LMFDB - Embedded newform 8042.2.a.a.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.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:	$64.2156933055$
Analytic rank:	$1$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.67548 q^{3} +1.00000 q^{4} -3.27092 q^{5} -1.67548 q^{6} +1.23025 q^{7} +1.00000 q^{8} -0.192762 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.67548 q^{3} +1.00000 q^{4} -3.27092 q^{5} -1.67548 q^{6} +1.23025 q^{7} +1.00000 q^{8} -0.192762 q^{9} -3.27092 q^{10} +1.03060 q^{11} -1.67548 q^{12} -5.77159 q^{13} +1.23025 q^{14} +5.48037 q^{15} +1.00000 q^{16} -1.01519 q^{17} -0.192762 q^{18} +5.51119 q^{19} -3.27092 q^{20} -2.06125 q^{21} +1.03060 q^{22} -2.52383 q^{23} -1.67548 q^{24} +5.69895 q^{25} -5.77159 q^{26} +5.34941 q^{27} +1.23025 q^{28} +5.31836 q^{29} +5.48037 q^{30} +1.59392 q^{31} +1.00000 q^{32} -1.72674 q^{33} -1.01519 q^{34} -4.02404 q^{35} -0.192762 q^{36} -4.35474 q^{37} +5.51119 q^{38} +9.67019 q^{39} -3.27092 q^{40} +9.80940 q^{41} -2.06125 q^{42} +4.54208 q^{43} +1.03060 q^{44} +0.630511 q^{45} -2.52383 q^{46} -3.35414 q^{47} -1.67548 q^{48} -5.48649 q^{49} +5.69895 q^{50} +1.70093 q^{51} -5.77159 q^{52} +1.72285 q^{53} +5.34941 q^{54} -3.37100 q^{55} +1.23025 q^{56} -9.23390 q^{57} +5.31836 q^{58} +14.0138 q^{59} +5.48037 q^{60} -2.49991 q^{61} +1.59392 q^{62} -0.237145 q^{63} +1.00000 q^{64} +18.8784 q^{65} -1.72674 q^{66} -3.50881 q^{67} -1.01519 q^{68} +4.22863 q^{69} -4.02404 q^{70} -8.10150 q^{71} -0.192762 q^{72} +14.1753 q^{73} -4.35474 q^{74} -9.54848 q^{75} +5.51119 q^{76} +1.26789 q^{77} +9.67019 q^{78} -15.9550 q^{79} -3.27092 q^{80} -8.38456 q^{81} +9.80940 q^{82} -17.4681 q^{83} -2.06125 q^{84} +3.32061 q^{85} +4.54208 q^{86} -8.91082 q^{87} +1.03060 q^{88} +4.46238 q^{89} +0.630511 q^{90} -7.10047 q^{91} -2.52383 q^{92} -2.67058 q^{93} -3.35414 q^{94} -18.0267 q^{95} -1.67548 q^{96} -5.30079 q^{97} -5.48649 q^{98} -0.198660 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.67548	−0.967340	−0.483670	−	0.875251i	$-0.660696\pi$
$3$	−1.67548	−0.967340	−0.483670	+	0.875251i	$0.660696\pi$
$4$	1.00000	0.500000
$5$	−3.27092	−1.46280	−0.731401	−	0.681948i	$-0.761133\pi$
$5$	−3.27092	−1.46280	−0.731401	+	0.681948i	$0.761133\pi$
$6$	−1.67548	−0.684012
$7$	1.23025	0.464989	0.232495	−	0.972598i	$-0.425311\pi$
$7$	1.23025	0.464989	0.232495	+	0.972598i	$0.425311\pi$
$8$	1.00000	0.353553
$9$	−0.192762	−0.0642541
$10$	−3.27092	−1.03436
$11$	1.03060	0.310736	0.155368	−	0.987857i	$-0.450344\pi$
$11$	1.03060	0.310736	0.155368	+	0.987857i	$0.450344\pi$
$12$	−1.67548	−0.483670
$13$	−5.77159	−1.60075	−0.800375	−	0.599499i	$-0.795366\pi$
$13$	−5.77159	−1.60075	−0.800375	+	0.599499i	$0.795366\pi$
$14$	1.23025	0.328797
$15$	5.48037	1.41503
$16$	1.00000	0.250000
$17$	−1.01519	−0.246220	−0.123110	−	0.992393i	$-0.539287\pi$
$17$	−1.01519	−0.246220	−0.123110	+	0.992393i	$0.539287\pi$
$18$	−0.192762	−0.0454345
$19$	5.51119	1.26435	0.632177	−	0.774824i	$-0.282162\pi$
$19$	5.51119	1.26435	0.632177	+	0.774824i	$0.282162\pi$
$20$	−3.27092	−0.731401
$21$	−2.06125	−0.449803
$22$	1.03060	0.219724
$23$	−2.52383	−0.526255	−0.263128	−	0.964761i	$-0.584754\pi$
$23$	−2.52383	−0.526255	−0.263128	+	0.964761i	$0.584754\pi$
$24$	−1.67548	−0.342006
$25$	5.69895	1.13979
$26$	−5.77159	−1.13190
$27$	5.34941	1.02950
$28$	1.23025	0.232495
$29$	5.31836	0.987595	0.493798	−	0.869577i	$-0.335608\pi$
$29$	5.31836	0.987595	0.493798	+	0.869577i	$0.335608\pi$
$30$	5.48037	1.00057
$31$	1.59392	0.286276	0.143138	−	0.989703i	$-0.454281\pi$
$31$	1.59392	0.286276	0.143138	+	0.989703i	$0.454281\pi$
$32$	1.00000	0.176777
$33$	−1.72674	−0.300588
$34$	−1.01519	−0.174104
$35$	−4.02404	−0.680187
$36$	−0.192762	−0.0321271
$37$	−4.35474	−0.715914	−0.357957	−	0.933738i	$-0.616527\pi$
$37$	−4.35474	−0.715914	−0.357957	+	0.933738i	$0.616527\pi$
$38$	5.51119	0.894033
$39$	9.67019	1.54847
$40$	−3.27092	−0.517179
$41$	9.80940	1.53197	0.765986	−	0.642858i	$-0.222251\pi$
$41$	9.80940	1.53197	0.765986	+	0.642858i	$0.222251\pi$
$42$	−2.06125	−0.318058
$43$	4.54208	0.692661	0.346331	−	0.938113i	$-0.387428\pi$
$43$	4.54208	0.692661	0.346331	+	0.938113i	$0.387428\pi$
$44$	1.03060	0.155368
$45$	0.630511	0.0939911
$46$	−2.52383	−0.372119
$47$	−3.35414	−0.489252	−0.244626	−	0.969617i	$-0.578665\pi$
$47$	−3.35414	−0.489252	−0.244626	+	0.969617i	$0.578665\pi$
$48$	−1.67548	−0.241835
$49$	−5.48649	−0.783785
$50$	5.69895	0.805953
$51$	1.70093	0.238178
$52$	−5.77159	−0.800375
$53$	1.72285	0.236652	0.118326	−	0.992975i	$-0.462247\pi$
$53$	1.72285	0.236652	0.118326	+	0.992975i	$0.462247\pi$
$54$	5.34941	0.727963
$55$	−3.37100	−0.454546
$56$	1.23025	0.164399
$57$	−9.23390	−1.22306
$58$	5.31836	0.698335
$59$	14.0138	1.82444	0.912220	−	0.409700i	$-0.134367\pi$
$59$	14.0138	1.82444	0.912220	+	0.409700i	$0.134367\pi$
$60$	5.48037	0.707513
$61$	−2.49991	−0.320081	−0.160040	−	0.987110i	$-0.551162\pi$
$61$	−2.49991	−0.320081	−0.160040	+	0.987110i	$0.551162\pi$
$62$	1.59392	0.202428
$63$	−0.237145	−0.0298775
$64$	1.00000	0.125000
$65$	18.8784	2.34158
$66$	−1.72674	−0.212548
$67$	−3.50881	−0.428669	−0.214335	−	0.976760i	$-0.568758\pi$
$67$	−3.50881	−0.428669	−0.214335	+	0.976760i	$0.568758\pi$
$68$	−1.01519	−0.123110
$69$	4.22863	0.509067
$70$	−4.02404	−0.480965
$71$	−8.10150	−0.961472	−0.480736	−	0.876865i	$-0.659630\pi$
$71$	−8.10150	−0.961472	−0.480736	+	0.876865i	$0.659630\pi$
$72$	−0.192762	−0.0227173
$73$	14.1753	1.65909	0.829545	−	0.558440i	$-0.188600\pi$
$73$	14.1753	1.65909	0.829545	+	0.558440i	$0.188600\pi$
$74$	−4.35474	−0.506228
$75$	−9.54848	−1.10256
$76$	5.51119	0.632177
$77$	1.26789	0.144489
$78$	9.67019	1.09493
$79$	−15.9550	−1.79508	−0.897538	−	0.440938i	$-0.854646\pi$
$79$	−15.9550	−1.79508	−0.897538	+	0.440938i	$0.854646\pi$
$80$	−3.27092	−0.365700
$81$	−8.38456	−0.931617
$82$	9.80940	1.08327
$83$	−17.4681	−1.91738	−0.958689	−	0.284457i	$-0.908187\pi$
$83$	−17.4681	−1.91738	−0.958689	+	0.284457i	$0.908187\pi$
$84$	−2.06125	−0.224901
$85$	3.32061	0.360171
$86$	4.54208	0.489785
$87$	−8.91082	−0.955340
$88$	1.03060	0.109862
$89$	4.46238	0.473011	0.236505	−	0.971630i	$-0.423998\pi$
$89$	4.46238	0.473011	0.236505	+	0.971630i	$0.423998\pi$
$90$	0.630511	0.0664617
$91$	−7.10047	−0.744332
$92$	−2.52383	−0.263128
$93$	−2.67058	−0.276926
$94$	−3.35414	−0.345954
$95$	−18.0267	−1.84950
$96$	−1.67548	−0.171003
$97$	−5.30079	−0.538213	−0.269107	−	0.963110i	$-0.586728\pi$
$97$	−5.30079	−0.538213	−0.269107	+	0.963110i	$0.586728\pi$
$98$	−5.48649	−0.554220
$99$	−0.198660	−0.0199661
$100$	5.69895	0.569895
$101$	4.01787	0.399793	0.199897	−	0.979817i	$-0.435939\pi$
$101$	4.01787	0.399793	0.199897	+	0.979817i	$0.435939\pi$
$102$	1.70093	0.168417
$103$	4.67937	0.461072	0.230536	−	0.973064i	$-0.425952\pi$
$103$	4.67937	0.461072	0.230536	+	0.973064i	$0.425952\pi$
$104$	−5.77159	−0.565951
$105$	6.74221	0.657972
$106$	1.72285	0.167338
$107$	−14.0413	−1.35742	−0.678712	−	0.734404i	$-0.737462\pi$
$107$	−14.0413	−1.35742	−0.678712	+	0.734404i	$0.737462\pi$
$108$	5.34941	0.514748
$109$	8.69881	0.833195	0.416598	−	0.909091i	$-0.363222\pi$
$109$	8.69881	0.833195	0.416598	+	0.909091i	$0.363222\pi$
$110$	−3.37100	−0.321412
$111$	7.29628	0.692532
$112$	1.23025	0.116247
$113$	5.08862	0.478697	0.239348	−	0.970934i	$-0.423066\pi$
$113$	5.08862	0.478697	0.239348	+	0.970934i	$0.423066\pi$
$114$	−9.23390	−0.864834
$115$	8.25526	0.769807
$116$	5.31836	0.493798
$117$	1.11255	0.102855
$118$	14.0138	1.29007
$119$	−1.24893	−0.114490
$120$	5.48037	0.500287
$121$	−9.93787	−0.903443
$122$	−2.49991	−0.226331
$123$	−16.4355	−1.48194
$124$	1.59392	0.143138
$125$	−2.28620	−0.204484
$126$	−0.237145	−0.0211266
$127$	7.81182	0.693187	0.346593	−	0.938016i	$-0.387338\pi$
$127$	7.81182	0.693187	0.346593	+	0.938016i	$0.387338\pi$
$128$	1.00000	0.0883883
$129$	−7.61018	−0.670039
$130$	18.8784	1.65575
$131$	5.85723	0.511748	0.255874	−	0.966710i	$-0.417637\pi$
$131$	5.85723	0.511748	0.255874	+	0.966710i	$0.417637\pi$
$132$	−1.72674	−0.150294
$133$	6.78012	0.587911
$134$	−3.50881	−0.303115
$135$	−17.4975	−1.50595
$136$	−1.01519	−0.0870519
$137$	−4.58587	−0.391798	−0.195899	−	0.980624i	$-0.562762\pi$
$137$	−4.58587	−0.391798	−0.195899	+	0.980624i	$0.562762\pi$
$138$	4.22863	0.359965
$139$	−9.70172	−0.822890	−0.411445	−	0.911435i	$-0.634976\pi$
$139$	−9.70172	−0.822890	−0.411445	+	0.911435i	$0.634976\pi$
$140$	−4.02404	−0.340094
$141$	5.61981	0.473273
$142$	−8.10150	−0.679863
$143$	−5.94817	−0.497411
$144$	−0.192762	−0.0160635
$145$	−17.3960	−1.44466
$146$	14.1753	1.17315
$147$	9.19252	0.758186
$148$	−4.35474	−0.357957
$149$	−6.58321	−0.539318	−0.269659	−	0.962956i	$-0.586911\pi$
$149$	−6.58321	−0.539318	−0.269659	+	0.962956i	$0.586911\pi$
$150$	−9.54848	−0.779630
$151$	21.8645	1.77931	0.889653	−	0.456636i	$-0.150946\pi$
$151$	21.8645	1.77931	0.889653	+	0.456636i	$0.150946\pi$
$152$	5.51119	0.447017
$153$	0.195691	0.0158207
$154$	1.26789	0.102169
$155$	−5.21358	−0.418765
$156$	9.67019	0.774235
$157$	−2.72442	−0.217432	−0.108716	−	0.994073i	$-0.534674\pi$
$157$	−2.72442	−0.217432	−0.108716	+	0.994073i	$0.534674\pi$
$158$	−15.9550	−1.26931
$159$	−2.88660	−0.228923
$160$	−3.27092	−0.258589
$161$	−3.10493	−0.244703
$162$	−8.38456	−0.658753
$163$	−1.91800	−0.150229	−0.0751147	−	0.997175i	$-0.523932\pi$
$163$	−1.91800	−0.150229	−0.0751147	+	0.997175i	$0.523932\pi$
$164$	9.80940	0.765986
$165$	5.64805	0.439700
$166$	−17.4681	−1.35579
$167$	−12.2075	−0.944644	−0.472322	−	0.881426i	$-0.656584\pi$
$167$	−12.2075	−0.944644	−0.472322	+	0.881426i	$0.656584\pi$
$168$	−2.06125	−0.159029
$169$	20.3112	1.56240
$170$	3.32061	0.254679
$171$	−1.06235	−0.0812400
$172$	4.54208	0.346331
$173$	5.60069	0.425812	0.212906	−	0.977073i	$-0.431707\pi$
$173$	5.60069	0.425812	0.212906	+	0.977073i	$0.431707\pi$
$174$	−8.91082	−0.675528
$175$	7.01111	0.529990
$176$	1.03060	0.0776841
$177$	−23.4798	−1.76485
$178$	4.46238	0.334469
$179$	−18.0700	−1.35062	−0.675309	−	0.737535i	$-0.735990\pi$
$179$	−18.0700	−1.35062	−0.675309	+	0.737535i	$0.735990\pi$
$180$	0.630511	0.0469955
$181$	8.65449	0.643284	0.321642	−	0.946861i	$-0.395765\pi$
$181$	8.65449	0.643284	0.321642	+	0.946861i	$0.395765\pi$
$182$	−7.10047	−0.526322
$183$	4.18855	0.309627
$184$	−2.52383	−0.186059
$185$	14.2440	1.04724
$186$	−2.67058	−0.195816
$187$	−1.04625	−0.0765095
$188$	−3.35414	−0.244626
$189$	6.58110	0.478704
$190$	−18.0267	−1.30779
$191$	−18.3055	−1.32454	−0.662269	−	0.749266i	$-0.730407\pi$
$191$	−18.3055	−1.32454	−0.662269	+	0.749266i	$0.730407\pi$
$192$	−1.67548	−0.120917
$193$	5.78081	0.416112	0.208056	−	0.978117i	$-0.433286\pi$
$193$	5.78081	0.416112	0.208056	+	0.978117i	$0.433286\pi$
$194$	−5.30079	−0.380574
$195$	−31.6305	−2.26510
$196$	−5.48649	−0.391892
$197$	0.432448	0.0308106	0.0154053	−	0.999881i	$-0.495096\pi$
$197$	0.432448	0.0308106	0.0154053	+	0.999881i	$0.495096\pi$
$198$	−0.198660	−0.0141182
$199$	−6.98443	−0.495113	−0.247557	−	0.968873i	$-0.579628\pi$
$199$	−6.98443	−0.495113	−0.247557	+	0.968873i	$0.579628\pi$
$200$	5.69895	0.402976
$201$	5.87894	0.414669
$202$	4.01787	0.282696
$203$	6.54290	0.459221
$204$	1.70093	0.119089
$205$	−32.0858	−2.24097
$206$	4.67937	0.326027
$207$	0.486500	0.0338141
$208$	−5.77159	−0.400188
$209$	5.67981	0.392881
$210$	6.74221	0.465256
$211$	−6.16999	−0.424759	−0.212380	−	0.977187i	$-0.568121\pi$
$211$	−6.16999	−0.424759	−0.212380	+	0.977187i	$0.568121\pi$
$212$	1.72285	0.118326
$213$	13.5739	0.930070
$214$	−14.0413	−0.959844
$215$	−14.8568	−1.01323
$216$	5.34941	0.363982
$217$	1.96091	0.133115
$218$	8.69881	0.589158
$219$	−23.7504	−1.60490
$220$	−3.37100	−0.227273
$221$	5.85926	0.394137
$222$	7.29628	0.489694
$223$	−21.9244	−1.46817	−0.734083	−	0.679059i	$-0.762388\pi$
$223$	−21.9244	−1.46817	−0.734083	+	0.679059i	$0.762388\pi$
$224$	1.23025	0.0821993
$225$	−1.09854	−0.0732362
$226$	5.08862	0.338490
$227$	11.0850	0.735739	0.367870	−	0.929877i	$-0.380087\pi$
$227$	11.0850	0.735739	0.367870	+	0.929877i	$0.380087\pi$
$228$	−9.23390	−0.611530
$229$	−10.7911	−0.713097	−0.356549	−	0.934277i	$-0.616047\pi$
$229$	−10.7911	−0.713097	−0.356549	+	0.934277i	$0.616047\pi$
$230$	8.25526	0.544336
$231$	−2.12432	−0.139770
$232$	5.31836	0.349168
$233$	−19.7448	−1.29353	−0.646763	−	0.762691i	$-0.723878\pi$
$233$	−19.7448	−1.29353	−0.646763	+	0.762691i	$0.723878\pi$
$234$	1.11255	0.0727294
$235$	10.9712	0.715679
$236$	14.0138	0.912220
$237$	26.7323	1.73645
$238$	−1.24893	−0.0809564
$239$	−10.7899	−0.697940	−0.348970	−	0.937134i	$-0.613469\pi$
$239$	−10.7899	−0.697940	−0.348970	+	0.937134i	$0.613469\pi$
$240$	5.48037	0.353757
$241$	−23.0471	−1.48459	−0.742296	−	0.670072i	$-0.766263\pi$
$241$	−23.0471	−1.48459	−0.742296	+	0.670072i	$0.766263\pi$
$242$	−9.93787	−0.638831
$243$	−2.00008	−0.128305
$244$	−2.49991	−0.160040
$245$	17.9459	1.14652
$246$	−16.4355	−1.04789
$247$	−31.8083	−2.02392
$248$	1.59392	0.101214
$249$	29.2675	1.85476
$250$	−2.28620	−0.144592
$251$	20.7041	1.30683	0.653417	−	0.756999i	$-0.273335\pi$
$251$	20.7041	1.30683	0.653417	+	0.756999i	$0.273335\pi$
$252$	−0.237145	−0.0149387
$253$	−2.60105	−0.163527
$254$	7.81182	0.490157
$255$	−5.56362	−0.348408
$256$	1.00000	0.0625000
$257$	−9.13962	−0.570114	−0.285057	−	0.958511i	$-0.592013\pi$
$257$	−9.13962	−0.570114	−0.285057	+	0.958511i	$0.592013\pi$
$258$	−7.61018	−0.473789
$259$	−5.35740	−0.332892
$260$	18.8784	1.17079
$261$	−1.02518	−0.0634571
$262$	5.85723	0.361861
$263$	−7.00486	−0.431938	−0.215969	−	0.976400i	$-0.569291\pi$
$263$	−7.00486	−0.431938	−0.215969	+	0.976400i	$0.569291\pi$
$264$	−1.72674	−0.106274
$265$	−5.63532	−0.346175
$266$	6.78012	0.415716
$267$	−7.47663	−0.457562
$268$	−3.50881	−0.214335
$269$	25.2611	1.54020	0.770098	−	0.637925i	$-0.220207\pi$
$269$	25.2611	1.54020	0.770098	+	0.637925i	$0.220207\pi$
$270$	−17.4975	−1.06487
$271$	17.9345	1.08944	0.544722	−	0.838616i	$-0.316635\pi$
$271$	17.9345	1.08944	0.544722	+	0.838616i	$0.316635\pi$
$272$	−1.01519	−0.0615550
$273$	11.8967	0.720022
$274$	−4.58587	−0.277043
$275$	5.87331	0.354174
$276$	4.22863	0.254534
$277$	−29.9909	−1.80198	−0.900991	−	0.433839i	$-0.857159\pi$
$277$	−29.9909	−1.80198	−0.900991	+	0.433839i	$0.857159\pi$
$278$	−9.70172	−0.581871
$279$	−0.307247	−0.0183944
$280$	−4.02404	−0.240482
$281$	16.9433	1.01075	0.505376	−	0.862899i	$-0.331354\pi$
$281$	16.9433	1.01075	0.505376	+	0.862899i	$0.331354\pi$
$282$	5.61981	0.334655
$283$	15.5126	0.922131	0.461065	−	0.887366i	$-0.347467\pi$
$283$	15.5126	0.922131	0.461065	+	0.887366i	$0.347467\pi$
$284$	−8.10150	−0.480736
$285$	30.2034	1.78909
$286$	−5.94817	−0.351723
$287$	12.0680	0.712350
$288$	−0.192762	−0.0113586
$289$	−15.9694	−0.939376
$290$	−17.3960	−1.02153
$291$	8.88137	0.520635
$292$	14.1753	0.829545
$293$	−14.3968	−0.841073	−0.420536	−	0.907276i	$-0.638158\pi$
$293$	−14.3968	−0.841073	−0.420536	+	0.907276i	$0.638158\pi$
$294$	9.19252	0.536119
$295$	−45.8380	−2.66879
$296$	−4.35474	−0.253114
$297$	5.51308	0.319902
$298$	−6.58321	−0.381355
$299$	14.5665	0.842403
$300$	−9.54848	−0.551282
$301$	5.58788	0.322080
$302$	21.8645	1.25816
$303$	−6.73187	−0.386736
$304$	5.51119	0.316089
$305$	8.17702	0.468215
$306$	0.195691	0.0111869
$307$	1.51151	0.0862665	0.0431332	−	0.999069i	$-0.486266\pi$
$307$	1.51151	0.0862665	0.0431332	+	0.999069i	$0.486266\pi$
$308$	1.26789	0.0722445
$309$	−7.84019	−0.446013
$310$	−5.21358	−0.296112
$311$	−12.4486	−0.705896	−0.352948	−	0.935643i	$-0.614821\pi$
$311$	−12.4486	−0.705896	−0.352948	+	0.935643i	$0.614821\pi$
$312$	9.67019	0.547467
$313$	4.24020	0.239670	0.119835	−	0.992794i	$-0.461763\pi$
$313$	4.24020	0.239670	0.119835	+	0.992794i	$0.461763\pi$
$314$	−2.72442	−0.153748
$315$	0.775684	0.0437048
$316$	−15.9550	−0.897538
$317$	−25.0957	−1.40952	−0.704758	−	0.709448i	$-0.748944\pi$
$317$	−25.0957	−1.40952	−0.704758	+	0.709448i	$0.748944\pi$
$318$	−2.88660	−0.161873
$319$	5.48108	0.306882
$320$	−3.27092	−0.182850
$321$	23.5260	1.31309
$322$	−3.10493	−0.173031
$323$	−5.59491	−0.311309
$324$	−8.38456	−0.465809
$325$	−32.8920	−1.82452
$326$	−1.91800	−0.106228
$327$	−14.5747	−0.805983
$328$	9.80940	0.541634
$329$	−4.12642	−0.227497
$330$	5.64805	0.310915
$331$	34.4643	1.89433	0.947165	−	0.320748i	$-0.103934\pi$
$331$	34.4643	1.89433	0.947165	+	0.320748i	$0.103934\pi$
$332$	−17.4681	−0.958689
$333$	0.839429	0.0460004
$334$	−12.2075	−0.667964
$335$	11.4770	0.627058
$336$	−2.06125	−0.112451
$337$	−31.3283	−1.70656	−0.853279	−	0.521454i	$-0.825390\pi$
$337$	−31.3283	−1.70656	−0.853279	+	0.521454i	$0.825390\pi$
$338$	20.3112	1.10479
$339$	−8.52588	−0.463062
$340$	3.32061	0.180085
$341$	1.64268	0.0889564
$342$	−1.06235	−0.0574453
$343$	−15.3615	−0.829441
$344$	4.54208	0.244893
$345$	−13.8315	−0.744665
$346$	5.60069	0.301095
$347$	−25.0147	−1.34286	−0.671429	−	0.741069i	$-0.734319\pi$
$347$	−25.0147	−1.34286	−0.671429	+	0.741069i	$0.734319\pi$
$348$	−8.91082	−0.477670
$349$	−10.6112	−0.568004	−0.284002	−	0.958824i	$-0.591662\pi$
$349$	−10.6112	−0.568004	−0.284002	+	0.958824i	$0.591662\pi$
$350$	7.01111	0.374759
$351$	−30.8746	−1.64796
$352$	1.03060	0.0549309
$353$	26.2076	1.39489	0.697446	−	0.716637i	$-0.254320\pi$
$353$	26.2076	1.39489	0.697446	+	0.716637i	$0.254320\pi$
$354$	−23.4798	−1.24794
$355$	26.4994	1.40644
$356$	4.46238	0.236505
$357$	2.09257	0.110750
$358$	−18.0700	−0.955031
$359$	−14.9153	−0.787199	−0.393600	−	0.919282i	$-0.628770\pi$
$359$	−14.9153	−0.787199	−0.393600	+	0.919282i	$0.628770\pi$
$360$	0.630511	0.0332309
$361$	11.3732	0.598591
$362$	8.65449	0.454870
$363$	16.6507	0.873936
$364$	−7.10047	−0.372166
$365$	−46.3662	−2.42692
$366$	4.18855	0.218939
$367$	−29.2307	−1.52583	−0.762914	−	0.646500i	$-0.776232\pi$
$367$	−29.2307	−1.52583	−0.762914	+	0.646500i	$0.776232\pi$
$368$	−2.52383	−0.131564
$369$	−1.89088	−0.0984355
$370$	14.2440	0.740511
$371$	2.11953	0.110041
$372$	−2.67058	−0.138463
$373$	−26.7645	−1.38582	−0.692908	−	0.721026i	$-0.743671\pi$
$373$	−26.7645	−1.38582	−0.692908	+	0.721026i	$0.743671\pi$
$374$	−1.04625	−0.0541004
$375$	3.83048	0.197805
$376$	−3.35414	−0.172977
$377$	−30.6954	−1.58089
$378$	6.58110	0.338495
$379$	0.719383	0.0369522	0.0184761	−	0.999829i	$-0.494119\pi$
$379$	0.719383	0.0369522	0.0184761	+	0.999829i	$0.494119\pi$
$380$	−18.0267	−0.924750
$381$	−13.0886	−0.670547
$382$	−18.3055	−0.936590
$383$	35.7180	1.82510	0.912552	−	0.408961i	$-0.134109\pi$
$383$	35.7180	1.82510	0.912552	+	0.408961i	$0.134109\pi$
$384$	−1.67548	−0.0855015
$385$	−4.14716	−0.211359
$386$	5.78081	0.294235
$387$	−0.875543	−0.0445064
$388$	−5.30079	−0.269107
$389$	−4.52778	−0.229568	−0.114784	−	0.993390i	$-0.536618\pi$
$389$	−4.52778	−0.229568	−0.114784	+	0.993390i	$0.536618\pi$
$390$	−31.6305	−1.60167
$391$	2.56217	0.129575
$392$	−5.48649	−0.277110
$393$	−9.81368	−0.495035
$394$	0.432448	0.0217864
$395$	52.1875	2.62584
$396$	−0.198660	−0.00998305
$397$	7.07292	0.354980	0.177490	−	0.984123i	$-0.443202\pi$
$397$	7.07292	0.354980	0.177490	+	0.984123i	$0.443202\pi$
$398$	−6.98443	−0.350098
$399$	−11.3600	−0.568710
$400$	5.69895	0.284947
$401$	11.4434	0.571457	0.285728	−	0.958311i	$-0.407765\pi$
$401$	11.4434	0.571457	0.285728	+	0.958311i	$0.407765\pi$
$402$	5.87894	0.293215
$403$	−9.19943	−0.458256
$404$	4.01787	0.199897
$405$	27.4252	1.36277
$406$	6.54290	0.324719
$407$	−4.48797	−0.222461
$408$	1.70093	0.0842087
$409$	−23.4154	−1.15782	−0.578909	−	0.815392i	$-0.696521\pi$
$409$	−23.4154	−1.15782	−0.578909	+	0.815392i	$0.696521\pi$
$410$	−32.0858	−1.58461
$411$	7.68355	0.379001
$412$	4.67937	0.230536
$413$	17.2404	0.848345
$414$	0.486500	0.0239102
$415$	57.1370	2.80474
$416$	−5.77159	−0.282975
$417$	16.2551	0.796014
$418$	5.67981	0.277809
$419$	35.3483	1.72688	0.863438	−	0.504455i	$-0.168307\pi$
$419$	35.3483	1.72688	0.863438	+	0.504455i	$0.168307\pi$
$420$	6.74221	0.328986
$421$	−39.0439	−1.90288	−0.951442	−	0.307827i	$-0.900398\pi$
$421$	−39.0439	−1.90288	−0.951442	+	0.307827i	$0.900398\pi$
$422$	−6.16999	−0.300350
$423$	0.646553	0.0314365
$424$	1.72285	0.0836690
$425$	−5.78552	−0.280639
$426$	13.5739	0.657658
$427$	−3.07551	−0.148834
$428$	−14.0413	−0.678712
$429$	9.96606	0.481166
$430$	−14.8568	−0.716459
$431$	11.8228	0.569482	0.284741	−	0.958604i	$-0.408092\pi$
$431$	11.8228	0.569482	0.284741	+	0.958604i	$0.408092\pi$
$432$	5.34941	0.257374
$433$	−14.0008	−0.672834	−0.336417	−	0.941713i	$-0.609215\pi$
$433$	−14.0008	−0.672834	−0.336417	+	0.941713i	$0.609215\pi$
$434$	1.96091	0.0941267
$435$	29.1466	1.39747
$436$	8.69881	0.416598
$437$	−13.9093	−0.665373
$438$	−23.7504	−1.13484
$439$	−36.1255	−1.72418	−0.862089	−	0.506757i	$-0.830844\pi$
$439$	−36.1255	−1.72418	−0.862089	+	0.506757i	$0.830844\pi$
$440$	−3.37100	−0.160706
$441$	1.05759	0.0503614
$442$	5.85926	0.278697
$443$	−20.5890	−0.978213	−0.489107	−	0.872224i	$-0.662677\pi$
$443$	−20.5890	−0.978213	−0.489107	+	0.872224i	$0.662677\pi$
$444$	7.29628	0.346266
$445$	−14.5961	−0.691921
$446$	−21.9244	−1.03815
$447$	11.0300	0.521703
$448$	1.23025	0.0581237
$449$	16.1228	0.760881	0.380441	−	0.924805i	$-0.375772\pi$
$449$	16.1228	0.760881	0.380441	+	0.924805i	$0.375772\pi$
$450$	−1.09854	−0.0517858
$451$	10.1095	0.476039
$452$	5.08862	0.239348
$453$	−36.6335	−1.72119
$454$	11.0850	0.520246
$455$	23.2251	1.08881
$456$	−9.23390	−0.432417
$457$	22.1163	1.03456	0.517280	−	0.855816i	$-0.326945\pi$
$457$	22.1163	1.03456	0.517280	+	0.855816i	$0.326945\pi$
$458$	−10.7911	−0.504236
$459$	−5.43068	−0.253482
$460$	8.25526	0.384903
$461$	15.6986	0.731158	0.365579	−	0.930780i	$-0.380871\pi$
$461$	15.6986	0.731158	0.365579	+	0.930780i	$0.380871\pi$
$462$	−2.12432	−0.0988323
$463$	−37.5351	−1.74440	−0.872201	−	0.489147i	$-0.837308\pi$
$463$	−37.5351	−1.74440	−0.872201	+	0.489147i	$0.837308\pi$
$464$	5.31836	0.246899
$465$	8.73526	0.405088
$466$	−19.7448	−0.914661
$467$	34.9749	1.61844	0.809222	−	0.587502i	$-0.199889\pi$
$467$	34.9749	1.61844	0.809222	+	0.587502i	$0.199889\pi$
$468$	1.11255	0.0514274
$469$	−4.31670	−0.199327
$470$	10.9712	0.506062
$471$	4.56471	0.210331
$472$	14.0138	0.645037
$473$	4.68105	0.215235
$474$	26.7323	1.22785
$475$	31.4080	1.44110
$476$	−1.24893	−0.0572448
$477$	−0.332101	−0.0152059
$478$	−10.7899	−0.493518
$479$	−39.9938	−1.82736	−0.913681	−	0.406433i	$-0.866773\pi$
$479$	−39.9938	−1.82736	−0.913681	+	0.406433i	$0.866773\pi$
$480$	5.48037	0.250144
$481$	25.1337	1.14600
$482$	−23.0471	−1.04977
$483$	5.20226	0.236711
$484$	−9.93787	−0.451721
$485$	17.3385	0.787299
$486$	−2.00008	−0.0907253
$487$	32.6062	1.47753	0.738764	−	0.673964i	$-0.235410\pi$
$487$	32.6062	1.47753	0.738764	+	0.673964i	$0.235410\pi$
$488$	−2.49991	−0.113166
$489$	3.21358	0.145323
$490$	17.9459	0.810714
$491$	1.15726	0.0522263	0.0261131	−	0.999659i	$-0.491687\pi$
$491$	1.15726	0.0522263	0.0261131	+	0.999659i	$0.491687\pi$
$492$	−16.4355	−0.740968
$493$	−5.39916	−0.243166
$494$	−31.8083	−1.43112
$495$	0.649802	0.0292064
$496$	1.59392	0.0715690
$497$	−9.96684	−0.447074
$498$	29.2675	1.31151
$499$	−0.648504	−0.0290310	−0.0145155	−	0.999895i	$-0.504621\pi$
$499$	−0.648504	−0.0290310	−0.0145155	+	0.999895i	$0.504621\pi$
$500$	−2.28620	−0.102242
$501$	20.4534	0.913792
$502$	20.7041	0.924071
$503$	−34.3028	−1.52949	−0.764743	−	0.644336i	$-0.777134\pi$
$503$	−34.3028	−1.52949	−0.764743	+	0.644336i	$0.777134\pi$
$504$	−0.237145	−0.0105633
$505$	−13.1422	−0.584818
$506$	−2.60105	−0.115631
$507$	−34.0311	−1.51137
$508$	7.81182	0.346593
$509$	−8.63106	−0.382565	−0.191283	−	0.981535i	$-0.561265\pi$
$509$	−8.63106	−0.382565	−0.191283	+	0.981535i	$0.561265\pi$
$510$	−5.56362	−0.246361
$511$	17.4391	0.771459
$512$	1.00000	0.0441942
$513$	29.4816	1.30165
$514$	−9.13962	−0.403132
$515$	−15.3059	−0.674457
$516$	−7.61018	−0.335019
$517$	−3.45677	−0.152028
$518$	−5.35740	−0.235390
$519$	−9.38384	−0.411905
$520$	18.8784	0.827874
$521$	6.11582	0.267939	0.133970	−	0.990985i	$-0.457228\pi$
$521$	6.11582	0.267939	0.133970	+	0.990985i	$0.457228\pi$
$522$	−1.02518	−0.0448709
$523$	−31.2719	−1.36743	−0.683713	−	0.729751i	$-0.739636\pi$
$523$	−31.2719	−1.36743	−0.683713	+	0.729751i	$0.739636\pi$
$524$	5.85723	0.255874
$525$	−11.7470	−0.512680
$526$	−7.00486	−0.305426
$527$	−1.61813	−0.0704869
$528$	−1.72674	−0.0751469
$529$	−16.6303	−0.723056
$530$	−5.63532	−0.244782
$531$	−2.70133	−0.117228
$532$	6.78012	0.293956
$533$	−56.6158	−2.45230
$534$	−7.47663	−0.323545
$535$	45.9281	1.98564
$536$	−3.50881	−0.151557
$537$	30.2760	1.30651
$538$	25.2611	1.08908
$539$	−5.65436	−0.243550
$540$	−17.4975	−0.752974
$541$	−4.39429	−0.188925	−0.0944626	−	0.995528i	$-0.530113\pi$
$541$	−4.39429	−0.188925	−0.0944626	+	0.995528i	$0.530113\pi$
$542$	17.9345	0.770354
$543$	−14.5004	−0.622274
$544$	−1.01519	−0.0435259
$545$	−28.4532	−1.21880
$546$	11.8967	0.509132
$547$	−12.2016	−0.521702	−0.260851	−	0.965379i	$-0.584003\pi$
$547$	−12.2016	−0.521702	−0.260851	+	0.965379i	$0.584003\pi$
$548$	−4.58587	−0.195899
$549$	0.481889	0.0205665
$550$	5.87331	0.250439
$551$	29.3105	1.24867
$552$	4.22863	0.179983
$553$	−19.6286	−0.834691
$554$	−29.9909	−1.27419
$555$	−23.8656	−1.01304
$556$	−9.70172	−0.411445
$557$	20.2470	0.857892	0.428946	−	0.903330i	$-0.358885\pi$
$557$	20.2470	0.857892	0.428946	+	0.903330i	$0.358885\pi$
$558$	−0.307247	−0.0130068
$559$	−26.2150	−1.10878
$560$	−4.02404	−0.170047
$561$	1.75297	0.0740107
$562$	16.9433	0.714709
$563$	9.06236	0.381933	0.190966	−	0.981597i	$-0.438838\pi$
$563$	9.06236	0.381933	0.190966	+	0.981597i	$0.438838\pi$
$564$	5.61981	0.236637
$565$	−16.6445	−0.700238
$566$	15.5126	0.652045
$567$	−10.3151	−0.433192
$568$	−8.10150	−0.339932
$569$	13.4297	0.563002	0.281501	−	0.959561i	$-0.409168\pi$
$569$	13.4297	0.563002	0.281501	+	0.959561i	$0.409168\pi$
$570$	30.2034	1.26508
$571$	29.8524	1.24929	0.624643	−	0.780911i	$-0.285245\pi$
$571$	29.8524	1.24929	0.624643	+	0.780911i	$0.285245\pi$
$572$	−5.94817	−0.248706
$573$	30.6705	1.28128
$574$	12.0680	0.503708
$575$	−14.3832	−0.599820
$576$	−0.192762	−0.00803177
$577$	42.7212	1.77850	0.889252	−	0.457417i	$-0.151225\pi$
$577$	42.7212	1.77850	0.889252	+	0.457417i	$0.151225\pi$
$578$	−15.9694	−0.664239
$579$	−9.68564	−0.402521
$580$	−17.3960	−0.722328
$581$	−21.4901	−0.891560
$582$	8.88137	0.368145
$583$	1.77556	0.0735363
$584$	14.1753	0.586577
$585$	−3.63905	−0.150456
$586$	−14.3968	−0.594728
$587$	−36.5357	−1.50799	−0.753994	−	0.656881i	$-0.771875\pi$
$587$	−36.5357	−1.50799	−0.753994	+	0.656881i	$0.771875\pi$
$588$	9.19252	0.379093
$589$	8.78438	0.361954
$590$	−45.8380	−1.88712
$591$	−0.724558	−0.0298044
$592$	−4.35474	−0.178979
$593$	−44.0021	−1.80695	−0.903475	−	0.428641i	$-0.858993\pi$
$593$	−44.0021	−1.80695	−0.903475	+	0.428641i	$0.858993\pi$
$594$	5.51308	0.226205
$595$	4.08517	0.167476
$596$	−6.58321	−0.269659
$597$	11.7023	0.478943
$598$	14.5665	0.595669
$599$	27.5899	1.12729	0.563646	−	0.826016i	$-0.309398\pi$
$599$	27.5899	1.12729	0.563646	+	0.826016i	$0.309398\pi$
$600$	−9.54848	−0.389815
$601$	−11.1967	−0.456725	−0.228362	−	0.973576i	$-0.573337\pi$
$601$	−11.1967	−0.456725	−0.228362	+	0.973576i	$0.573337\pi$
$602$	5.58788	0.227745
$603$	0.676367	0.0275438
$604$	21.8645	0.889653
$605$	32.5060	1.32156
$606$	−6.73187	−0.273463
$607$	17.8311	0.723742	0.361871	−	0.932228i	$-0.382138\pi$
$607$	17.8311	0.723742	0.361871	+	0.932228i	$0.382138\pi$
$608$	5.51119	0.223508
$609$	−10.9625	−0.444223
$610$	8.17702	0.331078
$611$	19.3587	0.783171
$612$	0.195691	0.00791033
$613$	−38.3484	−1.54888	−0.774438	−	0.632650i	$-0.781967\pi$
$613$	−38.3484	−1.54888	−0.774438	+	0.632650i	$0.781967\pi$
$614$	1.51151	0.0609996
$615$	53.7592	2.16778
$616$	1.26789	0.0510846
$617$	5.29385	0.213123	0.106561	−	0.994306i	$-0.466016\pi$
$617$	5.29385	0.213123	0.106561	+	0.994306i	$0.466016\pi$
$618$	−7.84019	−0.315379
$619$	−36.8198	−1.47991	−0.739955	−	0.672656i	$-0.765153\pi$
$619$	−36.8198	−1.47991	−0.739955	+	0.672656i	$0.765153\pi$
$620$	−5.21358	−0.209383
$621$	−13.5010	−0.541777
$622$	−12.4486	−0.499144
$623$	5.48982	0.219945
$624$	9.67019	0.387117
$625$	−21.0167	−0.840670
$626$	4.24020	0.169473
$627$	−9.51642	−0.380049
$628$	−2.72442	−0.108716
$629$	4.42089	0.176272
$630$	0.775684	0.0309040
$631$	6.39195	0.254460	0.127230	−	0.991873i	$-0.459391\pi$
$631$	6.39195	0.254460	0.127230	+	0.991873i	$0.459391\pi$
$632$	−15.9550	−0.634655
$633$	10.3377	0.410887
$634$	−25.0957	−0.996678
$635$	−25.5519	−1.01399
$636$	−2.88660	−0.114461
$637$	31.6658	1.25464
$638$	5.48108	0.216998
$639$	1.56167	0.0617785
$640$	−3.27092	−0.129295
$641$	−2.33826	−0.0923556	−0.0461778	−	0.998933i	$-0.514704\pi$
$641$	−2.33826	−0.0923556	−0.0461778	+	0.998933i	$0.514704\pi$
$642$	23.5260	0.928495
$643$	−1.92774	−0.0760227	−0.0380114	−	0.999277i	$-0.512102\pi$
$643$	−1.92774	−0.0760227	−0.0380114	+	0.999277i	$0.512102\pi$
$644$	−3.10493	−0.122352
$645$	24.8923	0.980134
$646$	−5.59491	−0.220129
$647$	−36.1916	−1.42284	−0.711419	−	0.702768i	$-0.751947\pi$
$647$	−36.1916	−1.42284	−0.711419	+	0.702768i	$0.751947\pi$
$648$	−8.38456	−0.329376
$649$	14.4426	0.566920
$650$	−32.8920	−1.29013
$651$	−3.28547	−0.128768
$652$	−1.91800	−0.0751147
$653$	6.01108	0.235232	0.117616	−	0.993059i	$-0.462475\pi$
$653$	6.01108	0.235232	0.117616	+	0.993059i	$0.462475\pi$
$654$	−14.5747	−0.569916
$655$	−19.1586	−0.748587
$656$	9.80940	0.382993
$657$	−2.73246	−0.106603
$658$	−4.12642	−0.160865
$659$	25.2506	0.983626	0.491813	−	0.870701i	$-0.336334\pi$
$659$	25.2506	0.983626	0.491813	+	0.870701i	$0.336334\pi$
$660$	5.64805	0.219850
$661$	−40.7724	−1.58586	−0.792932	−	0.609310i	$-0.791447\pi$
$661$	−40.7724	−1.58586	−0.792932	+	0.609310i	$0.791447\pi$
$662$	34.4643	1.33949
$663$	−9.81709	−0.381264
$664$	−17.4681	−0.677895
$665$	−22.1773	−0.859998
$666$	0.839429	0.0325272
$667$	−13.4227	−0.519727
$668$	−12.2075	−0.472322
$669$	36.7339	1.42022
$670$	11.4770	0.443397
$671$	−2.57640	−0.0994608
$672$	−2.06125	−0.0795146
$673$	−38.1955	−1.47233	−0.736165	−	0.676802i	$-0.763365\pi$
$673$	−38.1955	−1.47233	−0.736165	+	0.676802i	$0.763365\pi$
$674$	−31.3283	−1.20672
$675$	30.4860	1.17341
$676$	20.3112	0.781201
$677$	32.9447	1.26617	0.633084	−	0.774083i	$-0.281789\pi$
$677$	32.9447	1.26617	0.633084	+	0.774083i	$0.281789\pi$
$678$	−8.52588	−0.327435
$679$	−6.52127	−0.250263
$680$	3.32061	0.127340
$681$	−18.5728	−0.711710
$682$	1.64268	0.0629016
$683$	3.55360	0.135975	0.0679873	−	0.997686i	$-0.478342\pi$
$683$	3.55360	0.135975	0.0679873	+	0.997686i	$0.478342\pi$
$684$	−1.06235	−0.0406200
$685$	15.0000	0.573122
$686$	−15.3615	−0.586503
$687$	18.0803	0.689807
$688$	4.54208	0.173165
$689$	−9.94359	−0.378820
$690$	−13.8315	−0.526557
$691$	−3.58738	−0.136470	−0.0682352	−	0.997669i	$-0.521737\pi$
$691$	−3.58738	−0.136470	−0.0682352	+	0.997669i	$0.521737\pi$
$692$	5.60069	0.212906
$693$	−0.244401	−0.00928402
$694$	−25.0147	−0.949544
$695$	31.7336	1.20372
$696$	−8.91082	−0.337764
$697$	−9.95842	−0.377202
$698$	−10.6112	−0.401639
$699$	33.0821	1.25128
$700$	7.01111	0.264995
$701$	45.0840	1.70280	0.851399	−	0.524519i	$-0.175755\pi$
$701$	45.0840	1.70280	0.851399	+	0.524519i	$0.175755\pi$
$702$	−30.8746	−1.16529
$703$	−23.9998	−0.905169
$704$	1.03060	0.0388420
$705$	−18.3820	−0.692305
$706$	26.2076	0.986338
$707$	4.94297	0.185900
$708$	−23.4798	−0.882427
$709$	34.6108	1.29984	0.649919	−	0.760004i	$-0.274803\pi$
$709$	34.6108	1.29984	0.649919	+	0.760004i	$0.274803\pi$
$710$	26.4994	0.994505
$711$	3.07552	0.115341
$712$	4.46238	0.167235
$713$	−4.02278	−0.150654
$714$	2.09257	0.0783123
$715$	19.4560	0.727614
$716$	−18.0700	−0.675309
$717$	18.0783	0.675145
$718$	−14.9153	−0.556634
$719$	−32.9516	−1.22889	−0.614443	−	0.788961i	$-0.710619\pi$
$719$	−32.9516	−1.22889	−0.614443	+	0.788961i	$0.710619\pi$
$720$	0.630511	0.0234978
$721$	5.75677	0.214393
$722$	11.3732	0.423268
$723$	38.6149	1.43611
$724$	8.65449	0.321642
$725$	30.3091	1.12565
$726$	16.6507	0.617966
$727$	−2.89091	−0.107218	−0.0536089	−	0.998562i	$-0.517072\pi$
$727$	−2.89091	−0.107218	−0.0536089	+	0.998562i	$0.517072\pi$
$728$	−7.10047	−0.263161
$729$	28.5048	1.05573
$730$	−46.3662	−1.71609
$731$	−4.61108	−0.170547
$732$	4.18855	0.154813
$733$	51.0512	1.88562	0.942810	−	0.333331i	$-0.108172\pi$
$733$	51.0512	1.88562	0.942810	+	0.333331i	$0.108172\pi$
$734$	−29.2307	−1.07892
$735$	−30.0680	−1.10908
$736$	−2.52383	−0.0930296
$737$	−3.61616	−0.133203
$738$	−1.89088	−0.0696044
$739$	8.30753	0.305597	0.152799	−	0.988257i	$-0.451171\pi$
$739$	8.30753	0.305597	0.152799	+	0.988257i	$0.451171\pi$
$740$	14.2440	0.523620
$741$	53.2943	1.95781
$742$	2.11953	0.0778104
$743$	18.5948	0.682177	0.341088	−	0.940031i	$-0.389204\pi$
$743$	18.5948	0.682177	0.341088	+	0.940031i	$0.389204\pi$
$744$	−2.67058	−0.0979082
$745$	21.5332	0.788915
$746$	−26.7645	−0.979920
$747$	3.36720	0.123199
$748$	−1.04625	−0.0382547
$749$	−17.2743	−0.631188
$750$	3.83048	0.139869
$751$	−23.6909	−0.864492	−0.432246	−	0.901756i	$-0.642279\pi$
$751$	−23.6909	−0.864492	−0.432246	+	0.901756i	$0.642279\pi$
$752$	−3.35414	−0.122313
$753$	−34.6894	−1.26415
$754$	−30.6954	−1.11786
$755$	−71.5171	−2.60277
$756$	6.58110	0.239352
$757$	−8.22004	−0.298762	−0.149381	−	0.988780i	$-0.547728\pi$
$757$	−8.22004	−0.298762	−0.149381	+	0.988780i	$0.547728\pi$
$758$	0.719383	0.0261292
$759$	4.35801	0.158186
$760$	−18.0267	−0.653897
$761$	−37.3385	−1.35352	−0.676760	−	0.736203i	$-0.736617\pi$
$761$	−37.3385	−1.35352	−0.676760	+	0.736203i	$0.736617\pi$
$762$	−13.0886	−0.474148
$763$	10.7017	0.387427
$764$	−18.3055	−0.662269
$765$	−0.640089	−0.0231425
$766$	35.7180	1.29054
$767$	−80.8818	−2.92047
$768$	−1.67548	−0.0604587
$769$	23.9716	0.864438	0.432219	−	0.901769i	$-0.357731\pi$
$769$	23.9716	0.864438	0.432219	+	0.901769i	$0.357731\pi$
$770$	−4.14716	−0.149453
$771$	15.3133	0.551494
$772$	5.78081	0.208056
$773$	26.5916	0.956432	0.478216	−	0.878242i	$-0.341284\pi$
$773$	26.5916	0.956432	0.478216	+	0.878242i	$0.341284\pi$
$774$	−0.875543	−0.0314707
$775$	9.08365	0.326294
$776$	−5.30079	−0.190287
$777$	8.97622	0.322020
$778$	−4.52778	−0.162329
$779$	54.0615	1.93695
$780$	−31.6305	−1.13255
$781$	−8.34938	−0.298764
$782$	2.56217	0.0916230
$783$	28.4501	1.01672
$784$	−5.48649	−0.195946
$785$	8.91136	0.318060
$786$	−9.81368	−0.350042
$787$	3.73496	0.133137	0.0665685	−	0.997782i	$-0.478795\pi$
$787$	3.73496	0.133137	0.0665685	+	0.997782i	$0.478795\pi$
$788$	0.432448	0.0154053
$789$	11.7365	0.417831
$790$	52.1875	1.85675
$791$	6.26025	0.222589
$792$	−0.198660	−0.00705908
$793$	14.4285	0.512370
$794$	7.07292	0.251008
$795$	9.44187	0.334868
$796$	−6.98443	−0.247557
$797$	10.1986	0.361254	0.180627	−	0.983552i	$-0.442187\pi$
$797$	10.1986	0.361254	0.180627	+	0.983552i	$0.442187\pi$
$798$	−11.3600	−0.402139
$799$	3.40510	0.120464
$800$	5.69895	0.201488
$801$	−0.860178	−0.0303929
$802$	11.4434	0.404081
$803$	14.6090	0.515540
$804$	5.87894	0.207334
$805$	10.1560	0.357952
$806$	−9.19943	−0.324036
$807$	−42.3245	−1.48989
$808$	4.01787	0.141348
$809$	6.66281	0.234252	0.117126	−	0.993117i	$-0.462632\pi$
$809$	6.66281	0.234252	0.117126	+	0.993117i	$0.462632\pi$
$810$	27.4252	0.963625
$811$	−17.2188	−0.604633	−0.302317	−	0.953208i	$-0.597760\pi$
$811$	−17.2188	−0.604633	−0.302317	+	0.953208i	$0.597760\pi$
$812$	6.54290	0.229611
$813$	−30.0490	−1.05386
$814$	−4.48797	−0.157303
$815$	6.27364	0.219756
$816$	1.70093	0.0595446
$817$	25.0323	0.875769
$818$	−23.4154	−0.818701
$819$	1.36870	0.0478264
$820$	−32.0858	−1.12049
$821$	8.20969	0.286520	0.143260	−	0.989685i	$-0.454241\pi$
$821$	8.20969	0.286520	0.143260	+	0.989685i	$0.454241\pi$
$822$	7.68355	0.267994
$823$	−19.6814	−0.686050	−0.343025	−	0.939326i	$-0.611451\pi$
$823$	−19.6814	−0.686050	−0.343025	+	0.939326i	$0.611451\pi$
$824$	4.67937	0.163013
$825$	−9.84062	−0.342606
$826$	17.2404	0.599871
$827$	24.2739	0.844086	0.422043	−	0.906576i	$-0.361313\pi$
$827$	24.2739	0.844086	0.422043	+	0.906576i	$0.361313\pi$
$828$	0.486500	0.0169070
$829$	29.9937	1.04172	0.520862	−	0.853641i	$-0.325611\pi$
$829$	29.9937	1.04172	0.520862	+	0.853641i	$0.325611\pi$
$830$	57.1370	1.98325
$831$	50.2493	1.74313
$832$	−5.77159	−0.200094
$833$	5.56984	0.192983
$834$	16.2551	0.562867
$835$	39.9298	1.38183
$836$	5.67981	0.196440
$837$	8.52652	0.294720
$838$	35.3483	1.22109
$839$	−21.8066	−0.752848	−0.376424	−	0.926447i	$-0.622846\pi$
$839$	−21.8066	−0.752848	−0.376424	+	0.926447i	$0.622846\pi$
$840$	6.74221	0.232628
$841$	−0.714999	−0.0246552
$842$	−39.0439	−1.34554
$843$	−28.3881	−0.977740
$844$	−6.16999	−0.212380
$845$	−66.4365	−2.28548
$846$	0.646553	0.0222290
$847$	−12.2260	−0.420091
$848$	1.72285	0.0591629
$849$	−25.9911	−0.892014
$850$	−5.78552	−0.198442
$851$	10.9906	0.376753
$852$	13.5739	0.465035
$853$	25.6078	0.876794	0.438397	−	0.898782i	$-0.355546\pi$
$853$	25.6078	0.876794	0.438397	+	0.898782i	$0.355546\pi$
$854$	−3.07551	−0.105242
$855$	3.47487	0.118838
$856$	−14.0413	−0.479922
$857$	−16.3062	−0.557008	−0.278504	−	0.960435i	$-0.589839\pi$
$857$	−16.3062	−0.557008	−0.278504	+	0.960435i	$0.589839\pi$
$858$	9.96606	0.340236
$859$	42.1342	1.43760	0.718800	−	0.695217i	$-0.244692\pi$
$859$	42.1342	1.43760	0.718800	+	0.695217i	$0.244692\pi$
$860$	−14.8568	−0.506613
$861$	−20.2197	−0.689085
$862$	11.8228	0.402685
$863$	−39.0269	−1.32849	−0.664245	−	0.747515i	$-0.731247\pi$
$863$	−39.0269	−1.32849	−0.664245	+	0.747515i	$0.731247\pi$
$864$	5.34941	0.181991
$865$	−18.3194	−0.622879
$866$	−14.0008	−0.475765
$867$	26.7564	0.908695
$868$	1.96091	0.0665576
$869$	−16.4431	−0.557795
$870$	29.1466	0.988163
$871$	20.2514	0.686193
$872$	8.69881	0.294579
$873$	1.02179	0.0345824
$874$	−13.9093	−0.470490
$875$	−2.81259	−0.0950828
$876$	−23.7504	−0.802452
$877$	−28.7317	−0.970200	−0.485100	−	0.874459i	$-0.661217\pi$
$877$	−28.7317	−0.970200	−0.485100	+	0.874459i	$0.661217\pi$
$878$	−36.1255	−1.21918
$879$	24.1216	0.813603
$880$	−3.37100	−0.113636
$881$	−10.8550	−0.365715	−0.182858	−	0.983139i	$-0.558535\pi$
$881$	−10.8550	−0.365715	−0.182858	+	0.983139i	$0.558535\pi$
$882$	1.05759	0.0356109
$883$	10.2387	0.344559	0.172280	−	0.985048i	$-0.444887\pi$
$883$	10.2387	0.344559	0.172280	+	0.985048i	$0.444887\pi$
$884$	5.85926	0.197068
$885$	76.8008	2.58163
$886$	−20.5890	−0.691701
$887$	24.3808	0.818628	0.409314	−	0.912394i	$-0.365768\pi$
$887$	24.3808	0.818628	0.409314	+	0.912394i	$0.365768\pi$
$888$	7.29628	0.244847
$889$	9.61046	0.322324
$890$	−14.5961	−0.489262
$891$	−8.64109	−0.289487
$892$	−21.9244	−0.734083
$893$	−18.4853	−0.618588
$894$	11.0300	0.368900
$895$	59.1057	1.97569
$896$	1.23025	0.0410996
$897$	−24.4059	−0.814890
$898$	16.1228	0.538024
$899$	8.47703	0.282725
$900$	−1.09854	−0.0366181
$901$	−1.74902	−0.0582684
$902$	10.1095	0.336611
$903$	−9.36239	−0.311561
$904$	5.08862	0.169245
$905$	−28.3082	−0.940996
$906$	−36.6335	−1.21707
$907$	20.6024	0.684090	0.342045	−	0.939684i	$-0.388880\pi$
$907$	20.6024	0.684090	0.342045	+	0.939684i	$0.388880\pi$
$908$	11.0850	0.367870
$909$	−0.774495	−0.0256884
$910$	23.2251	0.769905
$911$	48.1552	1.59545	0.797727	−	0.603019i	$-0.206036\pi$
$911$	48.1552	1.59545	0.797727	+	0.603019i	$0.206036\pi$
$912$	−9.23390	−0.305765
$913$	−18.0026	−0.595799
$914$	22.1163	0.731544
$915$	−13.7004	−0.452923
$916$	−10.7911	−0.356549
$917$	7.20583	0.237958
$918$	−5.43068	−0.179239
$919$	−48.9924	−1.61611	−0.808055	−	0.589107i	$-0.799480\pi$
$919$	−48.9924	−1.61611	−0.808055	+	0.589107i	$0.799480\pi$
$920$	8.25526	0.272168
$921$	−2.53251	−0.0834490
$922$	15.6986	0.517007
$923$	46.7585	1.53908
$924$	−2.12432	−0.0698850
$925$	−24.8174	−0.815991
$926$	−37.5351	−1.23348
$927$	−0.902006	−0.0296258
$928$	5.31836	0.174584
$929$	32.4438	1.06445	0.532224	−	0.846604i	$-0.321356\pi$
$929$	32.4438	1.06445	0.532224	+	0.846604i	$0.321356\pi$
$930$	8.73526	0.286440
$931$	−30.2371	−0.990982
$932$	−19.7448	−0.646763
$933$	20.8574	0.682841
$934$	34.9749	1.14441
$935$	3.42221	0.111918
$936$	1.11255	0.0363647
$937$	0.965582	0.0315442	0.0157721	−	0.999876i	$-0.494979\pi$
$937$	0.965582	0.0315442	0.0157721	+	0.999876i	$0.494979\pi$
$938$	−4.31670	−0.140945
$939$	−7.10438	−0.231843
$940$	10.9712	0.357840
$941$	23.3676	0.761763	0.380882	−	0.924624i	$-0.375621\pi$
$941$	23.3676	0.761763	0.380882	+	0.924624i	$0.375621\pi$
$942$	4.56471	0.148726
$943$	−24.7573	−0.806208
$944$	14.0138	0.456110
$945$	−21.5263	−0.700249
$946$	4.68105	0.152194
$947$	−3.75900	−0.122151	−0.0610755	−	0.998133i	$-0.519453\pi$
$947$	−3.75900	−0.122151	−0.0610755	+	0.998133i	$0.519453\pi$
$948$	26.7323	0.868224
$949$	−81.8138	−2.65579
$950$	31.4080	1.01901
$951$	42.0474	1.36348
$952$	−1.24893	−0.0404782
$953$	33.5486	1.08674	0.543372	−	0.839492i	$-0.317147\pi$
$953$	33.5486	1.08674	0.543372	+	0.839492i	$0.317147\pi$
$954$	−0.332101	−0.0107522
$955$	59.8759	1.93754
$956$	−10.7899	−0.348970
$957$	−9.18346	−0.296859
$958$	−39.9938	−1.29214
$959$	−5.64175	−0.182182
$960$	5.48037	0.176878
$961$	−28.4594	−0.918046
$962$	25.1337	0.810344
$963$	2.70664	0.0872202
$964$	−23.0471	−0.742296
$965$	−18.9086	−0.608689
$966$	5.20226	0.167380
$967$	27.3858	0.880669	0.440335	−	0.897834i	$-0.354860\pi$
$967$	27.3858	0.880669	0.440335	+	0.897834i	$0.354860\pi$
$968$	−9.93787	−0.319415
$969$	9.37417	0.301142
$970$	17.3385	0.556705
$971$	−38.3483	−1.23065	−0.615327	−	0.788272i	$-0.710976\pi$
$971$	−38.3483	−1.23065	−0.615327	+	0.788272i	$0.710976\pi$
$972$	−2.00008	−0.0641525
$973$	−11.9355	−0.382635
$974$	32.6062	1.04477
$975$	55.1099	1.76493
$976$	−2.49991	−0.0800202
$977$	4.20089	0.134398	0.0671991	−	0.997740i	$-0.478594\pi$
$977$	4.20089	0.134398	0.0671991	+	0.997740i	$0.478594\pi$
$978$	3.21358	0.102759
$979$	4.59891	0.146982
$980$	17.9459	0.573261
$981$	−1.67680	−0.0535362
$982$	1.15726	0.0369295
$983$	46.6160	1.48682	0.743410	−	0.668835i	$-0.233207\pi$
$983$	46.6160	1.48682	0.743410	+	0.668835i	$0.233207\pi$
$984$	−16.4355	−0.523944
$985$	−1.41450	−0.0450699
$986$	−5.39916	−0.171944
$987$	6.91375	0.220067
$988$	−31.8083	−1.01196
$989$	−11.4635	−0.364517
$990$	0.649802	0.0206521
$991$	5.84863	0.185788	0.0928940	−	0.995676i	$-0.470388\pi$
$991$	5.84863	0.185788	0.0928940	+	0.995676i	$0.470388\pi$
$992$	1.59392	0.0506069
$993$	−57.7443	−1.83246
$994$	−9.96684	−0.316129
$995$	22.8456	0.724253
$996$	29.2675	0.927378
$997$	−13.8354	−0.438171	−0.219086	−	0.975706i	$-0.570307\pi$
$997$	−13.8354	−0.438171	−0.219086	+	0.975706i	$0.570307\pi$
$998$	−0.648504	−0.0205280
$999$	−23.2953	−0.737030

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.17	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.17	✓	67	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.67548 q^{3} +1.00000 q^{4} -3.27092 q^{5} -1.67548 q^{6} +1.23025 q^{7} +1.00000 q^{8} -0.192762 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.67548 q^{3} +1.00000 q^{4} -3.27092 q^{5} -1.67548 q^{6} +1.23025 q^{7} +1.00000 q^{8} -0.192762 q^{9} -3.27092 q^{10} +1.03060 q^{11} -1.67548 q^{12} -5.77159 q^{13} +1.23025 q^{14} +5.48037 q^{15} +1.00000 q^{16} -1.01519 q^{17} -0.192762 q^{18} +5.51119 q^{19} -3.27092 q^{20} -2.06125 q^{21} +1.03060 q^{22} -2.52383 q^{23} -1.67548 q^{24} +5.69895 q^{25} -5.77159 q^{26} +5.34941 q^{27} +1.23025 q^{28} +5.31836 q^{29} +5.48037 q^{30} +1.59392 q^{31} +1.00000 q^{32} -1.72674 q^{33} -1.01519 q^{34} -4.02404 q^{35} -0.192762 q^{36} -4.35474 q^{37} +5.51119 q^{38} +9.67019 q^{39} -3.27092 q^{40} +9.80940 q^{41} -2.06125 q^{42} +4.54208 q^{43} +1.03060 q^{44} +0.630511 q^{45} -2.52383 q^{46} -3.35414 q^{47} -1.67548 q^{48} -5.48649 q^{49} +5.69895 q^{50} +1.70093 q^{51} -5.77159 q^{52} +1.72285 q^{53} +5.34941 q^{54} -3.37100 q^{55} +1.23025 q^{56} -9.23390 q^{57} +5.31836 q^{58} +14.0138 q^{59} +5.48037 q^{60} -2.49991 q^{61} +1.59392 q^{62} -0.237145 q^{63} +1.00000 q^{64} +18.8784 q^{65} -1.72674 q^{66} -3.50881 q^{67} -1.01519 q^{68} +4.22863 q^{69} -4.02404 q^{70} -8.10150 q^{71} -0.192762 q^{72} +14.1753 q^{73} -4.35474 q^{74} -9.54848 q^{75} +5.51119 q^{76} +1.26789 q^{77} +9.67019 q^{78} -15.9550 q^{79} -3.27092 q^{80} -8.38456 q^{81} +9.80940 q^{82} -17.4681 q^{83} -2.06125 q^{84} +3.32061 q^{85} +4.54208 q^{86} -8.91082 q^{87} +1.03060 q^{88} +4.46238 q^{89} +0.630511 q^{90} -7.10047 q^{91} -2.52383 q^{92} -2.67058 q^{93} -3.35414 q^{94} -18.0267 q^{95} -1.67548 q^{96} -5.30079 q^{97} -5.48649 q^{98} -0.198660 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

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