LMFDB - Embedded newform 4003.2.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4003,2,Mod(1,4003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4003, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4003.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:	$31.9641159291$
Analytic rank:	$1$
Dimension:	$152$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	4003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.74603 q^{2} -0.470182 q^{3} +5.54067 q^{4} -1.60967 q^{5} +1.29113 q^{6} -2.65994 q^{7} -9.72280 q^{8} -2.77893 q^{9} +O(q^{10})$	\(q-2.74603 q^{2} -0.470182 q^{3} +5.54067 q^{4} -1.60967 q^{5} +1.29113 q^{6} -2.65994 q^{7} -9.72280 q^{8} -2.77893 q^{9} +4.42020 q^{10} -4.89664 q^{11} -2.60512 q^{12} +4.07842 q^{13} +7.30426 q^{14} +0.756837 q^{15} +15.6177 q^{16} -1.69505 q^{17} +7.63102 q^{18} +2.50475 q^{19} -8.91866 q^{20} +1.25065 q^{21} +13.4463 q^{22} -5.52567 q^{23} +4.57148 q^{24} -2.40896 q^{25} -11.1995 q^{26} +2.71715 q^{27} -14.7378 q^{28} -0.190228 q^{29} -2.07830 q^{30} +5.82448 q^{31} -23.4411 q^{32} +2.30231 q^{33} +4.65467 q^{34} +4.28162 q^{35} -15.3971 q^{36} +6.76398 q^{37} -6.87812 q^{38} -1.91760 q^{39} +15.6505 q^{40} +6.84648 q^{41} -3.43433 q^{42} +3.96451 q^{43} -27.1307 q^{44} +4.47316 q^{45} +15.1736 q^{46} +6.99841 q^{47} -7.34317 q^{48} +0.0752656 q^{49} +6.61507 q^{50} +0.796983 q^{51} +22.5972 q^{52} +0.426551 q^{53} -7.46136 q^{54} +7.88198 q^{55} +25.8620 q^{56} -1.17769 q^{57} +0.522372 q^{58} +10.7234 q^{59} +4.19339 q^{60} +11.5904 q^{61} -15.9942 q^{62} +7.39178 q^{63} +33.1346 q^{64} -6.56492 q^{65} -6.32221 q^{66} -9.73873 q^{67} -9.39174 q^{68} +2.59807 q^{69} -11.7575 q^{70} +11.1732 q^{71} +27.0190 q^{72} -3.34920 q^{73} -18.5741 q^{74} +1.13265 q^{75} +13.8780 q^{76} +13.0248 q^{77} +5.26578 q^{78} -8.54360 q^{79} -25.1394 q^{80} +7.05924 q^{81} -18.8006 q^{82} -9.08513 q^{83} +6.92946 q^{84} +2.72848 q^{85} -10.8867 q^{86} +0.0894418 q^{87} +47.6090 q^{88} -11.6083 q^{89} -12.2834 q^{90} -10.8483 q^{91} -30.6159 q^{92} -2.73856 q^{93} -19.2178 q^{94} -4.03182 q^{95} +11.0216 q^{96} -11.2327 q^{97} -0.206681 q^{98} +13.6074 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9}+O(q^{10}) $ 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9	\( 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9} - 15 q^{10} - 40 q^{11} - 53 q^{12} - 59 q^{13} - 36 q^{14} - 40 q^{15} + 118 q^{16} - 93 q^{17} - 59 q^{18} - 16 q^{19} - 108 q^{20} - 62 q^{21} - 37 q^{22} - 107 q^{23} - 31 q^{24} + 101 q^{25} - 64 q^{26} - 63 q^{27} - 53 q^{28} - 124 q^{29} - 68 q^{30} - 15 q^{31} - 129 q^{32} - 49 q^{33} - 76 q^{35} + 45 q^{36} - 98 q^{37} - 125 q^{38} - 47 q^{39} - 7 q^{40} - 56 q^{41} - 84 q^{42} - 62 q^{43} - 114 q^{44} - 142 q^{45} - 3 q^{46} - 111 q^{47} - 92 q^{48} + 117 q^{49} - 64 q^{50} - 21 q^{51} - 85 q^{52} - 347 q^{53} + 3 q^{54} - 16 q^{55} - 73 q^{56} - 115 q^{57} - 29 q^{58} - 50 q^{59} - 54 q^{60} - 62 q^{61} - 55 q^{62} - 70 q^{63} + 64 q^{64} - 147 q^{65} + 34 q^{66} - 86 q^{67} - 174 q^{68} - 104 q^{69} - 7 q^{70} - 86 q^{71} - 139 q^{72} - 27 q^{73} - 52 q^{74} - 49 q^{75} - 11 q^{76} - 346 q^{77} - 59 q^{78} - 17 q^{79} - 149 q^{80} - 8 q^{81} - 31 q^{82} - 106 q^{83} - 51 q^{84} - 69 q^{85} - 85 q^{86} - 32 q^{87} - 113 q^{88} - 59 q^{89} + 10 q^{90} - 9 q^{91} - 314 q^{92} - 230 q^{93} + 7 q^{94} - 74 q^{95} - 54 q^{96} - 60 q^{97} - 77 q^{98} - 96 q^{99}+O(q^{100}) \) 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9 - 15 * q^10 - 40 * q^11 - 53 * q^12 - 59 * q^13 - 36 * q^14 - 40 * q^15 + 118 * q^16 - 93 * q^17 - 59 * q^18 - 16 * q^19 - 108 * q^20 - 62 * q^21 - 37 * q^22 - 107 * q^23 - 31 * q^24 + 101 * q^25 - 64 * q^26 - 63 * q^27 - 53 * q^28 - 124 * q^29 - 68 * q^30 - 15 * q^31 - 129 * q^32 - 49 * q^33 - 76 * q^35 + 45 * q^36 - 98 * q^37 - 125 * q^38 - 47 * q^39 - 7 * q^40 - 56 * q^41 - 84 * q^42 - 62 * q^43 - 114 * q^44 - 142 * q^45 - 3 * q^46 - 111 * q^47 - 92 * q^48 + 117 * q^49 - 64 * q^50 - 21 * q^51 - 85 * q^52 - 347 * q^53 + 3 * q^54 - 16 * q^55 - 73 * q^56 - 115 * q^57 - 29 * q^58 - 50 * q^59 - 54 * q^60 - 62 * q^61 - 55 * q^62 - 70 * q^63 + 64 * q^64 - 147 * q^65 + 34 * q^66 - 86 * q^67 - 174 * q^68 - 104 * q^69 - 7 * q^70 - 86 * q^71 - 139 * q^72 - 27 * q^73 - 52 * q^74 - 49 * q^75 - 11 * q^76 - 346 * q^77 - 59 * q^78 - 17 * q^79 - 149 * q^80 - 8 * q^81 - 31 * q^82 - 106 * q^83 - 51 * q^84 - 69 * q^85 - 85 * q^86 - 32 * q^87 - 113 * q^88 - 59 * q^89 + 10 * q^90 - 9 * q^91 - 314 * q^92 - 230 * q^93 + 7 * q^94 - 74 * q^95 - 54 * q^96 - 60 * q^97 - 77 * q^98 - 96 * 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 $

$2$	−2.74603	−1.94174	−0.970868	−	0.239616i	$-0.922979\pi$
$2$	−2.74603	−1.94174	−0.970868	+	0.239616i	$0.922979\pi$
$3$	−0.470182	−0.271459	−0.135730	−	0.990746i	$-0.543338\pi$
$3$	−0.470182	−0.271459	−0.135730	+	0.990746i	$0.543338\pi$
$4$	5.54067	2.77034
$5$	−1.60967	−0.719867	−0.359933	−	0.932978i	$-0.617201\pi$
$5$	−1.60967	−0.719867	−0.359933	+	0.932978i	$0.617201\pi$
$6$	1.29113	0.527102
$7$	−2.65994	−1.00536	−0.502681	−	0.864472i	$-0.667653\pi$
$7$	−2.65994	−1.00536	−0.502681	+	0.864472i	$0.667653\pi$
$8$	−9.72280	−3.43753
$9$	−2.77893	−0.926310
$10$	4.42020	1.39779
$11$	−4.89664	−1.47639	−0.738196	−	0.674586i	$-0.764322\pi$
$11$	−4.89664	−1.47639	−0.738196	+	0.674586i	$0.764322\pi$
$12$	−2.60512	−0.752034
$13$	4.07842	1.13115	0.565575	−	0.824697i	$-0.308654\pi$
$13$	4.07842	1.13115	0.565575	+	0.824697i	$0.308654\pi$
$14$	7.30426	1.95215
$15$	0.756837	0.195415
$16$	15.6177	3.90443
$17$	−1.69505	−0.411111	−0.205555	−	0.978645i	$-0.565900\pi$
$17$	−1.69505	−0.411111	−0.205555	+	0.978645i	$0.565900\pi$
$18$	7.63102	1.79865
$19$	2.50475	0.574629	0.287315	−	0.957836i	$-0.407237\pi$
$19$	2.50475	0.574629	0.287315	+	0.957836i	$0.407237\pi$
$20$	−8.91866	−1.99427
$21$	1.25065	0.272915
$22$	13.4463	2.86676
$23$	−5.52567	−1.15218	−0.576091	−	0.817386i	$-0.695422\pi$
$23$	−5.52567	−1.15218	−0.576091	+	0.817386i	$0.695422\pi$
$24$	4.57148	0.933149
$25$	−2.40896	−0.481792
$26$	−11.1995	−2.19640
$27$	2.71715	0.522915
$28$	−14.7378	−2.78519
$29$	−0.190228	−0.0353245	−0.0176622	−	0.999844i	$-0.505622\pi$
$29$	−0.190228	−0.0353245	−0.0176622	+	0.999844i	$0.505622\pi$
$30$	−2.07830	−0.379443
$31$	5.82448	1.04611	0.523053	−	0.852300i	$-0.324793\pi$
$31$	5.82448	1.04611	0.523053	+	0.852300i	$0.324793\pi$
$32$	−23.4411	−4.14385
$33$	2.30231	0.400781
$34$	4.65467	0.798268
$35$	4.28162	0.723726
$36$	−15.3971	−2.56619
$37$	6.76398	1.11199	0.555996	−	0.831185i	$-0.312337\pi$
$37$	6.76398	1.11199	0.555996	+	0.831185i	$0.312337\pi$
$38$	−6.87812	−1.11578
$39$	−1.91760	−0.307062
$40$	15.6505	2.47456
$41$	6.84648	1.06924	0.534620	−	0.845092i	$-0.320455\pi$
$41$	6.84648	1.06924	0.534620	+	0.845092i	$0.320455\pi$
$42$	−3.43433	−0.529929
$43$	3.96451	0.604583	0.302291	−	0.953216i	$-0.402248\pi$
$43$	3.96451	0.604583	0.302291	+	0.953216i	$0.402248\pi$
$44$	−27.1307	−4.09010
$45$	4.47316	0.666819
$46$	15.1736	2.23723
$47$	6.99841	1.02082	0.510412	−	0.859930i	$-0.329493\pi$
$47$	6.99841	1.02082	0.510412	+	0.859930i	$0.329493\pi$
$48$	−7.34317	−1.05989
$49$	0.0752656	0.0107522
$50$	6.61507	0.935513
$51$	0.796983	0.111600
$52$	22.5972	3.13367
$53$	0.426551	0.0585913	0.0292956	−	0.999571i	$-0.490674\pi$
$53$	0.426551	0.0585913	0.0292956	+	0.999571i	$0.490674\pi$
$54$	−7.46136	−1.01536
$55$	7.88198	1.06281
$56$	25.8620	3.45596
$57$	−1.17769	−0.155988
$58$	0.522372	0.0685908
$59$	10.7234	1.39606	0.698031	−	0.716068i	$-0.254060\pi$
$59$	10.7234	1.39606	0.698031	+	0.716068i	$0.254060\pi$
$60$	4.19339	0.541364
$61$	11.5904	1.48400	0.742001	−	0.670399i	$-0.233877\pi$
$61$	11.5904	1.48400	0.742001	+	0.670399i	$0.233877\pi$
$62$	−15.9942	−2.03126
$63$	7.39178	0.931276
$64$	33.1346	4.14182
$65$	−6.56492	−0.814278
$66$	−6.32221	−0.778210
$67$	−9.73873	−1.18978	−0.594888	−	0.803809i	$-0.702803\pi$
$67$	−9.73873	−1.18978	−0.594888	+	0.803809i	$0.702803\pi$
$68$	−9.39174	−1.13892
$69$	2.59807	0.312770
$70$	−11.7575	−1.40529
$71$	11.1732	1.32601	0.663007	−	0.748614i	$-0.269280\pi$
$71$	11.1732	1.32601	0.663007	+	0.748614i	$0.269280\pi$
$72$	27.0190	3.18422
$73$	−3.34920	−0.391994	−0.195997	−	0.980604i	$-0.562794\pi$
$73$	−3.34920	−0.391994	−0.195997	+	0.980604i	$0.562794\pi$
$74$	−18.5741	−2.15919
$75$	1.13265	0.130787
$76$	13.8780	1.59192
$77$	13.0248	1.48431
$78$	5.26578	0.596232
$79$	−8.54360	−0.961230	−0.480615	−	0.876932i	$-0.659587\pi$
$79$	−8.54360	−0.961230	−0.480615	+	0.876932i	$0.659587\pi$
$80$	−25.1394	−2.81067
$81$	7.05924	0.784360
$82$	−18.8006	−2.07618
$83$	−9.08513	−0.997222	−0.498611	−	0.866826i	$-0.666156\pi$
$83$	−9.08513	−0.997222	−0.498611	+	0.866826i	$0.666156\pi$
$84$	6.92946	0.756066
$85$	2.72848	0.295945
$86$	−10.8867	−1.17394
$87$	0.0894418	0.00958917
$88$	47.6090	5.07514
$89$	−11.6083	−1.23048	−0.615241	−	0.788339i	$-0.710941\pi$
$89$	−11.6083	−1.23048	−0.615241	+	0.788339i	$0.710941\pi$
$90$	−12.2834	−1.29479
$91$	−10.8483	−1.13722
$92$	−30.6159	−3.19193
$93$	−2.73856	−0.283976
$94$	−19.2178	−1.98217
$95$	−4.03182	−0.413656
$96$	11.0216	1.12489
$97$	−11.2327	−1.14050	−0.570252	−	0.821470i	$-0.693154\pi$
$97$	−11.2327	−1.14050	−0.570252	+	0.821470i	$0.693154\pi$
$98$	−0.206681	−0.0208780
$99$	13.6074	1.36760
$100$	−13.3473	−1.33473
$101$	−6.24174	−0.621076	−0.310538	−	0.950561i	$-0.600509\pi$
$101$	−6.24174	−0.621076	−0.310538	+	0.950561i	$0.600509\pi$
$102$	−2.18854	−0.216697
$103$	4.94761	0.487502	0.243751	−	0.969838i	$-0.421622\pi$
$103$	4.94761	0.487502	0.243751	+	0.969838i	$0.421622\pi$
$104$	−39.6537	−3.88836
$105$	−2.01314	−0.196462
$106$	−1.17132	−0.113769
$107$	17.5616	1.69774	0.848872	−	0.528599i	$-0.177282\pi$
$107$	17.5616	1.69774	0.848872	+	0.528599i	$0.177282\pi$
$108$	15.0548	1.44865
$109$	−12.5718	−1.20416	−0.602079	−	0.798436i	$-0.705661\pi$
$109$	−12.5718	−1.20416	−0.602079	+	0.798436i	$0.705661\pi$
$110$	−21.6441	−2.06369
$111$	−3.18030	−0.301861
$112$	−41.5422	−3.92537
$113$	3.48385	0.327733	0.163867	−	0.986482i	$-0.447603\pi$
$113$	3.48385	0.327733	0.163867	+	0.986482i	$0.447603\pi$
$114$	3.23396	0.302888
$115$	8.89450	0.829417
$116$	−1.05399	−0.0978608
$117$	−11.3336	−1.04780
$118$	−29.4466	−2.71078
$119$	4.50873	0.413315
$120$	−7.35858	−0.671743
$121$	12.9771	1.17973
$122$	−31.8276	−2.88154
$123$	−3.21909	−0.290255
$124$	32.2715	2.89807
$125$	11.9260	1.06669
$126$	−20.2980	−1.80829
$127$	−14.8041	−1.31365	−0.656827	−	0.754041i	$-0.728102\pi$
$127$	−14.8041	−1.31365	−0.656827	+	0.754041i	$0.728102\pi$
$128$	−44.1063	−3.89848
$129$	−1.86404	−0.164120
$130$	18.0275	1.58111
$131$	−6.14475	−0.536869	−0.268435	−	0.963298i	$-0.586506\pi$
$131$	−6.14475	−0.536869	−0.268435	+	0.963298i	$0.586506\pi$
$132$	12.7563	1.11030
$133$	−6.66248	−0.577710
$134$	26.7428	2.31023
$135$	−4.37371	−0.376429
$136$	16.4807	1.41320
$137$	−5.66736	−0.484195	−0.242097	−	0.970252i	$-0.577835\pi$
$137$	−5.66736	−0.484195	−0.242097	+	0.970252i	$0.577835\pi$
$138$	−7.13436	−0.607317
$139$	14.0764	1.19394	0.596971	−	0.802263i	$-0.296371\pi$
$139$	14.0764	1.19394	0.596971	+	0.802263i	$0.296371\pi$
$140$	23.7231	2.00497
$141$	−3.29052	−0.277112
$142$	−30.6819	−2.57477
$143$	−19.9706	−1.67002
$144$	−43.4006	−3.61671
$145$	0.306205	0.0254289
$146$	9.19700	0.761149
$147$	−0.0353885	−0.00291879
$148$	37.4770	3.08059
$149$	9.59049	0.785684	0.392842	−	0.919606i	$-0.371492\pi$
$149$	9.59049	0.785684	0.392842	+	0.919606i	$0.371492\pi$
$150$	−3.11029	−0.253954
$151$	13.8690	1.12865	0.564323	−	0.825554i	$-0.309137\pi$
$151$	13.8690	1.12865	0.564323	+	0.825554i	$0.309137\pi$
$152$	−24.3532	−1.97530
$153$	4.71043	0.380816
$154$	−35.7663	−2.88213
$155$	−9.37549	−0.753057
$156$	−10.6248	−0.850664
$157$	−4.79420	−0.382619	−0.191310	−	0.981530i	$-0.561273\pi$
$157$	−4.79420	−0.382619	−0.191310	+	0.981530i	$0.561273\pi$
$158$	23.4610	1.86646
$159$	−0.200556	−0.0159052
$160$	37.7325	2.98302
$161$	14.6979	1.15836
$162$	−19.3849	−1.52302
$163$	−5.56014	−0.435504	−0.217752	−	0.976004i	$-0.569872\pi$
$163$	−5.56014	−0.435504	−0.217752	+	0.976004i	$0.569872\pi$
$164$	37.9341	2.96216
$165$	−3.70596	−0.288509
$166$	24.9480	1.93634
$167$	10.3760	0.802920	0.401460	−	0.915876i	$-0.368503\pi$
$167$	10.3760	0.802920	0.401460	+	0.915876i	$0.368503\pi$
$168$	−12.1598	−0.938152
$169$	3.63353	0.279502
$170$	−7.49248	−0.574647
$171$	−6.96052	−0.532285
$172$	21.9661	1.67490
$173$	−21.1798	−1.61027	−0.805134	−	0.593093i	$-0.797906\pi$
$173$	−21.1798	−1.61027	−0.805134	+	0.593093i	$0.797906\pi$
$174$	−0.245610	−0.0186196
$175$	6.40768	0.484375
$176$	−76.4744	−5.76447
$177$	−5.04192	−0.378974
$178$	31.8768	2.38927
$179$	13.7086	1.02463	0.512314	−	0.858798i	$-0.328788\pi$
$179$	13.7086	1.02463	0.512314	+	0.858798i	$0.328788\pi$
$180$	24.7843	1.84731
$181$	−18.1284	−1.34747	−0.673736	−	0.738973i	$-0.735311\pi$
$181$	−18.1284	−1.34747	−0.673736	+	0.738973i	$0.735311\pi$
$182$	29.7899	2.20817
$183$	−5.44960	−0.402846
$184$	53.7249	3.96065
$185$	−10.8878	−0.800486
$186$	7.52017	0.551405
$187$	8.30006	0.606961
$188$	38.7759	2.82802
$189$	−7.22744	−0.525719
$190$	11.0715	0.803211
$191$	1.02485	0.0741558	0.0370779	−	0.999312i	$-0.488195\pi$
$191$	1.02485	0.0741558	0.0370779	+	0.999312i	$0.488195\pi$
$192$	−15.5793	−1.12434
$193$	−4.00182	−0.288057	−0.144029	−	0.989574i	$-0.546006\pi$
$193$	−4.00182	−0.288057	−0.144029	+	0.989574i	$0.546006\pi$
$194$	30.8452	2.21456
$195$	3.08670	0.221043
$196$	0.417022	0.0297873
$197$	14.8251	1.05624	0.528122	−	0.849168i	$-0.322896\pi$
$197$	14.8251	1.05624	0.528122	+	0.849168i	$0.322896\pi$
$198$	−37.3663	−2.65551
$199$	−7.26216	−0.514801	−0.257400	−	0.966305i	$-0.582866\pi$
$199$	−7.26216	−0.514801	−0.257400	+	0.966305i	$0.582866\pi$
$200$	23.4218	1.65617
$201$	4.57897	0.322976
$202$	17.1400	1.20597
$203$	0.505995	0.0355139
$204$	4.41582	0.309169
$205$	−11.0206	−0.769711
$206$	−13.5863	−0.946601
$207$	15.3554	1.06728
$208$	63.6957	4.41650
$209$	−12.2649	−0.848378
$210$	5.52814	0.381478
$211$	−4.01457	−0.276375	−0.138187	−	0.990406i	$-0.544128\pi$
$211$	−4.01457	−0.276375	−0.138187	+	0.990406i	$0.544128\pi$
$212$	2.36338	0.162318
$213$	−5.25342	−0.359959
$214$	−48.2247	−3.29657
$215$	−6.38156	−0.435219
$216$	−26.4183	−1.79753
$217$	−15.4927	−1.05172
$218$	34.5225	2.33816
$219$	1.57473	0.106411
$220$	43.6715	2.94433
$221$	−6.91314	−0.465028
$222$	8.73319	0.586133
$223$	0.161528	0.0108167	0.00540835	−	0.999985i	$-0.498278\pi$
$223$	0.161528	0.0108167	0.00540835	+	0.999985i	$0.498278\pi$
$224$	62.3520	4.16607
$225$	6.69433	0.446289
$226$	−9.56676	−0.636372
$227$	−6.79610	−0.451073	−0.225536	−	0.974235i	$-0.572413\pi$
$227$	−6.79610	−0.451073	−0.225536	+	0.974235i	$0.572413\pi$
$228$	−6.52518	−0.432141
$229$	11.9247	0.788004	0.394002	−	0.919110i	$-0.371090\pi$
$229$	11.9247	0.788004	0.394002	+	0.919110i	$0.371090\pi$
$230$	−24.4246	−1.61051
$231$	−6.12400	−0.402929
$232$	1.84955	0.121429
$233$	−13.3036	−0.871549	−0.435774	−	0.900056i	$-0.643525\pi$
$233$	−13.3036	−0.871549	−0.435774	+	0.900056i	$0.643525\pi$
$234$	31.1225	2.03454
$235$	−11.2651	−0.734857
$236$	59.4146	3.86756
$237$	4.01704	0.260935
$238$	−12.3811	−0.802549
$239$	7.62197	0.493024	0.246512	−	0.969140i	$-0.420715\pi$
$239$	7.62197	0.493024	0.246512	+	0.969140i	$0.420715\pi$
$240$	11.8201	0.762983
$241$	−21.2907	−1.37146	−0.685728	−	0.727858i	$-0.740516\pi$
$241$	−21.2907	−1.37146	−0.685728	+	0.727858i	$0.740516\pi$
$242$	−35.6354	−2.29073
$243$	−11.4706	−0.735837
$244$	64.2187	4.11118
$245$	−0.121153	−0.00774017
$246$	8.83971	0.563599
$247$	10.2154	0.649992
$248$	−56.6302	−3.59602
$249$	4.27166	0.270705
$250$	−32.7491	−2.07124
$251$	−2.29506	−0.144863	−0.0724315	−	0.997373i	$-0.523076\pi$
$251$	−2.29506	−0.144863	−0.0724315	+	0.997373i	$0.523076\pi$
$252$	40.9554	2.57995
$253$	27.0572	1.70107
$254$	40.6526	2.55077
$255$	−1.28288	−0.0803370
$256$	54.8479	3.42800
$257$	23.0571	1.43826	0.719130	−	0.694876i	$-0.244541\pi$
$257$	23.0571	1.43826	0.719130	+	0.694876i	$0.244541\pi$
$258$	5.11871	0.318677
$259$	−17.9918	−1.11795
$260$	−36.3741	−2.25582
$261$	0.528631	0.0327214
$262$	16.8737	1.04246
$263$	−15.2074	−0.937727	−0.468864	−	0.883271i	$-0.655336\pi$
$263$	−15.2074	−0.937727	−0.468864	+	0.883271i	$0.655336\pi$
$264$	−22.3849	−1.37769
$265$	−0.686607	−0.0421779
$266$	18.2954	1.12176
$267$	5.45802	0.334026
$268$	−53.9591	−3.29608
$269$	30.0857	1.83435	0.917177	−	0.398479i	$-0.130462\pi$
$269$	30.0857	1.83435	0.917177	+	0.398479i	$0.130462\pi$
$270$	12.0103	0.730926
$271$	1.70814	0.103762	0.0518810	−	0.998653i	$-0.483478\pi$
$271$	1.70814	0.103762	0.0518810	+	0.998653i	$0.483478\pi$
$272$	−26.4729	−1.60515
$273$	5.10069	0.308708
$274$	15.5627	0.940178
$275$	11.7958	0.711314
$276$	14.3950	0.866480
$277$	16.1307	0.969202	0.484601	−	0.874735i	$-0.338965\pi$
$277$	16.1307	0.969202	0.484601	+	0.874735i	$0.338965\pi$
$278$	−38.6541	−2.31832
$279$	−16.1858	−0.969019
$280$	−41.6293	−2.48783
$281$	−22.1067	−1.31878	−0.659388	−	0.751803i	$-0.729184\pi$
$281$	−22.1067	−1.31878	−0.659388	+	0.751803i	$0.729184\pi$
$282$	9.03587	0.538078
$283$	9.10027	0.540955	0.270478	−	0.962726i	$-0.412818\pi$
$283$	9.10027	0.540955	0.270478	+	0.962726i	$0.412818\pi$
$284$	61.9070	3.67350
$285$	1.89569	0.112291
$286$	54.8397	3.24274
$287$	−18.2112	−1.07497
$288$	65.1413	3.83849
$289$	−14.1268	−0.830988
$290$	−0.840847	−0.0493762
$291$	5.28139	0.309601
$292$	−18.5568	−1.08596
$293$	5.55305	0.324413	0.162206	−	0.986757i	$-0.448139\pi$
$293$	5.55305	0.324413	0.162206	+	0.986757i	$0.448139\pi$
$294$	0.0971778	0.00566752
$295$	−17.2611	−1.00498
$296$	−65.7648	−3.82250
$297$	−13.3049	−0.772027
$298$	−26.3358	−1.52559
$299$	−22.5360	−1.30329
$300$	6.27564	0.362324
$301$	−10.5454	−0.607824
$302$	−38.0848	−2.19153
$303$	2.93475	0.168597
$304$	39.1185	2.24360
$305$	−18.6568	−1.06828
$306$	−12.9350	−0.739444
$307$	−2.10232	−0.119986	−0.0599929	−	0.998199i	$-0.519108\pi$
$307$	−2.10232	−0.119986	−0.0599929	+	0.998199i	$0.519108\pi$
$308$	72.1659	4.11203
$309$	−2.32627	−0.132337
$310$	25.7454	1.46224
$311$	7.69815	0.436522	0.218261	−	0.975890i	$-0.429962\pi$
$311$	7.69815	0.436522	0.218261	+	0.975890i	$0.429962\pi$
$312$	18.6444	1.05553
$313$	−15.8785	−0.897508	−0.448754	−	0.893655i	$-0.648132\pi$
$313$	−15.8785	−0.897508	−0.448754	+	0.893655i	$0.648132\pi$
$314$	13.1650	0.742945
$315$	−11.8983	−0.670395
$316$	−47.3373	−2.66293
$317$	−9.55406	−0.536610	−0.268305	−	0.963334i	$-0.586463\pi$
$317$	−9.55406	−0.536610	−0.268305	+	0.963334i	$0.586463\pi$
$318$	0.550734	0.0308836
$319$	0.931479	0.0521528
$320$	−53.3358	−2.98156
$321$	−8.25714	−0.460868
$322$	−40.3609	−2.24923
$323$	−4.24568	−0.236236
$324$	39.1129	2.17294
$325$	−9.82476	−0.544979
$326$	15.2683	0.845634
$327$	5.91102	0.326880
$328$	−66.5669	−3.67554
$329$	−18.6153	−1.02630
$330$	10.1767	0.560207
$331$	27.6748	1.52114	0.760572	−	0.649254i	$-0.224919\pi$
$331$	27.6748	1.52114	0.760572	+	0.649254i	$0.224919\pi$
$332$	−50.3377	−2.76264
$333$	−18.7966	−1.03005
$334$	−28.4928	−1.55906
$335$	15.6761	0.856479
$336$	19.5324	1.06558
$337$	−10.6232	−0.578680	−0.289340	−	0.957226i	$-0.593436\pi$
$337$	−10.6232	−0.578680	−0.289340	+	0.957226i	$0.593436\pi$
$338$	−9.97777	−0.542719
$339$	−1.63804	−0.0889663
$340$	15.1176	0.819867
$341$	−28.5204	−1.54446
$342$	19.1138	1.03356
$343$	18.4194	0.994552
$344$	−38.5462	−2.07827
$345$	−4.18203	−0.225153
$346$	58.1602	3.12671
$347$	−18.7687	−1.00756	−0.503778	−	0.863833i	$-0.668057\pi$
$347$	−18.7687	−1.00756	−0.503778	+	0.863833i	$0.668057\pi$
$348$	0.495568	0.0265652
$349$	24.7753	1.32619	0.663096	−	0.748534i	$-0.269242\pi$
$349$	24.7753	1.32619	0.663096	+	0.748534i	$0.269242\pi$
$350$	−17.5957	−0.940529
$351$	11.0817	0.591496
$352$	114.783	6.11794
$353$	−10.7918	−0.574390	−0.287195	−	0.957872i	$-0.592723\pi$
$353$	−10.7918	−0.574390	−0.287195	+	0.957872i	$0.592723\pi$
$354$	13.8453	0.735868
$355$	−17.9851	−0.954553
$356$	−64.3180	−3.40885
$357$	−2.11992	−0.112198
$358$	−37.6442	−1.98956
$359$	23.0128	1.21457	0.607284	−	0.794485i	$-0.292259\pi$
$359$	23.0128	1.21457	0.607284	+	0.794485i	$0.292259\pi$
$360$	−43.4916	−2.29221
$361$	−12.7262	−0.669801
$362$	49.7810	2.61643
$363$	−6.10158	−0.320250
$364$	−60.1072	−3.15047
$365$	5.39111	0.282184
$366$	14.9648	0.782221
$367$	−37.0853	−1.93584	−0.967920	−	0.251260i	$-0.919155\pi$
$367$	−37.0853	−1.93584	−0.967920	+	0.251260i	$0.919155\pi$
$368$	−86.2984	−4.49861
$369$	−19.0259	−0.990448
$370$	29.8982	1.55433
$371$	−1.13460	−0.0589054
$372$	−15.1735	−0.786708
$373$	−8.65496	−0.448137	−0.224068	−	0.974573i	$-0.571934\pi$
$373$	−8.65496	−0.448137	−0.224068	+	0.974573i	$0.571934\pi$
$374$	−22.7922	−1.17856
$375$	−5.60738	−0.289564
$376$	−68.0441	−3.50911
$377$	−0.775831	−0.0399573
$378$	19.8468	1.02081
$379$	27.2552	1.40000	0.700002	−	0.714141i	$-0.253182\pi$
$379$	27.2552	1.40000	0.700002	+	0.714141i	$0.253182\pi$
$380$	−22.3390	−1.14597
$381$	6.96063	0.356604
$382$	−2.81428	−0.143991
$383$	20.2303	1.03372	0.516860	−	0.856070i	$-0.327101\pi$
$383$	20.2303	1.03372	0.516860	+	0.856070i	$0.327101\pi$
$384$	20.7380	1.05828
$385$	−20.9656	−1.06850
$386$	10.9891	0.559331
$387$	−11.0171	−0.560031
$388$	−62.2365	−3.15958
$389$	7.56961	0.383795	0.191897	−	0.981415i	$-0.438536\pi$
$389$	7.56961	0.383795	0.191897	+	0.981415i	$0.438536\pi$
$390$	−8.47617	−0.429208
$391$	9.36630	0.473674
$392$	−0.731792	−0.0369611
$393$	2.88915	0.145738
$394$	−40.7102	−2.05095
$395$	13.7524	0.691958
$396$	75.3943	3.78870
$397$	8.45853	0.424522	0.212261	−	0.977213i	$-0.431917\pi$
$397$	8.45853	0.424522	0.212261	+	0.977213i	$0.431917\pi$
$398$	19.9421	0.999607
$399$	3.13257	0.156825
$400$	−37.6225	−1.88112
$401$	−0.861583	−0.0430254	−0.0215127	−	0.999769i	$-0.506848\pi$
$401$	−0.861583	−0.0430254	−0.0215127	+	0.999769i	$0.506848\pi$
$402$	−12.5740	−0.627133
$403$	23.7547	1.18330
$404$	−34.5835	−1.72059
$405$	−11.3630	−0.564634
$406$	−1.38948	−0.0689586
$407$	−33.1208	−1.64174
$408$	−7.74890	−0.383628
$409$	21.7678	1.07635	0.538173	−	0.842834i	$-0.319115\pi$
$409$	21.7678	1.07635	0.538173	+	0.842834i	$0.319115\pi$
$410$	30.2628	1.49457
$411$	2.66469	0.131439
$412$	27.4131	1.35055
$413$	−28.5234	−1.40355
$414$	−42.1665	−2.07237
$415$	14.6241	0.717867
$416$	−95.6029	−4.68732
$417$	−6.61845	−0.324107
$418$	33.6797	1.64733
$419$	−2.74815	−0.134256	−0.0671280	−	0.997744i	$-0.521384\pi$
$419$	−2.74815	−0.134256	−0.0671280	+	0.997744i	$0.521384\pi$
$420$	−11.1542	−0.544267
$421$	7.34156	0.357806	0.178903	−	0.983867i	$-0.442745\pi$
$421$	7.34156	0.357806	0.178903	+	0.983867i	$0.442745\pi$
$422$	11.0241	0.536647
$423$	−19.4481	−0.945599
$424$	−4.14727	−0.201409
$425$	4.08332	0.198070
$426$	14.4261	0.698945
$427$	−30.8298	−1.49196
$428$	97.3031	4.70332
$429$	9.38979	0.453343
$430$	17.5240	0.845080
$431$	−7.41589	−0.357211	−0.178605	−	0.983921i	$-0.557159\pi$
$431$	−7.41589	−0.357211	−0.178605	+	0.983921i	$0.557159\pi$
$432$	42.4356	2.04169
$433$	−34.7396	−1.66948	−0.834739	−	0.550645i	$-0.814382\pi$
$433$	−34.7396	−1.66948	−0.834739	+	0.550645i	$0.814382\pi$
$434$	42.5435	2.04215
$435$	−0.143972	−0.00690292
$436$	−69.6562	−3.33593
$437$	−13.8404	−0.662077
$438$	−4.32426	−0.206621
$439$	3.29480	0.157252	0.0786262	−	0.996904i	$-0.474947\pi$
$439$	3.29480	0.157252	0.0786262	+	0.996904i	$0.474947\pi$
$440$	−76.6348	−3.65342
$441$	−0.209158	−0.00995989
$442$	18.9837	0.902962
$443$	5.38213	0.255713	0.127856	−	0.991793i	$-0.459190\pi$
$443$	5.38213	0.255713	0.127856	+	0.991793i	$0.459190\pi$
$444$	−17.6210	−0.836256
$445$	18.6856	0.885782
$446$	−0.443560	−0.0210032
$447$	−4.50927	−0.213281
$448$	−88.1359	−4.16403
$449$	−11.3022	−0.533386	−0.266693	−	0.963782i	$-0.585931\pi$
$449$	−11.3022	−0.533386	−0.266693	+	0.963782i	$0.585931\pi$
$450$	−18.3828	−0.866575
$451$	−33.5247	−1.57862
$452$	19.3029	0.907932
$453$	−6.52096	−0.306381
$454$	18.6623	0.875864
$455$	17.4623	0.818644
$456$	11.4504	0.536215
$457$	1.49305	0.0698421	0.0349211	−	0.999390i	$-0.488882\pi$
$457$	1.49305	0.0698421	0.0349211	+	0.999390i	$0.488882\pi$
$458$	−32.7455	−1.53010
$459$	−4.60571	−0.214976
$460$	49.2815	2.29776
$461$	5.75443	0.268010	0.134005	−	0.990981i	$-0.457216\pi$
$461$	5.75443	0.268010	0.134005	+	0.990981i	$0.457216\pi$
$462$	16.8167	0.782382
$463$	−21.1701	−0.983860	−0.491930	−	0.870635i	$-0.663708\pi$
$463$	−21.1701	−0.983860	−0.491930	+	0.870635i	$0.663708\pi$
$464$	−2.97093	−0.137922
$465$	4.40818	0.204425
$466$	36.5321	1.69232
$467$	−16.4039	−0.759081	−0.379540	−	0.925175i	$-0.623918\pi$
$467$	−16.4039	−0.759081	−0.379540	+	0.925175i	$0.623918\pi$
$468$	−62.7961	−2.90275
$469$	25.9044	1.19615
$470$	30.9344	1.42690
$471$	2.25415	0.103866
$472$	−104.261	−4.79900
$473$	−19.4128	−0.892601
$474$	−11.0309	−0.506667
$475$	−6.03384	−0.276852
$476$	24.9814	1.14502
$477$	−1.18536	−0.0542737
$478$	−20.9302	−0.957323
$479$	9.35845	0.427598	0.213799	−	0.976878i	$-0.431416\pi$
$479$	9.35845	0.427598	0.213799	+	0.976878i	$0.431416\pi$
$480$	−17.7411	−0.809768
$481$	27.5864	1.25783
$482$	58.4650	2.66301
$483$	−6.91069	−0.314447
$484$	71.9017	3.26826
$485$	18.0809	0.821011
$486$	31.4985	1.42880
$487$	22.6800	1.02773	0.513864	−	0.857872i	$-0.328214\pi$
$487$	22.6800	1.02773	0.513864	+	0.857872i	$0.328214\pi$
$488$	−112.691	−5.10129
$489$	2.61428	0.118222
$490$	0.332689	0.0150294
$491$	16.9729	0.765977	0.382989	−	0.923753i	$-0.374895\pi$
$491$	16.9729	0.765977	0.382989	+	0.923753i	$0.374895\pi$
$492$	−17.8359	−0.804105
$493$	0.322447	0.0145223
$494$	−28.0519	−1.26211
$495$	−21.9035	−0.984487
$496$	90.9651	4.08445
$497$	−29.7200	−1.33312
$498$	−11.7301	−0.525638
$499$	28.3413	1.26873	0.634366	−	0.773033i	$-0.281261\pi$
$499$	28.3413	1.26873	0.634366	+	0.773033i	$0.281261\pi$
$500$	66.0780	2.95510
$501$	−4.87861	−0.217960
$502$	6.30230	0.281286
$503$	−3.42434	−0.152684	−0.0763418	−	0.997082i	$-0.524324\pi$
$503$	−3.42434	−0.152684	−0.0763418	+	0.997082i	$0.524324\pi$
$504$	−71.8687	−3.20129
$505$	10.0471	0.447092
$506$	−74.2998	−3.30303
$507$	−1.70842	−0.0758735
$508$	−82.0248	−3.63926
$509$	−14.1239	−0.626031	−0.313015	−	0.949748i	$-0.601339\pi$
$509$	−14.1239	−0.626031	−0.313015	+	0.949748i	$0.601339\pi$
$510$	3.52282	0.155993
$511$	8.90866	0.394096
$512$	−62.4014	−2.75778
$513$	6.80577	0.300482
$514$	−63.3154	−2.79272
$515$	−7.96402	−0.350937
$516$	−10.3280	−0.454667
$517$	−34.2687	−1.50714
$518$	49.4059	2.17077
$519$	9.95833	0.437122
$520$	63.8293	2.79910
$521$	−22.7648	−0.997344	−0.498672	−	0.866791i	$-0.666179\pi$
$521$	−22.7648	−0.997344	−0.498672	+	0.866791i	$0.666179\pi$
$522$	−1.45164	−0.0635363
$523$	−27.9751	−1.22327	−0.611633	−	0.791142i	$-0.709487\pi$
$523$	−27.9751	−1.22327	−0.611633	+	0.791142i	$0.709487\pi$
$524$	−34.0461	−1.48731
$525$	−3.01277	−0.131488
$526$	41.7599	1.82082
$527$	−9.87280	−0.430066
$528$	35.9568	1.56482
$529$	7.53299	0.327521
$530$	1.88544	0.0818984
$531$	−29.7994	−1.29319
$532$	−36.9146	−1.60045
$533$	27.9228	1.20947
$534$	−14.9879	−0.648590
$535$	−28.2684	−1.22215
$536$	94.6877	4.08989
$537$	−6.44553	−0.278145
$538$	−82.6161	−3.56183
$539$	−0.368548	−0.0158745
$540$	−24.2333	−1.04284
$541$	24.0526	1.03410	0.517052	−	0.855954i	$-0.327030\pi$
$541$	24.0526	1.03410	0.517052	+	0.855954i	$0.327030\pi$
$542$	−4.69060	−0.201478
$543$	8.52362	0.365784
$544$	39.7340	1.70358
$545$	20.2364	0.866833
$546$	−14.0066	−0.599429
$547$	38.0708	1.62779	0.813895	−	0.581011i	$-0.197343\pi$
$547$	38.0708	1.62779	0.813895	+	0.581011i	$0.197343\pi$
$548$	−31.4010	−1.34138
$549$	−32.2090	−1.37464
$550$	−32.3916	−1.38118
$551$	−0.476474	−0.0202985
$552$	−25.2605	−1.07516
$553$	22.7254	0.966384
$554$	−44.2955	−1.88193
$555$	5.11923	0.217299
$556$	77.9926	3.30762
$557$	−30.8288	−1.30626	−0.653129	−	0.757246i	$-0.726544\pi$
$557$	−30.8288	−1.30626	−0.653129	+	0.757246i	$0.726544\pi$
$558$	44.4467	1.88158
$559$	16.1690	0.683874
$560$	66.8692	2.82574
$561$	−3.90254	−0.164765
$562$	60.7056	2.56071
$563$	−31.0356	−1.30799	−0.653997	−	0.756497i	$-0.726909\pi$
$563$	−31.0356	−1.30799	−0.653997	+	0.756497i	$0.726909\pi$
$564$	−18.2317	−0.767694
$565$	−5.60786	−0.235924
$566$	−24.9896	−1.05039
$567$	−18.7771	−0.788565
$568$	−108.635	−4.55821
$569$	−10.8245	−0.453786	−0.226893	−	0.973920i	$-0.572857\pi$
$569$	−10.8245	−0.453786	−0.226893	+	0.973920i	$0.572857\pi$
$570$	−5.20562	−0.218039
$571$	−3.08269	−0.129007	−0.0645033	−	0.997917i	$-0.520546\pi$
$571$	−3.08269	−0.129007	−0.0645033	+	0.997917i	$0.520546\pi$
$572$	−110.650	−4.62652
$573$	−0.481867	−0.0201303
$574$	50.0085	2.08731
$575$	13.3111	0.555112
$576$	−92.0787	−3.83661
$577$	−32.6341	−1.35857	−0.679287	−	0.733873i	$-0.737711\pi$
$577$	−32.6341	−1.35857	−0.679287	+	0.733873i	$0.737711\pi$
$578$	38.7926	1.61356
$579$	1.88158	0.0781958
$580$	1.69658	0.0704467
$581$	24.1659	1.00257
$582$	−14.5029	−0.601162
$583$	−2.08867	−0.0865037
$584$	32.5636	1.34749
$585$	18.2434	0.754273
$586$	−15.2488	−0.629924
$587$	−0.246729	−0.0101836	−0.00509180	−	0.999987i	$-0.501621\pi$
$587$	−0.246729	−0.0101836	−0.00509180	+	0.999987i	$0.501621\pi$
$588$	−0.196076	−0.00808604
$589$	14.5889	0.601123
$590$	47.3994	1.95140
$591$	−6.97049	−0.286728
$592$	105.638	4.34170
$593$	−24.1964	−0.993627	−0.496813	−	0.867857i	$-0.665497\pi$
$593$	−24.1964	−0.993627	−0.496813	+	0.867857i	$0.665497\pi$
$594$	36.5356	1.49907
$595$	−7.25758	−0.297532
$596$	53.1378	2.17661
$597$	3.41453	0.139747
$598$	61.8845	2.53065
$599$	−44.7347	−1.82781	−0.913906	−	0.405927i	$-0.866949\pi$
$599$	−44.7347	−1.82781	−0.913906	+	0.405927i	$0.866949\pi$
$600$	−11.0125	−0.449584
$601$	34.4612	1.40570	0.702850	−	0.711338i	$-0.251910\pi$
$601$	34.4612	1.40570	0.702850	+	0.711338i	$0.251910\pi$
$602$	28.9579	1.18023
$603$	27.0632	1.10210
$604$	76.8438	3.12673
$605$	−20.8888	−0.849251
$606$	−8.05891	−0.327371
$607$	−14.3095	−0.580805	−0.290403	−	0.956905i	$-0.593789\pi$
$607$	−14.3095	−0.580805	−0.290403	+	0.956905i	$0.593789\pi$
$608$	−58.7142	−2.38118
$609$	−0.237910	−0.00964058
$610$	51.2320	2.07432
$611$	28.5425	1.15470
$612$	26.0990	1.05499
$613$	42.8703	1.73151	0.865757	−	0.500465i	$-0.166838\pi$
$613$	42.8703	1.73151	0.865757	+	0.500465i	$0.166838\pi$
$614$	5.77303	0.232981
$615$	5.18167	0.208945
$616$	−126.637	−5.10235
$617$	2.19462	0.0883521	0.0441761	−	0.999024i	$-0.485934\pi$
$617$	2.19462	0.0883521	0.0441761	+	0.999024i	$0.485934\pi$
$618$	6.38802	0.256964
$619$	−10.2708	−0.412818	−0.206409	−	0.978466i	$-0.566178\pi$
$619$	−10.2708	−0.412818	−0.206409	+	0.978466i	$0.566178\pi$
$620$	−51.9465	−2.08622
$621$	−15.0140	−0.602493
$622$	−21.1393	−0.847610
$623$	30.8774	1.23708
$624$	−29.9485	−1.19890
$625$	−7.15211	−0.286084
$626$	43.6029	1.74272
$627$	5.76671	0.230300
$628$	−26.5631	−1.05998
$629$	−11.4653	−0.457152
$630$	32.6732	1.30173
$631$	−34.0423	−1.35520	−0.677601	−	0.735430i	$-0.736980\pi$
$631$	−34.0423	−1.35520	−0.677601	+	0.735430i	$0.736980\pi$
$632$	83.0677	3.30426
$633$	1.88758	0.0750245
$634$	26.2357	1.04195
$635$	23.8298	0.945655
$636$	−1.11122	−0.0440626
$637$	0.306965	0.0121624
$638$	−2.55787	−0.101267
$639$	−31.0495	−1.22830
$640$	70.9966	2.80639
$641$	−25.3312	−1.00052	−0.500262	−	0.865874i	$-0.666763\pi$
$641$	−25.3312	−1.00052	−0.500262	+	0.865874i	$0.666763\pi$
$642$	22.6743	0.894885
$643$	−22.6069	−0.891528	−0.445764	−	0.895151i	$-0.647068\pi$
$643$	−22.6069	−0.891528	−0.445764	+	0.895151i	$0.647068\pi$
$644$	81.4364	3.20904
$645$	3.00049	0.118144
$646$	11.6588	0.458708
$647$	38.5836	1.51688	0.758438	−	0.651745i	$-0.225963\pi$
$647$	38.5836	1.51688	0.758438	+	0.651745i	$0.225963\pi$
$648$	−68.6355	−2.69626
$649$	−52.5084	−2.06113
$650$	26.9791	1.05821
$651$	7.28440	0.285498
$652$	−30.8069	−1.20649
$653$	30.6649	1.20001	0.600006	−	0.799996i	$-0.295165\pi$
$653$	30.6649	1.20001	0.600006	+	0.799996i	$0.295165\pi$
$654$	−16.2318	−0.634715
$655$	9.89103	0.386474
$656$	106.926	4.17478
$657$	9.30719	0.363108
$658$	51.1182	1.99280
$659$	26.1911	1.02026	0.510129	−	0.860098i	$-0.329598\pi$
$659$	26.1911	1.02026	0.510129	+	0.860098i	$0.329598\pi$
$660$	−20.5335	−0.799266
$661$	−28.8131	−1.12070	−0.560349	−	0.828257i	$-0.689333\pi$
$661$	−28.8131	−1.12070	−0.560349	+	0.828257i	$0.689333\pi$
$662$	−75.9958	−2.95366
$663$	3.25043	0.126236
$664$	88.3328	3.42798
$665$	10.7244	0.415874
$666$	51.6161	2.00008
$667$	1.05114	0.0407002
$668$	57.4901	2.22436
$669$	−0.0759474	−0.00293629
$670$	−43.0472	−1.66306
$671$	−56.7541	−2.19097
$672$	−29.3167	−1.13092
$673$	14.7278	0.567716	0.283858	−	0.958866i	$-0.408386\pi$
$673$	14.7278	0.567716	0.283858	+	0.958866i	$0.408386\pi$
$674$	29.1715	1.12364
$675$	−6.54550	−0.251936
$676$	20.1322	0.774315
$677$	−28.3284	−1.08875	−0.544375	−	0.838842i	$-0.683233\pi$
$677$	−28.3284	−1.08875	−0.544375	+	0.838842i	$0.683233\pi$
$678$	4.49812	0.172749
$679$	29.8782	1.14662
$680$	−26.5284	−1.01732
$681$	3.19540	0.122448
$682$	78.3177	2.99894
$683$	−32.4897	−1.24318	−0.621591	−	0.783342i	$-0.713513\pi$
$683$	−32.4897	−1.24318	−0.621591	+	0.783342i	$0.713513\pi$
$684$	−38.5660	−1.47461
$685$	9.12258	0.348556
$686$	−50.5801	−1.93116
$687$	−5.60676	−0.213911
$688$	61.9167	2.36055
$689$	1.73965	0.0662756
$690$	11.4840	0.437188
$691$	−47.5626	−1.80937	−0.904684	−	0.426084i	$-0.859893\pi$
$691$	−47.5626	−1.80937	−0.904684	+	0.426084i	$0.859893\pi$
$692$	−117.350	−4.46098
$693$	−36.1949	−1.37493
$694$	51.5393	1.95641
$695$	−22.6583	−0.859479
$696$	−0.869624	−0.0329630
$697$	−11.6051	−0.439576
$698$	−68.0338	−2.57512
$699$	6.25511	0.236590
$700$	35.5029	1.34188
$701$	−26.3336	−0.994605	−0.497303	−	0.867577i	$-0.665676\pi$
$701$	−26.3336	−0.994605	−0.497303	+	0.867577i	$0.665676\pi$
$702$	−30.4306	−1.14853
$703$	16.9421	0.638983
$704$	−162.248	−6.11496
$705$	5.29666	0.199484
$706$	29.6346	1.11531
$707$	16.6026	0.624406
$708$	−27.9357	−1.04989
$709$	−21.2688	−0.798766	−0.399383	−	0.916784i	$-0.630776\pi$
$709$	−21.2688	−0.798766	−0.399383	+	0.916784i	$0.630776\pi$
$710$	49.3877	1.85349
$711$	23.7421	0.890397
$712$	112.865	4.22981
$713$	−32.1841	−1.20530
$714$	5.82137	0.217859
$715$	32.1460	1.20219
$716$	75.9549	2.83857
$717$	−3.58371	−0.133836
$718$	−63.1937	−2.35837
$719$	28.8499	1.07592	0.537960	−	0.842970i	$-0.319195\pi$
$719$	28.8499	1.07592	0.537960	+	0.842970i	$0.319195\pi$
$720$	69.8606	2.60355
$721$	−13.1603	−0.490116
$722$	34.9466	1.30058
$723$	10.0105	0.372295
$724$	−100.443	−3.73295
$725$	0.458252	0.0170191
$726$	16.7551	0.621840
$727$	0.900002	0.0333792	0.0166896	−	0.999861i	$-0.494687\pi$
$727$	0.900002	0.0333792	0.0166896	+	0.999861i	$0.494687\pi$
$728$	105.476	3.90921
$729$	−15.7845	−0.584610
$730$	−14.8041	−0.547926
$731$	−6.72006	−0.248551
$732$	−30.1945	−1.11602
$733$	5.03562	0.185995	0.0929974	−	0.995666i	$-0.470355\pi$
$733$	5.03562	0.185995	0.0929974	+	0.995666i	$0.470355\pi$
$734$	101.837	3.75889
$735$	0.0569638	0.00210114
$736$	129.528	4.77446
$737$	47.6870	1.75657
$738$	52.2456	1.92319
$739$	17.6396	0.648884	0.324442	−	0.945906i	$-0.394823\pi$
$739$	17.6396	0.648884	0.324442	+	0.945906i	$0.394823\pi$
$740$	−60.3257	−2.21762
$741$	−4.80311	−0.176446
$742$	3.11564	0.114379
$743$	−41.9569	−1.53925	−0.769625	−	0.638497i	$-0.779557\pi$
$743$	−41.9569	−1.53925	−0.769625	+	0.638497i	$0.779557\pi$
$744$	26.6265	0.976174
$745$	−15.4375	−0.565587
$746$	23.7668	0.870163
$747$	25.2469	0.923737
$748$	45.9880	1.68149
$749$	−46.7127	−1.70685
$750$	15.3980	0.562256
$751$	−38.5387	−1.40630	−0.703148	−	0.711043i	$-0.748223\pi$
$751$	−38.5387	−1.40630	−0.703148	+	0.711043i	$0.748223\pi$
$752$	109.299	3.98573
$753$	1.07910	0.0393244
$754$	2.13045	0.0775866
$755$	−22.3246	−0.812474
$756$	−40.0449	−1.45642
$757$	−41.5869	−1.51150	−0.755750	−	0.654860i	$-0.772728\pi$
$757$	−41.5869	−1.51150	−0.755750	+	0.654860i	$0.772728\pi$
$758$	−74.8435	−2.71844
$759$	−12.7218	−0.461772
$760$	39.2006	1.42195
$761$	−45.5773	−1.65218	−0.826088	−	0.563541i	$-0.809439\pi$
$761$	−45.5773	−1.65218	−0.826088	+	0.563541i	$0.809439\pi$
$762$	−19.1141	−0.692430
$763$	33.4402	1.21061
$764$	5.67838	0.205437
$765$	−7.58225	−0.274137
$766$	−55.5530	−2.00721
$767$	43.7344	1.57916
$768$	−25.7885	−0.930562
$769$	3.56591	0.128590	0.0642950	−	0.997931i	$-0.479520\pi$
$769$	3.56591	0.128590	0.0642950	+	0.997931i	$0.479520\pi$
$770$	57.5720	2.07475
$771$	−10.8410	−0.390429
$772$	−22.1728	−0.798015
$773$	−36.8221	−1.32440	−0.662200	−	0.749328i	$-0.730377\pi$
$773$	−36.8221	−1.32440	−0.662200	+	0.749328i	$0.730377\pi$
$774$	30.2533	1.08743
$775$	−14.0309	−0.504006
$776$	109.213	3.92051
$777$	8.45939	0.303479
$778$	−20.7864	−0.745228
$779$	17.1487	0.614417
$780$	17.1024	0.612365
$781$	−54.7110	−1.95772
$782$	−25.7201	−0.919750
$783$	−0.516878	−0.0184717
$784$	1.17548	0.0419813
$785$	7.71709	0.275435
$786$	−7.93369	−0.282985
$787$	−29.7462	−1.06034	−0.530168	−	0.847892i	$-0.677871\pi$
$787$	−29.7462	−1.06034	−0.530168	+	0.847892i	$0.677871\pi$
$788$	82.1411	2.92615
$789$	7.15023	0.254555
$790$	−37.7645	−1.34360
$791$	−9.26683	−0.329491
$792$	−132.302	−4.70115
$793$	47.2706	1.67863
$794$	−23.2274	−0.824309
$795$	0.322830	0.0114496
$796$	−40.2373	−1.42617
$797$	18.0581	0.639652	0.319826	−	0.947476i	$-0.396376\pi$
$797$	18.0581	0.639652	0.319826	+	0.947476i	$0.396376\pi$
$798$	−8.60214	−0.304512
$799$	−11.8627	−0.419671
$800$	56.4688	1.99647
$801$	32.2587	1.13981
$802$	2.36593	0.0835440
$803$	16.3998	0.578737
$804$	25.3706	0.894752
$805$	−23.6588	−0.833864
$806$	−65.2310	−2.29766
$807$	−14.1457	−0.497953
$808$	60.6872	2.13497
$809$	−36.1766	−1.27190	−0.635951	−	0.771729i	$-0.719392\pi$
$809$	−36.1766	−1.27190	−0.635951	+	0.771729i	$0.719392\pi$
$810$	31.2033	1.09637
$811$	−3.75195	−0.131749	−0.0658744	−	0.997828i	$-0.520984\pi$
$811$	−3.75195	−0.131749	−0.0658744	+	0.997828i	$0.520984\pi$
$812$	2.80355	0.0983855
$813$	−0.803135	−0.0281672
$814$	90.9506	3.18782
$815$	8.95000	0.313505
$816$	12.4471	0.435734
$817$	9.93012	0.347411
$818$	−59.7749	−2.08998
$819$	30.1468	1.05341
$820$	−61.0614	−2.13236
$821$	43.5031	1.51827	0.759134	−	0.650935i	$-0.225623\pi$
$821$	43.5031	1.51827	0.759134	+	0.650935i	$0.225623\pi$
$822$	−7.31730	−0.255220
$823$	51.9001	1.80913	0.904563	−	0.426341i	$-0.140198\pi$
$823$	51.9001	1.80913	0.904563	+	0.426341i	$0.140198\pi$
$824$	−48.1046	−1.67580
$825$	−5.54617	−0.193093
$826$	78.3262	2.72532
$827$	−34.7769	−1.20931	−0.604655	−	0.796487i	$-0.706689\pi$
$827$	−34.7769	−1.20931	−0.604655	+	0.796487i	$0.706689\pi$
$828$	85.0795	2.95672
$829$	−50.2100	−1.74387	−0.871933	−	0.489625i	$-0.837134\pi$
$829$	−50.2100	−1.74387	−0.871933	+	0.489625i	$0.837134\pi$
$830$	−40.1581	−1.39391
$831$	−7.58437	−0.263099
$832$	135.137	4.68503
$833$	−0.127579	−0.00442036
$834$	18.1745	0.629330
$835$	−16.7020	−0.577996
$836$	−67.9556	−2.35029
$837$	15.8260	0.547025
$838$	7.54650	0.260689
$839$	46.5277	1.60632	0.803158	−	0.595766i	$-0.203152\pi$
$839$	46.5277	1.60632	0.803158	+	0.595766i	$0.203152\pi$
$840$	19.5733	0.675345
$841$	−28.9638	−0.998752
$842$	−20.1601	−0.694764
$843$	10.3942	0.357994
$844$	−22.2435	−0.765651
$845$	−5.84878	−0.201204
$846$	53.4050	1.83610
$847$	−34.5182	−1.18606
$848$	6.66176	0.228766
$849$	−4.27878	−0.146847
$850$	−11.2129	−0.384599
$851$	−37.3755	−1.28122
$852$	−29.1075	−0.997207
$853$	29.3807	1.00597	0.502987	−	0.864294i	$-0.332234\pi$
$853$	29.3807	1.00597	0.502987	+	0.864294i	$0.332234\pi$
$854$	84.6595	2.89699
$855$	11.2042	0.383174
$856$	−170.748	−5.83604
$857$	13.9617	0.476924	0.238462	−	0.971152i	$-0.423357\pi$
$857$	13.9617	0.476924	0.238462	+	0.971152i	$0.423357\pi$
$858$	−25.7846	−0.880273
$859$	−3.16770	−0.108081	−0.0540403	−	0.998539i	$-0.517210\pi$
$859$	−3.16770	−0.108081	−0.0540403	+	0.998539i	$0.517210\pi$
$860$	−35.3582	−1.20570
$861$	8.56257	0.291812
$862$	20.3643	0.693609
$863$	−9.71500	−0.330703	−0.165351	−	0.986235i	$-0.552876\pi$
$863$	−9.71500	−0.330703	−0.165351	+	0.986235i	$0.552876\pi$
$864$	−63.6930	−2.16688
$865$	34.0924	1.15918
$866$	95.3959	3.24169
$867$	6.64216	0.225579
$868$	−85.8402	−2.91361
$869$	41.8349	1.41915
$870$	0.395351	0.0134036
$871$	−39.7187	−1.34582
$872$	122.233	4.13933
$873$	31.2148	1.05646
$874$	38.0062	1.28558
$875$	−31.7224	−1.07241
$876$	8.72508	0.294793
$877$	47.7665	1.61296	0.806480	−	0.591261i	$-0.201370\pi$
$877$	47.7665	1.61296	0.806480	+	0.591261i	$0.201370\pi$
$878$	−9.04763	−0.305343
$879$	−2.61094	−0.0880649
$880$	123.099	4.14965
$881$	29.7879	1.00358	0.501790	−	0.864989i	$-0.332675\pi$
$881$	29.7879	1.00358	0.501790	+	0.864989i	$0.332675\pi$
$882$	0.574353	0.0193395
$883$	33.4635	1.12613	0.563067	−	0.826411i	$-0.309621\pi$
$883$	33.4635	1.12613	0.563067	+	0.826411i	$0.309621\pi$
$884$	−38.3035	−1.28829
$885$	8.11584	0.272811
$886$	−14.7795	−0.496526
$887$	26.8872	0.902783	0.451391	−	0.892326i	$-0.350928\pi$
$887$	26.8872	0.902783	0.451391	+	0.892326i	$0.350928\pi$
$888$	30.9214	1.03765
$889$	39.3780	1.32070
$890$	−51.3112	−1.71996
$891$	−34.5665	−1.15802
$892$	0.894973	0.0299659
$893$	17.5293	0.586595
$894$	12.3826	0.414136
$895$	−22.0663	−0.737596
$896$	117.320	3.91938
$897$	10.5960	0.353790
$898$	31.0363	1.03569
$899$	−1.10798	−0.0369532
$900$	37.0911	1.23637
$901$	−0.723027	−0.0240875
$902$	92.0599	3.06526
$903$	4.95823	0.165000
$904$	−33.8728	−1.12659
$905$	29.1807	0.969999
$906$	17.9067	0.594912
$907$	−31.3165	−1.03985	−0.519923	−	0.854213i	$-0.674040\pi$
$907$	−31.3165	−1.03985	−0.519923	+	0.854213i	$0.674040\pi$
$908$	−37.6550	−1.24962
$909$	17.3454	0.575309
$910$	−47.9519	−1.58959
$911$	−5.68168	−0.188243	−0.0941213	−	0.995561i	$-0.530004\pi$
$911$	−5.68168	−0.188243	−0.0941213	+	0.995561i	$0.530004\pi$
$912$	−18.3928	−0.609046
$913$	44.4866	1.47229
$914$	−4.09997	−0.135615
$915$	8.77206	0.289995
$916$	66.0707	2.18304
$917$	16.3447	0.539748
$918$	12.6474	0.417427
$919$	−12.5936	−0.415426	−0.207713	−	0.978190i	$-0.566602\pi$
$919$	−12.5936	−0.415426	−0.207713	+	0.978190i	$0.566602\pi$
$920$	−86.4794	−2.85114
$921$	0.988473	0.0325713
$922$	−15.8018	−0.520406
$923$	45.5690	1.49992
$924$	−33.9311	−1.11625
$925$	−16.2942	−0.535749
$926$	58.1338	1.91040
$927$	−13.7491	−0.451578
$928$	4.45917	0.146379
$929$	7.31497	0.239996	0.119998	−	0.992774i	$-0.461711\pi$
$929$	7.31497	0.239996	0.119998	+	0.992774i	$0.461711\pi$
$930$	−12.1050	−0.396938
$931$	0.188521	0.00617854
$932$	−73.7110	−2.41448
$933$	−3.61953	−0.118498
$934$	45.0455	1.47393
$935$	−13.3604	−0.436931
$936$	110.195	3.60183
$937$	58.1266	1.89891	0.949456	−	0.313899i	$-0.101635\pi$
$937$	58.1266	1.89891	0.949456	+	0.313899i	$0.101635\pi$
$938$	−71.1343	−2.32262
$939$	7.46580	0.243637
$940$	−62.4165	−2.03580
$941$	−18.5988	−0.606304	−0.303152	−	0.952942i	$-0.598039\pi$
$941$	−18.5988	−0.606304	−0.303152	+	0.952942i	$0.598039\pi$
$942$	−6.18995	−0.201679
$943$	−37.8314	−1.23196
$944$	167.474	5.45083
$945$	11.6338	0.378447
$946$	53.3081	1.73320
$947$	56.7319	1.84354	0.921770	−	0.387738i	$-0.126743\pi$
$947$	56.7319	1.84354	0.921770	+	0.387738i	$0.126743\pi$
$948$	22.2571	0.722878
$949$	−13.6595	−0.443405
$950$	16.5691	0.537573
$951$	4.49214	0.145668
$952$	−43.8375	−1.42078
$953$	19.3861	0.627979	0.313989	−	0.949427i	$-0.398334\pi$
$953$	19.3861	0.627979	0.313989	+	0.949427i	$0.398334\pi$
$954$	3.25502	0.105385
$955$	−1.64968	−0.0533823
$956$	42.2309	1.36584
$957$	−0.437964	−0.0141574
$958$	−25.6986	−0.830283
$959$	15.0748	0.486791
$960$	25.0775	0.809373
$961$	2.92453	0.0943396
$962$	−75.7530	−2.44237
$963$	−48.8024	−1.57264
$964$	−117.965	−3.79940
$965$	6.44161	0.207363
$966$	18.9770	0.610574
$967$	41.0883	1.32131	0.660655	−	0.750689i	$-0.270279\pi$
$967$	41.0883	1.32131	0.660655	+	0.750689i	$0.270279\pi$
$968$	−126.173	−4.05537
$969$	1.99624	0.0641285
$970$	−49.6506	−1.59419
$971$	−36.8027	−1.18105	−0.590527	−	0.807018i	$-0.701080\pi$
$971$	−36.8027	−1.18105	−0.590527	+	0.807018i	$0.701080\pi$
$972$	−63.5546	−2.03852
$973$	−37.4423	−1.20034
$974$	−62.2798	−1.99557
$975$	4.61942	0.147940
$976$	181.016	5.79418
$977$	−5.25283	−0.168053	−0.0840264	−	0.996464i	$-0.526778\pi$
$977$	−5.25283	−0.168053	−0.0840264	+	0.996464i	$0.526778\pi$
$978$	−7.17888	−0.229555
$979$	56.8418	1.81667
$980$	−0.671268	−0.0214429
$981$	34.9361	1.11542
$982$	−46.6081	−1.48733
$983$	−6.69589	−0.213566	−0.106783	−	0.994282i	$-0.534055\pi$
$983$	−6.69589	−0.213566	−0.106783	+	0.994282i	$0.534055\pi$
$984$	31.2985	0.997761
$985$	−23.8635	−0.760355
$986$	−0.885449	−0.0281984
$987$	8.75259	0.278598
$988$	56.6004	1.80070
$989$	−21.9066	−0.696589
$990$	60.1475	1.91161
$991$	34.1011	1.08326	0.541628	−	0.840618i	$-0.317808\pi$
$991$	34.1011	1.08326	0.541628	+	0.840618i	$0.317808\pi$
$992$	−136.532	−4.33491
$993$	−13.0122	−0.412929
$994$	81.6119	2.58857
$995$	11.6897	0.370588
$996$	23.6679	0.749945
$997$	−6.02981	−0.190966	−0.0954830	−	0.995431i	$-0.530440\pi$
$997$	−6.02981	−0.190966	−0.0954830	+	0.995431i	$0.530440\pi$
$998$	−77.8261	−2.46354
$999$	18.3787	0.581477

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4003.2.a.b.1.3	✓	152

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4003.2.a.b.1.3	✓	152	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:	\( N \)	\(=\)	\( 4003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4003.a (trivial)

\(f(q)\)	\(=\)	\(q-2.74603 q^{2} -0.470182 q^{3} +5.54067 q^{4} -1.60967 q^{5} +1.29113 q^{6} -2.65994 q^{7} -9.72280 q^{8} -2.77893 q^{9} +O(q^{10})\)	\(q-2.74603 q^{2} -0.470182 q^{3} +5.54067 q^{4} -1.60967 q^{5} +1.29113 q^{6} -2.65994 q^{7} -9.72280 q^{8} -2.77893 q^{9} +4.42020 q^{10} -4.89664 q^{11} -2.60512 q^{12} +4.07842 q^{13} +7.30426 q^{14} +0.756837 q^{15} +15.6177 q^{16} -1.69505 q^{17} +7.63102 q^{18} +2.50475 q^{19} -8.91866 q^{20} +1.25065 q^{21} +13.4463 q^{22} -5.52567 q^{23} +4.57148 q^{24} -2.40896 q^{25} -11.1995 q^{26} +2.71715 q^{27} -14.7378 q^{28} -0.190228 q^{29} -2.07830 q^{30} +5.82448 q^{31} -23.4411 q^{32} +2.30231 q^{33} +4.65467 q^{34} +4.28162 q^{35} -15.3971 q^{36} +6.76398 q^{37} -6.87812 q^{38} -1.91760 q^{39} +15.6505 q^{40} +6.84648 q^{41} -3.43433 q^{42} +3.96451 q^{43} -27.1307 q^{44} +4.47316 q^{45} +15.1736 q^{46} +6.99841 q^{47} -7.34317 q^{48} +0.0752656 q^{49} +6.61507 q^{50} +0.796983 q^{51} +22.5972 q^{52} +0.426551 q^{53} -7.46136 q^{54} +7.88198 q^{55} +25.8620 q^{56} -1.17769 q^{57} +0.522372 q^{58} +10.7234 q^{59} +4.19339 q^{60} +11.5904 q^{61} -15.9942 q^{62} +7.39178 q^{63} +33.1346 q^{64} -6.56492 q^{65} -6.32221 q^{66} -9.73873 q^{67} -9.39174 q^{68} +2.59807 q^{69} -11.7575 q^{70} +11.1732 q^{71} +27.0190 q^{72} -3.34920 q^{73} -18.5741 q^{74} +1.13265 q^{75} +13.8780 q^{76} +13.0248 q^{77} +5.26578 q^{78} -8.54360 q^{79} -25.1394 q^{80} +7.05924 q^{81} -18.8006 q^{82} -9.08513 q^{83} +6.92946 q^{84} +2.72848 q^{85} -10.8867 q^{86} +0.0894418 q^{87} +47.6090 q^{88} -11.6083 q^{89} -12.2834 q^{90} -10.8483 q^{91} -30.6159 q^{92} -2.73856 q^{93} -19.2178 q^{94} -4.03182 q^{95} +11.0216 q^{96} -11.2327 q^{97} -0.206681 q^{98} +13.6074 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9}+O(q^{10}) \) 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9	\( 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9} - 15 q^{10} - 40 q^{11} - 53 q^{12} - 59 q^{13} - 36 q^{14} - 40 q^{15} + 118 q^{16} - 93 q^{17} - 59 q^{18} - 16 q^{19} - 108 q^{20} - 62 q^{21} - 37 q^{22} - 107 q^{23} - 31 q^{24} + 101 q^{25} - 64 q^{26} - 63 q^{27} - 53 q^{28} - 124 q^{29} - 68 q^{30} - 15 q^{31} - 129 q^{32} - 49 q^{33} - 76 q^{35} + 45 q^{36} - 98 q^{37} - 125 q^{38} - 47 q^{39} - 7 q^{40} - 56 q^{41} - 84 q^{42} - 62 q^{43} - 114 q^{44} - 142 q^{45} - 3 q^{46} - 111 q^{47} - 92 q^{48} + 117 q^{49} - 64 q^{50} - 21 q^{51} - 85 q^{52} - 347 q^{53} + 3 q^{54} - 16 q^{55} - 73 q^{56} - 115 q^{57} - 29 q^{58} - 50 q^{59} - 54 q^{60} - 62 q^{61} - 55 q^{62} - 70 q^{63} + 64 q^{64} - 147 q^{65} + 34 q^{66} - 86 q^{67} - 174 q^{68} - 104 q^{69} - 7 q^{70} - 86 q^{71} - 139 q^{72} - 27 q^{73} - 52 q^{74} - 49 q^{75} - 11 q^{76} - 346 q^{77} - 59 q^{78} - 17 q^{79} - 149 q^{80} - 8 q^{81} - 31 q^{82} - 106 q^{83} - 51 q^{84} - 69 q^{85} - 85 q^{86} - 32 q^{87} - 113 q^{88} - 59 q^{89} + 10 q^{90} - 9 q^{91} - 314 q^{92} - 230 q^{93} + 7 q^{94} - 74 q^{95} - 54 q^{96} - 60 q^{97} - 77 q^{98} - 96 q^{99}+O(q^{100}) \) 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9 - 15 * q^10 - 40 * q^11 - 53 * q^12 - 59 * q^13 - 36 * q^14 - 40 * q^15 + 118 * q^16 - 93 * q^17 - 59 * q^18 - 16 * q^19 - 108 * q^20 - 62 * q^21 - 37 * q^22 - 107 * q^23 - 31 * q^24 + 101 * q^25 - 64 * q^26 - 63 * q^27 - 53 * q^28 - 124 * q^29 - 68 * q^30 - 15 * q^31 - 129 * q^32 - 49 * q^33 - 76 * q^35 + 45 * q^36 - 98 * q^37 - 125 * q^38 - 47 * q^39 - 7 * q^40 - 56 * q^41 - 84 * q^42 - 62 * q^43 - 114 * q^44 - 142 * q^45 - 3 * q^46 - 111 * q^47 - 92 * q^48 + 117 * q^49 - 64 * q^50 - 21 * q^51 - 85 * q^52 - 347 * q^53 + 3 * q^54 - 16 * q^55 - 73 * q^56 - 115 * q^57 - 29 * q^58 - 50 * q^59 - 54 * q^60 - 62 * q^61 - 55 * q^62 - 70 * q^63 + 64 * q^64 - 147 * q^65 + 34 * q^66 - 86 * q^67 - 174 * q^68 - 104 * q^69 - 7 * q^70 - 86 * q^71 - 139 * q^72 - 27 * q^73 - 52 * q^74 - 49 * q^75 - 11 * q^76 - 346 * q^77 - 59 * q^78 - 17 * q^79 - 149 * q^80 - 8 * q^81 - 31 * q^82 - 106 * q^83 - 51 * q^84 - 69 * q^85 - 85 * q^86 - 32 * q^87 - 113 * q^88 - 59 * q^89 + 10 * q^90 - 9 * q^91 - 314 * q^92 - 230 * q^93 + 7 * q^94 - 74 * q^95 - 54 * q^96 - 60 * q^97 - 77 * q^98 - 96 * q^99

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