LMFDB - Embedded newform 8047.2.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8047,2,Mod(1,8047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8047, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8047 = 13 \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8047.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:	$64.2556185065$
Analytic rank:	$0$
Dimension:	$156$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.71116 q^{2} -0.430887 q^{3} +5.35037 q^{4} +1.49097 q^{5} +1.16820 q^{6} +4.38752 q^{7} -9.08338 q^{8} -2.81434 q^{9} +O(q^{10})$	\(q-2.71116 q^{2} -0.430887 q^{3} +5.35037 q^{4} +1.49097 q^{5} +1.16820 q^{6} +4.38752 q^{7} -9.08338 q^{8} -2.81434 q^{9} -4.04226 q^{10} +3.44005 q^{11} -2.30541 q^{12} -1.00000 q^{13} -11.8952 q^{14} -0.642442 q^{15} +13.9257 q^{16} +4.34659 q^{17} +7.63011 q^{18} +2.19174 q^{19} +7.97726 q^{20} -1.89053 q^{21} -9.32653 q^{22} +0.742337 q^{23} +3.91391 q^{24} -2.77700 q^{25} +2.71116 q^{26} +2.50532 q^{27} +23.4748 q^{28} +8.04419 q^{29} +1.74176 q^{30} +1.15497 q^{31} -19.5880 q^{32} -1.48228 q^{33} -11.7843 q^{34} +6.54167 q^{35} -15.0577 q^{36} +2.25130 q^{37} -5.94216 q^{38} +0.430887 q^{39} -13.5431 q^{40} +10.1309 q^{41} +5.12551 q^{42} +4.54977 q^{43} +18.4056 q^{44} -4.19610 q^{45} -2.01259 q^{46} +2.42229 q^{47} -6.00042 q^{48} +12.2503 q^{49} +7.52888 q^{50} -1.87289 q^{51} -5.35037 q^{52} +10.6312 q^{53} -6.79233 q^{54} +5.12903 q^{55} -39.8535 q^{56} -0.944394 q^{57} -21.8090 q^{58} -6.70720 q^{59} -3.43730 q^{60} -8.43339 q^{61} -3.13132 q^{62} -12.3479 q^{63} +25.2548 q^{64} -1.49097 q^{65} +4.01868 q^{66} -11.7495 q^{67} +23.2559 q^{68} -0.319864 q^{69} -17.7355 q^{70} -2.92353 q^{71} +25.5637 q^{72} +10.7887 q^{73} -6.10362 q^{74} +1.19657 q^{75} +11.7266 q^{76} +15.0933 q^{77} -1.16820 q^{78} -0.0289694 q^{79} +20.7629 q^{80} +7.36350 q^{81} -27.4665 q^{82} +0.344703 q^{83} -10.1150 q^{84} +6.48065 q^{85} -12.3351 q^{86} -3.46614 q^{87} -31.2473 q^{88} -4.89175 q^{89} +11.3763 q^{90} -4.38752 q^{91} +3.97178 q^{92} -0.497664 q^{93} -6.56720 q^{94} +3.26783 q^{95} +8.44024 q^{96} -14.8984 q^{97} -33.2125 q^{98} -9.68147 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9}+O(q^{10}) $ 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9} + 11 q^{10} + 23 q^{11} + 57 q^{12} - 156 q^{13} + 18 q^{14} + 32 q^{15} + 159 q^{16} + 119 q^{17} + 36 q^{18} + 35 q^{19} + 109 q^{20} + 33 q^{21} + 11 q^{22} + 55 q^{23} + 63 q^{24} + 189 q^{25} - 13 q^{26} + 89 q^{27} + 54 q^{28} - 55 q^{29} + 47 q^{31} + 112 q^{32} + 109 q^{33} + 51 q^{34} + 25 q^{35} + 162 q^{36} + 53 q^{37} + 37 q^{38} - 23 q^{39} + 25 q^{40} + 113 q^{41} + 26 q^{42} + 31 q^{43} + 86 q^{44} + 144 q^{45} + 37 q^{46} + 115 q^{47} + 129 q^{48} + 189 q^{49} + 72 q^{50} - 4 q^{51} - 161 q^{52} + 51 q^{53} + 108 q^{54} + 22 q^{55} + 39 q^{56} + 102 q^{57} + 31 q^{58} + 75 q^{59} + 97 q^{60} + 7 q^{61} + 77 q^{62} + 94 q^{63} + 158 q^{64} - 39 q^{65} + 48 q^{66} + 37 q^{67} + 235 q^{68} + 27 q^{69} + 38 q^{70} + 70 q^{71} + 152 q^{72} + 155 q^{73} - 18 q^{74} + 80 q^{75} + 21 q^{76} + 101 q^{77} - 25 q^{78} + 10 q^{79} + 211 q^{80} + 220 q^{81} + 45 q^{82} + 132 q^{83} + 86 q^{84} + 74 q^{85} + 35 q^{86} + 53 q^{87} + 51 q^{88} + 190 q^{89} - 27 q^{90} - 19 q^{91} + 125 q^{92} + 96 q^{93} - 19 q^{94} + 72 q^{95} + 146 q^{96} + 155 q^{97} + 135 q^{98} + 89 q^{99}+O(q^{100}) \) 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9 + 11 * q^10 + 23 * q^11 + 57 * q^12 - 156 * q^13 + 18 * q^14 + 32 * q^15 + 159 * q^16 + 119 * q^17 + 36 * q^18 + 35 * q^19 + 109 * q^20 + 33 * q^21 + 11 * q^22 + 55 * q^23 + 63 * q^24 + 189 * q^25 - 13 * q^26 + 89 * q^27 + 54 * q^28 - 55 * q^29 + 47 * q^31 + 112 * q^32 + 109 * q^33 + 51 * q^34 + 25 * q^35 + 162 * q^36 + 53 * q^37 + 37 * q^38 - 23 * q^39 + 25 * q^40 + 113 * q^41 + 26 * q^42 + 31 * q^43 + 86 * q^44 + 144 * q^45 + 37 * q^46 + 115 * q^47 + 129 * q^48 + 189 * q^49 + 72 * q^50 - 4 * q^51 - 161 * q^52 + 51 * q^53 + 108 * q^54 + 22 * q^55 + 39 * q^56 + 102 * q^57 + 31 * q^58 + 75 * q^59 + 97 * q^60 + 7 * q^61 + 77 * q^62 + 94 * q^63 + 158 * q^64 - 39 * q^65 + 48 * q^66 + 37 * q^67 + 235 * q^68 + 27 * q^69 + 38 * q^70 + 70 * q^71 + 152 * q^72 + 155 * q^73 - 18 * q^74 + 80 * q^75 + 21 * q^76 + 101 * q^77 - 25 * q^78 + 10 * q^79 + 211 * q^80 + 220 * q^81 + 45 * q^82 + 132 * q^83 + 86 * q^84 + 74 * q^85 + 35 * q^86 + 53 * q^87 + 51 * q^88 + 190 * q^89 - 27 * q^90 - 19 * q^91 + 125 * q^92 + 96 * q^93 - 19 * q^94 + 72 * q^95 + 146 * q^96 + 155 * q^97 + 135 * q^98 + 89 * 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 $

$2$	−2.71116	−1.91708	−0.958539	−	0.284963i	$-0.908019\pi$
$2$	−2.71116	−1.91708	−0.958539	+	0.284963i	$0.908019\pi$
$3$	−0.430887	−0.248773	−0.124386	−	0.992234i	$-0.539696\pi$
$3$	−0.430887	−0.248773	−0.124386	+	0.992234i	$0.539696\pi$
$4$	5.35037	2.67519
$5$	1.49097	0.666784	0.333392	−	0.942788i	$-0.391807\pi$
$5$	1.49097	0.666784	0.333392	+	0.942788i	$0.391807\pi$
$6$	1.16820	0.476917
$7$	4.38752	1.65833	0.829163	−	0.559008i	$-0.188818\pi$
$7$	4.38752	1.65833	0.829163	+	0.559008i	$0.188818\pi$
$8$	−9.08338	−3.21146
$9$	−2.81434	−0.938112
$10$	−4.04226	−1.27828
$11$	3.44005	1.03722	0.518608	−	0.855012i	$-0.326451\pi$
$11$	3.44005	1.03722	0.518608	+	0.855012i	$0.326451\pi$
$12$	−2.30541	−0.665514
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−11.8952	−3.17914
$15$	−0.642442	−0.165878
$16$	13.9257	3.48143
$17$	4.34659	1.05420	0.527102	−	0.849802i	$-0.323279\pi$
$17$	4.34659	1.05420	0.527102	+	0.849802i	$0.323279\pi$
$18$	7.63011	1.79843
$19$	2.19174	0.502820	0.251410	−	0.967881i	$-0.419106\pi$
$19$	2.19174	0.502820	0.251410	+	0.967881i	$0.419106\pi$
$20$	7.97726	1.78377
$21$	−1.89053	−0.412546
$22$	−9.32653	−1.98842
$23$	0.742337	0.154788	0.0773940	−	0.997001i	$-0.475340\pi$
$23$	0.742337	0.154788	0.0773940	+	0.997001i	$0.475340\pi$
$24$	3.91391	0.798924
$25$	−2.77700	−0.555400
$26$	2.71116	0.531702
$27$	2.50532	0.482150
$28$	23.4748	4.43633
$29$	8.04419	1.49377	0.746884	−	0.664954i	$-0.231549\pi$
$29$	8.04419	1.49377	0.746884	+	0.664954i	$0.231549\pi$
$30$	1.74176	0.318000
$31$	1.15497	0.207439	0.103720	−	0.994607i	$-0.466926\pi$
$31$	1.15497	0.207439	0.103720	+	0.994607i	$0.466926\pi$
$32$	−19.5880	−3.46271
$33$	−1.48228	−0.258031
$34$	−11.7843	−2.02099
$35$	6.54167	1.10574
$36$	−15.0577	−2.50962
$37$	2.25130	0.370111	0.185055	−	0.982728i	$-0.440754\pi$
$37$	2.25130	0.370111	0.185055	+	0.982728i	$0.440754\pi$
$38$	−5.94216	−0.963945
$39$	0.430887	0.0689972
$40$	−13.5431	−2.14135
$41$	10.1309	1.58218	0.791092	−	0.611697i	$-0.209513\pi$
$41$	10.1309	1.58218	0.791092	+	0.611697i	$0.209513\pi$
$42$	5.12551	0.790883
$43$	4.54977	0.693834	0.346917	−	0.937896i	$-0.387229\pi$
$43$	4.54977	0.693834	0.346917	+	0.937896i	$0.387229\pi$
$44$	18.4056	2.77474
$45$	−4.19610	−0.625518
$46$	−2.01259	−0.296740
$47$	2.42229	0.353327	0.176663	−	0.984271i	$-0.443470\pi$
$47$	2.42229	0.353327	0.176663	+	0.984271i	$0.443470\pi$
$48$	−6.00042	−0.866086
$49$	12.2503	1.75004
$50$	7.52888	1.06474
$51$	−1.87289	−0.262257
$52$	−5.35037	−0.741963
$53$	10.6312	1.46030	0.730151	−	0.683286i	$-0.239450\pi$
$53$	10.6312	1.46030	0.730151	+	0.683286i	$0.239450\pi$
$54$	−6.79233	−0.924319
$55$	5.12903	0.691598
$56$	−39.8535	−5.32564
$57$	−0.944394	−0.125088
$58$	−21.8090	−2.86367
$59$	−6.70720	−0.873203	−0.436601	−	0.899655i	$-0.643818\pi$
$59$	−6.70720	−0.873203	−0.436601	+	0.899655i	$0.643818\pi$
$60$	−3.43730	−0.443754
$61$	−8.43339	−1.07978	−0.539892	−	0.841734i	$-0.681535\pi$
$61$	−8.43339	−1.07978	−0.539892	+	0.841734i	$0.681535\pi$
$62$	−3.13132	−0.397677
$63$	−12.3479	−1.55569
$64$	25.2548	3.15685
$65$	−1.49097	−0.184932
$66$	4.01868	0.494666
$67$	−11.7495	−1.43543	−0.717716	−	0.696336i	$-0.754812\pi$
$67$	−11.7495	−1.43543	−0.717716	+	0.696336i	$0.754812\pi$
$68$	23.2559	2.82019
$69$	−0.319864	−0.0385071
$70$	−17.7355	−2.11980
$71$	−2.92353	−0.346959	−0.173480	−	0.984837i	$-0.555501\pi$
$71$	−2.92353	−0.346959	−0.173480	+	0.984837i	$0.555501\pi$
$72$	25.5637	3.01271
$73$	10.7887	1.26272	0.631362	−	0.775489i	$-0.282496\pi$
$73$	10.7887	1.26272	0.631362	+	0.775489i	$0.282496\pi$
$74$	−6.10362	−0.709531
$75$	1.19657	0.138168
$76$	11.7266	1.34514
$77$	15.0933	1.72004
$78$	−1.16820	−0.132273
$79$	−0.0289694	−0.00325931	−0.00162965	−	0.999999i	$-0.500519\pi$
$79$	−0.0289694	−0.00325931	−0.00162965	+	0.999999i	$0.500519\pi$
$80$	20.7629	2.32136
$81$	7.36350	0.818166
$82$	−27.4665	−3.03317
$83$	0.344703	0.0378360	0.0189180	−	0.999821i	$-0.493978\pi$
$83$	0.344703	0.0378360	0.0189180	+	0.999821i	$0.493978\pi$
$84$	−10.1150	−1.10364
$85$	6.48065	0.702925
$86$	−12.3351	−1.33013
$87$	−3.46614	−0.371609
$88$	−31.2473	−3.33098
$89$	−4.89175	−0.518525	−0.259262	−	0.965807i	$-0.583479\pi$
$89$	−4.89175	−0.518525	−0.259262	+	0.965807i	$0.583479\pi$
$90$	11.3763	1.19917
$91$	−4.38752	−0.459937
$92$	3.97178	0.414086
$93$	−0.497664	−0.0516053
$94$	−6.56720	−0.677355
$95$	3.26783	0.335272
$96$	8.44024	0.861429
$97$	−14.8984	−1.51270	−0.756351	−	0.654166i	$-0.773020\pi$
$97$	−14.8984	−1.51270	−0.756351	+	0.654166i	$0.773020\pi$
$98$	−33.2125	−3.35497
$99$	−9.68147	−0.973024
$100$	−14.8580	−1.48580
$101$	6.18199	0.615131	0.307565	−	0.951527i	$-0.400486\pi$
$101$	6.18199	0.615131	0.307565	+	0.951527i	$0.400486\pi$
$102$	5.07770	0.502767
$103$	17.5136	1.72566	0.862831	−	0.505492i	$-0.168689\pi$
$103$	17.5136	1.72566	0.862831	+	0.505492i	$0.168689\pi$
$104$	9.08338	0.890698
$105$	−2.81872	−0.275079
$106$	−28.8227	−2.79951
$107$	−1.94206	−0.187746	−0.0938731	−	0.995584i	$-0.529925\pi$
$107$	−1.94206	−0.187746	−0.0938731	+	0.995584i	$0.529925\pi$
$108$	13.4044	1.28984
$109$	−14.5930	−1.39776	−0.698879	−	0.715240i	$-0.746317\pi$
$109$	−14.5930	−1.39776	−0.698879	+	0.715240i	$0.746317\pi$
$110$	−13.9056	−1.32585
$111$	−0.970055	−0.0920735
$112$	61.0993	5.77334
$113$	−13.8862	−1.30631	−0.653154	−	0.757225i	$-0.726554\pi$
$113$	−13.8862	−1.30631	−0.653154	+	0.757225i	$0.726554\pi$
$114$	2.56040	0.239804
$115$	1.10680	0.103210
$116$	43.0394	3.99611
$117$	2.81434	0.260185
$118$	18.1843	1.67400
$119$	19.0707	1.74821
$120$	5.83554	0.532709
$121$	0.833977	0.0758160
$122$	22.8642	2.07003
$123$	−4.36529	−0.393605
$124$	6.17954	0.554939
$125$	−11.5953	−1.03711
$126$	33.4772	2.98239
$127$	−3.13284	−0.277994	−0.138997	−	0.990293i	$-0.544388\pi$
$127$	−3.13284	−0.277994	−0.138997	+	0.990293i	$0.544388\pi$
$128$	−29.2937	−2.58922
$129$	−1.96044	−0.172607
$130$	4.04226	0.354530
$131$	−13.7369	−1.20020	−0.600100	−	0.799925i	$-0.704872\pi$
$131$	−13.7369	−1.20020	−0.600100	+	0.799925i	$0.704872\pi$
$132$	−7.93073	−0.690281
$133$	9.61631	0.833839
$134$	31.8548	2.75183
$135$	3.73537	0.321490
$136$	−39.4817	−3.38553
$137$	13.3795	1.14308	0.571542	−	0.820573i	$-0.306345\pi$
$137$	13.3795	1.14308	0.571542	+	0.820573i	$0.306345\pi$
$138$	0.867201	0.0738210
$139$	18.2587	1.54868	0.774340	−	0.632770i	$-0.218082\pi$
$139$	18.2587	1.54868	0.774340	+	0.632770i	$0.218082\pi$
$140$	35.0004	2.95807
$141$	−1.04373	−0.0878982
$142$	7.92615	0.665148
$143$	−3.44005	−0.287672
$144$	−39.1917	−3.26597
$145$	11.9937	0.996020
$146$	−29.2499	−2.42074
$147$	−5.27850	−0.435363
$148$	12.0453	0.990115
$149$	14.4093	1.18045	0.590227	−	0.807237i	$-0.299038\pi$
$149$	14.4093	1.18045	0.590227	+	0.807237i	$0.299038\pi$
$150$	−3.24410	−0.264880
$151$	2.21907	0.180586	0.0902928	−	0.995915i	$-0.471220\pi$
$151$	2.21907	0.180586	0.0902928	+	0.995915i	$0.471220\pi$
$152$	−19.9084	−1.61479
$153$	−12.2328	−0.988961
$154$	−40.9203	−3.29745
$155$	1.72204	0.138317
$156$	2.30541	0.184580
$157$	−15.1835	−1.21177	−0.605886	−	0.795551i	$-0.707181\pi$
$157$	−15.1835	−1.21177	−0.605886	+	0.795551i	$0.707181\pi$
$158$	0.0785405	0.00624834
$159$	−4.58083	−0.363284
$160$	−29.2053	−2.30888
$161$	3.25702	0.256689
$162$	−19.9636	−1.56849
$163$	1.71998	0.134719	0.0673595	−	0.997729i	$-0.478543\pi$
$163$	1.71998	0.134719	0.0673595	+	0.997729i	$0.478543\pi$
$164$	54.2042	4.23263
$165$	−2.21003	−0.172051
$166$	−0.934543	−0.0725346
$167$	21.1284	1.63497	0.817484	−	0.575951i	$-0.195368\pi$
$167$	21.1284	1.63497	0.817484	+	0.575951i	$0.195368\pi$
$168$	17.1724	1.32488
$169$	1.00000	0.0769231
$170$	−17.5701	−1.34756
$171$	−6.16830	−0.471702
$172$	24.3430	1.85613
$173$	12.1020	0.920102	0.460051	−	0.887893i	$-0.347831\pi$
$173$	12.1020	0.920102	0.460051	+	0.887893i	$0.347831\pi$
$174$	9.39724	0.712403
$175$	−12.1841	−0.921033
$176$	47.9052	3.61099
$177$	2.89005	0.217229
$178$	13.2623	0.994052
$179$	−18.6734	−1.39572	−0.697858	−	0.716237i	$-0.745863\pi$
$179$	−18.6734	−1.39572	−0.697858	+	0.716237i	$0.745863\pi$
$180$	−22.4507	−1.67338
$181$	−1.48667	−0.110504	−0.0552518	−	0.998472i	$-0.517596\pi$
$181$	−1.48667	−0.110504	−0.0552518	+	0.998472i	$0.517596\pi$
$182$	11.8952	0.881734
$183$	3.63384	0.268621
$184$	−6.74293	−0.497095
$185$	3.35662	0.246784
$186$	1.34924	0.0989314
$187$	14.9525	1.09344
$188$	12.9601	0.945215
$189$	10.9921	0.799561
$190$	−8.85960	−0.642743
$191$	14.4960	1.04889	0.524446	−	0.851444i	$-0.324272\pi$
$191$	14.4960	1.04889	0.524446	+	0.851444i	$0.324272\pi$
$192$	−10.8820	−0.785340
$193$	−9.51607	−0.684982	−0.342491	−	0.939521i	$-0.611271\pi$
$193$	−9.51607	−0.684982	−0.342491	+	0.939521i	$0.611271\pi$
$194$	40.3919	2.89997
$195$	0.642442	0.0460062
$196$	65.5436	4.68169
$197$	10.8477	0.772867	0.386434	−	0.922317i	$-0.373707\pi$
$197$	10.8477	0.772867	0.386434	+	0.922317i	$0.373707\pi$
$198$	26.2480	1.86536
$199$	11.2419	0.796919	0.398459	−	0.917186i	$-0.369545\pi$
$199$	11.2419	0.796919	0.398459	+	0.917186i	$0.369545\pi$
$200$	25.2245	1.78364
$201$	5.06272	0.357097
$202$	−16.7603	−1.17925
$203$	35.2940	2.47715
$204$	−10.0207	−0.701587
$205$	15.1049	1.05497
$206$	−47.4820	−3.30823
$207$	−2.08919	−0.145208
$208$	−13.9257	−0.965575
$209$	7.53971	0.521533
$210$	7.64200	0.527348
$211$	−13.7262	−0.944954	−0.472477	−	0.881343i	$-0.656640\pi$
$211$	−13.7262	−0.944954	−0.472477	+	0.881343i	$0.656640\pi$
$212$	56.8807	3.90658
$213$	1.25971	0.0863141
$214$	5.26523	0.359924
$215$	6.78359	0.462637
$216$	−22.7568	−1.54840
$217$	5.06747	0.344002
$218$	39.5640	2.67961
$219$	−4.64872	−0.314131
$220$	27.4422	1.85015
$221$	−4.34659	−0.292383
$222$	2.62997	0.176512
$223$	9.47067	0.634203	0.317102	−	0.948392i	$-0.397290\pi$
$223$	9.47067	0.634203	0.317102	+	0.948392i	$0.397290\pi$
$224$	−85.9429	−5.74230
$225$	7.81541	0.521027
$226$	37.6478	2.50429
$227$	20.5257	1.36234	0.681168	−	0.732127i	$-0.261472\pi$
$227$	20.5257	1.36234	0.681168	+	0.732127i	$0.261472\pi$
$228$	−5.05286	−0.334634
$229$	−28.4832	−1.88222	−0.941111	−	0.338098i	$-0.890217\pi$
$229$	−28.4832	−1.88222	−0.941111	+	0.338098i	$0.890217\pi$
$230$	−3.00072	−0.197862
$231$	−6.50351	−0.427900
$232$	−73.0684	−4.79717
$233$	−6.45194	−0.422681	−0.211340	−	0.977413i	$-0.567783\pi$
$233$	−6.45194	−0.422681	−0.211340	+	0.977413i	$0.567783\pi$
$234$	−7.63011	−0.498796
$235$	3.61156	0.235592
$236$	−35.8860	−2.33598
$237$	0.0124825	0.000810827 0
$238$	−51.7037	−3.35146
$239$	2.73474	0.176896	0.0884478	−	0.996081i	$-0.471809\pi$
$239$	2.73474	0.176896	0.0884478	+	0.996081i	$0.471809\pi$
$240$	−8.94646	−0.577492
$241$	1.46370	0.0942850	0.0471425	−	0.998888i	$-0.484989\pi$
$241$	1.46370	0.0942850	0.0471425	+	0.998888i	$0.484989\pi$
$242$	−2.26104	−0.145345
$243$	−10.6888	−0.685687
$244$	−45.1217	−2.88862
$245$	18.2649	1.16690
$246$	11.8350	0.754570
$247$	−2.19174	−0.139457
$248$	−10.4911	−0.666183
$249$	−0.148528	−0.00941258
$250$	31.4367	1.98823
$251$	7.08403	0.447140	0.223570	−	0.974688i	$-0.428229\pi$
$251$	7.08403	0.447140	0.223570	+	0.974688i	$0.428229\pi$
$252$	−66.0661	−4.16177
$253$	2.55368	0.160548
$254$	8.49361	0.532936
$255$	−2.79243	−0.174869
$256$	28.9102	1.80688
$257$	−5.32701	−0.332290	−0.166145	−	0.986101i	$-0.553132\pi$
$257$	−5.32701	−0.332290	−0.166145	+	0.986101i	$0.553132\pi$
$258$	5.31506	0.330901
$259$	9.87760	0.613764
$260$	−7.97726	−0.494729
$261$	−22.6390	−1.40132
$262$	37.2429	2.30087
$263$	−0.713788	−0.0440141	−0.0220070	−	0.999758i	$-0.507006\pi$
$263$	−0.713788	−0.0440141	−0.0220070	+	0.999758i	$0.507006\pi$
$264$	13.4641	0.828657
$265$	15.8508	0.973706
$266$	−26.0713	−1.59853
$267$	2.10780	0.128995
$268$	−62.8643	−3.84005
$269$	−10.3014	−0.628088	−0.314044	−	0.949408i	$-0.601684\pi$
$269$	−10.3014	−0.628088	−0.314044	+	0.949408i	$0.601684\pi$
$270$	−10.1272	−0.616320
$271$	−26.6136	−1.61666	−0.808331	−	0.588728i	$-0.799629\pi$
$271$	−26.6136	−1.61666	−0.808331	+	0.588728i	$0.799629\pi$
$272$	60.5294	3.67013
$273$	1.89053	0.114420
$274$	−36.2738	−2.19138
$275$	−9.55303	−0.576069
$276$	−1.71139	−0.103014
$277$	21.0560	1.26513	0.632567	−	0.774506i	$-0.282001\pi$
$277$	21.0560	1.26513	0.632567	+	0.774506i	$0.282001\pi$
$278$	−49.5021	−2.96894
$279$	−3.25048	−0.194601
$280$	−59.4205	−3.55105
$281$	8.91541	0.531849	0.265924	−	0.963994i	$-0.414323\pi$
$281$	8.91541	0.531849	0.265924	+	0.963994i	$0.414323\pi$
$282$	2.82972	0.168508
$283$	−10.8455	−0.644697	−0.322349	−	0.946621i	$-0.604472\pi$
$283$	−10.8455	−0.644697	−0.322349	+	0.946621i	$0.604472\pi$
$284$	−15.6420	−0.928180
$285$	−1.40807	−0.0834067
$286$	9.32653	0.551489
$287$	44.4496	2.62378
$288$	55.1274	3.24841
$289$	1.89285	0.111344
$290$	−32.5167	−1.90945
$291$	6.41953	0.376319
$292$	57.7236	3.37802
$293$	−22.3253	−1.30426	−0.652130	−	0.758107i	$-0.726124\pi$
$293$	−22.3253	−1.30426	−0.652130	+	0.758107i	$0.726124\pi$
$294$	14.3108	0.834625
$295$	−10.0003	−0.582237
$296$	−20.4494	−1.18860
$297$	8.61845	0.500093
$298$	−39.0658	−2.26302
$299$	−0.742337	−0.0429305
$300$	6.40211	0.369626
$301$	19.9622	1.15060
$302$	−6.01625	−0.346197
$303$	−2.66374	−0.153028
$304$	30.5216	1.75053
$305$	−12.5740	−0.719983
$306$	33.1649	1.89591
$307$	4.90812	0.280121	0.140061	−	0.990143i	$-0.455270\pi$
$307$	4.90812	0.280121	0.140061	+	0.990143i	$0.455270\pi$
$308$	80.7547	4.60143
$309$	−7.54637	−0.429298
$310$	−4.66871	−0.265165
$311$	25.0636	1.42123	0.710614	−	0.703582i	$-0.248417\pi$
$311$	25.0636	1.42123	0.710614	+	0.703582i	$0.248417\pi$
$312$	−3.91391	−0.221582
$313$	6.73258	0.380548	0.190274	−	0.981731i	$-0.439062\pi$
$313$	6.73258	0.380548	0.190274	+	0.981731i	$0.439062\pi$
$314$	41.1648	2.32306
$315$	−18.4105	−1.03731
$316$	−0.154997	−0.00871925
$317$	−20.8338	−1.17014	−0.585072	−	0.810982i	$-0.698934\pi$
$317$	−20.8338	−1.17014	−0.585072	+	0.810982i	$0.698934\pi$
$318$	12.4194	0.696443
$319$	27.6724	1.54936
$320$	37.6543	2.10494
$321$	0.836810	0.0467062
$322$	−8.83028	−0.492092
$323$	9.52661	0.530075
$324$	39.3974	2.18875
$325$	2.77700	0.154040
$326$	−4.66313	−0.258267
$327$	6.28795	0.347724
$328$	−92.0230	−5.08112
$329$	10.6278	0.585931
$330$	5.99175	0.329835
$331$	−23.3541	−1.28366	−0.641829	−	0.766848i	$-0.721824\pi$
$331$	−23.3541	−1.28366	−0.641829	+	0.766848i	$0.721824\pi$
$332$	1.84429	0.101218
$333$	−6.33590	−0.347205
$334$	−57.2825	−3.13436
$335$	−17.5182	−0.957123
$336$	−26.3269	−1.43625
$337$	−22.4978	−1.22553	−0.612766	−	0.790264i	$-0.709943\pi$
$337$	−22.4978	−1.22553	−0.612766	+	0.790264i	$0.709943\pi$
$338$	−2.71116	−0.147467
$339$	5.98340	0.324974
$340$	34.6739	1.88046
$341$	3.97317	0.215159
$342$	16.7232	0.904289
$343$	23.0358	1.24381
$344$	−41.3273	−2.22822
$345$	−0.476908	−0.0256759
$346$	−32.8105	−1.76391
$347$	−5.68612	−0.305247	−0.152623	−	0.988284i	$-0.548772\pi$
$347$	−5.68612	−0.305247	−0.152623	+	0.988284i	$0.548772\pi$
$348$	−18.5451	−0.994123
$349$	−37.0968	−1.98575	−0.992874	−	0.119168i	$-0.961977\pi$
$349$	−37.0968	−1.98575	−0.992874	+	0.119168i	$0.961977\pi$
$350$	33.0331	1.76569
$351$	−2.50532	−0.133724
$352$	−67.3840	−3.59158
$353$	−31.1014	−1.65536	−0.827681	−	0.561199i	$-0.810340\pi$
$353$	−31.1014	−1.65536	−0.827681	+	0.561199i	$0.810340\pi$
$354$	−7.83537	−0.416445
$355$	−4.35891	−0.231347
$356$	−26.1727	−1.38715
$357$	−8.21734	−0.434908
$358$	50.6265	2.67569
$359$	4.59765	0.242655	0.121327	−	0.992613i	$-0.461285\pi$
$359$	4.59765	0.242655	0.121327	+	0.992613i	$0.461285\pi$
$360$	38.1148	2.00882
$361$	−14.1963	−0.747172
$362$	4.03061	0.211844
$363$	−0.359350	−0.0188610
$364$	−23.4748	−1.23042
$365$	16.0857	0.841963
$366$	−9.85191	−0.514968
$367$	−12.0474	−0.628867	−0.314433	−	0.949280i	$-0.601814\pi$
$367$	−12.0474	−0.628867	−0.314433	+	0.949280i	$0.601814\pi$
$368$	10.3376	0.538883
$369$	−28.5118	−1.48427
$370$	−9.10033	−0.473104
$371$	46.6444	2.42166
$372$	−2.66269	−0.138054
$373$	18.8176	0.974340	0.487170	−	0.873307i	$-0.338029\pi$
$373$	18.8176	0.974340	0.487170	+	0.873307i	$0.338029\pi$
$374$	−40.5386	−2.09620
$375$	4.99627	0.258006
$376$	−22.0025	−1.13469
$377$	−8.04419	−0.414297
$378$	−29.8014	−1.53282
$379$	21.1461	1.08620	0.543102	−	0.839667i	$-0.317250\pi$
$379$	21.1461	1.08620	0.543102	+	0.839667i	$0.317250\pi$
$380$	17.4841	0.896915
$381$	1.34990	0.0691575
$382$	−39.3009	−2.01081
$383$	−2.44778	−0.125076	−0.0625378	−	0.998043i	$-0.519919\pi$
$383$	−2.44778	−0.125076	−0.0625378	+	0.998043i	$0.519919\pi$
$384$	12.6223	0.644128
$385$	22.5037	1.14689
$386$	25.7996	1.31316
$387$	−12.8046	−0.650894
$388$	−79.7119	−4.04676
$389$	−18.2934	−0.927515	−0.463757	−	0.885962i	$-0.653499\pi$
$389$	−18.2934	−0.927515	−0.463757	+	0.885962i	$0.653499\pi$
$390$	−1.74176	−0.0881974
$391$	3.22664	0.163178
$392$	−111.274	−5.62019
$393$	5.91906	0.298577
$394$	−29.4098	−1.48165
$395$	−0.0431925	−0.00217325
$396$	−51.7994	−2.60302
$397$	−11.9919	−0.601858	−0.300929	−	0.953647i	$-0.597297\pi$
$397$	−11.9919	−0.601858	−0.300929	+	0.953647i	$0.597297\pi$
$398$	−30.4786	−1.52776
$399$	−4.14354	−0.207437
$400$	−38.6717	−1.93358
$401$	25.2817	1.26251	0.631255	−	0.775575i	$-0.282540\pi$
$401$	25.2817	1.26251	0.631255	+	0.775575i	$0.282540\pi$
$402$	−13.7258	−0.684582
$403$	−1.15497	−0.0575334
$404$	33.0759	1.64559
$405$	10.9788	0.545540
$406$	−95.6875	−4.74889
$407$	7.74458	0.383885
$408$	17.0122	0.842228
$409$	1.71486	0.0847941	0.0423971	−	0.999101i	$-0.486501\pi$
$409$	1.71486	0.0847941	0.0423971	+	0.999101i	$0.486501\pi$
$410$	−40.9518	−2.02247
$411$	−5.76504	−0.284369
$412$	93.7040	4.61647
$413$	−29.4279	−1.44805
$414$	5.66411	0.278376
$415$	0.513942	0.0252284
$416$	19.5880	0.960383
$417$	−7.86743	−0.385270
$418$	−20.4413	−0.999819
$419$	−24.7950	−1.21131	−0.605657	−	0.795725i	$-0.707090\pi$
$419$	−24.7950	−1.21131	−0.605657	+	0.795725i	$0.707090\pi$
$420$	−15.0812	−0.735888
$421$	37.1210	1.80917	0.904584	−	0.426295i	$-0.140182\pi$
$421$	37.1210	1.80917	0.904584	+	0.426295i	$0.140182\pi$
$422$	37.2140	1.81155
$423$	−6.81713	−0.331460
$424$	−96.5669	−4.68970
$425$	−12.0705	−0.585504
$426$	−3.41528	−0.165471
$427$	−37.0016	−1.79063
$428$	−10.3907	−0.502256
$429$	1.48228	0.0715650
$430$	−18.3914	−0.886911
$431$	−11.2898	−0.543810	−0.271905	−	0.962324i	$-0.587654\pi$
$431$	−11.2898	−0.543810	−0.271905	+	0.962324i	$0.587654\pi$
$432$	34.8884	1.67857
$433$	−39.0350	−1.87590	−0.937952	−	0.346766i	$-0.887280\pi$
$433$	−39.0350	−1.87590	−0.937952	+	0.346766i	$0.887280\pi$
$434$	−13.7387	−0.659478
$435$	−5.16792	−0.247783
$436$	−78.0781	−3.73926
$437$	1.62701	0.0778305
$438$	12.6034	0.602214
$439$	−12.0482	−0.575030	−0.287515	−	0.957776i	$-0.592829\pi$
$439$	−12.0482	−0.575030	−0.287515	+	0.957776i	$0.592829\pi$
$440$	−46.5889	−2.22104
$441$	−34.4764	−1.64174
$442$	11.7843	0.560521
$443$	21.2116	1.00779	0.503896	−	0.863765i	$-0.331900\pi$
$443$	21.2116	1.00779	0.503896	+	0.863765i	$0.331900\pi$
$444$	−5.19015	−0.246314
$445$	−7.29348	−0.345744
$446$	−25.6765	−1.21582
$447$	−6.20877	−0.293665
$448$	110.806	5.23509
$449$	−18.9263	−0.893185	−0.446593	−	0.894737i	$-0.647363\pi$
$449$	−18.9263	−0.893185	−0.446593	+	0.894737i	$0.647363\pi$
$450$	−21.1888	−0.998849
$451$	34.8509	1.64107
$452$	−74.2965	−3.49461
$453$	−0.956171	−0.0449248
$454$	−55.6483	−2.61170
$455$	−6.54167	−0.306678
$456$	8.57829	0.401715
$457$	27.0819	1.26684	0.633419	−	0.773809i	$-0.281651\pi$
$457$	27.0819	1.26684	0.633419	+	0.773809i	$0.281651\pi$
$458$	77.2224	3.60837
$459$	10.8896	0.508284
$460$	5.92181	0.276106
$461$	33.4762	1.55914	0.779570	−	0.626315i	$-0.215438\pi$
$461$	33.4762	1.55914	0.779570	+	0.626315i	$0.215438\pi$
$462$	17.6320	0.820317
$463$	−30.2781	−1.40714	−0.703571	−	0.710625i	$-0.748412\pi$
$463$	−30.2781	−1.40714	−0.703571	+	0.710625i	$0.748412\pi$
$464$	112.021	5.20045
$465$	−0.742003	−0.0344096
$466$	17.4922	0.810311
$467$	40.7909	1.88758	0.943789	−	0.330547i	$-0.107233\pi$
$467$	40.7909	1.88758	0.943789	+	0.330547i	$0.107233\pi$
$468$	15.0577	0.696044
$469$	−51.5512	−2.38041
$470$	−9.79152	−0.451649
$471$	6.54236	0.301456
$472$	60.9240	2.80425
$473$	15.6515	0.719655
$474$	−0.0338421	−0.00155442
$475$	−6.08647	−0.279266
$476$	102.035	4.67679
$477$	−29.9197	−1.36993
$478$	−7.41431	−0.339122
$479$	32.2653	1.47424	0.737120	−	0.675762i	$-0.236185\pi$
$479$	32.2653	1.47424	0.737120	+	0.675762i	$0.236185\pi$
$480$	12.5842	0.574387
$481$	−2.25130	−0.102650
$482$	−3.96831	−0.180752
$483$	−1.40341	−0.0638572
$484$	4.46208	0.202822
$485$	−22.2131	−1.00865
$486$	28.9790	1.31452
$487$	−12.5111	−0.566934	−0.283467	−	0.958982i	$-0.591485\pi$
$487$	−12.5111	−0.566934	−0.283467	+	0.958982i	$0.591485\pi$
$488$	76.6036	3.46768
$489$	−0.741117	−0.0335145
$490$	−49.5189	−2.23704
$491$	−7.18296	−0.324162	−0.162081	−	0.986777i	$-0.551821\pi$
$491$	−7.18296	−0.324162	−0.162081	+	0.986777i	$0.551821\pi$
$492$	−23.3559	−1.05297
$493$	34.9648	1.57473
$494$	5.94216	0.267350
$495$	−14.4348	−0.648797
$496$	16.0838	0.722186
$497$	−12.8270	−0.575371
$498$	0.402683	0.0180446
$499$	19.2089	0.859907	0.429954	−	0.902851i	$-0.358530\pi$
$499$	19.2089	0.859907	0.429954	+	0.902851i	$0.358530\pi$
$500$	−62.0391	−2.77447
$501$	−9.10398	−0.406736
$502$	−19.2059	−0.857202
$503$	−38.3807	−1.71131	−0.855655	−	0.517547i	$-0.826845\pi$
$503$	−38.3807	−1.71131	−0.855655	+	0.517547i	$0.826845\pi$
$504$	112.161	4.99605
$505$	9.21718	0.410159
$506$	−6.92343	−0.307784
$507$	−0.430887	−0.0191364
$508$	−16.7618	−0.743686
$509$	−4.76292	−0.211113	−0.105556	−	0.994413i	$-0.533662\pi$
$509$	−4.76292	−0.211113	−0.105556	+	0.994413i	$0.533662\pi$
$510$	7.57072	0.335237
$511$	47.3356	2.09401
$512$	−19.7925	−0.874715
$513$	5.49103	0.242435
$514$	14.4424	0.637025
$515$	26.1123	1.15064
$516$	−10.4891	−0.461756
$517$	8.33280	0.366476
$518$	−26.7797	−1.17663
$519$	−5.21462	−0.228896
$520$	13.5431	0.593903
$521$	−16.8277	−0.737233	−0.368616	−	0.929582i	$-0.620168\pi$
$521$	−16.8277	−0.737233	−0.368616	+	0.929582i	$0.620168\pi$
$522$	61.3780	2.68644
$523$	−4.63146	−0.202519	−0.101260	−	0.994860i	$-0.532287\pi$
$523$	−4.63146	−0.202519	−0.101260	+	0.994860i	$0.532287\pi$
$524$	−73.4975	−3.21075
$525$	5.24999	0.229128
$526$	1.93519	0.0843784
$527$	5.02020	0.218683
$528$	−20.6418	−0.898317
$529$	−22.4489	−0.976041
$530$	−42.9739	−1.86667
$531$	18.8763	0.819162
$532$	51.4508	2.23067
$533$	−10.1309	−0.438819
$534$	−5.71456	−0.247293
$535$	−2.89556	−0.125186
$536$	106.725	4.60983
$537$	8.04613	0.347216
$538$	27.9287	1.20409
$539$	42.1417	1.81517
$540$	19.9856	0.860044
$541$	35.0919	1.50872	0.754360	−	0.656461i	$-0.227947\pi$
$541$	35.0919	1.50872	0.754360	+	0.656461i	$0.227947\pi$
$542$	72.1537	3.09927
$543$	0.640589	0.0274903
$544$	−85.1412	−3.65040
$545$	−21.7578	−0.932002
$546$	−5.12551	−0.219352
$547$	24.1160	1.03112	0.515562	−	0.856852i	$-0.327583\pi$
$547$	24.1160	1.03112	0.515562	+	0.856852i	$0.327583\pi$
$548$	71.5851	3.05796
$549$	23.7344	1.01296
$550$	25.8998	1.10437
$551$	17.6308	0.751097
$552$	2.90544	0.123664
$553$	−0.127103	−0.00540499
$554$	−57.0862	−2.42536
$555$	−1.44633	−0.0613931
$556$	97.6906	4.14300
$557$	16.0432	0.679771	0.339885	−	0.940467i	$-0.389612\pi$
$557$	16.0432	0.679771	0.339885	+	0.940467i	$0.389612\pi$
$558$	8.81257	0.373066
$559$	−4.54977	−0.192435
$560$	91.0974	3.84957
$561$	−6.44285	−0.272017
$562$	−24.1711	−1.01960
$563$	−21.8038	−0.918922	−0.459461	−	0.888198i	$-0.651957\pi$
$563$	−21.8038	−0.918922	−0.459461	+	0.888198i	$0.651957\pi$
$564$	−5.58436	−0.235144
$565$	−20.7040	−0.871024
$566$	29.4038	1.23593
$567$	32.3075	1.35679
$568$	26.5555	1.11425
$569$	42.5341	1.78312	0.891560	−	0.452902i	$-0.149611\pi$
$569$	42.5341	1.78312	0.891560	+	0.452902i	$0.149611\pi$
$570$	3.81749	0.159897
$571$	6.03851	0.252704	0.126352	−	0.991985i	$-0.459673\pi$
$571$	6.03851	0.252704	0.126352	+	0.991985i	$0.459673\pi$
$572$	−18.4056	−0.769575
$573$	−6.24613	−0.260936
$574$	−120.510	−5.02998
$575$	−2.06147	−0.0859692
$576$	−71.0756	−2.96148
$577$	−3.27258	−0.136239	−0.0681197	−	0.997677i	$-0.521700\pi$
$577$	−3.27258	−0.136239	−0.0681197	+	0.997677i	$0.521700\pi$
$578$	−5.13181	−0.213455
$579$	4.10035	0.170405
$580$	64.1706	2.66454
$581$	1.51239	0.0627444
$582$	−17.4043	−0.721433
$583$	36.5718	1.51465
$584$	−97.9979	−4.05518
$585$	4.19610	0.173487
$586$	60.5275	2.50037
$587$	−10.6014	−0.437567	−0.218783	−	0.975773i	$-0.570209\pi$
$587$	−10.6014	−0.437567	−0.218783	+	0.975773i	$0.570209\pi$
$588$	−28.2419	−1.16468
$589$	2.53141	0.104305
$590$	27.1122	1.11619
$591$	−4.67414	−0.192269
$592$	31.3509	1.28851
$593$	33.6695	1.38264	0.691321	−	0.722548i	$-0.257029\pi$
$593$	33.6695	1.38264	0.691321	+	0.722548i	$0.257029\pi$
$594$	−23.3660	−0.958718
$595$	28.4340	1.16568
$596$	77.0949	3.15793
$597$	−4.84400	−0.198252
$598$	2.01259	0.0823010
$599$	−7.99324	−0.326595	−0.163297	−	0.986577i	$-0.552213\pi$
$599$	−7.99324	−0.326595	−0.163297	+	0.986577i	$0.552213\pi$
$600$	−10.8689	−0.443722
$601$	−5.62405	−0.229410	−0.114705	−	0.993400i	$-0.536592\pi$
$601$	−5.62405	−0.229410	−0.114705	+	0.993400i	$0.536592\pi$
$602$	−54.1207	−2.20579
$603$	33.0671	1.34660
$604$	11.8729	0.483100
$605$	1.24344	0.0505529
$606$	7.22182	0.293366
$607$	28.5118	1.15726	0.578630	−	0.815590i	$-0.303588\pi$
$607$	28.5118	1.15726	0.578630	+	0.815590i	$0.303588\pi$
$608$	−42.9320	−1.74112
$609$	−15.2077	−0.616249
$610$	34.0900	1.38026
$611$	−2.42229	−0.0979952
$612$	−65.4498	−2.64565
$613$	35.7776	1.44504	0.722522	−	0.691348i	$-0.242983\pi$
$613$	35.7776	1.44504	0.722522	+	0.691348i	$0.242983\pi$
$614$	−13.3067	−0.537014
$615$	−6.50853	−0.262449
$616$	−137.098	−5.52384
$617$	13.3230	0.536364	0.268182	−	0.963368i	$-0.413577\pi$
$617$	13.3230	0.536364	0.268182	+	0.963368i	$0.413577\pi$
$618$	20.4594	0.822998
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	9.21353	0.370024
$621$	1.85979	0.0746310
$622$	−67.9514	−2.72460
$623$	−21.4626	−0.859883
$624$	6.00042	0.240209
$625$	−3.40329	−0.136132
$626$	−18.2531	−0.729540
$627$	−3.24877	−0.129743
$628$	−81.2372	−3.24172
$629$	9.78546	0.390172
$630$	49.9136	1.98861
$631$	32.0130	1.27442	0.637208	−	0.770692i	$-0.280089\pi$
$631$	32.0130	1.27442	0.637208	+	0.770692i	$0.280089\pi$
$632$	0.263140	0.0104671
$633$	5.91447	0.235079
$634$	56.4837	2.24325
$635$	−4.67098	−0.185362
$636$	−24.5092	−0.971851
$637$	−12.2503	−0.485374
$638$	−75.0243	−2.97024
$639$	8.22780	0.325487
$640$	−43.6761	−1.72645
$641$	−45.8171	−1.80967	−0.904833	−	0.425767i	$-0.860004\pi$
$641$	−45.8171	−1.80967	−0.904833	+	0.425767i	$0.860004\pi$
$642$	−2.26872	−0.0895393
$643$	21.9887	0.867149	0.433575	−	0.901118i	$-0.357252\pi$
$643$	21.9887	0.867149	0.433575	+	0.901118i	$0.357252\pi$
$644$	17.4262	0.686690
$645$	−2.92296	−0.115092
$646$	−25.8281	−1.01619
$647$	−19.3519	−0.760803	−0.380402	−	0.924821i	$-0.624214\pi$
$647$	−19.3519	−0.760803	−0.380402	+	0.924821i	$0.624214\pi$
$648$	−66.8854	−2.62751
$649$	−23.0731	−0.905699
$650$	−7.52888	−0.295307
$651$	−2.18351	−0.0855784
$652$	9.20252	0.360399
$653$	−15.2796	−0.597938	−0.298969	−	0.954263i	$-0.596643\pi$
$653$	−15.2796	−0.597938	−0.298969	+	0.954263i	$0.596643\pi$
$654$	−17.0476	−0.666614
$655$	−20.4814	−0.800273
$656$	141.080	5.50826
$657$	−30.3631	−1.18458
$658$	−28.8137	−1.12327
$659$	−39.8834	−1.55364	−0.776819	−	0.629724i	$-0.783168\pi$
$659$	−39.8834	−1.55364	−0.776819	+	0.629724i	$0.783168\pi$
$660$	−11.8245	−0.460268
$661$	−2.41433	−0.0939065	−0.0469532	−	0.998897i	$-0.514951\pi$
$661$	−2.41433	−0.0939065	−0.0469532	+	0.998897i	$0.514951\pi$
$662$	63.3166	2.46087
$663$	1.87289	0.0727371
$664$	−3.13106	−0.121509
$665$	14.3377	0.555990
$666$	17.1776	0.665619
$667$	5.97150	0.231217
$668$	113.045	4.37384
$669$	−4.08079	−0.157773
$670$	47.4946	1.83488
$671$	−29.0113	−1.11997
$672$	37.0317	1.42853
$673$	−48.0193	−1.85101	−0.925504	−	0.378739i	$-0.876358\pi$
$673$	−48.0193	−1.85101	−0.925504	+	0.378739i	$0.876358\pi$
$674$	60.9950	2.34944
$675$	−6.95728	−0.267786
$676$	5.35037	0.205783
$677$	−12.8934	−0.495533	−0.247767	−	0.968820i	$-0.579697\pi$
$677$	−12.8934	−0.495533	−0.247767	+	0.968820i	$0.579697\pi$
$678$	−16.2219	−0.623000
$679$	−65.3669	−2.50855
$680$	−58.8662	−2.25742
$681$	−8.84425	−0.338913
$682$	−10.7719	−0.412477
$683$	−1.19314	−0.0456544	−0.0228272	−	0.999739i	$-0.507267\pi$
$683$	−1.19314	−0.0456544	−0.0228272	+	0.999739i	$0.507267\pi$
$684$	−33.0027	−1.26189
$685$	19.9484	0.762190
$686$	−62.4535	−2.38449
$687$	12.2730	0.468246
$688$	63.3589	2.41553
$689$	−10.6312	−0.405015
$690$	1.29297	0.0492226
$691$	−9.78194	−0.372122	−0.186061	−	0.982538i	$-0.559572\pi$
$691$	−9.78194	−0.372122	−0.186061	+	0.982538i	$0.559572\pi$
$692$	64.7504	2.46144
$693$	−42.4776	−1.61359
$694$	15.4160	0.585182
$695$	27.2232	1.03263
$696$	31.4842	1.19341
$697$	44.0350	1.66794
$698$	100.575	3.80683
$699$	2.78006	0.105151
$700$	−65.1896	−2.46393
$701$	−36.9714	−1.39639	−0.698195	−	0.715908i	$-0.746013\pi$
$701$	−36.9714	−1.39639	−0.698195	+	0.715908i	$0.746013\pi$
$702$	6.79233	0.256360
$703$	4.93426	0.186099
$704$	86.8780	3.27434
$705$	−1.55618	−0.0586090
$706$	84.3208	3.17346
$707$	27.1236	1.02009
$708$	15.4628	0.581128
$709$	−30.1673	−1.13296	−0.566478	−	0.824077i	$-0.691694\pi$
$709$	−30.1673	−1.13296	−0.566478	+	0.824077i	$0.691694\pi$
$710$	11.8177	0.443510
$711$	0.0815295	0.00305759
$712$	44.4337	1.66522
$713$	0.857380	0.0321091
$714$	22.2785	0.833752
$715$	−5.12903	−0.191815
$716$	−99.9096	−3.73380
$717$	−1.17836	−0.0440068
$718$	−12.4650	−0.465188
$719$	−35.6620	−1.32997	−0.664984	−	0.746858i	$-0.731562\pi$
$719$	−35.6620	−1.32997	−0.664984	+	0.746858i	$0.731562\pi$
$720$	−58.4337	−2.17770
$721$	76.8410	2.86171
$722$	38.4883	1.43239
$723$	−0.630688	−0.0234555
$724$	−7.95426	−0.295618
$725$	−22.3387	−0.829638
$726$	0.974254	0.0361580
$727$	26.9424	0.999239	0.499619	−	0.866245i	$-0.333473\pi$
$727$	26.9424	0.999239	0.499619	+	0.866245i	$0.333473\pi$
$728$	39.8535	1.47707
$729$	−17.4848	−0.647586
$730$	−43.6108	−1.61411
$731$	19.7760	0.731442
$732$	19.4424	0.718611
$733$	36.3433	1.34237	0.671186	−	0.741289i	$-0.265785\pi$
$733$	36.3433	1.34237	0.671186	+	0.741289i	$0.265785\pi$
$734$	32.6623	1.20559
$735$	−7.87010	−0.290293
$736$	−14.5409	−0.535986
$737$	−40.4190	−1.48885
$738$	77.3000	2.84545
$739$	3.47442	0.127809	0.0639044	−	0.997956i	$-0.479645\pi$
$739$	3.47442	0.127809	0.0639044	+	0.997956i	$0.479645\pi$
$740$	17.9592	0.660192
$741$	0.944394	0.0346932
$742$	−126.460	−4.64250
$743$	11.4691	0.420761	0.210380	−	0.977620i	$-0.432530\pi$
$743$	11.4691	0.420761	0.210380	+	0.977620i	$0.432530\pi$
$744$	4.52047	0.165728
$745$	21.4838	0.787107
$746$	−51.0176	−1.86789
$747$	−0.970109	−0.0354944
$748$	80.0015	2.92514
$749$	−8.52082	−0.311344
$750$	−13.5457	−0.494618
$751$	7.29349	0.266143	0.133072	−	0.991106i	$-0.457516\pi$
$751$	7.29349	0.266143	0.133072	+	0.991106i	$0.457516\pi$
$752$	33.7321	1.23008
$753$	−3.05242	−0.111236
$754$	21.8090	0.794239
$755$	3.30858	0.120412
$756$	58.8121	2.13897
$757$	26.9368	0.979034	0.489517	−	0.871994i	$-0.337173\pi$
$757$	26.9368	0.979034	0.489517	+	0.871994i	$0.337173\pi$
$758$	−57.3305	−2.08234
$759$	−1.10035	−0.0399401
$760$	−29.6829	−1.07671
$761$	16.8859	0.612114	0.306057	−	0.952013i	$-0.400990\pi$
$761$	16.8859	0.612114	0.306057	+	0.952013i	$0.400990\pi$
$762$	−3.65979	−0.132580
$763$	−64.0271	−2.31794
$764$	77.5588	2.80598
$765$	−18.2387	−0.659423
$766$	6.63631	0.239780
$767$	6.70720	0.242183
$768$	−12.4570	−0.449504
$769$	−29.3353	−1.05786	−0.528929	−	0.848666i	$-0.677406\pi$
$769$	−29.3353	−1.05786	−0.528929	+	0.848666i	$0.677406\pi$
$770$	−61.0111	−2.19869
$771$	2.29534	0.0826647
$772$	−50.9145	−1.83245
$773$	18.3770	0.660974	0.330487	−	0.943810i	$-0.392787\pi$
$773$	18.3770	0.660974	0.330487	+	0.943810i	$0.392787\pi$
$774$	34.7153	1.24781
$775$	−3.20736	−0.115212
$776$	135.328	4.85798
$777$	−4.25613	−0.152688
$778$	49.5964	1.77812
$779$	22.2044	0.795554
$780$	3.43730	0.123075
$781$	−10.0571	−0.359872
$782$	−8.74791	−0.312825
$783$	20.1533	0.720220
$784$	170.594	6.09265
$785$	−22.6381	−0.807990
$786$	−16.0475	−0.572395
$787$	−49.7613	−1.77380	−0.886900	−	0.461962i	$-0.847146\pi$
$787$	−49.7613	−1.77380	−0.886900	+	0.461962i	$0.847146\pi$
$788$	58.0393	2.06756
$789$	0.307562	0.0109495
$790$	0.117102	0.00416629
$791$	−60.9261	−2.16628
$792$	87.9404	3.12483
$793$	8.43339	0.299478
$794$	32.5120	1.15381
$795$	−6.82990	−0.242232
$796$	60.1485	2.13191
$797$	12.9145	0.457454	0.228727	−	0.973491i	$-0.426544\pi$
$797$	12.9145	0.457454	0.228727	+	0.973491i	$0.426544\pi$
$798$	11.2338	0.397672
$799$	10.5287	0.372478
$800$	54.3960	1.92319
$801$	13.7670	0.486434
$802$	−68.5427	−2.42033
$803$	37.1138	1.30972
$804$	27.0874	0.955300
$805$	4.85612	0.171156
$806$	3.13132	0.110296
$807$	4.43875	0.156251
$808$	−56.1533	−1.97547
$809$	−11.8119	−0.415286	−0.207643	−	0.978205i	$-0.566579\pi$
$809$	−11.8119	−0.415286	−0.207643	+	0.978205i	$0.566579\pi$
$810$	−29.7652	−1.04584
$811$	−45.0680	−1.58255	−0.791276	−	0.611459i	$-0.790583\pi$
$811$	−45.0680	−1.58255	−0.791276	+	0.611459i	$0.790583\pi$
$812$	188.836	6.62684
$813$	11.4675	0.402182
$814$	−20.9968	−0.735936
$815$	2.56444	0.0898285
$816$	−26.0814	−0.913030
$817$	9.97193	0.348874
$818$	−4.64924	−0.162557
$819$	12.3479	0.431472
$820$	80.8170	2.82225
$821$	−26.9995	−0.942288	−0.471144	−	0.882056i	$-0.656159\pi$
$821$	−26.9995	−0.942288	−0.471144	+	0.882056i	$0.656159\pi$
$822$	15.6299	0.545156
$823$	−38.8064	−1.35271	−0.676354	−	0.736577i	$-0.736441\pi$
$823$	−38.8064	−1.35271	−0.676354	+	0.736577i	$0.736441\pi$
$824$	−159.082	−5.54189
$825$	4.11628	0.143310
$826$	79.7837	2.77603
$827$	−4.68769	−0.163007	−0.0815035	−	0.996673i	$-0.525972\pi$
$827$	−4.68769	−0.163007	−0.0815035	+	0.996673i	$0.525972\pi$
$828$	−11.1779	−0.388459
$829$	3.22766	0.112101	0.0560506	−	0.998428i	$-0.482149\pi$
$829$	3.22766	0.112101	0.0560506	+	0.998428i	$0.482149\pi$
$830$	−1.39338	−0.0483649
$831$	−9.07278	−0.314731
$832$	−25.2548	−0.875554
$833$	53.2470	1.84490
$834$	21.3298	0.738592
$835$	31.5019	1.09017
$836$	40.3403	1.39520
$837$	2.89358	0.100017
$838$	67.2231	2.32218
$839$	2.75696	0.0951807	0.0475904	−	0.998867i	$-0.484846\pi$
$839$	2.75696	0.0951807	0.0475904	+	0.998867i	$0.484846\pi$
$840$	25.6035	0.883406
$841$	35.7089	1.23134
$842$	−100.641	−3.46831
$843$	−3.84154	−0.132310
$844$	−73.4405	−2.52793
$845$	1.49097	0.0512910
$846$	18.4823	0.635435
$847$	3.65909	0.125728
$848$	148.047	5.08394
$849$	4.67318	0.160383
$850$	32.7249	1.12246
$851$	1.67122	0.0572887
$852$	6.73993	0.230906
$853$	42.0468	1.43965	0.719827	−	0.694154i	$-0.244221\pi$
$853$	42.0468	1.43965	0.719827	+	0.694154i	$0.244221\pi$
$854$	100.317	3.43278
$855$	−9.19677	−0.314523
$856$	17.6405	0.602939
$857$	51.6935	1.76582	0.882908	−	0.469546i	$-0.155582\pi$
$857$	51.6935	1.76582	0.882908	+	0.469546i	$0.155582\pi$
$858$	−4.01868	−0.137196
$859$	2.30879	0.0787749	0.0393874	−	0.999224i	$-0.487459\pi$
$859$	2.30879	0.0787749	0.0393874	+	0.999224i	$0.487459\pi$
$860$	36.2947	1.23764
$861$	−19.1528	−0.652724
$862$	30.6084	1.04253
$863$	42.0300	1.43072	0.715359	−	0.698758i	$-0.246263\pi$
$863$	42.0300	1.43072	0.715359	+	0.698758i	$0.246263\pi$
$864$	−49.0744	−1.66955
$865$	18.0438	0.613509
$866$	105.830	3.59625
$867$	−0.815604	−0.0276994
$868$	27.1128	0.920269
$869$	−0.0996562	−0.00338060
$870$	14.0110	0.475019
$871$	11.7495	0.398117
$872$	132.554	4.48884
$873$	41.9291	1.41908
$874$	−4.41108	−0.149207
$875$	−50.8746	−1.71987
$876$	−24.8724	−0.840360
$877$	30.4717	1.02896	0.514478	−	0.857504i	$-0.327986\pi$
$877$	30.4717	1.02896	0.514478	+	0.857504i	$0.327986\pi$
$878$	32.6646	1.10238
$879$	9.61970	0.324464
$880$	71.4254	2.40775
$881$	−2.42048	−0.0815479	−0.0407740	−	0.999168i	$-0.512982\pi$
$881$	−2.42048	−0.0815479	−0.0407740	+	0.999168i	$0.512982\pi$
$882$	93.4710	3.14733
$883$	−11.6138	−0.390834	−0.195417	−	0.980720i	$-0.562606\pi$
$883$	−11.6138	−0.390834	−0.195417	+	0.980720i	$0.562606\pi$
$884$	−23.2559	−0.782179
$885$	4.30898	0.144845
$886$	−57.5078	−1.93201
$887$	27.7553	0.931933	0.465966	−	0.884802i	$-0.345707\pi$
$887$	27.7553	0.931933	0.465966	+	0.884802i	$0.345707\pi$
$888$	8.81138	0.295690
$889$	−13.7454	−0.461005
$890$	19.7738	0.662818
$891$	25.3308	0.848615
$892$	50.6716	1.69661
$893$	5.30903	0.177660
$894$	16.8330	0.562978
$895$	−27.8415	−0.930640
$896$	−128.527	−4.29377
$897$	0.319864	0.0106799
$898$	51.3120	1.71230
$899$	9.29083	0.309866
$900$	41.8153	1.39384
$901$	46.2093	1.53946
$902$	−94.4863	−3.14605
$903$	−8.60146	−0.286239
$904$	126.134	4.19515
$905$	−2.21659	−0.0736820
$906$	2.59233	0.0861243
$907$	21.4195	0.711223	0.355612	−	0.934634i	$-0.384273\pi$
$907$	21.4195	0.711223	0.355612	+	0.934634i	$0.384273\pi$
$908$	109.820	3.64450
$909$	−17.3982	−0.577061
$910$	17.7355	0.587926
$911$	−0.805834	−0.0266985	−0.0133492	−	0.999911i	$-0.504249\pi$
$911$	−0.805834	−0.0266985	−0.0133492	+	0.999911i	$0.504249\pi$
$912$	−13.1514	−0.435485
$913$	1.18580	0.0392441
$914$	−73.4233	−2.42863
$915$	5.41796	0.179112
$916$	−152.396	−5.03529
$917$	−60.2709	−1.99032
$918$	−29.5235	−0.974419
$919$	−31.2873	−1.03207	−0.516036	−	0.856567i	$-0.672593\pi$
$919$	−31.2873	−1.03207	−0.516036	+	0.856567i	$0.672593\pi$
$920$	−10.0535	−0.331455
$921$	−2.11485	−0.0696866
$922$	−90.7591	−2.98899
$923$	2.92353	0.0962292
$924$	−34.7962	−1.14471
$925$	−6.25184	−0.205559
$926$	82.0887	2.69760
$927$	−49.2890	−1.61886
$928$	−157.570	−5.17249
$929$	27.2106	0.892752	0.446376	−	0.894846i	$-0.352714\pi$
$929$	27.2106	0.892752	0.446376	+	0.894846i	$0.352714\pi$
$930$	2.01169	0.0659658
$931$	26.8495	0.879957
$932$	−34.5203	−1.13075
$933$	−10.7996	−0.353563
$934$	−110.591	−3.61863
$935$	22.2938	0.729085
$936$	−25.5637	−0.835575
$937$	40.0416	1.30810	0.654051	−	0.756450i	$-0.273068\pi$
$937$	40.0416	1.30810	0.654051	+	0.756450i	$0.273068\pi$
$938$	139.763	4.56344
$939$	−2.90099	−0.0946701
$940$	19.3232	0.630254
$941$	−28.8387	−0.940114	−0.470057	−	0.882636i	$-0.655767\pi$
$941$	−28.8387	−0.940114	−0.470057	+	0.882636i	$0.655767\pi$
$942$	−17.7374	−0.577915
$943$	7.52056	0.244903
$944$	−93.4025	−3.03999
$945$	16.3890	0.533134
$946$	−42.4336	−1.37963
$947$	44.0072	1.43004	0.715021	−	0.699103i	$-0.246417\pi$
$947$	44.0072	1.43004	0.715021	+	0.699103i	$0.246417\pi$
$948$	0.0667862	0.00216911
$949$	−10.7887	−0.350216
$950$	16.5014	0.535375
$951$	8.97702	0.291100
$952$	−173.227	−5.61431
$953$	48.3902	1.56751	0.783757	−	0.621068i	$-0.213301\pi$
$953$	48.3902	1.56751	0.783757	+	0.621068i	$0.213301\pi$
$954$	81.1169	2.62626
$955$	21.6131	0.699384
$956$	14.6319	0.473228
$957$	−11.9237	−0.385439
$958$	−87.4764	−2.82623
$959$	58.7026	1.89561
$960$	−16.2248	−0.523652
$961$	−29.6660	−0.956969
$962$	6.10362	0.196788
$963$	5.46561	0.176127
$964$	7.83132	0.252230
$965$	−14.1882	−0.456735
$966$	3.80486	0.122419
$967$	−17.0974	−0.549816	−0.274908	−	0.961471i	$-0.588647\pi$
$967$	−17.0974	−0.549816	−0.274908	+	0.961471i	$0.588647\pi$
$968$	−7.57532	−0.243480
$969$	−4.10490	−0.131868
$970$	60.2232	1.93365
$971$	−21.7125	−0.696787	−0.348393	−	0.937348i	$-0.613273\pi$
$971$	−21.7125	−0.696787	−0.348393	+	0.937348i	$0.613273\pi$
$972$	−57.1891	−1.83434
$973$	80.1102	2.56821
$974$	33.9197	1.08686
$975$	−1.19657	−0.0383210
$976$	−117.441	−3.75919
$977$	−26.0372	−0.833002	−0.416501	−	0.909135i	$-0.636744\pi$
$977$	−26.0372	−0.833002	−0.416501	+	0.909135i	$0.636744\pi$
$978$	2.00928	0.0642498
$979$	−16.8279	−0.537822
$980$	97.7238	3.12167
$981$	41.0697	1.31125
$982$	19.4741	0.621444
$983$	−3.93981	−0.125661	−0.0628303	−	0.998024i	$-0.520013\pi$
$983$	−3.93981	−0.125661	−0.0628303	+	0.998024i	$0.520013\pi$
$984$	39.6515	1.26404
$985$	16.1736	0.515335
$986$	−94.7950	−3.01889
$987$	−4.57939	−0.145764
$988$	−11.7266	−0.373074
$989$	3.37747	0.107397
$990$	39.1350	1.24379
$991$	−9.66297	−0.306954	−0.153477	−	0.988152i	$-0.549047\pi$
$991$	−9.66297	−0.306954	−0.153477	+	0.988152i	$0.549047\pi$
$992$	−22.6237	−0.718303
$993$	10.0630	0.319339
$994$	34.7761	1.10303
$995$	16.7614	0.531372
$996$	−0.794680	−0.0251804
$997$	−4.02612	−0.127508	−0.0637542	−	0.997966i	$-0.520307\pi$
$997$	−4.02612	−0.127508	−0.0637542	+	0.997966i	$0.520307\pi$
$998$	−52.0783	−1.64851
$999$	5.64023	0.178449

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.d.1.3	✓	156

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.d.1.3	✓	156	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:	\( N \)	\(=\)	\( 8047 = 13 \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.71116 q^{2} -0.430887 q^{3} +5.35037 q^{4} +1.49097 q^{5} +1.16820 q^{6} +4.38752 q^{7} -9.08338 q^{8} -2.81434 q^{9} +O(q^{10})\)	\(q-2.71116 q^{2} -0.430887 q^{3} +5.35037 q^{4} +1.49097 q^{5} +1.16820 q^{6} +4.38752 q^{7} -9.08338 q^{8} -2.81434 q^{9} -4.04226 q^{10} +3.44005 q^{11} -2.30541 q^{12} -1.00000 q^{13} -11.8952 q^{14} -0.642442 q^{15} +13.9257 q^{16} +4.34659 q^{17} +7.63011 q^{18} +2.19174 q^{19} +7.97726 q^{20} -1.89053 q^{21} -9.32653 q^{22} +0.742337 q^{23} +3.91391 q^{24} -2.77700 q^{25} +2.71116 q^{26} +2.50532 q^{27} +23.4748 q^{28} +8.04419 q^{29} +1.74176 q^{30} +1.15497 q^{31} -19.5880 q^{32} -1.48228 q^{33} -11.7843 q^{34} +6.54167 q^{35} -15.0577 q^{36} +2.25130 q^{37} -5.94216 q^{38} +0.430887 q^{39} -13.5431 q^{40} +10.1309 q^{41} +5.12551 q^{42} +4.54977 q^{43} +18.4056 q^{44} -4.19610 q^{45} -2.01259 q^{46} +2.42229 q^{47} -6.00042 q^{48} +12.2503 q^{49} +7.52888 q^{50} -1.87289 q^{51} -5.35037 q^{52} +10.6312 q^{53} -6.79233 q^{54} +5.12903 q^{55} -39.8535 q^{56} -0.944394 q^{57} -21.8090 q^{58} -6.70720 q^{59} -3.43730 q^{60} -8.43339 q^{61} -3.13132 q^{62} -12.3479 q^{63} +25.2548 q^{64} -1.49097 q^{65} +4.01868 q^{66} -11.7495 q^{67} +23.2559 q^{68} -0.319864 q^{69} -17.7355 q^{70} -2.92353 q^{71} +25.5637 q^{72} +10.7887 q^{73} -6.10362 q^{74} +1.19657 q^{75} +11.7266 q^{76} +15.0933 q^{77} -1.16820 q^{78} -0.0289694 q^{79} +20.7629 q^{80} +7.36350 q^{81} -27.4665 q^{82} +0.344703 q^{83} -10.1150 q^{84} +6.48065 q^{85} -12.3351 q^{86} -3.46614 q^{87} -31.2473 q^{88} -4.89175 q^{89} +11.3763 q^{90} -4.38752 q^{91} +3.97178 q^{92} -0.497664 q^{93} -6.56720 q^{94} +3.26783 q^{95} +8.44024 q^{96} -14.8984 q^{97} -33.2125 q^{98} -9.68147 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9}+O(q^{10}) \) 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9} + 11 q^{10} + 23 q^{11} + 57 q^{12} - 156 q^{13} + 18 q^{14} + 32 q^{15} + 159 q^{16} + 119 q^{17} + 36 q^{18} + 35 q^{19} + 109 q^{20} + 33 q^{21} + 11 q^{22} + 55 q^{23} + 63 q^{24} + 189 q^{25} - 13 q^{26} + 89 q^{27} + 54 q^{28} - 55 q^{29} + 47 q^{31} + 112 q^{32} + 109 q^{33} + 51 q^{34} + 25 q^{35} + 162 q^{36} + 53 q^{37} + 37 q^{38} - 23 q^{39} + 25 q^{40} + 113 q^{41} + 26 q^{42} + 31 q^{43} + 86 q^{44} + 144 q^{45} + 37 q^{46} + 115 q^{47} + 129 q^{48} + 189 q^{49} + 72 q^{50} - 4 q^{51} - 161 q^{52} + 51 q^{53} + 108 q^{54} + 22 q^{55} + 39 q^{56} + 102 q^{57} + 31 q^{58} + 75 q^{59} + 97 q^{60} + 7 q^{61} + 77 q^{62} + 94 q^{63} + 158 q^{64} - 39 q^{65} + 48 q^{66} + 37 q^{67} + 235 q^{68} + 27 q^{69} + 38 q^{70} + 70 q^{71} + 152 q^{72} + 155 q^{73} - 18 q^{74} + 80 q^{75} + 21 q^{76} + 101 q^{77} - 25 q^{78} + 10 q^{79} + 211 q^{80} + 220 q^{81} + 45 q^{82} + 132 q^{83} + 86 q^{84} + 74 q^{85} + 35 q^{86} + 53 q^{87} + 51 q^{88} + 190 q^{89} - 27 q^{90} - 19 q^{91} + 125 q^{92} + 96 q^{93} - 19 q^{94} + 72 q^{95} + 146 q^{96} + 155 q^{97} + 135 q^{98} + 89 q^{99}+O(q^{100}) \) 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9 + 11 * q^10 + 23 * q^11 + 57 * q^12 - 156 * q^13 + 18 * q^14 + 32 * q^15 + 159 * q^16 + 119 * q^17 + 36 * q^18 + 35 * q^19 + 109 * q^20 + 33 * q^21 + 11 * q^22 + 55 * q^23 + 63 * q^24 + 189 * q^25 - 13 * q^26 + 89 * q^27 + 54 * q^28 - 55 * q^29 + 47 * q^31 + 112 * q^32 + 109 * q^33 + 51 * q^34 + 25 * q^35 + 162 * q^36 + 53 * q^37 + 37 * q^38 - 23 * q^39 + 25 * q^40 + 113 * q^41 + 26 * q^42 + 31 * q^43 + 86 * q^44 + 144 * q^45 + 37 * q^46 + 115 * q^47 + 129 * q^48 + 189 * q^49 + 72 * q^50 - 4 * q^51 - 161 * q^52 + 51 * q^53 + 108 * q^54 + 22 * q^55 + 39 * q^56 + 102 * q^57 + 31 * q^58 + 75 * q^59 + 97 * q^60 + 7 * q^61 + 77 * q^62 + 94 * q^63 + 158 * q^64 - 39 * q^65 + 48 * q^66 + 37 * q^67 + 235 * q^68 + 27 * q^69 + 38 * q^70 + 70 * q^71 + 152 * q^72 + 155 * q^73 - 18 * q^74 + 80 * q^75 + 21 * q^76 + 101 * q^77 - 25 * q^78 + 10 * q^79 + 211 * q^80 + 220 * q^81 + 45 * q^82 + 132 * q^83 + 86 * q^84 + 74 * q^85 + 35 * q^86 + 53 * q^87 + 51 * q^88 + 190 * q^89 - 27 * q^90 - 19 * q^91 + 125 * q^92 + 96 * q^93 - 19 * q^94 + 72 * q^95 + 146 * q^96 + 155 * q^97 + 135 * q^98 + 89 * q^99

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