LMFDB - Embedded newform 4027.2.a.b.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4027,2,Mod(1,4027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4027, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4027 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4027.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:	$32.1557568940$
Analytic rank:	$1$
Dimension:	$159$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	4027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.63499 q^{2} -1.76075 q^{3} +4.94319 q^{4} +1.57936 q^{5} +4.63957 q^{6} +3.18044 q^{7} -7.75529 q^{8} +0.100248 q^{9} +O(q^{10})$	\(q-2.63499 q^{2} -1.76075 q^{3} +4.94319 q^{4} +1.57936 q^{5} +4.63957 q^{6} +3.18044 q^{7} -7.75529 q^{8} +0.100248 q^{9} -4.16161 q^{10} -6.06898 q^{11} -8.70374 q^{12} -0.690586 q^{13} -8.38044 q^{14} -2.78087 q^{15} +10.5488 q^{16} -1.72528 q^{17} -0.264152 q^{18} -4.91229 q^{19} +7.80709 q^{20} -5.59997 q^{21} +15.9917 q^{22} +0.203285 q^{23} +13.6551 q^{24} -2.50562 q^{25} +1.81969 q^{26} +5.10574 q^{27} +15.7215 q^{28} +8.42978 q^{29} +7.32756 q^{30} +8.69812 q^{31} -12.2853 q^{32} +10.6860 q^{33} +4.54609 q^{34} +5.02307 q^{35} +0.495543 q^{36} -5.81830 q^{37} +12.9438 q^{38} +1.21595 q^{39} -12.2484 q^{40} +3.89329 q^{41} +14.7559 q^{42} +7.98793 q^{43} -30.0001 q^{44} +0.158327 q^{45} -0.535656 q^{46} +3.77642 q^{47} -18.5738 q^{48} +3.11521 q^{49} +6.60228 q^{50} +3.03778 q^{51} -3.41370 q^{52} -7.23258 q^{53} -13.4536 q^{54} -9.58511 q^{55} -24.6653 q^{56} +8.64932 q^{57} -22.2124 q^{58} -10.5298 q^{59} -13.7464 q^{60} +11.4648 q^{61} -22.9195 q^{62} +0.318831 q^{63} +11.2743 q^{64} -1.09068 q^{65} -28.1575 q^{66} -7.10973 q^{67} -8.52837 q^{68} -0.357935 q^{69} -13.2358 q^{70} -1.56902 q^{71} -0.777449 q^{72} +12.6648 q^{73} +15.3312 q^{74} +4.41177 q^{75} -24.2824 q^{76} -19.3020 q^{77} -3.20402 q^{78} +12.1826 q^{79} +16.6603 q^{80} -9.29069 q^{81} -10.2588 q^{82} +8.11678 q^{83} -27.6817 q^{84} -2.72484 q^{85} -21.0482 q^{86} -14.8428 q^{87} +47.0667 q^{88} +8.93589 q^{89} -0.417191 q^{90} -2.19637 q^{91} +1.00488 q^{92} -15.3152 q^{93} -9.95084 q^{94} -7.75828 q^{95} +21.6314 q^{96} +10.8343 q^{97} -8.20855 q^{98} -0.608400 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 159 q - 22 q^{2} - 19 q^{3} + 148 q^{4} - 70 q^{5} - 23 q^{6} - 19 q^{7} - 66 q^{8} + 126 q^{9}+O(q^{10}) $ 159 * q - 22 * q^2 - 19 * q^3 + 148 * q^4 - 70 * q^5 - 23 * q^6 - 19 * q^7 - 66 * q^8 + 126 * q^9	\( 159 q - 22 q^{2} - 19 q^{3} + 148 q^{4} - 70 q^{5} - 23 q^{6} - 19 q^{7} - 66 q^{8} + 126 q^{9} - 23 q^{10} - 33 q^{11} - 57 q^{12} - 90 q^{13} - 28 q^{14} - 22 q^{15} + 130 q^{16} - 145 q^{17} - 50 q^{18} - 28 q^{19} - 121 q^{20} - 69 q^{21} - 26 q^{22} - 79 q^{23} - 62 q^{24} + 123 q^{25} - 40 q^{26} - 70 q^{27} - 43 q^{28} - 109 q^{29} - 43 q^{30} - 21 q^{31} - 139 q^{32} - 83 q^{33} - 93 q^{35} + 75 q^{36} - 65 q^{37} - 122 q^{38} - 18 q^{39} - 43 q^{40} - 71 q^{41} - 88 q^{42} - 72 q^{43} - 79 q^{44} - 181 q^{45} - 11 q^{46} - 114 q^{47} - 118 q^{48} + 118 q^{49} - 77 q^{50} - 29 q^{51} - 169 q^{52} - 220 q^{53} - 80 q^{54} - 37 q^{55} - 72 q^{56} - 90 q^{57} - 8 q^{58} - 60 q^{59} - 42 q^{60} - 108 q^{61} - 152 q^{62} - 65 q^{63} + 114 q^{64} - 81 q^{65} - 40 q^{66} - 50 q^{67} - 319 q^{68} - 103 q^{69} + 4 q^{70} - 7 q^{71} - 129 q^{72} - 94 q^{73} - 79 q^{74} - 59 q^{75} - 46 q^{76} - 329 q^{77} + 8 q^{78} - 18 q^{79} - 190 q^{80} + 59 q^{81} - 56 q^{82} - 201 q^{83} - 71 q^{84} - 26 q^{85} - 52 q^{86} - 126 q^{87} - 66 q^{88} - 114 q^{89} - 33 q^{90} - 30 q^{91} - 204 q^{92} - 125 q^{93} + 9 q^{94} - 84 q^{95} - 88 q^{96} - 56 q^{97} - 110 q^{98} - 46 q^{99}+O(q^{100}) \) 159 * q - 22 * q^2 - 19 * q^3 + 148 * q^4 - 70 * q^5 - 23 * q^6 - 19 * q^7 - 66 * q^8 + 126 * q^9 - 23 * q^10 - 33 * q^11 - 57 * q^12 - 90 * q^13 - 28 * q^14 - 22 * q^15 + 130 * q^16 - 145 * q^17 - 50 * q^18 - 28 * q^19 - 121 * q^20 - 69 * q^21 - 26 * q^22 - 79 * q^23 - 62 * q^24 + 123 * q^25 - 40 * q^26 - 70 * q^27 - 43 * q^28 - 109 * q^29 - 43 * q^30 - 21 * q^31 - 139 * q^32 - 83 * q^33 - 93 * q^35 + 75 * q^36 - 65 * q^37 - 122 * q^38 - 18 * q^39 - 43 * q^40 - 71 * q^41 - 88 * q^42 - 72 * q^43 - 79 * q^44 - 181 * q^45 - 11 * q^46 - 114 * q^47 - 118 * q^48 + 118 * q^49 - 77 * q^50 - 29 * q^51 - 169 * q^52 - 220 * q^53 - 80 * q^54 - 37 * q^55 - 72 * q^56 - 90 * q^57 - 8 * q^58 - 60 * q^59 - 42 * q^60 - 108 * q^61 - 152 * q^62 - 65 * q^63 + 114 * q^64 - 81 * q^65 - 40 * q^66 - 50 * q^67 - 319 * q^68 - 103 * q^69 + 4 * q^70 - 7 * q^71 - 129 * q^72 - 94 * q^73 - 79 * q^74 - 59 * q^75 - 46 * q^76 - 329 * q^77 + 8 * q^78 - 18 * q^79 - 190 * q^80 + 59 * q^81 - 56 * q^82 - 201 * q^83 - 71 * q^84 - 26 * q^85 - 52 * q^86 - 126 * q^87 - 66 * q^88 - 114 * q^89 - 33 * q^90 - 30 * q^91 - 204 * q^92 - 125 * q^93 + 9 * q^94 - 84 * q^95 - 88 * q^96 - 56 * q^97 - 110 * q^98 - 46 * 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 $

$2$	−2.63499	−1.86322	−0.931611	−	0.363457i	$-0.881596\pi$
$2$	−2.63499	−1.86322	−0.931611	+	0.363457i	$0.881596\pi$
$3$	−1.76075	−1.01657	−0.508285	−	0.861189i	$-0.669720\pi$
$3$	−1.76075	−1.01657	−0.508285	+	0.861189i	$0.669720\pi$
$4$	4.94319	2.47160
$5$	1.57936	0.706312	0.353156	−	0.935564i	$-0.385108\pi$
$5$	1.57936	0.706312	0.353156	+	0.935564i	$0.385108\pi$
$6$	4.63957	1.89410
$7$	3.18044	1.20209	0.601047	−	0.799214i	$-0.294750\pi$
$7$	3.18044	1.20209	0.601047	+	0.799214i	$0.294750\pi$
$8$	−7.75529	−2.74191
$9$	0.100248	0.0334159
$10$	−4.16161	−1.31602
$11$	−6.06898	−1.82987	−0.914933	−	0.403606i	$-0.867757\pi$
$11$	−6.06898	−1.82987	−0.914933	+	0.403606i	$0.867757\pi$
$12$	−8.70374	−2.51255
$13$	−0.690586	−0.191534	−0.0957670	−	0.995404i	$-0.530530\pi$
$13$	−0.690586	−0.191534	−0.0957670	+	0.995404i	$0.530530\pi$
$14$	−8.38044	−2.23977
$15$	−2.78087	−0.718016
$16$	10.5488	2.63719
$17$	−1.72528	−0.418441	−0.209220	−	0.977869i	$-0.567093\pi$
$17$	−1.72528	−0.418441	−0.209220	+	0.977869i	$0.567093\pi$
$18$	−0.264152	−0.0622612
$19$	−4.91229	−1.12696	−0.563478	−	0.826131i	$-0.690537\pi$
$19$	−4.91229	−1.12696	−0.563478	+	0.826131i	$0.690537\pi$
$20$	7.80709	1.74572
$21$	−5.59997	−1.22201
$22$	15.9917	3.40945
$23$	0.203285	0.0423879	0.0211940	−	0.999775i	$-0.493253\pi$
$23$	0.203285	0.0423879	0.0211940	+	0.999775i	$0.493253\pi$
$24$	13.6551	2.78735
$25$	−2.50562	−0.501123
$26$	1.81969	0.356870
$27$	5.10574	0.982601
$28$	15.7215	2.97109
$29$	8.42978	1.56537	0.782686	−	0.622417i	$-0.213849\pi$
$29$	8.42978	1.56537	0.782686	+	0.622417i	$0.213849\pi$
$30$	7.32756	1.33782
$31$	8.69812	1.56223	0.781115	−	0.624388i	$-0.214651\pi$
$31$	8.69812	1.56223	0.781115	+	0.624388i	$0.214651\pi$
$32$	−12.2853	−2.17176
$33$	10.6860	1.86019
$34$	4.54609	0.779648
$35$	5.02307	0.849054
$36$	0.495543	0.0825905
$37$	−5.81830	−0.956523	−0.478261	−	0.878217i	$-0.658733\pi$
$37$	−5.81830	−0.956523	−0.478261	+	0.878217i	$0.658733\pi$
$38$	12.9438	2.09977
$39$	1.21595	0.194708
$40$	−12.2484	−1.93664
$41$	3.89329	0.608029	0.304015	−	0.952667i	$-0.401673\pi$
$41$	3.89329	0.608029	0.304015	+	0.952667i	$0.401673\pi$
$42$	14.7559	2.27688
$43$	7.98793	1.21815	0.609074	−	0.793113i	$-0.291541\pi$
$43$	7.98793	1.21815	0.609074	+	0.793113i	$0.291541\pi$
$44$	−30.0001	−4.52269
$45$	0.158327	0.0236020
$46$	−0.535656	−0.0789782
$47$	3.77642	0.550847	0.275424	−	0.961323i	$-0.411182\pi$
$47$	3.77642	0.550847	0.275424	+	0.961323i	$0.411182\pi$
$48$	−18.5738	−2.68089
$49$	3.11521	0.445030
$50$	6.60228	0.933703
$51$	3.03778	0.425375
$52$	−3.41370	−0.473395
$53$	−7.23258	−0.993472	−0.496736	−	0.867902i	$-0.665468\pi$
$53$	−7.23258	−0.993472	−0.496736	+	0.867902i	$0.665468\pi$
$54$	−13.4536	−1.83080
$55$	−9.58511	−1.29246
$56$	−24.6653	−3.29603
$57$	8.64932	1.14563
$58$	−22.2124	−2.91663
$59$	−10.5298	−1.37087	−0.685434	−	0.728135i	$-0.740387\pi$
$59$	−10.5298	−1.37087	−0.685434	+	0.728135i	$0.740387\pi$
$60$	−13.7464	−1.77465
$61$	11.4648	1.46791	0.733957	−	0.679195i	$-0.237671\pi$
$61$	11.4648	1.46791	0.733957	+	0.679195i	$0.237671\pi$
$62$	−22.9195	−2.91078
$63$	0.318831	0.0401690
$64$	11.2743	1.40928
$65$	−1.09068	−0.135283
$66$	−28.1575	−3.46594
$67$	−7.10973	−0.868592	−0.434296	−	0.900770i	$-0.643003\pi$
$67$	−7.10973	−0.868592	−0.434296	+	0.900770i	$0.643003\pi$
$68$	−8.52837	−1.03422
$69$	−0.357935	−0.0430903
$70$	−13.2358	−1.58198
$71$	−1.56902	−0.186208	−0.0931040	−	0.995656i	$-0.529679\pi$
$71$	−1.56902	−0.186208	−0.0931040	+	0.995656i	$0.529679\pi$
$72$	−0.777449	−0.0916233
$73$	12.6648	1.48230	0.741150	−	0.671339i	$-0.234281\pi$
$73$	12.6648	1.48230	0.741150	+	0.671339i	$0.234281\pi$
$74$	15.3312	1.78221
$75$	4.41177	0.509427
$76$	−24.2824	−2.78538
$77$	−19.3020	−2.19967
$78$	−3.20402	−0.362784
$79$	12.1826	1.37065	0.685325	−	0.728238i	$-0.259660\pi$
$79$	12.1826	1.37065	0.685325	+	0.728238i	$0.259660\pi$
$80$	16.6603	1.86268
$81$	−9.29069	−1.03230
$82$	−10.2588	−1.13289
$83$	8.11678	0.890932	0.445466	−	0.895299i	$-0.353038\pi$
$83$	8.11678	0.890932	0.445466	+	0.895299i	$0.353038\pi$
$84$	−27.6817	−3.02032
$85$	−2.72484	−0.295550
$86$	−21.0482	−2.26968
$87$	−14.8428	−1.59131
$88$	47.0667	5.01733
$89$	8.93589	0.947202	0.473601	−	0.880740i	$-0.342954\pi$
$89$	8.93589	0.947202	0.473601	+	0.880740i	$0.342954\pi$
$90$	−0.417191	−0.0439758
$91$	−2.19637	−0.230242
$92$	1.00488	0.104766
$93$	−15.3152	−1.58812
$94$	−9.95084	−1.02635
$95$	−7.75828	−0.795983
$96$	21.6314	2.20775
$97$	10.8343	1.10006	0.550028	−	0.835146i	$-0.314617\pi$
$97$	10.8343	1.10006	0.550028	+	0.835146i	$0.314617\pi$
$98$	−8.20855	−0.829189
$99$	−0.608400	−0.0611465
$100$	−12.3857	−1.23857
$101$	−15.3458	−1.52696	−0.763482	−	0.645829i	$-0.776512\pi$
$101$	−15.3458	−1.52696	−0.763482	+	0.645829i	$0.776512\pi$
$102$	−8.00454	−0.792567
$103$	−0.509116	−0.0501647	−0.0250824	−	0.999685i	$-0.507985\pi$
$103$	−0.509116	−0.0501647	−0.0250824	+	0.999685i	$0.507985\pi$
$104$	5.35569	0.525169
$105$	−8.84438	−0.863123
$106$	19.0578	1.85106
$107$	−5.56578	−0.538064	−0.269032	−	0.963131i	$-0.586704\pi$
$107$	−5.56578	−0.538064	−0.269032	+	0.963131i	$0.586704\pi$
$108$	25.2387	2.42859
$109$	−19.8179	−1.89821	−0.949104	−	0.314962i	$-0.898008\pi$
$109$	−19.8179	−1.89821	−0.949104	+	0.314962i	$0.898008\pi$
$110$	25.2567	2.40813
$111$	10.2446	0.972373
$112$	33.5497	3.17015
$113$	−16.0296	−1.50794	−0.753970	−	0.656909i	$-0.771864\pi$
$113$	−16.0296	−1.50794	−0.753970	+	0.656909i	$0.771864\pi$
$114$	−22.7909	−2.13456
$115$	0.321061	0.0299391
$116$	41.6700	3.86897
$117$	−0.0692295	−0.00640027
$118$	27.7460	2.55423
$119$	−5.48714	−0.503005
$120$	21.5664	1.96874
$121$	25.8325	2.34841
$122$	−30.2096	−2.73505
$123$	−6.85511	−0.618105
$124$	42.9965	3.86120
$125$	−11.8541	−1.06026
$126$	−0.840119	−0.0748438
$127$	4.43799	0.393808	0.196904	−	0.980423i	$-0.436911\pi$
$127$	4.43799	0.393808	0.196904	+	0.980423i	$0.436911\pi$
$128$	−5.13696	−0.454047
$129$	−14.0648	−1.23833
$130$	2.87395	0.252062
$131$	2.27447	0.198722	0.0993609	−	0.995051i	$-0.468320\pi$
$131$	2.27447	0.198722	0.0993609	+	0.995051i	$0.468320\pi$
$132$	52.8228	4.59763
$133$	−15.6232	−1.35471
$134$	18.7341	1.61838
$135$	8.06382	0.694023
$136$	13.3800	1.14733
$137$	−18.7981	−1.60603	−0.803017	−	0.595956i	$-0.796773\pi$
$137$	−18.7981	−1.60603	−0.803017	+	0.595956i	$0.796773\pi$
$138$	0.943157	0.0802869
$139$	8.47932	0.719206	0.359603	−	0.933105i	$-0.382912\pi$
$139$	8.47932	0.719206	0.359603	+	0.933105i	$0.382912\pi$
$140$	24.8300	2.09852
$141$	−6.64933	−0.559975
$142$	4.13435	0.346947
$143$	4.19115	0.350482
$144$	1.05749	0.0881240
$145$	13.3137	1.10564
$146$	−33.3716	−2.76185
$147$	−5.48511	−0.452404
$148$	−28.7610	−2.36414
$149$	1.83083	0.149988	0.0749938	−	0.997184i	$-0.476106\pi$
$149$	1.83083	0.149988	0.0749938	+	0.997184i	$0.476106\pi$
$150$	−11.6250	−0.949176
$151$	3.61386	0.294092	0.147046	−	0.989130i	$-0.453023\pi$
$151$	3.61386	0.294092	0.147046	+	0.989130i	$0.453023\pi$
$152$	38.0962	3.09001
$153$	−0.172955	−0.0139826
$154$	50.8607	4.09847
$155$	13.7375	1.10342
$156$	6.01068	0.481239
$157$	−15.3594	−1.22581	−0.612906	−	0.790156i	$-0.710000\pi$
$157$	−15.3594	−1.22581	−0.612906	+	0.790156i	$0.710000\pi$
$158$	−32.1011	−2.55382
$159$	12.7348	1.00993
$160$	−19.4030	−1.53394
$161$	0.646537	0.0509543
$162$	24.4809	1.92340
$163$	−8.40756	−0.658531	−0.329265	−	0.944237i	$-0.606801\pi$
$163$	−8.40756	−0.658531	−0.329265	+	0.944237i	$0.606801\pi$
$164$	19.2453	1.50280
$165$	16.8770	1.31387
$166$	−21.3877	−1.66000
$167$	3.96447	0.306780	0.153390	−	0.988166i	$-0.450981\pi$
$167$	3.96447	0.306780	0.153390	+	0.988166i	$0.450981\pi$
$168$	43.4294	3.35065
$169$	−12.5231	−0.963315
$170$	7.17992	0.550675
$171$	−0.492445	−0.0376582
$172$	39.4859	3.01077
$173$	−25.5751	−1.94444	−0.972218	−	0.234079i	$-0.924793\pi$
$173$	−25.5751	−1.94444	−0.972218	+	0.234079i	$0.924793\pi$
$174$	39.1106	2.96497
$175$	−7.96896	−0.602397
$176$	−64.0202	−4.82571
$177$	18.5404	1.39358
$178$	−23.5460	−1.76485
$179$	7.19902	0.538080	0.269040	−	0.963129i	$-0.413294\pi$
$179$	7.19902	0.538080	0.269040	+	0.963129i	$0.413294\pi$
$180$	0.782642	0.0583347
$181$	17.6484	1.31179	0.655897	−	0.754850i	$-0.272291\pi$
$181$	17.6484	1.31179	0.655897	+	0.754850i	$0.272291\pi$
$182$	5.78741	0.428992
$183$	−20.1866	−1.49224
$184$	−1.57654	−0.116224
$185$	−9.18921	−0.675604
$186$	40.3556	2.95901
$187$	10.4707	0.765691
$188$	18.6676	1.36147
$189$	16.2385	1.18118
$190$	20.4430	1.48309
$191$	−16.5801	−1.19969	−0.599846	−	0.800116i	$-0.704771\pi$
$191$	−16.5801	−1.19969	−0.599846	+	0.800116i	$0.704771\pi$
$192$	−19.8512	−1.43264
$193$	−3.30492	−0.237893	−0.118947	−	0.992901i	$-0.537952\pi$
$193$	−3.30492	−0.237893	−0.118947	+	0.992901i	$0.537952\pi$
$194$	−28.5483	−2.04965
$195$	1.92043	0.137525
$196$	15.3991	1.09993
$197$	−23.5285	−1.67633	−0.838167	−	0.545413i	$-0.816373\pi$
$197$	−23.5285	−1.67633	−0.838167	+	0.545413i	$0.816373\pi$
$198$	1.60313	0.113930
$199$	−3.72864	−0.264317	−0.132158	−	0.991229i	$-0.542191\pi$
$199$	−3.72864	−0.264317	−0.132158	+	0.991229i	$0.542191\pi$
$200$	19.4318	1.37403
$201$	12.5185	0.882985
$202$	40.4361	2.84507
$203$	26.8104	1.88172
$204$	15.0163	1.05135
$205$	6.14891	0.429458
$206$	1.34152	0.0934680
$207$	0.0203789	0.00141643
$208$	−7.28483	−0.505112
$209$	29.8126	2.06218
$210$	23.3049	1.60819
$211$	−19.4656	−1.34007	−0.670033	−	0.742331i	$-0.733720\pi$
$211$	−19.4656	−1.34007	−0.670033	+	0.742331i	$0.733720\pi$
$212$	−35.7520	−2.45546
$213$	2.76265	0.189294
$214$	14.6658	1.00253
$215$	12.6158	0.860393
$216$	−39.5965	−2.69420
$217$	27.6639	1.87795
$218$	52.2200	3.53678
$219$	−22.2995	−1.50686
$220$	−47.3811	−3.19443
$221$	1.19145	0.0801457
$222$	−26.9944	−1.81175
$223$	−8.63883	−0.578499	−0.289250	−	0.957254i	$-0.593406\pi$
$223$	−8.63883	−0.578499	−0.289250	+	0.957254i	$0.593406\pi$
$224$	−39.0728	−2.61066
$225$	−0.251182	−0.0167455
$226$	42.2379	2.80963
$227$	10.6009	0.703606	0.351803	−	0.936074i	$-0.385569\pi$
$227$	10.6009	0.703606	0.351803	+	0.936074i	$0.385569\pi$
$228$	42.7553	2.83154
$229$	22.4921	1.48632	0.743160	−	0.669114i	$-0.233326\pi$
$229$	22.4921	1.48632	0.743160	+	0.669114i	$0.233326\pi$
$230$	−0.845995	−0.0557832
$231$	33.9861	2.23612
$232$	−65.3755	−4.29211
$233$	21.8436	1.43102	0.715512	−	0.698601i	$-0.246194\pi$
$233$	21.8436	1.43102	0.715512	+	0.698601i	$0.246194\pi$
$234$	0.182419	0.0119251
$235$	5.96433	0.389070
$236$	−52.0510	−3.38823
$237$	−21.4505	−1.39336
$238$	14.4586	0.937210
$239$	1.45127	0.0938747	0.0469374	−	0.998898i	$-0.485054\pi$
$239$	1.45127	0.0938747	0.0469374	+	0.998898i	$0.485054\pi$
$240$	−29.3347	−1.89355
$241$	4.27408	0.275318	0.137659	−	0.990480i	$-0.456042\pi$
$241$	4.27408	0.275318	0.137659	+	0.990480i	$0.456042\pi$
$242$	−68.0685	−4.37561
$243$	1.04137	0.0668040
$244$	56.6726	3.62809
$245$	4.92004	0.314330
$246$	18.0632	1.15167
$247$	3.39236	0.215850
$248$	−67.4565	−4.28349
$249$	−14.2916	−0.905696
$250$	31.2354	1.97550
$251$	−12.6321	−0.797329	−0.398664	−	0.917097i	$-0.630526\pi$
$251$	−12.6321	−0.797329	−0.398664	+	0.917097i	$0.630526\pi$
$252$	1.57605	0.0992815
$253$	−1.23373	−0.0775642
$254$	−11.6941	−0.733752
$255$	4.79776	0.300447
$256$	−9.01270	−0.563294
$257$	−13.4777	−0.840717	−0.420358	−	0.907358i	$-0.638096\pi$
$257$	−13.4777	−0.840717	−0.420358	+	0.907358i	$0.638096\pi$
$258$	37.0606	2.30729
$259$	−18.5048	−1.14983
$260$	−5.39147	−0.334364
$261$	0.845065	0.0523082
$262$	−5.99322	−0.370263
$263$	5.63473	0.347452	0.173726	−	0.984794i	$-0.444419\pi$
$263$	5.63473	0.347452	0.173726	+	0.984794i	$0.444419\pi$
$264$	−82.8728	−5.10047
$265$	−11.4229	−0.701701
$266$	41.1672	2.52412
$267$	−15.7339	−0.962898
$268$	−35.1448	−2.14681
$269$	16.6569	1.01559	0.507794	−	0.861479i	$-0.330461\pi$
$269$	16.6569	1.01559	0.507794	+	0.861479i	$0.330461\pi$
$270$	−21.2481	−1.29312
$271$	−13.6909	−0.831661	−0.415830	−	0.909442i	$-0.636509\pi$
$271$	−13.6909	−0.831661	−0.415830	+	0.909442i	$0.636509\pi$
$272$	−18.1995	−1.10351
$273$	3.86726	0.234057
$274$	49.5330	2.99240
$275$	15.2065	0.916988
$276$	−1.76934	−0.106502
$277$	−20.7387	−1.24607	−0.623034	−	0.782195i	$-0.714100\pi$
$277$	−20.7387	−1.24607	−0.623034	+	0.782195i	$0.714100\pi$
$278$	−22.3429	−1.34004
$279$	0.871966	0.0522032
$280$	−38.9554	−2.32803
$281$	−5.96410	−0.355789	−0.177894	−	0.984050i	$-0.556929\pi$
$281$	−5.96410	−0.355789	−0.177894	+	0.984050i	$0.556929\pi$
$282$	17.5210	1.04336
$283$	−14.3267	−0.851635	−0.425818	−	0.904809i	$-0.640013\pi$
$283$	−14.3267	−0.851635	−0.425818	+	0.904809i	$0.640013\pi$
$284$	−7.75595	−0.460231
$285$	13.6604	0.809173
$286$	−11.0437	−0.653025
$287$	12.3824	0.730908
$288$	−1.23158	−0.0725713
$289$	−14.0234	−0.824907
$290$	−35.0815	−2.06005
$291$	−19.0765	−1.11829
$292$	62.6045	3.66365
$293$	−16.6335	−0.971741	−0.485871	−	0.874031i	$-0.661497\pi$
$293$	−16.6335	−0.971741	−0.485871	+	0.874031i	$0.661497\pi$
$294$	14.4532	0.842929
$295$	−16.6304	−0.968260
$296$	45.1226	2.62270
$297$	−30.9867	−1.79803
$298$	−4.82423	−0.279460
$299$	−0.140386	−0.00811873
$300$	21.8082	1.25910
$301$	25.4052	1.46433
$302$	−9.52251	−0.547959
$303$	27.0201	1.55227
$304$	−51.8186	−2.97200
$305$	18.1070	1.03681
$306$	0.455735	0.0260526
$307$	9.70093	0.553661	0.276831	−	0.960919i	$-0.410716\pi$
$307$	9.70093	0.553661	0.276831	+	0.960919i	$0.410716\pi$
$308$	−95.4136	−5.43670
$309$	0.896428	0.0509960
$310$	−36.1982	−2.05592
$311$	16.7808	0.951551	0.475776	−	0.879567i	$-0.342167\pi$
$311$	16.7808	0.951551	0.475776	+	0.879567i	$0.342167\pi$
$312$	−9.43005	−0.533871
$313$	−11.1678	−0.631243	−0.315622	−	0.948885i	$-0.602213\pi$
$313$	−11.1678	−0.631243	−0.315622	+	0.948885i	$0.602213\pi$
$314$	40.4719	2.28396
$315$	0.503550	0.0283719
$316$	60.2209	3.38769
$317$	11.6722	0.655576	0.327788	−	0.944751i	$-0.393697\pi$
$317$	11.6722	0.655576	0.327788	+	0.944751i	$0.393697\pi$
$318$	−33.5561	−1.88173
$319$	−51.1602	−2.86442
$320$	17.8062	0.995395
$321$	9.79995	0.546980
$322$	−1.70362	−0.0949392
$323$	8.47505	0.471564
$324$	−45.9257	−2.55143
$325$	1.73034	0.0959821
$326$	22.1539	1.22699
$327$	34.8944	1.92966
$328$	−30.1936	−1.66716
$329$	12.0107	0.662170
$330$	−44.4708	−2.44804
$331$	12.1761	0.669259	0.334629	−	0.942350i	$-0.391389\pi$
$331$	12.1761	0.669259	0.334629	+	0.942350i	$0.391389\pi$
$332$	40.1228	2.20203
$333$	−0.583271	−0.0319630
$334$	−10.4464	−0.571600
$335$	−11.2288	−0.613497
$336$	−59.0728	−3.22268
$337$	−33.0593	−1.80086	−0.900428	−	0.435005i	$-0.856747\pi$
$337$	−33.0593	−1.80086	−0.900428	+	0.435005i	$0.856747\pi$
$338$	32.9983	1.79487
$339$	28.2242	1.53293
$340$	−13.4694	−0.730480
$341$	−52.7887	−2.85867
$342$	1.29759	0.0701656
$343$	−12.3554	−0.667126
$344$	−61.9488	−3.34005
$345$	−0.565309	−0.0304352
$346$	67.3901	3.62291
$347$	−34.5825	−1.85649	−0.928243	−	0.371974i	$-0.878681\pi$
$347$	−34.5825	−1.85649	−0.928243	+	0.371974i	$0.878681\pi$
$348$	−73.3706	−3.93308
$349$	13.8994	0.744020	0.372010	−	0.928229i	$-0.378669\pi$
$349$	13.8994	0.744020	0.372010	+	0.928229i	$0.378669\pi$
$350$	20.9982	1.12240
$351$	−3.52595	−0.188202
$352$	74.5595	3.97403
$353$	−20.4841	−1.09026	−0.545130	−	0.838351i	$-0.683520\pi$
$353$	−20.4841	−1.09026	−0.545130	+	0.838351i	$0.683520\pi$
$354$	−48.8539	−2.59655
$355$	−2.47805	−0.131521
$356$	44.1718	2.34110
$357$	9.66149	0.511340
$358$	−18.9694	−1.00256
$359$	−27.1644	−1.43368	−0.716842	−	0.697236i	$-0.754413\pi$
$359$	−27.1644	−1.43368	−0.716842	+	0.697236i	$0.754413\pi$
$360$	−1.22787	−0.0647146
$361$	5.13057	0.270030
$362$	−46.5034	−2.44416
$363$	−45.4846	−2.38732
$364$	−10.8571	−0.569065
$365$	20.0023	1.04697
$366$	53.1917	2.78037
$367$	−1.21638	−0.0634943	−0.0317471	−	0.999496i	$-0.510107\pi$
$367$	−1.21638	−0.0634943	−0.0317471	+	0.999496i	$0.510107\pi$
$368$	2.14441	0.111785
$369$	0.390293	0.0203178
$370$	24.2135	1.25880
$371$	−23.0028	−1.19425
$372$	−75.7062	−3.92518
$373$	−26.2638	−1.35989	−0.679945	−	0.733263i	$-0.737996\pi$
$373$	−26.2638	−1.35989	−0.679945	+	0.733263i	$0.737996\pi$
$374$	−27.5901	−1.42665
$375$	20.8721	1.07783
$376$	−29.2872	−1.51037
$377$	−5.82149	−0.299822
$378$	−42.7884	−2.20080
$379$	5.61645	0.288497	0.144249	−	0.989541i	$-0.453923\pi$
$379$	5.61645	0.288497	0.144249	+	0.989541i	$0.453923\pi$
$380$	−38.3507	−1.96735
$381$	−7.81420	−0.400334
$382$	43.6884	2.23529
$383$	7.22280	0.369068	0.184534	−	0.982826i	$-0.440922\pi$
$383$	7.22280	0.369068	0.184534	+	0.982826i	$0.440922\pi$
$384$	9.04491	0.461571
$385$	−30.4849	−1.55365
$386$	8.70844	0.443248
$387$	0.800771	0.0407055
$388$	53.5561	2.71890
$389$	28.5937	1.44976	0.724879	−	0.688877i	$-0.241896\pi$
$389$	28.5937	1.44976	0.724879	+	0.688877i	$0.241896\pi$
$390$	−5.06031	−0.256239
$391$	−0.350723	−0.0177368
$392$	−24.1594	−1.22023
$393$	−4.00478	−0.202015
$394$	61.9974	3.12338
$395$	19.2407	0.968106
$396$	−3.00744	−0.151130
$397$	−11.5367	−0.579011	−0.289505	−	0.957176i	$-0.593491\pi$
$397$	−11.5367	−0.579011	−0.289505	+	0.957176i	$0.593491\pi$
$398$	9.82495	0.492480
$399$	27.5087	1.37716
$400$	−26.4311	−1.32156
$401$	18.4544	0.921570	0.460785	−	0.887512i	$-0.347568\pi$
$401$	18.4544	0.921570	0.460785	+	0.887512i	$0.347568\pi$
$402$	−32.9861	−1.64520
$403$	−6.00680	−0.299220
$404$	−75.8572	−3.77404
$405$	−14.6734	−0.729126
$406$	−70.6453	−3.50607
$407$	35.3111	1.75031
$408$	−23.5589	−1.16634
$409$	2.43445	0.120376	0.0601878	−	0.998187i	$-0.480830\pi$
$409$	2.43445	0.120376	0.0601878	+	0.998187i	$0.480830\pi$
$410$	−16.2023	−0.800177
$411$	33.0989	1.63265
$412$	−2.51666	−0.123987
$413$	−33.4895	−1.64791
$414$	−0.0536982	−0.00263912
$415$	12.8193	0.629276
$416$	8.48408	0.415966
$417$	−14.9300	−0.731124
$418$	−78.5559	−3.84230
$419$	−6.84795	−0.334544	−0.167272	−	0.985911i	$-0.553496\pi$
$419$	−6.84795	−0.334544	−0.167272	+	0.985911i	$0.553496\pi$
$420$	−43.7195	−2.13329
$421$	0.674531	0.0328746	0.0164373	−	0.999865i	$-0.494768\pi$
$421$	0.674531	0.0328746	0.0164373	+	0.999865i	$0.494768\pi$
$422$	51.2917	2.49684
$423$	0.378577	0.0184070
$424$	56.0908	2.72401
$425$	4.32288	0.209690
$426$	−7.27956	−0.352696
$427$	36.4631	1.76457
$428$	−27.5127	−1.32988
$429$	−7.37957	−0.356289
$430$	−33.2427	−1.60310
$431$	−0.789936	−0.0380499	−0.0190249	−	0.999819i	$-0.506056\pi$
$431$	−0.789936	−0.0380499	−0.0190249	+	0.999819i	$0.506056\pi$
$432$	53.8593	2.59131
$433$	−19.9064	−0.956642	−0.478321	−	0.878185i	$-0.658754\pi$
$433$	−19.9064	−0.956642	−0.478321	+	0.878185i	$0.658754\pi$
$434$	−72.8941	−3.49903
$435$	−23.4421	−1.12396
$436$	−97.9636	−4.69160
$437$	−0.998597	−0.0477694
$438$	58.7591	2.80762
$439$	8.43587	0.402622	0.201311	−	0.979527i	$-0.435480\pi$
$439$	8.43587	0.402622	0.201311	+	0.979527i	$0.435480\pi$
$440$	74.3354	3.54380
$441$	0.312292	0.0148710
$442$	−3.13947	−0.149329
$443$	−16.5946	−0.788432	−0.394216	−	0.919018i	$-0.628984\pi$
$443$	−16.5946	−0.788432	−0.394216	+	0.919018i	$0.628984\pi$
$444$	50.6410	2.40331
$445$	14.1130	0.669020
$446$	22.7633	1.07787
$447$	−3.22364	−0.152473
$448$	35.8572	1.69409
$449$	20.7209	0.977881	0.488941	−	0.872317i	$-0.337383\pi$
$449$	20.7209	0.977881	0.488941	+	0.872317i	$0.337383\pi$
$450$	0.661863	0.0312005
$451$	−23.6283	−1.11261
$452$	−79.2375	−3.72702
$453$	−6.36312	−0.298965
$454$	−27.9333	−1.31097
$455$	−3.46886	−0.162623
$456$	−67.0780	−3.14122
$457$	−20.9278	−0.978961	−0.489480	−	0.872014i	$-0.662814\pi$
$457$	−20.9278	−0.978961	−0.489480	+	0.872014i	$0.662814\pi$
$458$	−59.2666	−2.76934
$459$	−8.80882	−0.411160
$460$	1.58707	0.0739974
$461$	33.6152	1.56561	0.782807	−	0.622264i	$-0.213787\pi$
$461$	33.6152	1.56561	0.782807	+	0.622264i	$0.213787\pi$
$462$	−89.5531	−4.16639
$463$	32.4307	1.50718	0.753591	−	0.657344i	$-0.228320\pi$
$463$	32.4307	1.50718	0.753591	+	0.657344i	$0.228320\pi$
$464$	88.9238	4.12818
$465$	−24.1883	−1.12171
$466$	−57.5578	−2.66631
$467$	19.2088	0.888875	0.444438	−	0.895810i	$-0.353404\pi$
$467$	19.2088	0.888875	0.444438	+	0.895810i	$0.353404\pi$
$468$	−0.342215	−0.0158189
$469$	−22.6121	−1.04413
$470$	−15.7160	−0.724924
$471$	27.0441	1.24613
$472$	81.6619	3.75879
$473$	−48.4786	−2.22905
$474$	56.5220	2.59614
$475$	12.3083	0.564744
$476$	−27.1240	−1.24323
$477$	−0.725049	−0.0331977
$478$	−3.82408	−0.174909
$479$	18.9693	0.866731	0.433366	−	0.901218i	$-0.357326\pi$
$479$	18.9693	0.866731	0.433366	+	0.901218i	$0.357326\pi$
$480$	34.1639	1.55936
$481$	4.01804	0.183207
$482$	−11.2622	−0.512978
$483$	−1.13839	−0.0517986
$484$	127.695	5.80432
$485$	17.1113	0.776984
$486$	−2.74401	−0.124471
$487$	−16.8841	−0.765091	−0.382545	−	0.923937i	$-0.624952\pi$
$487$	−16.8841	−0.765091	−0.382545	+	0.923937i	$0.624952\pi$
$488$	−88.9127	−4.02489
$489$	14.8036	0.669443
$490$	−12.9643	−0.585666
$491$	−13.2835	−0.599476	−0.299738	−	0.954022i	$-0.596899\pi$
$491$	−13.2835	−0.599476	−0.299738	+	0.954022i	$0.596899\pi$
$492$	−33.8861	−1.52771
$493$	−14.5437	−0.655015
$494$	−8.93884	−0.402177
$495$	−0.960884	−0.0431885
$496$	91.7545	4.11990
$497$	−4.99017	−0.223840
$498$	37.6584	1.68751
$499$	18.7301	0.838474	0.419237	−	0.907877i	$-0.362298\pi$
$499$	18.7301	0.838474	0.419237	+	0.907877i	$0.362298\pi$
$500$	−58.5970	−2.62054
$501$	−6.98046	−0.311864
$502$	33.2854	1.48560
$503$	−29.7766	−1.32767	−0.663837	−	0.747877i	$-0.731073\pi$
$503$	−29.7766	−1.32767	−0.663837	+	0.747877i	$0.731073\pi$
$504$	−2.47263	−0.110140
$505$	−24.2366	−1.07851
$506$	3.25088	0.144519
$507$	22.0501	0.979277
$508$	21.9378	0.973334
$509$	−40.6858	−1.80337	−0.901683	−	0.432397i	$-0.857668\pi$
$509$	−40.6858	−1.80337	−0.901683	+	0.432397i	$0.857668\pi$
$510$	−12.6421	−0.559800
$511$	40.2796	1.78186
$512$	34.0223	1.50359
$513$	−25.0809	−1.10735
$514$	35.5137	1.56644
$515$	−0.804079	−0.0354320
$516$	−69.5248	−3.06066
$517$	−22.9190	−1.00798
$518$	48.7600	2.14239
$519$	45.0313	1.97666
$520$	8.45858	0.370933
$521$	8.36166	0.366331	0.183165	−	0.983082i	$-0.441366\pi$
$521$	8.36166	0.366331	0.183165	+	0.983082i	$0.441366\pi$
$522$	−2.22674	−0.0974618
$523$	−12.2171	−0.534214	−0.267107	−	0.963667i	$-0.586068\pi$
$523$	−12.2171	−0.534214	−0.267107	+	0.963667i	$0.586068\pi$
$524$	11.2432	0.491160
$525$	14.0314	0.612379
$526$	−14.8475	−0.647381
$527$	−15.0067	−0.653700
$528$	112.724	4.90567
$529$	−22.9587	−0.998203
$530$	30.0992	1.30742
$531$	−1.05559	−0.0458087
$532$	−77.2287	−3.34829
$533$	−2.68865	−0.116458
$534$	41.4587	1.79409
$535$	−8.79038	−0.380041
$536$	55.1380	2.38160
$537$	−12.6757	−0.546996
$538$	−43.8908	−1.89227
$539$	−18.9061	−0.814345
$540$	39.8610	1.71534
$541$	−16.9045	−0.726780	−0.363390	−	0.931637i	$-0.618381\pi$
$541$	−16.9045	−0.726780	−0.363390	+	0.931637i	$0.618381\pi$
$542$	36.0753	1.54957
$543$	−31.0744	−1.33353
$544$	21.1956	0.908754
$545$	−31.2996	−1.34073
$546$	−10.1902	−0.436100
$547$	26.5968	1.13720	0.568598	−	0.822615i	$-0.307486\pi$
$547$	26.5968	1.13720	0.568598	+	0.822615i	$0.307486\pi$
$548$	−92.9228	−3.96947
$549$	1.14932	0.0490516
$550$	−40.0691	−1.70855
$551$	−41.4095	−1.76411
$552$	2.77589	0.118150
$553$	38.7460	1.64765
$554$	54.6464	2.32170
$555$	16.1799	0.686799
$556$	41.9149	1.77759
$557$	17.0719	0.723359	0.361680	−	0.932302i	$-0.382203\pi$
$557$	17.0719	0.723359	0.361680	+	0.932302i	$0.382203\pi$
$558$	−2.29762	−0.0972662
$559$	−5.51635	−0.233317
$560$	52.9872	2.23912
$561$	−18.4362	−0.778378
$562$	15.7154	0.662913
$563$	−39.4811	−1.66393	−0.831966	−	0.554827i	$-0.812785\pi$
$563$	−39.4811	−1.66393	−0.831966	+	0.554827i	$0.812785\pi$
$564$	−32.8689	−1.38403
$565$	−25.3166	−1.06508
$566$	37.7508	1.58679
$567$	−29.5485	−1.24092
$568$	12.1682	0.510566
$569$	38.2544	1.60371	0.801854	−	0.597520i	$-0.203847\pi$
$569$	38.2544	1.60371	0.801854	+	0.597520i	$0.203847\pi$
$570$	−35.9951	−1.50767
$571$	−1.59764	−0.0668590	−0.0334295	−	0.999441i	$-0.510643\pi$
$571$	−1.59764	−0.0668590	−0.0334295	+	0.999441i	$0.510643\pi$
$572$	20.7177	0.866249
$573$	29.1934	1.21957
$574$	−32.6275	−1.36184
$575$	−0.509355	−0.0212416
$576$	1.13022	0.0470924
$577$	−6.73474	−0.280371	−0.140186	−	0.990125i	$-0.544770\pi$
$577$	−6.73474	−0.280371	−0.140186	+	0.990125i	$0.544770\pi$
$578$	36.9516	1.53699
$579$	5.81914	0.241835
$580$	65.8121	2.73270
$581$	25.8149	1.07098
$582$	50.2665	2.08361
$583$	43.8944	1.81792
$584$	−98.2191	−4.06434
$585$	−0.109339	−0.00452059
$586$	43.8293	1.81057
$587$	2.23525	0.0922588	0.0461294	−	0.998935i	$-0.485311\pi$
$587$	2.23525	0.0922588	0.0461294	+	0.998935i	$0.485311\pi$
$588$	−27.1139	−1.11816
$589$	−42.7277	−1.76056
$590$	43.8210	1.80408
$591$	41.4278	1.70411
$592$	−61.3759	−2.52253
$593$	8.14612	0.334521	0.167261	−	0.985913i	$-0.446508\pi$
$593$	8.14612	0.334521	0.167261	+	0.985913i	$0.446508\pi$
$594$	81.6496	3.35013
$595$	−8.66618	−0.355279
$596$	9.05015	0.370709
$597$	6.56522	0.268696
$598$	0.369916	0.0151270
$599$	38.0056	1.55287	0.776433	−	0.630200i	$-0.217027\pi$
$599$	38.0056	1.55287	0.776433	+	0.630200i	$0.217027\pi$
$600$	−34.2145	−1.39680
$601$	−33.2227	−1.35518	−0.677591	−	0.735439i	$-0.736976\pi$
$601$	−33.2227	−1.35518	−0.677591	+	0.735439i	$0.736976\pi$
$602$	−66.9424	−2.72837
$603$	−0.712733	−0.0290247
$604$	17.8640	0.726877
$605$	40.7989	1.65871
$606$	−71.1979	−2.89222
$607$	9.86139	0.400262	0.200131	−	0.979769i	$-0.435863\pi$
$607$	9.86139	0.400262	0.200131	+	0.979769i	$0.435863\pi$
$608$	60.3492	2.44748
$609$	−47.2065	−1.91291
$610$	−47.7119	−1.93180
$611$	−2.60794	−0.105506
$612$	−0.854948	−0.0345592
$613$	−28.8208	−1.16406	−0.582030	−	0.813167i	$-0.697741\pi$
$613$	−28.8208	−1.16406	−0.582030	+	0.813167i	$0.697741\pi$
$614$	−25.5619	−1.03159
$615$	−10.8267	−0.436575
$616$	149.693	6.03130
$617$	−18.2504	−0.734733	−0.367366	−	0.930076i	$-0.619740\pi$
$617$	−18.2504	−0.734733	−0.367366	+	0.930076i	$0.619740\pi$
$618$	−2.36208	−0.0950169
$619$	34.6389	1.39226	0.696128	−	0.717918i	$-0.254905\pi$
$619$	34.6389	1.39226	0.696128	+	0.717918i	$0.254905\pi$
$620$	67.9070	2.72721
$621$	1.03792	0.0416504
$622$	−44.2173	−1.77295
$623$	28.4201	1.13863
$624$	12.8268	0.513482
$625$	−6.19382	−0.247753
$626$	29.4272	1.17615
$627$	−52.4925	−2.09635
$628$	−75.9244	−3.02971
$629$	10.0382	0.400248
$630$	−1.32685	−0.0528631
$631$	−40.8105	−1.62464	−0.812320	−	0.583212i	$-0.801796\pi$
$631$	−40.8105	−1.62464	−0.812320	+	0.583212i	$0.801796\pi$
$632$	−94.4796	−3.75820
$633$	34.2741	1.36227
$634$	−30.7562	−1.22148
$635$	7.00919	0.278151
$636$	62.9505	2.49615
$637$	−2.15132	−0.0852383
$638$	134.807	5.33705
$639$	−0.157290	−0.00622230
$640$	−8.11312	−0.320699
$641$	−38.3016	−1.51282	−0.756412	−	0.654096i	$-0.773049\pi$
$641$	−38.3016	−1.51282	−0.756412	+	0.654096i	$0.773049\pi$
$642$	−25.8228	−1.01915
$643$	27.1802	1.07188	0.535940	−	0.844256i	$-0.319957\pi$
$643$	27.1802	1.07188	0.535940	+	0.844256i	$0.319957\pi$
$644$	3.19596	0.125938
$645$	−22.2134	−0.874650
$646$	−22.3317	−0.878629
$647$	33.8955	1.33257	0.666286	−	0.745697i	$-0.267883\pi$
$647$	33.8955	1.33257	0.666286	+	0.745697i	$0.267883\pi$
$648$	72.0521	2.83047
$649$	63.9053	2.50850
$650$	−4.55944	−0.178836
$651$	−48.7092	−1.90906
$652$	−41.5602	−1.62762
$653$	6.44156	0.252078	0.126039	−	0.992025i	$-0.459774\pi$
$653$	6.44156	0.252078	0.126039	+	0.992025i	$0.459774\pi$
$654$	−91.9464	−3.59539
$655$	3.59222	0.140360
$656$	41.0694	1.60349
$657$	1.26961	0.0495323
$658$	−31.6481	−1.23377
$659$	16.8978	0.658246	0.329123	−	0.944287i	$-0.393247\pi$
$659$	16.8978	0.658246	0.329123	+	0.944287i	$0.393247\pi$
$660$	83.4263	3.24736
$661$	21.2124	0.825068	0.412534	−	0.910942i	$-0.364644\pi$
$661$	21.2124	0.825068	0.412534	+	0.910942i	$0.364644\pi$
$662$	−32.0839	−1.24698
$663$	−2.09785	−0.0814737
$664$	−62.9480	−2.44286
$665$	−24.6748	−0.956846
$666$	1.53691	0.0595542
$667$	1.71365	0.0663529
$668$	19.5972	0.758237
$669$	15.2108	0.588085
$670$	29.5879	1.14308
$671$	−69.5795	−2.68609
$672$	68.7975	2.65392
$673$	−44.0878	−1.69946	−0.849731	−	0.527217i	$-0.823236\pi$
$673$	−44.0878	−1.69946	−0.849731	+	0.527217i	$0.823236\pi$
$674$	87.1111	3.35539
$675$	−12.7930	−0.492404
$676$	−61.9040	−2.38092
$677$	29.9916	1.15267	0.576336	−	0.817213i	$-0.304482\pi$
$677$	29.9916	1.15267	0.576336	+	0.817213i	$0.304482\pi$
$678$	−74.3705	−2.85618
$679$	34.4579	1.32237
$680$	21.1319	0.810371
$681$	−18.6655	−0.715265
$682$	139.098	5.32634
$683$	34.8100	1.33197	0.665983	−	0.745967i	$-0.268012\pi$
$683$	34.8100	1.33197	0.665983	+	0.745967i	$0.268012\pi$
$684$	−2.43425	−0.0930759
$685$	−29.6891	−1.13436
$686$	32.5563	1.24300
$687$	−39.6030	−1.51095
$688$	84.2628	3.21249
$689$	4.99472	0.190284
$690$	1.48959	0.0567076
$691$	17.2422	0.655926	0.327963	−	0.944691i	$-0.393638\pi$
$691$	17.2422	0.655926	0.327963	+	0.944691i	$0.393638\pi$
$692$	−126.422	−4.80586
$693$	−1.93498	−0.0735039
$694$	91.1247	3.45905
$695$	13.3919	0.507984
$696$	115.110	4.36323
$697$	−6.71699	−0.254424
$698$	−36.6249	−1.38627
$699$	−38.4612	−1.45474
$700$	−39.3921	−1.48888
$701$	−1.27307	−0.0480831	−0.0240416	−	0.999711i	$-0.507653\pi$
$701$	−1.27307	−0.0480831	−0.0240416	+	0.999711i	$0.507653\pi$
$702$	9.29087	0.350661
$703$	28.5812	1.07796
$704$	−68.4233	−2.57880
$705$	−10.5017	−0.395517
$706$	53.9756	2.03140
$707$	−48.8064	−1.83555
$708$	91.6489	3.44437
$709$	−42.7893	−1.60699	−0.803493	−	0.595314i	$-0.797027\pi$
$709$	−42.7893	−1.60699	−0.803493	+	0.595314i	$0.797027\pi$
$710$	6.52963	0.245053
$711$	1.22128	0.0458014
$712$	−69.3004	−2.59714
$713$	1.76820	0.0662197
$714$	−25.4580	−0.952740
$715$	6.61934	0.247549
$716$	35.5861	1.32992
$717$	−2.55532	−0.0954303
$718$	71.5781	2.67127
$719$	−6.13180	−0.228677	−0.114339	−	0.993442i	$-0.536475\pi$
$719$	−6.13180	−0.228677	−0.114339	+	0.993442i	$0.536475\pi$
$720$	1.67016	0.0622431
$721$	−1.61922	−0.0603027
$722$	−13.5190	−0.503126
$723$	−7.52559	−0.279880
$724$	87.2394	3.24223
$725$	−21.1218	−0.784444
$726$	119.852	4.44811
$727$	11.4981	0.426441	0.213221	−	0.977004i	$-0.431605\pi$
$727$	11.4981	0.426441	0.213221	+	0.977004i	$0.431605\pi$
$728$	17.0335	0.631303
$729$	26.0385	0.964388
$730$	−52.7059	−1.95073
$731$	−13.7814	−0.509723
$732$	−99.7864	−3.68821
$733$	4.52534	0.167147	0.0835737	−	0.996502i	$-0.473367\pi$
$733$	4.52534	0.167147	0.0835737	+	0.996502i	$0.473367\pi$
$734$	3.20514	0.118304
$735$	−8.66297	−0.319539
$736$	−2.49743	−0.0920566
$737$	43.1488	1.58941
$738$	−1.02842	−0.0378566
$739$	−50.6227	−1.86218	−0.931092	−	0.364784i	$-0.881143\pi$
$739$	−50.6227	−1.86218	−0.931092	+	0.364784i	$0.881143\pi$
$740$	−45.4240	−1.66982
$741$	−5.97310	−0.219427
$742$	60.6122	2.22515
$743$	−42.5777	−1.56203	−0.781013	−	0.624515i	$-0.785297\pi$
$743$	−42.5777	−1.56203	−0.781013	+	0.624515i	$0.785297\pi$
$744$	118.774	4.35447
$745$	2.89155	0.105938
$746$	69.2050	2.53378
$747$	0.813687	0.0297713
$748$	51.7585	1.89248
$749$	−17.7016	−0.646803
$750$	−54.9979	−2.00824
$751$	−50.6740	−1.84912	−0.924560	−	0.381037i	$-0.875567\pi$
$751$	−50.6740	−1.84912	−0.924560	+	0.381037i	$0.875567\pi$
$752$	39.8365	1.45269
$753$	22.2419	0.810541
$754$	15.3396	0.558635
$755$	5.70760	0.207721
$756$	80.2701	2.91940
$757$	−15.7119	−0.571059	−0.285530	−	0.958370i	$-0.592170\pi$
$757$	−15.7119	−0.571059	−0.285530	+	0.958370i	$0.592170\pi$
$758$	−14.7993	−0.537535
$759$	2.17230	0.0788495
$760$	60.1677	2.18251
$761$	−37.9058	−1.37408	−0.687042	−	0.726618i	$-0.741091\pi$
$761$	−37.9058	−1.37408	−0.687042	+	0.726618i	$0.741091\pi$
$762$	20.5904	0.745910
$763$	−63.0296	−2.28182
$764$	−81.9585	−2.96515
$765$	−0.273158	−0.00987605
$766$	−19.0320	−0.687655
$767$	7.27175	0.262568
$768$	15.8691	0.572628
$769$	5.73181	0.206694	0.103347	−	0.994645i	$-0.467045\pi$
$769$	5.73181	0.206694	0.103347	+	0.994645i	$0.467045\pi$
$770$	80.3275	2.89480
$771$	23.7309	0.854648
$772$	−16.3368	−0.587976
$773$	36.7816	1.32294	0.661471	−	0.749971i	$-0.269933\pi$
$773$	36.7816	1.32294	0.661471	+	0.749971i	$0.269933\pi$
$774$	−2.11003	−0.0758433
$775$	−21.7941	−0.782869
$776$	−84.0232	−3.01626
$777$	32.5823	1.16888
$778$	−75.3442	−2.70122
$779$	−19.1249	−0.685222
$780$	9.49303	0.339905
$781$	9.52233	0.340736
$782$	0.924154	0.0330477
$783$	43.0403	1.53814
$784$	32.8616	1.17363
$785$	−24.2580	−0.865806
$786$	10.5526	0.376398
$787$	35.5512	1.26726	0.633631	−	0.773635i	$-0.281564\pi$
$787$	35.5512	1.26726	0.633631	+	0.773635i	$0.281564\pi$
$788$	−116.306	−4.14322
$789$	−9.92136	−0.353210
$790$	−50.6992	−1.80380
$791$	−50.9813	−1.81269
$792$	4.71832	0.167658
$793$	−7.91741	−0.281156
$794$	30.3991	1.07883
$795$	20.1128	0.713329
$796$	−18.4314	−0.653284
$797$	6.30842	0.223456	0.111728	−	0.993739i	$-0.464362\pi$
$797$	6.30842	0.223456	0.111728	+	0.993739i	$0.464362\pi$
$798$	−72.4851	−2.56595
$799$	−6.51536	−0.230497
$800$	30.7823	1.08832
$801$	0.895801	0.0316516
$802$	−48.6273	−1.71709
$803$	−76.8623	−2.71241
$804$	61.8812	2.18238
$805$	1.02112	0.0359896
$806$	15.8279	0.557513
$807$	−29.3286	−1.03242
$808$	119.011	4.18680
$809$	−48.9936	−1.72252	−0.861261	−	0.508163i	$-0.830325\pi$
$809$	−48.9936	−1.72252	−0.861261	+	0.508163i	$0.830325\pi$
$810$	38.6642	1.35852
$811$	1.27861	0.0448981	0.0224490	−	0.999748i	$-0.492854\pi$
$811$	1.27861	0.0448981	0.0224490	+	0.999748i	$0.492854\pi$
$812$	132.529	4.65086
$813$	24.1062	0.845442
$814$	−93.0447	−3.26121
$815$	−13.2786	−0.465128
$816$	32.0449	1.12179
$817$	−39.2390	−1.37280
$818$	−6.41475	−0.224286
$819$	−0.220180	−0.00769373
$820$	30.3952	1.06145
$821$	−16.9953	−0.593141	−0.296570	−	0.955011i	$-0.595843\pi$
$821$	−16.9953	−0.593141	−0.296570	+	0.955011i	$0.595843\pi$
$822$	−87.2153	−3.04198
$823$	3.82831	0.133446	0.0667232	−	0.997772i	$-0.478746\pi$
$823$	3.82831	0.133446	0.0667232	+	0.997772i	$0.478746\pi$
$824$	3.94835	0.137547
$825$	−26.7749	−0.932183
$826$	88.2446	3.07042
$827$	−40.0812	−1.39376	−0.696880	−	0.717187i	$-0.745429\pi$
$827$	−40.0812	−1.39376	−0.696880	+	0.717187i	$0.745429\pi$
$828$	0.100737	0.00350084
$829$	−6.32801	−0.219781	−0.109890	−	0.993944i	$-0.535050\pi$
$829$	−6.32801	−0.219781	−0.109890	+	0.993944i	$0.535050\pi$
$830$	−33.7789	−1.17248
$831$	36.5157	1.26672
$832$	−7.78585	−0.269926
$833$	−5.37459	−0.186219
$834$	39.3404	1.36225
$835$	6.26134	0.216683
$836$	147.369	5.09687
$837$	44.4104	1.53505
$838$	18.0443	0.623330
$839$	28.5228	0.984716	0.492358	−	0.870393i	$-0.336135\pi$
$839$	28.5228	0.984716	0.492358	+	0.870393i	$0.336135\pi$
$840$	68.5908	2.36661
$841$	42.0613	1.45039
$842$	−1.77738	−0.0612527
$843$	10.5013	0.361684
$844$	−96.2222	−3.31210
$845$	−19.7785	−0.680401
$846$	−0.997547	−0.0342964
$847$	82.1587	2.82301
$848$	−76.2948	−2.61997
$849$	25.2258	0.865747
$850$	−11.3908	−0.390700
$851$	−1.18278	−0.0405450
$852$	13.6563	0.467857
$853$	−28.4246	−0.973241	−0.486621	−	0.873613i	$-0.661771\pi$
$853$	−28.4246	−0.973241	−0.486621	+	0.873613i	$0.661771\pi$
$854$	−96.0800	−3.28779
$855$	−0.777749	−0.0265984
$856$	43.1642	1.47532
$857$	0.722765	0.0246892	0.0123446	−	0.999924i	$-0.496070\pi$
$857$	0.722765	0.0246892	0.0123446	+	0.999924i	$0.496070\pi$
$858$	19.4451	0.663846
$859$	−23.0593	−0.786773	−0.393386	−	0.919373i	$-0.628697\pi$
$859$	−23.0593	−0.786773	−0.393386	+	0.919373i	$0.628697\pi$
$860$	62.3625	2.12654
$861$	−21.8023	−0.743020
$862$	2.08148	0.0708953
$863$	30.8548	1.05031	0.525155	−	0.851007i	$-0.324007\pi$
$863$	30.8548	1.05031	0.525155	+	0.851007i	$0.324007\pi$
$864$	−62.7258	−2.13398
$865$	−40.3923	−1.37338
$866$	52.4533	1.78244
$867$	24.6918	0.838577
$868$	136.748	4.64152
$869$	−73.9359	−2.50810
$870$	61.7698	2.09419
$871$	4.90988	0.166365
$872$	153.693	5.20472
$873$	1.08611	0.0367593
$874$	2.63130	0.0890049
$875$	−37.7012	−1.27453
$876$	−110.231	−3.72436
$877$	9.03935	0.305237	0.152619	−	0.988285i	$-0.451229\pi$
$877$	9.03935	0.305237	0.152619	+	0.988285i	$0.451229\pi$
$878$	−22.2285	−0.750175
$879$	29.2875	0.987844
$880$	−101.111	−3.40846
$881$	−8.61812	−0.290352	−0.145176	−	0.989406i	$-0.546375\pi$
$881$	−8.61812	−0.290352	−0.145176	+	0.989406i	$0.546375\pi$
$882$	−0.822888	−0.0277081
$883$	−30.9226	−1.04063	−0.520314	−	0.853975i	$-0.674185\pi$
$883$	−30.9226	−1.04063	−0.520314	+	0.853975i	$0.674185\pi$
$884$	5.88957	0.198088
$885$	29.2820	0.984305
$886$	43.7266	1.46902
$887$	30.2228	1.01478	0.507392	−	0.861716i	$-0.330610\pi$
$887$	30.2228	1.01478	0.507392	+	0.861716i	$0.330610\pi$
$888$	−79.4498	−2.66616
$889$	14.1148	0.473394
$890$	−37.1877	−1.24653
$891$	56.3850	1.88897
$892$	−42.7034	−1.42982
$893$	−18.5509	−0.620781
$894$	8.49427	0.284091
$895$	11.3699	0.380052
$896$	−16.3378	−0.545807
$897$	0.247185	0.00825327
$898$	−54.5995	−1.82201
$899$	73.3233	2.44547
$900$	−1.24164	−0.0413880
$901$	12.4782	0.415709
$902$	62.2604	2.07304
$903$	−44.7322	−1.48859
$904$	124.314	4.13464
$905$	27.8732	0.926536
$906$	16.7668	0.557039
$907$	47.2034	1.56736	0.783681	−	0.621164i	$-0.213340\pi$
$907$	47.2034	1.56736	0.783681	+	0.621164i	$0.213340\pi$
$908$	52.4023	1.73903
$909$	−1.53838	−0.0510248
$910$	9.14042	0.303002
$911$	−29.5680	−0.979631	−0.489815	−	0.871826i	$-0.662936\pi$
$911$	−29.5680	−0.979631	−0.489815	+	0.871826i	$0.662936\pi$
$912$	91.2397	3.02125
$913$	−49.2606	−1.63029
$914$	55.1446	1.82402
$915$	−31.8820	−1.05399
$916$	111.183	3.67358
$917$	7.23383	0.238882
$918$	23.2112	0.766083
$919$	2.23498	0.0737252	0.0368626	−	0.999320i	$-0.488264\pi$
$919$	2.23498	0.0737252	0.0368626	+	0.999320i	$0.488264\pi$
$920$	−2.48992	−0.0820904
$921$	−17.0809	−0.562836
$922$	−88.5758	−2.91709
$923$	1.08354	0.0356652
$924$	168.000	5.52679
$925$	14.5784	0.479336
$926$	−85.4547	−2.80821
$927$	−0.0510377	−0.00167630
$928$	−103.563	−3.39962
$929$	19.8852	0.652413	0.326206	−	0.945299i	$-0.394230\pi$
$929$	19.8852	0.652413	0.326206	+	0.945299i	$0.394230\pi$
$930$	63.7360	2.08999
$931$	−15.3028	−0.501529
$932$	107.977	3.53691
$933$	−29.5468	−0.967319
$934$	−50.6150	−1.65617
$935$	16.5370	0.540817
$936$	0.536895	0.0175490
$937$	−24.5241	−0.801168	−0.400584	−	0.916260i	$-0.631193\pi$
$937$	−24.5241	−0.801168	−0.400584	+	0.916260i	$0.631193\pi$
$938$	59.5827	1.94544
$939$	19.6638	0.641704
$940$	29.4828	0.961624
$941$	10.0587	0.327904	0.163952	−	0.986468i	$-0.447576\pi$
$941$	10.0587	0.327904	0.163952	+	0.986468i	$0.447576\pi$
$942$	−71.2610	−2.32181
$943$	0.791449	0.0257731
$944$	−111.077	−3.61524
$945$	25.6465	0.834281
$946$	127.741	4.15321
$947$	−27.0942	−0.880445	−0.440222	−	0.897889i	$-0.645100\pi$
$947$	−27.0942	−0.880445	−0.440222	+	0.897889i	$0.645100\pi$
$948$	−106.034	−3.44383
$949$	−8.74612	−0.283911
$950$	−32.4323	−1.05224
$951$	−20.5518	−0.666439
$952$	42.5544	1.37920
$953$	−10.3172	−0.334206	−0.167103	−	0.985939i	$-0.553441\pi$
$953$	−10.3172	−0.334206	−0.167103	+	0.985939i	$0.553441\pi$
$954$	1.91050	0.0618547
$955$	−26.1859	−0.847357
$956$	7.17390	0.232020
$957$	90.0804	2.91189
$958$	−49.9841	−1.61491
$959$	−59.7864	−1.93060
$960$	−31.3522	−1.01189
$961$	44.6573	1.44056
$962$	−10.5875	−0.341355
$963$	−0.557955	−0.0179799
$964$	21.1276	0.680474
$965$	−5.21966	−0.168027
$966$	2.99966	0.0965124
$967$	−21.8428	−0.702416	−0.351208	−	0.936297i	$-0.614229\pi$
$967$	−21.8428	−0.702416	−0.351208	+	0.936297i	$0.614229\pi$
$968$	−200.339	−6.43913
$969$	−14.9225	−0.479379
$970$	−45.0881	−1.44769
$971$	34.8913	1.11971	0.559857	−	0.828589i	$-0.310856\pi$
$971$	34.8913	1.11971	0.559857	+	0.828589i	$0.310856\pi$
$972$	5.14770	0.165113
$973$	26.9680	0.864553
$974$	44.4894	1.42553
$975$	−3.04670	−0.0975726
$976$	120.939	3.87117
$977$	31.4702	1.00682	0.503410	−	0.864048i	$-0.332079\pi$
$977$	31.4702	1.00682	0.503410	+	0.864048i	$0.332079\pi$
$978$	−39.0074	−1.24732
$979$	−54.2317	−1.73325
$980$	24.3207	0.776897
$981$	−1.98669	−0.0634303
$982$	35.0019	1.11696
$983$	25.1405	0.801859	0.400929	−	0.916109i	$-0.368687\pi$
$983$	25.1405	0.801859	0.400929	+	0.916109i	$0.368687\pi$
$984$	53.1634	1.69479
$985$	−37.1600	−1.18402
$986$	38.3226	1.22044
$987$	−21.1478	−0.673143
$988$	16.7691	0.533495
$989$	1.62383	0.0516348
$990$	2.53192	0.0804698
$991$	41.5088	1.31857	0.659285	−	0.751894i	$-0.270859\pi$
$991$	41.5088	1.31857	0.659285	+	0.751894i	$0.270859\pi$
$992$	−106.859	−3.39279
$993$	−21.4391	−0.680349
$994$	13.1491	0.417063
$995$	−5.88888	−0.186690
$996$	−70.6463	−2.23851
$997$	57.8631	1.83254	0.916272	−	0.400557i	$-0.131183\pi$
$997$	57.8631	1.83254	0.916272	+	0.400557i	$0.131183\pi$
$998$	−49.3537	−1.56226
$999$	−29.7068	−0.939880

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4027.2.a.b.1.9	✓	159

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4027.2.a.b.1.9	✓	159	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:	\( N \)	\(=\)	\( 4027 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4027.a (trivial)

\(f(q)\)	\(=\)	\(q-2.63499 q^{2} -1.76075 q^{3} +4.94319 q^{4} +1.57936 q^{5} +4.63957 q^{6} +3.18044 q^{7} -7.75529 q^{8} +0.100248 q^{9} +O(q^{10})\)	\(q-2.63499 q^{2} -1.76075 q^{3} +4.94319 q^{4} +1.57936 q^{5} +4.63957 q^{6} +3.18044 q^{7} -7.75529 q^{8} +0.100248 q^{9} -4.16161 q^{10} -6.06898 q^{11} -8.70374 q^{12} -0.690586 q^{13} -8.38044 q^{14} -2.78087 q^{15} +10.5488 q^{16} -1.72528 q^{17} -0.264152 q^{18} -4.91229 q^{19} +7.80709 q^{20} -5.59997 q^{21} +15.9917 q^{22} +0.203285 q^{23} +13.6551 q^{24} -2.50562 q^{25} +1.81969 q^{26} +5.10574 q^{27} +15.7215 q^{28} +8.42978 q^{29} +7.32756 q^{30} +8.69812 q^{31} -12.2853 q^{32} +10.6860 q^{33} +4.54609 q^{34} +5.02307 q^{35} +0.495543 q^{36} -5.81830 q^{37} +12.9438 q^{38} +1.21595 q^{39} -12.2484 q^{40} +3.89329 q^{41} +14.7559 q^{42} +7.98793 q^{43} -30.0001 q^{44} +0.158327 q^{45} -0.535656 q^{46} +3.77642 q^{47} -18.5738 q^{48} +3.11521 q^{49} +6.60228 q^{50} +3.03778 q^{51} -3.41370 q^{52} -7.23258 q^{53} -13.4536 q^{54} -9.58511 q^{55} -24.6653 q^{56} +8.64932 q^{57} -22.2124 q^{58} -10.5298 q^{59} -13.7464 q^{60} +11.4648 q^{61} -22.9195 q^{62} +0.318831 q^{63} +11.2743 q^{64} -1.09068 q^{65} -28.1575 q^{66} -7.10973 q^{67} -8.52837 q^{68} -0.357935 q^{69} -13.2358 q^{70} -1.56902 q^{71} -0.777449 q^{72} +12.6648 q^{73} +15.3312 q^{74} +4.41177 q^{75} -24.2824 q^{76} -19.3020 q^{77} -3.20402 q^{78} +12.1826 q^{79} +16.6603 q^{80} -9.29069 q^{81} -10.2588 q^{82} +8.11678 q^{83} -27.6817 q^{84} -2.72484 q^{85} -21.0482 q^{86} -14.8428 q^{87} +47.0667 q^{88} +8.93589 q^{89} -0.417191 q^{90} -2.19637 q^{91} +1.00488 q^{92} -15.3152 q^{93} -9.95084 q^{94} -7.75828 q^{95} +21.6314 q^{96} +10.8343 q^{97} -8.20855 q^{98} -0.608400 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 159 q - 22 q^{2} - 19 q^{3} + 148 q^{4} - 70 q^{5} - 23 q^{6} - 19 q^{7} - 66 q^{8} + 126 q^{9}+O(q^{10}) \) 159 * q - 22 * q^2 - 19 * q^3 + 148 * q^4 - 70 * q^5 - 23 * q^6 - 19 * q^7 - 66 * q^8 + 126 * q^9	\( 159 q - 22 q^{2} - 19 q^{3} + 148 q^{4} - 70 q^{5} - 23 q^{6} - 19 q^{7} - 66 q^{8} + 126 q^{9} - 23 q^{10} - 33 q^{11} - 57 q^{12} - 90 q^{13} - 28 q^{14} - 22 q^{15} + 130 q^{16} - 145 q^{17} - 50 q^{18} - 28 q^{19} - 121 q^{20} - 69 q^{21} - 26 q^{22} - 79 q^{23} - 62 q^{24} + 123 q^{25} - 40 q^{26} - 70 q^{27} - 43 q^{28} - 109 q^{29} - 43 q^{30} - 21 q^{31} - 139 q^{32} - 83 q^{33} - 93 q^{35} + 75 q^{36} - 65 q^{37} - 122 q^{38} - 18 q^{39} - 43 q^{40} - 71 q^{41} - 88 q^{42} - 72 q^{43} - 79 q^{44} - 181 q^{45} - 11 q^{46} - 114 q^{47} - 118 q^{48} + 118 q^{49} - 77 q^{50} - 29 q^{51} - 169 q^{52} - 220 q^{53} - 80 q^{54} - 37 q^{55} - 72 q^{56} - 90 q^{57} - 8 q^{58} - 60 q^{59} - 42 q^{60} - 108 q^{61} - 152 q^{62} - 65 q^{63} + 114 q^{64} - 81 q^{65} - 40 q^{66} - 50 q^{67} - 319 q^{68} - 103 q^{69} + 4 q^{70} - 7 q^{71} - 129 q^{72} - 94 q^{73} - 79 q^{74} - 59 q^{75} - 46 q^{76} - 329 q^{77} + 8 q^{78} - 18 q^{79} - 190 q^{80} + 59 q^{81} - 56 q^{82} - 201 q^{83} - 71 q^{84} - 26 q^{85} - 52 q^{86} - 126 q^{87} - 66 q^{88} - 114 q^{89} - 33 q^{90} - 30 q^{91} - 204 q^{92} - 125 q^{93} + 9 q^{94} - 84 q^{95} - 88 q^{96} - 56 q^{97} - 110 q^{98} - 46 q^{99}+O(q^{100}) \) 159 * q - 22 * q^2 - 19 * q^3 + 148 * q^4 - 70 * q^5 - 23 * q^6 - 19 * q^7 - 66 * q^8 + 126 * q^9 - 23 * q^10 - 33 * q^11 - 57 * q^12 - 90 * q^13 - 28 * q^14 - 22 * q^15 + 130 * q^16 - 145 * q^17 - 50 * q^18 - 28 * q^19 - 121 * q^20 - 69 * q^21 - 26 * q^22 - 79 * q^23 - 62 * q^24 + 123 * q^25 - 40 * q^26 - 70 * q^27 - 43 * q^28 - 109 * q^29 - 43 * q^30 - 21 * q^31 - 139 * q^32 - 83 * q^33 - 93 * q^35 + 75 * q^36 - 65 * q^37 - 122 * q^38 - 18 * q^39 - 43 * q^40 - 71 * q^41 - 88 * q^42 - 72 * q^43 - 79 * q^44 - 181 * q^45 - 11 * q^46 - 114 * q^47 - 118 * q^48 + 118 * q^49 - 77 * q^50 - 29 * q^51 - 169 * q^52 - 220 * q^53 - 80 * q^54 - 37 * q^55 - 72 * q^56 - 90 * q^57 - 8 * q^58 - 60 * q^59 - 42 * q^60 - 108 * q^61 - 152 * q^62 - 65 * q^63 + 114 * q^64 - 81 * q^65 - 40 * q^66 - 50 * q^67 - 319 * q^68 - 103 * q^69 + 4 * q^70 - 7 * q^71 - 129 * q^72 - 94 * q^73 - 79 * q^74 - 59 * q^75 - 46 * q^76 - 329 * q^77 + 8 * q^78 - 18 * q^79 - 190 * q^80 + 59 * q^81 - 56 * q^82 - 201 * q^83 - 71 * q^84 - 26 * q^85 - 52 * q^86 - 126 * q^87 - 66 * q^88 - 114 * q^89 - 33 * q^90 - 30 * q^91 - 204 * q^92 - 125 * q^93 + 9 * q^94 - 84 * q^95 - 88 * q^96 - 56 * q^97 - 110 * q^98 - 46 * q^99

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