−0.641811
$394$	4.18058	0.210615
$395$	−29.6723	−1.49297
$396$	3.96231	0.199114
$397$	15.7417	0.790051	0.395026	−	0.918670i	$-0.370736\pi$
$397$	15.7417	0.790051	0.395026	+	0.918670i	$0.370736\pi$
$398$	−0.0523907	−0.00262611
$399$	9.37423	0.469299
$400$	14.1718	0.708588
$401$	4.06914	0.203203	0.101602	−	0.994825i	$-0.467603\pi$
$401$	4.06914	0.203203	0.101602	+	0.994825i	$0.467603\pi$
$402$	1.44292	0.0719662
$403$	17.9836	0.895828
$404$	−8.55556	−0.425655
$405$	−2.96369	−0.147267
$406$	2.28613	0.113459
$407$	12.1892	0.604194
$408$	0.817567	0.0404756
$409$	32.9540	1.62947	0.814735	−	0.579834i	$-0.196882\pi$
$409$	32.9540	1.62947	0.814735	+	0.579834i	$0.196882\pi$
$410$	−3.37189	−0.166526
$411$	6.52930	0.322067
$412$	−21.1777	−1.04335
$413$	−0.00162268	−7.98471e−5 0
$414$	1.27085	0.0624591
$415$	30.3196	1.48833
$416$	10.8942	0.534134
$417$	16.1046	0.788644
$418$	1.18869	0.0581406
$419$	0.836509	0.0408662	0.0204331	−	0.999791i	$-0.493495\pi$
$419$	0.836509	0.0408662	0.0204331	+	0.999791i	$0.493495\pi$
$420$	19.1325	0.933571
$421$	−26.6633	−1.29949	−0.649745	−	0.760152i	$-0.725125\pi$
$421$	−26.6633	−1.29949	−0.649745	+	0.760152i	$0.725125\pi$
$422$	4.61783	0.224792
$423$	2.71134	0.131830
$424$	6.38723	0.310192
$425$	−3.78344	−0.183524
$426$	2.62580	0.127221
$427$	−33.9535	−1.64313
$428$	−26.0006	−1.25679
$429$	−9.15482	−0.441999
$430$	−0.493043	−0.0237766
$431$	15.4797	0.745633	0.372817	−	0.927905i	$-0.378392\pi$
$431$	15.4797	0.745633	0.372817	+	0.927905i	$0.378392\pi$
$432$	−3.74573	−0.180217
$433$	−15.4925	−0.744520	−0.372260	−	0.928129i	$-0.621417\pi$
$433$	−15.4925	−0.744520	−0.372260	+	0.928129i	$0.621417\pi$
$434$	−2.70966	−0.130068
$435$	9.94333	0.476746
$436$	−14.7178	−0.704855
$437$	−17.4836	−0.836354
$438$	−0.122942	−0.00587440
$439$	−27.3681	−1.30621	−0.653104	−	0.757268i	$-0.726533\pi$
$439$	−27.3681	−1.30621	−0.653104	+	0.757268i	$0.726533\pi$
$440$	4.90505	0.233839
$441$	3.87818	0.184675
$442$	−0.934296	−0.0444399
$443$	−16.7822	−0.797344	−0.398672	−	0.917093i	$-0.630529\pi$
$443$	−16.7822	−0.797344	−0.398672	+	0.917093i	$0.630529\pi$
$444$	−11.7855	−0.559314
$445$	−18.7221	−0.887512
$446$	−1.93746	−0.0917412
$447$	11.1683	0.528243
$448$	23.0669	1.08981
$449$	18.4042	0.868549	0.434275	−	0.900781i	$-0.357005\pi$
$449$	18.4042	0.868549	0.434275	+	0.900781i	$0.357005\pi$
$450$	−0.781645	−0.0368471
$451$	11.1482	0.524949
$452$	−13.3669	−0.628724
$453$	−8.50804	−0.399743
$454$	3.43323	0.161130
$455$	−44.2051	−2.07237
$456$	−2.32370	−0.108817
$457$	−1.54464	−0.0722551	−0.0361275	−	0.999347i	$-0.511502\pi$
$457$	−1.54464	−0.0722551	−0.0361275	+	0.999347i	$0.511502\pi$
$458$	−1.76519	−0.0824817
$459$	1.00000	0.0466760
$460$	−35.6835	−1.66375
$461$	30.9109	1.43967	0.719833	−	0.694148i	$-0.244219\pi$
$461$	30.9109	1.43967	0.719833	+	0.694148i	$0.244219\pi$
$462$	1.37939	0.0641752
$463$	−24.7695	−1.15113	−0.575567	−	0.817754i	$-0.695219\pi$
$463$	−24.7695	−1.15113	−0.575567	+	0.817754i	$0.695219\pi$
$464$	12.5671	0.583414
$465$	−11.7855	−0.546539
$466$	2.19243	0.101562
$467$	18.4197	0.852361	0.426181	−	0.904638i	$-0.359859\pi$
$467$	18.4197	0.852361	0.426181	+	0.904638i	$0.359859\pi$
$468$	8.85163	0.409167
$469$	−23.0355	−1.06368
$470$	1.66011	0.0765754
$471$	1.00000	0.0460776
$472$	0.000402234 0	1.85143e−5 0
$473$	1.63011	0.0749526
$474$	2.06843	0.0950062
$475$	10.7534	0.493399
$476$	−6.45564	−0.295894
$477$	7.81249	0.357709
$478$	−2.10316	−0.0961965
$479$	37.0653	1.69355	0.846777	−	0.531947i	$-0.178540\pi$
$479$	37.0653	1.69355	0.846777	+	0.531947i	$0.178540\pi$
$480$	−7.13948	−0.325871
$481$	27.2300	1.24158
$482$	1.24407	0.0566661
$483$	−20.2886	−0.923162
$484$	13.5094	0.614061
$485$	−19.6412	−0.891860
$486$	0.206596	0.00937139
$487$	−21.9334	−0.993895	−0.496948	−	0.867780i	$-0.665546\pi$
$487$	−21.9334	−0.993895	−0.496948	+	0.867780i	$0.665546\pi$
$488$	8.41647	0.380996
$489$	21.7533	0.983718
$490$	2.37456	0.107272
$491$	4.66692	0.210615	0.105308	−	0.994440i	$-0.466417\pi$
$491$	4.66692	0.210615	0.105308	+	0.994440i	$0.466417\pi$
$492$	−10.7790	−0.485956
$493$	−3.35506	−0.151104
$494$	2.65547	0.119475
$495$	5.99957	0.269660
$496$	−14.8954	−0.668822
$497$	−41.9197	−1.88035
$498$	−2.11355	−0.0947105
$499$	−0.286947	−0.0128455	−0.00642275	−	0.999979i	$-0.502044\pi$
$499$	−0.286947	−0.0128455	−0.00642275	+	0.999979i	$0.502044\pi$
$500$	−7.05710	−0.315603
$501$	2.07171	0.0925572
$502$	−2.04869	−0.0914373
$503$	12.3518	0.550741	0.275371	−	0.961338i	$-0.411199\pi$
$503$	12.3518	0.550741	0.275371	+	0.961338i	$0.411199\pi$
$504$	−2.69651	−0.120112
$505$	−12.9545	−0.576466
$506$	−2.57266	−0.114369
$507$	−7.45145	−0.330931
$508$	3.37422	0.149707
$509$	7.66157	0.339593	0.169796	−	0.985479i	$-0.445689\pi$
$509$	7.66157	0.339593	0.169796	+	0.985479i	$0.445689\pi$
$510$	0.612286	0.0271125
$511$	1.96271	0.0868252
$512$	−15.1482	−0.669461
$513$	−2.84222	−0.125487
$514$	3.01591	0.133026
$515$	−32.0664	−1.41302
$516$	−1.57613	−0.0693852
$517$	−5.48872	−0.241393
$518$	−4.10286	−0.180269
$519$	6.97136	0.306009
$520$	10.9577	0.480525
$521$	−5.42090	−0.237494	−0.118747	−	0.992925i	$-0.537888\pi$
$521$	−5.42090	−0.237494	−0.118747	+	0.992925i	$0.537888\pi$
$522$	−0.693142	−0.0303380
$523$	12.6758	0.554274	0.277137	−	0.960830i	$-0.410614\pi$
$523$	12.6758	0.554274	0.277137	+	0.960830i	$0.410614\pi$
$524$	−24.9038	−1.08793
$525$	12.4786	0.544610
$526$	−1.98931	−0.0867380
$527$	3.97663	0.173225
$528$	7.58270	0.329995
$529$	14.8396	0.645200
$530$	4.78348	0.207781
$531$	0.000491989 0	2.13505e−5 0
$532$	18.3484	0.795502
$533$	24.9046	1.07874
$534$	1.30510	0.0564773
$535$	−39.3690	−1.70207
$536$	5.71008	0.246638
$537$	15.8036	0.681977
$538$	5.00300	0.215695
$539$	−7.85083	−0.338159
$540$	−5.80088	−0.249630
$541$	13.3446	0.573730	0.286865	−	0.957971i	$-0.407387\pi$
$541$	13.3446	0.573730	0.286865	+	0.957971i	$0.407387\pi$
$542$	−0.193575	−0.00831474
$543$	−15.5742	−0.668354
$544$	2.40899	0.103284
$545$	−22.2851	−0.954588
$546$	3.08150	0.131876
$547$	−37.6872	−1.61139	−0.805693	−	0.592333i	$-0.798207\pi$
$547$	−37.6872	−1.61139	−0.805693	+	0.592333i	$0.798207\pi$
$548$	12.7799	0.545931
$549$	10.2945	0.439360
$550$	1.58233	0.0674708
$551$	9.53580	0.406239
$552$	5.02917	0.214056
$553$	−33.0215	−1.40422
$554$	2.92124	0.124112
$555$	−17.8451	−0.757481
$556$	31.5218	1.33682
$557$	−35.6177	−1.50917	−0.754586	−	0.656201i	$-0.772162\pi$
$557$	−35.6177	−1.50917	−0.754586	+	0.656201i	$0.772162\pi$
$558$	0.821556	0.0347792
$559$	3.64160	0.154023
$560$	36.6140	1.54722
$561$	−2.02436	−0.0854685
$562$	−6.68189	−0.281859
$563$	8.61111	0.362915	0.181457	−	0.983399i	$-0.441919\pi$
$563$	8.61111	0.362915	0.181457	+	0.983399i	$0.441919\pi$
$564$	5.30695	0.223463
$565$	−20.2395	−0.851483
$566$	−4.96953	−0.208885
$567$	−3.29821	−0.138512
$568$	10.3911	0.436002
$569$	39.4240	1.65274	0.826370	−	0.563127i	$-0.190402\pi$
$569$	39.4240	1.65274	0.826370	+	0.563127i	$0.190402\pi$
$570$	−1.74025	−0.0728912
$571$	−33.0428	−1.38280	−0.691399	−	0.722473i	$-0.743005\pi$
$571$	−33.0428	−1.38280	−0.691399	+	0.722473i	$0.743005\pi$
$572$	−17.9189	−0.749226
$573$	9.33763	0.390085
$574$	−3.75248	−0.156626
$575$	−23.2734	−0.970569
$576$	−6.99377	−0.291407
$577$	22.7303	0.946275	0.473137	−	0.880989i	$-0.343121\pi$
$577$	22.7303	0.946275	0.473137	+	0.880989i	$0.343121\pi$
$578$	−0.206596	−0.00859327
$579$	12.2457	0.508916
$580$	19.4623	0.808127
$581$	33.7418	1.39985
$582$	1.36917	0.0567539
$583$	−15.8153	−0.655002
$584$	−0.486521	−0.0201324
$585$	13.4028	0.554136
$586$	−5.77085	−0.238392
$587$	−13.8143	−0.570179	−0.285089	−	0.958501i	$-0.592023\pi$
$587$	−13.8143	−0.570179	−0.285089	+	0.958501i	$0.592023\pi$
$588$	7.59083	0.313041
$589$	−11.3024	−0.465709
$590$	0.000301238 0	1.24018e−5 0
$591$	20.2355	0.832378
$592$	−22.5539	−0.926961
$593$	28.1086	1.15428	0.577141	−	0.816644i	$-0.304168\pi$
$593$	28.1086	1.15428	0.577141	+	0.816644i	$0.304168\pi$
$594$	−0.418225	−0.0171600
$595$	−9.77486	−0.400730
$596$	21.8600	0.895419
$597$	−0.253590	−0.0103787
$598$	−5.74722	−0.235021
$599$	16.7476	0.684288	0.342144	−	0.939648i	$-0.388847\pi$
$599$	16.7476	0.684288	0.342144	+	0.939648i	$0.388847\pi$
$600$	−3.09322	−0.126280
$601$	−36.3144	−1.48130	−0.740649	−	0.671893i	$-0.765482\pi$
$601$	−36.3144	−1.48130	−0.740649	+	0.671893i	$0.765482\pi$
$602$	−0.548694	−0.0223631
$603$	6.98424	0.284420
$604$	−16.6529	−0.677599
$605$	20.4553	0.831626
$606$	0.903045	0.0366837
$607$	10.1764	0.413047	0.206523	−	0.978442i	$-0.433785\pi$
$607$	10.1764	0.413047	0.206523	+	0.978442i	$0.433785\pi$
$608$	−6.84687	−0.277677
$609$	11.0657	0.448404
$610$	6.30320	0.255209
$611$	−12.2616	−0.496049
$612$	1.95732	0.0791199
$613$	−34.9243	−1.41058	−0.705290	−	0.708919i	$-0.749183\pi$
$613$	−34.9243	−1.41058	−0.705290	+	0.708919i	$0.749183\pi$
$614$	−5.96257	−0.240630
$615$	−16.3211	−0.658132
$616$	5.45870	0.219937
$617$	−14.6332	−0.589112	−0.294556	−	0.955634i	$-0.595172\pi$
$617$	−14.6332	−0.589112	−0.294556	+	0.955634i	$0.595172\pi$
$618$	2.23532	0.0899179
$619$	26.5532	1.06726	0.533631	−	0.845717i	$-0.320827\pi$
$619$	26.5532	1.06726	0.533631	+	0.845717i	$0.320827\pi$
$620$	−23.0679	−0.926431
$621$	6.15139	0.246847
$622$	2.99654	0.120150
$623$	−20.8353	−0.834749
$624$	16.9394	0.678120
$625$	−29.6028	−1.18411
$626$	0.871953	0.0348503
$627$	5.75367	0.229780
$628$	1.95732	0.0781055
$629$	6.02124	0.240083
$630$	−2.01945	−0.0804568
$631$	4.91139	0.195519	0.0977596	−	0.995210i	$-0.468832\pi$
$631$	4.91139	0.195519	0.0977596	+	0.995210i	$0.468832\pi$
$632$	8.18544	0.325599
$633$	22.3520	0.888411
$634$	−1.11696	−0.0443602
$635$	5.10910	0.202748
$636$	15.2915	0.606349
$637$	−17.5384	−0.694897
$638$	1.40317	0.0555519
$639$	12.7098	0.502793
$640$	−18.5612	−0.733694
$641$	−10.3957	−0.410604	−0.205302	−	0.978699i	$-0.565818\pi$
$641$	−10.3957	−0.410604	−0.205302	+	0.978699i	$0.565818\pi$
$642$	2.74438	0.108312
$643$	−28.3287	−1.11718	−0.558588	−	0.829445i	$-0.688657\pi$
$643$	−28.3287	−1.11718	−0.558588	+	0.829445i	$0.688657\pi$
$644$	−39.7112	−1.56484
$645$	−2.38651	−0.0939686
$646$	0.587192	0.0231027
$647$	3.79379	0.149149	0.0745745	−	0.997215i	$-0.476240\pi$
$647$	3.79379	0.149149	0.0745745	+	0.997215i	$0.476240\pi$
$648$	0.817567	0.0321171
$649$	−0.000995963 0	−3.90950e−5 0
$650$	3.53485	0.138648
$651$	−13.1157	−0.514047
$652$	42.5781	1.66749
$653$	−1.93435	−0.0756971	−0.0378486	−	0.999283i	$-0.512050\pi$
$653$	−1.93435	−0.0756971	−0.0378486	+	0.999283i	$0.512050\pi$
$654$	1.55347	0.0607457
$655$	−37.7082	−1.47338
$656$	−20.6279	−0.805384
$657$	−0.595084	−0.0232164
$658$	1.84750	0.0720229
$659$	41.5604	1.61896	0.809481	−	0.587146i	$-0.199748\pi$
$659$	41.5604	1.61896	0.809481	+	0.587146i	$0.199748\pi$
$660$	11.7431	0.457098
$661$	7.16370	0.278635	0.139318	−	0.990248i	$-0.455509\pi$
$661$	7.16370	0.278635	0.139318	+	0.990248i	$0.455509\pi$
$662$	−4.88471	−0.189850
$663$	−4.52233	−0.175633
$664$	−8.36399	−0.324586
$665$	27.7823	1.07735
$666$	1.24397	0.0482027
$667$	−20.6383	−0.799116
$668$	4.05500	0.156893
$669$	−9.37799	−0.362574
$670$	4.27636	0.165210
$671$	−20.8398	−0.804513
$672$	−7.94534	−0.306498
$673$	−47.6487	−1.83672	−0.918362	−	0.395742i	$-0.870487\pi$
$673$	−47.6487	−1.83672	−0.918362	+	0.395742i	$0.870487\pi$
$674$	3.03421	0.116874
$675$	−3.78344	−0.145625
$676$	−14.5849	−0.560956
$677$	11.6955	0.449495	0.224747	−	0.974417i	$-0.427844\pi$
$677$	11.6955	0.449495	0.224747	+	0.974417i	$0.427844\pi$
$678$	1.41088	0.0541845
$679$	−21.8581	−0.838838
$680$	2.42301	0.0929183
$681$	16.6181	0.636806
$682$	−1.66312	−0.0636843
$683$	43.8188	1.67668	0.838340	−	0.545148i	$-0.183526\pi$
$683$	43.8188	1.67668	0.838340	+	0.545148i	$0.183526\pi$
$684$	−5.56313	−0.212711
$685$	19.3508	0.739356
$686$	−2.12720	−0.0812169
$687$	−8.54414	−0.325979
$688$	−3.01624	−0.114993
$689$	−35.3307	−1.34599
$690$	3.76641	0.143385
$691$	20.9087	0.795404	0.397702	−	0.917515i	$-0.369808\pi$
$691$	20.9087	0.795404	0.397702	+	0.917515i	$0.369808\pi$
$692$	13.6452	0.518711
$693$	6.67676	0.253629
$694$	0.0187611	0.000712163 0
$695$	47.7289	1.81046
$696$	−2.74298	−0.103972
$697$	5.50704	0.208594
$698$	5.93871	0.224784
$699$	10.6121	0.401388
$700$	24.4246	0.923161
$701$	−14.9812	−0.565831	−0.282915	−	0.959145i	$-0.591302\pi$
$701$	−14.9812	−0.565831	−0.282915	+	0.959145i	$0.591302\pi$
$702$	−0.934296	−0.0352627
$703$	−17.1137	−0.645455
$704$	14.1579	0.533596
$705$	8.03555	0.302636
$706$	0.721171	0.0271416
$707$	−14.4167	−0.542195
$708$	0.000962980 0	3.61910e−5 0
$709$	39.0216	1.46549	0.732743	−	0.680505i	$-0.238240\pi$
$709$	39.0216	1.46549	0.732743	+	0.680505i	$0.238240\pi$
$710$	7.78206	0.292055
$711$	10.0119	0.375477
$712$	5.16470	0.193555
$713$	24.4618	0.916101
$714$	0.681397	0.0255007
$715$	−27.1320	−1.01468
$716$	30.9327	1.15601
$717$	−10.1801	−0.380182
$718$	0.327036	0.0122049
$719$	0.467610	0.0174389	0.00871945	−	0.999962i	$-0.497224\pi$
$719$	0.467610	0.0174389	0.00871945	+	0.999962i	$0.497224\pi$
$720$	−11.1012	−0.413716
$721$	−35.6859	−1.32901
$722$	2.25640	0.0839745
$723$	6.02177	0.223952
$724$	−30.4837	−1.13292
$725$	12.6937	0.471431
$726$	−1.42592	−0.0529209
$727$	−17.1421	−0.635767	−0.317883	−	0.948130i	$-0.602972\pi$
$727$	−17.1421	−0.635767	−0.317883	+	0.948130i	$0.602972\pi$
$728$	12.1945	0.451958
$729$	1.00000	0.0370370
$730$	−0.364362	−0.0134856
$731$	0.805249	0.0297832
$732$	20.1497	0.744753
$733$	−10.4110	−0.384537	−0.192269	−	0.981342i	$-0.561584\pi$
$733$	−10.4110	−0.384537	−0.192269	+	0.981342i	$0.561584\pi$
$734$	−1.45881	−0.0538456
$735$	11.4937	0.423952
$736$	14.8186	0.546222
$737$	−14.1386	−0.520802
$738$	1.13773	0.0418806
$739$	5.63361	0.207236	0.103618	−	0.994617i	$-0.466958\pi$
$739$	5.63361	0.207236	0.103618	+	0.994617i	$0.466958\pi$
$740$	−34.9285	−1.28400
$741$	12.8534	0.472183
$742$	5.32341	0.195429
$743$	−28.7343	−1.05416	−0.527079	−	0.849816i	$-0.676713\pi$
$743$	−28.7343	−1.05416	−0.527079	+	0.849816i	$0.676713\pi$
$744$	3.25116	0.119193
$745$	33.0994	1.21267
$746$	0.312268	0.0114330
$747$	−10.2303	−0.374309
$748$	−3.96231	−0.144877
$749$	−43.8127	−1.60088
$750$	0.744882	0.0271992
$751$	−15.8107	−0.576941	−0.288470	−	0.957489i	$-0.593147\pi$
$751$	−15.8107	−0.576941	−0.288470	+	0.957489i	$0.593147\pi$
$752$	10.1559	0.370349
$753$	−9.91638	−0.361373
$754$	3.13461	0.114156
$755$	−25.2152	−0.917674
$756$	−6.45564	−0.234789
$757$	26.1004	0.948637	0.474318	−	0.880353i	$-0.342695\pi$
$757$	26.1004	0.948637	0.474318	+	0.880353i	$0.342695\pi$
$758$	6.71990	0.244078
$759$	−12.4526	−0.452001
$760$	−6.88673	−0.249808
$761$	−17.0426	−0.617794	−0.308897	−	0.951096i	$-0.599960\pi$
$761$	−17.0426	−0.617794	−0.308897	+	0.951096i	$0.599960\pi$
$762$	−0.356151	−0.0129020
$763$	−24.8005	−0.897837
$764$	18.2767	0.661228
$765$	2.96369	0.107152
$766$	−2.49278	−0.0900678
$767$	−0.00222494	−8.03378e−5 0
$768$	−12.6937	−0.458043
$769$	41.2812	1.48864	0.744320	−	0.667823i	$-0.232774\pi$
$769$	41.2812	1.48864	0.744320	+	0.667823i	$0.232774\pi$
$770$	4.08809	0.147324
$771$	14.5981	0.525738
$772$	23.9688	0.862656
$773$	−10.1049	−0.363447	−0.181723	−	0.983350i	$-0.558168\pi$
$773$	−10.1049	−0.363447	−0.181723	+	0.983350i	$0.558168\pi$
$774$	0.166361	0.00597973
$775$	−15.0453	−0.540445
$776$	5.41824	0.194503
$777$	−19.8593	−0.712449
$778$	−8.08034	−0.289694
$779$	−15.6522	−0.560799
$780$	26.2335	0.939309
$781$	−25.7292	−0.920665
$782$	−1.27085	−0.0454456
$783$	−3.35506	−0.119900
$784$	14.5266	0.518808
$785$	2.96369	0.105779
$786$	2.62861	0.0937593
$787$	38.9370	1.38795	0.693977	−	0.719997i	$-0.255857\pi$
$787$	38.9370	1.38795	0.693977	+	0.719997i	$0.255857\pi$
$788$	39.6074	1.41095
$789$	−9.62897	−0.342800
$790$	6.13018	0.218102
$791$	−22.5240	−0.800862
$792$	−1.65505	−0.0588096
$793$	−46.5552	−1.65323
$794$	−3.25217	−0.115415
$795$	23.1538	0.821180
$796$	−0.496356	−0.0175929
$797$	−5.96014	−0.211119	−0.105559	−	0.994413i	$-0.533663\pi$
$797$	−5.96014	−0.211119	−0.105559	+	0.994413i	$0.533663\pi$
$798$	−1.93668	−0.0685577
$799$	−2.71134	−0.0959202
$800$	−9.11426	−0.322238
$801$	6.31716	0.223206
$802$	−0.840668	−0.0296850
$803$	1.20466	0.0425116
$804$	13.6704	0.482117
$805$	−60.1290	−2.11927
$806$	−3.71535	−0.130868
$807$	24.2163	0.852456
$808$	3.57364	0.125720
$809$	45.2004	1.58916	0.794581	−	0.607159i	$-0.207691\pi$
$809$	45.2004	1.58916	0.794581	+	0.607159i	$0.207691\pi$
$810$	0.612286	0.0215136
$811$	−18.7209	−0.657380	−0.328690	−	0.944438i	$-0.606607\pi$
$811$	−18.7209	−0.657380	−0.328690	+	0.944438i	$0.606607\pi$
$812$	21.6590	0.760083
$813$	−0.936971	−0.0328610
$814$	−2.51823	−0.0882640
$815$	64.4700	2.25828
$816$	3.74573	0.131127
$817$	−2.28869	−0.0800712
$818$	−6.80817	−0.238042
$819$	14.9156	0.521193
$820$	−31.9457	−1.11559
$821$	−7.15064	−0.249559	−0.124780	−	0.992184i	$-0.539822\pi$
$821$	−7.15064	−0.249559	−0.124780	+	0.992184i	$0.539822\pi$
$822$	−1.34893	−0.0470493
$823$	13.0128	0.453598	0.226799	−	0.973942i	$-0.427174\pi$
$823$	13.0128	0.453598	0.226799	+	0.973942i	$0.427174\pi$
$824$	8.84589	0.308161
$825$	7.65904	0.266654
$826$	0.000335240 0	1.16645e−5 0
$827$	40.2504	1.39964	0.699821	−	0.714318i	$-0.253263\pi$
$827$	40.2504	1.39964	0.699821	+	0.714318i	$0.253263\pi$
$828$	12.0402	0.418427
$829$	22.0878	0.767142	0.383571	−	0.923511i	$-0.374694\pi$
$829$	22.0878	0.767142	0.383571	+	0.923511i	$0.374694\pi$
$830$	−6.26390	−0.217423
$831$	14.1399	0.490506
$832$	31.6281	1.09651
$833$	−3.87818	−0.134371
$834$	−3.32714	−0.115210
$835$	6.13990	0.212480
$836$	11.2618	0.389496
$837$	3.97663	0.137452
$838$	−0.172820	−0.00596995
$839$	24.3018	0.838990	0.419495	−	0.907758i	$-0.362207\pi$
$839$	24.3018	0.838990	0.419495	+	0.907758i	$0.362207\pi$
$840$	−7.99160	−0.275736
$841$	−17.7436	−0.611848
$842$	5.50854	0.189837
$843$	−32.3428	−1.11394
$844$	43.7499	1.50593
$845$	−22.0838	−0.759705
$846$	−0.560152	−0.0192584
$847$	22.7641	0.782185
$848$	29.2635	1.00491
$849$	−24.0543	−0.825541
$850$	0.781645	0.0268102
$851$	37.0390	1.26968
$852$	24.8772	0.852278
$853$	−1.23958	−0.0424423	−0.0212211	−	0.999775i	$-0.506755\pi$
$853$	−1.23958	−0.0424423	−0.0212211	+	0.999775i	$0.506755\pi$
$854$	7.01467	0.240037
$855$	−8.42345	−0.288076
$856$	10.8604	0.371201
$857$	−33.5881	−1.14735	−0.573674	−	0.819083i	$-0.694483\pi$
$857$	−33.5881	−1.14735	−0.573674	+	0.819083i	$0.694483\pi$
$858$	1.89135	0.0645696
$859$	−41.5601	−1.41801	−0.709006	−	0.705203i	$-0.750856\pi$
$859$	−41.5601	−1.41801	−0.709006	+	0.705203i	$0.750856\pi$
$860$	−4.67115	−0.159285
$861$	−18.1634	−0.619006
$862$	−3.19806	−0.108926
$863$	−8.15597	−0.277633	−0.138816	−	0.990318i	$-0.544330\pi$
$863$	−8.15597	−0.277633	−0.138816	+	0.990318i	$0.544330\pi$
$864$	2.40899	0.0819554
$865$	20.6609	0.702493
$866$	3.20068	0.108764
$867$	−1.00000	−0.0339618
$868$	−25.6717	−0.871354
$869$	−20.2678	−0.687537
$870$	−2.05425	−0.0696458
$871$	−31.5850	−1.07022
$872$	6.14759	0.208184
$873$	6.62727	0.224299
$874$	3.61204	0.122179
$875$	−11.8917	−0.402012
$876$	−1.16477	−0.0393539
$877$	−14.7098	−0.496716	−0.248358	−	0.968668i	$-0.579891\pi$
$877$	−14.7098	−0.496716	−0.248358	+	0.968668i	$0.579891\pi$
$878$	5.65414	0.190818
$879$	−27.9330	−0.942157
$880$	22.4728	0.757556
$881$	−35.4621	−1.19475	−0.597374	−	0.801963i	$-0.703789\pi$
$881$	−35.4621	−1.19475	−0.597374	+	0.801963i	$0.703789\pi$
$882$	−0.801217	−0.0269784
$883$	24.9029	0.838050	0.419025	−	0.907975i	$-0.362372\pi$
$883$	24.9029	0.838050	0.419025	+	0.907975i	$0.362372\pi$
$884$	−8.85163	−0.297713
$885$	0.00145810	4.90136e−5 0
$886$	3.46713	0.116481
$887$	17.0693	0.573132	0.286566	−	0.958061i	$-0.407486\pi$
$887$	17.0693	0.573132	0.286566	+	0.958061i	$0.407486\pi$
$888$	4.92277	0.165197
$889$	5.68578	0.190695
$890$	3.86791	0.129653
$891$	−2.02436	−0.0678186
$892$	−18.3557	−0.614594
$893$	7.70621	0.257879
$894$	−2.30733	−0.0771687
$895$	46.8370	1.56559
$896$	−20.6562	−0.690076
$897$	−27.8186	−0.928836
$898$	−3.80224	−0.126882
$899$	−13.3418	−0.444974
$900$	−7.40540	−0.246847
$901$	−7.81249	−0.260272
$902$	−2.30318	−0.0766875
$903$	−2.65588	−0.0883821
$904$	5.58330	0.185698
$905$	−46.1571	−1.53432
$906$	1.75773	0.0583966
$907$	−1.80384	−0.0598956	−0.0299478	−	0.999551i	$-0.509534\pi$
$907$	−1.80384	−0.0598956	−0.0299478	+	0.999551i	$0.509534\pi$
$908$	32.5269	1.07944
$909$	4.37106	0.144979
$910$	9.13261	0.302743
$911$	17.2375	0.571104	0.285552	−	0.958363i	$-0.407823\pi$
$911$	17.2375	0.571104	0.285552	+	0.958363i	$0.407823\pi$
$912$	−10.6462	−0.352530
$913$	20.7099	0.685398
$914$	0.319116	0.0105554
$915$	30.5098	1.00862
$916$	−16.7236	−0.552563
$917$	−41.9644	−1.38579
$918$	−0.206596	−0.00681869
$919$	29.7998	0.983006	0.491503	−	0.870876i	$-0.336448\pi$
$919$	29.7998	0.983006	0.491503	+	0.870876i	$0.336448\pi$
$920$	14.9049	0.491400
$921$	−28.8610	−0.951003
$922$	−6.38608	−0.210314
$923$	−57.4780	−1.89191
$924$	13.0685	0.429923
$925$	−22.7810	−0.749035
$926$	5.11728	0.168164
$927$	10.8198	0.355368
$928$	−8.08228	−0.265314
$929$	36.6222	1.20153	0.600767	−	0.799424i	$-0.294862\pi$
$929$	36.6222	1.20153	0.600767	+	0.799424i	$0.294862\pi$
$930$	2.43484	0.0798414
$931$	11.0226	0.361252
$932$	20.7713	0.680387
$933$	14.5043	0.474850
$934$	−3.80544	−0.124518
$935$	−5.99957	−0.196207
$936$	−3.69731	−0.120850
$937$	26.2106	0.856262	0.428131	−	0.903717i	$-0.359172\pi$
$937$	26.2106	0.856262	0.428131	+	0.903717i	$0.359172\pi$
$938$	4.75904	0.155388
$939$	4.22057	0.137733
$940$	15.7281	0.512995
$941$	31.3776	1.02288	0.511440	−	0.859319i	$-0.329112\pi$
$941$	31.3776	1.02288	0.511440	+	0.859319i	$0.329112\pi$
$942$	−0.206596	−0.00673127
$943$	33.8760	1.10315
$944$	0.00184286	5.99800e−5 0
$945$	−9.77486	−0.317976
$946$	−0.336775	−0.0109495
$947$	31.8858	1.03615	0.518075	−	0.855335i	$-0.326649\pi$
$947$	31.8858	1.03615	0.518075	+	0.855335i	$0.326649\pi$
$948$	19.5966	0.636467
$949$	2.69116	0.0873589
$950$	−2.22161	−0.0720784
$951$	−5.40649	−0.175318
$952$	2.69651	0.0873943
$953$	−32.6370	−1.05722	−0.528608	−	0.848866i	$-0.677286\pi$
$953$	−32.6370	−1.05722	−0.528608	+	0.848866i	$0.677286\pi$
$954$	−1.61403	−0.0522562
$955$	27.6738	0.895504
$956$	−19.9256	−0.644441
$957$	6.79184	0.219549
$958$	−7.65754	−0.247404
$959$	21.5350	0.695401
$960$	−20.7274	−0.668972
$961$	−15.1864	−0.489885
$962$	−5.62562	−0.181377
$963$	13.2838	0.428064
$964$	11.7865	0.379618
$965$	36.2926	1.16830
$966$	4.19154	0.134861
$967$	−0.913185	−0.0293661	−0.0146830	−	0.999892i	$-0.504674\pi$
$967$	−0.913185	−0.0293661	−0.0146830	+	0.999892i	$0.504674\pi$
$968$	−5.64282	−0.181367
$969$	2.84222	0.0913052
$970$	4.05779	0.130288
$971$	−21.3000	−0.683550	−0.341775	−	0.939782i	$-0.611028\pi$
$971$	−21.3000	−0.683550	−0.341775	+	0.939782i	$0.611028\pi$
$972$	1.95732	0.0627810
$973$	53.1162	1.70283
$974$	4.53135	0.145194
$975$	17.1100	0.547958
$976$	38.5605	1.23429
$977$	16.5788	0.530404	0.265202	−	0.964193i	$-0.414561\pi$
$977$	16.5788	0.530404	0.265202	+	0.964193i	$0.414561\pi$
$978$	−4.49415	−0.143707
$979$	−12.7882	−0.408713
$980$	22.4968	0.718635
$981$	7.51938	0.240075
$982$	−0.964168	−0.0307678
$983$	34.5744	1.10275	0.551376	−	0.834257i	$-0.314103\pi$
$983$	34.5744	1.10275	0.551376	+	0.834257i	$0.314103\pi$
$984$	4.50237	0.143530
$985$	59.9718	1.91086
$986$	0.693142	0.0220741
$987$	8.94255	0.284645
$988$	25.1583	0.800392
$989$	4.95340	0.157509
$990$	−1.23949	−0.0393935
$991$	13.4427	0.427020	0.213510	−	0.976941i	$-0.431510\pi$
$991$	13.4427	0.427020	0.213510	+	0.976941i	$0.431510\pi$
$992$	9.57965	0.304154
$993$	−23.6438	−0.750312
$994$	8.66044	0.274693
$995$	−0.751561	−0.0238261
$996$	−20.0240	−0.634486
$997$	30.8122	0.975831	0.487915	−	0.872891i	$-0.337757\pi$
$997$	30.8122	0.975831	0.487915	+	0.872891i	$0.337757\pi$
$998$	0.0592821	0.00187654
$999$	6.02124	0.190504

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.23	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.23	✓	48	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q-0.206596 q^{2} -1.00000 q^{3} -1.95732 q^{4} -2.96369 q^{5} +0.206596 q^{6} -3.29821 q^{7} +0.817567 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.206596 q^{2} -1.00000 q^{3} -1.95732 q^{4} -2.96369 q^{5} +0.206596 q^{6} -3.29821 q^{7} +0.817567 q^{8} +1.00000 q^{9} +0.612286 q^{10} -2.02436 q^{11} +1.95732 q^{12} -4.52233 q^{13} +0.681397 q^{14} +2.96369 q^{15} +3.74573 q^{16} -1.00000 q^{17} -0.206596 q^{18} +2.84222 q^{19} +5.80088 q^{20} +3.29821 q^{21} +0.418225 q^{22} -6.15139 q^{23} -0.817567 q^{24} +3.78344 q^{25} +0.934296 q^{26} -1.00000 q^{27} +6.45564 q^{28} +3.35506 q^{29} -0.612286 q^{30} -3.97663 q^{31} -2.40899 q^{32} +2.02436 q^{33} +0.206596 q^{34} +9.77486 q^{35} -1.95732 q^{36} -6.02124 q^{37} -0.587192 q^{38} +4.52233 q^{39} -2.42301 q^{40} -5.50704 q^{41} -0.681397 q^{42} -0.805249 q^{43} +3.96231 q^{44} -2.96369 q^{45} +1.27085 q^{46} +2.71134 q^{47} -3.74573 q^{48} +3.87818 q^{49} -0.781645 q^{50} +1.00000 q^{51} +8.85163 q^{52} +7.81249 q^{53} +0.206596 q^{54} +5.99957 q^{55} -2.69651 q^{56} -2.84222 q^{57} -0.693142 q^{58} +0.000491989 q^{59} -5.80088 q^{60} +10.2945 q^{61} +0.821556 q^{62} -3.29821 q^{63} -6.99377 q^{64} +13.4028 q^{65} -0.418225 q^{66} +6.98424 q^{67} +1.95732 q^{68} +6.15139 q^{69} -2.01945 q^{70} +12.7098 q^{71} +0.817567 q^{72} -0.595084 q^{73} +1.24397 q^{74} -3.78344 q^{75} -5.56313 q^{76} +6.67676 q^{77} -0.934296 q^{78} +10.0119 q^{79} -11.1012 q^{80} +1.00000 q^{81} +1.13773 q^{82} -10.2303 q^{83} -6.45564 q^{84} +2.96369 q^{85} +0.166361 q^{86} -3.35506 q^{87} -1.65505 q^{88} +6.31716 q^{89} +0.612286 q^{90} +14.9156 q^{91} +12.0402 q^{92} +3.97663 q^{93} -0.560152 q^{94} -8.42345 q^{95} +2.40899 q^{96} +6.62727 q^{97} -0.801217 q^{98} -2.02436 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * q^99

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