LMFDB - Embedded newform 8049.2.a.c.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8049,2,Mod(1,8049)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8049, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8049.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8049 = 3 \cdot 2683 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8049.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:	$64.2715885869$
Analytic rank:	$0$
Dimension:	$119$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8049.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.21831 q^{2} -1.00000 q^{3} +2.92089 q^{4} +1.70107 q^{5} +2.21831 q^{6} +0.0884207 q^{7} -2.04282 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.21831 q^{2} -1.00000 q^{3} +2.92089 q^{4} +1.70107 q^{5} +2.21831 q^{6} +0.0884207 q^{7} -2.04282 q^{8} +1.00000 q^{9} -3.77350 q^{10} -2.17452 q^{11} -2.92089 q^{12} -2.31214 q^{13} -0.196144 q^{14} -1.70107 q^{15} -1.31017 q^{16} +3.21902 q^{17} -2.21831 q^{18} +3.72011 q^{19} +4.96864 q^{20} -0.0884207 q^{21} +4.82375 q^{22} -7.72189 q^{23} +2.04282 q^{24} -2.10636 q^{25} +5.12905 q^{26} -1.00000 q^{27} +0.258267 q^{28} +5.82190 q^{29} +3.77350 q^{30} +4.47424 q^{31} +6.99201 q^{32} +2.17452 q^{33} -7.14078 q^{34} +0.150410 q^{35} +2.92089 q^{36} +0.540828 q^{37} -8.25236 q^{38} +2.31214 q^{39} -3.47499 q^{40} +6.26118 q^{41} +0.196144 q^{42} -1.74345 q^{43} -6.35153 q^{44} +1.70107 q^{45} +17.1295 q^{46} -8.20011 q^{47} +1.31017 q^{48} -6.99218 q^{49} +4.67255 q^{50} -3.21902 q^{51} -6.75352 q^{52} -8.44465 q^{53} +2.21831 q^{54} -3.69901 q^{55} -0.180628 q^{56} -3.72011 q^{57} -12.9148 q^{58} -0.0871124 q^{59} -4.96864 q^{60} +1.01579 q^{61} -9.92524 q^{62} +0.0884207 q^{63} -12.8901 q^{64} -3.93312 q^{65} -4.82375 q^{66} +5.47031 q^{67} +9.40241 q^{68} +7.72189 q^{69} -0.333655 q^{70} +0.943411 q^{71} -2.04282 q^{72} -16.0248 q^{73} -1.19972 q^{74} +2.10636 q^{75} +10.8660 q^{76} -0.192272 q^{77} -5.12905 q^{78} +11.9201 q^{79} -2.22870 q^{80} +1.00000 q^{81} -13.8892 q^{82} +5.49372 q^{83} -0.258267 q^{84} +5.47578 q^{85} +3.86751 q^{86} -5.82190 q^{87} +4.44215 q^{88} +6.14531 q^{89} -3.77350 q^{90} -0.204441 q^{91} -22.5548 q^{92} -4.47424 q^{93} +18.1904 q^{94} +6.32817 q^{95} -6.99201 q^{96} -1.84186 q^{97} +15.5108 q^{98} -2.17452 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 119 q + 11 q^{2} - 119 q^{3} + 137 q^{4} + 17 q^{5} - 11 q^{6} + 10 q^{7} + 33 q^{8} + 119 q^{9}+O(q^{10}) $ 119 * q + 11 * q^2 - 119 * q^3 + 137 * q^4 + 17 * q^5 - 11 * q^6 + 10 * q^7 + 33 * q^8 + 119 * q^9	\( 119 q + 11 q^{2} - 119 q^{3} + 137 q^{4} + 17 q^{5} - 11 q^{6} + 10 q^{7} + 33 q^{8} + 119 q^{9} - 10 q^{10} + 56 q^{11} - 137 q^{12} - 37 q^{13} + 31 q^{14} - 17 q^{15} + 173 q^{16} + 17 q^{17} + 11 q^{18} + 16 q^{19} + 61 q^{20} - 10 q^{21} - 3 q^{22} + 76 q^{23} - 33 q^{24} + 134 q^{25} + 47 q^{26} - 119 q^{27} - q^{28} + 47 q^{29} + 10 q^{30} + 51 q^{31} + 87 q^{32} - 56 q^{33} + 13 q^{34} + 58 q^{35} + 137 q^{36} - 67 q^{37} + 35 q^{38} + 37 q^{39} - 40 q^{40} + 47 q^{41} - 31 q^{42} + 12 q^{43} + 148 q^{44} + 17 q^{45} + 26 q^{46} + 107 q^{47} - 173 q^{48} + 163 q^{49} + 76 q^{50} - 17 q^{51} - 57 q^{52} + 64 q^{53} - 11 q^{54} + 71 q^{55} + 91 q^{56} - 16 q^{57} + 12 q^{58} + 98 q^{59} - 61 q^{60} - 50 q^{61} + 40 q^{62} + 10 q^{63} + 245 q^{64} + 40 q^{65} + 3 q^{66} + 12 q^{67} + 75 q^{68} - 76 q^{69} - 9 q^{70} + 194 q^{71} + 33 q^{72} - 79 q^{73} + 72 q^{74} - 134 q^{75} + 12 q^{76} + 71 q^{77} - 47 q^{78} + 127 q^{79} + 148 q^{80} + 119 q^{81} - 54 q^{82} + 77 q^{83} + q^{84} - 25 q^{85} + 142 q^{86} - 47 q^{87} + q^{88} + 93 q^{89} - 10 q^{90} + 61 q^{91} + 156 q^{92} - 51 q^{93} + 16 q^{94} + 138 q^{95} - 87 q^{96} - 110 q^{97} + 96 q^{98} + 56 q^{99}+O(q^{100}) \) 119 * q + 11 * q^2 - 119 * q^3 + 137 * q^4 + 17 * q^5 - 11 * q^6 + 10 * q^7 + 33 * q^8 + 119 * q^9 - 10 * q^10 + 56 * q^11 - 137 * q^12 - 37 * q^13 + 31 * q^14 - 17 * q^15 + 173 * q^16 + 17 * q^17 + 11 * q^18 + 16 * q^19 + 61 * q^20 - 10 * q^21 - 3 * q^22 + 76 * q^23 - 33 * q^24 + 134 * q^25 + 47 * q^26 - 119 * q^27 - q^28 + 47 * q^29 + 10 * q^30 + 51 * q^31 + 87 * q^32 - 56 * q^33 + 13 * q^34 + 58 * q^35 + 137 * q^36 - 67 * q^37 + 35 * q^38 + 37 * q^39 - 40 * q^40 + 47 * q^41 - 31 * q^42 + 12 * q^43 + 148 * q^44 + 17 * q^45 + 26 * q^46 + 107 * q^47 - 173 * q^48 + 163 * q^49 + 76 * q^50 - 17 * q^51 - 57 * q^52 + 64 * q^53 - 11 * q^54 + 71 * q^55 + 91 * q^56 - 16 * q^57 + 12 * q^58 + 98 * q^59 - 61 * q^60 - 50 * q^61 + 40 * q^62 + 10 * q^63 + 245 * q^64 + 40 * q^65 + 3 * q^66 + 12 * q^67 + 75 * q^68 - 76 * q^69 - 9 * q^70 + 194 * q^71 + 33 * q^72 - 79 * q^73 + 72 * q^74 - 134 * q^75 + 12 * q^76 + 71 * q^77 - 47 * q^78 + 127 * q^79 + 148 * q^80 + 119 * q^81 - 54 * q^82 + 77 * q^83 + q^84 - 25 * q^85 + 142 * q^86 - 47 * q^87 + q^88 + 93 * q^89 - 10 * q^90 + 61 * q^91 + 156 * q^92 - 51 * q^93 + 16 * q^94 + 138 * q^95 - 87 * q^96 - 110 * q^97 + 96 * q^98 + 56 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.21831	−1.56858	−0.784290	−	0.620394i	$-0.786973\pi$
$2$	−2.21831	−1.56858	−0.784290	+	0.620394i	$0.786973\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.92089	1.46045
$5$	1.70107	0.760742	0.380371	−	0.924834i	$-0.375796\pi$
$5$	1.70107	0.760742	0.380371	+	0.924834i	$0.375796\pi$
$6$	2.21831	0.905621
$7$	0.0884207	0.0334199	0.0167099	−	0.999860i	$-0.494681\pi$
$7$	0.0884207	0.0334199	0.0167099	+	0.999860i	$0.494681\pi$
$8$	−2.04282	−0.722247
$9$	1.00000	0.333333
$10$	−3.77350	−1.19329
$11$	−2.17452	−0.655642	−0.327821	−	0.944740i	$-0.606314\pi$
$11$	−2.17452	−0.655642	−0.327821	+	0.944740i	$0.606314\pi$
$12$	−2.92089	−0.843189
$13$	−2.31214	−0.641273	−0.320637	−	0.947202i	$-0.603897\pi$
$13$	−2.31214	−0.641273	−0.320637	+	0.947202i	$0.603897\pi$
$14$	−0.196144	−0.0524218
$15$	−1.70107	−0.439215
$16$	−1.31017	−0.327543
$17$	3.21902	0.780727	0.390364	−	0.920661i	$-0.372349\pi$
$17$	3.21902	0.780727	0.390364	+	0.920661i	$0.372349\pi$
$18$	−2.21831	−0.522860
$19$	3.72011	0.853452	0.426726	−	0.904381i	$-0.359667\pi$
$19$	3.72011	0.853452	0.426726	+	0.904381i	$0.359667\pi$
$20$	4.96864	1.11102
$21$	−0.0884207	−0.0192950
$22$	4.82375	1.02843
$23$	−7.72189	−1.61012	−0.805062	−	0.593190i	$-0.797868\pi$
$23$	−7.72189	−1.61012	−0.805062	+	0.593190i	$0.797868\pi$
$24$	2.04282	0.416989
$25$	−2.10636	−0.421272
$26$	5.12905	1.00589
$27$	−1.00000	−0.192450
$28$	0.258267	0.0488079
$29$	5.82190	1.08110	0.540550	−	0.841312i	$-0.318216\pi$
$29$	5.82190	1.08110	0.540550	+	0.841312i	$0.318216\pi$
$30$	3.77350	0.688943
$31$	4.47424	0.803597	0.401799	−	0.915728i	$-0.368385\pi$
$31$	4.47424	0.803597	0.401799	+	0.915728i	$0.368385\pi$
$32$	6.99201	1.23603
$33$	2.17452	0.378535
$34$	−7.14078	−1.22463
$35$	0.150410	0.0254239
$36$	2.92089	0.486815
$37$	0.540828	0.0889115	0.0444558	−	0.999011i	$-0.485845\pi$
$37$	0.540828	0.0889115	0.0444558	+	0.999011i	$0.485845\pi$
$38$	−8.25236	−1.33871
$39$	2.31214	0.370239
$40$	−3.47499	−0.549443
$41$	6.26118	0.977832	0.488916	−	0.872331i	$-0.337392\pi$
$41$	6.26118	0.977832	0.488916	+	0.872331i	$0.337392\pi$
$42$	0.196144	0.0302657
$43$	−1.74345	−0.265874	−0.132937	−	0.991125i	$-0.542441\pi$
$43$	−1.74345	−0.265874	−0.132937	+	0.991125i	$0.542441\pi$
$44$	−6.35153	−0.957529
$45$	1.70107	0.253581
$46$	17.1295	2.52561
$47$	−8.20011	−1.19611	−0.598054	−	0.801455i	$-0.704059\pi$
$47$	−8.20011	−1.19611	−0.598054	+	0.801455i	$0.704059\pi$
$48$	1.31017	0.189107
$49$	−6.99218	−0.998883
$50$	4.67255	0.660799
$51$	−3.21902	−0.450753
$52$	−6.75352	−0.936545
$53$	−8.44465	−1.15996	−0.579981	−	0.814630i	$-0.696940\pi$
$53$	−8.44465	−1.15996	−0.579981	+	0.814630i	$0.696940\pi$
$54$	2.21831	0.301874
$55$	−3.69901	−0.498774
$56$	−0.180628	−0.0241374
$57$	−3.72011	−0.492741
$58$	−12.9148	−1.69579
$59$	−0.0871124	−0.0113411	−0.00567054	−	0.999984i	$-0.501805\pi$
$59$	−0.0871124	−0.0113411	−0.00567054	+	0.999984i	$0.501805\pi$
$60$	−4.96864	−0.641449
$61$	1.01579	0.130059	0.0650294	−	0.997883i	$-0.479286\pi$
$61$	1.01579	0.130059	0.0650294	+	0.997883i	$0.479286\pi$
$62$	−9.92524	−1.26051
$63$	0.0884207	0.0111400
$64$	−12.8901	−1.61126
$65$	−3.93312	−0.487844
$66$	−4.82375	−0.593763
$67$	5.47031	0.668305	0.334152	−	0.942519i	$-0.391550\pi$
$67$	5.47031	0.668305	0.334152	+	0.942519i	$0.391550\pi$
$68$	9.40241	1.14021
$69$	7.72189	0.929606
$70$	−0.333655	−0.0398794
$71$	0.943411	0.111962	0.0559811	−	0.998432i	$-0.482171\pi$
$71$	0.943411	0.111962	0.0559811	+	0.998432i	$0.482171\pi$
$72$	−2.04282	−0.240749
$73$	−16.0248	−1.87556	−0.937779	−	0.347232i	$-0.887122\pi$
$73$	−16.0248	−1.87556	−0.937779	+	0.347232i	$0.887122\pi$
$74$	−1.19972	−0.139465
$75$	2.10636	0.243221
$76$	10.8660	1.24642
$77$	−0.192272	−0.0219115
$78$	−5.12905	−0.580750
$79$	11.9201	1.34111	0.670557	−	0.741858i	$-0.266055\pi$
$79$	11.9201	1.34111	0.670557	+	0.741858i	$0.266055\pi$
$80$	−2.22870	−0.249176
$81$	1.00000	0.111111
$82$	−13.8892	−1.53381
$83$	5.49372	0.603014	0.301507	−	0.953464i	$-0.402510\pi$
$83$	5.49372	0.603014	0.301507	+	0.953464i	$0.402510\pi$
$84$	−0.258267	−0.0281793
$85$	5.47578	0.593932
$86$	3.86751	0.417045
$87$	−5.82190	−0.624174
$88$	4.44215	0.473535
$89$	6.14531	0.651401	0.325701	−	0.945473i	$-0.394400\pi$
$89$	6.14531	0.651401	0.325701	+	0.945473i	$0.394400\pi$
$90$	−3.77350	−0.397762
$91$	−0.204441	−0.0214313
$92$	−22.5548	−2.35150
$93$	−4.47424	−0.463957
$94$	18.1904	1.87619
$95$	6.32817	0.649257
$96$	−6.99201	−0.713619
$97$	−1.84186	−0.187012	−0.0935060	−	0.995619i	$-0.529807\pi$
$97$	−1.84186	−0.187012	−0.0935060	+	0.995619i	$0.529807\pi$
$98$	15.5108	1.56683
$99$	−2.17452	−0.218547
$100$	−6.15245	−0.615245
$101$	−4.14891	−0.412832	−0.206416	−	0.978464i	$-0.566180\pi$
$101$	−4.14891	−0.412832	−0.206416	+	0.978464i	$0.566180\pi$
$102$	7.14078	0.707043
$103$	12.4339	1.22514	0.612572	−	0.790415i	$-0.290135\pi$
$103$	12.4339	1.22514	0.612572	+	0.790415i	$0.290135\pi$
$104$	4.72330	0.463158
$105$	−0.150410	−0.0146785
$106$	18.7328	1.81949
$107$	9.49262	0.917686	0.458843	−	0.888517i	$-0.348264\pi$
$107$	9.49262	0.917686	0.458843	+	0.888517i	$0.348264\pi$
$108$	−2.92089	−0.281063
$109$	−4.78246	−0.458077	−0.229038	−	0.973417i	$-0.573558\pi$
$109$	−4.78246	−0.458077	−0.229038	+	0.973417i	$0.573558\pi$
$110$	8.20554	0.782367
$111$	−0.540828	−0.0513331
$112$	−0.115846	−0.0109465
$113$	3.81246	0.358646	0.179323	−	0.983790i	$-0.442609\pi$
$113$	3.81246	0.358646	0.179323	+	0.983790i	$0.442609\pi$
$114$	8.25236	0.772904
$115$	−13.1355	−1.22489
$116$	17.0052	1.57889
$117$	−2.31214	−0.213758
$118$	0.193242	0.0177894
$119$	0.284628	0.0260918
$120$	3.47499	0.317221
$121$	−6.27147	−0.570134
$122$	−2.25334	−0.204008
$123$	−6.26118	−0.564552
$124$	13.0688	1.17361
$125$	−12.0884	−1.08122
$126$	−0.196144	−0.0174739
$127$	14.8742	1.31987	0.659936	−	0.751322i	$-0.270584\pi$
$127$	14.8742	1.31987	0.659936	+	0.751322i	$0.270584\pi$
$128$	14.6102	1.29137
$129$	1.74345	0.153502
$130$	8.72487	0.765222
$131$	17.5891	1.53677	0.768385	−	0.639987i	$-0.221060\pi$
$131$	17.5891	1.53677	0.768385	+	0.639987i	$0.221060\pi$
$132$	6.35153	0.552830
$133$	0.328935	0.0285223
$134$	−12.1348	−1.04829
$135$	−1.70107	−0.146405
$136$	−6.57589	−0.563878
$137$	16.5350	1.41268	0.706340	−	0.707873i	$-0.250345\pi$
$137$	16.5350	1.41268	0.706340	+	0.707873i	$0.250345\pi$
$138$	−17.1295	−1.45816
$139$	−10.4685	−0.887927	−0.443964	−	0.896045i	$-0.646428\pi$
$139$	−10.4685	−0.887927	−0.443964	+	0.896045i	$0.646428\pi$
$140$	0.439331	0.0371302
$141$	8.20011	0.690574
$142$	−2.09278	−0.175622
$143$	5.02780	0.420446
$144$	−1.31017	−0.109181
$145$	9.90347	0.822438
$146$	35.5479	2.94196
$147$	6.99218	0.576705
$148$	1.57970	0.129850
$149$	8.05964	0.660272	0.330136	−	0.943933i	$-0.392905\pi$
$149$	8.05964	0.660272	0.330136	+	0.943933i	$0.392905\pi$
$150$	−4.67255	−0.381512
$151$	−0.926458	−0.0753941	−0.0376970	−	0.999289i	$-0.512002\pi$
$151$	−0.926458	−0.0753941	−0.0376970	+	0.999289i	$0.512002\pi$
$152$	−7.59953	−0.616403
$153$	3.21902	0.260242
$154$	0.426519	0.0343699
$155$	7.61100	0.611330
$156$	6.75352	0.540715
$157$	−10.1575	−0.810653	−0.405326	−	0.914172i	$-0.632842\pi$
$157$	−10.1575	−0.810653	−0.405326	+	0.914172i	$0.632842\pi$
$158$	−26.4424	−2.10365
$159$	8.44465	0.669705
$160$	11.8939	0.940296
$161$	−0.682774	−0.0538101
$162$	−2.21831	−0.174287
$163$	−19.1164	−1.49731	−0.748656	−	0.662959i	$-0.769300\pi$
$163$	−19.1164	−1.49731	−0.748656	+	0.662959i	$0.769300\pi$
$164$	18.2882	1.42807
$165$	3.69901	0.287967
$166$	−12.1868	−0.945876
$167$	9.57003	0.740551	0.370276	−	0.928922i	$-0.379263\pi$
$167$	9.57003	0.740551	0.370276	+	0.928922i	$0.379263\pi$
$168$	0.180628	0.0139357
$169$	−7.65399	−0.588768
$170$	−12.1470	−0.931630
$171$	3.72011	0.284484
$172$	−5.09243	−0.388294
$173$	1.57305	0.119597	0.0597983	−	0.998210i	$-0.480954\pi$
$173$	1.57305	0.119597	0.0597983	+	0.998210i	$0.480954\pi$
$174$	12.9148	0.979067
$175$	−0.186246	−0.0140788
$176$	2.84899	0.214751
$177$	0.0871124	0.00654777
$178$	−13.6322	−1.02178
$179$	11.2694	0.842316	0.421158	−	0.906987i	$-0.361624\pi$
$179$	11.2694	0.842316	0.421158	+	0.906987i	$0.361624\pi$
$180$	4.96864	0.370341
$181$	−5.38817	−0.400500	−0.200250	−	0.979745i	$-0.564175\pi$
$181$	−5.38817	−0.400500	−0.200250	+	0.979745i	$0.564175\pi$
$182$	0.453514	0.0336167
$183$	−1.01579	−0.0750895
$184$	15.7744	1.16291
$185$	0.919986	0.0676387
$186$	9.92524	0.727754
$187$	−6.99982	−0.511877
$188$	−23.9516	−1.74685
$189$	−0.0884207	−0.00643166
$190$	−14.0378	−1.01841
$191$	15.9874	1.15680	0.578402	−	0.815752i	$-0.303676\pi$
$191$	15.9874	1.15680	0.578402	+	0.815752i	$0.303676\pi$
$192$	12.8901	0.930263
$193$	26.9354	1.93885	0.969427	−	0.245380i	$-0.0789128\pi$
$193$	26.9354	1.93885	0.969427	+	0.245380i	$0.0789128\pi$
$194$	4.08580	0.293344
$195$	3.93312	0.281657
$196$	−20.4234	−1.45881
$197$	−2.55757	−0.182220	−0.0911098	−	0.995841i	$-0.529041\pi$
$197$	−2.55757	−0.182220	−0.0911098	+	0.995841i	$0.529041\pi$
$198$	4.82375	0.342809
$199$	−7.00410	−0.496507	−0.248254	−	0.968695i	$-0.579857\pi$
$199$	−7.00410	−0.496507	−0.248254	+	0.968695i	$0.579857\pi$
$200$	4.30292	0.304262
$201$	−5.47031	−0.385846
$202$	9.20357	0.647561
$203$	0.514777	0.0361302
$204$	−9.40241	−0.658301
$205$	10.6507	0.743878
$206$	−27.5821	−1.92174
$207$	−7.72189	−0.536708
$208$	3.02931	0.210045
$209$	−8.08945	−0.559559
$210$	0.333655	0.0230244
$211$	−3.35007	−0.230629	−0.115314	−	0.993329i	$-0.536788\pi$
$211$	−3.35007	−0.230629	−0.115314	+	0.993329i	$0.536788\pi$
$212$	−24.6659	−1.69406
$213$	−0.943411	−0.0646414
$214$	−21.0576	−1.43946
$215$	−2.96573	−0.202261
$216$	2.04282	0.138996
$217$	0.395615	0.0268561
$218$	10.6090	0.718531
$219$	16.0248	1.08285
$220$	−10.8044	−0.728433
$221$	−7.44284	−0.500660
$222$	1.19972	0.0805201
$223$	−27.5822	−1.84704	−0.923519	−	0.383553i	$-0.874700\pi$
$223$	−27.5822	−1.84704	−0.923519	+	0.383553i	$0.874700\pi$
$224$	0.618238	0.0413078
$225$	−2.10636	−0.140424
$226$	−8.45720	−0.562565
$227$	11.5600	0.767263	0.383632	−	0.923486i	$-0.374673\pi$
$227$	11.5600	0.767263	0.383632	+	0.923486i	$0.374673\pi$
$228$	−10.8660	−0.719622
$229$	7.16928	0.473759	0.236880	−	0.971539i	$-0.423875\pi$
$229$	7.16928	0.473759	0.236880	+	0.971539i	$0.423875\pi$
$230$	29.1385	1.92134
$231$	0.192272	0.0126506
$232$	−11.8931	−0.780822
$233$	−8.20236	−0.537355	−0.268677	−	0.963230i	$-0.586587\pi$
$233$	−8.20236	−0.537355	−0.268677	+	0.963230i	$0.586587\pi$
$234$	5.12905	0.335296
$235$	−13.9490	−0.909930
$236$	−0.254446	−0.0165630
$237$	−11.9201	−0.774293
$238$	−0.631393	−0.0409271
$239$	6.53795	0.422905	0.211452	−	0.977388i	$-0.432181\pi$
$239$	6.53795	0.422905	0.211452	+	0.977388i	$0.432181\pi$
$240$	2.22870	0.143862
$241$	12.4492	0.801923	0.400962	−	0.916095i	$-0.368676\pi$
$241$	12.4492	0.801923	0.400962	+	0.916095i	$0.368676\pi$
$242$	13.9121	0.894301
$243$	−1.00000	−0.0641500
$244$	2.96702	0.189944
$245$	−11.8942	−0.759892
$246$	13.8892	0.885545
$247$	−8.60144	−0.547296
$248$	−9.14008	−0.580396
$249$	−5.49372	−0.348150
$250$	26.8158	1.69598
$251$	14.3234	0.904088	0.452044	−	0.891996i	$-0.350695\pi$
$251$	14.3234	0.904088	0.452044	+	0.891996i	$0.350695\pi$
$252$	0.258267	0.0162693
$253$	16.7914	1.05566
$254$	−32.9956	−2.07032
$255$	−5.47578	−0.342907
$256$	−6.62970	−0.414356
$257$	4.49086	0.280132	0.140066	−	0.990142i	$-0.455268\pi$
$257$	4.49086	0.280132	0.140066	+	0.990142i	$0.455268\pi$
$258$	−3.86751	−0.240781
$259$	0.0478203	0.00297141
$260$	−11.4882	−0.712469
$261$	5.82190	0.360367
$262$	−39.0181	−2.41055
$263$	16.8225	1.03732	0.518659	−	0.854981i	$-0.326432\pi$
$263$	16.8225	1.03732	0.518659	+	0.854981i	$0.326432\pi$
$264$	−4.44215	−0.273396
$265$	−14.3650	−0.882432
$266$	−0.729679	−0.0447395
$267$	−6.14531	−0.376087
$268$	15.9782	0.976023
$269$	−17.7555	−1.08257	−0.541286	−	0.840838i	$-0.682063\pi$
$269$	−17.7555	−1.08257	−0.541286	+	0.840838i	$0.682063\pi$
$270$	3.77350	0.229648
$271$	31.1355	1.89135	0.945674	−	0.325115i	$-0.105403\pi$
$271$	31.1355	1.89135	0.945674	+	0.325115i	$0.105403\pi$
$272$	−4.21748	−0.255722
$273$	0.204441	0.0123734
$274$	−36.6797	−2.21590
$275$	4.58031	0.276203
$276$	22.5548	1.35764
$277$	10.0897	0.606230	0.303115	−	0.952954i	$-0.401973\pi$
$277$	10.0897	0.606230	0.303115	+	0.952954i	$0.401973\pi$
$278$	23.2224	1.39279
$279$	4.47424	0.267866
$280$	−0.307261	−0.0183623
$281$	−14.2427	−0.849650	−0.424825	−	0.905276i	$-0.639664\pi$
$281$	−14.2427	−0.849650	−0.424825	+	0.905276i	$0.639664\pi$
$282$	−18.1904	−1.08322
$283$	3.58934	0.213364	0.106682	−	0.994293i	$-0.465977\pi$
$283$	3.58934	0.213364	0.106682	+	0.994293i	$0.465977\pi$
$284$	2.75560	0.163515
$285$	−6.32817	−0.374849
$286$	−11.1532	−0.659503
$287$	0.553618	0.0326790
$288$	6.99201	0.412008
$289$	−6.63790	−0.390465
$290$	−21.9689	−1.29006
$291$	1.84186	0.107971
$292$	−46.8066	−2.73915
$293$	−20.9945	−1.22651	−0.613255	−	0.789885i	$-0.710140\pi$
$293$	−20.9945	−1.22651	−0.613255	+	0.789885i	$0.710140\pi$
$294$	−15.5108	−0.904609
$295$	−0.148184	−0.00862763
$296$	−1.10482	−0.0642161
$297$	2.17452	0.126178
$298$	−17.8788	−1.03569
$299$	17.8541	1.03253
$300$	6.15245	0.355212
$301$	−0.154157	−0.00888547
$302$	2.05517	0.118262
$303$	4.14891	0.238349
$304$	−4.87399	−0.279543
$305$	1.72793	0.0989412
$306$	−7.14078	−0.408211
$307$	4.15659	0.237229	0.118614	−	0.992940i	$-0.462155\pi$
$307$	4.15659	0.237229	0.118614	+	0.992940i	$0.462155\pi$
$308$	−0.561607	−0.0320005
$309$	−12.4339	−0.707337
$310$	−16.8835	−0.958920
$311$	23.5559	1.33574	0.667868	−	0.744280i	$-0.267207\pi$
$311$	23.5559	1.33574	0.667868	+	0.744280i	$0.267207\pi$
$312$	−4.72330	−0.267404
$313$	−27.2808	−1.54200	−0.771001	−	0.636834i	$-0.780244\pi$
$313$	−27.2808	−1.54200	−0.771001	+	0.636834i	$0.780244\pi$
$314$	22.5324	1.27157
$315$	0.150410	0.00847463
$316$	34.8173	1.95862
$317$	18.5224	1.04032	0.520161	−	0.854068i	$-0.325872\pi$
$317$	18.5224	1.04032	0.520161	+	0.854068i	$0.325872\pi$
$318$	−18.7328	−1.05049
$319$	−12.6598	−0.708814
$320$	−21.9270	−1.22575
$321$	−9.49262	−0.529826
$322$	1.51460	0.0844056
$323$	11.9751	0.666314
$324$	2.92089	0.162272
$325$	4.87021	0.270150
$326$	42.4061	2.34865
$327$	4.78246	0.264471
$328$	−12.7905	−0.706236
$329$	−0.725059	−0.0399738
$330$	−8.20554	−0.451700
$331$	13.7598	0.756305	0.378152	−	0.925743i	$-0.376560\pi$
$331$	13.7598	0.756305	0.378152	+	0.925743i	$0.376560\pi$
$332$	16.0466	0.880670
$333$	0.540828	0.0296372
$334$	−21.2293	−1.16161
$335$	9.30538	0.508407
$336$	0.115846	0.00631994
$337$	11.3638	0.619026	0.309513	−	0.950895i	$-0.399834\pi$
$337$	11.3638	0.619026	0.309513	+	0.950895i	$0.399834\pi$
$338$	16.9789	0.923531
$339$	−3.81246	−0.207064
$340$	15.9942	0.867406
$341$	−9.72931	−0.526872
$342$	−8.25236	−0.446236
$343$	−1.23720	−0.0668024
$344$	3.56156	0.192027
$345$	13.1355	0.707190
$346$	−3.48951	−0.187597
$347$	−5.42531	−0.291246	−0.145623	−	0.989340i	$-0.546519\pi$
$347$	−5.42531	−0.291246	−0.145623	+	0.989340i	$0.546519\pi$
$348$	−17.0052	−0.911572
$349$	−1.37672	−0.0736939	−0.0368470	−	0.999321i	$-0.511731\pi$
$349$	−1.37672	−0.0736939	−0.0368470	+	0.999321i	$0.511731\pi$
$350$	0.413150	0.0220838
$351$	2.31214	0.123413
$352$	−15.2043	−0.810390
$353$	−14.9583	−0.796151	−0.398075	−	0.917353i	$-0.630322\pi$
$353$	−14.9583	−0.796151	−0.398075	+	0.917353i	$0.630322\pi$
$354$	−0.193242	−0.0102707
$355$	1.60481	0.0851744
$356$	17.9498	0.951336
$357$	−0.284628	−0.0150641
$358$	−24.9991	−1.32124
$359$	−21.7189	−1.14628	−0.573139	−	0.819458i	$-0.694275\pi$
$359$	−21.7189	−1.14628	−0.573139	+	0.819458i	$0.694275\pi$
$360$	−3.47499	−0.183148
$361$	−5.16076	−0.271619
$362$	11.9526	0.628216
$363$	6.27147	0.329167
$364$	−0.597151	−0.0312992
$365$	−27.2593	−1.42682
$366$	2.25334	0.117784
$367$	8.47860	0.442579	0.221290	−	0.975208i	$-0.428973\pi$
$367$	8.47860	0.442579	0.221290	+	0.975208i	$0.428973\pi$
$368$	10.1170	0.527386
$369$	6.26118	0.325944
$370$	−2.04081	−0.106097
$371$	−0.746682	−0.0387658
$372$	−13.0688	−0.677584
$373$	−28.3685	−1.46887	−0.734434	−	0.678680i	$-0.762552\pi$
$373$	−28.3685	−1.46887	−0.734434	+	0.678680i	$0.762552\pi$
$374$	15.5278	0.802921
$375$	12.0884	0.624243
$376$	16.7514	0.863886
$377$	−13.4611	−0.693281
$378$	0.196144	0.0100886
$379$	−19.3553	−0.994214	−0.497107	−	0.867689i	$-0.665604\pi$
$379$	−19.3553	−0.994214	−0.497107	+	0.867689i	$0.665604\pi$
$380$	18.4839	0.948205
$381$	−14.8742	−0.762028
$382$	−35.4649	−1.81454
$383$	−7.46800	−0.381597	−0.190798	−	0.981629i	$-0.561108\pi$
$383$	−7.46800	−0.381597	−0.190798	+	0.981629i	$0.561108\pi$
$384$	−14.6102	−0.745573
$385$	−0.327069	−0.0166690
$386$	−59.7511	−3.04125
$387$	−1.74345	−0.0886246
$388$	−5.37986	−0.273121
$389$	−21.0086	−1.06518	−0.532590	−	0.846373i	$-0.678781\pi$
$389$	−21.0086	−1.06518	−0.532590	+	0.846373i	$0.678781\pi$
$390$	−8.72487	−0.441801
$391$	−24.8569	−1.25707
$392$	14.2838	0.721440
$393$	−17.5891	−0.887255
$394$	5.67349	0.285826
$395$	20.2769	1.02024
$396$	−6.35153	−0.319176
$397$	15.0811	0.756899	0.378449	−	0.925622i	$-0.376457\pi$
$397$	15.0811	0.756899	0.378449	+	0.925622i	$0.376457\pi$
$398$	15.5373	0.778812
$399$	−0.328935	−0.0164673
$400$	2.75970	0.137985
$401$	−4.30253	−0.214858	−0.107429	−	0.994213i	$-0.534262\pi$
$401$	−4.30253	−0.214858	−0.107429	+	0.994213i	$0.534262\pi$
$402$	12.1348	0.605230
$403$	−10.3451	−0.515325
$404$	−12.1185	−0.602919
$405$	1.70107	0.0845269
$406$	−1.14193	−0.0566732
$407$	−1.17604	−0.0582941
$408$	6.57589	0.325555
$409$	−20.9300	−1.03492	−0.517461	−	0.855707i	$-0.673123\pi$
$409$	−20.9300	−1.03492	−0.517461	+	0.855707i	$0.673123\pi$
$410$	−23.6266	−1.16683
$411$	−16.5350	−0.815611
$412$	36.3180	1.78926
$413$	−0.00770254	−0.000379017 0
$414$	17.1295	0.841870
$415$	9.34520	0.458738
$416$	−16.1665	−0.792630
$417$	10.4685	0.512645
$418$	17.9449	0.877713
$419$	34.3271	1.67699	0.838494	−	0.544911i	$-0.183437\pi$
$419$	34.3271	1.67699	0.838494	+	0.544911i	$0.183437\pi$
$420$	−0.439331	−0.0214371
$421$	12.5176	0.610072	0.305036	−	0.952341i	$-0.401331\pi$
$421$	12.5176	0.610072	0.305036	+	0.952341i	$0.401331\pi$
$422$	7.43150	0.361760
$423$	−8.20011	−0.398703
$424$	17.2509	0.837779
$425$	−6.78042	−0.328898
$426$	2.09278	0.101395
$427$	0.0898170	0.00434655
$428$	27.7269	1.34023
$429$	−5.02780	−0.242744
$430$	6.57891	0.317263
$431$	20.1676	0.971440	0.485720	−	0.874114i	$-0.338557\pi$
$431$	20.1676	0.971440	0.485720	+	0.874114i	$0.338557\pi$
$432$	1.31017	0.0630357
$433$	−10.9114	−0.524371	−0.262185	−	0.965018i	$-0.584443\pi$
$433$	−10.9114	−0.524371	−0.262185	+	0.965018i	$0.584443\pi$
$434$	−0.877596	−0.0421260
$435$	−9.90347	−0.474835
$436$	−13.9691	−0.668997
$437$	−28.7263	−1.37416
$438$	−35.5479	−1.69854
$439$	−27.1168	−1.29422	−0.647108	−	0.762398i	$-0.724022\pi$
$439$	−27.1168	−1.29422	−0.647108	+	0.762398i	$0.724022\pi$
$440$	7.55642	0.360238
$441$	−6.99218	−0.332961
$442$	16.5105	0.785325
$443$	3.78412	0.179789	0.0898945	−	0.995951i	$-0.471347\pi$
$443$	3.78412	0.179789	0.0898945	+	0.995951i	$0.471347\pi$
$444$	−1.57970	−0.0749692
$445$	10.4536	0.495548
$446$	61.1857	2.89723
$447$	−8.05964	−0.381208
$448$	−1.13975	−0.0538482
$449$	−6.76391	−0.319209	−0.159604	−	0.987181i	$-0.551022\pi$
$449$	−6.76391	−0.319209	−0.159604	+	0.987181i	$0.551022\pi$
$450$	4.67255	0.220266
$451$	−13.6150	−0.641108
$452$	11.1358	0.523783
$453$	0.926458	0.0435288
$454$	−25.6436	−1.20351
$455$	−0.347769	−0.0163037
$456$	7.59953	0.355881
$457$	−2.54152	−0.118887	−0.0594437	−	0.998232i	$-0.518933\pi$
$457$	−2.54152	−0.118887	−0.0594437	+	0.998232i	$0.518933\pi$
$458$	−15.9037	−0.743129
$459$	−3.21902	−0.150251
$460$	−38.3673	−1.78888
$461$	36.2770	1.68959	0.844795	−	0.535090i	$-0.179722\pi$
$461$	36.2770	1.68959	0.844795	+	0.535090i	$0.179722\pi$
$462$	−0.426519	−0.0198435
$463$	−1.35344	−0.0628997	−0.0314498	−	0.999505i	$-0.510012\pi$
$463$	−1.35344	−0.0628997	−0.0314498	+	0.999505i	$0.510012\pi$
$464$	−7.62770	−0.354107
$465$	−7.61100	−0.352951
$466$	18.1954	0.842884
$467$	20.9877	0.971196	0.485598	−	0.874182i	$-0.338602\pi$
$467$	20.9877	0.971196	0.485598	+	0.874182i	$0.338602\pi$
$468$	−6.75352	−0.312182
$469$	0.483688	0.0223347
$470$	30.9431	1.42730
$471$	10.1575	0.468031
$472$	0.177955	0.00819106
$473$	3.79116	0.174318
$474$	26.4424	1.21454
$475$	−7.83589	−0.359535
$476$	0.831368	0.0381057
$477$	−8.44465	−0.386654
$478$	−14.5032	−0.663360
$479$	1.21934	0.0557131	0.0278566	−	0.999612i	$-0.491132\pi$
$479$	1.21934	0.0557131	0.0278566	+	0.999612i	$0.491132\pi$
$480$	−11.8939	−0.542880
$481$	−1.25047	−0.0570166
$482$	−27.6162	−1.25788
$483$	0.682774	0.0310673
$484$	−18.3183	−0.832650
$485$	−3.13313	−0.142268
$486$	2.21831	0.100625
$487$	−14.5839	−0.660858	−0.330429	−	0.943831i	$-0.607193\pi$
$487$	−14.5839	−0.660858	−0.330429	+	0.943831i	$0.607193\pi$
$488$	−2.07508	−0.0939346
$489$	19.1164	0.864473
$490$	26.3850	1.19195
$491$	1.53011	0.0690530	0.0345265	−	0.999404i	$-0.489008\pi$
$491$	1.53011	0.0690530	0.0345265	+	0.999404i	$0.489008\pi$
$492$	−18.2882	−0.824497
$493$	18.7408	0.844045
$494$	19.0806	0.858479
$495$	−3.69901	−0.166258
$496$	−5.86203	−0.263213
$497$	0.0834170	0.00374176
$498$	12.1868	0.546102
$499$	5.64712	0.252800	0.126400	−	0.991979i	$-0.459658\pi$
$499$	5.64712	0.252800	0.126400	+	0.991979i	$0.459658\pi$
$500$	−35.3090	−1.57906
$501$	−9.57003	−0.427557
$502$	−31.7738	−1.41813
$503$	−4.35636	−0.194241	−0.0971203	−	0.995273i	$-0.530963\pi$
$503$	−4.35636	−0.194241	−0.0971203	+	0.995273i	$0.530963\pi$
$504$	−0.180628	−0.00804580
$505$	−7.05759	−0.314059
$506$	−37.2484	−1.65590
$507$	7.65399	0.339926
$508$	43.4459	1.92760
$509$	−2.72218	−0.120658	−0.0603292	−	0.998179i	$-0.519215\pi$
$509$	−2.72218	−0.120658	−0.0603292	+	0.998179i	$0.519215\pi$
$510$	12.1470	0.537877
$511$	−1.41692	−0.0626809
$512$	−14.5137	−0.641419
$513$	−3.72011	−0.164247
$514$	−9.96212	−0.439410
$515$	21.1509	0.932019
$516$	5.09243	0.224182
$517$	17.8313	0.784219
$518$	−0.106080	−0.00466090
$519$	−1.57305	−0.0690492
$520$	8.03467	0.352344
$521$	27.5998	1.20917	0.604586	−	0.796540i	$-0.293339\pi$
$521$	27.5998	1.20917	0.604586	+	0.796540i	$0.293339\pi$
$522$	−12.9148	−0.565265
$523$	2.82000	0.123310	0.0616550	−	0.998098i	$-0.480362\pi$
$523$	2.82000	0.123310	0.0616550	+	0.998098i	$0.480362\pi$
$524$	51.3760	2.24437
$525$	0.186246	0.00812843
$526$	−37.3174	−1.62712
$527$	14.4027	0.627390
$528$	−2.84899	−0.123987
$529$	36.6275	1.59250
$530$	31.8659	1.38417
$531$	−0.0871124	−0.00378036
$532$	0.960783	0.0416552
$533$	−14.4768	−0.627058
$534$	13.6322	0.589922
$535$	16.1476	0.698122
$536$	−11.1749	−0.482681
$537$	−11.2694	−0.486311
$538$	39.3872	1.69810
$539$	15.2046	0.654909
$540$	−4.96864	−0.213816
$541$	39.8671	1.71402	0.857011	−	0.515299i	$-0.172319\pi$
$541$	39.8671	1.71402	0.857011	+	0.515299i	$0.172319\pi$
$542$	−69.0682	−2.96673
$543$	5.38817	0.231229
$544$	22.5074	0.964999
$545$	−8.13531	−0.348478
$546$	−0.453514	−0.0194086
$547$	38.7602	1.65726	0.828632	−	0.559793i	$-0.189119\pi$
$547$	38.7602	1.65726	0.828632	+	0.559793i	$0.189119\pi$
$548$	48.2969	2.06314
$549$	1.01579	0.0433529
$550$	−10.1606	−0.433247
$551$	21.6581	0.922668
$552$	−15.7744	−0.671405
$553$	1.05398	0.0448199
$554$	−22.3820	−0.950921
$555$	−0.919986	−0.0390512
$556$	−30.5774	−1.29677
$557$	−16.5786	−0.702458	−0.351229	−	0.936290i	$-0.614236\pi$
$557$	−16.5786	−0.702458	−0.351229	+	0.936290i	$0.614236\pi$
$558$	−9.92524	−0.420169
$559$	4.03111	0.170498
$560$	−0.197063	−0.00832743
$561$	6.99982	0.295533
$562$	31.5948	1.33274
$563$	37.3491	1.57408	0.787038	−	0.616905i	$-0.211614\pi$
$563$	37.3491	1.57408	0.787038	+	0.616905i	$0.211614\pi$
$564$	23.9516	1.00855
$565$	6.48526	0.272837
$566$	−7.96227	−0.334679
$567$	0.0884207	0.00371332
$568$	−1.92722	−0.0808644
$569$	5.03919	0.211254	0.105627	−	0.994406i	$-0.466315\pi$
$569$	5.03919	0.211254	0.105627	+	0.994406i	$0.466315\pi$
$570$	14.0378	0.587980
$571$	−14.0716	−0.588878	−0.294439	−	0.955670i	$-0.595133\pi$
$571$	−14.0716	−0.588878	−0.294439	+	0.955670i	$0.595133\pi$
$572$	14.6857	0.614038
$573$	−15.9874	−0.667882
$574$	−1.22810	−0.0512597
$575$	16.2651	0.678300
$576$	−12.8901	−0.537087
$577$	−44.6963	−1.86073	−0.930367	−	0.366631i	$-0.880511\pi$
$577$	−44.6963	−1.86073	−0.930367	+	0.366631i	$0.880511\pi$
$578$	14.7249	0.612476
$579$	−26.9354	−1.11940
$580$	28.9270	1.20113
$581$	0.485758	0.0201527
$582$	−4.08580	−0.169362
$583$	18.3630	0.760520
$584$	32.7358	1.35462
$585$	−3.93312	−0.162615
$586$	46.5722	1.92388
$587$	−19.4631	−0.803327	−0.401663	−	0.915787i	$-0.631568\pi$
$587$	−19.4631	−0.803327	−0.401663	+	0.915787i	$0.631568\pi$
$588$	20.4234	0.842247
$589$	16.6447	0.685832
$590$	0.328719	0.0135331
$591$	2.55757	0.105205
$592$	−0.708578	−0.0291224
$593$	42.0686	1.72755	0.863775	−	0.503878i	$-0.168094\pi$
$593$	42.0686	1.72755	0.863775	+	0.503878i	$0.168094\pi$
$594$	−4.82375	−0.197921
$595$	0.484172	0.0198491
$596$	23.5413	0.964291
$597$	7.00410	0.286659
$598$	−39.6059	−1.61961
$599$	−26.0388	−1.06392	−0.531958	−	0.846771i	$-0.678544\pi$
$599$	−26.0388	−1.06392	−0.531958	+	0.846771i	$0.678544\pi$
$600$	−4.30292	−0.175666
$601$	27.7188	1.13067	0.565337	−	0.824860i	$-0.308746\pi$
$601$	27.7188	1.13067	0.565337	+	0.824860i	$0.308746\pi$
$602$	0.341968	0.0139376
$603$	5.47031	0.222768
$604$	−2.70608	−0.110109
$605$	−10.6682	−0.433725
$606$	−9.20357	−0.373869
$607$	30.7415	1.24776	0.623879	−	0.781521i	$-0.285556\pi$
$607$	30.7415	1.24776	0.623879	+	0.781521i	$0.285556\pi$
$608$	26.0111	1.05489
$609$	−0.514777	−0.0208598
$610$	−3.83309	−0.155197
$611$	18.9598	0.767033
$612$	9.40241	0.380070
$613$	−14.4163	−0.582269	−0.291135	−	0.956682i	$-0.594033\pi$
$613$	−14.4163	−0.582269	−0.291135	+	0.956682i	$0.594033\pi$
$614$	−9.22059	−0.372113
$615$	−10.6507	−0.429478
$616$	0.392778	0.0158255
$617$	9.29751	0.374304	0.187152	−	0.982331i	$-0.440074\pi$
$617$	9.29751	0.374304	0.187152	+	0.982331i	$0.440074\pi$
$618$	27.5821	1.10952
$619$	24.6123	0.989252	0.494626	−	0.869106i	$-0.335305\pi$
$619$	24.6123	0.989252	0.494626	+	0.869106i	$0.335305\pi$
$620$	22.2309	0.892814
$621$	7.72189	0.309869
$622$	−52.2544	−2.09521
$623$	0.543372	0.0217697
$624$	−3.02931	−0.121269
$625$	−10.0315	−0.401258
$626$	60.5172	2.41876
$627$	8.08945	0.323061
$628$	−29.6688	−1.18391
$629$	1.74094	0.0694157
$630$	−0.333655	−0.0132931
$631$	−17.0584	−0.679084	−0.339542	−	0.940591i	$-0.610272\pi$
$631$	−17.0584	−0.679084	−0.339542	+	0.940591i	$0.610272\pi$
$632$	−24.3506	−0.968616
$633$	3.35007	0.133154
$634$	−41.0884	−1.63183
$635$	25.3021	1.00408
$636$	24.6659	0.978067
$637$	16.1669	0.640557
$638$	28.0834	1.11183
$639$	0.943411	0.0373207
$640$	24.8530	0.982399
$641$	−4.08517	−0.161354	−0.0806772	−	0.996740i	$-0.525708\pi$
$641$	−4.08517	−0.161354	−0.0806772	+	0.996740i	$0.525708\pi$
$642$	21.0576	0.831075
$643$	−32.9542	−1.29959	−0.649793	−	0.760111i	$-0.725144\pi$
$643$	−32.9542	−1.29959	−0.649793	+	0.760111i	$0.725144\pi$
$644$	−1.99431	−0.0785868
$645$	2.96573	0.116776
$646$	−26.5645	−1.04517
$647$	−0.926817	−0.0364369	−0.0182185	−	0.999834i	$-0.505799\pi$
$647$	−0.926817	−0.0364369	−0.0182185	+	0.999834i	$0.505799\pi$
$648$	−2.04282	−0.0802497
$649$	0.189427	0.00743568
$650$	−10.8036	−0.423753
$651$	−0.395615	−0.0155054
$652$	−55.8369	−2.18674
$653$	35.0633	1.37213	0.686066	−	0.727539i	$-0.259336\pi$
$653$	35.0633	1.37213	0.686066	+	0.727539i	$0.259336\pi$
$654$	−10.6090	−0.414844
$655$	29.9204	1.16909
$656$	−8.20323	−0.320282
$657$	−16.0248	−0.625186
$658$	1.60840	0.0627021
$659$	8.45054	0.329186	0.164593	−	0.986362i	$-0.447369\pi$
$659$	8.45054	0.329186	0.164593	+	0.986362i	$0.447369\pi$
$660$	10.8044	0.420561
$661$	24.4136	0.949578	0.474789	−	0.880100i	$-0.342524\pi$
$661$	24.4136	0.949578	0.474789	+	0.880100i	$0.342524\pi$
$662$	−30.5234	−1.18633
$663$	7.44284	0.289056
$664$	−11.2227	−0.435525
$665$	0.559541	0.0216981
$666$	−1.19972	−0.0464883
$667$	−44.9561	−1.74071
$668$	27.9530	1.08153
$669$	27.5822	1.06639
$670$	−20.6422	−0.797478
$671$	−2.20886	−0.0852720
$672$	−0.618238	−0.0238491
$673$	21.6369	0.834040	0.417020	−	0.908897i	$-0.363075\pi$
$673$	21.6369	0.834040	0.417020	+	0.908897i	$0.363075\pi$
$674$	−25.2084	−0.970993
$675$	2.10636	0.0810738
$676$	−22.3565	−0.859864
$677$	7.17478	0.275749	0.137875	−	0.990450i	$-0.455973\pi$
$677$	7.17478	0.275749	0.137875	+	0.990450i	$0.455973\pi$
$678$	8.45720	0.324797
$679$	−0.162858	−0.00624992
$680$	−11.1861	−0.428966
$681$	−11.5600	−0.442980
$682$	21.5826	0.826441
$683$	−10.4697	−0.400612	−0.200306	−	0.979733i	$-0.564194\pi$
$683$	−10.4697	−0.400612	−0.200306	+	0.979733i	$0.564194\pi$
$684$	10.8660	0.415474
$685$	28.1272	1.07468
$686$	2.74449	0.104785
$687$	−7.16928	−0.273525
$688$	2.28422	0.0870852
$689$	19.5253	0.743853
$690$	−29.1385	−1.10928
$691$	0.761300	0.0289612	0.0144806	−	0.999895i	$-0.495391\pi$
$691$	0.761300	0.0289612	0.0144806	+	0.999895i	$0.495391\pi$
$692$	4.59470	0.174664
$693$	−0.192272	−0.00730382
$694$	12.0350	0.456843
$695$	−17.8077	−0.675483
$696$	11.8931	0.450808
$697$	20.1549	0.763420
$698$	3.05398	0.115595
$699$	8.20236	0.310242
$700$	−0.544004	−0.0205614
$701$	21.7154	0.820178	0.410089	−	0.912045i	$-0.365498\pi$
$701$	21.7154	0.820178	0.410089	+	0.912045i	$0.365498\pi$
$702$	−5.12905	−0.193583
$703$	2.01194	0.0758817
$704$	28.0297	1.05641
$705$	13.9490	0.525348
$706$	33.1822	1.24883
$707$	−0.366850	−0.0137968
$708$	0.254446	0.00956267
$709$	−14.4154	−0.541384	−0.270692	−	0.962666i	$-0.587252\pi$
$709$	−14.4154	−0.541384	−0.270692	+	0.962666i	$0.587252\pi$
$710$	−3.55996	−0.133603
$711$	11.9201	0.447038
$712$	−12.5538	−0.470473
$713$	−34.5496	−1.29389
$714$	0.631393	0.0236293
$715$	8.55264	0.319851
$716$	32.9168	1.23016
$717$	−6.53795	−0.244164
$718$	48.1792	1.79803
$719$	−9.35356	−0.348829	−0.174414	−	0.984672i	$-0.555803\pi$
$719$	−9.35356	−0.348829	−0.174414	+	0.984672i	$0.555803\pi$
$720$	−2.22870	−0.0830586
$721$	1.09941	0.0409442
$722$	11.4482	0.426056
$723$	−12.4492	−0.462991
$724$	−15.7383	−0.584908
$725$	−12.2630	−0.455437
$726$	−13.9121	−0.516325
$727$	−37.1112	−1.37638	−0.688190	−	0.725531i	$-0.741594\pi$
$727$	−37.1112	−1.37638	−0.688190	+	0.725531i	$0.741594\pi$
$728$	0.417637	0.0154787
$729$	1.00000	0.0370370
$730$	60.4695	2.23808
$731$	−5.61221	−0.207575
$732$	−2.96702	−0.109664
$733$	44.1313	1.63003	0.815014	−	0.579442i	$-0.196729\pi$
$733$	44.1313	1.63003	0.815014	+	0.579442i	$0.196729\pi$
$734$	−18.8081	−0.694221
$735$	11.8942	0.438724
$736$	−53.9915	−1.99015
$737$	−11.8953	−0.438168
$738$	−13.8892	−0.511270
$739$	−44.3396	−1.63106	−0.815530	−	0.578715i	$-0.803555\pi$
$739$	−44.3396	−1.63106	−0.815530	+	0.578715i	$0.803555\pi$
$740$	2.68718	0.0987827
$741$	8.60144	0.315982
$742$	1.65637	0.0608073
$743$	35.4984	1.30231	0.651154	−	0.758945i	$-0.274285\pi$
$743$	35.4984	1.30231	0.651154	+	0.758945i	$0.274285\pi$
$744$	9.14008	0.335092
$745$	13.7100	0.502296
$746$	62.9302	2.30404
$747$	5.49372	0.201005
$748$	−20.4457	−0.747569
$749$	0.839344	0.0306689
$750$	−26.8158	−0.979176
$751$	37.6229	1.37288	0.686440	−	0.727187i	$-0.259173\pi$
$751$	37.6229	1.37288	0.686440	+	0.727187i	$0.259173\pi$
$752$	10.7436	0.391777
$753$	−14.3234	−0.521975
$754$	29.8608	1.08747
$755$	−1.57597	−0.0573554
$756$	−0.258267	−0.00939309
$757$	−20.6756	−0.751466	−0.375733	−	0.926728i	$-0.622609\pi$
$757$	−20.6756	−0.751466	−0.375733	+	0.926728i	$0.622609\pi$
$758$	42.9360	1.55950
$759$	−16.7914	−0.609488
$760$	−12.9273	−0.468924
$761$	12.7738	0.463051	0.231525	−	0.972829i	$-0.425628\pi$
$761$	12.7738	0.463051	0.231525	+	0.972829i	$0.425628\pi$
$762$	32.9956	1.19530
$763$	−0.422868	−0.0153089
$764$	46.6974	1.68945
$765$	5.47578	0.197977
$766$	16.5663	0.598566
$767$	0.201417	0.00727273
$768$	6.62970	0.239229
$769$	23.6374	0.852388	0.426194	−	0.904632i	$-0.359854\pi$
$769$	23.6374	0.852388	0.426194	+	0.904632i	$0.359854\pi$
$770$	0.725539	0.0261466
$771$	−4.49086	−0.161734
$772$	78.6755	2.83159
$773$	53.0448	1.90789	0.953944	−	0.299986i	$-0.0969819\pi$
$773$	53.0448	1.90789	0.953944	+	0.299986i	$0.0969819\pi$
$774$	3.86751	0.139015
$775$	−9.42435	−0.338533
$776$	3.76258	0.135069
$777$	−0.0478203	−0.00171555
$778$	46.6036	1.67082
$779$	23.2923	0.834533
$780$	11.4882	0.411344
$781$	−2.05146	−0.0734071
$782$	55.1403	1.97181
$783$	−5.82190	−0.208058
$784$	9.16097	0.327177
$785$	−17.2785	−0.616698
$786$	39.0181	1.39173
$787$	−3.26099	−0.116242	−0.0581209	−	0.998310i	$-0.518511\pi$
$787$	−3.26099	−0.116242	−0.0581209	+	0.998310i	$0.518511\pi$
$788$	−7.47040	−0.266122
$789$	−16.8225	−0.598895
$790$	−44.9804	−1.60033
$791$	0.337100	0.0119859
$792$	4.44215	0.157845
$793$	−2.34866	−0.0834032
$794$	−33.4545	−1.18726
$795$	14.3650	0.509472
$796$	−20.4582	−0.725122
$797$	54.6731	1.93662	0.968310	−	0.249750i	$-0.0803483\pi$
$797$	54.6731	1.93662	0.968310	+	0.249750i	$0.0803483\pi$
$798$	0.729679	0.0258304
$799$	−26.3963	−0.933835
$800$	−14.7277	−0.520703
$801$	6.14531	0.217134
$802$	9.54434	0.337022
$803$	34.8462	1.22969
$804$	−15.9782	−0.563507
$805$	−1.16145	−0.0409356
$806$	22.9486	0.808330
$807$	17.7555	0.625024
$808$	8.47549	0.298167
$809$	36.9504	1.29911	0.649553	−	0.760317i	$-0.274956\pi$
$809$	36.9504	1.29911	0.649553	+	0.760317i	$0.274956\pi$
$810$	−3.77350	−0.132587
$811$	12.9959	0.456348	0.228174	−	0.973620i	$-0.426725\pi$
$811$	12.9959	0.456348	0.228174	+	0.973620i	$0.426725\pi$
$812$	1.50361	0.0527663
$813$	−31.1355	−1.09197
$814$	2.60882	0.0914390
$815$	−32.5183	−1.13907
$816$	4.21748	0.147641
$817$	−6.48583	−0.226911
$818$	46.4292	1.62336
$819$	−0.204441	−0.00714376
$820$	31.1096	1.08639
$821$	38.2656	1.33548	0.667739	−	0.744395i	$-0.267262\pi$
$821$	38.2656	1.33548	0.667739	+	0.744395i	$0.267262\pi$
$822$	36.6797	1.27935
$823$	26.7320	0.931818	0.465909	−	0.884833i	$-0.345727\pi$
$823$	26.7320	0.931818	0.465909	+	0.884833i	$0.345727\pi$
$824$	−25.4002	−0.884857
$825$	−4.58031	−0.159466
$826$	0.0170866	0.000594519 0
$827$	3.07785	0.107027	0.0535137	−	0.998567i	$-0.482958\pi$
$827$	3.07785	0.107027	0.0535137	+	0.998567i	$0.482958\pi$
$828$	−22.5548	−0.783833
$829$	10.5705	0.367129	0.183564	−	0.983008i	$-0.441236\pi$
$829$	10.5705	0.367129	0.183564	+	0.983008i	$0.441236\pi$
$830$	−20.7305	−0.719568
$831$	−10.0897	−0.350007
$832$	29.8038	1.03326
$833$	−22.5080	−0.779855
$834$	−23.2224	−0.804125
$835$	16.2793	0.563368
$836$	−23.6284	−0.817206
$837$	−4.47424	−0.154652
$838$	−76.1481	−2.63049
$839$	46.2141	1.59549	0.797745	−	0.602995i	$-0.206026\pi$
$839$	46.2141	1.59549	0.797745	+	0.602995i	$0.206026\pi$
$840$	0.307261	0.0106015
$841$	4.89457	0.168778
$842$	−27.7680	−0.956948
$843$	14.2427	0.490546
$844$	−9.78521	−0.336821
$845$	−13.0200	−0.447901
$846$	18.1904	0.625398
$847$	−0.554528	−0.0190538
$848$	11.0640	0.379938
$849$	−3.58934	−0.123186
$850$	15.0411	0.515904
$851$	−4.17621	−0.143159
$852$	−2.75560	−0.0944053
$853$	−7.54508	−0.258339	−0.129169	−	0.991623i	$-0.541231\pi$
$853$	−7.54508	−0.258339	−0.129169	+	0.991623i	$0.541231\pi$
$854$	−0.199242	−0.00681791
$855$	6.32817	0.216419
$856$	−19.3917	−0.662796
$857$	−33.6119	−1.14816	−0.574080	−	0.818799i	$-0.694640\pi$
$857$	−33.6119	−1.14816	−0.574080	+	0.818799i	$0.694640\pi$
$858$	11.1532	0.380764
$859$	38.1628	1.30210	0.651048	−	0.759036i	$-0.274330\pi$
$859$	38.1628	1.30210	0.651048	+	0.759036i	$0.274330\pi$
$860$	−8.66259	−0.295392
$861$	−0.553618	−0.0188672
$862$	−44.7380	−1.52378
$863$	−3.54130	−0.120547	−0.0602736	−	0.998182i	$-0.519197\pi$
$863$	−3.54130	−0.120547	−0.0602736	+	0.998182i	$0.519197\pi$
$864$	−6.99201	−0.237873
$865$	2.67587	0.0909822
$866$	24.2049	0.822518
$867$	6.63790	0.225435
$868$	1.15555	0.0392219
$869$	−25.9204	−0.879290
$870$	21.9689	0.744817
$871$	−12.6481	−0.428566
$872$	9.76972	0.330845
$873$	−1.84186	−0.0623374
$874$	63.7238	2.15549
$875$	−1.06887	−0.0361343
$876$	46.8066	1.58145
$877$	−38.4786	−1.29933	−0.649666	−	0.760220i	$-0.725091\pi$
$877$	−38.4786	−1.29933	−0.649666	+	0.760220i	$0.725091\pi$
$878$	60.1535	2.03008
$879$	20.9945	0.708126
$880$	4.84634	0.163370
$881$	13.8343	0.466090	0.233045	−	0.972466i	$-0.425131\pi$
$881$	13.8343	0.466090	0.233045	+	0.972466i	$0.425131\pi$
$882$	15.5108	0.522276
$883$	16.7395	0.563329	0.281665	−	0.959513i	$-0.409113\pi$
$883$	16.7395	0.563329	0.281665	+	0.959513i	$0.409113\pi$
$884$	−21.7397	−0.731187
$885$	0.148184	0.00498116
$886$	−8.39435	−0.282014
$887$	26.1128	0.876782	0.438391	−	0.898784i	$-0.355548\pi$
$887$	26.1128	0.876782	0.438391	+	0.898784i	$0.355548\pi$
$888$	1.10482	0.0370752
$889$	1.31519	0.0441099
$890$	−23.1893	−0.777307
$891$	−2.17452	−0.0728491
$892$	−80.5645	−2.69750
$893$	−30.5053	−1.02082
$894$	17.8788	0.597956
$895$	19.1701	0.640785
$896$	1.29184	0.0431574
$897$	−17.8541	−0.596132
$898$	15.0044	0.500705
$899$	26.0486	0.868769
$900$	−6.15245	−0.205082
$901$	−27.1835	−0.905614
$902$	30.2024	1.00563
$903$	0.154157	0.00513003
$904$	−7.78817	−0.259031
$905$	−9.16566	−0.304677
$906$	−2.05517	−0.0682784
$907$	23.8517	0.791984	0.395992	−	0.918254i	$-0.370401\pi$
$907$	23.8517	0.791984	0.395992	+	0.918254i	$0.370401\pi$
$908$	33.7655	1.12055
$909$	−4.14891	−0.137611
$910$	0.771459	0.0255736
$911$	−27.4018	−0.907864	−0.453932	−	0.891036i	$-0.649979\pi$
$911$	−27.4018	−0.907864	−0.453932	+	0.891036i	$0.649979\pi$
$912$	4.87399	0.161394
$913$	−11.9462	−0.395361
$914$	5.63788	0.186484
$915$	−1.72793	−0.0571237
$916$	20.9407	0.691900
$917$	1.55524	0.0513587
$918$	7.14078	0.235681
$919$	26.1678	0.863197	0.431598	−	0.902066i	$-0.357950\pi$
$919$	26.1678	0.863197	0.431598	+	0.902066i	$0.357950\pi$
$920$	26.8334	0.884672
$921$	−4.15659	−0.136964
$922$	−80.4737	−2.65026
$923$	−2.18130	−0.0717984
$924$	0.561607	0.0184755
$925$	−1.13918	−0.0374559
$926$	3.00235	0.0986632
$927$	12.4339	0.408381
$928$	40.7068	1.33627
$929$	−50.5830	−1.65957	−0.829787	−	0.558081i	$-0.811538\pi$
$929$	−50.5830	−1.65957	−0.829787	+	0.558081i	$0.811538\pi$
$930$	16.8835	0.553633
$931$	−26.0117	−0.852499
$932$	−23.9582	−0.784778
$933$	−23.5559	−0.771187
$934$	−46.5572	−1.52340
$935$	−11.9072	−0.389407
$936$	4.72330	0.154386
$937$	19.8290	0.647784	0.323892	−	0.946094i	$-0.395009\pi$
$937$	19.8290	0.647784	0.323892	+	0.946094i	$0.395009\pi$
$938$	−1.07297	−0.0350337
$939$	27.2808	0.890275
$940$	−40.7434	−1.32890
$941$	32.7118	1.06637	0.533187	−	0.845997i	$-0.320994\pi$
$941$	32.7118	1.06637	0.533187	+	0.845997i	$0.320994\pi$
$942$	−22.5324	−0.734144
$943$	−48.3481	−1.57443
$944$	0.114132	0.00371469
$945$	−0.150410	−0.00489283
$946$	−8.40997	−0.273432
$947$	−5.90616	−0.191924	−0.0959622	−	0.995385i	$-0.530593\pi$
$947$	−5.90616	−0.191924	−0.0959622	+	0.995385i	$0.530593\pi$
$948$	−34.8173	−1.13081
$949$	37.0516	1.20275
$950$	17.3824	0.563960
$951$	−18.5224	−0.600630
$952$	−0.581445	−0.0188447
$953$	−24.9527	−0.808299	−0.404149	−	0.914693i	$-0.632432\pi$
$953$	−24.9527	−0.808299	−0.404149	+	0.914693i	$0.632432\pi$
$954$	18.7328	0.606498
$955$	27.1956	0.880030
$956$	19.0966	0.617629
$957$	12.6598	0.409234
$958$	−2.70487	−0.0873905
$959$	1.46204	0.0472116
$960$	21.9270	0.707690
$961$	−10.9812	−0.354232
$962$	2.77393	0.0894351
$963$	9.49262	0.305895
$964$	36.3628	1.17117
$965$	45.8190	1.47497
$966$	−1.51460	−0.0487316
$967$	16.3538	0.525904	0.262952	−	0.964809i	$-0.415304\pi$
$967$	16.3538	0.525904	0.262952	+	0.964809i	$0.415304\pi$
$968$	12.8115	0.411778
$969$	−11.9751	−0.384696
$970$	6.95024	0.223159
$971$	−13.3473	−0.428335	−0.214168	−	0.976797i	$-0.568704\pi$
$971$	−13.3473	−0.428335	−0.214168	+	0.976797i	$0.568704\pi$
$972$	−2.92089	−0.0936877
$973$	−0.925632	−0.0296744
$974$	32.3515	1.03661
$975$	−4.87021	−0.155971
$976$	−1.33086	−0.0425999
$977$	−51.4909	−1.64734	−0.823669	−	0.567070i	$-0.808077\pi$
$977$	−51.4909	−1.64734	−0.823669	+	0.567070i	$0.808077\pi$
$978$	−42.4061	−1.35600
$979$	−13.3631	−0.427086
$980$	−34.7417	−1.10978
$981$	−4.78246	−0.152692
$982$	−3.39426	−0.108315
$983$	−12.3627	−0.394308	−0.197154	−	0.980372i	$-0.563170\pi$
$983$	−12.3627	−0.394308	−0.197154	+	0.980372i	$0.563170\pi$
$984$	12.7905	0.407746
$985$	−4.35061	−0.138622
$986$	−41.5730	−1.32395
$987$	0.725059	0.0230789
$988$	−25.1239	−0.799297
$989$	13.4627	0.428090
$990$	8.20554	0.260789
$991$	4.24817	0.134948	0.0674738	−	0.997721i	$-0.478506\pi$
$991$	4.24817	0.134948	0.0674738	+	0.997721i	$0.478506\pi$
$992$	31.2839	0.993266
$993$	−13.7598	−0.436653
$994$	−0.185045	−0.00586926
$995$	−11.9145	−0.377714
$996$	−16.0466	−0.508455
$997$	17.8573	0.565546	0.282773	−	0.959187i	$-0.408746\pi$
$997$	17.8573	0.565546	0.282773	+	0.959187i	$0.408746\pi$
$998$	−12.5271	−0.396537
$999$	−0.540828	−0.0171110

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8049.2.a.c.1.19	✓	119

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8049.2.a.c.1.19	✓	119	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:	\( N \)	\(=\)	\( 8049 = 3 \cdot 2683 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8049.a (trivial)

\(f(q)\)	\(=\)	\(q-2.21831 q^{2} -1.00000 q^{3} +2.92089 q^{4} +1.70107 q^{5} +2.21831 q^{6} +0.0884207 q^{7} -2.04282 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.21831 q^{2} -1.00000 q^{3} +2.92089 q^{4} +1.70107 q^{5} +2.21831 q^{6} +0.0884207 q^{7} -2.04282 q^{8} +1.00000 q^{9} -3.77350 q^{10} -2.17452 q^{11} -2.92089 q^{12} -2.31214 q^{13} -0.196144 q^{14} -1.70107 q^{15} -1.31017 q^{16} +3.21902 q^{17} -2.21831 q^{18} +3.72011 q^{19} +4.96864 q^{20} -0.0884207 q^{21} +4.82375 q^{22} -7.72189 q^{23} +2.04282 q^{24} -2.10636 q^{25} +5.12905 q^{26} -1.00000 q^{27} +0.258267 q^{28} +5.82190 q^{29} +3.77350 q^{30} +4.47424 q^{31} +6.99201 q^{32} +2.17452 q^{33} -7.14078 q^{34} +0.150410 q^{35} +2.92089 q^{36} +0.540828 q^{37} -8.25236 q^{38} +2.31214 q^{39} -3.47499 q^{40} +6.26118 q^{41} +0.196144 q^{42} -1.74345 q^{43} -6.35153 q^{44} +1.70107 q^{45} +17.1295 q^{46} -8.20011 q^{47} +1.31017 q^{48} -6.99218 q^{49} +4.67255 q^{50} -3.21902 q^{51} -6.75352 q^{52} -8.44465 q^{53} +2.21831 q^{54} -3.69901 q^{55} -0.180628 q^{56} -3.72011 q^{57} -12.9148 q^{58} -0.0871124 q^{59} -4.96864 q^{60} +1.01579 q^{61} -9.92524 q^{62} +0.0884207 q^{63} -12.8901 q^{64} -3.93312 q^{65} -4.82375 q^{66} +5.47031 q^{67} +9.40241 q^{68} +7.72189 q^{69} -0.333655 q^{70} +0.943411 q^{71} -2.04282 q^{72} -16.0248 q^{73} -1.19972 q^{74} +2.10636 q^{75} +10.8660 q^{76} -0.192272 q^{77} -5.12905 q^{78} +11.9201 q^{79} -2.22870 q^{80} +1.00000 q^{81} -13.8892 q^{82} +5.49372 q^{83} -0.258267 q^{84} +5.47578 q^{85} +3.86751 q^{86} -5.82190 q^{87} +4.44215 q^{88} +6.14531 q^{89} -3.77350 q^{90} -0.204441 q^{91} -22.5548 q^{92} -4.47424 q^{93} +18.1904 q^{94} +6.32817 q^{95} -6.99201 q^{96} -1.84186 q^{97} +15.5108 q^{98} -2.17452 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 119 q + 11 q^{2} - 119 q^{3} + 137 q^{4} + 17 q^{5} - 11 q^{6} + 10 q^{7} + 33 q^{8} + 119 q^{9}+O(q^{10}) \) 119 * q + 11 * q^2 - 119 * q^3 + 137 * q^4 + 17 * q^5 - 11 * q^6 + 10 * q^7 + 33 * q^8 + 119 * q^9	\( 119 q + 11 q^{2} - 119 q^{3} + 137 q^{4} + 17 q^{5} - 11 q^{6} + 10 q^{7} + 33 q^{8} + 119 q^{9} - 10 q^{10} + 56 q^{11} - 137 q^{12} - 37 q^{13} + 31 q^{14} - 17 q^{15} + 173 q^{16} + 17 q^{17} + 11 q^{18} + 16 q^{19} + 61 q^{20} - 10 q^{21} - 3 q^{22} + 76 q^{23} - 33 q^{24} + 134 q^{25} + 47 q^{26} - 119 q^{27} - q^{28} + 47 q^{29} + 10 q^{30} + 51 q^{31} + 87 q^{32} - 56 q^{33} + 13 q^{34} + 58 q^{35} + 137 q^{36} - 67 q^{37} + 35 q^{38} + 37 q^{39} - 40 q^{40} + 47 q^{41} - 31 q^{42} + 12 q^{43} + 148 q^{44} + 17 q^{45} + 26 q^{46} + 107 q^{47} - 173 q^{48} + 163 q^{49} + 76 q^{50} - 17 q^{51} - 57 q^{52} + 64 q^{53} - 11 q^{54} + 71 q^{55} + 91 q^{56} - 16 q^{57} + 12 q^{58} + 98 q^{59} - 61 q^{60} - 50 q^{61} + 40 q^{62} + 10 q^{63} + 245 q^{64} + 40 q^{65} + 3 q^{66} + 12 q^{67} + 75 q^{68} - 76 q^{69} - 9 q^{70} + 194 q^{71} + 33 q^{72} - 79 q^{73} + 72 q^{74} - 134 q^{75} + 12 q^{76} + 71 q^{77} - 47 q^{78} + 127 q^{79} + 148 q^{80} + 119 q^{81} - 54 q^{82} + 77 q^{83} + q^{84} - 25 q^{85} + 142 q^{86} - 47 q^{87} + q^{88} + 93 q^{89} - 10 q^{90} + 61 q^{91} + 156 q^{92} - 51 q^{93} + 16 q^{94} + 138 q^{95} - 87 q^{96} - 110 q^{97} + 96 q^{98} + 56 q^{99}+O(q^{100}) \) 119 * q + 11 * q^2 - 119 * q^3 + 137 * q^4 + 17 * q^5 - 11 * q^6 + 10 * q^7 + 33 * q^8 + 119 * q^9 - 10 * q^10 + 56 * q^11 - 137 * q^12 - 37 * q^13 + 31 * q^14 - 17 * q^15 + 173 * q^16 + 17 * q^17 + 11 * q^18 + 16 * q^19 + 61 * q^20 - 10 * q^21 - 3 * q^22 + 76 * q^23 - 33 * q^24 + 134 * q^25 + 47 * q^26 - 119 * q^27 - q^28 + 47 * q^29 + 10 * q^30 + 51 * q^31 + 87 * q^32 - 56 * q^33 + 13 * q^34 + 58 * q^35 + 137 * q^36 - 67 * q^37 + 35 * q^38 + 37 * q^39 - 40 * q^40 + 47 * q^41 - 31 * q^42 + 12 * q^43 + 148 * q^44 + 17 * q^45 + 26 * q^46 + 107 * q^47 - 173 * q^48 + 163 * q^49 + 76 * q^50 - 17 * q^51 - 57 * q^52 + 64 * q^53 - 11 * q^54 + 71 * q^55 + 91 * q^56 - 16 * q^57 + 12 * q^58 + 98 * q^59 - 61 * q^60 - 50 * q^61 + 40 * q^62 + 10 * q^63 + 245 * q^64 + 40 * q^65 + 3 * q^66 + 12 * q^67 + 75 * q^68 - 76 * q^69 - 9 * q^70 + 194 * q^71 + 33 * q^72 - 79 * q^73 + 72 * q^74 - 134 * q^75 + 12 * q^76 + 71 * q^77 - 47 * q^78 + 127 * q^79 + 148 * q^80 + 119 * q^81 - 54 * q^82 + 77 * q^83 + q^84 - 25 * q^85 + 142 * q^86 - 47 * q^87 + q^88 + 93 * q^89 - 10 * q^90 + 61 * q^91 + 156 * q^92 - 51 * q^93 + 16 * q^94 + 138 * q^95 - 87 * q^96 - 110 * q^97 + 96 * q^98 + 56 * q^99

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