LMFDB - Embedded newform 8002.2.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, 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("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.8962916974$
Analytic rank:	$0$
Dimension:	$77$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.93461 q^{3} +1.00000 q^{4} +1.73622 q^{5} +2.93461 q^{6} -0.969734 q^{7} -1.00000 q^{8} +5.61192 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.93461 q^{3} +1.00000 q^{4} +1.73622 q^{5} +2.93461 q^{6} -0.969734 q^{7} -1.00000 q^{8} +5.61192 q^{9} -1.73622 q^{10} -5.59748 q^{11} -2.93461 q^{12} -5.07738 q^{13} +0.969734 q^{14} -5.09513 q^{15} +1.00000 q^{16} +4.63902 q^{17} -5.61192 q^{18} +1.16072 q^{19} +1.73622 q^{20} +2.84579 q^{21} +5.59748 q^{22} +8.33406 q^{23} +2.93461 q^{24} -1.98553 q^{25} +5.07738 q^{26} -7.66495 q^{27} -0.969734 q^{28} -5.44758 q^{29} +5.09513 q^{30} +6.78772 q^{31} -1.00000 q^{32} +16.4264 q^{33} -4.63902 q^{34} -1.68367 q^{35} +5.61192 q^{36} -11.1148 q^{37} -1.16072 q^{38} +14.9001 q^{39} -1.73622 q^{40} -5.20560 q^{41} -2.84579 q^{42} +0.841312 q^{43} -5.59748 q^{44} +9.74354 q^{45} -8.33406 q^{46} -0.998598 q^{47} -2.93461 q^{48} -6.05962 q^{49} +1.98553 q^{50} -13.6137 q^{51} -5.07738 q^{52} -2.85680 q^{53} +7.66495 q^{54} -9.71846 q^{55} +0.969734 q^{56} -3.40627 q^{57} +5.44758 q^{58} +6.29475 q^{59} -5.09513 q^{60} -3.58449 q^{61} -6.78772 q^{62} -5.44207 q^{63} +1.00000 q^{64} -8.81546 q^{65} -16.4264 q^{66} +8.13548 q^{67} +4.63902 q^{68} -24.4572 q^{69} +1.68367 q^{70} -9.51758 q^{71} -5.61192 q^{72} +8.69676 q^{73} +11.1148 q^{74} +5.82675 q^{75} +1.16072 q^{76} +5.42806 q^{77} -14.9001 q^{78} +13.9628 q^{79} +1.73622 q^{80} +5.65787 q^{81} +5.20560 q^{82} -17.3624 q^{83} +2.84579 q^{84} +8.05437 q^{85} -0.841312 q^{86} +15.9865 q^{87} +5.59748 q^{88} -7.79136 q^{89} -9.74354 q^{90} +4.92370 q^{91} +8.33406 q^{92} -19.9193 q^{93} +0.998598 q^{94} +2.01527 q^{95} +2.93461 q^{96} -1.51463 q^{97} +6.05962 q^{98} -31.4126 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) $ 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * 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
$3$	−2.93461	−1.69430	−0.847148	−	0.531357i	$-0.821682\pi$
$3$	−2.93461	−1.69430	−0.847148	+	0.531357i	$0.821682\pi$
$4$	1.00000	0.500000
$5$	1.73622	0.776462	0.388231	−	0.921562i	$-0.373086\pi$
$5$	1.73622	0.776462	0.388231	+	0.921562i	$0.373086\pi$
$6$	2.93461	1.19805
$7$	−0.969734	−0.366525	−0.183262	−	0.983064i	$-0.558666\pi$
$7$	−0.969734	−0.366525	−0.183262	+	0.983064i	$0.558666\pi$
$8$	−1.00000	−0.353553
$9$	5.61192	1.87064
$10$	−1.73622	−0.549042
$11$	−5.59748	−1.68770	−0.843851	−	0.536577i	$-0.819717\pi$
$11$	−5.59748	−1.68770	−0.843851	+	0.536577i	$0.819717\pi$
$12$	−2.93461	−0.847148
$13$	−5.07738	−1.40821	−0.704106	−	0.710095i	$-0.748652\pi$
$13$	−5.07738	−1.40821	−0.704106	+	0.710095i	$0.748652\pi$
$14$	0.969734	0.259172
$15$	−5.09513	−1.31556
$16$	1.00000	0.250000
$17$	4.63902	1.12513	0.562564	−	0.826754i	$-0.309815\pi$
$17$	4.63902	1.12513	0.562564	+	0.826754i	$0.309815\pi$
$18$	−5.61192	−1.32274
$19$	1.16072	0.266288	0.133144	−	0.991097i	$-0.457493\pi$
$19$	1.16072	0.266288	0.133144	+	0.991097i	$0.457493\pi$
$20$	1.73622	0.388231
$21$	2.84579	0.621002
$22$	5.59748	1.19339
$23$	8.33406	1.73777	0.868886	−	0.495012i	$-0.164836\pi$
$23$	8.33406	1.73777	0.868886	+	0.495012i	$0.164836\pi$
$24$	2.93461	0.599024
$25$	−1.98553	−0.397106
$26$	5.07738	0.995756
$27$	−7.66495	−1.47512
$28$	−0.969734	−0.183262
$29$	−5.44758	−1.01159	−0.505795	−	0.862654i	$-0.668801\pi$
$29$	−5.44758	−1.01159	−0.505795	+	0.862654i	$0.668801\pi$
$30$	5.09513	0.930239
$31$	6.78772	1.21911	0.609555	−	0.792744i	$-0.291348\pi$
$31$	6.78772	1.21911	0.609555	+	0.792744i	$0.291348\pi$
$32$	−1.00000	−0.176777
$33$	16.4264	2.85947
$34$	−4.63902	−0.795585
$35$	−1.68367	−0.284593
$36$	5.61192	0.935320
$37$	−11.1148	−1.82727	−0.913633	−	0.406539i	$-0.866735\pi$
$37$	−11.1148	−1.82727	−0.913633	+	0.406539i	$0.866735\pi$
$38$	−1.16072	−0.188294
$39$	14.9001	2.38593
$40$	−1.73622	−0.274521
$41$	−5.20560	−0.812978	−0.406489	−	0.913656i	$-0.633247\pi$
$41$	−5.20560	−0.812978	−0.406489	+	0.913656i	$0.633247\pi$
$42$	−2.84579	−0.439114
$43$	0.841312	0.128299	0.0641494	−	0.997940i	$-0.479567\pi$
$43$	0.841312	0.128299	0.0641494	+	0.997940i	$0.479567\pi$
$44$	−5.59748	−0.843851
$45$	9.74354	1.45248
$46$	−8.33406	−1.22879
$47$	−0.998598	−0.145660	−0.0728302	−	0.997344i	$-0.523203\pi$
$47$	−0.998598	−0.145660	−0.0728302	+	0.997344i	$0.523203\pi$
$48$	−2.93461	−0.423574
$49$	−6.05962	−0.865660
$50$	1.98553	0.280796
$51$	−13.6137	−1.90630
$52$	−5.07738	−0.704106
$53$	−2.85680	−0.392412	−0.196206	−	0.980563i	$-0.562862\pi$
$53$	−2.85680	−0.392412	−0.196206	+	0.980563i	$0.562862\pi$
$54$	7.66495	1.04307
$55$	−9.71846	−1.31044
$56$	0.969734	0.129586
$57$	−3.40627	−0.451171
$58$	5.44758	0.715303
$59$	6.29475	0.819507	0.409753	−	0.912196i	$-0.365615\pi$
$59$	6.29475	0.819507	0.409753	+	0.912196i	$0.365615\pi$
$60$	−5.09513	−0.657779
$61$	−3.58449	−0.458947	−0.229474	−	0.973315i	$-0.573700\pi$
$61$	−3.58449	−0.458947	−0.229474	+	0.973315i	$0.573700\pi$
$62$	−6.78772	−0.862041
$63$	−5.44207	−0.685636
$64$	1.00000	0.125000
$65$	−8.81546	−1.09342
$66$	−16.4264	−2.02195
$67$	8.13548	0.993907	0.496953	−	0.867777i	$-0.334452\pi$
$67$	8.13548	0.993907	0.496953	+	0.867777i	$0.334452\pi$
$68$	4.63902	0.562564
$69$	−24.4572	−2.94430
$70$	1.68367	0.201237
$71$	−9.51758	−1.12953	−0.564765	−	0.825252i	$-0.691033\pi$
$71$	−9.51758	−1.12953	−0.564765	+	0.825252i	$0.691033\pi$
$72$	−5.61192	−0.661371
$73$	8.69676	1.01788	0.508939	−	0.860802i	$-0.330038\pi$
$73$	8.69676	1.01788	0.508939	+	0.860802i	$0.330038\pi$
$74$	11.1148	1.29207
$75$	5.82675	0.672815
$76$	1.16072	0.133144
$77$	5.42806	0.618585
$78$	−14.9001	−1.68710
$79$	13.9628	1.57094	0.785469	−	0.618901i	$-0.212422\pi$
$79$	13.9628	1.57094	0.785469	+	0.618901i	$0.212422\pi$
$80$	1.73622	0.194116
$81$	5.65787	0.628652
$82$	5.20560	0.574862
$83$	−17.3624	−1.90577	−0.952884	−	0.303334i	$-0.901900\pi$
$83$	−17.3624	−1.90577	−0.952884	+	0.303334i	$0.901900\pi$
$84$	2.84579	0.310501
$85$	8.05437	0.873619
$86$	−0.841312	−0.0907210
$87$	15.9865	1.71393
$88$	5.59748	0.596693
$89$	−7.79136	−0.825882	−0.412941	−	0.910758i	$-0.635498\pi$
$89$	−7.79136	−0.825882	−0.412941	+	0.910758i	$0.635498\pi$
$90$	−9.74354	−1.02706
$91$	4.92370	0.516144
$92$	8.33406	0.868886
$93$	−19.9193	−2.06553
$94$	0.998598	0.102997
$95$	2.01527	0.206763
$96$	2.93461	0.299512
$97$	−1.51463	−0.153787	−0.0768937	−	0.997039i	$-0.524500\pi$
$97$	−1.51463	−0.153787	−0.0768937	+	0.997039i	$0.524500\pi$
$98$	6.05962	0.612114
$99$	−31.4126	−3.15708
$100$	−1.98553	−0.198553
$101$	−1.08327	−0.107789	−0.0538946	−	0.998547i	$-0.517164\pi$
$101$	−1.08327	−0.107789	−0.0538946	+	0.998547i	$0.517164\pi$
$102$	13.6137	1.34796
$103$	0.0218735	0.00215526	0.00107763	−	0.999999i	$-0.499657\pi$
$103$	0.0218735	0.00215526	0.00107763	+	0.999999i	$0.499657\pi$
$104$	5.07738	0.497878
$105$	4.94092	0.482184
$106$	2.85680	0.277477
$107$	12.2708	1.18626	0.593131	−	0.805106i	$-0.297892\pi$
$107$	12.2708	1.18626	0.593131	+	0.805106i	$0.297892\pi$
$108$	−7.66495	−0.737560
$109$	−1.36902	−0.131128	−0.0655641	−	0.997848i	$-0.520885\pi$
$109$	−1.36902	−0.131128	−0.0655641	+	0.997848i	$0.520885\pi$
$110$	9.71846	0.926619
$111$	32.6177	3.09593
$112$	−0.969734	−0.0916312
$113$	−2.41954	−0.227611	−0.113806	−	0.993503i	$-0.536304\pi$
$113$	−2.41954	−0.227611	−0.113806	+	0.993503i	$0.536304\pi$
$114$	3.40627	0.319026
$115$	14.4698	1.34932
$116$	−5.44758	−0.505795
$117$	−28.4938	−2.63425
$118$	−6.29475	−0.579479
$119$	−4.49861	−0.412387
$120$	5.09513	0.465120
$121$	20.3317	1.84834
$122$	3.58449	0.324525
$123$	15.2764	1.37742
$124$	6.78772	0.609555
$125$	−12.1284	−1.08480
$126$	5.44207	0.484818
$127$	2.56016	0.227178	0.113589	−	0.993528i	$-0.463765\pi$
$127$	2.56016	0.227178	0.113589	+	0.993528i	$0.463765\pi$
$128$	−1.00000	−0.0883883
$129$	−2.46892	−0.217376
$130$	8.81546	0.773167
$131$	−1.62071	−0.141602	−0.0708009	−	0.997490i	$-0.522555\pi$
$131$	−1.62071	−0.141602	−0.0708009	+	0.997490i	$0.522555\pi$
$132$	16.4264	1.42973
$133$	−1.12559	−0.0976013
$134$	−8.13548	−0.702798
$135$	−13.3081	−1.14538
$136$	−4.63902	−0.397792
$137$	4.22758	0.361187	0.180593	−	0.983558i	$-0.442198\pi$
$137$	4.22758	0.361187	0.180593	+	0.983558i	$0.442198\pi$
$138$	24.4572	2.08194
$139$	−12.2166	−1.03620	−0.518098	−	0.855321i	$-0.673360\pi$
$139$	−12.2166	−1.03620	−0.518098	+	0.855321i	$0.673360\pi$
$140$	−1.68367	−0.142296
$141$	2.93049	0.246792
$142$	9.51758	0.798698
$143$	28.4205	2.37664
$144$	5.61192	0.467660
$145$	−9.45822	−0.785462
$146$	−8.69676	−0.719749
$147$	17.7826	1.46668
$148$	−11.1148	−0.913633
$149$	−3.67818	−0.301328	−0.150664	−	0.988585i	$-0.548141\pi$
$149$	−3.67818	−0.301328	−0.150664	+	0.988585i	$0.548141\pi$
$150$	−5.82675	−0.475752
$151$	−0.832853	−0.0677766	−0.0338883	−	0.999426i	$-0.510789\pi$
$151$	−0.832853	−0.0677766	−0.0338883	+	0.999426i	$0.510789\pi$
$152$	−1.16072	−0.0941471
$153$	26.0338	2.10471
$154$	−5.42806	−0.437406
$155$	11.7850	0.946594
$156$	14.9001	1.19296
$157$	15.6537	1.24930	0.624650	−	0.780905i	$-0.285242\pi$
$157$	15.6537	1.24930	0.624650	+	0.780905i	$0.285242\pi$
$158$	−13.9628	−1.11082
$159$	8.38359	0.664862
$160$	−1.73622	−0.137260
$161$	−8.08182	−0.636937
$162$	−5.65787	−0.444524
$163$	−10.3750	−0.812637	−0.406318	−	0.913732i	$-0.633188\pi$
$163$	−10.3750	−0.812637	−0.406318	+	0.913732i	$0.633188\pi$
$164$	−5.20560	−0.406489
$165$	28.5199	2.22027
$166$	17.3624	1.34758
$167$	0.599480	0.0463892	0.0231946	−	0.999731i	$-0.492616\pi$
$167$	0.599480	0.0463892	0.0231946	+	0.999731i	$0.492616\pi$
$168$	−2.84579	−0.219557
$169$	12.7798	0.983058
$170$	−8.05437	−0.617742
$171$	6.51389	0.498129
$172$	0.841312	0.0641494
$173$	−13.4289	−1.02098	−0.510489	−	0.859884i	$-0.670536\pi$
$173$	−13.4289	−1.02098	−0.510489	+	0.859884i	$0.670536\pi$
$174$	−15.9865	−1.21193
$175$	1.92544	0.145549
$176$	−5.59748	−0.421926
$177$	−18.4726	−1.38849
$178$	7.79136	0.583987
$179$	−6.70467	−0.501131	−0.250565	−	0.968100i	$-0.580616\pi$
$179$	−6.70467	−0.501131	−0.250565	+	0.968100i	$0.580616\pi$
$180$	9.74354	0.726241
$181$	−21.0383	−1.56377	−0.781883	−	0.623425i	$-0.785741\pi$
$181$	−21.0383	−1.56377	−0.781883	+	0.623425i	$0.785741\pi$
$182$	−4.92370	−0.364969
$183$	10.5191	0.777593
$184$	−8.33406	−0.614395
$185$	−19.2978	−1.41880
$186$	19.9193	1.46055
$187$	−25.9668	−1.89888
$188$	−0.998598	−0.0728302
$189$	7.43296	0.540668
$190$	−2.01527	−0.146203
$191$	−17.6389	−1.27631	−0.638153	−	0.769910i	$-0.720301\pi$
$191$	−17.6389	−1.27631	−0.638153	+	0.769910i	$0.720301\pi$
$192$	−2.93461	−0.211787
$193$	−8.84040	−0.636346	−0.318173	−	0.948033i	$-0.603069\pi$
$193$	−8.84040	−0.636346	−0.318173	+	0.948033i	$0.603069\pi$
$194$	1.51463	0.108744
$195$	25.8699	1.85258
$196$	−6.05962	−0.432830
$197$	6.41973	0.457387	0.228693	−	0.973499i	$-0.426555\pi$
$197$	6.41973	0.457387	0.228693	+	0.973499i	$0.426555\pi$
$198$	31.4126	2.23239
$199$	16.7976	1.19075	0.595375	−	0.803448i	$-0.297003\pi$
$199$	16.7976	1.19075	0.595375	+	0.803448i	$0.297003\pi$
$200$	1.98553	0.140398
$201$	−23.8744	−1.68397
$202$	1.08327	0.0762185
$203$	5.28270	0.370773
$204$	−13.6137	−0.953149
$205$	−9.03807	−0.631246
$206$	−0.0218735	−0.00152400
$207$	46.7701	3.25075
$208$	−5.07738	−0.352053
$209$	−6.49712	−0.449415
$210$	−4.94092	−0.340956
$211$	15.0713	1.03755	0.518775	−	0.854911i	$-0.326388\pi$
$211$	15.0713	1.03755	0.518775	+	0.854911i	$0.326388\pi$
$212$	−2.85680	−0.196206
$213$	27.9304	1.91376
$214$	−12.2708	−0.838814
$215$	1.46070	0.0996192
$216$	7.66495	0.521534
$217$	−6.58228	−0.446834
$218$	1.36902	0.0927216
$219$	−25.5216	−1.72459
$220$	−9.71846	−0.655219
$221$	−23.5540	−1.58442
$222$	−32.6177	−2.18915
$223$	−20.7382	−1.38873	−0.694366	−	0.719622i	$-0.744315\pi$
$223$	−20.7382	−1.38873	−0.694366	+	0.719622i	$0.744315\pi$
$224$	0.969734	0.0647931
$225$	−11.1426	−0.742842
$226$	2.41954	0.160946
$227$	17.3460	1.15129	0.575646	−	0.817699i	$-0.304751\pi$
$227$	17.3460	1.15129	0.575646	+	0.817699i	$0.304751\pi$
$228$	−3.40627	−0.225586
$229$	20.7031	1.36810	0.684049	−	0.729436i	$-0.260217\pi$
$229$	20.7031	1.36810	0.684049	+	0.729436i	$0.260217\pi$
$230$	−14.4698	−0.954110
$231$	−15.9292	−1.04807
$232$	5.44758	0.357651
$233$	9.04071	0.592277	0.296138	−	0.955145i	$-0.404301\pi$
$233$	9.04071	0.592277	0.296138	+	0.955145i	$0.404301\pi$
$234$	28.4938	1.86270
$235$	−1.73379	−0.113100
$236$	6.29475	0.409753
$237$	−40.9753	−2.66163
$238$	4.49861	0.291602
$239$	−1.12309	−0.0726465	−0.0363232	−	0.999340i	$-0.511565\pi$
$239$	−1.12309	−0.0726465	−0.0363232	+	0.999340i	$0.511565\pi$
$240$	−5.09513	−0.328889
$241$	−26.8969	−1.73258	−0.866291	−	0.499540i	$-0.833502\pi$
$241$	−26.8969	−1.73258	−0.866291	+	0.499540i	$0.833502\pi$
$242$	−20.3317	−1.30697
$243$	6.39123	0.409998
$244$	−3.58449	−0.229474
$245$	−10.5208	−0.672152
$246$	−15.2764	−0.973986
$247$	−5.89343	−0.374990
$248$	−6.78772	−0.431021
$249$	50.9518	3.22894
$250$	12.1284	0.767070
$251$	−14.5477	−0.918246	−0.459123	−	0.888373i	$-0.651836\pi$
$251$	−14.5477	−0.918246	−0.459123	+	0.888373i	$0.651836\pi$
$252$	−5.44207	−0.342818
$253$	−46.6497	−2.93284
$254$	−2.56016	−0.160639
$255$	−23.6364	−1.48017
$256$	1.00000	0.0625000
$257$	−1.83563	−0.114503	−0.0572517	−	0.998360i	$-0.518234\pi$
$257$	−1.83563	−0.114503	−0.0572517	+	0.998360i	$0.518234\pi$
$258$	2.46892	0.153708
$259$	10.7784	0.669739
$260$	−8.81546	−0.546711
$261$	−30.5714	−1.89232
$262$	1.62071	0.100128
$263$	4.52022	0.278729	0.139364	−	0.990241i	$-0.455494\pi$
$263$	4.52022	0.278729	0.139364	+	0.990241i	$0.455494\pi$
$264$	−16.4264	−1.01097
$265$	−4.96004	−0.304693
$266$	1.12559	0.0690145
$267$	22.8646	1.39929
$268$	8.13548	0.496953
$269$	15.9214	0.970746	0.485373	−	0.874307i	$-0.338684\pi$
$269$	15.9214	0.970746	0.485373	+	0.874307i	$0.338684\pi$
$270$	13.3081	0.809903
$271$	26.9178	1.63514	0.817569	−	0.575831i	$-0.195321\pi$
$271$	26.9178	1.63514	0.817569	+	0.575831i	$0.195321\pi$
$272$	4.63902	0.281282
$273$	−14.4491	−0.874501
$274$	−4.22758	−0.255397
$275$	11.1140	0.670197
$276$	−24.4572	−1.47215
$277$	−19.0216	−1.14290	−0.571448	−	0.820638i	$-0.693618\pi$
$277$	−19.0216	−1.14290	−0.571448	+	0.820638i	$0.693618\pi$
$278$	12.2166	0.732701
$279$	38.0921	2.28052
$280$	1.68367	0.100619
$281$	33.4586	1.99597	0.997986	−	0.0634286i	$-0.0202035\pi$
$281$	33.4586	1.99597	0.997986	+	0.0634286i	$0.0202035\pi$
$282$	−2.93049	−0.174508
$283$	−12.1836	−0.724237	−0.362118	−	0.932132i	$-0.617946\pi$
$283$	−12.1836	−0.724237	−0.362118	+	0.932132i	$0.617946\pi$
$284$	−9.51758	−0.564765
$285$	−5.91404	−0.350317
$286$	−28.4205	−1.68054
$287$	5.04804	0.297976
$288$	−5.61192	−0.330685
$289$	4.52048	0.265911
$290$	9.45822	0.555406
$291$	4.44484	0.260561
$292$	8.69676	0.508939
$293$	9.28817	0.542620	0.271310	−	0.962492i	$-0.412543\pi$
$293$	9.28817	0.542620	0.271310	+	0.962492i	$0.412543\pi$
$294$	−17.7826	−1.03710
$295$	10.9291	0.636316
$296$	11.1148	0.646036
$297$	42.9044	2.48957
$298$	3.67818	0.213071
$299$	−42.3152	−2.44715
$300$	5.82675	0.336408
$301$	−0.815848	−0.0470247
$302$	0.832853	0.0479253
$303$	3.17897	0.182627
$304$	1.16072	0.0665721
$305$	−6.22348	−0.356355
$306$	−26.0338	−1.48825
$307$	−5.68571	−0.324501	−0.162250	−	0.986750i	$-0.551875\pi$
$307$	−5.68571	−0.324501	−0.162250	+	0.986750i	$0.551875\pi$
$308$	5.42806	0.309292
$309$	−0.0641901	−0.00365165
$310$	−11.7850	−0.669343
$311$	−18.2473	−1.03471	−0.517355	−	0.855771i	$-0.673083\pi$
$311$	−18.2473	−1.03471	−0.517355	+	0.855771i	$0.673083\pi$
$312$	−14.9001	−0.843552
$313$	8.12769	0.459404	0.229702	−	0.973261i	$-0.426225\pi$
$313$	8.12769	0.459404	0.229702	+	0.973261i	$0.426225\pi$
$314$	−15.6537	−0.883388
$315$	−9.44864	−0.532370
$316$	13.9628	0.785469
$317$	20.5473	1.15405	0.577026	−	0.816726i	$-0.304213\pi$
$317$	20.5473	1.15405	0.577026	+	0.816726i	$0.304213\pi$
$318$	−8.38359	−0.470128
$319$	30.4927	1.70726
$320$	1.73622	0.0970578
$321$	−36.0099	−2.00988
$322$	8.08182	0.450382
$323$	5.38462	0.299608
$324$	5.65787	0.314326
$325$	10.0813	0.559209
$326$	10.3750	0.574621
$327$	4.01753	0.222170
$328$	5.20560	0.287431
$329$	0.968374	0.0533882
$330$	−28.5199	−1.56997
$331$	−19.4503	−1.06908	−0.534541	−	0.845142i	$-0.679516\pi$
$331$	−19.4503	−1.06908	−0.534541	+	0.845142i	$0.679516\pi$
$332$	−17.3624	−0.952884
$333$	−62.3755	−3.41816
$334$	−0.599480	−0.0328021
$335$	14.1250	0.771731
$336$	2.84579	0.155250
$337$	−8.05826	−0.438961	−0.219481	−	0.975617i	$-0.570436\pi$
$337$	−8.05826	−0.438961	−0.219481	+	0.975617i	$0.570436\pi$
$338$	−12.7798	−0.695127
$339$	7.10041	0.385641
$340$	8.05437	0.436809
$341$	−37.9941	−2.05750
$342$	−6.51389	−0.352231
$343$	12.6643	0.683811
$344$	−0.841312	−0.0453605
$345$	−42.4632	−2.28614
$346$	13.4289	0.721941
$347$	27.7871	1.49169	0.745846	−	0.666119i	$-0.232046\pi$
$347$	27.7871	1.49169	0.745846	+	0.666119i	$0.232046\pi$
$348$	15.9865	0.856967
$349$	−22.2523	−1.19114	−0.595568	−	0.803305i	$-0.703073\pi$
$349$	−22.2523	−1.19114	−0.595568	+	0.803305i	$0.703073\pi$
$350$	−1.92544	−0.102919
$351$	38.9179	2.07728
$352$	5.59748	0.298346
$353$	17.8198	0.948451	0.474226	−	0.880403i	$-0.342728\pi$
$353$	17.8198	0.948451	0.474226	+	0.880403i	$0.342728\pi$
$354$	18.4726	0.981809
$355$	−16.5246	−0.877037
$356$	−7.79136	−0.412941
$357$	13.2017	0.698706
$358$	6.70467	0.354353
$359$	−32.7383	−1.72786	−0.863930	−	0.503611i	$-0.832004\pi$
$359$	−32.7383	−1.72786	−0.863930	+	0.503611i	$0.832004\pi$
$360$	−9.74354	−0.513530
$361$	−17.6527	−0.929091
$362$	21.0383	1.10575
$363$	−59.6656	−3.13163
$364$	4.92370	0.258072
$365$	15.0995	0.790344
$366$	−10.5191	−0.549841
$367$	−35.4344	−1.84966	−0.924829	−	0.380383i	$-0.875792\pi$
$367$	−35.4344	−1.84966	−0.924829	+	0.380383i	$0.875792\pi$
$368$	8.33406	0.434443
$369$	−29.2134	−1.52079
$370$	19.2978	1.00325
$371$	2.77034	0.143829
$372$	−19.9193	−1.03277
$373$	10.5647	0.547021	0.273510	−	0.961869i	$-0.411815\pi$
$373$	10.5647	0.547021	0.273510	+	0.961869i	$0.411815\pi$
$374$	25.9668	1.34271
$375$	35.5922	1.83797
$376$	0.998598	0.0514987
$377$	27.6594	1.42453
$378$	−7.43296	−0.382310
$379$	−17.8530	−0.917046	−0.458523	−	0.888683i	$-0.651621\pi$
$379$	−17.8530	−0.917046	−0.458523	+	0.888683i	$0.651621\pi$
$380$	2.01527	0.103381
$381$	−7.51307	−0.384906
$382$	17.6389	0.902484
$383$	21.5663	1.10199	0.550993	−	0.834510i	$-0.314249\pi$
$383$	21.5663	1.10199	0.550993	+	0.834510i	$0.314249\pi$
$384$	2.93461	0.149756
$385$	9.42432	0.480308
$386$	8.84040	0.449964
$387$	4.72137	0.240001
$388$	−1.51463	−0.0768937
$389$	−4.28700	−0.217359	−0.108680	−	0.994077i	$-0.534662\pi$
$389$	−4.28700	−0.217359	−0.108680	+	0.994077i	$0.534662\pi$
$390$	−25.8699	−1.30997
$391$	38.6619	1.95521
$392$	6.05962	0.306057
$393$	4.75614	0.239915
$394$	−6.41973	−0.323421
$395$	24.2425	1.21977
$396$	−31.4126	−1.57854
$397$	21.9150	1.09988	0.549942	−	0.835203i	$-0.314650\pi$
$397$	21.9150	1.09988	0.549942	+	0.835203i	$0.314650\pi$
$398$	−16.7976	−0.841988
$399$	3.30317	0.165365
$400$	−1.98553	−0.0992765
$401$	−12.2197	−0.610225	−0.305112	−	0.952316i	$-0.598694\pi$
$401$	−12.2197	−0.610225	−0.305112	+	0.952316i	$0.598694\pi$
$402$	23.8744	1.19075
$403$	−34.4638	−1.71677
$404$	−1.08327	−0.0538946
$405$	9.82332	0.488125
$406$	−5.28270	−0.262176
$407$	62.2150	3.08388
$408$	13.6137	0.673978
$409$	12.8807	0.636909	0.318454	−	0.947938i	$-0.396836\pi$
$409$	12.8807	0.636909	0.318454	+	0.947938i	$0.396836\pi$
$410$	9.03807	0.446359
$411$	−12.4063	−0.611957
$412$	0.0218735	0.00107763
$413$	−6.10423	−0.300370
$414$	−46.7701	−2.29862
$415$	−30.1450	−1.47976
$416$	5.07738	0.248939
$417$	35.8508	1.75562
$418$	6.49712	0.317785
$419$	20.2844	0.990957	0.495479	−	0.868620i	$-0.334993\pi$
$419$	20.2844	0.990957	0.495479	+	0.868620i	$0.334993\pi$
$420$	4.94092	0.241092
$421$	−18.3081	−0.892284	−0.446142	−	0.894962i	$-0.647202\pi$
$421$	−18.3081	−0.892284	−0.446142	+	0.894962i	$0.647202\pi$
$422$	−15.0713	−0.733659
$423$	−5.60405	−0.272478
$424$	2.85680	0.138739
$425$	−9.21091	−0.446795
$426$	−27.9304	−1.35323
$427$	3.47600	0.168216
$428$	12.2708	0.593131
$429$	−83.4030	−4.02673
$430$	−1.46070	−0.0704414
$431$	7.43296	0.358033	0.179017	−	0.983846i	$-0.442708\pi$
$431$	7.43296	0.358033	0.179017	+	0.983846i	$0.442708\pi$
$432$	−7.66495	−0.368780
$433$	−2.68774	−0.129165	−0.0645823	−	0.997912i	$-0.520571\pi$
$433$	−2.68774	−0.129165	−0.0645823	+	0.997912i	$0.520571\pi$
$434$	6.58228	0.315960
$435$	27.7561	1.33081
$436$	−1.36902	−0.0655641
$437$	9.67354	0.462748
$438$	25.5216	1.21947
$439$	−40.8831	−1.95124	−0.975621	−	0.219463i	$-0.929569\pi$
$439$	−40.8831	−1.95124	−0.975621	+	0.219463i	$0.929569\pi$
$440$	9.71846	0.463310
$441$	−34.0061	−1.61934
$442$	23.5540	1.12035
$443$	3.69104	0.175367	0.0876833	−	0.996148i	$-0.472054\pi$
$443$	3.69104	0.175367	0.0876833	+	0.996148i	$0.472054\pi$
$444$	32.6177	1.54797
$445$	−13.5275	−0.641266
$446$	20.7382	0.981982
$447$	10.7940	0.510539
$448$	−0.969734	−0.0458156
$449$	17.2332	0.813287	0.406644	−	0.913587i	$-0.366699\pi$
$449$	17.2332	0.813287	0.406644	+	0.913587i	$0.366699\pi$
$450$	11.1426	0.525269
$451$	29.1382	1.37206
$452$	−2.41954	−0.113806
$453$	2.44410	0.114834
$454$	−17.3460	−0.814087
$455$	8.54864	0.400767
$456$	3.40627	0.159513
$457$	39.1433	1.83105	0.915524	−	0.402264i	$-0.131777\pi$
$457$	39.1433	1.83105	0.915524	+	0.402264i	$0.131777\pi$
$458$	−20.7031	−0.967391
$459$	−35.5579	−1.65970
$460$	14.4698	0.674658
$461$	−7.39099	−0.344233	−0.172116	−	0.985077i	$-0.555061\pi$
$461$	−7.39099	−0.344233	−0.172116	+	0.985077i	$0.555061\pi$
$462$	15.9292	0.741095
$463$	26.8790	1.24917	0.624586	−	0.780956i	$-0.285268\pi$
$463$	26.8790	1.24917	0.624586	+	0.780956i	$0.285268\pi$
$464$	−5.44758	−0.252898
$465$	−34.5843	−1.60381
$466$	−9.04071	−0.418803
$467$	−21.2586	−0.983730	−0.491865	−	0.870671i	$-0.663685\pi$
$467$	−21.2586	−0.983730	−0.491865	+	0.870671i	$0.663685\pi$
$468$	−28.4938	−1.31713
$469$	−7.88924	−0.364291
$470$	1.73379	0.0799737
$471$	−45.9374	−2.11668
$472$	−6.29475	−0.289739
$473$	−4.70922	−0.216530
$474$	40.9753	1.88206
$475$	−2.30465	−0.105745
$476$	−4.49861	−0.206194
$477$	−16.0321	−0.734061
$478$	1.12309	0.0513688
$479$	32.8925	1.50290	0.751448	−	0.659792i	$-0.229356\pi$
$479$	32.8925	1.50290	0.751448	+	0.659792i	$0.229356\pi$
$480$	5.09513	0.232560
$481$	56.4342	2.57318
$482$	26.8969	1.22512
$483$	23.7170	1.07916
$484$	20.3317	0.924170
$485$	−2.62973	−0.119410
$486$	−6.39123	−0.289912
$487$	38.1817	1.73018	0.865088	−	0.501620i	$-0.167262\pi$
$487$	38.1817	1.73018	0.865088	+	0.501620i	$0.167262\pi$
$488$	3.58449	0.162262
$489$	30.4467	1.37685
$490$	10.5208	0.475283
$491$	37.9964	1.71475	0.857376	−	0.514691i	$-0.172093\pi$
$491$	37.9964	1.71475	0.857376	+	0.514691i	$0.172093\pi$
$492$	15.2764	0.688712
$493$	−25.2714	−1.13817
$494$	5.89343	0.265158
$495$	−54.5392	−2.45136
$496$	6.78772	0.304778
$497$	9.22952	0.414001
$498$	−50.9518	−2.28320
$499$	19.9074	0.891176	0.445588	−	0.895238i	$-0.352995\pi$
$499$	19.9074	0.891176	0.445588	+	0.895238i	$0.352995\pi$
$500$	−12.1284	−0.542400
$501$	−1.75924	−0.0785970
$502$	14.5477	0.649298
$503$	8.32446	0.371169	0.185585	−	0.982628i	$-0.440582\pi$
$503$	8.32446	0.371169	0.185585	+	0.982628i	$0.440582\pi$
$504$	5.44207	0.242409
$505$	−1.88080	−0.0836943
$506$	46.6497	2.07383
$507$	−37.5036	−1.66559
$508$	2.56016	0.113589
$509$	20.3976	0.904106	0.452053	−	0.891991i	$-0.350692\pi$
$509$	20.3976	0.904106	0.452053	+	0.891991i	$0.350692\pi$
$510$	23.6364	1.04664
$511$	−8.43354	−0.373078
$512$	−1.00000	−0.0441942
$513$	−8.89689	−0.392807
$514$	1.83563	0.0809661
$515$	0.0379773	0.00167348
$516$	−2.46892	−0.108688
$517$	5.58963	0.245831
$518$	−10.7784	−0.473577
$519$	39.4085	1.72984
$520$	8.81546	0.386583
$521$	−34.4011	−1.50714	−0.753569	−	0.657369i	$-0.771669\pi$
$521$	−34.4011	−1.50714	−0.753569	+	0.657369i	$0.771669\pi$
$522$	30.5714	1.33807
$523$	38.5727	1.68667	0.843333	−	0.537391i	$-0.180590\pi$
$523$	38.5727	1.68667	0.843333	+	0.537391i	$0.180590\pi$
$524$	−1.62071	−0.0708009
$525$	−5.65040	−0.246604
$526$	−4.52022	−0.197091
$527$	31.4884	1.37165
$528$	16.4264	0.714867
$529$	46.4566	2.01985
$530$	4.96004	0.215451
$531$	35.3256	1.53300
$532$	−1.12559	−0.0488006
$533$	26.4308	1.14484
$534$	−22.8646	−0.989447
$535$	21.3048	0.921087
$536$	−8.13548	−0.351399
$537$	19.6756	0.849064
$538$	−15.9214	−0.686421
$539$	33.9186	1.46098
$540$	−13.3081	−0.572688
$541$	15.3445	0.659712	0.329856	−	0.944031i	$-0.393000\pi$
$541$	15.3445	0.659712	0.329856	+	0.944031i	$0.393000\pi$
$542$	−26.9178	−1.15622
$543$	61.7392	2.64948
$544$	−4.63902	−0.198896
$545$	−2.37692	−0.101816
$546$	14.4491	0.618366
$547$	−21.9635	−0.939089	−0.469545	−	0.882909i	$-0.655582\pi$
$547$	−21.9635	−0.939089	−0.469545	+	0.882909i	$0.655582\pi$
$548$	4.22758	0.180593
$549$	−20.1159	−0.858525
$550$	−11.1140	−0.473901
$551$	−6.32314	−0.269375
$552$	24.4572	1.04097
$553$	−13.5402	−0.575788
$554$	19.0216	0.808150
$555$	56.6315	2.40387
$556$	−12.2166	−0.518098
$557$	15.5796	0.660130	0.330065	−	0.943958i	$-0.392929\pi$
$557$	15.5796	0.660130	0.330065	+	0.943958i	$0.392929\pi$
$558$	−38.0921	−1.61257
$559$	−4.27166	−0.180672
$560$	−1.68367	−0.0711482
$561$	76.2023	3.21726
$562$	−33.4586	−1.41137
$563$	6.33660	0.267056	0.133528	−	0.991045i	$-0.457369\pi$
$563$	6.33660	0.267056	0.133528	+	0.991045i	$0.457369\pi$
$564$	2.93049	0.123396
$565$	−4.20086	−0.176732
$566$	12.1836	0.512113
$567$	−5.48663	−0.230417
$568$	9.51758	0.399349
$569$	−7.83139	−0.328309	−0.164154	−	0.986435i	$-0.552490\pi$
$569$	−7.83139	−0.328309	−0.164154	+	0.986435i	$0.552490\pi$
$570$	5.91404	0.247712
$571$	−18.9693	−0.793839	−0.396919	−	0.917853i	$-0.629921\pi$
$571$	−18.9693	−0.793839	−0.396919	+	0.917853i	$0.629921\pi$
$572$	28.4205	1.18832
$573$	51.7632	2.16244
$574$	−5.04804	−0.210701
$575$	−16.5475	−0.690080
$576$	5.61192	0.233830
$577$	−24.5903	−1.02371	−0.511853	−	0.859073i	$-0.671041\pi$
$577$	−24.5903	−1.02371	−0.511853	+	0.859073i	$0.671041\pi$
$578$	−4.52048	−0.188027
$579$	25.9431	1.07816
$580$	−9.45822	−0.392731
$581$	16.8369	0.698511
$582$	−4.44484	−0.184245
$583$	15.9909	0.662275
$584$	−8.69676	−0.359874
$585$	−49.4716	−2.04540
$586$	−9.28817	−0.383691
$587$	47.0341	1.94131	0.970653	−	0.240483i	$-0.0773057\pi$
$587$	47.0341	1.94131	0.970653	+	0.240483i	$0.0773057\pi$
$588$	17.7826	0.733342
$589$	7.87867	0.324635
$590$	−10.9291	−0.449944
$591$	−18.8394	−0.774948
$592$	−11.1148	−0.456817
$593$	19.9052	0.817409	0.408705	−	0.912667i	$-0.365981\pi$
$593$	19.9052	0.817409	0.408705	+	0.912667i	$0.365981\pi$
$594$	−42.9044	−1.76039
$595$	−7.81059	−0.320203
$596$	−3.67818	−0.150664
$597$	−49.2944	−2.01748
$598$	42.3152	1.73040
$599$	−29.5929	−1.20913	−0.604566	−	0.796555i	$-0.706653\pi$
$599$	−29.5929	−1.20913	−0.604566	+	0.796555i	$0.706653\pi$
$600$	−5.82675	−0.237876
$601$	−17.5753	−0.716913	−0.358457	−	0.933546i	$-0.616697\pi$
$601$	−17.5753	−0.716913	−0.358457	+	0.933546i	$0.616697\pi$
$602$	0.815848	0.0332515
$603$	45.6556	1.85924
$604$	−0.832853	−0.0338883
$605$	35.3004	1.43517
$606$	−3.17897	−0.129137
$607$	34.7179	1.40916	0.704578	−	0.709627i	$-0.251136\pi$
$607$	34.7179	1.40916	0.704578	+	0.709627i	$0.251136\pi$
$608$	−1.16072	−0.0470736
$609$	−15.5027	−0.628199
$610$	6.22348	0.251981
$611$	5.07026	0.205121
$612$	26.0338	1.05235
$613$	−10.0265	−0.404967	−0.202484	−	0.979286i	$-0.564901\pi$
$613$	−10.0265	−0.404967	−0.202484	+	0.979286i	$0.564901\pi$
$614$	5.68571	0.229457
$615$	26.5232	1.06952
$616$	−5.42806	−0.218703
$617$	23.7495	0.956120	0.478060	−	0.878327i	$-0.341340\pi$
$617$	23.7495	0.956120	0.478060	+	0.878327i	$0.341340\pi$
$618$	0.0641901	0.00258211
$619$	19.9963	0.803720	0.401860	−	0.915701i	$-0.368364\pi$
$619$	19.9963	0.803720	0.401860	+	0.915701i	$0.368364\pi$
$620$	11.7850	0.473297
$621$	−63.8802	−2.56342
$622$	18.2473	0.731651
$623$	7.55554	0.302706
$624$	14.9001	0.596482
$625$	−11.1300	−0.445201
$626$	−8.12769	−0.324848
$627$	19.0665	0.761443
$628$	15.6537	0.624650
$629$	−51.5619	−2.05591
$630$	9.44864	0.376443
$631$	22.2279	0.884877	0.442439	−	0.896799i	$-0.354113\pi$
$631$	22.2279	0.884877	0.442439	+	0.896799i	$0.354113\pi$
$632$	−13.9628	−0.555410
$633$	−44.2283	−1.75792
$634$	−20.5473	−0.816037
$635$	4.44501	0.176395
$636$	8.38359	0.332431
$637$	30.7670	1.21903
$638$	−30.4927	−1.20722
$639$	−53.4119	−2.11294
$640$	−1.73622	−0.0686302
$641$	11.5431	0.455926	0.227963	−	0.973670i	$-0.426794\pi$
$641$	11.5431	0.455926	0.227963	+	0.973670i	$0.426794\pi$
$642$	36.0099	1.42120
$643$	−45.0879	−1.77809	−0.889047	−	0.457816i	$-0.848632\pi$
$643$	−45.0879	−1.77809	−0.889047	+	0.457816i	$0.848632\pi$
$644$	−8.08182	−0.318468
$645$	−4.28659	−0.168784
$646$	−5.38462	−0.211855
$647$	5.54638	0.218051	0.109025	−	0.994039i	$-0.465227\pi$
$647$	5.54638	0.218051	0.109025	+	0.994039i	$0.465227\pi$
$648$	−5.65787	−0.222262
$649$	−35.2347	−1.38308
$650$	−10.0813	−0.395421
$651$	19.3164	0.757070
$652$	−10.3750	−0.406318
$653$	46.1850	1.80736	0.903679	−	0.428211i	$-0.140856\pi$
$653$	46.1850	1.80736	0.903679	+	0.428211i	$0.140856\pi$
$654$	−4.01753	−0.157098
$655$	−2.81391	−0.109948
$656$	−5.20560	−0.203244
$657$	48.8055	1.90408
$658$	−0.968374	−0.0377511
$659$	−10.2559	−0.399514	−0.199757	−	0.979845i	$-0.564015\pi$
$659$	−10.2559	−0.399514	−0.199757	+	0.979845i	$0.564015\pi$
$660$	28.5199	1.11013
$661$	33.1065	1.28769	0.643847	−	0.765154i	$-0.277337\pi$
$661$	33.1065	1.28769	0.643847	+	0.765154i	$0.277337\pi$
$662$	19.4503	0.755956
$663$	69.1219	2.68447
$664$	17.3624	0.673791
$665$	−1.95428	−0.0757837
$666$	62.3755	2.41700
$667$	−45.4005	−1.75791
$668$	0.599480	0.0231946
$669$	60.8585	2.35292
$670$	−14.1250	−0.545696
$671$	20.0641	0.774566
$672$	−2.84579	−0.109779
$673$	39.6904	1.52995	0.764977	−	0.644057i	$-0.222750\pi$
$673$	39.6904	1.52995	0.764977	+	0.644057i	$0.222750\pi$
$674$	8.05826	0.310392
$675$	15.2190	0.585780
$676$	12.7798	0.491529
$677$	26.2019	1.00702	0.503510	−	0.863989i	$-0.332042\pi$
$677$	26.2019	1.00702	0.503510	+	0.863989i	$0.332042\pi$
$678$	−7.10041	−0.272689
$679$	1.46879	0.0563669
$680$	−8.05437	−0.308871
$681$	−50.9036	−1.95063
$682$	37.9941	1.45487
$683$	30.1594	1.15402	0.577009	−	0.816738i	$-0.304220\pi$
$683$	30.1594	1.15402	0.577009	+	0.816738i	$0.304220\pi$
$684$	6.51389	0.249065
$685$	7.34002	0.280448
$686$	−12.6643	−0.483527
$687$	−60.7554	−2.31796
$688$	0.841312	0.0320747
$689$	14.5051	0.552599
$690$	42.4632	1.61654
$691$	40.0267	1.52269	0.761343	−	0.648349i	$-0.224540\pi$
$691$	40.0267	1.52269	0.761343	+	0.648349i	$0.224540\pi$
$692$	−13.4289	−0.510489
$693$	30.4618	1.15715
$694$	−27.7871	−1.05479
$695$	−21.2107	−0.804567
$696$	−15.9865	−0.605967
$697$	−24.1489	−0.914703
$698$	22.2523	0.842260
$699$	−26.5309	−1.00349
$700$	1.92544	0.0727746
$701$	−29.3923	−1.11013	−0.555067	−	0.831806i	$-0.687307\pi$
$701$	−29.3923	−1.11013	−0.555067	+	0.831806i	$0.687307\pi$
$702$	−38.9179	−1.46886
$703$	−12.9012	−0.486580
$704$	−5.59748	−0.210963
$705$	5.08799	0.191625
$706$	−17.8198	−0.670656
$707$	1.05048	0.0395074
$708$	−18.4726	−0.694244
$709$	4.92938	0.185127	0.0925634	−	0.995707i	$-0.470494\pi$
$709$	4.92938	0.185127	0.0925634	+	0.995707i	$0.470494\pi$
$710$	16.5246	0.620159
$711$	78.3581	2.93866
$712$	7.79136	0.291993
$713$	56.5693	2.11854
$714$	−13.2017	−0.494060
$715$	49.3443	1.84537
$716$	−6.70467	−0.250565
$717$	3.29582	0.123085
$718$	32.7383	1.22178
$719$	41.5432	1.54930	0.774650	−	0.632390i	$-0.217926\pi$
$719$	41.5432	1.54930	0.774650	+	0.632390i	$0.217926\pi$
$720$	9.74354	0.363120
$721$	−0.0212115	−0.000789956 0
$722$	17.6527	0.656966
$723$	78.9318	2.93551
$724$	−21.0383	−0.781883
$725$	10.8163	0.401709
$726$	59.6656	2.21440
$727$	−41.9372	−1.55536	−0.777681	−	0.628659i	$-0.783604\pi$
$727$	−41.9372	−1.55536	−0.777681	+	0.628659i	$0.783604\pi$
$728$	−4.92370	−0.182485
$729$	−35.7294	−1.32331
$730$	−15.0995	−0.558858
$731$	3.90286	0.144352
$732$	10.5191	0.388796
$733$	−31.9386	−1.17968	−0.589839	−	0.807521i	$-0.700809\pi$
$733$	−31.9386	−1.17968	−0.589839	+	0.807521i	$0.700809\pi$
$734$	35.4344	1.30791
$735$	30.8745	1.13882
$736$	−8.33406	−0.307198
$737$	−45.5381	−1.67742
$738$	29.2134	1.07536
$739$	2.94365	0.108284	0.0541419	−	0.998533i	$-0.482758\pi$
$739$	2.94365	0.108284	0.0541419	+	0.998533i	$0.482758\pi$
$740$	−19.2978	−0.709402
$741$	17.2949	0.635344
$742$	−2.77034	−0.101702
$743$	28.7128	1.05337	0.526686	−	0.850060i	$-0.323434\pi$
$743$	28.7128	1.05337	0.526686	+	0.850060i	$0.323434\pi$
$744$	19.9193	0.730277
$745$	−6.38614	−0.233970
$746$	−10.5647	−0.386802
$747$	−97.4362	−3.56501
$748$	−25.9668	−0.949440
$749$	−11.8994	−0.434794
$750$	−35.5922	−1.29964
$751$	−38.8152	−1.41638	−0.708192	−	0.706019i	$-0.750489\pi$
$751$	−38.8152	−1.41638	−0.708192	+	0.706019i	$0.750489\pi$
$752$	−0.998598	−0.0364151
$753$	42.6919	1.55578
$754$	−27.6594	−1.00730
$755$	−1.44602	−0.0526260
$756$	7.43296	0.270334
$757$	5.40087	0.196298	0.0981490	−	0.995172i	$-0.468708\pi$
$757$	5.40087	0.196298	0.0981490	+	0.995172i	$0.468708\pi$
$758$	17.8530	0.648449
$759$	136.899	4.96910
$760$	−2.01527	−0.0731017
$761$	−33.5624	−1.21664	−0.608318	−	0.793693i	$-0.708155\pi$
$761$	−33.5624	−1.21664	−0.608318	+	0.793693i	$0.708155\pi$
$762$	7.51307	0.272170
$763$	1.32758	0.0480617
$764$	−17.6389	−0.638153
$765$	45.2005	1.63423
$766$	−21.5663	−0.779222
$767$	−31.9608	−1.15404
$768$	−2.93461	−0.105894
$769$	43.7374	1.57721	0.788605	−	0.614900i	$-0.210804\pi$
$769$	43.7374	1.57721	0.788605	+	0.614900i	$0.210804\pi$
$770$	−9.42432	−0.339629
$771$	5.38685	0.194003
$772$	−8.84040	−0.318173
$773$	−28.7498	−1.03406	−0.517029	−	0.855968i	$-0.672962\pi$
$773$	−28.7498	−1.03406	−0.517029	+	0.855968i	$0.672962\pi$
$774$	−4.72137	−0.169706
$775$	−13.4772	−0.484116
$776$	1.51463	0.0543720
$777$	−31.6304	−1.13474
$778$	4.28700	0.153696
$779$	−6.04226	−0.216486
$780$	25.8699	0.926291
$781$	53.2744	1.90631
$782$	−38.6619	−1.38255
$783$	41.7555	1.49222
$784$	−6.05962	−0.216415
$785$	27.1783	0.970034
$786$	−4.75614	−0.169646
$787$	0.501539	0.0178779	0.00893897	−	0.999960i	$-0.497155\pi$
$787$	0.501539	0.0178779	0.00893897	+	0.999960i	$0.497155\pi$
$788$	6.41973	0.228693
$789$	−13.2651	−0.472249
$790$	−24.2425	−0.862511
$791$	2.34631	0.0834252
$792$	31.4126	1.11620
$793$	18.1998	0.646295
$794$	−21.9150	−0.777736
$795$	14.5558	0.516240
$796$	16.7976	0.595375
$797$	2.00065	0.0708665	0.0354332	−	0.999372i	$-0.488719\pi$
$797$	2.00065	0.0708665	0.0354332	+	0.999372i	$0.488719\pi$
$798$	−3.30317	−0.116931
$799$	−4.63251	−0.163887
$800$	1.98553	0.0701991
$801$	−43.7245	−1.54493
$802$	12.2197	0.431494
$803$	−48.6799	−1.71788
$804$	−23.8744	−0.841986
$805$	−14.0318	−0.494557
$806$	34.4638	1.21394
$807$	−46.7231	−1.64473
$808$	1.08327	0.0381092
$809$	10.7594	0.378282	0.189141	−	0.981950i	$-0.439430\pi$
$809$	10.7594	0.378282	0.189141	+	0.981950i	$0.439430\pi$
$810$	−9.82332	−0.345156
$811$	−11.2000	−0.393284	−0.196642	−	0.980475i	$-0.563004\pi$
$811$	−11.2000	−0.393284	−0.196642	+	0.980475i	$0.563004\pi$
$812$	5.28270	0.185387
$813$	−78.9931	−2.77041
$814$	−62.2150	−2.18063
$815$	−18.0134	−0.630982
$816$	−13.6137	−0.476575
$817$	0.976530	0.0341645
$818$	−12.8807	−0.450363
$819$	27.6314	0.965520
$820$	−9.03807	−0.315623
$821$	37.9997	1.32620	0.663100	−	0.748531i	$-0.269241\pi$
$821$	37.9997	1.32620	0.663100	+	0.748531i	$0.269241\pi$
$822$	12.4063	0.432719
$823$	37.6727	1.31319	0.656594	−	0.754245i	$-0.271997\pi$
$823$	37.6727	1.31319	0.656594	+	0.754245i	$0.271997\pi$
$824$	−0.0218735	−0.000762000 0
$825$	−32.6151	−1.13551
$826$	6.10423	0.212393
$827$	−34.8576	−1.21212	−0.606059	−	0.795420i	$-0.707251\pi$
$827$	−34.8576	−1.21212	−0.606059	+	0.795420i	$0.707251\pi$
$828$	46.7701	1.62537
$829$	52.1624	1.81168	0.905838	−	0.423624i	$-0.139242\pi$
$829$	52.1624	1.81168	0.905838	+	0.423624i	$0.139242\pi$
$830$	30.1450	1.04635
$831$	55.8209	1.93640
$832$	−5.07738	−0.176026
$833$	−28.1107	−0.973977
$834$	−35.8508	−1.24141
$835$	1.04083	0.0360195
$836$	−6.49712	−0.224708
$837$	−52.0276	−1.79834
$838$	−20.2844	−0.700713
$839$	−20.4438	−0.705798	−0.352899	−	0.935661i	$-0.614804\pi$
$839$	−20.4438	−0.705798	−0.352899	+	0.935661i	$0.614804\pi$
$840$	−4.94092	−0.170478
$841$	0.676151	0.0233156
$842$	18.3081	0.630940
$843$	−98.1878	−3.38177
$844$	15.0713	0.518775
$845$	22.1885	0.763308
$846$	5.60405	0.192671
$847$	−19.7164	−0.677462
$848$	−2.85680	−0.0981030
$849$	35.7539	1.22707
$850$	9.21091	0.315932
$851$	−92.6317	−3.17537
$852$	27.9304	0.956879
$853$	10.2350	0.350438	0.175219	−	0.984529i	$-0.443937\pi$
$853$	10.2350	0.350438	0.175219	+	0.984529i	$0.443937\pi$
$854$	−3.47600	−0.118946
$855$	11.3096	0.386779
$856$	−12.2708	−0.419407
$857$	−31.4030	−1.07271	−0.536354	−	0.843993i	$-0.680199\pi$
$857$	−31.4030	−1.07271	−0.536354	+	0.843993i	$0.680199\pi$
$858$	83.4030	2.84733
$859$	−22.0748	−0.753184	−0.376592	−	0.926379i	$-0.622904\pi$
$859$	−22.0748	−0.753184	−0.376592	+	0.926379i	$0.622904\pi$
$860$	1.46070	0.0498096
$861$	−14.8140	−0.504860
$862$	−7.43296	−0.253168
$863$	−17.4248	−0.593147	−0.296574	−	0.955010i	$-0.595844\pi$
$863$	−17.4248	−0.593147	−0.296574	+	0.955010i	$0.595844\pi$
$864$	7.66495	0.260767
$865$	−23.3155	−0.792752
$866$	2.68774	0.0913331
$867$	−13.2658	−0.450532
$868$	−6.58228	−0.223417
$869$	−78.1564	−2.65128
$870$	−27.7561	−0.941021
$871$	−41.3069	−1.39963
$872$	1.36902	0.0463608
$873$	−8.49998	−0.287681
$874$	−9.67354	−0.327213
$875$	11.7614	0.397606
$876$	−25.5216	−0.862294
$877$	−1.96363	−0.0663069	−0.0331535	−	0.999450i	$-0.510555\pi$
$877$	−1.96363	−0.0663069	−0.0331535	+	0.999450i	$0.510555\pi$
$878$	40.8831	1.37974
$879$	−27.2571	−0.919360
$880$	−9.71846	−0.327609
$881$	13.1791	0.444014	0.222007	−	0.975045i	$-0.428739\pi$
$881$	13.1791	0.444014	0.222007	+	0.975045i	$0.428739\pi$
$882$	34.0061	1.14504
$883$	34.7747	1.17026	0.585131	−	0.810939i	$-0.301043\pi$
$883$	34.7747	1.17026	0.585131	+	0.810939i	$0.301043\pi$
$884$	−23.5540	−0.792208
$885$	−32.0726	−1.07811
$886$	−3.69104	−0.124003
$887$	41.7789	1.40280	0.701398	−	0.712770i	$-0.252559\pi$
$887$	41.7789	1.40280	0.701398	+	0.712770i	$0.252559\pi$
$888$	−32.6177	−1.09458
$889$	−2.48267	−0.0832662
$890$	13.5275	0.453444
$891$	−31.6698	−1.06098
$892$	−20.7382	−0.694366
$893$	−1.15910	−0.0387877
$894$	−10.7940	−0.361006
$895$	−11.6408	−0.389109
$896$	0.969734	0.0323965
$897$	124.178	4.14620
$898$	−17.2332	−0.575081
$899$	−36.9767	−1.23324
$900$	−11.1426	−0.371421
$901$	−13.2528	−0.441513
$902$	−29.1382	−0.970196
$903$	2.39419	0.0796738
$904$	2.41954	0.0804728
$905$	−36.5272	−1.21421
$906$	−2.44410	−0.0811996
$907$	−29.4064	−0.976423	−0.488211	−	0.872725i	$-0.662350\pi$
$907$	−29.4064	−0.976423	−0.488211	+	0.872725i	$0.662350\pi$
$908$	17.3460	0.575646
$909$	−6.07921	−0.201635
$910$	−8.54864	−0.283385
$911$	−12.9355	−0.428571	−0.214285	−	0.976771i	$-0.568742\pi$
$911$	−12.9355	−0.428571	−0.214285	+	0.976771i	$0.568742\pi$
$912$	−3.40627	−0.112793
$913$	97.1855	3.21637
$914$	−39.1433	−1.29475
$915$	18.2635	0.603771
$916$	20.7031	0.684049
$917$	1.57165	0.0519006
$918$	35.5579	1.17358
$919$	−0.578930	−0.0190971	−0.00954856	−	0.999954i	$-0.503039\pi$
$919$	−0.578930	−0.0190971	−0.00954856	+	0.999954i	$0.503039\pi$
$920$	−14.4698	−0.477055
$921$	16.6853	0.549800
$922$	7.39099	0.243409
$923$	48.3244	1.59062
$924$	−15.9292	−0.524033
$925$	22.0688	0.725619
$926$	−26.8790	−0.883298
$927$	0.122752	0.00403171
$928$	5.44758	0.178826
$929$	17.3036	0.567712	0.283856	−	0.958867i	$-0.408386\pi$
$929$	17.3036	0.567712	0.283856	+	0.958867i	$0.408386\pi$
$930$	34.5843	1.13406
$931$	−7.03354	−0.230515
$932$	9.04071	0.296138
$933$	53.5487	1.75311
$934$	21.2586	0.695602
$935$	−45.0841	−1.47441
$936$	28.4938	0.931350
$937$	−17.8150	−0.581989	−0.290995	−	0.956725i	$-0.593986\pi$
$937$	−17.8150	−0.581989	−0.290995	+	0.956725i	$0.593986\pi$
$938$	7.88924	0.257593
$939$	−23.8516	−0.778367
$940$	−1.73379	−0.0565499
$941$	−20.4653	−0.667150	−0.333575	−	0.942724i	$-0.608255\pi$
$941$	−20.4653	−0.667150	−0.333575	+	0.942724i	$0.608255\pi$
$942$	45.9374	1.49672
$943$	−43.3838	−1.41277
$944$	6.29475	0.204877
$945$	12.9053	0.419809
$946$	4.70922	0.153110
$947$	14.1382	0.459431	0.229716	−	0.973258i	$-0.426220\pi$
$947$	14.1382	0.459431	0.229716	+	0.973258i	$0.426220\pi$
$948$	−40.9753	−1.33082
$949$	−44.1567	−1.43339
$950$	2.30465	0.0747728
$951$	−60.2982	−1.95530
$952$	4.49861	0.145801
$953$	30.4788	0.987305	0.493652	−	0.869659i	$-0.335661\pi$
$953$	30.4788	0.987305	0.493652	+	0.869659i	$0.335661\pi$
$954$	16.0321	0.519060
$955$	−30.6250	−0.991003
$956$	−1.12309	−0.0363232
$957$	−89.4841	−2.89261
$958$	−32.8925	−1.06271
$959$	−4.09963	−0.132384
$960$	−5.09513	−0.164445
$961$	15.0732	0.486231
$962$	−56.4342	−1.81951
$963$	68.8626	2.21907
$964$	−26.8969	−0.866291
$965$	−15.3489	−0.494098
$966$	−23.7170	−0.763081
$967$	5.03801	0.162011	0.0810057	−	0.996714i	$-0.474187\pi$
$967$	5.03801	0.162011	0.0810057	+	0.996714i	$0.474187\pi$
$968$	−20.3317	−0.653487
$969$	−15.8017	−0.507625
$970$	2.62973	0.0844357
$971$	36.0266	1.15615	0.578074	−	0.815984i	$-0.303804\pi$
$971$	36.0266	1.15615	0.578074	+	0.815984i	$0.303804\pi$
$972$	6.39123	0.204999
$973$	11.8468	0.379791
$974$	−38.1817	−1.22342
$975$	−29.5846	−0.947466
$976$	−3.58449	−0.114737
$977$	−36.1702	−1.15719	−0.578594	−	0.815616i	$-0.696398\pi$
$977$	−36.1702	−1.15719	−0.578594	+	0.815616i	$0.696398\pi$
$978$	−30.4467	−0.973578
$979$	43.6119	1.39384
$980$	−10.5208	−0.336076
$981$	−7.68282	−0.245293
$982$	−37.9964	−1.21251
$983$	39.4426	1.25802	0.629012	−	0.777396i	$-0.283460\pi$
$983$	39.4426	1.25802	0.629012	+	0.777396i	$0.283460\pi$
$984$	−15.2764	−0.486993
$985$	11.1461	0.355143
$986$	25.2714	0.804806
$987$	−2.84180	−0.0904554
$988$	−5.89343	−0.187495
$989$	7.01154	0.222954
$990$	54.5392	1.73337
$991$	−50.6112	−1.60772	−0.803858	−	0.594821i	$-0.797223\pi$
$991$	−50.6112	−1.60772	−0.803858	+	0.594821i	$0.797223\pi$
$992$	−6.78772	−0.215510
$993$	57.0789	1.81134
$994$	−9.22952	−0.292743
$995$	29.1644	0.924573
$996$	50.9518	1.61447
$997$	39.3897	1.24748	0.623742	−	0.781631i	$-0.285612\pi$
$997$	39.3897	1.24748	0.623742	+	0.781631i	$0.285612\pi$
$998$	−19.9074	−0.630157
$999$	85.1946	2.69544

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.e.1.4	✓	77

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.e.1.4	✓	77	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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.93461 q^{3} +1.00000 q^{4} +1.73622 q^{5} +2.93461 q^{6} -0.969734 q^{7} -1.00000 q^{8} +5.61192 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.93461 q^{3} +1.00000 q^{4} +1.73622 q^{5} +2.93461 q^{6} -0.969734 q^{7} -1.00000 q^{8} +5.61192 q^{9} -1.73622 q^{10} -5.59748 q^{11} -2.93461 q^{12} -5.07738 q^{13} +0.969734 q^{14} -5.09513 q^{15} +1.00000 q^{16} +4.63902 q^{17} -5.61192 q^{18} +1.16072 q^{19} +1.73622 q^{20} +2.84579 q^{21} +5.59748 q^{22} +8.33406 q^{23} +2.93461 q^{24} -1.98553 q^{25} +5.07738 q^{26} -7.66495 q^{27} -0.969734 q^{28} -5.44758 q^{29} +5.09513 q^{30} +6.78772 q^{31} -1.00000 q^{32} +16.4264 q^{33} -4.63902 q^{34} -1.68367 q^{35} +5.61192 q^{36} -11.1148 q^{37} -1.16072 q^{38} +14.9001 q^{39} -1.73622 q^{40} -5.20560 q^{41} -2.84579 q^{42} +0.841312 q^{43} -5.59748 q^{44} +9.74354 q^{45} -8.33406 q^{46} -0.998598 q^{47} -2.93461 q^{48} -6.05962 q^{49} +1.98553 q^{50} -13.6137 q^{51} -5.07738 q^{52} -2.85680 q^{53} +7.66495 q^{54} -9.71846 q^{55} +0.969734 q^{56} -3.40627 q^{57} +5.44758 q^{58} +6.29475 q^{59} -5.09513 q^{60} -3.58449 q^{61} -6.78772 q^{62} -5.44207 q^{63} +1.00000 q^{64} -8.81546 q^{65} -16.4264 q^{66} +8.13548 q^{67} +4.63902 q^{68} -24.4572 q^{69} +1.68367 q^{70} -9.51758 q^{71} -5.61192 q^{72} +8.69676 q^{73} +11.1148 q^{74} +5.82675 q^{75} +1.16072 q^{76} +5.42806 q^{77} -14.9001 q^{78} +13.9628 q^{79} +1.73622 q^{80} +5.65787 q^{81} +5.20560 q^{82} -17.3624 q^{83} +2.84579 q^{84} +8.05437 q^{85} -0.841312 q^{86} +15.9865 q^{87} +5.59748 q^{88} -7.79136 q^{89} -9.74354 q^{90} +4.92370 q^{91} +8.33406 q^{92} -19.9193 q^{93} +0.998598 q^{94} +2.01527 q^{95} +2.93461 q^{96} -1.51463 q^{97} +6.05962 q^{98} -31.4126 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * q^99

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