LMFDB - Embedded newform 855.2.k.g.676.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,2,Mod(406,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("855.406");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.k (of order $3$, degree $2$, minimal)

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:	no
Analytic conductor:	$6.82720937282$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.3518667.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49 $ x^6 - x^5 + 7x^4 - 8x^3 + 43x^2 - 42x + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			676.1
Root			$-1.25351 + 2.17114i$ of defining polynomial
Character	$\chi$	$=$	855.676
Dual form			855.2.k.g.406.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.25351 + 2.17114i) q^{2} +(-2.14257 - 3.71104i) q^{4} +(-0.500000 + 0.866025i) q^{5} -0.221876 q^{7} +5.72889 q^{8} +O(q^{10})$	\(q+(-1.25351 + 2.17114i) q^{2} +(-2.14257 - 3.71104i) q^{4} +(-0.500000 + 0.866025i) q^{5} -0.221876 q^{7} +5.72889 q^{8} +(-1.25351 - 2.17114i) q^{10} +0.778124 q^{11} +(2.50000 + 4.33013i) q^{13} +(0.278124 - 0.481725i) q^{14} +(-2.89608 + 5.01616i) q^{16} +(3.53865 - 6.12912i) q^{17} +(1.33281 - 4.15013i) q^{19} +4.28514 q^{20} +(-0.975385 + 1.68942i) q^{22} +(4.03865 + 6.99515i) q^{23} +(-0.500000 - 0.866025i) q^{25} -12.5351 q^{26} +(0.475385 + 0.823392i) q^{28} +(0.110938 + 0.192150i) q^{29} +2.50702 q^{31} +(-1.53163 - 2.65287i) q^{32} +(8.87147 + 15.3658i) q^{34} +(0.110938 - 0.192150i) q^{35} -1.90466 q^{37} +(7.33983 + 8.09596i) q^{38} +(-2.86445 + 4.96137i) q^{40} +(-3.61796 + 6.26648i) q^{41} +(-3.64959 + 6.32128i) q^{43} +(-1.66719 - 2.88765i) q^{44} -20.2500 q^{46} +(-1.39608 - 2.41808i) q^{47} -6.95077 q^{49} +2.50702 q^{50} +(10.7129 - 18.5552i) q^{52} +(2.19024 + 3.79361i) q^{53} +(-0.389062 + 0.673875i) q^{55} -1.27111 q^{56} -0.556248 q^{58} +(-1.39608 + 2.41808i) q^{59} +(6.29216 + 10.8983i) q^{61} +(-3.14257 + 5.44309i) q^{62} -3.90466 q^{64} -5.00000 q^{65} +(5.28514 + 9.15414i) q^{67} -30.3273 q^{68} +(0.278124 + 0.481725i) q^{70} +(-4.92070 + 8.52289i) q^{71} +(7.03865 - 12.1913i) q^{73} +(2.38750 - 4.13528i) q^{74} +(-18.2570 + 3.94583i) q^{76} -0.172647 q^{77} +(-0.792161 + 1.37206i) q^{79} +(-2.89608 - 5.01616i) q^{80} +(-9.07028 - 15.7102i) q^{82} +9.52106 q^{83} +(3.53865 + 6.12912i) q^{85} +(-9.14959 - 15.8476i) q^{86} +4.45779 q^{88} +(1.57028 + 2.71981i) q^{89} +(-0.554690 - 0.960752i) q^{91} +(17.3062 - 29.9752i) q^{92} +7.00000 q^{94} +(2.92771 + 3.22932i) q^{95} +(3.18122 - 5.51004i) q^{97} +(8.71286 - 15.0911i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{2} - 7 q^{4} - 3 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10}) $ 6 * q + q^2 - 7 * q^4 - 3 * q^5 + 4 * q^7 + 12 * q^8	$ 6 q + q^{2} - 7 q^{4} - 3 q^{5} + 4 q^{7} + 12 q^{8} + q^{10} + 10 q^{11} + 15 q^{13} + 7 q^{14} - 3 q^{16} + q^{17} + 14 q^{20} + 8 q^{22} + 4 q^{23} - 3 q^{25} + 10 q^{26} - 11 q^{28} - 2 q^{29} - 2 q^{31} - 6 q^{32} + 25 q^{34} - 2 q^{35} - 4 q^{37} + 19 q^{38} - 6 q^{40} - 2 q^{41} + q^{43} - 18 q^{44} - 48 q^{46} + 6 q^{47} - 14 q^{49} - 2 q^{50} + 35 q^{52} + 11 q^{53} - 5 q^{55} - 30 q^{56} - 14 q^{58} + 6 q^{59} + 9 q^{61} - 13 q^{62} - 16 q^{64} - 30 q^{65} + 20 q^{67} - 68 q^{68} + 7 q^{70} - 29 q^{71} + 22 q^{73} - 7 q^{74} - 19 q^{76} + 32 q^{77} + 24 q^{79} - 3 q^{80} - 31 q^{82} + 6 q^{83} + q^{85} - 32 q^{86} - 18 q^{88} - 14 q^{89} + 10 q^{91} + 41 q^{92} + 42 q^{94} - 7 q^{97} + 23 q^{98}+O(q^{100}) $ 6 * q + q^2 - 7 * q^4 - 3 * q^5 + 4 * q^7 + 12 * q^8 + q^10 + 10 * q^11 + 15 * q^13 + 7 * q^14 - 3 * q^16 + q^17 + 14 * q^20 + 8 * q^22 + 4 * q^23 - 3 * q^25 + 10 * q^26 - 11 * q^28 - 2 * q^29 - 2 * q^31 - 6 * q^32 + 25 * q^34 - 2 * q^35 - 4 * q^37 + 19 * q^38 - 6 * q^40 - 2 * q^41 + q^43 - 18 * q^44 - 48 * q^46 + 6 * q^47 - 14 * q^49 - 2 * q^50 + 35 * q^52 + 11 * q^53 - 5 * q^55 - 30 * q^56 - 14 * q^58 + 6 * q^59 + 9 * q^61 - 13 * q^62 - 16 * q^64 - 30 * q^65 + 20 * q^67 - 68 * q^68 + 7 * q^70 - 29 * q^71 + 22 * q^73 - 7 * q^74 - 19 * q^76 + 32 * q^77 + 24 * q^79 - 3 * q^80 - 31 * q^82 + 6 * q^83 + q^85 - 32 * q^86 - 18 * q^88 - 14 * q^89 + 10 * q^91 + 41 * q^92 + 42 * q^94 - 7 * q^97 + 23 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/855\mathbb{Z}\right)^\times$.

$n$	$172$	$191$	$496$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−1.25351	+	2.17114i	−0.886365	+	1.53523i	−0.0422238	+	0.999108i	$0.513444\pi$
$2$	−1.25351	+	2.17114i	−0.886365	+	1.53523i	−0.844141	+	0.536121i	$0.819889\pi$
$3$		0			0
$3$		0			0
$4$	−2.14257	−	3.71104i	−1.07129	−	1.85552i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	−0.221876			−0.0838613			−0.0419307	−	0.999121i	$-0.513351\pi$
$7$	−0.221876			−0.0838613			−0.0419307	+	0.999121i	$0.513351\pi$
$8$	5.72889			2.02547
$9$		0			0
$10$	−1.25351	−	2.17114i	−0.396394	−	0.686575i
$11$	0.778124			0.234613			0.117307	−	0.993096i	$-0.462574\pi$
$11$	0.778124			0.234613			0.117307	+	0.993096i	$0.462574\pi$
$12$		0			0
$13$	2.50000	+	4.33013i	0.693375	+	1.20096i	0.970725	+	0.240192i	$0.0772105\pi$
$13$	2.50000	+	4.33013i	0.693375	+	1.20096i	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0.278124	−	0.481725i	0.0743317	−	0.128746i
$15$		0			0
$16$	−2.89608	+	5.01616i	−0.724020	+	1.25404i
$17$	3.53865	−	6.12912i	0.858249	−	1.48653i	−0.0153485	−	0.999882i	$-0.504886\pi$
$17$	3.53865	−	6.12912i	0.858249	−	1.48653i	0.873598	−	0.486649i	$-0.161781\pi$
$18$		0			0
$19$	1.33281	−	4.15013i	0.305769	−	0.952106i
$19$	1.33281	−	4.15013i	0.305769	−	0.952106i
$20$	4.28514			0.958187
$21$		0			0
$22$	−0.975385	+	1.68942i	−0.207953	+	0.360185i
$23$	4.03865	+	6.99515i	0.842117	+	1.45859i	0.888101	+	0.459647i	$0.152024\pi$
$23$	4.03865	+	6.99515i	0.842117	+	1.45859i	−0.0459843	+	0.998942i	$0.514642\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$	−12.5351			−2.45833
$27$		0			0
$28$	0.475385	+	0.823392i	0.0898394	+	0.155606i
$29$	0.110938	+	0.192150i	0.0206007	+	0.0356814i	0.876142	−	0.482053i	$-0.160109\pi$
$29$	0.110938	+	0.192150i	0.0206007	+	0.0356814i	−0.855541	+	0.517735i	$0.826775\pi$
$30$		0			0
$31$	2.50702			0.450274			0.225137	−	0.974327i	$-0.427717\pi$
$31$	2.50702			0.450274			0.225137	+	0.974327i	$0.427717\pi$
$32$	−1.53163	−	2.65287i	−0.270757	−	0.468965i
$33$		0			0
$34$	8.87147	+	15.3658i	1.52144	+	2.63522i
$35$	0.110938	−	0.192150i	0.0187520	−	0.0324793i
$36$		0			0
$37$	−1.90466			−0.313124			−0.156562	−	0.987668i	$-0.550041\pi$
$37$	−1.90466			−0.313124			−0.156562	+	0.987668i	$0.550041\pi$
$38$	7.33983	+	8.09596i	1.19068	+	1.31334i
$39$		0			0
$40$	−2.86445	+	4.96137i	−0.452909	+	0.784461i
$41$	−3.61796	+	6.26648i	−0.565030	+	0.978661i	0.432017	+	0.901865i	$0.357802\pi$
$41$	−3.61796	+	6.26648i	−0.565030	+	0.978661i	−0.997047	+	0.0767950i	$0.975531\pi$
$42$		0			0
$43$	−3.64959	+	6.32128i	−0.556557	+	0.963985i	0.441223	+	0.897397i	$0.354545\pi$
$43$	−3.64959	+	6.32128i	−0.556557	+	0.963985i	−0.997781	+	0.0665881i	$0.978789\pi$
$44$	−1.66719	−	2.88765i	−0.251338	−	0.435330i
$45$		0			0
$46$	−20.2500			−2.98569
$47$	−1.39608	−	2.41808i	−0.203639	−	0.352714i	0.746059	−	0.665880i	$-0.231944\pi$
$47$	−1.39608	−	2.41808i	−0.203639	−	0.352714i	−0.949698	+	0.313166i	$0.898610\pi$
$48$		0			0
$49$	−6.95077			−0.992967
$50$	2.50702			0.354546
$51$		0			0
$52$	10.7129	−	18.5552i	1.48561	−	2.57314i
$53$	2.19024	+	3.79361i	0.300853	+	0.521093i	0.976329	−	0.216289i	$-0.0693953\pi$
$53$	2.19024	+	3.79361i	0.300853	+	0.521093i	−0.675476	+	0.737382i	$0.736062\pi$
$54$		0			0
$55$	−0.389062	+	0.673875i	−0.0524611	+	0.0908653i
$56$	−1.27111			−0.169859
$57$		0			0
$58$	−0.556248			−0.0730389
$59$	−1.39608	+	2.41808i	−0.181754	+	0.314808i	−0.942478	−	0.334268i	$-0.891511\pi$
$59$	−1.39608	+	2.41808i	−0.181754	+	0.314808i	0.760724	+	0.649076i	$0.224844\pi$
$60$		0			0
$61$	6.29216	+	10.8983i	0.805629	+	1.39539i	0.915866	+	0.401484i	$0.131506\pi$
$61$	6.29216	+	10.8983i	0.805629	+	1.39539i	−0.110237	+	0.993905i	$0.535161\pi$
$62$	−3.14257	+	5.44309i	−0.399107	+	0.691274i
$63$		0			0
$64$	−3.90466			−0.488082
$65$	−5.00000			−0.620174
$66$		0			0
$67$	5.28514	+	9.15414i	0.645683	+	1.11836i	0.984143	+	0.177375i	$0.0567605\pi$
$67$	5.28514	+	9.15414i	0.645683	+	1.11836i	−0.338460	+	0.940981i	$0.609906\pi$
$68$	−30.3273			−3.67772
$69$		0			0
$70$	0.278124	+	0.481725i	0.0332422	+	0.0575771i
$71$	−4.92070	+	8.52289i	−0.583979	+	1.01148i	0.411023	+	0.911625i	$0.365172\pi$
$71$	−4.92070	+	8.52289i	−0.583979	+	1.01148i	−0.995002	+	0.0998563i	$0.968162\pi$
$72$		0			0
$73$	7.03865	−	12.1913i	0.823812	−	1.42688i	−0.0790121	−	0.996874i	$-0.525177\pi$
$73$	7.03865	−	12.1913i	0.823812	−	1.42688i	0.902824	−	0.430010i	$-0.141490\pi$
$74$	2.38750	−	4.13528i	0.277542	−	0.480716i
$75$		0			0
$76$	−18.2570	+	3.94583i	−2.09422	+	0.452617i
$77$	−0.172647			−0.0196750
$78$		0			0
$79$	−0.792161	+	1.37206i	−0.0891251	+	0.154369i	−0.907142	−	0.420826i	$-0.861740\pi$
$79$	−0.792161	+	1.37206i	−0.0891251	+	0.154369i	0.818016	+	0.575195i	$0.195074\pi$
$80$	−2.89608	−	5.01616i	−0.323792	−	0.560824i
$81$		0			0
$82$	−9.07028	−	15.7102i	−1.00165	−	1.73490i
$83$	9.52106			1.04507			0.522536	−	0.852617i	$-0.324986\pi$
$83$	9.52106			1.04507			0.522536	+	0.852617i	$0.324986\pi$
$84$		0			0
$85$	3.53865	+	6.12912i	0.383821	+	0.664797i
$86$	−9.14959	−	15.8476i	−0.986626	−	1.70889i
$87$		0			0
$88$	4.45779			0.475202
$89$	1.57028	+	2.71981i	0.166450	+	0.288300i	0.937169	−	0.348875i	$-0.113436\pi$
$89$	1.57028	+	2.71981i	0.166450	+	0.288300i	−0.770719	+	0.637175i	$0.780103\pi$
$90$		0			0
$91$	−0.554690	−	0.960752i	−0.0581474	−	0.100714i
$92$	17.3062	−	29.9752i	1.80430	−	3.12513i
$93$		0			0
$94$	7.00000			0.721995
$95$	2.92771	+	3.22932i	0.300377	+	0.331321i
$96$		0			0
$97$	3.18122	−	5.51004i	0.323004	−	0.559460i	−0.658102	−	0.752929i	$-0.728641\pi$
$97$	3.18122	−	5.51004i	0.323004	−	0.559460i	0.981106	+	0.193469i	$0.0619738\pi$
$98$	8.71286	−	15.0911i	0.880131	−	1.52443i
$99$		0			0
$100$	−2.14257	+	3.71104i	−0.214257	+	0.371104i
$101$	−3.69726	−	6.40385i	−0.367891	−	0.637206i	0.621344	−	0.783538i	$-0.286587\pi$
$101$	−3.69726	−	6.40385i	−0.367891	−	0.637206i	−0.989236	+	0.146331i	$0.953253\pi$
$102$		0			0
$103$	12.2038			1.20248			0.601240	−	0.799069i	$-0.294674\pi$
$103$	12.2038			1.20248			0.601240	+	0.799069i	$0.294674\pi$
$104$	14.3222	+	24.8068i	1.40441	+	2.43251i
$105$		0			0
$106$	−10.9820			−1.06666
$107$	1.63355			0.157921			0.0789607	−	0.996878i	$-0.474840\pi$
$107$	1.63355			0.157921			0.0789607	+	0.996878i	$0.474840\pi$
$108$		0			0
$109$	3.80820	−	6.59600i	0.364759	−	0.631782i	−0.623978	−	0.781442i	$-0.714485\pi$
$109$	3.80820	−	6.59600i	0.364759	−	0.631782i	0.988738	+	0.149660i	$0.0478179\pi$
$110$	−0.975385	−	1.68942i	−0.0929994	−	0.161080i
$111$		0			0
$112$	0.642571	−	1.11297i	0.0607173	−	0.105165i
$113$	−12.4890			−1.17486			−0.587432	−	0.809273i	$-0.699861\pi$
$113$	−12.4890			−1.17486			−0.587432	+	0.809273i	$0.699861\pi$
$114$		0			0
$115$	−8.07730			−0.753212
$116$	0.475385	−	0.823392i	0.0441384	−	0.0764500i
$117$		0			0
$118$	−3.50000	−	6.06218i	−0.322201	−	0.558069i
$119$	−0.785142	+	1.35991i	−0.0719739	+	0.124662i
$120$		0			0
$121$	−10.3945			−0.944957
$122$	−31.5491			−2.85632
$123$		0			0
$124$	−5.37147	−	9.30365i	−0.482372	−	0.835493i
$125$	1.00000			0.0894427
$126$		0			0
$127$	6.52106	+	11.2948i	0.578650	+	1.00225i	0.995635	+	0.0933378i	$0.0297536\pi$
$127$	6.52106	+	11.2948i	0.578650	+	1.00225i	−0.416984	+	0.908914i	$0.636913\pi$
$128$	7.95779	−	13.7833i	0.703376	−	1.21828i
$129$		0			0
$130$	6.26755	−	10.8557i	0.549700	−	0.952109i
$131$	3.55469	−	6.15690i	0.310575	−	0.537931i	−0.667912	−	0.744240i	$-0.732812\pi$
$131$	3.55469	−	6.15690i	0.310575	−	0.537931i	0.978487	+	0.206309i	$0.0661452\pi$
$132$		0			0
$133$	−0.295720	+	0.920816i	−0.0256422	+	0.0798448i
$134$	−26.4999			−2.28924
$135$		0			0
$136$	20.2726	−	35.1131i	1.73836	−	3.01092i
$137$	2.71286	+	4.69880i	0.231775	+	0.401446i	0.958331	−	0.285662i	$-0.0922134\pi$
$137$	2.71286	+	4.69880i	0.231775	+	0.401446i	−0.726556	+	0.687108i	$0.758880\pi$
$138$		0			0
$139$	−1.78514	−	3.09196i	−0.151414	−	0.262256i	0.780334	−	0.625363i	$-0.215049\pi$
$139$	−1.78514	−	3.09196i	−0.151414	−	0.262256i	−0.931747	+	0.363107i	$0.881716\pi$
$140$	−0.950771			−0.0803548
$141$		0			0
$142$	−12.3363	−	21.3671i	−1.03524	−	1.79308i
$143$	1.94531	+	3.36938i	0.162675	+	0.281761i
$144$		0			0
$145$	−0.221876			−0.0184258
$146$	17.6460	+	30.5638i	1.46040	+	2.52948i
$147$		0			0
$148$	4.08086	+	7.06826i	0.335445	+	0.581007i
$149$	−0.468367	+	0.811235i	−0.0383701	+	0.0664590i	−0.884573	−	0.466402i	$-0.845550\pi$
$149$	−0.468367	+	0.811235i	−0.0383701	+	0.0664590i	0.846203	+	0.532861i	$0.178883\pi$
$150$		0			0
$151$	−0.971925			−0.0790942			−0.0395471	−	0.999218i	$-0.512592\pi$
$151$	−0.971925			−0.0790942			−0.0395471	+	0.999218i	$0.512592\pi$
$152$	7.63555	−	23.7757i	0.619325	−	1.92846i
$153$		0			0
$154$	0.216415	−	0.374841i	0.0174392	−	0.0302056i
$155$	−1.25351	+	2.17114i	−0.100684	+	0.174390i
$156$		0			0
$157$	−1.35743	+	2.35114i	−0.108335	+	0.187641i	−0.915096	−	0.403237i	$-0.867885\pi$
$157$	−1.35743	+	2.35114i	−0.108335	+	0.187641i	0.806761	+	0.590878i	$0.201218\pi$
$158$	−1.98596	−	3.43979i	−0.157995	−	0.273655i
$159$		0			0
$160$	3.06327			0.242172
$161$	−0.896081	−	1.55206i	−0.0706210	−	0.122319i
$162$		0			0
$163$	−3.20384			−0.250944			−0.125472	−	0.992097i	$-0.540044\pi$
$163$	−3.20384			−0.250944			−0.125472	+	0.992097i	$0.540044\pi$
$164$	31.0069			2.42123
$165$		0			0
$166$	−11.9347	+	20.6716i	−0.926315	+	1.60442i
$167$	5.11094	+	8.85240i	0.395496	+	0.685020i	0.993164	−	0.116724i	$-0.0372393\pi$
$167$	5.11094	+	8.85240i	0.395496	+	0.685020i	−0.597668	+	0.801744i	$0.703906\pi$
$168$		0			0
$169$	−6.00000	+	10.3923i	−0.461538	+	0.799408i
$170$	−17.7429			−1.36082
$171$		0			0
$172$	31.2780			2.38493
$173$	−1.33281	+	2.30850i	−0.101332	+	0.175512i	−0.912234	−	0.409670i	$-0.865644\pi$
$173$	−1.33281	+	2.30850i	−0.101332	+	0.175512i	0.810902	+	0.585182i	$0.198977\pi$
$174$		0			0
$175$	0.110938	+	0.192150i	0.00838613	+	0.0145252i
$176$	−2.25351	+	3.90319i	−0.169865	+	0.294214i
$177$		0			0
$178$	−7.87347			−0.590141
$179$	12.9367			0.966937			0.483468	−	0.875362i	$-0.339377\pi$
$179$	12.9367			0.966937			0.483468	+	0.875362i	$0.339377\pi$
$180$		0			0
$181$	9.30620	+	16.1188i	0.691724	+	1.19810i	0.971273	+	0.237970i	$0.0764820\pi$
$181$	9.30620	+	16.1188i	0.691724	+	1.19810i	−0.279548	+	0.960132i	$0.590185\pi$
$182$	2.78124			0.206159
$183$		0			0
$184$	23.1370	+	40.0745i	1.70568	+	2.95433i
$185$	0.952328	−	1.64948i	0.0700166	−	0.121272i
$186$		0			0
$187$	2.75351	−	4.76922i	0.201357	−	0.348760i
$188$	−5.98240	+	10.3618i	−0.436312	+	0.755714i
$189$		0			0
$190$	−10.6812	+	2.30850i	−0.774897	+	0.167476i
$191$	23.0421			1.66727			0.833634	−	0.552317i	$-0.186256\pi$
$191$	23.0421			1.66727			0.833634	+	0.552317i	$0.186256\pi$
$192$		0			0
$193$	8.53865	−	14.7894i	0.614626	−	1.06456i	−0.375824	−	0.926691i	$-0.622640\pi$
$193$	8.53865	−	14.7894i	0.614626	−	1.06456i	0.990450	−	0.137872i	$-0.0440262\pi$
$194$	7.97539	+	13.8138i	0.572599	+	0.991771i
$195$		0			0
$196$	14.8925	+	25.7946i	1.06375	+	1.84247i
$197$	8.82024			0.628416			0.314208	−	0.949354i	$-0.398261\pi$
$197$	8.82024			0.628416			0.314208	+	0.949354i	$0.398261\pi$
$198$		0			0
$199$	1.42771	+	2.47287i	0.101208	+	0.175297i	0.912183	−	0.409784i	$-0.134396\pi$
$199$	1.42771	+	2.47287i	0.101208	+	0.175297i	−0.810975	+	0.585081i	$0.801063\pi$
$200$	−2.86445	−	4.96137i	−0.202547	−	0.350822i
$201$		0			0
$202$	18.5382			1.30434
$203$	−0.0246145	−	0.0426336i	−0.00172760	−	0.00299229i
$204$		0			0
$205$	−3.61796	−	6.26648i	−0.252689	−	0.437670i
$206$	−15.2976	+	26.4963i	−1.06584	+	1.84608i
$207$		0			0
$208$	−28.9608			−2.00807
$209$	1.03709	−	3.22932i	0.0717373	−	0.223377i
$210$		0			0
$211$	6.44375	−	11.1609i	0.443606	−	0.768348i	−0.554348	−	0.832285i	$-0.687032\pi$
$211$	6.44375	−	11.1609i	0.443606	−	0.768348i	0.997954	+	0.0639367i	$0.0203656\pi$
$212$	9.38550	−	16.2562i	0.644599	−	1.11648i
$213$		0			0
$214$	−2.04767	+	3.54667i	−0.139976	+	0.242445i
$215$	−3.64959	−	6.32128i	−0.248900	−	0.431107i
$216$		0			0
$217$	−0.556248			−0.0377606
$218$	9.54723	+	16.5363i	0.646620	+	1.11998i
$219$		0			0
$220$	3.33437			0.224803
$221$	35.3865			2.38035
$222$		0			0
$223$	−10.3539	+	17.9334i	−0.693346	+	1.20091i	0.277389	+	0.960758i	$0.410531\pi$
$223$	−10.3539	+	17.9334i	−0.693346	+	1.20091i	−0.970735	+	0.240153i	$0.922802\pi$
$224$	0.339833	+	0.588608i	0.0227060	+	0.0393280i
$225$		0			0
$226$	15.6551	−	27.1153i	1.04136	−	1.80369i
$227$	−4.00000			−0.265489			−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000			−0.265489			−0.132745	+	0.991150i	$0.542379\pi$
$228$		0			0
$229$	3.53910			0.233870			0.116935	−	0.993140i	$-0.462693\pi$
$229$	3.53910			0.233870			0.116935	+	0.993140i	$0.462693\pi$
$230$	10.1250	−	17.5370i	0.667621	−	1.15635i
$231$		0			0
$232$	0.635553	+	1.10081i	0.0417261	+	0.0722717i
$233$	−9.89252	+	17.1344i	−0.648081	+	1.12251i	0.335500	+	0.942040i	$0.391095\pi$
$233$	−9.89252	+	17.1344i	−0.648081	+	1.12251i	−0.983581	+	0.180468i	$0.942239\pi$
$234$		0			0
$235$	2.79216			0.182141
$236$	11.9648			0.778843
$237$		0			0
$238$	−1.96837	−	3.40931i	−0.127590	−	0.220993i
$239$	−23.7741			−1.53782			−0.768910	−	0.639357i	$-0.779201\pi$
$239$	−23.7741			−1.53782			−0.768910	+	0.639357i	$0.779201\pi$
$240$		0			0
$241$	2.27111	+	3.93367i	0.146295	+	0.253390i	0.929855	−	0.367926i	$-0.119932\pi$
$241$	2.27111	+	3.93367i	0.146295	+	0.253390i	−0.783561	+	0.621315i	$0.786599\pi$
$242$	13.0296	−	22.5680i	0.837576	−	1.45072i
$243$		0			0
$244$	26.9628	−	46.7010i	1.72612	−	2.98972i
$245$	3.47539	−	6.01954i	0.222034	−	0.384575i
$246$		0			0
$247$	21.3026	−	4.60408i	1.35545	−	0.292950i
$248$	14.3624			0.912016
$249$		0			0
$250$	−1.25351	+	2.17114i	−0.0792789	+	0.137315i
$251$	−9.75151	−	16.8901i	−0.615510	−	1.06609i	−0.990295	−	0.138982i	$-0.955617\pi$
$251$	−9.75151	−	16.8901i	−0.615510	−	1.06609i	0.374785	−	0.927112i	$-0.377716\pi$
$252$		0			0
$253$	3.14257	+	5.44309i	0.197572	+	0.342204i
$254$	−32.6968			−2.05158
$255$		0			0
$256$	16.0457	+	27.7919i	1.00285	+	1.73699i
$257$	0.882043	+	1.52774i	0.0550203	+	0.0952980i	0.892224	−	0.451594i	$-0.149144\pi$
$257$	0.882043	+	1.52774i	0.0550203	+	0.0952980i	−0.837203	+	0.546892i	$0.815811\pi$
$258$		0			0
$259$	0.422598			0.0262590
$260$	10.7129	+	18.5552i	0.664383	+	1.15075i
$261$		0			0
$262$	8.91168	+	15.4355i	0.550565	+	0.953607i
$263$	−3.23591	+	5.60477i	−0.199535	+	0.345605i	−0.948378	−	0.317143i	$-0.897276\pi$
$263$	−3.23591	+	5.60477i	−0.199535	+	0.345605i	0.748843	+	0.662748i	$0.230610\pi$
$264$		0			0
$265$	−4.38049			−0.269091
$266$	−1.62853	−	1.79630i	−0.0998518	−	0.110138i
$267$		0			0
$268$	22.6476	−	39.2268i	1.38342	−	2.39616i
$269$	14.2429	−	24.6695i	0.868407	−	1.50412i	0.00478280	−	0.999989i	$-0.498478\pi$
$269$	14.2429	−	24.6695i	0.868407	−	1.50412i	0.863624	−	0.504136i	$-0.168189\pi$
$270$		0			0
$271$	0.246491	−	0.426934i	0.0149732	−	0.0259344i	−0.858442	−	0.512911i	$-0.828567\pi$
$271$	0.246491	−	0.426934i	0.0149732	−	0.0259344i	0.873415	+	0.486977i	$0.161900\pi$
$272$	20.4964	+	35.5009i	1.24278	+	2.15256i
$273$		0			0
$274$	−13.6024			−0.821749
$275$	−0.389062	−	0.673875i	−0.0234613	−	0.0406362i
$276$		0			0
$277$	−8.78905			−0.528083			−0.264041	−	0.964511i	$-0.585056\pi$
$277$	−8.78905			−0.528083			−0.264041	+	0.964511i	$0.585056\pi$
$278$	8.95077			0.536832
$279$		0			0
$280$	0.635553	−	1.10081i	0.0379815	−	0.0657859i
$281$	−2.37147	−	4.10750i	−0.141470	−	0.245033i	0.786581	−	0.617488i	$-0.211849\pi$
$281$	−2.37147	−	4.10750i	−0.141470	−	0.245033i	−0.928050	+	0.372455i	$0.878516\pi$
$282$		0			0
$283$	−8.43473	+	14.6094i	−0.501393	+	0.868438i	0.498606	+	0.866829i	$0.333845\pi$
$283$	−8.43473	+	14.6094i	−0.501393	+	0.868438i	−0.999999	+	0.00160901i	$0.999488\pi$
$284$	42.1718			2.50243
$285$		0			0
$286$	−9.75385			−0.576758
$287$	0.802738	−	1.39038i	0.0473841	−	0.0820717i
$288$		0			0
$289$	−16.5441	−	28.6552i	−0.973183	−	1.68560i
$290$	0.278124	−	0.481725i	0.0163320	−	0.0282878i
$291$		0			0
$292$	−60.3233			−3.53015
$293$	−11.6336			−0.679639			−0.339820	−	0.940491i	$-0.610366\pi$
$293$	−11.6336			−0.679639			−0.339820	+	0.940491i	$0.610366\pi$
$294$		0			0
$295$	−1.39608	−	2.41808i	−0.0812830	−	0.140786i
$296$	−10.9116			−0.634223
$297$		0			0
$298$	−1.17420	−	2.03378i	−0.0680198	−	0.117814i
$299$	−20.1933	+	34.9758i	−1.16781	+	2.02270i
$300$		0			0
$301$	0.809757	−	1.40254i	0.0466736	−	0.0808411i
$302$	1.21832	−	2.11019i	0.0701063	−	0.121428i
$303$		0			0
$304$	16.9578	+	18.7047i	0.972596	+	1.07279i
$305$	−12.5843			−0.720576
$306$		0			0
$307$	−2.94531	+	5.10143i	−0.168098	+	0.291154i	−0.937751	−	0.347308i	$-0.887096\pi$
$307$	−2.94531	+	5.10143i	−0.168098	+	0.291154i	0.769653	+	0.638462i	$0.220429\pi$
$308$	0.369909	+	0.640701i	0.0210775	+	0.0365073i
$309$		0			0
$310$	−3.14257	−	5.44309i	−0.178486	−	0.309147i
$311$	−14.8202			−0.840378			−0.420189	−	0.907437i	$-0.638036\pi$
$311$	−14.8202			−0.840378			−0.420189	+	0.907437i	$0.638036\pi$
$312$		0			0
$313$	2.94731	+	5.10489i	0.166592	+	0.288546i	0.937219	−	0.348740i	$-0.113390\pi$
$313$	2.94731	+	5.10489i	0.166592	+	0.288546i	−0.770628	+	0.637286i	$0.780057\pi$
$314$	−3.40310	−	5.89434i	−0.192048	−	0.332637i
$315$		0			0
$316$	6.78905			0.381914
$317$	−2.07930	−	3.60146i	−0.116785	−	0.202278i	0.801707	−	0.597718i	$-0.203926\pi$
$317$	−2.07930	−	3.60146i	−0.116785	−	0.202278i	−0.918492	+	0.395439i	$0.870592\pi$
$318$		0			0
$319$	0.0863236	+	0.149517i	0.00483319	+	0.00837133i
$320$	1.95233	−	3.38153i	0.109138	−	0.189033i
$321$		0			0
$322$	4.49298			0.250384
$323$	−20.7203	−	22.8549i	−1.15291	−	1.27168i
$324$		0			0
$325$	2.50000	−	4.33013i	0.138675	−	0.240192i
$326$	4.01604	−	6.95598i	0.222428	−	0.385256i
$327$		0			0
$328$	−20.7269	+	35.9000i	−1.14445	+	1.98225i
$329$	0.309757	+	0.536515i	0.0170775	+	0.0295790i
$330$		0			0
$331$	10.6797			0.587008			0.293504	−	0.955958i	$-0.405179\pi$
$331$	10.6797			0.587008			0.293504	+	0.955958i	$0.405179\pi$
$332$	−20.3995	−	35.3330i	−1.11957	−	1.93915i
$333$		0			0
$334$	−25.6264			−1.40222
$335$	−10.5703			−0.577516
$336$		0			0
$337$	13.1726	−	22.8157i	0.717560	−	1.24285i	−0.244404	−	0.969673i	$-0.578592\pi$
$337$	13.1726	−	22.8157i	0.717560	−	1.24285i	0.961964	−	0.273177i	$-0.0880743\pi$
$338$	−15.0421	−	26.0537i	−0.818183	−	1.41713i
$339$		0			0
$340$	15.1636	−	26.2642i	0.822363	−	1.42437i
$341$	1.95077			0.105640
$342$		0			0
$343$	3.09534			0.167133
$344$	−20.9081	+	36.2139i	−1.12729	+	1.95252i
$345$		0			0
$346$	−3.34139	−	5.78746i	−0.179634	−	0.311136i
$347$	−8.44175	+	14.6215i	−0.453177	+	0.784925i	−0.998581	−	0.0532476i	$-0.983043\pi$
$347$	−8.44175	+	14.6215i	−0.453177	+	0.784925i	0.545404	+	0.838173i	$0.316376\pi$
$348$		0			0
$349$	4.61640			0.247110			0.123555	−	0.992338i	$-0.460570\pi$
$349$	4.61640			0.247110			0.123555	+	0.992338i	$0.460570\pi$
$350$	−0.556248			−0.0297327
$351$		0			0
$352$	−1.19180	−	2.06426i	−0.0635232	−	0.110025i
$353$	24.9508			1.32800			0.663998	−	0.747735i	$-0.268858\pi$
$353$	24.9508			1.32800			0.663998	+	0.747735i	$0.268858\pi$
$354$		0			0
$355$	−4.92070	−	8.52289i	−0.261163	−	0.452348i
$356$	6.72889	−	11.6548i	0.356631	−	0.617703i
$357$		0			0
$358$	−16.2163	+	28.0875i	−0.857059	+	1.48447i
$359$	5.90110	−	10.2210i	0.311448	−	0.539444i	−0.667228	−	0.744854i	$-0.732519\pi$
$359$	5.90110	−	10.2210i	0.311448	−	0.539444i	0.978676	+	0.205410i	$0.0658527\pi$
$360$		0			0
$361$	−15.4472	−	11.0627i	−0.813011	−	0.582248i
$362$	−46.6616			−2.45248
$363$		0			0
$364$	−2.37693	+	4.11696i	−0.124585	+	0.215787i
$365$	7.03865	+	12.1913i	0.368420	+	0.638122i
$366$		0			0
$367$	−13.1164	−	22.7183i	−0.684670	−	1.18588i	−0.973540	−	0.228516i	$-0.926613\pi$
$367$	−13.1164	−	22.7183i	−0.684670	−	1.18588i	0.288870	−	0.957368i	$-0.406721\pi$
$368$	−46.7850			−2.43884
$369$		0			0
$370$	2.38750	+	4.13528i	0.124120	+	0.214983i
$371$	−0.485963	−	0.841712i	−0.0252299	−	0.0436995i
$372$		0			0
$373$	11.9960			0.621129			0.310565	−	0.950552i	$-0.399482\pi$
$373$	11.9960			0.621129			0.310565	+	0.950552i	$0.399482\pi$
$374$	6.90310	+	11.9565i	0.356951	+	0.618257i
$375$		0			0
$376$	−7.99800	−	13.8529i	−0.412465	−	0.714411i
$377$	−0.554690	+	0.960752i	−0.0285680	+	0.0494812i
$378$		0			0
$379$	0.313217			0.0160889			0.00804444	−	0.999968i	$-0.497439\pi$
$379$	0.313217			0.0160889			0.00804444	+	0.999968i	$0.497439\pi$
$380$	5.71130	−	17.7839i	0.292983	−	0.912295i
$381$		0			0
$382$	−28.8835	+	50.0277i	−1.47781	+	2.55964i
$383$	15.0441	−	26.0572i	0.768718	−	1.33146i	−0.169540	−	0.985523i	$-0.554228\pi$
$383$	15.0441	−	26.0572i	0.768718	−	1.33146i	0.938258	−	0.345936i	$-0.112439\pi$
$384$		0			0
$385$	0.0863236	−	0.149517i	0.00439946	−	0.00762008i
$386$	21.4066	+	37.0772i	1.08957	+	1.88718i
$387$		0			0
$388$	−27.2640			−1.38412
$389$	−17.8609	−	30.9360i	−0.905583	−	1.56852i	−0.820133	−	0.572173i	$-0.806100\pi$
$389$	−17.8609	−	30.9360i	−0.905583	−	1.56852i	−0.0854503	−	0.996342i	$-0.527233\pi$
$390$		0			0
$391$	57.1655			2.89099
$392$	−39.8202			−2.01123
$393$		0			0
$394$	−11.0562	+	19.1500i	−0.557006	+	0.964762i
$395$	−0.792161	−	1.37206i	−0.0398580	−	0.0690360i
$396$		0			0
$397$	−4.77657	+	8.27326i	−0.239729	+	0.415223i	−0.960636	−	0.277809i	$-0.910392\pi$
$397$	−4.77657	+	8.27326i	−0.239729	+	0.415223i	0.720907	+	0.693031i	$0.243725\pi$
$398$	−7.15861			−0.358829
$399$		0			0
$400$	5.79216			0.289608
$401$	15.4418	−	26.7459i	0.771124	−	1.33563i	−0.165823	−	0.986156i	$-0.553028\pi$
$401$	15.4418	−	26.7459i	0.771124	−	1.33563i	0.936947	−	0.349471i	$-0.113639\pi$
$402$		0			0
$403$	6.26755	+	10.8557i	0.312209	+	0.540761i
$404$	−15.8433	+	27.4414i	−0.788233	+	1.36526i
$405$		0			0
$406$	0.123418			0.00612514
$407$	−1.48206			−0.0734629
$408$		0			0
$409$	−15.7816	−	27.3345i	−0.780349	−	1.35160i	−0.931738	−	0.363130i	$-0.881708\pi$
$409$	−15.7816	−	27.3345i	−0.780349	−	1.35160i	0.151389	−	0.988474i	$-0.451625\pi$
$410$	18.1406			0.895899
$411$		0			0
$412$	−26.1476	−	45.2890i	−1.28820	−	2.23123i
$413$	0.309757	−	0.536515i	0.0152421	−	0.0264002i
$414$		0			0
$415$	−4.76053	+	8.24548i	−0.233685	+	0.404755i
$416$	7.65817	−	13.2643i	0.375472	−	0.650337i
$417$		0			0
$418$	5.71130	+	6.29966i	0.279349	+	0.308126i
$419$	26.5070			1.29495			0.647476	−	0.762086i	$-0.275824\pi$
$419$	26.5070			1.29495			0.647476	+	0.762086i	$0.275824\pi$
$420$		0			0
$421$	10.1180	−	17.5248i	0.493119	−	0.854107i	−0.506850	−	0.862035i	$-0.669190\pi$
$421$	10.1180	−	17.5248i	0.493119	−	0.854107i	0.999969	+	0.00792731i	$0.00252337\pi$
$422$	16.1546	+	27.9806i	0.786394	+	1.36207i
$423$		0			0
$424$	12.5477	+	21.7332i	0.609369	+	1.05546i
$425$	−7.07730			−0.343300
$426$		0			0
$427$	−1.39608	−	2.41808i	−0.0675611	−	0.117019i
$428$	−3.50000	−	6.06218i	−0.169179	−	0.293026i
$429$		0			0
$430$	18.2992			0.882465
$431$	−5.73045	−	9.92543i	−0.276026	−	0.478091i	0.694367	−	0.719621i	$-0.255684\pi$
$431$	−5.73045	−	9.92543i	−0.276026	−	0.478091i	−0.970394	+	0.241529i	$0.922351\pi$
$432$		0			0
$433$	−12.9734	−	22.4706i	−0.623461	−	1.07987i	−0.988836	−	0.149006i	$-0.952393\pi$
$433$	−12.9734	−	22.4706i	−0.623461	−	1.07987i	0.365375	−	0.930860i	$-0.380941\pi$
$434$	0.697262	−	1.20769i	0.0334696	−	0.0579711i
$435$		0			0
$436$	−32.6374			−1.56305
$437$	34.4136	−	7.43771i	1.64622	−	0.355794i
$438$		0			0
$439$	−12.6562	+	21.9211i	−0.604046	+	1.04624i	0.388156	+	0.921594i	$0.373112\pi$
$439$	−12.6562	+	21.9211i	−0.604046	+	1.04624i	−0.992202	+	0.124644i	$0.960221\pi$
$440$	−2.22889	+	3.86056i	−0.106258	+	0.184045i
$441$		0			0
$442$	−44.3573	+	76.8291i	−2.10986	+	3.65439i
$443$	10.0125	+	17.3421i	0.475707	+	0.823949i	0.999613	−	0.0278272i	$-0.00885883\pi$
$443$	10.0125	+	17.3421i	0.475707	+	0.823949i	−0.523905	+	0.851776i	$0.675525\pi$
$444$		0			0
$445$	−3.14057			−0.148877
$446$	−25.9573	−	44.9594i	−1.22912	−	2.12889i
$447$		0			0
$448$	0.866350			0.0409312
$449$	−19.7601			−0.932536			−0.466268	−	0.884644i	$-0.654402\pi$
$449$	−19.7601			−0.932536			−0.466268	+	0.884644i	$0.654402\pi$
$450$		0			0
$451$	−2.81522	+	4.87610i	−0.132563	+	0.229607i
$452$	26.7585	+	46.3471i	1.25862	+	2.17999i
$453$		0			0
$454$	5.01404	−	8.68457i	0.235320	−	0.407587i
$455$	1.10938			0.0520086
$456$		0			0
$457$	−34.4647			−1.61219			−0.806096	−	0.591785i	$-0.798423\pi$
$457$	−34.4647			−1.61219			−0.806096	+	0.591785i	$0.798423\pi$
$458$	−4.43629	+	7.68388i	−0.207294	+	0.359044i
$459$		0			0
$460$	17.3062	+	29.9752i	0.806906	+	1.39760i
$461$	−3.96637	+	6.86995i	−0.184732	+	0.319965i	−0.943486	−	0.331412i	$-0.892475\pi$
$461$	−3.96637	+	6.86995i	−0.184732	+	0.319965i	0.758754	+	0.651377i	$0.225808\pi$
$462$		0			0
$463$	9.25395			0.430068			0.215034	−	0.976607i	$-0.431014\pi$
$463$	9.25395			0.430068			0.215034	+	0.976607i	$0.431014\pi$
$464$	−1.28514			−0.0596612
$465$		0			0
$466$	−24.8007	−	42.9561i	−1.14887	−	1.98990i
$467$	−9.47183			−0.438304			−0.219152	−	0.975691i	$-0.570329\pi$
$467$	−9.47183			−0.438304			−0.219152	+	0.975691i	$0.570329\pi$
$468$		0			0
$469$	−1.17265	−	2.03108i	−0.0541478	−	0.0937868i
$470$	−3.50000	+	6.06218i	−0.161443	+	0.279627i
$471$		0			0
$472$	−7.99800	+	13.8529i	−0.368138	+	0.637633i
$473$	−2.83983	+	4.91873i	−0.130576	+	0.226164i
$474$		0			0
$475$	−4.26053	+	0.920816i	−0.195486	+	0.0422499i
$476$	6.72889			0.308418
$477$		0			0
$478$	29.8011	−	51.6170i	1.36307	−	2.36091i
$479$	−17.3574	−	30.0639i	−0.793081	−	1.37366i	−0.924050	−	0.382270i	$-0.875142\pi$
$479$	−17.3574	−	30.0639i	−0.793081	−	1.37366i	0.130969	−	0.991386i	$-0.458191\pi$
$480$		0			0
$481$	−4.76164	−	8.24740i	−0.217112	−	0.376049i
$482$	−11.3874			−0.518682
$483$		0			0
$484$	22.2710	+	38.5745i	1.01232	+	1.75339i
$485$	3.18122	+	5.51004i	0.144452	+	0.250198i
$486$		0			0
$487$	−28.5070			−1.29178			−0.645888	−	0.763432i	$-0.723513\pi$
$487$	−28.5070			−1.29178			−0.645888	+	0.763432i	$0.723513\pi$
$488$	36.0471	+	62.4355i	1.63178	+	2.82632i
$489$		0			0
$490$	8.71286	+	15.0911i	0.393607	+	0.681747i
$491$	−15.5949	+	27.0112i	−0.703788	+	1.21900i	0.263339	+	0.964703i	$0.415176\pi$
$491$	−15.5949	+	27.0112i	−0.703788	+	1.21900i	−0.967127	+	0.254293i	$0.918157\pi$
$492$		0			0
$493$	1.57028			0.0707221
$494$	−16.7070	+	52.0223i	−0.751681	+	2.34059i
$495$		0			0
$496$	−7.26053	+	12.5756i	−0.326007	+	0.564661i
$497$	1.09178	−	1.89103i	0.0489732	−	0.0848242i
$498$		0			0
$499$	−8.09334	+	14.0181i	−0.362308	+	0.627535i	−0.988340	−	0.152262i	$-0.951344\pi$
$499$	−8.09334	+	14.0181i	−0.362308	+	0.627535i	0.626032	+	0.779797i	$0.284678\pi$
$500$	−2.14257	−	3.71104i	−0.0958187	−	0.165963i
$501$		0			0
$502$	48.8944			2.18226
$503$	8.61094	+	14.9146i	0.383943	+	0.665008i	0.991622	−	0.129174i	$-0.0412327\pi$
$503$	8.61094	+	14.9146i	0.383943	+	0.665008i	−0.607679	+	0.794183i	$0.707899\pi$
$504$		0			0
$505$	7.39452			0.329052
$506$	−15.7570			−0.700483
$507$		0			0
$508$	27.9437	−	48.3998i	1.23980	−	2.14740i
$509$	−18.2956	−	31.6889i	−0.810939	−	1.40459i	−0.912207	−	0.409729i	$-0.865623\pi$
$509$	−18.2956	−	31.6889i	−0.810939	−	1.40459i	0.101268	−	0.994859i	$-0.467710\pi$
$510$		0			0
$511$	−1.56171	+	2.70496i	−0.0690859	+	0.119660i
$512$	−48.6224			−2.14883
$513$		0			0
$514$	−4.42260			−0.195072
$515$	−6.10192	+	10.5688i	−0.268883	+	0.465718i
$516$		0			0
$517$	−1.08632	−	1.88157i	−0.0477765	−	0.0827512i
$518$	−0.529730	+	0.917520i	−0.0232750	+	0.0403135i
$519$		0			0
$520$	−28.6445			−1.25614
$521$	3.63667			0.159325			0.0796626	−	0.996822i	$-0.474616\pi$
$521$	3.63667			0.159325			0.0796626	+	0.996822i	$0.474616\pi$
$522$		0			0
$523$	17.8117	+	30.8507i	0.778850	+	1.34901i	0.932605	+	0.360898i	$0.117530\pi$
$523$	17.8117	+	30.8507i	0.778850	+	1.34901i	−0.153756	+	0.988109i	$0.549137\pi$
$524$	−30.4647			−1.33086
$525$		0			0
$526$	−8.11250	−	14.0513i	−0.353722	−	0.612664i
$527$	8.87147	−	15.3658i	0.386447	−	0.669346i
$528$		0			0
$529$	−21.1214	+	36.5834i	−0.918322	+	1.59058i
$530$	5.49098	−	9.51066i	0.238513	−	0.413117i
$531$		0			0
$532$	4.05079	−	0.875485i	0.175624	−	0.0379571i
$533$	−36.1796			−1.56711
$534$		0			0
$535$	−0.816776	+	1.41470i	−0.0353123	+	0.0611627i
$536$	30.2780	+	52.4431i	1.30781	+	2.26520i
$537$		0			0
$538$	35.7073	+	61.8469i	1.53945	+	2.66641i
$539$	−5.40856			−0.232963
$540$		0			0
$541$	−1.58632	−	2.74759i	−0.0682014	−	0.118128i	0.829908	−	0.557900i	$-0.188393\pi$
$541$	−1.58632	−	2.74759i	−0.0682014	−	0.118128i	−0.898110	+	0.439772i	$0.855059\pi$
$542$	0.617957	+	1.07033i	0.0265435	+	0.0459747i
$543$		0			0
$544$	−21.6797			−0.929508
$545$	3.80820	+	6.59600i	0.163125	+	0.282541i
$546$		0			0
$547$	14.1902	+	24.5782i	0.606731	+	1.05089i	0.991775	+	0.127991i	$0.0408528\pi$
$547$	14.1902	+	24.5782i	0.606731	+	1.05089i	−0.385044	+	0.922898i	$0.625814\pi$
$548$	11.6250	−	20.1350i	0.496594	−	0.860127i
$549$		0			0
$550$	1.95077			0.0831812
$551$	0.945310	−	0.204307i	0.0402715	−	0.00870377i
$552$		0			0
$553$	0.175762	−	0.304428i	0.00747415	−	0.0129456i
$554$	11.0172	−	19.0823i	0.468074	−	0.810728i
$555$		0			0
$556$	−7.64959	+	13.2495i	−0.324415	+	0.561903i
$557$	1.06527	+	1.84510i	0.0451368	+	0.0781793i	0.887711	−	0.460401i	$-0.152294\pi$
$557$	1.06527	+	1.84510i	0.0451368	+	0.0781793i	−0.842574	+	0.538580i	$0.818961\pi$
$558$		0			0
$559$	−36.4959			−1.54361
$560$	0.642571	+	1.11297i	0.0271536	+	0.0470314i
$561$		0			0
$562$	11.8906			0.501575
$563$	20.2609			0.853894			0.426947	−	0.904277i	$-0.359589\pi$
$563$	20.2609			0.853894			0.426947	+	0.904277i	$0.359589\pi$
$564$		0			0
$565$	6.24449	−	10.8158i	0.262708	−	0.455023i
$566$	−21.1460	−	36.6260i	−0.888834	−	1.53951i
$567$		0			0
$568$	−28.1901	+	48.8268i	−1.18283	+	2.04873i
$569$	34.7530			1.45692			0.728460	−	0.685088i	$-0.240236\pi$
$569$	34.7530			1.45692			0.728460	+	0.685088i	$0.240236\pi$
$570$		0			0
$571$	32.0702			1.34210			0.671048	−	0.741414i	$-0.265845\pi$
$571$	32.0702			1.34210			0.671048	+	0.741414i	$0.265845\pi$
$572$	8.33593	−	14.4383i	0.348543	−	0.603694i
$573$		0			0
$574$	2.01248	+	3.48572i	0.0839993	+	0.145491i
$575$	4.03865	−	6.99515i	0.168423	−	0.291718i
$576$		0			0
$577$	30.2740			1.26032			0.630162	−	0.776464i	$-0.282988\pi$
$577$	30.2740			1.26032			0.630162	+	0.776464i	$0.282988\pi$
$578$	82.9528			3.45038
$579$		0			0
$580$	0.475385	+	0.823392i	0.0197393	+	0.0341895i
$581$	−2.11250			−0.0876411
$582$		0			0
$583$	1.70428	+	2.95190i	0.0705841	+	0.122255i
$584$	40.3237	−	69.8427i	1.66861	−	2.89011i
$585$		0			0
$586$	14.5828	−	25.2581i	0.602408	−	1.04340i
$587$	1.24805	−	2.16168i	0.0515125	−	0.0892222i	−0.839119	−	0.543947i	$-0.816929\pi$
$587$	1.24805	−	2.16168i	0.0515125	−	0.0892222i	0.890632	+	0.454725i	$0.150262\pi$
$588$		0			0
$589$	3.34139	−	10.4045i	0.137680	−	0.428708i
$590$	7.00000			0.288185
$591$		0			0
$592$	5.51604	−	9.55406i	0.226708	−	0.392669i
$593$	−4.98196	−	8.62901i	−0.204585	−	0.354351i	0.745416	−	0.666600i	$-0.232251\pi$
$593$	−4.98196	−	8.62901i	−0.204585	−	0.354351i	−0.950000	+	0.312249i	$0.898918\pi$
$594$		0			0
$595$	−0.785142	−	1.35991i	−0.0321877	−	0.0557507i
$596$	4.01404			0.164421
$597$		0			0
$598$	−50.6249	−	87.6849i	−2.07021	−	3.58570i
$599$	16.0351	+	27.7736i	0.655176	+	1.13480i	0.981850	+	0.189661i	$0.0607389\pi$
$599$	16.0351	+	27.7736i	0.655176	+	1.13480i	−0.326673	+	0.945137i	$0.605928\pi$
$600$		0			0
$601$	3.68278			0.150224			0.0751119	−	0.997175i	$-0.476069\pi$
$601$	3.68278			0.150224			0.0751119	+	0.997175i	$0.476069\pi$
$602$	2.03008	+	3.51619i	0.0827397	+	0.143309i
$603$		0			0
$604$	2.08242	+	3.60686i	0.0847324	+	0.146761i
$605$	5.19726	−	9.00192i	0.211299	−	0.365980i
$606$		0			0
$607$	−20.4850			−0.831460			−0.415730	−	0.909488i	$-0.636474\pi$
$607$	−20.4850			−0.831460			−0.415730	+	0.909488i	$0.636474\pi$
$608$	−13.0511	+	2.82070i	−0.529293	+	0.114395i
$609$		0			0
$610$	15.7746	−	27.3223i	0.638693	−	1.10625i
$611$	6.98040	−	12.0904i	0.282397	−	0.489126i
$612$		0			0
$613$	−5.35587	+	9.27664i	−0.216322	+	0.374680i	−0.953681	−	0.300821i	$-0.902739\pi$
$613$	−5.35587	+	9.27664i	−0.216322	+	0.374680i	0.737359	+	0.675501i	$0.236073\pi$
$614$	−7.38395	−	12.7894i	−0.297992	−	0.516137i
$615$		0			0
$616$	−0.989077			−0.0398511
$617$	−8.86600	−	15.3564i	−0.356932	−	0.618224i	0.630515	−	0.776177i	$-0.282844\pi$
$617$	−8.86600	−	15.3564i	−0.356932	−	0.618224i	−0.987447	+	0.157953i	$0.949511\pi$
$618$		0			0
$619$	−13.2780			−0.533689			−0.266844	−	0.963740i	$-0.585981\pi$
$619$	−13.2780			−0.533689			−0.266844	+	0.963740i	$0.585981\pi$
$620$	10.7429			0.431447
$621$		0			0
$622$	18.5773	−	32.1768i	0.744882	−	1.29017i
$623$	−0.348409	−	0.603462i	−0.0139587	−	0.0241772i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$	−14.7779			−0.590645
$627$		0			0
$628$	11.6336			0.464229
$629$	−6.73992	+	11.6739i	−0.268738	+	0.465468i
$630$		0			0
$631$	−2.03208	−	3.51966i	−0.0808957	−	0.140115i	0.822739	−	0.568419i	$-0.192445\pi$
$631$	−2.03208	−	3.51966i	−0.0808957	−	0.140115i	−0.903635	+	0.428304i	$0.859111\pi$
$632$	−4.53821	+	7.86041i	−0.180520	+	0.312670i
$633$		0			0
$634$	10.4257			0.414058
$635$	−13.0421			−0.517560
$636$		0			0
$637$	−17.3769	−	30.0977i	−0.688499	−	1.19252i
$638$	−0.432830			−0.0171359
$639$		0			0
$640$	7.95779	+	13.7833i	0.314559	+	0.544833i
$641$	2.49254	−	4.31720i	0.0984493	−	0.170519i	−0.812594	−	0.582831i	$-0.801945\pi$
$641$	2.49254	−	4.31720i	0.0984493	−	0.170519i	0.911043	+	0.412311i	$0.135278\pi$
$642$		0			0
$643$	2.93829	−	5.08927i	0.115875	−	0.200701i	−0.802254	−	0.596983i	$-0.796366\pi$
$643$	2.93829	−	5.08927i	0.115875	−	0.200701i	0.918129	+	0.396281i	$0.129700\pi$
$644$	−3.83983	+	6.65079i	−0.151311	+	0.262078i
$645$		0			0
$646$	75.5943	−	16.3380i	2.97422	−	0.642809i
$647$	−16.9757			−0.667385			−0.333692	−	0.942682i	$-0.608295\pi$
$647$	−16.9757			−0.667385			−0.333692	+	0.942682i	$0.608295\pi$
$648$		0			0
$649$	−1.08632	+	1.88157i	−0.0426419	+	0.0738580i
$650$	6.26755	+	10.8557i	0.245833	+	0.425796i
$651$		0			0
$652$	6.86445	+	11.8896i	0.268833	+	0.465632i
$653$	−24.6797			−0.965790			−0.482895	−	0.875678i	$-0.660415\pi$
$653$	−24.6797			−0.965790			−0.482895	+	0.875678i	$0.660415\pi$
$654$		0			0
$655$	3.55469	+	6.15690i	0.138893	+	0.240570i
$656$	−20.9558	−	36.2965i	−0.818186	−	1.41714i
$657$		0			0
$658$	−1.55313			−0.0605474
$659$	0.943308	+	1.63386i	0.0367461	+	0.0636461i	0.883814	−	0.467839i	$-0.154967\pi$
$659$	0.943308	+	1.63386i	0.0367461	+	0.0636461i	−0.847067	+	0.531485i	$0.821634\pi$
$660$		0			0
$661$	−22.3749	−	38.7545i	−0.870284	−	1.50738i	−0.861703	−	0.507413i	$-0.830602\pi$
$661$	−22.3749	−	38.7545i	−0.870284	−	1.50738i	−0.00858048	−	0.999963i	$-0.502731\pi$
$662$	−13.3871	+	23.1871i	−0.520303	+	0.901191i
$663$		0			0
$664$	54.5451			2.11676
$665$	−0.649590	−	0.716509i	−0.0251900	−	0.0277850i
$666$		0			0
$667$	−0.896081	+	1.55206i	−0.0346964	+	0.0600959i
$668$	21.9011	−	37.9338i	0.847379	−	1.46770i
$669$		0			0
$670$	13.2500	−	22.9496i	0.511890	−	0.886620i
$671$	4.89608	+	8.48026i	0.189011	+	0.327377i
$672$		0			0
$673$	25.4046			0.979274			0.489637	−	0.871926i	$-0.337129\pi$
$673$	25.4046			0.979274			0.489637	+	0.871926i	$0.337129\pi$
$674$	33.0241	+	57.1994i	1.27204	+	2.20324i
$675$		0			0
$676$	51.4217			1.97776
$677$	3.86946			0.148716			0.0743578	−	0.997232i	$-0.476309\pi$
$677$	3.86946			0.148716			0.0743578	+	0.997232i	$0.476309\pi$
$678$		0			0
$679$	−0.705838	+	1.22255i	−0.0270876	+	0.0469170i
$680$	20.2726	+	35.1131i	0.777417	+	1.34653i
$681$		0			0
$682$	−2.44531	+	4.23540i	−0.0936357	+	0.162182i
$683$	48.5171			1.85645			0.928227	−	0.372015i	$-0.121333\pi$
$683$	48.5171			1.85645			0.928227	+	0.372015i	$0.121333\pi$
$684$		0			0
$685$	−5.42571			−0.207306
$686$	−3.88004	+	6.72043i	−0.148141	+	0.256587i
$687$		0			0
$688$	−21.1390	−	36.6138i	−0.805917	−	1.39589i
$689$	−10.9512	+	18.9681i	−0.417208	+	0.722626i
$690$		0			0
$691$	−47.6374			−1.81221			−0.906105	−	0.423052i	$-0.860959\pi$
$691$	−47.6374			−1.81221			−0.906105	+	0.423052i	$0.860959\pi$
$692$	11.4226			0.434222
$693$		0			0
$694$	−21.1636	−	36.6565i	−0.803360	−	1.39146i
$695$	3.57028			0.135429
$696$		0			0
$697$	25.6054	+	44.3498i	0.969873	+	1.67987i
$698$	−5.78670	+	10.0229i	−0.219030	+	0.379371i
$699$		0			0
$700$	0.475385	−	0.823392i	0.0179679	−	0.0311213i
$701$	14.2766	−	24.7277i	0.539218	−	0.933954i	−0.459728	−	0.888060i	$-0.652053\pi$
$701$	14.2766	−	24.7277i	0.539218	−	0.933954i	0.998946	−	0.0458939i	$-0.0146136\pi$
$702$		0			0
$703$	−2.53855	+	7.90458i	−0.0957434	+	0.298127i
$704$	−3.03831			−0.114510
$705$		0			0
$706$	−31.2760	+	54.1717i	−1.17709	+	2.03878i
$707$	0.820334	+	1.42086i	0.0308518	+	0.0534370i
$708$		0			0
$709$	−9.74137	−	16.8726i	−0.365845	−	0.633662i	0.623066	−	0.782169i	$-0.285887\pi$
$709$	−9.74137	−	16.8726i	−0.365845	−	0.633662i	−0.988911	+	0.148507i	$0.952553\pi$
$710$	24.6725			0.925944
$711$		0			0
$712$	8.99600	+	15.5815i	0.337139	+	0.583942i
$713$	10.1250	+	17.5370i	0.379183	+	0.656765i
$714$		0			0
$715$	−3.89062			−0.145501
$716$	−27.7179	−	48.0088i	−1.03587	−	1.79417i
$717$		0			0
$718$	14.7942	+	25.6242i	0.552113	+	0.956288i
$719$	2.05313	−	3.55613i	0.0765689	−	0.132621i	−0.825199	−	0.564843i	$-0.808937\pi$
$719$	2.05313	−	3.55613i	0.0765689	−	0.132621i	0.901767	+	0.432221i	$0.142270\pi$
$720$		0			0
$721$	−2.70774			−0.100842
$722$	43.3819	−	19.6709i	1.61451	−	0.732074i
$723$		0			0
$724$	39.8784	−	69.0714i	1.48207	−	2.56702i
$725$	0.110938	−	0.192150i	0.00412014	−	0.00713629i
$726$		0			0
$727$	6.20584	−	10.7488i	0.230162	−	0.398652i	−0.727694	−	0.685902i	$-0.759408\pi$
$727$	6.20584	−	10.7488i	0.230162	−	0.398652i	0.957856	+	0.287250i	$0.0927411\pi$
$728$	−3.17776	−	5.50405i	−0.117776	−	0.203994i
$729$		0			0
$730$	−35.2921			−1.30622
$731$	25.8293	+	44.7376i	0.955330	+	1.65468i
$732$		0			0
$733$	−16.3985			−0.605693			−0.302847	−	0.953039i	$-0.597937\pi$
$733$	−16.3985			−0.605693			−0.302847	+	0.953039i	$0.597937\pi$
$734$	65.7661			2.42747
$735$		0			0
$736$	12.3715	−	21.4280i	0.456018	−	0.789847i
$737$	4.11250	+	7.12305i	0.151486	+	0.262381i
$738$		0			0
$739$	23.1018	−	40.0135i	0.849814	−	1.47192i	−0.0315597	−	0.999502i	$-0.510047\pi$
$739$	23.1018	−	40.0135i	0.849814	−	1.47192i	0.881374	−	0.472419i	$-0.156619\pi$
$740$	−8.16172			−0.300031
$741$		0			0
$742$	2.43664			0.0894517
$743$	−12.4066	+	21.4888i	−0.455153	+	0.788347i	−0.998697	−	0.0510331i	$-0.983749\pi$
$743$	−12.4066	+	21.4888i	−0.455153	+	0.788347i	0.543544	+	0.839380i	$0.317082\pi$
$744$		0			0
$745$	−0.468367	−	0.811235i	−0.0171596	−	0.0297214i
$746$	−15.0371	+	26.0450i	−0.550547	+	0.953576i
$747$		0			0
$748$	−23.5984			−0.862841
$749$	−0.362446			−0.0132435
$750$		0			0
$751$	5.26955	+	9.12712i	0.192289	+	0.333054i	0.946008	−	0.324142i	$-0.105076\pi$
$751$	5.26955	+	9.12712i	0.192289	+	0.333054i	−0.753720	+	0.657196i	$0.771742\pi$
$752$	16.1726			0.589756
$753$		0			0
$754$	−1.39062	−	2.40862i	−0.0506434	−	0.0877169i
$755$	0.485963	−	0.841712i	0.0176860	−	0.0306330i
$756$		0			0
$757$	3.09836	−	5.36652i	0.112612	−	0.195049i	−0.804211	−	0.594344i	$-0.797412\pi$
$757$	3.09836	−	5.36652i	0.112612	−	0.195049i	0.916823	+	0.399295i	$0.130745\pi$
$758$	−0.392621	+	0.680039i	−0.0142606	+	0.0247001i
$759$		0			0
$760$	16.7726	+	18.5004i	0.608405	+	0.671081i
$761$	−35.6084			−1.29080			−0.645402	−	0.763843i	$-0.723310\pi$
$761$	−35.6084			−1.29080			−0.645402	+	0.763843i	$0.723310\pi$
$762$		0			0
$763$	−0.844949	+	1.46349i	−0.0305892	+	0.0529820i
$764$	−49.3694	−	85.5103i	−1.78612	−	3.09365i
$765$		0			0
$766$	37.7159	+	65.3258i	1.36273	+	2.36032i
$767$	−13.9608			−0.504095
$768$		0			0
$769$	0.0777477	+	0.134663i	0.00280365	+	0.00485607i	0.867424	−	0.497570i	$-0.165774\pi$
$769$	0.0777477	+	0.134663i	0.00280365	+	0.00485607i	−0.864620	+	0.502426i	$0.832441\pi$
$770$	0.216415	+	0.374841i	0.00779905	+	0.0135083i
$771$		0			0
$772$	−73.1787			−2.63376
$773$	−2.54411	−	4.40653i	−0.0915054	−	0.158492i	0.816639	−	0.577148i	$-0.195835\pi$
$773$	−2.54411	−	4.40653i	−0.0915054	−	0.158492i	−0.908145	+	0.418656i	$0.862501\pi$
$774$		0			0
$775$	−1.25351	−	2.17114i	−0.0450274	−	0.0779897i
$776$	18.2249	−	31.5664i	0.654236	−	1.13317i
$777$		0			0
$778$	89.5552			3.21071
$779$	21.1847	+	23.3671i	0.759020	+	0.837212i
$780$		0			0
$781$	−3.82891	+	6.63187i	−0.137009	+	0.237307i
$782$	−71.6575	+	124.114i	−2.56247	+	4.43832i
$783$		0

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:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.25351 + 2.17114i) q^{2} +(-2.14257 - 3.71104i) q^{4} +(-0.500000 + 0.866025i) q^{5} -0.221876 q^{7} +5.72889 q^{8} +O(q^{10})\)	\(q+(-1.25351 + 2.17114i) q^{2} +(-2.14257 - 3.71104i) q^{4} +(-0.500000 + 0.866025i) q^{5} -0.221876 q^{7} +5.72889 q^{8} +(-1.25351 - 2.17114i) q^{10} +0.778124 q^{11} +(2.50000 + 4.33013i) q^{13} +(0.278124 - 0.481725i) q^{14} +(-2.89608 + 5.01616i) q^{16} +(3.53865 - 6.12912i) q^{17} +(1.33281 - 4.15013i) q^{19} +4.28514 q^{20} +(-0.975385 + 1.68942i) q^{22} +(4.03865 + 6.99515i) q^{23} +(-0.500000 - 0.866025i) q^{25} -12.5351 q^{26} +(0.475385 + 0.823392i) q^{28} +(0.110938 + 0.192150i) q^{29} +2.50702 q^{31} +(-1.53163 - 2.65287i) q^{32} +(8.87147 + 15.3658i) q^{34} +(0.110938 - 0.192150i) q^{35} -1.90466 q^{37} +(7.33983 + 8.09596i) q^{38} +(-2.86445 + 4.96137i) q^{40} +(-3.61796 + 6.26648i) q^{41} +(-3.64959 + 6.32128i) q^{43} +(-1.66719 - 2.88765i) q^{44} -20.2500 q^{46} +(-1.39608 - 2.41808i) q^{47} -6.95077 q^{49} +2.50702 q^{50} +(10.7129 - 18.5552i) q^{52} +(2.19024 + 3.79361i) q^{53} +(-0.389062 + 0.673875i) q^{55} -1.27111 q^{56} -0.556248 q^{58} +(-1.39608 + 2.41808i) q^{59} +(6.29216 + 10.8983i) q^{61} +(-3.14257 + 5.44309i) q^{62} -3.90466 q^{64} -5.00000 q^{65} +(5.28514 + 9.15414i) q^{67} -30.3273 q^{68} +(0.278124 + 0.481725i) q^{70} +(-4.92070 + 8.52289i) q^{71} +(7.03865 - 12.1913i) q^{73} +(2.38750 - 4.13528i) q^{74} +(-18.2570 + 3.94583i) q^{76} -0.172647 q^{77} +(-0.792161 + 1.37206i) q^{79} +(-2.89608 - 5.01616i) q^{80} +(-9.07028 - 15.7102i) q^{82} +9.52106 q^{83} +(3.53865 + 6.12912i) q^{85} +(-9.14959 - 15.8476i) q^{86} +4.45779 q^{88} +(1.57028 + 2.71981i) q^{89} +(-0.554690 - 0.960752i) q^{91} +(17.3062 - 29.9752i) q^{92} +7.00000 q^{94} +(2.92771 + 3.22932i) q^{95} +(3.18122 - 5.51004i) q^{97} +(8.71286 - 15.0911i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} - 7 q^{4} - 3 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10}) \) 6 * q + q^2 - 7 * q^4 - 3 * q^5 + 4 * q^7 + 12 * q^8	\( 6 q + q^{2} - 7 q^{4} - 3 q^{5} + 4 q^{7} + 12 q^{8} + q^{10} + 10 q^{11} + 15 q^{13} + 7 q^{14} - 3 q^{16} + q^{17} + 14 q^{20} + 8 q^{22} + 4 q^{23} - 3 q^{25} + 10 q^{26} - 11 q^{28} - 2 q^{29} - 2 q^{31} - 6 q^{32} + 25 q^{34} - 2 q^{35} - 4 q^{37} + 19 q^{38} - 6 q^{40} - 2 q^{41} + q^{43} - 18 q^{44} - 48 q^{46} + 6 q^{47} - 14 q^{49} - 2 q^{50} + 35 q^{52} + 11 q^{53} - 5 q^{55} - 30 q^{56} - 14 q^{58} + 6 q^{59} + 9 q^{61} - 13 q^{62} - 16 q^{64} - 30 q^{65} + 20 q^{67} - 68 q^{68} + 7 q^{70} - 29 q^{71} + 22 q^{73} - 7 q^{74} - 19 q^{76} + 32 q^{77} + 24 q^{79} - 3 q^{80} - 31 q^{82} + 6 q^{83} + q^{85} - 32 q^{86} - 18 q^{88} - 14 q^{89} + 10 q^{91} + 41 q^{92} + 42 q^{94} - 7 q^{97} + 23 q^{98}+O(q^{100}) \) 6 * q + q^2 - 7 * q^4 - 3 * q^5 + 4 * q^7 + 12 * q^8 + q^10 + 10 * q^11 + 15 * q^13 + 7 * q^14 - 3 * q^16 + q^17 + 14 * q^20 + 8 * q^22 + 4 * q^23 - 3 * q^25 + 10 * q^26 - 11 * q^28 - 2 * q^29 - 2 * q^31 - 6 * q^32 + 25 * q^34 - 2 * q^35 - 4 * q^37 + 19 * q^38 - 6 * q^40 - 2 * q^41 + q^43 - 18 * q^44 - 48 * q^46 + 6 * q^47 - 14 * q^49 - 2 * q^50 + 35 * q^52 + 11 * q^53 - 5 * q^55 - 30 * q^56 - 14 * q^58 + 6 * q^59 + 9 * q^61 - 13 * q^62 - 16 * q^64 - 30 * q^65 + 20 * q^67 - 68 * q^68 + 7 * q^70 - 29 * q^71 + 22 * q^73 - 7 * q^74 - 19 * q^76 + 32 * q^77 + 24 * q^79 - 3 * q^80 - 31 * q^82 + 6 * q^83 + q^85 - 32 * q^86 - 18 * q^88 - 14 * q^89 + 10 * q^91 + 41 * q^92 + 42 * q^94 - 7 * q^97 + 23 * q^98

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