LMFDB - Embedded newform 8002.2.a.e.1.42

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			1.42
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)

$f(q)$	$=$	$q-1.00000 q^{2} +0.705213 q^{3} +1.00000 q^{4} -2.52692 q^{5} -0.705213 q^{6} +3.12362 q^{7} -1.00000 q^{8} -2.50268 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.705213 q^{3} +1.00000 q^{4} -2.52692 q^{5} -0.705213 q^{6} +3.12362 q^{7} -1.00000 q^{8} -2.50268 q^{9} +2.52692 q^{10} +0.877506 q^{11} +0.705213 q^{12} -6.10826 q^{13} -3.12362 q^{14} -1.78202 q^{15} +1.00000 q^{16} +0.902592 q^{17} +2.50268 q^{18} +7.31283 q^{19} -2.52692 q^{20} +2.20282 q^{21} -0.877506 q^{22} +7.35010 q^{23} -0.705213 q^{24} +1.38533 q^{25} +6.10826 q^{26} -3.88056 q^{27} +3.12362 q^{28} +3.76472 q^{29} +1.78202 q^{30} +8.06985 q^{31} -1.00000 q^{32} +0.618828 q^{33} -0.902592 q^{34} -7.89314 q^{35} -2.50268 q^{36} -10.0998 q^{37} -7.31283 q^{38} -4.30763 q^{39} +2.52692 q^{40} -4.47609 q^{41} -2.20282 q^{42} +5.12785 q^{43} +0.877506 q^{44} +6.32406 q^{45} -7.35010 q^{46} +1.24384 q^{47} +0.705213 q^{48} +2.75700 q^{49} -1.38533 q^{50} +0.636520 q^{51} -6.10826 q^{52} -0.435541 q^{53} +3.88056 q^{54} -2.21739 q^{55} -3.12362 q^{56} +5.15710 q^{57} -3.76472 q^{58} +10.0940 q^{59} -1.78202 q^{60} -7.77085 q^{61} -8.06985 q^{62} -7.81741 q^{63} +1.00000 q^{64} +15.4351 q^{65} -0.618828 q^{66} -6.45194 q^{67} +0.902592 q^{68} +5.18338 q^{69} +7.89314 q^{70} -8.05663 q^{71} +2.50268 q^{72} -4.59204 q^{73} +10.0998 q^{74} +0.976950 q^{75} +7.31283 q^{76} +2.74100 q^{77} +4.30763 q^{78} -15.3991 q^{79} -2.52692 q^{80} +4.77141 q^{81} +4.47609 q^{82} -1.05026 q^{83} +2.20282 q^{84} -2.28078 q^{85} -5.12785 q^{86} +2.65493 q^{87} -0.877506 q^{88} -7.84085 q^{89} -6.32406 q^{90} -19.0799 q^{91} +7.35010 q^{92} +5.69096 q^{93} -1.24384 q^{94} -18.4789 q^{95} -0.705213 q^{96} -9.03477 q^{97} -2.75700 q^{98} -2.19611 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.705213	0.407155	0.203577	−	0.979059i	$-0.434743\pi$
$3$	0.705213	0.407155	0.203577	+	0.979059i	$0.434743\pi$
$4$	1.00000	0.500000
$5$	−2.52692	−1.13007	−0.565037	−	0.825066i	$-0.691138\pi$
$5$	−2.52692	−1.13007	−0.565037	+	0.825066i	$0.691138\pi$
$6$	−0.705213	−0.287902
$7$	3.12362	1.18062	0.590309	−	0.807178i	$-0.299006\pi$
$7$	3.12362	1.18062	0.590309	+	0.807178i	$0.299006\pi$
$8$	−1.00000	−0.353553
$9$	−2.50268	−0.834225
$10$	2.52692	0.799082
$11$	0.877506	0.264578	0.132289	−	0.991211i	$-0.457767\pi$
$11$	0.877506	0.264578	0.132289	+	0.991211i	$0.457767\pi$
$12$	0.705213	0.203577
$13$	−6.10826	−1.69413	−0.847064	−	0.531491i	$-0.821632\pi$
$13$	−6.10826	−1.69413	−0.847064	+	0.531491i	$0.821632\pi$
$14$	−3.12362	−0.834823
$15$	−1.78202	−0.460115
$16$	1.00000	0.250000
$17$	0.902592	0.218911	0.109455	−	0.993992i	$-0.465089\pi$
$17$	0.902592	0.218911	0.109455	+	0.993992i	$0.465089\pi$
$18$	2.50268	0.589886
$19$	7.31283	1.67768	0.838839	−	0.544379i	$-0.183235\pi$
$19$	7.31283	1.67768	0.838839	+	0.544379i	$0.183235\pi$
$20$	−2.52692	−0.565037
$21$	2.20282	0.480694
$22$	−0.877506	−0.187085
$23$	7.35010	1.53260	0.766301	−	0.642482i	$-0.222095\pi$
$23$	7.35010	1.53260	0.766301	+	0.642482i	$0.222095\pi$
$24$	−0.705213	−0.143951
$25$	1.38533	0.277065
$26$	6.10826	1.19793
$27$	−3.88056	−0.746813
$28$	3.12362	0.590309
$29$	3.76472	0.699090	0.349545	−	0.936919i	$-0.386336\pi$
$29$	3.76472	0.699090	0.349545	+	0.936919i	$0.386336\pi$
$30$	1.78202	0.325350
$31$	8.06985	1.44939	0.724694	−	0.689071i	$-0.241981\pi$
$31$	8.06985	1.44939	0.724694	+	0.689071i	$0.241981\pi$
$32$	−1.00000	−0.176777
$33$	0.618828	0.107724
$34$	−0.902592	−0.154793
$35$	−7.89314	−1.33418
$36$	−2.50268	−0.417113
$37$	−10.0998	−1.66040	−0.830198	−	0.557468i	$-0.811773\pi$
$37$	−10.0998	−1.66040	−0.830198	+	0.557468i	$0.811773\pi$
$38$	−7.31283	−1.18630
$39$	−4.30763	−0.689772
$40$	2.52692	0.399541
$41$	−4.47609	−0.699048	−0.349524	−	0.936927i	$-0.613657\pi$
$41$	−4.47609	−0.699048	−0.349524	+	0.936927i	$0.613657\pi$
$42$	−2.20282	−0.339902
$43$	5.12785	0.781989	0.390995	−	0.920393i	$-0.372131\pi$
$43$	5.12785	0.781989	0.390995	+	0.920393i	$0.372131\pi$
$44$	0.877506	0.132289
$45$	6.32406	0.942735
$46$	−7.35010	−1.08371
$47$	1.24384	0.181433	0.0907166	−	0.995877i	$-0.471084\pi$
$47$	1.24384	0.181433	0.0907166	+	0.995877i	$0.471084\pi$
$48$	0.705213	0.101789
$49$	2.75700	0.393857
$50$	−1.38533	−0.195915
$51$	0.636520	0.0891306
$52$	−6.10826	−0.847064
$53$	−0.435541	−0.0598261	−0.0299131	−	0.999553i	$-0.509523\pi$
$53$	−0.435541	−0.0598261	−0.0299131	+	0.999553i	$0.509523\pi$
$54$	3.88056	0.528077
$55$	−2.21739	−0.298993
$56$	−3.12362	−0.417411
$57$	5.15710	0.683075
$58$	−3.76472	−0.494332
$59$	10.0940	1.31412	0.657060	−	0.753838i	$-0.271800\pi$
$59$	10.0940	1.31412	0.657060	+	0.753838i	$0.271800\pi$
$60$	−1.78202	−0.230057
$61$	−7.77085	−0.994956	−0.497478	−	0.867477i	$-0.665740\pi$
$61$	−7.77085	−0.994956	−0.497478	+	0.867477i	$0.665740\pi$
$62$	−8.06985	−1.02487
$63$	−7.81741	−0.984901
$64$	1.00000	0.125000
$65$	15.4351	1.91449
$66$	−0.618828	−0.0761725
$67$	−6.45194	−0.788230	−0.394115	−	0.919061i	$-0.628949\pi$
$67$	−6.45194	−0.788230	−0.394115	+	0.919061i	$0.628949\pi$
$68$	0.902592	0.109455
$69$	5.18338	0.624006
$70$	7.89314	0.943411
$71$	−8.05663	−0.956146	−0.478073	−	0.878320i	$-0.658665\pi$
$71$	−8.05663	−0.956146	−0.478073	+	0.878320i	$0.658665\pi$
$72$	2.50268	0.294943
$73$	−4.59204	−0.537457	−0.268729	−	0.963216i	$-0.586603\pi$
$73$	−4.59204	−0.537457	−0.268729	+	0.963216i	$0.586603\pi$
$74$	10.0998	1.17408
$75$	0.976950	0.112808
$76$	7.31283	0.838839
$77$	2.74100	0.312365
$78$	4.30763	0.487743
$79$	−15.3991	−1.73254	−0.866269	−	0.499578i	$-0.833488\pi$
$79$	−15.3991	−1.73254	−0.866269	+	0.499578i	$0.833488\pi$
$80$	−2.52692	−0.282518
$81$	4.77141	0.530156
$82$	4.47609	0.494302
$83$	−1.05026	−0.115281	−0.0576407	−	0.998337i	$-0.518358\pi$
$83$	−1.05026	−0.115281	−0.0576407	+	0.998337i	$0.518358\pi$
$84$	2.20282	0.240347
$85$	−2.28078	−0.247385
$86$	−5.12785	−0.552950
$87$	2.65493	0.284638
$88$	−0.877506	−0.0935425
$89$	−7.84085	−0.831129	−0.415564	−	0.909564i	$-0.636416\pi$
$89$	−7.84085	−0.831129	−0.415564	+	0.909564i	$0.636416\pi$
$90$	−6.32406	−0.666615
$91$	−19.0799	−2.00012
$92$	7.35010	0.766301
$93$	5.69096	0.590125
$94$	−1.24384	−0.128293
$95$	−18.4789	−1.89590
$96$	−0.705213	−0.0719755
$97$	−9.03477	−0.917342	−0.458671	−	0.888606i	$-0.651674\pi$
$97$	−9.03477	−0.917342	−0.458671	+	0.888606i	$0.651674\pi$
$98$	−2.75700	−0.278499
$99$	−2.19611	−0.220718
$100$	1.38533	0.138533
$101$	−4.18584	−0.416507	−0.208253	−	0.978075i	$-0.566778\pi$
$101$	−4.18584	−0.416507	−0.208253	+	0.978075i	$0.566778\pi$
$102$	−0.636520	−0.0630248
$103$	−1.48280	−0.146105	−0.0730524	−	0.997328i	$-0.523274\pi$
$103$	−1.48280	−0.146105	−0.0730524	+	0.997328i	$0.523274\pi$
$104$	6.10826	0.598965
$105$	−5.56634	−0.543219
$106$	0.435541	0.0423034
$107$	5.40919	0.522926	0.261463	−	0.965214i	$-0.415795\pi$
$107$	5.40919	0.522926	0.261463	+	0.965214i	$0.415795\pi$
$108$	−3.88056	−0.373407
$109$	17.5617	1.68211	0.841054	−	0.540952i	$-0.181936\pi$
$109$	17.5617	1.68211	0.841054	+	0.540952i	$0.181936\pi$
$110$	2.21739	0.211420
$111$	−7.12251	−0.676038
$112$	3.12362	0.295154
$113$	16.3647	1.53946	0.769730	−	0.638369i	$-0.220391\pi$
$113$	16.3647	1.53946	0.769730	+	0.638369i	$0.220391\pi$
$114$	−5.15710	−0.483007
$115$	−18.5731	−1.73195
$116$	3.76472	0.349545
$117$	15.2870	1.41328
$118$	−10.0940	−0.929224
$119$	2.81936	0.258450
$120$	1.78202	0.162675
$121$	−10.2300	−0.929998
$122$	7.77085	0.703540
$123$	−3.15660	−0.284621
$124$	8.06985	0.724694
$125$	9.13399	0.816969
$126$	7.81741	0.696430
$127$	19.1369	1.69812	0.849062	−	0.528293i	$-0.177168\pi$
$127$	19.1369	1.69812	0.849062	+	0.528293i	$0.177168\pi$
$128$	−1.00000	−0.0883883
$129$	3.61622	0.318391
$130$	−15.4351	−1.35375
$131$	−3.45848	−0.302168	−0.151084	−	0.988521i	$-0.548276\pi$
$131$	−3.45848	−0.302168	−0.151084	+	0.988521i	$0.548276\pi$
$132$	0.618828	0.0538621
$133$	22.8425	1.98070
$134$	6.45194	0.557363
$135$	9.80586	0.843954
$136$	−0.902592	−0.0773967
$137$	12.8066	1.09414	0.547069	−	0.837087i	$-0.315743\pi$
$137$	12.8066	1.09414	0.547069	+	0.837087i	$0.315743\pi$
$138$	−5.18338	−0.441239
$139$	−11.5081	−0.976108	−0.488054	−	0.872813i	$-0.662293\pi$
$139$	−11.5081	−0.976108	−0.488054	+	0.872813i	$0.662293\pi$
$140$	−7.89314	−0.667092
$141$	0.877174	0.0738714
$142$	8.05663	0.676098
$143$	−5.36004	−0.448229
$144$	−2.50268	−0.208556
$145$	−9.51314	−0.790023
$146$	4.59204	0.380040
$147$	1.94427	0.160361
$148$	−10.0998	−0.830198
$149$	3.44632	0.282333	0.141167	−	0.989986i	$-0.454915\pi$
$149$	3.44632	0.282333	0.141167	+	0.989986i	$0.454915\pi$
$150$	−0.976950	−0.0797677
$151$	−1.88964	−0.153776	−0.0768882	−	0.997040i	$-0.524498\pi$
$151$	−1.88964	−0.153776	−0.0768882	+	0.997040i	$0.524498\pi$
$152$	−7.31283	−0.593149
$153$	−2.25890	−0.182621
$154$	−2.74100	−0.220876
$155$	−20.3919	−1.63791
$156$	−4.30763	−0.344886
$157$	−0.407147	−0.0324939	−0.0162470	−	0.999868i	$-0.505172\pi$
$157$	−0.407147	−0.0324939	−0.0162470	+	0.999868i	$0.505172\pi$
$158$	15.3991	1.22509
$159$	−0.307149	−0.0243585
$160$	2.52692	0.199771
$161$	22.9589	1.80942
$162$	−4.77141	−0.374877
$163$	−11.2394	−0.880335	−0.440167	−	0.897916i	$-0.645081\pi$
$163$	−11.2394	−0.880335	−0.440167	+	0.897916i	$0.645081\pi$
$164$	−4.47609	−0.349524
$165$	−1.56373	−0.121736
$166$	1.05026	0.0815163
$167$	12.5925	0.974437	0.487219	−	0.873280i	$-0.338012\pi$
$167$	12.5925	0.974437	0.487219	+	0.873280i	$0.338012\pi$
$168$	−2.20282	−0.169951
$169$	24.3109	1.87007
$170$	2.28078	0.174928
$171$	−18.3016	−1.39956
$172$	5.12785	0.390995
$173$	17.4308	1.32524	0.662620	−	0.748956i	$-0.269445\pi$
$173$	17.4308	1.32524	0.662620	+	0.748956i	$0.269445\pi$
$174$	−2.65493	−0.201269
$175$	4.32724	0.327108
$176$	0.877506	0.0661445
$177$	7.11838	0.535050
$178$	7.84085	0.587697
$179$	23.2021	1.73421	0.867103	−	0.498128i	$-0.165979\pi$
$179$	23.2021	1.73421	0.867103	+	0.498128i	$0.165979\pi$
$180$	6.32406	0.471368
$181$	2.06892	0.153781	0.0768907	−	0.997040i	$-0.475501\pi$
$181$	2.06892	0.153781	0.0768907	+	0.997040i	$0.475501\pi$
$182$	19.0799	1.41430
$183$	−5.48011	−0.405101
$184$	−7.35010	−0.541856
$185$	25.5214	1.87637
$186$	−5.69096	−0.417282
$187$	0.792030	0.0579190
$188$	1.24384	0.0907166
$189$	−12.1214	−0.881701
$190$	18.4789	1.34060
$191$	11.5835	0.838151	0.419075	−	0.907951i	$-0.362354\pi$
$191$	11.5835	0.838151	0.419075	+	0.907951i	$0.362354\pi$
$192$	0.705213	0.0508943
$193$	11.5606	0.832150	0.416075	−	0.909330i	$-0.363405\pi$
$193$	11.5606	0.832150	0.416075	+	0.909330i	$0.363405\pi$
$194$	9.03477	0.648659
$195$	10.8850	0.779493
$196$	2.75700	0.196929
$197$	9.71843	0.692409	0.346205	−	0.938159i	$-0.387470\pi$
$197$	9.71843	0.692409	0.346205	+	0.938159i	$0.387470\pi$
$198$	2.19611	0.156071
$199$	25.8931	1.83552	0.917758	−	0.397140i	$-0.129997\pi$
$199$	25.8931	1.83552	0.917758	+	0.397140i	$0.129997\pi$
$200$	−1.38533	−0.0979574
$201$	−4.54999	−0.320931
$202$	4.18584	0.294515
$203$	11.7595	0.825358
$204$	0.636520	0.0445653
$205$	11.3107	0.789975
$206$	1.48280	0.103312
$207$	−18.3949	−1.27853
$208$	−6.10826	−0.423532
$209$	6.41705	0.443877
$210$	5.56634	0.384114
$211$	−13.7235	−0.944767	−0.472384	−	0.881393i	$-0.656606\pi$
$211$	−13.7235	−0.944767	−0.472384	+	0.881393i	$0.656606\pi$
$212$	−0.435541	−0.0299131
$213$	−5.68164	−0.389299
$214$	−5.40919	−0.369764
$215$	−12.9577	−0.883705
$216$	3.88056	0.264038
$217$	25.2071	1.71117
$218$	−17.5617	−1.18943
$219$	−3.23836	−0.218828
$220$	−2.21739	−0.149496
$221$	−5.51327	−0.370863
$222$	7.12251	0.478031
$223$	−3.92329	−0.262723	−0.131361	−	0.991335i	$-0.541935\pi$
$223$	−3.92329	−0.262723	−0.131361	+	0.991335i	$0.541935\pi$
$224$	−3.12362	−0.208706
$225$	−3.46702	−0.231135
$226$	−16.3647	−1.08856
$227$	22.8991	1.51987	0.759933	−	0.650002i	$-0.225232\pi$
$227$	22.8991	1.51987	0.759933	+	0.650002i	$0.225232\pi$
$228$	5.15710	0.341537
$229$	−13.6383	−0.901243	−0.450621	−	0.892715i	$-0.648798\pi$
$229$	−13.6383	−0.901243	−0.450621	+	0.892715i	$0.648798\pi$
$230$	18.5731	1.22467
$231$	1.93298	0.127181
$232$	−3.76472	−0.247166
$233$	−23.4669	−1.53737	−0.768683	−	0.639630i	$-0.779087\pi$
$233$	−23.4669	−1.53737	−0.768683	+	0.639630i	$0.779087\pi$
$234$	−15.2870	−0.999342
$235$	−3.14309	−0.205033
$236$	10.0940	0.657060
$237$	−10.8597	−0.705411
$238$	−2.81936	−0.182752
$239$	23.0848	1.49323	0.746616	−	0.665255i	$-0.231677\pi$
$239$	23.0848	1.49323	0.746616	+	0.665255i	$0.231677\pi$
$240$	−1.78202	−0.115029
$241$	15.7621	1.01533	0.507663	−	0.861556i	$-0.330510\pi$
$241$	15.7621	1.01533	0.507663	+	0.861556i	$0.330510\pi$
$242$	10.2300	0.657608
$243$	15.0065	0.962669
$244$	−7.77085	−0.497478
$245$	−6.96672	−0.445088
$246$	3.15660	0.201257
$247$	−44.6687	−2.84220
$248$	−8.06985	−0.512436
$249$	−0.740659	−0.0469374
$250$	−9.13399	−0.577684
$251$	24.7088	1.55961	0.779804	−	0.626024i	$-0.215319\pi$
$251$	24.7088	1.55961	0.779804	+	0.626024i	$0.215319\pi$
$252$	−7.81741	−0.492450
$253$	6.44976	0.405493
$254$	−19.1369	−1.20075
$255$	−1.60843	−0.100724
$256$	1.00000	0.0625000
$257$	−8.67353	−0.541040	−0.270520	−	0.962714i	$-0.587196\pi$
$257$	−8.67353	−0.541040	−0.270520	+	0.962714i	$0.587196\pi$
$258$	−3.61622	−0.225136
$259$	−31.5479	−1.96029
$260$	15.4351	0.957244
$261$	−9.42186	−0.583199
$262$	3.45848	0.213665
$263$	−10.6730	−0.658123	−0.329061	−	0.944309i	$-0.606732\pi$
$263$	−10.6730	−0.658123	−0.329061	+	0.944309i	$0.606732\pi$
$264$	−0.618828	−0.0380863
$265$	1.10058	0.0676079
$266$	−22.8425	−1.40056
$267$	−5.52947	−0.338398
$268$	−6.45194	−0.394115
$269$	−17.4085	−1.06141	−0.530706	−	0.847556i	$-0.678073\pi$
$269$	−17.4085	−1.06141	−0.530706	+	0.847556i	$0.678073\pi$
$270$	−9.80586	−0.596765
$271$	30.7659	1.86889	0.934446	−	0.356104i	$-0.115895\pi$
$271$	30.7659	1.86889	0.934446	+	0.356104i	$0.115895\pi$
$272$	0.902592	0.0547277
$273$	−13.4554	−0.814357
$274$	−12.8066	−0.773673
$275$	1.21563	0.0733054
$276$	5.18338	0.312003
$277$	6.11561	0.367452	0.183726	−	0.982978i	$-0.441184\pi$
$277$	6.11561	0.367452	0.183726	+	0.982978i	$0.441184\pi$
$278$	11.5081	0.690213
$279$	−20.1962	−1.20912
$280$	7.89314	0.471705
$281$	−21.7617	−1.29819	−0.649097	−	0.760705i	$-0.724853\pi$
$281$	−21.7617	−1.29819	−0.649097	+	0.760705i	$0.724853\pi$
$282$	−0.877174	−0.0522349
$283$	31.5210	1.87373	0.936863	−	0.349696i	$-0.113715\pi$
$283$	31.5210	1.87373	0.936863	+	0.349696i	$0.113715\pi$
$284$	−8.05663	−0.478073
$285$	−13.0316	−0.771924
$286$	5.36004	0.316946
$287$	−13.9816	−0.825308
$288$	2.50268	0.147472
$289$	−16.1853	−0.952078
$290$	9.51314	0.558631
$291$	−6.37143	−0.373500
$292$	−4.59204	−0.268729
$293$	24.5227	1.43263	0.716317	−	0.697775i	$-0.245827\pi$
$293$	24.5227	1.43263	0.716317	+	0.697775i	$0.245827\pi$
$294$	−1.94427	−0.113392
$295$	−25.5066	−1.48505
$296$	10.0998	0.587039
$297$	−3.40521	−0.197590
$298$	−3.44632	−0.199640
$299$	−44.8963	−2.59642
$300$	0.976950	0.0564042
$301$	16.0174	0.923230
$302$	1.88964	0.108736
$303$	−2.95191	−0.169583
$304$	7.31283	0.419420
$305$	19.6363	1.12437
$306$	2.25890	0.129132
$307$	6.64748	0.379392	0.189696	−	0.981843i	$-0.439250\pi$
$307$	6.64748	0.379392	0.189696	+	0.981843i	$0.439250\pi$
$308$	2.74100	0.156183
$309$	−1.04569	−0.0594873
$310$	20.3919	1.15818
$311$	−8.82355	−0.500338	−0.250169	−	0.968202i	$-0.580486\pi$
$311$	−8.82355	−0.500338	−0.250169	+	0.968202i	$0.580486\pi$
$312$	4.30763	0.243871
$313$	4.91596	0.277866	0.138933	−	0.990302i	$-0.455633\pi$
$313$	4.91596	0.277866	0.138933	+	0.990302i	$0.455633\pi$
$314$	0.407147	0.0229767
$315$	19.7540	1.11301
$316$	−15.3991	−0.866269
$317$	14.7796	0.830104	0.415052	−	0.909798i	$-0.363763\pi$
$317$	14.7796	0.830104	0.415052	+	0.909798i	$0.363763\pi$
$318$	0.307149	0.0172240
$319$	3.30356	0.184964
$320$	−2.52692	−0.141259
$321$	3.81463	0.212912
$322$	−22.9589	−1.27945
$323$	6.60051	0.367262
$324$	4.77141	0.265078
$325$	−8.46194	−0.469384
$326$	11.2394	0.622491
$327$	12.3847	0.684878
$328$	4.47609	0.247151
$329$	3.88529	0.214203
$330$	1.56373	0.0860805
$331$	−17.4316	−0.958127	−0.479063	−	0.877780i	$-0.659024\pi$
$331$	−17.4316	−0.958127	−0.479063	+	0.877780i	$0.659024\pi$
$332$	−1.05026	−0.0576407
$333$	25.2765	1.38514
$334$	−12.5925	−0.689031
$335$	16.3035	0.890757
$336$	2.20282	0.120173
$337$	22.1303	1.20551	0.602757	−	0.797925i	$-0.294069\pi$
$337$	22.1303	1.20551	0.602757	+	0.797925i	$0.294069\pi$
$338$	−24.3109	−1.32234
$339$	11.5406	0.626799
$340$	−2.28078	−0.123693
$341$	7.08134	0.383476
$342$	18.3016	0.989639
$343$	−13.2535	−0.715623
$344$	−5.12785	−0.276475
$345$	−13.0980	−0.705172
$346$	−17.4308	−0.937086
$347$	16.1874	0.868984	0.434492	−	0.900676i	$-0.356928\pi$
$347$	16.1874	0.868984	0.434492	+	0.900676i	$0.356928\pi$
$348$	2.65493	0.142319
$349$	−6.57125	−0.351751	−0.175875	−	0.984412i	$-0.556276\pi$
$349$	−6.57125	−0.351751	−0.175875	+	0.984412i	$0.556276\pi$
$350$	−4.32724	−0.231300
$351$	23.7035	1.26520
$352$	−0.877506	−0.0467712
$353$	−3.70339	−0.197112	−0.0985558	−	0.995132i	$-0.531422\pi$
$353$	−3.70339	−0.197112	−0.0985558	+	0.995132i	$0.531422\pi$
$354$	−7.11838	−0.378338
$355$	20.3585	1.08052
$356$	−7.84085	−0.415564
$357$	1.98825	0.105229
$358$	−23.2021	−1.22627
$359$	−12.6181	−0.665957	−0.332978	−	0.942934i	$-0.608054\pi$
$359$	−12.6181	−0.665957	−0.332978	+	0.942934i	$0.608054\pi$
$360$	−6.32406	−0.333307
$361$	34.4775	1.81460
$362$	−2.06892	−0.108740
$363$	−7.21431	−0.378653
$364$	−19.0799	−1.00006
$365$	11.6037	0.607366
$366$	5.48011	0.286450
$367$	−11.4213	−0.596185	−0.298092	−	0.954537i	$-0.596350\pi$
$367$	−11.4213	−0.596185	−0.298092	+	0.954537i	$0.596350\pi$
$368$	7.35010	0.383150
$369$	11.2022	0.583163
$370$	−25.5214	−1.32679
$371$	−1.36046	−0.0706317
$372$	5.69096	0.295063
$373$	−34.4473	−1.78361	−0.891806	−	0.452418i	$-0.850562\pi$
$373$	−34.4473	−1.78361	−0.891806	+	0.452418i	$0.850562\pi$
$374$	−0.792030	−0.0409549
$375$	6.44141	0.332633
$376$	−1.24384	−0.0641463
$377$	−22.9959	−1.18435
$378$	12.1214	0.623457
$379$	−31.0665	−1.59578	−0.797891	−	0.602802i	$-0.794051\pi$
$379$	−31.0665	−1.59578	−0.797891	+	0.602802i	$0.794051\pi$
$380$	−18.4789	−0.947950
$381$	13.4956	0.691399
$382$	−11.5835	−0.592662
$383$	36.7768	1.87921	0.939604	−	0.342263i	$-0.111194\pi$
$383$	36.7768	1.87921	0.939604	+	0.342263i	$0.111194\pi$
$384$	−0.705213	−0.0359877
$385$	−6.92628	−0.352996
$386$	−11.5606	−0.588419
$387$	−12.8333	−0.652355
$388$	−9.03477	−0.458671
$389$	2.96738	0.150452	0.0752260	−	0.997167i	$-0.476032\pi$
$389$	2.96738	0.150452	0.0752260	+	0.997167i	$0.476032\pi$
$390$	−10.8850	−0.551185
$391$	6.63414	0.335503
$392$	−2.75700	−0.139250
$393$	−2.43896	−0.123029
$394$	−9.71843	−0.489607
$395$	38.9124	1.95789
$396$	−2.19611	−0.110359
$397$	18.4477	0.925864	0.462932	−	0.886394i	$-0.346797\pi$
$397$	18.4477	0.925864	0.462932	+	0.886394i	$0.346797\pi$
$398$	−25.8931	−1.29791
$399$	16.1088	0.806450
$400$	1.38533	0.0692664
$401$	5.92055	0.295658	0.147829	−	0.989013i	$-0.452771\pi$
$401$	5.92055	0.295658	0.147829	+	0.989013i	$0.452771\pi$
$402$	4.54999	0.226933
$403$	−49.2928	−2.45545
$404$	−4.18584	−0.208253
$405$	−12.0570	−0.599116
$406$	−11.7595	−0.583616
$407$	−8.86264	−0.439305
$408$	−0.636520	−0.0315124
$409$	10.5693	0.522621	0.261310	−	0.965255i	$-0.415845\pi$
$409$	10.5693	0.522621	0.261310	+	0.965255i	$0.415845\pi$
$410$	−11.3107	−0.558597
$411$	9.03135	0.445484
$412$	−1.48280	−0.0730524
$413$	31.5297	1.55147
$414$	18.3949	0.904060
$415$	2.65393	0.130276
$416$	6.10826	0.299482
$417$	−8.11569	−0.397427
$418$	−6.41705	−0.313868
$419$	−37.3238	−1.82339	−0.911694	−	0.410870i	$-0.865225\pi$
$419$	−37.3238	−1.82339	−0.911694	+	0.410870i	$0.865225\pi$
$420$	−5.56634	−0.271610
$421$	13.8074	0.672932	0.336466	−	0.941696i	$-0.390768\pi$
$421$	13.8074	0.672932	0.336466	+	0.941696i	$0.390768\pi$
$422$	13.7235	0.668051
$423$	−3.11293	−0.151356
$424$	0.435541	0.0211517
$425$	1.25039	0.0606526
$426$	5.68164	0.275276
$427$	−24.2732	−1.17466
$428$	5.40919	0.261463
$429$	−3.77997	−0.182499
$430$	12.9577	0.624874
$431$	−4.42423	−0.213108	−0.106554	−	0.994307i	$-0.533982\pi$
$431$	−4.42423	−0.213108	−0.106554	+	0.994307i	$0.533982\pi$
$432$	−3.88056	−0.186703
$433$	39.2363	1.88557	0.942787	−	0.333395i	$-0.108194\pi$
$433$	39.2363	1.88557	0.942787	+	0.333395i	$0.108194\pi$
$434$	−25.2071	−1.20998
$435$	−6.70879	−0.321662
$436$	17.5617	0.841054
$437$	53.7500	2.57121
$438$	3.23836	0.154735
$439$	39.7070	1.89511	0.947555	−	0.319592i	$-0.103546\pi$
$439$	39.7070	1.89511	0.947555	+	0.319592i	$0.103546\pi$
$440$	2.21739	0.105710
$441$	−6.89988	−0.328566
$442$	5.51327	0.262240
$443$	7.03855	0.334412	0.167206	−	0.985922i	$-0.446526\pi$
$443$	7.03855	0.334412	0.167206	+	0.985922i	$0.446526\pi$
$444$	−7.12251	−0.338019
$445$	19.8132	0.939236
$446$	3.92329	0.185773
$447$	2.43039	0.114953
$448$	3.12362	0.147577
$449$	9.33948	0.440757	0.220379	−	0.975414i	$-0.429271\pi$
$449$	9.33948	0.440757	0.220379	+	0.975414i	$0.429271\pi$
$450$	3.46702	0.163437
$451$	−3.92780	−0.184953
$452$	16.3647	0.769730
$453$	−1.33259	−0.0626108
$454$	−22.8991	−1.07471
$455$	48.2134	2.26028
$456$	−5.15710	−0.241503
$457$	−0.653240	−0.0305573	−0.0152786	−	0.999883i	$-0.504864\pi$
$457$	−0.653240	−0.0305573	−0.0152786	+	0.999883i	$0.504864\pi$
$458$	13.6383	0.637275
$459$	−3.50256	−0.163486
$460$	−18.5731	−0.865976
$461$	−20.2411	−0.942724	−0.471362	−	0.881940i	$-0.656238\pi$
$461$	−20.2411	−0.942724	−0.471362	+	0.881940i	$0.656238\pi$
$462$	−1.93298	−0.0899306
$463$	39.6496	1.84267	0.921337	−	0.388764i	$-0.127098\pi$
$463$	39.6496	1.84267	0.921337	+	0.388764i	$0.127098\pi$
$464$	3.76472	0.174773
$465$	−14.3806	−0.666885
$466$	23.4669	1.08708
$467$	26.9268	1.24602	0.623011	−	0.782213i	$-0.285909\pi$
$467$	26.9268	1.24602	0.623011	+	0.782213i	$0.285909\pi$
$468$	15.2870	0.706642
$469$	−20.1534	−0.930598
$470$	3.14309	0.144980
$471$	−0.287126	−0.0132300
$472$	−10.0940	−0.464612
$473$	4.49972	0.206897
$474$	10.8597	0.498801
$475$	10.1307	0.464827
$476$	2.81936	0.129225
$477$	1.09002	0.0499084
$478$	−23.0848	−1.05588
$479$	−14.3587	−0.656068	−0.328034	−	0.944666i	$-0.606386\pi$
$479$	−14.3587	−0.656068	−0.328034	+	0.944666i	$0.606386\pi$
$480$	1.78202	0.0813375
$481$	61.6922	2.81292
$482$	−15.7621	−0.717943
$483$	16.1909	0.736712
$484$	−10.2300	−0.464999
$485$	22.8301	1.03666
$486$	−15.0065	−0.680710
$487$	40.4178	1.83150	0.915752	−	0.401744i	$-0.131596\pi$
$487$	40.4178	1.83150	0.915752	+	0.401744i	$0.131596\pi$
$488$	7.77085	0.351770
$489$	−7.92614	−0.358433
$490$	6.96672	0.314724
$491$	2.70819	0.122219	0.0611095	−	0.998131i	$-0.480536\pi$
$491$	2.70819	0.122219	0.0611095	+	0.998131i	$0.480536\pi$
$492$	−3.15660	−0.142310
$493$	3.39801	0.153038
$494$	44.6687	2.00974
$495$	5.54940	0.249427
$496$	8.06985	0.362347
$497$	−25.1659	−1.12884
$498$	0.740659	0.0331897
$499$	−16.4582	−0.736772	−0.368386	−	0.929673i	$-0.620089\pi$
$499$	−16.4582	−0.736772	−0.368386	+	0.929673i	$0.620089\pi$
$500$	9.13399	0.408484
$501$	8.88039	0.396747
$502$	−24.7088	−1.10281
$503$	−0.740939	−0.0330368	−0.0165184	−	0.999864i	$-0.505258\pi$
$503$	−0.740939	−0.0330368	−0.0165184	+	0.999864i	$0.505258\pi$
$504$	7.81741	0.348215
$505$	10.5773	0.470683
$506$	−6.44976	−0.286727
$507$	17.1443	0.761407
$508$	19.1369	0.849062
$509$	−13.2684	−0.588112	−0.294056	−	0.955788i	$-0.595005\pi$
$509$	−13.2684	−0.588112	−0.294056	+	0.955788i	$0.595005\pi$
$510$	1.60843	0.0712227
$511$	−14.3438	−0.634531
$512$	−1.00000	−0.0441942
$513$	−28.3778	−1.25291
$514$	8.67353	0.382573
$515$	3.74692	0.165109
$516$	3.61622	0.159195
$517$	1.09148	0.0480032
$518$	31.5479	1.38614
$519$	12.2924	0.539578
$520$	−15.4351	−0.676874
$521$	1.91569	0.0839279	0.0419640	−	0.999119i	$-0.486639\pi$
$521$	1.91569	0.0839279	0.0419640	+	0.999119i	$0.486639\pi$
$522$	9.42186	0.412384
$523$	−7.66993	−0.335383	−0.167691	−	0.985840i	$-0.553631\pi$
$523$	−7.66993	−0.335383	−0.167691	+	0.985840i	$0.553631\pi$
$524$	−3.45848	−0.151084
$525$	3.05162	0.133184
$526$	10.6730	0.465363
$527$	7.28379	0.317287
$528$	0.618828	0.0269311
$529$	31.0239	1.34887
$530$	−1.10058	−0.0478060
$531$	−25.2619	−1.09627
$532$	22.8425	0.990348
$533$	27.3411	1.18428
$534$	5.52947	0.239283
$535$	−13.6686	−0.590944
$536$	6.45194	0.278681
$537$	16.3624	0.706091
$538$	17.4085	0.750532
$539$	2.41929	0.104206
$540$	9.80586	0.421977
$541$	0.450491	0.0193681	0.00968406	−	0.999953i	$-0.496917\pi$
$541$	0.450491	0.0193681	0.00968406	+	0.999953i	$0.496917\pi$
$542$	−30.7659	−1.32151
$543$	1.45903	0.0626128
$544$	−0.902592	−0.0386983
$545$	−44.3771	−1.90090
$546$	13.4554	0.575837
$547$	−27.2595	−1.16553	−0.582766	−	0.812640i	$-0.698030\pi$
$547$	−27.2595	−1.16553	−0.582766	+	0.812640i	$0.698030\pi$
$548$	12.8066	0.547069
$549$	19.4479	0.830017
$550$	−1.21563	−0.0518348
$551$	27.5307	1.17285
$552$	−5.18338	−0.220619
$553$	−48.1010	−2.04546
$554$	−6.11561	−0.259828
$555$	17.9980	0.763973
$556$	−11.5081	−0.488054
$557$	−17.2991	−0.732987	−0.366494	−	0.930421i	$-0.619442\pi$
$557$	−17.2991	−0.732987	−0.366494	+	0.930421i	$0.619442\pi$
$558$	20.1962	0.854974
$559$	−31.3222	−1.32479
$560$	−7.89314	−0.333546
$561$	0.558550	0.0235820
$562$	21.7617	0.917962
$563$	8.99229	0.378980	0.189490	−	0.981883i	$-0.439317\pi$
$563$	8.99229	0.378980	0.189490	+	0.981883i	$0.439317\pi$
$564$	0.877174	0.0369357
$565$	−41.3523	−1.73970
$566$	−31.5210	−1.32492
$567$	14.9041	0.625912
$568$	8.05663	0.338049
$569$	−5.53875	−0.232197	−0.116098	−	0.993238i	$-0.537039\pi$
$569$	−5.53875	−0.232197	−0.116098	+	0.993238i	$0.537039\pi$
$570$	13.0316	0.545833
$571$	−15.1188	−0.632701	−0.316351	−	0.948642i	$-0.602458\pi$
$571$	−15.1188	−0.632701	−0.316351	+	0.948642i	$0.602458\pi$
$572$	−5.36004	−0.224115
$573$	8.16881	0.341257
$574$	13.9816	0.583581
$575$	10.1823	0.424631
$576$	−2.50268	−0.104278
$577$	24.5524	1.02213	0.511066	−	0.859542i	$-0.329251\pi$
$577$	24.5524	1.02213	0.511066	+	0.859542i	$0.329251\pi$
$578$	16.1853	0.673221
$579$	8.15268	0.338814
$580$	−9.51314	−0.395012
$581$	−3.28063	−0.136103
$582$	6.37143	0.264104
$583$	−0.382190	−0.0158287
$584$	4.59204	0.190020
$585$	−38.6290	−1.59711
$586$	−24.5227	−1.01302
$587$	−33.0743	−1.36512	−0.682562	−	0.730828i	$-0.739134\pi$
$587$	−33.0743	−1.36512	−0.682562	+	0.730828i	$0.739134\pi$
$588$	1.94427	0.0801804
$589$	59.0135	2.43161
$590$	25.5066	1.05009
$591$	6.85356	0.281918
$592$	−10.0998	−0.415099
$593$	−18.4892	−0.759262	−0.379631	−	0.925138i	$-0.623949\pi$
$593$	−18.4892	−0.759262	−0.379631	+	0.925138i	$0.623949\pi$
$594$	3.40521	0.139718
$595$	−7.12429	−0.292067
$596$	3.44632	0.141167
$597$	18.2602	0.747339
$598$	44.8963	1.83595
$599$	44.6655	1.82498	0.912491	−	0.409097i	$-0.134156\pi$
$599$	44.6655	1.82498	0.912491	+	0.409097i	$0.134156\pi$
$600$	−0.976950	−0.0398838
$601$	29.3370	1.19668	0.598340	−	0.801242i	$-0.295827\pi$
$601$	29.3370	1.19668	0.598340	+	0.801242i	$0.295827\pi$
$602$	−16.0174	−0.652822
$603$	16.1471	0.657561
$604$	−1.88964	−0.0768882
$605$	25.8504	1.05097
$606$	2.95191	0.119913
$607$	−35.1380	−1.42621	−0.713103	−	0.701059i	$-0.752711\pi$
$607$	−35.1380	−1.42621	−0.713103	+	0.701059i	$0.752711\pi$
$608$	−7.31283	−0.296574
$609$	8.29298	0.336049
$610$	−19.6363	−0.795052
$611$	−7.59772	−0.307371
$612$	−2.25890	−0.0913104
$613$	−10.2121	−0.412461	−0.206231	−	0.978503i	$-0.566120\pi$
$613$	−10.2121	−0.412461	−0.206231	+	0.978503i	$0.566120\pi$
$614$	−6.64748	−0.268270
$615$	7.97647	0.321642
$616$	−2.74100	−0.110438
$617$	−4.33454	−0.174502	−0.0872509	−	0.996186i	$-0.527808\pi$
$617$	−4.33454	−0.174502	−0.0872509	+	0.996186i	$0.527808\pi$
$618$	1.04569	0.0420639
$619$	24.3645	0.979291	0.489646	−	0.871922i	$-0.337126\pi$
$619$	24.3645	0.979291	0.489646	+	0.871922i	$0.337126\pi$
$620$	−20.3919	−0.818957
$621$	−28.5225	−1.14457
$622$	8.82355	0.353792
$623$	−24.4918	−0.981245
$624$	−4.30763	−0.172443
$625$	−30.0075	−1.20030
$626$	−4.91596	−0.196481
$627$	4.52539	0.180727
$628$	−0.407147	−0.0162470
$629$	−9.11600	−0.363479
$630$	−19.7540	−0.787017
$631$	−23.1102	−0.920004	−0.460002	−	0.887918i	$-0.652151\pi$
$631$	−23.1102	−0.920004	−0.460002	+	0.887918i	$0.652151\pi$
$632$	15.3991	0.612545
$633$	−9.67801	−0.384666
$634$	−14.7796	−0.586972
$635$	−48.3574	−1.91900
$636$	−0.307149	−0.0121792
$637$	−16.8405	−0.667244
$638$	−3.30356	−0.130789
$639$	20.1631	0.797641
$640$	2.52692	0.0998853
$641$	16.2706	0.642651	0.321326	−	0.946969i	$-0.395872\pi$
$641$	16.2706	0.642651	0.321326	+	0.946969i	$0.395872\pi$
$642$	−3.81463	−0.150551
$643$	−14.8216	−0.584507	−0.292253	−	0.956341i	$-0.594405\pi$
$643$	−14.8216	−0.584507	−0.292253	+	0.956341i	$0.594405\pi$
$644$	22.9589	0.904708
$645$	−9.13791	−0.359805
$646$	−6.60051	−0.259693
$647$	40.9521	1.60999	0.804996	−	0.593281i	$-0.202168\pi$
$647$	40.9521	1.60999	0.804996	+	0.593281i	$0.202168\pi$
$648$	−4.77141	−0.187439
$649$	8.85751	0.347688
$650$	8.46194	0.331905
$651$	17.7764	0.696712
$652$	−11.2394	−0.440167
$653$	−5.20215	−0.203576	−0.101788	−	0.994806i	$-0.532456\pi$
$653$	−5.20215	−0.203576	−0.101788	+	0.994806i	$0.532456\pi$
$654$	−12.3847	−0.484282
$655$	8.73929	0.341473
$656$	−4.47609	−0.174762
$657$	11.4924	0.448360
$658$	−3.88529	−0.151464
$659$	0.927369	0.0361252	0.0180626	−	0.999837i	$-0.494250\pi$
$659$	0.927369	0.0361252	0.0180626	+	0.999837i	$0.494250\pi$
$660$	−1.56373	−0.0608681
$661$	−16.8779	−0.656475	−0.328237	−	0.944595i	$-0.606455\pi$
$661$	−16.8779	−0.656475	−0.328237	+	0.944595i	$0.606455\pi$
$662$	17.4316	0.677498
$663$	−3.88803	−0.150999
$664$	1.05026	0.0407581
$665$	−57.7212	−2.23833
$666$	−25.2765	−0.979445
$667$	27.6710	1.07143
$668$	12.5925	0.487219
$669$	−2.76675	−0.106969
$670$	−16.3035	−0.629860
$671$	−6.81897	−0.263244
$672$	−2.20282	−0.0849755
$673$	28.2777	1.09002	0.545012	−	0.838428i	$-0.316525\pi$
$673$	28.2777	1.09002	0.545012	+	0.838428i	$0.316525\pi$
$674$	−22.1303	−0.852427
$675$	−5.37584	−0.206916
$676$	24.3109	0.935034
$677$	10.4374	0.401141	0.200571	−	0.979679i	$-0.435720\pi$
$677$	10.4374	0.401141	0.200571	+	0.979679i	$0.435720\pi$
$678$	−11.5406	−0.443214
$679$	−28.2212	−1.08303
$680$	2.28078	0.0874639
$681$	16.1487	0.618821
$682$	−7.08134	−0.271159
$683$	−34.6976	−1.32767	−0.663834	−	0.747880i	$-0.731072\pi$
$683$	−34.6976	−1.32767	−0.663834	+	0.747880i	$0.731072\pi$
$684$	−18.3016	−0.699781
$685$	−32.3612	−1.23646
$686$	13.2535	0.506022
$687$	−9.61789	−0.366945
$688$	5.12785	0.195497
$689$	2.66040	0.101353
$690$	13.0980	0.498632
$691$	−1.07718	−0.0409779	−0.0204890	−	0.999790i	$-0.506522\pi$
$691$	−1.07718	−0.0409779	−0.0204890	+	0.999790i	$0.506522\pi$
$692$	17.4308	0.662620
$693$	−6.85982	−0.260583
$694$	−16.1874	−0.614464
$695$	29.0802	1.10307
$696$	−2.65493	−0.100635
$697$	−4.04009	−0.153029
$698$	6.57125	0.248725
$699$	−16.5491	−0.625946
$700$	4.32724	0.163554
$701$	12.9615	0.489549	0.244774	−	0.969580i	$-0.421286\pi$
$701$	12.9615	0.489549	0.244774	+	0.969580i	$0.421286\pi$
$702$	−23.7035	−0.894630
$703$	−73.8581	−2.78561
$704$	0.877506	0.0330723
$705$	−2.21655	−0.0834800
$706$	3.70339	0.139379
$707$	−13.0750	−0.491735
$708$	7.11838	0.267525
$709$	−24.5201	−0.920873	−0.460437	−	0.887693i	$-0.652307\pi$
$709$	−24.5201	−0.920873	−0.460437	+	0.887693i	$0.652307\pi$
$710$	−20.3585	−0.764040
$711$	38.5390	1.44533
$712$	7.84085	0.293848
$713$	59.3142	2.22133
$714$	−1.98825	−0.0744082
$715$	13.5444	0.506532
$716$	23.2021	0.867103
$717$	16.2797	0.607977
$718$	12.6181	0.470903
$719$	−17.5399	−0.654128	−0.327064	−	0.945002i	$-0.606059\pi$
$719$	−17.5399	−0.654128	−0.327064	+	0.945002i	$0.606059\pi$
$720$	6.32406	0.235684
$721$	−4.63171	−0.172494
$722$	−34.4775	−1.28312
$723$	11.1156	0.413394
$724$	2.06892	0.0768907
$725$	5.21536	0.193694
$726$	7.21431	0.267748
$727$	17.8595	0.662374	0.331187	−	0.943565i	$-0.392551\pi$
$727$	17.8595	0.662374	0.331187	+	0.943565i	$0.392551\pi$
$728$	19.0799	0.707148
$729$	−3.73143	−0.138201
$730$	−11.6037	−0.429473
$731$	4.62835	0.171186
$732$	−5.48011	−0.202551
$733$	−30.6189	−1.13094	−0.565468	−	0.824770i	$-0.691304\pi$
$733$	−30.6189	−1.13094	−0.565468	+	0.824770i	$0.691304\pi$
$734$	11.4213	0.421566
$735$	−4.91302	−0.181219
$736$	−7.35010	−0.270928
$737$	−5.66162	−0.208548
$738$	−11.2022	−0.412359
$739$	−18.6997	−0.687879	−0.343940	−	0.938992i	$-0.611762\pi$
$739$	−18.6997	−0.687879	−0.343940	+	0.938992i	$0.611762\pi$
$740$	25.5214	0.938185
$741$	−31.5009	−1.15722
$742$	1.36046	0.0499442
$743$	−2.72895	−0.100116	−0.0500578	−	0.998746i	$-0.515941\pi$
$743$	−2.72895	−0.100116	−0.0500578	+	0.998746i	$0.515941\pi$
$744$	−5.69096	−0.208641
$745$	−8.70857	−0.319057
$746$	34.4473	1.26120
$747$	2.62847	0.0961707
$748$	0.792030	0.0289595
$749$	16.8962	0.617375
$750$	−6.44141	−0.235207
$751$	−0.263626	−0.00961984	−0.00480992	−	0.999988i	$-0.501531\pi$
$751$	−0.263626	−0.00961984	−0.00480992	+	0.999988i	$0.501531\pi$
$752$	1.24384	0.0453583
$753$	17.4250	0.635002
$754$	22.9959	0.837461
$755$	4.77496	0.173779
$756$	−12.1214	−0.440850
$757$	−11.1381	−0.404820	−0.202410	−	0.979301i	$-0.564877\pi$
$757$	−11.1381	−0.404820	−0.202410	+	0.979301i	$0.564877\pi$
$758$	31.0665	1.12839
$759$	4.54845	0.165098
$760$	18.4789	0.670302
$761$	13.5686	0.491863	0.245932	−	0.969287i	$-0.420906\pi$
$761$	13.5686	0.491863	0.245932	+	0.969287i	$0.420906\pi$
$762$	−13.4956	−0.488893
$763$	54.8561	1.98593
$764$	11.5835	0.419075
$765$	5.70805	0.206375
$766$	−36.7768	−1.32880
$767$	−61.6565	−2.22629
$768$	0.705213	0.0254472
$769$	13.2360	0.477302	0.238651	−	0.971105i	$-0.423295\pi$
$769$	13.2360	0.477302	0.238651	+	0.971105i	$0.423295\pi$
$770$	6.92628	0.249606
$771$	−6.11668	−0.220287
$772$	11.5606	0.416075
$773$	13.5198	0.486273	0.243137	−	0.969992i	$-0.421824\pi$
$773$	13.5198	0.486273	0.243137	+	0.969992i	$0.421824\pi$
$774$	12.8333	0.461285
$775$	11.1794	0.401575
$776$	9.03477	0.324329
$777$	−22.2480	−0.798143
$778$	−2.96738	−0.106386
$779$	−32.7329	−1.17278
$780$	10.8850	0.389746
$781$	−7.06974	−0.252975
$782$	−6.63414	−0.237236
$783$	−14.6092	−0.522090
$784$	2.75700	0.0984643
$785$	1.02883	0.0367205
$786$	2.43896	0.0869949
$787$	−33.8176	−1.20547	−0.602733	−	0.797943i	$-0.705922\pi$
$787$	−33.8176	−1.20547	−0.602733	+	0.797943i	$0.705922\pi$
$788$	9.71843	0.346205
$789$	−7.52670	−0.267958
$790$	−38.9124	−1.38444
$791$	51.1171	1.81751
$792$	2.19611	0.0780355
$793$	47.4664	1.68558
$794$	−18.4477	−0.654685
$795$	0.776141	0.0275269
$796$	25.8931	0.917758
$797$	−51.1728	−1.81263	−0.906316	−	0.422600i	$-0.861117\pi$
$797$	−51.1728	−1.81263	−0.906316	+	0.422600i	$0.861117\pi$
$798$	−16.1088	−0.570246
$799$	1.12268	0.0397177
$800$	−1.38533	−0.0489787
$801$	19.6231	0.693348
$802$	−5.92055	−0.209062
$803$	−4.02954	−0.142199
$804$	−4.54999	−0.160466
$805$	−58.0153	−2.04477
$806$	49.2928	1.73626
$807$	−12.2767	−0.432159
$808$	4.18584	0.147257
$809$	55.0603	1.93582	0.967909	−	0.251302i	$-0.0808587\pi$
$809$	55.0603	1.93582	0.967909	+	0.251302i	$0.0808587\pi$
$810$	12.0570	0.423639
$811$	25.5814	0.898285	0.449142	−	0.893460i	$-0.351730\pi$
$811$	25.5814	0.898285	0.449142	+	0.893460i	$0.351730\pi$
$812$	11.7595	0.412679
$813$	21.6965	0.760929
$814$	8.86264	0.310635
$815$	28.4010	0.994843
$816$	0.636520	0.0222826
$817$	37.4991	1.31193
$818$	−10.5693	−0.369549
$819$	47.7508	1.66855
$820$	11.3107	0.394988
$821$	40.3041	1.40662	0.703312	−	0.710882i	$-0.251704\pi$
$821$	40.3041	1.40662	0.703312	+	0.710882i	$0.251704\pi$
$822$	−9.03135	−0.315004
$823$	44.1497	1.53896	0.769481	−	0.638669i	$-0.220515\pi$
$823$	44.1497	1.53896	0.769481	+	0.638669i	$0.220515\pi$
$824$	1.48280	0.0516559
$825$	0.857280	0.0298467
$826$	−31.5297	−1.09706
$827$	31.4447	1.09344	0.546720	−	0.837315i	$-0.315876\pi$
$827$	31.4447	1.09344	0.546720	+	0.837315i	$0.315876\pi$
$828$	−18.3949	−0.639267
$829$	−28.7806	−0.999591	−0.499796	−	0.866143i	$-0.666592\pi$
$829$	−28.7806	−0.999591	−0.499796	+	0.866143i	$0.666592\pi$
$830$	−2.65393	−0.0921194
$831$	4.31281	0.149610
$832$	−6.10826	−0.211766
$833$	2.48845	0.0862196
$834$	8.11569	0.281023
$835$	−31.8203	−1.10119
$836$	6.41705	0.221938
$837$	−31.3155	−1.08242
$838$	37.3238	1.28933
$839$	−27.9221	−0.963977	−0.481988	−	0.876178i	$-0.660085\pi$
$839$	−27.9221	−0.963977	−0.481988	+	0.876178i	$0.660085\pi$
$840$	5.56634	0.192057
$841$	−14.8269	−0.511273
$842$	−13.8074	−0.475835
$843$	−15.3466	−0.528566
$844$	−13.7235	−0.472384
$845$	−61.4317	−2.11331
$846$	3.11293	0.107025
$847$	−31.9546	−1.09797
$848$	−0.435541	−0.0149565
$849$	22.2290	0.762897
$850$	−1.25039	−0.0428879
$851$	−74.2345	−2.54473
$852$	−5.68164	−0.194650
$853$	25.8517	0.885146	0.442573	−	0.896732i	$-0.354066\pi$
$853$	25.8517	0.885146	0.442573	+	0.896732i	$0.354066\pi$
$854$	24.2732	0.830612
$855$	46.2468	1.58161
$856$	−5.40919	−0.184882
$857$	16.7535	0.572287	0.286143	−	0.958187i	$-0.407627\pi$
$857$	16.7535	0.572287	0.286143	+	0.958187i	$0.407627\pi$
$858$	3.77997	0.129046
$859$	−8.15264	−0.278164	−0.139082	−	0.990281i	$-0.544415\pi$
$859$	−8.15264	−0.278164	−0.139082	+	0.990281i	$0.544415\pi$
$860$	−12.9577	−0.441853
$861$	−9.86001	−0.336028
$862$	4.42423	0.150690
$863$	8.40983	0.286274	0.143137	−	0.989703i	$-0.454281\pi$
$863$	8.40983	0.286274	0.143137	+	0.989703i	$0.454281\pi$
$864$	3.88056	0.132019
$865$	−44.0463	−1.49762
$866$	−39.2363	−1.33330
$867$	−11.4141	−0.387643
$868$	25.2071	0.855586
$869$	−13.5128	−0.458392
$870$	6.70879	0.227449
$871$	39.4101	1.33536
$872$	−17.5617	−0.594715
$873$	22.6111	0.765270
$874$	−53.7500	−1.81812
$875$	28.5311	0.964528
$876$	−3.23836	−0.109414
$877$	52.9897	1.78933	0.894667	−	0.446734i	$-0.147413\pi$
$877$	52.9897	1.78933	0.894667	+	0.446734i	$0.147413\pi$
$878$	−39.7070	−1.34005
$879$	17.2937	0.583304
$880$	−2.21739	−0.0747481
$881$	53.4257	1.79996	0.899979	−	0.435934i	$-0.143582\pi$
$881$	53.4257	1.79996	0.899979	+	0.435934i	$0.143582\pi$
$882$	6.89988	0.232331
$883$	−43.9814	−1.48009	−0.740045	−	0.672557i	$-0.765196\pi$
$883$	−43.9814	−1.48009	−0.740045	+	0.672557i	$0.765196\pi$
$884$	−5.51327	−0.185431
$885$	−17.9876	−0.604646
$886$	−7.03855	−0.236465
$887$	0.275807	0.00926071	0.00463035	−	0.999989i	$-0.498526\pi$
$887$	0.275807	0.00926071	0.00463035	+	0.999989i	$0.498526\pi$
$888$	7.12251	0.239016
$889$	59.7763	2.00483
$890$	−19.8132	−0.664140
$891$	4.18694	0.140268
$892$	−3.92329	−0.131361
$893$	9.09601	0.304386
$894$	−2.43039	−0.0812843
$895$	−58.6299	−1.95978
$896$	−3.12362	−0.104353
$897$	−31.6615	−1.05715
$898$	−9.33948	−0.311663
$899$	30.3807	1.01325
$900$	−3.46702	−0.115567
$901$	−0.393116	−0.0130966
$902$	3.92780	0.130781
$903$	11.2957	0.375897
$904$	−16.3647	−0.544281
$905$	−5.22799	−0.173784
$906$	1.33259	0.0442725
$907$	−7.34078	−0.243747	−0.121873	−	0.992546i	$-0.538890\pi$
$907$	−7.34078	−0.243747	−0.121873	+	0.992546i	$0.538890\pi$
$908$	22.8991	0.759933
$909$	10.4758	0.347461
$910$	−48.2134	−1.59826
$911$	25.2893	0.837871	0.418935	−	0.908016i	$-0.362403\pi$
$911$	25.2893	0.837871	0.418935	+	0.908016i	$0.362403\pi$
$912$	5.15710	0.170769
$913$	−0.921613	−0.0305009
$914$	0.653240	0.0216073
$915$	13.8478	0.457794
$916$	−13.6383	−0.450621
$917$	−10.8030	−0.356745
$918$	3.50256	0.115602
$919$	−9.29274	−0.306539	−0.153270	−	0.988184i	$-0.548980\pi$
$919$	−9.29274	−0.306539	−0.153270	+	0.988184i	$0.548980\pi$
$920$	18.5731	0.612337
$921$	4.68789	0.154471
$922$	20.2411	0.666607
$923$	49.2120	1.61983
$924$	1.93298	0.0635905
$925$	−13.9915	−0.460039
$926$	−39.6496	−1.30297
$927$	3.71097	0.121884
$928$	−3.76472	−0.123583
$929$	26.6844	0.875487	0.437743	−	0.899100i	$-0.355778\pi$
$929$	26.6844	0.875487	0.437743	+	0.899100i	$0.355778\pi$
$930$	14.3806	0.471559
$931$	20.1615	0.660766
$932$	−23.4669	−0.768683
$933$	−6.22248	−0.203715
$934$	−26.9268	−0.881070
$935$	−2.00140	−0.0654527
$936$	−15.2870	−0.499671
$937$	−27.8313	−0.909208	−0.454604	−	0.890694i	$-0.650219\pi$
$937$	−27.8313	−0.909208	−0.454604	+	0.890694i	$0.650219\pi$
$938$	20.1534	0.658032
$939$	3.46679	0.113135
$940$	−3.14309	−0.102516
$941$	−19.0204	−0.620049	−0.310024	−	0.950729i	$-0.600337\pi$
$941$	−19.0204	−0.620049	−0.310024	+	0.950729i	$0.600337\pi$
$942$	0.287126	0.00935506
$943$	−32.8997	−1.07136
$944$	10.0940	0.328530
$945$	30.6298	0.996386
$946$	−4.49972	−0.146298
$947$	−32.1398	−1.04440	−0.522202	−	0.852822i	$-0.674889\pi$
$947$	−32.1398	−1.04440	−0.522202	+	0.852822i	$0.674889\pi$
$948$	−10.8597	−0.352706
$949$	28.0494	0.910521
$950$	−10.1307	−0.328682
$951$	10.4228	0.337981
$952$	−2.81936	−0.0913758
$953$	−18.5772	−0.601775	−0.300887	−	0.953660i	$-0.597283\pi$
$953$	−18.5772	−0.601775	−0.300887	+	0.953660i	$0.597283\pi$
$954$	−1.09002	−0.0352906
$955$	−29.2705	−0.947172
$956$	23.0848	0.746616
$957$	2.32971	0.0753090
$958$	14.3587	0.463910
$959$	40.0028	1.29176
$960$	−1.78202	−0.0575143
$961$	34.1225	1.10073
$962$	−61.6922	−1.98904
$963$	−13.5374	−0.436238
$964$	15.7621	0.507663
$965$	−29.2127	−0.940390
$966$	−16.1909	−0.520934
$967$	−30.0439	−0.966146	−0.483073	−	0.875580i	$-0.660479\pi$
$967$	−30.0439	−0.966146	−0.483073	+	0.875580i	$0.660479\pi$
$968$	10.2300	0.328804
$969$	4.65476	0.149532
$970$	−22.8301	−0.733032
$971$	−35.5341	−1.14034	−0.570172	−	0.821526i	$-0.693123\pi$
$971$	−35.5341	−1.14034	−0.570172	+	0.821526i	$0.693123\pi$
$972$	15.0065	0.481335
$973$	−35.9471	−1.15241
$974$	−40.4178	−1.29507
$975$	−5.96747	−0.191112
$976$	−7.77085	−0.248739
$977$	58.3023	1.86526	0.932628	−	0.360839i	$-0.117510\pi$
$977$	58.3023	1.86526	0.932628	+	0.360839i	$0.117510\pi$
$978$	7.92614	0.253450
$979$	−6.88039	−0.219898
$980$	−6.96672	−0.222544
$981$	−43.9513	−1.40326
$982$	−2.70819	−0.0864218
$983$	16.0693	0.512530	0.256265	−	0.966607i	$-0.417508\pi$
$983$	16.0693	0.512530	0.256265	+	0.966607i	$0.417508\pi$
$984$	3.15660	0.100629
$985$	−24.5577	−0.782473
$986$	−3.39801	−0.108215
$987$	2.73996	0.0872138
$988$	−44.6687	−1.42110
$989$	37.6902	1.19848
$990$	−5.54940	−0.176372
$991$	−9.84429	−0.312714	−0.156357	−	0.987701i	$-0.549975\pi$
$991$	−9.84429	−0.312714	−0.156357	+	0.987701i	$0.549975\pi$
$992$	−8.06985	−0.256218
$993$	−12.2930	−0.390106
$994$	25.1659	0.798212
$995$	−65.4299	−2.07427
$996$	−0.740659	−0.0234687
$997$	−31.5021	−0.997680	−0.498840	−	0.866694i	$-0.666240\pi$
$997$	−31.5021	−0.997680	−0.498840	+	0.866694i	$0.666240\pi$
$998$	16.4582	0.520976
$999$	39.1928	1.24001

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.e.1.42	✓	77

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.e.1.42	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.705213 q^{3} +1.00000 q^{4} -2.52692 q^{5} -0.705213 q^{6} +3.12362 q^{7} -1.00000 q^{8} -2.50268 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.705213 q^{3} +1.00000 q^{4} -2.52692 q^{5} -0.705213 q^{6} +3.12362 q^{7} -1.00000 q^{8} -2.50268 q^{9} +2.52692 q^{10} +0.877506 q^{11} +0.705213 q^{12} -6.10826 q^{13} -3.12362 q^{14} -1.78202 q^{15} +1.00000 q^{16} +0.902592 q^{17} +2.50268 q^{18} +7.31283 q^{19} -2.52692 q^{20} +2.20282 q^{21} -0.877506 q^{22} +7.35010 q^{23} -0.705213 q^{24} +1.38533 q^{25} +6.10826 q^{26} -3.88056 q^{27} +3.12362 q^{28} +3.76472 q^{29} +1.78202 q^{30} +8.06985 q^{31} -1.00000 q^{32} +0.618828 q^{33} -0.902592 q^{34} -7.89314 q^{35} -2.50268 q^{36} -10.0998 q^{37} -7.31283 q^{38} -4.30763 q^{39} +2.52692 q^{40} -4.47609 q^{41} -2.20282 q^{42} +5.12785 q^{43} +0.877506 q^{44} +6.32406 q^{45} -7.35010 q^{46} +1.24384 q^{47} +0.705213 q^{48} +2.75700 q^{49} -1.38533 q^{50} +0.636520 q^{51} -6.10826 q^{52} -0.435541 q^{53} +3.88056 q^{54} -2.21739 q^{55} -3.12362 q^{56} +5.15710 q^{57} -3.76472 q^{58} +10.0940 q^{59} -1.78202 q^{60} -7.77085 q^{61} -8.06985 q^{62} -7.81741 q^{63} +1.00000 q^{64} +15.4351 q^{65} -0.618828 q^{66} -6.45194 q^{67} +0.902592 q^{68} +5.18338 q^{69} +7.89314 q^{70} -8.05663 q^{71} +2.50268 q^{72} -4.59204 q^{73} +10.0998 q^{74} +0.976950 q^{75} +7.31283 q^{76} +2.74100 q^{77} +4.30763 q^{78} -15.3991 q^{79} -2.52692 q^{80} +4.77141 q^{81} +4.47609 q^{82} -1.05026 q^{83} +2.20282 q^{84} -2.28078 q^{85} -5.12785 q^{86} +2.65493 q^{87} -0.877506 q^{88} -7.84085 q^{89} -6.32406 q^{90} -19.0799 q^{91} +7.35010 q^{92} +5.69096 q^{93} -1.24384 q^{94} -18.4789 q^{95} -0.705213 q^{96} -9.03477 q^{97} -2.75700 q^{98} -2.19611 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) \) 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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