LMFDB - Embedded newform 4009.2.a.c.1.24

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			1.24
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)

$f(q)$	$=$	$q-1.26802 q^{2} +1.92190 q^{3} -0.392130 q^{4} +0.425349 q^{5} -2.43700 q^{6} -0.607552 q^{7} +3.03326 q^{8} +0.693695 q^{9} +O(q^{10})$	\(q-1.26802 q^{2} +1.92190 q^{3} -0.392130 q^{4} +0.425349 q^{5} -2.43700 q^{6} -0.607552 q^{7} +3.03326 q^{8} +0.693695 q^{9} -0.539350 q^{10} -0.619317 q^{11} -0.753634 q^{12} -3.46565 q^{13} +0.770387 q^{14} +0.817477 q^{15} -3.06197 q^{16} +1.24879 q^{17} -0.879617 q^{18} +1.00000 q^{19} -0.166792 q^{20} -1.16765 q^{21} +0.785306 q^{22} +4.10560 q^{23} +5.82963 q^{24} -4.81908 q^{25} +4.39451 q^{26} -4.43249 q^{27} +0.238239 q^{28} +9.84019 q^{29} -1.03658 q^{30} -10.2472 q^{31} -2.18389 q^{32} -1.19027 q^{33} -1.58349 q^{34} -0.258421 q^{35} -0.272019 q^{36} +1.76737 q^{37} -1.26802 q^{38} -6.66064 q^{39} +1.29019 q^{40} +11.9062 q^{41} +1.48061 q^{42} -8.23855 q^{43} +0.242853 q^{44} +0.295062 q^{45} -5.20597 q^{46} +7.82219 q^{47} -5.88480 q^{48} -6.63088 q^{49} +6.11068 q^{50} +2.40005 q^{51} +1.35899 q^{52} -5.21067 q^{53} +5.62047 q^{54} -0.263426 q^{55} -1.84287 q^{56} +1.92190 q^{57} -12.4775 q^{58} -3.82208 q^{59} -0.320557 q^{60} -1.62154 q^{61} +12.9936 q^{62} -0.421456 q^{63} +8.89316 q^{64} -1.47411 q^{65} +1.50928 q^{66} -4.66338 q^{67} -0.489688 q^{68} +7.89054 q^{69} +0.327683 q^{70} -14.4021 q^{71} +2.10416 q^{72} -1.25636 q^{73} -2.24106 q^{74} -9.26178 q^{75} -0.392130 q^{76} +0.376267 q^{77} +8.44581 q^{78} +2.31903 q^{79} -1.30241 q^{80} -10.5999 q^{81} -15.0972 q^{82} -4.58337 q^{83} +0.457872 q^{84} +0.531171 q^{85} +10.4466 q^{86} +18.9119 q^{87} -1.87855 q^{88} -5.36849 q^{89} -0.374144 q^{90} +2.10556 q^{91} -1.60993 q^{92} -19.6941 q^{93} -9.91868 q^{94} +0.425349 q^{95} -4.19722 q^{96} +11.8776 q^{97} +8.40808 q^{98} -0.429617 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.26802	−0.896624	−0.448312	−	0.893877i	$-0.647975\pi$
$2$	−1.26802	−0.896624	−0.448312	+	0.893877i	$0.647975\pi$
$3$	1.92190	1.10961	0.554804	−	0.831981i	$-0.312793\pi$
$3$	1.92190	1.10961	0.554804	+	0.831981i	$0.312793\pi$
$4$	−0.392130	−0.196065
$5$	0.425349	0.190222	0.0951108	−	0.995467i	$-0.469679\pi$
$5$	0.425349	0.190222	0.0951108	+	0.995467i	$0.469679\pi$
$6$	−2.43700	−0.994902
$7$	−0.607552	−0.229633	−0.114817	−	0.993387i	$-0.536628\pi$
$7$	−0.607552	−0.229633	−0.114817	+	0.993387i	$0.536628\pi$
$8$	3.03326	1.07242
$9$	0.693695	0.231232
$10$	−0.539350	−0.170557
$11$	−0.619317	−0.186731	−0.0933656	−	0.995632i	$-0.529763\pi$
$11$	−0.619317	−0.186731	−0.0933656	+	0.995632i	$0.529763\pi$
$12$	−0.753634	−0.217555
$13$	−3.46565	−0.961199	−0.480600	−	0.876940i	$-0.659581\pi$
$13$	−3.46565	−0.961199	−0.480600	+	0.876940i	$0.659581\pi$
$14$	0.770387	0.205895
$15$	0.817477	0.211072
$16$	−3.06197	−0.765494
$17$	1.24879	0.302876	0.151438	−	0.988467i	$-0.451610\pi$
$17$	1.24879	0.302876	0.151438	+	0.988467i	$0.451610\pi$
$18$	−0.879617	−0.207328
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.166792	−0.0372958
$21$	−1.16765	−0.254803
$22$	0.785306	0.167428
$23$	4.10560	0.856076	0.428038	−	0.903761i	$-0.359205\pi$
$23$	4.10560	0.856076	0.428038	+	0.903761i	$0.359205\pi$
$24$	5.82963	1.18997
$25$	−4.81908	−0.963816
$26$	4.39451	0.861835
$27$	−4.43249	−0.853032
$28$	0.238239	0.0450230
$29$	9.84019	1.82728	0.913639	−	0.406527i	$-0.133260\pi$
$29$	9.84019	1.82728	0.913639	+	0.406527i	$0.133260\pi$
$30$	−1.03658	−0.189252
$31$	−10.2472	−1.84045	−0.920226	−	0.391386i	$-0.871996\pi$
$31$	−10.2472	−1.84045	−0.920226	+	0.391386i	$0.871996\pi$
$32$	−2.18389	−0.386061
$33$	−1.19027	−0.207199
$34$	−1.58349	−0.271566
$35$	−0.258421	−0.0436812
$36$	−0.272019	−0.0453364
$37$	1.76737	0.290554	0.145277	−	0.989391i	$-0.453593\pi$
$37$	1.76737	0.290554	0.145277	+	0.989391i	$0.453593\pi$
$38$	−1.26802	−0.205700
$39$	−6.66064	−1.06656
$40$	1.29019	0.203998
$41$	11.9062	1.85943	0.929716	−	0.368278i	$-0.120053\pi$
$41$	11.9062	1.85943	0.929716	+	0.368278i	$0.120053\pi$
$42$	1.48061	0.228462
$43$	−8.23855	−1.25637	−0.628184	−	0.778065i	$-0.716201\pi$
$43$	−8.23855	−1.25637	−0.628184	+	0.778065i	$0.716201\pi$
$44$	0.242853	0.0366115
$45$	0.295062	0.0439853
$46$	−5.20597	−0.767579
$47$	7.82219	1.14098	0.570492	−	0.821303i	$-0.306753\pi$
$47$	7.82219	1.14098	0.570492	+	0.821303i	$0.306753\pi$
$48$	−5.88480	−0.849398
$49$	−6.63088	−0.947269
$50$	6.11068	0.864181
$51$	2.40005	0.336074
$52$	1.35899	0.188458
$53$	−5.21067	−0.715741	−0.357870	−	0.933771i	$-0.616497\pi$
$53$	−5.21067	−0.715741	−0.357870	+	0.933771i	$0.616497\pi$
$54$	5.62047	0.764849
$55$	−0.263426	−0.0355203
$56$	−1.84287	−0.246263
$57$	1.92190	0.254562
$58$	−12.4775	−1.63838
$59$	−3.82208	−0.497593	−0.248796	−	0.968556i	$-0.580035\pi$
$59$	−3.82208	−0.497593	−0.248796	+	0.968556i	$0.580035\pi$
$60$	−0.320557	−0.0413838
$61$	−1.62154	−0.207617	−0.103809	−	0.994597i	$-0.533103\pi$
$61$	−1.62154	−0.207617	−0.103809	+	0.994597i	$0.533103\pi$
$62$	12.9936	1.65019
$63$	−0.421456	−0.0530984
$64$	8.89316	1.11165
$65$	−1.47411	−0.182841
$66$	1.50928	0.185779
$67$	−4.66338	−0.569723	−0.284862	−	0.958569i	$-0.591948\pi$
$67$	−4.66338	−0.569723	−0.284862	+	0.958569i	$0.591948\pi$
$68$	−0.489688	−0.0593833
$69$	7.89054	0.949910
$70$	0.327683	0.0391656
$71$	−14.4021	−1.70921	−0.854607	−	0.519276i	$-0.826202\pi$
$71$	−14.4021	−1.70921	−0.854607	+	0.519276i	$0.826202\pi$
$72$	2.10416	0.247978
$73$	−1.25636	−0.147046	−0.0735231	−	0.997294i	$-0.523424\pi$
$73$	−1.25636	−0.147046	−0.0735231	+	0.997294i	$0.523424\pi$
$74$	−2.24106	−0.260517
$75$	−9.26178	−1.06946
$76$	−0.392130	−0.0449804
$77$	0.376267	0.0428797
$78$	8.44581	0.956299
$79$	2.31903	0.260912	0.130456	−	0.991454i	$-0.458356\pi$
$79$	2.31903	0.260912	0.130456	+	0.991454i	$0.458356\pi$
$80$	−1.30241	−0.145613
$81$	−10.5999	−1.17776
$82$	−15.0972	−1.66721
$83$	−4.58337	−0.503090	−0.251545	−	0.967846i	$-0.580939\pi$
$83$	−4.58337	−0.503090	−0.251545	+	0.967846i	$0.580939\pi$
$84$	0.457872	0.0499579
$85$	0.531171	0.0576135
$86$	10.4466	1.12649
$87$	18.9119	2.02756
$88$	−1.87855	−0.200254
$89$	−5.36849	−0.569058	−0.284529	−	0.958667i	$-0.591837\pi$
$89$	−5.36849	−0.569058	−0.284529	+	0.958667i	$0.591837\pi$
$90$	−0.374144	−0.0394382
$91$	2.10556	0.220723
$92$	−1.60993	−0.167847
$93$	−19.6941	−2.04218
$94$	−9.91868	−1.02303
$95$	0.425349	0.0436398
$96$	−4.19722	−0.428376
$97$	11.8776	1.20598	0.602991	−	0.797748i	$-0.293975\pi$
$97$	11.8776	1.20598	0.602991	+	0.797748i	$0.293975\pi$
$98$	8.40808	0.849344
$99$	−0.429617	−0.0431782
$100$	1.88971	0.188971
$101$	9.11591	0.907067	0.453533	−	0.891239i	$-0.350163\pi$
$101$	9.11591	0.907067	0.453533	+	0.891239i	$0.350163\pi$
$102$	−3.04330	−0.301332
$103$	−20.0496	−1.97555	−0.987774	−	0.155893i	$-0.950174\pi$
$103$	−20.0496	−1.97555	−0.987774	+	0.155893i	$0.950174\pi$
$104$	−10.5122	−1.03081
$105$	−0.496660	−0.0484690
$106$	6.60723	0.641750
$107$	−11.6016	−1.12157	−0.560786	−	0.827961i	$-0.689501\pi$
$107$	−11.6016	−1.12157	−0.560786	+	0.827961i	$0.689501\pi$
$108$	1.73811	0.167250
$109$	−13.8932	−1.33072	−0.665362	−	0.746521i	$-0.731723\pi$
$109$	−13.8932	−1.33072	−0.665362	+	0.746521i	$0.731723\pi$
$110$	0.334029	0.0318484
$111$	3.39670	0.322401
$112$	1.86031	0.175783
$113$	17.5843	1.65419	0.827095	−	0.562062i	$-0.189992\pi$
$113$	17.5843	1.65419	0.827095	+	0.562062i	$0.189992\pi$
$114$	−2.43700	−0.228246
$115$	1.74631	0.162844
$116$	−3.85863	−0.358265
$117$	−2.40411	−0.222260
$118$	4.84647	0.446154
$119$	−0.758704	−0.0695503
$120$	2.47962	0.226358
$121$	−10.6164	−0.965131
$122$	2.05615	0.186155
$123$	22.8825	2.06324
$124$	4.01824	0.360848
$125$	−4.17653	−0.373560
$126$	0.534413	0.0476093
$127$	13.6766	1.21360	0.606801	−	0.794853i	$-0.292452\pi$
$127$	13.6766	1.21360	0.606801	+	0.794853i	$0.292452\pi$
$128$	−6.90891	−0.610667
$129$	−15.8337	−1.39408
$130$	1.86920	0.163940
$131$	−16.9845	−1.48395	−0.741973	−	0.670430i	$-0.766110\pi$
$131$	−16.9845	−1.48395	−0.741973	+	0.670430i	$0.766110\pi$
$132$	0.466739	0.0406244
$133$	−0.607552	−0.0526814
$134$	5.91326	0.510828
$135$	−1.88535	−0.162265
$136$	3.78791	0.324810
$137$	−14.8028	−1.26469	−0.632346	−	0.774686i	$-0.717908\pi$
$137$	−14.8028	−1.26469	−0.632346	+	0.774686i	$0.717908\pi$
$138$	−10.0053	−0.851712
$139$	6.38961	0.541960	0.270980	−	0.962585i	$-0.412652\pi$
$139$	6.38961	0.541960	0.270980	+	0.962585i	$0.412652\pi$
$140$	0.101335	0.00856435
$141$	15.0335	1.26605
$142$	18.2621	1.53252
$143$	2.14634	0.179486
$144$	−2.12408	−0.177006
$145$	4.18551	0.347588
$146$	1.59309	0.131845
$147$	−12.7439	−1.05110
$148$	−0.693038	−0.0569674
$149$	10.2819	0.842330	0.421165	−	0.906984i	$-0.361621\pi$
$149$	10.2819	0.842330	0.421165	+	0.906984i	$0.361621\pi$
$150$	11.7441	0.958902
$151$	−14.9454	−1.21624	−0.608118	−	0.793847i	$-0.708075\pi$
$151$	−14.9454	−1.21624	−0.608118	+	0.793847i	$0.708075\pi$
$152$	3.03326	0.246030
$153$	0.866278	0.0700344
$154$	−0.477114	−0.0384469
$155$	−4.35864	−0.350094
$156$	2.61184	0.209114
$157$	11.9772	0.955887	0.477943	−	0.878391i	$-0.341382\pi$
$157$	11.9772	0.955887	0.477943	+	0.878391i	$0.341382\pi$
$158$	−2.94058	−0.233940
$159$	−10.0144	−0.794192
$160$	−0.928915	−0.0734371
$161$	−2.49436	−0.196583
$162$	13.4408	1.05601
$163$	−1.13660	−0.0890256	−0.0445128	−	0.999009i	$-0.514174\pi$
$163$	−1.13660	−0.0890256	−0.0445128	+	0.999009i	$0.514174\pi$
$164$	−4.66877	−0.364569
$165$	−0.506278	−0.0394137
$166$	5.81180	0.451083
$167$	−9.61917	−0.744354	−0.372177	−	0.928162i	$-0.621388\pi$
$167$	−9.61917	−0.744354	−0.372177	+	0.928162i	$0.621388\pi$
$168$	−3.54180	−0.273256
$169$	−0.989245	−0.0760958
$170$	−0.673534	−0.0516577
$171$	0.693695	0.0530482
$172$	3.23058	0.246330
$173$	5.50743	0.418722	0.209361	−	0.977838i	$-0.432862\pi$
$173$	5.50743	0.418722	0.209361	+	0.977838i	$0.432862\pi$
$174$	−23.9806	−1.81796
$175$	2.92784	0.221324
$176$	1.89633	0.142942
$177$	−7.34566	−0.552133
$178$	6.80734	0.510231
$179$	16.1014	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$179$	16.1014	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$180$	−0.115703	−0.00862397
$181$	−11.6244	−0.864033	−0.432016	−	0.901866i	$-0.642198\pi$
$181$	−11.6244	−0.864033	−0.432016	+	0.901866i	$0.642198\pi$
$182$	−2.66989	−0.197906
$183$	−3.11644	−0.230374
$184$	12.4534	0.918074
$185$	0.751748	0.0552696
$186$	24.9725	1.83107
$187$	−0.773397	−0.0565564
$188$	−3.06732	−0.223707
$189$	2.69296	0.195884
$190$	−0.539350	−0.0391285
$191$	2.94751	0.213275	0.106637	−	0.994298i	$-0.465992\pi$
$191$	2.94751	0.213275	0.106637	+	0.994298i	$0.465992\pi$
$192$	17.0918	1.23349
$193$	10.3771	0.746960	0.373480	−	0.927638i	$-0.378165\pi$
$193$	10.3771	0.746960	0.373480	+	0.927638i	$0.378165\pi$
$194$	−15.0610	−1.08131
$195$	−2.83309	−0.202882
$196$	2.60017	0.185726
$197$	−14.8044	−1.05477	−0.527385	−	0.849626i	$-0.676827\pi$
$197$	−14.8044	−1.05477	−0.527385	+	0.849626i	$0.676827\pi$
$198$	0.544762	0.0387146
$199$	−21.0492	−1.49214	−0.746069	−	0.665868i	$-0.768061\pi$
$199$	−21.0492	−1.49214	−0.746069	+	0.665868i	$0.768061\pi$
$200$	−14.6175	−1.03362
$201$	−8.96255	−0.632170
$202$	−11.5591	−0.813298
$203$	−5.97843	−0.419603
$204$	−0.941130	−0.0658923
$205$	5.06427	0.353704
$206$	25.4233	1.77132
$207$	2.84803	0.197952
$208$	10.6117	0.735792
$209$	−0.619317	−0.0428391
$210$	0.629774	0.0434585
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	2.04326	0.140332
$213$	−27.6794	−1.89656
$214$	14.7111	1.00563
$215$	−3.50426	−0.238988
$216$	−13.4449	−0.914809
$217$	6.22571	0.422629
$218$	17.6168	1.19316
$219$	−2.41460	−0.163164
$220$	0.103297	0.00696429
$221$	−4.32787	−0.291124
$222$	−4.30708	−0.289072
$223$	−13.6712	−0.915489	−0.457744	−	0.889084i	$-0.651342\pi$
$223$	−13.6712	−0.915489	−0.457744	+	0.889084i	$0.651342\pi$
$224$	1.32683	0.0886523
$225$	−3.34297	−0.222865
$226$	−22.2972	−1.48319
$227$	2.85447	0.189458	0.0947289	−	0.995503i	$-0.469802\pi$
$227$	2.85447	0.189458	0.0947289	+	0.995503i	$0.469802\pi$
$228$	−0.753634	−0.0499106
$229$	−16.1420	−1.06669	−0.533347	−	0.845897i	$-0.679066\pi$
$229$	−16.1420	−1.06669	−0.533347	+	0.845897i	$0.679066\pi$
$230$	−2.21435	−0.146010
$231$	0.723148	0.0475796
$232$	29.8479	1.95961
$233$	−14.2184	−0.931477	−0.465739	−	0.884922i	$-0.654211\pi$
$233$	−14.2184	−0.931477	−0.465739	+	0.884922i	$0.654211\pi$
$234$	3.04845	0.199283
$235$	3.32716	0.217040
$236$	1.49875	0.0975605
$237$	4.45695	0.289510
$238$	0.962051	0.0623605
$239$	11.9919	0.775689	0.387844	−	0.921725i	$-0.373220\pi$
$239$	11.9919	0.775689	0.387844	+	0.921725i	$0.373220\pi$
$240$	−2.50309	−0.161574
$241$	11.4382	0.736799	0.368399	−	0.929668i	$-0.379906\pi$
$241$	11.4382	0.736799	0.368399	+	0.929668i	$0.379906\pi$
$242$	13.4618	0.865360
$243$	−7.07442	−0.453825
$244$	0.635855	0.0407065
$245$	−2.82044	−0.180191
$246$	−29.0154	−1.84995
$247$	−3.46565	−0.220514
$248$	−31.0825	−1.97374
$249$	−8.80877	−0.558233
$250$	5.29592	0.334943
$251$	16.4825	1.04037	0.520183	−	0.854055i	$-0.325864\pi$
$251$	16.4825	1.04037	0.520183	+	0.854055i	$0.325864\pi$
$252$	0.165265	0.0104107
$253$	−2.54267	−0.159856
$254$	−17.3422	−1.08815
$255$	1.02086	0.0639285
$256$	−9.02570	−0.564106
$257$	−24.6228	−1.53593	−0.767963	−	0.640494i	$-0.778730\pi$
$257$	−24.6228	−1.53593	−0.767963	+	0.640494i	$0.778730\pi$
$258$	20.0774	1.24996
$259$	−1.07377	−0.0667207
$260$	0.578043	0.0358487
$261$	6.82609	0.422524
$262$	21.5367	1.33054
$263$	−2.08983	−0.128865	−0.0644324	−	0.997922i	$-0.520524\pi$
$263$	−2.08983	−0.128865	−0.0644324	+	0.997922i	$0.520524\pi$
$264$	−3.61039	−0.222204
$265$	−2.21635	−0.136149
$266$	0.770387	0.0472354
$267$	−10.3177	−0.631432
$268$	1.82865	0.111703
$269$	−25.7480	−1.56988	−0.784940	−	0.619572i	$-0.787306\pi$
$269$	−25.7480	−1.56988	−0.784940	+	0.619572i	$0.787306\pi$
$270$	2.39066	0.145491
$271$	6.75437	0.410299	0.205149	−	0.978731i	$-0.434232\pi$
$271$	6.75437	0.410299	0.205149	+	0.978731i	$0.434232\pi$
$272$	−3.82376	−0.231849
$273$	4.04668	0.244916
$274$	18.7703	1.13395
$275$	2.98454	0.179974
$276$	−3.09412	−0.186244
$277$	18.0049	1.08181	0.540905	−	0.841084i	$-0.318082\pi$
$277$	18.0049	1.08181	0.540905	+	0.841084i	$0.318082\pi$
$278$	−8.10215	−0.485934
$279$	−7.10843	−0.425571
$280$	−0.783860	−0.0468446
$281$	10.8515	0.647347	0.323674	−	0.946169i	$-0.395082\pi$
$281$	10.8515	0.647347	0.323674	+	0.946169i	$0.395082\pi$
$282$	−19.0627	−1.13517
$283$	−32.1955	−1.91382	−0.956911	−	0.290381i	$-0.906218\pi$
$283$	−32.1955	−1.91382	−0.956911	+	0.290381i	$0.906218\pi$
$284$	5.64749	0.335117
$285$	0.817477	0.0484232
$286$	−2.72160	−0.160931
$287$	−7.23362	−0.426987
$288$	−1.51495	−0.0892695
$289$	−15.4405	−0.908266
$290$	−5.30731	−0.311656
$291$	22.8275	1.33817
$292$	0.492658	0.0288306
$293$	12.6932	0.741547	0.370774	−	0.928723i	$-0.379093\pi$
$293$	12.6932	0.741547	0.370774	+	0.928723i	$0.379093\pi$
$294$	16.1595	0.942440
$295$	−1.62572	−0.0946529
$296$	5.36090	0.311596
$297$	2.74512	0.159288
$298$	−13.0377	−0.755253
$299$	−14.2286	−0.822860
$300$	3.63182	0.209683
$301$	5.00535	0.288503
$302$	18.9510	1.09051
$303$	17.5199	1.00649
$304$	−3.06197	−0.175616
$305$	−0.689721	−0.0394933
$306$	−1.09846	−0.0627946
$307$	−8.92314	−0.509271	−0.254635	−	0.967037i	$-0.581955\pi$
$307$	−8.92314	−0.509271	−0.254635	+	0.967037i	$0.581955\pi$
$308$	−0.147546	−0.00840720
$309$	−38.5333	−2.19209
$310$	5.52683	0.313903
$311$	26.4068	1.49739	0.748696	−	0.662914i	$-0.230680\pi$
$311$	26.4068	1.49739	0.748696	+	0.662914i	$0.230680\pi$
$312$	−20.2035	−1.14380
$313$	−10.3524	−0.585154	−0.292577	−	0.956242i	$-0.594513\pi$
$313$	−10.3524	−0.585154	−0.292577	+	0.956242i	$0.594513\pi$
$314$	−15.1873	−0.857071
$315$	−0.179266	−0.0101005
$316$	−0.909363	−0.0511556
$317$	−25.5542	−1.43527	−0.717634	−	0.696420i	$-0.754775\pi$
$317$	−25.5542	−1.43527	−0.717634	+	0.696420i	$0.754775\pi$
$318$	12.6984	0.712092
$319$	−6.09420	−0.341210
$320$	3.78269	0.211459
$321$	−22.2972	−1.24451
$322$	3.16290	0.176261
$323$	1.24879	0.0694845
$324$	4.15653	0.230918
$325$	16.7013	0.926419
$326$	1.44123	0.0798225
$327$	−26.7012	−1.47658
$328$	36.1146	1.99409
$329$	−4.75239	−0.262008
$330$	0.641969	0.0353392
$331$	34.1223	1.87553	0.937766	−	0.347267i	$-0.112890\pi$
$331$	34.1223	1.87553	0.937766	+	0.347267i	$0.112890\pi$
$332$	1.79728	0.0986384
$333$	1.22601	0.0671852
$334$	12.1973	0.667405
$335$	−1.98356	−0.108374
$336$	3.57532	0.195050
$337$	−26.7619	−1.45782	−0.728908	−	0.684612i	$-0.759972\pi$
$337$	−26.7619	−1.45782	−0.728908	+	0.684612i	$0.759972\pi$
$338$	1.25438	0.0682293
$339$	33.7952	1.83550
$340$	−0.208288	−0.0112960
$341$	6.34627	0.343670
$342$	−0.879617	−0.0475643
$343$	8.28147	0.447157
$344$	−24.9897	−1.34735
$345$	3.35623	0.180693
$346$	−6.98352	−0.375437
$347$	−2.67557	−0.143632	−0.0718162	−	0.997418i	$-0.522879\pi$
$347$	−2.67557	−0.143632	−0.0718162	+	0.997418i	$0.522879\pi$
$348$	−7.41591	−0.397534
$349$	9.30260	0.497957	0.248978	−	0.968509i	$-0.419905\pi$
$349$	9.30260	0.497957	0.248978	+	0.968509i	$0.419905\pi$
$350$	−3.71255	−0.198444
$351$	15.3615	0.819934
$352$	1.35252	0.0720896
$353$	14.2206	0.756885	0.378442	−	0.925625i	$-0.376460\pi$
$353$	14.2206	0.756885	0.378442	+	0.925625i	$0.376460\pi$
$354$	9.31442	0.495056
$355$	−6.12591	−0.325129
$356$	2.10514	0.111572
$357$	−1.45815	−0.0771736
$358$	−20.4168	−1.07906
$359$	15.7957	0.833664	0.416832	−	0.908984i	$-0.363140\pi$
$359$	15.7957	0.833664	0.416832	+	0.908984i	$0.363140\pi$
$360$	0.895001	0.0471707
$361$	1.00000	0.0526316
$362$	14.7399	0.774713
$363$	−20.4037	−1.07092
$364$	−0.825655	−0.0432761
$365$	−0.534393	−0.0279714
$366$	3.95170	0.206559
$367$	16.5604	0.864444	0.432222	−	0.901767i	$-0.357730\pi$
$367$	16.5604	0.864444	0.432222	+	0.901767i	$0.357730\pi$
$368$	−12.5712	−0.655321
$369$	8.25925	0.429959
$370$	−0.953230	−0.0495561
$371$	3.16575	0.164358
$372$	7.72265	0.400401
$373$	30.4812	1.57826	0.789130	−	0.614227i	$-0.210532\pi$
$373$	30.4812	1.57826	0.789130	+	0.614227i	$0.210532\pi$
$374$	0.980681	0.0507098
$375$	−8.02687	−0.414506
$376$	23.7268	1.22361
$377$	−34.1027	−1.75638
$378$	−3.41473	−0.175635
$379$	−21.8715	−1.12346	−0.561732	−	0.827319i	$-0.689865\pi$
$379$	−21.8715	−1.12346	−0.561732	+	0.827319i	$0.689865\pi$
$380$	−0.166792	−0.00855625
$381$	26.2851	1.34662
$382$	−3.73750	−0.191227
$383$	26.7078	1.36470	0.682352	−	0.731024i	$-0.260957\pi$
$383$	26.7078	1.36470	0.682352	+	0.731024i	$0.260957\pi$
$384$	−13.2782	−0.677601
$385$	0.160045	0.00815664
$386$	−13.1583	−0.669742
$387$	−5.71504	−0.290512
$388$	−4.65755	−0.236451
$389$	−10.9031	−0.552808	−0.276404	−	0.961042i	$-0.589143\pi$
$389$	−10.9031	−0.552808	−0.276404	+	0.961042i	$0.589143\pi$
$390$	3.59241	0.181909
$391$	5.12702	0.259285
$392$	−20.1132	−1.01587
$393$	−32.6426	−1.64660
$394$	18.7722	0.945732
$395$	0.986398	0.0496311
$396$	0.168466	0.00846573
$397$	−6.48537	−0.325491	−0.162746	−	0.986668i	$-0.552035\pi$
$397$	−6.48537	−0.325491	−0.162746	+	0.986668i	$0.552035\pi$
$398$	26.6908	1.33789
$399$	−1.16765	−0.0584558
$400$	14.7559	0.737795
$401$	29.5454	1.47542	0.737712	−	0.675115i	$-0.235906\pi$
$401$	29.5454	1.47542	0.737712	+	0.675115i	$0.235906\pi$
$402$	11.3647	0.566819
$403$	35.5133	1.76904
$404$	−3.57462	−0.177844
$405$	−4.50864	−0.224036
$406$	7.58076	0.376227
$407$	−1.09456	−0.0542554
$408$	7.27997	0.360412
$409$	24.4517	1.20906	0.604529	−	0.796583i	$-0.293362\pi$
$409$	24.4517	1.20906	0.604529	+	0.796583i	$0.293362\pi$
$410$	−6.42159	−0.317140
$411$	−28.4495	−1.40331
$412$	7.86206	0.387336
$413$	2.32211	0.114264
$414$	−3.61135	−0.177488
$415$	−1.94953	−0.0956987
$416$	7.56861	0.371081
$417$	12.2802	0.601364
$418$	0.785306	0.0384106
$419$	1.34678	0.0657946	0.0328973	−	0.999459i	$-0.489527\pi$
$419$	1.34678	0.0657946	0.0328973	+	0.999459i	$0.489527\pi$
$420$	0.194755	0.00950308
$421$	−13.2879	−0.647614	−0.323807	−	0.946123i	$-0.604963\pi$
$421$	−13.2879	−0.647614	−0.323807	+	0.946123i	$0.604963\pi$
$422$	−1.26802	−0.0617262
$423$	5.42621	0.263831
$424$	−15.8053	−0.767575
$425$	−6.01801	−0.291916
$426$	35.0979	1.70050
$427$	0.985171	0.0476758
$428$	4.54935	0.219901
$429$	4.12505	0.199159
$430$	4.44346	0.214283
$431$	−10.0982	−0.486414	−0.243207	−	0.969974i	$-0.578199\pi$
$431$	−10.0982	−0.486414	−0.243207	+	0.969974i	$0.578199\pi$
$432$	13.5722	0.652991
$433$	−33.2018	−1.59558	−0.797789	−	0.602937i	$-0.793997\pi$
$433$	−33.2018	−1.59558	−0.797789	+	0.602937i	$0.793997\pi$
$434$	−7.89431	−0.378939
$435$	8.04413	0.385687
$436$	5.44792	0.260908
$437$	4.10560	0.196397
$438$	3.06176	0.146297
$439$	−34.2590	−1.63509	−0.817547	−	0.575861i	$-0.804667\pi$
$439$	−34.2590	−1.63509	−0.817547	+	0.575861i	$0.804667\pi$
$440$	−0.799040	−0.0380927
$441$	−4.59981	−0.219038
$442$	5.48782	0.261029
$443$	18.4025	0.874331	0.437166	−	0.899381i	$-0.355982\pi$
$443$	18.4025	0.874331	0.437166	+	0.899381i	$0.355982\pi$
$444$	−1.33195	−0.0632115
$445$	−2.28348	−0.108247
$446$	17.3353	0.820849
$447$	19.7609	0.934656
$448$	−5.40306	−0.255270
$449$	−8.63882	−0.407691	−0.203846	−	0.979003i	$-0.565344\pi$
$449$	−8.63882	−0.407691	−0.203846	+	0.979003i	$0.565344\pi$
$450$	4.23895	0.199826
$451$	−7.37370	−0.347214
$452$	−6.89532	−0.324329
$453$	−28.7235	−1.34955
$454$	−3.61952	−0.169872
$455$	0.895599	0.0419863
$456$	5.82963	0.272997
$457$	−22.9335	−1.07278	−0.536392	−	0.843969i	$-0.680213\pi$
$457$	−22.9335	−1.07278	−0.536392	+	0.843969i	$0.680213\pi$
$458$	20.4684	0.956424
$459$	−5.53524	−0.258363
$460$	−0.684781	−0.0319281
$461$	−34.9189	−1.62633	−0.813167	−	0.582030i	$-0.802259\pi$
$461$	−34.9189	−1.62633	−0.813167	+	0.582030i	$0.802259\pi$
$462$	−0.916965	−0.0426611
$463$	8.34051	0.387616	0.193808	−	0.981039i	$-0.437916\pi$
$463$	8.34051	0.387616	0.193808	+	0.981039i	$0.437916\pi$
$464$	−30.1304	−1.39877
$465$	−8.37686	−0.388467
$466$	18.0292	0.835185
$467$	−11.7550	−0.543957	−0.271978	−	0.962303i	$-0.587678\pi$
$467$	−11.7550	−0.543957	−0.271978	+	0.962303i	$0.587678\pi$
$468$	0.942722	0.0435773
$469$	2.83325	0.130827
$470$	−4.21890	−0.194603
$471$	23.0190	1.06066
$472$	−11.5934	−0.533629
$473$	5.10228	0.234603
$474$	−5.65149	−0.259582
$475$	−4.81908	−0.221114
$476$	0.297511	0.0136364
$477$	−3.61461	−0.165502
$478$	−15.2059	−0.695501
$479$	−38.5279	−1.76039	−0.880193	−	0.474616i	$-0.842587\pi$
$479$	−38.5279	−1.76039	−0.880193	+	0.474616i	$0.842587\pi$
$480$	−1.78528	−0.0814865
$481$	−6.12509	−0.279280
$482$	−14.5038	−0.660632
$483$	−4.79391	−0.218131
$484$	4.16303	0.189229
$485$	5.05210	0.229404
$486$	8.97050	0.406910
$487$	−24.7854	−1.12313	−0.561566	−	0.827432i	$-0.689801\pi$
$487$	−24.7854	−1.12313	−0.561566	+	0.827432i	$0.689801\pi$
$488$	−4.91857	−0.222653
$489$	−2.18443	−0.0987836
$490$	3.57636	0.161564
$491$	12.0581	0.544174	0.272087	−	0.962273i	$-0.412286\pi$
$491$	12.0581	0.544174	0.272087	+	0.962273i	$0.412286\pi$
$492$	−8.97290	−0.404529
$493$	12.2883	0.553438
$494$	4.39451	0.197718
$495$	−0.182737	−0.00821342
$496$	31.3767	1.40885
$497$	8.75002	0.392492
$498$	11.1697	0.500526
$499$	−35.6755	−1.59706	−0.798528	−	0.601958i	$-0.794388\pi$
$499$	−35.6755	−1.59706	−0.798528	+	0.601958i	$0.794388\pi$
$500$	1.63774	0.0732421
$501$	−18.4871	−0.825941
$502$	−20.9001	−0.932818
$503$	−12.9853	−0.578987	−0.289494	−	0.957180i	$-0.593487\pi$
$503$	−12.9853	−0.578987	−0.289494	+	0.957180i	$0.593487\pi$
$504$	−1.27839	−0.0569438
$505$	3.87744	0.172544
$506$	3.22415	0.143331
$507$	−1.90123	−0.0844365
$508$	−5.36301	−0.237945
$509$	2.29175	0.101580	0.0507901	−	0.998709i	$-0.483826\pi$
$509$	2.29175	0.101580	0.0507901	+	0.998709i	$0.483826\pi$
$510$	−1.29446	−0.0573198
$511$	0.763306	0.0337667
$512$	25.2626	1.11646
$513$	−4.43249	−0.195699
$514$	31.2221	1.37715
$515$	−8.52808	−0.375792
$516$	6.20885	0.273330
$517$	−4.84442	−0.213057
$518$	1.36156	0.0598234
$519$	10.5847	0.464618
$520$	−4.47137	−0.196082
$521$	36.7418	1.60969	0.804844	−	0.593486i	$-0.202249\pi$
$521$	36.7418	1.60969	0.804844	+	0.593486i	$0.202249\pi$
$522$	−8.65561	−0.378846
$523$	6.24915	0.273256	0.136628	−	0.990622i	$-0.456373\pi$
$523$	6.24915	0.273256	0.136628	+	0.990622i	$0.456373\pi$
$524$	6.66014	0.290950
$525$	5.62701	0.245583
$526$	2.64995	0.115543
$527$	−12.7966	−0.557429
$528$	3.64456	0.158609
$529$	−6.14407	−0.267134
$530$	2.81037	0.122075
$531$	−2.65136	−0.115059
$532$	0.238239	0.0103290
$533$	−41.2627	−1.78728
$534$	13.0830	0.566157
$535$	−4.93474	−0.213347
$536$	−14.1453	−0.610983
$537$	30.9452	1.33538
$538$	32.6489	1.40759
$539$	4.10662	0.176885
$540$	0.739303	0.0318145
$541$	13.7158	0.589690	0.294845	−	0.955545i	$-0.404732\pi$
$541$	13.7158	0.589690	0.294845	+	0.955545i	$0.404732\pi$
$542$	−8.56466	−0.367884
$543$	−22.3409	−0.958738
$544$	−2.72722	−0.116928
$545$	−5.90944	−0.253132
$546$	−5.13127	−0.219598
$547$	−5.87140	−0.251043	−0.125522	−	0.992091i	$-0.540060\pi$
$547$	−5.87140	−0.251043	−0.125522	+	0.992091i	$0.540060\pi$
$548$	5.80464	0.247962
$549$	−1.12486	−0.0480077
$550$	−3.78445	−0.161369
$551$	9.84019	0.419206
$552$	23.9341	1.01870
$553$	−1.40893	−0.0599139
$554$	−22.8305	−0.969976
$555$	1.44478	0.0613276
$556$	−2.50556	−0.106259
$557$	1.13896	0.0482592	0.0241296	−	0.999709i	$-0.492319\pi$
$557$	1.13896	0.0482592	0.0241296	+	0.999709i	$0.492319\pi$
$558$	9.01362	0.381577
$559$	28.5520	1.20762
$560$	0.791280	0.0334377
$561$	−1.48639	−0.0627554
$562$	−13.7599	−0.580427
$563$	18.1491	0.764895	0.382448	−	0.923977i	$-0.375081\pi$
$563$	18.1491	0.764895	0.382448	+	0.923977i	$0.375081\pi$
$564$	−5.89507	−0.248227
$565$	7.47945	0.314663
$566$	40.8244	1.71598
$567$	6.43997	0.270453
$568$	−43.6853	−1.83300
$569$	23.7463	0.995496	0.497748	−	0.867322i	$-0.334160\pi$
$569$	23.7463	0.995496	0.497748	+	0.867322i	$0.334160\pi$
$570$	−1.03658	−0.0434174
$571$	−9.47097	−0.396348	−0.198174	−	0.980167i	$-0.563501\pi$
$571$	−9.47097	−0.396348	−0.198174	+	0.980167i	$0.563501\pi$
$572$	−0.841644	−0.0351909
$573$	5.66482	0.236651
$574$	9.17236	0.382847
$575$	−19.7852	−0.825100
$576$	6.16914	0.257047
$577$	9.56620	0.398246	0.199123	−	0.979974i	$-0.436191\pi$
$577$	9.56620	0.398246	0.199123	+	0.979974i	$0.436191\pi$
$578$	19.5789	0.814374
$579$	19.9437	0.828833
$580$	−1.64126	−0.0681498
$581$	2.78464	0.115526
$582$	−28.9456	−1.19983
$583$	3.22706	0.133651
$584$	−3.81088	−0.157695
$585$	−1.02258	−0.0422786
$586$	−16.0953	−0.664889
$587$	1.86732	0.0770727	0.0385364	−	0.999257i	$-0.487730\pi$
$587$	1.86732	0.0770727	0.0385364	+	0.999257i	$0.487730\pi$
$588$	4.99726	0.206083
$589$	−10.2472	−0.422229
$590$	2.06144	0.0848681
$591$	−28.4526	−1.17038
$592$	−5.41164	−0.222417
$593$	−16.2074	−0.665560	−0.332780	−	0.943005i	$-0.607987\pi$
$593$	−16.2074	−0.665560	−0.332780	+	0.943005i	$0.607987\pi$
$594$	−3.48086	−0.142821
$595$	−0.322714	−0.0132300
$596$	−4.03186	−0.165151
$597$	−40.4544	−1.65569
$598$	18.0421	0.737796
$599$	0.975271	0.0398485	0.0199242	−	0.999801i	$-0.493657\pi$
$599$	0.975271	0.0398485	0.0199242	+	0.999801i	$0.493657\pi$
$600$	−28.0934	−1.14691
$601$	6.23496	0.254329	0.127165	−	0.991882i	$-0.459412\pi$
$601$	6.23496	0.254329	0.127165	+	0.991882i	$0.459412\pi$
$602$	−6.34687	−0.258679
$603$	−3.23497	−0.131738
$604$	5.86053	0.238461
$605$	−4.51569	−0.183589
$606$	−22.2155	−0.902443
$607$	−12.0197	−0.487866	−0.243933	−	0.969792i	$-0.578438\pi$
$607$	−12.0197	−0.487866	−0.243933	+	0.969792i	$0.578438\pi$
$608$	−2.18389	−0.0885684
$609$	−11.4899	−0.465596
$610$	0.874578	0.0354106
$611$	−27.1090	−1.09671
$612$	−0.339694	−0.0137313
$613$	−5.85147	−0.236339	−0.118169	−	0.992993i	$-0.537703\pi$
$613$	−5.85147	−0.236339	−0.118169	+	0.992993i	$0.537703\pi$
$614$	11.3147	0.456624
$615$	9.73302	0.392473
$616$	1.14132	0.0459850
$617$	22.0265	0.886752	0.443376	−	0.896336i	$-0.353781\pi$
$617$	22.0265	0.886752	0.443376	+	0.896336i	$0.353781\pi$
$618$	48.8610	1.96548
$619$	−39.4373	−1.58512	−0.792559	−	0.609795i	$-0.791252\pi$
$619$	−39.4373	−1.58512	−0.792559	+	0.609795i	$0.791252\pi$
$620$	1.70915	0.0686412
$621$	−18.1980	−0.730261
$622$	−33.4843	−1.34260
$623$	3.26163	0.130675
$624$	20.3947	0.816441
$625$	22.3189	0.892756
$626$	13.1271	0.524663
$627$	−1.19027	−0.0475346
$628$	−4.69663	−0.187416
$629$	2.20707	0.0880017
$630$	0.227312	0.00905632
$631$	−30.0265	−1.19534	−0.597669	−	0.801743i	$-0.703906\pi$
$631$	−30.0265	−1.19534	−0.597669	+	0.801743i	$0.703906\pi$
$632$	7.03424	0.279807
$633$	1.92190	0.0763886
$634$	32.4032	1.28690
$635$	5.81733	0.230854
$636$	3.92694	0.155713
$637$	22.9803	0.910514
$638$	7.72756	0.305937
$639$	−9.99065	−0.395224
$640$	−2.93869	−0.116162
$641$	−17.0604	−0.673844	−0.336922	−	0.941533i	$-0.609386\pi$
$641$	−17.0604	−0.673844	−0.336922	+	0.941533i	$0.609386\pi$
$642$	28.2732	1.11585
$643$	4.84669	0.191135	0.0955674	−	0.995423i	$-0.469533\pi$
$643$	4.84669	0.191135	0.0955674	+	0.995423i	$0.469533\pi$
$644$	0.978115	0.0385431
$645$	−6.73483	−0.265184
$646$	−1.58349	−0.0623015
$647$	−12.5749	−0.494369	−0.247184	−	0.968968i	$-0.579505\pi$
$647$	−12.5749	−0.494369	−0.247184	+	0.968968i	$0.579505\pi$
$648$	−32.1522	−1.26306
$649$	2.36708	0.0929161
$650$	−21.1775	−0.830650
$651$	11.9652	0.468953
$652$	0.445696	0.0174548
$653$	−30.2366	−1.18325	−0.591624	−	0.806214i	$-0.701513\pi$
$653$	−30.2366	−1.18325	−0.591624	+	0.806214i	$0.701513\pi$
$654$	33.8577	1.32394
$655$	−7.22435	−0.282279
$656$	−36.4564	−1.42338
$657$	−0.871533	−0.0340017
$658$	6.02611	0.234922
$659$	18.1465	0.706888	0.353444	−	0.935456i	$-0.385011\pi$
$659$	18.1465	0.706888	0.353444	+	0.935456i	$0.385011\pi$
$660$	0.198527	0.00772764
$661$	34.9262	1.35847	0.679236	−	0.733920i	$-0.262311\pi$
$661$	34.9262	1.35847	0.679236	+	0.733920i	$0.262311\pi$
$662$	−43.2677	−1.68165
$663$	−8.31773	−0.323034
$664$	−13.9026	−0.539525
$665$	−0.258421	−0.0100212
$666$	−1.55461	−0.0602398
$667$	40.3999	1.56429
$668$	3.77196	0.145942
$669$	−26.2746	−1.01583
$670$	2.51520	0.0971705
$671$	1.00425	0.0387686
$672$	2.55003	0.0983694
$673$	30.4338	1.17314	0.586568	−	0.809900i	$-0.300479\pi$
$673$	30.4338	1.17314	0.586568	+	0.809900i	$0.300479\pi$
$674$	33.9346	1.30711
$675$	21.3605	0.822166
$676$	0.387913	0.0149197
$677$	19.5120	0.749906	0.374953	−	0.927044i	$-0.377659\pi$
$677$	19.5120	0.749906	0.374953	+	0.927044i	$0.377659\pi$
$678$	−42.8529	−1.64576
$679$	−7.21623	−0.276933
$680$	1.61118	0.0617860
$681$	5.48600	0.210224
$682$	−8.04719	−0.308143
$683$	−25.1469	−0.962218	−0.481109	−	0.876661i	$-0.659766\pi$
$683$	−25.1469	−0.962218	−0.481109	+	0.876661i	$0.659766\pi$
$684$	−0.272019	−0.0104009
$685$	−6.29636	−0.240572
$686$	−10.5011	−0.400932
$687$	−31.0233	−1.18361
$688$	25.2262	0.961741
$689$	18.0584	0.687970
$690$	−4.25576	−0.162014
$691$	30.8193	1.17242	0.586211	−	0.810158i	$-0.300619\pi$
$691$	30.8193	1.17242	0.586211	+	0.810158i	$0.300619\pi$
$692$	−2.15963	−0.0820968
$693$	0.261015	0.00991513
$694$	3.39268	0.128784
$695$	2.71781	0.103093
$696$	57.3646	2.17440
$697$	14.8683	0.563177
$698$	−11.7959	−0.446480
$699$	−27.3263	−1.03358
$700$	−1.14809	−0.0433939
$701$	−15.9533	−0.602549	−0.301275	−	0.953537i	$-0.597412\pi$
$701$	−15.9533	−0.602549	−0.301275	+	0.953537i	$0.597412\pi$
$702$	−19.4786	−0.735173
$703$	1.76737	0.0666576
$704$	−5.50769	−0.207579
$705$	6.39446	0.240829
$706$	−18.0319	−0.678641
$707$	−5.53839	−0.208293
$708$	2.88045	0.108254
$709$	24.8794	0.934367	0.467184	−	0.884160i	$-0.345269\pi$
$709$	24.8794	0.934367	0.467184	+	0.884160i	$0.345269\pi$
$710$	7.76776	0.291519
$711$	1.60870	0.0603310
$712$	−16.2840	−0.610270
$713$	−42.0709	−1.57557
$714$	1.84896	0.0691957
$715$	0.912943	0.0341421
$716$	−6.31382	−0.235959
$717$	23.0471	0.860711
$718$	−20.0292	−0.747483
$719$	−5.15131	−0.192112	−0.0960558	−	0.995376i	$-0.530623\pi$
$719$	−5.15131	−0.192112	−0.0960558	+	0.995376i	$0.530623\pi$
$720$	−0.903472	−0.0336704
$721$	12.1812	0.453651
$722$	−1.26802	−0.0471907
$723$	21.9830	0.817558
$724$	4.55827	0.169407
$725$	−47.4207	−1.76116
$726$	25.8723	0.960211
$727$	26.9955	1.00121	0.500603	−	0.865677i	$-0.333112\pi$
$727$	26.9955	1.00121	0.500603	+	0.865677i	$0.333112\pi$
$728$	6.38673	0.236708
$729$	18.2033	0.674196
$730$	0.677620	0.0250798
$731$	−10.2882	−0.380523
$732$	1.22205	0.0451683
$733$	1.91163	0.0706078	0.0353039	−	0.999377i	$-0.488760\pi$
$733$	1.91163	0.0706078	0.0353039	+	0.999377i	$0.488760\pi$
$734$	−20.9989	−0.775082
$735$	−5.42059	−0.199942
$736$	−8.96617	−0.330497
$737$	2.88812	0.106385
$738$	−10.4729	−0.385512
$739$	−34.4898	−1.26873	−0.634364	−	0.773034i	$-0.718738\pi$
$739$	−34.4898	−1.26873	−0.634364	+	0.773034i	$0.718738\pi$
$740$	−0.294783	−0.0108364
$741$	−6.66064	−0.244685
$742$	−4.01423	−0.147367
$743$	−8.45429	−0.310158	−0.155079	−	0.987902i	$-0.549563\pi$
$743$	−8.45429	−0.310158	−0.155079	+	0.987902i	$0.549563\pi$
$744$	−59.7374	−2.19008
$745$	4.37341	0.160229
$746$	−38.6508	−1.41511
$747$	−3.17946	−0.116330
$748$	0.303272	0.0110887
$749$	7.04859	0.257550
$750$	10.1782	0.371656
$751$	24.5290	0.895076	0.447538	−	0.894265i	$-0.352301\pi$
$751$	24.5290	0.895076	0.447538	+	0.894265i	$0.352301\pi$
$752$	−23.9513	−0.873416
$753$	31.6777	1.15440
$754$	43.2428	1.57481
$755$	−6.35699	−0.231355
$756$	−1.05599	−0.0384061
$757$	41.7004	1.51563	0.757813	−	0.652472i	$-0.226268\pi$
$757$	41.7004	1.51563	0.757813	+	0.652472i	$0.226268\pi$
$758$	27.7335	1.00732
$759$	−4.88675	−0.177378
$760$	1.29019	0.0468003
$761$	8.51276	0.308587	0.154294	−	0.988025i	$-0.450690\pi$
$761$	8.51276	0.308587	0.154294	+	0.988025i	$0.450690\pi$
$762$	−33.3299	−1.20742
$763$	8.44081	0.305578
$764$	−1.15581	−0.0418157
$765$	0.368470	0.0133221
$766$	−33.8660	−1.22363
$767$	13.2460	0.478286
$768$	−17.3465	−0.625937
$769$	−29.5933	−1.06716	−0.533582	−	0.845749i	$-0.679154\pi$
$769$	−29.5933	−1.06716	−0.533582	+	0.845749i	$0.679154\pi$
$770$	−0.202940	−0.00731344
$771$	−47.3225	−1.70428
$772$	−4.06917	−0.146453
$773$	−26.0110	−0.935550	−0.467775	−	0.883848i	$-0.654944\pi$
$773$	−26.0110	−0.935550	−0.467775	+	0.883848i	$0.654944\pi$
$774$	7.24677	0.260480
$775$	49.3821	1.77386
$776$	36.0278	1.29332
$777$	−2.06367	−0.0740339
$778$	13.8253	0.495661
$779$	11.9062	0.426583
$780$	1.11094	0.0397780
$781$	8.91946	0.319164
$782$	−6.50116	−0.232481
$783$	−43.6165	−1.55873
$784$	20.3036	0.725128
$785$	5.09450	0.181830
$786$	41.3913	1.47638
$787$	35.9508	1.28151	0.640755	−	0.767746i	$-0.278622\pi$
$787$	35.9508	1.28151	0.640755	+	0.767746i	$0.278622\pi$
$788$	5.80525	0.206803
$789$	−4.01645	−0.142989
$790$	−1.25077	−0.0445004
$791$	−10.6834	−0.379857
$792$	−1.30314	−0.0463052
$793$	5.61970	0.199562
$794$	8.22357	0.291844
$795$	−4.25960	−0.151073
$796$	8.25403	0.292556
$797$	5.96064	0.211137	0.105568	−	0.994412i	$-0.466334\pi$
$797$	5.96064	0.211137	0.105568	+	0.994412i	$0.466334\pi$
$798$	1.48061	0.0524129
$799$	9.76826	0.345576
$800$	10.5243	0.372092
$801$	−3.72409	−0.131584
$802$	−37.4640	−1.32290
$803$	0.778088	0.0274581
$804$	3.51449	0.123946
$805$	−1.06097	−0.0373944
$806$	−45.0315	−1.58617
$807$	−49.4850	−1.74195
$808$	27.6510	0.972757
$809$	−7.95914	−0.279828	−0.139914	−	0.990164i	$-0.544683\pi$
$809$	−7.95914	−0.279828	−0.139914	+	0.990164i	$0.544683\pi$
$810$	5.71704	0.200876
$811$	11.6607	0.409462	0.204731	−	0.978818i	$-0.434368\pi$
$811$	11.6607	0.409462	0.204731	+	0.978818i	$0.434368\pi$
$812$	2.34432	0.0822695
$813$	12.9812	0.455271
$814$	1.38792	0.0486467
$815$	−0.483452	−0.0169346
$816$	−7.34888	−0.257262
$817$	−8.23855	−0.288230
$818$	−31.0052	−1.08407
$819$	1.46062	0.0510382
$820$	−1.98585	−0.0693490
$821$	13.8681	0.484001	0.242000	−	0.970276i	$-0.422196\pi$
$821$	13.8681	0.484001	0.242000	+	0.970276i	$0.422196\pi$
$822$	36.0745	1.25824
$823$	−21.8911	−0.763074	−0.381537	−	0.924354i	$-0.624605\pi$
$823$	−21.8911	−0.763074	−0.381537	+	0.924354i	$0.624605\pi$
$824$	−60.8158	−2.11862
$825$	5.73598	0.199701
$826$	−2.94448	−0.102452
$827$	38.3680	1.33418	0.667092	−	0.744975i	$-0.267539\pi$
$827$	38.3680	1.33418	0.667092	+	0.744975i	$0.267539\pi$
$828$	−1.11680	−0.0388114
$829$	4.41778	0.153436	0.0767179	−	0.997053i	$-0.475556\pi$
$829$	4.41778	0.153436	0.0767179	+	0.997053i	$0.475556\pi$
$830$	2.47204	0.0858058
$831$	34.6036	1.20038
$832$	−30.8206	−1.06851
$833$	−8.28057	−0.286905
$834$	−15.5715	−0.539197
$835$	−4.09150	−0.141592
$836$	0.242853	0.00839924
$837$	45.4206	1.56997
$838$	−1.70774	−0.0589930
$839$	32.8176	1.13299	0.566495	−	0.824065i	$-0.308299\pi$
$839$	32.8176	1.13299	0.566495	+	0.824065i	$0.308299\pi$
$840$	−1.50650	−0.0519792
$841$	67.8294	2.33894
$842$	16.8493	0.580667
$843$	20.8555	0.718302
$844$	−0.392130	−0.0134977
$845$	−0.420774	−0.0144751
$846$	−6.88054	−0.236558
$847$	6.45004	0.221626
$848$	15.9549	0.547895
$849$	−61.8764	−2.12359
$850$	7.63095	0.261739
$851$	7.25610	0.248736
$852$	10.8539	0.371849
$853$	−27.7351	−0.949632	−0.474816	−	0.880085i	$-0.657485\pi$
$853$	−27.7351	−0.949632	−0.474816	+	0.880085i	$0.657485\pi$
$854$	−1.24921	−0.0427473
$855$	0.295062	0.0100909
$856$	−35.1908	−1.20280
$857$	25.2808	0.863575	0.431787	−	0.901975i	$-0.357883\pi$
$857$	25.2808	0.863575	0.431787	+	0.901975i	$0.357883\pi$
$858$	−5.23064	−0.178571
$859$	25.7104	0.877228	0.438614	−	0.898676i	$-0.355470\pi$
$859$	25.7104	0.877228	0.438614	+	0.898676i	$0.355470\pi$
$860$	1.37412	0.0468572
$861$	−13.9023	−0.473788
$862$	12.8047	0.436131
$863$	−18.6569	−0.635089	−0.317544	−	0.948243i	$-0.602858\pi$
$863$	−18.6569	−0.635089	−0.317544	+	0.948243i	$0.602858\pi$
$864$	9.68006	0.329322
$865$	2.34258	0.0796501
$866$	42.1005	1.43063
$867$	−29.6751	−1.00782
$868$	−2.44129	−0.0828627
$869$	−1.43622	−0.0487204
$870$	−10.2001	−0.345816
$871$	16.1617	0.547618
$872$	−42.1416	−1.42710
$873$	8.23940	0.278861
$874$	−5.20597	−0.176095
$875$	2.53746	0.0857818
$876$	0.946839	0.0319907
$877$	−2.35210	−0.0794249	−0.0397124	−	0.999211i	$-0.512644\pi$
$877$	−2.35210	−0.0794249	−0.0397124	+	0.999211i	$0.512644\pi$
$878$	43.4411	1.46607
$879$	24.3951	0.822827
$880$	0.806603	0.0271906
$881$	52.6462	1.77370	0.886848	−	0.462061i	$-0.152890\pi$
$881$	52.6462	1.77370	0.886848	+	0.462061i	$0.152890\pi$
$882$	5.83264	0.196395
$883$	36.8479	1.24003	0.620014	−	0.784590i	$-0.287127\pi$
$883$	36.8479	1.24003	0.620014	+	0.784590i	$0.287127\pi$
$884$	1.69709	0.0570792
$885$	−3.12446	−0.105028
$886$	−23.3348	−0.783946
$887$	−9.16068	−0.307586	−0.153793	−	0.988103i	$-0.549149\pi$
$887$	−9.16068	−0.307586	−0.153793	+	0.988103i	$0.549149\pi$
$888$	10.3031	0.345749
$889$	−8.30925	−0.278683
$890$	2.89549	0.0970571
$891$	6.56469	0.219925
$892$	5.36087	0.179495
$893$	7.82219	0.261760
$894$	−25.0571	−0.838036
$895$	6.84869	0.228926
$896$	4.19752	0.140229
$897$	−27.3459	−0.913053
$898$	10.9542	0.365546
$899$	−100.835	−3.36302
$900$	1.31088	0.0436960
$901$	−6.50703	−0.216781
$902$	9.34998	0.311320
$903$	9.61977	0.320126
$904$	53.3378	1.77399
$905$	−4.94441	−0.164358
$906$	36.4219	1.21004
$907$	2.29998	0.0763695	0.0381847	−	0.999271i	$-0.487842\pi$
$907$	2.29998	0.0763695	0.0381847	+	0.999271i	$0.487842\pi$
$908$	−1.11932	−0.0371461
$909$	6.32366	0.209742
$910$	−1.13564	−0.0376460
$911$	−47.1813	−1.56319	−0.781594	−	0.623788i	$-0.785593\pi$
$911$	−47.1813	−1.56319	−0.781594	+	0.623788i	$0.785593\pi$
$912$	−5.88480	−0.194865
$913$	2.83856	0.0939427
$914$	29.0801	0.961884
$915$	−1.32557	−0.0438221
$916$	6.32977	0.209141
$917$	10.3190	0.340763
$918$	7.01878	0.231654
$919$	34.0268	1.12244	0.561220	−	0.827667i	$-0.310332\pi$
$919$	34.0268	1.12244	0.561220	+	0.827667i	$0.310332\pi$
$920$	5.29702	0.174638
$921$	−17.1494	−0.565091
$922$	44.2778	1.45821
$923$	49.9126	1.64289
$924$	−0.283568	−0.00932870
$925$	−8.51709	−0.280040
$926$	−10.5759	−0.347546
$927$	−13.9083	−0.456809
$928$	−21.4899	−0.705440
$929$	34.0386	1.11677	0.558385	−	0.829582i	$-0.311421\pi$
$929$	34.0386	1.11677	0.558385	+	0.829582i	$0.311421\pi$
$930$	10.6220	0.348309
$931$	−6.63088	−0.217318
$932$	5.57546	0.182630
$933$	50.7512	1.66152
$934$	14.9056	0.487725
$935$	−0.328963	−0.0107582
$936$	−7.29229	−0.238356
$937$	34.5653	1.12920	0.564599	−	0.825365i	$-0.309031\pi$
$937$	34.5653	1.12920	0.564599	+	0.825365i	$0.309031\pi$
$938$	−3.59261	−0.117303
$939$	−19.8963	−0.649292
$940$	−1.30468	−0.0425539
$941$	4.89097	0.159441	0.0797205	−	0.996817i	$-0.474597\pi$
$941$	4.89097	0.159441	0.0797205	+	0.996817i	$0.474597\pi$
$942$	−29.1885	−0.951014
$943$	48.8819	1.59181
$944$	11.7031	0.380904
$945$	1.14545	0.0372615
$946$	−6.46978	−0.210351
$947$	15.2982	0.497123	0.248562	−	0.968616i	$-0.420042\pi$
$947$	15.2982	0.497123	0.248562	+	0.968616i	$0.420042\pi$
$948$	−1.74770	−0.0567627
$949$	4.35412	0.141341
$950$	6.11068	0.198257
$951$	−49.1127	−1.59259
$952$	−2.30135	−0.0745872
$953$	−5.62811	−0.182312	−0.0911562	−	0.995837i	$-0.529056\pi$
$953$	−5.62811	−0.182312	−0.0911562	+	0.995837i	$0.529056\pi$
$954$	4.58340	0.148393
$955$	1.25372	0.0405695
$956$	−4.70237	−0.152085
$957$	−11.7124	−0.378609
$958$	48.8541	1.57840
$959$	8.99349	0.290415
$960$	7.26995	0.234637
$961$	74.0053	2.38727
$962$	7.76672	0.250409
$963$	−8.04799	−0.259343
$964$	−4.48526	−0.144460
$965$	4.41388	0.142088
$966$	6.07877	0.195581
$967$	30.2510	0.972808	0.486404	−	0.873734i	$-0.338308\pi$
$967$	30.2510	0.972808	0.486404	+	0.873734i	$0.338308\pi$
$968$	−32.2025	−1.03503
$969$	2.40005	0.0771006
$970$	−6.40616	−0.205689
$971$	18.7888	0.602961	0.301480	−	0.953472i	$-0.402519\pi$
$971$	18.7888	0.602961	0.301480	+	0.953472i	$0.402519\pi$
$972$	2.77409	0.0889791
$973$	−3.88202	−0.124452
$974$	31.4283	1.00703
$975$	32.0981	1.02796
$976$	4.96512	0.158930
$977$	10.4023	0.332799	0.166400	−	0.986058i	$-0.446786\pi$
$977$	10.4023	0.332799	0.166400	+	0.986058i	$0.446786\pi$
$978$	2.76990	0.0885717
$979$	3.32480	0.106261
$980$	1.10598	0.0353292
$981$	−9.63761	−0.307705
$982$	−15.2899	−0.487920
$983$	18.3650	0.585752	0.292876	−	0.956150i	$-0.405388\pi$
$983$	18.3650	0.585752	0.292876	+	0.956150i	$0.405388\pi$
$984$	69.4085	2.21266
$985$	−6.29703	−0.200640
$986$	−15.5818	−0.496226
$987$	−9.13361	−0.290726
$988$	1.35899	0.0432351
$989$	−33.8242	−1.07555
$990$	0.231714	0.00736435
$991$	−11.3202	−0.359599	−0.179800	−	0.983703i	$-0.557545\pi$
$991$	−11.3202	−0.359599	−0.179800	+	0.983703i	$0.557545\pi$
$992$	22.3788	0.710527
$993$	65.5797	2.08111
$994$	−11.0952	−0.351918
$995$	−8.95325	−0.283837
$996$	3.45419	0.109450
$997$	−14.0140	−0.443827	−0.221913	−	0.975066i	$-0.571230\pi$
$997$	−14.0140	−0.443827	−0.221913	+	0.975066i	$0.571230\pi$
$998$	45.2372	1.43196
$999$	−7.83384	−0.247852

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.24	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.24	✓	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)

\(f(q)\)	\(=\)	\(q-1.26802 q^{2} +1.92190 q^{3} -0.392130 q^{4} +0.425349 q^{5} -2.43700 q^{6} -0.607552 q^{7} +3.03326 q^{8} +0.693695 q^{9} +O(q^{10})\)	\(q-1.26802 q^{2} +1.92190 q^{3} -0.392130 q^{4} +0.425349 q^{5} -2.43700 q^{6} -0.607552 q^{7} +3.03326 q^{8} +0.693695 q^{9} -0.539350 q^{10} -0.619317 q^{11} -0.753634 q^{12} -3.46565 q^{13} +0.770387 q^{14} +0.817477 q^{15} -3.06197 q^{16} +1.24879 q^{17} -0.879617 q^{18} +1.00000 q^{19} -0.166792 q^{20} -1.16765 q^{21} +0.785306 q^{22} +4.10560 q^{23} +5.82963 q^{24} -4.81908 q^{25} +4.39451 q^{26} -4.43249 q^{27} +0.238239 q^{28} +9.84019 q^{29} -1.03658 q^{30} -10.2472 q^{31} -2.18389 q^{32} -1.19027 q^{33} -1.58349 q^{34} -0.258421 q^{35} -0.272019 q^{36} +1.76737 q^{37} -1.26802 q^{38} -6.66064 q^{39} +1.29019 q^{40} +11.9062 q^{41} +1.48061 q^{42} -8.23855 q^{43} +0.242853 q^{44} +0.295062 q^{45} -5.20597 q^{46} +7.82219 q^{47} -5.88480 q^{48} -6.63088 q^{49} +6.11068 q^{50} +2.40005 q^{51} +1.35899 q^{52} -5.21067 q^{53} +5.62047 q^{54} -0.263426 q^{55} -1.84287 q^{56} +1.92190 q^{57} -12.4775 q^{58} -3.82208 q^{59} -0.320557 q^{60} -1.62154 q^{61} +12.9936 q^{62} -0.421456 q^{63} +8.89316 q^{64} -1.47411 q^{65} +1.50928 q^{66} -4.66338 q^{67} -0.489688 q^{68} +7.89054 q^{69} +0.327683 q^{70} -14.4021 q^{71} +2.10416 q^{72} -1.25636 q^{73} -2.24106 q^{74} -9.26178 q^{75} -0.392130 q^{76} +0.376267 q^{77} +8.44581 q^{78} +2.31903 q^{79} -1.30241 q^{80} -10.5999 q^{81} -15.0972 q^{82} -4.58337 q^{83} +0.457872 q^{84} +0.531171 q^{85} +10.4466 q^{86} +18.9119 q^{87} -1.87855 q^{88} -5.36849 q^{89} -0.374144 q^{90} +2.10556 q^{91} -1.60993 q^{92} -19.6941 q^{93} -9.91868 q^{94} +0.425349 q^{95} -4.19722 q^{96} +11.8776 q^{97} +8.40808 q^{98} -0.429617 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 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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