LMFDB - Embedded newform 845.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,2,Mod(1,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, 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("845.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	845.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:	$6.74735897080$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 17x^{7} - 9x^{6} + 59x^{5} + 32x^{4} - 44x^{3} - 23x^{2} + x + 1 $ x^9 - 17x^7 - 9x^6 + 59x^5 + 32x^4 - 44x^3 - 23x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.88295$ of defining polynomial
Character	$\chi$	$=$	845.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.63597 q^{2} +1.98944 q^{3} +4.94835 q^{4} -1.00000 q^{5} -5.24412 q^{6} +3.28231 q^{7} -7.77176 q^{8} +0.957886 q^{9} +O(q^{10})$	\(q-2.63597 q^{2} +1.98944 q^{3} +4.94835 q^{4} -1.00000 q^{5} -5.24412 q^{6} +3.28231 q^{7} -7.77176 q^{8} +0.957886 q^{9} +2.63597 q^{10} +3.22850 q^{11} +9.84446 q^{12} -8.65206 q^{14} -1.98944 q^{15} +10.5894 q^{16} -4.25510 q^{17} -2.52496 q^{18} +2.87293 q^{19} -4.94835 q^{20} +6.52996 q^{21} -8.51023 q^{22} +6.09907 q^{23} -15.4615 q^{24} +1.00000 q^{25} -4.06267 q^{27} +16.2420 q^{28} +5.77280 q^{29} +5.24412 q^{30} +0.835435 q^{31} -12.3700 q^{32} +6.42292 q^{33} +11.2163 q^{34} -3.28231 q^{35} +4.73995 q^{36} +5.59588 q^{37} -7.57297 q^{38} +7.77176 q^{40} -2.18667 q^{41} -17.2128 q^{42} +2.48711 q^{43} +15.9757 q^{44} -0.957886 q^{45} -16.0770 q^{46} -10.5761 q^{47} +21.0671 q^{48} +3.77353 q^{49} -2.63597 q^{50} -8.46528 q^{51} -5.08351 q^{53} +10.7091 q^{54} -3.22850 q^{55} -25.5093 q^{56} +5.71554 q^{57} -15.2169 q^{58} +0.144765 q^{59} -9.84446 q^{60} +6.06541 q^{61} -2.20218 q^{62} +3.14407 q^{63} +11.4280 q^{64} -16.9306 q^{66} -12.6133 q^{67} -21.0557 q^{68} +12.1338 q^{69} +8.65206 q^{70} +9.02740 q^{71} -7.44446 q^{72} +5.70395 q^{73} -14.7506 q^{74} +1.98944 q^{75} +14.2163 q^{76} +10.5969 q^{77} +14.1043 q^{79} -10.5894 q^{80} -10.9561 q^{81} +5.76401 q^{82} -7.41467 q^{83} +32.3125 q^{84} +4.25510 q^{85} -6.55595 q^{86} +11.4847 q^{87} -25.0911 q^{88} +13.0912 q^{89} +2.52496 q^{90} +30.1803 q^{92} +1.66205 q^{93} +27.8783 q^{94} -2.87293 q^{95} -24.6093 q^{96} -2.97494 q^{97} -9.94691 q^{98} +3.09253 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 3 q^{2} + 7 q^{3} + 17 q^{4} - 9 q^{5} + 2 q^{6} - 7 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) $ 9 * q - 3 * q^2 + 7 * q^3 + 17 * q^4 - 9 * q^5 + 2 * q^6 - 7 * q^7 - 12 * q^8 + 16 * q^9	\( 9 q - 3 q^{2} + 7 q^{3} + 17 q^{4} - 9 q^{5} + 2 q^{6} - 7 q^{7} - 12 q^{8} + 16 q^{9} + 3 q^{10} + 9 q^{11} + 12 q^{12} - 2 q^{14} - 7 q^{15} + 37 q^{16} - q^{17} + 10 q^{18} + 4 q^{19} - 17 q^{20} + q^{21} + 12 q^{22} + 14 q^{23} + 35 q^{24} + 9 q^{25} + 22 q^{27} - 18 q^{28} + 12 q^{29} - 2 q^{30} + 7 q^{31} - 22 q^{32} + 8 q^{33} + 30 q^{34} + 7 q^{35} + 3 q^{36} + 5 q^{37} - 47 q^{38} + 12 q^{40} + 10 q^{41} - 11 q^{42} + 39 q^{43} + 25 q^{44} - 16 q^{45} - 6 q^{46} - 36 q^{47} - 3 q^{48} + 16 q^{49} - 3 q^{50} + 43 q^{51} - 8 q^{53} + 2 q^{54} - 9 q^{55} - 29 q^{56} + 32 q^{57} - 21 q^{58} + 21 q^{59} - 12 q^{60} - 3 q^{61} - 10 q^{62} - 35 q^{63} + 34 q^{64} - 49 q^{66} - q^{67} - 20 q^{68} - 13 q^{69} + 2 q^{70} + q^{71} + 3 q^{72} - 15 q^{74} + 7 q^{75} + 5 q^{76} - 4 q^{77} + 39 q^{79} - 37 q^{80} + 29 q^{81} - 4 q^{82} - 7 q^{83} - 12 q^{84} + q^{85} + 24 q^{86} + 16 q^{87} + 42 q^{88} + 19 q^{89} - 10 q^{90} - 27 q^{92} - 31 q^{93} + 16 q^{94} - 4 q^{95} + 7 q^{96} + 34 q^{97} - 48 q^{98} + q^{99}+O(q^{100}) \) 9 * q - 3 * q^2 + 7 * q^3 + 17 * q^4 - 9 * q^5 + 2 * q^6 - 7 * q^7 - 12 * q^8 + 16 * q^9 + 3 * q^10 + 9 * q^11 + 12 * q^12 - 2 * q^14 - 7 * q^15 + 37 * q^16 - q^17 + 10 * q^18 + 4 * q^19 - 17 * q^20 + q^21 + 12 * q^22 + 14 * q^23 + 35 * q^24 + 9 * q^25 + 22 * q^27 - 18 * q^28 + 12 * q^29 - 2 * q^30 + 7 * q^31 - 22 * q^32 + 8 * q^33 + 30 * q^34 + 7 * q^35 + 3 * q^36 + 5 * q^37 - 47 * q^38 + 12 * q^40 + 10 * q^41 - 11 * q^42 + 39 * q^43 + 25 * q^44 - 16 * q^45 - 6 * q^46 - 36 * q^47 - 3 * q^48 + 16 * q^49 - 3 * q^50 + 43 * q^51 - 8 * q^53 + 2 * q^54 - 9 * q^55 - 29 * q^56 + 32 * q^57 - 21 * q^58 + 21 * q^59 - 12 * q^60 - 3 * q^61 - 10 * q^62 - 35 * q^63 + 34 * q^64 - 49 * q^66 - q^67 - 20 * q^68 - 13 * q^69 + 2 * q^70 + q^71 + 3 * q^72 - 15 * q^74 + 7 * q^75 + 5 * q^76 - 4 * q^77 + 39 * q^79 - 37 * q^80 + 29 * q^81 - 4 * q^82 - 7 * q^83 - 12 * q^84 + q^85 + 24 * q^86 + 16 * q^87 + 42 * q^88 + 19 * q^89 - 10 * q^90 - 27 * q^92 - 31 * q^93 + 16 * q^94 - 4 * q^95 + 7 * q^96 + 34 * q^97 - 48 * q^98 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.63597	−1.86391	−0.931957	−	0.362569i	$-0.881900\pi$
$2$	−2.63597	−1.86391	−0.931957	+	0.362569i	$0.881900\pi$
$3$	1.98944	1.14861	0.574303	−	0.818643i	$-0.305273\pi$
$3$	1.98944	1.14861	0.574303	+	0.818643i	$0.305273\pi$
$4$	4.94835	2.47417
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−5.24412	−2.14090
$7$	3.28231	1.24059	0.620297	−	0.784367i	$-0.287012\pi$
$7$	3.28231	1.24059	0.620297	+	0.784367i	$0.287012\pi$
$8$	−7.77176	−2.74773
$9$	0.957886	0.319295
$10$	2.63597	0.833567
$11$	3.22850	0.973429	0.486715	−	0.873561i	$-0.338195\pi$
$11$	3.22850	0.973429	0.486715	+	0.873561i	$0.338195\pi$
$12$	9.84446	2.84185
$13$	0	0
$13$	0	0
$14$	−8.65206	−2.31236
$15$	−1.98944	−0.513672
$16$	10.5894	2.64736
$17$	−4.25510	−1.03201	−0.516007	−	0.856584i	$-0.672582\pi$
$17$	−4.25510	−1.03201	−0.516007	+	0.856584i	$0.672582\pi$
$18$	−2.52496	−0.595139
$19$	2.87293	0.659096	0.329548	−	0.944139i	$-0.393104\pi$
$19$	2.87293	0.659096	0.329548	+	0.944139i	$0.393104\pi$
$20$	−4.94835	−1.10648
$21$	6.52996	1.42495
$22$	−8.51023	−1.81439
$23$	6.09907	1.27174	0.635872	−	0.771795i	$-0.280641\pi$
$23$	6.09907	1.27174	0.635872	+	0.771795i	$0.280641\pi$
$24$	−15.4615	−3.15606
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.06267	−0.781861
$28$	16.2420	3.06945
$29$	5.77280	1.07198	0.535991	−	0.844224i	$-0.319938\pi$
$29$	5.77280	1.07198	0.535991	+	0.844224i	$0.319938\pi$
$30$	5.24412	0.957440
$31$	0.835435	0.150048	0.0750242	−	0.997182i	$-0.476097\pi$
$31$	0.835435	0.150048	0.0750242	+	0.997182i	$0.476097\pi$
$32$	−12.3700	−2.18672
$33$	6.42292	1.11809
$34$	11.2163	1.92358
$35$	−3.28231	−0.554811
$36$	4.73995	0.789992
$37$	5.59588	0.919957	0.459978	−	0.887930i	$-0.347857\pi$
$37$	5.59588	0.919957	0.459978	+	0.887930i	$0.347857\pi$
$38$	−7.57297	−1.22850
$39$	0	0
$40$	7.77176	1.22882
$41$	−2.18667	−0.341501	−0.170750	−	0.985314i	$-0.554619\pi$
$41$	−2.18667	−0.341501	−0.170750	+	0.985314i	$0.554619\pi$
$42$	−17.2128	−2.65599
$43$	2.48711	0.379280	0.189640	−	0.981854i	$-0.439268\pi$
$43$	2.48711	0.379280	0.189640	+	0.981854i	$0.439268\pi$
$44$	15.9757	2.40843
$45$	−0.957886	−0.142793
$46$	−16.0770	−2.37042
$47$	−10.5761	−1.54268	−0.771341	−	0.636422i	$-0.780414\pi$
$47$	−10.5761	−1.54268	−0.771341	+	0.636422i	$0.780414\pi$
$48$	21.0671	3.04078
$49$	3.77353	0.539075
$50$	−2.63597	−0.372783
$51$	−8.46528	−1.18538
$52$	0	0
$53$	−5.08351	−0.698273	−0.349137	−	0.937072i	$-0.613525\pi$
$53$	−5.08351	−0.698273	−0.349137	+	0.937072i	$0.613525\pi$
$54$	10.7091	1.45732
$55$	−3.22850	−0.435331
$56$	−25.5093	−3.40882
$57$	5.71554	0.757042
$58$	−15.2169	−1.99808
$59$	0.144765	0.0188468	0.00942338	−	0.999956i	$-0.497000\pi$
$59$	0.144765	0.0188468	0.00942338	+	0.999956i	$0.497000\pi$
$60$	−9.84446	−1.27091
$61$	6.06541	0.776596	0.388298	−	0.921534i	$-0.373063\pi$
$61$	6.06541	0.776596	0.388298	+	0.921534i	$0.373063\pi$
$62$	−2.20218	−0.279677
$63$	3.14407	0.396116
$64$	11.4280	1.42850
$65$	0	0
$66$	−16.9306	−2.08402
$67$	−12.6133	−1.54096	−0.770482	−	0.637461i	$-0.779985\pi$
$67$	−12.6133	−1.54096	−0.770482	+	0.637461i	$0.779985\pi$
$68$	−21.0557	−2.55338
$69$	12.1338	1.46073
$70$	8.65206	1.03412
$71$	9.02740	1.07136	0.535678	−	0.844423i	$-0.320056\pi$
$71$	9.02740	1.07136	0.535678	+	0.844423i	$0.320056\pi$
$72$	−7.44446	−0.877338
$73$	5.70395	0.667597	0.333798	−	0.942644i	$-0.391669\pi$
$73$	5.70395	0.667597	0.333798	+	0.942644i	$0.391669\pi$
$74$	−14.7506	−1.71472
$75$	1.98944	0.229721
$76$	14.2163	1.63072
$77$	10.5969	1.20763
$78$	0	0
$79$	14.1043	1.58686	0.793430	−	0.608662i	$-0.208293\pi$
$79$	14.1043	1.58686	0.793430	+	0.608662i	$0.208293\pi$
$80$	−10.5894	−1.18394
$81$	−10.9561	−1.21735
$82$	5.76401	0.636528
$83$	−7.41467	−0.813865	−0.406933	−	0.913458i	$-0.633402\pi$
$83$	−7.41467	−0.813865	−0.406933	+	0.913458i	$0.633402\pi$
$84$	32.3125	3.52558
$85$	4.25510	0.461531
$86$	−6.55595	−0.706946
$87$	11.4847	1.23128
$88$	−25.0911	−2.67472
$89$	13.0912	1.38766	0.693830	−	0.720138i	$-0.255922\pi$
$89$	13.0912	1.38766	0.693830	+	0.720138i	$0.255922\pi$
$90$	2.52496	0.266154
$91$	0	0
$92$	30.1803	3.14652
$93$	1.66205	0.172347
$94$	27.8783	2.87543
$95$	−2.87293	−0.294757
$96$	−24.6093	−2.51168
$97$	−2.97494	−0.302059	−0.151030	−	0.988529i	$-0.548259\pi$
$97$	−2.97494	−0.302059	−0.151030	+	0.988529i	$0.548259\pi$
$98$	−9.94691	−1.00479
$99$	3.09253	0.310811
$100$	4.94835	0.494835
$101$	2.24007	0.222896	0.111448	−	0.993770i	$-0.464451\pi$
$101$	2.24007	0.222896	0.111448	+	0.993770i	$0.464451\pi$
$102$	22.3143	2.20944
$103$	−12.6517	−1.24661	−0.623305	−	0.781979i	$-0.714210\pi$
$103$	−12.6517	−1.24661	−0.623305	+	0.781979i	$0.714210\pi$
$104$	0	0
$105$	−6.52996	−0.637259
$106$	13.4000	1.30152
$107$	5.51892	0.533534	0.266767	−	0.963761i	$-0.414045\pi$
$107$	5.51892	0.533534	0.266767	+	0.963761i	$0.414045\pi$
$108$	−20.1035	−1.93446
$109$	7.12142	0.682108	0.341054	−	0.940044i	$-0.389216\pi$
$109$	7.12142	0.682108	0.341054	+	0.940044i	$0.389216\pi$
$110$	8.51023	0.811419
$111$	11.1327	1.05667
$112$	34.7578	3.28430
$113$	18.7572	1.76453	0.882264	−	0.470756i	$-0.156019\pi$
$113$	18.7572	1.76453	0.882264	+	0.470756i	$0.156019\pi$
$114$	−15.0660	−1.41106
$115$	−6.09907	−0.568741
$116$	28.5658	2.65227
$117$	0	0
$118$	−0.381596	−0.0351287
$119$	−13.9665	−1.28031
$120$	15.4615	1.41143
$121$	−0.576790	−0.0524355
$122$	−15.9883	−1.44751
$123$	−4.35026	−0.392250
$124$	4.13402	0.371246
$125$	−1.00000	−0.0894427
$126$	−8.28769	−0.738326
$127$	0.368129	0.0326661	0.0163331	−	0.999867i	$-0.494801\pi$
$127$	0.368129	0.0326661	0.0163331	+	0.999867i	$0.494801\pi$
$128$	−5.38392	−0.475875
$129$	4.94796	0.435644
$130$	0	0
$131$	3.09282	0.270221	0.135110	−	0.990831i	$-0.456861\pi$
$131$	3.09282	0.270221	0.135110	+	0.990831i	$0.456861\pi$
$132$	31.7828	2.76634
$133$	9.42984	0.817671
$134$	33.2484	2.87222
$135$	4.06267	0.349659
$136$	33.0696	2.83570
$137$	15.5865	1.33164	0.665822	−	0.746111i	$-0.268081\pi$
$137$	15.5865	1.33164	0.665822	+	0.746111i	$0.268081\pi$
$138$	−31.9842	−2.72268
$139$	18.7547	1.59075	0.795376	−	0.606116i	$-0.207273\pi$
$139$	18.7547	1.59075	0.795376	+	0.606116i	$0.207273\pi$
$140$	−16.2420	−1.37270
$141$	−21.0405	−1.77193
$142$	−23.7960	−1.99691
$143$	0	0
$144$	10.1435	0.845290
$145$	−5.77280	−0.479405
$146$	−15.0355	−1.24434
$147$	7.50722	0.619185
$148$	27.6904	2.27613
$149$	−13.1350	−1.07606	−0.538032	−	0.842925i	$-0.680832\pi$
$149$	−13.1350	−1.07606	−0.538032	+	0.842925i	$0.680832\pi$
$150$	−5.24412	−0.428180
$151$	−1.66546	−0.135533	−0.0677665	−	0.997701i	$-0.521587\pi$
$151$	−1.66546	−0.135533	−0.0677665	+	0.997701i	$0.521587\pi$
$152$	−22.3278	−1.81102
$153$	−4.07590	−0.329517
$154$	−27.9332	−2.25092
$155$	−0.835435	−0.0671037
$156$	0	0
$157$	−20.6056	−1.64450	−0.822251	−	0.569125i	$-0.807282\pi$
$157$	−20.6056	−1.64450	−0.822251	+	0.569125i	$0.807282\pi$
$158$	−37.1786	−2.95777
$159$	−10.1134	−0.802041
$160$	12.3700	0.977932
$161$	20.0190	1.57772
$162$	28.8800	2.26903
$163$	−5.62628	−0.440684	−0.220342	−	0.975423i	$-0.570717\pi$
$163$	−5.62628	−0.440684	−0.220342	+	0.975423i	$0.570717\pi$
$164$	−10.8204	−0.844933
$165$	−6.42292	−0.500023
$166$	19.5449	1.51697
$167$	−14.1216	−1.09276	−0.546381	−	0.837537i	$-0.683995\pi$
$167$	−14.1216	−1.09276	−0.546381	+	0.837537i	$0.683995\pi$
$168$	−50.7493	−3.91539
$169$	0	0
$170$	−11.2163	−0.860253
$171$	2.75194	0.210446
$172$	12.3071	0.938406
$173$	−9.86557	−0.750066	−0.375033	−	0.927012i	$-0.622369\pi$
$173$	−9.86557	−0.750066	−0.375033	+	0.927012i	$0.622369\pi$
$174$	−30.2732	−2.29501
$175$	3.28231	0.248119
$176$	34.1880	2.57702
$177$	0.288001	0.0216475
$178$	−34.5079	−2.58648
$179$	9.25334	0.691627	0.345813	−	0.938303i	$-0.387603\pi$
$179$	9.25334	0.691627	0.345813	+	0.938303i	$0.387603\pi$
$180$	−4.73995	−0.353295
$181$	−21.6558	−1.60967	−0.804833	−	0.593501i	$-0.797745\pi$
$181$	−21.6558	−1.60967	−0.804833	+	0.593501i	$0.797745\pi$
$182$	0	0
$183$	12.0668	0.892003
$184$	−47.4005	−3.49441
$185$	−5.59588	−0.411417
$186$	−4.38112	−0.321239
$187$	−13.7376	−1.00459
$188$	−52.3342	−3.81686
$189$	−13.3349	−0.969973
$190$	7.57297	0.549401
$191$	−21.3578	−1.54539	−0.772697	−	0.634775i	$-0.781093\pi$
$191$	−21.3578	−1.54539	−0.772697	+	0.634775i	$0.781093\pi$
$192$	22.7353	1.64078
$193$	5.77649	0.415801	0.207901	−	0.978150i	$-0.433337\pi$
$193$	5.77649	0.415801	0.207901	+	0.978150i	$0.433337\pi$
$194$	7.84186	0.563013
$195$	0	0
$196$	18.6727	1.33377
$197$	12.4045	0.883787	0.441894	−	0.897067i	$-0.354307\pi$
$197$	12.4045	0.883787	0.441894	+	0.897067i	$0.354307\pi$
$198$	−8.15183	−0.579325
$199$	−6.03432	−0.427762	−0.213881	−	0.976860i	$-0.568610\pi$
$199$	−6.03432	−0.427762	−0.213881	+	0.976860i	$0.568610\pi$
$200$	−7.77176	−0.549547
$201$	−25.0935	−1.76996
$202$	−5.90477	−0.415458
$203$	18.9481	1.32989
$204$	−41.8892	−2.93283
$205$	2.18667	0.152724
$206$	33.3495	2.32357
$207$	5.84221	0.406062
$208$	0	0
$209$	9.27526	0.641583
$210$	17.2128	1.18780
$211$	−13.6840	−0.942049	−0.471024	−	0.882120i	$-0.656116\pi$
$211$	−13.6840	−0.942049	−0.471024	+	0.882120i	$0.656116\pi$
$212$	−25.1550	−1.72765
$213$	17.9595	1.23056
$214$	−14.5477	−0.994461
$215$	−2.48711	−0.169619
$216$	31.5741	2.14835
$217$	2.74215	0.186149
$218$	−18.7719	−1.27139
$219$	11.3477	0.766806
$220$	−15.9757	−1.07708
$221$	0	0
$222$	−29.3454	−1.96954
$223$	18.6882	1.25145	0.625727	−	0.780042i	$-0.284802\pi$
$223$	18.6882	1.25145	0.625727	+	0.780042i	$0.284802\pi$
$224$	−40.6020	−2.71284
$225$	0.957886	0.0638590
$226$	−49.4434	−3.28893
$227$	11.5082	0.763828	0.381914	−	0.924198i	$-0.375265\pi$
$227$	11.5082	0.763828	0.381914	+	0.924198i	$0.375265\pi$
$228$	28.2825	1.87305
$229$	−1.98065	−0.130885	−0.0654424	−	0.997856i	$-0.520846\pi$
$229$	−1.98065	−0.130885	−0.0654424	+	0.997856i	$0.520846\pi$
$230$	16.0770	1.06008
$231$	21.0820	1.38709
$232$	−44.8648	−2.94552
$233$	22.1334	1.45001	0.725005	−	0.688744i	$-0.241837\pi$
$233$	22.1334	1.45001	0.725005	+	0.688744i	$0.241837\pi$
$234$	0	0
$235$	10.5761	0.689909
$236$	0.716346	0.0466302
$237$	28.0597	1.82268
$238$	36.8154	2.38639
$239$	−11.2846	−0.729942	−0.364971	−	0.931019i	$-0.618921\pi$
$239$	−11.2846	−0.729942	−0.364971	+	0.931019i	$0.618921\pi$
$240$	−21.0671	−1.35988
$241$	−24.9029	−1.60414	−0.802068	−	0.597233i	$-0.796267\pi$
$241$	−24.9029	−1.60414	−0.802068	+	0.597233i	$0.796267\pi$
$242$	1.52040	0.0977352
$243$	−9.60855	−0.616389
$244$	30.0138	1.92143
$245$	−3.77353	−0.241082
$246$	11.4672	0.731120
$247$	0	0
$248$	−6.49280	−0.412293
$249$	−14.7511	−0.934810
$250$	2.63597	0.166713
$251$	−14.1454	−0.892848	−0.446424	−	0.894821i	$-0.647303\pi$
$251$	−14.1454	−0.892848	−0.446424	+	0.894821i	$0.647303\pi$
$252$	15.5580	0.980060
$253$	19.6908	1.23795
$254$	−0.970377	−0.0608868
$255$	8.46528	0.530117
$256$	−8.66412	−0.541508
$257$	3.53245	0.220348	0.110174	−	0.993912i	$-0.464859\pi$
$257$	3.53245	0.220348	0.110174	+	0.993912i	$0.464859\pi$
$258$	−13.0427	−0.812002
$259$	18.3674	1.14129
$260$	0	0
$261$	5.52968	0.342279
$262$	−8.15259	−0.503669
$263$	−12.2296	−0.754109	−0.377055	−	0.926191i	$-0.623063\pi$
$263$	−12.2296	−0.754109	−0.377055	+	0.926191i	$0.623063\pi$
$264$	−49.9174	−3.07220
$265$	5.08351	0.312277
$266$	−24.8568	−1.52407
$267$	26.0441	1.59388
$268$	−62.4152	−3.81261
$269$	−22.4563	−1.36919	−0.684593	−	0.728926i	$-0.740020\pi$
$269$	−22.4563	−1.36919	−0.684593	+	0.728926i	$0.740020\pi$
$270$	−10.7091	−0.651734
$271$	−19.3264	−1.17399	−0.586997	−	0.809589i	$-0.699690\pi$
$271$	−19.3264	−1.17399	−0.586997	+	0.809589i	$0.699690\pi$
$272$	−45.0592	−2.73211
$273$	0	0
$274$	−41.0855	−2.48207
$275$	3.22850	0.194686
$276$	60.0420	3.61411
$277$	−11.8931	−0.714587	−0.357294	−	0.933992i	$-0.616300\pi$
$277$	−11.8931	−0.714587	−0.357294	+	0.933992i	$0.616300\pi$
$278$	−49.4368	−2.96502
$279$	0.800251	0.0479098
$280$	25.5093	1.52447
$281$	22.7024	1.35431	0.677157	−	0.735838i	$-0.263212\pi$
$281$	22.7024	1.35431	0.677157	+	0.735838i	$0.263212\pi$
$282$	55.4623	3.30273
$283$	−8.86050	−0.526702	−0.263351	−	0.964700i	$-0.584828\pi$
$283$	−8.86050	−0.526702	−0.263351	+	0.964700i	$0.584828\pi$
$284$	44.6707	2.65072
$285$	−5.71554	−0.338559
$286$	0	0
$287$	−7.17733	−0.423664
$288$	−11.8490	−0.698210
$289$	1.10589	0.0650525
$290$	15.2169	0.893569
$291$	−5.91848	−0.346947
$292$	28.2251	1.65175
$293$	−6.40308	−0.374072	−0.187036	−	0.982353i	$-0.559888\pi$
$293$	−6.40308	−0.374072	−0.187036	+	0.982353i	$0.559888\pi$
$294$	−19.7888	−1.15411
$295$	−0.144765	−0.00842853
$296$	−43.4898	−2.52780
$297$	−13.1163	−0.761087
$298$	34.6236	2.00569
$299$	0	0
$300$	9.84446	0.568370
$301$	8.16345	0.470533
$302$	4.39010	0.252622
$303$	4.45650	0.256019
$304$	30.4228	1.74487
$305$	−6.06541	−0.347304
$306$	10.7440	0.614191
$307$	−25.8317	−1.47429	−0.737146	−	0.675733i	$-0.763827\pi$
$307$	−25.8317	−1.47429	−0.737146	+	0.675733i	$0.763827\pi$
$308$	52.4372	2.98789
$309$	−25.1698	−1.43186
$310$	2.20218	0.125076
$311$	−3.87202	−0.219562	−0.109781	−	0.993956i	$-0.535015\pi$
$311$	−3.87202	−0.219562	−0.109781	+	0.993956i	$0.535015\pi$
$312$	0	0
$313$	4.13361	0.233645	0.116823	−	0.993153i	$-0.462729\pi$
$313$	4.13361	0.233645	0.116823	+	0.993153i	$0.462729\pi$
$314$	54.3157	3.06521
$315$	−3.14407	−0.177148
$316$	69.7931	3.92617
$317$	−24.9158	−1.39941	−0.699706	−	0.714431i	$-0.746686\pi$
$317$	−24.9158	−1.39941	−0.699706	+	0.714431i	$0.746686\pi$
$318$	26.6585	1.49493
$319$	18.6375	1.04350
$320$	−11.4280	−0.638844
$321$	10.9796	0.612820
$322$	−52.7695	−2.94073
$323$	−12.2246	−0.680196
$324$	−54.2147	−3.01193
$325$	0	0
$326$	14.8307	0.821397
$327$	14.1677	0.783474
$328$	16.9943	0.938353
$329$	−34.7140	−1.91384
$330$	16.9306	0.932001
$331$	5.56860	0.306078	0.153039	−	0.988220i	$-0.451094\pi$
$331$	5.56860	0.306078	0.153039	+	0.988220i	$0.451094\pi$
$332$	−36.6903	−2.01364
$333$	5.36021	0.293738
$334$	37.2241	2.03681
$335$	12.6133	0.689140
$336$	69.1487	3.77237
$337$	16.8799	0.919507	0.459753	−	0.888047i	$-0.347938\pi$
$337$	16.8799	0.919507	0.459753	+	0.888047i	$0.347938\pi$
$338$	0	0
$339$	37.3164	2.02675
$340$	21.0557	1.14191
$341$	2.69720	0.146062
$342$	−7.25404	−0.392254
$343$	−10.5903	−0.571821
$344$	−19.3292	−1.04216
$345$	−12.1338	−0.653259
$346$	26.0054	1.39806
$347$	−12.4254	−0.667031	−0.333516	−	0.942745i	$-0.608235\pi$
$347$	−12.4254	−0.667031	−0.333516	+	0.942745i	$0.608235\pi$
$348$	56.8301	3.04641
$349$	2.73516	0.146410	0.0732050	−	0.997317i	$-0.476677\pi$
$349$	2.73516	0.146410	0.0732050	+	0.997317i	$0.476677\pi$
$350$	−8.65206	−0.462472
$351$	0	0
$352$	−39.9364	−2.12862
$353$	−22.7460	−1.21065	−0.605323	−	0.795980i	$-0.706956\pi$
$353$	−22.7460	−1.21065	−0.605323	+	0.795980i	$0.706956\pi$
$354$	−0.759163	−0.0403491
$355$	−9.02740	−0.479125
$356$	64.7796	3.43331
$357$	−27.7856	−1.47057
$358$	−24.3915	−1.28913
$359$	−11.5189	−0.607945	−0.303973	−	0.952681i	$-0.598313\pi$
$359$	−11.5189	−0.607945	−0.303973	+	0.952681i	$0.598313\pi$
$360$	7.44446	0.392357
$361$	−10.7463	−0.565592
$362$	57.0842	3.00028
$363$	−1.14749	−0.0602277
$364$	0	0
$365$	−5.70395	−0.298558
$366$	−31.8077	−1.66262
$367$	14.0733	0.734622	0.367311	−	0.930098i	$-0.380279\pi$
$367$	14.0733	0.734622	0.367311	+	0.930098i	$0.380279\pi$
$368$	64.5858	3.36677
$369$	−2.09458	−0.109040
$370$	14.7506	0.766846
$371$	−16.6856	−0.866274
$372$	8.22440	0.426415
$373$	18.4780	0.956753	0.478377	−	0.878155i	$-0.341225\pi$
$373$	18.4780	0.956753	0.478377	+	0.878155i	$0.341225\pi$
$374$	36.2119	1.87247
$375$	−1.98944	−0.102734
$376$	82.1949	4.23888
$377$	0	0
$378$	35.1505	1.80795
$379$	1.53391	0.0787918	0.0393959	−	0.999224i	$-0.487457\pi$
$379$	1.53391	0.0787918	0.0393959	+	0.999224i	$0.487457\pi$
$380$	−14.2163	−0.729279
$381$	0.732371	0.0375205
$382$	56.2985	2.88048
$383$	30.1620	1.54120	0.770602	−	0.637317i	$-0.219956\pi$
$383$	30.1620	1.54120	0.770602	+	0.637317i	$0.219956\pi$
$384$	−10.7110	−0.546593
$385$	−10.5969	−0.540069
$386$	−15.2267	−0.775017
$387$	2.38237	0.121102
$388$	−14.7210	−0.747348
$389$	19.8405	1.00595	0.502976	−	0.864301i	$-0.332239\pi$
$389$	19.8405	1.00595	0.502976	+	0.864301i	$0.332239\pi$
$390$	0	0
$391$	−25.9522	−1.31246
$392$	−29.3269	−1.48123
$393$	6.15299	0.310377
$394$	−32.6980	−1.64730
$395$	−14.1043	−0.709665
$396$	15.3029	0.769001
$397$	−5.70766	−0.286459	−0.143230	−	0.989689i	$-0.545749\pi$
$397$	−5.70766	−0.286459	−0.143230	+	0.989689i	$0.545749\pi$
$398$	15.9063	0.797311
$399$	18.7601	0.939182
$400$	10.5894	0.529472
$401$	−31.3372	−1.56491	−0.782454	−	0.622709i	$-0.786032\pi$
$401$	−31.3372	−1.56491	−0.782454	+	0.622709i	$0.786032\pi$
$402$	66.1458	3.29905
$403$	0	0
$404$	11.0847	0.551482
$405$	10.9561	0.544414
$406$	−49.9466	−2.47881
$407$	18.0663	0.895513
$408$	65.7902	3.25710
$409$	19.2508	0.951889	0.475944	−	0.879475i	$-0.342106\pi$
$409$	19.2508	0.951889	0.475944	+	0.879475i	$0.342106\pi$
$410$	−5.76401	−0.284664
$411$	31.0084	1.52953
$412$	−62.6050	−3.08433
$413$	0.475162	0.0233812
$414$	−15.3999	−0.756864
$415$	7.41467	0.363972
$416$	0	0
$417$	37.3114	1.82715
$418$	−24.4493	−1.19586
$419$	−12.8466	−0.627596	−0.313798	−	0.949490i	$-0.601601\pi$
$419$	−12.8466	−0.627596	−0.313798	+	0.949490i	$0.601601\pi$
$420$	−32.3125	−1.57669
$421$	15.5491	0.757819	0.378910	−	0.925434i	$-0.376299\pi$
$421$	15.5491	0.757819	0.378910	+	0.925434i	$0.376299\pi$
$422$	36.0708	1.75590
$423$	−10.1307	−0.492571
$424$	39.5078	1.91867
$425$	−4.25510	−0.206403
$426$	−47.3407	−2.29367
$427$	19.9085	0.963441
$428$	27.3095	1.32006
$429$	0	0
$430$	6.55595	0.316156
$431$	16.4219	0.791017	0.395509	−	0.918462i	$-0.370568\pi$
$431$	16.4219	0.791017	0.395509	+	0.918462i	$0.370568\pi$
$432$	−43.0214	−2.06987
$433$	−15.4378	−0.741895	−0.370948	−	0.928654i	$-0.620967\pi$
$433$	−15.4378	−0.741895	−0.370948	+	0.928654i	$0.620967\pi$
$434$	−7.22823	−0.346966
$435$	−11.4847	−0.550647
$436$	35.2393	1.68765
$437$	17.5222	0.838201
$438$	−29.9122	−1.42926
$439$	−39.6616	−1.89294	−0.946472	−	0.322785i	$-0.895381\pi$
$439$	−39.6616	−1.89294	−0.946472	+	0.322785i	$0.895381\pi$
$440$	25.0911	1.19617
$441$	3.61461	0.172124
$442$	0	0
$443$	4.90611	0.233097	0.116548	−	0.993185i	$-0.462817\pi$
$443$	4.90611	0.233097	0.116548	+	0.993185i	$0.462817\pi$
$444$	55.0884	2.61438
$445$	−13.0912	−0.620581
$446$	−49.2615	−2.33260
$447$	−26.1314	−1.23597
$448$	37.5101	1.77219
$449$	−28.2902	−1.33510	−0.667549	−	0.744566i	$-0.732657\pi$
$449$	−28.2902	−1.33510	−0.667549	+	0.744566i	$0.732657\pi$
$450$	−2.52496	−0.119028
$451$	−7.05967	−0.332427
$452$	92.8171	4.36575
$453$	−3.31333	−0.155674
$454$	−30.3354	−1.42371
$455$	0	0
$456$	−44.4198	−2.08015
$457$	−32.3338	−1.51251	−0.756257	−	0.654275i	$-0.772974\pi$
$457$	−32.3338	−1.51251	−0.756257	+	0.654275i	$0.772974\pi$
$458$	5.22093	0.243958
$459$	17.2871	0.806892
$460$	−30.1803	−1.40716
$461$	6.97401	0.324812	0.162406	−	0.986724i	$-0.448075\pi$
$461$	6.97401	0.324812	0.162406	+	0.986724i	$0.448075\pi$
$462$	−55.5715	−2.58542
$463$	−33.1964	−1.54277	−0.771384	−	0.636370i	$-0.780435\pi$
$463$	−33.1964	−1.54277	−0.771384	+	0.636370i	$0.780435\pi$
$464$	61.1307	2.83792
$465$	−1.66205	−0.0770757
$466$	−58.3431	−2.70269
$467$	−31.6108	−1.46277	−0.731386	−	0.681963i	$-0.761126\pi$
$467$	−31.6108	−1.46277	−0.731386	+	0.681963i	$0.761126\pi$
$468$	0	0
$469$	−41.4008	−1.91171
$470$	−27.8783	−1.28593
$471$	−40.9936	−1.88889
$472$	−1.12508	−0.0517859
$473$	8.02963	0.369203
$474$	−73.9647	−3.39731
$475$	2.87293	0.131819
$476$	−69.1113	−3.16771
$477$	−4.86942	−0.222955
$478$	29.7460	1.36055
$479$	−10.3683	−0.473742	−0.236871	−	0.971541i	$-0.576122\pi$
$479$	−10.3683	−0.473742	−0.236871	+	0.971541i	$0.576122\pi$
$480$	24.6093	1.12326
$481$	0	0
$482$	65.6433	2.98997
$483$	39.8267	1.81218
$484$	−2.85416	−0.129734
$485$	2.97494	0.135085
$486$	25.3279	1.14890
$487$	−34.9155	−1.58217	−0.791086	−	0.611705i	$-0.790484\pi$
$487$	−34.9155	−1.58217	−0.791086	+	0.611705i	$0.790484\pi$
$488$	−47.1389	−2.13388
$489$	−11.1932	−0.506172
$490$	9.94691	0.449356
$491$	17.5013	0.789824	0.394912	−	0.918719i	$-0.370775\pi$
$491$	17.5013	0.789824	0.394912	+	0.918719i	$0.370775\pi$
$492$	−21.5266	−0.970495
$493$	−24.5638	−1.10630
$494$	0	0
$495$	−3.09253	−0.138999
$496$	8.84679	0.397233
$497$	29.6307	1.32912
$498$	38.8834	1.74241
$499$	8.46542	0.378964	0.189482	−	0.981884i	$-0.439319\pi$
$499$	8.46542	0.378964	0.189482	+	0.981884i	$0.439319\pi$
$500$	−4.94835	−0.221297
$501$	−28.0941	−1.25515
$502$	37.2868	1.66419
$503$	−4.00003	−0.178353	−0.0891763	−	0.996016i	$-0.528423\pi$
$503$	−4.00003	−0.178353	−0.0891763	+	0.996016i	$0.528423\pi$
$504$	−24.4350	−1.08842
$505$	−2.24007	−0.0996819
$506$	−51.9045	−2.30744
$507$	0	0
$508$	1.82163	0.0808217
$509$	18.2954	0.810927	0.405464	−	0.914111i	$-0.367110\pi$
$509$	18.2954	0.810927	0.405464	+	0.914111i	$0.367110\pi$
$510$	−22.3143	−0.988092
$511$	18.7221	0.828217
$512$	33.6062	1.48520
$513$	−11.6718	−0.515322
$514$	−9.31144	−0.410710
$515$	12.6517	0.557501
$516$	24.4842	1.07786
$517$	−34.1449	−1.50169
$518$	−48.4159	−2.12727
$519$	−19.6270	−0.861530
$520$	0	0
$521$	−0.596886	−0.0261500	−0.0130750	−	0.999915i	$-0.504162\pi$
$521$	−0.596886	−0.0261500	−0.0130750	+	0.999915i	$0.504162\pi$
$522$	−14.5761	−0.637978
$523$	34.0800	1.49021	0.745107	−	0.666945i	$-0.232398\pi$
$523$	34.0800	1.49021	0.745107	+	0.666945i	$0.232398\pi$
$524$	15.3044	0.668574
$525$	6.52996	0.284991
$526$	32.2369	1.40559
$527$	−3.55486	−0.154852
$528$	68.0152	2.95998
$529$	14.1986	0.617332
$530$	−13.4000	−0.582058
$531$	0.138668	0.00601768
$532$	46.6621	2.02306
$533$	0	0
$534$	−68.6516	−2.97085
$535$	−5.51892	−0.238604
$536$	98.0278	4.23416
$537$	18.4090	0.794406
$538$	59.1942	2.55204
$539$	12.1828	0.524752
$540$	20.1035	0.865117
$541$	−42.5430	−1.82907	−0.914534	−	0.404510i	$-0.867442\pi$
$541$	−42.5430	−1.82907	−0.914534	+	0.404510i	$0.867442\pi$
$542$	50.9438	2.18822
$543$	−43.0831	−1.84887
$544$	52.6355	2.25673
$545$	−7.12142	−0.305048
$546$	0	0
$547$	42.0185	1.79658	0.898291	−	0.439401i	$-0.144809\pi$
$547$	42.0185	1.79658	0.898291	+	0.439401i	$0.144809\pi$
$548$	77.1274	3.29472
$549$	5.80997	0.247963
$550$	−8.51023	−0.362878
$551$	16.5849	0.706539
$552$	−94.3006	−4.01370
$553$	46.2947	1.96865
$554$	31.3499	1.33193
$555$	−11.1327	−0.472556
$556$	92.8047	3.93580
$557$	36.5703	1.54953	0.774767	−	0.632247i	$-0.217867\pi$
$557$	36.5703	1.54953	0.774767	+	0.632247i	$0.217867\pi$
$558$	−2.10944	−0.0892997
$559$	0	0
$560$	−34.7578	−1.46879
$561$	−27.3302	−1.15388
$562$	−59.8430	−2.52433
$563$	24.6898	1.04055	0.520277	−	0.853998i	$-0.325829\pi$
$563$	24.6898	1.04055	0.520277	+	0.853998i	$0.325829\pi$
$564$	−104.116	−4.38407
$565$	−18.7572	−0.789121
$566$	23.3560	0.981727
$567$	−35.9613	−1.51023
$568$	−70.1588	−2.94380
$569$	−21.7107	−0.910162	−0.455081	−	0.890450i	$-0.650390\pi$
$569$	−21.7107	−0.910162	−0.455081	+	0.890450i	$0.650390\pi$
$570$	15.0660	0.631045
$571$	30.8235	1.28992	0.644962	−	0.764215i	$-0.276873\pi$
$571$	30.8235	1.28992	0.644962	+	0.764215i	$0.276873\pi$
$572$	0	0
$573$	−42.4901	−1.77505
$574$	18.9192	0.789674
$575$	6.09907	0.254349
$576$	10.9467	0.456113
$577$	−32.2437	−1.34232	−0.671161	−	0.741311i	$-0.734204\pi$
$577$	−32.2437	−1.34232	−0.671161	+	0.741311i	$0.734204\pi$
$578$	−2.91510	−0.121252
$579$	11.4920	0.477592
$580$	−28.5658	−1.18613
$581$	−24.3372	−1.00968
$582$	15.6009	0.646680
$583$	−16.4121	−0.679720
$584$	−44.3297	−1.83438
$585$	0	0
$586$	16.8783	0.697238
$587$	7.71022	0.318235	0.159117	−	0.987260i	$-0.449135\pi$
$587$	7.71022	0.318235	0.159117	+	0.987260i	$0.449135\pi$
$588$	37.1483	1.53197
$589$	2.40015	0.0988964
$590$	0.381596	0.0157101
$591$	24.6781	1.01512
$592$	59.2573	2.43546
$593$	17.1507	0.704295	0.352147	−	0.935945i	$-0.385452\pi$
$593$	17.1507	0.704295	0.352147	+	0.935945i	$0.385452\pi$
$594$	34.5743	1.41860
$595$	13.9665	0.572572
$596$	−64.9967	−2.66237
$597$	−12.0049	−0.491330
$598$	0	0
$599$	−14.8111	−0.605163	−0.302582	−	0.953123i	$-0.597849\pi$
$599$	−14.8111	−0.605163	−0.302582	+	0.953123i	$0.597849\pi$
$600$	−15.4615	−0.631212
$601$	5.70203	0.232591	0.116295	−	0.993215i	$-0.462898\pi$
$601$	5.70203	0.232591	0.116295	+	0.993215i	$0.462898\pi$
$602$	−21.5186	−0.877033
$603$	−12.0821	−0.492023
$604$	−8.24126	−0.335332
$605$	0.576790	0.0234499
$606$	−11.7472	−0.477198
$607$	4.95359	0.201060	0.100530	−	0.994934i	$-0.467946\pi$
$607$	4.95359	0.201060	0.100530	+	0.994934i	$0.467946\pi$
$608$	−35.5381	−1.44126
$609$	37.6961	1.52752
$610$	15.9883	0.647345
$611$	0	0
$612$	−20.1690	−0.815282
$613$	−20.0157	−0.808427	−0.404213	−	0.914665i	$-0.632455\pi$
$613$	−20.0157	−0.808427	−0.404213	+	0.914665i	$0.632455\pi$
$614$	68.0916	2.74795
$615$	4.35026	0.175420
$616$	−82.3567	−3.31825
$617$	4.36915	0.175895	0.0879477	−	0.996125i	$-0.471969\pi$
$617$	4.36915	0.175895	0.0879477	+	0.996125i	$0.471969\pi$
$618$	66.3470	2.66887
$619$	−29.2465	−1.17552	−0.587758	−	0.809037i	$-0.699989\pi$
$619$	−29.2465	−1.17552	−0.587758	+	0.809037i	$0.699989\pi$
$620$	−4.13402	−0.166026
$621$	−24.7785	−0.994327
$622$	10.2065	0.409245
$623$	42.9692	1.72152
$624$	0	0
$625$	1.00000	0.0400000
$626$	−10.8961	−0.435495
$627$	18.4526	0.736926
$628$	−101.963	−4.06879
$629$	−23.8110	−0.949408
$630$	8.28769	0.330189
$631$	2.37642	0.0946038	0.0473019	−	0.998881i	$-0.484938\pi$
$631$	2.37642	0.0946038	0.0473019	+	0.998881i	$0.484938\pi$
$632$	−109.615	−4.36027
$633$	−27.2236	−1.08204
$634$	65.6774	2.60838
$635$	−0.368129	−0.0146087
$636$	−50.0444	−1.98439
$637$	0	0
$638$	−49.1279	−1.94499
$639$	8.64722	0.342079
$640$	5.38392	0.212818
$641$	−28.9687	−1.14419	−0.572097	−	0.820186i	$-0.693870\pi$
$641$	−28.9687	−1.14419	−0.572097	+	0.820186i	$0.693870\pi$
$642$	−28.9419	−1.14224
$643$	−14.0697	−0.554855	−0.277427	−	0.960747i	$-0.589482\pi$
$643$	−14.0697	−0.554855	−0.277427	+	0.960747i	$0.589482\pi$
$644$	99.0610	3.90355
$645$	−4.94796	−0.194826
$646$	32.2238	1.26783
$647$	8.25026	0.324351	0.162176	−	0.986762i	$-0.448149\pi$
$647$	8.25026	0.324351	0.162176	+	0.986762i	$0.448149\pi$
$648$	85.1483	3.34494
$649$	0.467373	0.0183460
$650$	0	0
$651$	5.45536	0.213812
$652$	−27.8408	−1.09033
$653$	−13.2659	−0.519136	−0.259568	−	0.965725i	$-0.583580\pi$
$653$	−13.2659	−0.519136	−0.259568	+	0.965725i	$0.583580\pi$
$654$	−37.3456	−1.46033
$655$	−3.09282	−0.120847
$656$	−23.1557	−0.904077
$657$	5.46373	0.213161
$658$	91.5051	3.56724
$659$	28.0173	1.09140	0.545699	−	0.837981i	$-0.316264\pi$
$659$	28.0173	1.09140	0.545699	+	0.837981i	$0.316264\pi$
$660$	−31.7828	−1.23714
$661$	30.0308	1.16806	0.584031	−	0.811731i	$-0.301474\pi$
$661$	30.0308	1.16806	0.584031	+	0.811731i	$0.301474\pi$
$662$	−14.6787	−0.570503
$663$	0	0
$664$	57.6250	2.23628
$665$	−9.42984	−0.365674
$666$	−14.1294	−0.547502
$667$	35.2087	1.36329
$668$	−69.8785	−2.70368
$669$	37.1791	1.43743
$670$	−33.2484	−1.28450
$671$	19.5822	0.755962
$672$	−80.7754	−3.11598
$673$	20.1701	0.777499	0.388749	−	0.921344i	$-0.372907\pi$
$673$	20.1701	0.777499	0.388749	+	0.921344i	$0.372907\pi$
$674$	−44.4949	−1.71388
$675$	−4.06267	−0.156372
$676$	0	0
$677$	−28.5492	−1.09723	−0.548616	−	0.836074i	$-0.684845\pi$
$677$	−28.5492	−1.09723	−0.548616	+	0.836074i	$0.684845\pi$
$678$	−98.3649	−3.77768
$679$	−9.76466	−0.374733
$680$	−33.0696	−1.26816
$681$	22.8950	0.877337
$682$	−7.10974	−0.272246
$683$	−12.4889	−0.477876	−0.238938	−	0.971035i	$-0.576799\pi$
$683$	−12.4889	−0.477876	−0.238938	+	0.971035i	$0.576799\pi$
$684$	13.6176	0.520681
$685$	−15.5865	−0.595529
$686$	27.9157	1.06582
$687$	−3.94038	−0.150335
$688$	26.3371	1.00409
$689$	0	0
$690$	31.9842	1.21762
$691$	45.5955	1.73453	0.867267	−	0.497843i	$-0.165874\pi$
$691$	45.5955	1.73453	0.867267	+	0.497843i	$0.165874\pi$
$692$	−48.8183	−1.85579
$693$	10.1506	0.385591
$694$	32.7530	1.24329
$695$	−18.7547	−0.711406
$696$	−89.2560	−3.38324
$697$	9.30452	0.352434
$698$	−7.20981	−0.272895
$699$	44.0332	1.66549
$700$	16.2420	0.613889
$701$	−4.57596	−0.172831	−0.0864157	−	0.996259i	$-0.527541\pi$
$701$	−4.57596	−0.172831	−0.0864157	+	0.996259i	$0.527541\pi$
$702$	0	0
$703$	16.0766	0.606340
$704$	36.8952	1.39054
$705$	21.0405	0.792433
$706$	59.9578	2.25654
$707$	7.35260	0.276523
$708$	1.42513	0.0535597
$709$	24.2827	0.911957	0.455979	−	0.889991i	$-0.349289\pi$
$709$	24.2827	0.911957	0.455979	+	0.889991i	$0.349289\pi$
$710$	23.7960	0.893047
$711$	13.5103	0.506677
$712$	−101.741	−3.81292
$713$	5.09537	0.190823
$714$	73.2422	2.74102
$715$	0	0
$716$	45.7887	1.71120
$717$	−22.4501	−0.838416
$718$	30.3635	1.13316
$719$	−13.0137	−0.485329	−0.242664	−	0.970110i	$-0.578021\pi$
$719$	−13.0137	−0.485329	−0.242664	+	0.970110i	$0.578021\pi$
$720$	−10.1435	−0.378025
$721$	−41.5267	−1.54654
$722$	28.3268	1.05422
$723$	−49.5429	−1.84252
$724$	−107.161	−3.98259
$725$	5.77280	0.214396
$726$	3.02476	0.112259
$727$	11.5596	0.428722	0.214361	−	0.976755i	$-0.431233\pi$
$727$	11.5596	0.428722	0.214361	+	0.976755i	$0.431233\pi$
$728$	0	0
$729$	13.7527	0.509358
$730$	15.0355	0.556487
$731$	−10.5829	−0.391423
$732$	59.7107	2.20697
$733$	13.7767	0.508855	0.254427	−	0.967092i	$-0.418113\pi$
$733$	13.7767	0.508855	0.254427	+	0.967092i	$0.418113\pi$
$734$	−37.0969	−1.36927
$735$	−7.50722	−0.276908
$736$	−75.4453	−2.78095
$737$	−40.7222	−1.50002
$738$	5.52126	0.203240
$739$	−5.83557	−0.214665	−0.107332	−	0.994223i	$-0.534231\pi$
$739$	−5.83557	−0.214665	−0.107332	+	0.994223i	$0.534231\pi$
$740$	−27.6904	−1.01792
$741$	0	0
$742$	43.9828	1.61466
$743$	−12.1353	−0.445202	−0.222601	−	0.974910i	$-0.571455\pi$
$743$	−12.1353	−0.445202	−0.222601	+	0.974910i	$0.571455\pi$
$744$	−12.9171	−0.473562
$745$	13.1350	0.481230
$746$	−48.7074	−1.78331
$747$	−7.10240	−0.259863
$748$	−67.9784	−2.48554
$749$	18.1148	0.661900
$750$	5.24412	0.191488
$751$	45.3878	1.65622	0.828112	−	0.560563i	$-0.189415\pi$
$751$	45.3878	1.65622	0.828112	+	0.560563i	$0.189415\pi$
$752$	−111.995	−4.08404
$753$	−28.1414	−1.02553
$754$	0	0
$755$	1.66546	0.0606122
$756$	−65.9859	−2.39988
$757$	19.5260	0.709684	0.354842	−	0.934926i	$-0.384535\pi$
$757$	19.5260	0.709684	0.354842	+	0.934926i	$0.384535\pi$
$758$	−4.04335	−0.146861
$759$	39.1738	1.42192
$760$	22.3278	0.809913
$761$	28.1496	1.02042	0.510210	−	0.860050i	$-0.329568\pi$
$761$	28.1496	1.02042	0.510210	+	0.860050i	$0.329568\pi$
$762$	−1.93051	−0.0699350
$763$	23.3747	0.846220
$764$	−105.686	−3.82357
$765$	4.07590	0.147365
$766$	−79.5061	−2.87267
$767$	0	0
$768$	−17.2368	−0.621979
$769$	1.42110	0.0512463	0.0256232	−	0.999672i	$-0.491843\pi$
$769$	1.42110	0.0512463	0.0256232	+	0.999672i	$0.491843\pi$
$770$	27.9332	1.00664
$771$	7.02761	0.253093
$772$	28.5841	1.02876
$773$	13.8186	0.497020	0.248510	−	0.968629i	$-0.420059\pi$
$773$	13.8186	0.497020	0.248510	+	0.968629i	$0.420059\pi$
$774$	−6.27985	−0.225724
$775$	0.835435	0.0300097
$776$	23.1205	0.829979
$777$	36.5409	1.31090
$778$	−52.2989	−1.87501
$779$	−6.28217	−0.225082
$780$	0	0
$781$	29.1450	1.04289
$782$	68.4092	2.44631
$783$	−23.4530	−0.838141
$784$	39.9596	1.42713
$785$	20.6056	0.735444
$786$	−16.2191	−0.578517
$787$	−9.96442	−0.355193	−0.177597	−	0.984103i	$-0.556832\pi$
$787$	−9.96442	−0.355193	−0.177597	+	0.984103i	$0.556832\pi$
$788$	61.3820	2.18664
$789$	−24.3301	−0.866174
$790$	37.1786	1.32275
$791$	61.5668	2.18906
$792$	−24.0344	−0.854026
$793$	0	0
$794$	15.0452	0.533935
$795$	10.1134	0.358684
$796$	−29.8599	−1.05836
$797$	−10.0651	−0.356523	−0.178262	−	0.983983i	$-0.557047\pi$
$797$	−10.0651	−0.356523	−0.178262	+	0.983983i	$0.557047\pi$
$798$	−49.4512	−1.75055
$799$	45.0024	1.59207
$800$	−12.3700	−0.437344
$801$	12.5398	0.443073
$802$	82.6041	2.91685
$803$	18.4152	0.649858
$804$	−124.171	−4.37919
$805$	−20.0190	−0.705577
$806$	0	0
$807$	−44.6756	−1.57265
$808$	−17.4093	−0.612457
$809$	−38.1107	−1.33990	−0.669950	−	0.742406i	$-0.733684\pi$
$809$	−38.1107	−1.33990	−0.669950	+	0.742406i	$0.733684\pi$
$810$	−28.8800	−1.01474
$811$	−3.57508	−0.125538	−0.0627690	−	0.998028i	$-0.519993\pi$
$811$	−3.57508	−0.125538	−0.0627690	+	0.998028i	$0.519993\pi$
$812$	93.7617	3.29039
$813$	−38.4488	−1.34846
$814$	−47.6222	−1.66916
$815$	5.62628	0.197080
$816$	−89.6427	−3.13812
$817$	7.14530	0.249982
$818$	−50.7445	−1.77424
$819$	0	0
$820$	10.8204	0.377865
$821$	43.5678	1.52053	0.760263	−	0.649615i	$-0.225070\pi$
$821$	43.5678	1.52053	0.760263	+	0.649615i	$0.225070\pi$
$822$	−81.7374	−2.85092
$823$	−4.31879	−0.150544	−0.0752718	−	0.997163i	$-0.523982\pi$
$823$	−4.31879	−0.150544	−0.0752718	+	0.997163i	$0.523982\pi$
$824$	98.3260	3.42535
$825$	6.42292	0.223617
$826$	−1.25251	−0.0435805
$827$	23.6479	0.822320	0.411160	−	0.911563i	$-0.365124\pi$
$827$	23.6479	0.822320	0.411160	+	0.911563i	$0.365124\pi$
$828$	28.9093	1.00467
$829$	−49.9657	−1.73538	−0.867691	−	0.497105i	$-0.834397\pi$
$829$	−49.9657	−1.73538	−0.867691	+	0.497105i	$0.834397\pi$
$830$	−19.5449	−0.678412
$831$	−23.6607	−0.820779
$832$	0	0
$833$	−16.0567	−0.556333
$834$	−98.3518	−3.40564
$835$	14.1216	0.488698
$836$	45.8972	1.58739
$837$	−3.39410	−0.117317
$838$	33.8632	1.16978
$839$	14.7181	0.508125	0.254063	−	0.967188i	$-0.418233\pi$
$839$	14.7181	0.508125	0.254063	+	0.967188i	$0.418233\pi$
$840$	50.7493	1.75102
$841$	4.32519	0.149145
$842$	−40.9871	−1.41251
$843$	45.1652	1.55557
$844$	−67.7134	−2.33079
$845$	0	0
$846$	26.7042	0.918110
$847$	−1.89320	−0.0650512
$848$	−53.8315	−1.84858
$849$	−17.6275	−0.604973
$850$	11.2163	0.384717
$851$	34.1297	1.16995
$852$	88.8699	3.04463
$853$	27.0196	0.925134	0.462567	−	0.886584i	$-0.346929\pi$
$853$	27.0196	0.925134	0.462567	+	0.886584i	$0.346929\pi$
$854$	−52.4783	−1.79577
$855$	−2.75194	−0.0941144
$856$	−42.8917	−1.46601
$857$	33.1663	1.13294	0.566470	−	0.824083i	$-0.308309\pi$
$857$	33.1663	1.13294	0.566470	+	0.824083i	$0.308309\pi$
$858$	0	0
$859$	−47.4781	−1.61993	−0.809966	−	0.586476i	$-0.800515\pi$
$859$	−47.4781	−1.61993	−0.809966	+	0.586476i	$0.800515\pi$
$860$	−12.3071	−0.419668
$861$	−14.2789	−0.486623
$862$	−43.2878	−1.47439
$863$	−3.65653	−0.124470	−0.0622349	−	0.998062i	$-0.519823\pi$
$863$	−3.65653	−0.124470	−0.0622349	+	0.998062i	$0.519823\pi$
$864$	50.2551	1.70971
$865$	9.86557	0.335440
$866$	40.6937	1.38283
$867$	2.20011	0.0747196
$868$	13.5691	0.460566
$869$	45.5358	1.54470
$870$	30.2732	1.02636
$871$	0	0
$872$	−55.3460	−1.87425
$873$	−2.84965	−0.0964461
$874$	−46.1881	−1.56234
$875$	−3.28231	−0.110962
$876$	56.1523	1.89721
$877$	−7.26069	−0.245176	−0.122588	−	0.992458i	$-0.539119\pi$
$877$	−7.26069	−0.245176	−0.122588	+	0.992458i	$0.539119\pi$
$878$	104.547	3.52828
$879$	−12.7386	−0.429661
$880$	−34.1880	−1.15248
$881$	−39.7044	−1.33767	−0.668837	−	0.743409i	$-0.733208\pi$
$881$	−39.7044	−1.33767	−0.668837	+	0.743409i	$0.733208\pi$
$882$	−9.52800	−0.320825
$883$	7.48855	0.252010	0.126005	−	0.992030i	$-0.459785\pi$
$883$	7.48855	0.252010	0.126005	+	0.992030i	$0.459785\pi$
$884$	0	0
$885$	−0.288001	−0.00968106
$886$	−12.9324	−0.434472
$887$	−3.24520	−0.108963	−0.0544816	−	0.998515i	$-0.517351\pi$
$887$	−3.24520	−0.108963	−0.0544816	+	0.998515i	$0.517351\pi$
$888$	−86.5206	−2.90344
$889$	1.20831	0.0405254
$890$	34.5079	1.15671
$891$	−35.3718	−1.18500
$892$	92.4757	3.09631
$893$	−30.3844	−1.01678
$894$	68.8816	2.30375
$895$	−9.25334	−0.309305
$896$	−17.6717	−0.590369
$897$	0	0
$898$	74.5722	2.48851
$899$	4.82279	0.160849
$900$	4.73995	0.157998
$901$	21.6308	0.720628
$902$	18.6091	0.619615
$903$	16.2407	0.540457
$904$	−145.776	−4.84845
$905$	21.6558	0.719865
$906$	8.73385	0.290163
$907$	28.5229	0.947087	0.473544	−	0.880770i	$-0.342975\pi$
$907$	28.5229	0.947087	0.473544	+	0.880770i	$0.342975\pi$
$908$	56.9467	1.88984
$909$	2.14573	0.0711695
$910$	0	0
$911$	−49.2789	−1.63268	−0.816341	−	0.577570i	$-0.804001\pi$
$911$	−49.2789	−1.63268	−0.816341	+	0.577570i	$0.804001\pi$
$912$	60.5244	2.00416
$913$	−23.9382	−0.792240
$914$	85.2311	2.81919
$915$	−12.0668	−0.398916
$916$	−9.80092	−0.323832
$917$	10.1516	0.335235
$918$	−45.5683	−1.50398
$919$	−9.83623	−0.324467	−0.162234	−	0.986752i	$-0.551870\pi$
$919$	−9.83623	−0.324467	−0.162234	+	0.986752i	$0.551870\pi$
$920$	47.4005	1.56275
$921$	−51.3907	−1.69338
$922$	−18.3833	−0.605421
$923$	0	0
$924$	104.321	3.43191
$925$	5.59588	0.183991
$926$	87.5048	2.87559
$927$	−12.1189	−0.398036
$928$	−71.4093	−2.34413
$929$	27.6765	0.908037	0.454018	−	0.890992i	$-0.349990\pi$
$929$	27.6765	0.908037	0.454018	+	0.890992i	$0.349990\pi$
$930$	4.38112	0.143662
$931$	10.8411	0.355302
$932$	109.524	3.58758
$933$	−7.70317	−0.252190
$934$	83.3251	2.72648
$935$	13.7376	0.449267
$936$	0	0
$937$	−39.8796	−1.30281	−0.651406	−	0.758730i	$-0.725820\pi$
$937$	−39.8796	−1.30281	−0.651406	+	0.758730i	$0.725820\pi$
$938$	109.131	3.56327
$939$	8.22358	0.268367
$940$	52.3342	1.70695
$941$	2.58047	0.0841210	0.0420605	−	0.999115i	$-0.486608\pi$
$941$	2.58047	0.0841210	0.0420605	+	0.999115i	$0.486608\pi$
$942$	108.058	3.52072
$943$	−13.3367	−0.434302
$944$	1.53298	0.0498942
$945$	13.3349	0.433785
$946$	−21.1659	−0.688162
$947$	5.28389	0.171703	0.0858517	−	0.996308i	$-0.472639\pi$
$947$	5.28389	0.171703	0.0858517	+	0.996308i	$0.472639\pi$
$948$	138.849	4.50962
$949$	0	0
$950$	−7.57297	−0.245700
$951$	−49.5686	−1.60737
$952$	108.545	3.51795
$953$	2.22577	0.0720999	0.0360499	−	0.999350i	$-0.488522\pi$
$953$	2.22577	0.0720999	0.0360499	+	0.999350i	$0.488522\pi$
$954$	12.8356	0.415570
$955$	21.3578	0.691121
$956$	−55.8403	−1.80600
$957$	37.0782	1.19857
$958$	27.3307	0.883014
$959$	51.1596	1.65203
$960$	−22.7353	−0.733780
$961$	−30.3020	−0.977485
$962$	0	0
$963$	5.28649	0.170355
$964$	−123.228	−3.96891
$965$	−5.77649	−0.185952
$966$	−104.982	−3.37774
$967$	27.6359	0.888712	0.444356	−	0.895850i	$-0.353433\pi$
$967$	27.6359	0.888712	0.444356	+	0.895850i	$0.353433\pi$
$968$	4.48268	0.144079
$969$	−24.3202	−0.781277
$970$	−7.84186	−0.251787
$971$	−49.9320	−1.60239	−0.801197	−	0.598400i	$-0.795803\pi$
$971$	−49.9320	−1.60239	−0.801197	+	0.598400i	$0.795803\pi$
$972$	−47.5465	−1.52505
$973$	61.5586	1.97348
$974$	92.0363	2.94903
$975$	0	0
$976$	64.2294	2.05593
$977$	35.2611	1.12810	0.564052	−	0.825739i	$-0.309242\pi$
$977$	35.2611	1.12810	0.564052	+	0.825739i	$0.309242\pi$
$978$	29.5049	0.943461
$979$	42.2648	1.35079
$980$	−18.6727	−0.596478
$981$	6.82151	0.217794
$982$	−46.1330	−1.47216
$983$	−20.6986	−0.660184	−0.330092	−	0.943949i	$-0.607080\pi$
$983$	−20.6986	−0.660184	−0.330092	+	0.943949i	$0.607080\pi$
$984$	33.8092	1.07780
$985$	−12.4045	−0.395242
$986$	64.7496	2.06205
$987$	−69.0615	−2.19825
$988$	0	0
$989$	15.1690	0.482348
$990$	8.15183	0.259082
$991$	12.6676	0.402400	0.201200	−	0.979550i	$-0.435516\pi$
$991$	12.6676	0.402400	0.201200	+	0.979550i	$0.435516\pi$
$992$	−10.3343	−0.328114
$993$	11.0784	0.351563
$994$	−78.1056	−2.47736
$995$	6.03432	0.191301
$996$	−72.9934	−2.31288
$997$	−47.7614	−1.51262	−0.756309	−	0.654214i	$-0.772999\pi$
$997$	−47.7614	−1.51262	−0.756309	+	0.654214i	$0.772999\pi$
$998$	−22.3146	−0.706356
$999$	−22.7342	−0.719279

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.a.n.1.2	✓	9
3.2	odd	2		7605.2.a.cs.1.8		9
5.4	even	2		4225.2.a.bt.1.8		9
13.2	odd	12		845.2.m.j.316.2		36
13.3	even	3		845.2.e.p.191.8		18
13.4	even	6		845.2.e.o.146.2		18
13.5	odd	4		845.2.c.h.506.17		18
13.6	odd	12		845.2.m.j.361.17		36
13.7	odd	12		845.2.m.j.361.2		36
13.8	odd	4		845.2.c.h.506.2		18
13.9	even	3		845.2.e.p.146.8		18
13.10	even	6		845.2.e.o.191.2		18
13.11	odd	12		845.2.m.j.316.17		36
13.12	even	2		845.2.a.o.1.8	yes	9
39.38	odd	2		7605.2.a.cp.1.2		9
65.64	even	2		4225.2.a.bs.1.2		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.a.n.1.2	✓	9	1.1	even	1	trivial
845.2.a.o.1.8	yes	9	13.12	even	2
845.2.c.h.506.2		18	13.8	odd	4
845.2.c.h.506.17		18	13.5	odd	4
845.2.e.o.146.2		18	13.4	even	6
845.2.e.o.191.2		18	13.10	even	6
845.2.e.p.146.8		18	13.9	even	3
845.2.e.p.191.8		18	13.3	even	3
845.2.m.j.316.2		36	13.2	odd	12
845.2.m.j.316.17		36	13.11	odd	12
845.2.m.j.361.2		36	13.7	odd	12
845.2.m.j.361.17		36	13.6	odd	12
4225.2.a.bs.1.2		9	65.64	even	2
4225.2.a.bt.1.8		9	5.4	even	2
7605.2.a.cp.1.2		9	39.38	odd	2
7605.2.a.cs.1.8		9	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:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.a (trivial)

\(f(q)\)	\(=\)	\(q-2.63597 q^{2} +1.98944 q^{3} +4.94835 q^{4} -1.00000 q^{5} -5.24412 q^{6} +3.28231 q^{7} -7.77176 q^{8} +0.957886 q^{9} +O(q^{10})\)	\(q-2.63597 q^{2} +1.98944 q^{3} +4.94835 q^{4} -1.00000 q^{5} -5.24412 q^{6} +3.28231 q^{7} -7.77176 q^{8} +0.957886 q^{9} +2.63597 q^{10} +3.22850 q^{11} +9.84446 q^{12} -8.65206 q^{14} -1.98944 q^{15} +10.5894 q^{16} -4.25510 q^{17} -2.52496 q^{18} +2.87293 q^{19} -4.94835 q^{20} +6.52996 q^{21} -8.51023 q^{22} +6.09907 q^{23} -15.4615 q^{24} +1.00000 q^{25} -4.06267 q^{27} +16.2420 q^{28} +5.77280 q^{29} +5.24412 q^{30} +0.835435 q^{31} -12.3700 q^{32} +6.42292 q^{33} +11.2163 q^{34} -3.28231 q^{35} +4.73995 q^{36} +5.59588 q^{37} -7.57297 q^{38} +7.77176 q^{40} -2.18667 q^{41} -17.2128 q^{42} +2.48711 q^{43} +15.9757 q^{44} -0.957886 q^{45} -16.0770 q^{46} -10.5761 q^{47} +21.0671 q^{48} +3.77353 q^{49} -2.63597 q^{50} -8.46528 q^{51} -5.08351 q^{53} +10.7091 q^{54} -3.22850 q^{55} -25.5093 q^{56} +5.71554 q^{57} -15.2169 q^{58} +0.144765 q^{59} -9.84446 q^{60} +6.06541 q^{61} -2.20218 q^{62} +3.14407 q^{63} +11.4280 q^{64} -16.9306 q^{66} -12.6133 q^{67} -21.0557 q^{68} +12.1338 q^{69} +8.65206 q^{70} +9.02740 q^{71} -7.44446 q^{72} +5.70395 q^{73} -14.7506 q^{74} +1.98944 q^{75} +14.2163 q^{76} +10.5969 q^{77} +14.1043 q^{79} -10.5894 q^{80} -10.9561 q^{81} +5.76401 q^{82} -7.41467 q^{83} +32.3125 q^{84} +4.25510 q^{85} -6.55595 q^{86} +11.4847 q^{87} -25.0911 q^{88} +13.0912 q^{89} +2.52496 q^{90} +30.1803 q^{92} +1.66205 q^{93} +27.8783 q^{94} -2.87293 q^{95} -24.6093 q^{96} -2.97494 q^{97} -9.94691 q^{98} +3.09253 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 3 q^{2} + 7 q^{3} + 17 q^{4} - 9 q^{5} + 2 q^{6} - 7 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \) 9 * q - 3 * q^2 + 7 * q^3 + 17 * q^4 - 9 * q^5 + 2 * q^6 - 7 * q^7 - 12 * q^8 + 16 * q^9	\( 9 q - 3 q^{2} + 7 q^{3} + 17 q^{4} - 9 q^{5} + 2 q^{6} - 7 q^{7} - 12 q^{8} + 16 q^{9} + 3 q^{10} + 9 q^{11} + 12 q^{12} - 2 q^{14} - 7 q^{15} + 37 q^{16} - q^{17} + 10 q^{18} + 4 q^{19} - 17 q^{20} + q^{21} + 12 q^{22} + 14 q^{23} + 35 q^{24} + 9 q^{25} + 22 q^{27} - 18 q^{28} + 12 q^{29} - 2 q^{30} + 7 q^{31} - 22 q^{32} + 8 q^{33} + 30 q^{34} + 7 q^{35} + 3 q^{36} + 5 q^{37} - 47 q^{38} + 12 q^{40} + 10 q^{41} - 11 q^{42} + 39 q^{43} + 25 q^{44} - 16 q^{45} - 6 q^{46} - 36 q^{47} - 3 q^{48} + 16 q^{49} - 3 q^{50} + 43 q^{51} - 8 q^{53} + 2 q^{54} - 9 q^{55} - 29 q^{56} + 32 q^{57} - 21 q^{58} + 21 q^{59} - 12 q^{60} - 3 q^{61} - 10 q^{62} - 35 q^{63} + 34 q^{64} - 49 q^{66} - q^{67} - 20 q^{68} - 13 q^{69} + 2 q^{70} + q^{71} + 3 q^{72} - 15 q^{74} + 7 q^{75} + 5 q^{76} - 4 q^{77} + 39 q^{79} - 37 q^{80} + 29 q^{81} - 4 q^{82} - 7 q^{83} - 12 q^{84} + q^{85} + 24 q^{86} + 16 q^{87} + 42 q^{88} + 19 q^{89} - 10 q^{90} - 27 q^{92} - 31 q^{93} + 16 q^{94} - 4 q^{95} + 7 q^{96} + 34 q^{97} - 48 q^{98} + q^{99}+O(q^{100}) \) 9 * q - 3 * q^2 + 7 * q^3 + 17 * q^4 - 9 * q^5 + 2 * q^6 - 7 * q^7 - 12 * q^8 + 16 * q^9 + 3 * q^10 + 9 * q^11 + 12 * q^12 - 2 * q^14 - 7 * q^15 + 37 * q^16 - q^17 + 10 * q^18 + 4 * q^19 - 17 * q^20 + q^21 + 12 * q^22 + 14 * q^23 + 35 * q^24 + 9 * q^25 + 22 * q^27 - 18 * q^28 + 12 * q^29 - 2 * q^30 + 7 * q^31 - 22 * q^32 + 8 * q^33 + 30 * q^34 + 7 * q^35 + 3 * q^36 + 5 * q^37 - 47 * q^38 + 12 * q^40 + 10 * q^41 - 11 * q^42 + 39 * q^43 + 25 * q^44 - 16 * q^45 - 6 * q^46 - 36 * q^47 - 3 * q^48 + 16 * q^49 - 3 * q^50 + 43 * q^51 - 8 * q^53 + 2 * q^54 - 9 * q^55 - 29 * q^56 + 32 * q^57 - 21 * q^58 + 21 * q^59 - 12 * q^60 - 3 * q^61 - 10 * q^62 - 35 * q^63 + 34 * q^64 - 49 * q^66 - q^67 - 20 * q^68 - 13 * q^69 + 2 * q^70 + q^71 + 3 * q^72 - 15 * q^74 + 7 * q^75 + 5 * q^76 - 4 * q^77 + 39 * q^79 - 37 * q^80 + 29 * q^81 - 4 * q^82 - 7 * q^83 - 12 * q^84 + q^85 + 24 * q^86 + 16 * q^87 + 42 * q^88 + 19 * q^89 - 10 * q^90 - 27 * q^92 - 31 * q^93 + 16 * q^94 - 4 * q^95 + 7 * q^96 + 34 * q^97 - 48 * q^98 + q^99

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