LMFDB - Embedded newform 8002.2.a.e.1.12

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			1.12
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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.37610 q^{3} +1.00000 q^{4} -3.32970 q^{5} +2.37610 q^{6} +3.18362 q^{7} -1.00000 q^{8} +2.64585 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.37610 q^{3} +1.00000 q^{4} -3.32970 q^{5} +2.37610 q^{6} +3.18362 q^{7} -1.00000 q^{8} +2.64585 q^{9} +3.32970 q^{10} -3.12572 q^{11} -2.37610 q^{12} -4.61345 q^{13} -3.18362 q^{14} +7.91169 q^{15} +1.00000 q^{16} +5.18964 q^{17} -2.64585 q^{18} +3.13483 q^{19} -3.32970 q^{20} -7.56459 q^{21} +3.12572 q^{22} +5.95882 q^{23} +2.37610 q^{24} +6.08687 q^{25} +4.61345 q^{26} +0.841493 q^{27} +3.18362 q^{28} +1.91697 q^{29} -7.91169 q^{30} -1.12111 q^{31} -1.00000 q^{32} +7.42702 q^{33} -5.18964 q^{34} -10.6005 q^{35} +2.64585 q^{36} -3.03323 q^{37} -3.13483 q^{38} +10.9620 q^{39} +3.32970 q^{40} +7.25292 q^{41} +7.56459 q^{42} -11.0303 q^{43} -3.12572 q^{44} -8.80988 q^{45} -5.95882 q^{46} -2.10688 q^{47} -2.37610 q^{48} +3.13541 q^{49} -6.08687 q^{50} -12.3311 q^{51} -4.61345 q^{52} +11.0192 q^{53} -0.841493 q^{54} +10.4077 q^{55} -3.18362 q^{56} -7.44867 q^{57} -1.91697 q^{58} -5.59792 q^{59} +7.91169 q^{60} -1.78705 q^{61} +1.12111 q^{62} +8.42338 q^{63} +1.00000 q^{64} +15.3614 q^{65} -7.42702 q^{66} +10.2713 q^{67} +5.18964 q^{68} -14.1588 q^{69} +10.6005 q^{70} +7.76365 q^{71} -2.64585 q^{72} -13.9399 q^{73} +3.03323 q^{74} -14.4630 q^{75} +3.13483 q^{76} -9.95109 q^{77} -10.9620 q^{78} +2.57202 q^{79} -3.32970 q^{80} -9.93702 q^{81} -7.25292 q^{82} +11.1175 q^{83} -7.56459 q^{84} -17.2799 q^{85} +11.0303 q^{86} -4.55491 q^{87} +3.12572 q^{88} +10.2553 q^{89} +8.80988 q^{90} -14.6875 q^{91} +5.95882 q^{92} +2.66387 q^{93} +2.10688 q^{94} -10.4380 q^{95} +2.37610 q^{96} +5.61516 q^{97} -3.13541 q^{98} -8.27019 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.37610	−1.37184	−0.685921	−	0.727676i	$-0.740600\pi$
$3$	−2.37610	−1.37184	−0.685921	+	0.727676i	$0.740600\pi$
$4$	1.00000	0.500000
$5$	−3.32970	−1.48908	−0.744542	−	0.667575i	$-0.767332\pi$
$5$	−3.32970	−1.48908	−0.744542	+	0.667575i	$0.767332\pi$
$6$	2.37610	0.970039
$7$	3.18362	1.20329	0.601647	−	0.798762i	$-0.294511\pi$
$7$	3.18362	1.20329	0.601647	+	0.798762i	$0.294511\pi$
$8$	−1.00000	−0.353553
$9$	2.64585	0.881950
$10$	3.32970	1.05294
$11$	−3.12572	−0.942440	−0.471220	−	0.882016i	$-0.656186\pi$
$11$	−3.12572	−0.942440	−0.471220	+	0.882016i	$0.656186\pi$
$12$	−2.37610	−0.685921
$13$	−4.61345	−1.27954	−0.639770	−	0.768566i	$-0.720971\pi$
$13$	−4.61345	−1.27954	−0.639770	+	0.768566i	$0.720971\pi$
$14$	−3.18362	−0.850857
$15$	7.91169	2.04279
$16$	1.00000	0.250000
$17$	5.18964	1.25867	0.629336	−	0.777134i	$-0.283327\pi$
$17$	5.18964	1.25867	0.629336	+	0.777134i	$0.283327\pi$
$18$	−2.64585	−0.623633
$19$	3.13483	0.719179	0.359590	−	0.933111i	$-0.382917\pi$
$19$	3.13483	0.719179	0.359590	+	0.933111i	$0.382917\pi$
$20$	−3.32970	−0.744542
$21$	−7.56459	−1.65073
$22$	3.12572	0.666405
$23$	5.95882	1.24250	0.621250	−	0.783612i	$-0.286625\pi$
$23$	5.95882	1.24250	0.621250	+	0.783612i	$0.286625\pi$
$24$	2.37610	0.485019
$25$	6.08687	1.21737
$26$	4.61345	0.904772
$27$	0.841493	0.161945
$28$	3.18362	0.601647
$29$	1.91697	0.355972	0.177986	−	0.984033i	$-0.443042\pi$
$29$	1.91697	0.355972	0.177986	+	0.984033i	$0.443042\pi$
$30$	−7.91169	−1.44447
$31$	−1.12111	−0.201357	−0.100679	−	0.994919i	$-0.532101\pi$
$31$	−1.12111	−0.201357	−0.100679	+	0.994919i	$0.532101\pi$
$32$	−1.00000	−0.176777
$33$	7.42702	1.29288
$34$	−5.18964	−0.890015
$35$	−10.6005	−1.79181
$36$	2.64585	0.440975
$37$	−3.03323	−0.498661	−0.249330	−	0.968418i	$-0.580210\pi$
$37$	−3.03323	−0.498661	−0.249330	+	0.968418i	$0.580210\pi$
$38$	−3.13483	−0.508537
$39$	10.9620	1.75533
$40$	3.32970	0.526471
$41$	7.25292	1.13272	0.566358	−	0.824160i	$-0.308352\pi$
$41$	7.25292	1.13272	0.566358	+	0.824160i	$0.308352\pi$
$42$	7.56459	1.16724
$43$	−11.0303	−1.68210	−0.841050	−	0.540957i	$-0.818062\pi$
$43$	−11.0303	−1.68210	−0.841050	+	0.540957i	$0.818062\pi$
$44$	−3.12572	−0.471220
$45$	−8.80988	−1.31330
$46$	−5.95882	−0.878580
$47$	−2.10688	−0.307320	−0.153660	−	0.988124i	$-0.549106\pi$
$47$	−2.10688	−0.307320	−0.153660	+	0.988124i	$0.549106\pi$
$48$	−2.37610	−0.342960
$49$	3.13541	0.447916
$50$	−6.08687	−0.860813
$51$	−12.3311	−1.72670
$52$	−4.61345	−0.639770
$53$	11.0192	1.51360	0.756802	−	0.653644i	$-0.226760\pi$
$53$	11.0192	1.51360	0.756802	+	0.653644i	$0.226760\pi$
$54$	−0.841493	−0.114513
$55$	10.4077	1.40337
$56$	−3.18362	−0.425429
$57$	−7.44867	−0.986600
$58$	−1.91697	−0.251710
$59$	−5.59792	−0.728788	−0.364394	−	0.931245i	$-0.618724\pi$
$59$	−5.59792	−0.728788	−0.364394	+	0.931245i	$0.618724\pi$
$60$	7.91169	1.02139
$61$	−1.78705	−0.228808	−0.114404	−	0.993434i	$-0.536496\pi$
$61$	−1.78705	−0.228808	−0.114404	+	0.993434i	$0.536496\pi$
$62$	1.12111	0.142381
$63$	8.42338	1.06125
$64$	1.00000	0.125000
$65$	15.3614	1.90534
$66$	−7.42702	−0.914203
$67$	10.2713	1.25484	0.627421	−	0.778680i	$-0.284110\pi$
$67$	10.2713	1.25484	0.627421	+	0.778680i	$0.284110\pi$
$68$	5.18964	0.629336
$69$	−14.1588	−1.70451
$70$	10.6005	1.26700
$71$	7.76365	0.921375	0.460688	−	0.887562i	$-0.347603\pi$
$71$	7.76365	0.921375	0.460688	+	0.887562i	$0.347603\pi$
$72$	−2.64585	−0.311817
$73$	−13.9399	−1.63155	−0.815773	−	0.578372i	$-0.803688\pi$
$73$	−13.9399	−1.63155	−0.815773	+	0.578372i	$0.803688\pi$
$74$	3.03323	0.352606
$75$	−14.4630	−1.67004
$76$	3.13483	0.359590
$77$	−9.95109	−1.13403
$78$	−10.9620	−1.24120
$79$	2.57202	0.289374	0.144687	−	0.989477i	$-0.453782\pi$
$79$	2.57202	0.289374	0.144687	+	0.989477i	$0.453782\pi$
$80$	−3.32970	−0.372271
$81$	−9.93702	−1.10411
$82$	−7.25292	−0.800951
$83$	11.1175	1.22031	0.610154	−	0.792282i	$-0.291107\pi$
$83$	11.1175	1.22031	0.610154	+	0.792282i	$0.291107\pi$
$84$	−7.56459	−0.825365
$85$	−17.2799	−1.87427
$86$	11.0303	1.18942
$87$	−4.55491	−0.488337
$88$	3.12572	0.333203
$89$	10.2553	1.08706	0.543532	−	0.839389i	$-0.317087\pi$
$89$	10.2553	1.08706	0.543532	+	0.839389i	$0.317087\pi$
$90$	8.80988	0.928643
$91$	−14.6875	−1.53966
$92$	5.95882	0.621250
$93$	2.66387	0.276230
$94$	2.10688	0.217308
$95$	−10.4380	−1.07092
$96$	2.37610	0.242510
$97$	5.61516	0.570133	0.285066	−	0.958508i	$-0.407984\pi$
$97$	5.61516	0.570133	0.285066	+	0.958508i	$0.407984\pi$
$98$	−3.13541	−0.316725
$99$	−8.27019	−0.831185
$100$	6.08687	0.608687
$101$	−3.24073	−0.322465	−0.161233	−	0.986916i	$-0.551547\pi$
$101$	−3.24073	−0.322465	−0.161233	+	0.986916i	$0.551547\pi$
$102$	12.3311	1.22096
$103$	−7.04994	−0.694651	−0.347325	−	0.937745i	$-0.612910\pi$
$103$	−7.04994	−0.694651	−0.347325	+	0.937745i	$0.612910\pi$
$104$	4.61345	0.452386
$105$	25.1878	2.45808
$106$	−11.0192	−1.07028
$107$	−5.95942	−0.576119	−0.288059	−	0.957613i	$-0.593010\pi$
$107$	−5.95942	−0.576119	−0.288059	+	0.957613i	$0.593010\pi$
$108$	0.841493	0.0809727
$109$	0.338190	0.0323927	0.0161964	−	0.999869i	$-0.494844\pi$
$109$	0.338190	0.0323927	0.0161964	+	0.999869i	$0.494844\pi$
$110$	−10.4077	−0.992334
$111$	7.20727	0.684084
$112$	3.18362	0.300823
$113$	−16.0462	−1.50950	−0.754752	−	0.656011i	$-0.772243\pi$
$113$	−16.0462	−1.50950	−0.754752	+	0.656011i	$0.772243\pi$
$114$	7.44867	0.697632
$115$	−19.8411	−1.85019
$116$	1.91697	0.177986
$117$	−12.2065	−1.12849
$118$	5.59792	0.515331
$119$	16.5218	1.51455
$120$	−7.91169	−0.722235
$121$	−1.22989	−0.111808
$122$	1.78705	0.161792
$123$	−17.2337	−1.55391
$124$	−1.12111	−0.100679
$125$	−3.61895	−0.323688
$126$	−8.42338	−0.750414
$127$	−13.1241	−1.16458	−0.582290	−	0.812981i	$-0.697843\pi$
$127$	−13.1241	−1.16458	−0.582290	+	0.812981i	$0.697843\pi$
$128$	−1.00000	−0.0883883
$129$	26.2090	2.30758
$130$	−15.3614	−1.34728
$131$	−6.74933	−0.589692	−0.294846	−	0.955545i	$-0.595268\pi$
$131$	−6.74933	−0.589692	−0.294846	+	0.955545i	$0.595268\pi$
$132$	7.42702	0.646439
$133$	9.98009	0.865384
$134$	−10.2713	−0.887308
$135$	−2.80191	−0.241150
$136$	−5.18964	−0.445008
$137$	11.4608	0.979166	0.489583	−	0.871957i	$-0.337149\pi$
$137$	11.4608	0.979166	0.489583	+	0.871957i	$0.337149\pi$
$138$	14.1588	1.20527
$139$	8.42584	0.714671	0.357335	−	0.933976i	$-0.383685\pi$
$139$	8.42584	0.714671	0.357335	+	0.933976i	$0.383685\pi$
$140$	−10.6005	−0.895903
$141$	5.00616	0.421595
$142$	−7.76365	−0.651511
$143$	14.4203	1.20589
$144$	2.64585	0.220488
$145$	−6.38292	−0.530073
$146$	13.9399	1.15368
$147$	−7.45006	−0.614470
$148$	−3.03323	−0.249330
$149$	−6.27409	−0.513994	−0.256997	−	0.966412i	$-0.582733\pi$
$149$	−6.27409	−0.513994	−0.256997	+	0.966412i	$0.582733\pi$
$150$	14.4630	1.18090
$151$	−15.8952	−1.29353	−0.646765	−	0.762690i	$-0.723878\pi$
$151$	−15.8952	−1.29353	−0.646765	+	0.762690i	$0.723878\pi$
$152$	−3.13483	−0.254268
$153$	13.7310	1.11009
$154$	9.95109	0.801882
$155$	3.73296	0.299838
$156$	10.9620	0.877664
$157$	7.20509	0.575029	0.287515	−	0.957776i	$-0.407171\pi$
$157$	7.20509	0.575029	0.287515	+	0.957776i	$0.407171\pi$
$158$	−2.57202	−0.204619
$159$	−26.1827	−2.07643
$160$	3.32970	0.263236
$161$	18.9706	1.49509
$162$	9.93702	0.780726
$163$	−18.7543	−1.46895	−0.734473	−	0.678638i	$-0.762571\pi$
$163$	−18.7543	−1.46895	−0.734473	+	0.678638i	$0.762571\pi$
$164$	7.25292	0.566358
$165$	−24.7297	−1.92521
$166$	−11.1175	−0.862889
$167$	18.4553	1.42812	0.714059	−	0.700086i	$-0.246855\pi$
$167$	18.4553	1.42812	0.714059	+	0.700086i	$0.246855\pi$
$168$	7.56459	0.583621
$169$	8.28391	0.637224
$170$	17.2799	1.32531
$171$	8.29429	0.634280
$172$	−11.0303	−0.841050
$173$	1.00758	0.0766045	0.0383023	−	0.999266i	$-0.487805\pi$
$173$	1.00758	0.0766045	0.0383023	+	0.999266i	$0.487805\pi$
$174$	4.55491	0.345307
$175$	19.3783	1.46486
$176$	−3.12572	−0.235610
$177$	13.3012	0.999781
$178$	−10.2553	−0.768670
$179$	7.43191	0.555487	0.277743	−	0.960655i	$-0.410413\pi$
$179$	7.43191	0.555487	0.277743	+	0.960655i	$0.410413\pi$
$180$	−8.80988	−0.656650
$181$	−0.732910	−0.0544768	−0.0272384	−	0.999629i	$-0.508671\pi$
$181$	−0.732910	−0.0544768	−0.0272384	+	0.999629i	$0.508671\pi$
$182$	14.6875	1.08871
$183$	4.24620	0.313888
$184$	−5.95882	−0.439290
$185$	10.0997	0.742548
$186$	−2.66387	−0.195324
$187$	−16.2213	−1.18622
$188$	−2.10688	−0.153660
$189$	2.67899	0.194868
$190$	10.4380	0.757254
$191$	−1.87680	−0.135801	−0.0679003	−	0.997692i	$-0.521630\pi$
$191$	−1.87680	−0.135801	−0.0679003	+	0.997692i	$0.521630\pi$
$192$	−2.37610	−0.171480
$193$	3.42745	0.246713	0.123357	−	0.992362i	$-0.460634\pi$
$193$	3.42745	0.246713	0.123357	+	0.992362i	$0.460634\pi$
$194$	−5.61516	−0.403145
$195$	−36.5002	−2.61383
$196$	3.13541	0.223958
$197$	−11.9950	−0.854608	−0.427304	−	0.904108i	$-0.640537\pi$
$197$	−11.9950	−0.854608	−0.427304	+	0.904108i	$0.640537\pi$
$198$	8.27019	0.587736
$199$	−0.888401	−0.0629771	−0.0314885	−	0.999504i	$-0.510025\pi$
$199$	−0.888401	−0.0629771	−0.0314885	+	0.999504i	$0.510025\pi$
$200$	−6.08687	−0.430407
$201$	−24.4057	−1.72145
$202$	3.24073	0.228017
$203$	6.10289	0.428339
$204$	−12.3311	−0.863349
$205$	−24.1500	−1.68671
$206$	7.04994	0.491192
$207$	15.7662	1.09582
$208$	−4.61345	−0.319885
$209$	−9.79859	−0.677783
$210$	−25.1878	−1.73812
$211$	−3.50907	−0.241574	−0.120787	−	0.992678i	$-0.538542\pi$
$211$	−3.50907	−0.241574	−0.120787	+	0.992678i	$0.538542\pi$
$212$	11.0192	0.756802
$213$	−18.4472	−1.26398
$214$	5.95942	0.407377
$215$	36.7274	2.50479
$216$	−0.841493	−0.0572563
$217$	−3.56919	−0.242292
$218$	−0.338190	−0.0229051
$219$	33.1227	2.23822
$220$	10.4077	0.701686
$221$	−23.9421	−1.61052
$222$	−7.20727	−0.483720
$223$	−11.1884	−0.749231	−0.374615	−	0.927180i	$-0.622225\pi$
$223$	−11.1884	−0.749231	−0.374615	+	0.927180i	$0.622225\pi$
$224$	−3.18362	−0.212714
$225$	16.1050	1.07366
$226$	16.0462	1.06738
$227$	−24.7115	−1.64016	−0.820079	−	0.572250i	$-0.806071\pi$
$227$	−24.7115	−1.64016	−0.820079	+	0.572250i	$0.806071\pi$
$228$	−7.44867	−0.493300
$229$	14.2933	0.944528	0.472264	−	0.881457i	$-0.343437\pi$
$229$	14.2933	0.944528	0.472264	+	0.881457i	$0.343437\pi$
$230$	19.8411	1.30828
$231$	23.6448	1.55571
$232$	−1.91697	−0.125855
$233$	24.1996	1.58537	0.792684	−	0.609633i	$-0.208683\pi$
$233$	24.1996	1.58537	0.792684	+	0.609633i	$0.208683\pi$
$234$	12.2065	0.797964
$235$	7.01527	0.457626
$236$	−5.59792	−0.364394
$237$	−6.11137	−0.396976
$238$	−16.5218	−1.07095
$239$	−20.3277	−1.31489	−0.657444	−	0.753503i	$-0.728363\pi$
$239$	−20.3277	−1.31489	−0.657444	+	0.753503i	$0.728363\pi$
$240$	7.91169	0.510697
$241$	19.8055	1.27578	0.637891	−	0.770126i	$-0.279807\pi$
$241$	19.8055	1.27578	0.637891	+	0.770126i	$0.279807\pi$
$242$	1.22989	0.0790600
$243$	21.0869	1.35272
$244$	−1.78705	−0.114404
$245$	−10.4400	−0.666985
$246$	17.2337	1.09878
$247$	−14.4624	−0.920219
$248$	1.12111	0.0711906
$249$	−26.4164	−1.67407
$250$	3.61895	0.228882
$251$	0.326101	0.0205833	0.0102917	−	0.999947i	$-0.496724\pi$
$251$	0.326101	0.0205833	0.0102917	+	0.999947i	$0.496724\pi$
$252$	8.42338	0.530623
$253$	−18.6256	−1.17098
$254$	13.1241	0.823482
$255$	41.0588	2.57120
$256$	1.00000	0.0625000
$257$	−6.63146	−0.413659	−0.206830	−	0.978377i	$-0.566315\pi$
$257$	−6.63146	−0.413659	−0.206830	+	0.978377i	$0.566315\pi$
$258$	−26.2090	−1.63170
$259$	−9.65665	−0.600035
$260$	15.3614	0.952672
$261$	5.07201	0.313950
$262$	6.74933	0.416975
$263$	−2.70731	−0.166940	−0.0834700	−	0.996510i	$-0.526600\pi$
$263$	−2.70731	−0.166940	−0.0834700	+	0.996510i	$0.526600\pi$
$264$	−7.42702	−0.457101
$265$	−36.6906	−2.25389
$266$	−9.98009	−0.611919
$267$	−24.3677	−1.49128
$268$	10.2713	0.627421
$269$	−15.3811	−0.937801	−0.468900	−	0.883251i	$-0.655350\pi$
$269$	−15.3811	−0.937801	−0.468900	+	0.883251i	$0.655350\pi$
$270$	2.80191	0.170519
$271$	−3.34226	−0.203028	−0.101514	−	0.994834i	$-0.532369\pi$
$271$	−3.34226	−0.203028	−0.101514	+	0.994834i	$0.532369\pi$
$272$	5.18964	0.314668
$273$	34.8989	2.11217
$274$	−11.4608	−0.692375
$275$	−19.0258	−1.14730
$276$	−14.1588	−0.852257
$277$	−12.6605	−0.760696	−0.380348	−	0.924844i	$-0.624196\pi$
$277$	−12.6605	−0.760696	−0.380348	+	0.924844i	$0.624196\pi$
$278$	−8.42584	−0.505348
$279$	−2.96629	−0.177587
$280$	10.6005	0.633499
$281$	−33.1853	−1.97967	−0.989835	−	0.142220i	$-0.954576\pi$
$281$	−33.1853	−1.97967	−0.989835	+	0.142220i	$0.954576\pi$
$282$	−5.00616	−0.298112
$283$	−5.52575	−0.328472	−0.164236	−	0.986421i	$-0.552516\pi$
$283$	−5.52575	−0.328472	−0.164236	+	0.986421i	$0.552516\pi$
$284$	7.76365	0.460688
$285$	24.8018	1.46913
$286$	−14.4203	−0.852693
$287$	23.0905	1.36299
$288$	−2.64585	−0.155908
$289$	9.93232	0.584254
$290$	6.38292	0.374818
$291$	−13.3422	−0.782132
$292$	−13.9399	−0.815773
$293$	−8.68084	−0.507140	−0.253570	−	0.967317i	$-0.581605\pi$
$293$	−8.68084	−0.507140	−0.253570	+	0.967317i	$0.581605\pi$
$294$	7.45006	0.434496
$295$	18.6394	1.08523
$296$	3.03323	0.176303
$297$	−2.63027	−0.152624
$298$	6.27409	0.363448
$299$	−27.4907	−1.58983
$300$	−14.4630	−0.835022
$301$	−35.1161	−2.02406
$302$	15.8952	0.914663
$303$	7.70031	0.442371
$304$	3.13483	0.179795
$305$	5.95032	0.340714
$306$	−13.7310	−0.784949
$307$	−14.0197	−0.800148	−0.400074	−	0.916483i	$-0.631016\pi$
$307$	−14.0197	−0.800148	−0.400074	+	0.916483i	$0.631016\pi$
$308$	−9.95109	−0.567016
$309$	16.7514	0.952951
$310$	−3.73296	−0.212018
$311$	−18.5350	−1.05102	−0.525510	−	0.850787i	$-0.676126\pi$
$311$	−18.5350	−1.05102	−0.525510	+	0.850787i	$0.676126\pi$
$312$	−10.9620	−0.620602
$313$	−14.8127	−0.837263	−0.418632	−	0.908156i	$-0.637490\pi$
$313$	−14.8127	−0.837263	−0.418632	+	0.908156i	$0.637490\pi$
$314$	−7.20509	−0.406607
$315$	−28.0473	−1.58028
$316$	2.57202	0.144687
$317$	24.5542	1.37910	0.689549	−	0.724239i	$-0.257809\pi$
$317$	24.5542	1.37910	0.689549	+	0.724239i	$0.257809\pi$
$318$	26.1827	1.46826
$319$	−5.99190	−0.335482
$320$	−3.32970	−0.186136
$321$	14.1602	0.790344
$322$	−18.9706	−1.05719
$323$	16.2686	0.905210
$324$	−9.93702	−0.552057
$325$	−28.0815	−1.55768
$326$	18.7543	1.03870
$327$	−0.803574	−0.0444377
$328$	−7.25292	−0.400475
$329$	−6.70750	−0.369796
$330$	24.7297	1.36133
$331$	30.3283	1.66700	0.833498	−	0.552523i	$-0.186335\pi$
$331$	30.3283	1.66700	0.833498	+	0.552523i	$0.186335\pi$
$332$	11.1175	0.610154
$333$	−8.02549	−0.439794
$334$	−18.4553	−1.00983
$335$	−34.2004	−1.86857
$336$	−7.56459	−0.412682
$337$	15.8510	0.863460	0.431730	−	0.902003i	$-0.357903\pi$
$337$	15.8510	0.863460	0.431730	+	0.902003i	$0.357903\pi$
$338$	−8.28391	−0.450585
$339$	38.1275	2.07080
$340$	−17.2799	−0.937134
$341$	3.50427	0.189767
$342$	−8.29429	−0.448504
$343$	−12.3034	−0.664319
$344$	11.0303	0.594712
$345$	47.1443	2.53817
$346$	−1.00758	−0.0541676
$347$	31.4368	1.68762	0.843808	−	0.536646i	$-0.180309\pi$
$347$	31.4368	1.68762	0.843808	+	0.536646i	$0.180309\pi$
$348$	−4.55491	−0.244169
$349$	34.2486	1.83329	0.916644	−	0.399705i	$-0.130887\pi$
$349$	34.2486	1.83329	0.916644	+	0.399705i	$0.130887\pi$
$350$	−19.3783	−1.03581
$351$	−3.88218	−0.207216
$352$	3.12572	0.166601
$353$	30.4436	1.62035	0.810174	−	0.586189i	$-0.199372\pi$
$353$	30.4436	1.62035	0.810174	+	0.586189i	$0.199372\pi$
$354$	−13.3012	−0.706952
$355$	−25.8506	−1.37201
$356$	10.2553	0.543532
$357$	−39.2575	−2.07773
$358$	−7.43191	−0.392789
$359$	9.77774	0.516049	0.258025	−	0.966138i	$-0.416928\pi$
$359$	9.77774	0.516049	0.258025	+	0.966138i	$0.416928\pi$
$360$	8.80988	0.464321
$361$	−9.17284	−0.482781
$362$	0.732910	0.0385209
$363$	2.92233	0.153383
$364$	−14.6875	−0.769832
$365$	46.4157	2.42951
$366$	−4.24620	−0.221952
$367$	13.5313	0.706330	0.353165	−	0.935561i	$-0.385105\pi$
$367$	13.5313	0.706330	0.353165	+	0.935561i	$0.385105\pi$
$368$	5.95882	0.310625
$369$	19.1901	0.998999
$370$	−10.0997	−0.525061
$371$	35.0809	1.82131
$372$	2.66387	0.138115
$373$	24.4204	1.26444	0.632219	−	0.774790i	$-0.282144\pi$
$373$	24.4204	1.26444	0.632219	+	0.774790i	$0.282144\pi$
$374$	16.2213	0.838785
$375$	8.59898	0.444049
$376$	2.10688	0.108654
$377$	−8.84384	−0.455481
$378$	−2.67899	−0.137792
$379$	5.27058	0.270731	0.135366	−	0.990796i	$-0.456779\pi$
$379$	5.27058	0.270731	0.135366	+	0.990796i	$0.456779\pi$
$380$	−10.4380	−0.535459
$381$	31.1843	1.59762
$382$	1.87680	0.0960255
$383$	−15.0420	−0.768609	−0.384304	−	0.923206i	$-0.625559\pi$
$383$	−15.0420	−0.768609	−0.384304	+	0.923206i	$0.625559\pi$
$384$	2.37610	0.121255
$385$	33.1341	1.68867
$386$	−3.42745	−0.174453
$387$	−29.1845	−1.48353
$388$	5.61516	0.285066
$389$	8.10304	0.410841	0.205420	−	0.978674i	$-0.434144\pi$
$389$	8.10304	0.410841	0.205420	+	0.978674i	$0.434144\pi$
$390$	36.5002	1.84826
$391$	30.9241	1.56390
$392$	−3.13541	−0.158362
$393$	16.0371	0.808964
$394$	11.9950	0.604299
$395$	−8.56403	−0.430903
$396$	−8.27019	−0.415592
$397$	−3.71395	−0.186398	−0.0931988	−	0.995648i	$-0.529709\pi$
$397$	−3.71395	−0.186398	−0.0931988	+	0.995648i	$0.529709\pi$
$398$	0.888401	0.0445315
$399$	−23.7137	−1.18717
$400$	6.08687	0.304343
$401$	−7.25391	−0.362243	−0.181121	−	0.983461i	$-0.557973\pi$
$401$	−7.25391	−0.362243	−0.181121	+	0.983461i	$0.557973\pi$
$402$	24.4057	1.21725
$403$	5.17218	0.257645
$404$	−3.24073	−0.161233
$405$	33.0873	1.64412
$406$	−6.10289	−0.302881
$407$	9.48104	0.469958
$408$	12.3311	0.610480
$409$	−6.89849	−0.341109	−0.170554	−	0.985348i	$-0.554556\pi$
$409$	−6.89849	−0.341109	−0.170554	+	0.985348i	$0.554556\pi$
$410$	24.1500	1.19268
$411$	−27.2321	−1.34326
$412$	−7.04994	−0.347325
$413$	−17.8216	−0.876946
$414$	−15.7662	−0.774864
$415$	−37.0180	−1.81714
$416$	4.61345	0.226193
$417$	−20.0206	−0.980415
$418$	9.79859	0.479265
$419$	−28.7368	−1.40389	−0.701943	−	0.712233i	$-0.747684\pi$
$419$	−28.7368	−1.40389	−0.701943	+	0.712233i	$0.747684\pi$
$420$	25.1878	1.22904
$421$	19.8686	0.968337	0.484169	−	0.874975i	$-0.339122\pi$
$421$	19.8686	0.968337	0.484169	+	0.874975i	$0.339122\pi$
$422$	3.50907	0.170819
$423$	−5.57449	−0.271041
$424$	−11.0192	−0.535140
$425$	31.5886	1.53227
$426$	18.4472	0.893770
$427$	−5.68927	−0.275323
$428$	−5.95942	−0.288059
$429$	−34.2642	−1.65429
$430$	−36.7274	−1.77115
$431$	20.8178	1.00276	0.501378	−	0.865228i	$-0.332827\pi$
$431$	20.8178	1.00276	0.501378	+	0.865228i	$0.332827\pi$
$432$	0.841493	0.0404863
$433$	2.54094	0.122110	0.0610550	−	0.998134i	$-0.480553\pi$
$433$	2.54094	0.122110	0.0610550	+	0.998134i	$0.480553\pi$
$434$	3.56919	0.171326
$435$	15.1665	0.727176
$436$	0.338190	0.0161964
$437$	18.6799	0.893580
$438$	−33.1227	−1.58266
$439$	−8.94228	−0.426792	−0.213396	−	0.976966i	$-0.568452\pi$
$439$	−8.94228	−0.426792	−0.213396	+	0.976966i	$0.568452\pi$
$440$	−10.4077	−0.496167
$441$	8.29584	0.395040
$442$	23.9421	1.13881
$443$	26.7326	1.27011	0.635053	−	0.772469i	$-0.280978\pi$
$443$	26.7326	1.27011	0.635053	+	0.772469i	$0.280978\pi$
$444$	7.20727	0.342042
$445$	−34.1471	−1.61873
$446$	11.1884	0.529786
$447$	14.9079	0.705118
$448$	3.18362	0.150412
$449$	9.24604	0.436347	0.218174	−	0.975910i	$-0.429990\pi$
$449$	9.24604	0.436347	0.218174	+	0.975910i	$0.429990\pi$
$450$	−16.1050	−0.759195
$451$	−22.6706	−1.06752
$452$	−16.0462	−0.754752
$453$	37.7685	1.77452
$454$	24.7115	1.15977
$455$	48.9047	2.29269
$456$	7.44867	0.348816
$457$	−3.05356	−0.142840	−0.0714198	−	0.997446i	$-0.522753\pi$
$457$	−3.05356	−0.142840	−0.0714198	+	0.997446i	$0.522753\pi$
$458$	−14.2933	−0.667882
$459$	4.36704	0.203836
$460$	−19.8411	−0.925094
$461$	−3.45615	−0.160969	−0.0804844	−	0.996756i	$-0.525647\pi$
$461$	−3.45615	−0.160969	−0.0804844	+	0.996756i	$0.525647\pi$
$462$	−23.6448	−1.10005
$463$	28.0621	1.30416	0.652079	−	0.758151i	$-0.273897\pi$
$463$	28.0621	1.30416	0.652079	+	0.758151i	$0.273897\pi$
$464$	1.91697	0.0889930
$465$	−8.86988	−0.411331
$466$	−24.1996	−1.12102
$467$	23.8285	1.10265	0.551325	−	0.834291i	$-0.314122\pi$
$467$	23.8285	1.10265	0.551325	+	0.834291i	$0.314122\pi$
$468$	−12.2065	−0.564246
$469$	32.7000	1.50994
$470$	−7.01527	−0.323590
$471$	−17.1200	−0.788849
$472$	5.59792	0.257665
$473$	34.4775	1.58528
$474$	6.11137	0.280704
$475$	19.0813	0.875510
$476$	16.5218	0.757276
$477$	29.1552	1.33492
$478$	20.3277	0.929767
$479$	17.5981	0.804077	0.402039	−	0.915623i	$-0.368302\pi$
$479$	17.5981	0.804077	0.402039	+	0.915623i	$0.368302\pi$
$480$	−7.91169	−0.361118
$481$	13.9937	0.638057
$482$	−19.8055	−0.902115
$483$	−45.0760	−2.05103
$484$	−1.22989	−0.0559039
$485$	−18.6968	−0.848976
$486$	−21.0869	−0.956521
$487$	−39.5420	−1.79182	−0.895909	−	0.444238i	$-0.853475\pi$
$487$	−39.5420	−1.79182	−0.895909	+	0.444238i	$0.853475\pi$
$488$	1.78705	0.0808958
$489$	44.5620	2.01516
$490$	10.4400	0.471630
$491$	−7.52025	−0.339384	−0.169692	−	0.985497i	$-0.554277\pi$
$491$	−7.52025	−0.339384	−0.169692	+	0.985497i	$0.554277\pi$
$492$	−17.2337	−0.776953
$493$	9.94837	0.448052
$494$	14.4624	0.650693
$495$	27.5372	1.23770
$496$	−1.12111	−0.0503393
$497$	24.7165	1.10869
$498$	26.4164	1.18375
$499$	−20.2139	−0.904899	−0.452450	−	0.891790i	$-0.649450\pi$
$499$	−20.2139	−0.904899	−0.452450	+	0.891790i	$0.649450\pi$
$500$	−3.61895	−0.161844
$501$	−43.8518	−1.95915
$502$	−0.326101	−0.0145546
$503$	43.3623	1.93343	0.966714	−	0.255858i	$-0.0823581\pi$
$503$	43.3623	1.93343	0.966714	+	0.255858i	$0.0823581\pi$
$504$	−8.42338	−0.375207
$505$	10.7907	0.480178
$506$	18.6256	0.828009
$507$	−19.6834	−0.874171
$508$	−13.1241	−0.582290
$509$	−1.64894	−0.0730878	−0.0365439	−	0.999332i	$-0.511635\pi$
$509$	−1.64894	−0.0730878	−0.0365439	+	0.999332i	$0.511635\pi$
$510$	−41.0588	−1.81811
$511$	−44.3794	−1.96323
$512$	−1.00000	−0.0441942
$513$	2.63794	0.116468
$514$	6.63146	0.292501
$515$	23.4741	1.03439
$516$	26.2090	1.15379
$517$	6.58551	0.289631
$518$	9.65665	0.424289
$519$	−2.39410	−0.105089
$520$	−15.3614	−0.673641
$521$	−11.8964	−0.521192	−0.260596	−	0.965448i	$-0.583919\pi$
$521$	−11.8964	−0.521192	−0.260596	+	0.965448i	$0.583919\pi$
$522$	−5.07201	−0.221996
$523$	2.33214	0.101977	0.0509886	−	0.998699i	$-0.483763\pi$
$523$	2.33214	0.101977	0.0509886	+	0.998699i	$0.483763\pi$
$524$	−6.74933	−0.294846
$525$	−46.0447	−2.00955
$526$	2.70731	0.118044
$527$	−5.81815	−0.253443
$528$	7.42702	0.323220
$529$	12.5075	0.543807
$530$	36.6906	1.59374
$531$	−14.8113	−0.642755
$532$	9.98009	0.432692
$533$	−33.4610	−1.44936
$534$	24.3677	1.05449
$535$	19.8430	0.857890
$536$	−10.2713	−0.443654
$537$	−17.6590	−0.762040
$538$	15.3811	0.663125
$539$	−9.80042	−0.422134
$540$	−2.80191	−0.120575
$541$	−25.2930	−1.08743	−0.543714	−	0.839270i	$-0.682983\pi$
$541$	−25.2930	−1.08743	−0.543714	+	0.839270i	$0.682983\pi$
$542$	3.34226	0.143562
$543$	1.74147	0.0747335
$544$	−5.18964	−0.222504
$545$	−1.12607	−0.0482355
$546$	−34.8989	−1.49353
$547$	33.9482	1.45152	0.725760	−	0.687947i	$-0.241488\pi$
$547$	33.9482	1.45152	0.725760	+	0.687947i	$0.241488\pi$
$548$	11.4608	0.489583
$549$	−4.72826	−0.201797
$550$	19.0258	0.811265
$551$	6.00937	0.256008
$552$	14.1588	0.602637
$553$	8.18831	0.348202
$554$	12.6605	0.537893
$555$	−23.9980	−1.01866
$556$	8.42584	0.357335
$557$	−38.9956	−1.65230	−0.826149	−	0.563452i	$-0.809473\pi$
$557$	−38.9956	−1.65230	−0.826149	+	0.563452i	$0.809473\pi$
$558$	2.96629	0.125573
$559$	50.8876	2.15232
$560$	−10.6005	−0.447952
$561$	38.5435	1.62731
$562$	33.1853	1.39984
$563$	34.3956	1.44960	0.724801	−	0.688958i	$-0.241932\pi$
$563$	34.3956	1.44960	0.724801	+	0.688958i	$0.241932\pi$
$564$	5.00616	0.210797
$565$	53.4291	2.24778
$566$	5.52575	0.232264
$567$	−31.6357	−1.32857
$568$	−7.76365	−0.325755
$569$	16.7879	0.703786	0.351893	−	0.936040i	$-0.385538\pi$
$569$	16.7879	0.703786	0.351893	+	0.936040i	$0.385538\pi$
$570$	−24.8018	−1.03883
$571$	17.8951	0.748888	0.374444	−	0.927250i	$-0.377834\pi$
$571$	17.8951	0.748888	0.374444	+	0.927250i	$0.377834\pi$
$572$	14.4203	0.602945
$573$	4.45947	0.186297
$574$	−23.0905	−0.963779
$575$	36.2706	1.51259
$576$	2.64585	0.110244
$577$	30.8130	1.28276	0.641380	−	0.767223i	$-0.278362\pi$
$577$	30.8130	1.28276	0.641380	+	0.767223i	$0.278362\pi$
$578$	−9.93232	−0.413130
$579$	−8.14397	−0.338452
$580$	−6.38292	−0.265036
$581$	35.3940	1.46839
$582$	13.3422	0.553051
$583$	−34.4429	−1.42648
$584$	13.9399	0.576838
$585$	40.6439	1.68042
$586$	8.68084	0.358602
$587$	−24.7381	−1.02105	−0.510525	−	0.859863i	$-0.670549\pi$
$587$	−24.7381	−1.02105	−0.510525	+	0.859863i	$0.670549\pi$
$588$	−7.45006	−0.307235
$589$	−3.51449	−0.144812
$590$	−18.6394	−0.767371
$591$	28.5013	1.17239
$592$	−3.03323	−0.124665
$593$	38.1659	1.56728	0.783642	−	0.621212i	$-0.213360\pi$
$593$	38.1659	1.56728	0.783642	+	0.621212i	$0.213360\pi$
$594$	2.63027	0.107921
$595$	−55.0126	−2.25530
$596$	−6.27409	−0.256997
$597$	2.11093	0.0863946
$598$	27.4907	1.12418
$599$	14.8644	0.607343	0.303671	−	0.952777i	$-0.401788\pi$
$599$	14.8644	0.607343	0.303671	+	0.952777i	$0.401788\pi$
$600$	14.4630	0.590450
$601$	−16.4384	−0.670538	−0.335269	−	0.942122i	$-0.608827\pi$
$601$	−16.4384	−0.670538	−0.335269	+	0.942122i	$0.608827\pi$
$602$	35.1161	1.43123
$603$	27.1764	1.10671
$604$	−15.8952	−0.646765
$605$	4.09514	0.166491
$606$	−7.70031	−0.312804
$607$	40.8144	1.65661	0.828303	−	0.560281i	$-0.189307\pi$
$607$	40.8144	1.65661	0.828303	+	0.560281i	$0.189307\pi$
$608$	−3.13483	−0.127134
$609$	−14.5011	−0.587613
$610$	−5.95032	−0.240921
$611$	9.71999	0.393229
$612$	13.7310	0.555043
$613$	16.4984	0.666366	0.333183	−	0.942862i	$-0.391877\pi$
$613$	16.4984	0.666366	0.333183	+	0.942862i	$0.391877\pi$
$614$	14.0197	0.565790
$615$	57.3828	2.31390
$616$	9.95109	0.400941
$617$	−11.7501	−0.473042	−0.236521	−	0.971626i	$-0.576007\pi$
$617$	−11.7501	−0.473042	−0.236521	+	0.971626i	$0.576007\pi$
$618$	−16.7514	−0.673838
$619$	23.0958	0.928297	0.464149	−	0.885757i	$-0.346360\pi$
$619$	23.0958	0.928297	0.464149	+	0.885757i	$0.346360\pi$
$620$	3.73296	0.149919
$621$	5.01431	0.201217
$622$	18.5350	0.743184
$623$	32.6491	1.30806
$624$	10.9620	0.438832
$625$	−18.3844	−0.735375
$626$	14.8127	0.592035
$627$	23.2824	0.929811
$628$	7.20509	0.287515
$629$	−15.7414	−0.627650
$630$	28.0473	1.11743
$631$	29.9156	1.19092	0.595461	−	0.803384i	$-0.296969\pi$
$631$	29.9156	1.19092	0.595461	+	0.803384i	$0.296969\pi$
$632$	−2.57202	−0.102309
$633$	8.33791	0.331402
$634$	−24.5542	−0.975170
$635$	43.6994	1.73416
$636$	−26.1827	−1.03821
$637$	−14.4651	−0.573127
$638$	5.99190	0.237222
$639$	20.5415	0.812607
$640$	3.32970	0.131618
$641$	−14.0426	−0.554650	−0.277325	−	0.960776i	$-0.589448\pi$
$641$	−14.0426	−0.554650	−0.277325	+	0.960776i	$0.589448\pi$
$642$	−14.1602	−0.558857
$643$	31.4250	1.23928	0.619641	−	0.784885i	$-0.287278\pi$
$643$	31.4250	1.23928	0.619641	+	0.784885i	$0.287278\pi$
$644$	18.9706	0.747546
$645$	−87.2681	−3.43618
$646$	−16.2686	−0.640080
$647$	43.6708	1.71688	0.858438	−	0.512918i	$-0.171435\pi$
$647$	43.6708	1.71688	0.858438	+	0.512918i	$0.171435\pi$
$648$	9.93702	0.390363
$649$	17.4975	0.686838
$650$	28.0815	1.10145
$651$	8.48074	0.332386
$652$	−18.7543	−0.734473
$653$	2.41506	0.0945087	0.0472543	−	0.998883i	$-0.484953\pi$
$653$	2.41506	0.0945087	0.0472543	+	0.998883i	$0.484953\pi$
$654$	0.803574	0.0314222
$655$	22.4732	0.878101
$656$	7.25292	0.283179
$657$	−36.8830	−1.43894
$658$	6.70750	0.261486
$659$	−10.5046	−0.409202	−0.204601	−	0.978845i	$-0.565590\pi$
$659$	−10.5046	−0.409202	−0.204601	+	0.978845i	$0.565590\pi$
$660$	−24.7297	−0.962603
$661$	0.552451	0.0214879	0.0107439	−	0.999942i	$-0.496580\pi$
$661$	0.552451	0.0214879	0.0107439	+	0.999942i	$0.496580\pi$
$662$	−30.3283	−1.17874
$663$	56.8889	2.20938
$664$	−11.1175	−0.431444
$665$	−33.2307	−1.28863
$666$	8.02549	0.310981
$667$	11.4229	0.442295
$668$	18.4553	0.714059
$669$	26.5848	1.02783
$670$	34.2004	1.32128
$671$	5.58580	0.215637
$672$	7.56459	0.291810
$673$	−4.36721	−0.168344	−0.0841718	−	0.996451i	$-0.526824\pi$
$673$	−4.36721	−0.168344	−0.0841718	+	0.996451i	$0.526824\pi$
$674$	−15.8510	−0.610559
$675$	5.12206	0.197148
$676$	8.28391	0.318612
$677$	31.0621	1.19381	0.596907	−	0.802311i	$-0.296396\pi$
$677$	31.0621	1.19381	0.596907	+	0.802311i	$0.296396\pi$
$678$	−38.1275	−1.46428
$679$	17.8765	0.686037
$680$	17.2799	0.662654
$681$	58.7169	2.25004
$682$	−3.50427	−0.134186
$683$	13.8271	0.529078	0.264539	−	0.964375i	$-0.414780\pi$
$683$	13.8271	0.529078	0.264539	+	0.964375i	$0.414780\pi$
$684$	8.29429	0.317140
$685$	−38.1611	−1.45806
$686$	12.3034	0.469744
$687$	−33.9623	−1.29574
$688$	−11.0303	−0.420525
$689$	−50.8366	−1.93672
$690$	−47.1443	−1.79475
$691$	−30.0337	−1.14254	−0.571269	−	0.820763i	$-0.693549\pi$
$691$	−30.0337	−1.14254	−0.571269	+	0.820763i	$0.693549\pi$
$692$	1.00758	0.0383023
$693$	−26.3291	−1.00016
$694$	−31.4368	−1.19332
$695$	−28.0555	−1.06421
$696$	4.55491	0.172653
$697$	37.6400	1.42572
$698$	−34.2486	−1.29633
$699$	−57.5006	−2.17487
$700$	19.3783	0.732429
$701$	−26.7792	−1.01144	−0.505718	−	0.862699i	$-0.668772\pi$
$701$	−26.7792	−1.01144	−0.505718	+	0.862699i	$0.668772\pi$
$702$	3.88218	0.146524
$703$	−9.50867	−0.358626
$704$	−3.12572	−0.117805
$705$	−16.6690	−0.627790
$706$	−30.4436	−1.14576
$707$	−10.3173	−0.388020
$708$	13.3012	0.499891
$709$	24.3031	0.912721	0.456360	−	0.889795i	$-0.349153\pi$
$709$	24.3031	0.912721	0.456360	+	0.889795i	$0.349153\pi$
$710$	25.8506	0.970155
$711$	6.80517	0.255214
$712$	−10.2553	−0.384335
$713$	−6.68050	−0.250187
$714$	39.2575	1.46917
$715$	−48.0153	−1.79567
$716$	7.43191	0.277743
$717$	48.3006	1.80382
$718$	−9.77774	−0.364902
$719$	2.74048	0.102203	0.0511014	−	0.998693i	$-0.483727\pi$
$719$	2.74048	0.102203	0.0511014	+	0.998693i	$0.483727\pi$
$720$	−8.80988	−0.328325
$721$	−22.4443	−0.835869
$722$	9.17284	0.341378
$723$	−47.0598	−1.75017
$724$	−0.732910	−0.0272384
$725$	11.6683	0.433351
$726$	−2.92233	−0.108458
$727$	30.0007	1.11267	0.556333	−	0.830960i	$-0.312208\pi$
$727$	30.0007	1.11267	0.556333	+	0.830960i	$0.312208\pi$
$728$	14.6875	0.544353
$729$	−20.2935	−0.751610
$730$	−46.4157	−1.71792
$731$	−57.2431	−2.11721
$732$	4.24620	0.156944
$733$	5.09470	0.188177	0.0940885	−	0.995564i	$-0.470006\pi$
$733$	5.09470	0.188177	0.0940885	+	0.995564i	$0.470006\pi$
$734$	−13.5313	−0.499450
$735$	24.8064	0.914998
$736$	−5.95882	−0.219645
$737$	−32.1053	−1.18261
$738$	−19.1901	−0.706399
$739$	37.7134	1.38731	0.693655	−	0.720307i	$-0.255999\pi$
$739$	37.7134	1.38731	0.693655	+	0.720307i	$0.255999\pi$
$740$	10.0997	0.371274
$741$	34.3641	1.26240
$742$	−35.0809	−1.28786
$743$	8.71086	0.319570	0.159785	−	0.987152i	$-0.448920\pi$
$743$	8.71086	0.319570	0.159785	+	0.987152i	$0.448920\pi$
$744$	−2.66387	−0.0976622
$745$	20.8908	0.765380
$746$	−24.4204	−0.894093
$747$	29.4154	1.07625
$748$	−16.2213	−0.593111
$749$	−18.9725	−0.693240
$750$	−8.59898	−0.313990
$751$	−24.4454	−0.892024	−0.446012	−	0.895027i	$-0.647156\pi$
$751$	−24.4454	−0.892024	−0.446012	+	0.895027i	$0.647156\pi$
$752$	−2.10688	−0.0768300
$753$	−0.774850	−0.0282371
$754$	8.84384	0.322073
$755$	52.9260	1.92617
$756$	2.67899	0.0974339
$757$	−42.0307	−1.52763	−0.763816	−	0.645434i	$-0.776677\pi$
$757$	−42.0307	−1.52763	−0.763816	+	0.645434i	$0.776677\pi$
$758$	−5.27058	−0.191436
$759$	44.2563	1.60640
$760$	10.4380	0.378627
$761$	14.6367	0.530579	0.265290	−	0.964169i	$-0.414532\pi$
$761$	14.6367	0.530579	0.265290	+	0.964169i	$0.414532\pi$
$762$	−31.1843	−1.12969
$763$	1.07667	0.0389780
$764$	−1.87680	−0.0679003
$765$	−45.7201	−1.65301
$766$	15.0420	0.543488
$767$	25.8257	0.932513
$768$	−2.37610	−0.0857401
$769$	16.5708	0.597559	0.298779	−	0.954322i	$-0.403421\pi$
$769$	16.5708	0.597559	0.298779	+	0.954322i	$0.403421\pi$
$770$	−33.1341	−1.19407
$771$	15.7570	0.567475
$772$	3.42745	0.123357
$773$	13.1010	0.471212	0.235606	−	0.971849i	$-0.424293\pi$
$773$	13.1010	0.471212	0.235606	+	0.971849i	$0.424293\pi$
$774$	29.1845	1.04901
$775$	−6.82405	−0.245127
$776$	−5.61516	−0.201572
$777$	22.9452	0.823154
$778$	−8.10304	−0.290508
$779$	22.7367	0.814625
$780$	−36.5002	−1.30692
$781$	−24.2670	−0.868341
$782$	−30.9241	−1.10584
$783$	1.61312	0.0576480
$784$	3.13541	0.111979
$785$	−23.9908	−0.856267
$786$	−16.0371	−0.572024
$787$	−16.1252	−0.574802	−0.287401	−	0.957810i	$-0.592791\pi$
$787$	−16.1252	−0.574802	−0.287401	+	0.957810i	$0.592791\pi$
$788$	−11.9950	−0.427304
$789$	6.43284	0.229015
$790$	8.56403	0.304694
$791$	−51.0851	−1.81638
$792$	8.27019	0.293868
$793$	8.24444	0.292769
$794$	3.71395	0.131803
$795$	87.1806	3.09198
$796$	−0.888401	−0.0314885
$797$	18.2281	0.645673	0.322837	−	0.946455i	$-0.395364\pi$
$797$	18.2281	0.645673	0.322837	+	0.946455i	$0.395364\pi$
$798$	23.7137	0.839456
$799$	−10.9339	−0.386815
$800$	−6.08687	−0.215203
$801$	27.1341	0.958736
$802$	7.25391	0.256144
$803$	43.5723	1.53763
$804$	−24.4057	−0.860723
$805$	−63.1663	−2.22632
$806$	−5.17218	−0.182182
$807$	36.5470	1.28651
$808$	3.24073	0.114009
$809$	−24.4286	−0.858864	−0.429432	−	0.903099i	$-0.641286\pi$
$809$	−24.4286	−0.858864	−0.429432	+	0.903099i	$0.641286\pi$
$810$	−33.0873	−1.16257
$811$	−8.12806	−0.285415	−0.142707	−	0.989765i	$-0.545581\pi$
$811$	−8.12806	−0.285415	−0.142707	+	0.989765i	$0.545581\pi$
$812$	6.10289	0.214170
$813$	7.94155	0.278522
$814$	−9.48104	−0.332310
$815$	62.4459	2.18739
$816$	−12.3311	−0.431675
$817$	−34.5780	−1.20973
$818$	6.89849	0.241200
$819$	−38.8608	−1.35791
$820$	−24.1500	−0.843355
$821$	−6.43954	−0.224741	−0.112371	−	0.993666i	$-0.535844\pi$
$821$	−6.43954	−0.224741	−0.112371	+	0.993666i	$0.535844\pi$
$822$	27.2321	0.949829
$823$	7.82533	0.272774	0.136387	−	0.990656i	$-0.456451\pi$
$823$	7.82533	0.272774	0.136387	+	0.990656i	$0.456451\pi$
$824$	7.04994	0.245596
$825$	45.2073	1.57392
$826$	17.8216	0.620094
$827$	−44.0561	−1.53198	−0.765990	−	0.642852i	$-0.777751\pi$
$827$	−44.0561	−1.53198	−0.765990	+	0.642852i	$0.777751\pi$
$828$	15.7662	0.547912
$829$	−14.4498	−0.501863	−0.250932	−	0.968005i	$-0.580737\pi$
$829$	−14.4498	−0.501863	−0.250932	+	0.968005i	$0.580737\pi$
$830$	37.0180	1.28491
$831$	30.0826	1.04355
$832$	−4.61345	−0.159943
$833$	16.2717	0.563779
$834$	20.0206	0.693258
$835$	−61.4507	−2.12659
$836$	−9.79859	−0.338891
$837$	−0.943406	−0.0326089
$838$	28.7368	0.992698
$839$	23.4747	0.810435	0.405217	−	0.914220i	$-0.367196\pi$
$839$	23.4747	0.810435	0.405217	+	0.914220i	$0.367196\pi$
$840$	−25.1878	−0.869061
$841$	−25.3252	−0.873284
$842$	−19.8686	−0.684718
$843$	78.8516	2.71579
$844$	−3.50907	−0.120787
$845$	−27.5829	−0.948881
$846$	5.57449	0.191655
$847$	−3.91548	−0.134538
$848$	11.0192	0.378401
$849$	13.1297	0.450611
$850$	−31.5886	−1.08348
$851$	−18.0745	−0.619586
$852$	−18.4472	−0.631991
$853$	−0.379924	−0.0130084	−0.00650418	−	0.999979i	$-0.502070\pi$
$853$	−0.379924	−0.0130084	−0.00650418	+	0.999979i	$0.502070\pi$
$854$	5.68927	0.194683
$855$	−27.6175	−0.944497
$856$	5.95942	0.203689
$857$	−1.05432	−0.0360151	−0.0180075	−	0.999838i	$-0.505732\pi$
$857$	−1.05432	−0.0360151	−0.0180075	+	0.999838i	$0.505732\pi$
$858$	34.2642	1.16976
$859$	8.89477	0.303486	0.151743	−	0.988420i	$-0.451511\pi$
$859$	8.89477	0.303486	0.151743	+	0.988420i	$0.451511\pi$
$860$	36.7274	1.25240
$861$	−54.8654	−1.86981
$862$	−20.8178	−0.709056
$863$	12.6455	0.430459	0.215229	−	0.976564i	$-0.430950\pi$
$863$	12.6455	0.430459	0.215229	+	0.976564i	$0.430950\pi$
$864$	−0.841493	−0.0286282
$865$	−3.35492	−0.114071
$866$	−2.54094	−0.0863448
$867$	−23.6002	−0.801504
$868$	−3.56919	−0.121146
$869$	−8.03940	−0.272718
$870$	−15.1665	−0.514191
$871$	−47.3862	−1.60562
$872$	−0.338190	−0.0114526
$873$	14.8569	0.502829
$874$	−18.6799	−0.631857
$875$	−11.5213	−0.389492
$876$	33.1227	1.11911
$877$	−17.1007	−0.577450	−0.288725	−	0.957412i	$-0.593231\pi$
$877$	−17.1007	−0.577450	−0.288725	+	0.957412i	$0.593231\pi$
$878$	8.94228	0.301787
$879$	20.6265	0.695716
$880$	10.4077	0.350843
$881$	−27.6613	−0.931934	−0.465967	−	0.884802i	$-0.654293\pi$
$881$	−27.6613	−0.931934	−0.465967	+	0.884802i	$0.654293\pi$
$882$	−8.29584	−0.279335
$883$	44.9933	1.51414	0.757072	−	0.653331i	$-0.226629\pi$
$883$	44.9933	1.51414	0.757072	+	0.653331i	$0.226629\pi$
$884$	−23.9421	−0.805261
$885$	−44.2890	−1.48876
$886$	−26.7326	−0.898100
$887$	14.6070	0.490454	0.245227	−	0.969466i	$-0.421137\pi$
$887$	14.6070	0.490454	0.245227	+	0.969466i	$0.421137\pi$
$888$	−7.20727	−0.241860
$889$	−41.7822	−1.40133
$890$	34.1471	1.14461
$891$	31.0603	1.04056
$892$	−11.1884	−0.374615
$893$	−6.60471	−0.221018
$894$	−14.9079	−0.498594
$895$	−24.7460	−0.827167
$896$	−3.18362	−0.106357
$897$	65.3207	2.18099
$898$	−9.24604	−0.308544
$899$	−2.14913	−0.0716776
$900$	16.1050	0.536832
$901$	57.1857	1.90513
$902$	22.6706	0.754848
$903$	83.4395	2.77669
$904$	16.0462	0.533690
$905$	2.44037	0.0811205
$906$	−37.7685	−1.25477
$907$	7.30025	0.242401	0.121200	−	0.992628i	$-0.461326\pi$
$907$	7.30025	0.242401	0.121200	+	0.992628i	$0.461326\pi$
$908$	−24.7115	−0.820079
$909$	−8.57450	−0.284398
$910$	−48.9047	−1.62118
$911$	52.3563	1.73464	0.867321	−	0.497749i	$-0.165840\pi$
$911$	52.3563	1.73464	0.867321	+	0.497749i	$0.165840\pi$
$912$	−7.44867	−0.246650
$913$	−34.7503	−1.15007
$914$	3.05356	0.101003
$915$	−14.1385	−0.467406
$916$	14.2933	0.472264
$917$	−21.4873	−0.709572
$918$	−4.36704	−0.144134
$919$	41.8351	1.38001	0.690007	−	0.723803i	$-0.257608\pi$
$919$	41.8351	1.38001	0.690007	+	0.723803i	$0.257608\pi$
$920$	19.8411	0.654140
$921$	33.3123	1.09768
$922$	3.45615	0.113822
$923$	−35.8172	−1.17894
$924$	23.6448	0.777856
$925$	−18.4629	−0.607056
$926$	−28.0621	−0.922179
$927$	−18.6531	−0.612648
$928$	−1.91697	−0.0629276
$929$	−30.2794	−0.993434	−0.496717	−	0.867912i	$-0.665461\pi$
$929$	−30.2794	−0.993434	−0.496717	+	0.867912i	$0.665461\pi$
$930$	8.86988	0.290855
$931$	9.82899	0.322132
$932$	24.1996	0.792684
$933$	44.0409	1.44183
$934$	−23.8285	−0.779691
$935$	54.0121	1.76638
$936$	12.2065	0.398982
$937$	−14.9054	−0.486939	−0.243469	−	0.969909i	$-0.578286\pi$
$937$	−14.9054	−0.486939	−0.243469	+	0.969909i	$0.578286\pi$
$938$	−32.7000	−1.06769
$939$	35.1965	1.14859
$940$	7.01527	0.228813
$941$	2.61608	0.0852819	0.0426409	−	0.999090i	$-0.486423\pi$
$941$	2.61608	0.0852819	0.0426409	+	0.999090i	$0.486423\pi$
$942$	17.1200	0.557801
$943$	43.2188	1.40740
$944$	−5.59792	−0.182197
$945$	−8.92022	−0.290175
$946$	−34.4775	−1.12096
$947$	−6.17747	−0.200741	−0.100370	−	0.994950i	$-0.532003\pi$
$947$	−6.17747	−0.200741	−0.100370	+	0.994950i	$0.532003\pi$
$948$	−6.11137	−0.198488
$949$	64.3112	2.08763
$950$	−19.0813	−0.619079
$951$	−58.3431	−1.89191
$952$	−16.5218	−0.535475
$953$	12.4695	0.403927	0.201964	−	0.979393i	$-0.435268\pi$
$953$	12.4695	0.403927	0.201964	+	0.979393i	$0.435268\pi$
$954$	−29.1552	−0.943934
$955$	6.24917	0.202218
$956$	−20.3277	−0.657444
$957$	14.2374	0.460228
$958$	−17.5981	−0.568569
$959$	36.4869	1.17823
$960$	7.91169	0.255349
$961$	−29.7431	−0.959455
$962$	−13.9937	−0.451174
$963$	−15.7677	−0.508108
$964$	19.8055	0.637891
$965$	−11.4124	−0.367377
$966$	45.0760	1.45030
$967$	−56.1967	−1.80716	−0.903582	−	0.428416i	$-0.859072\pi$
$967$	−56.1967	−1.80716	−0.903582	+	0.428416i	$0.859072\pi$
$968$	1.22989	0.0395300
$969$	−38.6559	−1.24181
$970$	18.6968	0.600317
$971$	42.2178	1.35483	0.677417	−	0.735600i	$-0.263100\pi$
$971$	42.2178	1.35483	0.677417	+	0.735600i	$0.263100\pi$
$972$	21.0869	0.676362
$973$	26.8246	0.859959
$974$	39.5420	1.26701
$975$	66.7244	2.13689
$976$	−1.78705	−0.0572019
$977$	−7.42062	−0.237407	−0.118703	−	0.992930i	$-0.537874\pi$
$977$	−7.42062	−0.237407	−0.118703	+	0.992930i	$0.537874\pi$
$978$	−44.5620	−1.42493
$979$	−32.0553	−1.02449
$980$	−10.4400	−0.333493
$981$	0.894801	0.0285688
$982$	7.52025	0.239981
$983$	34.1733	1.08996	0.544980	−	0.838449i	$-0.316537\pi$
$983$	34.1733	1.08996	0.544980	+	0.838449i	$0.316537\pi$
$984$	17.2337	0.549389
$985$	39.9397	1.27258
$986$	−9.94837	−0.316821
$987$	15.9377	0.507302
$988$	−14.4624	−0.460109
$989$	−65.7274	−2.09001
$990$	−27.5372	−0.875190
$991$	−37.5417	−1.19255	−0.596276	−	0.802779i	$-0.703354\pi$
$991$	−37.5417	−1.19255	−0.596276	+	0.802779i	$0.703354\pi$
$992$	1.12111	0.0355953
$993$	−72.0631	−2.28685
$994$	−24.7165	−0.783959
$995$	2.95810	0.0937782
$996$	−26.4164	−0.837036
$997$	30.6259	0.969933	0.484966	−	0.874533i	$-0.338832\pi$
$997$	30.6259	0.969933	0.484966	+	0.874533i	$0.338832\pi$
$998$	20.2139	0.639861
$999$	−2.55244	−0.0807558

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.e.1.12	✓	77

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.e.1.12	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.37610 q^{3} +1.00000 q^{4} -3.32970 q^{5} +2.37610 q^{6} +3.18362 q^{7} -1.00000 q^{8} +2.64585 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.37610 q^{3} +1.00000 q^{4} -3.32970 q^{5} +2.37610 q^{6} +3.18362 q^{7} -1.00000 q^{8} +2.64585 q^{9} +3.32970 q^{10} -3.12572 q^{11} -2.37610 q^{12} -4.61345 q^{13} -3.18362 q^{14} +7.91169 q^{15} +1.00000 q^{16} +5.18964 q^{17} -2.64585 q^{18} +3.13483 q^{19} -3.32970 q^{20} -7.56459 q^{21} +3.12572 q^{22} +5.95882 q^{23} +2.37610 q^{24} +6.08687 q^{25} +4.61345 q^{26} +0.841493 q^{27} +3.18362 q^{28} +1.91697 q^{29} -7.91169 q^{30} -1.12111 q^{31} -1.00000 q^{32} +7.42702 q^{33} -5.18964 q^{34} -10.6005 q^{35} +2.64585 q^{36} -3.03323 q^{37} -3.13483 q^{38} +10.9620 q^{39} +3.32970 q^{40} +7.25292 q^{41} +7.56459 q^{42} -11.0303 q^{43} -3.12572 q^{44} -8.80988 q^{45} -5.95882 q^{46} -2.10688 q^{47} -2.37610 q^{48} +3.13541 q^{49} -6.08687 q^{50} -12.3311 q^{51} -4.61345 q^{52} +11.0192 q^{53} -0.841493 q^{54} +10.4077 q^{55} -3.18362 q^{56} -7.44867 q^{57} -1.91697 q^{58} -5.59792 q^{59} +7.91169 q^{60} -1.78705 q^{61} +1.12111 q^{62} +8.42338 q^{63} +1.00000 q^{64} +15.3614 q^{65} -7.42702 q^{66} +10.2713 q^{67} +5.18964 q^{68} -14.1588 q^{69} +10.6005 q^{70} +7.76365 q^{71} -2.64585 q^{72} -13.9399 q^{73} +3.03323 q^{74} -14.4630 q^{75} +3.13483 q^{76} -9.95109 q^{77} -10.9620 q^{78} +2.57202 q^{79} -3.32970 q^{80} -9.93702 q^{81} -7.25292 q^{82} +11.1175 q^{83} -7.56459 q^{84} -17.2799 q^{85} +11.0303 q^{86} -4.55491 q^{87} +3.12572 q^{88} +10.2553 q^{89} +8.80988 q^{90} -14.6875 q^{91} +5.95882 q^{92} +2.66387 q^{93} +2.10688 q^{94} -10.4380 q^{95} +2.37610 q^{96} +5.61516 q^{97} -3.13541 q^{98} -8.27019 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) \) 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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