LMFDB - Embedded newform 6047.2.a.a.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6047,2,Mod(1,6047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6047, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6047 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6047.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:	$48.2855381023$
Analytic rank:	$1$
Dimension:	$217$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	6047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.50438 q^{2} +2.01082 q^{3} +4.27191 q^{4} +3.19046 q^{5} -5.03585 q^{6} +0.790850 q^{7} -5.68973 q^{8} +1.04339 q^{9} +O(q^{10})$	\(q-2.50438 q^{2} +2.01082 q^{3} +4.27191 q^{4} +3.19046 q^{5} -5.03585 q^{6} +0.790850 q^{7} -5.68973 q^{8} +1.04339 q^{9} -7.99013 q^{10} -2.85059 q^{11} +8.59004 q^{12} -2.42137 q^{13} -1.98059 q^{14} +6.41544 q^{15} +5.70542 q^{16} +2.08136 q^{17} -2.61304 q^{18} +0.238553 q^{19} +13.6294 q^{20} +1.59026 q^{21} +7.13896 q^{22} +0.00126801 q^{23} -11.4410 q^{24} +5.17905 q^{25} +6.06402 q^{26} -3.93439 q^{27} +3.37844 q^{28} -9.31384 q^{29} -16.0667 q^{30} -9.00065 q^{31} -2.90906 q^{32} -5.73202 q^{33} -5.21251 q^{34} +2.52318 q^{35} +4.45726 q^{36} -0.877078 q^{37} -0.597427 q^{38} -4.86893 q^{39} -18.1529 q^{40} -5.36108 q^{41} -3.98260 q^{42} -1.10802 q^{43} -12.1775 q^{44} +3.32889 q^{45} -0.00317557 q^{46} -9.52505 q^{47} +11.4726 q^{48} -6.37456 q^{49} -12.9703 q^{50} +4.18523 q^{51} -10.3439 q^{52} +10.9698 q^{53} +9.85320 q^{54} -9.09471 q^{55} -4.49973 q^{56} +0.479687 q^{57} +23.3254 q^{58} -1.84744 q^{59} +27.4062 q^{60} +1.35042 q^{61} +22.5410 q^{62} +0.825164 q^{63} -4.12544 q^{64} -7.72528 q^{65} +14.3552 q^{66} -9.34063 q^{67} +8.89138 q^{68} +0.00254973 q^{69} -6.31899 q^{70} +3.61122 q^{71} -5.93660 q^{72} +11.6743 q^{73} +2.19654 q^{74} +10.4141 q^{75} +1.01908 q^{76} -2.25439 q^{77} +12.1936 q^{78} +8.15452 q^{79} +18.2029 q^{80} -11.0415 q^{81} +13.4262 q^{82} +5.00880 q^{83} +6.79344 q^{84} +6.64049 q^{85} +2.77490 q^{86} -18.7284 q^{87} +16.2191 q^{88} -18.2144 q^{89} -8.33681 q^{90} -1.91494 q^{91} +0.00541681 q^{92} -18.0987 q^{93} +23.8543 q^{94} +0.761094 q^{95} -5.84960 q^{96} +6.52392 q^{97} +15.9643 q^{98} -2.97427 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 217 q - 20 q^{2} - 27 q^{3} + 184 q^{4} - 19 q^{5} - 17 q^{6} - 48 q^{7} - 57 q^{8} + 152 q^{9}+O(q^{10}) $ 217 * q - 20 * q^2 - 27 * q^3 + 184 * q^4 - 19 * q^5 - 17 * q^6 - 48 * q^7 - 57 * q^8 + 152 * q^9	\( 217 q - 20 q^{2} - 27 q^{3} + 184 q^{4} - 19 q^{5} - 17 q^{6} - 48 q^{7} - 57 q^{8} + 152 q^{9} - 46 q^{10} - 32 q^{11} - 72 q^{12} - 80 q^{13} - 22 q^{14} - 43 q^{15} + 122 q^{16} - 61 q^{17} - 88 q^{18} - 43 q^{19} - 41 q^{20} - 61 q^{21} - 93 q^{22} - 60 q^{23} - 41 q^{24} + 26 q^{25} - 9 q^{26} - 93 q^{27} - 126 q^{28} - 47 q^{29} - 36 q^{30} - 100 q^{31} - 114 q^{32} - 133 q^{33} - 75 q^{34} - 37 q^{35} + 75 q^{36} - 264 q^{37} - 35 q^{38} - 47 q^{39} - 118 q^{40} - 72 q^{41} - 64 q^{42} - 107 q^{43} - 59 q^{44} - 69 q^{45} - 111 q^{46} - 54 q^{47} - 135 q^{48} + 33 q^{49} - 42 q^{50} - 26 q^{51} - 173 q^{52} - 103 q^{53} - 28 q^{54} - 78 q^{55} - 44 q^{56} - 205 q^{57} - 189 q^{58} - 38 q^{59} - 105 q^{60} - 108 q^{61} - 14 q^{62} - 116 q^{63} + 39 q^{64} - 146 q^{65} + 5 q^{66} - 206 q^{67} - 62 q^{68} - 55 q^{69} - 125 q^{70} - 78 q^{71} - 225 q^{72} - 326 q^{73} + 3 q^{74} - 95 q^{75} - 84 q^{76} - 79 q^{77} - 86 q^{78} - 117 q^{79} - 39 q^{80} + q^{81} - 96 q^{82} - 23 q^{83} - 57 q^{84} - 224 q^{85} - 7 q^{86} - 45 q^{87} - 250 q^{88} - 104 q^{89} - 36 q^{90} - 96 q^{91} - 137 q^{92} - 155 q^{93} - 48 q^{94} - 38 q^{95} - 33 q^{96} - 447 q^{97} - 46 q^{98} - 94 q^{99}+O(q^{100}) \) 217 * q - 20 * q^2 - 27 * q^3 + 184 * q^4 - 19 * q^5 - 17 * q^6 - 48 * q^7 - 57 * q^8 + 152 * q^9 - 46 * q^10 - 32 * q^11 - 72 * q^12 - 80 * q^13 - 22 * q^14 - 43 * q^15 + 122 * q^16 - 61 * q^17 - 88 * q^18 - 43 * q^19 - 41 * q^20 - 61 * q^21 - 93 * q^22 - 60 * q^23 - 41 * q^24 + 26 * q^25 - 9 * q^26 - 93 * q^27 - 126 * q^28 - 47 * q^29 - 36 * q^30 - 100 * q^31 - 114 * q^32 - 133 * q^33 - 75 * q^34 - 37 * q^35 + 75 * q^36 - 264 * q^37 - 35 * q^38 - 47 * q^39 - 118 * q^40 - 72 * q^41 - 64 * q^42 - 107 * q^43 - 59 * q^44 - 69 * q^45 - 111 * q^46 - 54 * q^47 - 135 * q^48 + 33 * q^49 - 42 * q^50 - 26 * q^51 - 173 * q^52 - 103 * q^53 - 28 * q^54 - 78 * q^55 - 44 * q^56 - 205 * q^57 - 189 * q^58 - 38 * q^59 - 105 * q^60 - 108 * q^61 - 14 * q^62 - 116 * q^63 + 39 * q^64 - 146 * q^65 + 5 * q^66 - 206 * q^67 - 62 * q^68 - 55 * q^69 - 125 * q^70 - 78 * q^71 - 225 * q^72 - 326 * q^73 + 3 * q^74 - 95 * q^75 - 84 * q^76 - 79 * q^77 - 86 * q^78 - 117 * q^79 - 39 * q^80 + q^81 - 96 * q^82 - 23 * q^83 - 57 * q^84 - 224 * q^85 - 7 * q^86 - 45 * q^87 - 250 * q^88 - 104 * q^89 - 36 * q^90 - 96 * q^91 - 137 * q^92 - 155 * q^93 - 48 * q^94 - 38 * q^95 - 33 * q^96 - 447 * q^97 - 46 * q^98 - 94 * 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 $

$2$	−2.50438	−1.77086	−0.885432	−	0.464770i	$-0.846137\pi$
$2$	−2.50438	−1.77086	−0.885432	+	0.464770i	$0.846137\pi$
$3$	2.01082	1.16095	0.580473	−	0.814279i	$-0.302868\pi$
$3$	2.01082	1.16095	0.580473	+	0.814279i	$0.302868\pi$
$4$	4.27191	2.13596
$5$	3.19046	1.42682	0.713409	−	0.700748i	$-0.247150\pi$
$5$	3.19046	1.42682	0.713409	+	0.700748i	$0.247150\pi$
$6$	−5.03585	−2.05588
$7$	0.790850	0.298913	0.149457	−	0.988768i	$-0.452248\pi$
$7$	0.790850	0.298913	0.149457	+	0.988768i	$0.452248\pi$
$8$	−5.68973	−2.01162
$9$	1.04339	0.347796
$10$	−7.99013	−2.52670
$11$	−2.85059	−0.859486	−0.429743	−	0.902951i	$-0.641396\pi$
$11$	−2.85059	−0.859486	−0.429743	+	0.902951i	$0.641396\pi$
$12$	8.59004	2.47973
$13$	−2.42137	−0.671566	−0.335783	−	0.941939i	$-0.609001\pi$
$13$	−2.42137	−0.671566	−0.335783	+	0.941939i	$0.609001\pi$
$14$	−1.98059	−0.529335
$15$	6.41544	1.65646
$16$	5.70542	1.42635
$17$	2.08136	0.504803	0.252402	−	0.967623i	$-0.418780\pi$
$17$	2.08136	0.504803	0.252402	+	0.967623i	$0.418780\pi$
$18$	−2.61304	−0.615899
$19$	0.238553	0.0547278	0.0273639	−	0.999626i	$-0.491289\pi$
$19$	0.238553	0.0547278	0.0273639	+	0.999626i	$0.491289\pi$
$20$	13.6294	3.04762
$21$	1.59026	0.347022
$22$	7.13896	1.52203
$23$	0.00126801	0.000264397 0	0.000132199	−	1.00000i	$-0.499958\pi$
$23$	0.00126801	0.000264397 0	0.000132199		1.00000i	$0.499958\pi$
$24$	−11.4410	−2.33539
$25$	5.17905	1.03581
$26$	6.06402	1.18925
$27$	−3.93439	−0.757174
$28$	3.37844	0.638466
$29$	−9.31384	−1.72954	−0.864768	−	0.502171i	$-0.832535\pi$
$29$	−9.31384	−1.72954	−0.864768	+	0.502171i	$0.832535\pi$
$30$	−16.0667	−2.93336
$31$	−9.00065	−1.61656	−0.808282	−	0.588796i	$-0.799602\pi$
$31$	−9.00065	−1.61656	−0.808282	+	0.588796i	$0.799602\pi$
$32$	−2.90906	−0.514255
$33$	−5.73202	−0.997817
$34$	−5.21251	−0.893938
$35$	2.52318	0.426495
$36$	4.45726	0.742877
$37$	−0.877078	−0.144191	−0.0720954	−	0.997398i	$-0.522969\pi$
$37$	−0.877078	−0.144191	−0.0720954	+	0.997398i	$0.522969\pi$
$38$	−0.597427	−0.0969155
$39$	−4.86893	−0.779652
$40$	−18.1529	−2.87022
$41$	−5.36108	−0.837260	−0.418630	−	0.908157i	$-0.637490\pi$
$41$	−5.36108	−0.837260	−0.418630	+	0.908157i	$0.637490\pi$
$42$	−3.98260	−0.614529
$43$	−1.10802	−0.168971	−0.0844857	−	0.996425i	$-0.526925\pi$
$43$	−1.10802	−0.168971	−0.0844857	+	0.996425i	$0.526925\pi$
$44$	−12.1775	−1.83582
$45$	3.32889	0.496242
$46$	−0.00317557	−0.000468212 0
$47$	−9.52505	−1.38937	−0.694685	−	0.719314i	$-0.744456\pi$
$47$	−9.52505	−1.38937	−0.694685	+	0.719314i	$0.744456\pi$
$48$	11.4726	1.65592
$49$	−6.37456	−0.910651
$50$	−12.9703	−1.83428
$51$	4.18523	0.586049
$52$	−10.3439	−1.43444
$53$	10.9698	1.50681	0.753407	−	0.657554i	$-0.228409\pi$
$53$	10.9698	1.50681	0.753407	+	0.657554i	$0.228409\pi$
$54$	9.85320	1.34085
$55$	−9.09471	−1.22633
$56$	−4.49973	−0.601301
$57$	0.479687	0.0635360
$58$	23.3254	3.06277
$59$	−1.84744	−0.240516	−0.120258	−	0.992743i	$-0.538372\pi$
$59$	−1.84744	−0.240516	−0.120258	+	0.992743i	$0.538372\pi$
$60$	27.4062	3.53813
$61$	1.35042	0.172904	0.0864520	−	0.996256i	$-0.472447\pi$
$61$	1.35042	0.172904	0.0864520	+	0.996256i	$0.472447\pi$
$62$	22.5410	2.86271
$63$	0.825164	0.103961
$64$	−4.12544	−0.515680
$65$	−7.72528	−0.958203
$66$	14.3552	1.76700
$67$	−9.34063	−1.14114	−0.570570	−	0.821249i	$-0.693278\pi$
$67$	−9.34063	−1.14114	−0.570570	+	0.821249i	$0.693278\pi$
$68$	8.89138	1.07824
$69$	0.00254973	0.000306951 0
$70$	−6.31899	−0.755264
$71$	3.61122	0.428573	0.214286	−	0.976771i	$-0.431257\pi$
$71$	3.61122	0.428573	0.214286	+	0.976771i	$0.431257\pi$
$72$	−5.93660	−0.699635
$73$	11.6743	1.36638	0.683190	−	0.730241i	$-0.260592\pi$
$73$	11.6743	1.36638	0.683190	+	0.730241i	$0.260592\pi$
$74$	2.19654	0.255342
$75$	10.4141	1.20252
$76$	1.01908	0.116896
$77$	−2.25439	−0.256912
$78$	12.1936	1.38066
$79$	8.15452	0.917455	0.458728	−	0.888577i	$-0.348305\pi$
$79$	8.15452	0.917455	0.458728	+	0.888577i	$0.348305\pi$
$80$	18.2029	2.03515
$81$	−11.0415	−1.22683
$82$	13.4262	1.48267
$83$	5.00880	0.549787	0.274894	−	0.961475i	$-0.411357\pi$
$83$	5.00880	0.549787	0.274894	+	0.961475i	$0.411357\pi$
$84$	6.79344	0.741225
$85$	6.64049	0.720262
$86$	2.77490	0.299225
$87$	−18.7284	−2.00790
$88$	16.2191	1.72896
$89$	−18.2144	−1.93072	−0.965359	−	0.260926i	$-0.915972\pi$
$89$	−18.2144	−1.93072	−0.965359	+	0.260926i	$0.915972\pi$
$90$	−8.33681	−0.878776
$91$	−1.91494	−0.200740
$92$	0.00541681	0.000564741 0
$93$	−18.0987	−1.87674
$94$	23.8543	2.46039
$95$	0.761094	0.0780866
$96$	−5.84960	−0.597022
$97$	6.52392	0.662404	0.331202	−	0.943560i	$-0.392546\pi$
$97$	6.52392	0.662404	0.331202	+	0.943560i	$0.392546\pi$
$98$	15.9643	1.61264
$99$	−2.97427	−0.298926
$100$	22.1245	2.21245
$101$	−1.97455	−0.196475	−0.0982377	−	0.995163i	$-0.531321\pi$
$101$	−1.97455	−0.196475	−0.0982377	+	0.995163i	$0.531321\pi$
$102$	−10.4814	−1.03781
$103$	−8.44874	−0.832479	−0.416240	−	0.909255i	$-0.636652\pi$
$103$	−8.44874	−0.832479	−0.416240	+	0.909255i	$0.636652\pi$
$104$	13.7769	1.35094
$105$	5.07365	0.495138
$106$	−27.4725	−2.66836
$107$	18.8832	1.82550	0.912752	−	0.408514i	$-0.133953\pi$
$107$	18.8832	1.82550	0.912752	+	0.408514i	$0.133953\pi$
$108$	−16.8074	−1.61729
$109$	−12.2294	−1.17136	−0.585681	−	0.810541i	$-0.699173\pi$
$109$	−12.2294	−1.17136	−0.585681	+	0.810541i	$0.699173\pi$
$110$	22.7766	2.17166
$111$	−1.76364	−0.167398
$112$	4.51213	0.426356
$113$	−14.3697	−1.35179	−0.675894	−	0.736999i	$-0.736242\pi$
$113$	−14.3697	−1.35179	−0.675894	+	0.736999i	$0.736242\pi$
$114$	−1.20132	−0.112514
$115$	0.00404552	0.000377247 0
$116$	−39.7879	−3.69421
$117$	−2.52643	−0.233568
$118$	4.62668	0.425921
$119$	1.64604	0.150892
$120$	−36.5021	−3.33217
$121$	−2.87412	−0.261284
$122$	−3.38197	−0.306189
$123$	−10.7802	−0.972014
$124$	−38.4500	−3.45291
$125$	0.571256	0.0510947
$126$	−2.06652	−0.184101
$127$	−5.24885	−0.465760	−0.232880	−	0.972505i	$-0.574815\pi$
$127$	−5.24885	−0.465760	−0.232880	+	0.972505i	$0.574815\pi$
$128$	16.1498	1.42745
$129$	−2.22803	−0.196167
$130$	19.3470	1.69685
$131$	−2.89568	−0.252997	−0.126499	−	0.991967i	$-0.540374\pi$
$131$	−2.89568	−0.252997	−0.126499	+	0.991967i	$0.540374\pi$
$132$	−24.4867	−2.13129
$133$	0.188660	0.0163589
$134$	23.3925	2.02080
$135$	−12.5525	−1.08035
$136$	−11.8424	−1.01547
$137$	21.7114	1.85493	0.927463	−	0.373914i	$-0.121985\pi$
$137$	21.7114	1.85493	0.927463	+	0.373914i	$0.121985\pi$
$138$	−0.00638549	−0.000543569 0
$139$	1.43807	0.121975	0.0609876	−	0.998139i	$-0.480575\pi$
$139$	1.43807	0.121975	0.0609876	+	0.998139i	$0.480575\pi$
$140$	10.7788	0.910975
$141$	−19.1531	−1.61298
$142$	−9.04386	−0.758944
$143$	6.90233	0.577202
$144$	5.95297	0.496081
$145$	−29.7154	−2.46773
$146$	−29.2370	−2.41967
$147$	−12.8181	−1.05722
$148$	−3.74680	−0.307985
$149$	7.86532	0.644352	0.322176	−	0.946680i	$-0.395586\pi$
$149$	7.86532	0.644352	0.322176	+	0.946680i	$0.395586\pi$
$150$	−26.0809	−2.12950
$151$	0.0776229	0.00631687	0.00315843	−	0.999995i	$-0.498995\pi$
$151$	0.0776229	0.00631687	0.00315843	+	0.999995i	$0.498995\pi$
$152$	−1.35730	−0.110092
$153$	2.17166	0.175569
$154$	5.64585	0.454956
$155$	−28.7162	−2.30654
$156$	−20.7996	−1.66530
$157$	−12.9338	−1.03223	−0.516117	−	0.856518i	$-0.672623\pi$
$157$	−12.9338	−1.03223	−0.516117	+	0.856518i	$0.672623\pi$
$158$	−20.4220	−1.62469
$159$	22.0582	1.74933
$160$	−9.28126	−0.733748
$161$	0.00100280	7.90319e−5 0
$162$	27.6521	2.17256
$163$	1.78981	0.140188	0.0700942	−	0.997540i	$-0.477670\pi$
$163$	1.78981	0.140188	0.0700942	+	0.997540i	$0.477670\pi$
$164$	−22.9021	−1.78835
$165$	−18.2878	−1.42370
$166$	−12.5439	−0.973598
$167$	−12.0367	−0.931424	−0.465712	−	0.884936i	$-0.654202\pi$
$167$	−12.0367	−0.931424	−0.465712	+	0.884936i	$0.654202\pi$
$168$	−9.04813	−0.698078
$169$	−7.13698	−0.548999
$170$	−16.6303	−1.27549
$171$	0.248903	0.0190341
$172$	−4.73336	−0.360916
$173$	1.83863	0.139789	0.0698943	−	0.997554i	$-0.477734\pi$
$173$	1.83863	0.139789	0.0698943	+	0.997554i	$0.477734\pi$
$174$	46.9031	3.55571
$175$	4.09585	0.309617
$176$	−16.2638	−1.22593
$177$	−3.71486	−0.279226
$178$	45.6156	3.41904
$179$	−3.58725	−0.268123	−0.134062	−	0.990973i	$-0.542802\pi$
$179$	−3.58725	−0.268123	−0.134062	+	0.990973i	$0.542802\pi$
$180$	14.2207	1.05995
$181$	−14.7265	−1.09461	−0.547307	−	0.836932i	$-0.684347\pi$
$181$	−14.7265	−1.09461	−0.547307	+	0.836932i	$0.684347\pi$
$182$	4.79573	0.355483
$183$	2.71546	0.200732
$184$	−0.00721461	−0.000531868 0
$185$	−2.79829	−0.205734
$186$	45.3259	3.32346
$187$	−5.93310	−0.433871
$188$	−40.6902	−2.96764
$189$	−3.11151	−0.226329
$190$	−1.90607	−0.138281
$191$	14.9823	1.08408	0.542040	−	0.840353i	$-0.317652\pi$
$191$	14.9823	1.08408	0.542040	+	0.840353i	$0.317652\pi$
$192$	−8.29550	−0.598676
$193$	−21.3113	−1.53402	−0.767010	−	0.641635i	$-0.778257\pi$
$193$	−21.3113	−1.53402	−0.767010	+	0.641635i	$0.778257\pi$
$194$	−16.3384	−1.17303
$195$	−15.5341	−1.11242
$196$	−27.2316	−1.94511
$197$	4.43248	0.315801	0.157900	−	0.987455i	$-0.449528\pi$
$197$	4.43248	0.315801	0.157900	+	0.987455i	$0.449528\pi$
$198$	7.44871	0.529357
$199$	24.3863	1.72870	0.864350	−	0.502891i	$-0.167730\pi$
$199$	24.3863	1.72870	0.864350	+	0.502891i	$0.167730\pi$
$200$	−29.4674	−2.08366
$201$	−18.7823	−1.32480
$202$	4.94503	0.347931
$203$	−7.36585	−0.516981
$204$	17.8789	1.25178
$205$	−17.1043	−1.19462
$206$	21.1588	1.47421
$207$	0.00132302	9.19564e−5 0
$208$	−13.8149	−0.957892
$209$	−0.680017	−0.0470378
$210$	−12.7063	−0.876821
$211$	21.5620	1.48439	0.742193	−	0.670186i	$-0.233786\pi$
$211$	21.5620	1.48439	0.742193	+	0.670186i	$0.233786\pi$
$212$	46.8619	3.21849
$213$	7.26151	0.497550
$214$	−47.2906	−3.23272
$215$	−3.53510	−0.241091
$216$	22.3856	1.52315
$217$	−7.11816	−0.483212
$218$	30.6270	2.07432
$219$	23.4750	1.58629
$220$	−38.8518	−2.61939
$221$	−5.03973	−0.339009
$222$	4.41683	0.296439
$223$	−2.54423	−0.170374	−0.0851872	−	0.996365i	$-0.527149\pi$
$223$	−2.54423	−0.170374	−0.0851872	+	0.996365i	$0.527149\pi$
$224$	−2.30063	−0.153718
$225$	5.40376	0.360251
$226$	35.9872	2.39383
$227$	0.315809	0.0209610	0.0104805	−	0.999945i	$-0.496664\pi$
$227$	0.315809	0.0209610	0.0104805	+	0.999945i	$0.496664\pi$
$228$	2.04918	0.135710
$229$	4.74936	0.313847	0.156923	−	0.987611i	$-0.449842\pi$
$229$	4.74936	0.313847	0.156923	+	0.987611i	$0.449842\pi$
$230$	−0.0101315	−0.000668053 0
$231$	−4.53317	−0.298261
$232$	52.9932	3.47918
$233$	−13.0529	−0.855123	−0.427562	−	0.903986i	$-0.640627\pi$
$233$	−13.0529	−0.855123	−0.427562	+	0.903986i	$0.640627\pi$
$234$	6.32713	0.413617
$235$	−30.3893	−1.98238
$236$	−7.89209	−0.513732
$237$	16.3973	1.06512
$238$	−4.12231	−0.267210
$239$	25.4721	1.64766	0.823828	−	0.566840i	$-0.191834\pi$
$239$	25.4721	1.64766	0.823828	+	0.566840i	$0.191834\pi$
$240$	36.6028	2.36270
$241$	15.0431	0.969013	0.484506	−	0.874788i	$-0.338999\pi$
$241$	15.0431	0.969013	0.484506	+	0.874788i	$0.338999\pi$
$242$	7.19789	0.462698
$243$	−10.3993	−0.667115
$244$	5.76889	0.369316
$245$	−20.3378	−1.29933
$246$	26.9976	1.72130
$247$	−0.577624	−0.0367534
$248$	51.2113	3.25192
$249$	10.0718	0.638274
$250$	−1.43064	−0.0904817
$251$	−4.65176	−0.293616	−0.146808	−	0.989165i	$-0.546900\pi$
$251$	−4.65176	−0.293616	−0.146808	+	0.989165i	$0.546900\pi$
$252$	3.52503	0.222056
$253$	−0.00361457	−0.000227246 0
$254$	13.1451	0.824797
$255$	13.3528	0.836186
$256$	−32.1943	−2.01214
$257$	8.22994	0.513370	0.256685	−	0.966495i	$-0.417370\pi$
$257$	8.22994	0.513370	0.256685	+	0.966495i	$0.417370\pi$
$258$	5.57982	0.347384
$259$	−0.693638	−0.0431005
$260$	−33.0017	−2.04668
$261$	−9.71795	−0.601526
$262$	7.25189	0.448023
$263$	−12.2692	−0.756551	−0.378275	−	0.925693i	$-0.623483\pi$
$263$	−12.2692	−0.756551	−0.378275	+	0.925693i	$0.623483\pi$
$264$	32.6137	2.00723
$265$	34.9987	2.14995
$266$	−0.472475	−0.0289693
$267$	−36.6257	−2.24146
$268$	−39.9024	−2.43743
$269$	3.65920	0.223106	0.111553	−	0.993759i	$-0.464418\pi$
$269$	3.65920	0.223106	0.111553	+	0.993759i	$0.464418\pi$
$270$	31.4363	1.91315
$271$	−25.7237	−1.56261	−0.781303	−	0.624152i	$-0.785444\pi$
$271$	−25.7237	−1.56261	−0.781303	+	0.624152i	$0.785444\pi$
$272$	11.8750	0.720028
$273$	−3.85059	−0.233048
$274$	−54.3735	−3.28482
$275$	−14.7634	−0.890264
$276$	0.0108922	0.000655634 0
$277$	30.0391	1.80488	0.902438	−	0.430821i	$-0.141776\pi$
$277$	30.0391	1.80488	0.902438	+	0.430821i	$0.141776\pi$
$278$	−3.60146	−0.216001
$279$	−9.39117	−0.562235
$280$	−14.3562	−0.857948
$281$	8.88843	0.530239	0.265120	−	0.964216i	$-0.414589\pi$
$281$	8.88843	0.530239	0.265120	+	0.964216i	$0.414589\pi$
$282$	47.9667	2.85638
$283$	−2.24236	−0.133294	−0.0666471	−	0.997777i	$-0.521230\pi$
$283$	−2.24236	−0.133294	−0.0666471	+	0.997777i	$0.521230\pi$
$284$	15.4268	0.915413
$285$	1.53042	0.0906544
$286$	−17.2860	−1.02215
$287$	−4.23981	−0.250268
$288$	−3.03528	−0.178856
$289$	−12.6680	−0.745174
$290$	74.4187	4.37002
$291$	13.1184	0.769015
$292$	49.8718	2.91853
$293$	−16.8304	−0.983245	−0.491623	−	0.870808i	$-0.663596\pi$
$293$	−16.8304	−0.983245	−0.491623	+	0.870808i	$0.663596\pi$
$294$	32.1013	1.87219
$295$	−5.89418	−0.343173
$296$	4.99034	0.290058
$297$	11.2153	0.650780
$298$	−19.6978	−1.14106
$299$	−0.00307031	−0.000177560 0
$300$	44.4883	2.56853
$301$	−0.876278	−0.0505078
$302$	−0.194397	−0.0111863
$303$	−3.97047	−0.228097
$304$	1.36104	0.0780613
$305$	4.30848	0.246703
$306$	−5.43867	−0.310908
$307$	−13.8030	−0.787780	−0.393890	−	0.919158i	$-0.628871\pi$
$307$	−13.8030	−0.787780	−0.393890	+	0.919158i	$0.628871\pi$
$308$	−9.63057	−0.548753
$309$	−16.9889	−0.966464
$310$	71.9163	4.08457
$311$	8.82172	0.500234	0.250117	−	0.968216i	$-0.419531\pi$
$311$	8.82172	0.500234	0.250117	+	0.968216i	$0.419531\pi$
$312$	27.7029	1.56837
$313$	−18.4523	−1.04298	−0.521492	−	0.853256i	$-0.674624\pi$
$313$	−18.4523	−1.04298	−0.521492	+	0.853256i	$0.674624\pi$
$314$	32.3913	1.82794
$315$	2.63265	0.148333
$316$	34.8354	1.95964
$317$	−22.0470	−1.23828	−0.619142	−	0.785279i	$-0.712520\pi$
$317$	−22.0470	−1.23828	−0.619142	+	0.785279i	$0.712520\pi$
$318$	−55.2421	−3.09783
$319$	26.5500	1.48651
$320$	−13.1621	−0.735781
$321$	37.9706	2.11931
$322$	−0.00251140	−0.000139955 0
$323$	0.496514	0.0276268
$324$	−47.1684	−2.62046
$325$	−12.5404	−0.695615
$326$	−4.48235	−0.248255
$327$	−24.5911	−1.35989
$328$	30.5031	1.68425
$329$	−7.53288	−0.415301
$330$	45.7996	2.52118
$331$	−18.3796	−1.01023	−0.505116	−	0.863051i	$-0.668550\pi$
$331$	−18.3796	−1.01023	−0.505116	+	0.863051i	$0.668550\pi$
$332$	21.3972	1.17432
$333$	−0.915133	−0.0501490
$334$	30.1443	1.64943
$335$	−29.8009	−1.62820
$336$	9.07308	0.494977
$337$	28.9942	1.57941	0.789706	−	0.613485i	$-0.210233\pi$
$337$	28.9942	1.57941	0.789706	+	0.613485i	$0.210233\pi$
$338$	17.8737	0.972202
$339$	−28.8949	−1.56935
$340$	28.3676	1.53845
$341$	25.6572	1.38941
$342$	−0.623348	−0.0337068
$343$	−10.5773	−0.571119
$344$	6.30434	0.339907
$345$	0.00813481	0.000437964 0
$346$	−4.60463	−0.247546
$347$	−3.99988	−0.214725	−0.107362	−	0.994220i	$-0.534241\pi$
$347$	−3.99988	−0.214725	−0.107362	+	0.994220i	$0.534241\pi$
$348$	−80.0062	−4.28878
$349$	−22.7879	−1.21981	−0.609904	−	0.792475i	$-0.708792\pi$
$349$	−22.7879	−1.21981	−0.609904	+	0.792475i	$0.708792\pi$
$350$	−10.2576	−0.548290
$351$	9.52660	0.508492
$352$	8.29256	0.441995
$353$	11.8229	0.629272	0.314636	−	0.949212i	$-0.398118\pi$
$353$	11.8229	0.629272	0.314636	+	0.949212i	$0.398118\pi$
$354$	9.30342	0.494471
$355$	11.5215	0.611496
$356$	−77.8101	−4.12393
$357$	3.30989	0.175178
$358$	8.98382	0.474810
$359$	24.0575	1.26971	0.634853	−	0.772633i	$-0.281061\pi$
$359$	24.0575	1.26971	0.634853	+	0.772633i	$0.281061\pi$
$360$	−18.9405	−0.998252
$361$	−18.9431	−0.997005
$362$	36.8808	1.93841
$363$	−5.77934	−0.303337
$364$	−8.18045	−0.428772
$365$	37.2466	1.94957
$366$	−6.80053	−0.355469
$367$	11.8171	0.616846	0.308423	−	0.951249i	$-0.400199\pi$
$367$	11.8171	0.616846	0.308423	+	0.951249i	$0.400199\pi$
$368$	0.00723450	0.000377125 0
$369$	−5.59369	−0.291196
$370$	7.00797	0.364327
$371$	8.67545	0.450407
$372$	−77.3159	−4.00864
$373$	20.9736	1.08597	0.542987	−	0.839741i	$-0.317293\pi$
$373$	20.9736	1.08597	0.542987	+	0.839741i	$0.317293\pi$
$374$	14.8587	0.768327
$375$	1.14869	0.0593182
$376$	54.1950	2.79489
$377$	22.5522	1.16150
$378$	7.79241	0.400798
$379$	−22.7950	−1.17090	−0.585450	−	0.810708i	$-0.699082\pi$
$379$	−22.7950	−1.17090	−0.585450	+	0.810708i	$0.699082\pi$
$380$	3.25133	0.166790
$381$	−10.5545	−0.540722
$382$	−37.5213	−1.91976
$383$	36.1951	1.84948	0.924742	−	0.380594i	$-0.124281\pi$
$383$	36.1951	1.84948	0.924742	+	0.380594i	$0.124281\pi$
$384$	32.4743	1.65720
$385$	−7.19255	−0.366566
$386$	53.3716	2.71654
$387$	−1.15609	−0.0587676
$388$	27.8696	1.41487
$389$	19.3851	0.982865	0.491433	−	0.870916i	$-0.336473\pi$
$389$	19.3851	0.982865	0.491433	+	0.870916i	$0.336473\pi$
$390$	38.9033	1.96995
$391$	0.00263917	0.000133469 0
$392$	36.2695	1.83189
$393$	−5.82269	−0.293716
$394$	−11.1006	−0.559240
$395$	26.0167	1.30904
$396$	−12.7058	−0.638493
$397$	20.3149	1.01958	0.509789	−	0.860300i	$-0.329724\pi$
$397$	20.3149	1.01958	0.509789	+	0.860300i	$0.329724\pi$
$398$	−61.0726	−3.06129
$399$	0.379360	0.0189918
$400$	29.5487	1.47743
$401$	26.5143	1.32406	0.662031	−	0.749476i	$-0.269695\pi$
$401$	26.5143	1.32406	0.662031	+	0.749476i	$0.269695\pi$
$402$	47.0380	2.34604
$403$	21.7939	1.08563
$404$	−8.43512	−0.419663
$405$	−35.2275	−1.75047
$406$	18.4469	0.915503
$407$	2.50019	0.123930
$408$	−23.8128	−1.17891
$409$	4.15257	0.205331	0.102666	−	0.994716i	$-0.467263\pi$
$409$	4.15257	0.205331	0.102666	+	0.994716i	$0.467263\pi$
$410$	42.8357	2.11550
$411$	43.6576	2.15347
$412$	−36.0923	−1.77814
$413$	−1.46105	−0.0718934
$414$	−0.00331335	−0.000162842 0
$415$	15.9804	0.784447
$416$	7.04391	0.345356
$417$	2.89169	0.141607
$418$	1.70302	0.0832975
$419$	−16.3248	−0.797517	−0.398759	−	0.917056i	$-0.630559\pi$
$419$	−16.3248	−0.797517	−0.398759	+	0.917056i	$0.630559\pi$
$420$	21.6742	1.05759
$421$	−19.2407	−0.937734	−0.468867	−	0.883269i	$-0.655338\pi$
$421$	−19.2407	−0.937734	−0.468867	+	0.883269i	$0.655338\pi$
$422$	−53.9993	−2.62865
$423$	−9.93832	−0.483218
$424$	−62.4151	−3.03114
$425$	10.7795	0.522880
$426$	−18.1856	−0.881093
$427$	1.06798	0.0516833
$428$	80.6672	3.89920
$429$	13.8793	0.670100
$430$	8.85322	0.426940
$431$	6.00449	0.289226	0.144613	−	0.989488i	$-0.453806\pi$
$431$	6.00449	0.289226	0.144613	+	0.989488i	$0.453806\pi$
$432$	−22.4473	−1.08000
$433$	−21.2896	−1.02311	−0.511555	−	0.859250i	$-0.670931\pi$
$433$	−21.2896	−1.02311	−0.511555	+	0.859250i	$0.670931\pi$
$434$	17.8266	0.855703
$435$	−59.7524	−2.86491
$436$	−52.2429	−2.50198
$437$	0.000302487 0	1.44699e−5 0
$438$	−58.7903	−2.80911
$439$	−9.02088	−0.430543	−0.215272	−	0.976554i	$-0.569064\pi$
$439$	−9.02088	−0.430543	−0.215272	+	0.976554i	$0.569064\pi$
$440$	51.7465	2.46692
$441$	−6.65114	−0.316721
$442$	12.6214	0.600338
$443$	28.3534	1.34711	0.673555	−	0.739137i	$-0.264766\pi$
$443$	28.3534	1.34711	0.673555	+	0.739137i	$0.264766\pi$
$444$	−7.53414	−0.357554
$445$	−58.1122	−2.75478
$446$	6.37172	0.301710
$447$	15.8157	0.748059
$448$	−3.26260	−0.154143
$449$	17.9961	0.849286	0.424643	−	0.905361i	$-0.360400\pi$
$449$	17.9961	0.849286	0.424643	+	0.905361i	$0.360400\pi$
$450$	−13.5331	−0.637955
$451$	15.2822	0.719613
$452$	−61.3861	−2.88736
$453$	0.156086	0.00733354
$454$	−0.790904	−0.0371190
$455$	−6.10954	−0.286420
$456$	−2.72929	−0.127811
$457$	2.13183	0.0997230	0.0498615	−	0.998756i	$-0.484122\pi$
$457$	2.13183	0.0997230	0.0498615	+	0.998756i	$0.484122\pi$
$458$	−11.8942	−0.555780
$459$	−8.18887	−0.382224
$460$	0.0172821	0.000805783 0
$461$	−13.6079	−0.633784	−0.316892	−	0.948462i	$-0.602639\pi$
$461$	−13.6079	−0.633784	−0.316892	+	0.948462i	$0.602639\pi$
$462$	11.3528	0.528179
$463$	−4.13632	−0.192231	−0.0961155	−	0.995370i	$-0.530642\pi$
$463$	−4.13632	−0.192231	−0.0961155	+	0.995370i	$0.530642\pi$
$464$	−53.1393	−2.46693
$465$	−57.7431	−2.67777
$466$	32.6894	1.51431
$467$	−7.97191	−0.368896	−0.184448	−	0.982842i	$-0.559050\pi$
$467$	−7.97191	−0.368896	−0.184448	+	0.982842i	$0.559050\pi$
$468$	−10.7927	−0.498891
$469$	−7.38704	−0.341102
$470$	76.1063	3.51052
$471$	−26.0076	−1.19837
$472$	10.5114	0.483828
$473$	3.15851	0.145229
$474$	−41.0649	−1.88618
$475$	1.23548	0.0566876
$476$	7.03175	0.322300
$477$	11.4457	0.524064
$478$	−63.7919	−2.91777
$479$	32.6113	1.49005	0.745023	−	0.667038i	$-0.232438\pi$
$479$	32.6113	1.49005	0.745023	+	0.667038i	$0.232438\pi$
$480$	−18.6629	−0.851842
$481$	2.12373	0.0968337
$482$	−37.6737	−1.71599
$483$	0.00201645	9.17518e−5 0
$484$	−12.2780	−0.558091
$485$	20.8143	0.945130
$486$	26.0438	1.18137
$487$	27.1418	1.22991	0.614956	−	0.788562i	$-0.289174\pi$
$487$	27.1418	1.22991	0.614956	+	0.788562i	$0.289174\pi$
$488$	−7.68355	−0.347818
$489$	3.59897	0.162751
$490$	50.9335	2.30094
$491$	20.3805	0.919759	0.459880	−	0.887981i	$-0.347893\pi$
$491$	20.3805	0.919759	0.459880	+	0.887981i	$0.347893\pi$
$492$	−46.0519	−2.07618
$493$	−19.3854	−0.873075
$494$	1.44659	0.0650852
$495$	−9.48931	−0.426513
$496$	−51.3525	−2.30579
$497$	2.85593	0.128106
$498$	−25.2236	−1.13030
$499$	−12.8047	−0.573216	−0.286608	−	0.958048i	$-0.592528\pi$
$499$	−12.8047	−0.573216	−0.286608	+	0.958048i	$0.592528\pi$
$500$	2.44036	0.109136
$501$	−24.2035	−1.08133
$502$	11.6498	0.519955
$503$	−10.0681	−0.448916	−0.224458	−	0.974484i	$-0.572061\pi$
$503$	−10.0681	−0.448916	−0.224458	+	0.974484i	$0.572061\pi$
$504$	−4.69496	−0.209130
$505$	−6.29974	−0.280335
$506$	0.00905225	0.000402421 0
$507$	−14.3512	−0.637358
$508$	−22.4226	−0.994843
$509$	−14.7274	−0.652780	−0.326390	−	0.945235i	$-0.605832\pi$
$509$	−14.7274	−0.652780	−0.326390	+	0.945235i	$0.605832\pi$
$510$	−33.4405	−1.48077
$511$	9.23266	0.408429
$512$	48.3272	2.13578
$513$	−0.938560	−0.0414385
$514$	−20.6109	−0.909107
$515$	−26.9554	−1.18780
$516$	−9.51793	−0.419004
$517$	27.1520	1.19414
$518$	1.73713	0.0763252
$519$	3.69715	0.162287
$520$	43.9548	1.92754
$521$	6.75940	0.296135	0.148067	−	0.988977i	$-0.452695\pi$
$521$	6.75940	0.296135	0.148067	+	0.988977i	$0.452695\pi$
$522$	24.3374	1.06522
$523$	13.0797	0.571933	0.285967	−	0.958240i	$-0.407685\pi$
$523$	13.0797	0.571933	0.285967	+	0.958240i	$0.407685\pi$
$524$	−12.3701	−0.540391
$525$	8.23602	0.359449
$526$	30.7267	1.33975
$527$	−18.7336	−0.816047
$528$	−32.7036	−1.42324
$529$	−23.0000	−1.00000
$530$	−87.6499	−3.80727
$531$	−1.92760	−0.0836505
$532$	0.805938	0.0349418
$533$	12.9811	0.562275
$534$	91.7248	3.96932
$535$	60.2460	2.60466
$536$	53.1457	2.29554
$537$	−7.21330	−0.311277
$538$	−9.16403	−0.395089
$539$	18.1713	0.782692
$540$	−53.6233	−2.30758
$541$	−4.39811	−0.189089	−0.0945447	−	0.995521i	$-0.530140\pi$
$541$	−4.39811	−0.189089	−0.0945447	+	0.995521i	$0.530140\pi$
$542$	64.4220	2.76716
$543$	−29.6124	−1.27079
$544$	−6.05480	−0.259598
$545$	−39.0174	−1.67132
$546$	9.64334	0.412697
$547$	−14.4596	−0.618249	−0.309125	−	0.951022i	$-0.600036\pi$
$547$	−14.4596	−0.618249	−0.309125	+	0.951022i	$0.600036\pi$
$548$	92.7490	3.96204
$549$	1.40902	0.0601353
$550$	36.9731	1.57654
$551$	−2.22184	−0.0946537
$552$	−0.0145073	−0.000617470 0
$553$	6.44901	0.274240
$554$	−75.2293	−3.19619
$555$	−5.62684	−0.238846
$556$	6.14329	0.260534
$557$	2.06157	0.0873516	0.0436758	−	0.999046i	$-0.486093\pi$
$557$	2.06157	0.0873516	0.0436758	+	0.999046i	$0.486093\pi$
$558$	23.5190	0.995641
$559$	2.68292	0.113476
$560$	14.3958	0.608333
$561$	−11.9304	−0.503701
$562$	−22.2600	−0.938981
$563$	20.3908	0.859371	0.429686	−	0.902979i	$-0.358624\pi$
$563$	20.3908	0.859371	0.429686	+	0.902979i	$0.358624\pi$
$564$	−81.8205	−3.44527
$565$	−45.8460	−1.92876
$566$	5.61571	0.236046
$567$	−8.73218	−0.366717
$568$	−20.5469	−0.862128
$569$	19.1866	0.804344	0.402172	−	0.915564i	$-0.368255\pi$
$569$	19.1866	0.804344	0.402172	+	0.915564i	$0.368255\pi$
$570$	−3.83276	−0.160537
$571$	21.0680	0.881668	0.440834	−	0.897589i	$-0.354683\pi$
$571$	21.0680	0.881668	0.440834	+	0.897589i	$0.354683\pi$
$572$	29.4862	1.23288
$573$	30.1267	1.25856
$574$	10.6181	0.443191
$575$	0.00656707	0.000273866 0
$576$	−4.30443	−0.179351
$577$	−25.4188	−1.05820	−0.529099	−	0.848560i	$-0.677470\pi$
$577$	−25.4188	−1.05820	−0.529099	+	0.848560i	$0.677470\pi$
$578$	31.7254	1.31960
$579$	−42.8531	−1.78092
$580$	−126.942	−5.27097
$581$	3.96121	0.164339
$582$	−32.8535	−1.36182
$583$	−31.2704	−1.29509
$584$	−66.4239	−2.74864
$585$	−8.06047	−0.333259
$586$	42.1498	1.74119
$587$	−7.85314	−0.324134	−0.162067	−	0.986780i	$-0.551816\pi$
$587$	−7.85314	−0.324134	−0.162067	+	0.986780i	$0.551816\pi$
$588$	−54.7577	−2.25817
$589$	−2.14713	−0.0884710
$590$	14.7613	0.607712
$591$	8.91290	0.366628
$592$	−5.00410	−0.205667
$593$	23.0173	0.945209	0.472604	−	0.881275i	$-0.343314\pi$
$593$	23.0173	0.945209	0.472604	+	0.881275i	$0.343314\pi$
$594$	−28.0875	−1.15244
$595$	5.25163	0.215296
$596$	33.6000	1.37631
$597$	49.0364	2.00693
$598$	0.00768921	0.000314435 0
$599$	−20.2450	−0.827188	−0.413594	−	0.910461i	$-0.635727\pi$
$599$	−20.2450	−0.827188	−0.413594	+	0.910461i	$0.635727\pi$
$600$	−59.2536	−2.41902
$601$	19.1068	0.779381	0.389690	−	0.920946i	$-0.372582\pi$
$601$	19.1068	0.779381	0.389690	+	0.920946i	$0.372582\pi$
$602$	2.19453	0.0894424
$603$	−9.74591	−0.396884
$604$	0.331599	0.0134926
$605$	−9.16978	−0.372805
$606$	9.94355	0.403929
$607$	14.8454	0.602557	0.301279	−	0.953536i	$-0.402587\pi$
$607$	14.8454	0.602557	0.301279	+	0.953536i	$0.402587\pi$
$608$	−0.693966	−0.0281440
$609$	−14.8114	−0.600188
$610$	−10.7901	−0.436877
$611$	23.0636	0.933055
$612$	9.27716	0.375007
$613$	28.7113	1.15964	0.579820	−	0.814745i	$-0.303123\pi$
$613$	28.7113	1.15964	0.579820	+	0.814745i	$0.303123\pi$
$614$	34.5680	1.39505
$615$	−34.3937	−1.38689
$616$	12.8269	0.516810
$617$	−40.4138	−1.62700	−0.813498	−	0.581567i	$-0.802440\pi$
$617$	−40.4138	−1.62700	−0.813498	+	0.581567i	$0.802440\pi$
$618$	42.5466	1.71147
$619$	35.2003	1.41482	0.707409	−	0.706805i	$-0.249864\pi$
$619$	35.2003	1.41482	0.707409	+	0.706805i	$0.249864\pi$
$620$	−122.673	−4.92668
$621$	−0.00498883	−0.000200195 0
$622$	−22.0929	−0.885846
$623$	−14.4048	−0.577117
$624$	−27.7793	−1.11206
$625$	−24.0727	−0.962907
$626$	46.2115	1.84698
$627$	−1.36739	−0.0546083
$628$	−55.2523	−2.20481
$629$	−1.82551	−0.0727880
$630$	−6.59316	−0.262678
$631$	8.37319	0.333331	0.166666	−	0.986013i	$-0.446700\pi$
$631$	8.37319	0.333331	0.166666	+	0.986013i	$0.446700\pi$
$632$	−46.3970	−1.84558
$633$	43.3572	1.72329
$634$	55.2141	2.19283
$635$	−16.7462	−0.664555
$636$	94.2308	3.73649
$637$	15.4351	0.611562
$638$	−66.4911	−2.63241
$639$	3.76791	0.149056
$640$	51.5253	2.03672
$641$	23.5308	0.929412	0.464706	−	0.885465i	$-0.346160\pi$
$641$	23.5308	0.929412	0.464706	+	0.885465i	$0.346160\pi$
$642$	−95.0928	−3.75301
$643$	−13.2398	−0.522128	−0.261064	−	0.965322i	$-0.584073\pi$
$643$	−13.2398	−0.522128	−0.261064	+	0.965322i	$0.584073\pi$
$644$	0.00428389	0.000168809 0
$645$	−7.10843	−0.279894
$646$	−1.24346	−0.0489232
$647$	−26.2806	−1.03320	−0.516599	−	0.856227i	$-0.672802\pi$
$647$	−26.2806	−1.03320	−0.516599	+	0.856227i	$0.672802\pi$
$648$	62.8232	2.46793
$649$	5.26629	0.206720
$650$	31.4059	1.23184
$651$	−14.3133	−0.560984
$652$	7.64590	0.299436
$653$	22.6774	0.887434	0.443717	−	0.896167i	$-0.353659\pi$
$653$	22.6774	0.887434	0.443717	+	0.896167i	$0.353659\pi$
$654$	61.5854	2.40818
$655$	−9.23857	−0.360981
$656$	−30.5872	−1.19423
$657$	12.1809	0.475221
$658$	18.8652	0.735442
$659$	−22.0958	−0.860732	−0.430366	−	0.902654i	$-0.641616\pi$
$659$	−22.0958	−0.860732	−0.430366	+	0.902654i	$0.641616\pi$
$660$	−78.1239	−3.04097
$661$	28.1485	1.09485	0.547425	−	0.836855i	$-0.315608\pi$
$661$	28.1485	1.09485	0.547425	+	0.836855i	$0.315608\pi$
$662$	46.0294	1.78898
$663$	−10.1340	−0.393571
$664$	−28.4987	−1.10597
$665$	0.601912	0.0233411
$666$	2.29184	0.0888070
$667$	−0.0118100	−0.000457285 0
$668$	−51.4196	−1.98948
$669$	−5.11599	−0.197796
$670$	74.6328	2.88332
$671$	−3.84951	−0.148609
$672$	−4.62616	−0.178458
$673$	−43.9580	−1.69446	−0.847229	−	0.531228i	$-0.821731\pi$
$673$	−43.9580	−1.69446	−0.847229	+	0.531228i	$0.821731\pi$
$674$	−72.6124	−2.79692
$675$	−20.3764	−0.784288
$676$	−30.4886	−1.17264
$677$	9.46161	0.363639	0.181820	−	0.983332i	$-0.441801\pi$
$677$	9.46161	0.363639	0.181820	+	0.983332i	$0.441801\pi$
$678$	72.3637	2.77911
$679$	5.15945	0.198001
$680$	−37.7826	−1.44890
$681$	0.635034	0.0243345
$682$	−64.2553	−2.46046
$683$	−35.4647	−1.35702	−0.678510	−	0.734591i	$-0.737374\pi$
$683$	−35.4647	−1.35702	−0.678510	+	0.734591i	$0.737374\pi$
$684$	1.06329	0.0406561
$685$	69.2693	2.64664
$686$	26.4895	1.01137
$687$	9.55010	0.364359
$688$	−6.32172	−0.241013
$689$	−26.5619	−1.01193
$690$	−0.0203727	−0.000775574 0
$691$	31.7405	1.20746	0.603732	−	0.797187i	$-0.293680\pi$
$691$	31.7405	1.20746	0.603732	+	0.797187i	$0.293680\pi$
$692$	7.85448	0.298582
$693$	−2.35221	−0.0893529
$694$	10.0172	0.380248
$695$	4.58809	0.174036
$696$	106.560	4.03914
$697$	−11.1583	−0.422651
$698$	57.0695	2.16011
$699$	−26.2470	−0.992752
$700$	17.4971	0.661330
$701$	−16.5673	−0.625738	−0.312869	−	0.949796i	$-0.601290\pi$
$701$	−16.5673	−0.625738	−0.312869	+	0.949796i	$0.601290\pi$
$702$	−23.8582	−0.900470
$703$	−0.209230	−0.00789125
$704$	11.7599	0.443219
$705$	−61.1073	−2.30144
$706$	−29.6091	−1.11435
$707$	−1.56158	−0.0587291
$708$	−15.8696	−0.596415
$709$	−49.1851	−1.84719	−0.923593	−	0.383374i	$-0.874762\pi$
$709$	−49.1851	−1.84719	−0.923593	+	0.383374i	$0.874762\pi$
$710$	−28.8541	−1.08288
$711$	8.50833	0.319087
$712$	103.635	3.88388
$713$	−0.0114129	−0.000427415 0
$714$	−8.28922	−0.310216
$715$	22.0216	0.823562
$716$	−15.3244	−0.572700
$717$	51.2198	1.91284
$718$	−60.2491	−2.24848
$719$	31.3364	1.16865	0.584326	−	0.811519i	$-0.301359\pi$
$719$	31.3364	1.16865	0.584326	+	0.811519i	$0.301359\pi$
$720$	18.9927	0.707817
$721$	−6.68169	−0.248839
$722$	47.4407	1.76556
$723$	30.2490	1.12497
$724$	−62.9105	−2.33805
$725$	−48.2368	−1.79147
$726$	14.4736	0.537168
$727$	32.3337	1.19919	0.599596	−	0.800303i	$-0.295328\pi$
$727$	32.3337	1.19919	0.599596	+	0.800303i	$0.295328\pi$
$728$	10.8955	0.403814
$729$	12.2134	0.452350
$730$	−93.2795	−3.45243
$731$	−2.30618	−0.0852973
$732$	11.6002	0.428755
$733$	−4.07873	−0.150651	−0.0753256	−	0.997159i	$-0.524000\pi$
$733$	−4.07873	−0.150651	−0.0753256	+	0.997159i	$0.524000\pi$
$734$	−29.5944	−1.09235
$735$	−40.8956	−1.50846
$736$	−0.00368871	−0.000135968 0
$737$	26.6263	0.980794
$738$	14.0087	0.515668
$739$	−42.2560	−1.55441	−0.777206	−	0.629246i	$-0.783364\pi$
$739$	−42.2560	−1.55441	−0.777206	+	0.629246i	$0.783364\pi$
$740$	−11.9540	−0.439439
$741$	−1.16150	−0.0426687
$742$	−21.7266	−0.797609
$743$	−45.2132	−1.65871	−0.829356	−	0.558721i	$-0.811292\pi$
$743$	−45.2132	−1.65871	−0.829356	+	0.558721i	$0.811292\pi$
$744$	102.977	3.77530
$745$	25.0940	0.919374
$746$	−52.5260	−1.92311
$747$	5.22613	0.191214
$748$	−25.3457	−0.926730
$749$	14.9338	0.545668
$750$	−2.87676	−0.105044
$751$	−39.2731	−1.43309	−0.716547	−	0.697539i	$-0.754279\pi$
$751$	−39.2731	−1.43309	−0.716547	+	0.697539i	$0.754279\pi$
$752$	−54.3444	−1.98174
$753$	−9.35384	−0.340873
$754$	−56.4793	−2.05685
$755$	0.247653	0.00901302
$756$	−13.2921	−0.483430
$757$	25.7317	0.935234	0.467617	−	0.883931i	$-0.345113\pi$
$757$	25.7317	0.935234	0.467617	+	0.883931i	$0.345113\pi$
$758$	57.0873	2.07350
$759$	−0.00726824	−0.000263820 0
$760$	−4.33042	−0.157081
$761$	−6.65903	−0.241390	−0.120695	−	0.992690i	$-0.538512\pi$
$761$	−6.65903	−0.241390	−0.120695	+	0.992690i	$0.538512\pi$
$762$	26.4324	0.957545
$763$	−9.67161	−0.350136
$764$	64.0031	2.31555
$765$	6.92861	0.250504
$766$	−90.6463	−3.27518
$767$	4.47332	0.161522
$768$	−64.7369	−2.33599
$769$	−38.3908	−1.38441	−0.692204	−	0.721702i	$-0.743360\pi$
$769$	−38.3908	−1.38441	−0.692204	+	0.721702i	$0.743360\pi$
$770$	18.0129	0.649139
$771$	16.5489	0.595995
$772$	−91.0400	−3.27660
$773$	−47.6369	−1.71338	−0.856690	−	0.515831i	$-0.827483\pi$
$773$	−47.6369	−1.71338	−0.856690	+	0.515831i	$0.827483\pi$
$774$	2.89530	0.104069
$775$	−46.6148	−1.67445
$776$	−37.1194	−1.33251
$777$	−1.39478	−0.0500374
$778$	−48.5477	−1.74052
$779$	−1.27890	−0.0458214
$780$	−66.3605	−2.37609
$781$	−10.2941	−0.368352
$782$	−0.00660949	−0.000236355 0
$783$	36.6443	1.30956
$784$	−36.3695	−1.29891
$785$	−41.2650	−1.47281
$786$	14.5822	0.520131
$787$	−7.55189	−0.269196	−0.134598	−	0.990900i	$-0.542974\pi$
$787$	−7.55189	−0.269196	−0.134598	+	0.990900i	$0.542974\pi$
$788$	18.9352	0.674537
$789$	−24.6711	−0.878315
$790$	−65.1557	−2.31813
$791$	−11.3643	−0.404067
$792$	16.9228	0.601327
$793$	−3.26987	−0.116117
$794$	−50.8763	−1.80553
$795$	70.3759	2.49598
$796$	104.176	3.69243
$797$	4.73439	0.167701	0.0838503	−	0.996478i	$-0.473278\pi$
$797$	4.73439	0.167701	0.0838503	+	0.996478i	$0.473278\pi$
$798$	−0.950062	−0.0336318
$799$	−19.8250	−0.701359
$800$	−15.0662	−0.532670
$801$	−19.0046	−0.671496
$802$	−66.4019	−2.34473
$803$	−33.2788	−1.17438
$804$	−80.2364	−2.82972
$805$	0.00319940	0.000112764 0
$806$	−54.5801	−1.92250
$807$	7.35799	0.259014
$808$	11.2347	0.395235
$809$	8.95744	0.314927	0.157463	−	0.987525i	$-0.449668\pi$
$809$	8.95744	0.314927	0.157463	+	0.987525i	$0.449668\pi$
$810$	88.2230	3.09984
$811$	27.3832	0.961555	0.480777	−	0.876843i	$-0.340355\pi$
$811$	27.3832	0.961555	0.480777	+	0.876843i	$0.340355\pi$
$812$	−31.4663	−1.10425
$813$	−51.7257	−1.81410
$814$	−6.26143	−0.219463
$815$	5.71031	0.200023
$816$	23.8785	0.835914
$817$	−0.264321	−0.00924744
$818$	−10.3996	−0.363614
$819$	−1.99802	−0.0698166
$820$	−73.0682	−2.55165
$821$	16.7438	0.584364	0.292182	−	0.956363i	$-0.405619\pi$
$821$	16.7438	0.584364	0.292182	+	0.956363i	$0.405619\pi$
$822$	−109.335	−3.81350
$823$	−40.4236	−1.40908	−0.704540	−	0.709665i	$-0.748846\pi$
$823$	−40.4236	−1.40908	−0.704540	+	0.709665i	$0.748846\pi$
$824$	48.0711	1.67464
$825$	−29.6864	−1.03355
$826$	3.65901	0.127313
$827$	−29.1603	−1.01400	−0.507001	−	0.861945i	$-0.669246\pi$
$827$	−29.1603	−1.01400	−0.507001	+	0.861945i	$0.669246\pi$
$828$	0.00565184	0.000196415 0
$829$	−42.6598	−1.48164	−0.740818	−	0.671706i	$-0.765562\pi$
$829$	−42.6598	−1.48164	−0.740818	+	0.671706i	$0.765562\pi$
$830$	−40.0210	−1.38915
$831$	60.4032	2.09536
$832$	9.98920	0.346313
$833$	−13.2677	−0.459699
$834$	−7.24188	−0.250766
$835$	−38.4025	−1.32897
$836$	−2.90498	−0.100471
$837$	35.4120	1.22402
$838$	40.8834	1.41229
$839$	−18.9427	−0.653976	−0.326988	−	0.945028i	$-0.606034\pi$
$839$	−18.9427	−0.653976	−0.326988	+	0.945028i	$0.606034\pi$
$840$	−28.8677	−0.996031
$841$	57.7476	1.99130
$842$	48.1860	1.66060
$843$	17.8730	0.615579
$844$	92.1108	3.17058
$845$	−22.7703	−0.783321
$846$	24.8893	0.855713
$847$	−2.27300	−0.0781012
$848$	62.5872	2.14925
$849$	−4.50897	−0.154747
$850$	−26.9958	−0.925950
$851$	−0.00111214	−3.81237e−5 0
$852$	31.0205	1.06275
$853$	−34.6981	−1.18804	−0.594020	−	0.804451i	$-0.702460\pi$
$853$	−34.6981	−1.18804	−0.594020	+	0.804451i	$0.702460\pi$
$854$	−2.67463	−0.0915241
$855$	0.794117	0.0271582
$856$	−107.440	−3.67223
$857$	54.0805	1.84735	0.923677	−	0.383172i	$-0.125168\pi$
$857$	54.0805	1.84735	0.923677	+	0.383172i	$0.125168\pi$
$858$	−34.7591	−1.18666
$859$	−52.1629	−1.77978	−0.889888	−	0.456178i	$-0.849218\pi$
$859$	−52.1629	−1.77978	−0.889888	+	0.456178i	$0.849218\pi$
$860$	−15.1016	−0.514961
$861$	−8.52549	−0.290548
$862$	−15.0375	−0.512180
$863$	52.3833	1.78315	0.891574	−	0.452876i	$-0.149602\pi$
$863$	52.3833	1.78315	0.891574	+	0.452876i	$0.149602\pi$
$864$	11.4454	0.389380
$865$	5.86609	0.199453
$866$	53.3171	1.81179
$867$	−25.4729	−0.865107
$868$	−30.4082	−1.03212
$869$	−23.2452	−0.788540
$870$	149.643	5.07336
$871$	22.6171	0.766351
$872$	69.5819	2.35634
$873$	6.80698	0.230381
$874$	−0.000757541 0	−2.56242e−5 0
$875$	0.451778	0.0152729
$876$	100.283	3.38825
$877$	−6.12184	−0.206720	−0.103360	−	0.994644i	$-0.532959\pi$
$877$	−6.12184	−0.206720	−0.103360	+	0.994644i	$0.532959\pi$
$878$	22.5917	0.762433
$879$	−33.8430	−1.14149
$880$	−51.8891	−1.74918
$881$	56.3363	1.89802	0.949009	−	0.315249i	$-0.102088\pi$
$881$	56.3363	1.89802	0.949009	+	0.315249i	$0.102088\pi$
$882$	16.6570	0.560869
$883$	8.55118	0.287770	0.143885	−	0.989594i	$-0.454040\pi$
$883$	8.55118	0.287770	0.143885	+	0.989594i	$0.454040\pi$
$884$	−21.5293	−0.724108
$885$	−11.8521	−0.398405
$886$	−71.0076	−2.38555
$887$	−37.0462	−1.24389	−0.621945	−	0.783061i	$-0.713657\pi$
$887$	−37.0462	−1.24389	−0.621945	+	0.783061i	$0.713657\pi$
$888$	10.0347	0.336741
$889$	−4.15105	−0.139222
$890$	145.535	4.87834
$891$	31.4748	1.05445
$892$	−10.8687	−0.363912
$893$	−2.27223	−0.0760372
$894$	−39.6086	−1.32471
$895$	−11.4450	−0.382563
$896$	12.7721	0.426685
$897$	−0.00617383	−0.000206138 0
$898$	−45.0689	−1.50397
$899$	83.8305	2.79591
$900$	23.0844	0.769480
$901$	22.8320	0.760645
$902$	−38.2725	−1.27434
$903$	−1.76203	−0.0586368
$904$	81.7598	2.71929
$905$	−46.9845	−1.56182
$906$	−0.390897	−0.0129867
$907$	40.2951	1.33798	0.668988	−	0.743273i	$-0.266728\pi$
$907$	40.2951	1.33798	0.668988	+	0.743273i	$0.266728\pi$
$908$	1.34911	0.0447717
$909$	−2.06023	−0.0683334
$910$	15.3006	0.507210
$911$	−46.7577	−1.54915	−0.774576	−	0.632481i	$-0.782036\pi$
$911$	−46.7577	−1.54915	−0.774576	+	0.632481i	$0.782036\pi$
$912$	2.73681	0.0906249
$913$	−14.2781	−0.472535
$914$	−5.33892	−0.176596
$915$	8.66356	0.286408
$916$	20.2889	0.670363
$917$	−2.29005	−0.0756242
$918$	20.5080	0.676866
$919$	−33.9069	−1.11848	−0.559242	−	0.829004i	$-0.688908\pi$
$919$	−33.9069	−1.11848	−0.559242	+	0.829004i	$0.688908\pi$
$920$	−0.0230179	−0.000758879 0
$921$	−27.7554	−0.914570
$922$	34.0794	1.12234
$923$	−8.74409	−0.287815
$924$	−19.3653	−0.637072
$925$	−4.54243	−0.149354
$926$	10.3589	0.340415
$927$	−8.81532	−0.289533
$928$	27.0946	0.889422
$929$	30.1936	0.990620	0.495310	−	0.868716i	$-0.335054\pi$
$929$	30.1936	0.990620	0.495310	+	0.868716i	$0.335054\pi$
$930$	144.611	4.74197
$931$	−1.52067	−0.0498379
$932$	−55.7608	−1.82651
$933$	17.7389	0.580745
$934$	19.9647	0.653264
$935$	−18.9293	−0.619055
$936$	14.3747	0.469851
$937$	−54.1007	−1.76739	−0.883696	−	0.468061i	$-0.844953\pi$
$937$	−54.1007	−1.76739	−0.883696	+	0.468061i	$0.844953\pi$
$938$	18.4999	0.604045
$939$	−37.1042	−1.21085
$940$	−129.820	−4.23428
$941$	−51.3561	−1.67416	−0.837081	−	0.547079i	$-0.815740\pi$
$941$	−51.3561	−1.67416	−0.837081	+	0.547079i	$0.815740\pi$
$942$	65.1329	2.12214
$943$	−0.00679788	−0.000221369 0
$944$	−10.5404	−0.343061
$945$	−9.92717	−0.322931
$946$	−7.91011	−0.257180
$947$	46.7235	1.51831	0.759154	−	0.650911i	$-0.225613\pi$
$947$	46.7235	1.51831	0.759154	+	0.650911i	$0.225613\pi$
$948$	70.0477	2.27504
$949$	−28.2679	−0.917614
$950$	−3.09411	−0.100386
$951$	−44.3326	−1.43758
$952$	−9.36554	−0.303539
$953$	25.8745	0.838158	0.419079	−	0.907950i	$-0.362353\pi$
$953$	25.8745	0.838158	0.419079	+	0.907950i	$0.362353\pi$
$954$	−28.6645	−0.928046
$955$	47.8004	1.54679
$956$	108.815	3.51932
$957$	53.3871	1.72576
$958$	−81.6710	−2.63867
$959$	17.1704	0.554462
$960$	−26.4665	−0.854202
$961$	50.0116	1.61328
$962$	−5.31862	−0.171479
$963$	19.7025	0.634903
$964$	64.2629	2.06977
$965$	−67.9929	−2.18877
$966$	−0.00504996	−0.000162480 0
$967$	−40.4484	−1.30073	−0.650367	−	0.759620i	$-0.725385\pi$
$967$	−40.4484	−1.30073	−0.650367	+	0.759620i	$0.725385\pi$
$968$	16.3530	0.525605
$969$	0.998399	0.0320732
$970$	−52.1270	−1.67370
$971$	0.0749970	0.00240677	0.00120338	−	0.999999i	$-0.499617\pi$
$971$	0.0749970	0.00240677	0.00120338	+	0.999999i	$0.499617\pi$
$972$	−44.4249	−1.42493
$973$	1.13729	0.0364600
$974$	−67.9733	−2.17801
$975$	−25.2164	−0.807572
$976$	7.70473	0.246622
$977$	−11.8737	−0.379873	−0.189937	−	0.981796i	$-0.560828\pi$
$977$	−11.8737	−0.379873	−0.189937	+	0.981796i	$0.560828\pi$
$978$	−9.01319	−0.288210
$979$	51.9217	1.65942
$980$	−86.8812	−2.77532
$981$	−12.7600	−0.407395
$982$	−51.0405	−1.62877
$983$	43.1551	1.37643	0.688217	−	0.725505i	$-0.258394\pi$
$983$	43.1551	1.37643	0.688217	+	0.725505i	$0.258394\pi$
$984$	61.3362	1.95533
$985$	14.1416	0.450590
$986$	48.5484	1.54610
$987$	−15.1473	−0.482143
$988$	−2.46756	−0.0785036
$989$	−0.00140498	−4.46756e−5 0
$990$	23.7648	0.755296
$991$	33.6711	1.06960	0.534798	−	0.844980i	$-0.320388\pi$
$991$	33.6711	1.06960	0.534798	+	0.844980i	$0.320388\pi$
$992$	26.1835	0.831326
$993$	−36.9580	−1.17283
$994$	−7.15234	−0.226859
$995$	77.8036	2.46654
$996$	43.0258	1.36332
$997$	32.1354	1.01774	0.508870	−	0.860844i	$-0.330064\pi$
$997$	32.1354	1.01774	0.508870	+	0.860844i	$0.330064\pi$
$998$	32.0678	1.01509
$999$	3.45077	0.109177

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6047.2.a.a.1.15	✓	217

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6047.2.a.a.1.15	✓	217	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:	\( N \)	\(=\)	\( 6047 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.50438 q^{2} +2.01082 q^{3} +4.27191 q^{4} +3.19046 q^{5} -5.03585 q^{6} +0.790850 q^{7} -5.68973 q^{8} +1.04339 q^{9} +O(q^{10})\)	\(q-2.50438 q^{2} +2.01082 q^{3} +4.27191 q^{4} +3.19046 q^{5} -5.03585 q^{6} +0.790850 q^{7} -5.68973 q^{8} +1.04339 q^{9} -7.99013 q^{10} -2.85059 q^{11} +8.59004 q^{12} -2.42137 q^{13} -1.98059 q^{14} +6.41544 q^{15} +5.70542 q^{16} +2.08136 q^{17} -2.61304 q^{18} +0.238553 q^{19} +13.6294 q^{20} +1.59026 q^{21} +7.13896 q^{22} +0.00126801 q^{23} -11.4410 q^{24} +5.17905 q^{25} +6.06402 q^{26} -3.93439 q^{27} +3.37844 q^{28} -9.31384 q^{29} -16.0667 q^{30} -9.00065 q^{31} -2.90906 q^{32} -5.73202 q^{33} -5.21251 q^{34} +2.52318 q^{35} +4.45726 q^{36} -0.877078 q^{37} -0.597427 q^{38} -4.86893 q^{39} -18.1529 q^{40} -5.36108 q^{41} -3.98260 q^{42} -1.10802 q^{43} -12.1775 q^{44} +3.32889 q^{45} -0.00317557 q^{46} -9.52505 q^{47} +11.4726 q^{48} -6.37456 q^{49} -12.9703 q^{50} +4.18523 q^{51} -10.3439 q^{52} +10.9698 q^{53} +9.85320 q^{54} -9.09471 q^{55} -4.49973 q^{56} +0.479687 q^{57} +23.3254 q^{58} -1.84744 q^{59} +27.4062 q^{60} +1.35042 q^{61} +22.5410 q^{62} +0.825164 q^{63} -4.12544 q^{64} -7.72528 q^{65} +14.3552 q^{66} -9.34063 q^{67} +8.89138 q^{68} +0.00254973 q^{69} -6.31899 q^{70} +3.61122 q^{71} -5.93660 q^{72} +11.6743 q^{73} +2.19654 q^{74} +10.4141 q^{75} +1.01908 q^{76} -2.25439 q^{77} +12.1936 q^{78} +8.15452 q^{79} +18.2029 q^{80} -11.0415 q^{81} +13.4262 q^{82} +5.00880 q^{83} +6.79344 q^{84} +6.64049 q^{85} +2.77490 q^{86} -18.7284 q^{87} +16.2191 q^{88} -18.2144 q^{89} -8.33681 q^{90} -1.91494 q^{91} +0.00541681 q^{92} -18.0987 q^{93} +23.8543 q^{94} +0.761094 q^{95} -5.84960 q^{96} +6.52392 q^{97} +15.9643 q^{98} -2.97427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 217 q - 20 q^{2} - 27 q^{3} + 184 q^{4} - 19 q^{5} - 17 q^{6} - 48 q^{7} - 57 q^{8} + 152 q^{9}+O(q^{10}) \) 217 * q - 20 * q^2 - 27 * q^3 + 184 * q^4 - 19 * q^5 - 17 * q^6 - 48 * q^7 - 57 * q^8 + 152 * q^9	\( 217 q - 20 q^{2} - 27 q^{3} + 184 q^{4} - 19 q^{5} - 17 q^{6} - 48 q^{7} - 57 q^{8} + 152 q^{9} - 46 q^{10} - 32 q^{11} - 72 q^{12} - 80 q^{13} - 22 q^{14} - 43 q^{15} + 122 q^{16} - 61 q^{17} - 88 q^{18} - 43 q^{19} - 41 q^{20} - 61 q^{21} - 93 q^{22} - 60 q^{23} - 41 q^{24} + 26 q^{25} - 9 q^{26} - 93 q^{27} - 126 q^{28} - 47 q^{29} - 36 q^{30} - 100 q^{31} - 114 q^{32} - 133 q^{33} - 75 q^{34} - 37 q^{35} + 75 q^{36} - 264 q^{37} - 35 q^{38} - 47 q^{39} - 118 q^{40} - 72 q^{41} - 64 q^{42} - 107 q^{43} - 59 q^{44} - 69 q^{45} - 111 q^{46} - 54 q^{47} - 135 q^{48} + 33 q^{49} - 42 q^{50} - 26 q^{51} - 173 q^{52} - 103 q^{53} - 28 q^{54} - 78 q^{55} - 44 q^{56} - 205 q^{57} - 189 q^{58} - 38 q^{59} - 105 q^{60} - 108 q^{61} - 14 q^{62} - 116 q^{63} + 39 q^{64} - 146 q^{65} + 5 q^{66} - 206 q^{67} - 62 q^{68} - 55 q^{69} - 125 q^{70} - 78 q^{71} - 225 q^{72} - 326 q^{73} + 3 q^{74} - 95 q^{75} - 84 q^{76} - 79 q^{77} - 86 q^{78} - 117 q^{79} - 39 q^{80} + q^{81} - 96 q^{82} - 23 q^{83} - 57 q^{84} - 224 q^{85} - 7 q^{86} - 45 q^{87} - 250 q^{88} - 104 q^{89} - 36 q^{90} - 96 q^{91} - 137 q^{92} - 155 q^{93} - 48 q^{94} - 38 q^{95} - 33 q^{96} - 447 q^{97} - 46 q^{98} - 94 q^{99}+O(q^{100}) \) 217 * q - 20 * q^2 - 27 * q^3 + 184 * q^4 - 19 * q^5 - 17 * q^6 - 48 * q^7 - 57 * q^8 + 152 * q^9 - 46 * q^10 - 32 * q^11 - 72 * q^12 - 80 * q^13 - 22 * q^14 - 43 * q^15 + 122 * q^16 - 61 * q^17 - 88 * q^18 - 43 * q^19 - 41 * q^20 - 61 * q^21 - 93 * q^22 - 60 * q^23 - 41 * q^24 + 26 * q^25 - 9 * q^26 - 93 * q^27 - 126 * q^28 - 47 * q^29 - 36 * q^30 - 100 * q^31 - 114 * q^32 - 133 * q^33 - 75 * q^34 - 37 * q^35 + 75 * q^36 - 264 * q^37 - 35 * q^38 - 47 * q^39 - 118 * q^40 - 72 * q^41 - 64 * q^42 - 107 * q^43 - 59 * q^44 - 69 * q^45 - 111 * q^46 - 54 * q^47 - 135 * q^48 + 33 * q^49 - 42 * q^50 - 26 * q^51 - 173 * q^52 - 103 * q^53 - 28 * q^54 - 78 * q^55 - 44 * q^56 - 205 * q^57 - 189 * q^58 - 38 * q^59 - 105 * q^60 - 108 * q^61 - 14 * q^62 - 116 * q^63 + 39 * q^64 - 146 * q^65 + 5 * q^66 - 206 * q^67 - 62 * q^68 - 55 * q^69 - 125 * q^70 - 78 * q^71 - 225 * q^72 - 326 * q^73 + 3 * q^74 - 95 * q^75 - 84 * q^76 - 79 * q^77 - 86 * q^78 - 117 * q^79 - 39 * q^80 + q^81 - 96 * q^82 - 23 * q^83 - 57 * q^84 - 224 * q^85 - 7 * q^86 - 45 * q^87 - 250 * q^88 - 104 * q^89 - 36 * q^90 - 96 * q^91 - 137 * q^92 - 155 * q^93 - 48 * q^94 - 38 * q^95 - 33 * q^96 - 447 * q^97 - 46 * q^98 - 94 * q^99

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