LMFDB - Embedded newform 847.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,2,Mod(1,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, 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("847.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	847.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:	$6.76332905120$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2525.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 5 $ x^4 - 2x^3 - 4x^2 + 5*x + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.77748$ of defining polynomial
Character	$\chi$	$=$	847.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.77748 q^{2} +0.618034 q^{3} +1.15945 q^{4} -2.77748 q^{5} -1.09855 q^{6} +1.00000 q^{7} +1.49406 q^{8} -2.61803 q^{9} +O(q^{10})$	\(q-1.77748 q^{2} +0.618034 q^{3} +1.15945 q^{4} -2.77748 q^{5} -1.09855 q^{6} +1.00000 q^{7} +1.49406 q^{8} -2.61803 q^{9} +4.93693 q^{10} +0.716580 q^{12} +4.29348 q^{13} -1.77748 q^{14} -1.71658 q^{15} -4.97458 q^{16} +2.75556 q^{17} +4.65351 q^{18} -1.93910 q^{19} -3.22035 q^{20} +0.618034 q^{21} +4.37009 q^{23} +0.923382 q^{24} +2.71442 q^{25} -7.63159 q^{26} -3.47214 q^{27} +1.15945 q^{28} -8.62809 q^{29} +3.05119 q^{30} -0.200588 q^{31} +5.85410 q^{32} -4.89796 q^{34} -2.77748 q^{35} -3.03548 q^{36} +1.03548 q^{37} +3.44671 q^{38} +2.65351 q^{39} -4.14974 q^{40} -9.60616 q^{41} -1.09855 q^{42} +4.70820 q^{43} +7.27155 q^{45} -7.76777 q^{46} -13.0455 q^{47} -3.07446 q^{48} +1.00000 q^{49} -4.82484 q^{50} +1.70303 q^{51} +4.97807 q^{52} -3.90012 q^{53} +6.17167 q^{54} +1.49406 q^{56} -1.19843 q^{57} +15.3363 q^{58} -8.55713 q^{59} -1.99029 q^{60} +0.988609 q^{61} +0.356542 q^{62} -2.61803 q^{63} -0.456423 q^{64} -11.9251 q^{65} -5.41745 q^{67} +3.19493 q^{68} +2.70087 q^{69} +4.93693 q^{70} -2.01705 q^{71} -3.91151 q^{72} +9.97108 q^{73} -1.84055 q^{74} +1.67760 q^{75} -2.24828 q^{76} -4.71658 q^{78} -6.29348 q^{79} +13.8168 q^{80} +5.70820 q^{81} +17.0748 q^{82} +1.72146 q^{83} +0.716580 q^{84} -7.65351 q^{85} -8.36876 q^{86} -5.33245 q^{87} -15.3035 q^{89} -12.9251 q^{90} +4.29348 q^{91} +5.06691 q^{92} -0.123970 q^{93} +23.1882 q^{94} +5.38581 q^{95} +3.61803 q^{96} +11.6162 q^{97} -1.77748 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{5} + q^{6} + 4 q^{7} - 9 q^{8} - 6 q^{9}+O(q^{10}) $ 4 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 6 * q^5 + q^6 + 4 * q^7 - 9 * q^8 - 6 * q^9	$ 4 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{5} + q^{6} + 4 q^{7} - 9 q^{8} - 6 q^{9} + 14 q^{10} - 7 q^{12} - 2 q^{14} + 3 q^{15} - 4 q^{16} + 3 q^{17} + 3 q^{18} - 3 q^{19} - 17 q^{20} - 2 q^{21} - 8 q^{23} + 12 q^{24} - 12 q^{26} + 4 q^{27} + 4 q^{28} - 3 q^{29} - 12 q^{30} - 3 q^{31} + 10 q^{32} - 12 q^{34} - 6 q^{35} - q^{36} - 7 q^{37} - 20 q^{38} - 5 q^{39} + 13 q^{40} - 4 q^{41} + q^{42} - 8 q^{43} + 9 q^{45} + 3 q^{46} - 14 q^{47} - 3 q^{48} + 4 q^{49} - 33 q^{50} + 11 q^{51} + 17 q^{52} - 9 q^{53} - 2 q^{54} - 9 q^{56} - 6 q^{57} + 3 q^{58} - 25 q^{59} + 21 q^{60} + 19 q^{61} - 10 q^{62} - 6 q^{63} + 3 q^{64} - 12 q^{65} - 15 q^{67} + q^{68} + 14 q^{69} + 14 q^{70} - 7 q^{71} + 6 q^{72} + 11 q^{73} - 8 q^{74} - 5 q^{75} + 26 q^{76} - 9 q^{78} - 8 q^{79} - 4 q^{80} - 4 q^{81} + 3 q^{82} + q^{83} - 7 q^{84} - 15 q^{85} + 4 q^{86} - 6 q^{87} - 17 q^{89} - 16 q^{90} - 17 q^{92} - 11 q^{93} + 20 q^{94} - 17 q^{95} + 10 q^{96} - 15 q^{97} - 2 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 6 * q^5 + q^6 + 4 * q^7 - 9 * q^8 - 6 * q^9 + 14 * q^10 - 7 * q^12 - 2 * q^14 + 3 * q^15 - 4 * q^16 + 3 * q^17 + 3 * q^18 - 3 * q^19 - 17 * q^20 - 2 * q^21 - 8 * q^23 + 12 * q^24 - 12 * q^26 + 4 * q^27 + 4 * q^28 - 3 * q^29 - 12 * q^30 - 3 * q^31 + 10 * q^32 - 12 * q^34 - 6 * q^35 - q^36 - 7 * q^37 - 20 * q^38 - 5 * q^39 + 13 * q^40 - 4 * q^41 + q^42 - 8 * q^43 + 9 * q^45 + 3 * q^46 - 14 * q^47 - 3 * q^48 + 4 * q^49 - 33 * q^50 + 11 * q^51 + 17 * q^52 - 9 * q^53 - 2 * q^54 - 9 * q^56 - 6 * q^57 + 3 * q^58 - 25 * q^59 + 21 * q^60 + 19 * q^61 - 10 * q^62 - 6 * q^63 + 3 * q^64 - 12 * q^65 - 15 * q^67 + q^68 + 14 * q^69 + 14 * q^70 - 7 * q^71 + 6 * q^72 + 11 * q^73 - 8 * q^74 - 5 * q^75 + 26 * q^76 - 9 * q^78 - 8 * q^79 - 4 * q^80 - 4 * q^81 + 3 * q^82 + q^83 - 7 * q^84 - 15 * q^85 + 4 * q^86 - 6 * q^87 - 17 * q^89 - 16 * q^90 - 17 * q^92 - 11 * q^93 + 20 * q^94 - 17 * q^95 + 10 * q^96 - 15 * q^97 - 2 * q^98

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 $

$2$	−1.77748	−1.25687	−0.628436	−	0.777862i	$-0.716304\pi$
$2$	−1.77748	−1.25687	−0.628436	+	0.777862i	$0.716304\pi$
$3$	0.618034	0.356822	0.178411	−	0.983956i	$-0.442904\pi$
$3$	0.618034	0.356822	0.178411	+	0.983956i	$0.442904\pi$
$4$	1.15945	0.579725
$5$	−2.77748	−1.24213	−0.621064	−	0.783760i	$-0.713299\pi$
$5$	−2.77748	−1.24213	−0.621064	+	0.783760i	$0.713299\pi$
$6$	−1.09855	−0.448479
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.49406	0.528231
$9$	−2.61803	−0.872678
$10$	4.93693	1.56120
$11$	0	0
$11$	0	0
$12$	0.716580	0.206859
$13$	4.29348	1.19080	0.595398	−	0.803431i	$-0.296994\pi$
$13$	4.29348	1.19080	0.595398	+	0.803431i	$0.296994\pi$
$14$	−1.77748	−0.475053
$15$	−1.71658	−0.443219
$16$	−4.97458	−1.24364
$17$	2.75556	0.668321	0.334160	−	0.942516i	$-0.391547\pi$
$17$	2.75556	0.668321	0.334160	+	0.942516i	$0.391547\pi$
$18$	4.65351	1.09684
$19$	−1.93910	−0.444859	−0.222429	−	0.974949i	$-0.571399\pi$
$19$	−1.93910	−0.444859	−0.222429	+	0.974949i	$0.571399\pi$
$20$	−3.22035	−0.720093
$21$	0.618034	0.134866
$22$	0	0
$23$	4.37009	0.911228	0.455614	−	0.890178i	$-0.349420\pi$
$23$	4.37009	0.911228	0.455614	+	0.890178i	$0.349420\pi$
$24$	0.923382	0.188485
$25$	2.71442	0.542884
$26$	−7.63159	−1.49668
$27$	−3.47214	−0.668213
$28$	1.15945	0.219116
$29$	−8.62809	−1.60220	−0.801098	−	0.598533i	$-0.795750\pi$
$29$	−8.62809	−1.60220	−0.801098	+	0.598533i	$0.795750\pi$
$30$	3.05119	0.557069
$31$	−0.200588	−0.0360266	−0.0180133	−	0.999838i	$-0.505734\pi$
$31$	−0.200588	−0.0360266	−0.0180133	+	0.999838i	$0.505734\pi$
$32$	5.85410	1.03487
$33$	0	0
$34$	−4.89796	−0.839993
$35$	−2.77748	−0.469481
$36$	−3.03548	−0.505913
$37$	1.03548	0.170232	0.0851159	−	0.996371i	$-0.472874\pi$
$37$	1.03548	0.170232	0.0851159	+	0.996371i	$0.472874\pi$
$38$	3.44671	0.559130
$39$	2.65351	0.424902
$40$	−4.14974	−0.656131
$41$	−9.60616	−1.50023	−0.750115	−	0.661307i	$-0.770002\pi$
$41$	−9.60616	−1.50023	−0.750115	+	0.661307i	$0.770002\pi$
$42$	−1.09855	−0.169509
$43$	4.70820	0.717994	0.358997	−	0.933339i	$-0.383119\pi$
$43$	4.70820	0.717994	0.358997	+	0.933339i	$0.383119\pi$
$44$	0	0
$45$	7.27155	1.08398
$46$	−7.76777	−1.14530
$47$	−13.0455	−1.90289	−0.951443	−	0.307823i	$-0.900399\pi$
$47$	−13.0455	−1.90289	−0.951443	+	0.307823i	$0.900399\pi$
$48$	−3.07446	−0.443760
$49$	1.00000	0.142857
$50$	−4.82484	−0.682335
$51$	1.70303	0.238472
$52$	4.97807	0.690334
$53$	−3.90012	−0.535723	−0.267861	−	0.963457i	$-0.586317\pi$
$53$	−3.90012	−0.535723	−0.267861	+	0.963457i	$0.586317\pi$
$54$	6.17167	0.839857
$55$	0	0
$56$	1.49406	0.199653
$57$	−1.19843	−0.158736
$58$	15.3363	2.01375
$59$	−8.55713	−1.11404	−0.557022	−	0.830498i	$-0.688056\pi$
$59$	−8.55713	−1.11404	−0.557022	+	0.830498i	$0.688056\pi$
$60$	−1.99029	−0.256945
$61$	0.988609	0.126578	0.0632892	−	0.997995i	$-0.479841\pi$
$61$	0.988609	0.126578	0.0632892	+	0.997995i	$0.479841\pi$
$62$	0.356542	0.0452808
$63$	−2.61803	−0.329841
$64$	−0.456423	−0.0570529
$65$	−11.9251	−1.47912
$66$	0	0
$67$	−5.41745	−0.661846	−0.330923	−	0.943658i	$-0.607360\pi$
$67$	−5.41745	−0.661846	−0.330923	+	0.943658i	$0.607360\pi$
$68$	3.19493	0.387442
$69$	2.70087	0.325146
$70$	4.93693	0.590077
$71$	−2.01705	−0.239380	−0.119690	−	0.992811i	$-0.538190\pi$
$71$	−2.01705	−0.239380	−0.119690	+	0.992811i	$0.538190\pi$
$72$	−3.91151	−0.460976
$73$	9.97108	1.16703	0.583513	−	0.812104i	$-0.301678\pi$
$73$	9.97108	1.16703	0.583513	+	0.812104i	$0.301678\pi$
$74$	−1.84055	−0.213960
$75$	1.67760	0.193713
$76$	−2.24828	−0.257896
$77$	0	0
$78$	−4.71658	−0.534047
$79$	−6.29348	−0.708071	−0.354036	−	0.935232i	$-0.615191\pi$
$79$	−6.29348	−0.708071	−0.354036	+	0.935232i	$0.615191\pi$
$80$	13.8168	1.54477
$81$	5.70820	0.634245
$82$	17.0748	1.88560
$83$	1.72146	0.188955	0.0944773	−	0.995527i	$-0.469882\pi$
$83$	1.72146	0.188955	0.0944773	+	0.995527i	$0.469882\pi$
$84$	0.716580	0.0781853
$85$	−7.65351	−0.830140
$86$	−8.36876	−0.902426
$87$	−5.33245	−0.571699
$88$	0	0
$89$	−15.3035	−1.62217	−0.811086	−	0.584928i	$-0.801123\pi$
$89$	−15.3035	−1.62217	−0.811086	+	0.584928i	$0.801123\pi$
$90$	−12.9251	−1.36242
$91$	4.29348	0.450079
$92$	5.06691	0.528262
$93$	−0.123970	−0.0128551
$94$	23.1882	2.39168
$95$	5.38581	0.552572
$96$	3.61803	0.369264
$97$	11.6162	1.17945	0.589724	−	0.807605i	$-0.299236\pi$
$97$	11.6162	1.17945	0.589724	+	0.807605i	$0.299236\pi$
$98$	−1.77748	−0.179553
$99$	0	0
$100$	3.14723	0.314723
$101$	−3.41179	−0.339486	−0.169743	−	0.985488i	$-0.554294\pi$
$101$	−3.41179	−0.339486	−0.169743	+	0.985488i	$0.554294\pi$
$102$	−3.02710	−0.299728
$103$	−18.1826	−1.79158	−0.895791	−	0.444475i	$-0.853390\pi$
$103$	−18.1826	−1.79158	−0.895791	+	0.444475i	$0.853390\pi$
$104$	6.41473	0.629016
$105$	−1.71658	−0.167521
$106$	6.93240	0.673334
$107$	−3.24746	−0.313944	−0.156972	−	0.987603i	$-0.550173\pi$
$107$	−3.24746	−0.313944	−0.156972	+	0.987603i	$0.550173\pi$
$108$	−4.02577	−0.387380
$109$	−12.6912	−1.21559	−0.607796	−	0.794093i	$-0.707946\pi$
$109$	−12.6912	−1.21559	−0.607796	+	0.794093i	$0.707946\pi$
$110$	0	0
$111$	0.639962	0.0607425
$112$	−4.97458	−0.470053
$113$	−18.4480	−1.73545	−0.867723	−	0.497048i	$-0.834417\pi$
$113$	−18.4480	−1.73545	−0.867723	+	0.497048i	$0.834417\pi$
$114$	2.13018	0.199510
$115$	−12.1379	−1.13186
$116$	−10.0038	−0.928833
$117$	−11.2405	−1.03918
$118$	15.2102	1.40021
$119$	2.75556	0.252601
$120$	−2.56468	−0.234122
$121$	0	0
$122$	−1.75724	−0.159093
$123$	−5.93693	−0.535315
$124$	−0.232572	−0.0208855
$125$	6.34817	0.567797
$126$	4.65351	0.414568
$127$	−19.5093	−1.73117	−0.865585	−	0.500762i	$-0.833053\pi$
$127$	−19.5093	−1.73117	−0.865585	+	0.500762i	$0.833053\pi$
$128$	−10.8969	−0.963161
$129$	2.90983	0.256196
$130$	21.1966	1.85907
$131$	6.89796	0.602677	0.301339	−	0.953517i	$-0.402567\pi$
$131$	6.89796	0.602677	0.301339	+	0.953517i	$0.402567\pi$
$132$	0	0
$133$	−1.93910	−0.168141
$134$	9.62943	0.831856
$135$	9.64380	0.830006
$136$	4.11698	0.353028
$137$	−2.61070	−0.223047	−0.111523	−	0.993762i	$-0.535573\pi$
$137$	−2.61070	−0.223047	−0.111523	+	0.993762i	$0.535573\pi$
$138$	−4.80075	−0.408667
$139$	1.00134	0.0849322	0.0424661	−	0.999098i	$-0.486479\pi$
$139$	1.00134	0.0849322	0.0424661	+	0.999098i	$0.486479\pi$
$140$	−3.22035	−0.272170
$141$	−8.06258	−0.678992
$142$	3.58527	0.300869
$143$	0	0
$144$	13.0236	1.08530
$145$	23.9644	1.99013
$146$	−17.7234	−1.46680
$147$	0.618034	0.0509746
$148$	1.20059	0.0986877
$149$	14.8948	1.22023	0.610113	−	0.792314i	$-0.291124\pi$
$149$	14.8948	1.22023	0.610113	+	0.792314i	$0.291124\pi$
$150$	−2.98191	−0.243472
$151$	−15.8046	−1.28616	−0.643080	−	0.765799i	$-0.722344\pi$
$151$	−15.8046	−1.28616	−0.643080	+	0.765799i	$0.722344\pi$
$152$	−2.89713	−0.234988
$153$	−7.21414	−0.583229
$154$	0	0
$155$	0.557129	0.0447497
$156$	3.07662	0.246327
$157$	4.59123	0.366420	0.183210	−	0.983074i	$-0.441351\pi$
$157$	4.59123	0.366420	0.183210	+	0.983074i	$0.441351\pi$
$158$	11.1866	0.889955
$159$	−2.41041	−0.191158
$160$	−16.2597	−1.28544
$161$	4.37009	0.344412
$162$	−10.1462	−0.797164
$163$	8.03764	0.629557	0.314778	−	0.949165i	$-0.398070\pi$
$163$	8.03764	0.629557	0.314778	+	0.949165i	$0.398070\pi$
$164$	−11.1379	−0.869721
$165$	0	0
$166$	−3.05987	−0.237492
$167$	13.4069	1.03746	0.518729	−	0.854939i	$-0.326405\pi$
$167$	13.4069	1.03746	0.518729	+	0.854939i	$0.326405\pi$
$168$	0.923382	0.0712405
$169$	5.43394	0.417995
$170$	13.6040	1.04338
$171$	5.07662	0.388219
$172$	5.45893	0.416239
$173$	20.5821	1.56483	0.782413	−	0.622760i	$-0.213989\pi$
$173$	20.5821	1.56483	0.782413	+	0.622760i	$0.213989\pi$
$174$	9.47835	0.718552
$175$	2.71442	0.205191
$176$	0	0
$177$	−5.28860	−0.397515
$178$	27.2018	2.03886
$179$	−3.71520	−0.277687	−0.138843	−	0.990314i	$-0.544338\pi$
$179$	−3.71520	−0.277687	−0.138843	+	0.990314i	$0.544338\pi$
$180$	8.43100	0.628410
$181$	4.77183	0.354687	0.177344	−	0.984149i	$-0.443250\pi$
$181$	4.77183	0.354687	0.177344	+	0.984149i	$0.443250\pi$
$182$	−7.63159	−0.565691
$183$	0.610994	0.0451660
$184$	6.52920	0.481339
$185$	−2.87603	−0.211450
$186$	0.220355	0.0161572
$187$	0	0
$188$	−15.1257	−1.10315
$189$	−3.47214	−0.252561
$190$	−9.57319	−0.694512
$191$	−0.829158	−0.0599958	−0.0299979	−	0.999550i	$-0.509550\pi$
$191$	−0.829158	−0.0599958	−0.0299979	+	0.999550i	$0.509550\pi$
$192$	−0.282085	−0.0203577
$193$	6.73803	0.485014	0.242507	−	0.970150i	$-0.422030\pi$
$193$	6.73803	0.485014	0.242507	+	0.970150i	$0.422030\pi$
$194$	−20.6476	−1.48241
$195$	−7.37009	−0.527783
$196$	1.15945	0.0828179
$197$	10.9216	0.778129	0.389065	−	0.921210i	$-0.372798\pi$
$197$	10.9216	0.778129	0.389065	+	0.921210i	$0.372798\pi$
$198$	0	0
$199$	20.9746	1.48685	0.743424	−	0.668820i	$-0.233200\pi$
$199$	20.9746	1.48685	0.743424	+	0.668820i	$0.233200\pi$
$200$	4.05552	0.286768
$201$	−3.34817	−0.236161
$202$	6.06440	0.426690
$203$	−8.62809	−0.605573
$204$	1.97458	0.138248
$205$	26.6810	1.86348
$206$	32.3192	2.25179
$207$	−11.4411	−0.795208
$208$	−21.3582	−1.48093
$209$	0	0
$210$	3.05119	0.210552
$211$	6.02361	0.414682	0.207341	−	0.978269i	$-0.433519\pi$
$211$	6.02361	0.414682	0.207341	+	0.978269i	$0.433519\pi$
$212$	−4.52199	−0.310572
$213$	−1.24660	−0.0854159
$214$	5.77231	0.394587
$215$	−13.0770	−0.891841
$216$	−5.18759	−0.352971
$217$	−0.200588	−0.0136168
$218$	22.5583	1.52784
$219$	6.16247	0.416421
$220$	0	0
$221$	11.8309	0.795833
$222$	−1.13752	−0.0763455
$223$	−4.62507	−0.309718	−0.154859	−	0.987937i	$-0.549492\pi$
$223$	−4.62507	−0.309718	−0.154859	+	0.987937i	$0.549492\pi$
$224$	5.85410	0.391144
$225$	−7.10644	−0.473763
$226$	32.7911	2.18123
$227$	−17.3003	−1.14826	−0.574132	−	0.818763i	$-0.694660\pi$
$227$	−17.3003	−1.14826	−0.574132	+	0.818763i	$0.694660\pi$
$228$	−1.38952	−0.0920230
$229$	−4.42612	−0.292486	−0.146243	−	0.989249i	$-0.546718\pi$
$229$	−4.42612	−0.292486	−0.146243	+	0.989249i	$0.546718\pi$
$230$	21.5749	1.42260
$231$	0	0
$232$	−12.8909	−0.846330
$233$	10.5330	0.690042	0.345021	−	0.938595i	$-0.387872\pi$
$233$	10.5330	0.690042	0.345021	+	0.938595i	$0.387872\pi$
$234$	19.9798	1.30612
$235$	36.2338	2.36363
$236$	−9.92157	−0.645839
$237$	−3.88958	−0.252656
$238$	−4.89796	−0.317487
$239$	9.75646	0.631093	0.315546	−	0.948910i	$-0.397812\pi$
$239$	9.75646	0.631093	0.315546	+	0.948910i	$0.397812\pi$
$240$	8.53926	0.551207
$241$	−12.5501	−0.808422	−0.404211	−	0.914666i	$-0.632454\pi$
$241$	−12.5501	−0.808422	−0.404211	+	0.914666i	$0.632454\pi$
$242$	0	0
$243$	13.9443	0.894525
$244$	1.14624	0.0733807
$245$	−2.77748	−0.177447
$246$	10.5528	0.672822
$247$	−8.32546	−0.529736
$248$	−0.299691	−0.0190304
$249$	1.06392	0.0674232
$250$	−11.2838	−0.713648
$251$	−10.9912	−0.693758	−0.346879	−	0.937910i	$-0.612759\pi$
$251$	−10.9912	−0.693758	−0.346879	+	0.937910i	$0.612759\pi$
$252$	−3.03548	−0.191217
$253$	0	0
$254$	34.6775	2.17586
$255$	−4.73013	−0.296212
$256$	20.2819	1.26762
$257$	5.33280	0.332651	0.166325	−	0.986071i	$-0.446810\pi$
$257$	5.33280	0.332651	0.166325	+	0.986071i	$0.446810\pi$
$258$	−5.17218	−0.322006
$259$	1.03548	0.0643416
$260$	−13.8265	−0.857484
$261$	22.5886	1.39820
$262$	−12.2610	−0.757488
$263$	8.18034	0.504421	0.252211	−	0.967672i	$-0.418842\pi$
$263$	8.18034	0.504421	0.252211	+	0.967672i	$0.418842\pi$
$264$	0	0
$265$	10.8325	0.665436
$266$	3.44671	0.211331
$267$	−9.45810	−0.578826
$268$	−6.28126	−0.383689
$269$	12.8439	0.783107	0.391554	−	0.920155i	$-0.371938\pi$
$269$	12.8439	0.783107	0.391554	+	0.920155i	$0.371938\pi$
$270$	−17.1417	−1.04321
$271$	22.0472	1.33927	0.669636	−	0.742689i	$-0.266450\pi$
$271$	22.0472	1.33927	0.669636	+	0.742689i	$0.266450\pi$
$272$	−13.7077	−0.831153
$273$	2.65351	0.160598
$274$	4.64047	0.280341
$275$	0	0
$276$	3.13152	0.188495
$277$	−24.0770	−1.44664	−0.723322	−	0.690511i	$-0.757386\pi$
$277$	−24.0770	−1.44664	−0.723322	+	0.690511i	$0.757386\pi$
$278$	−1.77986	−0.106749
$279$	0.525146	0.0314396
$280$	−4.14974	−0.247994
$281$	−15.4418	−0.921182	−0.460591	−	0.887612i	$-0.652363\pi$
$281$	−15.4418	−0.921182	−0.460591	+	0.887612i	$0.652363\pi$
$282$	14.3311	0.853406
$283$	3.35732	0.199572	0.0997860	−	0.995009i	$-0.468184\pi$
$283$	3.35732	0.199572	0.0997860	+	0.995009i	$0.468184\pi$
$284$	−2.33867	−0.138774
$285$	3.32861	0.197170
$286$	0	0
$287$	−9.60616	−0.567034
$288$	−15.3262	−0.903107
$289$	−9.40691	−0.553348
$290$	−42.5963	−2.50134
$291$	7.17922	0.420853
$292$	11.5610	0.676555
$293$	−6.83705	−0.399425	−0.199712	−	0.979855i	$-0.564001\pi$
$293$	−6.83705	−0.399425	−0.199712	+	0.979855i	$0.564001\pi$
$294$	−1.09855	−0.0640685
$295$	23.7673	1.38379
$296$	1.54707	0.0899218
$297$	0	0
$298$	−26.4752	−1.53367
$299$	18.7629	1.08509
$300$	1.94510	0.112300
$301$	4.70820	0.271376
$302$	28.0924	1.61654
$303$	−2.10860	−0.121136
$304$	9.64618	0.553246
$305$	−2.74585	−0.157227
$306$	12.8230	0.733043
$307$	11.7970	0.673293	0.336646	−	0.941631i	$-0.390707\pi$
$307$	11.7970	0.673293	0.336646	+	0.941631i	$0.390707\pi$
$308$	0	0
$309$	−11.2375	−0.639276
$310$	−0.990289	−0.0562446
$311$	25.3139	1.43542	0.717709	−	0.696343i	$-0.245191\pi$
$311$	25.3139	1.43542	0.717709	+	0.696343i	$0.245191\pi$
$312$	3.96452	0.224447
$313$	−19.6543	−1.11093	−0.555464	−	0.831540i	$-0.687460\pi$
$313$	−19.6543	−1.11093	−0.555464	+	0.831540i	$0.687460\pi$
$314$	−8.16083	−0.460542
$315$	7.27155	0.409705
$316$	−7.29697	−0.410487
$317$	3.61056	0.202789	0.101395	−	0.994846i	$-0.467670\pi$
$317$	3.61056	0.202789	0.101395	+	0.994846i	$0.467670\pi$
$318$	4.28446	0.240261
$319$	0	0
$320$	1.26771	0.0708670
$321$	−2.00704	−0.112022
$322$	−7.76777	−0.432881
$323$	−5.34329	−0.297308
$324$	6.61838	0.367688
$325$	11.6543	0.646464
$326$	−14.2868	−0.791272
$327$	−7.84357	−0.433750
$328$	−14.3522	−0.792469
$329$	−13.0455	−0.719224
$330$	0	0
$331$	−26.5335	−1.45841	−0.729205	−	0.684295i	$-0.760110\pi$
$331$	−26.5335	−1.45841	−0.729205	+	0.684295i	$0.760110\pi$
$332$	1.99595	0.109542
$333$	−2.71092	−0.148558
$334$	−23.8306	−1.30395
$335$	15.0469	0.822098
$336$	−3.07446	−0.167725
$337$	0.685979	0.0373676	0.0186838	−	0.999825i	$-0.494052\pi$
$337$	0.685979	0.0373676	0.0186838	+	0.999825i	$0.494052\pi$
$338$	−9.65874	−0.525366
$339$	−11.4015	−0.619246
$340$	−8.87387	−0.481253
$341$	0	0
$342$	−9.02361	−0.487941
$343$	1.00000	0.0539949
$344$	7.03436	0.379267
$345$	−7.50161	−0.403873
$346$	−36.5843	−1.96678
$347$	−21.5055	−1.15447	−0.577237	−	0.816577i	$-0.695869\pi$
$347$	−21.5055	−1.15447	−0.577237	+	0.816577i	$0.695869\pi$
$348$	−6.18271	−0.331428
$349$	−19.4429	−1.04075	−0.520377	−	0.853937i	$-0.674208\pi$
$349$	−19.4429	−1.04075	−0.520377	+	0.853937i	$0.674208\pi$
$350$	−4.82484	−0.257898
$351$	−14.9075	−0.795705
$352$	0	0
$353$	−20.9307	−1.11403	−0.557015	−	0.830502i	$-0.688053\pi$
$353$	−20.9307	−1.11403	−0.557015	+	0.830502i	$0.688053\pi$
$354$	9.40040	0.499625
$355$	5.60232	0.297340
$356$	−17.7437	−0.940413
$357$	1.70303	0.0901338
$358$	6.60370	0.349017
$359$	−9.77127	−0.515708	−0.257854	−	0.966184i	$-0.583015\pi$
$359$	−9.77127	−0.515708	−0.257854	+	0.966184i	$0.583015\pi$
$360$	10.8642	0.572591
$361$	−15.2399	−0.802100
$362$	−8.48185	−0.445796
$363$	0	0
$364$	4.97807	0.260922
$365$	−27.6945	−1.44960
$366$	−1.08603	−0.0567678
$367$	9.89969	0.516759	0.258380	−	0.966043i	$-0.416811\pi$
$367$	9.89969	0.516759	0.258380	+	0.966043i	$0.416811\pi$
$368$	−21.7394	−1.13324
$369$	25.1493	1.30922
$370$	5.11210	0.265765
$371$	−3.90012	−0.202484
$372$	−0.143737	−0.00745242
$373$	4.27475	0.221338	0.110669	−	0.993857i	$-0.464701\pi$
$373$	4.27475	0.221338	0.110669	+	0.993857i	$0.464701\pi$
$374$	0	0
$375$	3.92338	0.202603
$376$	−19.4909	−1.00516
$377$	−37.0445	−1.90789
$378$	6.17167	0.317436
$379$	4.32594	0.222209	0.111104	−	0.993809i	$-0.464561\pi$
$379$	4.32594	0.222209	0.111104	+	0.993809i	$0.464561\pi$
$380$	6.24458	0.320340
$381$	−12.0574	−0.617720
$382$	1.47382	0.0754070
$383$	−1.32210	−0.0675561	−0.0337781	−	0.999429i	$-0.510754\pi$
$383$	−1.32210	−0.0675561	−0.0337781	+	0.999429i	$0.510754\pi$
$384$	−6.73467	−0.343677
$385$	0	0
$386$	−11.9767	−0.609600
$387$	−12.3262	−0.626578
$388$	13.4684	0.683756
$389$	−38.5092	−1.95249	−0.976246	−	0.216665i	$-0.930482\pi$
$389$	−38.5092	−1.95249	−0.976246	+	0.216665i	$0.930482\pi$
$390$	13.1002	0.663356
$391$	12.0420	0.608992
$392$	1.49406	0.0754616
$393$	4.26317	0.215049
$394$	−19.4129	−0.978008
$395$	17.4800	0.879516
$396$	0	0
$397$	−0.410109	−0.0205828	−0.0102914	−	0.999947i	$-0.503276\pi$
$397$	−0.410109	−0.0205828	−0.0102914	+	0.999947i	$0.503276\pi$
$398$	−37.2820	−1.86878
$399$	−1.19843	−0.0599964
$400$	−13.5031	−0.675154
$401$	−1.56684	−0.0782443	−0.0391221	−	0.999234i	$-0.512456\pi$
$401$	−1.56684	−0.0782443	−0.0391221	+	0.999234i	$0.512456\pi$
$402$	5.95131	0.296824
$403$	−0.861219	−0.0429004
$404$	−3.95580	−0.196808
$405$	−15.8544	−0.787814
$406$	15.3363	0.761127
$407$	0	0
$408$	2.54443	0.125968
$409$	6.77852	0.335176	0.167588	−	0.985857i	$-0.446402\pi$
$409$	6.77852	0.335176	0.167588	+	0.985857i	$0.446402\pi$
$410$	−47.4250	−2.34215
$411$	−1.61350	−0.0795880
$412$	−21.0818	−1.03863
$413$	−8.55713	−0.421069
$414$	20.3363	0.999474
$415$	−4.78133	−0.234706
$416$	25.1344	1.23232
$417$	0.618859	0.0303057
$418$	0	0
$419$	28.7218	1.40315	0.701577	−	0.712594i	$-0.252480\pi$
$419$	28.7218	1.40315	0.701577	+	0.712594i	$0.252480\pi$
$420$	−1.99029	−0.0971161
$421$	12.2256	0.595838	0.297919	−	0.954591i	$-0.403707\pi$
$421$	12.2256	0.595838	0.297919	+	0.954591i	$0.403707\pi$
$422$	−10.7069	−0.521202
$423$	34.1537	1.66061
$424$	−5.82703	−0.282985
$425$	7.47973	0.362820
$426$	2.21582	0.107357
$427$	0.988609	0.0478421
$428$	−3.76527	−0.182001
$429$	0	0
$430$	23.2441	1.12093
$431$	−3.69129	−0.177803	−0.0889016	−	0.996040i	$-0.528336\pi$
$431$	−3.69129	−0.177803	−0.0889016	+	0.996040i	$0.528336\pi$
$432$	17.2724	0.831019
$433$	29.0749	1.39725	0.698625	−	0.715488i	$-0.253796\pi$
$433$	29.0749	1.39725	0.698625	+	0.715488i	$0.253796\pi$
$434$	0.356542	0.0171145
$435$	14.8108	0.710124
$436$	−14.7148	−0.704709
$437$	−8.47403	−0.405368
$438$	−10.9537	−0.523387
$439$	14.2017	0.677811	0.338905	−	0.940820i	$-0.389943\pi$
$439$	14.2017	0.677811	0.338905	+	0.940820i	$0.389943\pi$
$440$	0	0
$441$	−2.61803	−0.124668
$442$	−21.0293	−1.00026
$443$	27.2356	1.29400	0.647001	−	0.762489i	$-0.276023\pi$
$443$	27.2356	1.29400	0.647001	+	0.762489i	$0.276023\pi$
$444$	0.742004	0.0352140
$445$	42.5053	2.01495
$446$	8.22100	0.389275
$447$	9.20547	0.435404
$448$	−0.456423	−0.0215640
$449$	−41.9159	−1.97813	−0.989067	−	0.147467i	$-0.952888\pi$
$449$	−41.9159	−1.97813	−0.989067	+	0.147467i	$0.952888\pi$
$450$	12.6316	0.595459
$451$	0	0
$452$	−21.3896	−1.00608
$453$	−9.76777	−0.458930
$454$	30.7511	1.44322
$455$	−11.9251	−0.559056
$456$	−1.79053	−0.0838491
$457$	−19.4977	−0.912063	−0.456032	−	0.889964i	$-0.650730\pi$
$457$	−19.4977	−0.912063	−0.456032	+	0.889964i	$0.650730\pi$
$458$	7.86736	0.367617
$459$	−9.56767	−0.446580
$460$	−14.0733	−0.656169
$461$	−12.2251	−0.569380	−0.284690	−	0.958620i	$-0.591891\pi$
$461$	−12.2251	−0.569380	−0.284690	+	0.958620i	$0.591891\pi$
$462$	0	0
$463$	13.8550	0.643894	0.321947	−	0.946758i	$-0.395663\pi$
$463$	13.8550	0.643894	0.321947	+	0.946758i	$0.395663\pi$
$464$	42.9211	1.99256
$465$	0.344325	0.0159677
$466$	−18.7223	−0.867294
$467$	−16.4207	−0.759861	−0.379930	−	0.925015i	$-0.624052\pi$
$467$	−16.4207	−0.759861	−0.379930	+	0.925015i	$0.624052\pi$
$468$	−13.0328	−0.602440
$469$	−5.41745	−0.250154
$470$	−64.4050	−2.97078
$471$	2.83753	0.130747
$472$	−12.7849	−0.588473
$473$	0	0
$474$	6.91367	0.317555
$475$	−5.26352	−0.241507
$476$	3.19493	0.146439
$477$	10.2106	0.467513
$478$	−17.3420	−0.793202
$479$	24.7914	1.13275	0.566374	−	0.824149i	$-0.308346\pi$
$479$	24.7914	1.13275	0.566374	+	0.824149i	$0.308346\pi$
$480$	−10.0490	−0.458673
$481$	4.44581	0.202711
$482$	22.3076	1.01608
$483$	2.70087	0.122894
$484$	0	0
$485$	−32.2639	−1.46503
$486$	−24.7857	−1.12430
$487$	−14.3342	−0.649544	−0.324772	−	0.945792i	$-0.605288\pi$
$487$	−14.3342	−0.649544	−0.324772	+	0.945792i	$0.605288\pi$
$488$	1.47704	0.0668627
$489$	4.96754	0.224640
$490$	4.93693	0.223028
$491$	22.0193	0.993719	0.496859	−	0.867831i	$-0.334486\pi$
$491$	22.0193	0.993719	0.496859	+	0.867831i	$0.334486\pi$
$492$	−6.88358	−0.310336
$493$	−23.7752	−1.07078
$494$	14.7984	0.665810
$495$	0	0
$496$	0.997839	0.0448043
$497$	−2.01705	−0.0904770
$498$	−1.89110	−0.0847423
$499$	−25.3117	−1.13311	−0.566554	−	0.824024i	$-0.691724\pi$
$499$	−25.3117	−1.13311	−0.566554	+	0.824024i	$0.691724\pi$
$500$	7.36038	0.329166
$501$	8.28593	0.370188
$502$	19.5367	0.871964
$503$	26.0214	1.16024	0.580119	−	0.814531i	$-0.303006\pi$
$503$	26.0214	1.16024	0.580119	+	0.814531i	$0.303006\pi$
$504$	−3.91151	−0.174233
$505$	9.47619	0.421685
$506$	0	0
$507$	3.35836	0.149150
$508$	−22.6201	−1.00360
$509$	38.1269	1.68994	0.844972	−	0.534810i	$-0.179617\pi$
$509$	38.1269	1.68994	0.844972	+	0.534810i	$0.179617\pi$
$510$	8.40773	0.372301
$511$	9.97108	0.441095
$512$	−14.2570	−0.630077
$513$	6.73280	0.297261
$514$	−9.47896	−0.418099
$515$	50.5018	2.22538
$516$	3.37380	0.148523
$517$	0	0
$518$	−1.84055	−0.0808691
$519$	12.7204	0.558364
$520$	−17.8168	−0.781319
$521$	−26.0420	−1.14092	−0.570460	−	0.821325i	$-0.693235\pi$
$521$	−26.0420	−1.14092	−0.570460	+	0.821325i	$0.693235\pi$
$522$	−40.1509	−1.75736
$523$	−19.1230	−0.836189	−0.418095	−	0.908403i	$-0.637302\pi$
$523$	−19.1230	−0.836189	−0.418095	+	0.908403i	$0.637302\pi$
$524$	7.99784	0.349387
$525$	1.67760	0.0732166
$526$	−14.5404	−0.633993
$527$	−0.552731	−0.0240773
$528$	0	0
$529$	−3.90228	−0.169664
$530$	−19.2546	−0.836368
$531$	22.4029	0.972201
$532$	−2.24828	−0.0974755
$533$	−41.2438	−1.78647
$534$	16.8116	0.727510
$535$	9.01977	0.389959
$536$	−8.09401	−0.349608
$537$	−2.29612	−0.0990848
$538$	−22.8298	−0.984265
$539$	0	0
$540$	11.1815	0.481176
$541$	−10.8860	−0.468027	−0.234014	−	0.972233i	$-0.575186\pi$
$541$	−10.8860	−0.468027	−0.234014	+	0.972233i	$0.575186\pi$
$542$	−39.1886	−1.68329
$543$	2.94915	0.126560
$544$	16.1313	0.691624
$545$	35.2495	1.50992
$546$	−4.71658	−0.201851
$547$	12.9091	0.551951	0.275976	−	0.961165i	$-0.410999\pi$
$547$	12.9091	0.551951	0.275976	+	0.961165i	$0.410999\pi$
$548$	−3.02697	−0.129306
$549$	−2.58821	−0.110462
$550$	0	0
$551$	16.7307	0.712751
$552$	4.03527	0.171752
$553$	−6.29348	−0.267626
$554$	42.7964	1.81825
$555$	−1.77748	−0.0754500
$556$	1.16100	0.0492373
$557$	0.762626	0.0323135	0.0161567	−	0.999869i	$-0.494857\pi$
$557$	0.762626	0.0323135	0.0161567	+	0.999869i	$0.494857\pi$
$558$	−0.933438	−0.0395156
$559$	20.2146	0.854985
$560$	13.8168	0.583867
$561$	0	0
$562$	27.4476	1.15781
$563$	22.2281	0.936803	0.468402	−	0.883516i	$-0.344830\pi$
$563$	22.2281	0.936803	0.468402	+	0.883516i	$0.344830\pi$
$564$	−9.34817	−0.393629
$565$	51.2392	2.15565
$566$	−5.96758	−0.250836
$567$	5.70820	0.239722
$568$	−3.01360	−0.126448
$569$	20.5131	0.859955	0.429978	−	0.902839i	$-0.358521\pi$
$569$	20.5131	0.859955	0.429978	+	0.902839i	$0.358521\pi$
$570$	−5.91656	−0.247817
$571$	19.5654	0.818785	0.409393	−	0.912358i	$-0.365741\pi$
$571$	19.5654	0.818785	0.409393	+	0.912358i	$0.365741\pi$
$572$	0	0
$573$	−0.512448	−0.0214078
$574$	17.0748	0.712688
$575$	11.8623	0.494691
$576$	1.19493	0.0497888
$577$	−19.5716	−0.814776	−0.407388	−	0.913255i	$-0.633560\pi$
$577$	−19.5716	−0.814776	−0.407388	+	0.913255i	$0.633560\pi$
$578$	16.7206	0.695487
$579$	4.16433	0.173064
$580$	27.7855	1.15373
$581$	1.72146	0.0714182
$582$	−12.7609	−0.528958
$583$	0	0
$584$	14.8974	0.616460
$585$	31.2202	1.29080
$586$	12.1528	0.502026
$587$	1.57068	0.0648290	0.0324145	−	0.999475i	$-0.489680\pi$
$587$	1.57068	0.0648290	0.0324145	+	0.999475i	$0.489680\pi$
$588$	0.716580	0.0295512
$589$	0.388959	0.0160268
$590$	−42.2460	−1.73924
$591$	6.74990	0.277654
$592$	−5.15107	−0.211708
$593$	30.1230	1.23700	0.618502	−	0.785783i	$-0.287740\pi$
$593$	30.1230	1.23700	0.618502	+	0.785783i	$0.287740\pi$
$594$	0	0
$595$	−7.65351	−0.313763
$596$	17.2697	0.707396
$597$	12.9630	0.530540
$598$	−33.3507	−1.36381
$599$	5.77792	0.236079	0.118040	−	0.993009i	$-0.462339\pi$
$599$	5.77792	0.236079	0.118040	+	0.993009i	$0.462339\pi$
$600$	2.50645	0.102325
$601$	−45.5645	−1.85862	−0.929308	−	0.369305i	$-0.879596\pi$
$601$	−45.5645	−1.85862	−0.929308	+	0.369305i	$0.879596\pi$
$602$	−8.36876	−0.341085
$603$	14.1831	0.577579
$604$	−18.3246	−0.745619
$605$	0	0
$606$	3.74801	0.152252
$607$	−34.7211	−1.40928	−0.704642	−	0.709563i	$-0.748892\pi$
$607$	−34.7211	−1.40928	−0.704642	+	0.709563i	$0.748892\pi$
$608$	−11.3517	−0.460371
$609$	−5.33245	−0.216082
$610$	4.88070	0.197614
$611$	−56.0107	−2.26595
$612$	−8.36444	−0.338112
$613$	24.1423	0.975097	0.487548	−	0.873096i	$-0.337891\pi$
$613$	24.1423	0.975097	0.487548	+	0.873096i	$0.337891\pi$
$614$	−20.9690	−0.846242
$615$	16.4897	0.664931
$616$	0	0
$617$	−13.4967	−0.543358	−0.271679	−	0.962388i	$-0.587579\pi$
$617$	−13.4967	−0.543358	−0.271679	+	0.962388i	$0.587579\pi$
$618$	19.9744	0.803488
$619$	−43.4856	−1.74783	−0.873917	−	0.486074i	$-0.838428\pi$
$619$	−43.4856	−1.74783	−0.873917	+	0.486074i	$0.838428\pi$
$620$	0.645964	0.0259425
$621$	−15.1736	−0.608894
$622$	−44.9950	−1.80414
$623$	−15.3035	−0.613123
$624$	−13.2001	−0.528427
$625$	−31.2040	−1.24816
$626$	34.9353	1.39629
$627$	0	0
$628$	5.32330	0.212423
$629$	2.85332	0.113769
$630$	−12.9251	−0.514947
$631$	6.42012	0.255581	0.127790	−	0.991801i	$-0.459212\pi$
$631$	6.42012	0.255581	0.127790	+	0.991801i	$0.459212\pi$
$632$	−9.40286	−0.374026
$633$	3.72279	0.147968
$634$	−6.41771	−0.254880
$635$	54.1868	2.15034
$636$	−2.79475	−0.110819
$637$	4.29348	0.170114
$638$	0	0
$639$	5.28070	0.208901
$640$	30.2660	1.19637
$641$	−6.39600	−0.252627	−0.126313	−	0.991990i	$-0.540314\pi$
$641$	−6.39600	−0.252627	−0.126313	+	0.991990i	$0.540314\pi$
$642$	3.56748	0.140797
$643$	0.652660	0.0257384	0.0128692	−	0.999917i	$-0.495903\pi$
$643$	0.652660	0.0257384	0.0128692	+	0.999917i	$0.495903\pi$
$644$	5.06691	0.199664
$645$	−8.08201	−0.318229
$646$	9.49761	0.373678
$647$	17.9621	0.706165	0.353082	−	0.935592i	$-0.385134\pi$
$647$	17.9621	0.706165	0.353082	+	0.935592i	$0.385134\pi$
$648$	8.52842	0.335028
$649$	0	0
$650$	−20.7153	−0.812522
$651$	−0.123970	−0.00485877
$652$	9.31925	0.364970
$653$	−16.9181	−0.662055	−0.331028	−	0.943621i	$-0.607395\pi$
$653$	−16.9181	−0.662055	−0.331028	+	0.943621i	$0.607395\pi$
$654$	13.9418	0.545168
$655$	−19.1590	−0.748603
$656$	47.7866	1.86575
$657$	−26.1046	−1.01844
$658$	23.1882	0.903972
$659$	−23.6249	−0.920297	−0.460148	−	0.887842i	$-0.652204\pi$
$659$	−23.6249	−0.920297	−0.460148	+	0.887842i	$0.652204\pi$
$660$	0	0
$661$	−20.9819	−0.816103	−0.408051	−	0.912959i	$-0.633792\pi$
$661$	−20.9819	−0.816103	−0.408051	+	0.912959i	$0.633792\pi$
$662$	47.1628	1.83303
$663$	7.31191	0.283971
$664$	2.57197	0.0998118
$665$	5.38581	0.208853
$666$	4.81862	0.186718
$667$	−37.7056	−1.45997
$668$	15.5446	0.601440
$669$	−2.85845	−0.110514
$670$	−26.7456	−1.03327
$671$	0	0
$672$	3.61803	0.139569
$673$	−12.0986	−0.466368	−0.233184	−	0.972433i	$-0.574914\pi$
$673$	−12.0986	−0.466368	−0.233184	+	0.972433i	$0.574914\pi$
$674$	−1.21932	−0.0469663
$675$	−9.42483	−0.362762
$676$	6.30038	0.242322
$677$	9.42988	0.362420	0.181210	−	0.983444i	$-0.441999\pi$
$677$	9.42988	0.362420	0.181210	+	0.983444i	$0.441999\pi$
$678$	20.2660	0.778312
$679$	11.6162	0.445790
$680$	−11.4348	−0.438506
$681$	−10.6922	−0.409726
$682$	0	0
$683$	−15.2986	−0.585385	−0.292692	−	0.956207i	$-0.594551\pi$
$683$	−15.2986	−0.585385	−0.292692	+	0.956207i	$0.594551\pi$
$684$	5.88609	0.225060
$685$	7.25117	0.277053
$686$	−1.77748	−0.0678647
$687$	−2.73549	−0.104366
$688$	−23.4213	−0.892929
$689$	−16.7451	−0.637936
$690$	13.3340	0.507617
$691$	22.8193	0.868086	0.434043	−	0.900892i	$-0.357087\pi$
$691$	22.8193	0.868086	0.434043	+	0.900892i	$0.357087\pi$
$692$	23.8639	0.907169
$693$	0	0
$694$	38.2256	1.45102
$695$	−2.78119	−0.105497
$696$	−7.96703	−0.301989
$697$	−26.4703	−1.00263
$698$	34.5594	1.30809
$699$	6.50978	0.246222
$700$	3.14723	0.118954
$701$	32.3242	1.22087	0.610433	−	0.792068i	$-0.290995\pi$
$701$	32.3242	1.22087	0.610433	+	0.792068i	$0.290995\pi$
$702$	26.4979	1.00010
$703$	−2.00789	−0.0757292
$704$	0	0
$705$	22.3937	0.843396
$706$	37.2040	1.40019
$707$	−3.41179	−0.128314
$708$	−6.13186	−0.230450
$709$	14.5598	0.546804	0.273402	−	0.961900i	$-0.411851\pi$
$709$	14.5598	0.546804	0.273402	+	0.961900i	$0.411851\pi$
$710$	−9.95804	−0.373718
$711$	16.4765	0.617918
$712$	−22.8645	−0.856882
$713$	−0.876587	−0.0328285
$714$	−3.02710	−0.113287
$715$	0	0
$716$	−4.30759	−0.160982
$717$	6.02982	0.225188
$718$	17.3683	0.648178
$719$	44.8602	1.67300	0.836501	−	0.547965i	$-0.184597\pi$
$719$	44.8602	1.67300	0.836501	+	0.547965i	$0.184597\pi$
$720$	−36.1729	−1.34808
$721$	−18.1826	−0.677155
$722$	27.0887	1.00814
$723$	−7.75638	−0.288463
$724$	5.53270	0.205621
$725$	−23.4202	−0.869806
$726$	0	0
$727$	28.3582	1.05175	0.525874	−	0.850562i	$-0.323738\pi$
$727$	28.3582	1.05175	0.525874	+	0.850562i	$0.323738\pi$
$728$	6.41473	0.237746
$729$	−8.50658	−0.315058
$730$	49.2266	1.82196
$731$	12.9737	0.479850
$732$	0.708417	0.0261838
$733$	−6.25696	−0.231106	−0.115553	−	0.993301i	$-0.536864\pi$
$733$	−6.25696	−0.231106	−0.115553	+	0.993301i	$0.536864\pi$
$734$	−17.5965	−0.649500
$735$	−1.71658	−0.0633170
$736$	25.5830	0.943001
$737$	0	0
$738$	−44.7024	−1.64552
$739$	−9.35201	−0.344019	−0.172010	−	0.985095i	$-0.555026\pi$
$739$	−9.35201	−0.344019	−0.172010	+	0.985095i	$0.555026\pi$
$740$	−3.33461	−0.122583
$741$	−5.14542	−0.189022
$742$	6.93240	0.254496
$743$	−25.2066	−0.924740	−0.462370	−	0.886687i	$-0.653001\pi$
$743$	−25.2066	−0.924740	−0.462370	+	0.886687i	$0.653001\pi$
$744$	−0.185219	−0.00679047
$745$	−41.3700	−1.51568
$746$	−7.59830	−0.278193
$747$	−4.50684	−0.164897
$748$	0	0
$749$	−3.24746	−0.118660
$750$	−6.97375	−0.254645
$751$	−34.7493	−1.26802	−0.634010	−	0.773325i	$-0.718592\pi$
$751$	−34.7493	−1.26802	−0.634010	+	0.773325i	$0.718592\pi$
$752$	64.8960	2.36651
$753$	−6.79293	−0.247548
$754$	65.8460	2.39797
$755$	43.8970	1.59758
$756$	−4.02577	−0.146416
$757$	34.7960	1.26468	0.632341	−	0.774690i	$-0.282094\pi$
$757$	34.7960	1.26468	0.632341	+	0.774690i	$0.282094\pi$
$758$	−7.68929	−0.279288
$759$	0	0
$760$	8.04674	0.291886
$761$	12.2255	0.443172	0.221586	−	0.975141i	$-0.428877\pi$
$761$	12.2255	0.443172	0.221586	+	0.975141i	$0.428877\pi$
$762$	21.4319	0.776394
$763$	−12.6912	−0.459451
$764$	−0.961368	−0.0347811
$765$	20.0372	0.724445
$766$	2.35001	0.0849094
$767$	−36.7398	−1.32660
$768$	12.5349	0.452315
$769$	2.61946	0.0944603	0.0472301	−	0.998884i	$-0.484961\pi$
$769$	2.61946	0.0944603	0.0472301	+	0.998884i	$0.484961\pi$
$770$	0	0
$771$	3.29585	0.118697
$772$	7.81241	0.281175
$773$	−0.212804	−0.00765405	−0.00382702	−	0.999993i	$-0.501218\pi$
$773$	−0.212804	−0.00765405	−0.00382702	+	0.999993i	$0.501218\pi$
$774$	21.9097	0.787528
$775$	−0.544479	−0.0195583
$776$	17.3554	0.623022
$777$	0.639962	0.0229585
$778$	68.4494	2.45403
$779$	18.6273	0.667391
$780$	−8.54526	−0.305969
$781$	0	0
$782$	−21.4045	−0.765425
$783$	29.9579	1.07061
$784$	−4.97458	−0.177663
$785$	−12.7521	−0.455141
$786$	−7.57772	−0.270288
$787$	−29.1375	−1.03864	−0.519320	−	0.854580i	$-0.673815\pi$
$787$	−29.1375	−1.03864	−0.519320	+	0.854580i	$0.673815\pi$
$788$	12.6630	0.451101
$789$	5.05573	0.179989
$790$	−31.0705	−1.10544
$791$	−18.4480	−0.655937
$792$	0	0
$793$	4.24457	0.150729
$794$	0.728962	0.0258699
$795$	6.69486	0.237442
$796$	24.3190	0.861963
$797$	32.2284	1.14159	0.570794	−	0.821093i	$-0.306635\pi$
$797$	32.2284	1.14159	0.570794	+	0.821093i	$0.306635\pi$
$798$	2.13018	0.0754077
$799$	−35.9477	−1.27174
$800$	15.8905	0.561813
$801$	40.0652	1.41563
$802$	2.78503	0.0983430
$803$	0	0
$804$	−3.88203	−0.136909
$805$	−12.1379	−0.427804
$806$	1.53080	0.0539202
$807$	7.93797	0.279430
$808$	−5.09743	−0.179327
$809$	29.3461	1.03175	0.515877	−	0.856663i	$-0.327466\pi$
$809$	29.3461	1.03175	0.515877	+	0.856663i	$0.327466\pi$
$810$	28.1810	0.990180
$811$	−1.03940	−0.0364982	−0.0182491	−	0.999833i	$-0.505809\pi$
$811$	−1.03940	−0.0364982	−0.0182491	+	0.999833i	$0.505809\pi$
$812$	−10.0038	−0.351066
$813$	13.6259	0.477882
$814$	0	0
$815$	−22.3244	−0.781990
$816$	−8.47184	−0.296574
$817$	−9.12966	−0.319406
$818$	−12.0487	−0.421274
$819$	−11.2405	−0.392774
$820$	30.9352	1.08031
$821$	−28.5323	−0.995786	−0.497893	−	0.867239i	$-0.665893\pi$
$821$	−28.5323	−0.995786	−0.497893	+	0.867239i	$0.665893\pi$
$822$	2.86797	0.100032
$823$	26.6577	0.929227	0.464614	−	0.885513i	$-0.346193\pi$
$823$	26.6577	0.929227	0.464614	+	0.885513i	$0.346193\pi$
$824$	−27.1659	−0.946370
$825$	0	0
$826$	15.2102	0.529229
$827$	−1.71964	−0.0597978	−0.0298989	−	0.999553i	$-0.509519\pi$
$827$	−1.71964	−0.0597978	−0.0298989	+	0.999553i	$0.509519\pi$
$828$	−13.2653	−0.461002
$829$	27.9605	0.971107	0.485554	−	0.874207i	$-0.338618\pi$
$829$	27.9605	0.971107	0.485554	+	0.874207i	$0.338618\pi$
$830$	8.49873	0.294995
$831$	−14.8804	−0.516195
$832$	−1.95964	−0.0679383
$833$	2.75556	0.0954744
$834$	−1.10001	−0.0380903
$835$	−37.2375	−1.28866
$836$	0	0
$837$	0.696468	0.0240735
$838$	−51.0526	−1.76358
$839$	35.8624	1.23811	0.619054	−	0.785348i	$-0.287516\pi$
$839$	35.8624	1.23811	0.619054	+	0.785348i	$0.287516\pi$
$840$	−2.56468	−0.0884899
$841$	45.4439	1.56703
$842$	−21.7308	−0.748892
$843$	−9.54358	−0.328698
$844$	6.98407	0.240402
$845$	−15.0927	−0.519204
$846$	−60.7076	−2.08717
$847$	0	0
$848$	19.4014	0.666248
$849$	2.07494	0.0712117
$850$	−13.2951	−0.456018
$851$	4.52515	0.155120
$852$	−1.44538	−0.0495178
$853$	15.1427	0.518478	0.259239	−	0.965813i	$-0.416528\pi$
$853$	15.1427	0.518478	0.259239	+	0.965813i	$0.416528\pi$
$854$	−1.75724	−0.0601314
$855$	−14.1002	−0.482218
$856$	−4.85191	−0.165835
$857$	−25.3267	−0.865142	−0.432571	−	0.901600i	$-0.642394\pi$
$857$	−25.3267	−0.865142	−0.432571	+	0.901600i	$0.642394\pi$
$858$	0	0
$859$	41.5291	1.41696	0.708478	−	0.705733i	$-0.249382\pi$
$859$	41.5291	1.41696	0.708478	+	0.705733i	$0.249382\pi$
$860$	−15.1621	−0.517023
$861$	−5.93693	−0.202330
$862$	6.56121	0.223476
$863$	−24.0504	−0.818684	−0.409342	−	0.912381i	$-0.634242\pi$
$863$	−24.0504	−0.818684	−0.409342	+	0.912381i	$0.634242\pi$
$864$	−20.3262	−0.691513
$865$	−57.1664	−1.94372
$866$	−51.6801	−1.75616
$867$	−5.81379	−0.197447
$868$	−0.232572	−0.00789399
$869$	0	0
$870$	−26.3260	−0.892534
$871$	−23.2597	−0.788124
$872$	−18.9614	−0.642114
$873$	−30.4117	−1.02928
$874$	15.0625	0.509495
$875$	6.34817	0.214607
$876$	7.14507	0.241410
$877$	−33.8052	−1.14152	−0.570760	−	0.821117i	$-0.693351\pi$
$877$	−33.8052	−1.14152	−0.570760	+	0.821117i	$0.693351\pi$
$878$	−25.2433	−0.851921
$879$	−4.22553	−0.142524
$880$	0	0
$881$	−36.8296	−1.24082	−0.620410	−	0.784278i	$-0.713034\pi$
$881$	−36.8296	−1.24082	−0.620410	+	0.784278i	$0.713034\pi$
$882$	4.65351	0.156692
$883$	53.4103	1.79740	0.898701	−	0.438563i	$-0.144512\pi$
$883$	53.4103	1.79740	0.898701	+	0.438563i	$0.144512\pi$
$884$	13.7174	0.461365
$885$	14.6890	0.493765
$886$	−48.4108	−1.62639
$887$	−11.4616	−0.384843	−0.192421	−	0.981312i	$-0.561634\pi$
$887$	−11.4616	−0.384843	−0.192421	+	0.981312i	$0.561634\pi$
$888$	0.956144	0.0320861
$889$	−19.5093	−0.654321
$890$	−75.5525	−2.53253
$891$	0	0
$892$	−5.36254	−0.179551
$893$	25.2965	0.846516
$894$	−16.3626	−0.547246
$895$	10.3189	0.344923
$896$	−10.8969	−0.364041
$897$	11.5961	0.387183
$898$	74.5049	2.48626
$899$	1.73069	0.0577217
$900$	−8.23956	−0.274652
$901$	−10.7470	−0.358034
$902$	0	0
$903$	2.90983	0.0968331
$904$	−27.5626	−0.916717
$905$	−13.2537	−0.440567
$906$	17.3621	0.576816
$907$	57.0582	1.89459	0.947293	−	0.320369i	$-0.103807\pi$
$907$	57.0582	1.89459	0.947293	+	0.320369i	$0.103807\pi$
$908$	−20.0589	−0.665677
$909$	8.93218	0.296262
$910$	21.1966	0.702661
$911$	6.67566	0.221175	0.110587	−	0.993866i	$-0.464727\pi$
$911$	6.67566	0.221175	0.110587	+	0.993866i	$0.464727\pi$
$912$	5.96167	0.197410
$913$	0	0
$914$	34.6568	1.14635
$915$	−1.69703	−0.0561019
$916$	−5.13186	−0.169562
$917$	6.89796	0.227791
$918$	17.0064	0.561294
$919$	29.1339	0.961038	0.480519	−	0.876984i	$-0.340448\pi$
$919$	29.1339	0.961038	0.480519	+	0.876984i	$0.340448\pi$
$920$	−18.1347	−0.597885
$921$	7.29097	0.240246
$922$	21.7299	0.715637
$923$	−8.66015	−0.285052
$924$	0	0
$925$	2.81073	0.0924161
$926$	−24.6270	−0.809292
$927$	47.6026	1.56347
$928$	−50.5097	−1.65806
$929$	2.34472	0.0769277	0.0384638	−	0.999260i	$-0.487754\pi$
$929$	2.34472	0.0769277	0.0384638	+	0.999260i	$0.487754\pi$
$930$	−0.612032	−0.0200693
$931$	−1.93910	−0.0635513
$932$	12.2125	0.400035
$933$	15.6448	0.512189
$934$	29.1876	0.955047
$935$	0	0
$936$	−16.7940	−0.548928
$937$	−8.78780	−0.287085	−0.143542	−	0.989644i	$-0.545849\pi$
$937$	−8.78780	−0.287085	−0.143542	+	0.989644i	$0.545849\pi$
$938$	9.62943	0.314412
$939$	−12.1470	−0.396404
$940$	42.0113	1.37026
$941$	−8.88871	−0.289764	−0.144882	−	0.989449i	$-0.546280\pi$
$941$	−8.88871	−0.289764	−0.144882	+	0.989449i	$0.546280\pi$
$942$	−5.04367	−0.164332
$943$	−41.9798	−1.36705
$944$	42.5681	1.38547
$945$	9.64380	0.313713
$946$	0	0
$947$	−7.86275	−0.255505	−0.127752	−	0.991806i	$-0.540776\pi$
$947$	−7.86275	−0.255505	−0.127752	+	0.991806i	$0.540776\pi$
$948$	−4.50978	−0.146471
$949$	42.8106	1.38969
$950$	9.35582	0.303543
$951$	2.23145	0.0723597
$952$	4.11698	0.133432
$953$	17.4644	0.565727	0.282864	−	0.959160i	$-0.408716\pi$
$953$	17.4644	0.565727	0.282864	+	0.959160i	$0.408716\pi$
$954$	−18.1493	−0.587604
$955$	2.30297	0.0745225
$956$	11.3121	0.365860
$957$	0	0
$958$	−44.0663	−1.42372
$959$	−2.61070	−0.0843038
$960$	0.783486	0.0252869
$961$	−30.9598	−0.998702
$962$	−7.90236	−0.254782
$963$	8.50196	0.273972
$964$	−14.5512	−0.468663
$965$	−18.7148	−0.602450
$966$	−4.80075	−0.154462
$967$	45.6122	1.46679	0.733395	−	0.679802i	$-0.237934\pi$
$967$	45.6122	1.46679	0.733395	+	0.679802i	$0.237934\pi$
$968$	0	0
$969$	−3.30233	−0.106086
$970$	57.3485	1.84135
$971$	2.38378	0.0764992	0.0382496	−	0.999268i	$-0.487822\pi$
$971$	2.38378	0.0764992	0.0382496	+	0.999268i	$0.487822\pi$
$972$	16.1677	0.518579
$973$	1.00134	0.0321013
$974$	25.4788	0.816393
$975$	7.20275	0.230673
$976$	−4.91791	−0.157418
$977$	−19.2662	−0.616380	−0.308190	−	0.951325i	$-0.599723\pi$
$977$	−19.2662	−0.616380	−0.308190	+	0.951325i	$0.599723\pi$
$978$	−8.82972	−0.282343
$979$	0	0
$980$	−3.22035	−0.102870
$981$	33.2259	1.06082
$982$	−39.1390	−1.24898
$983$	−16.2256	−0.517517	−0.258759	−	0.965942i	$-0.583313\pi$
$983$	−16.2256	−0.517517	−0.258759	+	0.965942i	$0.583313\pi$
$984$	−8.87016	−0.282770
$985$	−30.3345	−0.966537
$986$	42.2600	1.34583
$987$	−8.06258	−0.256635
$988$	−9.65296	−0.307101
$989$	20.5753	0.654256
$990$	0	0
$991$	−50.5214	−1.60487	−0.802433	−	0.596743i	$-0.796461\pi$
$991$	−50.5214	−1.60487	−0.802433	+	0.596743i	$0.796461\pi$
$992$	−1.17426	−0.0372828
$993$	−16.3986	−0.520393
$994$	3.58527	0.113718
$995$	−58.2566	−1.84686
$996$	1.23356	0.0390869
$997$	45.0206	1.42582	0.712909	−	0.701257i	$-0.247377\pi$
$997$	45.0206	1.42582	0.712909	+	0.701257i	$0.247377\pi$
$998$	44.9912	1.42417
$999$	−3.59533	−0.113751

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.a.k.1.2		4
3.2	odd	2		7623.2.a.co.1.3		4
7.6	odd	2		5929.2.a.bb.1.2		4
11.2	odd	10		847.2.f.p.323.1		8
11.3	even	5		847.2.f.q.372.1		8
11.4	even	5		847.2.f.q.148.1		8
11.5	even	5		847.2.f.s.729.2		8
11.6	odd	10		847.2.f.p.729.1		8
11.7	odd	10		77.2.f.a.71.2	yes	8
11.8	odd	10		77.2.f.a.64.2	✓	8
11.9	even	5		847.2.f.s.323.2		8
11.10	odd	2		847.2.a.l.1.3		4
33.8	even	10		693.2.m.g.64.1		8
33.29	even	10		693.2.m.g.379.1		8
33.32	even	2		7623.2.a.ch.1.2		4
77.18	odd	30		539.2.q.c.324.1		16
77.19	even	30		539.2.q.b.361.1		16
77.30	odd	30		539.2.q.c.361.1		16
77.40	even	30		539.2.q.b.214.2		16
77.41	even	10		539.2.f.d.295.2		8
77.51	odd	30		539.2.q.c.214.2		16
77.52	even	30		539.2.q.b.471.2		16
77.62	even	10		539.2.f.d.148.2		8
77.73	even	30		539.2.q.b.324.1		16
77.74	odd	30		539.2.q.c.471.2		16
77.76	even	2		5929.2.a.bi.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.a.64.2	✓	8	11.8	odd	10
77.2.f.a.71.2	yes	8	11.7	odd	10
539.2.f.d.148.2		8	77.62	even	10
539.2.f.d.295.2		8	77.41	even	10
539.2.q.b.214.2		16	77.40	even	30
539.2.q.b.324.1		16	77.73	even	30
539.2.q.b.361.1		16	77.19	even	30
539.2.q.b.471.2		16	77.52	even	30
539.2.q.c.214.2		16	77.51	odd	30
539.2.q.c.324.1		16	77.18	odd	30
539.2.q.c.361.1		16	77.30	odd	30
539.2.q.c.471.2		16	77.74	odd	30
693.2.m.g.64.1		8	33.8	even	10
693.2.m.g.379.1		8	33.29	even	10
847.2.a.k.1.2		4	1.1	even	1	trivial
847.2.a.l.1.3		4	11.10	odd	2
847.2.f.p.323.1		8	11.2	odd	10
847.2.f.p.729.1		8	11.6	odd	10
847.2.f.q.148.1		8	11.4	even	5
847.2.f.q.372.1		8	11.3	even	5
847.2.f.s.323.2		8	11.9	even	5
847.2.f.s.729.2		8	11.5	even	5
5929.2.a.bb.1.2		4	7.6	odd	2
5929.2.a.bi.1.3		4	77.76	even	2
7623.2.a.ch.1.2		4	33.32	even	2
7623.2.a.co.1.3		4	3.2	odd	2

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:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.a (trivial)

\(f(q)\)	\(=\)	\(q-1.77748 q^{2} +0.618034 q^{3} +1.15945 q^{4} -2.77748 q^{5} -1.09855 q^{6} +1.00000 q^{7} +1.49406 q^{8} -2.61803 q^{9} +O(q^{10})\)	\(q-1.77748 q^{2} +0.618034 q^{3} +1.15945 q^{4} -2.77748 q^{5} -1.09855 q^{6} +1.00000 q^{7} +1.49406 q^{8} -2.61803 q^{9} +4.93693 q^{10} +0.716580 q^{12} +4.29348 q^{13} -1.77748 q^{14} -1.71658 q^{15} -4.97458 q^{16} +2.75556 q^{17} +4.65351 q^{18} -1.93910 q^{19} -3.22035 q^{20} +0.618034 q^{21} +4.37009 q^{23} +0.923382 q^{24} +2.71442 q^{25} -7.63159 q^{26} -3.47214 q^{27} +1.15945 q^{28} -8.62809 q^{29} +3.05119 q^{30} -0.200588 q^{31} +5.85410 q^{32} -4.89796 q^{34} -2.77748 q^{35} -3.03548 q^{36} +1.03548 q^{37} +3.44671 q^{38} +2.65351 q^{39} -4.14974 q^{40} -9.60616 q^{41} -1.09855 q^{42} +4.70820 q^{43} +7.27155 q^{45} -7.76777 q^{46} -13.0455 q^{47} -3.07446 q^{48} +1.00000 q^{49} -4.82484 q^{50} +1.70303 q^{51} +4.97807 q^{52} -3.90012 q^{53} +6.17167 q^{54} +1.49406 q^{56} -1.19843 q^{57} +15.3363 q^{58} -8.55713 q^{59} -1.99029 q^{60} +0.988609 q^{61} +0.356542 q^{62} -2.61803 q^{63} -0.456423 q^{64} -11.9251 q^{65} -5.41745 q^{67} +3.19493 q^{68} +2.70087 q^{69} +4.93693 q^{70} -2.01705 q^{71} -3.91151 q^{72} +9.97108 q^{73} -1.84055 q^{74} +1.67760 q^{75} -2.24828 q^{76} -4.71658 q^{78} -6.29348 q^{79} +13.8168 q^{80} +5.70820 q^{81} +17.0748 q^{82} +1.72146 q^{83} +0.716580 q^{84} -7.65351 q^{85} -8.36876 q^{86} -5.33245 q^{87} -15.3035 q^{89} -12.9251 q^{90} +4.29348 q^{91} +5.06691 q^{92} -0.123970 q^{93} +23.1882 q^{94} +5.38581 q^{95} +3.61803 q^{96} +11.6162 q^{97} -1.77748 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{5} + q^{6} + 4 q^{7} - 9 q^{8} - 6 q^{9}+O(q^{10}) \) 4 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 6 * q^5 + q^6 + 4 * q^7 - 9 * q^8 - 6 * q^9	\( 4 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{5} + q^{6} + 4 q^{7} - 9 q^{8} - 6 q^{9} + 14 q^{10} - 7 q^{12} - 2 q^{14} + 3 q^{15} - 4 q^{16} + 3 q^{17} + 3 q^{18} - 3 q^{19} - 17 q^{20} - 2 q^{21} - 8 q^{23} + 12 q^{24} - 12 q^{26} + 4 q^{27} + 4 q^{28} - 3 q^{29} - 12 q^{30} - 3 q^{31} + 10 q^{32} - 12 q^{34} - 6 q^{35} - q^{36} - 7 q^{37} - 20 q^{38} - 5 q^{39} + 13 q^{40} - 4 q^{41} + q^{42} - 8 q^{43} + 9 q^{45} + 3 q^{46} - 14 q^{47} - 3 q^{48} + 4 q^{49} - 33 q^{50} + 11 q^{51} + 17 q^{52} - 9 q^{53} - 2 q^{54} - 9 q^{56} - 6 q^{57} + 3 q^{58} - 25 q^{59} + 21 q^{60} + 19 q^{61} - 10 q^{62} - 6 q^{63} + 3 q^{64} - 12 q^{65} - 15 q^{67} + q^{68} + 14 q^{69} + 14 q^{70} - 7 q^{71} + 6 q^{72} + 11 q^{73} - 8 q^{74} - 5 q^{75} + 26 q^{76} - 9 q^{78} - 8 q^{79} - 4 q^{80} - 4 q^{81} + 3 q^{82} + q^{83} - 7 q^{84} - 15 q^{85} + 4 q^{86} - 6 q^{87} - 17 q^{89} - 16 q^{90} - 17 q^{92} - 11 q^{93} + 20 q^{94} - 17 q^{95} + 10 q^{96} - 15 q^{97} - 2 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 6 * q^5 + q^6 + 4 * q^7 - 9 * q^8 - 6 * q^9 + 14 * q^10 - 7 * q^12 - 2 * q^14 + 3 * q^15 - 4 * q^16 + 3 * q^17 + 3 * q^18 - 3 * q^19 - 17 * q^20 - 2 * q^21 - 8 * q^23 + 12 * q^24 - 12 * q^26 + 4 * q^27 + 4 * q^28 - 3 * q^29 - 12 * q^30 - 3 * q^31 + 10 * q^32 - 12 * q^34 - 6 * q^35 - q^36 - 7 * q^37 - 20 * q^38 - 5 * q^39 + 13 * q^40 - 4 * q^41 + q^42 - 8 * q^43 + 9 * q^45 + 3 * q^46 - 14 * q^47 - 3 * q^48 + 4 * q^49 - 33 * q^50 + 11 * q^51 + 17 * q^52 - 9 * q^53 - 2 * q^54 - 9 * q^56 - 6 * q^57 + 3 * q^58 - 25 * q^59 + 21 * q^60 + 19 * q^61 - 10 * q^62 - 6 * q^63 + 3 * q^64 - 12 * q^65 - 15 * q^67 + q^68 + 14 * q^69 + 14 * q^70 - 7 * q^71 + 6 * q^72 + 11 * q^73 - 8 * q^74 - 5 * q^75 + 26 * q^76 - 9 * q^78 - 8 * q^79 - 4 * q^80 - 4 * q^81 + 3 * q^82 + q^83 - 7 * q^84 - 15 * q^85 + 4 * q^86 - 6 * q^87 - 17 * q^89 - 16 * q^90 - 17 * q^92 - 11 * q^93 + 20 * q^94 - 17 * q^95 + 10 * q^96 - 15 * q^97 - 2 * q^98

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