LMFDB - Embedded newform 8007.2.a.f.1.42

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.9362168984$
Analytic rank:	$1$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.42
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.18706 q^{2} -1.00000 q^{3} +2.78321 q^{4} +0.653611 q^{5} -2.18706 q^{6} +0.500468 q^{7} +1.71293 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.18706 q^{2} -1.00000 q^{3} +2.78321 q^{4} +0.653611 q^{5} -2.18706 q^{6} +0.500468 q^{7} +1.71293 q^{8} +1.00000 q^{9} +1.42948 q^{10} -2.54739 q^{11} -2.78321 q^{12} +4.91435 q^{13} +1.09455 q^{14} -0.653611 q^{15} -1.82016 q^{16} -1.00000 q^{17} +2.18706 q^{18} -5.72996 q^{19} +1.81914 q^{20} -0.500468 q^{21} -5.57128 q^{22} +1.05121 q^{23} -1.71293 q^{24} -4.57279 q^{25} +10.7480 q^{26} -1.00000 q^{27} +1.39291 q^{28} -6.02019 q^{29} -1.42948 q^{30} +5.10176 q^{31} -7.40664 q^{32} +2.54739 q^{33} -2.18706 q^{34} +0.327111 q^{35} +2.78321 q^{36} -4.78322 q^{37} -12.5317 q^{38} -4.91435 q^{39} +1.11959 q^{40} -5.46543 q^{41} -1.09455 q^{42} -4.78285 q^{43} -7.08993 q^{44} +0.653611 q^{45} +2.29907 q^{46} -11.0936 q^{47} +1.82016 q^{48} -6.74953 q^{49} -10.0010 q^{50} +1.00000 q^{51} +13.6777 q^{52} +6.18793 q^{53} -2.18706 q^{54} -1.66500 q^{55} +0.857265 q^{56} +5.72996 q^{57} -13.1665 q^{58} +3.15201 q^{59} -1.81914 q^{60} -1.40256 q^{61} +11.1578 q^{62} +0.500468 q^{63} -12.5584 q^{64} +3.21207 q^{65} +5.57128 q^{66} +12.7398 q^{67} -2.78321 q^{68} -1.05121 q^{69} +0.715410 q^{70} +12.0679 q^{71} +1.71293 q^{72} -12.6838 q^{73} -10.4612 q^{74} +4.57279 q^{75} -15.9477 q^{76} -1.27489 q^{77} -10.7480 q^{78} +5.61999 q^{79} -1.18967 q^{80} +1.00000 q^{81} -11.9532 q^{82} -7.83774 q^{83} -1.39291 q^{84} -0.653611 q^{85} -10.4604 q^{86} +6.02019 q^{87} -4.36349 q^{88} +14.2306 q^{89} +1.42948 q^{90} +2.45947 q^{91} +2.92575 q^{92} -5.10176 q^{93} -24.2623 q^{94} -3.74516 q^{95} +7.40664 q^{96} -9.67368 q^{97} -14.7616 q^{98} -2.54739 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) $ 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.18706	1.54648	0.773241	−	0.634112i	$-0.218634\pi$
$2$	2.18706	1.54648	0.773241	+	0.634112i	$0.218634\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.78321	1.39161
$5$	0.653611	0.292304	0.146152	−	0.989262i	$-0.453311\pi$
$5$	0.653611	0.292304	0.146152	+	0.989262i	$0.453311\pi$
$6$	−2.18706	−0.892862
$7$	0.500468	0.189159	0.0945795	−	0.995517i	$-0.469849\pi$
$7$	0.500468	0.189159	0.0945795	+	0.995517i	$0.469849\pi$
$8$	1.71293	0.605611
$9$	1.00000	0.333333
$10$	1.42948	0.452042
$11$	−2.54739	−0.768067	−0.384034	−	0.923319i	$-0.625465\pi$
$11$	−2.54739	−0.768067	−0.384034	+	0.923319i	$0.625465\pi$
$12$	−2.78321	−0.803444
$13$	4.91435	1.36300	0.681498	−	0.731820i	$-0.261329\pi$
$13$	4.91435	1.36300	0.681498	+	0.731820i	$0.261329\pi$
$14$	1.09455	0.292531
$15$	−0.653611	−0.168762
$16$	−1.82016	−0.455039
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	2.18706	0.515494
$19$	−5.72996	−1.31454	−0.657271	−	0.753654i	$-0.728289\pi$
$19$	−5.72996	−1.31454	−0.657271	+	0.753654i	$0.728289\pi$
$20$	1.81914	0.406771
$21$	−0.500468	−0.109211
$22$	−5.57128	−1.18780
$23$	1.05121	0.219193	0.109597	−	0.993976i	$-0.465044\pi$
$23$	1.05121	0.219193	0.109597	+	0.993976i	$0.465044\pi$
$24$	−1.71293	−0.349650
$25$	−4.57279	−0.914559
$26$	10.7480	2.10785
$27$	−1.00000	−0.192450
$28$	1.39291	0.263235
$29$	−6.02019	−1.11792	−0.558961	−	0.829194i	$-0.688800\pi$
$29$	−6.02019	−1.11792	−0.558961	+	0.829194i	$0.688800\pi$
$30$	−1.42948	−0.260987
$31$	5.10176	0.916302	0.458151	−	0.888874i	$-0.348512\pi$
$31$	5.10176	0.916302	0.458151	+	0.888874i	$0.348512\pi$
$32$	−7.40664	−1.30932
$33$	2.54739	0.443444
$34$	−2.18706	−0.375077
$35$	0.327111	0.0552919
$36$	2.78321	0.463869
$37$	−4.78322	−0.786357	−0.393179	−	0.919462i	$-0.628625\pi$
$37$	−4.78322	−0.786357	−0.393179	+	0.919462i	$0.628625\pi$
$38$	−12.5317	−2.03292
$39$	−4.91435	−0.786926
$40$	1.11959	0.177022
$41$	−5.46543	−0.853557	−0.426779	−	0.904356i	$-0.640352\pi$
$41$	−5.46543	−0.853557	−0.426779	+	0.904356i	$0.640352\pi$
$42$	−1.09455	−0.168893
$43$	−4.78285	−0.729377	−0.364689	−	0.931130i	$-0.618825\pi$
$43$	−4.78285	−0.729377	−0.364689	+	0.931130i	$0.618825\pi$
$44$	−7.08993	−1.06885
$45$	0.653611	0.0974345
$46$	2.29907	0.338979
$47$	−11.0936	−1.61817	−0.809084	−	0.587692i	$-0.800037\pi$
$47$	−11.0936	−1.61817	−0.809084	+	0.587692i	$0.800037\pi$
$48$	1.82016	0.262717
$49$	−6.74953	−0.964219
$50$	−10.0010	−1.41435
$51$	1.00000	0.140028
$52$	13.6777	1.89675
$53$	6.18793	0.849978	0.424989	−	0.905199i	$-0.360278\pi$
$53$	6.18793	0.849978	0.424989	+	0.905199i	$0.360278\pi$
$54$	−2.18706	−0.297621
$55$	−1.66500	−0.224509
$56$	0.857265	0.114557
$57$	5.72996	0.758951
$58$	−13.1665	−1.72884
$59$	3.15201	0.410357	0.205178	−	0.978725i	$-0.434223\pi$
$59$	3.15201	0.410357	0.205178	+	0.978725i	$0.434223\pi$
$60$	−1.81914	−0.234850
$61$	−1.40256	−0.179580	−0.0897899	−	0.995961i	$-0.528620\pi$
$61$	−1.40256	−0.179580	−0.0897899	+	0.995961i	$0.528620\pi$
$62$	11.1578	1.41705
$63$	0.500468	0.0630530
$64$	−12.5584	−1.56980
$65$	3.21207	0.398409
$66$	5.57128	0.685778
$67$	12.7398	1.55641	0.778207	−	0.628007i	$-0.216129\pi$
$67$	12.7398	1.55641	0.778207	+	0.628007i	$0.216129\pi$
$68$	−2.78321	−0.337514
$69$	−1.05121	−0.126551
$70$	0.715410	0.0855078
$71$	12.0679	1.43220	0.716100	−	0.697998i	$-0.245925\pi$
$71$	12.0679	1.43220	0.716100	+	0.697998i	$0.245925\pi$
$72$	1.71293	0.201870
$73$	−12.6838	−1.48453	−0.742264	−	0.670107i	$-0.766248\pi$
$73$	−12.6838	−1.48453	−0.742264	+	0.670107i	$0.766248\pi$
$74$	−10.4612	−1.21609
$75$	4.57279	0.528021
$76$	−15.9477	−1.82932
$77$	−1.27489	−0.145287
$78$	−10.7480	−1.21697
$79$	5.61999	0.632298	0.316149	−	0.948710i	$-0.397610\pi$
$79$	5.61999	0.632298	0.316149	+	0.948710i	$0.397610\pi$
$80$	−1.18967	−0.133010
$81$	1.00000	0.111111
$82$	−11.9532	−1.32001
$83$	−7.83774	−0.860304	−0.430152	−	0.902757i	$-0.641540\pi$
$83$	−7.83774	−0.860304	−0.430152	+	0.902757i	$0.641540\pi$
$84$	−1.39291	−0.151979
$85$	−0.653611	−0.0708940
$86$	−10.4604	−1.12797
$87$	6.02019	0.645432
$88$	−4.36349	−0.465150
$89$	14.2306	1.50844	0.754220	−	0.656622i	$-0.228015\pi$
$89$	14.2306	1.50844	0.754220	+	0.656622i	$0.228015\pi$
$90$	1.42948	0.150681
$91$	2.45947	0.257823
$92$	2.92575	0.305031
$93$	−5.10176	−0.529027
$94$	−24.2623	−2.50247
$95$	−3.74516	−0.384245
$96$	7.40664	0.755937
$97$	−9.67368	−0.982214	−0.491107	−	0.871099i	$-0.663408\pi$
$97$	−9.67368	−0.982214	−0.491107	+	0.871099i	$0.663408\pi$
$98$	−14.7616	−1.49115
$99$	−2.54739	−0.256022
$100$	−12.7271	−1.27271
$101$	8.58394	0.854134	0.427067	−	0.904220i	$-0.359547\pi$
$101$	8.58394	0.854134	0.427067	+	0.904220i	$0.359547\pi$
$102$	2.18706	0.216551
$103$	2.72907	0.268903	0.134451	−	0.990920i	$-0.457073\pi$
$103$	2.72907	0.268903	0.134451	+	0.990920i	$0.457073\pi$
$104$	8.41793	0.825446
$105$	−0.327111	−0.0319228
$106$	13.5334	1.31448
$107$	7.52449	0.727420	0.363710	−	0.931512i	$-0.381510\pi$
$107$	7.52449	0.727420	0.363710	+	0.931512i	$0.381510\pi$
$108$	−2.78321	−0.267815
$109$	10.9829	1.05197	0.525984	−	0.850495i	$-0.323697\pi$
$109$	10.9829	1.05197	0.525984	+	0.850495i	$0.323697\pi$
$110$	−3.64145	−0.347199
$111$	4.78322	0.454004
$112$	−0.910929	−0.0860747
$113$	−16.2874	−1.53219	−0.766096	−	0.642726i	$-0.777803\pi$
$113$	−16.2874	−1.53219	−0.766096	+	0.642726i	$0.777803\pi$
$114$	12.5317	1.17370
$115$	0.687085	0.0640710
$116$	−16.7555	−1.55571
$117$	4.91435	0.454332
$118$	6.89362	0.634609
$119$	−0.500468	−0.0458778
$120$	−1.11959	−0.102204
$121$	−4.51080	−0.410073
$122$	−3.06748	−0.277717
$123$	5.46543	0.492802
$124$	14.1993	1.27513
$125$	−6.25688	−0.559632
$126$	1.09455	0.0975103
$127$	−4.90528	−0.435274	−0.217637	−	0.976030i	$-0.569835\pi$
$127$	−4.90528	−0.435274	−0.217637	+	0.976030i	$0.569835\pi$
$128$	−12.6527	−1.11835
$129$	4.78285	0.421106
$130$	7.02499	0.616132
$131$	−20.1165	−1.75759	−0.878794	−	0.477202i	$-0.841651\pi$
$131$	−20.1165	−1.75759	−0.878794	+	0.477202i	$0.841651\pi$
$132$	7.08993	0.617099
$133$	−2.86766	−0.248657
$134$	27.8627	2.40697
$135$	−0.653611	−0.0562539
$136$	−1.71293	−0.146882
$137$	19.7766	1.68963	0.844813	−	0.535062i	$-0.179712\pi$
$137$	19.7766	1.68963	0.844813	+	0.535062i	$0.179712\pi$
$138$	−2.29907	−0.195709
$139$	1.33900	0.113573	0.0567863	−	0.998386i	$-0.481915\pi$
$139$	1.33900	0.113573	0.0567863	+	0.998386i	$0.481915\pi$
$140$	0.910419	0.0769445
$141$	11.0936	0.934250
$142$	26.3932	2.21487
$143$	−12.5188	−1.04687
$144$	−1.82016	−0.151680
$145$	−3.93486	−0.326772
$146$	−27.7402	−2.29580
$147$	6.74953	0.556692
$148$	−13.3127	−1.09430
$149$	−14.0402	−1.15022	−0.575108	−	0.818078i	$-0.695040\pi$
$149$	−14.0402	−1.15022	−0.575108	+	0.818078i	$0.695040\pi$
$150$	10.0010	0.816574
$151$	−4.90072	−0.398815	−0.199407	−	0.979917i	$-0.563902\pi$
$151$	−4.90072	−0.398815	−0.199407	+	0.979917i	$0.563902\pi$
$152$	−9.81500	−0.796102
$153$	−1.00000	−0.0808452
$154$	−2.78825	−0.224683
$155$	3.33456	0.267839
$156$	−13.6777	−1.09509
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	12.2912	0.977837
$159$	−6.18793	−0.490735
$160$	−4.84106	−0.382719
$161$	0.526099	0.0414624
$162$	2.18706	0.171831
$163$	−13.2937	−1.04125	−0.520623	−	0.853787i	$-0.674300\pi$
$163$	−13.2937	−1.04125	−0.520623	+	0.853787i	$0.674300\pi$
$164$	−15.2115	−1.18782
$165$	1.66500	0.129620
$166$	−17.1416	−1.33044
$167$	2.38099	0.184246	0.0921231	−	0.995748i	$-0.470635\pi$
$167$	2.38099	0.184246	0.0921231	+	0.995748i	$0.470635\pi$
$168$	−0.857265	−0.0661394
$169$	11.1509	0.857759
$170$	−1.42948	−0.109636
$171$	−5.72996	−0.438181
$172$	−13.3117	−1.01501
$173$	13.3967	1.01853	0.509264	−	0.860610i	$-0.329918\pi$
$173$	13.3967	1.01853	0.509264	+	0.860610i	$0.329918\pi$
$174$	13.1665	0.998149
$175$	−2.28853	−0.172997
$176$	4.63665	0.349500
$177$	−3.15201	−0.236920
$178$	31.1231	2.33277
$179$	−25.4522	−1.90239	−0.951193	−	0.308596i	$-0.900141\pi$
$179$	−25.4522	−1.90239	−0.951193	+	0.308596i	$0.900141\pi$
$180$	1.81914	0.135590
$181$	−16.9026	−1.25636	−0.628179	−	0.778069i	$-0.716199\pi$
$181$	−16.9026	−1.25636	−0.628179	+	0.778069i	$0.716199\pi$
$182$	5.37901	0.398719
$183$	1.40256	0.103680
$184$	1.80065	0.132746
$185$	−3.12637	−0.229855
$186$	−11.1578	−0.818131
$187$	2.54739	0.186284
$188$	−30.8759	−2.25185
$189$	−0.500468	−0.0364037
$190$	−8.19088	−0.594229
$191$	−10.0403	−0.726493	−0.363246	−	0.931693i	$-0.618332\pi$
$191$	−10.0403	−0.726493	−0.363246	+	0.931693i	$0.618332\pi$
$192$	12.5584	0.906325
$193$	−25.7474	−1.85334	−0.926670	−	0.375876i	$-0.877342\pi$
$193$	−25.7474	−1.85334	−0.926670	+	0.375876i	$0.877342\pi$
$194$	−21.1569	−1.51898
$195$	−3.21207	−0.230021
$196$	−18.7854	−1.34181
$197$	−12.4216	−0.885004	−0.442502	−	0.896768i	$-0.645909\pi$
$197$	−12.4216	−0.885004	−0.442502	+	0.896768i	$0.645909\pi$
$198$	−5.57128	−0.395934
$199$	−5.06040	−0.358722	−0.179361	−	0.983783i	$-0.557403\pi$
$199$	−5.06040	−0.358722	−0.179361	+	0.983783i	$0.557403\pi$
$200$	−7.83286	−0.553867
$201$	−12.7398	−0.898596
$202$	18.7735	1.32090
$203$	−3.01291	−0.211465
$204$	2.78321	0.194864
$205$	−3.57227	−0.249498
$206$	5.96862	0.415853
$207$	1.05121	0.0730645
$208$	−8.94489	−0.620217
$209$	14.5964	1.00966
$210$	−0.715410	−0.0493680
$211$	0.403603	0.0277852	0.0138926	−	0.999903i	$-0.495578\pi$
$211$	0.403603	0.0277852	0.0138926	+	0.999903i	$0.495578\pi$
$212$	17.2223	1.18283
$213$	−12.0679	−0.826881
$214$	16.4565	1.12494
$215$	−3.12612	−0.213200
$216$	−1.71293	−0.116550
$217$	2.55326	0.173327
$218$	24.0201	1.62685
$219$	12.6838	0.857093
$220$	−4.63405	−0.312428
$221$	−4.91435	−0.330575
$222$	10.4612	0.702108
$223$	3.09427	0.207208	0.103604	−	0.994619i	$-0.466963\pi$
$223$	3.09427	0.207208	0.103604	+	0.994619i	$0.466963\pi$
$224$	−3.70678	−0.247670
$225$	−4.57279	−0.304853
$226$	−35.6215	−2.36951
$227$	−13.5590	−0.899942	−0.449971	−	0.893043i	$-0.648566\pi$
$227$	−13.5590	−0.899942	−0.449971	+	0.893043i	$0.648566\pi$
$228$	15.9477	1.05616
$229$	−0.0238185	−0.00157397	−0.000786985	−	1.00000i	$-0.500251\pi$
$229$	−0.0238185	−0.00157397	−0.000786985		1.00000i	$0.500251\pi$
$230$	1.50269	0.0990847
$231$	1.27489	0.0838814
$232$	−10.3121	−0.677026
$233$	8.96522	0.587331	0.293666	−	0.955908i	$-0.405125\pi$
$233$	8.96522	0.587331	0.293666	+	0.955908i	$0.405125\pi$
$234$	10.7480	0.702616
$235$	−7.25090	−0.472997
$236$	8.77271	0.571055
$237$	−5.61999	−0.365057
$238$	−1.09455	−0.0709492
$239$	19.3263	1.25011	0.625057	−	0.780579i	$-0.285076\pi$
$239$	19.3263	1.25011	0.625057	+	0.780579i	$0.285076\pi$
$240$	1.18967	0.0767931
$241$	−8.63121	−0.555985	−0.277992	−	0.960583i	$-0.589669\pi$
$241$	−8.63121	−0.555985	−0.277992	+	0.960583i	$0.589669\pi$
$242$	−9.86538	−0.634170
$243$	−1.00000	−0.0641500
$244$	−3.90363	−0.249904
$245$	−4.41157	−0.281845
$246$	11.9532	0.762109
$247$	−28.1590	−1.79172
$248$	8.73894	0.554923
$249$	7.83774	0.496697
$250$	−13.6841	−0.865461
$251$	17.1195	1.08057	0.540286	−	0.841482i	$-0.318316\pi$
$251$	17.1195	1.08057	0.540286	+	0.841482i	$0.318316\pi$
$252$	1.39291	0.0877449
$253$	−2.67785	−0.168355
$254$	−10.7281	−0.673143
$255$	0.653611	0.0409307
$256$	−2.55527	−0.159705
$257$	9.60054	0.598865	0.299433	−	0.954117i	$-0.403203\pi$
$257$	9.60054	0.598865	0.299433	+	0.954117i	$0.403203\pi$
$258$	10.4604	0.651233
$259$	−2.39385	−0.148747
$260$	8.93988	0.554428
$261$	−6.02019	−0.372640
$262$	−43.9959	−2.71808
$263$	10.1839	0.627968	0.313984	−	0.949428i	$-0.398336\pi$
$263$	10.1839	0.627968	0.313984	+	0.949428i	$0.398336\pi$
$264$	4.36349	0.268555
$265$	4.04450	0.248452
$266$	−6.27173	−0.384544
$267$	−14.2306	−0.870898
$268$	35.4576	2.16592
$269$	9.29959	0.567006	0.283503	−	0.958971i	$-0.408503\pi$
$269$	9.29959	0.567006	0.283503	+	0.958971i	$0.408503\pi$
$270$	−1.42948	−0.0869956
$271$	−14.8849	−0.904191	−0.452096	−	0.891969i	$-0.649323\pi$
$271$	−14.8849	−0.904191	−0.452096	+	0.891969i	$0.649323\pi$
$272$	1.82016	0.110363
$273$	−2.45947	−0.148854
$274$	43.2524	2.61298
$275$	11.6487	0.702442
$276$	−2.92575	−0.176110
$277$	1.34206	0.0806365	0.0403183	−	0.999187i	$-0.487163\pi$
$277$	1.34206	0.0806365	0.0403183	+	0.999187i	$0.487163\pi$
$278$	2.92847	0.175638
$279$	5.10176	0.305434
$280$	0.560317	0.0334854
$281$	21.8140	1.30132	0.650658	−	0.759371i	$-0.274493\pi$
$281$	21.8140	1.30132	0.650658	+	0.759371i	$0.274493\pi$
$282$	24.2623	1.44480
$283$	31.5233	1.87387	0.936933	−	0.349508i	$-0.113651\pi$
$283$	31.5233	1.87387	0.936933	+	0.349508i	$0.113651\pi$
$284$	33.5876	1.99306
$285$	3.74516	0.221844
$286$	−27.3793	−1.61897
$287$	−2.73527	−0.161458
$288$	−7.40664	−0.436440
$289$	1.00000	0.0588235
$290$	−8.60576	−0.505348
$291$	9.67368	0.567081
$292$	−35.3018	−2.06588
$293$	−4.29913	−0.251158	−0.125579	−	0.992084i	$-0.540079\pi$
$293$	−4.29913	−0.251158	−0.125579	+	0.992084i	$0.540079\pi$
$294$	14.7616	0.860914
$295$	2.06019	0.119949
$296$	−8.19332	−0.476227
$297$	2.54739	0.147815
$298$	−30.7066	−1.77879
$299$	5.16604	0.298760
$300$	12.7271	0.734797
$301$	−2.39366	−0.137968
$302$	−10.7181	−0.616760
$303$	−8.58394	−0.493134
$304$	10.4294	0.598168
$305$	−0.916730	−0.0524918
$306$	−2.18706	−0.125026
$307$	−1.20554	−0.0688039	−0.0344020	−	0.999408i	$-0.510953\pi$
$307$	−1.20554	−0.0688039	−0.0344020	+	0.999408i	$0.510953\pi$
$308$	−3.54828	−0.202182
$309$	−2.72907	−0.155251
$310$	7.29287	0.414207
$311$	4.70857	0.266998	0.133499	−	0.991049i	$-0.457379\pi$
$311$	4.70857	0.266998	0.133499	+	0.991049i	$0.457379\pi$
$312$	−8.41793	−0.476572
$313$	−13.8515	−0.782935	−0.391468	−	0.920192i	$-0.628033\pi$
$313$	−13.8515	−0.782935	−0.391468	+	0.920192i	$0.628033\pi$
$314$	−2.18706	−0.123423
$315$	0.327111	0.0184306
$316$	15.6416	0.879910
$317$	4.06176	0.228131	0.114066	−	0.993473i	$-0.463613\pi$
$317$	4.06176	0.228131	0.114066	+	0.993473i	$0.463613\pi$
$318$	−13.5334	−0.758913
$319$	15.3358	0.858639
$320$	−8.20832	−0.458859
$321$	−7.52449	−0.419976
$322$	1.15061	0.0641209
$323$	5.72996	0.318823
$324$	2.78321	0.154623
$325$	−22.4723	−1.24654
$326$	−29.0741	−1.61027
$327$	−10.9829	−0.607354
$328$	−9.36189	−0.516924
$329$	−5.55199	−0.306091
$330$	3.64145	0.200455
$331$	19.0799	1.04872	0.524362	−	0.851495i	$-0.324304\pi$
$331$	19.0799	1.04872	0.524362	+	0.851495i	$0.324304\pi$
$332$	−21.8141	−1.19720
$333$	−4.78322	−0.262119
$334$	5.20735	0.284934
$335$	8.32687	0.454946
$336$	0.910929	0.0496953
$337$	−21.4920	−1.17074	−0.585372	−	0.810765i	$-0.699051\pi$
$337$	−21.4920	−1.17074	−0.585372	+	0.810765i	$0.699051\pi$
$338$	24.3876	1.32651
$339$	16.2874	0.884611
$340$	−1.81914	−0.0986566
$341$	−12.9962	−0.703782
$342$	−12.5317	−0.677638
$343$	−6.88120	−0.371550
$344$	−8.19267	−0.441719
$345$	−0.687085	−0.0369914
$346$	29.2992	1.57514
$347$	30.1618	1.61917	0.809584	−	0.587004i	$-0.199693\pi$
$347$	30.1618	1.61917	0.809584	+	0.587004i	$0.199693\pi$
$348$	16.7555	0.898187
$349$	−29.9483	−1.60310	−0.801549	−	0.597929i	$-0.795990\pi$
$349$	−29.9483	−1.60310	−0.801549	+	0.597929i	$0.795990\pi$
$350$	−5.00515	−0.267537
$351$	−4.91435	−0.262309
$352$	18.8676	1.00565
$353$	17.6190	0.937767	0.468883	−	0.883260i	$-0.344656\pi$
$353$	17.6190	0.937767	0.468883	+	0.883260i	$0.344656\pi$
$354$	−6.89362	−0.366392
$355$	7.88773	0.418637
$356$	39.6067	2.09915
$357$	0.500468	0.0264876
$358$	−55.6654	−2.94201
$359$	31.4114	1.65783	0.828915	−	0.559375i	$-0.188959\pi$
$359$	31.4114	1.65783	0.828915	+	0.559375i	$0.188959\pi$
$360$	1.11959	0.0590075
$361$	13.8324	0.728021
$362$	−36.9668	−1.94293
$363$	4.51080	0.236756
$364$	6.84524	0.358788
$365$	−8.29028	−0.433933
$366$	3.06748	0.160340
$367$	−11.0200	−0.575238	−0.287619	−	0.957745i	$-0.592864\pi$
$367$	−11.0200	−0.575238	−0.287619	+	0.957745i	$0.592864\pi$
$368$	−1.91338	−0.0997416
$369$	−5.46543	−0.284519
$370$	−6.83754	−0.355467
$371$	3.09686	0.160781
$372$	−14.1993	−0.736198
$373$	−30.1282	−1.55998	−0.779989	−	0.625793i	$-0.784775\pi$
$373$	−30.1282	−1.55998	−0.779989	+	0.625793i	$0.784775\pi$
$374$	5.57128	0.288084
$375$	6.25688	0.323104
$376$	−19.0025	−0.979981
$377$	−29.5853	−1.52372
$378$	−1.09455	−0.0562976
$379$	14.2591	0.732440	0.366220	−	0.930528i	$-0.380652\pi$
$379$	14.2591	0.732440	0.366220	+	0.930528i	$0.380652\pi$
$380$	−10.4236	−0.534718
$381$	4.90528	0.251305
$382$	−21.9588	−1.12351
$383$	36.1063	1.84494	0.922472	−	0.386063i	$-0.126165\pi$
$383$	36.1063	1.84494	0.922472	+	0.386063i	$0.126165\pi$
$384$	12.6527	0.645679
$385$	−0.833279	−0.0424679
$386$	−56.3111	−2.86616
$387$	−4.78285	−0.243126
$388$	−26.9239	−1.36685
$389$	−32.6465	−1.65524	−0.827621	−	0.561288i	$-0.810306\pi$
$389$	−32.6465	−1.65524	−0.827621	+	0.561288i	$0.810306\pi$
$390$	−7.02499	−0.355724
$391$	−1.05121	−0.0531622
$392$	−11.5615	−0.583942
$393$	20.1165	1.01474
$394$	−27.1668	−1.36864
$395$	3.67328	0.184823
$396$	−7.08993	−0.356282
$397$	−3.41525	−0.171407	−0.0857033	−	0.996321i	$-0.527314\pi$
$397$	−3.41525	−0.171407	−0.0857033	+	0.996321i	$0.527314\pi$
$398$	−11.0674	−0.554757
$399$	2.86766	0.143562
$400$	8.32320	0.416160
$401$	−11.0896	−0.553789	−0.276894	−	0.960900i	$-0.589305\pi$
$401$	−11.0896	−0.553789	−0.276894	+	0.960900i	$0.589305\pi$
$402$	−27.8627	−1.38966
$403$	25.0718	1.24892
$404$	23.8909	1.18862
$405$	0.653611	0.0324782
$406$	−6.58940	−0.327027
$407$	12.1847	0.603975
$408$	1.71293	0.0848025
$409$	−7.63041	−0.377300	−0.188650	−	0.982044i	$-0.560411\pi$
$409$	−7.63041	−0.377300	−0.188650	+	0.982044i	$0.560411\pi$
$410$	−7.81275	−0.385844
$411$	−19.7766	−0.975506
$412$	7.59557	0.374207
$413$	1.57748	0.0776226
$414$	2.29907	0.112993
$415$	−5.12283	−0.251470
$416$	−36.3988	−1.78460
$417$	−1.33900	−0.0655712
$418$	31.9232	1.56142
$419$	−27.3956	−1.33836	−0.669180	−	0.743100i	$-0.733355\pi$
$419$	−27.3956	−1.33836	−0.669180	+	0.743100i	$0.733355\pi$
$420$	−0.910419	−0.0444239
$421$	9.41363	0.458792	0.229396	−	0.973333i	$-0.426325\pi$
$421$	9.41363	0.458792	0.229396	+	0.973333i	$0.426325\pi$
$422$	0.882702	0.0429693
$423$	−11.0936	−0.539390
$424$	10.5995	0.514756
$425$	4.57279	0.221813
$426$	−26.3932	−1.27876
$427$	−0.701937	−0.0339691
$428$	20.9422	1.01228
$429$	12.5188	0.604412
$430$	−6.83700	−0.329709
$431$	−35.4419	−1.70718	−0.853589	−	0.520947i	$-0.825579\pi$
$431$	−35.4419	−1.70718	−0.853589	+	0.520947i	$0.825579\pi$
$432$	1.82016	0.0875723
$433$	18.0352	0.866716	0.433358	−	0.901222i	$-0.357329\pi$
$433$	18.0352	0.866716	0.433358	+	0.901222i	$0.357329\pi$
$434$	5.58413	0.268047
$435$	3.93486	0.188662
$436$	30.5676	1.46392
$437$	−6.02342	−0.288139
$438$	27.7402	1.32548
$439$	19.7182	0.941096	0.470548	−	0.882374i	$-0.344056\pi$
$439$	19.7182	0.941096	0.470548	+	0.882374i	$0.344056\pi$
$440$	−2.85203	−0.135965
$441$	−6.74953	−0.321406
$442$	−10.7480	−0.511229
$443$	24.9002	1.18304	0.591522	−	0.806289i	$-0.298527\pi$
$443$	24.9002	1.18304	0.591522	+	0.806289i	$0.298527\pi$
$444$	13.3127	0.631794
$445$	9.30127	0.440922
$446$	6.76734	0.320443
$447$	14.0402	0.664077
$448$	−6.28508	−0.296942
$449$	−20.9682	−0.989549	−0.494775	−	0.869021i	$-0.664749\pi$
$449$	−20.9682	−0.989549	−0.494775	+	0.869021i	$0.664749\pi$
$450$	−10.0010	−0.471449
$451$	13.9226	0.655589
$452$	−45.3313	−2.13221
$453$	4.90072	0.230256
$454$	−29.6543	−1.39174
$455$	1.60754	0.0753626
$456$	9.81500	0.459629
$457$	33.6239	1.57286	0.786430	−	0.617679i	$-0.211927\pi$
$457$	33.6239	1.57286	0.786430	+	0.617679i	$0.211927\pi$
$458$	−0.0520923	−0.00243412
$459$	1.00000	0.0466760
$460$	1.91230	0.0891616
$461$	−6.85536	−0.319286	−0.159643	−	0.987175i	$-0.551034\pi$
$461$	−6.85536	−0.319286	−0.159643	+	0.987175i	$0.551034\pi$
$462$	2.78825	0.129721
$463$	−3.21493	−0.149410	−0.0747052	−	0.997206i	$-0.523802\pi$
$463$	−3.21493	−0.149410	−0.0747052	+	0.997206i	$0.523802\pi$
$464$	10.9577	0.508698
$465$	−3.33456	−0.154637
$466$	19.6074	0.908297
$467$	−23.2115	−1.07410	−0.537051	−	0.843550i	$-0.680462\pi$
$467$	−23.2115	−1.07410	−0.537051	+	0.843550i	$0.680462\pi$
$468$	13.6777	0.632251
$469$	6.37586	0.294410
$470$	−15.8581	−0.731481
$471$	1.00000	0.0460776
$472$	5.39916	0.248517
$473$	12.1838	0.560211
$474$	−12.2912	−0.564555
$475$	26.2019	1.20223
$476$	−1.39291	−0.0638438
$477$	6.18793	0.283326
$478$	42.2677	1.93328
$479$	5.27796	0.241156	0.120578	−	0.992704i	$-0.461525\pi$
$479$	5.27796	0.241156	0.120578	+	0.992704i	$0.461525\pi$
$480$	4.84106	0.220963
$481$	−23.5065	−1.07180
$482$	−18.8769	−0.859820
$483$	−0.526099	−0.0239383
$484$	−12.5545	−0.570660
$485$	−6.32282	−0.287105
$486$	−2.18706	−0.0992069
$487$	−19.4554	−0.881607	−0.440804	−	0.897604i	$-0.645307\pi$
$487$	−19.4554	−0.881607	−0.440804	+	0.897604i	$0.645307\pi$
$488$	−2.40249	−0.108756
$489$	13.2937	0.601164
$490$	−9.64834	−0.435868
$491$	23.3226	1.05254	0.526268	−	0.850319i	$-0.323591\pi$
$491$	23.3226	1.05254	0.526268	+	0.850319i	$0.323591\pi$
$492$	15.2115	0.685786
$493$	6.02019	0.271136
$494$	−61.5854	−2.77086
$495$	−1.66500	−0.0748363
$496$	−9.28599	−0.416953
$497$	6.03961	0.270913
$498$	17.1416	0.768132
$499$	−6.07229	−0.271833	−0.135917	−	0.990720i	$-0.543398\pi$
$499$	−6.07229	−0.271833	−0.135917	+	0.990720i	$0.543398\pi$
$500$	−17.4142	−0.778788
$501$	−2.38099	−0.106375
$502$	37.4412	1.67108
$503$	−12.6015	−0.561875	−0.280937	−	0.959726i	$-0.590645\pi$
$503$	−12.6015	−0.561875	−0.280937	+	0.959726i	$0.590645\pi$
$504$	0.857265	0.0381856
$505$	5.61055	0.249666
$506$	−5.85662	−0.260358
$507$	−11.1509	−0.495228
$508$	−13.6524	−0.605729
$509$	−26.1701	−1.15997	−0.579985	−	0.814627i	$-0.696942\pi$
$509$	−26.1701	−1.15997	−0.579985	+	0.814627i	$0.696942\pi$
$510$	1.42948	0.0632986
$511$	−6.34784	−0.280812
$512$	19.7168	0.871369
$513$	5.72996	0.252984
$514$	20.9969	0.926134
$515$	1.78375	0.0786013
$516$	13.3117	0.586014
$517$	28.2597	1.24286
$518$	−5.23548	−0.230034
$519$	−13.3967	−0.588048
$520$	5.50205	0.241281
$521$	−8.79900	−0.385491	−0.192746	−	0.981249i	$-0.561739\pi$
$521$	−8.79900	−0.385491	−0.192746	+	0.981249i	$0.561739\pi$
$522$	−13.1665	−0.576282
$523$	14.0243	0.613240	0.306620	−	0.951832i	$-0.400802\pi$
$523$	14.0243	0.613240	0.306620	+	0.951832i	$0.400802\pi$
$524$	−55.9885	−2.44587
$525$	2.28853	0.0998798
$526$	22.2728	0.971141
$527$	−5.10176	−0.222236
$528$	−4.63665	−0.201784
$529$	−21.8949	−0.951954
$530$	8.84555	0.384226
$531$	3.15201	0.136786
$532$	−7.98130	−0.346033
$533$	−26.8591	−1.16340
$534$	−31.1231	−1.34683
$535$	4.91809	0.212627
$536$	21.8224	0.942582
$537$	25.4522	1.09834
$538$	20.3387	0.876864
$539$	17.1937	0.740585
$540$	−1.81914	−0.0782832
$541$	−7.01524	−0.301609	−0.150804	−	0.988564i	$-0.548186\pi$
$541$	−7.01524	−0.301609	−0.150804	+	0.988564i	$0.548186\pi$
$542$	−32.5540	−1.39832
$543$	16.9026	0.725358
$544$	7.40664	0.317557
$545$	7.17852	0.307494
$546$	−5.37901	−0.230200
$547$	11.8495	0.506647	0.253323	−	0.967382i	$-0.418476\pi$
$547$	11.8495	0.506647	0.253323	+	0.967382i	$0.418476\pi$
$548$	55.0424	2.35129
$549$	−1.40256	−0.0598599
$550$	25.4763	1.08631
$551$	34.4954	1.46955
$552$	−1.80065	−0.0766410
$553$	2.81262	0.119605
$554$	2.93516	0.124703
$555$	3.12637	0.132707
$556$	3.72673	0.158048
$557$	6.81213	0.288639	0.144320	−	0.989531i	$-0.453901\pi$
$557$	6.81213	0.288639	0.144320	+	0.989531i	$0.453901\pi$
$558$	11.1578	0.472348
$559$	−23.5046	−0.994139
$560$	−0.595393	−0.0251600
$561$	−2.54739	−0.107551
$562$	47.7085	2.01246
$563$	−1.00815	−0.0424883	−0.0212442	−	0.999774i	$-0.506763\pi$
$563$	−1.00815	−0.0424883	−0.0212442	+	0.999774i	$0.506763\pi$
$564$	30.8759	1.30011
$565$	−10.6456	−0.447865
$566$	68.9432	2.89790
$567$	0.500468	0.0210177
$568$	20.6715	0.867356
$569$	25.1668	1.05505	0.527523	−	0.849541i	$-0.323121\pi$
$569$	25.1668	1.05505	0.527523	+	0.849541i	$0.323121\pi$
$570$	8.19088	0.343078
$571$	−34.1475	−1.42903	−0.714513	−	0.699622i	$-0.753352\pi$
$571$	−34.1475	−1.42903	−0.714513	+	0.699622i	$0.753352\pi$
$572$	−34.8424	−1.45683
$573$	10.0403	0.419441
$574$	−5.98219	−0.249692
$575$	−4.80699	−0.200465
$576$	−12.5584	−0.523267
$577$	9.56030	0.398001	0.199000	−	0.979999i	$-0.436231\pi$
$577$	9.56030	0.398001	0.199000	+	0.979999i	$0.436231\pi$
$578$	2.18706	0.0909695
$579$	25.7474	1.07003
$580$	−10.9516	−0.454738
$581$	−3.92254	−0.162734
$582$	21.1569	0.876981
$583$	−15.7631	−0.652840
$584$	−21.7265	−0.899047
$585$	3.21207	0.132803
$586$	−9.40243	−0.388411
$587$	39.6225	1.63540	0.817698	−	0.575647i	$-0.195250\pi$
$587$	39.6225	1.63540	0.817698	+	0.575647i	$0.195250\pi$
$588$	18.7854	0.774696
$589$	−29.2328	−1.20452
$590$	4.50574	0.185499
$591$	12.4216	0.510957
$592$	8.70621	0.357823
$593$	−43.8980	−1.80268	−0.901338	−	0.433117i	$-0.857414\pi$
$593$	−43.8980	−1.80268	−0.901338	+	0.433117i	$0.857414\pi$
$594$	5.57128	0.228593
$595$	−0.327111	−0.0134102
$596$	−39.0768	−1.60065
$597$	5.06040	0.207108
$598$	11.2984	0.462027
$599$	26.4191	1.07946	0.539729	−	0.841839i	$-0.318527\pi$
$599$	26.4191	1.07946	0.539729	+	0.841839i	$0.318527\pi$
$600$	7.83286	0.319775
$601$	−2.62631	−0.107129	−0.0535647	−	0.998564i	$-0.517058\pi$
$601$	−2.62631	−0.107129	−0.0535647	+	0.998564i	$0.517058\pi$
$602$	−5.23507	−0.213365
$603$	12.7398	0.518805
$604$	−13.6397	−0.554993
$605$	−2.94831	−0.119866
$606$	−18.7735	−0.762623
$607$	7.86184	0.319102	0.159551	−	0.987190i	$-0.448995\pi$
$607$	7.86184	0.319102	0.159551	+	0.987190i	$0.448995\pi$
$608$	42.4397	1.72116
$609$	3.01291	0.122089
$610$	−2.00494	−0.0811777
$611$	−54.5179	−2.20556
$612$	−2.78321	−0.112505
$613$	27.7663	1.12147	0.560735	−	0.827995i	$-0.310519\pi$
$613$	27.7663	1.12147	0.560735	+	0.827995i	$0.310519\pi$
$614$	−2.63659	−0.106404
$615$	3.57227	0.144048
$616$	−2.18379	−0.0879873
$617$	−30.1083	−1.21212	−0.606058	−	0.795420i	$-0.707250\pi$
$617$	−30.1083	−1.21212	−0.606058	+	0.795420i	$0.707250\pi$
$618$	−5.96862	−0.240093
$619$	−18.4697	−0.742358	−0.371179	−	0.928561i	$-0.621046\pi$
$619$	−18.4697	−0.742358	−0.371179	+	0.928561i	$0.621046\pi$
$620$	9.28079	0.372726
$621$	−1.05121	−0.0421838
$622$	10.2979	0.412908
$623$	7.12195	0.285335
$624$	8.94489	0.358082
$625$	18.7744	0.750976
$626$	−30.2941	−1.21080
$627$	−14.5964	−0.582925
$628$	−2.78321	−0.111062
$629$	4.78322	0.190720
$630$	0.715410	0.0285026
$631$	−9.33882	−0.371772	−0.185886	−	0.982571i	$-0.559516\pi$
$631$	−9.33882	−0.371772	−0.185886	+	0.982571i	$0.559516\pi$
$632$	9.62663	0.382927
$633$	−0.403603	−0.0160418
$634$	8.88329	0.352800
$635$	−3.20615	−0.127232
$636$	−17.2223	−0.682910
$637$	−33.1696	−1.31423
$638$	33.5402	1.32787
$639$	12.0679	0.477400
$640$	−8.26992	−0.326897
$641$	−22.0757	−0.871938	−0.435969	−	0.899962i	$-0.643594\pi$
$641$	−22.0757	−0.871938	−0.435969	+	0.899962i	$0.643594\pi$
$642$	−16.4565	−0.649485
$643$	31.1748	1.22942	0.614708	−	0.788755i	$-0.289274\pi$
$643$	31.1748	1.22942	0.614708	+	0.788755i	$0.289274\pi$
$644$	1.46424	0.0576993
$645$	3.12612	0.123091
$646$	12.5317	0.493054
$647$	4.10824	0.161512	0.0807558	−	0.996734i	$-0.474267\pi$
$647$	4.10824	0.161512	0.0807558	+	0.996734i	$0.474267\pi$
$648$	1.71293	0.0672901
$649$	−8.02940	−0.315181
$650$	−49.1482	−1.92775
$651$	−2.55326	−0.100070
$652$	−36.9993	−1.44900
$653$	8.91722	0.348958	0.174479	−	0.984661i	$-0.444176\pi$
$653$	8.91722	0.348958	0.174479	+	0.984661i	$0.444176\pi$
$654$	−24.0201	−0.939261
$655$	−13.1484	−0.513749
$656$	9.94794	0.388402
$657$	−12.6838	−0.494843
$658$	−12.1425	−0.473364
$659$	−30.0441	−1.17035	−0.585177	−	0.810906i	$-0.698975\pi$
$659$	−30.0441	−1.17035	−0.585177	+	0.810906i	$0.698975\pi$
$660$	4.63405	0.180380
$661$	26.2419	1.02069	0.510346	−	0.859969i	$-0.329517\pi$
$661$	26.2419	1.02069	0.510346	+	0.859969i	$0.329517\pi$
$662$	41.7287	1.62183
$663$	4.91435	0.190858
$664$	−13.4255	−0.521010
$665$	−1.87433	−0.0726835
$666$	−10.4612	−0.405362
$667$	−6.32851	−0.245041
$668$	6.62679	0.256398
$669$	−3.09427	−0.119631
$670$	18.2113	0.703565
$671$	3.57288	0.137929
$672$	3.70678	0.142992
$673$	−26.1504	−1.00803	−0.504013	−	0.863696i	$-0.668144\pi$
$673$	−26.1504	−1.00803	−0.504013	+	0.863696i	$0.668144\pi$
$674$	−47.0042	−1.81053
$675$	4.57279	0.176007
$676$	31.0352	1.19366
$677$	28.8649	1.10937	0.554684	−	0.832061i	$-0.312839\pi$
$677$	28.8649	1.10937	0.554684	+	0.832061i	$0.312839\pi$
$678$	35.6215	1.36804
$679$	−4.84136	−0.185795
$680$	−1.11959	−0.0429342
$681$	13.5590	0.519582
$682$	−28.4233	−1.08839
$683$	−24.4882	−0.937013	−0.468507	−	0.883460i	$-0.655208\pi$
$683$	−24.4882	−0.937013	−0.468507	+	0.883460i	$0.655208\pi$
$684$	−15.9477	−0.609775
$685$	12.9262	0.493884
$686$	−15.0496	−0.574595
$687$	0.0238185	0.000908732 0
$688$	8.70553	0.331895
$689$	30.4097	1.15852
$690$	−1.50269	−0.0572066
$691$	18.8134	0.715694	0.357847	−	0.933780i	$-0.383511\pi$
$691$	18.8134	0.715694	0.357847	+	0.933780i	$0.383511\pi$
$692$	37.2857	1.41739
$693$	−1.27489	−0.0484289
$694$	65.9654	2.50401
$695$	0.875186	0.0331977
$696$	10.3121	0.390881
$697$	5.46543	0.207018
$698$	−65.4987	−2.47916
$699$	−8.96522	−0.339096
$700$	−6.36948	−0.240744
$701$	46.1116	1.74161	0.870805	−	0.491629i	$-0.163598\pi$
$701$	46.1116	1.74161	0.870805	+	0.491629i	$0.163598\pi$
$702$	−10.7480	−0.405656
$703$	27.4077	1.03370
$704$	31.9912	1.20571
$705$	7.25090	0.273085
$706$	38.5338	1.45024
$707$	4.29598	0.161567
$708$	−8.77271	−0.329699
$709$	7.03007	0.264020	0.132010	−	0.991248i	$-0.457857\pi$
$709$	7.03007	0.264020	0.132010	+	0.991248i	$0.457857\pi$
$710$	17.2509	0.647415
$711$	5.61999	0.210766
$712$	24.3760	0.913528
$713$	5.36304	0.200848
$714$	1.09455	0.0409625
$715$	−8.18241	−0.306005
$716$	−70.8388	−2.64737
$717$	−19.3263	−0.721754
$718$	68.6984	2.56380
$719$	28.9808	1.08080	0.540401	−	0.841408i	$-0.318273\pi$
$719$	28.9808	1.08080	0.540401	+	0.841408i	$0.318273\pi$
$720$	−1.18967	−0.0443365
$721$	1.36581	0.0508654
$722$	30.2522	1.12587
$723$	8.63121	0.320998
$724$	−47.0434	−1.74835
$725$	27.5291	1.02240
$726$	9.86538	0.366138
$727$	27.5855	1.02309	0.511544	−	0.859257i	$-0.329074\pi$
$727$	27.5855	1.02309	0.511544	+	0.859257i	$0.329074\pi$
$728$	4.21290	0.156141
$729$	1.00000	0.0370370
$730$	−18.1313	−0.671070
$731$	4.78285	0.176900
$732$	3.90363	0.144282
$733$	−20.9034	−0.772084	−0.386042	−	0.922481i	$-0.626158\pi$
$733$	−20.9034	−0.772084	−0.386042	+	0.922481i	$0.626158\pi$
$734$	−24.1013	−0.889595
$735$	4.41157	0.162723
$736$	−7.78597	−0.286995
$737$	−32.4533	−1.19543
$738$	−11.9532	−0.440004
$739$	24.2896	0.893507	0.446753	−	0.894657i	$-0.352580\pi$
$739$	24.2896	0.893507	0.446753	+	0.894657i	$0.352580\pi$
$740$	−8.70134	−0.319868
$741$	28.1590	1.03445
$742$	6.77300	0.248645
$743$	0.789729	0.0289724	0.0144862	−	0.999895i	$-0.495389\pi$
$743$	0.789729	0.0289724	0.0144862	+	0.999895i	$0.495389\pi$
$744$	−8.73894	−0.320385
$745$	−9.17681	−0.336212
$746$	−65.8920	−2.41248
$747$	−7.83774	−0.286768
$748$	7.08993	0.259233
$749$	3.76576	0.137598
$750$	13.6841	0.499674
$751$	19.8560	0.724557	0.362279	−	0.932070i	$-0.381999\pi$
$751$	19.8560	0.724557	0.362279	+	0.932070i	$0.381999\pi$
$752$	20.1921	0.736330
$753$	−17.1195	−0.623868
$754$	−64.7048	−2.35641
$755$	−3.20316	−0.116575
$756$	−1.39291	−0.0506595
$757$	−5.47545	−0.199009	−0.0995043	−	0.995037i	$-0.531726\pi$
$757$	−5.47545	−0.199009	−0.0995043	+	0.995037i	$0.531726\pi$
$758$	31.1854	1.13271
$759$	2.67785	0.0972000
$760$	−6.41519	−0.232703
$761$	26.6792	0.967120	0.483560	−	0.875311i	$-0.339343\pi$
$761$	26.6792	0.967120	0.483560	+	0.875311i	$0.339343\pi$
$762$	10.7281	0.388639
$763$	5.49657	0.198989
$764$	−27.9444	−1.01099
$765$	−0.653611	−0.0236313
$766$	78.9664	2.85317
$767$	15.4901	0.559315
$768$	2.55527	0.0922055
$769$	43.3183	1.56210	0.781049	−	0.624469i	$-0.214685\pi$
$769$	43.3183	1.56210	0.781049	+	0.624469i	$0.214685\pi$
$770$	−1.82243	−0.0656758
$771$	−9.60054	−0.345755
$772$	−71.6605	−2.57912
$773$	−6.44371	−0.231764	−0.115882	−	0.993263i	$-0.536969\pi$
$773$	−6.44371	−0.231764	−0.115882	+	0.993263i	$0.536969\pi$
$774$	−10.4604	−0.375990
$775$	−23.3293	−0.838012
$776$	−16.5703	−0.594840
$777$	2.39385	0.0858788
$778$	−71.3996	−2.55980
$779$	31.3167	1.12204
$780$	−8.93988	−0.320099
$781$	−30.7417	−1.10003
$782$	−2.29907	−0.0822144
$783$	6.02019	0.215144
$784$	12.2852	0.438757
$785$	−0.653611	−0.0233284
$786$	43.9959	1.56928
$787$	−5.95214	−0.212171	−0.106086	−	0.994357i	$-0.533832\pi$
$787$	−5.95214	−0.212171	−0.106086	+	0.994357i	$0.533832\pi$
$788$	−34.5720	−1.23158
$789$	−10.1839	−0.362557
$790$	8.03368	0.285825
$791$	−8.15133	−0.289828
$792$	−4.36349	−0.155050
$793$	−6.89269	−0.244767
$794$	−7.46935	−0.265077
$795$	−4.04450	−0.143444
$796$	−14.0842	−0.499200
$797$	−45.2399	−1.60248	−0.801240	−	0.598344i	$-0.795826\pi$
$797$	−45.2399	−1.60248	−0.801240	+	0.598344i	$0.795826\pi$
$798$	6.27173	0.222017
$799$	11.0936	0.392464
$800$	33.8690	1.19745
$801$	14.2306	0.502813
$802$	−24.2536	−0.856424
$803$	32.3106	1.14022
$804$	−35.4576	−1.25049
$805$	0.343864	0.0121196
$806$	54.8335	1.93143
$807$	−9.29959	−0.327361
$808$	14.7037	0.517273
$809$	−39.1783	−1.37744	−0.688718	−	0.725029i	$-0.741826\pi$
$809$	−39.1783	−1.37744	−0.688718	+	0.725029i	$0.741826\pi$
$810$	1.42948	0.0502269
$811$	−29.2032	−1.02546	−0.512731	−	0.858549i	$-0.671366\pi$
$811$	−29.2032	−1.02546	−0.512731	+	0.858549i	$0.671366\pi$
$812$	−8.38557	−0.294276
$813$	14.8849	0.522035
$814$	26.6487	0.934036
$815$	−8.68893	−0.304360
$816$	−1.82016	−0.0637182
$817$	27.4055	0.958797
$818$	−16.6881	−0.583487
$819$	2.45947	0.0859410
$820$	−9.94237	−0.347203
$821$	8.97149	0.313107	0.156553	−	0.987669i	$-0.449962\pi$
$821$	8.97149	0.313107	0.156553	+	0.987669i	$0.449962\pi$
$822$	−43.2524	−1.50860
$823$	16.7956	0.585459	0.292729	−	0.956195i	$-0.405436\pi$
$823$	16.7956	0.585459	0.292729	+	0.956195i	$0.405436\pi$
$824$	4.67469	0.162851
$825$	−11.6487	−0.405555
$826$	3.45003	0.120042
$827$	41.3024	1.43623	0.718113	−	0.695927i	$-0.245006\pi$
$827$	41.3024	1.43623	0.718113	+	0.695927i	$0.245006\pi$
$828$	2.92575	0.101677
$829$	7.29659	0.253421	0.126710	−	0.991940i	$-0.459558\pi$
$829$	7.29659	0.253421	0.126710	+	0.991940i	$0.459558\pi$
$830$	−11.2039	−0.388894
$831$	−1.34206	−0.0465555
$832$	−61.7165	−2.13963
$833$	6.74953	0.233857
$834$	−2.92847	−0.101405
$835$	1.55624	0.0538559
$836$	40.6250	1.40504
$837$	−5.10176	−0.176342
$838$	−59.9156	−2.06975
$839$	42.9510	1.48283	0.741417	−	0.671045i	$-0.234154\pi$
$839$	42.9510	1.48283	0.741417	+	0.671045i	$0.234154\pi$
$840$	−0.560317	−0.0193328
$841$	7.24270	0.249748
$842$	20.5881	0.709514
$843$	−21.8140	−0.751315
$844$	1.12331	0.0386660
$845$	7.28833	0.250726
$846$	−24.2623	−0.834156
$847$	−2.25751	−0.0775690
$848$	−11.2630	−0.386773
$849$	−31.5233	−1.08188
$850$	10.0010	0.343030
$851$	−5.02820	−0.172364
$852$	−33.5876	−1.15069
$853$	−8.25293	−0.282575	−0.141288	−	0.989969i	$-0.545124\pi$
$853$	−8.25293	−0.282575	−0.141288	+	0.989969i	$0.545124\pi$
$854$	−1.53518	−0.0525326
$855$	−3.74516	−0.128082
$856$	12.8889	0.440534
$857$	−16.0591	−0.548570	−0.274285	−	0.961648i	$-0.588441\pi$
$857$	−16.0591	−0.548570	−0.274285	+	0.961648i	$0.588441\pi$
$858$	27.3793	0.934712
$859$	−29.7192	−1.01401	−0.507003	−	0.861944i	$-0.669247\pi$
$859$	−29.7192	−1.01401	−0.507003	+	0.861944i	$0.669247\pi$
$860$	−8.70065	−0.296690
$861$	2.73527	0.0932178
$862$	−77.5135	−2.64012
$863$	−1.93361	−0.0658207	−0.0329104	−	0.999458i	$-0.510478\pi$
$863$	−1.93361	−0.0658207	−0.0329104	+	0.999458i	$0.510478\pi$
$864$	7.40664	0.251979
$865$	8.75620	0.297720
$866$	39.4440	1.34036
$867$	−1.00000	−0.0339618
$868$	7.10627	0.241203
$869$	−14.3163	−0.485647
$870$	8.60576	0.291763
$871$	62.6079	2.12139
$872$	18.8128	0.637083
$873$	−9.67368	−0.327405
$874$	−13.1735	−0.445602
$875$	−3.13137	−0.105859
$876$	35.3018	1.19274
$877$	−6.34577	−0.214281	−0.107141	−	0.994244i	$-0.534170\pi$
$877$	−6.34577	−0.214281	−0.107141	+	0.994244i	$0.534170\pi$
$878$	43.1247	1.45539
$879$	4.29913	0.145006
$880$	3.03056	0.102160
$881$	−5.77600	−0.194598	−0.0972992	−	0.995255i	$-0.531020\pi$
$881$	−5.77600	−0.194598	−0.0972992	+	0.995255i	$0.531020\pi$
$882$	−14.7616	−0.497049
$883$	−5.47783	−0.184344	−0.0921718	−	0.995743i	$-0.529381\pi$
$883$	−5.47783	−0.184344	−0.0921718	+	0.995743i	$0.529381\pi$
$884$	−13.6777	−0.460030
$885$	−2.06019	−0.0692524
$886$	54.4581	1.82956
$887$	−23.5316	−0.790114	−0.395057	−	0.918657i	$-0.629275\pi$
$887$	−23.5316	−0.790114	−0.395057	+	0.918657i	$0.629275\pi$
$888$	8.19332	0.274950
$889$	−2.45494	−0.0823359
$890$	20.3424	0.681878
$891$	−2.54739	−0.0853408
$892$	8.61200	0.288351
$893$	63.5659	2.12715
$894$	30.7066	1.02698
$895$	−16.6358	−0.556074
$896$	−6.33225	−0.211546
$897$	−5.16604	−0.172489
$898$	−45.8586	−1.53032
$899$	−30.7135	−1.02435
$900$	−12.7271	−0.424235
$901$	−6.18793	−0.206150
$902$	30.4495	1.01386
$903$	2.39366	0.0796560
$904$	−27.8992	−0.927913
$905$	−11.0477	−0.367238
$906$	10.7181	0.356086
$907$	48.3117	1.60416	0.802082	−	0.597213i	$-0.203725\pi$
$907$	48.3117	1.60416	0.802082	+	0.597213i	$0.203725\pi$
$908$	−37.7376	−1.25236
$909$	8.58394	0.284711
$910$	3.51578	0.116547
$911$	13.8089	0.457510	0.228755	−	0.973484i	$-0.426535\pi$
$911$	13.8089	0.457510	0.228755	+	0.973484i	$0.426535\pi$
$912$	−10.4294	−0.345352
$913$	19.9658	0.660771
$914$	73.5374	2.43240
$915$	0.916730	0.0303062
$916$	−0.0662919	−0.00219035
$917$	−10.0677	−0.332463
$918$	2.18706	0.0721836
$919$	−42.7045	−1.40869	−0.704345	−	0.709858i	$-0.748759\pi$
$919$	−42.7045	−1.40869	−0.704345	+	0.709858i	$0.748759\pi$
$920$	1.17693	0.0388022
$921$	1.20554	0.0397240
$922$	−14.9930	−0.493770
$923$	59.3061	1.95208
$924$	3.54828	0.116730
$925$	21.8727	0.719170
$926$	−7.03122	−0.231060
$927$	2.72907	0.0896343
$928$	44.5894	1.46372
$929$	−1.57874	−0.0517968	−0.0258984	−	0.999665i	$-0.508245\pi$
$929$	−1.57874	−0.0517968	−0.0258984	+	0.999665i	$0.508245\pi$
$930$	−7.29287	−0.239143
$931$	38.6745	1.26751
$932$	24.9521	0.817333
$933$	−4.70857	−0.154152
$934$	−50.7649	−1.66108
$935$	1.66500	0.0544514
$936$	8.41793	0.275149
$937$	39.0306	1.27507	0.637536	−	0.770420i	$-0.279954\pi$
$937$	39.0306	1.27507	0.637536	+	0.770420i	$0.279954\pi$
$938$	13.9444	0.455299
$939$	13.8515	0.452028
$940$	−20.1808	−0.658225
$941$	38.6143	1.25879	0.629394	−	0.777086i	$-0.283303\pi$
$941$	38.6143	1.25879	0.629394	+	0.777086i	$0.283303\pi$
$942$	2.18706	0.0712581
$943$	−5.74535	−0.187094
$944$	−5.73715	−0.186728
$945$	−0.327111	−0.0106409
$946$	26.6466	0.866356
$947$	36.5778	1.18862	0.594310	−	0.804236i	$-0.297425\pi$
$947$	36.5778	1.18862	0.594310	+	0.804236i	$0.297425\pi$
$948$	−15.6416	−0.508016
$949$	−62.3328	−2.02341
$950$	57.3050	1.85922
$951$	−4.06176	−0.131712
$952$	−0.857265	−0.0277841
$953$	13.4318	0.435100	0.217550	−	0.976049i	$-0.430194\pi$
$953$	13.4318	0.435100	0.217550	+	0.976049i	$0.430194\pi$
$954$	13.5334	0.438158
$955$	−6.56247	−0.212356
$956$	53.7892	1.73967
$957$	−15.3358	−0.495735
$958$	11.5432	0.372944
$959$	9.89753	0.319608
$960$	8.20832	0.264922
$961$	−4.97208	−0.160390
$962$	−51.4099	−1.65752
$963$	7.52449	0.242473
$964$	−24.0225	−0.773712
$965$	−16.8288	−0.541738
$966$	−1.15061	−0.0370202
$967$	−26.1474	−0.840844	−0.420422	−	0.907329i	$-0.638118\pi$
$967$	−26.1474	−0.840844	−0.420422	+	0.907329i	$0.638118\pi$
$968$	−7.72668	−0.248345
$969$	−5.72996	−0.184073
$970$	−13.8284	−0.444002
$971$	30.1983	0.969111	0.484555	−	0.874761i	$-0.338981\pi$
$971$	30.1983	0.969111	0.484555	+	0.874761i	$0.338981\pi$
$972$	−2.78321	−0.0892716
$973$	0.670127	0.0214833
$974$	−42.5500	−1.36339
$975$	22.4723	0.719690
$976$	2.55288	0.0817158
$977$	−14.2672	−0.456449	−0.228225	−	0.973608i	$-0.573292\pi$
$977$	−14.2672	−0.456449	−0.228225	+	0.973608i	$0.573292\pi$
$978$	29.0741	0.929689
$979$	−36.2509	−1.15858
$980$	−12.2783	−0.392217
$981$	10.9829	0.350656
$982$	51.0079	1.62773
$983$	−54.8646	−1.74991	−0.874955	−	0.484205i	$-0.839109\pi$
$983$	−54.8646	−1.74991	−0.874955	+	0.484205i	$0.839109\pi$
$984$	9.36189	0.298446
$985$	−8.11891	−0.258690
$986$	13.1665	0.419307
$987$	5.55199	0.176722
$988$	−78.3725	−2.49336
$989$	−5.02780	−0.159875
$990$	−3.64145	−0.115733
$991$	−38.0931	−1.21007	−0.605034	−	0.796199i	$-0.706841\pi$
$991$	−38.0931	−1.21007	−0.605034	+	0.796199i	$0.706841\pi$
$992$	−37.7869	−1.19973
$993$	−19.0799	−0.605482
$994$	13.2090	0.418963
$995$	−3.30753	−0.104856
$996$	21.8141	0.691206
$997$	36.0885	1.14293	0.571467	−	0.820625i	$-0.306374\pi$
$997$	36.0885	1.14293	0.571467	+	0.820625i	$0.306374\pi$
$998$	−13.2804	−0.420385
$999$	4.78322	0.151335

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.42	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.42	✓	48	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q+2.18706 q^{2} -1.00000 q^{3} +2.78321 q^{4} +0.653611 q^{5} -2.18706 q^{6} +0.500468 q^{7} +1.71293 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.18706 q^{2} -1.00000 q^{3} +2.78321 q^{4} +0.653611 q^{5} -2.18706 q^{6} +0.500468 q^{7} +1.71293 q^{8} +1.00000 q^{9} +1.42948 q^{10} -2.54739 q^{11} -2.78321 q^{12} +4.91435 q^{13} +1.09455 q^{14} -0.653611 q^{15} -1.82016 q^{16} -1.00000 q^{17} +2.18706 q^{18} -5.72996 q^{19} +1.81914 q^{20} -0.500468 q^{21} -5.57128 q^{22} +1.05121 q^{23} -1.71293 q^{24} -4.57279 q^{25} +10.7480 q^{26} -1.00000 q^{27} +1.39291 q^{28} -6.02019 q^{29} -1.42948 q^{30} +5.10176 q^{31} -7.40664 q^{32} +2.54739 q^{33} -2.18706 q^{34} +0.327111 q^{35} +2.78321 q^{36} -4.78322 q^{37} -12.5317 q^{38} -4.91435 q^{39} +1.11959 q^{40} -5.46543 q^{41} -1.09455 q^{42} -4.78285 q^{43} -7.08993 q^{44} +0.653611 q^{45} +2.29907 q^{46} -11.0936 q^{47} +1.82016 q^{48} -6.74953 q^{49} -10.0010 q^{50} +1.00000 q^{51} +13.6777 q^{52} +6.18793 q^{53} -2.18706 q^{54} -1.66500 q^{55} +0.857265 q^{56} +5.72996 q^{57} -13.1665 q^{58} +3.15201 q^{59} -1.81914 q^{60} -1.40256 q^{61} +11.1578 q^{62} +0.500468 q^{63} -12.5584 q^{64} +3.21207 q^{65} +5.57128 q^{66} +12.7398 q^{67} -2.78321 q^{68} -1.05121 q^{69} +0.715410 q^{70} +12.0679 q^{71} +1.71293 q^{72} -12.6838 q^{73} -10.4612 q^{74} +4.57279 q^{75} -15.9477 q^{76} -1.27489 q^{77} -10.7480 q^{78} +5.61999 q^{79} -1.18967 q^{80} +1.00000 q^{81} -11.9532 q^{82} -7.83774 q^{83} -1.39291 q^{84} -0.653611 q^{85} -10.4604 q^{86} +6.02019 q^{87} -4.36349 q^{88} +14.2306 q^{89} +1.42948 q^{90} +2.45947 q^{91} +2.92575 q^{92} -5.10176 q^{93} -24.2623 q^{94} -3.74516 q^{95} +7.40664 q^{96} -9.67368 q^{97} -14.7616 q^{98} -2.54739 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * q^99

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