LMFDB - Embedded newform 8013.2.a.c.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8013,2,Mod(1,8013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8013, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8013.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8013 = 3 \cdot 2671 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8013.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.9841271397$
Analytic rank:	$1$
Dimension:	$116$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.39685 q^{2} -1.00000 q^{3} +3.74487 q^{4} +3.43258 q^{5} +2.39685 q^{6} -5.25948 q^{7} -4.18219 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.39685 q^{2} -1.00000 q^{3} +3.74487 q^{4} +3.43258 q^{5} +2.39685 q^{6} -5.25948 q^{7} -4.18219 q^{8} +1.00000 q^{9} -8.22737 q^{10} +0.722035 q^{11} -3.74487 q^{12} -4.39764 q^{13} +12.6062 q^{14} -3.43258 q^{15} +2.53432 q^{16} -0.834425 q^{17} -2.39685 q^{18} -2.62604 q^{19} +12.8546 q^{20} +5.25948 q^{21} -1.73061 q^{22} +4.35570 q^{23} +4.18219 q^{24} +6.78262 q^{25} +10.5405 q^{26} -1.00000 q^{27} -19.6961 q^{28} +3.58965 q^{29} +8.22737 q^{30} -3.98553 q^{31} +2.29000 q^{32} -0.722035 q^{33} +1.99999 q^{34} -18.0536 q^{35} +3.74487 q^{36} -5.59637 q^{37} +6.29421 q^{38} +4.39764 q^{39} -14.3557 q^{40} -2.79323 q^{41} -12.6062 q^{42} -7.73366 q^{43} +2.70393 q^{44} +3.43258 q^{45} -10.4400 q^{46} +1.64671 q^{47} -2.53432 q^{48} +20.6622 q^{49} -16.2569 q^{50} +0.834425 q^{51} -16.4686 q^{52} +12.6145 q^{53} +2.39685 q^{54} +2.47844 q^{55} +21.9961 q^{56} +2.62604 q^{57} -8.60384 q^{58} +14.5785 q^{59} -12.8546 q^{60} -1.74393 q^{61} +9.55270 q^{62} -5.25948 q^{63} -10.5574 q^{64} -15.0953 q^{65} +1.73061 q^{66} +11.8024 q^{67} -3.12481 q^{68} -4.35570 q^{69} +43.2717 q^{70} +5.12181 q^{71} -4.18219 q^{72} +15.0335 q^{73} +13.4136 q^{74} -6.78262 q^{75} -9.83418 q^{76} -3.79753 q^{77} -10.5405 q^{78} -11.2521 q^{79} +8.69926 q^{80} +1.00000 q^{81} +6.69493 q^{82} -11.0963 q^{83} +19.6961 q^{84} -2.86423 q^{85} +18.5364 q^{86} -3.58965 q^{87} -3.01969 q^{88} +17.5609 q^{89} -8.22737 q^{90} +23.1293 q^{91} +16.3116 q^{92} +3.98553 q^{93} -3.94691 q^{94} -9.01409 q^{95} -2.29000 q^{96} -16.3757 q^{97} -49.5240 q^{98} +0.722035 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 116 q - 16 q^{2} - 116 q^{3} + 116 q^{4} - 20 q^{5} + 16 q^{6} - 33 q^{7} - 45 q^{8} + 116 q^{9}+O(q^{10}) $ 116 * q - 16 * q^2 - 116 * q^3 + 116 * q^4 - 20 * q^5 + 16 * q^6 - 33 * q^7 - 45 * q^8 + 116 * q^9	\( 116 q - 16 q^{2} - 116 q^{3} + 116 q^{4} - 20 q^{5} + 16 q^{6} - 33 q^{7} - 45 q^{8} + 116 q^{9} + 3 q^{10} - 57 q^{11} - 116 q^{12} + 6 q^{13} - 9 q^{14} + 20 q^{15} + 112 q^{16} - 30 q^{17} - 16 q^{18} + 3 q^{19} - 54 q^{20} + 33 q^{21} - 22 q^{22} - 58 q^{23} + 45 q^{24} + 126 q^{25} - 21 q^{26} - 116 q^{27} - 77 q^{28} - 38 q^{29} - 3 q^{30} + 17 q^{31} - 106 q^{32} + 57 q^{33} + 35 q^{34} - 72 q^{35} + 116 q^{36} - 41 q^{37} - 45 q^{38} - 6 q^{39} + 5 q^{40} - 39 q^{41} + 9 q^{42} - 118 q^{43} - 103 q^{44} - 20 q^{45} - 8 q^{46} - 65 q^{47} - 112 q^{48} + 165 q^{49} - 72 q^{50} + 30 q^{51} - 10 q^{52} - 58 q^{53} + 16 q^{54} + 14 q^{55} - 23 q^{56} - 3 q^{57} - 27 q^{58} - 75 q^{59} + 54 q^{60} + 45 q^{61} - 73 q^{62} - 33 q^{63} + 111 q^{64} - 86 q^{65} + 22 q^{66} - 127 q^{67} - 94 q^{68} + 58 q^{69} - 7 q^{70} - 61 q^{71} - 45 q^{72} + 15 q^{73} - 51 q^{74} - 126 q^{75} + 96 q^{76} - 57 q^{77} + 21 q^{78} + 7 q^{79} - 144 q^{80} + 116 q^{81} - 37 q^{82} - 194 q^{83} + 77 q^{84} + 3 q^{85} - 57 q^{86} + 38 q^{87} - 42 q^{88} - 56 q^{89} + 3 q^{90} - 39 q^{91} - 138 q^{92} - 17 q^{93} + 51 q^{94} - 127 q^{95} + 106 q^{96} + 57 q^{97} - 105 q^{98} - 57 q^{99}+O(q^{100}) \) 116 * q - 16 * q^2 - 116 * q^3 + 116 * q^4 - 20 * q^5 + 16 * q^6 - 33 * q^7 - 45 * q^8 + 116 * q^9 + 3 * q^10 - 57 * q^11 - 116 * q^12 + 6 * q^13 - 9 * q^14 + 20 * q^15 + 112 * q^16 - 30 * q^17 - 16 * q^18 + 3 * q^19 - 54 * q^20 + 33 * q^21 - 22 * q^22 - 58 * q^23 + 45 * q^24 + 126 * q^25 - 21 * q^26 - 116 * q^27 - 77 * q^28 - 38 * q^29 - 3 * q^30 + 17 * q^31 - 106 * q^32 + 57 * q^33 + 35 * q^34 - 72 * q^35 + 116 * q^36 - 41 * q^37 - 45 * q^38 - 6 * q^39 + 5 * q^40 - 39 * q^41 + 9 * q^42 - 118 * q^43 - 103 * q^44 - 20 * q^45 - 8 * q^46 - 65 * q^47 - 112 * q^48 + 165 * q^49 - 72 * q^50 + 30 * q^51 - 10 * q^52 - 58 * q^53 + 16 * q^54 + 14 * q^55 - 23 * q^56 - 3 * q^57 - 27 * q^58 - 75 * q^59 + 54 * q^60 + 45 * q^61 - 73 * q^62 - 33 * q^63 + 111 * q^64 - 86 * q^65 + 22 * q^66 - 127 * q^67 - 94 * q^68 + 58 * q^69 - 7 * q^70 - 61 * q^71 - 45 * q^72 + 15 * q^73 - 51 * q^74 - 126 * q^75 + 96 * q^76 - 57 * q^77 + 21 * q^78 + 7 * q^79 - 144 * q^80 + 116 * q^81 - 37 * q^82 - 194 * q^83 + 77 * q^84 + 3 * q^85 - 57 * q^86 + 38 * q^87 - 42 * q^88 - 56 * q^89 + 3 * q^90 - 39 * q^91 - 138 * q^92 - 17 * q^93 + 51 * q^94 - 127 * q^95 + 106 * q^96 + 57 * q^97 - 105 * q^98 - 57 * 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 $

$2$	−2.39685	−1.69483	−0.847413	−	0.530934i	$-0.821841\pi$
$2$	−2.39685	−1.69483	−0.847413	+	0.530934i	$0.821841\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.74487	1.87244
$5$	3.43258	1.53510	0.767549	−	0.640991i	$-0.221476\pi$
$5$	3.43258	1.53510	0.767549	+	0.640991i	$0.221476\pi$
$6$	2.39685	0.978508
$7$	−5.25948	−1.98790	−0.993949	−	0.109844i	$-0.964965\pi$
$7$	−5.25948	−1.98790	−0.993949	+	0.109844i	$0.964965\pi$
$8$	−4.18219	−1.47863
$9$	1.00000	0.333333
$10$	−8.22737	−2.60172
$11$	0.722035	0.217702	0.108851	−	0.994058i	$-0.465283\pi$
$11$	0.722035	0.217702	0.108851	+	0.994058i	$0.465283\pi$
$12$	−3.74487	−1.08105
$13$	−4.39764	−1.21969	−0.609843	−	0.792522i	$-0.708768\pi$
$13$	−4.39764	−1.21969	−0.609843	+	0.792522i	$0.708768\pi$
$14$	12.6062	3.36914
$15$	−3.43258	−0.886289
$16$	2.53432	0.633580
$17$	−0.834425	−0.202378	−0.101189	−	0.994867i	$-0.532265\pi$
$17$	−0.834425	−0.202378	−0.101189	+	0.994867i	$0.532265\pi$
$18$	−2.39685	−0.564942
$19$	−2.62604	−0.602455	−0.301227	−	0.953552i	$-0.597396\pi$
$19$	−2.62604	−0.602455	−0.301227	+	0.953552i	$0.597396\pi$
$20$	12.8546	2.87437
$21$	5.25948	1.14771
$22$	−1.73061	−0.368967
$23$	4.35570	0.908227	0.454114	−	0.890944i	$-0.349956\pi$
$23$	4.35570	0.908227	0.454114	+	0.890944i	$0.349956\pi$
$24$	4.18219	0.853686
$25$	6.78262	1.35652
$26$	10.5405	2.06716
$27$	−1.00000	−0.192450
$28$	−19.6961	−3.72221
$29$	3.58965	0.666582	0.333291	−	0.942824i	$-0.391841\pi$
$29$	3.58965	0.666582	0.333291	+	0.942824i	$0.391841\pi$
$30$	8.22737	1.50211
$31$	−3.98553	−0.715822	−0.357911	−	0.933756i	$-0.616511\pi$
$31$	−3.98553	−0.715822	−0.357911	+	0.933756i	$0.616511\pi$
$32$	2.29000	0.404819
$33$	−0.722035	−0.125690
$34$	1.99999	0.342995
$35$	−18.0536	−3.05162
$36$	3.74487	0.624145
$37$	−5.59637	−0.920038	−0.460019	−	0.887909i	$-0.652157\pi$
$37$	−5.59637	−0.920038	−0.460019	+	0.887909i	$0.652157\pi$
$38$	6.29421	1.02106
$39$	4.39764	0.704186
$40$	−14.3557	−2.26984
$41$	−2.79323	−0.436229	−0.218114	−	0.975923i	$-0.569991\pi$
$41$	−2.79323	−0.436229	−0.218114	+	0.975923i	$0.569991\pi$
$42$	−12.6062	−1.94517
$43$	−7.73366	−1.17937	−0.589686	−	0.807633i	$-0.700748\pi$
$43$	−7.73366	−1.17937	−0.589686	+	0.807633i	$0.700748\pi$
$44$	2.70393	0.407633
$45$	3.43258	0.511699
$46$	−10.4400	−1.53929
$47$	1.64671	0.240197	0.120099	−	0.992762i	$-0.461679\pi$
$47$	1.64671	0.240197	0.120099	+	0.992762i	$0.461679\pi$
$48$	−2.53432	−0.365797
$49$	20.6622	2.95174
$50$	−16.2569	−2.29907
$51$	0.834425	0.116843
$52$	−16.4686	−2.28378
$53$	12.6145	1.73274	0.866371	−	0.499402i	$-0.166447\pi$
$53$	12.6145	1.73274	0.866371	+	0.499402i	$0.166447\pi$
$54$	2.39685	0.326169
$55$	2.47844	0.334193
$56$	21.9961	2.93936
$57$	2.62604	0.347827
$58$	−8.60384	−1.12974
$59$	14.5785	1.89796	0.948981	−	0.315332i	$-0.102116\pi$
$59$	14.5785	1.89796	0.948981	+	0.315332i	$0.102116\pi$
$60$	−12.8546	−1.65952
$61$	−1.74393	−0.223288	−0.111644	−	0.993748i	$-0.535612\pi$
$61$	−1.74393	−0.223288	−0.111644	+	0.993748i	$0.535612\pi$
$62$	9.55270	1.21319
$63$	−5.25948	−0.662633
$64$	−10.5574	−1.31968
$65$	−15.0953	−1.87234
$66$	1.73061	0.213023
$67$	11.8024	1.44189	0.720946	−	0.692991i	$-0.243708\pi$
$67$	11.8024	1.44189	0.720946	+	0.692991i	$0.243708\pi$
$68$	−3.12481	−0.378939
$69$	−4.35570	−0.524365
$70$	43.2717	5.17196
$71$	5.12181	0.607847	0.303924	−	0.952696i	$-0.401703\pi$
$71$	5.12181	0.607847	0.303924	+	0.952696i	$0.401703\pi$
$72$	−4.18219	−0.492876
$73$	15.0335	1.75954	0.879769	−	0.475402i	$-0.157697\pi$
$73$	15.0335	1.75954	0.879769	+	0.475402i	$0.157697\pi$
$74$	13.4136	1.55930
$75$	−6.78262	−0.783189
$76$	−9.83418	−1.12806
$77$	−3.79753	−0.432769
$78$	−10.5405	−1.19347
$79$	−11.2521	−1.26596	−0.632982	−	0.774166i	$-0.718169\pi$
$79$	−11.2521	−1.26596	−0.632982	+	0.774166i	$0.718169\pi$
$80$	8.69926	0.972607
$81$	1.00000	0.111111
$82$	6.69493	0.739332
$83$	−11.0963	−1.21798	−0.608991	−	0.793177i	$-0.708426\pi$
$83$	−11.0963	−1.21798	−0.608991	+	0.793177i	$0.708426\pi$
$84$	19.6961	2.14902
$85$	−2.86423	−0.310670
$86$	18.5364	1.99883
$87$	−3.58965	−0.384851
$88$	−3.01969	−0.321900
$89$	17.5609	1.86145	0.930727	−	0.365716i	$-0.119176\pi$
$89$	17.5609	1.86145	0.930727	+	0.365716i	$0.119176\pi$
$90$	−8.22737	−0.867241
$91$	23.1293	2.42461
$92$	16.3116	1.70060
$93$	3.98553	0.413280
$94$	−3.94691	−0.407093
$95$	−9.01409	−0.924826
$96$	−2.29000	−0.233723
$97$	−16.3757	−1.66270	−0.831352	−	0.555746i	$-0.812433\pi$
$97$	−16.3757	−1.66270	−0.831352	+	0.555746i	$0.812433\pi$
$98$	−49.5240	−5.00268
$99$	0.722035	0.0725672
$100$	25.4000	2.54000
$101$	3.53832	0.352076	0.176038	−	0.984383i	$-0.443672\pi$
$101$	3.53832	0.352076	0.176038	+	0.984383i	$0.443672\pi$
$102$	−1.99999	−0.198028
$103$	7.27622	0.716947	0.358474	−	0.933540i	$-0.383297\pi$
$103$	7.27622	0.716947	0.358474	+	0.933540i	$0.383297\pi$
$104$	18.3918	1.80346
$105$	18.0536	1.76185
$106$	−30.2351	−2.93670
$107$	−2.43832	−0.235721	−0.117861	−	0.993030i	$-0.537604\pi$
$107$	−2.43832	−0.235721	−0.117861	+	0.993030i	$0.537604\pi$
$108$	−3.74487	−0.360350
$109$	13.9140	1.33272	0.666362	−	0.745629i	$-0.267851\pi$
$109$	13.9140	1.33272	0.666362	+	0.745629i	$0.267851\pi$
$110$	−5.94045	−0.566400
$111$	5.59637	0.531184
$112$	−13.3292	−1.25949
$113$	3.05004	0.286924	0.143462	−	0.989656i	$-0.454177\pi$
$113$	3.05004	0.286924	0.143462	+	0.989656i	$0.454177\pi$
$114$	−6.29421	−0.589507
$115$	14.9513	1.39422
$116$	13.4428	1.24813
$117$	−4.39764	−0.406562
$118$	−34.9425	−3.21672
$119$	4.38864	0.402306
$120$	14.3557	1.31049
$121$	−10.4787	−0.952606
$122$	4.17994	0.378434
$123$	2.79323	0.251857
$124$	−14.9253	−1.34033
$125$	6.11898	0.547299
$126$	12.6062	1.12305
$127$	−10.5922	−0.939910	−0.469955	−	0.882691i	$-0.655730\pi$
$127$	−10.5922	−0.939910	−0.469955	+	0.882691i	$0.655730\pi$
$128$	20.7245	1.83181
$129$	7.73366	0.680911
$130$	36.1810	3.17329
$131$	16.6656	1.45608	0.728042	−	0.685533i	$-0.240431\pi$
$131$	16.6656	1.45608	0.728042	+	0.685533i	$0.240431\pi$
$132$	−2.70393	−0.235347
$133$	13.8116	1.19762
$134$	−28.2885	−2.44376
$135$	−3.43258	−0.295430
$136$	3.48972	0.299241
$137$	−8.94228	−0.763990	−0.381995	−	0.924164i	$-0.624763\pi$
$137$	−8.94228	−0.763990	−0.381995	+	0.924164i	$0.624763\pi$
$138$	10.4400	0.888708
$139$	0.147857	0.0125410	0.00627051	−	0.999980i	$-0.498004\pi$
$139$	0.147857	0.0125410	0.00627051	+	0.999980i	$0.498004\pi$
$140$	−67.6084	−5.71396
$141$	−1.64671	−0.138678
$142$	−12.2762	−1.03020
$143$	−3.17525	−0.265528
$144$	2.53432	0.211193
$145$	12.3218	1.02327
$146$	−36.0330	−2.98211
$147$	−20.6622	−1.70419
$148$	−20.9577	−1.72271
$149$	7.19313	0.589284	0.294642	−	0.955608i	$-0.404800\pi$
$149$	7.19313	0.589284	0.294642	+	0.955608i	$0.404800\pi$
$150$	16.2569	1.32737
$151$	−10.3645	−0.843448	−0.421724	−	0.906724i	$-0.638575\pi$
$151$	−10.3645	−0.843448	−0.421724	+	0.906724i	$0.638575\pi$
$152$	10.9826	0.890805
$153$	−0.834425	−0.0674593
$154$	9.10210	0.733468
$155$	−13.6807	−1.09886
$156$	16.4686	1.31854
$157$	−0.599296	−0.0478291	−0.0239145	−	0.999714i	$-0.507613\pi$
$157$	−0.599296	−0.0478291	−0.0239145	+	0.999714i	$0.507613\pi$
$158$	26.9696	2.14559
$159$	−12.6145	−1.00040
$160$	7.86062	0.621437
$161$	−22.9088	−1.80546
$162$	−2.39685	−0.188314
$163$	0.660446	0.0517301	0.0258651	−	0.999665i	$-0.491766\pi$
$163$	0.660446	0.0517301	0.0258651	+	0.999665i	$0.491766\pi$
$164$	−10.4603	−0.816810
$165$	−2.47844	−0.192947
$166$	26.5962	2.06427
$167$	−5.76566	−0.446160	−0.223080	−	0.974800i	$-0.571611\pi$
$167$	−5.76566	−0.446160	−0.223080	+	0.974800i	$0.571611\pi$
$168$	−21.9961	−1.69704
$169$	6.33925	0.487634
$170$	6.86512	0.526531
$171$	−2.62604	−0.200818
$172$	−28.9616	−2.20830
$173$	−11.9983	−0.912212	−0.456106	−	0.889925i	$-0.650756\pi$
$173$	−11.9983	−0.912212	−0.456106	+	0.889925i	$0.650756\pi$
$174$	8.60384	0.652256
$175$	−35.6731	−2.69663
$176$	1.82987	0.137931
$177$	−14.5785	−1.09579
$178$	−42.0908	−3.15484
$179$	−22.2694	−1.66450	−0.832248	−	0.554404i	$-0.812946\pi$
$179$	−22.2694	−1.66450	−0.832248	+	0.554404i	$0.812946\pi$
$180$	12.8546	0.958124
$181$	−10.7357	−0.797980	−0.398990	−	0.916955i	$-0.630639\pi$
$181$	−10.7357	−0.797980	−0.398990	+	0.916955i	$0.630639\pi$
$182$	−55.4374	−4.10929
$183$	1.74393	0.128915
$184$	−18.2164	−1.34293
$185$	−19.2100	−1.41235
$186$	−9.55270	−0.700438
$187$	−0.602484	−0.0440580
$188$	6.16672	0.449754
$189$	5.25948	0.382571
$190$	21.6054	1.56742
$191$	−7.92988	−0.573786	−0.286893	−	0.957963i	$-0.592622\pi$
$191$	−7.92988	−0.573786	−0.286893	+	0.957963i	$0.592622\pi$
$192$	10.5574	0.761916
$193$	−3.80388	−0.273809	−0.136905	−	0.990584i	$-0.543715\pi$
$193$	−3.80388	−0.273809	−0.136905	+	0.990584i	$0.543715\pi$
$194$	39.2501	2.81799
$195$	15.0953	1.08099
$196$	77.3771	5.52694
$197$	−26.0327	−1.85476	−0.927378	−	0.374126i	$-0.877943\pi$
$197$	−26.0327	−1.85476	−0.927378	+	0.374126i	$0.877943\pi$
$198$	−1.73061	−0.122989
$199$	3.24172	0.229800	0.114900	−	0.993377i	$-0.463345\pi$
$199$	3.24172	0.229800	0.114900	+	0.993377i	$0.463345\pi$
$200$	−28.3662	−2.00579
$201$	−11.8024	−0.832477
$202$	−8.48081	−0.596708
$203$	−18.8797	−1.32510
$204$	3.12481	0.218781
$205$	−9.58798	−0.669653
$206$	−17.4400	−1.21510
$207$	4.35570	0.302742
$208$	−11.1450	−0.772769
$209$	−1.89609	−0.131155
$210$	−43.2717	−2.98603
$211$	11.4163	0.785929	0.392965	−	0.919554i	$-0.371449\pi$
$211$	11.4163	0.785929	0.392965	+	0.919554i	$0.371449\pi$
$212$	47.2399	3.24445
$213$	−5.12181	−0.350941
$214$	5.84427	0.399506
$215$	−26.5464	−1.81045
$216$	4.18219	0.284562
$217$	20.9618	1.42298
$218$	−33.3498	−2.25873
$219$	−15.0335	−1.01587
$220$	9.28145	0.625756
$221$	3.66950	0.246837
$222$	−13.4136	−0.900265
$223$	−7.02006	−0.470098	−0.235049	−	0.971983i	$-0.575525\pi$
$223$	−7.02006	−0.470098	−0.235049	+	0.971983i	$0.575525\pi$
$224$	−12.0442	−0.804739
$225$	6.78262	0.452175
$226$	−7.31048	−0.486286
$227$	−17.5295	−1.16347	−0.581737	−	0.813377i	$-0.697627\pi$
$227$	−17.5295	−1.16347	−0.581737	+	0.813377i	$0.697627\pi$
$228$	9.83418	0.651284
$229$	28.2329	1.86568	0.932840	−	0.360291i	$-0.117322\pi$
$229$	28.2329	1.86568	0.932840	+	0.360291i	$0.117322\pi$
$230$	−35.8360	−2.36296
$231$	3.79753	0.249859
$232$	−15.0126	−0.985625
$233$	22.5210	1.47540	0.737699	−	0.675130i	$-0.235912\pi$
$233$	22.5210	1.47540	0.737699	+	0.675130i	$0.235912\pi$
$234$	10.5405	0.689052
$235$	5.65247	0.368726
$236$	54.5947	3.55381
$237$	11.2521	0.730905
$238$	−10.5189	−0.681839
$239$	−9.36051	−0.605481	−0.302741	−	0.953073i	$-0.597902\pi$
$239$	−9.36051	−0.605481	−0.302741	+	0.953073i	$0.597902\pi$
$240$	−8.69926	−0.561535
$241$	−0.453290	−0.0291990	−0.0145995	−	0.999893i	$-0.504647\pi$
$241$	−0.453290	−0.0291990	−0.0145995	+	0.999893i	$0.504647\pi$
$242$	25.1157	1.61450
$243$	−1.00000	−0.0641500
$244$	−6.53080	−0.418092
$245$	70.9246	4.53120
$246$	−6.69493	−0.426853
$247$	11.5484	0.734805
$248$	16.6682	1.05843
$249$	11.0963	0.703203
$250$	−14.6663	−0.927576
$251$	−30.5919	−1.93094	−0.965472	−	0.260508i	$-0.916110\pi$
$251$	−30.5919	−1.93094	−0.965472	+	0.260508i	$0.916110\pi$
$252$	−19.6961	−1.24074
$253$	3.14497	0.197723
$254$	25.3880	1.59298
$255$	2.86423	0.179365
$256$	−28.5586	−1.78491
$257$	−22.8660	−1.42634	−0.713171	−	0.700990i	$-0.752742\pi$
$257$	−22.8660	−1.42634	−0.713171	+	0.700990i	$0.752742\pi$
$258$	−18.5364	−1.15403
$259$	29.4340	1.82894
$260$	−56.5298	−3.50583
$261$	3.58965	0.222194
$262$	−39.9450	−2.46781
$263$	10.3009	0.635183	0.317592	−	0.948228i	$-0.397126\pi$
$263$	10.3009	0.635183	0.317592	+	0.948228i	$0.397126\pi$
$264$	3.01969	0.185849
$265$	43.3005	2.65993
$266$	−33.1043	−2.02975
$267$	−17.5609	−1.07471
$268$	44.1984	2.69985
$269$	25.9750	1.58372	0.791862	−	0.610700i	$-0.209112\pi$
$269$	25.9750	1.58372	0.791862	+	0.610700i	$0.209112\pi$
$270$	8.22737	0.500702
$271$	8.73429	0.530570	0.265285	−	0.964170i	$-0.414534\pi$
$271$	8.73429	0.530570	0.265285	+	0.964170i	$0.414534\pi$
$272$	−2.11470	−0.128222
$273$	−23.1293	−1.39985
$274$	21.4333	1.29483
$275$	4.89729	0.295318
$276$	−16.3116	−0.981840
$277$	−13.5930	−0.816726	−0.408363	−	0.912820i	$-0.633900\pi$
$277$	−13.5930	−0.816726	−0.408363	+	0.912820i	$0.633900\pi$
$278$	−0.354389	−0.0212549
$279$	−3.98553	−0.238607
$280$	75.5036	4.51220
$281$	−13.1286	−0.783188	−0.391594	−	0.920138i	$-0.628076\pi$
$281$	−13.1286	−0.783188	−0.391594	+	0.920138i	$0.628076\pi$
$282$	3.94691	0.235035
$283$	−17.1819	−1.02136	−0.510679	−	0.859771i	$-0.670606\pi$
$283$	−17.1819	−1.02136	−0.510679	+	0.859771i	$0.670606\pi$
$284$	19.1805	1.13815
$285$	9.01409	0.533949
$286$	7.61059	0.450023
$287$	14.6909	0.867178
$288$	2.29000	0.134940
$289$	−16.3037	−0.959043
$290$	−29.5334	−1.73426
$291$	16.3757	0.959962
$292$	56.2985	3.29462
$293$	−5.63489	−0.329194	−0.164597	−	0.986361i	$-0.552632\pi$
$293$	−5.63489	−0.329194	−0.164597	+	0.986361i	$0.552632\pi$
$294$	49.5240	2.88830
$295$	50.0420	2.91356
$296$	23.4051	1.36039
$297$	−0.722035	−0.0418967
$298$	−17.2408	−0.998734
$299$	−19.1548	−1.10775
$300$	−25.4000	−1.46647
$301$	40.6751	2.34447
$302$	24.8420	1.42950
$303$	−3.53832	−0.203271
$304$	−6.65522	−0.381703
$305$	−5.98619	−0.342768
$306$	1.99999	0.114332
$307$	−29.1843	−1.66564	−0.832818	−	0.553547i	$-0.813274\pi$
$307$	−29.1843	−1.66564	−0.832818	+	0.553547i	$0.813274\pi$
$308$	−14.2213	−0.810332
$309$	−7.27622	−0.413930
$310$	32.7904	1.86237
$311$	−3.07593	−0.174420	−0.0872099	−	0.996190i	$-0.527795\pi$
$311$	−3.07593	−0.174420	−0.0872099	+	0.996190i	$0.527795\pi$
$312$	−18.3918	−1.04123
$313$	−2.70499	−0.152895	−0.0764475	−	0.997074i	$-0.524358\pi$
$313$	−2.70499	−0.152895	−0.0764475	+	0.997074i	$0.524358\pi$
$314$	1.43642	0.0810619
$315$	−18.0536	−1.01721
$316$	−42.1378	−2.37044
$317$	15.5168	0.871511	0.435756	−	0.900065i	$-0.356481\pi$
$317$	15.5168	0.871511	0.435756	+	0.900065i	$0.356481\pi$
$318$	30.2351	1.69550
$319$	2.59185	0.145116
$320$	−36.2392	−2.02583
$321$	2.43832	0.136094
$322$	54.9088	3.05995
$323$	2.19123	0.121923
$324$	3.74487	0.208048
$325$	−29.8275	−1.65453
$326$	−1.58299	−0.0876735
$327$	−13.9140	−0.769448
$328$	11.6818	0.645019
$329$	−8.66084	−0.477488
$330$	5.94045	0.327011
$331$	−32.8112	−1.80346	−0.901732	−	0.432295i	$-0.857704\pi$
$331$	−32.8112	−1.80346	−0.901732	+	0.432295i	$0.857704\pi$
$332$	−41.5544	−2.28059
$333$	−5.59637	−0.306679
$334$	13.8194	0.756164
$335$	40.5127	2.21344
$336$	13.3292	0.727168
$337$	30.1549	1.64264	0.821322	−	0.570464i	$-0.193237\pi$
$337$	30.1549	1.64264	0.821322	+	0.570464i	$0.193237\pi$
$338$	−15.1942	−0.826456
$339$	−3.05004	−0.165656
$340$	−10.7262	−0.581709
$341$	−2.87769	−0.155836
$342$	6.29421	0.340352
$343$	−71.8559	−3.87985
$344$	32.3436	1.74385
$345$	−14.9513	−0.804952
$346$	28.7580	1.54604
$347$	33.3444	1.79002	0.895010	−	0.446046i	$-0.147168\pi$
$347$	33.3444	1.79002	0.895010	+	0.446046i	$0.147168\pi$
$348$	−13.4428	−0.720609
$349$	−19.1561	−1.02540	−0.512700	−	0.858568i	$-0.671355\pi$
$349$	−19.1561	−1.02540	−0.512700	+	0.858568i	$0.671355\pi$
$350$	85.5029	4.57032
$351$	4.39764	0.234729
$352$	1.65346	0.0881299
$353$	27.4712	1.46214	0.731072	−	0.682300i	$-0.239020\pi$
$353$	27.4712	1.46214	0.731072	+	0.682300i	$0.239020\pi$
$354$	34.9425	1.85717
$355$	17.5810	0.933104
$356$	65.7634	3.48545
$357$	−4.38864	−0.232272
$358$	53.3764	2.82103
$359$	25.7452	1.35878	0.679389	−	0.733778i	$-0.262245\pi$
$359$	25.7452	1.35878	0.679389	+	0.733778i	$0.262245\pi$
$360$	−14.3557	−0.756612
$361$	−12.1039	−0.637049
$362$	25.7319	1.35244
$363$	10.4787	0.549987
$364$	86.6163	4.53993
$365$	51.6037	2.70106
$366$	−4.17994	−0.218489
$367$	−26.1417	−1.36459	−0.682293	−	0.731079i	$-0.739017\pi$
$367$	−26.1417	−1.36459	−0.682293	+	0.731079i	$0.739017\pi$
$368$	11.0387	0.575434
$369$	−2.79323	−0.145410
$370$	46.0434	2.39368
$371$	−66.3460	−3.44451
$372$	14.9253	0.773840
$373$	19.1258	0.990299	0.495149	−	0.868808i	$-0.335113\pi$
$373$	19.1258	0.990299	0.495149	+	0.868808i	$0.335113\pi$
$374$	1.44406	0.0746707
$375$	−6.11898	−0.315983
$376$	−6.88685	−0.355162
$377$	−15.7860	−0.813020
$378$	−12.6062	−0.648392
$379$	5.84152	0.300059	0.150029	−	0.988682i	$-0.452063\pi$
$379$	5.84152	0.300059	0.150029	+	0.988682i	$0.452063\pi$
$380$	−33.7566	−1.73168
$381$	10.5922	0.542657
$382$	19.0067	0.972468
$383$	−2.42883	−0.124107	−0.0620536	−	0.998073i	$-0.519765\pi$
$383$	−2.42883	−0.124107	−0.0620536	+	0.998073i	$0.519765\pi$
$384$	−20.7245	−1.05759
$385$	−13.0353	−0.664342
$386$	9.11732	0.464059
$387$	−7.73366	−0.393124
$388$	−61.3250	−3.11331
$389$	−15.9998	−0.811224	−0.405612	−	0.914045i	$-0.632942\pi$
$389$	−15.9998	−0.811224	−0.405612	+	0.914045i	$0.632942\pi$
$390$	−36.1810	−1.83210
$391$	−3.63451	−0.183805
$392$	−86.4131	−4.36452
$393$	−16.6656	−0.840670
$394$	62.3965	3.14349
$395$	−38.6239	−1.94338
$396$	2.70393	0.135878
$397$	−14.7455	−0.740053	−0.370027	−	0.929021i	$-0.620651\pi$
$397$	−14.7455	−0.740053	−0.370027	+	0.929021i	$0.620651\pi$
$398$	−7.76991	−0.389470
$399$	−13.8116	−0.691445
$400$	17.1893	0.859466
$401$	−7.78496	−0.388762	−0.194381	−	0.980926i	$-0.562270\pi$
$401$	−7.78496	−0.388762	−0.194381	+	0.980926i	$0.562270\pi$
$402$	28.2885	1.41090
$403$	17.5269	0.873078
$404$	13.2506	0.659240
$405$	3.43258	0.170566
$406$	45.2518	2.24581
$407$	−4.04078	−0.200294
$408$	−3.48972	−0.172767
$409$	15.9738	0.789853	0.394926	−	0.918713i	$-0.370770\pi$
$409$	15.9738	0.789853	0.394926	+	0.918713i	$0.370770\pi$
$410$	22.9809	1.13495
$411$	8.94228	0.441090
$412$	27.2485	1.34244
$413$	−76.6755	−3.77296
$414$	−10.4400	−0.513096
$415$	−38.0891	−1.86972
$416$	−10.0706	−0.493752
$417$	−0.147857	−0.00724057
$418$	4.54464	0.222286
$419$	−21.8572	−1.06780	−0.533898	−	0.845549i	$-0.679273\pi$
$419$	−21.8572	−1.06780	−0.533898	+	0.845549i	$0.679273\pi$
$420$	67.6084	3.29895
$421$	21.7574	1.06039	0.530195	−	0.847876i	$-0.322119\pi$
$421$	21.7574	1.06039	0.530195	+	0.847876i	$0.322119\pi$
$422$	−27.3631	−1.33201
$423$	1.64671	0.0800658
$424$	−52.7564	−2.56208
$425$	−5.65959	−0.274530
$426$	12.2762	0.594783
$427$	9.17218	0.443873
$428$	−9.13119	−0.441373
$429$	3.17525	0.153303
$430$	63.6277	3.06840
$431$	27.8251	1.34029	0.670144	−	0.742231i	$-0.266232\pi$
$431$	27.8251	1.34029	0.670144	+	0.742231i	$0.266232\pi$
$432$	−2.53432	−0.121932
$433$	21.5141	1.03390	0.516951	−	0.856015i	$-0.327067\pi$
$433$	21.5141	1.03390	0.516951	+	0.856015i	$0.327067\pi$
$434$	−50.2423	−2.41171
$435$	−12.3218	−0.590784
$436$	52.1063	2.49544
$437$	−11.4382	−0.547166
$438$	36.0330	1.72172
$439$	20.9562	1.00018	0.500092	−	0.865972i	$-0.333299\pi$
$439$	20.9562	1.00018	0.500092	+	0.865972i	$0.333299\pi$
$440$	−10.3653	−0.494147
$441$	20.6622	0.983913
$442$	−8.79523	−0.418347
$443$	−27.8914	−1.32516	−0.662579	−	0.748992i	$-0.730538\pi$
$443$	−27.8914	−1.32516	−0.662579	+	0.748992i	$0.730538\pi$
$444$	20.9577	0.994608
$445$	60.2793	2.85751
$446$	16.8260	0.796735
$447$	−7.19313	−0.340223
$448$	55.5266	2.62339
$449$	6.91577	0.326376	0.163188	−	0.986595i	$-0.447822\pi$
$449$	6.91577	0.326376	0.163188	+	0.986595i	$0.447822\pi$
$450$	−16.2569	−0.766357
$451$	−2.01681	−0.0949677
$452$	11.4220	0.537247
$453$	10.3645	0.486965
$454$	42.0155	1.97189
$455$	79.3933	3.72201
$456$	−10.9826	−0.514307
$457$	−29.7065	−1.38961	−0.694807	−	0.719197i	$-0.744510\pi$
$457$	−29.7065	−1.38961	−0.694807	+	0.719197i	$0.744510\pi$
$458$	−67.6698	−3.16200
$459$	0.834425	0.0389476
$460$	55.9907	2.61058
$461$	10.0803	0.469488	0.234744	−	0.972057i	$-0.424575\pi$
$461$	10.0803	0.469488	0.234744	+	0.972057i	$0.424575\pi$
$462$	−9.10210	−0.423468
$463$	2.54399	0.118229	0.0591146	−	0.998251i	$-0.481172\pi$
$463$	2.54399	0.118229	0.0591146	+	0.998251i	$0.481172\pi$
$464$	9.09732	0.422333
$465$	13.6807	0.634425
$466$	−53.9793	−2.50054
$467$	−13.9494	−0.645501	−0.322750	−	0.946484i	$-0.604607\pi$
$467$	−13.9494	−0.645501	−0.322750	+	0.946484i	$0.604607\pi$
$468$	−16.4686	−0.761261
$469$	−62.0745	−2.86633
$470$	−13.5481	−0.624927
$471$	0.599296	0.0276141
$472$	−60.9702	−2.80638
$473$	−5.58397	−0.256751
$474$	−26.9696	−1.23876
$475$	−17.8114	−0.817244
$476$	16.4349	0.753293
$477$	12.6145	0.577580
$478$	22.4357	1.02619
$479$	37.6555	1.72052	0.860261	−	0.509853i	$-0.170300\pi$
$479$	37.6555	1.72052	0.860261	+	0.509853i	$0.170300\pi$
$480$	−7.86062	−0.358787
$481$	24.6108	1.12216
$482$	1.08647	0.0494872
$483$	22.9088	1.04238
$484$	−39.2413	−1.78369
$485$	−56.2110	−2.55241
$486$	2.39685	0.108723
$487$	−18.4619	−0.836588	−0.418294	−	0.908312i	$-0.637372\pi$
$487$	−18.4619	−0.836588	−0.418294	+	0.908312i	$0.637372\pi$
$488$	7.29345	0.330159
$489$	−0.660446	−0.0298664
$490$	−169.995	−7.67960
$491$	14.9799	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$491$	14.9799	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$492$	10.4603	0.471585
$493$	−2.99530	−0.134901
$494$	−27.6797	−1.24537
$495$	2.47844	0.111398
$496$	−10.1006	−0.453530
$497$	−26.9381	−1.20834
$498$	−26.5962	−1.19181
$499$	−21.6681	−0.969999	−0.484999	−	0.874515i	$-0.661180\pi$
$499$	−21.6681	−0.969999	−0.484999	+	0.874515i	$0.661180\pi$
$500$	22.9148	1.02478
$501$	5.76566	0.257591
$502$	73.3241	3.27261
$503$	−0.921442	−0.0410851	−0.0205425	−	0.999789i	$-0.506539\pi$
$503$	−0.921442	−0.0410851	−0.0205425	+	0.999789i	$0.506539\pi$
$504$	21.9961	0.979786
$505$	12.1456	0.540471
$506$	−7.53801	−0.335105
$507$	−6.33925	−0.281536
$508$	−39.6666	−1.75992
$509$	−28.7965	−1.27638	−0.638191	−	0.769878i	$-0.720317\pi$
$509$	−28.7965	−1.27638	−0.638191	+	0.769878i	$0.720317\pi$
$510$	−6.86512	−0.303993
$511$	−79.0684	−3.49778
$512$	27.0016	1.19331
$513$	2.62604	0.115942
$514$	54.8063	2.41740
$515$	24.9762	1.10058
$516$	28.9616	1.27496
$517$	1.18898	0.0522914
$518$	−70.5488	−3.09974
$519$	11.9983	0.526666
$520$	63.1312	2.76849
$521$	5.59568	0.245151	0.122576	−	0.992459i	$-0.460885\pi$
$521$	5.59568	0.245151	0.122576	+	0.992459i	$0.460885\pi$
$522$	−8.60384	−0.376580
$523$	12.8422	0.561550	0.280775	−	0.959774i	$-0.409409\pi$
$523$	12.8422	0.561550	0.280775	+	0.959774i	$0.409409\pi$
$524$	62.4107	2.72642
$525$	35.6731	1.55690
$526$	−24.6898	−1.07653
$527$	3.32563	0.144867
$528$	−1.82987	−0.0796347
$529$	−4.02784	−0.175123
$530$	−103.785	−4.50811
$531$	14.5785	0.632654
$532$	51.7227	2.24246
$533$	12.2836	0.532062
$534$	42.0908	1.82145
$535$	−8.36973	−0.361855
$536$	−49.3598	−2.13202
$537$	22.2694	0.960997
$538$	−62.2581	−2.68414
$539$	14.9188	0.642598
$540$	−12.8546	−0.553173
$541$	0.607183	0.0261049	0.0130524	−	0.999915i	$-0.495845\pi$
$541$	0.607183	0.0261049	0.0130524	+	0.999915i	$0.495845\pi$
$542$	−20.9347	−0.899224
$543$	10.7357	0.460714
$544$	−1.91084	−0.0819264
$545$	47.7611	2.04586
$546$	55.4374	2.37250
$547$	−21.1875	−0.905913	−0.452957	−	0.891533i	$-0.649631\pi$
$547$	−21.1875	−0.905913	−0.452957	+	0.891533i	$0.649631\pi$
$548$	−33.4877	−1.43052
$549$	−1.74393	−0.0744292
$550$	−11.7380	−0.500512
$551$	−9.42656	−0.401585
$552$	18.2164	0.775340
$553$	59.1804	2.51661
$554$	32.5804	1.38421
$555$	19.2100	0.815419
$556$	0.553704	0.0234823
$557$	29.0490	1.23085	0.615424	−	0.788197i	$-0.288985\pi$
$557$	29.0490	1.23085	0.615424	+	0.788197i	$0.288985\pi$
$558$	9.55270	0.404398
$559$	34.0099	1.43846
$560$	−45.7536	−1.93344
$561$	0.602484	0.0254369
$562$	31.4673	1.32737
$563$	3.05751	0.128859	0.0644294	−	0.997922i	$-0.479477\pi$
$563$	3.05751	0.128859	0.0644294	+	0.997922i	$0.479477\pi$
$564$	−6.16672	−0.259666
$565$	10.4695	0.440456
$566$	41.1824	1.73103
$567$	−5.25948	−0.220878
$568$	−21.4204	−0.898779
$569$	−12.9630	−0.543437	−0.271718	−	0.962377i	$-0.587592\pi$
$569$	−12.9630	−0.543437	−0.271718	+	0.962377i	$0.587592\pi$
$570$	−21.6054	−0.904950
$571$	−1.63725	−0.0685168	−0.0342584	−	0.999413i	$-0.510907\pi$
$571$	−1.63725	−0.0685168	−0.0342584	+	0.999413i	$0.510907\pi$
$572$	−11.8909	−0.497184
$573$	7.92988	0.331276
$574$	−35.2119	−1.46972
$575$	29.5431	1.23203
$576$	−10.5574	−0.439893
$577$	5.70844	0.237645	0.118823	−	0.992915i	$-0.462088\pi$
$577$	5.70844	0.237645	0.118823	+	0.992915i	$0.462088\pi$
$578$	39.0775	1.62541
$579$	3.80388	0.158084
$580$	46.1435	1.91600
$581$	58.3611	2.42122
$582$	−39.2501	−1.62697
$583$	9.10814	0.377221
$584$	−62.8729	−2.60170
$585$	−15.0953	−0.624112
$586$	13.5060	0.557926
$587$	4.66413	0.192509	0.0962546	−	0.995357i	$-0.469314\pi$
$587$	4.66413	0.192509	0.0962546	+	0.995357i	$0.469314\pi$
$588$	−77.3771	−3.19098
$589$	10.4662	0.431250
$590$	−119.943	−4.93797
$591$	26.0327	1.07084
$592$	−14.1830	−0.582917
$593$	−5.84210	−0.239906	−0.119953	−	0.992780i	$-0.538274\pi$
$593$	−5.84210	−0.239906	−0.119953	+	0.992780i	$0.538274\pi$
$594$	1.73061	0.0710077
$595$	15.0644	0.617579
$596$	26.9373	1.10340
$597$	−3.24172	−0.132675
$598$	45.9112	1.87745
$599$	−1.47322	−0.0601942	−0.0300971	−	0.999547i	$-0.509582\pi$
$599$	−1.47322	−0.0601942	−0.0300971	+	0.999547i	$0.509582\pi$
$600$	28.3662	1.15804
$601$	23.5110	0.959033	0.479516	−	0.877533i	$-0.340812\pi$
$601$	23.5110	0.959033	0.479516	+	0.877533i	$0.340812\pi$
$602$	−97.4918	−3.97347
$603$	11.8024	0.480631
$604$	−38.8136	−1.57930
$605$	−35.9689	−1.46234
$606$	8.48081	0.344509
$607$	−43.1136	−1.74993	−0.874965	−	0.484187i	$-0.839115\pi$
$607$	−43.1136	−1.74993	−0.874965	+	0.484187i	$0.839115\pi$
$608$	−6.01364	−0.243885
$609$	18.8797	0.765045
$610$	14.3480	0.580933
$611$	−7.24164	−0.292965
$612$	−3.12481	−0.126313
$613$	11.8338	0.477961	0.238981	−	0.971024i	$-0.423187\pi$
$613$	11.8338	0.477961	0.238981	+	0.971024i	$0.423187\pi$
$614$	69.9503	2.82296
$615$	9.58798	0.386624
$616$	15.8820	0.639904
$617$	−36.1511	−1.45539	−0.727694	−	0.685902i	$-0.759408\pi$
$617$	−36.1511	−1.45539	−0.727694	+	0.685902i	$0.759408\pi$
$618$	17.4400	0.701539
$619$	−38.5392	−1.54902	−0.774510	−	0.632562i	$-0.782004\pi$
$619$	−38.5392	−1.54902	−0.774510	+	0.632562i	$0.782004\pi$
$620$	−51.2323	−2.05754
$621$	−4.35570	−0.174788
$622$	7.37252	0.295611
$623$	−92.3613	−3.70038
$624$	11.1450	0.446158
$625$	−12.9092	−0.516367
$626$	6.48344	0.259130
$627$	1.89609	0.0757226
$628$	−2.24429	−0.0895568
$629$	4.66975	0.186195
$630$	43.2717	1.72399
$631$	−14.3842	−0.572628	−0.286314	−	0.958136i	$-0.592430\pi$
$631$	−14.3842	−0.572628	−0.286314	+	0.958136i	$0.592430\pi$
$632$	47.0585	1.87189
$633$	−11.4163	−0.453756
$634$	−37.1914	−1.47706
$635$	−36.3587	−1.44285
$636$	−47.2399	−1.87318
$637$	−90.8648	−3.60019
$638$	−6.21228	−0.245946
$639$	5.12181	0.202616
$640$	71.1386	2.81200
$641$	−1.48956	−0.0588340	−0.0294170	−	0.999567i	$-0.509365\pi$
$641$	−1.48956	−0.0588340	−0.0294170	+	0.999567i	$0.509365\pi$
$642$	−5.84427	−0.230655
$643$	−12.5674	−0.495612	−0.247806	−	0.968810i	$-0.579710\pi$
$643$	−12.5674	−0.495612	−0.247806	+	0.968810i	$0.579710\pi$
$644$	−85.7903	−3.38061
$645$	26.5464	1.04526
$646$	−5.25205	−0.206639
$647$	19.3717	0.761582	0.380791	−	0.924661i	$-0.375652\pi$
$647$	19.3717	0.761582	0.380791	+	0.924661i	$0.375652\pi$
$648$	−4.18219	−0.164292
$649$	10.5262	0.413190
$650$	71.4920	2.80415
$651$	−20.9618	−0.821558
$652$	2.47328	0.0968613
$653$	−17.4102	−0.681315	−0.340657	−	0.940187i	$-0.610650\pi$
$653$	−17.4102	−0.681315	−0.340657	+	0.940187i	$0.610650\pi$
$654$	33.3498	1.30408
$655$	57.2062	2.23523
$656$	−7.07892	−0.276386
$657$	15.0335	0.586513
$658$	20.7587	0.809259
$659$	21.4945	0.837307	0.418653	−	0.908146i	$-0.362502\pi$
$659$	21.4945	0.837307	0.418653	+	0.908146i	$0.362502\pi$
$660$	−9.28145	−0.361280
$661$	−35.8662	−1.39503	−0.697517	−	0.716568i	$-0.745712\pi$
$661$	−35.8662	−1.39503	−0.697517	+	0.716568i	$0.745712\pi$
$662$	78.6433	3.05656
$663$	−3.66950	−0.142512
$664$	46.4070	1.80094
$665$	47.4095	1.83846
$666$	13.4136	0.519768
$667$	15.6355	0.605407
$668$	−21.5917	−0.835407
$669$	7.02006	0.271411
$670$	−97.1027	−3.75140
$671$	−1.25918	−0.0486101
$672$	12.0442	0.464616
$673$	−47.4759	−1.83006	−0.915031	−	0.403383i	$-0.867834\pi$
$673$	−47.4759	−1.83006	−0.915031	+	0.403383i	$0.867834\pi$
$674$	−72.2768	−2.78400
$675$	−6.78262	−0.261063
$676$	23.7397	0.913064
$677$	13.0849	0.502893	0.251447	−	0.967871i	$-0.419094\pi$
$677$	13.0849	0.502893	0.251447	+	0.967871i	$0.419094\pi$
$678$	7.31048	0.280757
$679$	86.1279	3.30529
$680$	11.9788	0.459364
$681$	17.5295	0.671732
$682$	6.89738	0.264114
$683$	38.9533	1.49051	0.745253	−	0.666781i	$-0.232328\pi$
$683$	38.9533	1.49051	0.745253	+	0.666781i	$0.232328\pi$
$684$	−9.83418	−0.376019
$685$	−30.6951	−1.17280
$686$	172.228	6.57568
$687$	−28.2329	−1.07715
$688$	−19.5996	−0.747226
$689$	−55.4743	−2.11340
$690$	35.8360	1.36425
$691$	35.1011	1.33531	0.667654	−	0.744472i	$-0.267299\pi$
$691$	35.1011	1.33531	0.667654	+	0.744472i	$0.267299\pi$
$692$	−44.9320	−1.70806
$693$	−3.79753	−0.144256
$694$	−79.9213	−3.03377
$695$	0.507530	0.0192517
$696$	15.0126	0.569051
$697$	2.33074	0.0882830
$698$	45.9142	1.73788
$699$	−22.5210	−0.851822
$700$	−133.591	−5.04927
$701$	14.8994	0.562741	0.281371	−	0.959599i	$-0.409211\pi$
$701$	14.8994	0.562741	0.281371	+	0.959599i	$0.409211\pi$
$702$	−10.5405	−0.397824
$703$	14.6963	0.554281
$704$	−7.62283	−0.287296
$705$	−5.65247	−0.212884
$706$	−65.8442	−2.47808
$707$	−18.6097	−0.699891
$708$	−54.5947	−2.05180
$709$	−37.9536	−1.42538	−0.712688	−	0.701481i	$-0.752523\pi$
$709$	−37.9536	−1.42538	−0.712688	+	0.701481i	$0.752523\pi$
$710$	−42.1390	−1.58145
$711$	−11.2521	−0.421988
$712$	−73.4431	−2.75239
$713$	−17.3598	−0.650129
$714$	10.5189	0.393660
$715$	−10.8993	−0.407611
$716$	−83.3962	−3.11666
$717$	9.36051	0.349575
$718$	−61.7072	−2.30289
$719$	−25.5585	−0.953170	−0.476585	−	0.879129i	$-0.658125\pi$
$719$	−25.5585	−0.953170	−0.476585	+	0.879129i	$0.658125\pi$
$720$	8.69926	0.324202
$721$	−38.2691	−1.42522
$722$	29.0112	1.07969
$723$	0.453290	0.0168580
$724$	−40.2039	−1.49417
$725$	24.3472	0.904234
$726$	−25.1157	−0.932133
$727$	2.29958	0.0852868	0.0426434	−	0.999090i	$-0.486422\pi$
$727$	2.29958	0.0852868	0.0426434	+	0.999090i	$0.486422\pi$
$728$	−96.7312	−3.58510
$729$	1.00000	0.0370370
$730$	−123.686	−4.57783
$731$	6.45316	0.238679
$732$	6.53080	0.241385
$733$	13.1255	0.484800	0.242400	−	0.970176i	$-0.422065\pi$
$733$	13.1255	0.484800	0.242400	+	0.970176i	$0.422065\pi$
$734$	62.6576	2.31273
$735$	−70.9246	−2.61609
$736$	9.97458	0.367668
$737$	8.52174	0.313902
$738$	6.69493	0.246444
$739$	−3.57408	−0.131475	−0.0657373	−	0.997837i	$-0.520940\pi$
$739$	−3.57408	−0.131475	−0.0657373	+	0.997837i	$0.520940\pi$
$740$	−71.9390	−2.64453
$741$	−11.5484	−0.424240
$742$	159.021	5.83785
$743$	−7.63841	−0.280226	−0.140113	−	0.990136i	$-0.544747\pi$
$743$	−7.63841	−0.280226	−0.140113	+	0.990136i	$0.544747\pi$
$744$	−16.6682	−0.611087
$745$	24.6910	0.904608
$746$	−45.8417	−1.67838
$747$	−11.0963	−0.405994
$748$	−2.25623	−0.0824958
$749$	12.8243	0.468589
$750$	14.6663	0.535536
$751$	27.9158	1.01866	0.509331	−	0.860571i	$-0.329893\pi$
$751$	27.9158	1.01866	0.509331	+	0.860571i	$0.329893\pi$
$752$	4.17329	0.152184
$753$	30.5919	1.11483
$754$	37.8366	1.37793
$755$	−35.5769	−1.29477
$756$	19.6961	0.716340
$757$	−33.2560	−1.20871	−0.604356	−	0.796715i	$-0.706569\pi$
$757$	−33.2560	−1.20871	−0.604356	+	0.796715i	$0.706569\pi$
$758$	−14.0012	−0.508548
$759$	−3.14497	−0.114155
$760$	37.6986	1.36747
$761$	20.3648	0.738223	0.369111	−	0.929385i	$-0.379662\pi$
$761$	20.3648	0.738223	0.369111	+	0.929385i	$0.379662\pi$
$762$	−25.3880	−0.919709
$763$	−73.1807	−2.64932
$764$	−29.6964	−1.07438
$765$	−2.86423	−0.103557
$766$	5.82152	0.210340
$767$	−64.1111	−2.31492
$768$	28.5586	1.03052
$769$	0.390210	0.0140713	0.00703567	−	0.999975i	$-0.497760\pi$
$769$	0.390210	0.0140713	0.00703567	+	0.999975i	$0.497760\pi$
$770$	31.2437	1.12594
$771$	22.8660	0.823499
$772$	−14.2450	−0.512691
$773$	7.89969	0.284132	0.142066	−	0.989857i	$-0.454625\pi$
$773$	7.89969	0.284132	0.142066	+	0.989857i	$0.454625\pi$
$774$	18.5364	0.666277
$775$	−27.0323	−0.971030
$776$	68.4864	2.45852
$777$	−29.4340	−1.05594
$778$	38.3492	1.37488
$779$	7.33512	0.262808
$780$	56.5298	2.02409
$781$	3.69813	0.132329
$782$	8.71136	0.311518
$783$	−3.58965	−0.128284
$784$	52.3645	1.87016
$785$	−2.05713	−0.0734223
$786$	39.9450	1.42479
$787$	−43.9445	−1.56645	−0.783226	−	0.621737i	$-0.786427\pi$
$787$	−43.9445	−1.56645	−0.783226	+	0.621737i	$0.786427\pi$
$788$	−97.4893	−3.47291
$789$	−10.3009	−0.366723
$790$	92.5755	3.29369
$791$	−16.0417	−0.570375
$792$	−3.01969	−0.107300
$793$	7.66919	0.272341
$794$	35.3426	1.25426
$795$	−43.3005	−1.53571
$796$	12.1398	0.430285
$797$	−0.712411	−0.0252349	−0.0126175	−	0.999920i	$-0.504016\pi$
$797$	−0.712411	−0.0252349	−0.0126175	+	0.999920i	$0.504016\pi$
$798$	33.1043	1.17188
$799$	−1.37406	−0.0486106
$800$	15.5322	0.549147
$801$	17.5609	0.620484
$802$	18.6594	0.658885
$803$	10.8547	0.383054
$804$	−44.1984	−1.55876
$805$	−78.6362	−2.77156
$806$	−42.0093	−1.47972
$807$	−25.9750	−0.914364
$808$	−14.7979	−0.520589
$809$	30.5503	1.07409	0.537046	−	0.843553i	$-0.319540\pi$
$809$	30.5503	1.07409	0.537046	+	0.843553i	$0.319540\pi$
$810$	−8.22737	−0.289080
$811$	−12.8844	−0.452432	−0.226216	−	0.974077i	$-0.572636\pi$
$811$	−12.8844	−0.452432	−0.226216	+	0.974077i	$0.572636\pi$
$812$	−70.7021	−2.48116
$813$	−8.73429	−0.306325
$814$	9.68512	0.339463
$815$	2.26703	0.0794107
$816$	2.11470	0.0740293
$817$	20.3089	0.710518
$818$	−38.2867	−1.33866
$819$	23.1293	0.808204
$820$	−35.9057	−1.25388
$821$	1.68378	0.0587643	0.0293822	−	0.999568i	$-0.490646\pi$
$821$	1.68378	0.0587643	0.0293822	+	0.999568i	$0.490646\pi$
$822$	−21.4333	−0.747571
$823$	−20.8509	−0.726818	−0.363409	−	0.931630i	$-0.618387\pi$
$823$	−20.8509	−0.726818	−0.363409	+	0.931630i	$0.618387\pi$
$824$	−30.4305	−1.06010
$825$	−4.89729	−0.170502
$826$	183.779	6.39450
$827$	6.63658	0.230776	0.115388	−	0.993320i	$-0.463189\pi$
$827$	6.63658	0.230776	0.115388	+	0.993320i	$0.463189\pi$
$828$	16.3116	0.566866
$829$	5.21787	0.181224	0.0906120	−	0.995886i	$-0.471118\pi$
$829$	5.21787	0.181224	0.0906120	+	0.995886i	$0.471118\pi$
$830$	91.2938	3.16885
$831$	13.5930	0.471537
$832$	46.4278	1.60959
$833$	−17.2410	−0.597366
$834$	0.354389	0.0122715
$835$	−19.7911	−0.684900
$836$	−7.10062	−0.245580
$837$	3.98553	0.137760
$838$	52.3884	1.80973
$839$	−10.0726	−0.347744	−0.173872	−	0.984768i	$-0.555628\pi$
$839$	−10.0726	−0.347744	−0.173872	+	0.984768i	$0.555628\pi$
$840$	−75.5036	−2.60512
$841$	−16.1144	−0.555669
$842$	−52.1491	−1.79718
$843$	13.1286	0.452174
$844$	42.7525	1.47160
$845$	21.7600	0.748566
$846$	−3.94691	−0.135698
$847$	55.1124	1.89368
$848$	31.9693	1.09783
$849$	17.1819	0.589682
$850$	13.5652	0.465281
$851$	−24.3761	−0.835603
$852$	−19.1805	−0.657114
$853$	−8.31036	−0.284541	−0.142271	−	0.989828i	$-0.545440\pi$
$853$	−8.31036	−0.284541	−0.142271	+	0.989828i	$0.545440\pi$
$854$	−21.9843	−0.752288
$855$	−9.01409	−0.308275
$856$	10.1975	0.348544
$857$	13.3656	0.456561	0.228280	−	0.973595i	$-0.426690\pi$
$857$	13.3656	0.456561	0.228280	+	0.973595i	$0.426690\pi$
$858$	−7.61059	−0.259821
$859$	−30.0902	−1.02666	−0.513332	−	0.858190i	$-0.671589\pi$
$859$	−30.0902	−1.02666	−0.513332	+	0.858190i	$0.671589\pi$
$860$	−99.4129	−3.38995
$861$	−14.6909	−0.500665
$862$	−66.6925	−2.27156
$863$	4.32381	0.147184	0.0735920	−	0.997288i	$-0.476554\pi$
$863$	4.32381	0.147184	0.0735920	+	0.997288i	$0.476554\pi$
$864$	−2.29000	−0.0779075
$865$	−41.1851	−1.40033
$866$	−51.5660	−1.75228
$867$	16.3037	0.553704
$868$	78.4993	2.66444
$869$	−8.12444	−0.275603
$870$	29.5334	1.00128
$871$	−51.9027	−1.75866
$872$	−58.1911	−1.97060
$873$	−16.3757	−0.554235
$874$	27.4157	0.927351
$875$	−32.1827	−1.08797
$876$	−56.2985	−1.90215
$877$	−53.5666	−1.80882	−0.904408	−	0.426670i	$-0.859687\pi$
$877$	−53.5666	−1.80882	−0.904408	+	0.426670i	$0.859687\pi$
$878$	−50.2288	−1.69514
$879$	5.63489	0.190060
$880$	6.28117	0.211738
$881$	34.1650	1.15105	0.575524	−	0.817785i	$-0.304798\pi$
$881$	34.1650	1.15105	0.575524	+	0.817785i	$0.304798\pi$
$882$	−49.5240	−1.66756
$883$	16.9463	0.570288	0.285144	−	0.958485i	$-0.407959\pi$
$883$	16.9463	0.570288	0.285144	+	0.958485i	$0.407959\pi$
$884$	13.7418	0.462187
$885$	−50.0420	−1.68214
$886$	66.8513	2.24591
$887$	34.0563	1.14350	0.571750	−	0.820428i	$-0.306265\pi$
$887$	34.0563	1.14350	0.571750	+	0.820428i	$0.306265\pi$
$888$	−23.4051	−0.785423
$889$	55.7097	1.86844
$890$	−144.480	−4.84299
$891$	0.722035	0.0241891
$892$	−26.2892	−0.880229
$893$	−4.32432	−0.144708
$894$	17.2408	0.576619
$895$	−76.4416	−2.55516
$896$	−109.000	−3.64144
$897$	19.1548	0.639561
$898$	−16.5760	−0.553150
$899$	−14.3067	−0.477154
$900$	25.4000	0.846668
$901$	−10.5259	−0.350668
$902$	4.83397	0.160954
$903$	−40.6751	−1.35358
$904$	−12.7559	−0.424253
$905$	−36.8512	−1.22498
$906$	−24.8420	−0.825321
$907$	41.2727	1.37044	0.685219	−	0.728337i	$-0.259706\pi$
$907$	41.2727	1.37044	0.685219	+	0.728337i	$0.259706\pi$
$908$	−65.6457	−2.17853
$909$	3.53832	0.117359
$910$	−190.293	−6.30817
$911$	−25.8619	−0.856844	−0.428422	−	0.903579i	$-0.640930\pi$
$911$	−25.8619	−0.856844	−0.428422	+	0.903579i	$0.640930\pi$
$912$	6.65522	0.220376
$913$	−8.01195	−0.265157
$914$	71.2020	2.35515
$915$	5.98619	0.197897
$916$	105.728	3.49337
$917$	−87.6527	−2.89455
$918$	−1.99999	−0.0660095
$919$	−56.8558	−1.87550	−0.937749	−	0.347313i	$-0.887094\pi$
$919$	−56.8558	−1.87550	−0.937749	+	0.347313i	$0.887094\pi$
$920$	−62.5292	−2.06153
$921$	29.1843	0.961655
$922$	−24.1610	−0.795700
$923$	−22.5239	−0.741383
$924$	14.2213	0.467845
$925$	−37.9581	−1.24805
$926$	−6.09755	−0.200378
$927$	7.27622	0.238982
$928$	8.22032	0.269845
$929$	−15.9586	−0.523586	−0.261793	−	0.965124i	$-0.584314\pi$
$929$	−15.9586	−0.523586	−0.261793	+	0.965124i	$0.584314\pi$
$930$	−32.7904	−1.07524
$931$	−54.2596	−1.77829
$932$	84.3382	2.76259
$933$	3.07593	0.100701
$934$	33.4345	1.09401
$935$	−2.06808	−0.0676333
$936$	18.3918	0.601154
$937$	23.0881	0.754254	0.377127	−	0.926162i	$-0.376912\pi$
$937$	23.0881	0.754254	0.377127	+	0.926162i	$0.376912\pi$
$938$	148.783	4.85794
$939$	2.70499	0.0882739
$940$	21.1678	0.690416
$941$	−46.5453	−1.51733	−0.758666	−	0.651480i	$-0.774149\pi$
$941$	−46.5453	−1.51733	−0.758666	+	0.651480i	$0.774149\pi$
$942$	−1.43642	−0.0468011
$943$	−12.1665	−0.396195
$944$	36.9466	1.20251
$945$	18.0536	0.587284
$946$	13.3839	0.435149
$947$	49.0324	1.59334	0.796669	−	0.604416i	$-0.206593\pi$
$947$	49.0324	1.59334	0.796669	+	0.604416i	$0.206593\pi$
$948$	42.1378	1.36857
$949$	−66.1119	−2.14608
$950$	42.6912	1.38509
$951$	−15.5168	−0.503167
$952$	−18.3541	−0.594861
$953$	−6.56555	−0.212679	−0.106339	−	0.994330i	$-0.533913\pi$
$953$	−6.56555	−0.212679	−0.106339	+	0.994330i	$0.533913\pi$
$954$	−30.2351	−0.978898
$955$	−27.2200	−0.880817
$956$	−35.0539	−1.13372
$957$	−2.59185	−0.0837827
$958$	−90.2544	−2.91599
$959$	47.0318	1.51873
$960$	36.2392	1.16962
$961$	−15.1156	−0.487599
$962$	−58.9884	−1.90186
$963$	−2.43832	−0.0785737
$964$	−1.69751	−0.0546732
$965$	−13.0571	−0.420324
$966$	−54.9088	−1.76666
$967$	30.2944	0.974203	0.487101	−	0.873345i	$-0.338054\pi$
$967$	30.2944	0.974203	0.487101	+	0.873345i	$0.338054\pi$
$968$	43.8238	1.40855
$969$	−2.19123	−0.0703925
$970$	134.729	4.32589
$971$	−7.49646	−0.240573	−0.120286	−	0.992739i	$-0.538381\pi$
$971$	−7.49646	−0.240573	−0.120286	+	0.992739i	$0.538381\pi$
$972$	−3.74487	−0.120117
$973$	−0.777649	−0.0249303
$974$	44.2503	1.41787
$975$	29.8275	0.955245
$976$	−4.41968	−0.141471
$977$	−28.2259	−0.903027	−0.451513	−	0.892264i	$-0.649116\pi$
$977$	−28.2259	−0.903027	−0.451513	+	0.892264i	$0.649116\pi$
$978$	1.58299	0.0506183
$979$	12.6796	0.405242
$980$	265.603	8.48439
$981$	13.9140	0.444241
$982$	−35.9046	−1.14576
$983$	−10.3800	−0.331072	−0.165536	−	0.986204i	$-0.552935\pi$
$983$	−10.3800	−0.331072	−0.165536	+	0.986204i	$0.552935\pi$
$984$	−11.6818	−0.372402
$985$	−89.3595	−2.84723
$986$	7.17926	0.228634
$987$	8.66084	0.275678
$988$	43.2472	1.37588
$989$	−33.6855	−1.07114
$990$	−5.94045	−0.188800
$991$	−5.97020	−0.189650	−0.0948248	−	0.995494i	$-0.530229\pi$
$991$	−5.97020	−0.189650	−0.0948248	+	0.995494i	$0.530229\pi$
$992$	−9.12687	−0.289779
$993$	32.8112	1.04123
$994$	64.5664	2.04792
$995$	11.1275	0.352765
$996$	41.5544	1.31670
$997$	−11.8418	−0.375033	−0.187516	−	0.982261i	$-0.560044\pi$
$997$	−11.8418	−0.375033	−0.187516	+	0.982261i	$0.560044\pi$
$998$	51.9352	1.64398
$999$	5.59637	0.177061

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8013.2.a.c.1.14	✓	116

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8013.2.a.c.1.14	✓	116	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:	\( N \)	\(=\)	\( 8013 = 3 \cdot 2671 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8013.a (trivial)

\(f(q)\)	\(=\)	\(q-2.39685 q^{2} -1.00000 q^{3} +3.74487 q^{4} +3.43258 q^{5} +2.39685 q^{6} -5.25948 q^{7} -4.18219 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.39685 q^{2} -1.00000 q^{3} +3.74487 q^{4} +3.43258 q^{5} +2.39685 q^{6} -5.25948 q^{7} -4.18219 q^{8} +1.00000 q^{9} -8.22737 q^{10} +0.722035 q^{11} -3.74487 q^{12} -4.39764 q^{13} +12.6062 q^{14} -3.43258 q^{15} +2.53432 q^{16} -0.834425 q^{17} -2.39685 q^{18} -2.62604 q^{19} +12.8546 q^{20} +5.25948 q^{21} -1.73061 q^{22} +4.35570 q^{23} +4.18219 q^{24} +6.78262 q^{25} +10.5405 q^{26} -1.00000 q^{27} -19.6961 q^{28} +3.58965 q^{29} +8.22737 q^{30} -3.98553 q^{31} +2.29000 q^{32} -0.722035 q^{33} +1.99999 q^{34} -18.0536 q^{35} +3.74487 q^{36} -5.59637 q^{37} +6.29421 q^{38} +4.39764 q^{39} -14.3557 q^{40} -2.79323 q^{41} -12.6062 q^{42} -7.73366 q^{43} +2.70393 q^{44} +3.43258 q^{45} -10.4400 q^{46} +1.64671 q^{47} -2.53432 q^{48} +20.6622 q^{49} -16.2569 q^{50} +0.834425 q^{51} -16.4686 q^{52} +12.6145 q^{53} +2.39685 q^{54} +2.47844 q^{55} +21.9961 q^{56} +2.62604 q^{57} -8.60384 q^{58} +14.5785 q^{59} -12.8546 q^{60} -1.74393 q^{61} +9.55270 q^{62} -5.25948 q^{63} -10.5574 q^{64} -15.0953 q^{65} +1.73061 q^{66} +11.8024 q^{67} -3.12481 q^{68} -4.35570 q^{69} +43.2717 q^{70} +5.12181 q^{71} -4.18219 q^{72} +15.0335 q^{73} +13.4136 q^{74} -6.78262 q^{75} -9.83418 q^{76} -3.79753 q^{77} -10.5405 q^{78} -11.2521 q^{79} +8.69926 q^{80} +1.00000 q^{81} +6.69493 q^{82} -11.0963 q^{83} +19.6961 q^{84} -2.86423 q^{85} +18.5364 q^{86} -3.58965 q^{87} -3.01969 q^{88} +17.5609 q^{89} -8.22737 q^{90} +23.1293 q^{91} +16.3116 q^{92} +3.98553 q^{93} -3.94691 q^{94} -9.01409 q^{95} -2.29000 q^{96} -16.3757 q^{97} -49.5240 q^{98} +0.722035 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 16 q^{2} - 116 q^{3} + 116 q^{4} - 20 q^{5} + 16 q^{6} - 33 q^{7} - 45 q^{8} + 116 q^{9}+O(q^{10}) \) 116 * q - 16 * q^2 - 116 * q^3 + 116 * q^4 - 20 * q^5 + 16 * q^6 - 33 * q^7 - 45 * q^8 + 116 * q^9	\( 116 q - 16 q^{2} - 116 q^{3} + 116 q^{4} - 20 q^{5} + 16 q^{6} - 33 q^{7} - 45 q^{8} + 116 q^{9} + 3 q^{10} - 57 q^{11} - 116 q^{12} + 6 q^{13} - 9 q^{14} + 20 q^{15} + 112 q^{16} - 30 q^{17} - 16 q^{18} + 3 q^{19} - 54 q^{20} + 33 q^{21} - 22 q^{22} - 58 q^{23} + 45 q^{24} + 126 q^{25} - 21 q^{26} - 116 q^{27} - 77 q^{28} - 38 q^{29} - 3 q^{30} + 17 q^{31} - 106 q^{32} + 57 q^{33} + 35 q^{34} - 72 q^{35} + 116 q^{36} - 41 q^{37} - 45 q^{38} - 6 q^{39} + 5 q^{40} - 39 q^{41} + 9 q^{42} - 118 q^{43} - 103 q^{44} - 20 q^{45} - 8 q^{46} - 65 q^{47} - 112 q^{48} + 165 q^{49} - 72 q^{50} + 30 q^{51} - 10 q^{52} - 58 q^{53} + 16 q^{54} + 14 q^{55} - 23 q^{56} - 3 q^{57} - 27 q^{58} - 75 q^{59} + 54 q^{60} + 45 q^{61} - 73 q^{62} - 33 q^{63} + 111 q^{64} - 86 q^{65} + 22 q^{66} - 127 q^{67} - 94 q^{68} + 58 q^{69} - 7 q^{70} - 61 q^{71} - 45 q^{72} + 15 q^{73} - 51 q^{74} - 126 q^{75} + 96 q^{76} - 57 q^{77} + 21 q^{78} + 7 q^{79} - 144 q^{80} + 116 q^{81} - 37 q^{82} - 194 q^{83} + 77 q^{84} + 3 q^{85} - 57 q^{86} + 38 q^{87} - 42 q^{88} - 56 q^{89} + 3 q^{90} - 39 q^{91} - 138 q^{92} - 17 q^{93} + 51 q^{94} - 127 q^{95} + 106 q^{96} + 57 q^{97} - 105 q^{98} - 57 q^{99}+O(q^{100}) \) 116 * q - 16 * q^2 - 116 * q^3 + 116 * q^4 - 20 * q^5 + 16 * q^6 - 33 * q^7 - 45 * q^8 + 116 * q^9 + 3 * q^10 - 57 * q^11 - 116 * q^12 + 6 * q^13 - 9 * q^14 + 20 * q^15 + 112 * q^16 - 30 * q^17 - 16 * q^18 + 3 * q^19 - 54 * q^20 + 33 * q^21 - 22 * q^22 - 58 * q^23 + 45 * q^24 + 126 * q^25 - 21 * q^26 - 116 * q^27 - 77 * q^28 - 38 * q^29 - 3 * q^30 + 17 * q^31 - 106 * q^32 + 57 * q^33 + 35 * q^34 - 72 * q^35 + 116 * q^36 - 41 * q^37 - 45 * q^38 - 6 * q^39 + 5 * q^40 - 39 * q^41 + 9 * q^42 - 118 * q^43 - 103 * q^44 - 20 * q^45 - 8 * q^46 - 65 * q^47 - 112 * q^48 + 165 * q^49 - 72 * q^50 + 30 * q^51 - 10 * q^52 - 58 * q^53 + 16 * q^54 + 14 * q^55 - 23 * q^56 - 3 * q^57 - 27 * q^58 - 75 * q^59 + 54 * q^60 + 45 * q^61 - 73 * q^62 - 33 * q^63 + 111 * q^64 - 86 * q^65 + 22 * q^66 - 127 * q^67 - 94 * q^68 + 58 * q^69 - 7 * q^70 - 61 * q^71 - 45 * q^72 + 15 * q^73 - 51 * q^74 - 126 * q^75 + 96 * q^76 - 57 * q^77 + 21 * q^78 + 7 * q^79 - 144 * q^80 + 116 * q^81 - 37 * q^82 - 194 * q^83 + 77 * q^84 + 3 * q^85 - 57 * q^86 + 38 * q^87 - 42 * q^88 - 56 * q^89 + 3 * q^90 - 39 * q^91 - 138 * q^92 - 17 * q^93 + 51 * q^94 - 127 * q^95 + 106 * q^96 + 57 * q^97 - 105 * q^98 - 57 * q^99

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