LMFDB - Embedded newform 8007.2.a.f.1.28

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			1.28
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.498945 q^{2} -1.00000 q^{3} -1.75105 q^{4} +0.622534 q^{5} -0.498945 q^{6} +0.110017 q^{7} -1.87157 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.498945 q^{2} -1.00000 q^{3} -1.75105 q^{4} +0.622534 q^{5} -0.498945 q^{6} +0.110017 q^{7} -1.87157 q^{8} +1.00000 q^{9} +0.310610 q^{10} +0.189879 q^{11} +1.75105 q^{12} +6.61917 q^{13} +0.0548922 q^{14} -0.622534 q^{15} +2.56830 q^{16} -1.00000 q^{17} +0.498945 q^{18} -0.0616562 q^{19} -1.09009 q^{20} -0.110017 q^{21} +0.0947393 q^{22} -5.46866 q^{23} +1.87157 q^{24} -4.61245 q^{25} +3.30260 q^{26} -1.00000 q^{27} -0.192645 q^{28} -5.28863 q^{29} -0.310610 q^{30} -4.08323 q^{31} +5.02458 q^{32} -0.189879 q^{33} -0.498945 q^{34} +0.0684891 q^{35} -1.75105 q^{36} +7.61954 q^{37} -0.0307630 q^{38} -6.61917 q^{39} -1.16512 q^{40} -8.53971 q^{41} -0.0548922 q^{42} -1.22669 q^{43} -0.332489 q^{44} +0.622534 q^{45} -2.72856 q^{46} +11.0065 q^{47} -2.56830 q^{48} -6.98790 q^{49} -2.30136 q^{50} +1.00000 q^{51} -11.5905 q^{52} -0.0568805 q^{53} -0.498945 q^{54} +0.118206 q^{55} -0.205904 q^{56} +0.0616562 q^{57} -2.63873 q^{58} +3.88617 q^{59} +1.09009 q^{60} +3.47610 q^{61} -2.03731 q^{62} +0.110017 q^{63} -2.62961 q^{64} +4.12066 q^{65} -0.0947393 q^{66} -10.7555 q^{67} +1.75105 q^{68} +5.46866 q^{69} +0.0341723 q^{70} +6.87507 q^{71} -1.87157 q^{72} -6.33728 q^{73} +3.80173 q^{74} +4.61245 q^{75} +0.107963 q^{76} +0.0208899 q^{77} -3.30260 q^{78} -2.39038 q^{79} +1.59885 q^{80} +1.00000 q^{81} -4.26084 q^{82} +17.9437 q^{83} +0.192645 q^{84} -0.622534 q^{85} -0.612050 q^{86} +5.28863 q^{87} -0.355372 q^{88} +12.7186 q^{89} +0.310610 q^{90} +0.728219 q^{91} +9.57592 q^{92} +4.08323 q^{93} +5.49162 q^{94} -0.0383831 q^{95} -5.02458 q^{96} +9.70831 q^{97} -3.48657 q^{98} +0.189879 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) $ 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.498945	0.352807	0.176404	−	0.984318i	$-0.443554\pi$
$2$	0.498945	0.352807	0.176404	+	0.984318i	$0.443554\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.75105	−0.875527
$5$	0.622534	0.278406	0.139203	−	0.990264i	$-0.455546\pi$
$5$	0.622534	0.278406	0.139203	+	0.990264i	$0.455546\pi$
$6$	−0.498945	−0.203693
$7$	0.110017	0.0415824	0.0207912	−	0.999784i	$-0.493381\pi$
$7$	0.110017	0.0415824	0.0207912	+	0.999784i	$0.493381\pi$
$8$	−1.87157	−0.661699
$9$	1.00000	0.333333
$10$	0.310610	0.0982235
$11$	0.189879	0.0572508	0.0286254	−	0.999590i	$-0.490887\pi$
$11$	0.189879	0.0572508	0.0286254	+	0.999590i	$0.490887\pi$
$12$	1.75105	0.505486
$13$	6.61917	1.83583	0.917913	−	0.396781i	$-0.129873\pi$
$13$	6.61917	1.83583	0.917913	+	0.396781i	$0.129873\pi$
$14$	0.0548922	0.0146706
$15$	−0.622534	−0.160738
$16$	2.56830	0.642075
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0.498945	0.117602
$19$	−0.0616562	−0.0141449	−0.00707245	−	0.999975i	$-0.502251\pi$
$19$	−0.0616562	−0.0141449	−0.00707245	+	0.999975i	$0.502251\pi$
$20$	−1.09009	−0.243752
$21$	−0.110017	−0.0240076
$22$	0.0947393	0.0201985
$23$	−5.46866	−1.14029	−0.570147	−	0.821543i	$-0.693114\pi$
$23$	−5.46866	−1.14029	−0.570147	+	0.821543i	$0.693114\pi$
$24$	1.87157	0.382032
$25$	−4.61245	−0.922490
$26$	3.30260	0.647693
$27$	−1.00000	−0.192450
$28$	−0.192645	−0.0364065
$29$	−5.28863	−0.982073	−0.491037	−	0.871139i	$-0.663382\pi$
$29$	−5.28863	−0.982073	−0.491037	+	0.871139i	$0.663382\pi$
$30$	−0.310610	−0.0567094
$31$	−4.08323	−0.733370	−0.366685	−	0.930345i	$-0.619507\pi$
$31$	−4.08323	−0.733370	−0.366685	+	0.930345i	$0.619507\pi$
$32$	5.02458	0.888228
$33$	−0.189879	−0.0330537
$34$	−0.498945	−0.0855683
$35$	0.0684891	0.0115768
$36$	−1.75105	−0.291842
$37$	7.61954	1.25264	0.626322	−	0.779565i	$-0.284560\pi$
$37$	7.61954	1.25264	0.626322	+	0.779565i	$0.284560\pi$
$38$	−0.0307630	−0.00499042
$39$	−6.61917	−1.05991
$40$	−1.16512	−0.184221
$41$	−8.53971	−1.33368	−0.666840	−	0.745201i	$-0.732353\pi$
$41$	−8.53971	−1.33368	−0.666840	+	0.745201i	$0.732353\pi$
$42$	−0.0548922	−0.00847006
$43$	−1.22669	−0.187068	−0.0935342	−	0.995616i	$-0.529816\pi$
$43$	−1.22669	−0.187068	−0.0935342	+	0.995616i	$0.529816\pi$
$44$	−0.332489	−0.0501246
$45$	0.622534	0.0928019
$46$	−2.72856	−0.402304
$47$	11.0065	1.60546	0.802730	−	0.596343i	$-0.203380\pi$
$47$	11.0065	1.60546	0.802730	+	0.596343i	$0.203380\pi$
$48$	−2.56830	−0.370702
$49$	−6.98790	−0.998271
$50$	−2.30136	−0.325461
$51$	1.00000	0.140028
$52$	−11.5905	−1.60732
$53$	−0.0568805	−0.00781314	−0.00390657	−	0.999992i	$-0.501244\pi$
$53$	−0.0568805	−0.00781314	−0.00390657	+	0.999992i	$0.501244\pi$
$54$	−0.498945	−0.0678978
$55$	0.118206	0.0159389
$56$	−0.205904	−0.0275150
$57$	0.0616562	0.00816656
$58$	−2.63873	−0.346483
$59$	3.88617	0.505936	0.252968	−	0.967475i	$-0.418593\pi$
$59$	3.88617	0.505936	0.252968	+	0.967475i	$0.418593\pi$
$60$	1.09009	0.140730
$61$	3.47610	0.445069	0.222535	−	0.974925i	$-0.428567\pi$
$61$	3.47610	0.445069	0.222535	+	0.974925i	$0.428567\pi$
$62$	−2.03731	−0.258738
$63$	0.110017	0.0138608
$64$	−2.62961	−0.328702
$65$	4.12066	0.511105
$66$	−0.0947393	−0.0116616
$67$	−10.7555	−1.31399	−0.656994	−	0.753896i	$-0.728172\pi$
$67$	−10.7555	−1.31399	−0.656994	+	0.753896i	$0.728172\pi$
$68$	1.75105	0.212347
$69$	5.46866	0.658349
$70$	0.0341723	0.00408437
$71$	6.87507	0.815921	0.407960	−	0.913000i	$-0.366240\pi$
$71$	6.87507	0.815921	0.407960	+	0.913000i	$0.366240\pi$
$72$	−1.87157	−0.220566
$73$	−6.33728	−0.741723	−0.370861	−	0.928688i	$-0.620937\pi$
$73$	−6.33728	−0.741723	−0.370861	+	0.928688i	$0.620937\pi$
$74$	3.80173	0.441942
$75$	4.61245	0.532600
$76$	0.107963	0.0123842
$77$	0.0208899	0.00238062
$78$	−3.30260	−0.373946
$79$	−2.39038	−0.268939	−0.134469	−	0.990918i	$-0.542933\pi$
$79$	−2.39038	−0.268939	−0.134469	+	0.990918i	$0.542933\pi$
$80$	1.59885	0.178757
$81$	1.00000	0.111111
$82$	−4.26084	−0.470532
$83$	17.9437	1.96958	0.984790	−	0.173746i	$-0.0555872\pi$
$83$	17.9437	1.96958	0.984790	+	0.173746i	$0.0555872\pi$
$84$	0.192645	0.0210193
$85$	−0.622534	−0.0675233
$86$	−0.612050	−0.0659991
$87$	5.28863	0.567000
$88$	−0.355372	−0.0378828
$89$	12.7186	1.34817	0.674084	−	0.738655i	$-0.264539\pi$
$89$	12.7186	1.34817	0.674084	+	0.738655i	$0.264539\pi$
$90$	0.310610	0.0327412
$91$	0.728219	0.0763381
$92$	9.57592	0.998359
$93$	4.08323	0.423411
$94$	5.49162	0.566418
$95$	−0.0383831	−0.00393802
$96$	−5.02458	−0.512819
$97$	9.70831	0.985730	0.492865	−	0.870106i	$-0.335950\pi$
$97$	9.70831	0.985730	0.492865	+	0.870106i	$0.335950\pi$
$98$	−3.48657	−0.352197
$99$	0.189879	0.0190836
$100$	8.07665	0.807665
$101$	−1.38802	−0.138114	−0.0690568	−	0.997613i	$-0.521999\pi$
$101$	−1.38802	−0.138114	−0.0690568	+	0.997613i	$0.521999\pi$
$102$	0.498945	0.0494029
$103$	−1.57120	−0.154815	−0.0774073	−	0.997000i	$-0.524664\pi$
$103$	−1.57120	−0.154815	−0.0774073	+	0.997000i	$0.524664\pi$
$104$	−12.3882	−1.21477
$105$	−0.0684891	−0.00668386
$106$	−0.0283802	−0.00275653
$107$	7.18961	0.695046	0.347523	−	0.937672i	$-0.387023\pi$
$107$	7.18961	0.695046	0.347523	+	0.937672i	$0.387023\pi$
$108$	1.75105	0.168495
$109$	−11.7603	−1.12643	−0.563216	−	0.826310i	$-0.690436\pi$
$109$	−11.7603	−1.12643	−0.563216	+	0.826310i	$0.690436\pi$
$110$	0.0589784	0.00562337
$111$	−7.61954	−0.723214
$112$	0.282556	0.0266990
$113$	5.36524	0.504719	0.252360	−	0.967633i	$-0.418793\pi$
$113$	5.36524	0.504719	0.252360	+	0.967633i	$0.418793\pi$
$114$	0.0307630	0.00288122
$115$	−3.40443	−0.317465
$116$	9.26067	0.859832
$117$	6.61917	0.611942
$118$	1.93898	0.178498
$119$	−0.110017	−0.0100852
$120$	1.16512	0.106360
$121$	−10.9639	−0.996722
$122$	1.73438	0.157024
$123$	8.53971	0.770000
$124$	7.14996	0.642085
$125$	−5.98408	−0.535232
$126$	0.0548922	0.00489019
$127$	11.9742	1.06254	0.531269	−	0.847203i	$-0.321715\pi$
$127$	11.9742	1.06254	0.531269	+	0.847203i	$0.321715\pi$
$128$	−11.3612	−1.00420
$129$	1.22669	0.108004
$130$	2.05598	0.180321
$131$	−5.31945	−0.464763	−0.232381	−	0.972625i	$-0.574652\pi$
$131$	−5.31945	−0.464763	−0.232381	+	0.972625i	$0.574652\pi$
$132$	0.332489	0.0289395
$133$	−0.00678321	−0.000588179 0
$134$	−5.36638	−0.463584
$135$	−0.622534	−0.0535792
$136$	1.87157	0.160486
$137$	−4.30047	−0.367414	−0.183707	−	0.982981i	$-0.558810\pi$
$137$	−4.30047	−0.367414	−0.183707	+	0.982981i	$0.558810\pi$
$138$	2.72856	0.232270
$139$	−21.3093	−1.80743	−0.903716	−	0.428133i	$-0.859172\pi$
$139$	−21.3093	−1.80743	−0.903716	+	0.428133i	$0.859172\pi$
$140$	−0.119928	−0.0101358
$141$	−11.0065	−0.926913
$142$	3.43028	0.287863
$143$	1.25684	0.105102
$144$	2.56830	0.214025
$145$	−3.29235	−0.273415
$146$	−3.16195	−0.261685
$147$	6.98790	0.576352
$148$	−13.3422	−1.09672
$149$	−14.5703	−1.19365	−0.596823	−	0.802373i	$-0.703570\pi$
$149$	−14.5703	−1.19365	−0.596823	+	0.802373i	$0.703570\pi$
$150$	2.30136	0.187905
$151$	7.31852	0.595573	0.297786	−	0.954633i	$-0.403752\pi$
$151$	7.31852	0.595573	0.297786	+	0.954633i	$0.403752\pi$
$152$	0.115394	0.00935967
$153$	−1.00000	−0.0808452
$154$	0.0104229	0.000839901 0
$155$	−2.54195	−0.204174
$156$	11.5905	0.927984
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	−1.19267	−0.0948836
$159$	0.0568805	0.00451092
$160$	3.12797	0.247288
$161$	−0.601644	−0.0474162
$162$	0.498945	0.0392008
$163$	10.7686	0.843463	0.421731	−	0.906721i	$-0.361423\pi$
$163$	10.7686	0.843463	0.421731	+	0.906721i	$0.361423\pi$
$164$	14.9535	1.16767
$165$	−0.118206	−0.00920235
$166$	8.95293	0.694882
$167$	−5.12239	−0.396382	−0.198191	−	0.980163i	$-0.563507\pi$
$167$	−5.12239	−0.396382	−0.198191	+	0.980163i	$0.563507\pi$
$168$	0.205904	0.0158858
$169$	30.8134	2.37026
$170$	−0.310610	−0.0238227
$171$	−0.0616562	−0.00471496
$172$	2.14800	0.163783
$173$	−17.7180	−1.34707	−0.673536	−	0.739154i	$-0.735225\pi$
$173$	−17.7180	−1.34707	−0.673536	+	0.739154i	$0.735225\pi$
$174$	2.63873	0.200042
$175$	−0.507447	−0.0383594
$176$	0.487667	0.0367593
$177$	−3.88617	−0.292102
$178$	6.34587	0.475643
$179$	−7.02296	−0.524921	−0.262460	−	0.964943i	$-0.584534\pi$
$179$	−7.02296	−0.524921	−0.262460	+	0.964943i	$0.584534\pi$
$180$	−1.09009	−0.0812506
$181$	−11.5729	−0.860207	−0.430103	−	0.902780i	$-0.641523\pi$
$181$	−11.5729	−0.860207	−0.430103	+	0.902780i	$0.641523\pi$
$182$	0.363341	0.0269326
$183$	−3.47610	−0.256961
$184$	10.2350	0.754532
$185$	4.74342	0.348743
$186$	2.03731	0.149382
$187$	−0.189879	−0.0138854
$188$	−19.2729	−1.40562
$189$	−0.110017	−0.00800254
$190$	−0.0191510	−0.00138936
$191$	4.83937	0.350164	0.175082	−	0.984554i	$-0.443981\pi$
$191$	4.83937	0.350164	0.175082	+	0.984554i	$0.443981\pi$
$192$	2.62961	0.189776
$193$	−10.6120	−0.763865	−0.381933	−	0.924190i	$-0.624741\pi$
$193$	−10.6120	−0.763865	−0.381933	+	0.924190i	$0.624741\pi$
$194$	4.84391	0.347772
$195$	−4.12066	−0.295086
$196$	12.2362	0.874013
$197$	1.33752	0.0952941	0.0476470	−	0.998864i	$-0.484828\pi$
$197$	1.33752	0.0952941	0.0476470	+	0.998864i	$0.484828\pi$
$198$	0.0947393	0.00673283
$199$	−13.6565	−0.968080	−0.484040	−	0.875046i	$-0.660831\pi$
$199$	−13.6565	−0.968080	−0.484040	+	0.875046i	$0.660831\pi$
$200$	8.63252	0.610411
$201$	10.7555	0.758631
$202$	−0.692548	−0.0487275
$203$	−0.581837	−0.0408370
$204$	−1.75105	−0.122598
$205$	−5.31626	−0.371304
$206$	−0.783940	−0.0546197
$207$	−5.46866	−0.380098
$208$	17.0000	1.17874
$209$	−0.0117072	−0.000809806 0
$210$	−0.0341723	−0.00235811
$211$	−7.25466	−0.499432	−0.249716	−	0.968319i	$-0.580337\pi$
$211$	−7.25466	−0.499432	−0.249716	+	0.968319i	$0.580337\pi$
$212$	0.0996009	0.00684062
$213$	−6.87507	−0.471072
$214$	3.58722	0.245217
$215$	−0.763656	−0.0520809
$216$	1.87157	0.127344
$217$	−0.449223	−0.0304953
$218$	−5.86773	−0.397413
$219$	6.33728	0.428234
$220$	−0.206986	−0.0139550
$221$	−6.61917	−0.445253
$222$	−3.80173	−0.255155
$223$	−25.7018	−1.72112	−0.860562	−	0.509347i	$-0.829887\pi$
$223$	−25.7018	−1.72112	−0.860562	+	0.509347i	$0.829887\pi$
$224$	0.552787	0.0369346
$225$	−4.61245	−0.307497
$226$	2.67696	0.178069
$227$	−0.0456610	−0.00303063	−0.00151531	−	0.999999i	$-0.500482\pi$
$227$	−0.0456610	−0.00303063	−0.00151531	+	0.999999i	$0.500482\pi$
$228$	−0.107963	−0.00715004
$229$	−5.83691	−0.385714	−0.192857	−	0.981227i	$-0.561775\pi$
$229$	−5.83691	−0.385714	−0.192857	+	0.981227i	$0.561775\pi$
$230$	−1.69862	−0.112004
$231$	−0.0208899	−0.00137445
$232$	9.89803	0.649837
$233$	−7.68536	−0.503485	−0.251742	−	0.967794i	$-0.581004\pi$
$233$	−7.68536	−0.503485	−0.251742	+	0.967794i	$0.581004\pi$
$234$	3.30260	0.215898
$235$	6.85191	0.446969
$236$	−6.80489	−0.442961
$237$	2.39038	0.155272
$238$	−0.0548922	−0.00355813
$239$	6.81710	0.440962	0.220481	−	0.975391i	$-0.429237\pi$
$239$	6.81710	0.440962	0.220481	+	0.975391i	$0.429237\pi$
$240$	−1.59885	−0.103206
$241$	−23.3281	−1.50270	−0.751348	−	0.659907i	$-0.770596\pi$
$241$	−23.3281	−1.50270	−0.751348	+	0.659907i	$0.770596\pi$
$242$	−5.47040	−0.351651
$243$	−1.00000	−0.0641500
$244$	−6.08684	−0.389670
$245$	−4.35020	−0.277924
$246$	4.26084	0.271662
$247$	−0.408112	−0.0259676
$248$	7.64204	0.485270
$249$	−17.9437	−1.13714
$250$	−2.98572	−0.188834
$251$	2.30713	0.145625	0.0728124	−	0.997346i	$-0.476803\pi$
$251$	2.30713	0.145625	0.0728124	+	0.997346i	$0.476803\pi$
$252$	−0.192645	−0.0121355
$253$	−1.03839	−0.0652827
$254$	5.97446	0.374871
$255$	0.622534	0.0389846
$256$	−0.409375	−0.0255859
$257$	−11.9923	−0.748059	−0.374030	−	0.927417i	$-0.622024\pi$
$257$	−11.9923	−0.748059	−0.374030	+	0.927417i	$0.622024\pi$
$258$	0.612050	0.0381046
$259$	0.838276	0.0520879
$260$	−7.21549	−0.447486
$261$	−5.28863	−0.327358
$262$	−2.65411	−0.163972
$263$	20.1830	1.24454	0.622270	−	0.782803i	$-0.286211\pi$
$263$	20.1830	1.24454	0.622270	+	0.782803i	$0.286211\pi$
$264$	0.355372	0.0218716
$265$	−0.0354101	−0.00217522
$266$	−0.00338444	−0.000207514 0
$267$	−12.7186	−0.778365
$268$	18.8334	1.15043
$269$	−29.9514	−1.82617	−0.913085	−	0.407769i	$-0.866307\pi$
$269$	−29.9514	−1.82617	−0.913085	+	0.407769i	$0.866307\pi$
$270$	−0.310610	−0.0189031
$271$	−6.64163	−0.403450	−0.201725	−	0.979442i	$-0.564655\pi$
$271$	−6.64163	−0.403450	−0.201725	+	0.979442i	$0.564655\pi$
$272$	−2.56830	−0.155726
$273$	−0.728219	−0.0440738
$274$	−2.14570	−0.129626
$275$	−0.875809	−0.0528133
$276$	−9.57592	−0.576403
$277$	−0.752550	−0.0452163	−0.0226082	−	0.999744i	$-0.507197\pi$
$277$	−0.752550	−0.0452163	−0.0226082	+	0.999744i	$0.507197\pi$
$278$	−10.6322	−0.637675
$279$	−4.08323	−0.244457
$280$	−0.128182	−0.00766035
$281$	−17.1481	−1.02297	−0.511484	−	0.859293i	$-0.670904\pi$
$281$	−17.1481	−1.02297	−0.511484	+	0.859293i	$0.670904\pi$
$282$	−5.49162	−0.327021
$283$	−19.3556	−1.15057	−0.575285	−	0.817953i	$-0.695109\pi$
$283$	−19.3556	−1.15057	−0.575285	+	0.817953i	$0.695109\pi$
$284$	−12.0386	−0.714361
$285$	0.0383831	0.00227362
$286$	0.627095	0.0370809
$287$	−0.939511	−0.0554576
$288$	5.02458	0.296076
$289$	1.00000	0.0588235
$290$	−1.64270	−0.0964627
$291$	−9.70831	−0.569111
$292$	11.0969	0.649398
$293$	31.7072	1.85235	0.926177	−	0.377089i	$-0.123075\pi$
$293$	31.7072	1.85235	0.926177	+	0.377089i	$0.123075\pi$
$294$	3.48657	0.203341
$295$	2.41927	0.140855
$296$	−14.2605	−0.828874
$297$	−0.189879	−0.0110179
$298$	−7.26977	−0.421126
$299$	−36.1980	−2.09338
$300$	−8.07665	−0.466306
$301$	−0.134956	−0.00777875
$302$	3.65154	0.210122
$303$	1.38802	0.0797399
$304$	−0.158351	−0.00908208
$305$	2.16399	0.123910
$306$	−0.498945	−0.0285228
$307$	15.3128	0.873948	0.436974	−	0.899474i	$-0.356050\pi$
$307$	15.3128	0.873948	0.436974	+	0.899474i	$0.356050\pi$
$308$	−0.0365793	−0.00208430
$309$	1.57120	0.0893822
$310$	−1.26829	−0.0720342
$311$	20.6581	1.17141	0.585705	−	0.810524i	$-0.300818\pi$
$311$	20.6581	1.17141	0.585705	+	0.810524i	$0.300818\pi$
$312$	12.3882	0.701345
$313$	−17.8717	−1.01017	−0.505085	−	0.863070i	$-0.668539\pi$
$313$	−17.8717	−1.01017	−0.505085	+	0.863070i	$0.668539\pi$
$314$	−0.498945	−0.0281571
$315$	0.0684891	0.00385893
$316$	4.18569	0.235463
$317$	−16.8959	−0.948969	−0.474485	−	0.880264i	$-0.657366\pi$
$317$	−16.8959	−0.948969	−0.474485	+	0.880264i	$0.657366\pi$
$318$	0.0283802	0.00159148
$319$	−1.00420	−0.0562245
$320$	−1.63702	−0.0915125
$321$	−7.18961	−0.401285
$322$	−0.300187	−0.0167288
$323$	0.0616562	0.00343064
$324$	−1.75105	−0.0972808
$325$	−30.5306	−1.69353
$326$	5.37294	0.297580
$327$	11.7603	0.650345
$328$	15.9827	0.882495
$329$	1.21090	0.0667589
$330$	−0.0589784	−0.00324666
$331$	−30.6643	−1.68546	−0.842731	−	0.538335i	$-0.819053\pi$
$331$	−30.6643	−1.68546	−0.842731	+	0.538335i	$0.819053\pi$
$332$	−31.4205	−1.72442
$333$	7.61954	0.417548
$334$	−2.55579	−0.139846
$335$	−6.69564	−0.365822
$336$	−0.282556	−0.0154147
$337$	5.54873	0.302259	0.151129	−	0.988514i	$-0.451709\pi$
$337$	5.54873	0.302259	0.151129	+	0.988514i	$0.451709\pi$
$338$	15.3742	0.836244
$339$	−5.36524	−0.291400
$340$	1.09009	0.0591185
$341$	−0.775321	−0.0419860
$342$	−0.0307630	−0.00166347
$343$	−1.53890	−0.0830929
$344$	2.29583	0.123783
$345$	3.40443	0.183288
$346$	−8.84028	−0.475257
$347$	−17.2327	−0.925100	−0.462550	−	0.886593i	$-0.653065\pi$
$347$	−17.2327	−0.925100	−0.462550	+	0.886593i	$0.653065\pi$
$348$	−9.26067	−0.496424
$349$	−5.90035	−0.315838	−0.157919	−	0.987452i	$-0.550479\pi$
$349$	−5.90035	−0.315838	−0.157919	+	0.987452i	$0.550479\pi$
$350$	−0.253188	−0.0135335
$351$	−6.61917	−0.353305
$352$	0.954063	0.0508517
$353$	−22.4944	−1.19725	−0.598627	−	0.801028i	$-0.704287\pi$
$353$	−22.4944	−1.19725	−0.598627	+	0.801028i	$0.704287\pi$
$354$	−1.93898	−0.103056
$355$	4.27997	0.227157
$356$	−22.2709	−1.18036
$357$	0.110017	0.00582270
$358$	−3.50407	−0.185196
$359$	5.47971	0.289208	0.144604	−	0.989490i	$-0.453809\pi$
$359$	5.47971	0.289208	0.144604	+	0.989490i	$0.453809\pi$
$360$	−1.16512	−0.0614070
$361$	−18.9962	−0.999800
$362$	−5.77424	−0.303487
$363$	10.9639	0.575458
$364$	−1.27515	−0.0668360
$365$	−3.94517	−0.206500
$366$	−1.73438	−0.0906576
$367$	−19.8785	−1.03765	−0.518824	−	0.854881i	$-0.673630\pi$
$367$	−19.8785	−1.03765	−0.518824	+	0.854881i	$0.673630\pi$
$368$	−14.0452	−0.732154
$369$	−8.53971	−0.444560
$370$	2.36670	0.123039
$371$	−0.00625781	−0.000324889 0
$372$	−7.14996	−0.370708
$373$	25.6785	1.32958	0.664791	−	0.747030i	$-0.268521\pi$
$373$	25.6785	1.32958	0.664791	+	0.747030i	$0.268521\pi$
$374$	−0.0947393	−0.00489885
$375$	5.98408	0.309017
$376$	−20.5994	−1.06233
$377$	−35.0063	−1.80292
$378$	−0.0548922	−0.00282335
$379$	34.8480	1.79002	0.895012	−	0.446043i	$-0.147167\pi$
$379$	34.8480	1.79002	0.895012	+	0.446043i	$0.147167\pi$
$380$	0.0672108	0.00344784
$381$	−11.9742	−0.613457
$382$	2.41458	0.123540
$383$	27.5142	1.40591	0.702955	−	0.711235i	$-0.251864\pi$
$383$	27.5142	1.40591	0.702955	+	0.711235i	$0.251864\pi$
$384$	11.3612	0.579773
$385$	0.0130047	0.000662779 0
$386$	−5.29478	−0.269497
$387$	−1.22669	−0.0623561
$388$	−16.9998	−0.863033
$389$	−8.10655	−0.411019	−0.205509	−	0.978655i	$-0.565885\pi$
$389$	−8.10655	−0.411019	−0.205509	+	0.978655i	$0.565885\pi$
$390$	−2.05598	−0.104109
$391$	5.46866	0.276562
$392$	13.0783	0.660555
$393$	5.31945	0.268331
$394$	0.667346	0.0336204
$395$	−1.48809	−0.0748741
$396$	−0.332489	−0.0167082
$397$	16.7794	0.842133	0.421066	−	0.907030i	$-0.361656\pi$
$397$	16.7794	0.842133	0.421066	+	0.907030i	$0.361656\pi$
$398$	−6.81382	−0.341546
$399$	0.00678321	0.000339585 0
$400$	−11.8462	−0.592308
$401$	32.2773	1.61185	0.805927	−	0.592015i	$-0.201668\pi$
$401$	32.2773	1.61185	0.805927	+	0.592015i	$0.201668\pi$
$402$	5.36638	0.267650
$403$	−27.0276	−1.34634
$404$	2.43051	0.120922
$405$	0.622534	0.0309340
$406$	−0.290305	−0.0144076
$407$	1.44679	0.0717148
$408$	−1.87157	−0.0926564
$409$	−7.42351	−0.367069	−0.183534	−	0.983013i	$-0.558754\pi$
$409$	−7.42351	−0.367069	−0.183534	+	0.983013i	$0.558754\pi$
$410$	−2.65252	−0.130999
$411$	4.30047	0.212127
$412$	2.75125	0.135544
$413$	0.427543	0.0210380
$414$	−2.72856	−0.134101
$415$	11.1706	0.548343
$416$	33.2585	1.63063
$417$	21.3093	1.04352
$418$	−0.00584126	−0.000285705 0
$419$	−18.9566	−0.926090	−0.463045	−	0.886335i	$-0.653243\pi$
$419$	−18.9566	−0.926090	−0.463045	+	0.886335i	$0.653243\pi$
$420$	0.119928	0.00585190
$421$	−23.4129	−1.14107	−0.570536	−	0.821272i	$-0.693265\pi$
$421$	−23.4129	−1.14107	−0.570536	+	0.821272i	$0.693265\pi$
$422$	−3.61967	−0.176203
$423$	11.0065	0.535153
$424$	0.106456	0.00516995
$425$	4.61245	0.223737
$426$	−3.43028	−0.166198
$427$	0.382429	0.0185071
$428$	−12.5894	−0.608531
$429$	−1.25684	−0.0606809
$430$	−0.381022	−0.0183745
$431$	−25.4864	−1.22764	−0.613818	−	0.789448i	$-0.710367\pi$
$431$	−25.4864	−1.22764	−0.613818	+	0.789448i	$0.710367\pi$
$432$	−2.56830	−0.123567
$433$	−24.5265	−1.17867	−0.589335	−	0.807889i	$-0.700610\pi$
$433$	−24.5265	−1.17867	−0.589335	+	0.807889i	$0.700610\pi$
$434$	−0.224138	−0.0107589
$435$	3.29235	0.157856
$436$	20.5929	0.986221
$437$	0.337177	0.0161293
$438$	3.16195	0.151084
$439$	−9.14738	−0.436581	−0.218290	−	0.975884i	$-0.570048\pi$
$439$	−9.14738	−0.436581	−0.218290	+	0.975884i	$0.570048\pi$
$440$	−0.221231	−0.0105468
$441$	−6.98790	−0.332757
$442$	−3.30260	−0.157089
$443$	31.4276	1.49317	0.746586	−	0.665289i	$-0.231692\pi$
$443$	31.4276	1.49317	0.746586	+	0.665289i	$0.231692\pi$
$444$	13.3422	0.633194
$445$	7.91776	0.375338
$446$	−12.8238	−0.607224
$447$	14.5703	0.689151
$448$	−0.289301	−0.0136682
$449$	−39.3579	−1.85742	−0.928708	−	0.370812i	$-0.879079\pi$
$449$	−39.3579	−1.85742	−0.928708	+	0.370812i	$0.879079\pi$
$450$	−2.30136	−0.108487
$451$	−1.62152	−0.0763542
$452$	−9.39483	−0.441896
$453$	−7.31852	−0.343854
$454$	−0.0227823	−0.00106923
$455$	0.453341	0.0212530
$456$	−0.115394	−0.00540381
$457$	6.47916	0.303082	0.151541	−	0.988451i	$-0.451576\pi$
$457$	6.47916	0.303082	0.151541	+	0.988451i	$0.451576\pi$
$458$	−2.91230	−0.136083
$459$	1.00000	0.0466760
$460$	5.96134	0.277949
$461$	−31.1170	−1.44926	−0.724632	−	0.689136i	$-0.757990\pi$
$461$	−31.1170	−1.44926	−0.724632	+	0.689136i	$0.757990\pi$
$462$	−0.0104229	−0.000484917 0
$463$	−30.5942	−1.42183	−0.710916	−	0.703277i	$-0.751720\pi$
$463$	−30.5942	−1.42183	−0.710916	+	0.703277i	$0.751720\pi$
$464$	−13.5828	−0.630565
$465$	2.54195	0.117880
$466$	−3.83457	−0.177633
$467$	26.2418	1.21432	0.607162	−	0.794578i	$-0.292308\pi$
$467$	26.2418	1.21432	0.607162	+	0.794578i	$0.292308\pi$
$468$	−11.5905	−0.535772
$469$	−1.18328	−0.0546388
$470$	3.41872	0.157694
$471$	1.00000	0.0460776
$472$	−7.27323	−0.334777
$473$	−0.232923	−0.0107098
$474$	1.19267	0.0547811
$475$	0.284386	0.0130485
$476$	0.192645	0.00882988
$477$	−0.0568805	−0.00260438
$478$	3.40136	0.155574
$479$	33.8380	1.54610	0.773050	−	0.634346i	$-0.218730\pi$
$479$	33.8380	1.54610	0.773050	+	0.634346i	$0.218730\pi$
$480$	−3.12797	−0.142772
$481$	50.4350	2.29964
$482$	−11.6394	−0.530162
$483$	0.601644	0.0273757
$484$	19.1985	0.872657
$485$	6.04376	0.274433
$486$	−0.498945	−0.0226326
$487$	11.2538	0.509959	0.254980	−	0.966946i	$-0.417931\pi$
$487$	11.2538	0.509959	0.254980	+	0.966946i	$0.417931\pi$
$488$	−6.50576	−0.294502
$489$	−10.7686	−0.486973
$490$	−2.17051	−0.0980537
$491$	−28.9139	−1.30487	−0.652433	−	0.757846i	$-0.726252\pi$
$491$	−28.9139	−1.30487	−0.652433	+	0.757846i	$0.726252\pi$
$492$	−14.9535	−0.674156
$493$	5.28863	0.238188
$494$	−0.203625	−0.00916154
$495$	0.118206	0.00531298
$496$	−10.4870	−0.470878
$497$	0.756372	0.0339279
$498$	−8.95293	−0.401190
$499$	37.0687	1.65942	0.829712	−	0.558192i	$-0.188505\pi$
$499$	37.0687	1.65942	0.829712	+	0.558192i	$0.188505\pi$
$500$	10.4784	0.468610
$501$	5.12239	0.228851
$502$	1.15113	0.0513774
$503$	9.27716	0.413648	0.206824	−	0.978378i	$-0.433687\pi$
$503$	9.27716	0.413648	0.206824	+	0.978378i	$0.433687\pi$
$504$	−0.205904	−0.00917168
$505$	−0.864093	−0.0384516
$506$	−0.518097	−0.0230322
$507$	−30.8134	−1.36847
$508$	−20.9675	−0.930281
$509$	−3.67145	−0.162734	−0.0813672	−	0.996684i	$-0.525929\pi$
$509$	−3.67145	−0.162734	−0.0813672	+	0.996684i	$0.525929\pi$
$510$	0.310610	0.0137540
$511$	−0.697207	−0.0308426
$512$	22.5181	0.995169
$513$	0.0616562	0.00272219
$514$	−5.98349	−0.263921
$515$	−0.978124	−0.0431013
$516$	−2.14800	−0.0945604
$517$	2.08990	0.0919138
$518$	0.418253	0.0183770
$519$	17.7180	0.777732
$520$	−7.71209	−0.338198
$521$	31.1280	1.36374	0.681871	−	0.731473i	$-0.261167\pi$
$521$	31.1280	1.36374	0.681871	+	0.731473i	$0.261167\pi$
$522$	−2.63873	−0.115494
$523$	4.66119	0.203820	0.101910	−	0.994794i	$-0.467505\pi$
$523$	4.66119	0.203820	0.101910	+	0.994794i	$0.467505\pi$
$524$	9.31465	0.406913
$525$	0.507447	0.0221468
$526$	10.0702	0.439082
$527$	4.08323	0.177868
$528$	−0.487667	−0.0212230
$529$	6.90624	0.300271
$530$	−0.0176677	−0.000767434 0
$531$	3.88617	0.168645
$532$	0.0118778	0.000514966 0
$533$	−56.5258	−2.44840
$534$	−6.34587	−0.274613
$535$	4.47578	0.193505
$536$	20.1296	0.869465
$537$	7.02296	0.303063
$538$	−14.9441	−0.644286
$539$	−1.32686	−0.0571518
$540$	1.09009	0.0469101
$541$	−7.36605	−0.316691	−0.158346	−	0.987384i	$-0.550616\pi$
$541$	−7.36605	−0.316691	−0.158346	+	0.987384i	$0.550616\pi$
$542$	−3.31380	−0.142340
$543$	11.5729	0.496641
$544$	−5.02458	−0.215427
$545$	−7.32118	−0.313605
$546$	−0.363341	−0.0155495
$547$	21.8338	0.933544	0.466772	−	0.884378i	$-0.345417\pi$
$547$	21.8338	0.933544	0.466772	+	0.884378i	$0.345417\pi$
$548$	7.53036	0.321681
$549$	3.47610	0.148356
$550$	−0.436980	−0.0186329
$551$	0.326076	0.0138913
$552$	−10.2350	−0.435629
$553$	−0.262982	−0.0111831
$554$	−0.375481	−0.0159526
$555$	−4.74342	−0.201347
$556$	37.3138	1.58246
$557$	4.08349	0.173023	0.0865116	−	0.996251i	$-0.472428\pi$
$557$	4.08349	0.173023	0.0865116	+	0.996251i	$0.472428\pi$
$558$	−2.03731	−0.0862460
$559$	−8.11966	−0.343425
$560$	0.175901	0.00743316
$561$	0.189879	0.00801671
$562$	−8.55594	−0.360910
$563$	−30.8749	−1.30122	−0.650610	−	0.759412i	$-0.725487\pi$
$563$	−30.8749	−1.30122	−0.650610	+	0.759412i	$0.725487\pi$
$564$	19.2729	0.811537
$565$	3.34005	0.140517
$566$	−9.65736	−0.405929
$567$	0.110017	0.00462027
$568$	−12.8672	−0.539894
$569$	24.1268	1.01145	0.505724	−	0.862695i	$-0.331225\pi$
$569$	24.1268	1.01145	0.505724	+	0.862695i	$0.331225\pi$
$570$	0.0191510	0.000802148 0
$571$	8.49513	0.355510	0.177755	−	0.984075i	$-0.443117\pi$
$571$	8.49513	0.355510	0.177755	+	0.984075i	$0.443117\pi$
$572$	−2.20080	−0.0920201
$573$	−4.83937	−0.202167
$574$	−0.468764	−0.0195658
$575$	25.2239	1.05191
$576$	−2.62961	−0.109567
$577$	36.8184	1.53277	0.766385	−	0.642381i	$-0.222053\pi$
$577$	36.8184	1.53277	0.766385	+	0.642381i	$0.222053\pi$
$578$	0.498945	0.0207534
$579$	10.6120	0.441018
$580$	5.76509	0.239382
$581$	1.97411	0.0818999
$582$	−4.84391	−0.200787
$583$	−0.0108004	−0.000447308 0
$584$	11.8607	0.490797
$585$	4.12066	0.170368
$586$	15.8201	0.653524
$587$	−15.5905	−0.643488	−0.321744	−	0.946827i	$-0.604269\pi$
$587$	−15.5905	−0.643488	−0.321744	+	0.946827i	$0.604269\pi$
$588$	−12.2362	−0.504612
$589$	0.251756	0.0103734
$590$	1.20708	0.0496948
$591$	−1.33752	−0.0550181
$592$	19.5692	0.804291
$593$	−35.5948	−1.46170	−0.730852	−	0.682536i	$-0.760877\pi$
$593$	−35.5948	−1.46170	−0.730852	+	0.682536i	$0.760877\pi$
$594$	−0.0947393	−0.00388720
$595$	−0.0684891	−0.00280778
$596$	25.5134	1.04507
$597$	13.6565	0.558922
$598$	−18.0608	−0.738560
$599$	−18.9779	−0.775417	−0.387708	−	0.921782i	$-0.626733\pi$
$599$	−18.9779	−0.775417	−0.387708	+	0.921782i	$0.626733\pi$
$600$	−8.63252	−0.352421
$601$	35.2668	1.43856	0.719282	−	0.694718i	$-0.244471\pi$
$601$	35.2668	1.43856	0.719282	+	0.694718i	$0.244471\pi$
$602$	−0.0673357	−0.00274440
$603$	−10.7555	−0.437996
$604$	−12.8151	−0.521440
$605$	−6.82543	−0.277493
$606$	0.692548	0.0281328
$607$	−9.22942	−0.374611	−0.187305	−	0.982302i	$-0.559975\pi$
$607$	−9.22942	−0.374611	−0.187305	+	0.982302i	$0.559975\pi$
$608$	−0.309796	−0.0125639
$609$	0.581837	0.0235772
$610$	1.07971	0.0437163
$611$	72.8537	2.94735
$612$	1.75105	0.0707822
$613$	−9.69926	−0.391750	−0.195875	−	0.980629i	$-0.562755\pi$
$613$	−9.69926	−0.391750	−0.195875	+	0.980629i	$0.562755\pi$
$614$	7.64025	0.308335
$615$	5.31626	0.214372
$616$	−0.0390969	−0.00157526
$617$	−1.65175	−0.0664970	−0.0332485	−	0.999447i	$-0.510585\pi$
$617$	−1.65175	−0.0664970	−0.0332485	+	0.999447i	$0.510585\pi$
$618$	0.783940	0.0315347
$619$	13.8044	0.554845	0.277422	−	0.960748i	$-0.410520\pi$
$619$	13.8044	0.554845	0.277422	+	0.960748i	$0.410520\pi$
$620$	4.45109	0.178760
$621$	5.46866	0.219450
$622$	10.3072	0.413282
$623$	1.39926	0.0560600
$624$	−17.0000	−0.680545
$625$	19.3370	0.773478
$626$	−8.91700	−0.356395
$627$	0.0117072	0.000467542 0
$628$	1.75105	0.0698747
$629$	−7.61954	−0.303811
$630$	0.0341723	0.00136146
$631$	−47.9318	−1.90814	−0.954068	−	0.299591i	$-0.903150\pi$
$631$	−47.9318	−1.90814	−0.954068	+	0.299591i	$0.903150\pi$
$632$	4.47376	0.177957
$633$	7.25466	0.288347
$634$	−8.43013	−0.334803
$635$	7.45435	0.295817
$636$	−0.0996009	−0.00394943
$637$	−46.2540	−1.83265
$638$	−0.501041	−0.0198364
$639$	6.87507	0.271974
$640$	−7.07273	−0.279574
$641$	15.4994	0.612189	0.306095	−	0.952001i	$-0.400978\pi$
$641$	15.4994	0.612189	0.306095	+	0.952001i	$0.400978\pi$
$642$	−3.58722	−0.141576
$643$	−12.6980	−0.500762	−0.250381	−	0.968147i	$-0.580556\pi$
$643$	−12.6980	−0.500762	−0.250381	+	0.968147i	$0.580556\pi$
$644$	1.05351	0.0415141
$645$	0.763656	0.0300689
$646$	0.0307630	0.00121035
$647$	20.5803	0.809097	0.404548	−	0.914517i	$-0.367429\pi$
$647$	20.5803	0.809097	0.404548	+	0.914517i	$0.367429\pi$
$648$	−1.87157	−0.0735222
$649$	0.737903	0.0289652
$650$	−15.2331	−0.597490
$651$	0.449223	0.0176065
$652$	−18.8564	−0.738475
$653$	−3.97714	−0.155638	−0.0778188	−	0.996968i	$-0.524796\pi$
$653$	−3.97714	−0.155638	−0.0778188	+	0.996968i	$0.524796\pi$
$654$	5.86773	0.229447
$655$	−3.31154	−0.129393
$656$	−21.9325	−0.856322
$657$	−6.33728	−0.247241
$658$	0.604170	0.0235530
$659$	10.1460	0.395231	0.197615	−	0.980280i	$-0.436680\pi$
$659$	10.1460	0.395231	0.197615	+	0.980280i	$0.436680\pi$
$660$	0.206986	0.00805691
$661$	−49.2701	−1.91639	−0.958193	−	0.286121i	$-0.907634\pi$
$661$	−49.2701	−1.91639	−0.958193	+	0.286121i	$0.907634\pi$
$662$	−15.2998	−0.594643
$663$	6.61917	0.257067
$664$	−33.5829	−1.30327
$665$	−0.00422278	−0.000163752 0
$666$	3.80173	0.147314
$667$	28.9217	1.11985
$668$	8.96958	0.347043
$669$	25.7018	0.993691
$670$	−3.34075	−0.129065
$671$	0.660040	0.0254806
$672$	−0.552787	−0.0213242
$673$	41.3149	1.59257	0.796286	−	0.604921i	$-0.206795\pi$
$673$	41.3149	1.59257	0.796286	+	0.604921i	$0.206795\pi$
$674$	2.76851	0.106639
$675$	4.61245	0.177533
$676$	−53.9559	−2.07523
$677$	−35.3660	−1.35923	−0.679614	−	0.733570i	$-0.737853\pi$
$677$	−35.3660	−1.35923	−0.679614	+	0.733570i	$0.737853\pi$
$678$	−2.67696	−0.102808
$679$	1.06808	0.0409890
$680$	1.16512	0.0446801
$681$	0.0456610	0.00174973
$682$	−0.386842	−0.0148130
$683$	−28.1193	−1.07595	−0.537977	−	0.842960i	$-0.680811\pi$
$683$	−28.1193	−1.07595	−0.537977	+	0.842960i	$0.680811\pi$
$684$	0.107963	0.00412808
$685$	−2.67719	−0.102290
$686$	−0.767827	−0.0293158
$687$	5.83691	0.222692
$688$	−3.15051	−0.120112
$689$	−0.376502	−0.0143436
$690$	1.69862	0.0646654
$691$	−23.5978	−0.897704	−0.448852	−	0.893606i	$-0.648167\pi$
$691$	−23.5978	−0.897704	−0.448852	+	0.893606i	$0.648167\pi$
$692$	31.0251	1.17940
$693$	0.0208899	0.000793541 0
$694$	−8.59816	−0.326382
$695$	−13.2658	−0.503199
$696$	−9.89803	−0.375184
$697$	8.53971	0.323465
$698$	−2.94395	−0.111430
$699$	7.68536	0.290687
$700$	0.888566	0.0335847
$701$	−37.4447	−1.41427	−0.707133	−	0.707081i	$-0.750012\pi$
$701$	−37.4447	−1.41427	−0.707133	+	0.707081i	$0.750012\pi$
$702$	−3.30260	−0.124649
$703$	−0.469791	−0.0177185
$704$	−0.499309	−0.0188184
$705$	−6.85191	−0.258058
$706$	−11.2234	−0.422400
$707$	−0.152706	−0.00574310
$708$	6.80489	0.255743
$709$	−43.9110	−1.64911	−0.824555	−	0.565781i	$-0.808575\pi$
$709$	−43.9110	−1.64911	−0.824555	+	0.565781i	$0.808575\pi$
$710$	2.13547	0.0801426
$711$	−2.39038	−0.0896463
$712$	−23.8037	−0.892082
$713$	22.3298	0.836257
$714$	0.0548922	0.00205429
$715$	0.782428	0.0292611
$716$	12.2976	0.459582
$717$	−6.81710	−0.254589
$718$	2.73407	0.102035
$719$	−15.7143	−0.586046	−0.293023	−	0.956105i	$-0.594661\pi$
$719$	−15.7143	−0.586046	−0.293023	+	0.956105i	$0.594661\pi$
$720$	1.59885	0.0595858
$721$	−0.172858	−0.00643756
$722$	−9.47805	−0.352737
$723$	23.3281	0.867581
$724$	20.2648	0.753134
$725$	24.3935	0.905953
$726$	5.47040	0.203026
$727$	18.7455	0.695233	0.347617	−	0.937637i	$-0.386991\pi$
$727$	18.7455	0.695233	0.347617	+	0.937637i	$0.386991\pi$
$728$	−1.36291	−0.0505128
$729$	1.00000	0.0370370
$730$	−1.96842	−0.0728546
$731$	1.22669	0.0453708
$732$	6.08684	0.224976
$733$	−40.9054	−1.51087	−0.755437	−	0.655221i	$-0.772576\pi$
$733$	−40.9054	−1.51087	−0.755437	+	0.655221i	$0.772576\pi$
$734$	−9.91825	−0.366089
$735$	4.35020	0.160460
$736$	−27.4777	−1.01284
$737$	−2.04224	−0.0752268
$738$	−4.26084	−0.156844
$739$	47.1152	1.73316	0.866580	−	0.499038i	$-0.166313\pi$
$739$	47.1152	1.73316	0.866580	+	0.499038i	$0.166313\pi$
$740$	−8.30599	−0.305334
$741$	0.408112	0.0149924
$742$	−0.00312230	−0.000114623 0
$743$	22.3958	0.821620	0.410810	−	0.911721i	$-0.365246\pi$
$743$	22.3958	0.821620	0.410810	+	0.911721i	$0.365246\pi$
$744$	−7.64204	−0.280171
$745$	−9.07051	−0.332318
$746$	12.8121	0.469086
$747$	17.9437	0.656527
$748$	0.332489	0.0121570
$749$	0.790977	0.0289017
$750$	2.98572	0.109023
$751$	−2.90469	−0.105994	−0.0529969	−	0.998595i	$-0.516877\pi$
$751$	−2.90469	−0.105994	−0.0529969	+	0.998595i	$0.516877\pi$
$752$	28.2679	1.03083
$753$	−2.30713	−0.0840765
$754$	−17.4662	−0.636082
$755$	4.55603	0.165811
$756$	0.192645	0.00700644
$757$	−27.5912	−1.00282	−0.501410	−	0.865210i	$-0.667185\pi$
$757$	−27.5912	−1.00282	−0.501410	+	0.865210i	$0.667185\pi$
$758$	17.3872	0.631533
$759$	1.03839	0.0376910
$760$	0.0718365	0.00260579
$761$	29.3935	1.06551	0.532757	−	0.846268i	$-0.321156\pi$
$761$	29.3935	1.06551	0.532757	+	0.846268i	$0.321156\pi$
$762$	−5.97446	−0.216432
$763$	−1.29383	−0.0468397
$764$	−8.47399	−0.306578
$765$	−0.622534	−0.0225078
$766$	13.7281	0.496015
$767$	25.7232	0.928810
$768$	0.409375	0.0147720
$769$	10.5904	0.381899	0.190950	−	0.981600i	$-0.438843\pi$
$769$	10.5904	0.381899	0.190950	+	0.981600i	$0.438843\pi$
$770$	0.00648861	0.000233833 0
$771$	11.9923	0.431892
$772$	18.5821	0.668785
$773$	41.9637	1.50933	0.754665	−	0.656110i	$-0.227799\pi$
$773$	41.9637	1.50933	0.754665	+	0.656110i	$0.227799\pi$
$774$	−0.612050	−0.0219997
$775$	18.8337	0.676526
$776$	−18.1698	−0.652257
$777$	−0.838276	−0.0300730
$778$	−4.04472	−0.145010
$779$	0.526526	0.0188647
$780$	7.21549	0.258356
$781$	1.30543	0.0467121
$782$	2.72856	0.0975731
$783$	5.28863	0.189000
$784$	−17.9470	−0.640965
$785$	−0.622534	−0.0222192
$786$	2.65411	0.0946691
$787$	−12.7855	−0.455755	−0.227877	−	0.973690i	$-0.573179\pi$
$787$	−12.7855	−0.455755	−0.227877	+	0.973690i	$0.573179\pi$
$788$	−2.34206	−0.0834325
$789$	−20.1830	−0.718535
$790$	−0.742477	−0.0264161
$791$	0.590266	0.0209874
$792$	−0.355372	−0.0126276
$793$	23.0089	0.817070
$794$	8.37198	0.297110
$795$	0.0354101	0.00125587
$796$	23.9132	0.847581
$797$	30.3721	1.07583	0.537917	−	0.842997i	$-0.319211\pi$
$797$	30.3721	1.07583	0.537917	+	0.842997i	$0.319211\pi$
$798$	0.00338444	0.000119808 0
$799$	−11.0065	−0.389381
$800$	−23.1756	−0.819382
$801$	12.7186	0.449389
$802$	16.1046	0.568673
$803$	−1.20332	−0.0424642
$804$	−18.8334	−0.664202
$805$	−0.374544	−0.0132009
$806$	−13.4853	−0.474998
$807$	29.9514	1.05434
$808$	2.59778	0.0913897
$809$	−19.7563	−0.694594	−0.347297	−	0.937755i	$-0.612900\pi$
$809$	−19.7563	−0.694594	−0.347297	+	0.937755i	$0.612900\pi$
$810$	0.310610	0.0109137
$811$	9.69439	0.340416	0.170208	−	0.985408i	$-0.445556\pi$
$811$	9.69439	0.340416	0.170208	+	0.985408i	$0.445556\pi$
$812$	1.01883	0.0357539
$813$	6.64163	0.232932
$814$	0.721869	0.0253015
$815$	6.70383	0.234825
$816$	2.56830	0.0899085
$817$	0.0756330	0.00264606
$818$	−3.70392	−0.129505
$819$	0.728219	0.0254460
$820$	9.30907	0.325087
$821$	11.9684	0.417700	0.208850	−	0.977948i	$-0.433028\pi$
$821$	11.9684	0.417700	0.208850	+	0.977948i	$0.433028\pi$
$822$	2.14570	0.0748398
$823$	27.7894	0.968676	0.484338	−	0.874881i	$-0.339061\pi$
$823$	27.7894	0.968676	0.484338	+	0.874881i	$0.339061\pi$
$824$	2.94060	0.102441
$825$	0.875809	0.0304918
$826$	0.213320	0.00742237
$827$	−23.8914	−0.830786	−0.415393	−	0.909642i	$-0.636356\pi$
$827$	−23.8914	−0.830786	−0.415393	+	0.909642i	$0.636356\pi$
$828$	9.57592	0.332786
$829$	−51.7532	−1.79746	−0.898732	−	0.438498i	$-0.855510\pi$
$829$	−51.7532	−1.79746	−0.898732	+	0.438498i	$0.855510\pi$
$830$	5.57351	0.193459
$831$	0.752550	0.0261057
$832$	−17.4058	−0.603439
$833$	6.98790	0.242116
$834$	10.6322	0.368162
$835$	−3.18886	−0.110355
$836$	0.0205000	0.000709007 0
$837$	4.08323	0.141137
$838$	−9.45829	−0.326731
$839$	−3.45559	−0.119300	−0.0596500	−	0.998219i	$-0.518998\pi$
$839$	−3.45559	−0.119300	−0.0596500	+	0.998219i	$0.518998\pi$
$840$	0.128182	0.00442270
$841$	−1.03042	−0.0355319
$842$	−11.6817	−0.402579
$843$	17.1481	0.590611
$844$	12.7033	0.437266
$845$	19.1824	0.659894
$846$	5.49162	0.188806
$847$	−1.20622	−0.0414461
$848$	−0.146086	−0.00501662
$849$	19.3556	0.664282
$850$	2.30136	0.0789359
$851$	−41.6686	−1.42838
$852$	12.0386	0.412436
$853$	−21.5188	−0.736789	−0.368394	−	0.929670i	$-0.620092\pi$
$853$	−21.5188	−0.736789	−0.368394	+	0.929670i	$0.620092\pi$
$854$	0.190811	0.00652942
$855$	−0.0383831	−0.00131267
$856$	−13.4558	−0.459911
$857$	−20.6835	−0.706535	−0.353267	−	0.935522i	$-0.614929\pi$
$857$	−20.6835	−0.706535	−0.353267	+	0.935522i	$0.614929\pi$
$858$	−0.627095	−0.0214087
$859$	19.2852	0.658002	0.329001	−	0.944329i	$-0.393288\pi$
$859$	19.2852	0.658002	0.329001	+	0.944329i	$0.393288\pi$
$860$	1.33720	0.0455983
$861$	0.939511	0.0320184
$862$	−12.7163	−0.433118
$863$	−42.0720	−1.43215	−0.716075	−	0.698024i	$-0.754063\pi$
$863$	−42.0720	−1.43215	−0.716075	+	0.698024i	$0.754063\pi$
$864$	−5.02458	−0.170940
$865$	−11.0300	−0.375033
$866$	−12.2374	−0.415843
$867$	−1.00000	−0.0339618
$868$	0.786614	0.0266994
$869$	−0.453884	−0.0153970
$870$	1.64270	0.0556928
$871$	−71.1921	−2.41225
$872$	22.0102	0.745359
$873$	9.70831	0.328577
$874$	0.168232	0.00569055
$875$	−0.658349	−0.0222562
$876$	−11.0969	−0.374930
$877$	19.3400	0.653066	0.326533	−	0.945186i	$-0.394120\pi$
$877$	19.3400	0.653066	0.326533	+	0.945186i	$0.394120\pi$
$878$	−4.56404	−0.154029
$879$	−31.7072	−1.06946
$880$	0.303589	0.0102340
$881$	−20.6559	−0.695915	−0.347957	−	0.937510i	$-0.613125\pi$
$881$	−20.6559	−0.695915	−0.347957	+	0.937510i	$0.613125\pi$
$882$	−3.48657	−0.117399
$883$	−57.3331	−1.92941	−0.964705	−	0.263331i	$-0.915179\pi$
$883$	−57.3331	−1.92941	−0.964705	+	0.263331i	$0.915179\pi$
$884$	11.5905	0.389831
$885$	−2.41927	−0.0813229
$886$	15.6806	0.526802
$887$	−1.67229	−0.0561500	−0.0280750	−	0.999606i	$-0.508938\pi$
$887$	−1.67229	−0.0561500	−0.0280750	+	0.999606i	$0.508938\pi$
$888$	14.2605	0.478550
$889$	1.31736	0.0441829
$890$	3.95052	0.132422
$891$	0.189879	0.00636120
$892$	45.0053	1.50689
$893$	−0.678617	−0.0227091
$894$	7.26977	0.243137
$895$	−4.37203	−0.146141
$896$	−1.24992	−0.0417569
$897$	36.1980	1.20861
$898$	−19.6374	−0.655310
$899$	21.5947	0.720223
$900$	8.07665	0.269222
$901$	0.0568805	0.00189496
$902$	−0.809046	−0.0269383
$903$	0.134956	0.00449107
$904$	−10.0414	−0.333973
$905$	−7.20452	−0.239487
$906$	−3.65154	−0.121314
$907$	−59.1439	−1.96384	−0.981921	−	0.189292i	$-0.939381\pi$
$907$	−59.1439	−1.96384	−0.981921	+	0.189292i	$0.939381\pi$
$908$	0.0799549	0.00265340
$909$	−1.38802	−0.0460379
$910$	0.226192	0.00749819
$911$	23.3755	0.774465	0.387232	−	0.921982i	$-0.373431\pi$
$911$	23.3755	0.774465	0.387232	+	0.921982i	$0.373431\pi$
$912$	0.158351	0.00524354
$913$	3.40714	0.112760
$914$	3.23274	0.106930
$915$	−2.16399	−0.0715394
$916$	10.2207	0.337703
$917$	−0.585229	−0.0193260
$918$	0.498945	0.0164676
$919$	−51.9293	−1.71299	−0.856494	−	0.516157i	$-0.827362\pi$
$919$	−51.9293	−1.71299	−0.856494	+	0.516157i	$0.827362\pi$
$920$	6.37162	0.210066
$921$	−15.3128	−0.504574
$922$	−15.5257	−0.511311
$923$	45.5072	1.49789
$924$	0.0365793	0.00120337
$925$	−35.1447	−1.15555
$926$	−15.2648	−0.501633
$927$	−1.57120	−0.0516049
$928$	−26.5731	−0.872305
$929$	1.07154	0.0351560	0.0175780	−	0.999845i	$-0.494404\pi$
$929$	1.07154	0.0351560	0.0175780	+	0.999845i	$0.494404\pi$
$930$	1.26829	0.0415889
$931$	0.430847	0.0141204
$932$	13.4575	0.440815
$933$	−20.6581	−0.676314
$934$	13.0932	0.428422
$935$	−0.118206	−0.00386576
$936$	−12.3882	−0.404922
$937$	−2.80111	−0.0915084	−0.0457542	−	0.998953i	$-0.514569\pi$
$937$	−2.80111	−0.0915084	−0.0457542	+	0.998953i	$0.514569\pi$
$938$	−0.590391	−0.0192769
$939$	17.8717	0.583222
$940$	−11.9981	−0.391334
$941$	0.510951	0.0166565	0.00832827	−	0.999965i	$-0.497349\pi$
$941$	0.510951	0.0166565	0.00832827	+	0.999965i	$0.497349\pi$
$942$	0.498945	0.0162565
$943$	46.7008	1.52079
$944$	9.98084	0.324849
$945$	−0.0684891	−0.00222795
$946$	−0.116216	−0.00377850
$947$	14.4806	0.470557	0.235278	−	0.971928i	$-0.424400\pi$
$947$	14.4806	0.470557	0.235278	+	0.971928i	$0.424400\pi$
$948$	−4.18569	−0.135945
$949$	−41.9475	−1.36167
$950$	0.141893	0.00460361
$951$	16.8959	0.547888
$952$	0.205904	0.00667338
$953$	37.3143	1.20873	0.604365	−	0.796708i	$-0.293427\pi$
$953$	37.3143	1.20873	0.604365	+	0.796708i	$0.293427\pi$
$954$	−0.0283802	−0.000918844 0
$955$	3.01267	0.0974877
$956$	−11.9371	−0.386074
$957$	1.00420	0.0324612
$958$	16.8833	0.545475
$959$	−0.473124	−0.0152780
$960$	1.63702	0.0528347
$961$	−14.3272	−0.462169
$962$	25.1643	0.811328
$963$	7.18961	0.231682
$964$	40.8488	1.31565
$965$	−6.60630	−0.212664
$966$	0.300187	0.00965836
$967$	−8.80310	−0.283089	−0.141544	−	0.989932i	$-0.545207\pi$
$967$	−8.80310	−0.283089	−0.141544	+	0.989932i	$0.545207\pi$
$968$	20.5198	0.659531
$969$	−0.0616562	−0.00198068
$970$	3.01550	0.0968219
$971$	8.92922	0.286552	0.143276	−	0.989683i	$-0.454236\pi$
$971$	8.92922	0.286552	0.143276	+	0.989683i	$0.454236\pi$
$972$	1.75105	0.0561651
$973$	−2.34438	−0.0751573
$974$	5.61503	0.179917
$975$	30.5306	0.977761
$976$	8.92767	0.285768
$977$	−24.3957	−0.780488	−0.390244	−	0.920711i	$-0.627609\pi$
$977$	−24.3957	−0.780488	−0.390244	+	0.920711i	$0.627609\pi$
$978$	−5.37294	−0.171808
$979$	2.41500	0.0771836
$980$	7.61744	0.243330
$981$	−11.7603	−0.375477
$982$	−14.4264	−0.460366
$983$	52.1478	1.66326	0.831628	−	0.555333i	$-0.187409\pi$
$983$	52.1478	1.66326	0.831628	+	0.555333i	$0.187409\pi$
$984$	−15.9827	−0.509509
$985$	0.832649	0.0265304
$986$	2.63873	0.0840344
$987$	−1.21090	−0.0385433
$988$	0.714627	0.0227353
$989$	6.70835	0.213313
$990$	0.0589784	0.00187446
$991$	−21.5669	−0.685095	−0.342547	−	0.939501i	$-0.611290\pi$
$991$	−21.5669	−0.685095	−0.342547	+	0.939501i	$0.611290\pi$
$992$	−20.5165	−0.651399
$993$	30.6643	0.973102
$994$	0.377388	0.0119700
$995$	−8.50161	−0.269519
$996$	31.4205	0.995595
$997$	−46.4744	−1.47186	−0.735930	−	0.677058i	$-0.763255\pi$
$997$	−46.4744	−1.47186	−0.735930	+	0.677058i	$0.763255\pi$
$998$	18.4952	0.585457
$999$	−7.61954	−0.241071

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.28	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.28	✓	48	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.498945 q^{2} -1.00000 q^{3} -1.75105 q^{4} +0.622534 q^{5} -0.498945 q^{6} +0.110017 q^{7} -1.87157 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.498945 q^{2} -1.00000 q^{3} -1.75105 q^{4} +0.622534 q^{5} -0.498945 q^{6} +0.110017 q^{7} -1.87157 q^{8} +1.00000 q^{9} +0.310610 q^{10} +0.189879 q^{11} +1.75105 q^{12} +6.61917 q^{13} +0.0548922 q^{14} -0.622534 q^{15} +2.56830 q^{16} -1.00000 q^{17} +0.498945 q^{18} -0.0616562 q^{19} -1.09009 q^{20} -0.110017 q^{21} +0.0947393 q^{22} -5.46866 q^{23} +1.87157 q^{24} -4.61245 q^{25} +3.30260 q^{26} -1.00000 q^{27} -0.192645 q^{28} -5.28863 q^{29} -0.310610 q^{30} -4.08323 q^{31} +5.02458 q^{32} -0.189879 q^{33} -0.498945 q^{34} +0.0684891 q^{35} -1.75105 q^{36} +7.61954 q^{37} -0.0307630 q^{38} -6.61917 q^{39} -1.16512 q^{40} -8.53971 q^{41} -0.0548922 q^{42} -1.22669 q^{43} -0.332489 q^{44} +0.622534 q^{45} -2.72856 q^{46} +11.0065 q^{47} -2.56830 q^{48} -6.98790 q^{49} -2.30136 q^{50} +1.00000 q^{51} -11.5905 q^{52} -0.0568805 q^{53} -0.498945 q^{54} +0.118206 q^{55} -0.205904 q^{56} +0.0616562 q^{57} -2.63873 q^{58} +3.88617 q^{59} +1.09009 q^{60} +3.47610 q^{61} -2.03731 q^{62} +0.110017 q^{63} -2.62961 q^{64} +4.12066 q^{65} -0.0947393 q^{66} -10.7555 q^{67} +1.75105 q^{68} +5.46866 q^{69} +0.0341723 q^{70} +6.87507 q^{71} -1.87157 q^{72} -6.33728 q^{73} +3.80173 q^{74} +4.61245 q^{75} +0.107963 q^{76} +0.0208899 q^{77} -3.30260 q^{78} -2.39038 q^{79} +1.59885 q^{80} +1.00000 q^{81} -4.26084 q^{82} +17.9437 q^{83} +0.192645 q^{84} -0.622534 q^{85} -0.612050 q^{86} +5.28863 q^{87} -0.355372 q^{88} +12.7186 q^{89} +0.310610 q^{90} +0.728219 q^{91} +9.57592 q^{92} +4.08323 q^{93} +5.49162 q^{94} -0.0383831 q^{95} -5.02458 q^{96} +9.70831 q^{97} -3.48657 q^{98} +0.189879 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 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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