LMFDB - Embedded newform 4009.2.a.c.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$71$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.69413 q^{2} -0.649307 q^{3} +0.870061 q^{4} -0.0469641 q^{5} +1.10001 q^{6} +3.97959 q^{7} +1.91426 q^{8} -2.57840 q^{9} +O(q^{10})$	\(q-1.69413 q^{2} -0.649307 q^{3} +0.870061 q^{4} -0.0469641 q^{5} +1.10001 q^{6} +3.97959 q^{7} +1.91426 q^{8} -2.57840 q^{9} +0.0795630 q^{10} -5.59218 q^{11} -0.564937 q^{12} +4.37998 q^{13} -6.74193 q^{14} +0.0304941 q^{15} -4.98312 q^{16} +0.0418001 q^{17} +4.36813 q^{18} +1.00000 q^{19} -0.0408616 q^{20} -2.58398 q^{21} +9.47386 q^{22} +3.81507 q^{23} -1.24294 q^{24} -4.99779 q^{25} -7.42024 q^{26} +3.62210 q^{27} +3.46249 q^{28} +2.20513 q^{29} -0.0516608 q^{30} -1.50758 q^{31} +4.61351 q^{32} +3.63105 q^{33} -0.0708146 q^{34} -0.186898 q^{35} -2.24336 q^{36} +2.23622 q^{37} -1.69413 q^{38} -2.84395 q^{39} -0.0899014 q^{40} +1.18167 q^{41} +4.37758 q^{42} -7.41488 q^{43} -4.86554 q^{44} +0.121092 q^{45} -6.46321 q^{46} -9.98963 q^{47} +3.23557 q^{48} +8.83716 q^{49} +8.46689 q^{50} -0.0271411 q^{51} +3.81085 q^{52} -11.3478 q^{53} -6.13629 q^{54} +0.262632 q^{55} +7.61797 q^{56} -0.649307 q^{57} -3.73577 q^{58} +10.0208 q^{59} +0.0265317 q^{60} -5.26917 q^{61} +2.55403 q^{62} -10.2610 q^{63} +2.15037 q^{64} -0.205702 q^{65} -6.15145 q^{66} -0.395036 q^{67} +0.0363686 q^{68} -2.47715 q^{69} +0.316628 q^{70} +2.82311 q^{71} -4.93572 q^{72} +3.30543 q^{73} -3.78845 q^{74} +3.24510 q^{75} +0.870061 q^{76} -22.2546 q^{77} +4.81802 q^{78} -1.69175 q^{79} +0.234027 q^{80} +5.38335 q^{81} -2.00189 q^{82} -12.4114 q^{83} -2.24822 q^{84} -0.00196310 q^{85} +12.5617 q^{86} -1.43181 q^{87} -10.7049 q^{88} -3.52728 q^{89} -0.205145 q^{90} +17.4305 q^{91} +3.31935 q^{92} +0.978881 q^{93} +16.9237 q^{94} -0.0469641 q^{95} -2.99558 q^{96} +12.5900 q^{97} -14.9713 q^{98} +14.4189 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) $ 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.69413	−1.19793	−0.598964	−	0.800776i	$-0.704421\pi$
$2$	−1.69413	−1.19793	−0.598964	+	0.800776i	$0.704421\pi$
$3$	−0.649307	−0.374878	−0.187439	−	0.982276i	$-0.560019\pi$
$3$	−0.649307	−0.374878	−0.187439	+	0.982276i	$0.560019\pi$
$4$	0.870061	0.435030
$5$	−0.0469641	−0.0210030	−0.0105015	−	0.999945i	$-0.503343\pi$
$5$	−0.0469641	−0.0210030	−0.0105015	+	0.999945i	$0.503343\pi$
$6$	1.10001	0.449076
$7$	3.97959	1.50414	0.752072	−	0.659081i	$-0.229054\pi$
$7$	3.97959	1.50414	0.752072	+	0.659081i	$0.229054\pi$
$8$	1.91426	0.676793
$9$	−2.57840	−0.859467
$10$	0.0795630	0.0251600
$11$	−5.59218	−1.68611	−0.843053	−	0.537830i	$-0.819244\pi$
$11$	−5.59218	−1.68611	−0.843053	+	0.537830i	$0.819244\pi$
$12$	−0.564937	−0.163083
$13$	4.37998	1.21479	0.607394	−	0.794401i	$-0.292215\pi$
$13$	4.37998	1.21479	0.607394	+	0.794401i	$0.292215\pi$
$14$	−6.74193	−1.80186
$15$	0.0304941	0.00787355
$16$	−4.98312	−1.24578
$17$	0.0418001	0.0101380	0.00506901	−	0.999987i	$-0.498386\pi$
$17$	0.0418001	0.0101380	0.00506901	+	0.999987i	$0.498386\pi$
$18$	4.36813	1.02958
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.0408616	−0.00913693
$21$	−2.58398	−0.563870
$22$	9.47386	2.01983
$23$	3.81507	0.795498	0.397749	−	0.917494i	$-0.369792\pi$
$23$	3.81507	0.795498	0.397749	+	0.917494i	$0.369792\pi$
$24$	−1.24294	−0.253715
$25$	−4.99779	−0.999559
$26$	−7.42024	−1.45523
$27$	3.62210	0.697073
$28$	3.46249	0.654349
$29$	2.20513	0.409483	0.204742	−	0.978816i	$-0.434365\pi$
$29$	2.20513	0.409483	0.204742	+	0.978816i	$0.434365\pi$
$30$	−0.0516608	−0.00943194
$31$	−1.50758	−0.270769	−0.135384	−	0.990793i	$-0.543227\pi$
$31$	−1.50758	−0.270769	−0.135384	+	0.990793i	$0.543227\pi$
$32$	4.61351	0.815560
$33$	3.63105	0.632084
$34$	−0.0708146	−0.0121446
$35$	−0.186898	−0.0315915
$36$	−2.24336	−0.373894
$37$	2.23622	0.367633	0.183817	−	0.982961i	$-0.441155\pi$
$37$	2.23622	0.367633	0.183817	+	0.982961i	$0.441155\pi$
$38$	−1.69413	−0.274823
$39$	−2.84395	−0.455397
$40$	−0.0899014	−0.0142147
$41$	1.18167	0.184546	0.0922728	−	0.995734i	$-0.470587\pi$
$41$	1.18167	0.184546	0.0922728	+	0.995734i	$0.470587\pi$
$42$	4.37758	0.675476
$43$	−7.41488	−1.13076	−0.565379	−	0.824831i	$-0.691270\pi$
$43$	−7.41488	−1.13076	−0.565379	+	0.824831i	$0.691270\pi$
$44$	−4.86554	−0.733508
$45$	0.121092	0.0180513
$46$	−6.46321	−0.952948
$47$	−9.98963	−1.45714	−0.728568	−	0.684973i	$-0.759814\pi$
$47$	−9.98963	−1.45714	−0.728568	+	0.684973i	$0.759814\pi$
$48$	3.23557	0.467015
$49$	8.83716	1.26245
$50$	8.46689	1.19740
$51$	−0.0271411	−0.00380052
$52$	3.81085	0.528470
$53$	−11.3478	−1.55875	−0.779373	−	0.626560i	$-0.784462\pi$
$53$	−11.3478	−1.55875	−0.779373	+	0.626560i	$0.784462\pi$
$54$	−6.13629	−0.835043
$55$	0.262632	0.0354132
$56$	7.61797	1.01799
$57$	−0.649307	−0.0860029
$58$	−3.73577	−0.490531
$59$	10.0208	1.30459	0.652296	−	0.757964i	$-0.273806\pi$
$59$	10.0208	1.30459	0.652296	+	0.757964i	$0.273806\pi$
$60$	0.0265317	0.00342523
$61$	−5.26917	−0.674648	−0.337324	−	0.941389i	$-0.609522\pi$
$61$	−5.26917	−0.674648	−0.337324	+	0.941389i	$0.609522\pi$
$62$	2.55403	0.324362
$63$	−10.2610	−1.29276
$64$	2.15037	0.268797
$65$	−0.205702	−0.0255142
$66$	−6.15145	−0.757191
$67$	−0.395036	−0.0482613	−0.0241307	−	0.999709i	$-0.507682\pi$
$67$	−0.395036	−0.0482613	−0.0241307	+	0.999709i	$0.507682\pi$
$68$	0.0363686	0.00441035
$69$	−2.47715	−0.298214
$70$	0.316628	0.0378443
$71$	2.82311	0.335041	0.167521	−	0.985869i	$-0.446424\pi$
$71$	2.82311	0.335041	0.167521	+	0.985869i	$0.446424\pi$
$72$	−4.93572	−0.581681
$73$	3.30543	0.386871	0.193436	−	0.981113i	$-0.438037\pi$
$73$	3.30543	0.386871	0.193436	+	0.981113i	$0.438037\pi$
$74$	−3.78845	−0.440398
$75$	3.24510	0.374712
$76$	0.870061	0.0998028
$77$	−22.2546	−2.53615
$78$	4.81802	0.545533
$79$	−1.69175	−0.190336	−0.0951682	−	0.995461i	$-0.530339\pi$
$79$	−1.69175	−0.190336	−0.0951682	+	0.995461i	$0.530339\pi$
$80$	0.234027	0.0261651
$81$	5.38335	0.598150
$82$	−2.00189	−0.221072
$83$	−12.4114	−1.36232	−0.681162	−	0.732133i	$-0.738525\pi$
$83$	−12.4114	−1.36232	−0.681162	+	0.732133i	$0.738525\pi$
$84$	−2.24822	−0.245301
$85$	−0.00196310	−0.000212928 0
$86$	12.5617	1.35457
$87$	−1.43181	−0.153506
$88$	−10.7049	−1.14114
$89$	−3.52728	−0.373891	−0.186946	−	0.982370i	$-0.559859\pi$
$89$	−3.52728	−0.373891	−0.186946	+	0.982370i	$0.559859\pi$
$90$	−0.205145	−0.0216242
$91$	17.4305	1.82722
$92$	3.31935	0.346066
$93$	0.978881	0.101505
$94$	16.9237	1.74554
$95$	−0.0469641	−0.00481841
$96$	−2.99558	−0.305735
$97$	12.5900	1.27832	0.639158	−	0.769075i	$-0.279283\pi$
$97$	12.5900	1.27832	0.639158	+	0.769075i	$0.279283\pi$
$98$	−14.9713	−1.51232
$99$	14.4189	1.44915
$100$	−4.34839	−0.434839
$101$	−13.7879	−1.37195	−0.685974	−	0.727626i	$-0.740623\pi$
$101$	−13.7879	−1.37195	−0.685974	+	0.727626i	$0.740623\pi$
$102$	0.0459805	0.00455274
$103$	9.21485	0.907966	0.453983	−	0.891010i	$-0.350003\pi$
$103$	9.21485	0.907966	0.453983	+	0.891010i	$0.350003\pi$
$104$	8.38442	0.822160
$105$	0.121354	0.0118430
$106$	19.2247	1.86726
$107$	−2.88604	−0.279004	−0.139502	−	0.990222i	$-0.544550\pi$
$107$	−2.88604	−0.279004	−0.139502	+	0.990222i	$0.544550\pi$
$108$	3.15144	0.303248
$109$	−4.00971	−0.384061	−0.192030	−	0.981389i	$-0.561507\pi$
$109$	−4.00971	−0.384061	−0.192030	+	0.981389i	$0.561507\pi$
$110$	−0.444931	−0.0424225
$111$	−1.45200	−0.137817
$112$	−19.8308	−1.87383
$113$	−9.39425	−0.883737	−0.441869	−	0.897080i	$-0.645684\pi$
$113$	−9.39425	−0.883737	−0.441869	+	0.897080i	$0.645684\pi$
$114$	1.10001	0.103025
$115$	−0.179171	−0.0167078
$116$	1.91860	0.178138
$117$	−11.2933	−1.04407
$118$	−16.9764	−1.56281
$119$	0.166347	0.0152490
$120$	0.0583736	0.00532876
$121$	20.2725	1.84296
$122$	8.92664	0.808180
$123$	−0.767266	−0.0691820
$124$	−1.31168	−0.117793
$125$	0.469537	0.0419967
$126$	17.3834	1.54864
$127$	−1.34897	−0.119702	−0.0598508	−	0.998207i	$-0.519063\pi$
$127$	−1.34897	−0.119702	−0.0598508	+	0.998207i	$0.519063\pi$
$128$	−12.8700	−1.13756
$129$	4.81454	0.423896
$130$	0.348485	0.0305641
$131$	11.3864	0.994831	0.497416	−	0.867512i	$-0.334282\pi$
$131$	11.3864	0.994831	0.497416	+	0.867512i	$0.334282\pi$
$132$	3.15923	0.274976
$133$	3.97959	0.345074
$134$	0.669241	0.0578136
$135$	−0.170108	−0.0146406
$136$	0.0800162	0.00686133
$137$	−2.44545	−0.208929	−0.104465	−	0.994529i	$-0.533313\pi$
$137$	−2.44545	−0.208929	−0.104465	+	0.994529i	$0.533313\pi$
$138$	4.19661	0.357239
$139$	6.69864	0.568171	0.284086	−	0.958799i	$-0.408310\pi$
$139$	6.69864	0.568171	0.284086	+	0.958799i	$0.408310\pi$
$140$	−0.162612	−0.0137433
$141$	6.48634	0.546248
$142$	−4.78270	−0.401355
$143$	−24.4937	−2.04826
$144$	12.8485	1.07071
$145$	−0.103562	−0.00860036
$146$	−5.59981	−0.463443
$147$	−5.73803	−0.473265
$148$	1.94565	0.159932
$149$	5.75888	0.471786	0.235893	−	0.971779i	$-0.424199\pi$
$149$	5.75888	0.471786	0.235893	+	0.971779i	$0.424199\pi$
$150$	−5.49761	−0.448878
$151$	3.03478	0.246967	0.123484	−	0.992347i	$-0.460593\pi$
$151$	3.03478	0.246967	0.123484	+	0.992347i	$0.460593\pi$
$152$	1.91426	0.155267
$153$	−0.107777	−0.00871329
$154$	37.7021	3.03812
$155$	0.0708020	0.00568695
$156$	−2.47441	−0.198112
$157$	19.1663	1.52964	0.764819	−	0.644245i	$-0.222828\pi$
$157$	19.1663	1.52964	0.764819	+	0.644245i	$0.222828\pi$
$158$	2.86603	0.228009
$159$	7.36824	0.584339
$160$	−0.216669	−0.0171292
$161$	15.1824	1.19654
$162$	−9.12006	−0.716540
$163$	10.7188	0.839559	0.419780	−	0.907626i	$-0.362107\pi$
$163$	10.7188	0.839559	0.419780	+	0.907626i	$0.362107\pi$
$164$	1.02812	0.0802829
$165$	−0.170529	−0.0132756
$166$	21.0264	1.63197
$167$	0.0602231	0.00466020	0.00233010	−	0.999997i	$-0.499258\pi$
$167$	0.0602231	0.00466020	0.00233010	+	0.999997i	$0.499258\pi$
$168$	−4.94640	−0.381623
$169$	6.18425	0.475711
$170$	0.00332574	0.000255073 0
$171$	−2.57840	−0.197175
$172$	−6.45140	−0.491915
$173$	6.97111	0.530004	0.265002	−	0.964248i	$-0.414627\pi$
$173$	6.97111	0.530004	0.265002	+	0.964248i	$0.414627\pi$
$174$	2.42567	0.183889
$175$	−19.8892	−1.50348
$176$	27.8665	2.10052
$177$	−6.50656	−0.489063
$178$	5.97566	0.447895
$179$	−15.8642	−1.18575	−0.592875	−	0.805295i	$-0.702007\pi$
$179$	−15.8642	−1.18575	−0.592875	+	0.805295i	$0.702007\pi$
$180$	0.105358	0.00785289
$181$	6.35385	0.472278	0.236139	−	0.971719i	$-0.424118\pi$
$181$	6.35385	0.472278	0.236139	+	0.971719i	$0.424118\pi$
$182$	−29.5295	−2.18887
$183$	3.42131	0.252911
$184$	7.30303	0.538387
$185$	−0.105022	−0.00772139
$186$	−1.65835	−0.121596
$187$	−0.233754	−0.0170938
$188$	−8.69158	−0.633899
$189$	14.4145	1.04850
$190$	0.0795630	0.00577211
$191$	−26.6079	−1.92528	−0.962638	−	0.270791i	$-0.912715\pi$
$191$	−26.6079	−1.92528	−0.962638	+	0.270791i	$0.912715\pi$
$192$	−1.39625	−0.100766
$193$	−5.31613	−0.382664	−0.191332	−	0.981525i	$-0.561281\pi$
$193$	−5.31613	−0.382664	−0.191332	+	0.981525i	$0.561281\pi$
$194$	−21.3290	−1.53133
$195$	0.133564	0.00956469
$196$	7.68886	0.549205
$197$	25.0274	1.78313	0.891565	−	0.452893i	$-0.149608\pi$
$197$	25.0274	1.78313	0.891565	+	0.452893i	$0.149608\pi$
$198$	−24.4274	−1.73598
$199$	−14.6298	−1.03708	−0.518538	−	0.855054i	$-0.673524\pi$
$199$	−14.6298	−1.03708	−0.518538	+	0.855054i	$0.673524\pi$
$200$	−9.56707	−0.676494
$201$	0.256500	0.0180921
$202$	23.3584	1.64349
$203$	8.77553	0.615922
$204$	−0.0236144	−0.00165334
$205$	−0.0554959	−0.00387600
$206$	−15.6111	−1.08768
$207$	−9.83678	−0.683704
$208$	−21.8260	−1.51336
$209$	−5.59218	−0.386819
$210$	−0.205589	−0.0141870
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	−9.87331	−0.678102
$213$	−1.83306	−0.125599
$214$	4.88932	0.334227
$215$	0.348233	0.0237493
$216$	6.93363	0.471774
$217$	−5.99955	−0.407276
$218$	6.79295	0.460077
$219$	−2.14624	−0.145029
$220$	0.228506	0.0154058
$221$	0.183084	0.0123155
$222$	2.45987	0.165095
$223$	−18.2330	−1.22098	−0.610488	−	0.792026i	$-0.709026\pi$
$223$	−18.2330	−1.22098	−0.610488	+	0.792026i	$0.709026\pi$
$224$	18.3599	1.22672
$225$	12.8863	0.859088
$226$	15.9150	1.05865
$227$	−6.13636	−0.407285	−0.203642	−	0.979045i	$-0.565278\pi$
$227$	−6.13636	−0.407285	−0.203642	+	0.979045i	$0.565278\pi$
$228$	−0.564937	−0.0374139
$229$	0.785484	0.0519062	0.0259531	−	0.999663i	$-0.491738\pi$
$229$	0.785484	0.0519062	0.0259531	+	0.999663i	$0.491738\pi$
$230$	0.303539	0.0200147
$231$	14.4501	0.950746
$232$	4.22120	0.277135
$233$	−9.74435	−0.638374	−0.319187	−	0.947692i	$-0.603410\pi$
$233$	−9.74435	−0.638374	−0.319187	+	0.947692i	$0.603410\pi$
$234$	19.1323	1.25072
$235$	0.469153	0.0306042
$236$	8.71867	0.567537
$237$	1.09846	0.0713529
$238$	−0.281813	−0.0182672
$239$	−24.5938	−1.59084	−0.795422	−	0.606056i	$-0.792751\pi$
$239$	−24.5938	−1.59084	−0.795422	+	0.606056i	$0.792751\pi$
$240$	−0.151956	−0.00980870
$241$	−24.7850	−1.59654	−0.798272	−	0.602297i	$-0.794252\pi$
$241$	−24.7850	−1.59654	−0.798272	+	0.602297i	$0.794252\pi$
$242$	−34.3442	−2.20773
$243$	−14.3617	−0.921306
$244$	−4.58450	−0.293493
$245$	−0.415029	−0.0265152
$246$	1.29984	0.0828751
$247$	4.37998	0.278692
$248$	−2.88589	−0.183254
$249$	8.05879	0.510705
$250$	−0.795455	−0.0503090
$251$	−5.49044	−0.346553	−0.173277	−	0.984873i	$-0.555435\pi$
$251$	−5.49044	−0.346553	−0.173277	+	0.984873i	$0.555435\pi$
$252$	−8.92768	−0.562391
$253$	−21.3346	−1.34129
$254$	2.28532	0.143394
$255$	0.00127466	7.98221e−5 0
$256$	17.5027	1.09392
$257$	−24.1694	−1.50764	−0.753822	−	0.657079i	$-0.771792\pi$
$257$	−24.1694	−1.50764	−0.753822	+	0.657079i	$0.771792\pi$
$258$	−8.15643	−0.507797
$259$	8.89926	0.552973
$260$	−0.178973	−0.0110994
$261$	−5.68572	−0.351937
$262$	−19.2899	−1.19174
$263$	−12.7144	−0.784001	−0.392001	−	0.919965i	$-0.628217\pi$
$263$	−12.7144	−0.784001	−0.392001	+	0.919965i	$0.628217\pi$
$264$	6.95076	0.427790
$265$	0.532941	0.0327383
$266$	−6.74193	−0.413374
$267$	2.29029	0.140164
$268$	−0.343705	−0.0209952
$269$	−8.39307	−0.511735	−0.255867	−	0.966712i	$-0.582361\pi$
$269$	−8.39307	−0.511735	−0.255867	+	0.966712i	$0.582361\pi$
$270$	0.288185	0.0175384
$271$	23.8246	1.44724	0.723620	−	0.690199i	$-0.242477\pi$
$271$	23.8246	1.44724	0.723620	+	0.690199i	$0.242477\pi$
$272$	−0.208295	−0.0126297
$273$	−11.3178	−0.684983
$274$	4.14290	0.250282
$275$	27.9486	1.68536
$276$	−2.15528	−0.129732
$277$	0.490616	0.0294783	0.0147391	−	0.999891i	$-0.495308\pi$
$277$	0.490616	0.0294783	0.0147391	+	0.999891i	$0.495308\pi$
$278$	−11.3483	−0.680628
$279$	3.88714	0.232717
$280$	−0.357771	−0.0213809
$281$	−1.91950	−0.114508	−0.0572539	−	0.998360i	$-0.518234\pi$
$281$	−1.91950	−0.114508	−0.0572539	+	0.998360i	$0.518234\pi$
$282$	−10.9887	−0.654366
$283$	3.13947	0.186622	0.0933110	−	0.995637i	$-0.470255\pi$
$283$	3.13947	0.186622	0.0933110	+	0.995637i	$0.470255\pi$
$284$	2.45628	0.145753
$285$	0.0304941	0.00180632
$286$	41.4953	2.45367
$287$	4.70256	0.277583
$288$	−11.8955	−0.700947
$289$	−16.9983	−0.999897
$290$	0.175447	0.0103026
$291$	−8.17475	−0.479212
$292$	2.87592	0.168301
$293$	22.5970	1.32013	0.660067	−	0.751207i	$-0.270528\pi$
$293$	22.5970	1.32013	0.660067	+	0.751207i	$0.270528\pi$
$294$	9.72094	0.566937
$295$	−0.470616	−0.0274003
$296$	4.28071	0.248811
$297$	−20.2554	−1.17534
$298$	−9.75627	−0.565165
$299$	16.7100	0.966361
$300$	2.82344	0.163011
$301$	−29.5082	−1.70082
$302$	−5.14130	−0.295849
$303$	8.95258	0.514313
$304$	−4.98312	−0.285801
$305$	0.247462	0.0141696
$306$	0.182588	0.0104379
$307$	−31.2785	−1.78516	−0.892580	−	0.450888i	$-0.851107\pi$
$307$	−31.2785	−1.78516	−0.892580	+	0.450888i	$0.851107\pi$
$308$	−19.3629	−1.10330
$309$	−5.98327	−0.340376
$310$	−0.119947	−0.00681256
$311$	−8.01458	−0.454465	−0.227232	−	0.973841i	$-0.572968\pi$
$311$	−8.01458	−0.454465	−0.227232	+	0.973841i	$0.572968\pi$
$312$	−5.44407	−0.308209
$313$	−30.1947	−1.70670	−0.853352	−	0.521335i	$-0.825434\pi$
$313$	−30.1947	−1.70670	−0.853352	+	0.521335i	$0.825434\pi$
$314$	−32.4701	−1.83240
$315$	0.481897	0.0271518
$316$	−1.47192	−0.0828021
$317$	−18.6900	−1.04973	−0.524866	−	0.851185i	$-0.675885\pi$
$317$	−18.6900	−1.04973	−0.524866	+	0.851185i	$0.675885\pi$
$318$	−12.4827	−0.699996
$319$	−12.3315	−0.690432
$320$	−0.100990	−0.00564553
$321$	1.87393	0.104593
$322$	−25.7209	−1.43337
$323$	0.0418001	0.00232582
$324$	4.68384	0.260213
$325$	−21.8903	−1.21425
$326$	−18.1590	−1.00573
$327$	2.60353	0.143976
$328$	2.26202	0.124899
$329$	−39.7546	−2.19174
$330$	0.288897	0.0159033
$331$	25.0126	1.37482	0.687409	−	0.726270i	$-0.258748\pi$
$331$	25.0126	1.37482	0.687409	+	0.726270i	$0.258748\pi$
$332$	−10.7986	−0.592653
$333$	−5.76588	−0.315968
$334$	−0.102025	−0.00558258
$335$	0.0185525	0.00101363
$336$	12.8763	0.702458
$337$	−4.69437	−0.255719	−0.127859	−	0.991792i	$-0.540811\pi$
$337$	−4.69437	−0.255719	−0.127859	+	0.991792i	$0.540811\pi$
$338$	−10.4769	−0.569868
$339$	6.09976	0.331293
$340$	−0.00170802	−9.26303e−5 0
$341$	8.43065	0.456545
$342$	4.36813	0.236202
$343$	7.31114	0.394764
$344$	−14.1940	−0.765289
$345$	0.116337	0.00626339
$346$	−11.8099	−0.634906
$347$	−1.44912	−0.0777927	−0.0388963	−	0.999243i	$-0.512384\pi$
$347$	−1.44912	−0.0777927	−0.0388963	+	0.999243i	$0.512384\pi$
$348$	−1.24576	−0.0667798
$349$	14.1974	0.759968	0.379984	−	0.924993i	$-0.375929\pi$
$349$	14.1974	0.759968	0.379984	+	0.924993i	$0.375929\pi$
$350$	33.6948	1.80106
$351$	15.8647	0.846796
$352$	−25.7996	−1.37512
$353$	−3.09965	−0.164978	−0.0824889	−	0.996592i	$-0.526287\pi$
$353$	−3.09965	−0.164978	−0.0824889	+	0.996592i	$0.526287\pi$
$354$	11.0229	0.585862
$355$	−0.132585	−0.00703686
$356$	−3.06895	−0.162654
$357$	−0.108011	−0.00571653
$358$	26.8760	1.42044
$359$	12.0498	0.635963	0.317982	−	0.948097i	$-0.396995\pi$
$359$	12.0498	0.635963	0.317982	+	0.948097i	$0.396995\pi$
$360$	0.231802	0.0122170
$361$	1.00000	0.0526316
$362$	−10.7642	−0.565755
$363$	−13.1631	−0.690883
$364$	15.1656	0.794895
$365$	−0.155236	−0.00812544
$366$	−5.79613	−0.302969
$367$	−2.02994	−0.105962	−0.0529810	−	0.998596i	$-0.516872\pi$
$367$	−2.02994	−0.105962	−0.0529810	+	0.998596i	$0.516872\pi$
$368$	−19.0109	−0.991014
$369$	−3.04681	−0.158611
$370$	0.177921	0.00924966
$371$	−45.1598	−2.34458
$372$	0.851686	0.0441579
$373$	−25.2477	−1.30728	−0.653639	−	0.756807i	$-0.726758\pi$
$373$	−25.2477	−1.30728	−0.653639	+	0.756807i	$0.726758\pi$
$374$	0.396008	0.0204771
$375$	−0.304874	−0.0157436
$376$	−19.1227	−0.986180
$377$	9.65845	0.497435
$378$	−24.4199	−1.25602
$379$	14.8010	0.760278	0.380139	−	0.924929i	$-0.375876\pi$
$379$	14.8010	0.760278	0.380139	+	0.924929i	$0.375876\pi$
$380$	−0.0408616	−0.00209616
$381$	0.875896	0.0448735
$382$	45.0770	2.30634
$383$	−22.9268	−1.17150	−0.585752	−	0.810491i	$-0.699201\pi$
$383$	−22.9268	−1.17150	−0.585752	+	0.810491i	$0.699201\pi$
$384$	8.35660	0.426446
$385$	1.04517	0.0532666
$386$	9.00620	0.458403
$387$	19.1185	0.971850
$388$	10.9540	0.556106
$389$	14.0991	0.714852	0.357426	−	0.933941i	$-0.383654\pi$
$389$	14.0991	0.714852	0.357426	+	0.933941i	$0.383654\pi$
$390$	−0.226274	−0.0114578
$391$	0.159470	0.00806477
$392$	16.9166	0.854417
$393$	−7.39325	−0.372940
$394$	−42.3996	−2.13606
$395$	0.0794513	0.00399763
$396$	12.5453	0.630425
$397$	−26.1372	−1.31179	−0.655894	−	0.754853i	$-0.727708\pi$
$397$	−26.1372	−1.31179	−0.655894	+	0.754853i	$0.727708\pi$
$398$	24.7847	1.24234
$399$	−2.58398	−0.129361
$400$	24.9046	1.24523
$401$	−14.1607	−0.707151	−0.353576	−	0.935406i	$-0.615034\pi$
$401$	−14.1607	−0.707151	−0.353576	+	0.935406i	$0.615034\pi$
$402$	−0.434543	−0.0216730
$403$	−6.60317	−0.328927
$404$	−11.9963	−0.596839
$405$	−0.252824	−0.0125629
$406$	−14.8669	−0.737830
$407$	−12.5054	−0.619869
$408$	−0.0519551	−0.00257216
$409$	−22.3803	−1.10663	−0.553316	−	0.832971i	$-0.686638\pi$
$409$	−22.3803	−1.10663	−0.553316	+	0.832971i	$0.686638\pi$
$410$	0.0940171	0.00464317
$411$	1.58785	0.0783229
$412$	8.01748	0.394993
$413$	39.8786	1.96230
$414$	16.6647	0.819027
$415$	0.582888	0.0286129
$416$	20.2071	0.990733
$417$	−4.34948	−0.212995
$418$	9.47386	0.463382
$419$	−21.3786	−1.04441	−0.522206	−	0.852819i	$-0.674891\pi$
$419$	−21.3786	−1.04441	−0.522206	+	0.852819i	$0.674891\pi$
$420$	0.105585	0.00515204
$421$	−24.4088	−1.18961	−0.594807	−	0.803869i	$-0.702771\pi$
$421$	−24.4088	−1.18961	−0.594807	+	0.803869i	$0.702771\pi$
$422$	−1.69413	−0.0824687
$423$	25.7573	1.25236
$424$	−21.7227	−1.05495
$425$	−0.208908	−0.0101335
$426$	3.10544	0.150459
$427$	−20.9692	−1.01477
$428$	−2.51103	−0.121375
$429$	15.9039	0.767848
$430$	−0.589950	−0.0284499
$431$	13.6747	0.658688	0.329344	−	0.944210i	$-0.393172\pi$
$431$	13.6747	0.658688	0.329344	+	0.944210i	$0.393172\pi$
$432$	−18.0493	−0.868399
$433$	29.6189	1.42339	0.711697	−	0.702487i	$-0.247927\pi$
$433$	29.6189	1.42339	0.711697	+	0.702487i	$0.247927\pi$
$434$	10.1640	0.487887
$435$	0.0672436	0.00322408
$436$	−3.48869	−0.167078
$437$	3.81507	0.182500
$438$	3.63600	0.173735
$439$	0.515975	0.0246261	0.0123131	−	0.999924i	$-0.496081\pi$
$439$	0.515975	0.0246261	0.0123131	+	0.999924i	$0.496081\pi$
$440$	0.502745	0.0239674
$441$	−22.7857	−1.08503
$442$	−0.310167	−0.0147531
$443$	13.8425	0.657677	0.328839	−	0.944386i	$-0.393343\pi$
$443$	13.8425	0.657677	0.328839	+	0.944386i	$0.393343\pi$
$444$	−1.26333	−0.0599548
$445$	0.165656	0.00785283
$446$	30.8891	1.46264
$447$	−3.73928	−0.176862
$448$	8.55761	0.404309
$449$	5.47108	0.258196	0.129098	−	0.991632i	$-0.458792\pi$
$449$	5.47108	0.258196	0.129098	+	0.991632i	$0.458792\pi$
$450$	−21.8310	−1.02912
$451$	−6.60810	−0.311163
$452$	−8.17357	−0.384453
$453$	−1.97051	−0.0925825
$454$	10.3958	0.487897
$455$	−0.818609	−0.0383770
$456$	−1.24294	−0.0582061
$457$	−26.0373	−1.21797	−0.608987	−	0.793180i	$-0.708424\pi$
$457$	−26.0373	−1.21797	−0.608987	+	0.793180i	$0.708424\pi$
$458$	−1.33071	−0.0621799
$459$	0.151404	0.00706693
$460$	−0.155890	−0.00726841
$461$	5.99343	0.279142	0.139571	−	0.990212i	$-0.455428\pi$
$461$	5.99343	0.279142	0.139571	+	0.990212i	$0.455428\pi$
$462$	−24.4803	−1.13892
$463$	0.921382	0.0428202	0.0214101	−	0.999771i	$-0.493184\pi$
$463$	0.921382	0.0428202	0.0214101	+	0.999771i	$0.493184\pi$
$464$	−10.9884	−0.510125
$465$	−0.0459722	−0.00213191
$466$	16.5082	0.764725
$467$	−33.1359	−1.53335	−0.766674	−	0.642037i	$-0.778090\pi$
$467$	−33.1359	−1.53335	−0.766674	+	0.642037i	$0.778090\pi$
$468$	−9.82590	−0.454202
$469$	−1.57208	−0.0725920
$470$	−0.794805	−0.0366616
$471$	−12.4448	−0.573427
$472$	19.1823	0.882938
$473$	41.4654	1.90658
$474$	−1.86094	−0.0854756
$475$	−4.99779	−0.229315
$476$	0.144732	0.00663380
$477$	29.2593	1.33969
$478$	41.6651	1.90572
$479$	9.55220	0.436451	0.218226	−	0.975898i	$-0.429973\pi$
$479$	9.55220	0.436451	0.218226	+	0.975898i	$0.429973\pi$
$480$	0.140685	0.00642135
$481$	9.79463	0.446597
$482$	41.9889	1.91254
$483$	−9.85807	−0.448558
$484$	17.6383	0.801742
$485$	−0.591275	−0.0268484
$486$	24.3306	1.10366
$487$	27.3831	1.24085	0.620423	−	0.784267i	$-0.286961\pi$
$487$	27.3831	1.24085	0.620423	+	0.784267i	$0.286961\pi$
$488$	−10.0866	−0.456597
$489$	−6.95978	−0.314732
$490$	0.703111	0.0317633
$491$	7.24260	0.326854	0.163427	−	0.986555i	$-0.447745\pi$
$491$	7.24260	0.326854	0.163427	+	0.986555i	$0.447745\pi$
$492$	−0.667568	−0.0300963
$493$	0.0921748	0.00415135
$494$	−7.42024	−0.333852
$495$	−0.677169	−0.0304365
$496$	7.51244	0.337318
$497$	11.2348	0.503950
$498$	−13.6526	−0.611788
$499$	−9.02606	−0.404062	−0.202031	−	0.979379i	$-0.564754\pi$
$499$	−9.02606	−0.404062	−0.202031	+	0.979379i	$0.564754\pi$
$500$	0.408526	0.0182698
$501$	−0.0391033	−0.00174701
$502$	9.30149	0.415146
$503$	8.98411	0.400582	0.200291	−	0.979736i	$-0.435811\pi$
$503$	8.98411	0.400582	0.200291	+	0.979736i	$0.435811\pi$
$504$	−19.6422	−0.874932
$505$	0.647536	0.0288150
$506$	36.1435	1.60677
$507$	−4.01548	−0.178334
$508$	−1.17369	−0.0520739
$509$	−27.4140	−1.21510	−0.607551	−	0.794281i	$-0.707848\pi$
$509$	−27.4140	−1.21510	−0.607551	+	0.794281i	$0.707848\pi$
$510$	−0.00215943	−9.56211e−5 0
$511$	13.1543	0.581910
$512$	−3.91169	−0.172874
$513$	3.62210	0.159919
$514$	40.9459	1.80605
$515$	−0.432767	−0.0190700
$516$	4.18894	0.184408
$517$	55.8638	2.45689
$518$	−15.0765	−0.662422
$519$	−4.52640	−0.198687
$520$	−0.393766	−0.0172678
$521$	−23.0462	−1.00967	−0.504836	−	0.863215i	$-0.668447\pi$
$521$	−23.0462	−1.00967	−0.504836	+	0.863215i	$0.668447\pi$
$522$	9.63232	0.421595
$523$	7.81240	0.341612	0.170806	−	0.985305i	$-0.445363\pi$
$523$	7.81240	0.341612	0.170806	+	0.985305i	$0.445363\pi$
$524$	9.90683	0.432782
$525$	12.9142	0.563622
$526$	21.5397	0.939177
$527$	−0.0630169	−0.00274506
$528$	−18.0939	−0.787437
$529$	−8.44522	−0.367184
$530$	−0.902868	−0.0392181
$531$	−25.8375	−1.12125
$532$	3.46249	0.150118
$533$	5.17569	0.224184
$534$	−3.88004	−0.167906
$535$	0.135540	0.00585992
$536$	−0.756201	−0.0326629
$537$	10.3008	0.444511
$538$	14.2189	0.613021
$539$	−49.4190	−2.12863
$540$	−0.148005	−0.00636910
$541$	5.86306	0.252072	0.126036	−	0.992026i	$-0.459774\pi$
$541$	5.86306	0.252072	0.126036	+	0.992026i	$0.459774\pi$
$542$	−40.3618	−1.73369
$543$	−4.12560	−0.177046
$544$	0.192845	0.00826816
$545$	0.188312	0.00806641
$546$	19.1737	0.820560
$547$	−32.4104	−1.38577	−0.692885	−	0.721048i	$-0.743661\pi$
$547$	−32.4104	−1.38577	−0.692885	+	0.721048i	$0.743661\pi$
$548$	−2.12769	−0.0908905
$549$	13.5860	0.579838
$550$	−47.3484	−2.01894
$551$	2.20513	0.0939419
$552$	−4.74191	−0.201829
$553$	−6.73246	−0.286294
$554$	−0.831165	−0.0353128
$555$	0.0681917	0.00289458
$556$	5.82822	0.247172
$557$	−7.69091	−0.325874	−0.162937	−	0.986636i	$-0.552097\pi$
$557$	−7.69091	−0.325874	−0.162937	+	0.986636i	$0.552097\pi$
$558$	−6.58530	−0.278778
$559$	−32.4771	−1.37363
$560$	0.931334	0.0393560
$561$	0.151778	0.00640808
$562$	3.25187	0.137172
$563$	18.5397	0.781354	0.390677	−	0.920528i	$-0.372241\pi$
$563$	18.5397	0.781354	0.390677	+	0.920528i	$0.372241\pi$
$564$	5.64351	0.237635
$565$	0.441192	0.0185611
$566$	−5.31865	−0.223560
$567$	21.4235	0.899703
$568$	5.40416	0.226753
$569$	22.0856	0.925877	0.462938	−	0.886390i	$-0.346795\pi$
$569$	22.0856	0.925877	0.462938	+	0.886390i	$0.346795\pi$
$570$	−0.0516608	−0.00216383
$571$	−33.7721	−1.41332	−0.706659	−	0.707554i	$-0.749798\pi$
$571$	−33.7721	−1.41332	−0.706659	+	0.707554i	$0.749798\pi$
$572$	−21.3110	−0.891057
$573$	17.2767	0.721743
$574$	−7.96672	−0.332525
$575$	−19.0669	−0.795147
$576$	−5.54452	−0.231022
$577$	−0.378368	−0.0157517	−0.00787584	−	0.999969i	$-0.502507\pi$
$577$	−0.378368	−0.0157517	−0.00787584	+	0.999969i	$0.502507\pi$
$578$	28.7972	1.19780
$579$	3.45180	0.143452
$580$	−0.0901053	−0.00374142
$581$	−49.3922	−2.04913
$582$	13.8490	0.574062
$583$	63.4592	2.62821
$584$	6.32744	0.261831
$585$	0.530381	0.0219286
$586$	−38.2822	−1.58142
$587$	−41.4042	−1.70893	−0.854467	−	0.519505i	$-0.826116\pi$
$587$	−41.4042	−1.70893	−0.854467	+	0.519505i	$0.826116\pi$
$588$	−4.99244	−0.205885
$589$	−1.50758	−0.0621187
$590$	0.797282	0.0328236
$591$	−16.2505	−0.668456
$592$	−11.1434	−0.457990
$593$	−19.4339	−0.798054	−0.399027	−	0.916939i	$-0.630652\pi$
$593$	−19.4339	−0.798054	−0.399027	+	0.916939i	$0.630652\pi$
$594$	34.3152	1.40797
$595$	−0.00781235	−0.000320275 0
$596$	5.01058	0.205241
$597$	9.49921	0.388777
$598$	−28.3088	−1.15763
$599$	−2.27832	−0.0930898	−0.0465449	−	0.998916i	$-0.514821\pi$
$599$	−2.27832	−0.0930898	−0.0465449	+	0.998916i	$0.514821\pi$
$600$	6.21197	0.253603
$601$	23.2388	0.947929	0.473964	−	0.880544i	$-0.342823\pi$
$601$	23.2388	0.947929	0.473964	+	0.880544i	$0.342823\pi$
$602$	49.9906	2.03747
$603$	1.01856	0.0414790
$604$	2.64045	0.107438
$605$	−0.952080	−0.0387075
$606$	−15.1668	−0.616109
$607$	18.5306	0.752135	0.376067	−	0.926592i	$-0.377276\pi$
$607$	18.5306	0.752135	0.376067	+	0.926592i	$0.377276\pi$
$608$	4.61351	0.187102
$609$	−5.69802	−0.230895
$610$	−0.419231	−0.0169742
$611$	−43.7544	−1.77011
$612$	−0.0937729	−0.00379054
$613$	−10.0127	−0.404409	−0.202204	−	0.979343i	$-0.564811\pi$
$613$	−10.0127	−0.404409	−0.202204	+	0.979343i	$0.564811\pi$
$614$	52.9898	2.13849
$615$	0.0360339	0.00145303
$616$	−42.6011	−1.71645
$617$	4.34970	0.175112	0.0875562	−	0.996160i	$-0.472094\pi$
$617$	4.34970	0.175112	0.0875562	+	0.996160i	$0.472094\pi$
$618$	10.1364	0.407746
$619$	−37.9140	−1.52389	−0.761946	−	0.647641i	$-0.775756\pi$
$619$	−37.9140	−1.52389	−0.761946	+	0.647641i	$0.775756\pi$
$620$	0.0616020	0.00247400
$621$	13.8186	0.554520
$622$	13.5777	0.544416
$623$	−14.0372	−0.562387
$624$	14.1718	0.567324
$625$	24.9669	0.998677
$626$	51.1536	2.04451
$627$	3.63105	0.145010
$628$	16.6759	0.665439
$629$	0.0934744	0.00372707
$630$	−0.816395	−0.0325259
$631$	−32.0592	−1.27626	−0.638128	−	0.769931i	$-0.720291\pi$
$631$	−32.0592	−1.27626	−0.638128	+	0.769931i	$0.720291\pi$
$632$	−3.23844	−0.128818
$633$	−0.649307	−0.0258077
$634$	31.6631	1.25750
$635$	0.0633531	0.00251409
$636$	6.41082	0.254205
$637$	38.7066	1.53361
$638$	20.8911	0.827088
$639$	−7.27910	−0.287957
$640$	0.604428	0.0238921
$641$	−15.2515	−0.602399	−0.301199	−	0.953561i	$-0.597387\pi$
$641$	−15.2515	−0.602399	−0.301199	+	0.953561i	$0.597387\pi$
$642$	−3.17467	−0.125294
$643$	22.4121	0.883846	0.441923	−	0.897053i	$-0.354296\pi$
$643$	22.4121	0.883846	0.441923	+	0.897053i	$0.354296\pi$
$644$	13.2096	0.520533
$645$	−0.226110	−0.00890308
$646$	−0.0708146	−0.00278616
$647$	−7.31327	−0.287514	−0.143757	−	0.989613i	$-0.545918\pi$
$647$	−7.31327	−0.287514	−0.143757	+	0.989613i	$0.545918\pi$
$648$	10.3051	0.404823
$649$	−56.0379	−2.19968
$650$	37.0848	1.45459
$651$	3.89555	0.152679
$652$	9.32599	0.365234
$653$	24.7168	0.967244	0.483622	−	0.875277i	$-0.339321\pi$
$653$	24.7168	0.967244	0.483622	+	0.875277i	$0.339321\pi$
$654$	−4.41071	−0.172473
$655$	−0.534750	−0.0208944
$656$	−5.88839	−0.229903
$657$	−8.52271	−0.332503
$658$	67.3494	2.62555
$659$	36.6580	1.42799	0.713997	−	0.700149i	$-0.246883\pi$
$659$	36.6580	1.42799	0.713997	+	0.700149i	$0.246883\pi$
$660$	−0.148370	−0.00577531
$661$	−28.1898	−1.09645	−0.548227	−	0.836329i	$-0.684697\pi$
$661$	−28.1898	−1.09645	−0.548227	+	0.836329i	$0.684697\pi$
$662$	−42.3745	−1.64693
$663$	−0.118878	−0.00461682
$664$	−23.7586	−0.922011
$665$	−0.186898	−0.00724759
$666$	9.76813	0.378507
$667$	8.41275	0.325743
$668$	0.0523977	0.00202733
$669$	11.8389	0.457716
$670$	−0.0314303	−0.00121426
$671$	29.4662	1.13753
$672$	−11.9212	−0.459870
$673$	0.223753	0.00862506	0.00431253	−	0.999991i	$-0.498627\pi$
$673$	0.223753	0.00862506	0.00431253	+	0.999991i	$0.498627\pi$
$674$	7.95285	0.306332
$675$	−18.1025	−0.696765
$676$	5.38067	0.206949
$677$	−51.3445	−1.97333	−0.986665	−	0.162762i	$-0.947960\pi$
$677$	−51.3445	−1.97333	−0.986665	+	0.162762i	$0.947960\pi$
$678$	−10.3338	−0.396865
$679$	50.1029	1.92277
$680$	−0.00375789	−0.000144108 0
$681$	3.98438	0.152682
$682$	−14.2826	−0.546908
$683$	4.55170	0.174166	0.0870831	−	0.996201i	$-0.472245\pi$
$683$	4.55170	0.174166	0.0870831	+	0.996201i	$0.472245\pi$
$684$	−2.24336	−0.0857772
$685$	0.114848	0.00438813
$686$	−12.3860	−0.472899
$687$	−0.510020	−0.0194585
$688$	36.9492	1.40868
$689$	−49.7034	−1.89355
$690$	−0.197090	−0.00750308
$691$	1.70657	0.0649210	0.0324605	−	0.999473i	$-0.489666\pi$
$691$	1.70657	0.0649210	0.0324605	+	0.999473i	$0.489666\pi$
$692$	6.06529	0.230568
$693$	57.3813	2.17973
$694$	2.45499	0.0931900
$695$	−0.314595	−0.0119333
$696$	−2.74085	−0.103892
$697$	0.0493939	0.00187093
$698$	−24.0521	−0.910387
$699$	6.32708	0.239312
$700$	−17.3048	−0.654060
$701$	0.850966	0.0321405	0.0160703	−	0.999871i	$-0.494884\pi$
$701$	0.850966	0.0321405	0.0160703	+	0.999871i	$0.494884\pi$
$702$	−26.8768	−1.01440
$703$	2.23622	0.0843408
$704$	−12.0253	−0.453220
$705$	−0.304625	−0.0114728
$706$	5.25120	0.197632
$707$	−54.8702	−2.06361
$708$	−5.66110	−0.212757
$709$	34.8616	1.30925	0.654627	−	0.755952i	$-0.272826\pi$
$709$	34.8616	1.30925	0.654627	+	0.755952i	$0.272826\pi$
$710$	0.224615	0.00842965
$711$	4.36200	0.163588
$712$	−6.75213	−0.253047
$713$	−5.75152	−0.215396
$714$	0.182983	0.00684799
$715$	1.15032	0.0430196
$716$	−13.8029	−0.515837
$717$	15.9690	0.596372
$718$	−20.4138	−0.761838
$719$	2.86737	0.106935	0.0534675	−	0.998570i	$-0.482973\pi$
$719$	2.86737	0.106935	0.0534675	+	0.998570i	$0.482973\pi$
$720$	−0.603416	−0.0224880
$721$	36.6713	1.36571
$722$	−1.69413	−0.0630488
$723$	16.0931	0.598509
$724$	5.52824	0.205455
$725$	−11.0208	−0.409302
$726$	22.2999	0.827628
$727$	−6.14997	−0.228090	−0.114045	−	0.993476i	$-0.536381\pi$
$727$	−6.14997	−0.228090	−0.114045	+	0.993476i	$0.536381\pi$
$728$	33.3666	1.23665
$729$	−6.82486	−0.252773
$730$	0.262990	0.00973369
$731$	−0.309943	−0.0114637
$732$	2.97675	0.110024
$733$	43.6844	1.61352	0.806761	−	0.590879i	$-0.201219\pi$
$733$	43.6844	1.61352	0.806761	+	0.590879i	$0.201219\pi$
$734$	3.43897	0.126935
$735$	0.269481	0.00993997
$736$	17.6009	0.648776
$737$	2.20911	0.0813738
$738$	5.16168	0.190004
$739$	−52.8011	−1.94232	−0.971161	−	0.238425i	$-0.923369\pi$
$739$	−52.8011	−1.94232	−0.971161	+	0.238425i	$0.923369\pi$
$740$	−0.0913757	−0.00335904
$741$	−2.84395	−0.104475
$742$	76.5063	2.80864
$743$	−20.2633	−0.743389	−0.371695	−	0.928355i	$-0.621223\pi$
$743$	−20.2633	−0.743389	−0.371695	+	0.928355i	$0.621223\pi$
$744$	1.87383	0.0686980
$745$	−0.270461	−0.00990891
$746$	42.7728	1.56602
$747$	32.0015	1.17087
$748$	−0.203380	−0.00743631
$749$	−11.4853	−0.419663
$750$	0.516495	0.0188597
$751$	26.9787	0.984469	0.492234	−	0.870463i	$-0.336180\pi$
$751$	26.9787	0.984469	0.492234	+	0.870463i	$0.336180\pi$
$752$	49.7795	1.81527
$753$	3.56498	0.129915
$754$	−16.3626	−0.595892
$755$	−0.142526	−0.00518704
$756$	12.5415	0.456129
$757$	10.2291	0.371783	0.185892	−	0.982570i	$-0.440483\pi$
$757$	10.2291	0.371783	0.185892	+	0.982570i	$0.440483\pi$
$758$	−25.0748	−0.910758
$759$	13.8527	0.502821
$760$	−0.0899014	−0.00326106
$761$	15.6756	0.568239	0.284120	−	0.958789i	$-0.408299\pi$
$761$	15.6756	0.568239	0.284120	+	0.958789i	$0.408299\pi$
$762$	−1.48388	−0.0537552
$763$	−15.9570	−0.577683
$764$	−23.1505	−0.837554
$765$	0.00506166	0.000183005 0
$766$	38.8408	1.40338
$767$	43.8908	1.58480
$768$	−11.3646	−0.410085
$769$	24.2371	0.874012	0.437006	−	0.899459i	$-0.356039\pi$
$769$	24.2371	0.874012	0.437006	+	0.899459i	$0.356039\pi$
$770$	−1.77064	−0.0638096
$771$	15.6933	0.565182
$772$	−4.62536	−0.166470
$773$	−21.2591	−0.764638	−0.382319	−	0.924030i	$-0.624874\pi$
$773$	−21.2591	−0.764638	−0.382319	+	0.924030i	$0.624874\pi$
$774$	−32.3892	−1.16421
$775$	7.53456	0.270650
$776$	24.1004	0.865155
$777$	−5.77836	−0.207297
$778$	−23.8856	−0.856341
$779$	1.18167	0.0423377
$780$	0.116209	0.00416093
$781$	−15.7873	−0.564915
$782$	−0.270163	−0.00966101
$783$	7.98721	0.285439
$784$	−44.0366	−1.57273
$785$	−0.900128	−0.0321269
$786$	12.5251	0.446755
$787$	−43.3431	−1.54501	−0.772507	−	0.635007i	$-0.780997\pi$
$787$	−43.3431	−1.54501	−0.772507	+	0.635007i	$0.780997\pi$
$788$	21.7754	0.775716
$789$	8.25553	0.293905
$790$	−0.134600	−0.00478887
$791$	−37.3853	−1.32927
$792$	27.6015	0.980776
$793$	−23.0789	−0.819555
$794$	44.2797	1.57143
$795$	−0.346042	−0.0122729
$796$	−12.7288	−0.451160
$797$	−18.2133	−0.645147	−0.322574	−	0.946544i	$-0.604548\pi$
$797$	−18.2133	−0.645147	−0.322574	+	0.946544i	$0.604548\pi$
$798$	4.37758	0.154965
$799$	−0.417568	−0.0147725
$800$	−23.0574	−0.815201
$801$	9.09475	0.321347
$802$	23.9900	0.847116
$803$	−18.4846	−0.652306
$804$	0.223170	0.00787062
$805$	−0.713029	−0.0251310
$806$	11.1866	0.394031
$807$	5.44968	0.191838
$808$	−26.3936	−0.928524
$809$	19.1842	0.674480	0.337240	−	0.941419i	$-0.390507\pi$
$809$	19.1842	0.674480	0.337240	+	0.941419i	$0.390507\pi$
$810$	0.428315	0.0150495
$811$	11.1031	0.389883	0.194942	−	0.980815i	$-0.437548\pi$
$811$	11.1031	0.389883	0.194942	+	0.980815i	$0.437548\pi$
$812$	7.63525	0.267945
$813$	−15.4695	−0.542538
$814$	21.1857	0.742558
$815$	−0.503397	−0.0176332
$816$	0.135247	0.00473460
$817$	−7.41488	−0.259414
$818$	37.9150	1.32567
$819$	−44.9429	−1.57043
$820$	−0.0482848	−0.00168618
$821$	48.1530	1.68055	0.840276	−	0.542159i	$-0.182393\pi$
$821$	48.1530	1.68055	0.840276	+	0.542159i	$0.182393\pi$
$822$	−2.69002	−0.0938251
$823$	−41.4378	−1.44443	−0.722216	−	0.691667i	$-0.756876\pi$
$823$	−41.4378	−1.44443	−0.722216	+	0.691667i	$0.756876\pi$
$824$	17.6396	0.614504
$825$	−18.1472	−0.631805
$826$	−67.5593	−2.35069
$827$	−19.0787	−0.663431	−0.331716	−	0.943379i	$-0.607627\pi$
$827$	−19.0787	−0.663431	−0.331716	+	0.943379i	$0.607627\pi$
$828$	−8.55860	−0.297432
$829$	20.5643	0.714228	0.357114	−	0.934061i	$-0.383761\pi$
$829$	20.5643	0.714228	0.357114	+	0.934061i	$0.383761\pi$
$830$	−0.987486	−0.0342761
$831$	−0.318561	−0.0110507
$832$	9.41860	0.326531
$833$	0.369394	0.0127987
$834$	7.36856	0.255152
$835$	−0.00282832	−9.78780e−5 0
$836$	−4.86554	−0.168278
$837$	−5.46059	−0.188746
$838$	36.2180	1.25113
$839$	−0.920456	−0.0317777	−0.0158888	−	0.999874i	$-0.505058\pi$
$839$	−0.920456	−0.0317777	−0.0158888	+	0.999874i	$0.505058\pi$
$840$	0.232303	0.00801522
$841$	−24.1374	−0.832324
$842$	41.3516	1.42507
$843$	1.24635	0.0429264
$844$	0.870061	0.0299487
$845$	−0.290437	−0.00999135
$846$	−43.6360	−1.50024
$847$	80.6764	2.77207
$848$	56.5476	1.94185
$849$	−2.03848	−0.0699604
$850$	0.353917	0.0121393
$851$	8.53136	0.292451
$852$	−1.59488	−0.0546396
$853$	53.8465	1.84367	0.921834	−	0.387584i	$-0.126690\pi$
$853$	53.8465	1.84367	0.921834	+	0.387584i	$0.126690\pi$
$854$	35.5244	1.21562
$855$	0.121092	0.00414126
$856$	−5.52463	−0.188828
$857$	−32.4862	−1.10971	−0.554854	−	0.831948i	$-0.687226\pi$
$857$	−32.4862	−1.10971	−0.554854	+	0.831948i	$0.687226\pi$
$858$	−26.9432	−0.919827
$859$	−45.7593	−1.56129	−0.780643	−	0.624977i	$-0.785108\pi$
$859$	−45.7593	−1.56129	−0.780643	+	0.624977i	$0.785108\pi$
$860$	0.302984	0.0103317
$861$	−3.05341	−0.104060
$862$	−23.1667	−0.789061
$863$	23.4558	0.798444	0.399222	−	0.916854i	$-0.369280\pi$
$863$	23.4558	0.798444	0.399222	+	0.916854i	$0.369280\pi$
$864$	16.7106	0.568505
$865$	−0.327392	−0.0111317
$866$	−50.1781	−1.70512
$867$	11.0371	0.374839
$868$	−5.21997	−0.177177
$869$	9.46056	0.320928
$870$	−0.113919	−0.00386222
$871$	−1.73025	−0.0586273
$872$	−7.67562	−0.259929
$873$	−32.4619	−1.09867
$874$	−6.46321	−0.218621
$875$	1.86857	0.0631691
$876$	−1.86736	−0.0630922
$877$	−16.9046	−0.570829	−0.285414	−	0.958404i	$-0.592131\pi$
$877$	−16.9046	−0.570829	−0.285414	+	0.958404i	$0.592131\pi$
$878$	−0.874127	−0.0295003
$879$	−14.6724	−0.494889
$880$	−1.30872	−0.0441171
$881$	50.1948	1.69111	0.845554	−	0.533891i	$-0.179271\pi$
$881$	50.1948	1.69111	0.845554	+	0.533891i	$0.179271\pi$
$882$	38.6019	1.29979
$883$	−9.03340	−0.303998	−0.151999	−	0.988381i	$-0.548571\pi$
$883$	−9.03340	−0.303998	−0.151999	+	0.988381i	$0.548571\pi$
$884$	0.159294	0.00535764
$885$	0.305574	0.0102718
$886$	−23.4509	−0.787850
$887$	−30.9520	−1.03927	−0.519634	−	0.854389i	$-0.673932\pi$
$887$	−30.9520	−1.03927	−0.519634	+	0.854389i	$0.673932\pi$
$888$	−2.77950	−0.0932739
$889$	−5.36835	−0.180049
$890$	−0.280641	−0.00940712
$891$	−30.1047	−1.00854
$892$	−15.8639	−0.531161
$893$	−9.98963	−0.334290
$894$	6.33482	0.211868
$895$	0.745049	0.0249042
$896$	−51.2174	−1.71105
$897$	−10.8499	−0.362267
$898$	−9.26869	−0.309300
$899$	−3.32441	−0.110875
$900$	11.2119	0.373729
$901$	−0.474341	−0.0158026
$902$	11.1950	0.372751
$903$	19.1599	0.637601
$904$	−17.9830	−0.598107
$905$	−0.298403	−0.00991924
$906$	3.33828	0.110907
$907$	26.4279	0.877524	0.438762	−	0.898603i	$-0.355417\pi$
$907$	26.4279	0.877524	0.438762	+	0.898603i	$0.355417\pi$
$908$	−5.33901	−0.177181
$909$	35.5507	1.17914
$910$	1.38683	0.0459729
$911$	18.9340	0.627310	0.313655	−	0.949537i	$-0.398446\pi$
$911$	18.9340	0.627310	0.313655	+	0.949537i	$0.398446\pi$
$912$	3.23557	0.107141
$913$	69.4066	2.29702
$914$	44.1105	1.45904
$915$	−0.160679	−0.00531188
$916$	0.683419	0.0225808
$917$	45.3131	1.49637
$918$	−0.256497	−0.00846568
$919$	−23.5153	−0.775697	−0.387848	−	0.921723i	$-0.626782\pi$
$919$	−23.5153	−0.775697	−0.387848	+	0.921723i	$0.626782\pi$
$920$	−0.342980	−0.0113077
$921$	20.3094	0.669217
$922$	−10.1536	−0.334392
$923$	12.3652	0.407004
$924$	12.5725	0.413603
$925$	−11.1762	−0.367471
$926$	−1.56094	−0.0512955
$927$	−23.7596	−0.780366
$928$	10.1734	0.333958
$929$	0.813057	0.0266755	0.0133378	−	0.999911i	$-0.495754\pi$
$929$	0.813057	0.0266755	0.0133378	+	0.999911i	$0.495754\pi$
$930$	0.0778828	0.00255388
$931$	8.83716	0.289626
$932$	−8.47818	−0.277712
$933$	5.20392	0.170369
$934$	56.1364	1.83684
$935$	0.0109780	0.000359020 0
$936$	−21.6184	−0.706619
$937$	−7.29291	−0.238249	−0.119124	−	0.992879i	$-0.538009\pi$
$937$	−7.29291	−0.238249	−0.119124	+	0.992879i	$0.538009\pi$
$938$	2.66330	0.0869600
$939$	19.6056	0.639806
$940$	0.408192	0.0133138
$941$	−8.70623	−0.283815	−0.141908	−	0.989880i	$-0.545324\pi$
$941$	−8.70623	−0.283815	−0.141908	+	0.989880i	$0.545324\pi$
$942$	21.0831	0.686924
$943$	4.50815	0.146806
$944$	−49.9346	−1.62523
$945$	−0.676962	−0.0220216
$946$	−70.2476	−2.28394
$947$	−28.3874	−0.922465	−0.461233	−	0.887279i	$-0.652593\pi$
$947$	−28.3874	−0.922465	−0.461233	+	0.887279i	$0.652593\pi$
$948$	0.955730	0.0310407
$949$	14.4777	0.469966
$950$	8.46689	0.274702
$951$	12.1355	0.393521
$952$	0.318432	0.0103204
$953$	−50.4152	−1.63311	−0.816554	−	0.577270i	$-0.804118\pi$
$953$	−50.4152	−1.63311	−0.816554	+	0.577270i	$0.804118\pi$
$954$	−49.5689	−1.60485
$955$	1.24961	0.0404365
$956$	−21.3981	−0.692065
$957$	8.00694	0.258828
$958$	−16.1826	−0.522837
$959$	−9.73190	−0.314260
$960$	0.0655737	0.00211638
$961$	−28.7272	−0.926684
$962$	−16.5933	−0.534990
$963$	7.44137	0.239795
$964$	−21.5645	−0.694545
$965$	0.249667	0.00803707
$966$	16.7008	0.537339
$967$	52.8099	1.69825	0.849126	−	0.528190i	$-0.177129\pi$
$967$	52.8099	1.69825	0.849126	+	0.528190i	$0.177129\pi$
$968$	38.8068	1.24730
$969$	−0.0271411	−0.000871898 0
$970$	1.00169	0.0321625
$971$	−4.55841	−0.146286	−0.0731432	−	0.997321i	$-0.523303\pi$
$971$	−4.55841	−0.146286	−0.0731432	+	0.997321i	$0.523303\pi$
$972$	−12.4956	−0.400796
$973$	26.6579	0.854612
$974$	−46.3904	−1.48644
$975$	14.2135	0.455196
$976$	26.2569	0.840463
$977$	48.8332	1.56231	0.781157	−	0.624335i	$-0.214630\pi$
$977$	48.8332	1.56231	0.781157	+	0.624335i	$0.214630\pi$
$978$	11.7907	0.377026
$979$	19.7252	0.630421
$980$	−0.361100	−0.0115349
$981$	10.3386	0.330087
$982$	−12.2699	−0.391548
$983$	48.1842	1.53684	0.768419	−	0.639947i	$-0.221044\pi$
$983$	48.1842	1.53684	0.768419	+	0.639947i	$0.221044\pi$
$984$	−1.46875	−0.0468219
$985$	−1.17539	−0.0374510
$986$	−0.156156	−0.00497301
$987$	25.8130	0.821636
$988$	3.81085	0.121239
$989$	−28.2883	−0.899516
$990$	1.14721	0.0364607
$991$	−42.5023	−1.35013	−0.675065	−	0.737758i	$-0.735885\pi$
$991$	−42.5023	−1.35013	−0.675065	+	0.737758i	$0.735885\pi$
$992$	−6.95522	−0.220828
$993$	−16.2409	−0.515389
$994$	−19.0332	−0.603696
$995$	0.687073	0.0217817
$996$	7.01164	0.222172
$997$	12.1169	0.383746	0.191873	−	0.981420i	$-0.438544\pi$
$997$	12.1169	0.383746	0.191873	+	0.981420i	$0.438544\pi$
$998$	15.2913	0.484037
$999$	8.09982	0.256267

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.20	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.20	✓	71	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-1.69413 q^{2} -0.649307 q^{3} +0.870061 q^{4} -0.0469641 q^{5} +1.10001 q^{6} +3.97959 q^{7} +1.91426 q^{8} -2.57840 q^{9} +O(q^{10})\)	\(q-1.69413 q^{2} -0.649307 q^{3} +0.870061 q^{4} -0.0469641 q^{5} +1.10001 q^{6} +3.97959 q^{7} +1.91426 q^{8} -2.57840 q^{9} +0.0795630 q^{10} -5.59218 q^{11} -0.564937 q^{12} +4.37998 q^{13} -6.74193 q^{14} +0.0304941 q^{15} -4.98312 q^{16} +0.0418001 q^{17} +4.36813 q^{18} +1.00000 q^{19} -0.0408616 q^{20} -2.58398 q^{21} +9.47386 q^{22} +3.81507 q^{23} -1.24294 q^{24} -4.99779 q^{25} -7.42024 q^{26} +3.62210 q^{27} +3.46249 q^{28} +2.20513 q^{29} -0.0516608 q^{30} -1.50758 q^{31} +4.61351 q^{32} +3.63105 q^{33} -0.0708146 q^{34} -0.186898 q^{35} -2.24336 q^{36} +2.23622 q^{37} -1.69413 q^{38} -2.84395 q^{39} -0.0899014 q^{40} +1.18167 q^{41} +4.37758 q^{42} -7.41488 q^{43} -4.86554 q^{44} +0.121092 q^{45} -6.46321 q^{46} -9.98963 q^{47} +3.23557 q^{48} +8.83716 q^{49} +8.46689 q^{50} -0.0271411 q^{51} +3.81085 q^{52} -11.3478 q^{53} -6.13629 q^{54} +0.262632 q^{55} +7.61797 q^{56} -0.649307 q^{57} -3.73577 q^{58} +10.0208 q^{59} +0.0265317 q^{60} -5.26917 q^{61} +2.55403 q^{62} -10.2610 q^{63} +2.15037 q^{64} -0.205702 q^{65} -6.15145 q^{66} -0.395036 q^{67} +0.0363686 q^{68} -2.47715 q^{69} +0.316628 q^{70} +2.82311 q^{71} -4.93572 q^{72} +3.30543 q^{73} -3.78845 q^{74} +3.24510 q^{75} +0.870061 q^{76} -22.2546 q^{77} +4.81802 q^{78} -1.69175 q^{79} +0.234027 q^{80} +5.38335 q^{81} -2.00189 q^{82} -12.4114 q^{83} -2.24822 q^{84} -0.00196310 q^{85} +12.5617 q^{86} -1.43181 q^{87} -10.7049 q^{88} -3.52728 q^{89} -0.205145 q^{90} +17.4305 q^{91} +3.31935 q^{92} +0.978881 q^{93} +16.9237 q^{94} -0.0469641 q^{95} -2.99558 q^{96} +12.5900 q^{97} -14.9713 q^{98} +14.4189 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

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