LMFDB - Embedded newform 8038.2.a.b.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, 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("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.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:	$64.1837531447$
Analytic rank:	$0$
Dimension:	$83$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.90051 q^{3} +1.00000 q^{4} +2.46896 q^{5} +2.90051 q^{6} +0.644361 q^{7} -1.00000 q^{8} +5.41299 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.90051 q^{3} +1.00000 q^{4} +2.46896 q^{5} +2.90051 q^{6} +0.644361 q^{7} -1.00000 q^{8} +5.41299 q^{9} -2.46896 q^{10} -6.47212 q^{11} -2.90051 q^{12} -1.48266 q^{13} -0.644361 q^{14} -7.16124 q^{15} +1.00000 q^{16} +4.06541 q^{17} -5.41299 q^{18} +0.353534 q^{19} +2.46896 q^{20} -1.86898 q^{21} +6.47212 q^{22} -5.50199 q^{23} +2.90051 q^{24} +1.09574 q^{25} +1.48266 q^{26} -6.99890 q^{27} +0.644361 q^{28} +2.19452 q^{29} +7.16124 q^{30} -3.17645 q^{31} -1.00000 q^{32} +18.7725 q^{33} -4.06541 q^{34} +1.59090 q^{35} +5.41299 q^{36} -4.16995 q^{37} -0.353534 q^{38} +4.30047 q^{39} -2.46896 q^{40} -6.84841 q^{41} +1.86898 q^{42} -2.65712 q^{43} -6.47212 q^{44} +13.3644 q^{45} +5.50199 q^{46} -7.57521 q^{47} -2.90051 q^{48} -6.58480 q^{49} -1.09574 q^{50} -11.7918 q^{51} -1.48266 q^{52} +0.916397 q^{53} +6.99890 q^{54} -15.9794 q^{55} -0.644361 q^{56} -1.02543 q^{57} -2.19452 q^{58} +3.61786 q^{59} -7.16124 q^{60} +0.561897 q^{61} +3.17645 q^{62} +3.48792 q^{63} +1.00000 q^{64} -3.66062 q^{65} -18.7725 q^{66} +0.459976 q^{67} +4.06541 q^{68} +15.9586 q^{69} -1.59090 q^{70} +6.43753 q^{71} -5.41299 q^{72} +10.8109 q^{73} +4.16995 q^{74} -3.17821 q^{75} +0.353534 q^{76} -4.17038 q^{77} -4.30047 q^{78} +2.75253 q^{79} +2.46896 q^{80} +4.06146 q^{81} +6.84841 q^{82} +13.9474 q^{83} -1.86898 q^{84} +10.0373 q^{85} +2.65712 q^{86} -6.36525 q^{87} +6.47212 q^{88} -4.04771 q^{89} -13.3644 q^{90} -0.955368 q^{91} -5.50199 q^{92} +9.21335 q^{93} +7.57521 q^{94} +0.872860 q^{95} +2.90051 q^{96} +1.32634 q^{97} +6.58480 q^{98} -35.0335 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) $ 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * 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
$3$	−2.90051	−1.67461	−0.837306	−	0.546734i	$-0.815871\pi$
$3$	−2.90051	−1.67461	−0.837306	+	0.546734i	$0.815871\pi$
$4$	1.00000	0.500000
$5$	2.46896	1.10415	0.552075	−	0.833794i	$-0.313836\pi$
$5$	2.46896	1.10415	0.552075	+	0.833794i	$0.313836\pi$
$6$	2.90051	1.18413
$7$	0.644361	0.243546	0.121773	−	0.992558i	$-0.461142\pi$
$7$	0.644361	0.243546	0.121773	+	0.992558i	$0.461142\pi$
$8$	−1.00000	−0.353553
$9$	5.41299	1.80433
$10$	−2.46896	−0.780752
$11$	−6.47212	−1.95142	−0.975708	−	0.219073i	$-0.929697\pi$
$11$	−6.47212	−1.95142	−0.975708	+	0.219073i	$0.929697\pi$
$12$	−2.90051	−0.837306
$13$	−1.48266	−0.411216	−0.205608	−	0.978634i	$-0.565917\pi$
$13$	−1.48266	−0.411216	−0.205608	+	0.978634i	$0.565917\pi$
$14$	−0.644361	−0.172213
$15$	−7.16124	−1.84902
$16$	1.00000	0.250000
$17$	4.06541	0.986007	0.493003	−	0.870027i	$-0.335899\pi$
$17$	4.06541	0.986007	0.493003	+	0.870027i	$0.335899\pi$
$18$	−5.41299	−1.27585
$19$	0.353534	0.0811063	0.0405531	−	0.999177i	$-0.487088\pi$
$19$	0.353534	0.0811063	0.0405531	+	0.999177i	$0.487088\pi$
$20$	2.46896	0.552075
$21$	−1.86898	−0.407845
$22$	6.47212	1.37986
$23$	−5.50199	−1.14724	−0.573622	−	0.819120i	$-0.694462\pi$
$23$	−5.50199	−1.14724	−0.573622	+	0.819120i	$0.694462\pi$
$24$	2.90051	0.592065
$25$	1.09574	0.219148
$26$	1.48266	0.290773
$27$	−6.99890	−1.34694
$28$	0.644361	0.121773
$29$	2.19452	0.407513	0.203756	−	0.979022i	$-0.434685\pi$
$29$	2.19452	0.407513	0.203756	+	0.979022i	$0.434685\pi$
$30$	7.16124	1.30746
$31$	−3.17645	−0.570508	−0.285254	−	0.958452i	$-0.592078\pi$
$31$	−3.17645	−0.570508	−0.285254	+	0.958452i	$0.592078\pi$
$32$	−1.00000	−0.176777
$33$	18.7725	3.26787
$34$	−4.06541	−0.697212
$35$	1.59090	0.268911
$36$	5.41299	0.902164
$37$	−4.16995	−0.685536	−0.342768	−	0.939420i	$-0.611364\pi$
$37$	−4.16995	−0.685536	−0.342768	+	0.939420i	$0.611364\pi$
$38$	−0.353534	−0.0573508
$39$	4.30047	0.688627
$40$	−2.46896	−0.390376
$41$	−6.84841	−1.06954	−0.534771	−	0.844997i	$-0.679602\pi$
$41$	−6.84841	−1.06954	−0.534771	+	0.844997i	$0.679602\pi$
$42$	1.86898	0.288390
$43$	−2.65712	−0.405207	−0.202603	−	0.979261i	$-0.564940\pi$
$43$	−2.65712	−0.405207	−0.202603	+	0.979261i	$0.564940\pi$
$44$	−6.47212	−0.975708
$45$	13.3644	1.99225
$46$	5.50199	0.811224
$47$	−7.57521	−1.10496	−0.552479	−	0.833527i	$-0.686318\pi$
$47$	−7.57521	−1.10496	−0.552479	+	0.833527i	$0.686318\pi$
$48$	−2.90051	−0.418653
$49$	−6.58480	−0.940685
$50$	−1.09574	−0.154961
$51$	−11.7918	−1.65118
$52$	−1.48266	−0.205608
$53$	0.916397	0.125877	0.0629384	−	0.998017i	$-0.479953\pi$
$53$	0.916397	0.125877	0.0629384	+	0.998017i	$0.479953\pi$
$54$	6.99890	0.952430
$55$	−15.9794	−2.15466
$56$	−0.644361	−0.0861064
$57$	−1.02543	−0.135822
$58$	−2.19452	−0.288155
$59$	3.61786	0.471005	0.235502	−	0.971874i	$-0.424326\pi$
$59$	3.61786	0.471005	0.235502	+	0.971874i	$0.424326\pi$
$60$	−7.16124	−0.924512
$61$	0.561897	0.0719436	0.0359718	−	0.999353i	$-0.488547\pi$
$61$	0.561897	0.0719436	0.0359718	+	0.999353i	$0.488547\pi$
$62$	3.17645	0.403410
$63$	3.48792	0.439437
$64$	1.00000	0.125000
$65$	−3.66062	−0.454044
$66$	−18.7725	−2.31073
$67$	0.459976	0.0561950	0.0280975	−	0.999605i	$-0.491055\pi$
$67$	0.459976	0.0561950	0.0280975	+	0.999605i	$0.491055\pi$
$68$	4.06541	0.493003
$69$	15.9586	1.92119
$70$	−1.59090	−0.190149
$71$	6.43753	0.763994	0.381997	−	0.924164i	$-0.375237\pi$
$71$	6.43753	0.763994	0.381997	+	0.924164i	$0.375237\pi$
$72$	−5.41299	−0.637927
$73$	10.8109	1.26532	0.632661	−	0.774429i	$-0.281963\pi$
$73$	10.8109	1.26532	0.632661	+	0.774429i	$0.281963\pi$
$74$	4.16995	0.484747
$75$	−3.17821	−0.366988
$76$	0.353534	0.0405531
$77$	−4.17038	−0.475259
$78$	−4.30047	−0.486933
$79$	2.75253	0.309684	0.154842	−	0.987939i	$-0.450513\pi$
$79$	2.75253	0.309684	0.154842	+	0.987939i	$0.450513\pi$
$80$	2.46896	0.276038
$81$	4.06146	0.451273
$82$	6.84841	0.756280
$83$	13.9474	1.53093	0.765464	−	0.643478i	$-0.222509\pi$
$83$	13.9474	1.53093	0.765464	+	0.643478i	$0.222509\pi$
$84$	−1.86898	−0.203922
$85$	10.0373	1.08870
$86$	2.65712	0.286524
$87$	−6.36525	−0.682426
$88$	6.47212	0.689930
$89$	−4.04771	−0.429056	−0.214528	−	0.976718i	$-0.568821\pi$
$89$	−4.04771	−0.429056	−0.214528	+	0.976718i	$0.568821\pi$
$90$	−13.3644	−1.40873
$91$	−0.955368	−0.100150
$92$	−5.50199	−0.573622
$93$	9.21335	0.955379
$94$	7.57521	0.781323
$95$	0.872860	0.0895535
$96$	2.90051	0.296033
$97$	1.32634	0.134669	0.0673346	−	0.997730i	$-0.478551\pi$
$97$	1.32634	0.134669	0.0673346	+	0.997730i	$0.478551\pi$
$98$	6.58480	0.665165
$99$	−35.0335	−3.52100
$100$	1.09574	0.109574
$101$	−2.05837	−0.204815	−0.102408	−	0.994743i	$-0.532655\pi$
$101$	−2.05837	−0.204815	−0.102408	+	0.994743i	$0.532655\pi$
$102$	11.7918	1.16756
$103$	−16.4864	−1.62445	−0.812227	−	0.583342i	$-0.801745\pi$
$103$	−16.4864	−1.62445	−0.812227	+	0.583342i	$0.801745\pi$
$104$	1.48266	0.145387
$105$	−4.61443	−0.450322
$106$	−0.916397	−0.0890083
$107$	4.30523	0.416203	0.208101	−	0.978107i	$-0.433272\pi$
$107$	4.30523	0.416203	0.208101	+	0.978107i	$0.433272\pi$
$108$	−6.99890	−0.673470
$109$	5.19034	0.497144	0.248572	−	0.968613i	$-0.420039\pi$
$109$	5.19034	0.497144	0.248572	+	0.968613i	$0.420039\pi$
$110$	15.9794	1.52357
$111$	12.0950	1.14801
$112$	0.644361	0.0608864
$113$	7.77113	0.731046	0.365523	−	0.930802i	$-0.380890\pi$
$113$	7.77113	0.731046	0.365523	+	0.930802i	$0.380890\pi$
$114$	1.02543	0.0960404
$115$	−13.5842	−1.26673
$116$	2.19452	0.203756
$117$	−8.02561	−0.741968
$118$	−3.61786	−0.333051
$119$	2.61959	0.240138
$120$	7.16124	0.653729
$121$	30.8883	2.80803
$122$	−0.561897	−0.0508718
$123$	19.8639	1.79107
$124$	−3.17645	−0.285254
$125$	−9.63944	−0.862178
$126$	−3.48792	−0.310729
$127$	−17.8217	−1.58142	−0.790711	−	0.612190i	$-0.790289\pi$
$127$	−17.8217	−1.58142	−0.790711	+	0.612190i	$0.790289\pi$
$128$	−1.00000	−0.0883883
$129$	7.70701	0.678564
$130$	3.66062	0.321058
$131$	5.20759	0.454989	0.227495	−	0.973779i	$-0.426947\pi$
$131$	5.20759	0.454989	0.227495	+	0.973779i	$0.426947\pi$
$132$	18.7725	1.63393
$133$	0.227804	0.0197531
$134$	−0.459976	−0.0397358
$135$	−17.2800	−1.48722
$136$	−4.06541	−0.348606
$137$	2.06810	0.176689	0.0883447	−	0.996090i	$-0.471842\pi$
$137$	2.06810	0.176689	0.0883447	+	0.996090i	$0.471842\pi$
$138$	−15.9586	−1.35849
$139$	17.4915	1.48361	0.741804	−	0.670617i	$-0.233971\pi$
$139$	17.4915	1.48361	0.741804	+	0.670617i	$0.233971\pi$
$140$	1.59090	0.134456
$141$	21.9720	1.85038
$142$	−6.43753	−0.540225
$143$	9.59594	0.802453
$144$	5.41299	0.451082
$145$	5.41818	0.449955
$146$	−10.8109	−0.894717
$147$	19.0993	1.57528
$148$	−4.16995	−0.342768
$149$	4.41967	0.362073	0.181037	−	0.983476i	$-0.442055\pi$
$149$	4.41967	0.362073	0.181037	+	0.983476i	$0.442055\pi$
$150$	3.17821	0.259500
$151$	15.0994	1.22877	0.614384	−	0.789007i	$-0.289404\pi$
$151$	15.0994	1.22877	0.614384	+	0.789007i	$0.289404\pi$
$152$	−0.353534	−0.0286754
$153$	22.0060	1.77908
$154$	4.17038	0.336059
$155$	−7.84252	−0.629926
$156$	4.30047	0.344314
$157$	15.3223	1.22285	0.611427	−	0.791301i	$-0.290596\pi$
$157$	15.3223	1.22285	0.611427	+	0.791301i	$0.290596\pi$
$158$	−2.75253	−0.218980
$159$	−2.65802	−0.210795
$160$	−2.46896	−0.195188
$161$	−3.54527	−0.279406
$162$	−4.06146	−0.319098
$163$	−20.3281	−1.59222	−0.796112	−	0.605150i	$-0.793113\pi$
$163$	−20.3281	−1.59222	−0.796112	+	0.605150i	$0.793113\pi$
$164$	−6.84841	−0.534771
$165$	46.3484	3.60822
$166$	−13.9474	−1.08253
$167$	21.1732	1.63843	0.819215	−	0.573486i	$-0.194409\pi$
$167$	21.1732	1.63843	0.819215	+	0.573486i	$0.194409\pi$
$168$	1.86898	0.144195
$169$	−10.8017	−0.830902
$170$	−10.0373	−0.769827
$171$	1.91367	0.146342
$172$	−2.65712	−0.202603
$173$	7.80435	0.593354	0.296677	−	0.954978i	$-0.404122\pi$
$173$	7.80435	0.593354	0.296677	+	0.954978i	$0.404122\pi$
$174$	6.36525	0.482548
$175$	0.706053	0.0533726
$176$	−6.47212	−0.487854
$177$	−10.4936	−0.788750
$178$	4.04771	0.303388
$179$	−15.2448	−1.13945	−0.569725	−	0.821835i	$-0.692950\pi$
$179$	−15.2448	−1.13945	−0.569725	+	0.821835i	$0.692950\pi$
$180$	13.3644	0.996125
$181$	−4.67729	−0.347660	−0.173830	−	0.984776i	$-0.555614\pi$
$181$	−4.67729	−0.347660	−0.173830	+	0.984776i	$0.555614\pi$
$182$	0.955368	0.0708166
$183$	−1.62979	−0.120478
$184$	5.50199	0.405612
$185$	−10.2954	−0.756935
$186$	−9.21335	−0.675555
$187$	−26.3118	−1.92411
$188$	−7.57521	−0.552479
$189$	−4.50982	−0.328041
$190$	−0.872860	−0.0633239
$191$	5.57172	0.403156	0.201578	−	0.979472i	$-0.435393\pi$
$191$	5.57172	0.403156	0.201578	+	0.979472i	$0.435393\pi$
$192$	−2.90051	−0.209327
$193$	11.1491	0.802531	0.401266	−	0.915962i	$-0.368570\pi$
$193$	11.1491	0.802531	0.401266	+	0.915962i	$0.368570\pi$
$194$	−1.32634	−0.0952255
$195$	10.6177	0.760348
$196$	−6.58480	−0.470343
$197$	9.78084	0.696856	0.348428	−	0.937336i	$-0.386716\pi$
$197$	9.78084	0.696856	0.348428	+	0.937336i	$0.386716\pi$
$198$	35.0335	2.48972
$199$	−27.0126	−1.91487	−0.957436	−	0.288646i	$-0.906795\pi$
$199$	−27.0126	−1.91487	−0.957436	+	0.288646i	$0.906795\pi$
$200$	−1.09574	−0.0774805
$201$	−1.33417	−0.0941048
$202$	2.05837	0.144826
$203$	1.41407	0.0992480
$204$	−11.7918	−0.825590
$205$	−16.9084	−1.18093
$206$	16.4864	1.14866
$207$	−29.7822	−2.07001
$208$	−1.48266	−0.102804
$209$	−2.28811	−0.158272
$210$	4.61443	0.318426
$211$	−8.40055	−0.578318	−0.289159	−	0.957281i	$-0.593376\pi$
$211$	−8.40055	−0.578318	−0.289159	+	0.957281i	$0.593376\pi$
$212$	0.916397	0.0629384
$213$	−18.6721	−1.27939
$214$	−4.30523	−0.294300
$215$	−6.56031	−0.447409
$216$	6.99890	0.476215
$217$	−2.04678	−0.138945
$218$	−5.19034	−0.351534
$219$	−31.3572	−2.11892
$220$	−15.9794	−1.07733
$221$	−6.02762	−0.405461
$222$	−12.0950	−0.811764
$223$	20.5840	1.37841	0.689203	−	0.724568i	$-0.257961\pi$
$223$	20.5840	1.37841	0.689203	+	0.724568i	$0.257961\pi$
$224$	−0.644361	−0.0430532
$225$	5.93123	0.395415
$226$	−7.77113	−0.516928
$227$	−11.1138	−0.737646	−0.368823	−	0.929500i	$-0.620239\pi$
$227$	−11.1138	−0.737646	−0.368823	+	0.929500i	$0.620239\pi$
$228$	−1.02543	−0.0679108
$229$	−19.5569	−1.29236	−0.646178	−	0.763186i	$-0.723634\pi$
$229$	−19.5569	−1.29236	−0.646178	+	0.763186i	$0.723634\pi$
$230$	13.5842	0.895714
$231$	12.0963	0.795875
$232$	−2.19452	−0.144077
$233$	5.67802	0.371979	0.185990	−	0.982552i	$-0.440451\pi$
$233$	5.67802	0.371979	0.185990	+	0.982552i	$0.440451\pi$
$234$	8.02561	0.524651
$235$	−18.7028	−1.22004
$236$	3.61786	0.235502
$237$	−7.98375	−0.518601
$238$	−2.61959	−0.169803
$239$	−16.2048	−1.04820	−0.524100	−	0.851657i	$-0.675598\pi$
$239$	−16.2048	−1.04820	−0.524100	+	0.851657i	$0.675598\pi$
$240$	−7.16124	−0.462256
$241$	−10.9535	−0.705578	−0.352789	−	0.935703i	$-0.614767\pi$
$241$	−10.9535	−0.705578	−0.352789	+	0.935703i	$0.614767\pi$
$242$	−30.8883	−1.98558
$243$	9.21638	0.591231
$244$	0.561897	0.0359718
$245$	−16.2576	−1.03866
$246$	−19.8639	−1.26648
$247$	−0.524170	−0.0333522
$248$	3.17645	0.201705
$249$	−40.4547	−2.56371
$250$	9.63944	0.609652
$251$	28.9378	1.82654	0.913269	−	0.407358i	$-0.133550\pi$
$251$	28.9378	1.82654	0.913269	+	0.407358i	$0.133550\pi$
$252$	3.48792	0.219718
$253$	35.6095	2.23875
$254$	17.8217	1.11823
$255$	−29.1134	−1.82315
$256$	1.00000	0.0625000
$257$	−3.29557	−0.205572	−0.102786	−	0.994703i	$-0.532776\pi$
$257$	−3.29557	−0.205572	−0.102786	+	0.994703i	$0.532776\pi$
$258$	−7.70701	−0.479817
$259$	−2.68696	−0.166959
$260$	−3.66062	−0.227022
$261$	11.8789	0.735287
$262$	−5.20759	−0.321726
$263$	−13.4302	−0.828142	−0.414071	−	0.910245i	$-0.635894\pi$
$263$	−13.4302	−0.828142	−0.414071	+	0.910245i	$0.635894\pi$
$264$	−18.7725	−1.15537
$265$	2.26254	0.138987
$266$	−0.227804	−0.0139675
$267$	11.7404	0.718503
$268$	0.459976	0.0280975
$269$	−20.2780	−1.23637	−0.618186	−	0.786032i	$-0.712132\pi$
$269$	−20.2780	−1.23637	−0.618186	+	0.786032i	$0.712132\pi$
$270$	17.2800	1.05163
$271$	−4.91109	−0.298327	−0.149164	−	0.988813i	$-0.547658\pi$
$271$	−4.91109	−0.298327	−0.149164	+	0.988813i	$0.547658\pi$
$272$	4.06541	0.246502
$273$	2.77106	0.167712
$274$	−2.06810	−0.124938
$275$	−7.09176	−0.427649
$276$	15.9586	0.960595
$277$	31.0298	1.86440	0.932200	−	0.361943i	$-0.117886\pi$
$277$	31.0298	1.86440	0.932200	+	0.361943i	$0.117886\pi$
$278$	−17.4915	−1.04907
$279$	−17.1941	−1.02938
$280$	−1.59090	−0.0950744
$281$	18.7272	1.11717	0.558585	−	0.829447i	$-0.311344\pi$
$281$	18.7272	1.11717	0.558585	+	0.829447i	$0.311344\pi$
$282$	−21.9720	−1.30841
$283$	−26.0270	−1.54714	−0.773572	−	0.633709i	$-0.781532\pi$
$283$	−26.0270	−1.54714	−0.773572	+	0.633709i	$0.781532\pi$
$284$	6.43753	0.381997
$285$	−2.53174	−0.149967
$286$	−9.59594	−0.567420
$287$	−4.41285	−0.260482
$288$	−5.41299	−0.318963
$289$	−0.472440	−0.0277906
$290$	−5.41818	−0.318166
$291$	−3.84706	−0.225519
$292$	10.8109	0.632661
$293$	−8.26098	−0.482611	−0.241306	−	0.970449i	$-0.577576\pi$
$293$	−8.26098	−0.482611	−0.241306	+	0.970449i	$0.577576\pi$
$294$	−19.0993	−1.11389
$295$	8.93232	0.520060
$296$	4.16995	0.242374
$297$	45.2977	2.62844
$298$	−4.41967	−0.256024
$299$	8.15758	0.471765
$300$	−3.17821	−0.183494
$301$	−1.71214	−0.0986864
$302$	−15.0994	−0.868871
$303$	5.97032	0.342986
$304$	0.353534	0.0202766
$305$	1.38730	0.0794365
$306$	−22.0060	−1.25800
$307$	8.42222	0.480682	0.240341	−	0.970689i	$-0.422741\pi$
$307$	8.42222	0.480682	0.240341	+	0.970689i	$0.422741\pi$
$308$	−4.17038	−0.237630
$309$	47.8190	2.72033
$310$	7.84252	0.445425
$311$	−16.0435	−0.909741	−0.454871	−	0.890558i	$-0.650314\pi$
$311$	−16.0435	−0.909741	−0.454871	+	0.890558i	$0.650314\pi$
$312$	−4.30047	−0.243466
$313$	1.71903	0.0971653	0.0485826	−	0.998819i	$-0.484530\pi$
$313$	1.71903	0.0971653	0.0485826	+	0.998819i	$0.484530\pi$
$314$	−15.3223	−0.864688
$315$	8.61152	0.485204
$316$	2.75253	0.154842
$317$	31.2030	1.75253	0.876266	−	0.481827i	$-0.160027\pi$
$317$	31.2030	1.75253	0.876266	+	0.481827i	$0.160027\pi$
$318$	2.65802	0.149054
$319$	−14.2032	−0.795227
$320$	2.46896	0.138019
$321$	−12.4874	−0.696978
$322$	3.54527	0.197570
$323$	1.43726	0.0799713
$324$	4.06146	0.225637
$325$	−1.62461	−0.0901171
$326$	20.3281	1.12587
$327$	−15.0546	−0.832524
$328$	6.84841	0.378140
$329$	−4.88117	−0.269108
$330$	−46.3484	−2.55140
$331$	18.5722	1.02082	0.510410	−	0.859931i	$-0.329494\pi$
$331$	18.5722	1.02082	0.510410	+	0.859931i	$0.329494\pi$
$332$	13.9474	0.765464
$333$	−22.5719	−1.23693
$334$	−21.1732	−1.15855
$335$	1.13566	0.0620477
$336$	−1.86898	−0.101961
$337$	13.6674	0.744510	0.372255	−	0.928130i	$-0.378585\pi$
$337$	13.6674	0.744510	0.372255	+	0.928130i	$0.378585\pi$
$338$	10.8017	0.587536
$339$	−22.5403	−1.22422
$340$	10.0373	0.544350
$341$	20.5584	1.11330
$342$	−1.91367	−0.103480
$343$	−8.75352	−0.472646
$344$	2.65712	0.143262
$345$	39.4011	2.12128
$346$	−7.80435	−0.419564
$347$	12.2762	0.659020	0.329510	−	0.944152i	$-0.393116\pi$
$347$	12.2762	0.659020	0.329510	+	0.944152i	$0.393116\pi$
$348$	−6.36525	−0.341213
$349$	−15.2141	−0.814394	−0.407197	−	0.913340i	$-0.633494\pi$
$349$	−15.2141	−0.814394	−0.407197	+	0.913340i	$0.633494\pi$
$350$	−0.706053	−0.0377401
$351$	10.3770	0.553882
$352$	6.47212	0.344965
$353$	4.79807	0.255375	0.127688	−	0.991814i	$-0.459244\pi$
$353$	4.79807	0.255375	0.127688	+	0.991814i	$0.459244\pi$
$354$	10.4936	0.557731
$355$	15.8940	0.843564
$356$	−4.04771	−0.214528
$357$	−7.59817	−0.402138
$358$	15.2448	0.805713
$359$	12.3948	0.654170	0.327085	−	0.944995i	$-0.393934\pi$
$359$	12.3948	0.654170	0.327085	+	0.944995i	$0.393934\pi$
$360$	−13.3644	−0.704367
$361$	−18.8750	−0.993422
$362$	4.67729	0.245833
$363$	−89.5920	−4.70236
$364$	−0.955368	−0.0500749
$365$	26.6916	1.39710
$366$	1.62979	0.0851906
$367$	−8.23152	−0.429682	−0.214841	−	0.976649i	$-0.568923\pi$
$367$	−8.23152	−0.429682	−0.214841	+	0.976649i	$0.568923\pi$
$368$	−5.50199	−0.286811
$369$	−37.0703	−1.92980
$370$	10.2954	0.535234
$371$	0.590491	0.0306568
$372$	9.21335	0.477690
$373$	11.2867	0.584401	0.292201	−	0.956357i	$-0.405612\pi$
$373$	11.2867	0.584401	0.292201	+	0.956357i	$0.405612\pi$
$374$	26.3118	1.36055
$375$	27.9593	1.44381
$376$	7.57521	0.390661
$377$	−3.25373	−0.167576
$378$	4.50982	0.231960
$379$	−21.2915	−1.09367	−0.546836	−	0.837240i	$-0.684168\pi$
$379$	−21.2915	−1.09367	−0.546836	+	0.837240i	$0.684168\pi$
$380$	0.872860	0.0447768
$381$	51.6922	2.64827
$382$	−5.57172	−0.285074
$383$	20.1648	1.03038	0.515188	−	0.857077i	$-0.327722\pi$
$383$	20.1648	1.03038	0.515188	+	0.857077i	$0.327722\pi$
$384$	2.90051	0.148016
$385$	−10.2965	−0.524758
$386$	−11.1491	−0.567475
$387$	−14.3829	−0.731126
$388$	1.32634	0.0673346
$389$	−8.39011	−0.425395	−0.212698	−	0.977118i	$-0.568225\pi$
$389$	−8.39011	−0.425395	−0.212698	+	0.977118i	$0.568225\pi$
$390$	−10.6177	−0.537647
$391$	−22.3679	−1.13119
$392$	6.58480	0.332583
$393$	−15.1047	−0.761931
$394$	−9.78084	−0.492751
$395$	6.79587	0.341938
$396$	−35.0335	−1.76050
$397$	−23.9424	−1.20163	−0.600817	−	0.799386i	$-0.705158\pi$
$397$	−23.9424	−1.20163	−0.600817	+	0.799386i	$0.705158\pi$
$398$	27.0126	1.35402
$399$	−0.660748	−0.0330788
$400$	1.09574	0.0547870
$401$	−28.2696	−1.41172	−0.705859	−	0.708352i	$-0.749439\pi$
$401$	−28.2696	−1.41172	−0.705859	+	0.708352i	$0.749439\pi$
$402$	1.33417	0.0665422
$403$	4.70959	0.234602
$404$	−2.05837	−0.102408
$405$	10.0276	0.498273
$406$	−1.41407	−0.0701789
$407$	26.9884	1.33777
$408$	11.7918	0.583780
$409$	36.8963	1.82441	0.912203	−	0.409739i	$-0.134380\pi$
$409$	36.8963	1.82441	0.912203	+	0.409739i	$0.134380\pi$
$410$	16.9084	0.835047
$411$	−5.99854	−0.295886
$412$	−16.4864	−0.812227
$413$	2.33121	0.114711
$414$	29.7822	1.46372
$415$	34.4356	1.69038
$416$	1.48266	0.0726933
$417$	−50.7343	−2.48447
$418$	2.28811	0.111915
$419$	−6.18939	−0.302371	−0.151186	−	0.988505i	$-0.548309\pi$
$419$	−6.18939	−0.302371	−0.151186	+	0.988505i	$0.548309\pi$
$420$	−4.61443	−0.225161
$421$	6.62811	0.323034	0.161517	−	0.986870i	$-0.448361\pi$
$421$	6.62811	0.323034	0.161517	+	0.986870i	$0.448361\pi$
$422$	8.40055	0.408933
$423$	−41.0045	−1.99371
$424$	−0.916397	−0.0445042
$425$	4.45463	0.216081
$426$	18.6721	0.904668
$427$	0.362065	0.0175216
$428$	4.30523	0.208101
$429$	−27.8332	−1.34380
$430$	6.56031	0.316366
$431$	21.1521	1.01886	0.509431	−	0.860511i	$-0.329856\pi$
$431$	21.1521	1.01886	0.509431	+	0.860511i	$0.329856\pi$
$432$	−6.99890	−0.336735
$433$	−29.7118	−1.42786	−0.713928	−	0.700219i	$-0.753086\pi$
$433$	−29.7118	−1.42786	−0.713928	+	0.700219i	$0.753086\pi$
$434$	2.04678	0.0982487
$435$	−15.7155	−0.753501
$436$	5.19034	0.248572
$437$	−1.94514	−0.0930487
$438$	31.3572	1.49830
$439$	36.6737	1.75034	0.875171	−	0.483814i	$-0.160749\pi$
$439$	36.6737	1.75034	0.875171	+	0.483814i	$0.160749\pi$
$440$	15.9794	0.761787
$441$	−35.6434	−1.69731
$442$	6.02762	0.286705
$443$	−1.81456	−0.0862125	−0.0431063	−	0.999070i	$-0.513725\pi$
$443$	−1.81456	−0.0862125	−0.0431063	+	0.999070i	$0.513725\pi$
$444$	12.0950	0.574004
$445$	−9.99360	−0.473742
$446$	−20.5840	−0.974680
$447$	−12.8193	−0.606332
$448$	0.644361	0.0304432
$449$	11.2539	0.531102	0.265551	−	0.964097i	$-0.414446\pi$
$449$	11.2539	0.531102	0.265551	+	0.964097i	$0.414446\pi$
$450$	−5.93123	−0.279601
$451$	44.3237	2.08712
$452$	7.77113	0.365523
$453$	−43.7959	−2.05771
$454$	11.1138	0.521594
$455$	−2.35876	−0.110580
$456$	1.02543	0.0480202
$457$	−2.26377	−0.105895	−0.0529474	−	0.998597i	$-0.516862\pi$
$457$	−2.26377	−0.105895	−0.0529474	+	0.998597i	$0.516862\pi$
$458$	19.5569	0.913834
$459$	−28.4534	−1.32809
$460$	−13.5842	−0.633365
$461$	2.93395	0.136648	0.0683239	−	0.997663i	$-0.478235\pi$
$461$	2.93395	0.136648	0.0683239	+	0.997663i	$0.478235\pi$
$462$	−12.0963	−0.562769
$463$	−32.4908	−1.50998	−0.754988	−	0.655739i	$-0.772357\pi$
$463$	−32.4908	−1.50998	−0.754988	+	0.655739i	$0.772357\pi$
$464$	2.19452	0.101878
$465$	22.7473	1.05488
$466$	−5.67802	−0.263029
$467$	24.8501	1.14992	0.574962	−	0.818180i	$-0.305017\pi$
$467$	24.8501	1.14992	0.574962	+	0.818180i	$0.305017\pi$
$468$	−8.02561	−0.370984
$469$	0.296391	0.0136860
$470$	18.7028	0.862698
$471$	−44.4426	−2.04781
$472$	−3.61786	−0.166525
$473$	17.1972	0.790727
$474$	7.98375	0.366706
$475$	0.387381	0.0177743
$476$	2.61959	0.120069
$477$	4.96044	0.227123
$478$	16.2048	0.741189
$479$	13.1100	0.599011	0.299505	−	0.954095i	$-0.403178\pi$
$479$	13.1100	0.599011	0.299505	+	0.954095i	$0.403178\pi$
$480$	7.16124	0.326864
$481$	6.18262	0.281903
$482$	10.9535	0.498919
$483$	10.2831	0.467898
$484$	30.8883	1.40401
$485$	3.27467	0.148695
$486$	−9.21638	−0.418064
$487$	34.4123	1.55937	0.779686	−	0.626171i	$-0.215379\pi$
$487$	34.4123	1.55937	0.779686	+	0.626171i	$0.215379\pi$
$488$	−0.561897	−0.0254359
$489$	58.9621	2.66636
$490$	16.2576	0.734442
$491$	26.8212	1.21043	0.605213	−	0.796064i	$-0.293088\pi$
$491$	26.8212	1.21043	0.605213	+	0.796064i	$0.293088\pi$
$492$	19.8639	0.895534
$493$	8.92164	0.401810
$494$	0.524170	0.0235835
$495$	−86.4961	−3.88771
$496$	−3.17645	−0.142627
$497$	4.14809	0.186067
$498$	40.4547	1.81282
$499$	34.4468	1.54205	0.771025	−	0.636805i	$-0.219744\pi$
$499$	34.4468	1.54205	0.771025	+	0.636805i	$0.219744\pi$
$500$	−9.63944	−0.431089
$501$	−61.4131	−2.74374
$502$	−28.9378	−1.29156
$503$	−7.94665	−0.354324	−0.177162	−	0.984182i	$-0.556692\pi$
$503$	−7.94665	−0.354324	−0.177162	+	0.984182i	$0.556692\pi$
$504$	−3.48792	−0.155364
$505$	−5.08201	−0.226147
$506$	−35.6095	−1.58304
$507$	31.3306	1.39144
$508$	−17.8217	−0.790711
$509$	12.7370	0.564559	0.282280	−	0.959332i	$-0.408909\pi$
$509$	12.7370	0.564559	0.282280	+	0.959332i	$0.408909\pi$
$510$	29.1134	1.28916
$511$	6.96613	0.308164
$512$	−1.00000	−0.0441942
$513$	−2.47435	−0.109245
$514$	3.29557	0.145362
$515$	−40.7042	−1.79364
$516$	7.70701	0.339282
$517$	49.0276	2.15623
$518$	2.68696	0.118058
$519$	−22.6366	−0.993638
$520$	3.66062	0.160529
$521$	−18.8055	−0.823885	−0.411942	−	0.911210i	$-0.635150\pi$
$521$	−18.8055	−0.823885	−0.411942	+	0.911210i	$0.635150\pi$
$522$	−11.8789	−0.519926
$523$	15.1363	0.661864	0.330932	−	0.943655i	$-0.392637\pi$
$523$	15.1363	0.661864	0.330932	+	0.943655i	$0.392637\pi$
$524$	5.20759	0.227495
$525$	−2.04792	−0.0893784
$526$	13.4302	0.585585
$527$	−12.9136	−0.562524
$528$	18.7725	0.816967
$529$	7.27190	0.316170
$530$	−2.26254	−0.0982786
$531$	19.5834	0.849847
$532$	0.227804	0.00987654
$533$	10.1539	0.439812
$534$	−11.7404	−0.508058
$535$	10.6294	0.459550
$536$	−0.459976	−0.0198679
$537$	44.2178	1.90814
$538$	20.2780	0.874247
$539$	42.6176	1.83567
$540$	−17.2800	−0.743612
$541$	33.5548	1.44263	0.721316	−	0.692606i	$-0.243537\pi$
$541$	33.5548	1.44263	0.721316	+	0.692606i	$0.243537\pi$
$542$	4.91109	0.210949
$543$	13.5666	0.582196
$544$	−4.06541	−0.174303
$545$	12.8147	0.548922
$546$	−2.77106	−0.118590
$547$	12.8396	0.548981	0.274491	−	0.961590i	$-0.411491\pi$
$547$	12.8396	0.548981	0.274491	+	0.961590i	$0.411491\pi$
$548$	2.06810	0.0883447
$549$	3.04154	0.129810
$550$	7.09176	0.302394
$551$	0.775839	0.0330518
$552$	−15.9586	−0.679243
$553$	1.77362	0.0754222
$554$	−31.0298	−1.31833
$555$	29.8620	1.26757
$556$	17.4915	0.741804
$557$	6.46379	0.273880	0.136940	−	0.990579i	$-0.456273\pi$
$557$	6.46379	0.273880	0.136940	+	0.990579i	$0.456273\pi$
$558$	17.1941	0.727884
$559$	3.93960	0.166627
$560$	1.59090	0.0672278
$561$	76.3178	3.22214
$562$	−18.7272	−0.789959
$563$	8.32722	0.350951	0.175475	−	0.984484i	$-0.443854\pi$
$563$	8.32722	0.350951	0.175475	+	0.984484i	$0.443854\pi$
$564$	21.9720	0.925188
$565$	19.1866	0.807185
$566$	26.0270	1.09400
$567$	2.61705	0.109906
$568$	−6.43753	−0.270113
$569$	1.12454	0.0471431	0.0235716	−	0.999722i	$-0.492496\pi$
$569$	1.12454	0.0471431	0.0235716	+	0.999722i	$0.492496\pi$
$570$	2.53174	0.106043
$571$	1.99124	0.0833310	0.0416655	−	0.999132i	$-0.486734\pi$
$571$	1.99124	0.0833310	0.0416655	+	0.999132i	$0.486734\pi$
$572$	9.59594	0.401227
$573$	−16.1609	−0.675130
$574$	4.41285	0.184189
$575$	−6.02875	−0.251416
$576$	5.41299	0.225541
$577$	−1.95412	−0.0813510	−0.0406755	−	0.999172i	$-0.512951\pi$
$577$	−1.95412	−0.0813510	−0.0406755	+	0.999172i	$0.512951\pi$
$578$	0.472440	0.0196509
$579$	−32.3382	−1.34393
$580$	5.41818	0.224978
$581$	8.98718	0.372851
$582$	3.84706	0.159466
$583$	−5.93103	−0.245638
$584$	−10.8109	−0.447359
$585$	−19.8149	−0.819244
$586$	8.26098	0.341258
$587$	−9.17440	−0.378668	−0.189334	−	0.981913i	$-0.560633\pi$
$587$	−9.17440	−0.378668	−0.189334	+	0.981913i	$0.560633\pi$
$588$	19.0993	0.787642
$589$	−1.12298	−0.0462717
$590$	−8.93232	−0.367738
$591$	−28.3695	−1.16696
$592$	−4.16995	−0.171384
$593$	25.9378	1.06514	0.532569	−	0.846387i	$-0.321227\pi$
$593$	25.9378	1.06514	0.532569	+	0.846387i	$0.321227\pi$
$594$	−45.2977	−1.85859
$595$	6.46766	0.265148
$596$	4.41967	0.181037
$597$	78.3504	3.20667
$598$	−8.15758	−0.333588
$599$	−20.1789	−0.824489	−0.412245	−	0.911073i	$-0.635255\pi$
$599$	−20.1789	−0.824489	−0.412245	+	0.911073i	$0.635255\pi$
$600$	3.17821	0.129750
$601$	−45.6102	−1.86048	−0.930239	−	0.366954i	$-0.880401\pi$
$601$	−45.6102	−1.86048	−0.930239	+	0.366954i	$0.880401\pi$
$602$	1.71214	0.0697818
$603$	2.48984	0.101394
$604$	15.0994	0.614384
$605$	76.2618	3.10048
$606$	−5.97032	−0.242528
$607$	−8.10852	−0.329115	−0.164557	−	0.986368i	$-0.552620\pi$
$607$	−8.10852	−0.329115	−0.164557	+	0.986368i	$0.552620\pi$
$608$	−0.353534	−0.0143377
$609$	−4.10152	−0.166202
$610$	−1.38730	−0.0561701
$611$	11.2314	0.454376
$612$	22.0060	0.889540
$613$	28.1123	1.13545	0.567723	−	0.823220i	$-0.307825\pi$
$613$	28.1123	1.13545	0.567723	+	0.823220i	$0.307825\pi$
$614$	−8.42222	−0.339893
$615$	49.0431	1.97761
$616$	4.17038	0.168030
$617$	−34.2225	−1.37774	−0.688872	−	0.724883i	$-0.741894\pi$
$617$	−34.2225	−1.37774	−0.688872	+	0.724883i	$0.741894\pi$
$618$	−47.8190	−1.92356
$619$	−34.4298	−1.38385	−0.691925	−	0.721970i	$-0.743237\pi$
$619$	−34.4298	−1.38385	−0.691925	+	0.721970i	$0.743237\pi$
$620$	−7.84252	−0.314963
$621$	38.5079	1.54527
$622$	16.0435	0.643284
$623$	−2.60819	−0.104495
$624$	4.30047	0.172157
$625$	−29.2781	−1.17112
$626$	−1.71903	−0.0687062
$627$	6.63671	0.265045
$628$	15.3223	0.611427
$629$	−16.9526	−0.675943
$630$	−8.61152	−0.343091
$631$	19.2830	0.767642	0.383821	−	0.923407i	$-0.374608\pi$
$631$	19.2830	0.767642	0.383821	+	0.923407i	$0.374608\pi$
$632$	−2.75253	−0.109490
$633$	24.3659	0.968459
$634$	−31.2030	−1.23923
$635$	−44.0010	−1.74613
$636$	−2.65802	−0.105397
$637$	9.76301	0.386825
$638$	14.2032	0.562310
$639$	34.8462	1.37850
$640$	−2.46896	−0.0975940
$641$	2.64035	0.104288	0.0521438	−	0.998640i	$-0.483395\pi$
$641$	2.64035	0.104288	0.0521438	+	0.998640i	$0.483395\pi$
$642$	12.4874	0.492838
$643$	26.0557	1.02754	0.513769	−	0.857929i	$-0.328249\pi$
$643$	26.0557	1.02754	0.513769	+	0.857929i	$0.328249\pi$
$644$	−3.54527	−0.139703
$645$	19.0283	0.749237
$646$	−1.43726	−0.0565483
$647$	18.8220	0.739969	0.369985	−	0.929038i	$-0.379363\pi$
$647$	18.8220	0.739969	0.369985	+	0.929038i	$0.379363\pi$
$648$	−4.06146	−0.159549
$649$	−23.4152	−0.919126
$650$	1.62461	0.0637224
$651$	5.93672	0.232679
$652$	−20.3281	−0.796112
$653$	19.5931	0.766738	0.383369	−	0.923595i	$-0.374764\pi$
$653$	19.5931	0.766738	0.383369	+	0.923595i	$0.374764\pi$
$654$	15.0546	0.588683
$655$	12.8573	0.502377
$656$	−6.84841	−0.267385
$657$	58.5193	2.28306
$658$	4.88117	0.190288
$659$	6.35306	0.247480	0.123740	−	0.992315i	$-0.460511\pi$
$659$	6.35306	0.247480	0.123740	+	0.992315i	$0.460511\pi$
$660$	46.3484	1.80411
$661$	29.2653	1.13829	0.569144	−	0.822238i	$-0.307275\pi$
$661$	29.2653	1.13829	0.569144	+	0.822238i	$0.307275\pi$
$662$	−18.5722	−0.721828
$663$	17.4832	0.678991
$664$	−13.9474	−0.541265
$665$	0.562437	0.0218104
$666$	22.5719	0.874643
$667$	−12.0742	−0.467517
$668$	21.1732	0.819215
$669$	−59.7042	−2.30830
$670$	−1.13566	−0.0438743
$671$	−3.63667	−0.140392
$672$	1.86898	0.0720975
$673$	−0.574022	−0.0221269	−0.0110635	−	0.999939i	$-0.503522\pi$
$673$	−0.574022	−0.0221269	−0.0110635	+	0.999939i	$0.503522\pi$
$674$	−13.6674	−0.526448
$675$	−7.66898	−0.295179
$676$	−10.8017	−0.415451
$677$	30.6518	1.17805	0.589023	−	0.808116i	$-0.299513\pi$
$677$	30.6518	1.17805	0.589023	+	0.808116i	$0.299513\pi$
$678$	22.5403	0.865654
$679$	0.854640	0.0327981
$680$	−10.0373	−0.384914
$681$	32.2356	1.23527
$682$	−20.5584	−0.787221
$683$	−3.43802	−0.131552	−0.0657761	−	0.997834i	$-0.520952\pi$
$683$	−3.43802	−0.131552	−0.0657761	+	0.997834i	$0.520952\pi$
$684$	1.91367	0.0731712
$685$	5.10604	0.195092
$686$	8.75352	0.334211
$687$	56.7251	2.16420
$688$	−2.65712	−0.101302
$689$	−1.35870	−0.0517625
$690$	−39.4011	−1.49997
$691$	16.8218	0.639932	0.319966	−	0.947429i	$-0.396329\pi$
$691$	16.8218	0.639932	0.319966	+	0.947429i	$0.396329\pi$
$692$	7.80435	0.296677
$693$	−22.5742	−0.857524
$694$	−12.2762	−0.465998
$695$	43.1857	1.63813
$696$	6.36525	0.241274
$697$	−27.8416	−1.05458
$698$	15.2141	0.575864
$699$	−16.4692	−0.622921
$700$	0.706053	0.0266863
$701$	−3.13382	−0.118363	−0.0591814	−	0.998247i	$-0.518849\pi$
$701$	−3.13382	−0.118363	−0.0591814	+	0.998247i	$0.518849\pi$
$702$	−10.3770	−0.391654
$703$	−1.47422	−0.0556013
$704$	−6.47212	−0.243927
$705$	54.2479	2.04309
$706$	−4.79807	−0.180578
$707$	−1.32633	−0.0498818
$708$	−10.4936	−0.394375
$709$	34.8326	1.30816	0.654082	−	0.756423i	$-0.273055\pi$
$709$	34.8326	1.30816	0.654082	+	0.756423i	$0.273055\pi$
$710$	−15.8940	−0.596490
$711$	14.8994	0.558771
$712$	4.04771	0.151694
$713$	17.4768	0.654512
$714$	7.59817	0.284354
$715$	23.6920	0.886029
$716$	−15.2448	−0.569725
$717$	47.0022	1.75533
$718$	−12.3948	−0.462568
$719$	−1.34992	−0.0503437	−0.0251718	−	0.999683i	$-0.508013\pi$
$719$	−1.34992	−0.0503437	−0.0251718	+	0.999683i	$0.508013\pi$
$720$	13.3644	0.498063
$721$	−10.6232	−0.395629
$722$	18.8750	0.702455
$723$	31.7709	1.18157
$724$	−4.67729	−0.173830
$725$	2.40463	0.0893056
$726$	89.5920	3.32507
$727$	6.11365	0.226743	0.113371	−	0.993553i	$-0.463835\pi$
$727$	6.11365	0.226743	0.113371	+	0.993553i	$0.463835\pi$
$728$	0.955368	0.0354083
$729$	−38.9166	−1.44136
$730$	−26.6916	−0.987902
$731$	−10.8023	−0.399537
$732$	−1.62979	−0.0602388
$733$	1.87582	0.0692851	0.0346426	−	0.999400i	$-0.488971\pi$
$733$	1.87582	0.0692851	0.0346426	+	0.999400i	$0.488971\pi$
$734$	8.23152	0.303831
$735$	47.1553	1.73935
$736$	5.50199	0.202806
$737$	−2.97702	−0.109660
$738$	37.0703	1.36458
$739$	0.374594	0.0137797	0.00688983	−	0.999976i	$-0.497807\pi$
$739$	0.374594	0.0137797	0.00688983	+	0.999976i	$0.497807\pi$
$740$	−10.2954	−0.378467
$741$	1.52036	0.0558520
$742$	−0.590491	−0.0216776
$743$	44.9826	1.65025	0.825125	−	0.564950i	$-0.191104\pi$
$743$	44.9826	1.65025	0.825125	+	0.564950i	$0.191104\pi$
$744$	−9.21335	−0.337778
$745$	10.9120	0.399783
$746$	−11.2867	−0.413234
$747$	75.4972	2.76230
$748$	−26.3118	−0.962055
$749$	2.77413	0.101364
$750$	−27.9593	−1.02093
$751$	33.6309	1.22721	0.613605	−	0.789613i	$-0.289719\pi$
$751$	33.6309	1.22721	0.613605	+	0.789613i	$0.289719\pi$
$752$	−7.57521	−0.276239
$753$	−83.9345	−3.05874
$754$	3.25373	0.118494
$755$	37.2797	1.35675
$756$	−4.50982	−0.164021
$757$	11.7992	0.428850	0.214425	−	0.976740i	$-0.431212\pi$
$757$	11.7992	0.428850	0.214425	+	0.976740i	$0.431212\pi$
$758$	21.2915	0.773342
$759$	−103.286	−3.74904
$760$	−0.872860	−0.0316619
$761$	39.9363	1.44769	0.723845	−	0.689962i	$-0.242373\pi$
$761$	39.9363	1.44769	0.723845	+	0.689962i	$0.242373\pi$
$762$	−51.6922	−1.87261
$763$	3.34445	0.121077
$764$	5.57172	0.201578
$765$	54.3319	1.96437
$766$	−20.1648	−0.728585
$767$	−5.36405	−0.193684
$768$	−2.90051	−0.104663
$769$	−13.1516	−0.474258	−0.237129	−	0.971478i	$-0.576206\pi$
$769$	−13.1516	−0.474258	−0.237129	+	0.971478i	$0.576206\pi$
$770$	10.2965	0.371060
$771$	9.55886	0.344254
$772$	11.1491	0.401266
$773$	12.6722	0.455787	0.227894	−	0.973686i	$-0.426816\pi$
$773$	12.6722	0.455787	0.227894	+	0.973686i	$0.426816\pi$
$774$	14.3829	0.516984
$775$	−3.48057	−0.125026
$776$	−1.32634	−0.0476127
$777$	7.79356	0.279592
$778$	8.39011	0.300800
$779$	−2.42114	−0.0867465
$780$	10.6177	0.380174
$781$	−41.6644	−1.49087
$782$	22.3679	0.799873
$783$	−15.3593	−0.548895
$784$	−6.58480	−0.235171
$785$	37.8301	1.35021
$786$	15.1047	0.538767
$787$	5.75774	0.205241	0.102621	−	0.994721i	$-0.467277\pi$
$787$	5.75774	0.205241	0.102621	+	0.994721i	$0.467277\pi$
$788$	9.78084	0.348428
$789$	38.9545	1.38682
$790$	−6.79587	−0.241786
$791$	5.00742	0.178043
$792$	35.0335	1.24486
$793$	−0.833102	−0.0295843
$794$	23.9424	0.849684
$795$	−6.56254	−0.232749
$796$	−27.0126	−0.957436
$797$	−4.74576	−0.168103	−0.0840517	−	0.996461i	$-0.526786\pi$
$797$	−4.74576	−0.168103	−0.0840517	+	0.996461i	$0.526786\pi$
$798$	0.660748	0.0233902
$799$	−30.7963	−1.08950
$800$	−1.09574	−0.0387403
$801$	−21.9102	−0.774158
$802$	28.2696	0.998236
$803$	−69.9695	−2.46917
$804$	−1.33417	−0.0470524
$805$	−8.75312	−0.308507
$806$	−4.70959	−0.165888
$807$	58.8166	2.07044
$808$	2.05837	0.0724131
$809$	−24.1049	−0.847483	−0.423741	−	0.905783i	$-0.639283\pi$
$809$	−24.1049	−0.847483	−0.423741	+	0.905783i	$0.639283\pi$
$810$	−10.0276	−0.352333
$811$	−18.4487	−0.647823	−0.323911	−	0.946087i	$-0.604998\pi$
$811$	−18.4487	−0.647823	−0.323911	+	0.946087i	$0.604998\pi$
$812$	1.41407	0.0496240
$813$	14.2447	0.499583
$814$	−26.9884	−0.945944
$815$	−50.1893	−1.75805
$816$	−11.7918	−0.412795
$817$	−0.939382	−0.0328648
$818$	−36.8963	−1.29005
$819$	−5.17140	−0.180703
$820$	−16.9084	−0.590467
$821$	−42.7814	−1.49308	−0.746540	−	0.665340i	$-0.768286\pi$
$821$	−42.7814	−1.49308	−0.746540	+	0.665340i	$0.768286\pi$
$822$	5.99854	0.209223
$823$	3.18502	0.111023	0.0555114	−	0.998458i	$-0.482321\pi$
$823$	3.18502	0.111023	0.0555114	+	0.998458i	$0.482321\pi$
$824$	16.4864	0.574331
$825$	20.5698	0.716147
$826$	−2.33121	−0.0811130
$827$	−14.6982	−0.511106	−0.255553	−	0.966795i	$-0.582257\pi$
$827$	−14.6982	−0.511106	−0.255553	+	0.966795i	$0.582257\pi$
$828$	−29.7822	−1.03500
$829$	−38.3649	−1.33247	−0.666235	−	0.745742i	$-0.732095\pi$
$829$	−38.3649	−1.33247	−0.666235	+	0.745742i	$0.732095\pi$
$830$	−34.4356	−1.19528
$831$	−90.0024	−3.12215
$832$	−1.48266	−0.0514020
$833$	−26.7699	−0.927522
$834$	50.7343	1.75678
$835$	52.2757	1.80907
$836$	−2.28811	−0.0791361
$837$	22.2317	0.768439
$838$	6.18939	0.213809
$839$	−24.1988	−0.835435	−0.417717	−	0.908577i	$-0.637170\pi$
$839$	−24.1988	−0.835435	−0.417717	+	0.908577i	$0.637170\pi$
$840$	4.61443	0.159213
$841$	−24.1841	−0.833933
$842$	−6.62811	−0.228420
$843$	−54.3185	−1.87083
$844$	−8.40055	−0.289159
$845$	−26.6690	−0.917440
$846$	41.0045	1.40976
$847$	19.9032	0.683883
$848$	0.916397	0.0314692
$849$	75.4916	2.59087
$850$	−4.45463	−0.152793
$851$	22.9430	0.786477
$852$	−18.6721	−0.639697
$853$	20.3895	0.698123	0.349061	−	0.937100i	$-0.386500\pi$
$853$	20.3895	0.698123	0.349061	+	0.937100i	$0.386500\pi$
$854$	−0.362065	−0.0123896
$855$	4.72478	0.161584
$856$	−4.30523	−0.147150
$857$	44.4938	1.51988	0.759939	−	0.649994i	$-0.225229\pi$
$857$	44.4938	1.51988	0.759939	+	0.649994i	$0.225229\pi$
$858$	27.8332	0.950209
$859$	8.32126	0.283918	0.141959	−	0.989873i	$-0.454660\pi$
$859$	8.32126	0.283918	0.141959	+	0.989873i	$0.454660\pi$
$860$	−6.56031	−0.223705
$861$	12.7995	0.436207
$862$	−21.1521	−0.720445
$863$	36.7448	1.25081	0.625403	−	0.780302i	$-0.284934\pi$
$863$	36.7448	1.25081	0.625403	+	0.780302i	$0.284934\pi$
$864$	6.99890	0.238107
$865$	19.2686	0.655152
$866$	29.7118	1.00965
$867$	1.37032	0.0465385
$868$	−2.04678	−0.0694723
$869$	−17.8147	−0.604322
$870$	15.7155	0.532806
$871$	−0.681987	−0.0231082
$872$	−5.19034	−0.175767
$873$	7.17944	0.242987
$874$	1.94514	0.0657954
$875$	−6.21129	−0.209980
$876$	−31.3572	−1.05946
$877$	30.3855	1.02605	0.513023	−	0.858375i	$-0.328526\pi$
$877$	30.3855	1.02605	0.513023	+	0.858375i	$0.328526\pi$
$878$	−36.6737	−1.23768
$879$	23.9611	0.808187
$880$	−15.9794	−0.538664
$881$	−37.7526	−1.27192	−0.635959	−	0.771723i	$-0.719395\pi$
$881$	−37.7526	−1.27192	−0.635959	+	0.771723i	$0.719395\pi$
$882$	35.6434	1.20018
$883$	2.14197	0.0720829	0.0360414	−	0.999350i	$-0.488525\pi$
$883$	2.14197	0.0720829	0.0360414	+	0.999350i	$0.488525\pi$
$884$	−6.02762	−0.202731
$885$	−25.9083	−0.870899
$886$	1.81456	0.0609615
$887$	3.62858	0.121836	0.0609178	−	0.998143i	$-0.480597\pi$
$887$	3.62858	0.121836	0.0609178	+	0.998143i	$0.480597\pi$
$888$	−12.0950	−0.405882
$889$	−11.4836	−0.385149
$890$	9.99360	0.334986
$891$	−26.2862	−0.880622
$892$	20.5840	0.689203
$893$	−2.67809	−0.0896190
$894$	12.8193	0.428742
$895$	−37.6387	−1.25812
$896$	−0.644361	−0.0215266
$897$	−23.6612	−0.790024
$898$	−11.2539	−0.375546
$899$	−6.97080	−0.232489
$900$	5.93123	0.197708
$901$	3.72553	0.124115
$902$	−44.3237	−1.47582
$903$	4.96610	0.165261
$904$	−7.77113	−0.258464
$905$	−11.5480	−0.383869
$906$	43.7959	1.45502
$907$	31.8937	1.05901	0.529507	−	0.848306i	$-0.322377\pi$
$907$	31.8937	1.05901	0.529507	+	0.848306i	$0.322377\pi$
$908$	−11.1138	−0.368823
$909$	−11.1419	−0.369554
$910$	2.35876	0.0781922
$911$	−6.57146	−0.217722	−0.108861	−	0.994057i	$-0.534720\pi$
$911$	−6.57146	−0.217722	−0.108861	+	0.994057i	$0.534720\pi$
$912$	−1.02543	−0.0339554
$913$	−90.2694	−2.98748
$914$	2.26377	0.0748790
$915$	−4.02388	−0.133025
$916$	−19.5569	−0.646178
$917$	3.35557	0.110811
$918$	28.4534	0.939102
$919$	26.3267	0.868438	0.434219	−	0.900807i	$-0.357024\pi$
$919$	26.3267	0.868438	0.434219	+	0.900807i	$0.357024\pi$
$920$	13.5842	0.447857
$921$	−24.4288	−0.804956
$922$	−2.93395	−0.0966246
$923$	−9.54466	−0.314166
$924$	12.0963	0.397938
$925$	−4.56919	−0.150234
$926$	32.4908	1.06771
$927$	−89.2407	−2.93105
$928$	−2.19452	−0.0720387
$929$	25.7519	0.844891	0.422446	−	0.906388i	$-0.361172\pi$
$929$	25.7519	0.844891	0.422446	+	0.906388i	$0.361172\pi$
$930$	−22.7473	−0.745915
$931$	−2.32795	−0.0762955
$932$	5.67802	0.185990
$933$	46.5343	1.52346
$934$	−24.8501	−0.813119
$935$	−64.9627	−2.12451
$936$	8.02561	0.262325
$937$	35.8496	1.17116	0.585578	−	0.810616i	$-0.300868\pi$
$937$	35.8496	1.17116	0.585578	+	0.810616i	$0.300868\pi$
$938$	−0.296391	−0.00967749
$939$	−4.98607	−0.162714
$940$	−18.7028	−0.610020
$941$	16.0374	0.522806	0.261403	−	0.965230i	$-0.415815\pi$
$941$	16.0374	0.522806	0.261403	+	0.965230i	$0.415815\pi$
$942$	44.4426	1.44802
$943$	37.6799	1.22703
$944$	3.61786	0.117751
$945$	−11.1345	−0.362207
$946$	−17.1972	−0.559129
$947$	52.2631	1.69832	0.849162	−	0.528132i	$-0.177108\pi$
$947$	52.2631	1.69832	0.849162	+	0.528132i	$0.177108\pi$
$948$	−7.98375	−0.259300
$949$	−16.0289	−0.520320
$950$	−0.387381	−0.0125683
$951$	−90.5047	−2.93481
$952$	−2.61959	−0.0849015
$953$	10.0320	0.324969	0.162484	−	0.986711i	$-0.448049\pi$
$953$	10.0320	0.324969	0.162484	+	0.986711i	$0.448049\pi$
$954$	−4.96044	−0.160600
$955$	13.7563	0.445145
$956$	−16.2048	−0.524100
$957$	41.1966	1.33170
$958$	−13.1100	−0.423564
$959$	1.33260	0.0430319
$960$	−7.16124	−0.231128
$961$	−20.9102	−0.674521
$962$	−6.18262	−0.199336
$963$	23.3042	0.750966
$964$	−10.9535	−0.352789
$965$	27.5267	0.886115
$966$	−10.2831	−0.330854
$967$	3.93234	0.126456	0.0632278	−	0.997999i	$-0.479861\pi$
$967$	3.93234	0.126456	0.0632278	+	0.997999i	$0.479861\pi$
$968$	−30.8883	−0.992788
$969$	−4.16880	−0.133921
$970$	−3.27467	−0.105143
$971$	36.1258	1.15933	0.579666	−	0.814854i	$-0.303183\pi$
$971$	36.1258	1.15933	0.579666	+	0.814854i	$0.303183\pi$
$972$	9.21638	0.295616
$973$	11.2708	0.361326
$974$	−34.4123	−1.10264
$975$	4.71220	0.150911
$976$	0.561897	0.0179859
$977$	13.6089	0.435387	0.217694	−	0.976017i	$-0.430147\pi$
$977$	13.6089	0.435387	0.217694	+	0.976017i	$0.430147\pi$
$978$	−58.9621	−1.88540
$979$	26.1972	0.837267
$980$	−16.2576	−0.519329
$981$	28.0952	0.897011
$982$	−26.8212	−0.855900
$983$	25.6395	0.817772	0.408886	−	0.912585i	$-0.365917\pi$
$983$	25.6395	0.817772	0.408886	+	0.912585i	$0.365917\pi$
$984$	−19.8639	−0.633238
$985$	24.1484	0.769434
$986$	−8.92164	−0.284123
$987$	14.1579	0.450651
$988$	−0.524170	−0.0166761
$989$	14.6194	0.464871
$990$	86.4961	2.74903
$991$	47.2469	1.50085	0.750423	−	0.660957i	$-0.229850\pi$
$991$	47.2469	1.50085	0.750423	+	0.660957i	$0.229850\pi$
$992$	3.17645	0.100852
$993$	−53.8689	−1.70948
$994$	−4.14809	−0.131570
$995$	−66.6929	−2.11431
$996$	−40.4547	−1.28186
$997$	−26.6182	−0.843005	−0.421503	−	0.906827i	$-0.638497\pi$
$997$	−26.6182	−0.843005	−0.421503	+	0.906827i	$0.638497\pi$
$998$	−34.4468	−1.09039
$999$	29.1851	0.923375

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.b.1.6	✓	83

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.b.1.6	✓	83	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.90051 q^{3} +1.00000 q^{4} +2.46896 q^{5} +2.90051 q^{6} +0.644361 q^{7} -1.00000 q^{8} +5.41299 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.90051 q^{3} +1.00000 q^{4} +2.46896 q^{5} +2.90051 q^{6} +0.644361 q^{7} -1.00000 q^{8} +5.41299 q^{9} -2.46896 q^{10} -6.47212 q^{11} -2.90051 q^{12} -1.48266 q^{13} -0.644361 q^{14} -7.16124 q^{15} +1.00000 q^{16} +4.06541 q^{17} -5.41299 q^{18} +0.353534 q^{19} +2.46896 q^{20} -1.86898 q^{21} +6.47212 q^{22} -5.50199 q^{23} +2.90051 q^{24} +1.09574 q^{25} +1.48266 q^{26} -6.99890 q^{27} +0.644361 q^{28} +2.19452 q^{29} +7.16124 q^{30} -3.17645 q^{31} -1.00000 q^{32} +18.7725 q^{33} -4.06541 q^{34} +1.59090 q^{35} +5.41299 q^{36} -4.16995 q^{37} -0.353534 q^{38} +4.30047 q^{39} -2.46896 q^{40} -6.84841 q^{41} +1.86898 q^{42} -2.65712 q^{43} -6.47212 q^{44} +13.3644 q^{45} +5.50199 q^{46} -7.57521 q^{47} -2.90051 q^{48} -6.58480 q^{49} -1.09574 q^{50} -11.7918 q^{51} -1.48266 q^{52} +0.916397 q^{53} +6.99890 q^{54} -15.9794 q^{55} -0.644361 q^{56} -1.02543 q^{57} -2.19452 q^{58} +3.61786 q^{59} -7.16124 q^{60} +0.561897 q^{61} +3.17645 q^{62} +3.48792 q^{63} +1.00000 q^{64} -3.66062 q^{65} -18.7725 q^{66} +0.459976 q^{67} +4.06541 q^{68} +15.9586 q^{69} -1.59090 q^{70} +6.43753 q^{71} -5.41299 q^{72} +10.8109 q^{73} +4.16995 q^{74} -3.17821 q^{75} +0.353534 q^{76} -4.17038 q^{77} -4.30047 q^{78} +2.75253 q^{79} +2.46896 q^{80} +4.06146 q^{81} +6.84841 q^{82} +13.9474 q^{83} -1.86898 q^{84} +10.0373 q^{85} +2.65712 q^{86} -6.36525 q^{87} +6.47212 q^{88} -4.04771 q^{89} -13.3644 q^{90} -0.955368 q^{91} -5.50199 q^{92} +9.21335 q^{93} +7.57521 q^{94} +0.872860 q^{95} +2.90051 q^{96} +1.32634 q^{97} +6.58480 q^{98} -35.0335 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * q^99

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