LMFDB - Embedded newform 8043.2.a.t.1.31

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, 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("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.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:	$64.2236783457$
Analytic rank:	$0$
Dimension:	$52$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.31
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.510081 q^{2} -1.00000 q^{3} -1.73982 q^{4} -0.173032 q^{5} -0.510081 q^{6} +1.00000 q^{7} -1.90761 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.510081 q^{2} -1.00000 q^{3} -1.73982 q^{4} -0.173032 q^{5} -0.510081 q^{6} +1.00000 q^{7} -1.90761 q^{8} +1.00000 q^{9} -0.0882604 q^{10} -5.39973 q^{11} +1.73982 q^{12} -2.95869 q^{13} +0.510081 q^{14} +0.173032 q^{15} +2.50660 q^{16} -6.33645 q^{17} +0.510081 q^{18} -6.38888 q^{19} +0.301044 q^{20} -1.00000 q^{21} -2.75430 q^{22} -3.60244 q^{23} +1.90761 q^{24} -4.97006 q^{25} -1.50917 q^{26} -1.00000 q^{27} -1.73982 q^{28} +2.63960 q^{29} +0.0882604 q^{30} -6.86185 q^{31} +5.09379 q^{32} +5.39973 q^{33} -3.23210 q^{34} -0.173032 q^{35} -1.73982 q^{36} +2.16825 q^{37} -3.25884 q^{38} +2.95869 q^{39} +0.330078 q^{40} -9.92746 q^{41} -0.510081 q^{42} +2.19406 q^{43} +9.39454 q^{44} -0.173032 q^{45} -1.83753 q^{46} -3.66311 q^{47} -2.50660 q^{48} +1.00000 q^{49} -2.53513 q^{50} +6.33645 q^{51} +5.14757 q^{52} +14.2253 q^{53} -0.510081 q^{54} +0.934327 q^{55} -1.90761 q^{56} +6.38888 q^{57} +1.34641 q^{58} -11.5824 q^{59} -0.301044 q^{60} -4.06514 q^{61} -3.50010 q^{62} +1.00000 q^{63} -2.41495 q^{64} +0.511948 q^{65} +2.75430 q^{66} -13.4997 q^{67} +11.0243 q^{68} +3.60244 q^{69} -0.0882604 q^{70} -0.424899 q^{71} -1.90761 q^{72} +6.50543 q^{73} +1.10598 q^{74} +4.97006 q^{75} +11.1155 q^{76} -5.39973 q^{77} +1.50917 q^{78} +9.44015 q^{79} -0.433722 q^{80} +1.00000 q^{81} -5.06381 q^{82} -14.3937 q^{83} +1.73982 q^{84} +1.09641 q^{85} +1.11915 q^{86} -2.63960 q^{87} +10.3006 q^{88} +6.77066 q^{89} -0.0882604 q^{90} -2.95869 q^{91} +6.26758 q^{92} +6.86185 q^{93} -1.86849 q^{94} +1.10548 q^{95} -5.09379 q^{96} -7.21520 q^{97} +0.510081 q^{98} -5.39973 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) $ 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9 - 2 * q^10 + 9 * q^11 - 61 * q^12 + 44 * q^13 + 3 * q^14 + 7 * q^15 + 95 * q^16 - 6 * q^17 + 3 * q^18 + 7 * q^19 - 21 * q^20 - 52 * q^21 + 19 * q^22 - 4 * q^23 - 24 * q^24 + 83 * q^25 - 5 * q^26 - 52 * q^27 + 61 * q^28 + 31 * q^29 + 2 * q^30 + 11 * q^31 + 71 * q^32 - 9 * q^33 + 17 * q^34 - 7 * q^35 + 61 * q^36 + 71 * q^37 - 8 * q^38 - 44 * q^39 + 20 * q^40 - 25 * q^41 - 3 * q^42 + 75 * q^43 + 14 * q^44 - 7 * q^45 + 36 * q^46 - 20 * q^47 - 95 * q^48 + 52 * q^49 + 26 * q^50 + 6 * q^51 + 88 * q^52 + 70 * q^53 - 3 * q^54 + 12 * q^55 + 24 * q^56 - 7 * q^57 + 48 * q^58 - 27 * q^59 + 21 * q^60 + 59 * q^61 - 23 * q^62 + 52 * q^63 + 138 * q^64 + 44 * q^65 - 19 * q^66 + 65 * q^67 - 8 * q^68 + 4 * q^69 - 2 * q^70 - 11 * q^71 + 24 * q^72 + 34 * q^73 + 38 * q^74 - 83 * q^75 + 31 * q^76 + 9 * q^77 + 5 * q^78 + 74 * q^79 - 5 * q^80 + 52 * q^81 + 51 * q^82 - 30 * q^83 - 61 * q^84 + 70 * q^85 + 29 * q^86 - 31 * q^87 + 90 * q^88 - q^89 - 2 * q^90 + 44 * q^91 + 34 * q^92 - 11 * q^93 + 27 * q^94 + 9 * q^95 - 71 * q^96 + 73 * q^97 + 3 * q^98 + 9 * 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 $

$2$	0.510081	0.360682	0.180341	−	0.983604i	$-0.442280\pi$
$2$	0.510081	0.360682	0.180341	+	0.983604i	$0.442280\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.73982	−0.869909
$5$	−0.173032	−0.0773824	−0.0386912	−	0.999251i	$-0.512319\pi$
$5$	−0.173032	−0.0773824	−0.0386912	+	0.999251i	$0.512319\pi$
$6$	−0.510081	−0.208240
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.90761	−0.674442
$9$	1.00000	0.333333
$10$	−0.0882604	−0.0279104
$11$	−5.39973	−1.62808	−0.814039	−	0.580810i	$-0.802736\pi$
$11$	−5.39973	−1.62808	−0.814039	+	0.580810i	$0.802736\pi$
$12$	1.73982	0.502242
$13$	−2.95869	−0.820592	−0.410296	−	0.911952i	$-0.634575\pi$
$13$	−2.95869	−0.820592	−0.410296	+	0.911952i	$0.634575\pi$
$14$	0.510081	0.136325
$15$	0.173032	0.0446767
$16$	2.50660	0.626650
$17$	−6.33645	−1.53681	−0.768407	−	0.639961i	$-0.778950\pi$
$17$	−6.33645	−1.53681	−0.768407	+	0.639961i	$0.778950\pi$
$18$	0.510081	0.120227
$19$	−6.38888	−1.46571	−0.732854	−	0.680385i	$-0.761812\pi$
$19$	−6.38888	−1.46571	−0.732854	+	0.680385i	$0.761812\pi$
$20$	0.301044	0.0673156
$21$	−1.00000	−0.218218
$22$	−2.75430	−0.587218
$23$	−3.60244	−0.751160	−0.375580	−	0.926790i	$-0.622556\pi$
$23$	−3.60244	−0.751160	−0.375580	+	0.926790i	$0.622556\pi$
$24$	1.90761	0.389389
$25$	−4.97006	−0.994012
$26$	−1.50917	−0.295972
$27$	−1.00000	−0.192450
$28$	−1.73982	−0.328795
$29$	2.63960	0.490162	0.245081	−	0.969503i	$-0.421185\pi$
$29$	2.63960	0.490162	0.245081	+	0.969503i	$0.421185\pi$
$30$	0.0882604	0.0161141
$31$	−6.86185	−1.23243	−0.616213	−	0.787580i	$-0.711334\pi$
$31$	−6.86185	−1.23243	−0.616213	+	0.787580i	$0.711334\pi$
$32$	5.09379	0.900463
$33$	5.39973	0.939972
$34$	−3.23210	−0.554301
$35$	−0.173032	−0.0292478
$36$	−1.73982	−0.289970
$37$	2.16825	0.356458	0.178229	−	0.983989i	$-0.442963\pi$
$37$	2.16825	0.356458	0.178229	+	0.983989i	$0.442963\pi$
$38$	−3.25884	−0.528654
$39$	2.95869	0.473769
$40$	0.330078	0.0521899
$41$	−9.92746	−1.55041	−0.775204	−	0.631711i	$-0.782353\pi$
$41$	−9.92746	−1.55041	−0.775204	+	0.631711i	$0.782353\pi$
$42$	−0.510081	−0.0787072
$43$	2.19406	0.334591	0.167296	−	0.985907i	$-0.446497\pi$
$43$	2.19406	0.334591	0.167296	+	0.985907i	$0.446497\pi$
$44$	9.39454	1.41628
$45$	−0.173032	−0.0257941
$46$	−1.83753	−0.270930
$47$	−3.66311	−0.534320	−0.267160	−	0.963652i	$-0.586085\pi$
$47$	−3.66311	−0.534320	−0.267160	+	0.963652i	$0.586085\pi$
$48$	−2.50660	−0.361796
$49$	1.00000	0.142857
$50$	−2.53513	−0.358522
$51$	6.33645	0.887280
$52$	5.14757	0.713840
$53$	14.2253	1.95400	0.976998	−	0.213249i	$-0.0684046\pi$
$53$	14.2253	1.95400	0.976998	+	0.213249i	$0.0684046\pi$
$54$	−0.510081	−0.0694132
$55$	0.934327	0.125985
$56$	−1.90761	−0.254915
$57$	6.38888	0.846227
$58$	1.34641	0.176792
$59$	−11.5824	−1.50790	−0.753952	−	0.656929i	$-0.771855\pi$
$59$	−11.5824	−1.50790	−0.753952	+	0.656929i	$0.771855\pi$
$60$	−0.301044	−0.0388647
$61$	−4.06514	−0.520488	−0.260244	−	0.965543i	$-0.583803\pi$
$61$	−4.06514	−0.520488	−0.260244	+	0.965543i	$0.583803\pi$
$62$	−3.50010	−0.444513
$63$	1.00000	0.125988
$64$	−2.41495	−0.301869
$65$	0.511948	0.0634993
$66$	2.75430	0.339031
$67$	−13.4997	−1.64925	−0.824626	−	0.565678i	$-0.808615\pi$
$67$	−13.4997	−1.64925	−0.824626	+	0.565678i	$0.808615\pi$
$68$	11.0243	1.33689
$69$	3.60244	0.433682
$70$	−0.0882604	−0.0105491
$71$	−0.424899	−0.0504262	−0.0252131	−	0.999682i	$-0.508026\pi$
$71$	−0.424899	−0.0504262	−0.0252131	+	0.999682i	$0.508026\pi$
$72$	−1.90761	−0.224814
$73$	6.50543	0.761403	0.380702	−	0.924698i	$-0.375682\pi$
$73$	6.50543	0.761403	0.380702	+	0.924698i	$0.375682\pi$
$74$	1.10598	0.128568
$75$	4.97006	0.573893
$76$	11.1155	1.27503
$77$	−5.39973	−0.615356
$78$	1.50917	0.170880
$79$	9.44015	1.06210	0.531050	−	0.847341i	$-0.321798\pi$
$79$	9.44015	1.06210	0.531050	+	0.847341i	$0.321798\pi$
$80$	−0.433722	−0.0484916
$81$	1.00000	0.111111
$82$	−5.06381	−0.559204
$83$	−14.3937	−1.57991	−0.789955	−	0.613165i	$-0.789896\pi$
$83$	−14.3937	−1.57991	−0.789955	+	0.613165i	$0.789896\pi$
$84$	1.73982	0.189830
$85$	1.09641	0.118922
$86$	1.11915	0.120681
$87$	−2.63960	−0.282995
$88$	10.3006	1.09804
$89$	6.77066	0.717688	0.358844	−	0.933398i	$-0.383171\pi$
$89$	6.77066	0.717688	0.358844	+	0.933398i	$0.383171\pi$
$90$	−0.0882604	−0.00930347
$91$	−2.95869	−0.310155
$92$	6.26758	0.653440
$93$	6.86185	0.711541
$94$	−1.86849	−0.192720
$95$	1.10548	0.113420
$96$	−5.09379	−0.519883
$97$	−7.21520	−0.732592	−0.366296	−	0.930498i	$-0.619374\pi$
$97$	−7.21520	−0.732592	−0.366296	+	0.930498i	$0.619374\pi$
$98$	0.510081	0.0515260
$99$	−5.39973	−0.542693
$100$	8.64700	0.864700
$101$	−2.31345	−0.230197	−0.115099	−	0.993354i	$-0.536718\pi$
$101$	−2.31345	−0.230197	−0.115099	+	0.993354i	$0.536718\pi$
$102$	3.23210	0.320026
$103$	−3.36778	−0.331837	−0.165919	−	0.986139i	$-0.553059\pi$
$103$	−3.36778	−0.331837	−0.165919	+	0.986139i	$0.553059\pi$
$104$	5.64402	0.553442
$105$	0.173032	0.0168862
$106$	7.25606	0.704770
$107$	2.26876	0.219329	0.109664	−	0.993969i	$-0.465022\pi$
$107$	2.26876	0.219329	0.109664	+	0.993969i	$0.465022\pi$
$108$	1.73982	0.167414
$109$	4.22761	0.404932	0.202466	−	0.979289i	$-0.435104\pi$
$109$	4.22761	0.404932	0.202466	+	0.979289i	$0.435104\pi$
$110$	0.476582	0.0454403
$111$	−2.16825	−0.205801
$112$	2.50660	0.236851
$113$	7.23074	0.680211	0.340106	−	0.940387i	$-0.389537\pi$
$113$	7.23074	0.680211	0.340106	+	0.940387i	$0.389537\pi$
$114$	3.25884	0.305219
$115$	0.623337	0.0581265
$116$	−4.59243	−0.426396
$117$	−2.95869	−0.273531
$118$	−5.90798	−0.543874
$119$	−6.33645	−0.580861
$120$	−0.330078	−0.0301319
$121$	18.1571	1.65064
$122$	−2.07355	−0.187730
$123$	9.92746	0.895129
$124$	11.9384	1.07210
$125$	1.72514	0.154301
$126$	0.510081	0.0454416
$127$	−5.99074	−0.531592	−0.265796	−	0.964029i	$-0.585635\pi$
$127$	−5.99074	−0.531592	−0.265796	+	0.964029i	$0.585635\pi$
$128$	−11.4194	−1.00934
$129$	−2.19406	−0.193176
$130$	0.261135	0.0229031
$131$	11.2387	0.981926	0.490963	−	0.871180i	$-0.336645\pi$
$131$	11.2387	0.981926	0.490963	+	0.871180i	$0.336645\pi$
$132$	−9.39454	−0.817690
$133$	−6.38888	−0.553986
$134$	−6.88595	−0.594855
$135$	0.173032	0.0148922
$136$	12.0875	1.03649
$137$	−21.4960	−1.83652	−0.918262	−	0.395974i	$-0.870407\pi$
$137$	−21.4960	−1.83652	−0.918262	+	0.395974i	$0.870407\pi$
$138$	1.83753	0.156421
$139$	−4.14516	−0.351588	−0.175794	−	0.984427i	$-0.556249\pi$
$139$	−4.14516	−0.351588	−0.175794	+	0.984427i	$0.556249\pi$
$140$	0.301044	0.0254429
$141$	3.66311	0.308490
$142$	−0.216733	−0.0181878
$143$	15.9761	1.33599
$144$	2.50660	0.208883
$145$	−0.456736	−0.0379299
$146$	3.31830	0.274624
$147$	−1.00000	−0.0824786
$148$	−3.77235	−0.310086
$149$	−21.2788	−1.74322	−0.871612	−	0.490197i	$-0.836925\pi$
$149$	−21.2788	−1.74322	−0.871612	+	0.490197i	$0.836925\pi$
$150$	2.53513	0.206993
$151$	12.5776	1.02355	0.511776	−	0.859119i	$-0.328988\pi$
$151$	12.5776	1.02355	0.511776	+	0.859119i	$0.328988\pi$
$152$	12.1875	0.988535
$153$	−6.33645	−0.512271
$154$	−2.75430	−0.221948
$155$	1.18732	0.0953680
$156$	−5.14757	−0.412136
$157$	8.28699	0.661374	0.330687	−	0.943741i	$-0.392720\pi$
$157$	8.28699	0.661374	0.330687	+	0.943741i	$0.392720\pi$
$158$	4.81524	0.383080
$159$	−14.2253	−1.12814
$160$	−0.881389	−0.0696800
$161$	−3.60244	−0.283912
$162$	0.510081	0.0400757
$163$	−0.183756	−0.0143929	−0.00719645	−	0.999974i	$-0.502291\pi$
$163$	−0.183756	−0.0143929	−0.00719645	+	0.999974i	$0.502291\pi$
$164$	17.2720	1.34871
$165$	−0.934327	−0.0727372
$166$	−7.34193	−0.569845
$167$	16.4217	1.27075	0.635376	−	0.772203i	$-0.280845\pi$
$167$	16.4217	1.27075	0.635376	+	0.772203i	$0.280845\pi$
$168$	1.90761	0.147175
$169$	−4.24618	−0.326629
$170$	0.559258	0.0428931
$171$	−6.38888	−0.488570
$172$	−3.81727	−0.291064
$173$	−10.3600	−0.787658	−0.393829	−	0.919184i	$-0.628850\pi$
$173$	−10.3600	−0.787658	−0.393829	+	0.919184i	$0.628850\pi$
$174$	−1.34641	−0.102071
$175$	−4.97006	−0.375701
$176$	−13.5350	−1.02024
$177$	11.5824	0.870589
$178$	3.45358	0.258857
$179$	−21.3947	−1.59911	−0.799556	−	0.600592i	$-0.794932\pi$
$179$	−21.3947	−1.59911	−0.799556	+	0.600592i	$0.794932\pi$
$180$	0.301044	0.0224385
$181$	13.0172	0.967559	0.483779	−	0.875190i	$-0.339264\pi$
$181$	13.0172	0.967559	0.483779	+	0.875190i	$0.339264\pi$
$182$	−1.50917	−0.111867
$183$	4.06514	0.300504
$184$	6.87204	0.506614
$185$	−0.375177	−0.0275835
$186$	3.50010	0.256640
$187$	34.2151	2.50206
$188$	6.37315	0.464810
$189$	−1.00000	−0.0727393
$190$	0.563885	0.0409085
$191$	−15.5040	−1.12183	−0.560915	−	0.827873i	$-0.689551\pi$
$191$	−15.5040	−1.12183	−0.560915	+	0.827873i	$0.689551\pi$
$192$	2.41495	0.174284
$193$	−18.9609	−1.36483	−0.682417	−	0.730963i	$-0.739071\pi$
$193$	−18.9609	−1.36483	−0.682417	+	0.730963i	$0.739071\pi$
$194$	−3.68033	−0.264233
$195$	−0.511948	−0.0366614
$196$	−1.73982	−0.124273
$197$	−12.4409	−0.886375	−0.443188	−	0.896429i	$-0.646152\pi$
$197$	−12.4409	−0.886375	−0.443188	+	0.896429i	$0.646152\pi$
$198$	−2.75430	−0.195739
$199$	−1.12583	−0.0798079	−0.0399040	−	0.999204i	$-0.512705\pi$
$199$	−1.12583	−0.0798079	−0.0399040	+	0.999204i	$0.512705\pi$
$200$	9.48093	0.670403
$201$	13.4997	0.952196
$202$	−1.18005	−0.0830279
$203$	2.63960	0.185264
$204$	−11.0243	−0.771853
$205$	1.71777	0.119974
$206$	−1.71784	−0.119688
$207$	−3.60244	−0.250387
$208$	−7.41624	−0.514224
$209$	34.4982	2.38629
$210$	0.0882604	0.00609055
$211$	0.986764	0.0679316	0.0339658	−	0.999423i	$-0.489186\pi$
$211$	0.986764	0.0679316	0.0339658	+	0.999423i	$0.489186\pi$
$212$	−24.7494	−1.69980
$213$	0.424899	0.0291136
$214$	1.15725	0.0791079
$215$	−0.379643	−0.0258915
$216$	1.90761	0.129796
$217$	−6.86185	−0.465813
$218$	2.15642	0.146051
$219$	−6.50543	−0.439596
$220$	−1.62556	−0.109595
$221$	18.7476	1.26110
$222$	−1.10598	−0.0742286
$223$	−21.5846	−1.44541	−0.722706	−	0.691156i	$-0.757102\pi$
$223$	−21.5846	−1.44541	−0.722706	+	0.691156i	$0.757102\pi$
$224$	5.09379	0.340343
$225$	−4.97006	−0.331337
$226$	3.68827	0.245340
$227$	−2.06191	−0.136854	−0.0684270	−	0.997656i	$-0.521798\pi$
$227$	−2.06191	−0.136854	−0.0684270	+	0.997656i	$0.521798\pi$
$228$	−11.1155	−0.736141
$229$	14.7253	0.973074	0.486537	−	0.873660i	$-0.338260\pi$
$229$	14.7253	0.973074	0.486537	+	0.873660i	$0.338260\pi$
$230$	0.317953	0.0209652
$231$	5.39973	0.355276
$232$	−5.03533	−0.330586
$233$	23.9114	1.56649	0.783244	−	0.621714i	$-0.213563\pi$
$233$	23.9114	1.56649	0.783244	+	0.621714i	$0.213563\pi$
$234$	−1.50917	−0.0986575
$235$	0.633837	0.0413470
$236$	20.1513	1.31174
$237$	−9.44015	−0.613203
$238$	−3.23210	−0.209506
$239$	−8.87055	−0.573788	−0.286894	−	0.957962i	$-0.592623\pi$
$239$	−8.87055	−0.573788	−0.286894	+	0.957962i	$0.592623\pi$
$240$	0.433722	0.0279967
$241$	13.7017	0.882602	0.441301	−	0.897359i	$-0.354517\pi$
$241$	13.7017	0.882602	0.441301	+	0.897359i	$0.354517\pi$
$242$	9.26157	0.595356
$243$	−1.00000	−0.0641500
$244$	7.07260	0.452777
$245$	−0.173032	−0.0110546
$246$	5.06381	0.322857
$247$	18.9027	1.20275
$248$	13.0897	0.831199
$249$	14.3937	0.912161
$250$	0.879962	0.0556537
$251$	−16.7312	−1.05606	−0.528031	−	0.849225i	$-0.677069\pi$
$251$	−16.7312	−1.05606	−0.528031	+	0.849225i	$0.677069\pi$
$252$	−1.73982	−0.109598
$253$	19.4522	1.22295
$254$	−3.05576	−0.191736
$255$	−1.09641	−0.0686598
$256$	−0.994909	−0.0621818
$257$	−19.1083	−1.19194	−0.595970	−	0.803007i	$-0.703232\pi$
$257$	−19.1083	−1.19194	−0.595970	+	0.803007i	$0.703232\pi$
$258$	−1.11915	−0.0696752
$259$	2.16825	0.134728
$260$	−0.890696	−0.0552386
$261$	2.63960	0.163387
$262$	5.73263	0.354163
$263$	−19.7744	−1.21934	−0.609672	−	0.792654i	$-0.708699\pi$
$263$	−19.7744	−1.21934	−0.609672	+	0.792654i	$0.708699\pi$
$264$	−10.3006	−0.633956
$265$	−2.46144	−0.151205
$266$	−3.25884	−0.199813
$267$	−6.77066	−0.414357
$268$	23.4870	1.43470
$269$	7.82715	0.477230	0.238615	−	0.971114i	$-0.423307\pi$
$269$	7.82715	0.477230	0.238615	+	0.971114i	$0.423307\pi$
$270$	0.0882604	0.00537136
$271$	−15.5408	−0.944035	−0.472018	−	0.881589i	$-0.656474\pi$
$271$	−15.5408	−0.944035	−0.472018	+	0.881589i	$0.656474\pi$
$272$	−15.8829	−0.963045
$273$	2.95869	0.179068
$274$	−10.9647	−0.662400
$275$	26.8370	1.61833
$276$	−6.26758	−0.377264
$277$	10.7149	0.643798	0.321899	−	0.946774i	$-0.395679\pi$
$277$	10.7149	0.643798	0.321899	+	0.946774i	$0.395679\pi$
$278$	−2.11437	−0.126811
$279$	−6.86185	−0.410808
$280$	0.330078	0.0197259
$281$	15.8331	0.944521	0.472260	−	0.881459i	$-0.343438\pi$
$281$	15.8331	0.944521	0.472260	+	0.881459i	$0.343438\pi$
$282$	1.86849	0.111267
$283$	23.9954	1.42638	0.713188	−	0.700973i	$-0.247250\pi$
$283$	23.9954	1.42638	0.713188	+	0.700973i	$0.247250\pi$
$284$	0.739247	0.0438662
$285$	−1.10548	−0.0654831
$286$	8.14910	0.481867
$287$	−9.92746	−0.585999
$288$	5.09379	0.300154
$289$	23.1506	1.36180
$290$	−0.232972	−0.0136806
$291$	7.21520	0.422962
$292$	−11.3183	−0.662351
$293$	−20.9577	−1.22436	−0.612180	−	0.790718i	$-0.709707\pi$
$293$	−20.9577	−1.22436	−0.612180	+	0.790718i	$0.709707\pi$
$294$	−0.510081	−0.0297485
$295$	2.00413	0.116685
$296$	−4.13617	−0.240410
$297$	5.39973	0.313324
$298$	−10.8539	−0.628749
$299$	10.6585	0.616396
$300$	−8.64700	−0.499235
$301$	2.19406	0.126464
$302$	6.41560	0.369176
$303$	2.31345	0.132904
$304$	−16.0144	−0.918486
$305$	0.703400	0.0402766
$306$	−3.23210	−0.184767
$307$	−3.74333	−0.213643	−0.106821	−	0.994278i	$-0.534067\pi$
$307$	−3.74333	−0.213643	−0.106821	+	0.994278i	$0.534067\pi$
$308$	9.39454	0.535304
$309$	3.36778	0.191586
$310$	0.605630	0.0343975
$311$	−4.97700	−0.282220	−0.141110	−	0.989994i	$-0.545067\pi$
$311$	−4.97700	−0.282220	−0.141110	+	0.989994i	$0.545067\pi$
$312$	−5.64402	−0.319530
$313$	−13.7030	−0.774542	−0.387271	−	0.921966i	$-0.626582\pi$
$313$	−13.7030	−0.774542	−0.387271	+	0.921966i	$0.626582\pi$
$314$	4.22704	0.238545
$315$	−0.173032	−0.00974926
$316$	−16.4241	−0.923930
$317$	13.0646	0.733780	0.366890	−	0.930264i	$-0.380423\pi$
$317$	13.0646	0.733780	0.366890	+	0.930264i	$0.380423\pi$
$318$	−7.25606	−0.406899
$319$	−14.2531	−0.798022
$320$	0.417865	0.0233594
$321$	−2.26876	−0.126630
$322$	−1.83753	−0.102402
$323$	40.4828	2.25252
$324$	−1.73982	−0.0966565
$325$	14.7048	0.815678
$326$	−0.0937306	−0.00519126
$327$	−4.22761	−0.233787
$328$	18.9377	1.04566
$329$	−3.66311	−0.201954
$330$	−0.476582	−0.0262350
$331$	25.7545	1.41559	0.707797	−	0.706416i	$-0.249689\pi$
$331$	25.7545	1.41559	0.707797	+	0.706416i	$0.249689\pi$
$332$	25.0423	1.37438
$333$	2.16825	0.118819
$334$	8.37642	0.458337
$335$	2.33589	0.127623
$336$	−2.50660	−0.136746
$337$	18.0221	0.981726	0.490863	−	0.871237i	$-0.336682\pi$
$337$	18.0221	0.981726	0.490863	+	0.871237i	$0.336682\pi$
$338$	−2.16589	−0.117809
$339$	−7.23074	−0.392720
$340$	−1.90755	−0.103452
$341$	37.0521	2.00649
$342$	−3.25884	−0.176218
$343$	1.00000	0.0539949
$344$	−4.18541	−0.225662
$345$	−0.623337	−0.0335594
$346$	−5.28445	−0.284094
$347$	19.5204	1.04791	0.523955	−	0.851746i	$-0.324456\pi$
$347$	19.5204	1.04791	0.523955	+	0.851746i	$0.324456\pi$
$348$	4.59243	0.246180
$349$	−13.6242	−0.729287	−0.364643	−	0.931147i	$-0.618809\pi$
$349$	−13.6242	−0.729287	−0.364643	+	0.931147i	$0.618809\pi$
$350$	−2.53513	−0.135509
$351$	2.95869	0.157923
$352$	−27.5051	−1.46602
$353$	19.4715	1.03637	0.518183	−	0.855270i	$-0.326609\pi$
$353$	19.4715	1.03637	0.518183	+	0.855270i	$0.326609\pi$
$354$	5.90798	0.314006
$355$	0.0735212	0.00390210
$356$	−11.7797	−0.624323
$357$	6.33645	0.335360
$358$	−10.9130	−0.576770
$359$	15.2045	0.802463	0.401231	−	0.915977i	$-0.368582\pi$
$359$	15.2045	0.802463	0.401231	+	0.915977i	$0.368582\pi$
$360$	0.330078	0.0173966
$361$	21.8177	1.14830
$362$	6.63981	0.348981
$363$	−18.1571	−0.952998
$364$	5.14757	0.269806
$365$	−1.12565	−0.0589192
$366$	2.07355	0.108386
$367$	−32.1897	−1.68029	−0.840145	−	0.542362i	$-0.817530\pi$
$367$	−32.1897	−1.68029	−0.840145	+	0.542362i	$0.817530\pi$
$368$	−9.02986	−0.470714
$369$	−9.92746	−0.516803
$370$	−0.191370	−0.00994888
$371$	14.2253	0.738541
$372$	−11.9384	−0.618976
$373$	−11.7880	−0.610359	−0.305180	−	0.952295i	$-0.598716\pi$
$373$	−11.7880	−0.610359	−0.305180	+	0.952295i	$0.598716\pi$
$374$	17.4525	0.902446
$375$	−1.72514	−0.0890859
$376$	6.98779	0.360368
$377$	−7.80976	−0.402223
$378$	−0.510081	−0.0262357
$379$	18.6570	0.958346	0.479173	−	0.877721i	$-0.340937\pi$
$379$	18.6570	0.958346	0.479173	+	0.877721i	$0.340937\pi$
$380$	−1.92334	−0.0986651
$381$	5.99074	0.306915
$382$	−7.90829	−0.404624
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	11.4194	0.582744
$385$	0.934327	0.0476177
$386$	−9.67159	−0.492271
$387$	2.19406	0.111530
$388$	12.5531	0.637288
$389$	−27.7012	−1.40451	−0.702254	−	0.711927i	$-0.747823\pi$
$389$	−27.7012	−1.40451	−0.702254	+	0.711927i	$0.747823\pi$
$390$	−0.261135	−0.0132231
$391$	22.8266	1.15439
$392$	−1.90761	−0.0963488
$393$	−11.2387	−0.566915
$394$	−6.34585	−0.319699
$395$	−1.63345	−0.0821878
$396$	9.39454	0.472093
$397$	27.6301	1.38672	0.693358	−	0.720593i	$-0.256131\pi$
$397$	27.6301	1.38672	0.693358	+	0.720593i	$0.256131\pi$
$398$	−0.574264	−0.0287853
$399$	6.38888	0.319844
$400$	−12.4579	−0.622897
$401$	12.4791	0.623176	0.311588	−	0.950217i	$-0.399139\pi$
$401$	12.4791	0.623176	0.311588	+	0.950217i	$0.399139\pi$
$402$	6.88595	0.343440
$403$	20.3021	1.01132
$404$	4.02499	0.200251
$405$	−0.173032	−0.00859804
$406$	1.34641	0.0668213
$407$	−11.7079	−0.580341
$408$	−12.0875	−0.598419
$409$	−12.5137	−0.618765	−0.309383	−	0.950938i	$-0.600122\pi$
$409$	−12.5137	−0.618765	−0.309383	+	0.950938i	$0.600122\pi$
$410$	0.876202	0.0432725
$411$	21.4960	1.06032
$412$	5.85932	0.288668
$413$	−11.5824	−0.569934
$414$	−1.83753	−0.0903099
$415$	2.49057	0.122257
$416$	−15.0709	−0.738913
$417$	4.14516	0.202989
$418$	17.5969	0.860691
$419$	26.5551	1.29730	0.648651	−	0.761086i	$-0.275333\pi$
$419$	26.5551	1.29730	0.648651	+	0.761086i	$0.275333\pi$
$420$	−0.301044	−0.0146895
$421$	−28.1660	−1.37272	−0.686362	−	0.727260i	$-0.740794\pi$
$421$	−28.1660	−1.37272	−0.686362	+	0.727260i	$0.740794\pi$
$422$	0.503329	0.0245017
$423$	−3.66311	−0.178107
$424$	−27.1363	−1.31786
$425$	31.4925	1.52761
$426$	0.216733	0.0105007
$427$	−4.06514	−0.196726
$428$	−3.94722	−0.190796
$429$	−15.9761	−0.771333
$430$	−0.193649	−0.00933858
$431$	26.1656	1.26035	0.630176	−	0.776452i	$-0.282983\pi$
$431$	26.1656	1.26035	0.630176	+	0.776452i	$0.282983\pi$
$432$	−2.50660	−0.120599
$433$	−11.2669	−0.541453	−0.270727	−	0.962656i	$-0.587264\pi$
$433$	−11.2669	−0.541453	−0.270727	+	0.962656i	$0.587264\pi$
$434$	−3.50010	−0.168010
$435$	0.456736	0.0218988
$436$	−7.35527	−0.352254
$437$	23.0155	1.10098
$438$	−3.31830	−0.158554
$439$	10.2624	0.489796	0.244898	−	0.969549i	$-0.421246\pi$
$439$	10.2624	0.489796	0.244898	+	0.969549i	$0.421246\pi$
$440$	−1.78233	−0.0849693
$441$	1.00000	0.0476190
$442$	9.56277	0.454855
$443$	15.2780	0.725881	0.362940	−	0.931812i	$-0.381773\pi$
$443$	15.2780	0.725881	0.362940	+	0.931812i	$0.381773\pi$
$444$	3.77235	0.179028
$445$	−1.17154	−0.0555364
$446$	−11.0099	−0.521334
$447$	21.2788	1.00645
$448$	−2.41495	−0.114096
$449$	−14.3266	−0.676114	−0.338057	−	0.941126i	$-0.609770\pi$
$449$	−14.3266	−0.676114	−0.338057	+	0.941126i	$0.609770\pi$
$450$	−2.53513	−0.119507
$451$	53.6056	2.52419
$452$	−12.5802	−0.591722
$453$	−12.5776	−0.590948
$454$	−1.05174	−0.0493607
$455$	0.511948	0.0240005
$456$	−12.1875	−0.570731
$457$	−15.6363	−0.731433	−0.365717	−	0.930726i	$-0.619176\pi$
$457$	−15.6363	−0.731433	−0.365717	+	0.930726i	$0.619176\pi$
$458$	7.51108	0.350970
$459$	6.33645	0.295760
$460$	−1.08449	−0.0505648
$461$	−15.3342	−0.714186	−0.357093	−	0.934069i	$-0.616232\pi$
$461$	−15.3342	−0.714186	−0.357093	+	0.934069i	$0.616232\pi$
$462$	2.75430	0.128142
$463$	−10.8352	−0.503554	−0.251777	−	0.967785i	$-0.581015\pi$
$463$	−10.8352	−0.503554	−0.251777	+	0.967785i	$0.581015\pi$
$464$	6.61643	0.307160
$465$	−1.18732	−0.0550607
$466$	12.1968	0.565004
$467$	31.5577	1.46031	0.730157	−	0.683279i	$-0.239447\pi$
$467$	31.5577	1.46031	0.730157	+	0.683279i	$0.239447\pi$
$468$	5.14757	0.237947
$469$	−13.4997	−0.623359
$470$	0.323308	0.0149131
$471$	−8.28699	−0.381844
$472$	22.0948	1.01699
$473$	−11.8473	−0.544741
$474$	−4.81524	−0.221171
$475$	31.7531	1.45693
$476$	11.0243	0.505296
$477$	14.2253	0.651332
$478$	−4.52470	−0.206955
$479$	14.1512	0.646584	0.323292	−	0.946299i	$-0.395210\pi$
$479$	14.1512	0.646584	0.323292	+	0.946299i	$0.395210\pi$
$480$	0.881389	0.0402297
$481$	−6.41516	−0.292506
$482$	6.98896	0.318338
$483$	3.60244	0.163916
$484$	−31.5900	−1.43591
$485$	1.24846	0.0566897
$486$	−0.510081	−0.0231377
$487$	13.8197	0.626233	0.313116	−	0.949715i	$-0.398627\pi$
$487$	13.8197	0.626233	0.313116	+	0.949715i	$0.398627\pi$
$488$	7.75470	0.351039
$489$	0.183756	0.00830975
$490$	−0.0882604	−0.00398720
$491$	7.37510	0.332834	0.166417	−	0.986055i	$-0.446780\pi$
$491$	7.37510	0.332834	0.166417	+	0.986055i	$0.446780\pi$
$492$	−17.2720	−0.778680
$493$	−16.7257	−0.753288
$494$	9.64190	0.433810
$495$	0.934327	0.0419949
$496$	−17.1999	−0.772299
$497$	−0.424899	−0.0190593
$498$	7.34193	0.329000
$499$	29.5844	1.32438	0.662190	−	0.749336i	$-0.269627\pi$
$499$	29.5844	1.32438	0.662190	+	0.749336i	$0.269627\pi$
$500$	−3.00143	−0.134228
$501$	−16.4217	−0.733669
$502$	−8.53424	−0.380902
$503$	−31.8584	−1.42050	−0.710249	−	0.703951i	$-0.751418\pi$
$503$	−31.8584	−1.42050	−0.710249	+	0.703951i	$0.751418\pi$
$504$	−1.90761	−0.0849717
$505$	0.400302	0.0178132
$506$	9.92218	0.441095
$507$	4.24618	0.188579
$508$	10.4228	0.462436
$509$	−29.5508	−1.30982	−0.654908	−	0.755709i	$-0.727292\pi$
$509$	−29.5508	−1.30982	−0.654908	+	0.755709i	$0.727292\pi$
$510$	−0.559258	−0.0247643
$511$	6.50543	0.287783
$512$	22.3313	0.986914
$513$	6.38888	0.282076
$514$	−9.74676	−0.429911
$515$	0.582735	0.0256784
$516$	3.81727	0.168046
$517$	19.7798	0.869916
$518$	1.10598	0.0485940
$519$	10.3600	0.454755
$520$	−0.976597	−0.0428266
$521$	11.3643	0.497879	0.248940	−	0.968519i	$-0.419918\pi$
$521$	11.3643	0.497879	0.248940	+	0.968519i	$0.419918\pi$
$522$	1.34641	0.0589308
$523$	−31.6301	−1.38309	−0.691544	−	0.722334i	$-0.743069\pi$
$523$	−31.6301	−1.38309	−0.691544	+	0.722334i	$0.743069\pi$
$524$	−19.5532	−0.854186
$525$	4.97006	0.216911
$526$	−10.0866	−0.439795
$527$	43.4798	1.89401
$528$	13.5350	0.589033
$529$	−10.0225	−0.435759
$530$	−1.25553	−0.0545368
$531$	−11.5824	−0.502635
$532$	11.1155	0.481917
$533$	29.3722	1.27225
$534$	−3.45358	−0.149451
$535$	−0.392568	−0.0169722
$536$	25.7522	1.11232
$537$	21.3947	0.923248
$538$	3.99248	0.172128
$539$	−5.39973	−0.232583
$540$	−0.301044	−0.0129549
$541$	11.1746	0.480434	0.240217	−	0.970719i	$-0.422781\pi$
$541$	11.1746	0.480434	0.240217	+	0.970719i	$0.422781\pi$
$542$	−7.92706	−0.340496
$543$	−13.0172	−0.558620
$544$	−32.2765	−1.38384
$545$	−0.731513	−0.0313346
$546$	1.50917	0.0645865
$547$	−1.57123	−0.0671808	−0.0335904	−	0.999436i	$-0.510694\pi$
$547$	−1.57123	−0.0671808	−0.0335904	+	0.999436i	$0.510694\pi$
$548$	37.3990	1.59761
$549$	−4.06514	−0.173496
$550$	13.6890	0.583702
$551$	−16.8641	−0.718435
$552$	−6.87204	−0.292493
$553$	9.44015	0.401436
$554$	5.46549	0.232206
$555$	0.375177	0.0159254
$556$	7.21182	0.305849
$557$	−3.94463	−0.167139	−0.0835696	−	0.996502i	$-0.526632\pi$
$557$	−3.94463	−0.167139	−0.0835696	+	0.996502i	$0.526632\pi$
$558$	−3.50010	−0.148171
$559$	−6.49154	−0.274563
$560$	−0.433722	−0.0183281
$561$	−34.2151	−1.44456
$562$	8.07614	0.340671
$563$	−19.7014	−0.830314	−0.415157	−	0.909750i	$-0.636273\pi$
$563$	−19.7014	−0.830314	−0.415157	+	0.909750i	$0.636273\pi$
$564$	−6.37315	−0.268358
$565$	−1.25115	−0.0526364
$566$	12.2396	0.514468
$567$	1.00000	0.0419961
$568$	0.810541	0.0340096
$569$	37.9104	1.58929	0.794644	−	0.607076i	$-0.207658\pi$
$569$	37.9104	1.58929	0.794644	+	0.607076i	$0.207658\pi$
$570$	−0.563885	−0.0236185
$571$	−41.1392	−1.72162	−0.860810	−	0.508926i	$-0.830043\pi$
$571$	−41.1392	−1.72162	−0.860810	+	0.508926i	$0.830043\pi$
$572$	−27.7955	−1.16219
$573$	15.5040	0.647689
$574$	−5.06381	−0.211359
$575$	17.9043	0.746662
$576$	−2.41495	−0.100623
$577$	3.54331	0.147510	0.0737549	−	0.997276i	$-0.476502\pi$
$577$	3.54331	0.147510	0.0737549	+	0.997276i	$0.476502\pi$
$578$	11.8087	0.491176
$579$	18.9609	0.787988
$580$	0.794638	0.0329955
$581$	−14.3937	−0.597150
$582$	3.68033	0.152555
$583$	−76.8128	−3.18126
$584$	−12.4098	−0.513522
$585$	0.511948	0.0211664
$586$	−10.6901	−0.441604
$587$	−25.8672	−1.06765	−0.533827	−	0.845593i	$-0.679247\pi$
$587$	−25.8672	−1.06765	−0.533827	+	0.845593i	$0.679247\pi$
$588$	1.73982	0.0717489
$589$	43.8395	1.80638
$590$	1.02227	0.0420862
$591$	12.4409	0.511749
$592$	5.43493	0.223374
$593$	−0.871088	−0.0357713	−0.0178857	−	0.999840i	$-0.505693\pi$
$593$	−0.871088	−0.0357713	−0.0178857	+	0.999840i	$0.505693\pi$
$594$	2.75430	0.113010
$595$	1.09641	0.0449484
$596$	37.0212	1.51645
$597$	1.12583	0.0460771
$598$	5.43669	0.222323
$599$	−1.03900	−0.0424525	−0.0212262	−	0.999775i	$-0.506757\pi$
$599$	−1.03900	−0.0424525	−0.0212262	+	0.999775i	$0.506757\pi$
$600$	−9.48093	−0.387058
$601$	19.7481	0.805540	0.402770	−	0.915301i	$-0.368047\pi$
$601$	19.7481	0.805540	0.402770	+	0.915301i	$0.368047\pi$
$602$	1.11915	0.0456131
$603$	−13.4997	−0.549751
$604$	−21.8827	−0.890396
$605$	−3.14175	−0.127730
$606$	1.18005	0.0479362
$607$	31.0339	1.25963	0.629814	−	0.776746i	$-0.283131\pi$
$607$	31.0339	1.25963	0.629814	+	0.776746i	$0.283131\pi$
$608$	−32.5436	−1.31982
$609$	−2.63960	−0.106962
$610$	0.358791	0.0145270
$611$	10.8380	0.438459
$612$	11.0243	0.445629
$613$	−29.9198	−1.20845	−0.604224	−	0.796815i	$-0.706517\pi$
$613$	−29.9198	−1.20845	−0.604224	+	0.796815i	$0.706517\pi$
$614$	−1.90940	−0.0770571
$615$	−1.71777	−0.0692672
$616$	10.3006	0.415022
$617$	−7.58683	−0.305434	−0.152717	−	0.988270i	$-0.548802\pi$
$617$	−7.58683	−0.305434	−0.152717	+	0.988270i	$0.548802\pi$
$618$	1.71784	0.0691017
$619$	−12.2730	−0.493294	−0.246647	−	0.969105i	$-0.579329\pi$
$619$	−12.2730	−0.493294	−0.246647	+	0.969105i	$0.579329\pi$
$620$	−2.06572	−0.0829614
$621$	3.60244	0.144561
$622$	−2.53867	−0.101791
$623$	6.77066	0.271261
$624$	7.41624	0.296887
$625$	24.5518	0.982072
$626$	−6.98966	−0.279363
$627$	−34.4982	−1.37772
$628$	−14.4179	−0.575335
$629$	−13.7390	−0.547809
$630$	−0.0882604	−0.00351638
$631$	0.926387	0.0368789	0.0184394	−	0.999830i	$-0.494130\pi$
$631$	0.926387	0.0368789	0.0184394	+	0.999830i	$0.494130\pi$
$632$	−18.0081	−0.716324
$633$	−0.986764	−0.0392203
$634$	6.66399	0.264661
$635$	1.03659	0.0411358
$636$	24.7494	0.981379
$637$	−2.95869	−0.117227
$638$	−7.27025	−0.287832
$639$	−0.424899	−0.0168087
$640$	1.97592	0.0781053
$641$	−48.7515	−1.92557	−0.962784	−	0.270272i	$-0.912886\pi$
$641$	−48.7515	−1.92557	−0.962784	+	0.270272i	$0.912886\pi$
$642$	−1.15725	−0.0456730
$643$	−20.0141	−0.789278	−0.394639	−	0.918836i	$-0.629130\pi$
$643$	−20.0141	−0.789278	−0.394639	+	0.918836i	$0.629130\pi$
$644$	6.26758	0.246977
$645$	0.379643	0.0149484
$646$	20.6495	0.812444
$647$	−28.2542	−1.11079	−0.555394	−	0.831587i	$-0.687433\pi$
$647$	−28.2542	−1.11079	−0.555394	+	0.831587i	$0.687433\pi$
$648$	−1.90761	−0.0749380
$649$	62.5420	2.45499
$650$	7.50066	0.294200
$651$	6.86185	0.268937
$652$	0.319702	0.0125205
$653$	16.1454	0.631817	0.315909	−	0.948790i	$-0.397691\pi$
$653$	16.1454	0.631817	0.315909	+	0.948790i	$0.397691\pi$
$654$	−2.15642	−0.0843229
$655$	−1.94465	−0.0759838
$656$	−24.8842	−0.971563
$657$	6.50543	0.253801
$658$	−1.86849	−0.0728411
$659$	−19.1277	−0.745108	−0.372554	−	0.928010i	$-0.621518\pi$
$659$	−19.1277	−0.745108	−0.372554	+	0.928010i	$0.621518\pi$
$660$	1.62556	0.0632748
$661$	8.65920	0.336804	0.168402	−	0.985718i	$-0.446139\pi$
$661$	8.65920	0.336804	0.168402	+	0.985718i	$0.446139\pi$
$662$	13.1369	0.510579
$663$	−18.7476	−0.728095
$664$	27.4575	1.06556
$665$	1.10548	0.0428687
$666$	1.10598	0.0428559
$667$	−9.50900	−0.368190
$668$	−28.5708	−1.10544
$669$	21.5846	0.834509
$670$	1.19149	0.0460313
$671$	21.9506	0.847395
$672$	−5.09379	−0.196497
$673$	−24.0273	−0.926186	−0.463093	−	0.886310i	$-0.653260\pi$
$673$	−24.0273	−0.926186	−0.463093	+	0.886310i	$0.653260\pi$
$674$	9.19273	0.354091
$675$	4.97006	0.191298
$676$	7.38757	0.284137
$677$	−22.9001	−0.880121	−0.440060	−	0.897968i	$-0.645043\pi$
$677$	−22.9001	−0.880121	−0.440060	+	0.897968i	$0.645043\pi$
$678$	−3.68827	−0.141647
$679$	−7.21520	−0.276894
$680$	−2.09152	−0.0802062
$681$	2.06191	0.0790127
$682$	18.8996	0.723703
$683$	−4.76274	−0.182241	−0.0911205	−	0.995840i	$-0.529045\pi$
$683$	−4.76274	−0.182241	−0.0911205	+	0.995840i	$0.529045\pi$
$684$	11.1155	0.425011
$685$	3.71949	0.142115
$686$	0.510081	0.0194750
$687$	−14.7253	−0.561804
$688$	5.49963	0.209672
$689$	−42.0882	−1.60343
$690$	−0.317953	−0.0121042
$691$	−20.4627	−0.778438	−0.389219	−	0.921145i	$-0.627255\pi$
$691$	−20.4627	−0.778438	−0.389219	+	0.921145i	$0.627255\pi$
$692$	18.0246	0.685191
$693$	−5.39973	−0.205119
$694$	9.95699	0.377962
$695$	0.717246	0.0272067
$696$	5.03533	0.190864
$697$	62.9048	2.38269
$698$	−6.94945	−0.263040
$699$	−23.9114	−0.904413
$700$	8.64700	0.326826
$701$	17.6398	0.666245	0.333123	−	0.942884i	$-0.391898\pi$
$701$	17.6398	0.666245	0.333123	+	0.942884i	$0.391898\pi$
$702$	1.50917	0.0569599
$703$	−13.8527	−0.522463
$704$	13.0401	0.491467
$705$	−0.633837	−0.0238717
$706$	9.93206	0.373798
$707$	−2.31345	−0.0870064
$708$	−20.1513	−0.757333
$709$	−26.9970	−1.01389	−0.506947	−	0.861977i	$-0.669226\pi$
$709$	−26.9970	−1.01389	−0.506947	+	0.861977i	$0.669226\pi$
$710$	0.0375018	0.00140742
$711$	9.44015	0.354033
$712$	−12.9158	−0.484039
$713$	24.7194	0.925748
$714$	3.23210	0.120958
$715$	−2.76438	−0.103382
$716$	37.2228	1.39108
$717$	8.87055	0.331277
$718$	7.75552	0.289434
$719$	23.6074	0.880408	0.440204	−	0.897898i	$-0.354906\pi$
$719$	23.6074	0.880408	0.440204	+	0.897898i	$0.354906\pi$
$720$	−0.433722	−0.0161639
$721$	−3.36778	−0.125423
$722$	11.1288	0.414172
$723$	−13.7017	−0.509570
$724$	−22.6475	−0.841688
$725$	−13.1190	−0.487227
$726$	−9.26157	−0.343729
$727$	32.3927	1.20138	0.600689	−	0.799483i	$-0.294893\pi$
$727$	32.3927	1.20138	0.600689	+	0.799483i	$0.294893\pi$
$728$	5.64402	0.209181
$729$	1.00000	0.0370370
$730$	−0.574172	−0.0212511
$731$	−13.9026	−0.514205
$732$	−7.07260	−0.261411
$733$	30.8495	1.13945	0.569726	−	0.821835i	$-0.307049\pi$
$733$	30.8495	1.13945	0.569726	+	0.821835i	$0.307049\pi$
$734$	−16.4194	−0.606050
$735$	0.173032	0.00638239
$736$	−18.3500	−0.676392
$737$	72.8948	2.68511
$738$	−5.06381	−0.186401
$739$	16.3768	0.602429	0.301214	−	0.953556i	$-0.402608\pi$
$739$	16.3768	0.602429	0.301214	+	0.953556i	$0.402608\pi$
$740$	0.652739	0.0239952
$741$	−18.9027	−0.694407
$742$	7.25606	0.266378
$743$	−18.8766	−0.692515	−0.346257	−	0.938140i	$-0.612548\pi$
$743$	−18.8766	−0.692515	−0.346257	+	0.938140i	$0.612548\pi$
$744$	−13.0897	−0.479893
$745$	3.68191	0.134895
$746$	−6.01283	−0.220145
$747$	−14.3937	−0.526637
$748$	−59.5280	−2.17656
$749$	2.26876	0.0828985
$750$	−0.879962	−0.0321317
$751$	20.3697	0.743299	0.371650	−	0.928373i	$-0.378792\pi$
$751$	20.3697	0.743299	0.371650	+	0.928373i	$0.378792\pi$
$752$	−9.18196	−0.334832
$753$	16.7312	0.609717
$754$	−3.98361	−0.145074
$755$	−2.17633	−0.0792048
$756$	1.73982	0.0632765
$757$	18.1057	0.658062	0.329031	−	0.944319i	$-0.393278\pi$
$757$	18.1057	0.658062	0.329031	+	0.944319i	$0.393278\pi$
$758$	9.51658	0.345658
$759$	−19.4522	−0.706069
$760$	−2.10883	−0.0764952
$761$	−28.8009	−1.04403	−0.522015	−	0.852936i	$-0.674820\pi$
$761$	−28.8009	−1.04403	−0.522015	+	0.852936i	$0.674820\pi$
$762$	3.05576	0.110699
$763$	4.22761	0.153050
$764$	26.9741	0.975890
$765$	1.09641	0.0396408
$766$	−0.510081	−0.0184300
$767$	34.2688	1.23737
$768$	0.994909	0.0359007
$769$	33.7219	1.21604	0.608021	−	0.793921i	$-0.291964\pi$
$769$	33.7219	1.21604	0.608021	+	0.793921i	$0.291964\pi$
$770$	0.476582	0.0171748
$771$	19.1083	0.688167
$772$	32.9885	1.18728
$773$	12.4923	0.449317	0.224659	−	0.974438i	$-0.427873\pi$
$773$	12.4923	0.449317	0.224659	+	0.974438i	$0.427873\pi$
$774$	1.11915	0.0402270
$775$	34.1038	1.22505
$776$	13.7638	0.494091
$777$	−2.16825	−0.0777854
$778$	−14.1299	−0.506580
$779$	63.4253	2.27245
$780$	0.890696	0.0318920
$781$	2.29434	0.0820979
$782$	11.6434	0.416368
$783$	−2.63960	−0.0943317
$784$	2.50660	0.0895214
$785$	−1.43392	−0.0511787
$786$	−5.73263	−0.204476
$787$	17.5397	0.625223	0.312612	−	0.949881i	$-0.398796\pi$
$787$	17.5397	0.625223	0.312612	+	0.949881i	$0.398796\pi$
$788$	21.6448	0.771066
$789$	19.7744	0.703989
$790$	−0.833191	−0.0296436
$791$	7.23074	0.257096
$792$	10.3006	0.366015
$793$	12.0275	0.427108
$794$	14.0936	0.500163
$795$	2.46144	0.0872981
$796$	1.95874	0.0694256
$797$	25.4142	0.900216	0.450108	−	0.892974i	$-0.351385\pi$
$797$	25.4142	0.900216	0.450108	+	0.892974i	$0.351385\pi$
$798$	3.25884	0.115362
$799$	23.2111	0.821151
$800$	−25.3164	−0.895071
$801$	6.77066	0.239229
$802$	6.36535	0.224768
$803$	−35.1276	−1.23962
$804$	−23.4870	−0.828324
$805$	0.623337	0.0219698
$806$	10.3557	0.364764
$807$	−7.82715	−0.275529
$808$	4.41317	0.155255
$809$	12.7856	0.449516	0.224758	−	0.974415i	$-0.427841\pi$
$809$	12.7856	0.449516	0.224758	+	0.974415i	$0.427841\pi$
$810$	−0.0882604	−0.00310116
$811$	−53.1258	−1.86550	−0.932749	−	0.360527i	$-0.882597\pi$
$811$	−53.1258	−1.86550	−0.932749	+	0.360527i	$0.882597\pi$
$812$	−4.59243	−0.161163
$813$	15.5408	0.545039
$814$	−5.97200	−0.209318
$815$	0.0317958	0.00111376
$816$	15.8829	0.556014
$817$	−14.0176	−0.490413
$818$	−6.38303	−0.223177
$819$	−2.95869	−0.103385
$820$	−2.98861	−0.104367
$821$	−25.5319	−0.891069	−0.445534	−	0.895265i	$-0.646986\pi$
$821$	−25.5319	−0.891069	−0.445534	+	0.895265i	$0.646986\pi$
$822$	10.9647	0.382437
$823$	−19.0959	−0.665641	−0.332821	−	0.942990i	$-0.608000\pi$
$823$	−19.0959	−0.665641	−0.332821	+	0.942990i	$0.608000\pi$
$824$	6.42441	0.223805
$825$	−26.8370	−0.934343
$826$	−5.90798	−0.205565
$827$	−28.9862	−1.00795	−0.503975	−	0.863718i	$-0.668130\pi$
$827$	−28.9862	−1.00795	−0.503975	+	0.863718i	$0.668130\pi$
$828$	6.26758	0.217813
$829$	−55.1434	−1.91521	−0.957604	−	0.288087i	$-0.906981\pi$
$829$	−55.1434	−1.91521	−0.957604	+	0.288087i	$0.906981\pi$
$830$	1.27039	0.0440959
$831$	−10.7149	−0.371697
$832$	7.14509	0.247712
$833$	−6.33645	−0.219545
$834$	2.11437	0.0732145
$835$	−2.84149	−0.0983338
$836$	−60.0206	−2.07585
$837$	6.86185	0.237180
$838$	13.5453	0.467913
$839$	−33.9225	−1.17114	−0.585568	−	0.810623i	$-0.699128\pi$
$839$	−33.9225	−1.17114	−0.585568	+	0.810623i	$0.699128\pi$
$840$	−0.330078	−0.0113888
$841$	−22.0325	−0.759741
$842$	−14.3669	−0.495117
$843$	−15.8331	−0.545319
$844$	−1.71679	−0.0590943
$845$	0.734725	0.0252753
$846$	−1.86849	−0.0642398
$847$	18.1571	0.623884
$848$	35.6571	1.22447
$849$	−23.9954	−0.823519
$850$	16.0637	0.550982
$851$	−7.81097	−0.267757
$852$	−0.739247	−0.0253262
$853$	27.2647	0.933527	0.466763	−	0.884382i	$-0.345420\pi$
$853$	27.2647	0.933527	0.466763	+	0.884382i	$0.345420\pi$
$854$	−2.07355	−0.0709555
$855$	1.10548	0.0378067
$856$	−4.32790	−0.147925
$857$	−44.3209	−1.51397	−0.756987	−	0.653430i	$-0.773329\pi$
$857$	−44.3209	−1.51397	−0.756987	+	0.653430i	$0.773329\pi$
$858$	−8.14910	−0.278206
$859$	35.7180	1.21868	0.609341	−	0.792909i	$-0.291434\pi$
$859$	35.7180	1.21868	0.609341	+	0.792909i	$0.291434\pi$
$860$	0.660510	0.0225232
$861$	9.92746	0.338327
$862$	13.3466	0.454586
$863$	−27.2165	−0.926462	−0.463231	−	0.886238i	$-0.653310\pi$
$863$	−27.2165	−0.926462	−0.463231	+	0.886238i	$0.653310\pi$
$864$	−5.09379	−0.173294
$865$	1.79262	0.0609509
$866$	−5.74704	−0.195292
$867$	−23.1506	−0.786235
$868$	11.9384	0.405215
$869$	−50.9742	−1.72918
$870$	0.232972	0.00789851
$871$	39.9414	1.35336
$872$	−8.06463	−0.273103
$873$	−7.21520	−0.244197
$874$	11.7398	0.397104
$875$	1.72514	0.0583204
$876$	11.3183	0.382409
$877$	36.4726	1.23159	0.615797	−	0.787905i	$-0.288834\pi$
$877$	36.4726	1.23159	0.615797	+	0.787905i	$0.288834\pi$
$878$	5.23464	0.176660
$879$	20.9577	0.706885
$880$	2.34198	0.0789482
$881$	20.9291	0.705119	0.352559	−	0.935789i	$-0.385311\pi$
$881$	20.9291	0.705119	0.352559	+	0.935789i	$0.385311\pi$
$882$	0.510081	0.0171753
$883$	−30.0053	−1.00976	−0.504879	−	0.863190i	$-0.668463\pi$
$883$	−30.0053	−1.00976	−0.504879	+	0.863190i	$0.668463\pi$
$884$	−32.6173	−1.09704
$885$	−2.00413	−0.0673682
$886$	7.79303	0.261812
$887$	−41.1129	−1.38044	−0.690218	−	0.723601i	$-0.742485\pi$
$887$	−41.1129	−1.38044	−0.690218	+	0.723601i	$0.742485\pi$
$888$	4.13617	0.138801
$889$	−5.99074	−0.200923
$890$	−0.597581	−0.0200310
$891$	−5.39973	−0.180898
$892$	37.5533	1.25738
$893$	23.4032	0.783158
$894$	10.8539	0.363008
$895$	3.70197	0.123743
$896$	−11.4194	−0.381495
$897$	−10.6585	−0.355876
$898$	−7.30773	−0.243862
$899$	−18.1126	−0.604088
$900$	8.64700	0.288233
$901$	−90.1379	−3.00293
$902$	27.3432	0.910428
$903$	−2.19406	−0.0730138
$904$	−13.7934	−0.458763
$905$	−2.25239	−0.0748720
$906$	−6.41560	−0.213144
$907$	32.2169	1.06975	0.534873	−	0.844932i	$-0.320359\pi$
$907$	32.2169	1.06975	0.534873	+	0.844932i	$0.320359\pi$
$908$	3.58735	0.119050
$909$	−2.31345	−0.0767324
$910$	0.261135	0.00865654
$911$	−32.4221	−1.07419	−0.537096	−	0.843521i	$-0.680479\pi$
$911$	−32.4221	−1.07419	−0.537096	+	0.843521i	$0.680479\pi$
$912$	16.0144	0.530288
$913$	77.7218	2.57222
$914$	−7.97575	−0.263815
$915$	−0.703400	−0.0232537
$916$	−25.6193	−0.846485
$917$	11.2387	0.371133
$918$	3.23210	0.106675
$919$	−8.39654	−0.276976	−0.138488	−	0.990364i	$-0.544224\pi$
$919$	−8.39654	−0.276976	−0.138488	+	0.990364i	$0.544224\pi$
$920$	−1.18908	−0.0392030
$921$	3.74333	0.123347
$922$	−7.82170	−0.257594
$923$	1.25714	0.0413794
$924$	−9.39454	−0.309058
$925$	−10.7763	−0.354323
$926$	−5.52682	−0.181623
$927$	−3.36778	−0.110612
$928$	13.4456	0.441373
$929$	9.49681	0.311580	0.155790	−	0.987790i	$-0.450208\pi$
$929$	9.49681	0.311580	0.155790	+	0.987790i	$0.450208\pi$
$930$	−0.605630	−0.0198594
$931$	−6.38888	−0.209387
$932$	−41.6015	−1.36270
$933$	4.97700	0.162940
$934$	16.0970	0.526709
$935$	−5.92031	−0.193615
$936$	5.64402	0.184481
$937$	13.7888	0.450461	0.225230	−	0.974306i	$-0.427687\pi$
$937$	13.7888	0.450461	0.225230	+	0.974306i	$0.427687\pi$
$938$	−6.88595	−0.224834
$939$	13.7030	0.447182
$940$	−1.10276	−0.0359681
$941$	11.7648	0.383520	0.191760	−	0.981442i	$-0.438580\pi$
$941$	11.7648	0.383520	0.191760	+	0.981442i	$0.438580\pi$
$942$	−4.22704	−0.137724
$943$	35.7630	1.16460
$944$	−29.0325	−0.944928
$945$	0.173032	0.00562874
$946$	−6.04310	−0.196478
$947$	−44.9020	−1.45912	−0.729560	−	0.683917i	$-0.760275\pi$
$947$	−44.9020	−1.45912	−0.729560	+	0.683917i	$0.760275\pi$
$948$	16.4241	0.533431
$949$	−19.2475	−0.624801
$950$	16.1967	0.525489
$951$	−13.0646	−0.423648
$952$	12.0875	0.391757
$953$	−13.6746	−0.442964	−0.221482	−	0.975164i	$-0.571089\pi$
$953$	−13.6746	−0.442964	−0.221482	+	0.975164i	$0.571089\pi$
$954$	7.25606	0.234923
$955$	2.68269	0.0868099
$956$	15.4331	0.499144
$957$	14.2531	0.460738
$958$	7.21825	0.233211
$959$	−21.4960	−0.694141
$960$	−0.417865	−0.0134865
$961$	16.0850	0.518872
$962$	−3.27225	−0.105502
$963$	2.26876	0.0731096
$964$	−23.8384	−0.767783
$965$	3.28084	0.105614
$966$	1.83753	0.0591217
$967$	40.8409	1.31335	0.656677	−	0.754172i	$-0.271961\pi$
$967$	40.8409	1.31335	0.656677	+	0.754172i	$0.271961\pi$
$968$	−34.6366	−1.11326
$969$	−40.4828	−1.30049
$970$	0.636816	0.0204469
$971$	3.35157	0.107557	0.0537784	−	0.998553i	$-0.482874\pi$
$971$	3.35157	0.107557	0.0537784	+	0.998553i	$0.482874\pi$
$972$	1.73982	0.0558047
$973$	−4.14516	−0.132888
$974$	7.04919	0.225871
$975$	−14.7048	−0.470932
$976$	−10.1897	−0.326164
$977$	−52.6350	−1.68394	−0.841972	−	0.539521i	$-0.818605\pi$
$977$	−52.6350	−1.68394	−0.841972	+	0.539521i	$0.818605\pi$
$978$	0.0937306	0.00299717
$979$	−36.5597	−1.16845
$980$	0.301044	0.00961651
$981$	4.22761	0.134977
$982$	3.76190	0.120047
$983$	−44.3969	−1.41604	−0.708020	−	0.706192i	$-0.750411\pi$
$983$	−44.3969	−1.41604	−0.708020	+	0.706192i	$0.750411\pi$
$984$	−18.9377	−0.603712
$985$	2.15267	0.0685898
$986$	−8.53146	−0.271697
$987$	3.66311	0.116598
$988$	−32.8872	−1.04628
$989$	−7.90397	−0.251331
$990$	0.476582	0.0151468
$991$	−17.9140	−0.569059	−0.284529	−	0.958667i	$-0.591837\pi$
$991$	−17.9140	−0.569059	−0.284529	+	0.958667i	$0.591837\pi$
$992$	−34.9528	−1.10975
$993$	−25.7545	−0.817293
$994$	−0.216733	−0.00687435
$995$	0.194805	0.00617573
$996$	−25.0423	−0.793497
$997$	42.2817	1.33908	0.669538	−	0.742778i	$-0.266492\pi$
$997$	42.2817	1.33908	0.669538	+	0.742778i	$0.266492\pi$
$998$	15.0904	0.477679
$999$	−2.16825	−0.0686003

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.t.1.31	✓	52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.t.1.31	✓	52	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q+0.510081 q^{2} -1.00000 q^{3} -1.73982 q^{4} -0.173032 q^{5} -0.510081 q^{6} +1.00000 q^{7} -1.90761 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.510081 q^{2} -1.00000 q^{3} -1.73982 q^{4} -0.173032 q^{5} -0.510081 q^{6} +1.00000 q^{7} -1.90761 q^{8} +1.00000 q^{9} -0.0882604 q^{10} -5.39973 q^{11} +1.73982 q^{12} -2.95869 q^{13} +0.510081 q^{14} +0.173032 q^{15} +2.50660 q^{16} -6.33645 q^{17} +0.510081 q^{18} -6.38888 q^{19} +0.301044 q^{20} -1.00000 q^{21} -2.75430 q^{22} -3.60244 q^{23} +1.90761 q^{24} -4.97006 q^{25} -1.50917 q^{26} -1.00000 q^{27} -1.73982 q^{28} +2.63960 q^{29} +0.0882604 q^{30} -6.86185 q^{31} +5.09379 q^{32} +5.39973 q^{33} -3.23210 q^{34} -0.173032 q^{35} -1.73982 q^{36} +2.16825 q^{37} -3.25884 q^{38} +2.95869 q^{39} +0.330078 q^{40} -9.92746 q^{41} -0.510081 q^{42} +2.19406 q^{43} +9.39454 q^{44} -0.173032 q^{45} -1.83753 q^{46} -3.66311 q^{47} -2.50660 q^{48} +1.00000 q^{49} -2.53513 q^{50} +6.33645 q^{51} +5.14757 q^{52} +14.2253 q^{53} -0.510081 q^{54} +0.934327 q^{55} -1.90761 q^{56} +6.38888 q^{57} +1.34641 q^{58} -11.5824 q^{59} -0.301044 q^{60} -4.06514 q^{61} -3.50010 q^{62} +1.00000 q^{63} -2.41495 q^{64} +0.511948 q^{65} +2.75430 q^{66} -13.4997 q^{67} +11.0243 q^{68} +3.60244 q^{69} -0.0882604 q^{70} -0.424899 q^{71} -1.90761 q^{72} +6.50543 q^{73} +1.10598 q^{74} +4.97006 q^{75} +11.1155 q^{76} -5.39973 q^{77} +1.50917 q^{78} +9.44015 q^{79} -0.433722 q^{80} +1.00000 q^{81} -5.06381 q^{82} -14.3937 q^{83} +1.73982 q^{84} +1.09641 q^{85} +1.11915 q^{86} -2.63960 q^{87} +10.3006 q^{88} +6.77066 q^{89} -0.0882604 q^{90} -2.95869 q^{91} +6.26758 q^{92} +6.86185 q^{93} -1.86849 q^{94} +1.10548 q^{95} -5.09379 q^{96} -7.21520 q^{97} +0.510081 q^{98} -5.39973 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9 - 2 * q^10 + 9 * q^11 - 61 * q^12 + 44 * q^13 + 3 * q^14 + 7 * q^15 + 95 * q^16 - 6 * q^17 + 3 * q^18 + 7 * q^19 - 21 * q^20 - 52 * q^21 + 19 * q^22 - 4 * q^23 - 24 * q^24 + 83 * q^25 - 5 * q^26 - 52 * q^27 + 61 * q^28 + 31 * q^29 + 2 * q^30 + 11 * q^31 + 71 * q^32 - 9 * q^33 + 17 * q^34 - 7 * q^35 + 61 * q^36 + 71 * q^37 - 8 * q^38 - 44 * q^39 + 20 * q^40 - 25 * q^41 - 3 * q^42 + 75 * q^43 + 14 * q^44 - 7 * q^45 + 36 * q^46 - 20 * q^47 - 95 * q^48 + 52 * q^49 + 26 * q^50 + 6 * q^51 + 88 * q^52 + 70 * q^53 - 3 * q^54 + 12 * q^55 + 24 * q^56 - 7 * q^57 + 48 * q^58 - 27 * q^59 + 21 * q^60 + 59 * q^61 - 23 * q^62 + 52 * q^63 + 138 * q^64 + 44 * q^65 - 19 * q^66 + 65 * q^67 - 8 * q^68 + 4 * q^69 - 2 * q^70 - 11 * q^71 + 24 * q^72 + 34 * q^73 + 38 * q^74 - 83 * q^75 + 31 * q^76 + 9 * q^77 + 5 * q^78 + 74 * q^79 - 5 * q^80 + 52 * q^81 + 51 * q^82 - 30 * q^83 - 61 * q^84 + 70 * q^85 + 29 * q^86 - 31 * q^87 + 90 * q^88 - q^89 - 2 * q^90 + 44 * q^91 + 34 * q^92 - 11 * q^93 + 27 * q^94 + 9 * q^95 - 71 * q^96 + 73 * q^97 + 3 * q^98 + 9 * q^99

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