LMFDB - Embedded newform 4009.2.a.d.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, 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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$75$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.72632 q^{2} +0.426642 q^{3} +0.980187 q^{4} +1.18315 q^{5} -0.736521 q^{6} -0.668631 q^{7} +1.76053 q^{8} -2.81798 q^{9} +O(q^{10})$	\(q-1.72632 q^{2} +0.426642 q^{3} +0.980187 q^{4} +1.18315 q^{5} -0.736521 q^{6} -0.668631 q^{7} +1.76053 q^{8} -2.81798 q^{9} -2.04249 q^{10} -0.941497 q^{11} +0.418189 q^{12} +2.79731 q^{13} +1.15427 q^{14} +0.504780 q^{15} -4.99961 q^{16} +4.02064 q^{17} +4.86473 q^{18} -1.00000 q^{19} +1.15971 q^{20} -0.285266 q^{21} +1.62533 q^{22} +0.0115843 q^{23} +0.751114 q^{24} -3.60016 q^{25} -4.82905 q^{26} -2.48219 q^{27} -0.655383 q^{28} -6.25615 q^{29} -0.871413 q^{30} +3.93935 q^{31} +5.10988 q^{32} -0.401682 q^{33} -6.94092 q^{34} -0.791089 q^{35} -2.76214 q^{36} -1.23336 q^{37} +1.72632 q^{38} +1.19345 q^{39} +2.08296 q^{40} +7.74226 q^{41} +0.492461 q^{42} -7.16027 q^{43} -0.922843 q^{44} -3.33408 q^{45} -0.0199983 q^{46} +8.54757 q^{47} -2.13304 q^{48} -6.55293 q^{49} +6.21504 q^{50} +1.71537 q^{51} +2.74188 q^{52} -3.41386 q^{53} +4.28506 q^{54} -1.11393 q^{55} -1.17714 q^{56} -0.426642 q^{57} +10.8001 q^{58} -5.10680 q^{59} +0.494779 q^{60} +6.66409 q^{61} -6.80058 q^{62} +1.88419 q^{63} +1.17792 q^{64} +3.30963 q^{65} +0.693433 q^{66} -12.2848 q^{67} +3.94098 q^{68} +0.00494235 q^{69} +1.36567 q^{70} -6.00846 q^{71} -4.96112 q^{72} +11.6267 q^{73} +2.12918 q^{74} -1.53598 q^{75} -0.980187 q^{76} +0.629514 q^{77} -2.06028 q^{78} +7.68516 q^{79} -5.91527 q^{80} +7.39492 q^{81} -13.3656 q^{82} -8.17730 q^{83} -0.279614 q^{84} +4.75701 q^{85} +12.3609 q^{86} -2.66914 q^{87} -1.65753 q^{88} -0.138071 q^{89} +5.75570 q^{90} -1.87037 q^{91} +0.0113548 q^{92} +1.68069 q^{93} -14.7559 q^{94} -1.18315 q^{95} +2.18009 q^{96} -11.2973 q^{97} +11.3125 q^{98} +2.65312 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9}+O(q^{10}) $ 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9} - 48 q^{11} - 14 q^{12} - 3 q^{13} - 4 q^{14} - 39 q^{15} + 59 q^{16} - 23 q^{17} - 24 q^{18} - 75 q^{19} - 62 q^{20} - 3 q^{21} - 6 q^{22} - 73 q^{23} - 64 q^{24} + 57 q^{25} - 46 q^{26} - 22 q^{27} - 26 q^{28} - 39 q^{29} - 14 q^{30} - 44 q^{31} - 71 q^{32} - 3 q^{33} - 9 q^{34} - 49 q^{35} + 20 q^{36} - 12 q^{37} + 11 q^{38} - 90 q^{39} - 8 q^{40} - 42 q^{41} - 45 q^{42} - 24 q^{43} - 120 q^{44} - 63 q^{45} - 39 q^{46} - 59 q^{47} - 4 q^{48} + 48 q^{49} - 100 q^{50} - 55 q^{51} + 2 q^{52} + 13 q^{53} - 87 q^{54} - 36 q^{55} - 12 q^{56} + 4 q^{57} - 17 q^{58} - 47 q^{59} - 45 q^{60} - 35 q^{61} - 40 q^{62} - 69 q^{63} + 26 q^{64} - 44 q^{65} + 33 q^{66} - 39 q^{67} - 63 q^{68} + 42 q^{69} + 40 q^{70} - 154 q^{71} - 51 q^{72} - 29 q^{73} - 95 q^{74} + 37 q^{75} - 67 q^{76} - 24 q^{77} - 19 q^{78} - 95 q^{79} - 146 q^{80} + 23 q^{81} + 7 q^{82} - 52 q^{83} - 72 q^{84} - 36 q^{85} - 44 q^{86} - 103 q^{87} + 67 q^{88} + q^{89} - 2 q^{90} - 64 q^{91} - 183 q^{92} - 49 q^{93} + 5 q^{94} + 18 q^{95} - 69 q^{96} - 7 q^{97} - 23 q^{98} - 100 q^{99}+O(q^{100}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9 - 48 * q^11 - 14 * q^12 - 3 * q^13 - 4 * q^14 - 39 * q^15 + 59 * q^16 - 23 * q^17 - 24 * q^18 - 75 * q^19 - 62 * q^20 - 3 * q^21 - 6 * q^22 - 73 * q^23 - 64 * q^24 + 57 * q^25 - 46 * q^26 - 22 * q^27 - 26 * q^28 - 39 * q^29 - 14 * q^30 - 44 * q^31 - 71 * q^32 - 3 * q^33 - 9 * q^34 - 49 * q^35 + 20 * q^36 - 12 * q^37 + 11 * q^38 - 90 * q^39 - 8 * q^40 - 42 * q^41 - 45 * q^42 - 24 * q^43 - 120 * q^44 - 63 * q^45 - 39 * q^46 - 59 * q^47 - 4 * q^48 + 48 * q^49 - 100 * q^50 - 55 * q^51 + 2 * q^52 + 13 * q^53 - 87 * q^54 - 36 * q^55 - 12 * q^56 + 4 * q^57 - 17 * q^58 - 47 * q^59 - 45 * q^60 - 35 * q^61 - 40 * q^62 - 69 * q^63 + 26 * q^64 - 44 * q^65 + 33 * q^66 - 39 * q^67 - 63 * q^68 + 42 * q^69 + 40 * q^70 - 154 * q^71 - 51 * q^72 - 29 * q^73 - 95 * q^74 + 37 * q^75 - 67 * q^76 - 24 * q^77 - 19 * q^78 - 95 * q^79 - 146 * q^80 + 23 * q^81 + 7 * q^82 - 52 * q^83 - 72 * q^84 - 36 * q^85 - 44 * q^86 - 103 * q^87 + 67 * q^88 + q^89 - 2 * q^90 - 64 * q^91 - 183 * q^92 - 49 * q^93 + 5 * q^94 + 18 * q^95 - 69 * q^96 - 7 * q^97 - 23 * q^98 - 100 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.72632	−1.22069	−0.610347	−	0.792134i	$-0.708970\pi$
$2$	−1.72632	−1.22069	−0.610347	+	0.792134i	$0.708970\pi$
$3$	0.426642	0.246322	0.123161	−	0.992387i	$-0.460697\pi$
$3$	0.426642	0.246322	0.123161	+	0.992387i	$0.460697\pi$
$4$	0.980187	0.490093
$5$	1.18315	0.529120	0.264560	−	0.964369i	$-0.414773\pi$
$5$	1.18315	0.529120	0.264560	+	0.964369i	$0.414773\pi$
$6$	−0.736521	−0.300683
$7$	−0.668631	−0.252719	−0.126359	−	0.991985i	$-0.540329\pi$
$7$	−0.668631	−0.252719	−0.126359	+	0.991985i	$0.540329\pi$
$8$	1.76053	0.622440
$9$	−2.81798	−0.939326
$10$	−2.04249	−0.645893
$11$	−0.941497	−0.283872	−0.141936	−	0.989876i	$-0.545333\pi$
$11$	−0.941497	−0.283872	−0.141936	+	0.989876i	$0.545333\pi$
$12$	0.418189	0.120721
$13$	2.79731	0.775833	0.387917	−	0.921694i	$-0.373195\pi$
$13$	2.79731	0.775833	0.387917	+	0.921694i	$0.373195\pi$
$14$	1.15427	0.308492
$15$	0.504780	0.130334
$16$	−4.99961	−1.24990
$17$	4.02064	0.975149	0.487574	−	0.873081i	$-0.337882\pi$
$17$	4.02064	0.975149	0.487574	+	0.873081i	$0.337882\pi$
$18$	4.86473	1.14663
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.15971	0.259318
$21$	−0.285266	−0.0622501
$22$	1.62533	0.346521
$23$	0.0115843	0.00241550	0.00120775	−	0.999999i	$-0.499616\pi$
$23$	0.0115843	0.00241550	0.00120775	+	0.999999i	$0.499616\pi$
$24$	0.751114	0.153320
$25$	−3.60016	−0.720032
$26$	−4.82905	−0.947055
$27$	−2.48219	−0.477698
$28$	−0.655383	−0.123856
$29$	−6.25615	−1.16174	−0.580869	−	0.813997i	$-0.697287\pi$
$29$	−6.25615	−1.16174	−0.580869	+	0.813997i	$0.697287\pi$
$30$	−0.871413	−0.159098
$31$	3.93935	0.707528	0.353764	−	0.935335i	$-0.384902\pi$
$31$	3.93935	0.707528	0.353764	+	0.935335i	$0.384902\pi$
$32$	5.10988	0.903308
$33$	−0.401682	−0.0699239
$34$	−6.94092	−1.19036
$35$	−0.791089	−0.133718
$36$	−2.76214	−0.460357
$37$	−1.23336	−0.202763	−0.101382	−	0.994848i	$-0.532326\pi$
$37$	−1.23336	−0.202763	−0.101382	+	0.994848i	$0.532326\pi$
$38$	1.72632	0.280046
$39$	1.19345	0.191105
$40$	2.08296	0.329345
$41$	7.74226	1.20914	0.604569	−	0.796553i	$-0.293345\pi$
$41$	7.74226	1.20914	0.604569	+	0.796553i	$0.293345\pi$
$42$	0.492461	0.0759883
$43$	−7.16027	−1.09193	−0.545966	−	0.837808i	$-0.683837\pi$
$43$	−7.16027	−1.09193	−0.545966	+	0.837808i	$0.683837\pi$
$44$	−0.922843	−0.139124
$45$	−3.33408	−0.497016
$46$	−0.0199983	−0.00294858
$47$	8.54757	1.24679	0.623395	−	0.781907i	$-0.285753\pi$
$47$	8.54757	1.24679	0.623395	+	0.781907i	$0.285753\pi$
$48$	−2.13304	−0.307878
$49$	−6.55293	−0.936133
$50$	6.21504	0.878939
$51$	1.71537	0.240200
$52$	2.74188	0.380231
$53$	−3.41386	−0.468929	−0.234465	−	0.972125i	$-0.575334\pi$
$53$	−3.41386	−0.468929	−0.234465	+	0.972125i	$0.575334\pi$
$54$	4.28506	0.583123
$55$	−1.11393	−0.150202
$56$	−1.17714	−0.157302
$57$	−0.426642	−0.0565101
$58$	10.8001	1.41813
$59$	−5.10680	−0.664849	−0.332425	−	0.943130i	$-0.607867\pi$
$59$	−5.10680	−0.664849	−0.332425	+	0.943130i	$0.607867\pi$
$60$	0.494779	0.0638757
$61$	6.66409	0.853249	0.426624	−	0.904429i	$-0.359703\pi$
$61$	6.66409	0.853249	0.426624	+	0.904429i	$0.359703\pi$
$62$	−6.80058	−0.863675
$63$	1.88419	0.237385
$64$	1.17792	0.147240
$65$	3.30963	0.410509
$66$	0.693433	0.0853557
$67$	−12.2848	−1.50082	−0.750411	−	0.660972i	$-0.770144\pi$
$67$	−12.2848	−1.50082	−0.750411	+	0.660972i	$0.770144\pi$
$68$	3.94098	0.477914
$69$	0.00494235	0.000594990 0
$70$	1.36567	0.163229
$71$	−6.00846	−0.713073	−0.356536	−	0.934281i	$-0.616042\pi$
$71$	−6.00846	−0.713073	−0.356536	+	0.934281i	$0.616042\pi$
$72$	−4.96112	−0.584674
$73$	11.6267	1.36080	0.680401	−	0.732840i	$-0.261806\pi$
$73$	11.6267	1.36080	0.680401	+	0.732840i	$0.261806\pi$
$74$	2.12918	0.247512
$75$	−1.53598	−0.177360
$76$	−0.980187	−0.112435
$77$	0.629514	0.0717398
$78$	−2.06028	−0.233280
$79$	7.68516	0.864648	0.432324	−	0.901718i	$-0.357694\pi$
$79$	7.68516	0.864648	0.432324	+	0.901718i	$0.357694\pi$
$80$	−5.91527	−0.661348
$81$	7.39492	0.821658
$82$	−13.3656	−1.47599
$83$	−8.17730	−0.897575	−0.448788	−	0.893638i	$-0.648144\pi$
$83$	−8.17730	−0.897575	−0.448788	+	0.893638i	$0.648144\pi$
$84$	−0.279614	−0.0305084
$85$	4.75701	0.515971
$86$	12.3609	1.33291
$87$	−2.66914	−0.286161
$88$	−1.65753	−0.176693
$89$	−0.138071	−0.0146355	−0.00731773	−	0.999973i	$-0.502329\pi$
$89$	−0.138071	−0.0146355	−0.00731773	+	0.999973i	$0.502329\pi$
$90$	5.75570	0.606704
$91$	−1.87037	−0.196068
$92$	0.0113548	0.00118382
$93$	1.68069	0.174279
$94$	−14.7559	−1.52195
$95$	−1.18315	−0.121388
$96$	2.18009	0.222504
$97$	−11.2973	−1.14706	−0.573532	−	0.819183i	$-0.694427\pi$
$97$	−11.2973	−1.14706	−0.573532	+	0.819183i	$0.694427\pi$
$98$	11.3125	1.14273
$99$	2.65312	0.266648
$100$	−3.52883	−0.352883
$101$	5.09767	0.507237	0.253618	−	0.967304i	$-0.418379\pi$
$101$	5.09767	0.507237	0.253618	+	0.967304i	$0.418379\pi$
$102$	−2.96129	−0.293211
$103$	−12.2595	−1.20796	−0.603981	−	0.796999i	$-0.706420\pi$
$103$	−12.2595	−1.20796	−0.603981	+	0.796999i	$0.706420\pi$
$104$	4.92473	0.482910
$105$	−0.337512	−0.0329378
$106$	5.89341	0.572419
$107$	3.32610	0.321547	0.160773	−	0.986991i	$-0.448601\pi$
$107$	3.32610	0.321547	0.160773	+	0.986991i	$0.448601\pi$
$108$	−2.43301	−0.234117
$109$	4.28129	0.410073	0.205036	−	0.978754i	$-0.434269\pi$
$109$	4.28129	0.410073	0.205036	+	0.978754i	$0.434269\pi$
$110$	1.92300	0.183351
$111$	−0.526203	−0.0499450
$112$	3.34289	0.315874
$113$	−14.4252	−1.35700	−0.678502	−	0.734598i	$-0.737371\pi$
$113$	−14.4252	−1.35700	−0.678502	+	0.734598i	$0.737371\pi$
$114$	0.736521	0.0689815
$115$	0.0137060	0.00127809
$116$	−6.13220	−0.569360
$117$	−7.88275	−0.728760
$118$	8.81599	0.811577
$119$	−2.68832	−0.246438
$120$	0.888679	0.0811249
$121$	−10.1136	−0.919417
$122$	−11.5044	−1.04156
$123$	3.30317	0.297837
$124$	3.86130	0.346755
$125$	−10.1753	−0.910103
$126$	−3.25271	−0.289775
$127$	−20.6881	−1.83577	−0.917887	−	0.396842i	$-0.870106\pi$
$127$	−20.6881	−1.83577	−0.917887	+	0.396842i	$0.870106\pi$
$128$	−12.2532	−1.08304
$129$	−3.05487	−0.268966
$130$	−5.71348	−0.501105
$131$	2.14900	0.187759	0.0938794	−	0.995584i	$-0.470073\pi$
$131$	2.14900	0.187759	0.0938794	+	0.995584i	$0.470073\pi$
$132$	−0.393724	−0.0342692
$133$	0.668631	0.0579776
$134$	21.2074	1.83204
$135$	−2.93680	−0.252760
$136$	7.07844	0.606972
$137$	−11.4956	−0.982136	−0.491068	−	0.871121i	$-0.663393\pi$
$137$	−11.4956	−0.982136	−0.491068	+	0.871121i	$0.663393\pi$
$138$	−0.00853209	−0.000726300 0
$139$	9.78357	0.829831	0.414916	−	0.909860i	$-0.363811\pi$
$139$	9.78357	0.829831	0.414916	+	0.909860i	$0.363811\pi$
$140$	−0.775415	−0.0655345
$141$	3.64675	0.307112
$142$	10.3725	0.870444
$143$	−2.63366	−0.220237
$144$	14.0888	1.17406
$145$	−7.40195	−0.614698
$146$	−20.0714	−1.66112
$147$	−2.79576	−0.230590
$148$	−1.20892	−0.0993729
$149$	7.81524	0.640250	0.320125	−	0.947375i	$-0.396275\pi$
$149$	7.81524	0.640250	0.320125	+	0.947375i	$0.396275\pi$
$150$	2.65159	0.216502
$151$	15.6772	1.27579	0.637897	−	0.770122i	$-0.279805\pi$
$151$	15.6772	1.27579	0.637897	+	0.770122i	$0.279805\pi$
$152$	−1.76053	−0.142797
$153$	−11.3301	−0.915982
$154$	−1.08674	−0.0875723
$155$	4.66083	0.374367
$156$	1.16980	0.0936591
$157$	7.55464	0.602926	0.301463	−	0.953478i	$-0.402525\pi$
$157$	7.55464	0.602926	0.301463	+	0.953478i	$0.402525\pi$
$158$	−13.2671	−1.05547
$159$	−1.45649	−0.115507
$160$	6.04574	0.477958
$161$	−0.00774563	−0.000610441 0
$162$	−12.7660	−1.00299
$163$	4.58417	0.359060	0.179530	−	0.983753i	$-0.442542\pi$
$163$	4.58417	0.359060	0.179530	+	0.983753i	$0.442542\pi$
$164$	7.58886	0.592590
$165$	−0.475249	−0.0369981
$166$	14.1167	1.09566
$167$	11.0208	0.852815	0.426407	−	0.904531i	$-0.359779\pi$
$167$	11.0208	0.852815	0.426407	+	0.904531i	$0.359779\pi$
$168$	−0.502218	−0.0387469
$169$	−5.17507	−0.398083
$170$	−8.21214	−0.629842
$171$	2.81798	0.215496
$172$	−7.01841	−0.535148
$173$	−4.05250	−0.308106	−0.154053	−	0.988063i	$-0.549233\pi$
$173$	−4.05250	−0.308106	−0.154053	+	0.988063i	$0.549233\pi$
$174$	4.60779	0.349315
$175$	2.40718	0.181966
$176$	4.70712	0.354812
$177$	−2.17878	−0.163767
$178$	0.238354	0.0178654
$179$	−4.35975	−0.325863	−0.162932	−	0.986637i	$-0.552095\pi$
$179$	−4.35975	−0.325863	−0.162932	+	0.986637i	$0.552095\pi$
$180$	−3.26802	−0.243584
$181$	5.26141	0.391078	0.195539	−	0.980696i	$-0.437354\pi$
$181$	5.26141	0.391078	0.195539	+	0.980696i	$0.437354\pi$
$182$	3.22885	0.239338
$183$	2.84318	0.210174
$184$	0.0203945	0.00150350
$185$	−1.45925	−0.107286
$186$	−2.90141	−0.212742
$187$	−3.78542	−0.276818
$188$	8.37821	0.611044
$189$	1.65967	0.120723
$190$	2.04249	0.148178
$191$	−2.92990	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$191$	−2.92990	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$192$	0.502549	0.0362684
$193$	1.40372	0.101042	0.0505211	−	0.998723i	$-0.483912\pi$
$193$	1.40372	0.101042	0.0505211	+	0.998723i	$0.483912\pi$
$194$	19.5027	1.40021
$195$	1.41203	0.101117
$196$	−6.42310	−0.458793
$197$	20.9158	1.49019	0.745093	−	0.666960i	$-0.232405\pi$
$197$	20.9158	1.49019	0.745093	+	0.666960i	$0.232405\pi$
$198$	−4.58014	−0.325496
$199$	3.24540	0.230061	0.115030	−	0.993362i	$-0.463303\pi$
$199$	3.24540	0.230061	0.115030	+	0.993362i	$0.463303\pi$
$200$	−6.33818	−0.448177
$201$	−5.24119	−0.369685
$202$	−8.80021	−0.619181
$203$	4.18305	0.293593
$204$	1.68139	0.117721
$205$	9.16023	0.639778
$206$	21.1638	1.47455
$207$	−0.0326443	−0.00226894
$208$	−13.9854	−0.969716
$209$	0.941497	0.0651247
$210$	0.582654	0.0402069
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	−3.34622	−0.229819
$213$	−2.56346	−0.175645
$214$	−5.74193	−0.392510
$215$	−8.47166	−0.577762
$216$	−4.36996	−0.297338
$217$	−2.63397	−0.178805
$218$	−7.39088	−0.500573
$219$	4.96044	0.335195
$220$	−1.09186	−0.0736132
$221$	11.2470	0.756553
$222$	0.908396	0.0609675
$223$	12.2244	0.818608	0.409304	−	0.912398i	$-0.365772\pi$
$223$	12.2244	0.818608	0.409304	+	0.912398i	$0.365772\pi$
$224$	−3.41662	−0.228283
$225$	10.1452	0.676345
$226$	24.9025	1.65649
$227$	−13.9649	−0.926882	−0.463441	−	0.886128i	$-0.653385\pi$
$227$	−13.9649	−0.926882	−0.463441	+	0.886128i	$0.653385\pi$
$228$	−0.418189	−0.0276952
$229$	−11.9929	−0.792513	−0.396257	−	0.918140i	$-0.629691\pi$
$229$	−11.9929	−0.792513	−0.396257	+	0.918140i	$0.629691\pi$
$230$	−0.0236609	−0.00156015
$231$	0.268577	0.0176711
$232$	−11.0141	−0.723112
$233$	−19.5124	−1.27830	−0.639151	−	0.769081i	$-0.720714\pi$
$233$	−19.5124	−1.27830	−0.639151	+	0.769081i	$0.720714\pi$
$234$	13.6082	0.889593
$235$	10.1130	0.659702
$236$	−5.00562	−0.325838
$237$	3.27881	0.212982
$238$	4.64091	0.300826
$239$	−14.0088	−0.906151	−0.453075	−	0.891472i	$-0.649673\pi$
$239$	−14.0088	−0.906151	−0.453075	+	0.891472i	$0.649673\pi$
$240$	−2.52370	−0.162904
$241$	−10.8251	−0.697308	−0.348654	−	0.937251i	$-0.613361\pi$
$241$	−10.8251	−0.697308	−0.348654	+	0.937251i	$0.613361\pi$
$242$	17.4593	1.12233
$243$	10.6016	0.680090
$244$	6.53205	0.418172
$245$	−7.75309	−0.495327
$246$	−5.70234	−0.363568
$247$	−2.79731	−0.177988
$248$	6.93532	0.440393
$249$	−3.48878	−0.221092
$250$	17.5658	1.11096
$251$	−8.09956	−0.511240	−0.255620	−	0.966777i	$-0.582280\pi$
$251$	−8.09956	−0.511240	−0.255620	+	0.966777i	$0.582280\pi$
$252$	1.84685	0.116341
$253$	−0.0109066	−0.000685692 0
$254$	35.7144	2.24092
$255$	2.02954	0.127095
$256$	18.7972	1.17482
$257$	29.9422	1.86774	0.933872	−	0.357608i	$-0.116407\pi$
$257$	29.9422	1.86774	0.933872	+	0.357608i	$0.116407\pi$
$258$	5.27369	0.328326
$259$	0.824663	0.0512420
$260$	3.24405	0.201188
$261$	17.6297	1.09125
$262$	−3.70986	−0.229196
$263$	−12.8133	−0.790105	−0.395052	−	0.918659i	$-0.629274\pi$
$263$	−12.8133	−0.790105	−0.395052	+	0.918659i	$0.629274\pi$
$264$	−0.707172	−0.0435234
$265$	−4.03909	−0.248120
$266$	−1.15427	−0.0707729
$267$	−0.0589067	−0.00360503
$268$	−12.0413	−0.735542
$269$	−30.2177	−1.84240	−0.921202	−	0.389086i	$-0.872791\pi$
$269$	−30.2177	−1.84240	−0.921202	+	0.389086i	$0.872791\pi$
$270$	5.06986	0.308542
$271$	−13.8954	−0.844085	−0.422043	−	0.906576i	$-0.638687\pi$
$271$	−13.8954	−0.844085	−0.422043	+	0.906576i	$0.638687\pi$
$272$	−20.1016	−1.21884
$273$	−0.797976	−0.0482957
$274$	19.8451	1.19889
$275$	3.38954	0.204397
$276$	0.00484443	0.000291600 0
$277$	−23.5161	−1.41295	−0.706473	−	0.707740i	$-0.749715\pi$
$277$	−23.5161	−1.41295	−0.706473	+	0.707740i	$0.749715\pi$
$278$	−16.8896	−1.01297
$279$	−11.1010	−0.664599
$280$	−1.39273	−0.0832317
$281$	−24.0005	−1.43175	−0.715875	−	0.698228i	$-0.753972\pi$
$281$	−24.0005	−1.43175	−0.715875	+	0.698228i	$0.753972\pi$
$282$	−6.29546	−0.374889
$283$	−21.2255	−1.26172	−0.630862	−	0.775895i	$-0.717298\pi$
$283$	−21.2255	−1.26172	−0.630862	+	0.775895i	$0.717298\pi$
$284$	−5.88941	−0.349472
$285$	−0.504780	−0.0299006
$286$	4.54654	0.268843
$287$	−5.17671	−0.305572
$288$	−14.3995	−0.848500
$289$	−0.834439	−0.0490846
$290$	12.7781	0.750359
$291$	−4.81989	−0.282547
$292$	11.3963	0.666920
$293$	−3.10121	−0.181175	−0.0905874	−	0.995889i	$-0.528874\pi$
$293$	−3.10121	−0.181175	−0.0905874	+	0.995889i	$0.528874\pi$
$294$	4.82637	0.281480
$295$	−6.04210	−0.351785
$296$	−2.17136	−0.126208
$297$	2.33698	0.135605
$298$	−13.4916	−0.781549
$299$	0.0324049	0.00187402
$300$	−1.50555	−0.0869228
$301$	4.78758	0.275951
$302$	−27.0639	−1.55735
$303$	2.17488	0.124943
$304$	4.99961	0.286747
$305$	7.88460	0.451471
$306$	19.5594	1.11813
$307$	31.7279	1.81081	0.905404	−	0.424550i	$-0.139568\pi$
$307$	31.7279	1.81081	0.905404	+	0.424550i	$0.139568\pi$
$308$	0.617041	0.0351592
$309$	−5.23040	−0.297547
$310$	−8.04609	−0.456987
$311$	−12.5210	−0.710003	−0.355001	−	0.934866i	$-0.615520\pi$
$311$	−12.5210	−0.710003	−0.355001	+	0.934866i	$0.615520\pi$
$312$	2.10110	0.118951
$313$	−20.4762	−1.15738	−0.578692	−	0.815546i	$-0.696437\pi$
$313$	−20.4762	−1.15738	−0.578692	+	0.815546i	$0.696437\pi$
$314$	−13.0417	−0.735988
$315$	2.22927	0.125605
$316$	7.53289	0.423758
$317$	24.9060	1.39886	0.699431	−	0.714700i	$-0.253437\pi$
$317$	24.9060	1.39886	0.699431	+	0.714700i	$0.253437\pi$
$318$	2.51438	0.140999
$319$	5.89015	0.329785
$320$	1.39365	0.0779075
$321$	1.41906	0.0792039
$322$	0.0133715	0.000745162 0
$323$	−4.02064	−0.223715
$324$	7.24841	0.402689
$325$	−10.0708	−0.558625
$326$	−7.91375	−0.438302
$327$	1.82658	0.101010
$328$	13.6304	0.752615
$329$	−5.71517	−0.315087
$330$	0.820433	0.0451634
$331$	−24.0160	−1.32004	−0.660020	−	0.751248i	$-0.729452\pi$
$331$	−24.0160	−1.32004	−0.660020	+	0.751248i	$0.729452\pi$
$332$	−8.01528	−0.439896
$333$	3.47558	0.190461
$334$	−19.0254	−1.04103
$335$	−14.5347	−0.794114
$336$	1.42622	0.0778065
$337$	9.47208	0.515977	0.257988	−	0.966148i	$-0.416940\pi$
$337$	9.47208	0.515977	0.257988	+	0.966148i	$0.416940\pi$
$338$	8.93384	0.485937
$339$	−6.15437	−0.334260
$340$	4.66276	0.252874
$341$	−3.70889	−0.200847
$342$	−4.86473	−0.263055
$343$	9.06191	0.489297
$344$	−12.6058	−0.679662
$345$	0.00584754	0.000314821 0
$346$	6.99593	0.376103
$347$	−20.4473	−1.09767	−0.548833	−	0.835932i	$-0.684928\pi$
$347$	−20.4473	−1.09767	−0.548833	+	0.835932i	$0.684928\pi$
$348$	−2.61625	−0.140246
$349$	4.86742	0.260547	0.130273	−	0.991478i	$-0.458414\pi$
$349$	4.86742	0.260547	0.130273	+	0.991478i	$0.458414\pi$
$350$	−4.15556	−0.222124
$351$	−6.94345	−0.370614
$352$	−4.81094	−0.256424
$353$	−22.7227	−1.20940	−0.604702	−	0.796451i	$-0.706708\pi$
$353$	−22.7227	−1.20940	−0.604702	+	0.796451i	$0.706708\pi$
$354$	3.76127	0.199909
$355$	−7.10889	−0.377301
$356$	−0.135335	−0.00717274
$357$	−1.14695	−0.0607031
$358$	7.52634	0.397779
$359$	1.17683	0.0621105	0.0310552	−	0.999518i	$-0.490113\pi$
$359$	1.17683	0.0621105	0.0310552	+	0.999518i	$0.490113\pi$
$360$	−5.86974	−0.309362
$361$	1.00000	0.0526316
$362$	−9.08289	−0.477386
$363$	−4.31488	−0.226472
$364$	−1.83331	−0.0960914
$365$	13.7561	0.720027
$366$	−4.90824	−0.256558
$367$	−33.1272	−1.72922	−0.864612	−	0.502441i	$-0.832435\pi$
$367$	−33.1272	−1.72922	−0.864612	+	0.502441i	$0.832435\pi$
$368$	−0.0579170	−0.00301913
$369$	−21.8175	−1.13577
$370$	2.51913	0.130963
$371$	2.28261	0.118507
$372$	1.64739	0.0854132
$373$	29.0440	1.50384	0.751920	−	0.659254i	$-0.229128\pi$
$373$	29.0440	1.50384	0.751920	+	0.659254i	$0.229128\pi$
$374$	6.53486	0.337910
$375$	−4.34119	−0.224178
$376$	15.0482	0.776052
$377$	−17.5004	−0.901315
$378$	−2.86512	−0.147366
$379$	−1.06942	−0.0549323	−0.0274661	−	0.999623i	$-0.508744\pi$
$379$	−1.06942	−0.0549323	−0.0274661	+	0.999623i	$0.508744\pi$
$380$	−1.15971	−0.0594916
$381$	−8.82642	−0.452191
$382$	5.05794	0.258787
$383$	−31.1381	−1.59108	−0.795541	−	0.605900i	$-0.792813\pi$
$383$	−31.1381	−1.59108	−0.795541	+	0.605900i	$0.792813\pi$
$384$	−5.22774	−0.266777
$385$	0.744808	0.0379589
$386$	−2.42328	−0.123341
$387$	20.1775	1.02568
$388$	−11.0734	−0.562168
$389$	−30.1486	−1.52859	−0.764297	−	0.644864i	$-0.776914\pi$
$389$	−30.1486	−1.52859	−0.764297	+	0.644864i	$0.776914\pi$
$390$	−2.43761	−0.123433
$391$	0.0465764	0.00235547
$392$	−11.5366	−0.582687
$393$	0.916852	0.0462491
$394$	−36.1073	−1.81906
$395$	9.09267	0.457502
$396$	2.60055	0.130683
$397$	20.3441	1.02104	0.510521	−	0.859865i	$-0.329453\pi$
$397$	20.3441	1.02104	0.510521	+	0.859865i	$0.329453\pi$
$398$	−5.60261	−0.280833
$399$	0.285266	0.0142812
$400$	17.9994	0.899970
$401$	16.8859	0.843239	0.421620	−	0.906773i	$-0.361462\pi$
$401$	16.8859	0.843239	0.421620	+	0.906773i	$0.361462\pi$
$402$	9.04798	0.451272
$403$	11.0196	0.548924
$404$	4.99667	0.248593
$405$	8.74929	0.434756
$406$	−7.22130	−0.358387
$407$	1.16121	0.0575588
$408$	3.01996	0.149510
$409$	22.3175	1.10353	0.551765	−	0.834000i	$-0.313955\pi$
$409$	22.3175	1.10353	0.551765	+	0.834000i	$0.313955\pi$
$410$	−15.8135	−0.780974
$411$	−4.90451	−0.241921
$412$	−12.0166	−0.592014
$413$	3.41457	0.168020
$414$	0.0563546	0.00276968
$415$	−9.67496	−0.474925
$416$	14.2939	0.700816
$417$	4.17408	0.204406
$418$	−1.62533	−0.0794974
$419$	−31.3522	−1.53165	−0.765827	−	0.643047i	$-0.777670\pi$
$419$	−31.3522	−1.53165	−0.765827	+	0.643047i	$0.777670\pi$
$420$	−0.330824	−0.0161426
$421$	24.6160	1.19971	0.599855	−	0.800109i	$-0.295225\pi$
$421$	24.6160	1.19971	0.599855	+	0.800109i	$0.295225\pi$
$422$	1.72632	0.0840360
$423$	−24.0868	−1.17114
$424$	−6.01018	−0.291880
$425$	−14.4750	−0.702139
$426$	4.42536	0.214409
$427$	−4.45581	−0.215632
$428$	3.26020	0.157588
$429$	−1.12363	−0.0542493
$430$	14.6248	0.705271
$431$	14.4070	0.693963	0.346981	−	0.937872i	$-0.387207\pi$
$431$	14.4070	0.693963	0.346981	+	0.937872i	$0.387207\pi$
$432$	12.4100	0.597076
$433$	−7.54457	−0.362569	−0.181285	−	0.983431i	$-0.558025\pi$
$433$	−7.54457	−0.362569	−0.181285	+	0.983431i	$0.558025\pi$
$434$	4.54708	0.218267
$435$	−3.15798	−0.151414
$436$	4.19646	0.200974
$437$	−0.0115843	−0.000554153 0
$438$	−8.56331	−0.409171
$439$	−39.2136	−1.87156	−0.935781	−	0.352581i	$-0.885304\pi$
$439$	−39.2136	−1.87156	−0.935781	+	0.352581i	$0.885304\pi$
$440$	−1.96110	−0.0934919
$441$	18.4660	0.879334
$442$	−19.4159	−0.923520
$443$	1.17627	0.0558863	0.0279431	−	0.999610i	$-0.491104\pi$
$443$	1.17627	0.0558863	0.0279431	+	0.999610i	$0.491104\pi$
$444$	−0.515777	−0.0244777
$445$	−0.163358	−0.00774391
$446$	−21.1033	−0.999270
$447$	3.33431	0.157707
$448$	−0.787593	−0.0372103
$449$	−15.7960	−0.745457	−0.372729	−	0.927940i	$-0.621578\pi$
$449$	−15.7960	−0.745457	−0.372729	+	0.927940i	$0.621578\pi$
$450$	−17.5138	−0.825610
$451$	−7.28932	−0.343240
$452$	−14.1393	−0.665059
$453$	6.68855	0.314256
$454$	24.1079	1.13144
$455$	−2.21292	−0.103743
$456$	−0.751114	−0.0351741
$457$	36.1745	1.69217	0.846085	−	0.533048i	$-0.178953\pi$
$457$	36.1745	1.69217	0.846085	+	0.533048i	$0.178953\pi$
$458$	20.7036	0.967416
$459$	−9.98001	−0.465827
$460$	0.0134344	0.000626382 0
$461$	28.1638	1.31172	0.655860	−	0.754883i	$-0.272306\pi$
$461$	28.1638	1.31172	0.655860	+	0.754883i	$0.272306\pi$
$462$	−0.463650	−0.0215710
$463$	5.24537	0.243773	0.121887	−	0.992544i	$-0.461106\pi$
$463$	5.24537	0.243773	0.121887	+	0.992544i	$0.461106\pi$
$464$	31.2783	1.45206
$465$	1.98850	0.0922147
$466$	33.6847	1.56042
$467$	14.9705	0.692750	0.346375	−	0.938096i	$-0.387413\pi$
$467$	14.9705	0.692750	0.346375	+	0.938096i	$0.387413\pi$
$468$	−7.72656	−0.357160
$469$	8.21396	0.379286
$470$	−17.4584	−0.805294
$471$	3.22313	0.148514
$472$	−8.99066	−0.413829
$473$	6.74138	0.309969
$474$	−5.66028	−0.259985
$475$	3.60016	0.165187
$476$	−2.63506	−0.120778
$477$	9.62016	0.440477
$478$	24.1836	1.10613
$479$	−18.5876	−0.849290	−0.424645	−	0.905360i	$-0.639601\pi$
$479$	−18.5876	−0.849290	−0.424645	+	0.905360i	$0.639601\pi$
$480$	2.57937	0.117731
$481$	−3.45009	−0.157310
$482$	18.6877	0.851200
$483$	−0.00330461	−0.000150365 0
$484$	−9.91320	−0.450600
$485$	−13.3663	−0.606934
$486$	−18.3017	−0.830182
$487$	−25.6107	−1.16053	−0.580265	−	0.814428i	$-0.697051\pi$
$487$	−25.6107	−1.16053	−0.580265	+	0.814428i	$0.697051\pi$
$488$	11.7323	0.531096
$489$	1.95580	0.0884442
$490$	13.3843	0.604642
$491$	−28.1502	−1.27040	−0.635201	−	0.772347i	$-0.719083\pi$
$491$	−28.1502	−1.27040	−0.635201	+	0.772347i	$0.719083\pi$
$492$	3.23772	0.145968
$493$	−25.1537	−1.13287
$494$	4.82905	0.217269
$495$	3.13903	0.141089
$496$	−19.6952	−0.884340
$497$	4.01744	0.180207
$498$	6.02276	0.269886
$499$	−11.8108	−0.528726	−0.264363	−	0.964423i	$-0.585162\pi$
$499$	−11.8108	−0.528726	−0.264363	+	0.964423i	$0.585162\pi$
$500$	−9.97366	−0.446035
$501$	4.70193	0.210067
$502$	13.9824	0.624067
$503$	−9.02923	−0.402594	−0.201297	−	0.979530i	$-0.564516\pi$
$503$	−9.02923	−0.402594	−0.201297	+	0.979530i	$0.564516\pi$
$504$	3.31716	0.147758
$505$	6.03129	0.268389
$506$	0.0188283	0.000837020 0
$507$	−2.20790	−0.0980564
$508$	−20.2782	−0.899701
$509$	7.81393	0.346346	0.173173	−	0.984891i	$-0.444598\pi$
$509$	7.81393	0.346346	0.173173	+	0.984891i	$0.444598\pi$
$510$	−3.50364	−0.155144
$511$	−7.77397	−0.343900
$512$	−7.94352	−0.351057
$513$	2.48219	0.109591
$514$	−51.6899	−2.27994
$515$	−14.5048	−0.639156
$516$	−2.99435	−0.131819
$517$	−8.04751	−0.353929
$518$	−1.42363	−0.0625509
$519$	−1.72897	−0.0758933
$520$	5.82668	0.255517
$521$	−19.3919	−0.849573	−0.424786	−	0.905294i	$-0.639651\pi$
$521$	−19.3919	−0.849573	−0.424786	+	0.905294i	$0.639651\pi$
$522$	−30.4345	−1.33208
$523$	−25.5317	−1.11642	−0.558212	−	0.829698i	$-0.688512\pi$
$523$	−25.5317	−1.11642	−0.558212	+	0.829698i	$0.688512\pi$
$524$	2.10642	0.0920193
$525$	1.02700	0.0448221
$526$	22.1200	0.964476
$527$	15.8387	0.689945
$528$	2.00825	0.0873980
$529$	−22.9999	−0.999994
$530$	6.97278	0.302878
$531$	14.3909	0.624510
$532$	0.655383	0.0284145
$533$	21.6575	0.938089
$534$	0.101692	0.00440064
$535$	3.93527	0.170137
$536$	−21.6276	−0.934171
$537$	−1.86005	−0.0802672
$538$	52.1654	2.24901
$539$	6.16957	0.265742
$540$	−2.87861	−0.123876
$541$	−38.6411	−1.66131	−0.830655	−	0.556788i	$-0.812034\pi$
$541$	−38.6411	−1.66131	−0.830655	+	0.556788i	$0.812034\pi$
$542$	23.9879	1.03037
$543$	2.24474	0.0963309
$544$	20.5450	0.880859
$545$	5.06539	0.216978
$546$	1.37756	0.0589543
$547$	−13.9687	−0.597257	−0.298628	−	0.954370i	$-0.596529\pi$
$547$	−13.9687	−0.597257	−0.298628	+	0.954370i	$0.596529\pi$
$548$	−11.2678	−0.481338
$549$	−18.7792	−0.801478
$550$	−5.85144	−0.249506
$551$	6.25615	0.266521
$552$	0.00870114	0.000370345 0
$553$	−5.13853	−0.218513
$554$	40.5963	1.72477
$555$	−0.622576	−0.0264269
$556$	9.58972	0.406695
$557$	7.12552	0.301918	0.150959	−	0.988540i	$-0.451764\pi$
$557$	7.12552	0.301918	0.150959	+	0.988540i	$0.451764\pi$
$558$	19.1639	0.811272
$559$	−20.0295	−0.847157
$560$	3.95513	0.167135
$561$	−1.61502	−0.0681862
$562$	41.4326	1.74773
$563$	−21.5898	−0.909900	−0.454950	−	0.890517i	$-0.650343\pi$
$563$	−21.5898	−0.909900	−0.454950	+	0.890517i	$0.650343\pi$
$564$	3.57450	0.150513
$565$	−17.0671	−0.718018
$566$	36.6420	1.54018
$567$	−4.94447	−0.207648
$568$	−10.5780	−0.443845
$569$	−26.4408	−1.10846	−0.554228	−	0.832365i	$-0.686986\pi$
$569$	−26.4408	−1.10846	−0.554228	+	0.832365i	$0.686986\pi$
$570$	0.871413	0.0364995
$571$	38.0777	1.59350	0.796750	−	0.604308i	$-0.206551\pi$
$571$	38.0777	1.59350	0.796750	+	0.604308i	$0.206551\pi$
$572$	−2.58148	−0.107937
$573$	−1.25002	−0.0522202
$574$	8.93667	0.373009
$575$	−0.0417054	−0.00173924
$576$	−3.31935	−0.138306
$577$	−42.2812	−1.76019	−0.880095	−	0.474798i	$-0.842521\pi$
$577$	−42.2812	−1.76019	−0.880095	+	0.474798i	$0.842521\pi$
$578$	1.44051	0.0599173
$579$	0.598886	0.0248889
$580$	−7.25529	−0.301260
$581$	5.46760	0.226834
$582$	8.32068	0.344903
$583$	3.21414	0.133116
$584$	20.4691	0.847018
$585$	−9.32645	−0.385601
$586$	5.35369	0.221159
$587$	−33.4758	−1.38169	−0.690846	−	0.723001i	$-0.742762\pi$
$587$	−33.4758	−1.38169	−0.690846	+	0.723001i	$0.742762\pi$
$588$	−2.74036	−0.113011
$589$	−3.93935	−0.162318
$590$	10.4306	0.429422
$591$	8.92354	0.367065
$592$	6.16632	0.253434
$593$	−3.46058	−0.142109	−0.0710545	−	0.997472i	$-0.522636\pi$
$593$	−3.46058	−0.142109	−0.0710545	+	0.997472i	$0.522636\pi$
$594$	−4.03438	−0.165532
$595$	−3.18069	−0.130395
$596$	7.66040	0.313782
$597$	1.38462	0.0566689
$598$	−0.0559413	−0.00228761
$599$	−34.7432	−1.41957	−0.709784	−	0.704420i	$-0.751207\pi$
$599$	−34.7432	−1.41957	−0.709784	+	0.704420i	$0.751207\pi$
$600$	−2.70413	−0.110396
$601$	22.9619	0.936633	0.468317	−	0.883561i	$-0.344861\pi$
$601$	22.9619	0.936633	0.468317	+	0.883561i	$0.344861\pi$
$602$	−8.26490	−0.336852
$603$	34.6181	1.40976
$604$	15.3666	0.625258
$605$	−11.9659	−0.486481
$606$	−3.75454	−0.152518
$607$	8.82753	0.358298	0.179149	−	0.983822i	$-0.442666\pi$
$607$	8.82753	0.358298	0.179149	+	0.983822i	$0.442666\pi$
$608$	−5.10988	−0.207233
$609$	1.78467	0.0723183
$610$	−13.6114	−0.551108
$611$	23.9102	0.967302
$612$	−11.1056	−0.448917
$613$	5.37287	0.217008	0.108504	−	0.994096i	$-0.465394\pi$
$613$	5.37287	0.217008	0.108504	+	0.994096i	$0.465394\pi$
$614$	−54.7726	−2.21044
$615$	3.90814	0.157591
$616$	1.10828	0.0446537
$617$	18.7854	0.756272	0.378136	−	0.925750i	$-0.376565\pi$
$617$	18.7854	0.756272	0.378136	+	0.925750i	$0.376565\pi$
$618$	9.02936	0.363214
$619$	−8.70168	−0.349750	−0.174875	−	0.984591i	$-0.555952\pi$
$619$	−8.70168	−0.349750	−0.174875	+	0.984591i	$0.555952\pi$
$620$	4.56848	0.183475
$621$	−0.0287545	−0.00115388
$622$	21.6153	0.866696
$623$	0.0923183	0.00369866
$624$	−5.96677	−0.238862
$625$	5.96197	0.238479
$626$	35.3486	1.41281
$627$	0.401682	0.0160416
$628$	7.40496	0.295490
$629$	−4.95890	−0.197724
$630$	−3.84844	−0.153325
$631$	34.5307	1.37465	0.687324	−	0.726351i	$-0.258785\pi$
$631$	34.5307	1.37465	0.687324	+	0.726351i	$0.258785\pi$
$632$	13.5299	0.538191
$633$	−0.426642	−0.0169575
$634$	−42.9958	−1.70758
$635$	−24.4771	−0.971344
$636$	−1.42764	−0.0566094
$637$	−18.3306	−0.726283
$638$	−10.1683	−0.402567
$639$	16.9317	0.669807
$640$	−14.4974	−0.573059
$641$	−13.9680	−0.551703	−0.275851	−	0.961200i	$-0.588960\pi$
$641$	−13.9680	−0.551703	−0.275851	+	0.961200i	$0.588960\pi$
$642$	−2.44975	−0.0966838
$643$	48.9972	1.93226	0.966131	−	0.258053i	$-0.0830811\pi$
$643$	48.9972	1.93226	0.966131	+	0.258053i	$0.0830811\pi$
$644$	−0.00759216	−0.000299173 0
$645$	−3.61436	−0.142315
$646$	6.94092	0.273087
$647$	−27.6437	−1.08679	−0.543394	−	0.839478i	$-0.682861\pi$
$647$	−27.6437	−1.08679	−0.543394	+	0.839478i	$0.682861\pi$
$648$	13.0190	0.511433
$649$	4.80804	0.188732
$650$	17.3854	0.681910
$651$	−1.12376	−0.0440437
$652$	4.49334	0.175973
$653$	23.3162	0.912435	0.456217	−	0.889868i	$-0.349204\pi$
$653$	23.3162	0.912435	0.456217	+	0.889868i	$0.349204\pi$
$654$	−3.15326	−0.123302
$655$	2.54258	0.0993469
$656$	−38.7082	−1.51130
$657$	−32.7638	−1.27824
$658$	9.86622	0.384625
$659$	−10.4465	−0.406936	−0.203468	−	0.979082i	$-0.565221\pi$
$659$	−10.4465	−0.406936	−0.203468	+	0.979082i	$0.565221\pi$
$660$	−0.465833	−0.0181325
$661$	29.7693	1.15789	0.578945	−	0.815367i	$-0.303465\pi$
$661$	29.7693	1.15789	0.578945	+	0.815367i	$0.303465\pi$
$662$	41.4594	1.61137
$663$	4.79843	0.186355
$664$	−14.3964	−0.558687
$665$	0.791089	0.0306771
$666$	−5.99997	−0.232494
$667$	−0.0724732	−0.00280617
$668$	10.8024	0.417959
$669$	5.21545	0.201641
$670$	25.0915	0.969370
$671$	−6.27422	−0.242214
$672$	−1.45767	−0.0562310
$673$	3.96346	0.152780	0.0763901	−	0.997078i	$-0.475661\pi$
$673$	3.96346	0.152780	0.0763901	+	0.997078i	$0.475661\pi$
$674$	−16.3519	−0.629850
$675$	8.93629	0.343958
$676$	−5.07254	−0.195098
$677$	−23.2911	−0.895149	−0.447575	−	0.894247i	$-0.647712\pi$
$677$	−23.2911	−0.895149	−0.447575	+	0.894247i	$0.647712\pi$
$678$	10.6244	0.408029
$679$	7.55370	0.289884
$680$	8.37484	0.321161
$681$	−5.95800	−0.228311
$682$	6.40273	0.245173
$683$	−37.2915	−1.42692	−0.713460	−	0.700696i	$-0.752873\pi$
$683$	−37.2915	−1.42692	−0.713460	+	0.700696i	$0.752873\pi$
$684$	2.76214	0.105613
$685$	−13.6010	−0.519668
$686$	−15.6438	−0.597282
$687$	−5.11667	−0.195213
$688$	35.7986	1.36481
$689$	−9.54960	−0.363811
$690$	−0.0100947	−0.000384300 0
$691$	−3.21731	−0.122392	−0.0611961	−	0.998126i	$-0.519492\pi$
$691$	−3.21731	−0.122392	−0.0611961	+	0.998126i	$0.519492\pi$
$692$	−3.97221	−0.151001
$693$	−1.77396	−0.0673870
$694$	35.2985	1.33991
$695$	11.5754	0.439080
$696$	−4.69908	−0.178118
$697$	31.1288	1.17909
$698$	−8.40272	−0.318048
$699$	−8.32482	−0.314874
$700$	2.35948	0.0891801
$701$	−6.89967	−0.260597	−0.130298	−	0.991475i	$-0.541594\pi$
$701$	−6.89967	−0.260597	−0.130298	+	0.991475i	$0.541594\pi$
$702$	11.9866	0.452406
$703$	1.23336	0.0465171
$704$	−1.10901	−0.0417973
$705$	4.31464	0.162499
$706$	39.2266	1.47631
$707$	−3.40846	−0.128188
$708$	−2.13561	−0.0802610
$709$	23.9547	0.899636	0.449818	−	0.893120i	$-0.351489\pi$
$709$	23.9547	0.899636	0.449818	+	0.893120i	$0.351489\pi$
$710$	12.2722	0.460569
$711$	−21.6566	−0.812186
$712$	−0.243077	−0.00910970
$713$	0.0456346	0.00170903
$714$	1.98001	0.0740999
$715$	−3.11601	−0.116532
$716$	−4.27337	−0.159703
$717$	−5.97672	−0.223205
$718$	−2.03158	−0.0758179
$719$	4.81263	0.179481	0.0897404	−	0.995965i	$-0.471396\pi$
$719$	4.81263	0.179481	0.0897404	+	0.995965i	$0.471396\pi$
$720$	16.6691	0.621221
$721$	8.19706	0.305274
$722$	−1.72632	−0.0642470
$723$	−4.61846	−0.171762
$724$	5.15716	0.191665
$725$	22.5231	0.836489
$726$	7.44887	0.276453
$727$	9.89734	0.367072	0.183536	−	0.983013i	$-0.441246\pi$
$727$	9.89734	0.367072	0.183536	+	0.983013i	$0.441246\pi$
$728$	−3.29283	−0.122040
$729$	−17.6617	−0.654137
$730$	−23.7475	−0.878933
$731$	−28.7889	−1.06480
$732$	2.78685	0.103005
$733$	−1.41584	−0.0522953	−0.0261477	−	0.999658i	$-0.508324\pi$
$733$	−1.41584	−0.0522953	−0.0261477	+	0.999658i	$0.508324\pi$
$734$	57.1881	2.11085
$735$	−3.30779	−0.122010
$736$	0.0591945	0.00218194
$737$	11.5661	0.426041
$738$	37.6640	1.38643
$739$	−3.00325	−0.110476	−0.0552382	−	0.998473i	$-0.517592\pi$
$739$	−3.00325	−0.110476	−0.0552382	+	0.998473i	$0.517592\pi$
$740$	−1.43034	−0.0525802
$741$	−1.19345	−0.0438424
$742$	−3.94052	−0.144661
$743$	45.5675	1.67171	0.835855	−	0.548950i	$-0.184972\pi$
$743$	45.5675	1.67171	0.835855	+	0.548950i	$0.184972\pi$
$744$	2.95890	0.108478
$745$	9.24659	0.338769
$746$	−50.1393	−1.83573
$747$	23.0434	0.843116
$748$	−3.71042	−0.135666
$749$	−2.22394	−0.0812608
$750$	7.49429	0.273653
$751$	48.0729	1.75421	0.877103	−	0.480303i	$-0.159473\pi$
$751$	48.0729	1.75421	0.877103	+	0.480303i	$0.159473\pi$
$752$	−42.7345	−1.55837
$753$	−3.45561	−0.125929
$754$	30.2113	1.10023
$755$	18.5485	0.675047
$756$	1.62679	0.0591657
$757$	10.0773	0.366265	0.183132	−	0.983088i	$-0.441376\pi$
$757$	10.0773	0.366265	0.183132	+	0.983088i	$0.441376\pi$
$758$	1.84616	0.0670555
$759$	−0.00465321	−0.000168901 0
$760$	−2.08296	−0.0755570
$761$	18.1176	0.656764	0.328382	−	0.944545i	$-0.393497\pi$
$761$	18.1176	0.656764	0.328382	+	0.944545i	$0.393497\pi$
$762$	15.2372	0.551987
$763$	−2.86260	−0.103633
$764$	−2.87185	−0.103900
$765$	−13.4052	−0.484664
$766$	53.7543	1.94222
$767$	−14.2853	−0.515812
$768$	8.01966	0.289385
$769$	−11.6989	−0.421872	−0.210936	−	0.977500i	$-0.567651\pi$
$769$	−11.6989	−0.421872	−0.210936	+	0.977500i	$0.567651\pi$
$770$	−1.28578	−0.0463362
$771$	12.7746	0.460066
$772$	1.37591	0.0495201
$773$	−47.5328	−1.70964	−0.854819	−	0.518927i	$-0.826332\pi$
$773$	−47.5328	−1.70964	−0.854819	+	0.518927i	$0.826332\pi$
$774$	−34.8328	−1.25204
$775$	−14.1823	−0.509443
$776$	−19.8891	−0.713978
$777$	0.351836	0.0126220
$778$	52.0462	1.86595
$779$	−7.74226	−0.277395
$780$	1.38405	0.0495569
$781$	5.65695	0.202422
$782$	−0.0804058	−0.00287531
$783$	15.5290	0.554960
$784$	32.7621	1.17007
$785$	8.93826	0.319020
$786$	−1.58278	−0.0564560
$787$	17.7412	0.632407	0.316204	−	0.948691i	$-0.397592\pi$
$787$	17.7412	0.632407	0.316204	+	0.948691i	$0.397592\pi$
$788$	20.5014	0.730330
$789$	−5.46671	−0.194620
$790$	−15.6969	−0.558470
$791$	9.64510	0.342940
$792$	4.67088	0.165973
$793$	18.6415	0.661979
$794$	−35.1205	−1.24638
$795$	−1.72325	−0.0611173
$796$	3.18110	0.112751
$797$	0.506199	0.0179305	0.00896525	−	0.999960i	$-0.497146\pi$
$797$	0.506199	0.0179305	0.00896525	+	0.999960i	$0.497146\pi$
$798$	−0.492461	−0.0174329
$799$	34.3667	1.21581
$800$	−18.3964	−0.650411
$801$	0.389080	0.0137475
$802$	−29.1504	−1.02934
$803$	−10.9465	−0.386294
$804$	−5.13734	−0.181180
$805$	−0.00916422	−0.000322996 0
$806$	−19.0233	−0.670068
$807$	−12.8921	−0.453824
$808$	8.97457	0.315724
$809$	−37.3116	−1.31180	−0.655902	−	0.754846i	$-0.727712\pi$
$809$	−37.3116	−1.31180	−0.655902	+	0.754846i	$0.727712\pi$
$810$	−15.1041	−0.530703
$811$	39.3136	1.38049	0.690243	−	0.723577i	$-0.257503\pi$
$811$	39.3136	1.38049	0.690243	+	0.723577i	$0.257503\pi$
$812$	4.10017	0.143888
$813$	−5.92836	−0.207917
$814$	−2.00461	−0.0702617
$815$	5.42375	0.189986
$816$	−8.57620	−0.300227
$817$	7.16027	0.250506
$818$	−38.5272	−1.34707
$819$	5.27065	0.184171
$820$	8.97874	0.313551
$821$	12.3001	0.429276	0.214638	−	0.976694i	$-0.431143\pi$
$821$	12.3001	0.429276	0.214638	+	0.976694i	$0.431143\pi$
$822$	8.46676	0.295312
$823$	41.5001	1.44660	0.723301	−	0.690533i	$-0.242624\pi$
$823$	41.5001	1.44660	0.723301	+	0.690533i	$0.242624\pi$
$824$	−21.5831	−0.751883
$825$	1.44612	0.0503475
$826$	−5.89464	−0.205101
$827$	−30.3946	−1.05693	−0.528463	−	0.848957i	$-0.677231\pi$
$827$	−30.3946	−1.05693	−0.528463	+	0.848957i	$0.677231\pi$
$828$	−0.0319975	−0.00111199
$829$	−26.1818	−0.909333	−0.454666	−	0.890662i	$-0.650242\pi$
$829$	−26.1818	−0.909333	−0.454666	+	0.890662i	$0.650242\pi$
$830$	16.7021	0.579738
$831$	−10.0330	−0.348039
$832$	3.29500	0.114234
$833$	−26.3470	−0.912869
$834$	−7.20580	−0.249517
$835$	13.0392	0.451241
$836$	0.922843	0.0319172
$837$	−9.77822	−0.337985
$838$	54.1239	1.86968
$839$	30.0926	1.03891	0.519456	−	0.854497i	$-0.326135\pi$
$839$	30.0926	1.03891	0.519456	+	0.854497i	$0.326135\pi$
$840$	−0.594198	−0.0205018
$841$	10.1394	0.349635
$842$	−42.4951	−1.46448
$843$	−10.2396	−0.352671
$844$	−0.980187	−0.0337394
$845$	−6.12288	−0.210633
$846$	41.5816	1.42961
$847$	6.76225	0.232354
$848$	17.0679	0.586115
$849$	−9.05567	−0.310790
$850$	24.9884	0.857096
$851$	−0.0142876	−0.000489774 0
$852$	−2.51267	−0.0860826
$853$	30.2914	1.03716	0.518579	−	0.855030i	$-0.326461\pi$
$853$	30.2914	1.03716	0.518579	+	0.855030i	$0.326461\pi$
$854$	7.69217	0.263221
$855$	3.33408	0.114023
$856$	5.85569	0.200143
$857$	30.9428	1.05699	0.528493	−	0.848938i	$-0.322757\pi$
$857$	30.9428	1.05699	0.528493	+	0.848938i	$0.322757\pi$
$858$	1.93974	0.0662218
$859$	52.9828	1.80775	0.903875	−	0.427796i	$-0.140710\pi$
$859$	52.9828	1.80775	0.903875	+	0.427796i	$0.140710\pi$
$860$	−8.30381	−0.283158
$861$	−2.20860	−0.0752689
$862$	−24.8712	−0.847116
$863$	48.6879	1.65735	0.828677	−	0.559727i	$-0.189094\pi$
$863$	48.6879	1.65735	0.828677	+	0.559727i	$0.189094\pi$
$864$	−12.6837	−0.431508
$865$	−4.79471	−0.163025
$866$	13.0244	0.442586
$867$	−0.356006	−0.0120906
$868$	−2.58178	−0.0876314
$869$	−7.23555	−0.245449
$870$	5.45169	0.184830
$871$	−34.3642	−1.16439
$872$	7.53732	0.255246
$873$	31.8354	1.07747
$874$	0.0199983	0.000676451 0
$875$	6.80349	0.230000
$876$	4.86215	0.164277
$877$	3.56052	0.120230	0.0601150	−	0.998191i	$-0.480853\pi$
$877$	3.56052	0.120230	0.0601150	+	0.998191i	$0.480853\pi$
$878$	67.6953	2.28460
$879$	−1.32311	−0.0446273
$880$	5.56922	0.187738
$881$	34.7421	1.17049	0.585246	−	0.810856i	$-0.300998\pi$
$881$	34.7421	1.17049	0.585246	+	0.810856i	$0.300998\pi$
$882$	−31.8783	−1.07340
$883$	−22.3979	−0.753748	−0.376874	−	0.926264i	$-0.623001\pi$
$883$	−22.3979	−0.753748	−0.376874	+	0.926264i	$0.623001\pi$
$884$	11.0241	0.370782
$885$	−2.57781	−0.0866523
$886$	−2.03062	−0.0682200
$887$	45.5107	1.52810	0.764051	−	0.645156i	$-0.223208\pi$
$887$	45.5107	1.52810	0.764051	+	0.645156i	$0.223208\pi$
$888$	−0.926394	−0.0310878
$889$	13.8327	0.463934
$890$	0.282009	0.00945295
$891$	−6.96230	−0.233246
$892$	11.9822	0.401194
$893$	−8.54757	−0.286033
$894$	−5.75609	−0.192512
$895$	−5.15823	−0.172421
$896$	8.19288	0.273705
$897$	0.0138253	0.000461613 0
$898$	27.2689	0.909975
$899$	−24.6451	−0.821962
$900$	9.94416	0.331472
$901$	−13.7259	−0.457276
$902$	12.5837	0.418991
$903$	2.04258	0.0679729
$904$	−25.3959	−0.844653
$905$	6.22503	0.206927
$906$	−11.5466	−0.383610
$907$	13.8925	0.461292	0.230646	−	0.973038i	$-0.425916\pi$
$907$	13.8925	0.461292	0.230646	+	0.973038i	$0.425916\pi$
$908$	−13.6882	−0.454259
$909$	−14.3651	−0.476461
$910$	3.82021	0.126639
$911$	19.2621	0.638181	0.319090	−	0.947724i	$-0.396623\pi$
$911$	19.2621	0.638181	0.319090	+	0.947724i	$0.396623\pi$
$912$	2.13304	0.0706321
$913$	7.69891	0.254797
$914$	−62.4488	−2.06562
$915$	3.36390	0.111207
$916$	−11.7553	−0.388405
$917$	−1.43689	−0.0474501
$918$	17.2287	0.568632
$919$	−2.23987	−0.0738866	−0.0369433	−	0.999317i	$-0.511762\pi$
$919$	−2.23987	−0.0738866	−0.0369433	+	0.999317i	$0.511762\pi$
$920$	0.0241297	0.000795532 0
$921$	13.5365	0.446042
$922$	−48.6198	−1.60121
$923$	−16.8075	−0.553226
$924$	0.263256	0.00866048
$925$	4.44030	0.145996
$926$	−9.05520	−0.297572
$927$	34.5469	1.13467
$928$	−31.9682	−1.04941
$929$	−10.7707	−0.353376	−0.176688	−	0.984267i	$-0.556538\pi$
$929$	−10.7707	−0.353376	−0.176688	+	0.984267i	$0.556538\pi$
$930$	−3.43280	−0.112566
$931$	6.55293	0.214764
$932$	−19.1258	−0.626487
$933$	−5.34200	−0.174889
$934$	−25.8438	−0.845635
$935$	−4.47872	−0.146470
$936$	−13.8778	−0.453609
$937$	45.7104	1.49329	0.746647	−	0.665221i	$-0.231663\pi$
$937$	45.7104	1.49329	0.746647	+	0.665221i	$0.231663\pi$
$938$	−14.1799	−0.462991
$939$	−8.73601	−0.285089
$940$	9.91266	0.323315
$941$	1.94711	0.0634739	0.0317369	−	0.999496i	$-0.489896\pi$
$941$	1.94711	0.0634739	0.0317369	+	0.999496i	$0.489896\pi$
$942$	−5.56415	−0.181290
$943$	0.0896888	0.00292067
$944$	25.5320	0.830996
$945$	1.96363	0.0638770
$946$	−11.6378	−0.378377
$947$	−21.3107	−0.692505	−0.346252	−	0.938141i	$-0.612546\pi$
$947$	−21.3107	−0.692505	−0.346252	+	0.938141i	$0.612546\pi$
$948$	3.21385	0.104381
$949$	32.5234	1.05576
$950$	−6.21504	−0.201642
$951$	10.6260	0.344570
$952$	−4.73286	−0.153393
$953$	−0.862901	−0.0279521	−0.0139761	−	0.999902i	$-0.504449\pi$
$953$	−0.862901	−0.0279521	−0.0139761	+	0.999902i	$0.504449\pi$
$954$	−16.6075	−0.537688
$955$	−3.46650	−0.112173
$956$	−13.7312	−0.444099
$957$	2.51298	0.0812332
$958$	32.0882	1.03672
$959$	7.68632	0.248204
$960$	0.594590	0.0191903
$961$	−15.4815	−0.499405
$962$	5.95596	0.192028
$963$	−9.37288	−0.302037
$964$	−10.6107	−0.341746
$965$	1.66081	0.0534634
$966$	0.00570482	0.000183550 0
$967$	−27.6127	−0.887966	−0.443983	−	0.896035i	$-0.646435\pi$
$967$	−27.6127	−0.887966	−0.443983	+	0.896035i	$0.646435\pi$
$968$	−17.8052	−0.572282
$969$	−1.71537	−0.0551058
$970$	23.0746	0.740881
$971$	−20.9955	−0.673778	−0.336889	−	0.941544i	$-0.609375\pi$
$971$	−20.9955	−0.673778	−0.336889	+	0.941544i	$0.609375\pi$
$972$	10.3915	0.333308
$973$	−6.54159	−0.209714
$974$	44.2123	1.41665
$975$	−4.29661	−0.137602
$976$	−33.3178	−1.06648
$977$	27.1337	0.868083	0.434042	−	0.900893i	$-0.357087\pi$
$977$	27.1337	0.868083	0.434042	+	0.900893i	$0.357087\pi$
$978$	−3.37634	−0.107963
$979$	0.129993	0.00415460
$980$	−7.59947	−0.242756
$981$	−12.0646	−0.385192
$982$	48.5964	1.55077
$983$	−30.0378	−0.958056	−0.479028	−	0.877800i	$-0.659011\pi$
$983$	−30.0378	−0.958056	−0.479028	+	0.877800i	$0.659011\pi$
$984$	5.81532	0.185386
$985$	24.7464	0.788487
$986$	43.4234	1.38288
$987$	−2.43833	−0.0776129
$988$	−2.74188	−0.0872309
$989$	−0.0829469	−0.00263756
$990$	−5.41898	−0.172226
$991$	−0.430103	−0.0136627	−0.00683133	−	0.999977i	$-0.502174\pi$
$991$	−0.430103	−0.0136627	−0.00683133	+	0.999977i	$0.502174\pi$
$992$	20.1296	0.639115
$993$	−10.2462	−0.325155
$994$	−6.93539	−0.219977
$995$	3.83979	0.121730
$996$	−3.41966	−0.108356
$997$	40.1780	1.27245	0.636225	−	0.771503i	$-0.280495\pi$
$997$	40.1780	1.27245	0.636225	+	0.771503i	$0.280495\pi$
$998$	20.3893	0.645412
$999$	3.06144	0.0968596

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.d.1.18	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.d.1.18	✓	75	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-1.72632 q^{2} +0.426642 q^{3} +0.980187 q^{4} +1.18315 q^{5} -0.736521 q^{6} -0.668631 q^{7} +1.76053 q^{8} -2.81798 q^{9} +O(q^{10})\)	\(q-1.72632 q^{2} +0.426642 q^{3} +0.980187 q^{4} +1.18315 q^{5} -0.736521 q^{6} -0.668631 q^{7} +1.76053 q^{8} -2.81798 q^{9} -2.04249 q^{10} -0.941497 q^{11} +0.418189 q^{12} +2.79731 q^{13} +1.15427 q^{14} +0.504780 q^{15} -4.99961 q^{16} +4.02064 q^{17} +4.86473 q^{18} -1.00000 q^{19} +1.15971 q^{20} -0.285266 q^{21} +1.62533 q^{22} +0.0115843 q^{23} +0.751114 q^{24} -3.60016 q^{25} -4.82905 q^{26} -2.48219 q^{27} -0.655383 q^{28} -6.25615 q^{29} -0.871413 q^{30} +3.93935 q^{31} +5.10988 q^{32} -0.401682 q^{33} -6.94092 q^{34} -0.791089 q^{35} -2.76214 q^{36} -1.23336 q^{37} +1.72632 q^{38} +1.19345 q^{39} +2.08296 q^{40} +7.74226 q^{41} +0.492461 q^{42} -7.16027 q^{43} -0.922843 q^{44} -3.33408 q^{45} -0.0199983 q^{46} +8.54757 q^{47} -2.13304 q^{48} -6.55293 q^{49} +6.21504 q^{50} +1.71537 q^{51} +2.74188 q^{52} -3.41386 q^{53} +4.28506 q^{54} -1.11393 q^{55} -1.17714 q^{56} -0.426642 q^{57} +10.8001 q^{58} -5.10680 q^{59} +0.494779 q^{60} +6.66409 q^{61} -6.80058 q^{62} +1.88419 q^{63} +1.17792 q^{64} +3.30963 q^{65} +0.693433 q^{66} -12.2848 q^{67} +3.94098 q^{68} +0.00494235 q^{69} +1.36567 q^{70} -6.00846 q^{71} -4.96112 q^{72} +11.6267 q^{73} +2.12918 q^{74} -1.53598 q^{75} -0.980187 q^{76} +0.629514 q^{77} -2.06028 q^{78} +7.68516 q^{79} -5.91527 q^{80} +7.39492 q^{81} -13.3656 q^{82} -8.17730 q^{83} -0.279614 q^{84} +4.75701 q^{85} +12.3609 q^{86} -2.66914 q^{87} -1.65753 q^{88} -0.138071 q^{89} +5.75570 q^{90} -1.87037 q^{91} +0.0113548 q^{92} +1.68069 q^{93} -14.7559 q^{94} -1.18315 q^{95} +2.18009 q^{96} -11.2973 q^{97} +11.3125 q^{98} +2.65312 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9}+O(q^{10}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9} - 48 q^{11} - 14 q^{12} - 3 q^{13} - 4 q^{14} - 39 q^{15} + 59 q^{16} - 23 q^{17} - 24 q^{18} - 75 q^{19} - 62 q^{20} - 3 q^{21} - 6 q^{22} - 73 q^{23} - 64 q^{24} + 57 q^{25} - 46 q^{26} - 22 q^{27} - 26 q^{28} - 39 q^{29} - 14 q^{30} - 44 q^{31} - 71 q^{32} - 3 q^{33} - 9 q^{34} - 49 q^{35} + 20 q^{36} - 12 q^{37} + 11 q^{38} - 90 q^{39} - 8 q^{40} - 42 q^{41} - 45 q^{42} - 24 q^{43} - 120 q^{44} - 63 q^{45} - 39 q^{46} - 59 q^{47} - 4 q^{48} + 48 q^{49} - 100 q^{50} - 55 q^{51} + 2 q^{52} + 13 q^{53} - 87 q^{54} - 36 q^{55} - 12 q^{56} + 4 q^{57} - 17 q^{58} - 47 q^{59} - 45 q^{60} - 35 q^{61} - 40 q^{62} - 69 q^{63} + 26 q^{64} - 44 q^{65} + 33 q^{66} - 39 q^{67} - 63 q^{68} + 42 q^{69} + 40 q^{70} - 154 q^{71} - 51 q^{72} - 29 q^{73} - 95 q^{74} + 37 q^{75} - 67 q^{76} - 24 q^{77} - 19 q^{78} - 95 q^{79} - 146 q^{80} + 23 q^{81} + 7 q^{82} - 52 q^{83} - 72 q^{84} - 36 q^{85} - 44 q^{86} - 103 q^{87} + 67 q^{88} + q^{89} - 2 q^{90} - 64 q^{91} - 183 q^{92} - 49 q^{93} + 5 q^{94} + 18 q^{95} - 69 q^{96} - 7 q^{97} - 23 q^{98} - 100 q^{99}+O(q^{100}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9 - 48 * q^11 - 14 * q^12 - 3 * q^13 - 4 * q^14 - 39 * q^15 + 59 * q^16 - 23 * q^17 - 24 * q^18 - 75 * q^19 - 62 * q^20 - 3 * q^21 - 6 * q^22 - 73 * q^23 - 64 * q^24 + 57 * q^25 - 46 * q^26 - 22 * q^27 - 26 * q^28 - 39 * q^29 - 14 * q^30 - 44 * q^31 - 71 * q^32 - 3 * q^33 - 9 * q^34 - 49 * q^35 + 20 * q^36 - 12 * q^37 + 11 * q^38 - 90 * q^39 - 8 * q^40 - 42 * q^41 - 45 * q^42 - 24 * q^43 - 120 * q^44 - 63 * q^45 - 39 * q^46 - 59 * q^47 - 4 * q^48 + 48 * q^49 - 100 * q^50 - 55 * q^51 + 2 * q^52 + 13 * q^53 - 87 * q^54 - 36 * q^55 - 12 * q^56 + 4 * q^57 - 17 * q^58 - 47 * q^59 - 45 * q^60 - 35 * q^61 - 40 * q^62 - 69 * q^63 + 26 * q^64 - 44 * q^65 + 33 * q^66 - 39 * q^67 - 63 * q^68 + 42 * q^69 + 40 * q^70 - 154 * q^71 - 51 * q^72 - 29 * q^73 - 95 * q^74 + 37 * q^75 - 67 * q^76 - 24 * q^77 - 19 * q^78 - 95 * q^79 - 146 * q^80 + 23 * q^81 + 7 * q^82 - 52 * q^83 - 72 * q^84 - 36 * q^85 - 44 * q^86 - 103 * q^87 + 67 * q^88 + q^89 - 2 * q^90 - 64 * q^91 - 183 * q^92 - 49 * q^93 + 5 * q^94 + 18 * q^95 - 69 * q^96 - 7 * q^97 - 23 * q^98 - 100 * q^99

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