LMFDB - Embedded newform 847.2.a.o.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("847.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$6.76332905120$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 11x^{6} + 10x^{5} + 35x^{4} - 30x^{3} - 30x^{2} + 30x - 5 $ x^8 - x^7 - 11x^6 + 10x^5 + 35x^4 - 30x^3 - 30x^2 + 30x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.669744$ of defining polynomial
Character	$\chi$	$=$	847.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.669744 q^{2} +3.13977 q^{3} -1.55144 q^{4} +2.14378 q^{5} -2.10284 q^{6} -1.00000 q^{7} +2.37856 q^{8} +6.85818 q^{9} +O(q^{10})$	\(q-0.669744 q^{2} +3.13977 q^{3} -1.55144 q^{4} +2.14378 q^{5} -2.10284 q^{6} -1.00000 q^{7} +2.37856 q^{8} +6.85818 q^{9} -1.43578 q^{10} -4.87118 q^{12} -2.52826 q^{13} +0.669744 q^{14} +6.73098 q^{15} +1.50986 q^{16} +1.79092 q^{17} -4.59322 q^{18} +6.72740 q^{19} -3.32595 q^{20} -3.13977 q^{21} -3.16429 q^{23} +7.46813 q^{24} -0.404214 q^{25} +1.69329 q^{26} +12.1138 q^{27} +1.55144 q^{28} +0.924170 q^{29} -4.50803 q^{30} +3.00178 q^{31} -5.76834 q^{32} -1.19946 q^{34} -2.14378 q^{35} -10.6401 q^{36} -1.50718 q^{37} -4.50564 q^{38} -7.93817 q^{39} +5.09910 q^{40} -5.56104 q^{41} +2.10284 q^{42} +8.42985 q^{43} +14.7024 q^{45} +2.11926 q^{46} +4.39771 q^{47} +4.74062 q^{48} +1.00000 q^{49} +0.270720 q^{50} +5.62308 q^{51} +3.92246 q^{52} +0.667421 q^{53} -8.11314 q^{54} -2.37856 q^{56} +21.1225 q^{57} -0.618957 q^{58} -0.368360 q^{59} -10.4427 q^{60} -5.01149 q^{61} -2.01043 q^{62} -6.85818 q^{63} +0.843584 q^{64} -5.42003 q^{65} -0.902129 q^{67} -2.77851 q^{68} -9.93514 q^{69} +1.43578 q^{70} -14.8694 q^{71} +16.3126 q^{72} -8.03816 q^{73} +1.00942 q^{74} -1.26914 q^{75} -10.4372 q^{76} +5.31654 q^{78} -4.05668 q^{79} +3.23681 q^{80} +17.4600 q^{81} +3.72447 q^{82} +4.05134 q^{83} +4.87118 q^{84} +3.83934 q^{85} -5.64584 q^{86} +2.90168 q^{87} -8.30727 q^{89} -9.84685 q^{90} +2.52826 q^{91} +4.90921 q^{92} +9.42492 q^{93} -2.94534 q^{94} +14.4221 q^{95} -18.1113 q^{96} +8.51583 q^{97} -0.669744 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} + q^{6} - 8 q^{7} + 14 q^{9}+O(q^{10}) $ 8 * q - q^2 + 4 * q^3 + 7 * q^4 + 10 * q^5 + q^6 - 8 * q^7 + 14 * q^9	$ 8 q - q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} + q^{6} - 8 q^{7} + 14 q^{9} - 6 q^{10} + 9 q^{12} + 6 q^{13} + q^{14} + 11 q^{15} + q^{16} + 5 q^{17} - 8 q^{18} + 13 q^{19} + 23 q^{20} - 4 q^{21} + 16 q^{23} + 10 q^{24} + 16 q^{25} - 6 q^{26} + 10 q^{27} - 7 q^{28} - 9 q^{29} + 36 q^{30} + 9 q^{31} - 16 q^{32} - 12 q^{34} - 10 q^{35} - 14 q^{36} + 7 q^{37} - 10 q^{38} - 13 q^{39} - 5 q^{40} + 10 q^{41} - q^{42} + 4 q^{43} + 35 q^{45} - 4 q^{46} + 16 q^{47} - 3 q^{48} + 8 q^{49} - 6 q^{50} - 13 q^{51} + 41 q^{52} + 37 q^{53} - 30 q^{54} - 2 q^{57} - 15 q^{58} + q^{59} + 5 q^{60} - 19 q^{61} + 18 q^{62} - 14 q^{63} - 4 q^{64} + 4 q^{65} - 19 q^{67} - 9 q^{68} + 20 q^{69} + 6 q^{70} + 13 q^{71} + 35 q^{72} + 25 q^{73} - 33 q^{74} - 13 q^{75} - 26 q^{76} - 29 q^{78} + 4 q^{80} + 8 q^{81} - 13 q^{82} + 25 q^{83} - 9 q^{84} - 3 q^{85} + 4 q^{86} + 36 q^{87} + 37 q^{89} + 2 q^{90} - 6 q^{91} + 35 q^{92} + 21 q^{93} + 42 q^{94} - 21 q^{95} + 6 q^{96} + 15 q^{97} - q^{98}+O(q^{100}) $ 8 * q - q^2 + 4 * q^3 + 7 * q^4 + 10 * q^5 + q^6 - 8 * q^7 + 14 * q^9 - 6 * q^10 + 9 * q^12 + 6 * q^13 + q^14 + 11 * q^15 + q^16 + 5 * q^17 - 8 * q^18 + 13 * q^19 + 23 * q^20 - 4 * q^21 + 16 * q^23 + 10 * q^24 + 16 * q^25 - 6 * q^26 + 10 * q^27 - 7 * q^28 - 9 * q^29 + 36 * q^30 + 9 * q^31 - 16 * q^32 - 12 * q^34 - 10 * q^35 - 14 * q^36 + 7 * q^37 - 10 * q^38 - 13 * q^39 - 5 * q^40 + 10 * q^41 - q^42 + 4 * q^43 + 35 * q^45 - 4 * q^46 + 16 * q^47 - 3 * q^48 + 8 * q^49 - 6 * q^50 - 13 * q^51 + 41 * q^52 + 37 * q^53 - 30 * q^54 - 2 * q^57 - 15 * q^58 + q^59 + 5 * q^60 - 19 * q^61 + 18 * q^62 - 14 * q^63 - 4 * q^64 + 4 * q^65 - 19 * q^67 - 9 * q^68 + 20 * q^69 + 6 * q^70 + 13 * q^71 + 35 * q^72 + 25 * q^73 - 33 * q^74 - 13 * q^75 - 26 * q^76 - 29 * q^78 + 4 * q^80 + 8 * q^81 - 13 * q^82 + 25 * q^83 - 9 * q^84 - 3 * q^85 + 4 * q^86 + 36 * q^87 + 37 * q^89 + 2 * q^90 - 6 * q^91 + 35 * q^92 + 21 * q^93 + 42 * q^94 - 21 * q^95 + 6 * q^96 + 15 * q^97 - q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.669744	−0.473580	−0.236790	−	0.971561i	$-0.576095\pi$
$2$	−0.669744	−0.473580	−0.236790	+	0.971561i	$0.576095\pi$
$3$	3.13977	1.81275	0.906374	−	0.422475i	$-0.138839\pi$
$3$	3.13977	1.81275	0.906374	+	0.422475i	$0.138839\pi$
$4$	−1.55144	−0.775722
$5$	2.14378	0.958727	0.479363	−	0.877616i	$-0.340868\pi$
$5$	2.14378	0.958727	0.479363	+	0.877616i	$0.340868\pi$
$6$	−2.10284	−0.858482
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.37856	0.840947
$9$	6.85818	2.28606
$10$	−1.43578	−0.454034
$11$	0	0
$11$	0	0
$12$	−4.87118	−1.40619
$13$	−2.52826	−0.701214	−0.350607	−	0.936523i	$-0.614025\pi$
$13$	−2.52826	−0.701214	−0.350607	+	0.936523i	$0.614025\pi$
$14$	0.669744	0.178997
$15$	6.73098	1.73793
$16$	1.50986	0.377465
$17$	1.79092	0.434362	0.217181	−	0.976131i	$-0.430314\pi$
$17$	1.79092	0.434362	0.217181	+	0.976131i	$0.430314\pi$
$18$	−4.59322	−1.08263
$19$	6.72740	1.54337	0.771686	−	0.636004i	$-0.219414\pi$
$19$	6.72740	1.54337	0.771686	+	0.636004i	$0.219414\pi$
$20$	−3.32595	−0.743705
$21$	−3.13977	−0.685155
$22$	0	0
$23$	−3.16429	−0.659799	−0.329900	−	0.944016i	$-0.607015\pi$
$23$	−3.16429	−0.659799	−0.329900	+	0.944016i	$0.607015\pi$
$24$	7.46813	1.52443
$25$	−0.404214	−0.0808427
$26$	1.69329	0.332081
$27$	12.1138	2.33130
$28$	1.55144	0.293195
$29$	0.924170	0.171614	0.0858070	−	0.996312i	$-0.472653\pi$
$29$	0.924170	0.171614	0.0858070	+	0.996312i	$0.472653\pi$
$30$	−4.50803	−0.823050
$31$	3.00178	0.539136	0.269568	−	0.962981i	$-0.413119\pi$
$31$	3.00178	0.539136	0.269568	+	0.962981i	$0.413119\pi$
$32$	−5.76834	−1.01971
$33$	0	0
$34$	−1.19946	−0.205705
$35$	−2.14378	−0.362365
$36$	−10.6401	−1.77334
$37$	−1.50718	−0.247779	−0.123889	−	0.992296i	$-0.539537\pi$
$37$	−1.50718	−0.247779	−0.123889	+	0.992296i	$0.539537\pi$
$38$	−4.50564	−0.730911
$39$	−7.93817	−1.27112
$40$	5.09910	0.806239
$41$	−5.56104	−0.868488	−0.434244	−	0.900795i	$-0.642984\pi$
$41$	−5.56104	−0.868488	−0.434244	+	0.900795i	$0.642984\pi$
$42$	2.10284	0.324476
$43$	8.42985	1.28554	0.642770	−	0.766059i	$-0.277785\pi$
$43$	8.42985	1.28554	0.642770	+	0.766059i	$0.277785\pi$
$44$	0	0
$45$	14.7024	2.19171
$46$	2.11926	0.312468
$47$	4.39771	0.641471	0.320736	−	0.947169i	$-0.396070\pi$
$47$	4.39771	0.641471	0.320736	+	0.947169i	$0.396070\pi$
$48$	4.74062	0.684250
$49$	1.00000	0.142857
$50$	0.270720	0.0382855
$51$	5.62308	0.787389
$52$	3.92246	0.543947
$53$	0.667421	0.0916773	0.0458387	−	0.998949i	$-0.485404\pi$
$53$	0.667421	0.0916773	0.0458387	+	0.998949i	$0.485404\pi$
$54$	−8.11314	−1.10406
$55$	0	0
$56$	−2.37856	−0.317848
$57$	21.1225	2.79775
$58$	−0.618957	−0.0812731
$59$	−0.368360	−0.0479564	−0.0239782	−	0.999712i	$-0.507633\pi$
$59$	−0.368360	−0.0479564	−0.0239782	+	0.999712i	$0.507633\pi$
$60$	−10.4427	−1.34815
$61$	−5.01149	−0.641655	−0.320828	−	0.947138i	$-0.603961\pi$
$61$	−5.01149	−0.641655	−0.320828	+	0.947138i	$0.603961\pi$
$62$	−2.01043	−0.255324
$63$	−6.85818	−0.864049
$64$	0.843584	0.105448
$65$	−5.42003	−0.672273
$66$	0	0
$67$	−0.902129	−0.110213	−0.0551063	−	0.998480i	$-0.517550\pi$
$67$	−0.902129	−0.110213	−0.0551063	+	0.998480i	$0.517550\pi$
$68$	−2.77851	−0.336944
$69$	−9.93514	−1.19605
$70$	1.43578	0.171609
$71$	−14.8694	−1.76467	−0.882335	−	0.470623i	$-0.844029\pi$
$71$	−14.8694	−1.76467	−0.882335	+	0.470623i	$0.844029\pi$
$72$	16.3126	1.92245
$73$	−8.03816	−0.940795	−0.470398	−	0.882455i	$-0.655890\pi$
$73$	−8.03816	−0.940795	−0.470398	+	0.882455i	$0.655890\pi$
$74$	1.00942	0.117343
$75$	−1.26914	−0.146548
$76$	−10.4372	−1.19723
$77$	0	0
$78$	5.31654	0.601980
$79$	−4.05668	−0.456412	−0.228206	−	0.973613i	$-0.573286\pi$
$79$	−4.05668	−0.456412	−0.228206	+	0.973613i	$0.573286\pi$
$80$	3.23681	0.361886
$81$	17.4600	1.94001
$82$	3.72447	0.411299
$83$	4.05134	0.444692	0.222346	−	0.974968i	$-0.428628\pi$
$83$	4.05134	0.444692	0.222346	+	0.974968i	$0.428628\pi$
$84$	4.87118	0.531489
$85$	3.83934	0.416434
$86$	−5.64584	−0.608807
$87$	2.90168	0.311093
$88$	0	0
$89$	−8.30727	−0.880569	−0.440284	−	0.897858i	$-0.645122\pi$
$89$	−8.30727	−0.880569	−0.440284	+	0.897858i	$0.645122\pi$
$90$	−9.84685	−1.03795
$91$	2.52826	0.265034
$92$	4.90921	0.511820
$93$	9.42492	0.977318
$94$	−2.94534	−0.303788
$95$	14.4221	1.47967
$96$	−18.1113	−1.84847
$97$	8.51583	0.864652	0.432326	−	0.901717i	$-0.357693\pi$
$97$	8.51583	0.864652	0.432326	+	0.901717i	$0.357693\pi$
$98$	−0.669744	−0.0676544
$99$	0	0
$100$	0.627115	0.0627115
$101$	4.02707	0.400708	0.200354	−	0.979724i	$-0.435791\pi$
$101$	4.02707	0.400708	0.200354	+	0.979724i	$0.435791\pi$
$102$	−3.76602	−0.372892
$103$	−17.6258	−1.73672	−0.868362	−	0.495930i	$-0.834827\pi$
$103$	−17.6258	−1.73672	−0.868362	+	0.495930i	$0.834827\pi$
$104$	−6.01362	−0.589684
$105$	−6.73098	−0.656876
$106$	−0.447001	−0.0434166
$107$	15.4037	1.48913	0.744567	−	0.667548i	$-0.232656\pi$
$107$	15.4037	1.48913	0.744567	+	0.667548i	$0.232656\pi$
$108$	−18.7939	−1.80844
$109$	18.9265	1.81283	0.906416	−	0.422386i	$-0.138807\pi$
$109$	18.9265	1.81283	0.906416	+	0.422386i	$0.138807\pi$
$110$	0	0
$111$	−4.73220	−0.449161
$112$	−1.50986	−0.142669
$113$	1.68062	0.158100	0.0790498	−	0.996871i	$-0.474811\pi$
$113$	1.68062	0.158100	0.0790498	+	0.996871i	$0.474811\pi$
$114$	−14.1467	−1.32496
$115$	−6.78353	−0.632567
$116$	−1.43380	−0.133125
$117$	−17.3393	−1.60302
$118$	0.246707	0.0227112
$119$	−1.79092	−0.164173
$120$	16.0100	1.46151
$121$	0	0
$122$	3.35641	0.303875
$123$	−17.4604	−1.57435
$124$	−4.65710	−0.418220
$125$	−11.5854	−1.03623
$126$	4.59322	0.409197
$127$	−17.5669	−1.55881	−0.779406	−	0.626519i	$-0.784479\pi$
$127$	−17.5669	−1.55881	−0.779406	+	0.626519i	$0.784479\pi$
$128$	10.9717	0.969769
$129$	26.4678	2.33036
$130$	3.63003	0.318375
$131$	−6.72557	−0.587616	−0.293808	−	0.955865i	$-0.594923\pi$
$131$	−6.72557	−0.587616	−0.293808	+	0.955865i	$0.594923\pi$
$132$	0	0
$133$	−6.72740	−0.583340
$134$	0.604195	0.0521945
$135$	25.9693	2.23508
$136$	4.25981	0.365275
$137$	13.8676	1.18479	0.592396	−	0.805647i	$-0.298182\pi$
$137$	13.8676	1.18479	0.592396	+	0.805647i	$0.298182\pi$
$138$	6.65400	0.566426
$139$	−14.2707	−1.21043	−0.605214	−	0.796063i	$-0.706912\pi$
$139$	−14.2707	−1.21043	−0.605214	+	0.796063i	$0.706912\pi$
$140$	3.32595	0.281094
$141$	13.8078	1.16283
$142$	9.95867	0.835713
$143$	0	0
$144$	10.3549	0.862908
$145$	1.98122	0.164531
$146$	5.38351	0.445542
$147$	3.13977	0.258964
$148$	2.33830	0.192207
$149$	−2.62292	−0.214878	−0.107439	−	0.994212i	$-0.534265\pi$
$149$	−2.62292	−0.214878	−0.107439	+	0.994212i	$0.534265\pi$
$150$	0.849998	0.0694021
$151$	2.98960	0.243290	0.121645	−	0.992574i	$-0.461183\pi$
$151$	2.98960	0.243290	0.121645	+	0.992574i	$0.461183\pi$
$152$	16.0015	1.29789
$153$	12.2824	0.992977
$154$	0	0
$155$	6.43516	0.516884
$156$	12.3156	0.986039
$157$	11.2547	0.898222	0.449111	−	0.893476i	$-0.351741\pi$
$157$	11.2547	0.898222	0.449111	+	0.893476i	$0.351741\pi$
$158$	2.71694	0.216148
$159$	2.09555	0.166188
$160$	−12.3660	−0.977621
$161$	3.16429	0.249381
$162$	−11.6938	−0.918749
$163$	−18.2030	−1.42577	−0.712883	−	0.701283i	$-0.752611\pi$
$163$	−18.2030	−1.42577	−0.712883	+	0.701283i	$0.752611\pi$
$164$	8.62763	0.673705
$165$	0	0
$166$	−2.71336	−0.210598
$167$	−19.9312	−1.54233	−0.771163	−	0.636638i	$-0.780325\pi$
$167$	−19.9312	−1.54233	−0.771163	+	0.636638i	$0.780325\pi$
$168$	−7.46813	−0.576179
$169$	−6.60789	−0.508299
$170$	−2.57137	−0.197215
$171$	46.1377	3.52824
$172$	−13.0784	−0.997221
$173$	−5.99008	−0.455417	−0.227709	−	0.973729i	$-0.573123\pi$
$173$	−5.99008	−0.455417	−0.227709	+	0.973729i	$0.573123\pi$
$174$	−1.94339	−0.147328
$175$	0.404214	0.0305557
$176$	0	0
$177$	−1.15657	−0.0869329
$178$	5.56374	0.417020
$179$	−1.61894	−0.121005	−0.0605026	−	0.998168i	$-0.519270\pi$
$179$	−1.61894	−0.121005	−0.0605026	+	0.998168i	$0.519270\pi$
$180$	−22.8100	−1.70015
$181$	2.42682	0.180384	0.0901921	−	0.995924i	$-0.471252\pi$
$181$	2.42682	0.180384	0.0901921	+	0.995924i	$0.471252\pi$
$182$	−1.69329	−0.125515
$183$	−15.7349	−1.16316
$184$	−7.52643	−0.554856
$185$	−3.23106	−0.237552
$186$	−6.31228	−0.462839
$187$	0	0
$188$	−6.82279	−0.497603
$189$	−12.1138	−0.881149
$190$	−9.65909	−0.700744
$191$	12.7009	0.919006	0.459503	−	0.888176i	$-0.348027\pi$
$191$	12.7009	0.919006	0.459503	+	0.888176i	$0.348027\pi$
$192$	2.64866	0.191151
$193$	1.75931	0.126638	0.0633190	−	0.997993i	$-0.479831\pi$
$193$	1.75931	0.126638	0.0633190	+	0.997993i	$0.479831\pi$
$194$	−5.70343	−0.409482
$195$	−17.0177	−1.21866
$196$	−1.55144	−0.110817
$197$	0.903053	0.0643399	0.0321699	−	0.999482i	$-0.489758\pi$
$197$	0.903053	0.0643399	0.0321699	+	0.999482i	$0.489758\pi$
$198$	0	0
$199$	−15.6296	−1.10795	−0.553976	−	0.832533i	$-0.686890\pi$
$199$	−15.6296	−1.10795	−0.553976	+	0.832533i	$0.686890\pi$
$200$	−0.961445	−0.0679845
$201$	−2.83248	−0.199788
$202$	−2.69710	−0.189768
$203$	−0.924170	−0.0648640
$204$	−8.72389	−0.610795
$205$	−11.9216	−0.832643
$206$	11.8048	0.822479
$207$	−21.7012	−1.50834
$208$	−3.81733	−0.264684
$209$	0	0
$210$	4.50803	0.311084
$211$	−14.7687	−1.01672	−0.508360	−	0.861145i	$-0.669748\pi$
$211$	−14.7687	−1.01672	−0.508360	+	0.861145i	$0.669748\pi$
$212$	−1.03547	−0.0711161
$213$	−46.6864	−3.19890
$214$	−10.3165	−0.705225
$215$	18.0717	1.23248
$216$	28.8134	1.96050
$217$	−3.00178	−0.203774
$218$	−12.6759	−0.858522
$219$	−25.2380	−1.70543
$220$	0	0
$221$	−4.52791	−0.304581
$222$	3.16936	0.212714
$223$	−1.84537	−0.123575	−0.0617875	−	0.998089i	$-0.519680\pi$
$223$	−1.84537	−0.123575	−0.0617875	+	0.998089i	$0.519680\pi$
$224$	5.76834	0.385413
$225$	−2.77217	−0.184811
$226$	−1.12559	−0.0748728
$227$	21.9399	1.45620	0.728102	−	0.685468i	$-0.240403\pi$
$227$	21.9399	1.45620	0.728102	+	0.685468i	$0.240403\pi$
$228$	−32.7704	−2.17027
$229$	−20.2518	−1.33828	−0.669138	−	0.743138i	$-0.733337\pi$
$229$	−20.2518	−1.33828	−0.669138	+	0.743138i	$0.733337\pi$
$230$	4.54323	0.299571
$231$	0	0
$232$	2.19819	0.144318
$233$	−21.0626	−1.37986	−0.689928	−	0.723878i	$-0.742358\pi$
$233$	−21.0626	−1.37986	−0.689928	+	0.723878i	$0.742358\pi$
$234$	11.6129	0.759157
$235$	9.42771	0.614996
$236$	0.571490	0.0372008
$237$	−12.7371	−0.827360
$238$	1.19946	0.0777493
$239$	−15.4965	−1.00239	−0.501194	−	0.865335i	$-0.667106\pi$
$239$	−15.4965	−1.00239	−0.501194	+	0.865335i	$0.667106\pi$
$240$	10.1628	0.656009
$241$	−14.0848	−0.907283	−0.453641	−	0.891184i	$-0.649875\pi$
$241$	−14.0848	−0.907283	−0.453641	+	0.891184i	$0.649875\pi$
$242$	0	0
$243$	18.4792	1.18544
$244$	7.77504	0.497746
$245$	2.14378	0.136961
$246$	11.6940	0.745582
$247$	−17.0086	−1.08223
$248$	7.13991	0.453385
$249$	12.7203	0.806116
$250$	7.75928	0.490740
$251$	−1.07727	−0.0679966	−0.0339983	−	0.999422i	$-0.510824\pi$
$251$	−1.07727	−0.0679966	−0.0339983	+	0.999422i	$0.510824\pi$
$252$	10.6401	0.670261
$253$	0	0
$254$	11.7653	0.738223
$255$	12.0546	0.754891
$256$	−9.03539	−0.564712
$257$	13.0810	0.815971	0.407986	−	0.912988i	$-0.366231\pi$
$257$	13.0810	0.815971	0.407986	+	0.912988i	$0.366231\pi$
$258$	−17.7267	−1.10361
$259$	1.50718	0.0936516
$260$	8.40887	0.521496
$261$	6.33812	0.392320
$262$	4.50441	0.278283
$263$	9.57216	0.590245	0.295122	−	0.955459i	$-0.404640\pi$
$263$	9.57216	0.590245	0.295122	+	0.955459i	$0.404640\pi$
$264$	0	0
$265$	1.43080	0.0878935
$266$	4.50564	0.276258
$267$	−26.0829	−1.59625
$268$	1.39960	0.0854943
$269$	−4.76853	−0.290743	−0.145371	−	0.989377i	$-0.546438\pi$
$269$	−4.76853	−0.290743	−0.145371	+	0.989377i	$0.546438\pi$
$270$	−17.3928	−1.05849
$271$	20.0523	1.21809	0.609045	−	0.793136i	$-0.291553\pi$
$271$	20.0523	1.21809	0.609045	+	0.793136i	$0.291553\pi$
$272$	2.70404	0.163957
$273$	7.93817	0.480440
$274$	−9.28776	−0.561094
$275$	0	0
$276$	15.4138	0.927802
$277$	11.6024	0.697124	0.348562	−	0.937286i	$-0.386670\pi$
$277$	11.6024	0.697124	0.348562	+	0.937286i	$0.386670\pi$
$278$	9.55773	0.573235
$279$	20.5868	1.23250
$280$	−5.09910	−0.304730
$281$	−12.0788	−0.720562	−0.360281	−	0.932844i	$-0.617319\pi$
$281$	−12.0788	−0.720562	−0.360281	+	0.932844i	$0.617319\pi$
$282$	−9.24769	−0.550692
$283$	−21.9932	−1.30736	−0.653681	−	0.756771i	$-0.726776\pi$
$283$	−21.9932	−1.30736	−0.653681	+	0.756771i	$0.726776\pi$
$284$	23.0690	1.36889
$285$	45.2820	2.68227
$286$	0	0
$287$	5.56104	0.328258
$288$	−39.5603	−2.33111
$289$	−13.7926	−0.811330
$290$	−1.32691	−0.0779187
$291$	26.7378	1.56740
$292$	12.4707	0.729795
$293$	−1.41721	−0.0827941	−0.0413971	−	0.999143i	$-0.513181\pi$
$293$	−1.41721	−0.0827941	−0.0413971	+	0.999143i	$0.513181\pi$
$294$	−2.10284	−0.122640
$295$	−0.789683	−0.0459771
$296$	−3.58491	−0.208369
$297$	0	0
$298$	1.75669	0.101762
$299$	8.00014	0.462660
$300$	1.96900	0.113680
$301$	−8.42985	−0.485888
$302$	−2.00227	−0.115217
$303$	12.6441	0.726383
$304$	10.1574	0.582570
$305$	−10.7435	−0.615172
$306$	−8.22609	−0.470254
$307$	−29.4646	−1.68163	−0.840817	−	0.541319i	$-0.817925\pi$
$307$	−29.4646	−1.68163	−0.840817	+	0.541319i	$0.817925\pi$
$308$	0	0
$309$	−55.3411	−3.14825
$310$	−4.30991	−0.244786
$311$	26.8787	1.52415	0.762075	−	0.647489i	$-0.224181\pi$
$311$	26.8787	1.52415	0.762075	+	0.647489i	$0.224181\pi$
$312$	−18.8814	−1.06895
$313$	−4.49270	−0.253942	−0.126971	−	0.991906i	$-0.540526\pi$
$313$	−4.49270	−0.253942	−0.126971	+	0.991906i	$0.540526\pi$
$314$	−7.53776	−0.425381
$315$	−14.7024	−0.828387
$316$	6.29371	0.354049
$317$	−10.9501	−0.615017	−0.307508	−	0.951545i	$-0.599495\pi$
$317$	−10.9501	−0.615017	−0.307508	+	0.951545i	$0.599495\pi$
$318$	−1.40348	−0.0787034
$319$	0	0
$320$	1.80846	0.101096
$321$	48.3642	2.69943
$322$	−2.11926	−0.118102
$323$	12.0482	0.670382
$324$	−27.0883	−1.50490
$325$	1.02196	0.0566880
$326$	12.1913	0.675215
$327$	59.4250	3.28621
$328$	−13.2272	−0.730353
$329$	−4.39771	−0.242453
$330$	0	0
$331$	16.5226	0.908166	0.454083	−	0.890959i	$-0.349967\pi$
$331$	16.5226	0.908166	0.454083	+	0.890959i	$0.349967\pi$
$332$	−6.28543	−0.344958
$333$	−10.3365	−0.566437
$334$	13.3488	0.730415
$335$	−1.93397	−0.105664
$336$	−4.74062	−0.258622
$337$	12.1207	0.660258	0.330129	−	0.943936i	$-0.392908\pi$
$337$	12.1207	0.660258	0.330129	+	0.943936i	$0.392908\pi$
$338$	4.42559	0.240721
$339$	5.27677	0.286595
$340$	−5.95651	−0.323037
$341$	0	0
$342$	−30.9004	−1.67090
$343$	−1.00000	−0.0539949
$344$	20.0509	1.08107
$345$	−21.2987	−1.14669
$346$	4.01182	0.215677
$347$	24.2851	1.30369	0.651847	−	0.758350i	$-0.273994\pi$
$347$	24.2851	1.30369	0.651847	+	0.758350i	$0.273994\pi$
$348$	−4.50180	−0.241322
$349$	3.31880	0.177651	0.0888257	−	0.996047i	$-0.471689\pi$
$349$	3.31880	0.177651	0.0888257	+	0.996047i	$0.471689\pi$
$350$	−0.270720	−0.0144706
$351$	−30.6269	−1.63474
$352$	0	0
$353$	20.3272	1.08191	0.540955	−	0.841051i	$-0.318063\pi$
$353$	20.3272	1.08191	0.540955	+	0.841051i	$0.318063\pi$
$354$	0.774604	0.0411697
$355$	−31.8766	−1.69184
$356$	12.8883	0.683076
$357$	−5.62308	−0.297605
$358$	1.08427	0.0573057
$359$	−29.0412	−1.53274	−0.766369	−	0.642401i	$-0.777938\pi$
$359$	−29.0412	−1.53274	−0.766369	+	0.642401i	$0.777938\pi$
$360$	34.9705	1.84311
$361$	26.2579	1.38200
$362$	−1.62535	−0.0854264
$363$	0	0
$364$	−3.92246	−0.205593
$365$	−17.2320	−0.901966
$366$	10.5384	0.550850
$367$	−22.7588	−1.18800	−0.593999	−	0.804465i	$-0.702452\pi$
$367$	−22.7588	−1.18800	−0.593999	+	0.804465i	$0.702452\pi$
$368$	−4.77763	−0.249051
$369$	−38.1386	−1.98542
$370$	2.16398	0.112500
$371$	−0.667421	−0.0346508
$372$	−14.6222	−0.758127
$373$	22.2412	1.15160	0.575802	−	0.817589i	$-0.304690\pi$
$373$	22.2412	1.15160	0.575802	+	0.817589i	$0.304690\pi$
$374$	0	0
$375$	−36.3756	−1.87843
$376$	10.4602	0.539444
$377$	−2.33654	−0.120338
$378$	8.11314	0.417295
$379$	33.4707	1.71927	0.859637	−	0.510906i	$-0.170690\pi$
$379$	33.4707	1.71927	0.859637	+	0.510906i	$0.170690\pi$
$380$	−22.3750	−1.14781
$381$	−55.1561	−2.82573
$382$	−8.50637	−0.435224
$383$	−6.14592	−0.314042	−0.157021	−	0.987595i	$-0.550189\pi$
$383$	−6.14592	−0.314042	−0.157021	+	0.987595i	$0.550189\pi$
$384$	34.4486	1.75795
$385$	0	0
$386$	−1.17829	−0.0599733
$387$	57.8134	2.93882
$388$	−13.2118	−0.670729
$389$	7.40364	0.375380	0.187690	−	0.982228i	$-0.439900\pi$
$389$	7.40364	0.375380	0.187690	+	0.982228i	$0.439900\pi$
$390$	11.3975	0.577134
$391$	−5.66698	−0.286592
$392$	2.37856	0.120135
$393$	−21.1168	−1.06520
$394$	−0.604814	−0.0304701
$395$	−8.69662	−0.437575
$396$	0	0
$397$	17.8079	0.893752	0.446876	−	0.894596i	$-0.352537\pi$
$397$	17.8079	0.893752	0.446876	+	0.894596i	$0.352537\pi$
$398$	10.4678	0.524704
$399$	−21.1225	−1.05745
$400$	−0.610307	−0.0305153
$401$	25.6789	1.28234	0.641170	−	0.767398i	$-0.278449\pi$
$401$	25.6789	1.28234	0.641170	+	0.767398i	$0.278449\pi$
$402$	1.89704	0.0946156
$403$	−7.58929	−0.378050
$404$	−6.24777	−0.310838
$405$	37.4305	1.85994
$406$	0.618957	0.0307183
$407$	0	0
$408$	13.3748	0.662152
$409$	−1.33754	−0.0661373	−0.0330687	−	0.999453i	$-0.510528\pi$
$409$	−1.33754	−0.0661373	−0.0330687	+	0.999453i	$0.510528\pi$
$410$	7.98444	0.394323
$411$	43.5412	2.14773
$412$	27.3455	1.34722
$413$	0.368360	0.0181258
$414$	14.5343	0.714320
$415$	8.68518	0.426339
$416$	14.5839	0.715033
$417$	−44.8068	−2.19420
$418$	0	0
$419$	37.4618	1.83013	0.915064	−	0.403310i	$-0.132140\pi$
$419$	37.4618	1.83013	0.915064	+	0.403310i	$0.132140\pi$
$420$	10.4427	0.509553
$421$	8.26156	0.402644	0.201322	−	0.979525i	$-0.435476\pi$
$421$	8.26156	0.402644	0.201322	+	0.979525i	$0.435476\pi$
$422$	9.89125	0.481499
$423$	30.1602	1.46644
$424$	1.58750	0.0770958
$425$	−0.723914	−0.0351150
$426$	31.2680	1.51494
$427$	5.01149	0.242523
$428$	−23.8980	−1.15515
$429$	0	0
$430$	−12.1034	−0.583679
$431$	−32.6564	−1.57301	−0.786503	−	0.617587i	$-0.788110\pi$
$431$	−32.6564	−1.57301	−0.786503	+	0.617587i	$0.788110\pi$
$432$	18.2902	0.879986
$433$	15.7953	0.759072	0.379536	−	0.925177i	$-0.376084\pi$
$433$	15.7953	0.759072	0.379536	+	0.925177i	$0.376084\pi$
$434$	2.01043	0.0965035
$435$	6.22057	0.298253
$436$	−29.3634	−1.40625
$437$	−21.2874	−1.01832
$438$	16.9030	0.807656
$439$	20.6942	0.987678	0.493839	−	0.869553i	$-0.335593\pi$
$439$	20.6942	0.987678	0.493839	+	0.869553i	$0.335593\pi$
$440$	0	0
$441$	6.85818	0.326580
$442$	3.03254	0.144243
$443$	30.0955	1.42988	0.714939	−	0.699186i	$-0.246454\pi$
$443$	30.0955	1.42988	0.714939	+	0.699186i	$0.246454\pi$
$444$	7.34174	0.348424
$445$	−17.8089	−0.844225
$446$	1.23592	0.0585227
$447$	−8.23538	−0.389520
$448$	−0.843584	−0.0398556
$449$	36.3944	1.71756	0.858779	−	0.512345i	$-0.171223\pi$
$449$	36.3944	1.71756	0.858779	+	0.512345i	$0.171223\pi$
$450$	1.85664	0.0875230
$451$	0	0
$452$	−2.60739	−0.122641
$453$	9.38666	0.441024
$454$	−14.6941	−0.689630
$455$	5.42003	0.254095
$456$	50.2411	2.35276
$457$	9.79401	0.458145	0.229072	−	0.973409i	$-0.426431\pi$
$457$	9.79401	0.458145	0.229072	+	0.973409i	$0.426431\pi$
$458$	13.5635	0.633781
$459$	21.6948	1.01263
$460$	10.5243	0.490696
$461$	21.8596	1.01810	0.509052	−	0.860736i	$-0.329996\pi$
$461$	21.8596	1.01810	0.509052	+	0.860736i	$0.329996\pi$
$462$	0	0
$463$	6.75889	0.314112	0.157056	−	0.987590i	$-0.449800\pi$
$463$	6.75889	0.314112	0.157056	+	0.987590i	$0.449800\pi$
$464$	1.39537	0.0647784
$465$	20.2049	0.936981
$466$	14.1065	0.653473
$467$	−30.0232	−1.38931	−0.694654	−	0.719344i	$-0.744442\pi$
$467$	−30.0232	−1.38931	−0.694654	+	0.719344i	$0.744442\pi$
$468$	26.9009	1.24349
$469$	0.902129	0.0416565
$470$	−6.31415	−0.291250
$471$	35.3372	1.62825
$472$	−0.876166	−0.0403288
$473$	0	0
$474$	8.53056	0.391822
$475$	−2.71931	−0.124770
$476$	2.77851	0.127353
$477$	4.57729	0.209580
$478$	10.3787	0.474711
$479$	6.00353	0.274308	0.137154	−	0.990550i	$-0.456204\pi$
$479$	6.00353	0.274308	0.137154	+	0.990550i	$0.456204\pi$
$480$	−38.8265	−1.77218
$481$	3.81054	0.173746
$482$	9.43322	0.429671
$483$	9.93514	0.452064
$484$	0	0
$485$	18.2561	0.828965
$486$	−12.3763	−0.561402
$487$	6.36723	0.288527	0.144263	−	0.989539i	$-0.453919\pi$
$487$	6.36723	0.288527	0.144263	+	0.989539i	$0.453919\pi$
$488$	−11.9201	−0.539598
$489$	−57.1531	−2.58455
$490$	−1.43578	−0.0648620
$491$	−12.3769	−0.558561	−0.279281	−	0.960210i	$-0.590096\pi$
$491$	−12.3769	−0.558561	−0.279281	+	0.960210i	$0.590096\pi$
$492$	27.0888	1.22126
$493$	1.65511	0.0745426
$494$	11.3914	0.512525
$495$	0	0
$496$	4.53228	0.203505
$497$	14.8694	0.666982
$498$	−8.51934	−0.381761
$499$	−14.2797	−0.639245	−0.319623	−	0.947545i	$-0.603556\pi$
$499$	−14.2797	−0.639245	−0.319623	+	0.947545i	$0.603556\pi$
$500$	17.9741	0.803828
$501$	−62.5796	−2.79585
$502$	0.721494	0.0322019
$503$	−5.92573	−0.264215	−0.132108	−	0.991235i	$-0.542174\pi$
$503$	−5.92573	−0.264215	−0.132108	+	0.991235i	$0.542174\pi$
$504$	−16.3126	−0.726619
$505$	8.63314	0.384170
$506$	0	0
$507$	−20.7473	−0.921419
$508$	27.2541	1.20920
$509$	30.8735	1.36844	0.684222	−	0.729274i	$-0.260142\pi$
$509$	30.8735	1.36844	0.684222	+	0.729274i	$0.260142\pi$
$510$	−8.07352	−0.357502
$511$	8.03816	0.355587
$512$	−15.8920	−0.702333
$513$	81.4944	3.59806
$514$	−8.76093	−0.386428
$515$	−37.7859	−1.66504
$516$	−41.0633	−1.80771
$517$	0	0
$518$	−1.00942	−0.0443516
$519$	−18.8075	−0.825557
$520$	−12.8919	−0.565346
$521$	18.8870	0.827453	0.413726	−	0.910401i	$-0.364227\pi$
$521$	18.8870	0.827453	0.413726	+	0.910401i	$0.364227\pi$
$522$	−4.24492	−0.185795
$523$	7.94209	0.347283	0.173642	−	0.984809i	$-0.444447\pi$
$523$	7.94209	0.347283	0.173642	+	0.984809i	$0.444447\pi$
$524$	10.4343	0.455826
$525$	1.26914	0.0553898
$526$	−6.41090	−0.279528
$527$	5.37595	0.234180
$528$	0	0
$529$	−12.9873	−0.564665
$530$	−0.958271	−0.0416246
$531$	−2.52628	−0.109631
$532$	10.4372	0.452509
$533$	14.0598	0.608996
$534$	17.4689	0.755953
$535$	33.0222	1.42767
$536$	−2.14577	−0.0926830
$537$	−5.08310	−0.219352
$538$	3.19370	0.137690
$539$	0	0
$540$	−40.2899	−1.73380
$541$	−7.60165	−0.326820	−0.163410	−	0.986558i	$-0.552249\pi$
$541$	−7.60165	−0.326820	−0.163410	+	0.986558i	$0.552249\pi$
$542$	−13.4299	−0.576863
$543$	7.61967	0.326991
$544$	−10.3306	−0.442922
$545$	40.5743	1.73801
$546$	−5.31654	−0.227527
$547$	−21.6989	−0.927780	−0.463890	−	0.885893i	$-0.653547\pi$
$547$	−21.6989	−0.927780	−0.463890	+	0.885893i	$0.653547\pi$
$548$	−21.5148	−0.919068
$549$	−34.3697	−1.46686
$550$	0	0
$551$	6.21726	0.264864
$552$	−23.6313	−1.00581
$553$	4.05668	0.172508
$554$	−7.77067	−0.330144
$555$	−10.1448	−0.430622
$556$	22.1402	0.938955
$557$	40.6134	1.72084	0.860422	−	0.509582i	$-0.170200\pi$
$557$	40.6134	1.72084	0.860422	+	0.509582i	$0.170200\pi$
$558$	−13.7879	−0.583686
$559$	−21.3129	−0.901438
$560$	−3.23681	−0.136780
$561$	0	0
$562$	8.08972	0.341244
$563$	23.4293	0.987425	0.493713	−	0.869625i	$-0.335639\pi$
$563$	23.4293	0.987425	0.493713	+	0.869625i	$0.335639\pi$
$564$	−21.4220	−0.902030
$565$	3.60288	0.151574
$566$	14.7298	0.619141
$567$	−17.4600	−0.733253
$568$	−35.3676	−1.48399
$569$	11.1453	0.467237	0.233619	−	0.972328i	$-0.424943\pi$
$569$	11.1453	0.467237	0.233619	+	0.972328i	$0.424943\pi$
$570$	−30.3273	−1.27027
$571$	6.15846	0.257724	0.128862	−	0.991663i	$-0.458868\pi$
$571$	6.15846	0.257724	0.128862	+	0.991663i	$0.458868\pi$
$572$	0	0
$573$	39.8780	1.66593
$574$	−3.72447	−0.155456
$575$	1.27905	0.0533400
$576$	5.78545	0.241060
$577$	−14.4842	−0.602983	−0.301492	−	0.953469i	$-0.597485\pi$
$577$	−14.4842	−0.602983	−0.301492	+	0.953469i	$0.597485\pi$
$578$	9.23751	0.384230
$579$	5.52384	0.229563
$580$	−3.07374	−0.127630
$581$	−4.05134	−0.168078
$582$	−17.9075	−0.742288
$583$	0	0
$584$	−19.1192	−0.791159
$585$	−37.1715	−1.53685
$586$	0.949166	0.0392097
$587$	15.8570	0.654487	0.327243	−	0.944940i	$-0.393880\pi$
$587$	15.8570	0.654487	0.327243	+	0.944940i	$0.393880\pi$
$588$	−4.87118	−0.200884
$589$	20.1942	0.832087
$590$	0.528885	0.0217739
$591$	2.83538	0.116632
$592$	−2.27563	−0.0935279
$593$	−22.9285	−0.941560	−0.470780	−	0.882251i	$-0.656027\pi$
$593$	−22.9285	−0.941560	−0.470780	+	0.882251i	$0.656027\pi$
$594$	0	0
$595$	−3.83934	−0.157397
$596$	4.06931	0.166686
$597$	−49.0734	−2.00844
$598$	−5.35805	−0.219107
$599$	13.1512	0.537344	0.268672	−	0.963232i	$-0.413415\pi$
$599$	13.1512	0.537344	0.268672	+	0.963232i	$0.413415\pi$
$600$	−3.01872	−0.123239
$601$	−27.3525	−1.11573	−0.557865	−	0.829932i	$-0.688379\pi$
$601$	−27.3525	−1.11573	−0.557865	+	0.829932i	$0.688379\pi$
$602$	5.64584	0.230107
$603$	−6.18696	−0.251952
$604$	−4.63819	−0.188725
$605$	0	0
$606$	−8.46830	−0.344001
$607$	46.8743	1.90257	0.951285	−	0.308313i	$-0.0997643\pi$
$607$	46.8743	1.90257	0.951285	+	0.308313i	$0.0997643\pi$
$608$	−38.8059	−1.57379
$609$	−2.90168	−0.117582
$610$	7.19540	0.291333
$611$	−11.1186	−0.449809
$612$	−19.0555	−0.770273
$613$	20.4694	0.826752	0.413376	−	0.910560i	$-0.364349\pi$
$613$	20.4694	0.826752	0.413376	+	0.910560i	$0.364349\pi$
$614$	19.7337	0.796389
$615$	−37.4312	−1.50937
$616$	0	0
$617$	44.1691	1.77818	0.889090	−	0.457733i	$-0.151338\pi$
$617$	44.1691	1.77818	0.889090	+	0.457733i	$0.151338\pi$
$618$	37.0644	1.49095
$619$	0.681584	0.0273952	0.0136976	−	0.999906i	$-0.495640\pi$
$619$	0.681584	0.0273952	0.0136976	+	0.999906i	$0.495640\pi$
$620$	−9.98378	−0.400958
$621$	−38.3315	−1.53819
$622$	−18.0018	−0.721808
$623$	8.30727	0.332824
$624$	−11.9855	−0.479806
$625$	−22.8155	−0.912622
$626$	3.00896	0.120262
$627$	0	0
$628$	−17.4610	−0.696770
$629$	−2.69924	−0.107626
$630$	9.84685	0.392308
$631$	−36.8718	−1.46784	−0.733921	−	0.679234i	$-0.762312\pi$
$631$	−36.8718	−1.46784	−0.733921	+	0.679234i	$0.762312\pi$
$632$	−9.64904	−0.383818
$633$	−46.3704	−1.84306
$634$	7.33374	0.291260
$635$	−37.6596	−1.49447
$636$	−3.25113	−0.128916
$637$	−2.52826	−0.100173
$638$	0	0
$639$	−101.977	−4.03414
$640$	23.5209	0.929744
$641$	−20.5126	−0.810201	−0.405100	−	0.914272i	$-0.632763\pi$
$641$	−20.5126	−0.810201	−0.405100	+	0.914272i	$0.632763\pi$
$642$	−32.3916	−1.27840
$643$	−7.63660	−0.301158	−0.150579	−	0.988598i	$-0.548114\pi$
$643$	−7.63660	−0.301158	−0.150579	+	0.988598i	$0.548114\pi$
$644$	−4.90921	−0.193450
$645$	56.7411	2.23418
$646$	−8.06923	−0.317480
$647$	14.6828	0.577241	0.288621	−	0.957444i	$-0.406803\pi$
$647$	14.6828	0.577241	0.288621	+	0.957444i	$0.406803\pi$
$648$	41.5297	1.63144
$649$	0	0
$650$	−0.684450	−0.0268463
$651$	−9.42492	−0.369392
$652$	28.2408	1.10600
$653$	1.12703	0.0441043	0.0220521	−	0.999757i	$-0.492980\pi$
$653$	1.12703	0.0441043	0.0220521	+	0.999757i	$0.492980\pi$
$654$	−39.7995	−1.55628
$655$	−14.4181	−0.563363
$656$	−8.39640	−0.327824
$657$	−55.1271	−2.15071
$658$	2.94534	0.114821
$659$	−10.0215	−0.390384	−0.195192	−	0.980765i	$-0.562533\pi$
$659$	−10.0215	−0.390384	−0.195192	+	0.980765i	$0.562533\pi$
$660$	0	0
$661$	15.7371	0.612101	0.306050	−	0.952015i	$-0.400992\pi$
$661$	15.7371	0.612101	0.306050	+	0.952015i	$0.400992\pi$
$662$	−11.0659	−0.430090
$663$	−14.2166	−0.552128
$664$	9.63635	0.373963
$665$	−14.4221	−0.559263
$666$	6.92281	0.268253
$667$	−2.92434	−0.113231
$668$	30.9222	1.19642
$669$	−5.79404	−0.224010
$670$	1.29526	0.0500403
$671$	0	0
$672$	18.1113	0.698657
$673$	−32.0120	−1.23397	−0.616986	−	0.786974i	$-0.711647\pi$
$673$	−32.0120	−1.23397	−0.616986	+	0.786974i	$0.711647\pi$
$674$	−8.11779	−0.312686
$675$	−4.89656	−0.188469
$676$	10.2518	0.394299
$677$	15.3400	0.589566	0.294783	−	0.955564i	$-0.404753\pi$
$677$	15.3400	0.589566	0.294783	+	0.955564i	$0.404753\pi$
$678$	−3.53408	−0.135726
$679$	−8.51583	−0.326808
$680$	9.13208	0.350199
$681$	68.8864	2.63973
$682$	0	0
$683$	1.04764	0.0400868	0.0200434	−	0.999799i	$-0.493620\pi$
$683$	1.04764	0.0400868	0.0200434	+	0.999799i	$0.493620\pi$
$684$	−71.5800	−2.73693
$685$	29.7291	1.13589
$686$	0.669744	0.0255709
$687$	−63.5860	−2.42596
$688$	12.7279	0.485247
$689$	−1.68742	−0.0642854
$690$	14.2647	0.543048
$691$	29.1883	1.11038	0.555188	−	0.831725i	$-0.312646\pi$
$691$	29.1883	1.11038	0.555188	+	0.831725i	$0.312646\pi$
$692$	9.29326	0.353277
$693$	0	0
$694$	−16.2648	−0.617404
$695$	−30.5933	−1.16047
$696$	6.90182	0.261613
$697$	−9.95937	−0.377238
$698$	−2.22275	−0.0841322
$699$	−66.1317	−2.50133
$700$	−0.627115	−0.0237027
$701$	−12.7785	−0.482637	−0.241318	−	0.970446i	$-0.577580\pi$
$701$	−12.7785	−0.482637	−0.241318	+	0.970446i	$0.577580\pi$
$702$	20.5121	0.774181
$703$	−10.1394	−0.382415
$704$	0	0
$705$	29.6009	1.11483
$706$	−13.6141	−0.512372
$707$	−4.02707	−0.151453
$708$	1.79435	0.0674358
$709$	38.0710	1.42978	0.714892	−	0.699234i	$-0.246476\pi$
$709$	38.0710	1.42978	0.714892	+	0.699234i	$0.246476\pi$
$710$	21.3492	0.801220
$711$	−27.8214	−1.04338
$712$	−19.7593	−0.740512
$713$	−9.49850	−0.355722
$714$	3.76602	0.140940
$715$	0	0
$716$	2.51169	0.0938663
$717$	−48.6556	−1.81708
$718$	19.4502	0.725875
$719$	−39.2694	−1.46450	−0.732250	−	0.681036i	$-0.761530\pi$
$719$	−39.2694	−1.46450	−0.732250	+	0.681036i	$0.761530\pi$
$720$	22.1986	0.827293
$721$	17.6258	0.656420
$722$	−17.5861	−0.654486
$723$	−44.2231	−1.64468
$724$	−3.76507	−0.139928
$725$	−0.373562	−0.0138737
$726$	0	0
$727$	−28.4699	−1.05589	−0.527946	−	0.849278i	$-0.677037\pi$
$727$	−28.4699	−1.05589	−0.527946	+	0.849278i	$0.677037\pi$
$728$	6.01362	0.222879
$729$	5.64035	0.208902
$730$	11.5410	0.427153
$731$	15.0972	0.558389
$732$	24.4118	0.902288
$733$	2.99337	0.110563	0.0552813	−	0.998471i	$-0.482394\pi$
$733$	2.99337	0.110563	0.0552813	+	0.998471i	$0.482394\pi$
$734$	15.2426	0.562613
$735$	6.73098	0.248276
$736$	18.2527	0.672802
$737$	0	0
$738$	25.5431	0.940254
$739$	50.7329	1.86624	0.933120	−	0.359566i	$-0.117075\pi$
$739$	50.7329	1.86624	0.933120	+	0.359566i	$0.117075\pi$
$740$	5.01280	0.184274
$741$	−53.4033	−1.96182
$742$	0.447001	0.0164099
$743$	0.383877	0.0140831	0.00704154	−	0.999975i	$-0.497759\pi$
$743$	0.383877	0.0140831	0.00704154	+	0.999975i	$0.497759\pi$
$744$	22.4177	0.821873
$745$	−5.62296	−0.206009
$746$	−14.8959	−0.545378
$747$	27.7848	1.01659
$748$	0	0
$749$	−15.4037	−0.562840
$750$	24.3624	0.889588
$751$	−39.3570	−1.43616	−0.718078	−	0.695963i	$-0.754978\pi$
$751$	−39.3570	−1.43616	−0.718078	+	0.695963i	$0.754978\pi$
$752$	6.63993	0.242133
$753$	−3.38238	−0.123261
$754$	1.56489	0.0569898
$755$	6.40904	0.233249
$756$	18.7939	0.683526
$757$	11.3867	0.413856	0.206928	−	0.978356i	$-0.433653\pi$
$757$	11.3867	0.413856	0.206928	+	0.978356i	$0.433653\pi$
$758$	−22.4168	−0.814214
$759$	0	0
$760$	34.3037	1.24433
$761$	7.48165	0.271209	0.135605	−	0.990763i	$-0.456702\pi$
$761$	7.48165	0.271209	0.135605	+	0.990763i	$0.456702\pi$
$762$	36.9405	1.33821
$763$	−18.9265	−0.685186
$764$	−19.7048	−0.712893
$765$	26.3308	0.951993
$766$	4.11619	0.148724
$767$	0.931311	0.0336277
$768$	−28.3691	−1.02368
$769$	26.8378	0.967798	0.483899	−	0.875124i	$-0.339220\pi$
$769$	26.8378	0.967798	0.483899	+	0.875124i	$0.339220\pi$
$770$	0	0
$771$	41.0714	1.47915
$772$	−2.72947	−0.0982358
$773$	4.46781	0.160696	0.0803479	−	0.996767i	$-0.474397\pi$
$773$	4.46781	0.160696	0.0803479	+	0.996767i	$0.474397\pi$
$774$	−38.7202	−1.39177
$775$	−1.21336	−0.0435852
$776$	20.2554	0.727126
$777$	4.73220	0.169767
$778$	−4.95855	−0.177772
$779$	−37.4113	−1.34040
$780$	26.4020	0.945342
$781$	0	0
$782$	3.79543	0.135724
$783$	11.1952	0.400084
$784$	1.50986	0.0539236
$785$	24.1276	0.861150
$786$	14.1428	0.504458
$787$	13.4859	0.480721	0.240361	−	0.970684i	$-0.422734\pi$
$787$	13.4859	0.480721	0.240361	+	0.970684i	$0.422734\pi$
$788$	−1.40104	−0.0499098
$789$	30.0544	1.06997
$790$	5.82451	0.207227
$791$	−1.68062	−0.0597560
$792$	0	0
$793$	12.6704	0.449937
$794$	−11.9267	−0.423263
$795$	4.49240	0.159329
$796$	24.2484	0.859462
$797$	52.5052	1.85983	0.929915	−	0.367775i	$-0.119880\pi$
$797$	52.5052	1.85983	0.929915	+	0.367775i	$0.119880\pi$
$798$	14.1467	0.500787
$799$	7.87594	0.278631
$800$	2.33164	0.0824359
$801$	−56.9727	−2.01303
$802$	−17.1983	−0.607292
$803$	0	0
$804$	4.39443	0.154980
$805$	6.78353	0.239088
$806$	5.08288	0.179037
$807$	−14.9721	−0.527043
$808$	9.57861	0.336974
$809$	1.28984	0.0453484	0.0226742	−	0.999743i	$-0.492782\pi$
$809$	1.28984	0.0453484	0.0226742	+	0.999743i	$0.492782\pi$
$810$	−25.0688	−0.880829
$811$	34.1878	1.20049	0.600247	−	0.799815i	$-0.295069\pi$
$811$	34.1878	1.20049	0.600247	+	0.799815i	$0.295069\pi$
$812$	1.43380	0.0503164
$813$	62.9596	2.20809
$814$	0	0
$815$	−39.0231	−1.36692
$816$	8.49008	0.297212
$817$	56.7110	1.98407
$818$	0.895813	0.0313214
$819$	17.3393	0.605883
$820$	18.4957	0.645899
$821$	−39.6599	−1.38414	−0.692071	−	0.721830i	$-0.743301\pi$
$821$	−39.6599	−1.38414	−0.692071	+	0.721830i	$0.743301\pi$
$822$	−29.1615	−1.01712
$823$	−9.22714	−0.321638	−0.160819	−	0.986984i	$-0.551414\pi$
$823$	−9.22714	−0.321638	−0.160819	+	0.986984i	$0.551414\pi$
$824$	−41.9241	−1.46049
$825$	0	0
$826$	−0.246707	−0.00858403
$827$	25.6958	0.893530	0.446765	−	0.894651i	$-0.352576\pi$
$827$	25.6958	0.893530	0.446765	+	0.894651i	$0.352576\pi$
$828$	33.6682	1.17005
$829$	10.9385	0.379911	0.189955	−	0.981793i	$-0.439166\pi$
$829$	10.9385	0.379911	0.189955	+	0.981793i	$0.439166\pi$
$830$	−5.81685	−0.201906
$831$	36.4291	1.26371
$832$	−2.13280	−0.0739416
$833$	1.79092	0.0620517
$834$	30.0091	1.03913
$835$	−42.7282	−1.47867
$836$	0	0
$837$	36.3630	1.25689
$838$	−25.0898	−0.866712
$839$	−34.7374	−1.19927	−0.599634	−	0.800275i	$-0.704687\pi$
$839$	−34.7374	−1.19927	−0.599634	+	0.800275i	$0.704687\pi$
$840$	−16.0100	−0.552398
$841$	−28.1459	−0.970549
$842$	−5.53313	−0.190684
$843$	−37.9248	−1.30620
$844$	22.9128	0.788691
$845$	−14.1659	−0.487320
$846$	−20.1996	−0.694478
$847$	0	0
$848$	1.00771	0.0346050
$849$	−69.0537	−2.36992
$850$	0.484837	0.0166298
$851$	4.76915	0.163484
$852$	72.4314	2.48146
$853$	20.6422	0.706777	0.353389	−	0.935477i	$-0.385029\pi$
$853$	20.6422	0.706777	0.353389	+	0.935477i	$0.385029\pi$
$854$	−3.35641	−0.114854
$855$	98.9090	3.38262
$856$	36.6386	1.25228
$857$	34.1512	1.16658	0.583291	−	0.812263i	$-0.301765\pi$
$857$	34.1512	1.16658	0.583291	+	0.812263i	$0.301765\pi$
$858$	0	0
$859$	−33.4493	−1.14127	−0.570637	−	0.821202i	$-0.693304\pi$
$859$	−33.4493	−1.14127	−0.570637	+	0.821202i	$0.693304\pi$
$860$	−28.0373	−0.956063
$861$	17.4604	0.595049
$862$	21.8715	0.744945
$863$	−34.0340	−1.15853	−0.579265	−	0.815139i	$-0.696660\pi$
$863$	−34.0340	−1.15853	−0.579265	+	0.815139i	$0.696660\pi$
$864$	−69.8764	−2.37724
$865$	−12.8414	−0.436621
$866$	−10.5788	−0.359482
$867$	−43.3057	−1.47074
$868$	4.65710	0.158072
$869$	0	0
$870$	−4.16619	−0.141247
$871$	2.28082	0.0772826
$872$	45.0178	1.52450
$873$	58.4031	1.97664
$874$	14.2571	0.482254
$875$	11.5854	0.391659
$876$	39.1553	1.32294
$877$	−7.86706	−0.265652	−0.132826	−	0.991139i	$-0.542405\pi$
$877$	−7.86706	−0.265652	−0.132826	+	0.991139i	$0.542405\pi$
$878$	−13.8598	−0.467745
$879$	−4.44971	−0.150085
$880$	0	0
$881$	13.3289	0.449063	0.224531	−	0.974467i	$-0.427915\pi$
$881$	13.3289	0.449063	0.224531	+	0.974467i	$0.427915\pi$
$882$	−4.59322	−0.154662
$883$	−17.0998	−0.575454	−0.287727	−	0.957712i	$-0.592900\pi$
$883$	−17.0998	−0.575454	−0.287727	+	0.957712i	$0.592900\pi$
$884$	7.02480	0.236270
$885$	−2.47942	−0.0833449
$886$	−20.1563	−0.677163
$887$	26.5185	0.890403	0.445201	−	0.895430i	$-0.353132\pi$
$887$	26.5185	0.890403	0.445201	+	0.895430i	$0.353132\pi$
$888$	−11.2558	−0.377720
$889$	17.5669	0.589176
$890$	11.9274	0.399808
$891$	0	0
$892$	2.86298	0.0958598
$893$	29.5851	0.990029
$894$	5.51560	0.184469
$895$	−3.47065	−0.116011
$896$	−10.9717	−0.366538
$897$	25.1186	0.838687
$898$	−24.3749	−0.813402
$899$	2.77416	0.0925233
$900$	4.30086	0.143362
$901$	1.19530	0.0398211
$902$	0	0
$903$	−26.4678	−0.880794
$904$	3.99745	0.132953
$905$	5.20257	0.172939
$906$	−6.28666	−0.208860
$907$	8.85256	0.293944	0.146972	−	0.989141i	$-0.453047\pi$
$907$	8.85256	0.293944	0.146972	+	0.989141i	$0.453047\pi$
$908$	−34.0386	−1.12961
$909$	27.6183	0.916043
$910$	−3.63003	−0.120334
$911$	−55.8998	−1.85204	−0.926022	−	0.377469i	$-0.876794\pi$
$911$	−55.8998	−1.85204	−0.926022	+	0.377469i	$0.876794\pi$
$912$	31.8921	1.05605
$913$	0	0
$914$	−6.55948	−0.216968
$915$	−33.7322	−1.11515
$916$	31.4195	1.03813
$917$	6.72557	0.222098
$918$	−14.5300	−0.479561
$919$	−48.3776	−1.59583	−0.797915	−	0.602770i	$-0.794064\pi$
$919$	−48.3776	−1.59583	−0.797915	+	0.602770i	$0.794064\pi$
$920$	−16.1350	−0.531955
$921$	−92.5122	−3.04838
$922$	−14.6404	−0.482155
$923$	37.5937	1.23741
$924$	0	0
$925$	0.609223	0.0200311
$926$	−4.52673	−0.148757
$927$	−120.881	−3.97026
$928$	−5.33092	−0.174996
$929$	0.864295	0.0283566	0.0141783	−	0.999899i	$-0.495487\pi$
$929$	0.864295	0.0283566	0.0141783	+	0.999899i	$0.495487\pi$
$930$	−13.5321	−0.443736
$931$	6.72740	0.220482
$932$	32.6774	1.07038
$933$	84.3930	2.76290
$934$	20.1079	0.657949
$935$	0	0
$936$	−41.2424	−1.34805
$937$	−53.4580	−1.74640	−0.873199	−	0.487364i	$-0.837958\pi$
$937$	−53.4580	−1.74640	−0.873199	+	0.487364i	$0.837958\pi$
$938$	−0.604195	−0.0197277
$939$	−14.1061	−0.460334
$940$	−14.6266	−0.477066
$941$	−13.7000	−0.446607	−0.223304	−	0.974749i	$-0.571684\pi$
$941$	−13.7000	−0.446607	−0.223304	+	0.974749i	$0.571684\pi$
$942$	−23.6669	−0.771108
$943$	17.5967	0.573028
$944$	−0.556173	−0.0181019
$945$	−25.9693	−0.844781
$946$	0	0
$947$	31.6444	1.02830	0.514152	−	0.857699i	$-0.328107\pi$
$947$	31.6444	1.02830	0.514152	+	0.857699i	$0.328107\pi$
$948$	19.7608	0.641801
$949$	20.3226	0.659699
$950$	1.82124	0.0590888
$951$	−34.3807	−1.11487
$952$	−4.25981	−0.138061
$953$	−32.7379	−1.06049	−0.530243	−	0.847846i	$-0.677899\pi$
$953$	−32.7379	−1.06049	−0.530243	+	0.847846i	$0.677899\pi$
$954$	−3.06561	−0.0992529
$955$	27.2280	0.881076
$956$	24.0420	0.777573
$957$	0	0
$958$	−4.02083	−0.129907
$959$	−13.8676	−0.447809
$960$	5.67814	0.183261
$961$	−21.9893	−0.709332
$962$	−2.55209	−0.0822827
$963$	105.641	3.40425
$964$	21.8518	0.703799
$965$	3.77157	0.121411
$966$	−6.65400	−0.214089
$967$	32.3487	1.04026	0.520132	−	0.854086i	$-0.325883\pi$
$967$	32.3487	1.04026	0.520132	+	0.854086i	$0.325883\pi$
$968$	0	0
$969$	37.8287	1.21523
$970$	−12.2269	−0.392582
$971$	−29.4270	−0.944358	−0.472179	−	0.881503i	$-0.656532\pi$
$971$	−29.4270	−0.944358	−0.472179	+	0.881503i	$0.656532\pi$
$972$	−28.6694	−0.919572
$973$	14.2707	0.457498
$974$	−4.26441	−0.136641
$975$	3.20872	0.102761
$976$	−7.56665	−0.242203
$977$	13.6944	0.438124	0.219062	−	0.975711i	$-0.429700\pi$
$977$	13.6944	0.438124	0.219062	+	0.975711i	$0.429700\pi$
$978$	38.2780	1.22399
$979$	0	0
$980$	−3.32595	−0.106244
$981$	129.801	4.14424
$982$	8.28935	0.264524
$983$	17.0319	0.543234	0.271617	−	0.962405i	$-0.412442\pi$
$983$	17.0319	0.543234	0.271617	+	0.962405i	$0.412442\pi$
$984$	−41.5306	−1.32395
$985$	1.93595	0.0616844
$986$	−1.10850	−0.0353019
$987$	−13.8078	−0.439507
$988$	26.3879	0.839512
$989$	−26.6744	−0.848198
$990$	0	0
$991$	23.2202	0.737614	0.368807	−	0.929506i	$-0.379766\pi$
$991$	23.2202	0.737614	0.368807	+	0.929506i	$0.379766\pi$
$992$	−17.3153	−0.549761
$993$	51.8773	1.64628
$994$	−9.95867	−0.315870
$995$	−33.5064	−1.06222
$996$	−19.7348	−0.625321
$997$	18.3776	0.582025	0.291012	−	0.956719i	$-0.406008\pi$
$997$	18.3776	0.582025	0.291012	+	0.956719i	$0.406008\pi$
$998$	9.56371	0.302734
$999$	−18.2577	−0.577647

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.a.o.1.4		8
3.2	odd	2		7623.2.a.cw.1.5		8
7.6	odd	2		5929.2.a.bs.1.4		8
11.2	odd	10		847.2.f.w.323.2		16
11.3	even	5		847.2.f.x.372.2		16
11.4	even	5		847.2.f.x.148.2		16
11.5	even	5		847.2.f.v.729.3		16
11.6	odd	10		847.2.f.w.729.2		16
11.7	odd	10		77.2.f.b.71.3	yes	16
11.8	odd	10		77.2.f.b.64.3	✓	16
11.9	even	5		847.2.f.v.323.3		16
11.10	odd	2		847.2.a.p.1.5		8
33.8	even	10		693.2.m.i.64.2		16
33.29	even	10		693.2.m.i.379.2		16
33.32	even	2		7623.2.a.ct.1.4		8
77.18	odd	30		539.2.q.g.324.2		32
77.19	even	30		539.2.q.f.361.2		32
77.30	odd	30		539.2.q.g.361.2		32
77.40	even	30		539.2.q.f.214.3		32
77.41	even	10		539.2.f.e.295.3		16
77.51	odd	30		539.2.q.g.214.3		32
77.52	even	30		539.2.q.f.471.3		32
77.62	even	10		539.2.f.e.148.3		16
77.73	even	30		539.2.q.f.324.2		32
77.74	odd	30		539.2.q.g.471.3		32
77.76	even	2		5929.2.a.bt.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.b.64.3	✓	16	11.8	odd	10
77.2.f.b.71.3	yes	16	11.7	odd	10
539.2.f.e.148.3		16	77.62	even	10
539.2.f.e.295.3		16	77.41	even	10
539.2.q.f.214.3		32	77.40	even	30
539.2.q.f.324.2		32	77.73	even	30
539.2.q.f.361.2		32	77.19	even	30
539.2.q.f.471.3		32	77.52	even	30
539.2.q.g.214.3		32	77.51	odd	30
539.2.q.g.324.2		32	77.18	odd	30
539.2.q.g.361.2		32	77.30	odd	30
539.2.q.g.471.3		32	77.74	odd	30
693.2.m.i.64.2		16	33.8	even	10
693.2.m.i.379.2		16	33.29	even	10
847.2.a.o.1.4		8	1.1	even	1	trivial
847.2.a.p.1.5		8	11.10	odd	2
847.2.f.v.323.3		16	11.9	even	5
847.2.f.v.729.3		16	11.5	even	5
847.2.f.w.323.2		16	11.2	odd	10
847.2.f.w.729.2		16	11.6	odd	10
847.2.f.x.148.2		16	11.4	even	5
847.2.f.x.372.2		16	11.3	even	5
5929.2.a.bs.1.4		8	7.6	odd	2
5929.2.a.bt.1.5		8	77.76	even	2
7623.2.a.ct.1.4		8	33.32	even	2
7623.2.a.cw.1.5		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-0.669744 q^{2} +3.13977 q^{3} -1.55144 q^{4} +2.14378 q^{5} -2.10284 q^{6} -1.00000 q^{7} +2.37856 q^{8} +6.85818 q^{9} +O(q^{10})\)	\(q-0.669744 q^{2} +3.13977 q^{3} -1.55144 q^{4} +2.14378 q^{5} -2.10284 q^{6} -1.00000 q^{7} +2.37856 q^{8} +6.85818 q^{9} -1.43578 q^{10} -4.87118 q^{12} -2.52826 q^{13} +0.669744 q^{14} +6.73098 q^{15} +1.50986 q^{16} +1.79092 q^{17} -4.59322 q^{18} +6.72740 q^{19} -3.32595 q^{20} -3.13977 q^{21} -3.16429 q^{23} +7.46813 q^{24} -0.404214 q^{25} +1.69329 q^{26} +12.1138 q^{27} +1.55144 q^{28} +0.924170 q^{29} -4.50803 q^{30} +3.00178 q^{31} -5.76834 q^{32} -1.19946 q^{34} -2.14378 q^{35} -10.6401 q^{36} -1.50718 q^{37} -4.50564 q^{38} -7.93817 q^{39} +5.09910 q^{40} -5.56104 q^{41} +2.10284 q^{42} +8.42985 q^{43} +14.7024 q^{45} +2.11926 q^{46} +4.39771 q^{47} +4.74062 q^{48} +1.00000 q^{49} +0.270720 q^{50} +5.62308 q^{51} +3.92246 q^{52} +0.667421 q^{53} -8.11314 q^{54} -2.37856 q^{56} +21.1225 q^{57} -0.618957 q^{58} -0.368360 q^{59} -10.4427 q^{60} -5.01149 q^{61} -2.01043 q^{62} -6.85818 q^{63} +0.843584 q^{64} -5.42003 q^{65} -0.902129 q^{67} -2.77851 q^{68} -9.93514 q^{69} +1.43578 q^{70} -14.8694 q^{71} +16.3126 q^{72} -8.03816 q^{73} +1.00942 q^{74} -1.26914 q^{75} -10.4372 q^{76} +5.31654 q^{78} -4.05668 q^{79} +3.23681 q^{80} +17.4600 q^{81} +3.72447 q^{82} +4.05134 q^{83} +4.87118 q^{84} +3.83934 q^{85} -5.64584 q^{86} +2.90168 q^{87} -8.30727 q^{89} -9.84685 q^{90} +2.52826 q^{91} +4.90921 q^{92} +9.42492 q^{93} -2.94534 q^{94} +14.4221 q^{95} -18.1113 q^{96} +8.51583 q^{97} -0.669744 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} + q^{6} - 8 q^{7} + 14 q^{9}+O(q^{10}) \) 8 * q - q^2 + 4 * q^3 + 7 * q^4 + 10 * q^5 + q^6 - 8 * q^7 + 14 * q^9	\( 8 q - q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} + q^{6} - 8 q^{7} + 14 q^{9} - 6 q^{10} + 9 q^{12} + 6 q^{13} + q^{14} + 11 q^{15} + q^{16} + 5 q^{17} - 8 q^{18} + 13 q^{19} + 23 q^{20} - 4 q^{21} + 16 q^{23} + 10 q^{24} + 16 q^{25} - 6 q^{26} + 10 q^{27} - 7 q^{28} - 9 q^{29} + 36 q^{30} + 9 q^{31} - 16 q^{32} - 12 q^{34} - 10 q^{35} - 14 q^{36} + 7 q^{37} - 10 q^{38} - 13 q^{39} - 5 q^{40} + 10 q^{41} - q^{42} + 4 q^{43} + 35 q^{45} - 4 q^{46} + 16 q^{47} - 3 q^{48} + 8 q^{49} - 6 q^{50} - 13 q^{51} + 41 q^{52} + 37 q^{53} - 30 q^{54} - 2 q^{57} - 15 q^{58} + q^{59} + 5 q^{60} - 19 q^{61} + 18 q^{62} - 14 q^{63} - 4 q^{64} + 4 q^{65} - 19 q^{67} - 9 q^{68} + 20 q^{69} + 6 q^{70} + 13 q^{71} + 35 q^{72} + 25 q^{73} - 33 q^{74} - 13 q^{75} - 26 q^{76} - 29 q^{78} + 4 q^{80} + 8 q^{81} - 13 q^{82} + 25 q^{83} - 9 q^{84} - 3 q^{85} + 4 q^{86} + 36 q^{87} + 37 q^{89} + 2 q^{90} - 6 q^{91} + 35 q^{92} + 21 q^{93} + 42 q^{94} - 21 q^{95} + 6 q^{96} + 15 q^{97} - q^{98}+O(q^{100}) \) 8 * q - q^2 + 4 * q^3 + 7 * q^4 + 10 * q^5 + q^6 - 8 * q^7 + 14 * q^9 - 6 * q^10 + 9 * q^12 + 6 * q^13 + q^14 + 11 * q^15 + q^16 + 5 * q^17 - 8 * q^18 + 13 * q^19 + 23 * q^20 - 4 * q^21 + 16 * q^23 + 10 * q^24 + 16 * q^25 - 6 * q^26 + 10 * q^27 - 7 * q^28 - 9 * q^29 + 36 * q^30 + 9 * q^31 - 16 * q^32 - 12 * q^34 - 10 * q^35 - 14 * q^36 + 7 * q^37 - 10 * q^38 - 13 * q^39 - 5 * q^40 + 10 * q^41 - q^42 + 4 * q^43 + 35 * q^45 - 4 * q^46 + 16 * q^47 - 3 * q^48 + 8 * q^49 - 6 * q^50 - 13 * q^51 + 41 * q^52 + 37 * q^53 - 30 * q^54 - 2 * q^57 - 15 * q^58 + q^59 + 5 * q^60 - 19 * q^61 + 18 * q^62 - 14 * q^63 - 4 * q^64 + 4 * q^65 - 19 * q^67 - 9 * q^68 + 20 * q^69 + 6 * q^70 + 13 * q^71 + 35 * q^72 + 25 * q^73 - 33 * q^74 - 13 * q^75 - 26 * q^76 - 29 * q^78 + 4 * q^80 + 8 * q^81 - 13 * q^82 + 25 * q^83 - 9 * q^84 - 3 * q^85 + 4 * q^86 + 36 * q^87 + 37 * q^89 + 2 * q^90 - 6 * q^91 + 35 * q^92 + 21 * q^93 + 42 * q^94 - 21 * q^95 + 6 * q^96 + 15 * q^97 - q^98

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