LMFDB - Embedded newform 8042.2.a.a.1.47

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.2156933055$
Analytic rank:	$1$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.47
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.23546 q^{3} +1.00000 q^{4} -2.31218 q^{5} +1.23546 q^{6} -1.71376 q^{7} +1.00000 q^{8} -1.47363 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.23546 q^{3} +1.00000 q^{4} -2.31218 q^{5} +1.23546 q^{6} -1.71376 q^{7} +1.00000 q^{8} -1.47363 q^{9} -2.31218 q^{10} -4.55740 q^{11} +1.23546 q^{12} +5.58097 q^{13} -1.71376 q^{14} -2.85661 q^{15} +1.00000 q^{16} +1.18021 q^{17} -1.47363 q^{18} +4.84833 q^{19} -2.31218 q^{20} -2.11729 q^{21} -4.55740 q^{22} +8.87799 q^{23} +1.23546 q^{24} +0.346159 q^{25} +5.58097 q^{26} -5.52701 q^{27} -1.71376 q^{28} -2.15748 q^{29} -2.85661 q^{30} -1.90623 q^{31} +1.00000 q^{32} -5.63050 q^{33} +1.18021 q^{34} +3.96252 q^{35} -1.47363 q^{36} -7.10499 q^{37} +4.84833 q^{38} +6.89508 q^{39} -2.31218 q^{40} +4.09791 q^{41} -2.11729 q^{42} -8.51215 q^{43} -4.55740 q^{44} +3.40729 q^{45} +8.87799 q^{46} +8.24675 q^{47} +1.23546 q^{48} -4.06302 q^{49} +0.346159 q^{50} +1.45811 q^{51} +5.58097 q^{52} -0.803768 q^{53} -5.52701 q^{54} +10.5375 q^{55} -1.71376 q^{56} +5.98994 q^{57} -2.15748 q^{58} -11.0130 q^{59} -2.85661 q^{60} +5.83604 q^{61} -1.90623 q^{62} +2.52545 q^{63} +1.00000 q^{64} -12.9042 q^{65} -5.63050 q^{66} -6.34356 q^{67} +1.18021 q^{68} +10.9684 q^{69} +3.96252 q^{70} -7.01278 q^{71} -1.47363 q^{72} -10.8892 q^{73} -7.10499 q^{74} +0.427666 q^{75} +4.84833 q^{76} +7.81029 q^{77} +6.89508 q^{78} +1.40598 q^{79} -2.31218 q^{80} -2.40752 q^{81} +4.09791 q^{82} -13.2592 q^{83} -2.11729 q^{84} -2.72886 q^{85} -8.51215 q^{86} -2.66548 q^{87} -4.55740 q^{88} -11.7396 q^{89} +3.40729 q^{90} -9.56444 q^{91} +8.87799 q^{92} -2.35507 q^{93} +8.24675 q^{94} -11.2102 q^{95} +1.23546 q^{96} -13.5409 q^{97} -4.06302 q^{98} +6.71592 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.23546	0.713295	0.356648	−	0.934239i	$-0.383920\pi$
$3$	1.23546	0.713295	0.356648	+	0.934239i	$0.383920\pi$
$4$	1.00000	0.500000
$5$	−2.31218	−1.03404	−0.517018	−	0.855974i	$-0.672958\pi$
$5$	−2.31218	−1.03404	−0.517018	+	0.855974i	$0.672958\pi$
$6$	1.23546	0.504376
$7$	−1.71376	−0.647741	−0.323870	−	0.946101i	$-0.604984\pi$
$7$	−1.71376	−0.647741	−0.323870	+	0.946101i	$0.604984\pi$
$8$	1.00000	0.353553
$9$	−1.47363	−0.491210
$10$	−2.31218	−0.731174
$11$	−4.55740	−1.37411	−0.687053	−	0.726607i	$-0.741096\pi$
$11$	−4.55740	−1.37411	−0.687053	+	0.726607i	$0.741096\pi$
$12$	1.23546	0.356648
$13$	5.58097	1.54788	0.773941	−	0.633258i	$-0.218283\pi$
$13$	5.58097	1.54788	0.773941	+	0.633258i	$0.218283\pi$
$14$	−1.71376	−0.458022
$15$	−2.85661	−0.737573
$16$	1.00000	0.250000
$17$	1.18021	0.286244	0.143122	−	0.989705i	$-0.454286\pi$
$17$	1.18021	0.286244	0.143122	+	0.989705i	$0.454286\pi$
$18$	−1.47363	−0.347338
$19$	4.84833	1.11228	0.556142	−	0.831087i	$-0.312281\pi$
$19$	4.84833	1.11228	0.556142	+	0.831087i	$0.312281\pi$
$20$	−2.31218	−0.517018
$21$	−2.11729	−0.462030
$22$	−4.55740	−0.971640
$23$	8.87799	1.85119	0.925595	−	0.378516i	$-0.123565\pi$
$23$	8.87799	1.85119	0.925595	+	0.378516i	$0.123565\pi$
$24$	1.23546	0.252188
$25$	0.346159	0.0692318
$26$	5.58097	1.09452
$27$	−5.52701	−1.06367
$28$	−1.71376	−0.323870
$29$	−2.15748	−0.400633	−0.200317	−	0.979731i	$-0.564197\pi$
$29$	−2.15748	−0.400633	−0.200317	+	0.979731i	$0.564197\pi$
$30$	−2.85661	−0.521543
$31$	−1.90623	−0.342368	−0.171184	−	0.985239i	$-0.554759\pi$
$31$	−1.90623	−0.342368	−0.171184	+	0.985239i	$0.554759\pi$
$32$	1.00000	0.176777
$33$	−5.63050	−0.980144
$34$	1.18021	0.202405
$35$	3.96252	0.669788
$36$	−1.47363	−0.245605
$37$	−7.10499	−1.16805	−0.584027	−	0.811734i	$-0.698524\pi$
$37$	−7.10499	−1.16805	−0.584027	+	0.811734i	$0.698524\pi$
$38$	4.84833	0.786504
$39$	6.89508	1.10410
$40$	−2.31218	−0.365587
$41$	4.09791	0.639986	0.319993	−	0.947420i	$-0.396319\pi$
$41$	4.09791	0.639986	0.319993	+	0.947420i	$0.396319\pi$
$42$	−2.11729	−0.326705
$43$	−8.51215	−1.29809	−0.649045	−	0.760750i	$-0.724831\pi$
$43$	−8.51215	−1.29809	−0.649045	+	0.760750i	$0.724831\pi$
$44$	−4.55740	−0.687053
$45$	3.40729	0.507929
$46$	8.87799	1.30899
$47$	8.24675	1.20291	0.601456	−	0.798906i	$-0.294587\pi$
$47$	8.24675	1.20291	0.601456	+	0.798906i	$0.294587\pi$
$48$	1.23546	0.178324
$49$	−4.06302	−0.580432
$50$	0.346159	0.0489542
$51$	1.45811	0.204176
$52$	5.58097	0.773941
$53$	−0.803768	−0.110406	−0.0552030	−	0.998475i	$-0.517581\pi$
$53$	−0.803768	−0.110406	−0.0552030	+	0.998475i	$0.517581\pi$
$54$	−5.52701	−0.752130
$55$	10.5375	1.42088
$56$	−1.71376	−0.229011
$57$	5.98994	0.793387
$58$	−2.15748	−0.283290
$59$	−11.0130	−1.43378	−0.716888	−	0.697188i	$-0.754434\pi$
$59$	−11.0130	−1.43378	−0.716888	+	0.697188i	$0.754434\pi$
$60$	−2.85661	−0.368787
$61$	5.83604	0.747228	0.373614	−	0.927584i	$-0.378118\pi$
$61$	5.83604	0.747228	0.373614	+	0.927584i	$0.378118\pi$
$62$	−1.90623	−0.242091
$63$	2.52545	0.318177
$64$	1.00000	0.125000
$65$	−12.9042	−1.60057
$66$	−5.63050	−0.693066
$67$	−6.34356	−0.774989	−0.387495	−	0.921872i	$-0.626659\pi$
$67$	−6.34356	−0.774989	−0.387495	+	0.921872i	$0.626659\pi$
$68$	1.18021	0.143122
$69$	10.9684	1.32044
$70$	3.96252	0.473611
$71$	−7.01278	−0.832263	−0.416132	−	0.909304i	$-0.636614\pi$
$71$	−7.01278	−0.832263	−0.416132	+	0.909304i	$0.636614\pi$
$72$	−1.47363	−0.173669
$73$	−10.8892	−1.27448	−0.637240	−	0.770665i	$-0.719924\pi$
$73$	−10.8892	−1.27448	−0.637240	+	0.770665i	$0.719924\pi$
$74$	−7.10499	−0.825938
$75$	0.427666	0.0493827
$76$	4.84833	0.556142
$77$	7.81029	0.890065
$78$	6.89508	0.780714
$79$	1.40598	0.158185	0.0790926	−	0.996867i	$-0.474798\pi$
$79$	1.40598	0.158185	0.0790926	+	0.996867i	$0.474798\pi$
$80$	−2.31218	−0.258509
$81$	−2.40752	−0.267503
$82$	4.09791	0.452538
$83$	−13.2592	−1.45538	−0.727692	−	0.685904i	$-0.759407\pi$
$83$	−13.2592	−1.45538	−0.727692	+	0.685904i	$0.759407\pi$
$84$	−2.11729	−0.231015
$85$	−2.72886	−0.295986
$86$	−8.51215	−0.917888
$87$	−2.66548	−0.285770
$88$	−4.55740	−0.485820
$89$	−11.7396	−1.24439	−0.622196	−	0.782861i	$-0.713759\pi$
$89$	−11.7396	−1.24439	−0.622196	+	0.782861i	$0.713759\pi$
$90$	3.40729	0.359160
$91$	−9.56444	−1.00263
$92$	8.87799	0.925595
$93$	−2.35507	−0.244210
$94$	8.24675	0.850588
$95$	−11.2102	−1.15014
$96$	1.23546	0.126094
$97$	−13.5409	−1.37487	−0.687434	−	0.726247i	$-0.741263\pi$
$97$	−13.5409	−1.37487	−0.687434	+	0.726247i	$0.741263\pi$
$98$	−4.06302	−0.410427
$99$	6.71592	0.674975
$100$	0.346159	0.0346159
$101$	3.54352	0.352593	0.176297	−	0.984337i	$-0.443588\pi$
$101$	3.54352	0.352593	0.176297	+	0.984337i	$0.443588\pi$
$102$	1.45811	0.144374
$103$	4.20258	0.414093	0.207046	−	0.978331i	$-0.433615\pi$
$103$	4.20258	0.414093	0.207046	+	0.978331i	$0.433615\pi$
$104$	5.58097	0.547259
$105$	4.89555	0.477756
$106$	−0.803768	−0.0780688
$107$	−5.05564	−0.488747	−0.244374	−	0.969681i	$-0.578582\pi$
$107$	−5.05564	−0.488747	−0.244374	+	0.969681i	$0.578582\pi$
$108$	−5.52701	−0.531836
$109$	−1.85741	−0.177908	−0.0889538	−	0.996036i	$-0.528352\pi$
$109$	−1.85741	−0.177908	−0.0889538	+	0.996036i	$0.528352\pi$
$110$	10.5375	1.00471
$111$	−8.77796	−0.833167
$112$	−1.71376	−0.161935
$113$	15.7609	1.48266	0.741330	−	0.671140i	$-0.234195\pi$
$113$	15.7609	1.48266	0.741330	+	0.671140i	$0.234195\pi$
$114$	5.98994	0.561009
$115$	−20.5275	−1.91420
$116$	−2.15748	−0.200317
$117$	−8.22428	−0.760335
$118$	−11.0130	−1.01383
$119$	−2.02260	−0.185412
$120$	−2.85661	−0.260772
$121$	9.76986	0.888169
$122$	5.83604	0.528370
$123$	5.06282	0.456499
$124$	−1.90623	−0.171184
$125$	10.7605	0.962448
$126$	2.52545	0.224985
$127$	−5.64176	−0.500625	−0.250313	−	0.968165i	$-0.580533\pi$
$127$	−5.64176	−0.500625	−0.250313	+	0.968165i	$0.580533\pi$
$128$	1.00000	0.0883883
$129$	−10.5164	−0.925921
$130$	−12.9042	−1.13177
$131$	4.09116	0.357446	0.178723	−	0.983899i	$-0.442803\pi$
$131$	4.09116	0.357446	0.178723	+	0.983899i	$0.442803\pi$
$132$	−5.63050	−0.490072
$133$	−8.30889	−0.720472
$134$	−6.34356	−0.548000
$135$	12.7794	1.09988
$136$	1.18021	0.101202
$137$	−6.32269	−0.540184	−0.270092	−	0.962834i	$-0.587054\pi$
$137$	−6.32269	−0.540184	−0.270092	+	0.962834i	$0.587054\pi$
$138$	10.9684	0.933695
$139$	−10.4799	−0.888897	−0.444448	−	0.895804i	$-0.646600\pi$
$139$	−10.4799	−0.888897	−0.444448	+	0.895804i	$0.646600\pi$
$140$	3.96252	0.334894
$141$	10.1886	0.858032
$142$	−7.01278	−0.588499
$143$	−25.4347	−2.12695
$144$	−1.47363	−0.122803
$145$	4.98846	0.414269
$146$	−10.8892	−0.901194
$147$	−5.01972	−0.414019
$148$	−7.10499	−0.584027
$149$	8.23024	0.674247	0.337124	−	0.941460i	$-0.390546\pi$
$149$	8.23024	0.674247	0.337124	+	0.941460i	$0.390546\pi$
$150$	0.427666	0.0349188
$151$	5.98701	0.487216	0.243608	−	0.969874i	$-0.421669\pi$
$151$	5.98701	0.487216	0.243608	+	0.969874i	$0.421669\pi$
$152$	4.84833	0.393252
$153$	−1.73920	−0.140606
$154$	7.81029	0.629371
$155$	4.40753	0.354021
$156$	6.89508	0.552048
$157$	13.1509	1.04956	0.524778	−	0.851239i	$-0.324148\pi$
$157$	13.1509	1.04956	0.524778	+	0.851239i	$0.324148\pi$
$158$	1.40598	0.111854
$159$	−0.993026	−0.0787520
$160$	−2.31218	−0.182794
$161$	−15.2148	−1.19909
$162$	−2.40752	−0.189153
$163$	12.7248	0.996684	0.498342	−	0.866980i	$-0.333942\pi$
$163$	12.7248	0.996684	0.498342	+	0.866980i	$0.333942\pi$
$164$	4.09791	0.319993
$165$	13.0187	1.01350
$166$	−13.2592	−1.02911
$167$	−17.1390	−1.32625	−0.663126	−	0.748508i	$-0.730771\pi$
$167$	−17.1390	−1.32625	−0.663126	+	0.748508i	$0.730771\pi$
$168$	−2.11729	−0.163352
$169$	18.1472	1.39594
$170$	−2.72886	−0.209294
$171$	−7.14465	−0.546365
$172$	−8.51215	−0.649045
$173$	−8.56498	−0.651183	−0.325592	−	0.945511i	$-0.605563\pi$
$173$	−8.56498	−0.651183	−0.325592	+	0.945511i	$0.605563\pi$
$174$	−2.66548	−0.202070
$175$	−0.593233	−0.0448442
$176$	−4.55740	−0.343527
$177$	−13.6062	−1.02271
$178$	−11.7396	−0.879918
$179$	17.0777	1.27645	0.638225	−	0.769850i	$-0.279669\pi$
$179$	17.0777	1.27645	0.638225	+	0.769850i	$0.279669\pi$
$180$	3.40729	0.253965
$181$	−15.4803	−1.15064	−0.575320	−	0.817929i	$-0.695122\pi$
$181$	−15.4803	−1.15064	−0.575320	+	0.817929i	$0.695122\pi$
$182$	−9.56444	−0.708963
$183$	7.21021	0.532994
$184$	8.87799	0.654494
$185$	16.4280	1.20781
$186$	−2.35507	−0.172682
$187$	−5.37870	−0.393329
$188$	8.24675	0.601456
$189$	9.47197	0.688984
$190$	−11.2102	−0.813274
$191$	−21.7072	−1.57068	−0.785338	−	0.619067i	$-0.787511\pi$
$191$	−21.7072	−1.57068	−0.785338	+	0.619067i	$0.787511\pi$
$192$	1.23546	0.0891619
$193$	−23.2266	−1.67189	−0.835945	−	0.548814i	$-0.815080\pi$
$193$	−23.2266	−1.67189	−0.835945	+	0.548814i	$0.815080\pi$
$194$	−13.5409	−0.972179
$195$	−15.9426	−1.14168
$196$	−4.06302	−0.290216
$197$	−9.97743	−0.710863	−0.355431	−	0.934702i	$-0.615666\pi$
$197$	−9.97743	−0.710863	−0.355431	+	0.934702i	$0.615666\pi$
$198$	6.71592	0.477279
$199$	13.6459	0.967332	0.483666	−	0.875253i	$-0.339305\pi$
$199$	13.6459	0.967332	0.483666	+	0.875253i	$0.339305\pi$
$200$	0.346159	0.0244771
$201$	−7.83724	−0.552796
$202$	3.54352	0.249321
$203$	3.69740	0.259506
$204$	1.45811	0.102088
$205$	−9.47509	−0.661769
$206$	4.20258	0.292808
$207$	−13.0829	−0.909323
$208$	5.58097	0.386970
$209$	−22.0958	−1.52840
$210$	4.89555	0.337825
$211$	2.29058	0.157690	0.0788449	−	0.996887i	$-0.474877\pi$
$211$	2.29058	0.157690	0.0788449	+	0.996887i	$0.474877\pi$
$212$	−0.803768	−0.0552030
$213$	−8.66403	−0.593649
$214$	−5.05564	−0.345597
$215$	19.6816	1.34227
$216$	−5.52701	−0.376065
$217$	3.26681	0.221766
$218$	−1.85741	−0.125800
$219$	−13.4532	−0.909081
$220$	10.5375	0.710438
$221$	6.58673	0.443071
$222$	−8.77796	−0.589138
$223$	−11.3042	−0.756982	−0.378491	−	0.925605i	$-0.623557\pi$
$223$	−11.3042	−0.756982	−0.378491	+	0.925605i	$0.623557\pi$
$224$	−1.71376	−0.114505
$225$	−0.510110	−0.0340073
$226$	15.7609	1.04840
$227$	15.1654	1.00656	0.503281	−	0.864123i	$-0.332126\pi$
$227$	15.1654	1.00656	0.503281	+	0.864123i	$0.332126\pi$
$228$	5.98994	0.396693
$229$	−22.9898	−1.51921	−0.759603	−	0.650387i	$-0.774607\pi$
$229$	−22.9898	−1.51921	−0.759603	+	0.650387i	$0.774607\pi$
$230$	−20.5275	−1.35354
$231$	9.64933	0.634879
$232$	−2.15748	−0.141645
$233$	−11.3619	−0.744341	−0.372170	−	0.928164i	$-0.621386\pi$
$233$	−11.3619	−0.744341	−0.372170	+	0.928164i	$0.621386\pi$
$234$	−8.22428	−0.537638
$235$	−19.0679	−1.24386
$236$	−11.0130	−0.716888
$237$	1.73704	0.112833
$238$	−2.02260	−0.131106
$239$	−23.6275	−1.52834	−0.764168	−	0.645017i	$-0.776850\pi$
$239$	−23.6275	−1.52834	−0.764168	+	0.645017i	$0.776850\pi$
$240$	−2.85661	−0.184393
$241$	−9.30362	−0.599299	−0.299649	−	0.954049i	$-0.596870\pi$
$241$	−9.30362	−0.599299	−0.299649	+	0.954049i	$0.596870\pi$
$242$	9.76986	0.628031
$243$	13.6066	0.872865
$244$	5.83604	0.373614
$245$	9.39443	0.600188
$246$	5.06282	0.322793
$247$	27.0584	1.72168
$248$	−1.90623	−0.121045
$249$	−16.3812	−1.03812
$250$	10.7605	0.680554
$251$	−7.02958	−0.443703	−0.221852	−	0.975080i	$-0.571210\pi$
$251$	−7.02958	−0.443703	−0.221852	+	0.975080i	$0.571210\pi$
$252$	2.52545	0.159088
$253$	−40.4605	−2.54373
$254$	−5.64176	−0.353996
$255$	−3.37141	−0.211126
$256$	1.00000	0.0625000
$257$	−22.7193	−1.41719	−0.708597	−	0.705614i	$-0.750671\pi$
$257$	−22.7193	−1.41719	−0.708597	+	0.705614i	$0.750671\pi$
$258$	−10.5164	−0.654725
$259$	12.1763	0.756596
$260$	−12.9042	−0.800283
$261$	3.17932	0.196795
$262$	4.09116	0.252753
$263$	16.0875	0.991999	0.495999	−	0.868323i	$-0.334802\pi$
$263$	16.0875	0.991999	0.495999	+	0.868323i	$0.334802\pi$
$264$	−5.63050	−0.346533
$265$	1.85845	0.114164
$266$	−8.30889	−0.509450
$267$	−14.5038	−0.887619
$268$	−6.34356	−0.387495
$269$	−3.66110	−0.223221	−0.111611	−	0.993752i	$-0.535601\pi$
$269$	−3.66110	−0.223221	−0.111611	+	0.993752i	$0.535601\pi$
$270$	12.7794	0.777730
$271$	−8.96808	−0.544772	−0.272386	−	0.962188i	$-0.587813\pi$
$271$	−8.96808	−0.544772	−0.272386	+	0.962188i	$0.587813\pi$
$272$	1.18021	0.0715609
$273$	−11.8165	−0.715168
$274$	−6.32269	−0.381968
$275$	−1.57758	−0.0951318
$276$	10.9684	0.660222
$277$	10.5454	0.633613	0.316807	−	0.948490i	$-0.397389\pi$
$277$	10.5454	0.633613	0.316807	+	0.948490i	$0.397389\pi$
$278$	−10.4799	−0.628545
$279$	2.80907	0.168175
$280$	3.96252	0.236806
$281$	−12.1311	−0.723683	−0.361842	−	0.932240i	$-0.617852\pi$
$281$	−12.1311	−0.723683	−0.361842	+	0.932240i	$0.617852\pi$
$282$	10.1886	0.606720
$283$	18.4003	1.09379	0.546894	−	0.837202i	$-0.315810\pi$
$283$	18.4003	1.09379	0.546894	+	0.837202i	$0.315810\pi$
$284$	−7.01278	−0.416132
$285$	−13.8498	−0.820391
$286$	−25.4347	−1.50398
$287$	−7.02283	−0.414545
$288$	−1.47363	−0.0868345
$289$	−15.6071	−0.918065
$290$	4.98846	0.292933
$291$	−16.7293	−0.980687
$292$	−10.8892	−0.637240
$293$	2.16591	0.126534	0.0632670	−	0.997997i	$-0.479848\pi$
$293$	2.16591	0.126534	0.0632670	+	0.997997i	$0.479848\pi$
$294$	−5.01972	−0.292756
$295$	25.4641	1.48258
$296$	−7.10499	−0.412969
$297$	25.1888	1.46160
$298$	8.23024	0.476765
$299$	49.5478	2.86542
$300$	0.427666	0.0246913
$301$	14.5878	0.840826
$302$	5.98701	0.344514
$303$	4.37789	0.251503
$304$	4.84833	0.278071
$305$	−13.4939	−0.772661
$306$	−1.73920	−0.0994233
$307$	−28.0694	−1.60201	−0.801003	−	0.598660i	$-0.795700\pi$
$307$	−28.0694	−1.60201	−0.801003	+	0.598660i	$0.795700\pi$
$308$	7.81029	0.445032
$309$	5.19214	0.295370
$310$	4.40753	0.250331
$311$	−23.1670	−1.31368	−0.656840	−	0.754030i	$-0.728107\pi$
$311$	−23.1670	−1.31368	−0.656840	+	0.754030i	$0.728107\pi$
$312$	6.89508	0.390357
$313$	26.1288	1.47689	0.738444	−	0.674315i	$-0.235561\pi$
$313$	26.1288	1.47689	0.738444	+	0.674315i	$0.235561\pi$
$314$	13.1509	0.742148
$315$	−5.83928	−0.329006
$316$	1.40598	0.0790926
$317$	32.1504	1.80575	0.902873	−	0.429907i	$-0.141454\pi$
$317$	32.1504	1.80575	0.902873	+	0.429907i	$0.141454\pi$
$318$	−0.993026	−0.0556861
$319$	9.83247	0.550513
$320$	−2.31218	−0.129255
$321$	−6.24606	−0.348621
$322$	−15.2148	−0.847885
$323$	5.72207	0.318384
$324$	−2.40752	−0.133751
$325$	1.93190	0.107163
$326$	12.7248	0.704762
$327$	−2.29476	−0.126901
$328$	4.09791	0.226269
$329$	−14.1330	−0.779176
$330$	13.0187	0.716656
$331$	−21.4153	−1.17709	−0.588545	−	0.808464i	$-0.700299\pi$
$331$	−21.4153	−1.17709	−0.588545	+	0.808464i	$0.700299\pi$
$332$	−13.2592	−0.727692
$333$	10.4701	0.573760
$334$	−17.1390	−0.937802
$335$	14.6674	0.801367
$336$	−2.11729	−0.115508
$337$	−6.54283	−0.356411	−0.178205	−	0.983993i	$-0.557029\pi$
$337$	−6.54283	−0.356411	−0.178205	+	0.983993i	$0.557029\pi$
$338$	18.1472	0.987076
$339$	19.4720	1.05757
$340$	−2.72886	−0.147993
$341$	8.68742	0.470450
$342$	−7.14465	−0.386338
$343$	18.9594	1.02371
$344$	−8.51215	−0.458944
$345$	−25.3610	−1.36539
$346$	−8.56498	−0.460456
$347$	−6.25781	−0.335937	−0.167969	−	0.985792i	$-0.553721\pi$
$347$	−6.25781	−0.335937	−0.167969	+	0.985792i	$0.553721\pi$
$348$	−2.66548	−0.142885
$349$	−14.4380	−0.772846	−0.386423	−	0.922322i	$-0.626289\pi$
$349$	−14.4380	−0.772846	−0.386423	+	0.922322i	$0.626289\pi$
$350$	−0.593233	−0.0317097
$351$	−30.8460	−1.64644
$352$	−4.55740	−0.242910
$353$	5.57594	0.296777	0.148389	−	0.988929i	$-0.452591\pi$
$353$	5.57594	0.296777	0.148389	+	0.988929i	$0.452591\pi$
$354$	−13.6062	−0.723162
$355$	16.2148	0.860591
$356$	−11.7396	−0.622196
$357$	−2.49885	−0.132253
$358$	17.0777	0.902587
$359$	22.7423	1.20029	0.600147	−	0.799889i	$-0.295109\pi$
$359$	22.7423	1.20029	0.600147	+	0.799889i	$0.295109\pi$
$360$	3.40729	0.179580
$361$	4.50634	0.237176
$362$	−15.4803	−0.813625
$363$	12.0703	0.633527
$364$	−9.56444	−0.501313
$365$	25.1777	1.31786
$366$	7.21021	0.376884
$367$	−15.0692	−0.786604	−0.393302	−	0.919409i	$-0.628667\pi$
$367$	−15.0692	−0.786604	−0.393302	+	0.919409i	$0.628667\pi$
$368$	8.87799	0.462797
$369$	−6.03880	−0.314367
$370$	16.4280	0.854051
$371$	1.37747	0.0715144
$372$	−2.35507	−0.122105
$373$	12.2786	0.635759	0.317880	−	0.948131i	$-0.397029\pi$
$373$	12.2786	0.635759	0.317880	+	0.948131i	$0.397029\pi$
$374$	−5.37870	−0.278126
$375$	13.2942	0.686510
$376$	8.24675	0.425294
$377$	−12.0408	−0.620132
$378$	9.47197	0.487185
$379$	12.2012	0.626732	0.313366	−	0.949632i	$-0.398543\pi$
$379$	12.2012	0.626732	0.313366	+	0.949632i	$0.398543\pi$
$380$	−11.2102	−0.575071
$381$	−6.97019	−0.357094
$382$	−21.7072	−1.11064
$383$	−2.98629	−0.152592	−0.0762961	−	0.997085i	$-0.524309\pi$
$383$	−2.98629	−0.152592	−0.0762961	+	0.997085i	$0.524309\pi$
$384$	1.23546	0.0630470
$385$	−18.0588	−0.920360
$386$	−23.2266	−1.18220
$387$	12.5438	0.637635
$388$	−13.5409	−0.687434
$389$	6.03304	0.305887	0.152944	−	0.988235i	$-0.451125\pi$
$389$	6.03304	0.305887	0.152944	+	0.988235i	$0.451125\pi$
$390$	−15.9426	−0.807287
$391$	10.4779	0.529891
$392$	−4.06302	−0.205214
$393$	5.05448	0.254965
$394$	−9.97743	−0.502656
$395$	−3.25088	−0.163569
$396$	6.71592	0.337487
$397$	−5.09018	−0.255469	−0.127734	−	0.991808i	$-0.540771\pi$
$397$	−5.09018	−0.255469	−0.127734	+	0.991808i	$0.540771\pi$
$398$	13.6459	0.684007
$399$	−10.2653	−0.513909
$400$	0.346159	0.0173079
$401$	33.8749	1.69163	0.845816	−	0.533475i	$-0.179114\pi$
$401$	33.8749	1.69163	0.845816	+	0.533475i	$0.179114\pi$
$402$	−7.83724	−0.390886
$403$	−10.6386	−0.529945
$404$	3.54352	0.176297
$405$	5.56662	0.276608
$406$	3.69740	0.183499
$407$	32.3803	1.60503
$408$	1.45811	0.0721872
$409$	23.6324	1.16855	0.584273	−	0.811557i	$-0.301380\pi$
$409$	23.6324	1.16855	0.584273	+	0.811557i	$0.301380\pi$
$410$	−9.47509	−0.467941
$411$	−7.81146	−0.385311
$412$	4.20258	0.207046
$413$	18.8737	0.928716
$414$	−13.0829	−0.642988
$415$	30.6576	1.50492
$416$	5.58097	0.273629
$417$	−12.9476	−0.634046
$418$	−22.0958	−1.08074
$419$	29.7920	1.45544	0.727718	−	0.685876i	$-0.240581\pi$
$419$	29.7920	1.45544	0.727718	+	0.685876i	$0.240581\pi$
$420$	4.89555	0.238878
$421$	5.90773	0.287925	0.143963	−	0.989583i	$-0.454016\pi$
$421$	5.90773	0.287925	0.143963	+	0.989583i	$0.454016\pi$
$422$	2.29058	0.111504
$423$	−12.1527	−0.590883
$424$	−0.803768	−0.0390344
$425$	0.408541	0.0198172
$426$	−8.66403	−0.419774
$427$	−10.0016	−0.484010
$428$	−5.05564	−0.244374
$429$	−31.4236	−1.51715
$430$	19.6816	0.949130
$431$	19.0279	0.916541	0.458271	−	0.888813i	$-0.348469\pi$
$431$	19.0279	0.916541	0.458271	+	0.888813i	$0.348469\pi$
$432$	−5.52701	−0.265918
$433$	5.50141	0.264381	0.132190	−	0.991224i	$-0.457799\pi$
$433$	5.50141	0.264381	0.132190	+	0.991224i	$0.457799\pi$
$434$	3.26681	0.156812
$435$	6.16306	0.295496
$436$	−1.85741	−0.0889538
$437$	43.0435	2.05905
$438$	−13.4532	−0.642817
$439$	−25.6898	−1.22611	−0.613053	−	0.790041i	$-0.710059\pi$
$439$	−25.6898	−1.22611	−0.613053	+	0.790041i	$0.710059\pi$
$440$	10.5375	0.502356
$441$	5.98739	0.285114
$442$	6.58673	0.313299
$443$	21.2590	1.01004	0.505022	−	0.863107i	$-0.331485\pi$
$443$	21.2590	1.01004	0.505022	+	0.863107i	$0.331485\pi$
$444$	−8.77796	−0.416583
$445$	27.1440	1.28675
$446$	−11.3042	−0.535267
$447$	10.1682	0.480937
$448$	−1.71376	−0.0809676
$449$	−14.0982	−0.665337	−0.332669	−	0.943044i	$-0.607949\pi$
$449$	−14.0982	−0.665337	−0.332669	+	0.943044i	$0.607949\pi$
$450$	−0.510110	−0.0240468
$451$	−18.6758	−0.879409
$452$	15.7609	0.741330
$453$	7.39673	0.347529
$454$	15.1654	0.711746
$455$	22.1147	1.03675
$456$	5.98994	0.280505
$457$	−12.0123	−0.561910	−0.280955	−	0.959721i	$-0.590651\pi$
$457$	−12.0123	−0.561910	−0.280955	+	0.959721i	$0.590651\pi$
$458$	−22.9898	−1.07424
$459$	−6.52304	−0.304470
$460$	−20.5275	−0.957099
$461$	−11.9355	−0.555891	−0.277946	−	0.960597i	$-0.589654\pi$
$461$	−11.9355	−0.555891	−0.277946	+	0.960597i	$0.589654\pi$
$462$	9.64933	0.448927
$463$	−1.47585	−0.0685886	−0.0342943	−	0.999412i	$-0.510918\pi$
$463$	−1.47585	−0.0685886	−0.0342943	+	0.999412i	$0.510918\pi$
$464$	−2.15748	−0.100158
$465$	5.44534	0.252522
$466$	−11.3619	−0.526329
$467$	−20.3849	−0.943300	−0.471650	−	0.881786i	$-0.656341\pi$
$467$	−20.3849	−0.943300	−0.471650	+	0.881786i	$0.656341\pi$
$468$	−8.22428	−0.380167
$469$	10.8713	0.501992
$470$	−19.0679	−0.879539
$471$	16.2475	0.748643
$472$	−11.0130	−0.506917
$473$	38.7932	1.78371
$474$	1.73704	0.0797848
$475$	1.67829	0.0770054
$476$	−2.02260	−0.0927058
$477$	1.18446	0.0542325
$478$	−23.6275	−1.08070
$479$	26.5638	1.21373	0.606865	−	0.794805i	$-0.292427\pi$
$479$	26.5638	1.21373	0.606865	+	0.794805i	$0.292427\pi$
$480$	−2.85661	−0.130386
$481$	−39.6527	−1.80801
$482$	−9.30362	−0.423768
$483$	−18.7973	−0.855306
$484$	9.76986	0.444085
$485$	31.3089	1.42166
$486$	13.6066	0.617208
$487$	−23.1254	−1.04791	−0.523955	−	0.851746i	$-0.675544\pi$
$487$	−23.1254	−1.04791	−0.523955	+	0.851746i	$0.675544\pi$
$488$	5.83604	0.264185
$489$	15.7210	0.710930
$490$	9.39443	0.424397
$491$	14.7563	0.665944	0.332972	−	0.942937i	$-0.391948\pi$
$491$	14.7563	0.665944	0.332972	+	0.942937i	$0.391948\pi$
$492$	5.06282	0.228249
$493$	−2.54628	−0.114679
$494$	27.0584	1.21741
$495$	−15.5284	−0.697949
$496$	−1.90623	−0.0855920
$497$	12.0182	0.539091
$498$	−16.3812	−0.734061
$499$	7.39220	0.330920	0.165460	−	0.986216i	$-0.447089\pi$
$499$	7.39220	0.330920	0.165460	+	0.986216i	$0.447089\pi$
$500$	10.7605	0.481224
$501$	−21.1745	−0.946009
$502$	−7.02958	−0.313746
$503$	4.55116	0.202926	0.101463	−	0.994839i	$-0.467648\pi$
$503$	4.55116	0.202926	0.101463	+	0.994839i	$0.467648\pi$
$504$	2.52545	0.112492
$505$	−8.19324	−0.364595
$506$	−40.4605	−1.79869
$507$	22.4202	0.995715
$508$	−5.64176	−0.250313
$509$	−24.8982	−1.10359	−0.551796	−	0.833979i	$-0.686057\pi$
$509$	−24.8982	−1.10359	−0.551796	+	0.833979i	$0.686057\pi$
$510$	−3.37141	−0.149288
$511$	18.6614	0.825533
$512$	1.00000	0.0441942
$513$	−26.7968	−1.18311
$514$	−22.7193	−1.00211
$515$	−9.71711	−0.428187
$516$	−10.5164	−0.462961
$517$	−37.5837	−1.65293
$518$	12.1763	0.534994
$519$	−10.5817	−0.464486
$520$	−12.9042	−0.565886
$521$	−17.4336	−0.763782	−0.381891	−	0.924207i	$-0.624727\pi$
$521$	−17.4336	−0.763782	−0.381891	+	0.924207i	$0.624727\pi$
$522$	3.17932	0.139155
$523$	−10.7798	−0.471369	−0.235684	−	0.971830i	$-0.575733\pi$
$523$	−10.7798	−0.471369	−0.235684	+	0.971830i	$0.575733\pi$
$524$	4.09116	0.178723
$525$	−0.732918	−0.0319872
$526$	16.0875	0.701449
$527$	−2.24975	−0.0980007
$528$	−5.63050	−0.245036
$529$	55.8188	2.42690
$530$	1.85845	0.0807260
$531$	16.2292	0.704286
$532$	−8.30889	−0.360236
$533$	22.8703	0.990622
$534$	−14.5038	−0.627642
$535$	11.6895	0.505383
$536$	−6.34356	−0.274000
$537$	21.0989	0.910486
$538$	−3.66110	−0.157841
$539$	18.5168	0.797575
$540$	12.7794	0.549938
$541$	28.2600	1.21499	0.607497	−	0.794322i	$-0.292174\pi$
$541$	28.2600	1.21499	0.607497	+	0.794322i	$0.292174\pi$
$542$	−8.96808	−0.385212
$543$	−19.1253	−0.820745
$544$	1.18021	0.0506012
$545$	4.29466	0.183963
$546$	−11.8165	−0.505700
$547$	−26.1138	−1.11654	−0.558272	−	0.829658i	$-0.688536\pi$
$547$	−26.1138	−1.11654	−0.558272	+	0.829658i	$0.688536\pi$
$548$	−6.32269	−0.270092
$549$	−8.60016	−0.367046
$550$	−1.57758	−0.0672684
$551$	−10.4602	−0.445618
$552$	10.9684	0.466848
$553$	−2.40951	−0.102463
$554$	10.5454	0.448032
$555$	20.2962	0.861525
$556$	−10.4799	−0.444448
$557$	15.4019	0.652601	0.326301	−	0.945266i	$-0.394198\pi$
$557$	15.4019	0.652601	0.326301	+	0.945266i	$0.394198\pi$
$558$	2.80907	0.118917
$559$	−47.5060	−2.00929
$560$	3.96252	0.167447
$561$	−6.64519	−0.280560
$562$	−12.1311	−0.511721
$563$	41.5076	1.74933	0.874667	−	0.484724i	$-0.161080\pi$
$563$	41.5076	1.74933	0.874667	+	0.484724i	$0.161080\pi$
$564$	10.1886	0.429016
$565$	−36.4420	−1.53313
$566$	18.4003	0.773424
$567$	4.12592	0.173272
$568$	−7.01278	−0.294250
$569$	−11.4386	−0.479533	−0.239766	−	0.970831i	$-0.577071\pi$
$569$	−11.4386	−0.479533	−0.239766	+	0.970831i	$0.577071\pi$
$570$	−13.8498	−0.580104
$571$	22.3241	0.934233	0.467116	−	0.884196i	$-0.345293\pi$
$571$	22.3241	0.934233	0.467116	+	0.884196i	$0.345293\pi$
$572$	−25.4347	−1.06348
$573$	−26.8184	−1.12036
$574$	−7.02283	−0.293127
$575$	3.07320	0.128161
$576$	−1.47363	−0.0614013
$577$	−23.5978	−0.982391	−0.491195	−	0.871049i	$-0.663440\pi$
$577$	−23.5978	−0.982391	−0.491195	+	0.871049i	$0.663440\pi$
$578$	−15.6071	−0.649170
$579$	−28.6956	−1.19255
$580$	4.98846	0.207135
$581$	22.7231	0.942712
$582$	−16.7293	−0.693451
$583$	3.66309	0.151710
$584$	−10.8892	−0.450597
$585$	19.0160	0.786214
$586$	2.16591	0.0894730
$587$	25.4214	1.04926	0.524628	−	0.851332i	$-0.324205\pi$
$587$	25.4214	1.04926	0.524628	+	0.851332i	$0.324205\pi$
$588$	−5.01972	−0.207010
$589$	−9.24202	−0.380811
$590$	25.4641	1.04834
$591$	−12.3268	−0.507055
$592$	−7.10499	−0.292013
$593$	21.1695	0.869328	0.434664	−	0.900593i	$-0.356867\pi$
$593$	21.1695	0.869328	0.434664	+	0.900593i	$0.356867\pi$
$594$	25.1888	1.03351
$595$	4.67661	0.191722
$596$	8.23024	0.337124
$597$	16.8590	0.689993
$598$	49.5478	2.02616
$599$	−7.66055	−0.313002	−0.156501	−	0.987678i	$-0.550021\pi$
$599$	−7.66055	−0.313002	−0.156501	+	0.987678i	$0.550021\pi$
$600$	0.427666	0.0174594
$601$	18.2816	0.745722	0.372861	−	0.927887i	$-0.378377\pi$
$601$	18.2816	0.745722	0.372861	+	0.927887i	$0.378377\pi$
$602$	14.5878	0.594554
$603$	9.34806	0.380683
$604$	5.98701	0.243608
$605$	−22.5896	−0.918400
$606$	4.37789	0.177840
$607$	2.00683	0.0814546	0.0407273	−	0.999170i	$-0.487033\pi$
$607$	2.00683	0.0814546	0.0407273	+	0.999170i	$0.487033\pi$
$608$	4.84833	0.196626
$609$	4.56800	0.185105
$610$	−13.4939	−0.546354
$611$	46.0248	1.86197
$612$	−1.73920	−0.0703029
$613$	−7.52366	−0.303878	−0.151939	−	0.988390i	$-0.548552\pi$
$613$	−7.52366	−0.303878	−0.151939	+	0.988390i	$0.548552\pi$
$614$	−28.0694	−1.13279
$615$	−11.7061	−0.472036
$616$	7.81029	0.314685
$617$	−16.8509	−0.678393	−0.339197	−	0.940716i	$-0.610155\pi$
$617$	−16.8509	−0.678393	−0.339197	+	0.940716i	$0.610155\pi$
$618$	5.19214	0.208858
$619$	25.7852	1.03640	0.518198	−	0.855260i	$-0.326603\pi$
$619$	25.7852	1.03640	0.518198	+	0.855260i	$0.326603\pi$
$620$	4.40753	0.177011
$621$	−49.0687	−1.96906
$622$	−23.1670	−0.928912
$623$	20.1188	0.806044
$624$	6.89508	0.276024
$625$	−26.6110	−1.06444
$626$	26.1288	1.04432
$627$	−27.2985	−1.09020
$628$	13.1509	0.524778
$629$	−8.38540	−0.334348
$630$	−5.83928	−0.232643
$631$	−8.81651	−0.350980	−0.175490	−	0.984481i	$-0.556151\pi$
$631$	−8.81651	−0.350980	−0.175490	+	0.984481i	$0.556151\pi$
$632$	1.40598	0.0559269
$633$	2.82992	0.112479
$634$	32.1504	1.27686
$635$	13.0447	0.517665
$636$	−0.993026	−0.0393760
$637$	−22.6756	−0.898440
$638$	9.83247	0.389271
$639$	10.3342	0.408816
$640$	−2.31218	−0.0913968
$641$	11.1674	0.441087	0.220544	−	0.975377i	$-0.429217\pi$
$641$	11.1674	0.441087	0.220544	+	0.975377i	$0.429217\pi$
$642$	−6.24606	−0.246512
$643$	−5.64482	−0.222610	−0.111305	−	0.993786i	$-0.535503\pi$
$643$	−5.64482	−0.222610	−0.111305	+	0.993786i	$0.535503\pi$
$644$	−15.2148	−0.599545
$645$	24.3159	0.957437
$646$	5.72207	0.225132
$647$	17.3110	0.680565	0.340282	−	0.940323i	$-0.389477\pi$
$647$	17.3110	0.680565	0.340282	+	0.940323i	$0.389477\pi$
$648$	−2.40752	−0.0945765
$649$	50.1908	1.97016
$650$	1.93190	0.0757754
$651$	4.03603	0.158184
$652$	12.7248	0.498342
$653$	49.2476	1.92721	0.963604	−	0.267333i	$-0.0861422\pi$
$653$	49.2476	1.92721	0.963604	+	0.267333i	$0.0861422\pi$
$654$	−2.29476	−0.0897323
$655$	−9.45948	−0.369613
$656$	4.09791	0.159996
$657$	16.0466	0.626038
$658$	−14.1330	−0.550960
$659$	27.8810	1.08609	0.543045	−	0.839704i	$-0.317271\pi$
$659$	27.8810	1.08609	0.543045	+	0.839704i	$0.317271\pi$
$660$	13.0187	0.506752
$661$	−44.8140	−1.74306	−0.871532	−	0.490339i	$-0.836873\pi$
$661$	−44.8140	−1.74306	−0.871532	+	0.490339i	$0.836873\pi$
$662$	−21.4153	−0.832329
$663$	8.13766	0.316041
$664$	−13.2592	−0.514556
$665$	19.2116	0.744994
$666$	10.4701	0.405709
$667$	−19.1541	−0.741648
$668$	−17.1390	−0.663126
$669$	−13.9659	−0.539952
$670$	14.6674	0.566652
$671$	−26.5971	−1.02677
$672$	−2.11729	−0.0816762
$673$	34.4069	1.32629	0.663144	−	0.748492i	$-0.269222\pi$
$673$	34.4069	1.32629	0.663144	+	0.748492i	$0.269222\pi$
$674$	−6.54283	−0.252020
$675$	−1.91322	−0.0736399
$676$	18.1472	0.697968
$677$	10.5263	0.404559	0.202280	−	0.979328i	$-0.435165\pi$
$677$	10.5263	0.404559	0.202280	+	0.979328i	$0.435165\pi$
$678$	19.4720	0.747818
$679$	23.2058	0.890558
$680$	−2.72886	−0.104647
$681$	18.7363	0.717975
$682$	8.68742	0.332659
$683$	33.0394	1.26422	0.632109	−	0.774880i	$-0.282190\pi$
$683$	33.0394	1.26422	0.632109	+	0.774880i	$0.282190\pi$
$684$	−7.14465	−0.273183
$685$	14.6192	0.558570
$686$	18.9594	0.723872
$687$	−28.4030	−1.08364
$688$	−8.51215	−0.324523
$689$	−4.48580	−0.170895
$690$	−25.3610	−0.965475
$691$	−19.5334	−0.743084	−0.371542	−	0.928416i	$-0.621171\pi$
$691$	−19.5334	−0.743084	−0.371542	+	0.928416i	$0.621171\pi$
$692$	−8.56498	−0.325592
$693$	−11.5095	−0.437209
$694$	−6.25781	−0.237543
$695$	24.2315	0.919152
$696$	−2.66548	−0.101035
$697$	4.83640	0.183192
$698$	−14.4380	−0.546485
$699$	−14.0372	−0.530935
$700$	−0.593233	−0.0224221
$701$	−8.49634	−0.320902	−0.160451	−	0.987044i	$-0.551295\pi$
$701$	−8.49634	−0.320902	−0.160451	+	0.987044i	$0.551295\pi$
$702$	−30.8460	−1.16421
$703$	−34.4474	−1.29921
$704$	−4.55740	−0.171763
$705$	−23.5578	−0.887236
$706$	5.57594	0.209853
$707$	−6.07275	−0.228389
$708$	−13.6062	−0.511353
$709$	−7.84591	−0.294659	−0.147330	−	0.989087i	$-0.547068\pi$
$709$	−7.84591	−0.294659	−0.147330	+	0.989087i	$0.547068\pi$
$710$	16.2148	0.608530
$711$	−2.07190	−0.0777022
$712$	−11.7396	−0.439959
$713$	−16.9235	−0.633788
$714$	−2.49885	−0.0935172
$715$	58.8094	2.19935
$716$	17.0777	0.638225
$717$	−29.1909	−1.09015
$718$	22.7423	0.848737
$719$	24.3467	0.907980	0.453990	−	0.891007i	$-0.350000\pi$
$719$	24.3467	0.907980	0.453990	+	0.891007i	$0.350000\pi$
$720$	3.40729	0.126982
$721$	−7.20222	−0.268225
$722$	4.50634	0.167709
$723$	−11.4943	−0.427477
$724$	−15.4803	−0.575320
$725$	−0.746829	−0.0277365
$726$	12.0703	0.447971
$727$	−20.3018	−0.752952	−0.376476	−	0.926426i	$-0.622864\pi$
$727$	−20.3018	−0.752952	−0.376476	+	0.926426i	$0.622864\pi$
$728$	−9.56444	−0.354482
$729$	24.0330	0.890113
$730$	25.1777	0.931868
$731$	−10.0461	−0.371570
$732$	7.21021	0.266497
$733$	19.4670	0.719029	0.359515	−	0.933139i	$-0.382942\pi$
$733$	19.4670	0.719029	0.359515	+	0.933139i	$0.382942\pi$
$734$	−15.0692	−0.556213
$735$	11.6065	0.428111
$736$	8.87799	0.327247
$737$	28.9101	1.06492
$738$	−6.03880	−0.222291
$739$	−36.1816	−1.33096	−0.665481	−	0.746415i	$-0.731773\pi$
$739$	−36.1816	−1.33096	−0.665481	+	0.746415i	$0.731773\pi$
$740$	16.4280	0.603905
$741$	33.4296	1.22807
$742$	1.37747	0.0505684
$743$	29.4596	1.08077	0.540383	−	0.841419i	$-0.318279\pi$
$743$	29.4596	1.08077	0.540383	+	0.841419i	$0.318279\pi$
$744$	−2.35507	−0.0863411
$745$	−19.0298	−0.697196
$746$	12.2786	0.449550
$747$	19.5391	0.714900
$748$	−5.37870	−0.196665
$749$	8.66416	0.316582
$750$	13.2942	0.485436
$751$	−19.2628	−0.702908	−0.351454	−	0.936205i	$-0.614313\pi$
$751$	−19.2628	−0.702908	−0.351454	+	0.936205i	$0.614313\pi$
$752$	8.24675	0.300728
$753$	−8.68479	−0.316491
$754$	−12.0408	−0.438500
$755$	−13.8430	−0.503799
$756$	9.47197	0.344492
$757$	−35.2742	−1.28206	−0.641031	−	0.767515i	$-0.721493\pi$
$757$	−35.2742	−1.28206	−0.641031	+	0.767515i	$0.721493\pi$
$758$	12.2012	0.443167
$759$	−49.9875	−1.81443
$760$	−11.2102	−0.406637
$761$	2.09715	0.0760217	0.0380109	−	0.999277i	$-0.487898\pi$
$761$	2.09715	0.0760217	0.0380109	+	0.999277i	$0.487898\pi$
$762$	−6.97019	−0.252503
$763$	3.18316	0.115238
$764$	−21.7072	−0.785338
$765$	4.02133	0.145392
$766$	−2.98629	−0.107899
$767$	−61.4634	−2.21932
$768$	1.23546	0.0445809
$769$	45.0698	1.62526	0.812629	−	0.582782i	$-0.198036\pi$
$769$	45.0698	1.62526	0.812629	+	0.582782i	$0.198036\pi$
$770$	−18.0588	−0.650793
$771$	−28.0689	−1.01088
$772$	−23.2266	−0.835945
$773$	−12.5556	−0.451594	−0.225797	−	0.974174i	$-0.572499\pi$
$773$	−12.5556	−0.451594	−0.225797	+	0.974174i	$0.572499\pi$
$774$	12.5438	0.450876
$775$	−0.659857	−0.0237028
$776$	−13.5409	−0.486089
$777$	15.0433	0.539676
$778$	6.03304	0.216295
$779$	19.8680	0.711846
$780$	−15.9426	−0.570838
$781$	31.9600	1.14362
$782$	10.4779	0.374690
$783$	11.9244	0.426143
$784$	−4.06302	−0.145108
$785$	−30.4072	−1.08528
$786$	5.05448	0.180287
$787$	49.5559	1.76648	0.883239	−	0.468922i	$-0.155358\pi$
$787$	49.5559	1.76648	0.883239	+	0.468922i	$0.155358\pi$
$788$	−9.97743	−0.355431
$789$	19.8755	0.707588
$790$	−3.25088	−0.115661
$791$	−27.0104	−0.960380
$792$	6.71592	0.238640
$793$	32.5707	1.15662
$794$	−5.09018	−0.180644
$795$	2.29605	0.0814325
$796$	13.6459	0.483666
$797$	28.5230	1.01034	0.505168	−	0.863021i	$-0.331430\pi$
$797$	28.5230	1.01034	0.505168	+	0.863021i	$0.331430\pi$
$798$	−10.2653	−0.363389
$799$	9.73293	0.344326
$800$	0.346159	0.0122386
$801$	17.2998	0.611258
$802$	33.8749	1.19616
$803$	49.6263	1.75127
$804$	−7.83724	−0.276398
$805$	35.1792	1.23990
$806$	−10.6386	−0.374728
$807$	−4.52316	−0.159223
$808$	3.54352	0.124661
$809$	−44.1752	−1.55312	−0.776559	−	0.630045i	$-0.783037\pi$
$809$	−44.1752	−1.55312	−0.776559	+	0.630045i	$0.783037\pi$
$810$	5.56662	0.195591
$811$	−21.7637	−0.764226	−0.382113	−	0.924116i	$-0.624803\pi$
$811$	−21.7637	−0.764226	−0.382113	+	0.924116i	$0.624803\pi$
$812$	3.69740	0.129753
$813$	−11.0797	−0.388583
$814$	32.3803	1.13493
$815$	−29.4220	−1.03061
$816$	1.45811	0.0510441
$817$	−41.2697	−1.44385
$818$	23.6324	0.826286
$819$	14.0944	0.492500
$820$	−9.47509	−0.330884
$821$	−5.76002	−0.201026	−0.100513	−	0.994936i	$-0.532048\pi$
$821$	−5.76002	−0.201026	−0.100513	+	0.994936i	$0.532048\pi$
$822$	−7.81146	−0.272456
$823$	−27.7514	−0.967354	−0.483677	−	0.875247i	$-0.660699\pi$
$823$	−27.7514	−0.967354	−0.483677	+	0.875247i	$0.660699\pi$
$824$	4.20258	0.146404
$825$	−1.94905	−0.0678571
$826$	18.8737	0.656701
$827$	−43.2908	−1.50537	−0.752684	−	0.658382i	$-0.771241\pi$
$827$	−43.2908	−1.50537	−0.752684	+	0.658382i	$0.771241\pi$
$828$	−13.0829	−0.454661
$829$	−38.0928	−1.32302	−0.661509	−	0.749937i	$-0.730084\pi$
$829$	−38.0928	−1.32302	−0.661509	+	0.749937i	$0.730084\pi$
$830$	30.6576	1.06414
$831$	13.0285	0.451953
$832$	5.58097	0.193485
$833$	−4.79523	−0.166145
$834$	−12.9476	−0.448338
$835$	39.6283	1.37139
$836$	−22.0958	−0.764199
$837$	10.5357	0.364168
$838$	29.7920	1.02915
$839$	33.4064	1.15332	0.576659	−	0.816985i	$-0.304356\pi$
$839$	33.4064	1.15332	0.576659	+	0.816985i	$0.304356\pi$
$840$	4.89555	0.168912
$841$	−24.3453	−0.839493
$842$	5.90773	0.203594
$843$	−14.9876	−0.516200
$844$	2.29058	0.0788449
$845$	−41.9595	−1.44345
$846$	−12.1527	−0.417817
$847$	−16.7432	−0.575303
$848$	−0.803768	−0.0276015
$849$	22.7330	0.780193
$850$	0.408541	0.0140128
$851$	−63.0781	−2.16229
$852$	−8.66403	−0.296825
$853$	56.6571	1.93990	0.969952	−	0.243298i	$-0.0782293\pi$
$853$	56.6571	1.93990	0.969952	+	0.243298i	$0.0782293\pi$
$854$	−10.0016	−0.342247
$855$	16.5197	0.564962
$856$	−5.05564	−0.172798
$857$	−26.4630	−0.903958	−0.451979	−	0.892029i	$-0.649282\pi$
$857$	−26.4630	−0.903958	−0.451979	+	0.892029i	$0.649282\pi$
$858$	−31.4236	−1.07278
$859$	−49.1611	−1.67736	−0.838678	−	0.544628i	$-0.816671\pi$
$859$	−49.1611	−1.67736	−0.838678	+	0.544628i	$0.816671\pi$
$860$	19.6816	0.671136
$861$	−8.67646	−0.295693
$862$	19.0279	0.648092
$863$	19.7934	0.673774	0.336887	−	0.941545i	$-0.390626\pi$
$863$	19.7934	0.673774	0.336887	+	0.941545i	$0.390626\pi$
$864$	−5.52701	−0.188033
$865$	19.8037	0.673347
$866$	5.50141	0.186946
$867$	−19.2820	−0.654851
$868$	3.26681	0.110883
$869$	−6.40761	−0.217363
$870$	6.16306	0.208947
$871$	−35.4032	−1.19959
$872$	−1.85741	−0.0628998
$873$	19.9543	0.675349
$874$	43.0435	1.45597
$875$	−18.4409	−0.623417
$876$	−13.4532	−0.454541
$877$	−48.6492	−1.64277	−0.821383	−	0.570377i	$-0.806797\pi$
$877$	−48.6492	−1.64277	−0.821383	+	0.570377i	$0.806797\pi$
$878$	−25.6898	−0.866988
$879$	2.67591	0.0902561
$880$	10.5375	0.355219
$881$	17.5120	0.589993	0.294997	−	0.955498i	$-0.404681\pi$
$881$	17.5120	0.589993	0.294997	+	0.955498i	$0.404681\pi$
$882$	5.98739	0.201606
$883$	6.44041	0.216737	0.108368	−	0.994111i	$-0.465437\pi$
$883$	6.44041	0.216737	0.108368	+	0.994111i	$0.465437\pi$
$884$	6.58673	0.221536
$885$	31.4600	1.05752
$886$	21.2590	0.714208
$887$	−55.7840	−1.87304	−0.936522	−	0.350608i	$-0.885975\pi$
$887$	−55.7840	−1.87304	−0.936522	+	0.350608i	$0.885975\pi$
$888$	−8.77796	−0.294569
$889$	9.66863	0.324275
$890$	27.1440	0.909868
$891$	10.9720	0.367577
$892$	−11.3042	−0.378491
$893$	39.9830	1.33798
$894$	10.1682	0.340074
$895$	−39.4868	−1.31990
$896$	−1.71376	−0.0572527
$897$	61.2145	2.04389
$898$	−14.0982	−0.470464
$899$	4.11263	0.137164
$900$	−0.510110	−0.0170037
$901$	−0.948617	−0.0316030
$902$	−18.6758	−0.621836
$903$	18.0227	0.599757
$904$	15.7609	0.524200
$905$	35.7931	1.18980
$906$	7.39673	0.245740
$907$	11.4217	0.379253	0.189626	−	0.981856i	$-0.439272\pi$
$907$	11.4217	0.379253	0.189626	+	0.981856i	$0.439272\pi$
$908$	15.1654	0.503281
$909$	−5.22184	−0.173197
$910$	22.1147	0.733094
$911$	53.4452	1.77072	0.885359	−	0.464907i	$-0.153912\pi$
$911$	53.4452	1.77072	0.885359	+	0.464907i	$0.153912\pi$
$912$	5.98994	0.198347
$913$	60.4274	1.99985
$914$	−12.0123	−0.397331
$915$	−16.6713	−0.551135
$916$	−22.9898	−0.759603
$917$	−7.01127	−0.231533
$918$	−6.52304	−0.215293
$919$	0.628942	0.0207469	0.0103734	−	0.999946i	$-0.496698\pi$
$919$	0.628942	0.0207469	0.0103734	+	0.999946i	$0.496698\pi$
$920$	−20.5275	−0.676771
$921$	−34.6787	−1.14270
$922$	−11.9355	−0.393075
$923$	−39.1381	−1.28824
$924$	9.64933	0.317439
$925$	−2.45945	−0.0808664
$926$	−1.47585	−0.0484995
$927$	−6.19305	−0.203406
$928$	−2.15748	−0.0708226
$929$	−14.9197	−0.489498	−0.244749	−	0.969586i	$-0.578706\pi$
$929$	−14.9197	−0.489498	−0.244749	+	0.969586i	$0.578706\pi$
$930$	5.44534	0.178560
$931$	−19.6989	−0.645605
$932$	−11.3619	−0.372170
$933$	−28.6220	−0.937041
$934$	−20.3849	−0.667014
$935$	12.4365	0.406717
$936$	−8.22428	−0.268819
$937$	53.5661	1.74993	0.874965	−	0.484187i	$-0.160884\pi$
$937$	53.5661	1.74993	0.874965	+	0.484187i	$0.160884\pi$
$938$	10.8713	0.354962
$939$	32.2812	1.05346
$940$	−19.0679	−0.621928
$941$	−9.68921	−0.315859	−0.157930	−	0.987450i	$-0.550482\pi$
$941$	−9.68921	−0.315859	−0.157930	+	0.987450i	$0.550482\pi$
$942$	16.2475	0.529371
$943$	36.3812	1.18473
$944$	−11.0130	−0.358444
$945$	−21.9009	−0.712435
$946$	38.7932	1.26128
$947$	10.1402	0.329513	0.164757	−	0.986334i	$-0.447316\pi$
$947$	10.1402	0.329513	0.164757	+	0.986334i	$0.447316\pi$
$948$	1.73704	0.0564164
$949$	−60.7721	−1.97275
$950$	1.67829	0.0544510
$951$	39.7206	1.28803
$952$	−2.02260	−0.0655529
$953$	9.51032	0.308069	0.154035	−	0.988065i	$-0.450773\pi$
$953$	9.51032	0.308069	0.154035	+	0.988065i	$0.450773\pi$
$954$	1.18446	0.0383482
$955$	50.1908	1.62414
$956$	−23.6275	−0.764168
$957$	12.1477	0.392678
$958$	26.5638	0.858237
$959$	10.8356	0.349899
$960$	−2.85661	−0.0921967
$961$	−27.3663	−0.882784
$962$	−39.6527	−1.27845
$963$	7.45015	0.240078
$964$	−9.30362	−0.299649
$965$	53.7040	1.72879
$966$	−18.7973	−0.604792
$967$	−35.9542	−1.15621	−0.578105	−	0.815962i	$-0.696208\pi$
$967$	−35.9542	−1.15621	−0.578105	+	0.815962i	$0.696208\pi$
$968$	9.76986	0.314015
$969$	7.06940	0.227102
$970$	31.3089	1.00527
$971$	−28.4183	−0.911988	−0.455994	−	0.889983i	$-0.650716\pi$
$971$	−28.4183	−0.911988	−0.455994	+	0.889983i	$0.650716\pi$
$972$	13.6066	0.436432
$973$	17.9601	0.575775
$974$	−23.1254	−0.740984
$975$	2.38679	0.0764385
$976$	5.83604	0.186807
$977$	9.70161	0.310382	0.155191	−	0.987884i	$-0.450401\pi$
$977$	9.70161	0.310382	0.155191	+	0.987884i	$0.450401\pi$
$978$	15.7210	0.502704
$979$	53.5019	1.70993
$980$	9.39443	0.300094
$981$	2.73713	0.0873900
$982$	14.7563	0.470894
$983$	12.0010	0.382771	0.191386	−	0.981515i	$-0.438702\pi$
$983$	12.0010	0.382771	0.191386	+	0.981515i	$0.438702\pi$
$984$	5.06282	0.161397
$985$	23.0696	0.735058
$986$	−2.54628	−0.0810901
$987$	−17.4608	−0.555782
$988$	27.0584	0.860842
$989$	−75.5708	−2.40301
$990$	−15.5284	−0.493524
$991$	24.2710	0.770993	0.385496	−	0.922709i	$-0.374030\pi$
$991$	24.2710	0.770993	0.385496	+	0.922709i	$0.374030\pi$
$992$	−1.90623	−0.0605227
$993$	−26.4578	−0.839613
$994$	12.0182	0.381195
$995$	−31.5517	−1.00026
$996$	−16.3812	−0.519060
$997$	3.58716	0.113606	0.0568032	−	0.998385i	$-0.481909\pi$
$997$	3.58716	0.113606	0.0568032	+	0.998385i	$0.481909\pi$
$998$	7.39220	0.233996
$999$	39.2693	1.24243

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.47	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.47	✓	67	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.23546 q^{3} +1.00000 q^{4} -2.31218 q^{5} +1.23546 q^{6} -1.71376 q^{7} +1.00000 q^{8} -1.47363 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.23546 q^{3} +1.00000 q^{4} -2.31218 q^{5} +1.23546 q^{6} -1.71376 q^{7} +1.00000 q^{8} -1.47363 q^{9} -2.31218 q^{10} -4.55740 q^{11} +1.23546 q^{12} +5.58097 q^{13} -1.71376 q^{14} -2.85661 q^{15} +1.00000 q^{16} +1.18021 q^{17} -1.47363 q^{18} +4.84833 q^{19} -2.31218 q^{20} -2.11729 q^{21} -4.55740 q^{22} +8.87799 q^{23} +1.23546 q^{24} +0.346159 q^{25} +5.58097 q^{26} -5.52701 q^{27} -1.71376 q^{28} -2.15748 q^{29} -2.85661 q^{30} -1.90623 q^{31} +1.00000 q^{32} -5.63050 q^{33} +1.18021 q^{34} +3.96252 q^{35} -1.47363 q^{36} -7.10499 q^{37} +4.84833 q^{38} +6.89508 q^{39} -2.31218 q^{40} +4.09791 q^{41} -2.11729 q^{42} -8.51215 q^{43} -4.55740 q^{44} +3.40729 q^{45} +8.87799 q^{46} +8.24675 q^{47} +1.23546 q^{48} -4.06302 q^{49} +0.346159 q^{50} +1.45811 q^{51} +5.58097 q^{52} -0.803768 q^{53} -5.52701 q^{54} +10.5375 q^{55} -1.71376 q^{56} +5.98994 q^{57} -2.15748 q^{58} -11.0130 q^{59} -2.85661 q^{60} +5.83604 q^{61} -1.90623 q^{62} +2.52545 q^{63} +1.00000 q^{64} -12.9042 q^{65} -5.63050 q^{66} -6.34356 q^{67} +1.18021 q^{68} +10.9684 q^{69} +3.96252 q^{70} -7.01278 q^{71} -1.47363 q^{72} -10.8892 q^{73} -7.10499 q^{74} +0.427666 q^{75} +4.84833 q^{76} +7.81029 q^{77} +6.89508 q^{78} +1.40598 q^{79} -2.31218 q^{80} -2.40752 q^{81} +4.09791 q^{82} -13.2592 q^{83} -2.11729 q^{84} -2.72886 q^{85} -8.51215 q^{86} -2.66548 q^{87} -4.55740 q^{88} -11.7396 q^{89} +3.40729 q^{90} -9.56444 q^{91} +8.87799 q^{92} -2.35507 q^{93} +8.24675 q^{94} -11.2102 q^{95} +1.23546 q^{96} -13.5409 q^{97} -4.06302 q^{98} +6.71592 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

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