LMFDB - Embedded newform 8043.2.a.p.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.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:	$64.2236783457$
Analytic rank:	$0$
Dimension:	$41$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.415908 q^{2} +1.00000 q^{3} -1.82702 q^{4} -0.732616 q^{5} +0.415908 q^{6} -1.00000 q^{7} -1.59169 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.415908 q^{2} +1.00000 q^{3} -1.82702 q^{4} -0.732616 q^{5} +0.415908 q^{6} -1.00000 q^{7} -1.59169 q^{8} +1.00000 q^{9} -0.304701 q^{10} +5.01464 q^{11} -1.82702 q^{12} +6.10366 q^{13} -0.415908 q^{14} -0.732616 q^{15} +2.99205 q^{16} +1.31191 q^{17} +0.415908 q^{18} +3.22546 q^{19} +1.33851 q^{20} -1.00000 q^{21} +2.08563 q^{22} +7.49235 q^{23} -1.59169 q^{24} -4.46327 q^{25} +2.53856 q^{26} +1.00000 q^{27} +1.82702 q^{28} -3.76875 q^{29} -0.304701 q^{30} +7.30755 q^{31} +4.42779 q^{32} +5.01464 q^{33} +0.545632 q^{34} +0.732616 q^{35} -1.82702 q^{36} -0.374382 q^{37} +1.34149 q^{38} +6.10366 q^{39} +1.16610 q^{40} -9.78161 q^{41} -0.415908 q^{42} -10.4509 q^{43} -9.16184 q^{44} -0.732616 q^{45} +3.11613 q^{46} -7.00765 q^{47} +2.99205 q^{48} +1.00000 q^{49} -1.85631 q^{50} +1.31191 q^{51} -11.1515 q^{52} +10.0903 q^{53} +0.415908 q^{54} -3.67380 q^{55} +1.59169 q^{56} +3.22546 q^{57} -1.56745 q^{58} +13.7564 q^{59} +1.33851 q^{60} -6.71860 q^{61} +3.03927 q^{62} -1.00000 q^{63} -4.14254 q^{64} -4.47164 q^{65} +2.08563 q^{66} -4.70870 q^{67} -2.39688 q^{68} +7.49235 q^{69} +0.304701 q^{70} -8.10217 q^{71} -1.59169 q^{72} +11.7348 q^{73} -0.155708 q^{74} -4.46327 q^{75} -5.89298 q^{76} -5.01464 q^{77} +2.53856 q^{78} +14.1893 q^{79} -2.19202 q^{80} +1.00000 q^{81} -4.06825 q^{82} -9.55958 q^{83} +1.82702 q^{84} -0.961124 q^{85} -4.34661 q^{86} -3.76875 q^{87} -7.98174 q^{88} -6.43857 q^{89} -0.304701 q^{90} -6.10366 q^{91} -13.6887 q^{92} +7.30755 q^{93} -2.91454 q^{94} -2.36302 q^{95} +4.42779 q^{96} +3.64236 q^{97} +0.415908 q^{98} +5.01464 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 41 q + 7 q^{2} + 41 q^{3} + 45 q^{4} + 17 q^{5} + 7 q^{6} - 41 q^{7} + 12 q^{8} + 41 q^{9}+O(q^{10}) $ 41 * q + 7 * q^2 + 41 * q^3 + 45 * q^4 + 17 * q^5 + 7 * q^6 - 41 * q^7 + 12 * q^8 + 41 * q^9	\( 41 q + 7 q^{2} + 41 q^{3} + 45 q^{4} + 17 q^{5} + 7 q^{6} - 41 q^{7} + 12 q^{8} + 41 q^{9} + 18 q^{10} + 8 q^{11} + 45 q^{12} + 23 q^{13} - 7 q^{14} + 17 q^{15} + 37 q^{16} + 15 q^{17} + 7 q^{18} + 15 q^{19} + 53 q^{20} - 41 q^{21} + 13 q^{22} + 44 q^{23} + 12 q^{24} + 58 q^{25} + 9 q^{26} + 41 q^{27} - 45 q^{28} + 21 q^{29} + 18 q^{30} + 39 q^{31} + 61 q^{32} + 8 q^{33} + 9 q^{34} - 17 q^{35} + 45 q^{36} + 11 q^{37} + 44 q^{38} + 23 q^{39} + 24 q^{40} + 17 q^{41} - 7 q^{42} + 7 q^{43} + 30 q^{44} + 17 q^{45} - 12 q^{46} + 36 q^{47} + 37 q^{48} + 41 q^{49} + 28 q^{50} + 15 q^{51} + 58 q^{52} + 26 q^{53} + 7 q^{54} + 32 q^{55} - 12 q^{56} + 15 q^{57} - 4 q^{58} + 33 q^{59} + 53 q^{60} + 59 q^{61} - q^{62} - 41 q^{63} + 16 q^{64} + 72 q^{65} + 13 q^{66} + 12 q^{67} + 52 q^{68} + 44 q^{69} - 18 q^{70} + 33 q^{71} + 12 q^{72} + 18 q^{73} + 42 q^{74} + 58 q^{75} + 7 q^{76} - 8 q^{77} + 9 q^{78} + 22 q^{79} + 69 q^{80} + 41 q^{81} + 41 q^{82} + 32 q^{83} - 45 q^{84} - 44 q^{85} + 11 q^{86} + 21 q^{87} + 52 q^{88} + 63 q^{89} + 18 q^{90} - 23 q^{91} + 52 q^{92} + 39 q^{93} + 17 q^{94} + 37 q^{95} + 61 q^{96} + 8 q^{97} + 7 q^{98} + 8 q^{99}+O(q^{100}) \) 41 * q + 7 * q^2 + 41 * q^3 + 45 * q^4 + 17 * q^5 + 7 * q^6 - 41 * q^7 + 12 * q^8 + 41 * q^9 + 18 * q^10 + 8 * q^11 + 45 * q^12 + 23 * q^13 - 7 * q^14 + 17 * q^15 + 37 * q^16 + 15 * q^17 + 7 * q^18 + 15 * q^19 + 53 * q^20 - 41 * q^21 + 13 * q^22 + 44 * q^23 + 12 * q^24 + 58 * q^25 + 9 * q^26 + 41 * q^27 - 45 * q^28 + 21 * q^29 + 18 * q^30 + 39 * q^31 + 61 * q^32 + 8 * q^33 + 9 * q^34 - 17 * q^35 + 45 * q^36 + 11 * q^37 + 44 * q^38 + 23 * q^39 + 24 * q^40 + 17 * q^41 - 7 * q^42 + 7 * q^43 + 30 * q^44 + 17 * q^45 - 12 * q^46 + 36 * q^47 + 37 * q^48 + 41 * q^49 + 28 * q^50 + 15 * q^51 + 58 * q^52 + 26 * q^53 + 7 * q^54 + 32 * q^55 - 12 * q^56 + 15 * q^57 - 4 * q^58 + 33 * q^59 + 53 * q^60 + 59 * q^61 - q^62 - 41 * q^63 + 16 * q^64 + 72 * q^65 + 13 * q^66 + 12 * q^67 + 52 * q^68 + 44 * q^69 - 18 * q^70 + 33 * q^71 + 12 * q^72 + 18 * q^73 + 42 * q^74 + 58 * q^75 + 7 * q^76 - 8 * q^77 + 9 * q^78 + 22 * q^79 + 69 * q^80 + 41 * q^81 + 41 * q^82 + 32 * q^83 - 45 * q^84 - 44 * q^85 + 11 * q^86 + 21 * q^87 + 52 * q^88 + 63 * q^89 + 18 * q^90 - 23 * q^91 + 52 * q^92 + 39 * q^93 + 17 * q^94 + 37 * q^95 + 61 * q^96 + 8 * q^97 + 7 * q^98 + 8 * 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 $

$2$	0.415908	0.294091	0.147046	−	0.989130i	$-0.453024\pi$
$2$	0.415908	0.294091	0.147046	+	0.989130i	$0.453024\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.82702	−0.913510
$5$	−0.732616	−0.327636	−0.163818	−	0.986491i	$-0.552381\pi$
$5$	−0.732616	−0.327636	−0.163818	+	0.986491i	$0.552381\pi$
$6$	0.415908	0.169794
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.59169	−0.562747
$9$	1.00000	0.333333
$10$	−0.304701	−0.0963549
$11$	5.01464	1.51197	0.755985	−	0.654589i	$-0.227158\pi$
$11$	5.01464	1.51197	0.755985	+	0.654589i	$0.227158\pi$
$12$	−1.82702	−0.527415
$13$	6.10366	1.69285	0.846425	−	0.532507i	$-0.178750\pi$
$13$	6.10366	1.69285	0.846425	+	0.532507i	$0.178750\pi$
$14$	−0.415908	−0.111156
$15$	−0.732616	−0.189161
$16$	2.99205	0.748011
$17$	1.31191	0.318184	0.159092	−	0.987264i	$-0.449143\pi$
$17$	1.31191	0.318184	0.159092	+	0.987264i	$0.449143\pi$
$18$	0.415908	0.0980305
$19$	3.22546	0.739971	0.369985	−	0.929038i	$-0.379363\pi$
$19$	3.22546	0.739971	0.369985	+	0.929038i	$0.379363\pi$
$20$	1.33851	0.299299
$21$	−1.00000	−0.218218
$22$	2.08563	0.444657
$23$	7.49235	1.56226	0.781132	−	0.624366i	$-0.214643\pi$
$23$	7.49235	1.56226	0.781132	+	0.624366i	$0.214643\pi$
$24$	−1.59169	−0.324902
$25$	−4.46327	−0.892655
$26$	2.53856	0.497853
$27$	1.00000	0.192450
$28$	1.82702	0.345274
$29$	−3.76875	−0.699840	−0.349920	−	0.936780i	$-0.613791\pi$
$29$	−3.76875	−0.699840	−0.349920	+	0.936780i	$0.613791\pi$
$30$	−0.304701	−0.0556305
$31$	7.30755	1.31247	0.656237	−	0.754555i	$-0.272147\pi$
$31$	7.30755	1.31247	0.656237	+	0.754555i	$0.272147\pi$
$32$	4.42779	0.782730
$33$	5.01464	0.872936
$34$	0.545632	0.0935752
$35$	0.732616	0.123835
$36$	−1.82702	−0.304503
$37$	−0.374382	−0.0615480	−0.0307740	−	0.999526i	$-0.509797\pi$
$37$	−0.374382	−0.0615480	−0.0307740	+	0.999526i	$0.509797\pi$
$38$	1.34149	0.217619
$39$	6.10366	0.977368
$40$	1.16610	0.184376
$41$	−9.78161	−1.52763	−0.763815	−	0.645435i	$-0.776676\pi$
$41$	−9.78161	−1.52763	−0.763815	+	0.645435i	$0.776676\pi$
$42$	−0.415908	−0.0641760
$43$	−10.4509	−1.59375	−0.796873	−	0.604147i	$-0.793514\pi$
$43$	−10.4509	−1.59375	−0.796873	+	0.604147i	$0.793514\pi$
$44$	−9.16184	−1.38120
$45$	−0.732616	−0.109212
$46$	3.11613	0.459448
$47$	−7.00765	−1.02217	−0.511086	−	0.859530i	$-0.670757\pi$
$47$	−7.00765	−1.02217	−0.511086	+	0.859530i	$0.670757\pi$
$48$	2.99205	0.431865
$49$	1.00000	0.142857
$50$	−1.85631	−0.262522
$51$	1.31191	0.183704
$52$	−11.1515	−1.54644
$53$	10.0903	1.38601	0.693004	−	0.720934i	$-0.256287\pi$
$53$	10.0903	1.38601	0.693004	+	0.720934i	$0.256287\pi$
$54$	0.415908	0.0565979
$55$	−3.67380	−0.495376
$56$	1.59169	0.212698
$57$	3.22546	0.427222
$58$	−1.56745	−0.205817
$59$	13.7564	1.79093	0.895466	−	0.445129i	$-0.146842\pi$
$59$	13.7564	1.79093	0.895466	+	0.445129i	$0.146842\pi$
$60$	1.33851	0.172800
$61$	−6.71860	−0.860228	−0.430114	−	0.902775i	$-0.641527\pi$
$61$	−6.71860	−0.860228	−0.430114	+	0.902775i	$0.641527\pi$
$62$	3.03927	0.385987
$63$	−1.00000	−0.125988
$64$	−4.14254	−0.517817
$65$	−4.47164	−0.554639
$66$	2.08563	0.256723
$67$	−4.70870	−0.575260	−0.287630	−	0.957742i	$-0.592867\pi$
$67$	−4.70870	−0.575260	−0.287630	+	0.957742i	$0.592867\pi$
$68$	−2.39688	−0.290664
$69$	7.49235	0.901973
$70$	0.304701	0.0364187
$71$	−8.10217	−0.961551	−0.480775	−	0.876844i	$-0.659645\pi$
$71$	−8.10217	−0.961551	−0.480775	+	0.876844i	$0.659645\pi$
$72$	−1.59169	−0.187582
$73$	11.7348	1.37346	0.686728	−	0.726914i	$-0.259046\pi$
$73$	11.7348	1.37346	0.686728	+	0.726914i	$0.259046\pi$
$74$	−0.155708	−0.0181007
$75$	−4.46327	−0.515374
$76$	−5.89298	−0.675971
$77$	−5.01464	−0.571471
$78$	2.53856	0.287435
$79$	14.1893	1.59642	0.798212	−	0.602376i	$-0.205779\pi$
$79$	14.1893	1.59642	0.798212	+	0.602376i	$0.205779\pi$
$80$	−2.19202	−0.245075
$81$	1.00000	0.111111
$82$	−4.06825	−0.449263
$83$	−9.55958	−1.04930	−0.524650	−	0.851318i	$-0.675804\pi$
$83$	−9.55958	−1.04930	−0.524650	+	0.851318i	$0.675804\pi$
$84$	1.82702	0.199344
$85$	−0.961124	−0.104249
$86$	−4.34661	−0.468707
$87$	−3.76875	−0.404053
$88$	−7.98174	−0.850856
$89$	−6.43857	−0.682487	−0.341243	−	0.939975i	$-0.610848\pi$
$89$	−6.43857	−0.682487	−0.341243	+	0.939975i	$0.610848\pi$
$90$	−0.304701	−0.0321183
$91$	−6.10366	−0.639837
$92$	−13.6887	−1.42714
$93$	7.30755	0.757758
$94$	−2.91454	−0.300612
$95$	−2.36302	−0.242441
$96$	4.42779	0.451910
$97$	3.64236	0.369825	0.184913	−	0.982755i	$-0.440800\pi$
$97$	3.64236	0.369825	0.184913	+	0.982755i	$0.440800\pi$
$98$	0.415908	0.0420131
$99$	5.01464	0.503990
$100$	8.15449	0.815449
$101$	−6.71547	−0.668214	−0.334107	−	0.942535i	$-0.608435\pi$
$101$	−6.71547	−0.668214	−0.334107	+	0.942535i	$0.608435\pi$
$102$	0.545632	0.0540257
$103$	−14.0137	−1.38081	−0.690405	−	0.723423i	$-0.742568\pi$
$103$	−14.0137	−1.38081	−0.690405	+	0.723423i	$0.742568\pi$
$104$	−9.71512	−0.952646
$105$	0.732616	0.0714960
$106$	4.19663	0.407613
$107$	16.7305	1.61740	0.808701	−	0.588220i	$-0.200171\pi$
$107$	16.7305	1.61740	0.808701	+	0.588220i	$0.200171\pi$
$108$	−1.82702	−0.175805
$109$	1.29229	0.123779	0.0618893	−	0.998083i	$-0.480287\pi$
$109$	1.29229	0.123779	0.0618893	+	0.998083i	$0.480287\pi$
$110$	−1.52796	−0.145686
$111$	−0.374382	−0.0355348
$112$	−2.99205	−0.282722
$113$	16.9215	1.59184	0.795920	−	0.605402i	$-0.206987\pi$
$113$	16.9215	1.59184	0.795920	+	0.605402i	$0.206987\pi$
$114$	1.34149	0.125642
$115$	−5.48902	−0.511854
$116$	6.88559	0.639311
$117$	6.10366	0.564284
$118$	5.72140	0.526698
$119$	−1.31191	−0.120262
$120$	1.16610	0.106450
$121$	14.1466	1.28605
$122$	−2.79432	−0.252986
$123$	−9.78161	−0.881978
$124$	−13.3510	−1.19896
$125$	6.93295	0.620102
$126$	−0.415908	−0.0370520
$127$	3.43146	0.304493	0.152246	−	0.988343i	$-0.451349\pi$
$127$	3.43146	0.304493	0.152246	+	0.988343i	$0.451349\pi$
$128$	−10.5785	−0.935016
$129$	−10.4509	−0.920149
$130$	−1.85979	−0.163115
$131$	14.5385	1.27024	0.635119	−	0.772414i	$-0.280951\pi$
$131$	14.5385	1.27024	0.635119	+	0.772414i	$0.280951\pi$
$132$	−9.16184	−0.797436
$133$	−3.22546	−0.279683
$134$	−1.95839	−0.169179
$135$	−0.732616	−0.0630536
$136$	−2.08815	−0.179057
$137$	−9.76569	−0.834340	−0.417170	−	0.908829i	$-0.636978\pi$
$137$	−9.76569	−0.834340	−0.417170	+	0.908829i	$0.636978\pi$
$138$	3.11613	0.265263
$139$	6.66554	0.565364	0.282682	−	0.959214i	$-0.408776\pi$
$139$	6.66554	0.565364	0.282682	+	0.959214i	$0.408776\pi$
$140$	−1.33851	−0.113124
$141$	−7.00765	−0.590151
$142$	−3.36976	−0.282784
$143$	30.6076	2.55954
$144$	2.99205	0.249337
$145$	2.76105	0.229293
$146$	4.88060	0.403922
$147$	1.00000	0.0824786
$148$	0.684004	0.0562248
$149$	−6.85468	−0.561557	−0.280779	−	0.959773i	$-0.590593\pi$
$149$	−6.85468	−0.561557	−0.280779	+	0.959773i	$0.590593\pi$
$150$	−1.85631	−0.151567
$151$	−9.54376	−0.776660	−0.388330	−	0.921520i	$-0.626948\pi$
$151$	−9.54376	−0.776660	−0.388330	+	0.921520i	$0.626948\pi$
$152$	−5.13392	−0.416416
$153$	1.31191	0.106061
$154$	−2.08563	−0.168065
$155$	−5.35363	−0.430014
$156$	−11.1515	−0.892836
$157$	0.270712	0.0216052	0.0108026	−	0.999942i	$-0.496561\pi$
$157$	0.270712	0.0216052	0.0108026	+	0.999942i	$0.496561\pi$
$158$	5.90146	0.469495
$159$	10.0903	0.800212
$160$	−3.24387	−0.256451
$161$	−7.49235	−0.590480
$162$	0.415908	0.0326768
$163$	4.13571	0.323934	0.161967	−	0.986796i	$-0.448216\pi$
$163$	4.13571	0.323934	0.161967	+	0.986796i	$0.448216\pi$
$164$	17.8712	1.39551
$165$	−3.67380	−0.286005
$166$	−3.97591	−0.308590
$167$	17.2841	1.33748	0.668741	−	0.743495i	$-0.266833\pi$
$167$	17.2841	1.33748	0.668741	+	0.743495i	$0.266833\pi$
$168$	1.59169	0.122801
$169$	24.2547	1.86574
$170$	−0.399739	−0.0306586
$171$	3.22546	0.246657
$172$	19.0940	1.45590
$173$	−14.5520	−1.10637	−0.553184	−	0.833059i	$-0.686587\pi$
$173$	−14.5520	−1.10637	−0.553184	+	0.833059i	$0.686587\pi$
$174$	−1.56745	−0.118828
$175$	4.46327	0.337392
$176$	15.0040	1.13097
$177$	13.7564	1.03400
$178$	−2.67785	−0.200713
$179$	5.36006	0.400630	0.200315	−	0.979732i	$-0.435804\pi$
$179$	5.36006	0.400630	0.200315	+	0.979732i	$0.435804\pi$
$180$	1.33851	0.0997663
$181$	−3.01524	−0.224121	−0.112060	−	0.993701i	$-0.535745\pi$
$181$	−3.01524	−0.224121	−0.112060	+	0.993701i	$0.535745\pi$
$182$	−2.53856	−0.188171
$183$	−6.71860	−0.496653
$184$	−11.9255	−0.879159
$185$	0.274278	0.0201654
$186$	3.03927	0.222850
$187$	6.57873	0.481085
$188$	12.8031	0.933764
$189$	−1.00000	−0.0727393
$190$	−0.982800	−0.0712998
$191$	8.91221	0.644865	0.322432	−	0.946592i	$-0.395499\pi$
$191$	8.91221	0.644865	0.322432	+	0.946592i	$0.395499\pi$
$192$	−4.14254	−0.298962
$193$	−1.45259	−0.104560	−0.0522800	−	0.998632i	$-0.516649\pi$
$193$	−1.45259	−0.104560	−0.0522800	+	0.998632i	$0.516649\pi$
$194$	1.51488	0.108762
$195$	−4.47164	−0.320221
$196$	−1.82702	−0.130501
$197$	0.605296	0.0431256	0.0215628	−	0.999767i	$-0.493136\pi$
$197$	0.605296	0.0431256	0.0215628	+	0.999767i	$0.493136\pi$
$198$	2.08563	0.148219
$199$	−3.75015	−0.265841	−0.132920	−	0.991127i	$-0.542435\pi$
$199$	−3.75015	−0.265841	−0.132920	+	0.991127i	$0.542435\pi$
$200$	7.10414	0.502339
$201$	−4.70870	−0.332126
$202$	−2.79302	−0.196516
$203$	3.76875	0.264515
$204$	−2.39688	−0.167815
$205$	7.16617	0.500507
$206$	−5.82841	−0.406084
$207$	7.49235	0.520755
$208$	18.2624	1.26627
$209$	16.1745	1.11881
$210$	0.304701	0.0210264
$211$	5.43133	0.373908	0.186954	−	0.982369i	$-0.440138\pi$
$211$	5.43133	0.373908	0.186954	+	0.982369i	$0.440138\pi$
$212$	−18.4352	−1.26613
$213$	−8.10217	−0.555152
$214$	6.95837	0.475664
$215$	7.65649	0.522168
$216$	−1.59169	−0.108301
$217$	−7.30755	−0.496069
$218$	0.537472	0.0364022
$219$	11.7348	0.792965
$220$	6.71212	0.452531
$221$	8.00743	0.538638
$222$	−0.155708	−0.0104505
$223$	24.4431	1.63683	0.818415	−	0.574627i	$-0.194853\pi$
$223$	24.4431	1.63683	0.818415	+	0.574627i	$0.194853\pi$
$224$	−4.42779	−0.295844
$225$	−4.46327	−0.297552
$226$	7.03778	0.468146
$227$	−8.40236	−0.557684	−0.278842	−	0.960337i	$-0.589951\pi$
$227$	−8.40236	−0.557684	−0.278842	+	0.960337i	$0.589951\pi$
$228$	−5.89298	−0.390272
$229$	27.6791	1.82909	0.914545	−	0.404485i	$-0.132549\pi$
$229$	27.6791	1.82909	0.914545	+	0.404485i	$0.132549\pi$
$230$	−2.28293	−0.150532
$231$	−5.01464	−0.329939
$232$	5.99868	0.393833
$233$	−6.51272	−0.426663	−0.213331	−	0.976980i	$-0.568431\pi$
$233$	−6.51272	−0.426663	−0.213331	+	0.976980i	$0.568431\pi$
$234$	2.53856	0.165951
$235$	5.13392	0.334900
$236$	−25.1332	−1.63604
$237$	14.1893	0.921696
$238$	−0.545632	−0.0353681
$239$	−4.95110	−0.320260	−0.160130	−	0.987096i	$-0.551191\pi$
$239$	−4.95110	−0.320260	−0.160130	+	0.987096i	$0.551191\pi$
$240$	−2.19202	−0.141494
$241$	−18.0207	−1.16082	−0.580408	−	0.814326i	$-0.697107\pi$
$241$	−18.0207	−1.16082	−0.580408	+	0.814326i	$0.697107\pi$
$242$	5.88367	0.378217
$243$	1.00000	0.0641500
$244$	12.2750	0.785827
$245$	−0.732616	−0.0468051
$246$	−4.06825	−0.259382
$247$	19.6871	1.25266
$248$	−11.6313	−0.738591
$249$	−9.55958	−0.605814
$250$	2.88347	0.182367
$251$	14.0612	0.887535	0.443768	−	0.896142i	$-0.353642\pi$
$251$	14.0612	0.887535	0.443768	+	0.896142i	$0.353642\pi$
$252$	1.82702	0.115091
$253$	37.5714	2.36209
$254$	1.42717	0.0895486
$255$	−0.961124	−0.0601879
$256$	3.88539	0.242837
$257$	−4.77554	−0.297890	−0.148945	−	0.988845i	$-0.547588\pi$
$257$	−4.77554	−0.297890	−0.148945	+	0.988845i	$0.547588\pi$
$258$	−4.34661	−0.270608
$259$	0.374382	0.0232630
$260$	8.16978	0.506668
$261$	−3.76875	−0.233280
$262$	6.04669	0.373566
$263$	−5.01172	−0.309036	−0.154518	−	0.987990i	$-0.549382\pi$
$263$	−5.01172	−0.309036	−0.154518	+	0.987990i	$0.549382\pi$
$264$	−7.98174	−0.491242
$265$	−7.39231	−0.454106
$266$	−1.34149	−0.0822523
$267$	−6.43857	−0.394034
$268$	8.60290	0.525506
$269$	−9.26215	−0.564723	−0.282362	−	0.959308i	$-0.591118\pi$
$269$	−9.26215	−0.564723	−0.282362	+	0.959308i	$0.591118\pi$
$270$	−0.304701	−0.0185435
$271$	−27.4600	−1.66808	−0.834038	−	0.551707i	$-0.813977\pi$
$271$	−27.4600	−1.66808	−0.834038	+	0.551707i	$0.813977\pi$
$272$	3.92528	0.238005
$273$	−6.10366	−0.369410
$274$	−4.06163	−0.245372
$275$	−22.3817	−1.34967
$276$	−13.6887	−0.823962
$277$	1.81901	0.109294	0.0546469	−	0.998506i	$-0.482597\pi$
$277$	1.81901	0.109294	0.0546469	+	0.998506i	$0.482597\pi$
$278$	2.77225	0.166269
$279$	7.30755	0.437492
$280$	−1.16610	−0.0696876
$281$	−2.91237	−0.173737	−0.0868687	−	0.996220i	$-0.527686\pi$
$281$	−2.91237	−0.173737	−0.0868687	+	0.996220i	$0.527686\pi$
$282$	−2.91454	−0.173558
$283$	−11.2278	−0.667421	−0.333710	−	0.942676i	$-0.608301\pi$
$283$	−11.2278	−0.667421	−0.333710	+	0.942676i	$0.608301\pi$
$284$	14.8028	0.878386
$285$	−2.36302	−0.139973
$286$	12.7300	0.752738
$287$	9.78161	0.577390
$288$	4.42779	0.260910
$289$	−15.2789	−0.898759
$290$	1.14834	0.0674330
$291$	3.64236	0.213519
$292$	−21.4398	−1.25467
$293$	13.0193	0.760597	0.380299	−	0.924864i	$-0.375821\pi$
$293$	13.0193	0.760597	0.380299	+	0.924864i	$0.375821\pi$
$294$	0.415908	0.0242562
$295$	−10.0782	−0.586774
$296$	0.595900	0.0346360
$297$	5.01464	0.290979
$298$	−2.85092	−0.165149
$299$	45.7308	2.64468
$300$	8.15449	0.470800
$301$	10.4509	0.602379
$302$	−3.96932	−0.228409
$303$	−6.71547	−0.385794
$304$	9.65072	0.553507
$305$	4.92215	0.281842
$306$	0.545632	0.0311917
$307$	14.6178	0.834284	0.417142	−	0.908841i	$-0.363032\pi$
$307$	14.6178	0.834284	0.417142	+	0.908841i	$0.363032\pi$
$308$	9.16184	0.522044
$309$	−14.0137	−0.797211
$310$	−2.22662	−0.126463
$311$	3.81586	0.216378	0.108189	−	0.994130i	$-0.465495\pi$
$311$	3.81586	0.216378	0.108189	+	0.994130i	$0.465495\pi$
$312$	−9.71512	−0.550011
$313$	1.65784	0.0937066	0.0468533	−	0.998902i	$-0.485081\pi$
$313$	1.65784	0.0937066	0.0468533	+	0.998902i	$0.485081\pi$
$314$	0.112591	0.00635391
$315$	0.732616	0.0412783
$316$	−25.9242	−1.45835
$317$	−32.2719	−1.81257	−0.906285	−	0.422668i	$-0.861094\pi$
$317$	−32.2719	−1.81257	−0.906285	+	0.422668i	$0.861094\pi$
$318$	4.19663	0.235336
$319$	−18.8989	−1.05814
$320$	3.03489	0.169656
$321$	16.7305	0.933808
$322$	−3.11613	−0.173655
$323$	4.23150	0.235447
$324$	−1.82702	−0.101501
$325$	−27.2423	−1.51113
$326$	1.72007	0.0952660
$327$	1.29229	0.0714636
$328$	15.5693	0.859669
$329$	7.00765	0.386345
$330$	−1.52796	−0.0841117
$331$	31.8978	1.75326	0.876632	−	0.481162i	$-0.159785\pi$
$331$	31.8978	1.75326	0.876632	+	0.481162i	$0.159785\pi$
$332$	17.4655	0.958546
$333$	−0.374382	−0.0205160
$334$	7.18859	0.393342
$335$	3.44967	0.188476
$336$	−2.99205	−0.163229
$337$	21.2251	1.15620	0.578101	−	0.815965i	$-0.303794\pi$
$337$	21.2251	1.15620	0.578101	+	0.815965i	$0.303794\pi$
$338$	10.0877	0.548699
$339$	16.9215	0.919049
$340$	1.75599	0.0952321
$341$	36.6447	1.98442
$342$	1.34149	0.0725397
$343$	−1.00000	−0.0539949
$344$	16.6346	0.896875
$345$	−5.48902	−0.295519
$346$	−6.05229	−0.325373
$347$	−19.0031	−1.02014	−0.510071	−	0.860132i	$-0.670381\pi$
$347$	−19.0031	−1.02014	−0.510071	+	0.860132i	$0.670381\pi$
$348$	6.88559	0.369106
$349$	15.9724	0.854983	0.427491	−	0.904019i	$-0.359397\pi$
$349$	15.9724	0.854983	0.427491	+	0.904019i	$0.359397\pi$
$350$	1.85631	0.0992240
$351$	6.10366	0.325789
$352$	22.2038	1.18346
$353$	19.6029	1.04336	0.521680	−	0.853141i	$-0.325306\pi$
$353$	19.6029	1.04336	0.521680	+	0.853141i	$0.325306\pi$
$354$	5.72140	0.304089
$355$	5.93578	0.315039
$356$	11.7634	0.623459
$357$	−1.31191	−0.0694335
$358$	2.22929	0.117822
$359$	25.4224	1.34174	0.670870	−	0.741575i	$-0.265921\pi$
$359$	25.4224	1.34174	0.670870	+	0.741575i	$0.265921\pi$
$360$	1.16610	0.0614587
$361$	−8.59642	−0.452443
$362$	−1.25406	−0.0659120
$363$	14.1466	0.742502
$364$	11.1515	0.584498
$365$	−8.59712	−0.449994
$366$	−2.79432	−0.146061
$367$	6.90269	0.360318	0.180159	−	0.983638i	$-0.442339\pi$
$367$	6.90269	0.360318	0.180159	+	0.983638i	$0.442339\pi$
$368$	22.4175	1.16859
$369$	−9.78161	−0.509210
$370$	0.114075	0.00593046
$371$	−10.0903	−0.523862
$372$	−13.3510	−0.692219
$373$	20.3121	1.05172	0.525860	−	0.850571i	$-0.323744\pi$
$373$	20.3121	1.05172	0.525860	+	0.850571i	$0.323744\pi$
$374$	2.73615	0.141483
$375$	6.93295	0.358016
$376$	11.1540	0.575224
$377$	−23.0032	−1.18472
$378$	−0.415908	−0.0213920
$379$	6.07718	0.312164	0.156082	−	0.987744i	$-0.450114\pi$
$379$	6.07718	0.312164	0.156082	+	0.987744i	$0.450114\pi$
$380$	4.31729	0.221472
$381$	3.43146	0.175799
$382$	3.70666	0.189649
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	−10.5785	−0.539832
$385$	3.67380	0.187234
$386$	−0.604145	−0.0307502
$387$	−10.4509	−0.531248
$388$	−6.65466	−0.337839
$389$	2.37458	0.120396	0.0601980	−	0.998186i	$-0.480827\pi$
$389$	2.37458	0.120396	0.0601980	+	0.998186i	$0.480827\pi$
$390$	−1.85979	−0.0941742
$391$	9.82927	0.497087
$392$	−1.59169	−0.0803924
$393$	14.5385	0.733372
$394$	0.251747	0.0126829
$395$	−10.3953	−0.523046
$396$	−9.16184	−0.460400
$397$	−8.04099	−0.403566	−0.201783	−	0.979430i	$-0.564674\pi$
$397$	−8.04099	−0.403566	−0.201783	+	0.979430i	$0.564674\pi$
$398$	−1.55972	−0.0781815
$399$	−3.22546	−0.161475
$400$	−13.3543	−0.667716
$401$	−3.23697	−0.161647	−0.0808233	−	0.996728i	$-0.525755\pi$
$401$	−3.23697	−0.161647	−0.0808233	+	0.996728i	$0.525755\pi$
$402$	−1.95839	−0.0976755
$403$	44.6028	2.22182
$404$	12.2693	0.610421
$405$	−0.732616	−0.0364040
$406$	1.56745	0.0777914
$407$	−1.87739	−0.0930587
$408$	−2.08815	−0.103379
$409$	−31.3892	−1.55209	−0.776047	−	0.630675i	$-0.782778\pi$
$409$	−31.3892	−1.55209	−0.776047	+	0.630675i	$0.782778\pi$
$410$	2.98047	0.147195
$411$	−9.76569	−0.481706
$412$	25.6033	1.26138
$413$	−13.7564	−0.676909
$414$	3.11613	0.153149
$415$	7.00350	0.343789
$416$	27.0257	1.32505
$417$	6.66554	0.326413
$418$	6.72710	0.329033
$419$	32.4151	1.58358	0.791791	−	0.610792i	$-0.209149\pi$
$419$	32.4151	1.58358	0.791791	+	0.610792i	$0.209149\pi$
$420$	−1.33851	−0.0653124
$421$	−14.9340	−0.727841	−0.363920	−	0.931430i	$-0.618562\pi$
$421$	−14.9340	−0.727841	−0.363920	+	0.931430i	$0.618562\pi$
$422$	2.25893	0.109963
$423$	−7.00765	−0.340724
$424$	−16.0606	−0.779972
$425$	−5.85540	−0.284028
$426$	−3.36976	−0.163265
$427$	6.71860	0.325136
$428$	−30.5670	−1.47751
$429$	30.6076	1.47775
$430$	3.18440	0.153565
$431$	4.94785	0.238330	0.119165	−	0.992874i	$-0.461978\pi$
$431$	4.94785	0.238330	0.119165	+	0.992874i	$0.461978\pi$
$432$	2.99205	0.143955
$433$	−30.9116	−1.48551	−0.742757	−	0.669561i	$-0.766482\pi$
$433$	−30.9116	−1.48551	−0.742757	+	0.669561i	$0.766482\pi$
$434$	−3.03927	−0.145890
$435$	2.76105	0.132382
$436$	−2.36103	−0.113073
$437$	24.1663	1.15603
$438$	4.88060	0.233204
$439$	15.5864	0.743897	0.371948	−	0.928253i	$-0.378690\pi$
$439$	15.5864	0.743897	0.371948	+	0.928253i	$0.378690\pi$
$440$	5.84755	0.278771
$441$	1.00000	0.0476190
$442$	3.33035	0.158409
$443$	18.1876	0.864120	0.432060	−	0.901845i	$-0.357787\pi$
$443$	18.1876	0.864120	0.432060	+	0.901845i	$0.357787\pi$
$444$	0.684004	0.0324614
$445$	4.71700	0.223607
$446$	10.1661	0.481378
$447$	−6.85468	−0.324215
$448$	4.14254	0.195716
$449$	23.3685	1.10283	0.551414	−	0.834232i	$-0.314088\pi$
$449$	23.3685	1.10283	0.551414	+	0.834232i	$0.314088\pi$
$450$	−1.85631	−0.0875073
$451$	−49.0512	−2.30973
$452$	−30.9159	−1.45416
$453$	−9.54376	−0.448405
$454$	−3.49461	−0.164010
$455$	4.47164	0.209634
$456$	−5.13392	−0.240418
$457$	−34.6478	−1.62075	−0.810377	−	0.585908i	$-0.800738\pi$
$457$	−34.6478	−1.62075	−0.810377	+	0.585908i	$0.800738\pi$
$458$	11.5120	0.537919
$459$	1.31191	0.0612345
$460$	10.0286	0.467584
$461$	−20.0258	−0.932694	−0.466347	−	0.884602i	$-0.654430\pi$
$461$	−20.0258	−0.932694	−0.466347	+	0.884602i	$0.654430\pi$
$462$	−2.08563	−0.0970321
$463$	−18.2924	−0.850120	−0.425060	−	0.905165i	$-0.639747\pi$
$463$	−18.2924	−0.850120	−0.425060	+	0.905165i	$0.639747\pi$
$464$	−11.2763	−0.523488
$465$	−5.35363	−0.248269
$466$	−2.70869	−0.125478
$467$	12.0492	0.557568	0.278784	−	0.960354i	$-0.410069\pi$
$467$	12.0492	0.557568	0.278784	+	0.960354i	$0.410069\pi$
$468$	−11.1515	−0.515479
$469$	4.70870	0.217428
$470$	2.13524	0.0984913
$471$	0.270712	0.0124738
$472$	−21.8959	−1.00784
$473$	−52.4074	−2.40969
$474$	5.90146	0.271063
$475$	−14.3961	−0.660538
$476$	2.39688	0.109861
$477$	10.0903	0.462003
$478$	−2.05920	−0.0941857
$479$	15.0927	0.689604	0.344802	−	0.938676i	$-0.387946\pi$
$479$	15.0927	0.689604	0.344802	+	0.938676i	$0.387946\pi$
$480$	−3.24387	−0.148062
$481$	−2.28510	−0.104192
$482$	−7.49496	−0.341386
$483$	−7.49235	−0.340914
$484$	−25.8461	−1.17482
$485$	−2.66845	−0.121168
$486$	0.415908	0.0188660
$487$	30.0000	1.35943	0.679714	−	0.733477i	$-0.262104\pi$
$487$	30.0000	1.35943	0.679714	+	0.733477i	$0.262104\pi$
$488$	10.6939	0.484091
$489$	4.13571	0.187023
$490$	−0.304701	−0.0137650
$491$	26.9312	1.21539	0.607693	−	0.794172i	$-0.292095\pi$
$491$	26.9312	1.21539	0.607693	+	0.794172i	$0.292095\pi$
$492$	17.8712	0.805696
$493$	−4.94425	−0.222678
$494$	8.18802	0.368396
$495$	−3.67380	−0.165125
$496$	21.8645	0.981746
$497$	8.10217	0.363432
$498$	−3.97591	−0.178165
$499$	−3.73375	−0.167146	−0.0835728	−	0.996502i	$-0.526633\pi$
$499$	−3.73375	−0.167146	−0.0835728	+	0.996502i	$0.526633\pi$
$500$	−12.6666	−0.566469
$501$	17.2841	0.772196
$502$	5.84816	0.261016
$503$	−37.1978	−1.65857	−0.829284	−	0.558827i	$-0.811251\pi$
$503$	−37.1978	−1.65857	−0.829284	+	0.558827i	$0.811251\pi$
$504$	1.59169	0.0708994
$505$	4.91986	0.218931
$506$	15.6263	0.694672
$507$	24.2547	1.07719
$508$	−6.26934	−0.278157
$509$	13.8427	0.613566	0.306783	−	0.951779i	$-0.400747\pi$
$509$	13.8427	0.613566	0.306783	+	0.951779i	$0.400747\pi$
$510$	−0.399739	−0.0177008
$511$	−11.7348	−0.519118
$512$	22.7730	1.00643
$513$	3.22546	0.142407
$514$	−1.98618	−0.0876068
$515$	10.2667	0.452403
$516$	19.0940	0.840566
$517$	−35.1408	−1.54549
$518$	0.155708	0.00684144
$519$	−14.5520	−0.638761
$520$	7.11746	0.312121
$521$	−12.5220	−0.548598	−0.274299	−	0.961644i	$-0.588446\pi$
$521$	−12.5220	−0.548598	−0.274299	+	0.961644i	$0.588446\pi$
$522$	−1.56745	−0.0686056
$523$	15.2492	0.666800	0.333400	−	0.942785i	$-0.391804\pi$
$523$	15.2492	0.666800	0.333400	+	0.942785i	$0.391804\pi$
$524$	−26.5622	−1.16038
$525$	4.46327	0.194793
$526$	−2.08441	−0.0908848
$527$	9.58682	0.417609
$528$	15.0040	0.652966
$529$	33.1354	1.44067
$530$	−3.07452	−0.133549
$531$	13.7564	0.596977
$532$	5.89298	0.255493
$533$	−59.7036	−2.58605
$534$	−2.67785	−0.115882
$535$	−12.2571	−0.529919
$536$	7.49479	0.323726
$537$	5.36006	0.231304
$538$	−3.85220	−0.166080
$539$	5.01464	0.215996
$540$	1.33851	0.0576001
$541$	−8.47417	−0.364333	−0.182166	−	0.983268i	$-0.558311\pi$
$541$	−8.47417	−0.364333	−0.182166	+	0.983268i	$0.558311\pi$
$542$	−11.4208	−0.490567
$543$	−3.01524	−0.129396
$544$	5.80885	0.249052
$545$	−0.946750	−0.0405543
$546$	−2.53856	−0.108640
$547$	−5.08211	−0.217295	−0.108648	−	0.994080i	$-0.534652\pi$
$547$	−5.08211	−0.217295	−0.108648	+	0.994080i	$0.534652\pi$
$548$	17.8421	0.762178
$549$	−6.71860	−0.286743
$550$	−9.30872	−0.396925
$551$	−12.1560	−0.517861
$552$	−11.9255	−0.507583
$553$	−14.1893	−0.603392
$554$	0.756542	0.0321424
$555$	0.274278	0.0116425
$556$	−12.1781	−0.516466
$557$	−2.86792	−0.121518	−0.0607589	−	0.998152i	$-0.519352\pi$
$557$	−2.86792	−0.121518	−0.0607589	+	0.998152i	$0.519352\pi$
$558$	3.03927	0.128662
$559$	−63.7887	−2.69797
$560$	2.19202	0.0926298
$561$	6.57873	0.277754
$562$	−1.21128	−0.0510946
$563$	−42.6529	−1.79760	−0.898802	−	0.438355i	$-0.855561\pi$
$563$	−42.6529	−1.79760	−0.898802	+	0.438355i	$0.855561\pi$
$564$	12.8031	0.539109
$565$	−12.3970	−0.521544
$566$	−4.66971	−0.196283
$567$	−1.00000	−0.0419961
$568$	12.8961	0.541110
$569$	15.2669	0.640022	0.320011	−	0.947414i	$-0.396313\pi$
$569$	15.2669	0.640022	0.320011	+	0.947414i	$0.396313\pi$
$570$	−0.982800	−0.0411650
$571$	−10.5965	−0.443449	−0.221724	−	0.975109i	$-0.571169\pi$
$571$	−10.5965	−0.443449	−0.221724	+	0.975109i	$0.571169\pi$
$572$	−55.9208	−2.33816
$573$	8.91221	0.372313
$574$	4.06825	0.169805
$575$	−33.4404	−1.39456
$576$	−4.14254	−0.172606
$577$	11.2995	0.470406	0.235203	−	0.971946i	$-0.424425\pi$
$577$	11.2995	0.470406	0.235203	+	0.971946i	$0.424425\pi$
$578$	−6.35462	−0.264317
$579$	−1.45259	−0.0603677
$580$	−5.04450	−0.209461
$581$	9.55958	0.396598
$582$	1.51488	0.0627940
$583$	50.5991	2.09560
$584$	−18.6782	−0.772908
$585$	−4.47164	−0.184880
$586$	5.41484	0.223685
$587$	−29.3501	−1.21141	−0.605704	−	0.795690i	$-0.707108\pi$
$587$	−29.3501	−1.21141	−0.605704	+	0.795690i	$0.707108\pi$
$588$	−1.82702	−0.0753451
$589$	23.5702	0.971193
$590$	−4.19159	−0.172565
$591$	0.605296	0.0248986
$592$	−1.12017	−0.0460386
$593$	−13.2763	−0.545192	−0.272596	−	0.962129i	$-0.587882\pi$
$593$	−13.2763	−0.545192	−0.272596	+	0.962129i	$0.587882\pi$
$594$	2.08563	0.0855743
$595$	0.961124	0.0394023
$596$	12.5236	0.512988
$597$	−3.75015	−0.153483
$598$	19.0198	0.777777
$599$	−22.4660	−0.917936	−0.458968	−	0.888453i	$-0.651781\pi$
$599$	−22.4660	−0.917936	−0.458968	+	0.888453i	$0.651781\pi$
$600$	7.10414	0.290025
$601$	0.0585652	0.00238893	0.00119446	−	0.999999i	$-0.499620\pi$
$601$	0.0585652	0.00238893	0.00119446	+	0.999999i	$0.499620\pi$
$602$	4.34661	0.177154
$603$	−4.70870	−0.191753
$604$	17.4366	0.709487
$605$	−10.3640	−0.421357
$606$	−2.79302	−0.113459
$607$	44.6784	1.81344	0.906720	−	0.421734i	$-0.138578\pi$
$607$	44.6784	1.81344	0.906720	+	0.421734i	$0.138578\pi$
$608$	14.2817	0.579198
$609$	3.76875	0.152718
$610$	2.04716	0.0828872
$611$	−42.7723	−1.73038
$612$	−2.39688	−0.0968881
$613$	−21.4661	−0.867009	−0.433504	−	0.901151i	$-0.642723\pi$
$613$	−21.4661	−0.867009	−0.433504	+	0.901151i	$0.642723\pi$
$614$	6.07967	0.245356
$615$	7.16617	0.288968
$616$	7.98174	0.321593
$617$	45.9541	1.85004	0.925021	−	0.379915i	$-0.124047\pi$
$617$	45.9541	1.85004	0.925021	+	0.379915i	$0.124047\pi$
$618$	−5.82841	−0.234453
$619$	−19.8924	−0.799542	−0.399771	−	0.916615i	$-0.630910\pi$
$619$	−19.8924	−0.799542	−0.399771	+	0.916615i	$0.630910\pi$
$620$	9.78119	0.392822
$621$	7.49235	0.300658
$622$	1.58705	0.0636348
$623$	6.43857	0.257956
$624$	18.2624	0.731082
$625$	17.2372	0.689487
$626$	0.689509	0.0275583
$627$	16.1745	0.645947
$628$	−0.494597	−0.0197366
$629$	−0.491154	−0.0195836
$630$	0.304701	0.0121396
$631$	−6.52952	−0.259936	−0.129968	−	0.991518i	$-0.541488\pi$
$631$	−6.52952	−0.259936	−0.129968	+	0.991518i	$0.541488\pi$
$632$	−22.5850	−0.898383
$633$	5.43133	0.215876
$634$	−13.4221	−0.533061
$635$	−2.51394	−0.0997627
$636$	−18.4352	−0.731002
$637$	6.10366	0.241836
$638$	−7.86021	−0.311189
$639$	−8.10217	−0.320517
$640$	7.74998	0.306345
$641$	−39.3019	−1.55233	−0.776166	−	0.630528i	$-0.782838\pi$
$641$	−39.3019	−1.55233	−0.776166	+	0.630528i	$0.782838\pi$
$642$	6.95837	0.274625
$643$	−23.1296	−0.912142	−0.456071	−	0.889943i	$-0.650744\pi$
$643$	−23.1296	−0.912142	−0.456071	+	0.889943i	$0.650744\pi$
$644$	13.6887	0.539410
$645$	7.65649	0.301474
$646$	1.75991	0.0692429
$647$	29.9355	1.17689	0.588443	−	0.808538i	$-0.299741\pi$
$647$	29.9355	1.17689	0.588443	+	0.808538i	$0.299741\pi$
$648$	−1.59169	−0.0625274
$649$	68.9834	2.70784
$650$	−11.3303	−0.444411
$651$	−7.30755	−0.286405
$652$	−7.55602	−0.295917
$653$	−28.2952	−1.10728	−0.553638	−	0.832757i	$-0.686761\pi$
$653$	−28.2952	−1.10728	−0.553638	+	0.832757i	$0.686761\pi$
$654$	0.537472	0.0210168
$655$	−10.6512	−0.416176
$656$	−29.2670	−1.14268
$657$	11.7348	0.457819
$658$	2.91454	0.113621
$659$	−27.4900	−1.07086	−0.535429	−	0.844580i	$-0.679850\pi$
$659$	−27.4900	−1.07086	−0.535429	+	0.844580i	$0.679850\pi$
$660$	6.71212	0.261269
$661$	−38.2206	−1.48661	−0.743305	−	0.668953i	$-0.766743\pi$
$661$	−38.2206	−1.48661	−0.743305	+	0.668953i	$0.766743\pi$
$662$	13.2666	0.515620
$663$	8.00743	0.310983
$664$	15.2159	0.590490
$665$	2.36302	0.0916341
$666$	−0.155708	−0.00603358
$667$	−28.2368	−1.09333
$668$	−31.5784	−1.22180
$669$	24.4431	0.945025
$670$	1.43475	0.0554291
$671$	−33.6913	−1.30064
$672$	−4.42779	−0.170806
$673$	18.9635	0.730988	0.365494	−	0.930814i	$-0.380900\pi$
$673$	18.9635	0.730988	0.365494	+	0.930814i	$0.380900\pi$
$674$	8.82767	0.340029
$675$	−4.46327	−0.171791
$676$	−44.3138	−1.70438
$677$	17.5534	0.674634	0.337317	−	0.941391i	$-0.390481\pi$
$677$	17.5534	0.674634	0.337317	+	0.941391i	$0.390481\pi$
$678$	7.03778	0.270284
$679$	−3.64236	−0.139781
$680$	1.52981	0.0586655
$681$	−8.40236	−0.321979
$682$	15.2408	0.583601
$683$	0.208426	0.00797519	0.00398759	−	0.999992i	$-0.498731\pi$
$683$	0.208426	0.00797519	0.00398759	+	0.999992i	$0.498731\pi$
$684$	−5.89298	−0.225324
$685$	7.15451	0.273360
$686$	−0.415908	−0.0158794
$687$	27.6791	1.05603
$688$	−31.2695	−1.19214
$689$	61.5877	2.34631
$690$	−2.28293	−0.0869096
$691$	−30.0864	−1.14454	−0.572271	−	0.820064i	$-0.693938\pi$
$691$	−30.0864	−1.14454	−0.572271	+	0.820064i	$0.693938\pi$
$692$	26.5868	1.01068
$693$	−5.01464	−0.190490
$694$	−7.90356	−0.300015
$695$	−4.88329	−0.185234
$696$	5.99868	0.227379
$697$	−12.8326	−0.486068
$698$	6.64305	0.251443
$699$	−6.51272	−0.246334
$700$	−8.15449	−0.308211
$701$	16.2779	0.614807	0.307404	−	0.951579i	$-0.400540\pi$
$701$	16.2779	0.614807	0.307404	+	0.951579i	$0.400540\pi$
$702$	2.53856	0.0958118
$703$	−1.20755	−0.0455437
$704$	−20.7733	−0.782924
$705$	5.13392	0.193355
$706$	8.15302	0.306843
$707$	6.71547	0.252561
$708$	−25.1332	−0.944565
$709$	47.0591	1.76734	0.883670	−	0.468110i	$-0.155065\pi$
$709$	47.0591	1.76734	0.883670	+	0.468110i	$0.155065\pi$
$710$	2.46874	0.0926501
$711$	14.1893	0.532142
$712$	10.2482	0.384067
$713$	54.7507	2.05043
$714$	−0.545632	−0.0204198
$715$	−22.4237	−0.838597
$716$	−9.79294	−0.365979
$717$	−4.95110	−0.184902
$718$	10.5734	0.394594
$719$	32.3958	1.20816	0.604080	−	0.796924i	$-0.293541\pi$
$719$	32.3958	1.20816	0.604080	+	0.796924i	$0.293541\pi$
$720$	−2.19202	−0.0816918
$721$	14.0137	0.521897
$722$	−3.57532	−0.133060
$723$	−18.0207	−0.670197
$724$	5.50890	0.204737
$725$	16.8210	0.624715
$726$	5.88367	0.218364
$727$	−0.889718	−0.0329978	−0.0164989	−	0.999864i	$-0.505252\pi$
$727$	−0.889718	−0.0329978	−0.0164989	+	0.999864i	$0.505252\pi$
$728$	9.71512	0.360066
$729$	1.00000	0.0370370
$730$	−3.57561	−0.132339
$731$	−13.7106	−0.507104
$732$	12.2750	0.453698
$733$	−25.6122	−0.946008	−0.473004	−	0.881060i	$-0.656830\pi$
$733$	−25.6122	−0.946008	−0.473004	+	0.881060i	$0.656830\pi$
$734$	2.87089	0.105966
$735$	−0.732616	−0.0270230
$736$	33.1746	1.22283
$737$	−23.6124	−0.869775
$738$	−4.06825	−0.149754
$739$	−39.7629	−1.46270	−0.731351	−	0.682001i	$-0.761110\pi$
$739$	−39.7629	−1.46270	−0.731351	+	0.682001i	$0.761110\pi$
$740$	−0.501112	−0.0184213
$741$	19.6871	0.723224
$742$	−4.19663	−0.154063
$743$	38.9029	1.42721	0.713604	−	0.700550i	$-0.247062\pi$
$743$	38.9029	1.42721	0.713604	+	0.700550i	$0.247062\pi$
$744$	−11.6313	−0.426426
$745$	5.02185	0.183986
$746$	8.44796	0.309302
$747$	−9.55958	−0.349767
$748$	−12.0195	−0.439476
$749$	−16.7305	−0.611321
$750$	2.88347	0.105289
$751$	37.2046	1.35762	0.678808	−	0.734316i	$-0.262497\pi$
$751$	37.2046	1.35762	0.678808	+	0.734316i	$0.262497\pi$
$752$	−20.9672	−0.764596
$753$	14.0612	0.512419
$754$	−9.56721	−0.348417
$755$	6.99191	0.254462
$756$	1.82702	0.0664481
$757$	−34.5870	−1.25709	−0.628543	−	0.777775i	$-0.716349\pi$
$757$	−34.5870	−1.25709	−0.628543	+	0.777775i	$0.716349\pi$
$758$	2.52755	0.0918046
$759$	37.5714	1.36376
$760$	3.76120	0.136433
$761$	−21.5966	−0.782875	−0.391437	−	0.920205i	$-0.628022\pi$
$761$	−21.5966	−0.782875	−0.391437	+	0.920205i	$0.628022\pi$
$762$	1.42717	0.0517009
$763$	−1.29229	−0.0467839
$764$	−16.2828	−0.589091
$765$	−0.961124	−0.0347495
$766$	−0.415908	−0.0150274
$767$	83.9645	3.03178
$768$	3.88539	0.140202
$769$	18.6659	0.673109	0.336554	−	0.941664i	$-0.390738\pi$
$769$	18.6659	0.673109	0.336554	+	0.941664i	$0.390738\pi$
$770$	1.52796	0.0550640
$771$	−4.77554	−0.171987
$772$	2.65392	0.0955166
$773$	53.0525	1.90817	0.954083	−	0.299543i	$-0.0968343\pi$
$773$	53.0525	1.90817	0.954083	+	0.299543i	$0.0968343\pi$
$774$	−4.34661	−0.156236
$775$	−32.6156	−1.17159
$776$	−5.79750	−0.208118
$777$	0.374382	0.0134309
$778$	0.987606	0.0354074
$779$	−31.5502	−1.13040
$780$	8.16978	0.292525
$781$	−40.6294	−1.45384
$782$	4.08807	0.146189
$783$	−3.76875	−0.134684
$784$	2.99205	0.106859
$785$	−0.198328	−0.00707864
$786$	6.04669	0.215678
$787$	16.8134	0.599334	0.299667	−	0.954044i	$-0.403124\pi$
$787$	16.8134	0.599334	0.299667	+	0.954044i	$0.403124\pi$
$788$	−1.10589	−0.0393956
$789$	−5.01172	−0.178422
$790$	−4.32350	−0.153823
$791$	−16.9215	−0.601659
$792$	−7.98174	−0.283619
$793$	−41.0080	−1.45624
$794$	−3.34431	−0.118685
$795$	−7.39231	−0.262178
$796$	6.85159	0.242848
$797$	−0.424180	−0.0150252	−0.00751261	−	0.999972i	$-0.502391\pi$
$797$	−0.424180	−0.0150252	−0.00751261	+	0.999972i	$0.502391\pi$
$798$	−1.34149	−0.0474884
$799$	−9.19339	−0.325239
$800$	−19.7624	−0.698708
$801$	−6.43857	−0.227496
$802$	−1.34628	−0.0475388
$803$	58.8458	2.07662
$804$	8.60290	0.303401
$805$	5.48902	0.193463
$806$	18.5507	0.653419
$807$	−9.26215	−0.326043
$808$	10.6889	0.376036
$809$	37.1611	1.30652	0.653258	−	0.757135i	$-0.273402\pi$
$809$	37.1611	1.30652	0.653258	+	0.757135i	$0.273402\pi$
$810$	−0.304701	−0.0107061
$811$	14.8323	0.520834	0.260417	−	0.965496i	$-0.416140\pi$
$811$	14.8323	0.520834	0.260417	+	0.965496i	$0.416140\pi$
$812$	−6.88559	−0.241637
$813$	−27.4600	−0.963064
$814$	−0.780821	−0.0273678
$815$	−3.02989	−0.106132
$816$	3.92528	0.137412
$817$	−33.7089	−1.17932
$818$	−13.0550	−0.456458
$819$	−6.10366	−0.213279
$820$	−13.0927	−0.457218
$821$	−26.5228	−0.925653	−0.462827	−	0.886449i	$-0.653165\pi$
$821$	−26.5228	−0.925653	−0.462827	+	0.886449i	$0.653165\pi$
$822$	−4.06163	−0.141666
$823$	−33.7945	−1.17800	−0.589002	−	0.808132i	$-0.700479\pi$
$823$	−33.7945	−1.17800	−0.589002	+	0.808132i	$0.700479\pi$
$824$	22.3054	0.777047
$825$	−22.3817	−0.779230
$826$	−5.72140	−0.199073
$827$	6.97569	0.242568	0.121284	−	0.992618i	$-0.461299\pi$
$827$	6.97569	0.242568	0.121284	+	0.992618i	$0.461299\pi$
$828$	−13.6887	−0.475715
$829$	21.2194	0.736981	0.368490	−	0.929632i	$-0.379875\pi$
$829$	21.2194	0.736981	0.368490	+	0.929632i	$0.379875\pi$
$830$	2.91281	0.101105
$831$	1.81901	0.0631009
$832$	−25.2846	−0.876587
$833$	1.31191	0.0454549
$834$	2.77225	0.0959952
$835$	−12.6626	−0.438207
$836$	−29.5511	−1.02205
$837$	7.30755	0.252586
$838$	13.4817	0.465718
$839$	30.0492	1.03741	0.518707	−	0.854952i	$-0.326414\pi$
$839$	30.0492	1.03741	0.518707	+	0.854952i	$0.326414\pi$
$840$	−1.16610	−0.0402342
$841$	−14.7965	−0.510224
$842$	−6.21119	−0.214052
$843$	−2.91237	−0.100307
$844$	−9.92315	−0.341569
$845$	−17.7694	−0.611285
$846$	−2.91454	−0.100204
$847$	−14.1466	−0.486082
$848$	30.1906	1.03675
$849$	−11.2278	−0.385335
$850$	−2.43531	−0.0835303
$851$	−2.80500	−0.0961542
$852$	14.8028	0.507137
$853$	3.51976	0.120514	0.0602572	−	0.998183i	$-0.480808\pi$
$853$	3.51976	0.120514	0.0602572	+	0.998183i	$0.480808\pi$
$854$	2.79432	0.0956196
$855$	−2.36302	−0.0808137
$856$	−26.6298	−0.910188
$857$	41.8986	1.43123	0.715614	−	0.698496i	$-0.246147\pi$
$857$	41.8986	1.43123	0.715614	+	0.698496i	$0.246147\pi$
$858$	12.7300	0.434594
$859$	−18.2582	−0.622961	−0.311480	−	0.950253i	$-0.600825\pi$
$859$	−18.2582	−0.622961	−0.311480	+	0.950253i	$0.600825\pi$
$860$	−13.9886	−0.477006
$861$	9.78161	0.333356
$862$	2.05785	0.0700907
$863$	−20.9315	−0.712517	−0.356258	−	0.934387i	$-0.615948\pi$
$863$	−20.9315	−0.712517	−0.356258	+	0.934387i	$0.615948\pi$
$864$	4.42779	0.150637
$865$	10.6610	0.362486
$866$	−12.8564	−0.436877
$867$	−15.2789	−0.518899
$868$	13.3510	0.453164
$869$	71.1543	2.41375
$870$	1.14834	0.0389325
$871$	−28.7403	−0.973829
$872$	−2.05692	−0.0696560
$873$	3.64236	0.123275
$874$	10.0509	0.339978
$875$	−6.93295	−0.234376
$876$	−21.4398	−0.724382
$877$	−34.1414	−1.15287	−0.576436	−	0.817142i	$-0.695557\pi$
$877$	−34.1414	−1.15287	−0.576436	+	0.817142i	$0.695557\pi$
$878$	6.48249	0.218774
$879$	13.0193	0.439131
$880$	−10.9922	−0.370547
$881$	35.4047	1.19282	0.596408	−	0.802682i	$-0.296594\pi$
$881$	35.4047	1.19282	0.596408	+	0.802682i	$0.296594\pi$
$882$	0.415908	0.0140044
$883$	29.5485	0.994385	0.497192	−	0.867640i	$-0.334364\pi$
$883$	29.5485	0.994385	0.497192	+	0.867640i	$0.334364\pi$
$884$	−14.6297	−0.492051
$885$	−10.0782	−0.338774
$886$	7.56438	0.254130
$887$	−44.0598	−1.47938	−0.739692	−	0.672945i	$-0.765029\pi$
$887$	−44.0598	−1.47938	−0.739692	+	0.672945i	$0.765029\pi$
$888$	0.595900	0.0199971
$889$	−3.43146	−0.115087
$890$	1.96184	0.0657610
$891$	5.01464	0.167997
$892$	−44.6580	−1.49526
$893$	−22.6029	−0.756377
$894$	−2.85092	−0.0953489
$895$	−3.92687	−0.131261
$896$	10.5785	0.353403
$897$	45.7308	1.52691
$898$	9.71915	0.324332
$899$	−27.5403	−0.918522
$900$	8.15449	0.271816
$901$	13.2375	0.441006
$902$	−20.4008	−0.679272
$903$	10.4509	0.347784
$904$	−26.9337	−0.895803
$905$	2.20901	0.0734301
$906$	−3.96932	−0.131872
$907$	−33.7986	−1.12227	−0.561133	−	0.827726i	$-0.689634\pi$
$907$	−33.7986	−1.12227	−0.561133	+	0.827726i	$0.689634\pi$
$908$	15.3513	0.509450
$909$	−6.71547	−0.222738
$910$	1.85979	0.0616515
$911$	15.7365	0.521375	0.260687	−	0.965423i	$-0.416051\pi$
$911$	15.7365	0.521375	0.260687	+	0.965423i	$0.416051\pi$
$912$	9.65072	0.319567
$913$	−47.9378	−1.58651
$914$	−14.4103	−0.476650
$915$	4.92215	0.162721
$916$	−50.5704	−1.67089
$917$	−14.5385	−0.480105
$918$	0.545632	0.0180086
$919$	−49.6336	−1.63726	−0.818630	−	0.574321i	$-0.805266\pi$
$919$	−49.6336	−1.63726	−0.818630	+	0.574321i	$0.805266\pi$
$920$	8.73681	0.288044
$921$	14.6178	0.481674
$922$	−8.32888	−0.274297
$923$	−49.4529	−1.62776
$924$	9.16184	0.301402
$925$	1.67097	0.0549411
$926$	−7.60795	−0.250013
$927$	−14.0137	−0.460270
$928$	−16.6873	−0.547786
$929$	46.9320	1.53979	0.769895	−	0.638171i	$-0.220309\pi$
$929$	46.9320	1.53979	0.769895	+	0.638171i	$0.220309\pi$
$930$	−2.22662	−0.0730137
$931$	3.22546	0.105710
$932$	11.8989	0.389761
$933$	3.81586	0.124926
$934$	5.01134	0.163976
$935$	−4.81969	−0.157621
$936$	−9.71512	−0.317549
$937$	−15.7073	−0.513134	−0.256567	−	0.966526i	$-0.582591\pi$
$937$	−15.7073	−0.513134	−0.256567	+	0.966526i	$0.582591\pi$
$938$	1.95839	0.0639436
$939$	1.65784	0.0541015
$940$	−9.37978	−0.305935
$941$	−47.0794	−1.53474	−0.767372	−	0.641202i	$-0.778436\pi$
$941$	−47.0794	−1.53474	−0.767372	+	0.641202i	$0.778436\pi$
$942$	0.112591	0.00366843
$943$	−73.2873	−2.38656
$944$	41.1598	1.33964
$945$	0.732616	0.0238320
$946$	−21.7966	−0.708670
$947$	0.556813	0.0180940	0.00904700	−	0.999959i	$-0.497120\pi$
$947$	0.556813	0.0180940	0.00904700	+	0.999959i	$0.497120\pi$
$948$	−25.9242	−0.841979
$949$	71.6253	2.32506
$950$	−5.98745	−0.194259
$951$	−32.2719	−1.04649
$952$	2.08815	0.0676772
$953$	−55.6127	−1.80147	−0.900736	−	0.434367i	$-0.856972\pi$
$953$	−55.6127	−1.80147	−0.900736	+	0.434367i	$0.856972\pi$
$954$	4.19663	0.135871
$955$	−6.52923	−0.211281
$956$	9.04576	0.292561
$957$	−18.8989	−0.610915
$958$	6.27718	0.202806
$959$	9.76569	0.315351
$960$	3.03489	0.0979507
$961$	22.4003	0.722590
$962$	−0.950392	−0.0306419
$963$	16.7305	0.539134
$964$	32.9242	1.06042
$965$	1.06419	0.0342576
$966$	−3.11613	−0.100260
$967$	6.71127	0.215820	0.107910	−	0.994161i	$-0.465584\pi$
$967$	6.71127	0.215820	0.107910	+	0.994161i	$0.465584\pi$
$968$	−22.5169	−0.723722
$969$	4.23150	0.135935
$970$	−1.10983	−0.0356345
$971$	32.2804	1.03593	0.517963	−	0.855403i	$-0.326690\pi$
$971$	32.2804	1.03593	0.517963	+	0.855403i	$0.326690\pi$
$972$	−1.82702	−0.0586017
$973$	−6.66554	−0.213687
$974$	12.4772	0.399796
$975$	−27.2423	−0.872452
$976$	−20.1023	−0.643460
$977$	18.8293	0.602402	0.301201	−	0.953561i	$-0.402612\pi$
$977$	18.8293	0.602402	0.301201	+	0.953561i	$0.402612\pi$
$978$	1.72007	0.0550019
$979$	−32.2871	−1.03190
$980$	1.33851	0.0427570
$981$	1.29229	0.0412595
$982$	11.2009	0.357435
$983$	−9.64167	−0.307522	−0.153761	−	0.988108i	$-0.549139\pi$
$983$	−9.64167	−0.307522	−0.153761	+	0.988108i	$0.549139\pi$
$984$	15.5693	0.496330
$985$	−0.443450	−0.0141295
$986$	−2.05635	−0.0654876
$987$	7.00765	0.223056
$988$	−35.9687	−1.14432
$989$	−78.3017	−2.48985
$990$	−1.52796	−0.0485619
$991$	−9.33663	−0.296588	−0.148294	−	0.988943i	$-0.547378\pi$
$991$	−9.33663	−0.296588	−0.148294	+	0.988943i	$0.547378\pi$
$992$	32.3563	1.02731
$993$	31.8978	1.01225
$994$	3.36976	0.106882
$995$	2.74742	0.0870990
$996$	17.4655	0.553417
$997$	32.6734	1.03478	0.517389	−	0.855751i	$-0.326904\pi$
$997$	32.6734	1.03478	0.517389	+	0.855751i	$0.326904\pi$
$998$	−1.55290	−0.0491561
$999$	−0.374382	−0.0118449

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.p.1.20	✓	41

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.p.1.20	✓	41	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q+0.415908 q^{2} +1.00000 q^{3} -1.82702 q^{4} -0.732616 q^{5} +0.415908 q^{6} -1.00000 q^{7} -1.59169 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.415908 q^{2} +1.00000 q^{3} -1.82702 q^{4} -0.732616 q^{5} +0.415908 q^{6} -1.00000 q^{7} -1.59169 q^{8} +1.00000 q^{9} -0.304701 q^{10} +5.01464 q^{11} -1.82702 q^{12} +6.10366 q^{13} -0.415908 q^{14} -0.732616 q^{15} +2.99205 q^{16} +1.31191 q^{17} +0.415908 q^{18} +3.22546 q^{19} +1.33851 q^{20} -1.00000 q^{21} +2.08563 q^{22} +7.49235 q^{23} -1.59169 q^{24} -4.46327 q^{25} +2.53856 q^{26} +1.00000 q^{27} +1.82702 q^{28} -3.76875 q^{29} -0.304701 q^{30} +7.30755 q^{31} +4.42779 q^{32} +5.01464 q^{33} +0.545632 q^{34} +0.732616 q^{35} -1.82702 q^{36} -0.374382 q^{37} +1.34149 q^{38} +6.10366 q^{39} +1.16610 q^{40} -9.78161 q^{41} -0.415908 q^{42} -10.4509 q^{43} -9.16184 q^{44} -0.732616 q^{45} +3.11613 q^{46} -7.00765 q^{47} +2.99205 q^{48} +1.00000 q^{49} -1.85631 q^{50} +1.31191 q^{51} -11.1515 q^{52} +10.0903 q^{53} +0.415908 q^{54} -3.67380 q^{55} +1.59169 q^{56} +3.22546 q^{57} -1.56745 q^{58} +13.7564 q^{59} +1.33851 q^{60} -6.71860 q^{61} +3.03927 q^{62} -1.00000 q^{63} -4.14254 q^{64} -4.47164 q^{65} +2.08563 q^{66} -4.70870 q^{67} -2.39688 q^{68} +7.49235 q^{69} +0.304701 q^{70} -8.10217 q^{71} -1.59169 q^{72} +11.7348 q^{73} -0.155708 q^{74} -4.46327 q^{75} -5.89298 q^{76} -5.01464 q^{77} +2.53856 q^{78} +14.1893 q^{79} -2.19202 q^{80} +1.00000 q^{81} -4.06825 q^{82} -9.55958 q^{83} +1.82702 q^{84} -0.961124 q^{85} -4.34661 q^{86} -3.76875 q^{87} -7.98174 q^{88} -6.43857 q^{89} -0.304701 q^{90} -6.10366 q^{91} -13.6887 q^{92} +7.30755 q^{93} -2.91454 q^{94} -2.36302 q^{95} +4.42779 q^{96} +3.64236 q^{97} +0.415908 q^{98} +5.01464 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 41 q + 7 q^{2} + 41 q^{3} + 45 q^{4} + 17 q^{5} + 7 q^{6} - 41 q^{7} + 12 q^{8} + 41 q^{9}+O(q^{10}) \) 41 * q + 7 * q^2 + 41 * q^3 + 45 * q^4 + 17 * q^5 + 7 * q^6 - 41 * q^7 + 12 * q^8 + 41 * q^9	\( 41 q + 7 q^{2} + 41 q^{3} + 45 q^{4} + 17 q^{5} + 7 q^{6} - 41 q^{7} + 12 q^{8} + 41 q^{9} + 18 q^{10} + 8 q^{11} + 45 q^{12} + 23 q^{13} - 7 q^{14} + 17 q^{15} + 37 q^{16} + 15 q^{17} + 7 q^{18} + 15 q^{19} + 53 q^{20} - 41 q^{21} + 13 q^{22} + 44 q^{23} + 12 q^{24} + 58 q^{25} + 9 q^{26} + 41 q^{27} - 45 q^{28} + 21 q^{29} + 18 q^{30} + 39 q^{31} + 61 q^{32} + 8 q^{33} + 9 q^{34} - 17 q^{35} + 45 q^{36} + 11 q^{37} + 44 q^{38} + 23 q^{39} + 24 q^{40} + 17 q^{41} - 7 q^{42} + 7 q^{43} + 30 q^{44} + 17 q^{45} - 12 q^{46} + 36 q^{47} + 37 q^{48} + 41 q^{49} + 28 q^{50} + 15 q^{51} + 58 q^{52} + 26 q^{53} + 7 q^{54} + 32 q^{55} - 12 q^{56} + 15 q^{57} - 4 q^{58} + 33 q^{59} + 53 q^{60} + 59 q^{61} - q^{62} - 41 q^{63} + 16 q^{64} + 72 q^{65} + 13 q^{66} + 12 q^{67} + 52 q^{68} + 44 q^{69} - 18 q^{70} + 33 q^{71} + 12 q^{72} + 18 q^{73} + 42 q^{74} + 58 q^{75} + 7 q^{76} - 8 q^{77} + 9 q^{78} + 22 q^{79} + 69 q^{80} + 41 q^{81} + 41 q^{82} + 32 q^{83} - 45 q^{84} - 44 q^{85} + 11 q^{86} + 21 q^{87} + 52 q^{88} + 63 q^{89} + 18 q^{90} - 23 q^{91} + 52 q^{92} + 39 q^{93} + 17 q^{94} + 37 q^{95} + 61 q^{96} + 8 q^{97} + 7 q^{98} + 8 q^{99}+O(q^{100}) \) 41 * q + 7 * q^2 + 41 * q^3 + 45 * q^4 + 17 * q^5 + 7 * q^6 - 41 * q^7 + 12 * q^8 + 41 * q^9 + 18 * q^10 + 8 * q^11 + 45 * q^12 + 23 * q^13 - 7 * q^14 + 17 * q^15 + 37 * q^16 + 15 * q^17 + 7 * q^18 + 15 * q^19 + 53 * q^20 - 41 * q^21 + 13 * q^22 + 44 * q^23 + 12 * q^24 + 58 * q^25 + 9 * q^26 + 41 * q^27 - 45 * q^28 + 21 * q^29 + 18 * q^30 + 39 * q^31 + 61 * q^32 + 8 * q^33 + 9 * q^34 - 17 * q^35 + 45 * q^36 + 11 * q^37 + 44 * q^38 + 23 * q^39 + 24 * q^40 + 17 * q^41 - 7 * q^42 + 7 * q^43 + 30 * q^44 + 17 * q^45 - 12 * q^46 + 36 * q^47 + 37 * q^48 + 41 * q^49 + 28 * q^50 + 15 * q^51 + 58 * q^52 + 26 * q^53 + 7 * q^54 + 32 * q^55 - 12 * q^56 + 15 * q^57 - 4 * q^58 + 33 * q^59 + 53 * q^60 + 59 * q^61 - q^62 - 41 * q^63 + 16 * q^64 + 72 * q^65 + 13 * q^66 + 12 * q^67 + 52 * q^68 + 44 * q^69 - 18 * q^70 + 33 * q^71 + 12 * q^72 + 18 * q^73 + 42 * q^74 + 58 * q^75 + 7 * q^76 - 8 * q^77 + 9 * q^78 + 22 * q^79 + 69 * q^80 + 41 * q^81 + 41 * q^82 + 32 * q^83 - 45 * q^84 - 44 * q^85 + 11 * q^86 + 21 * q^87 + 52 * q^88 + 63 * q^89 + 18 * q^90 - 23 * q^91 + 52 * q^92 + 39 * q^93 + 17 * q^94 + 37 * q^95 + 61 * q^96 + 8 * q^97 + 7 * q^98 + 8 * q^99

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