LMFDB - Embedded newform 847.2.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("847.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$6.76332905120$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2525.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 5 $ x^4 - 2x^3 - 4x^2 + 5*x + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.777484$ 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)

$f(q)$	$=$	$q+0.777484 q^{2} +0.618034 q^{3} -1.39552 q^{4} -0.222516 q^{5} +0.480512 q^{6} +1.00000 q^{7} -2.63996 q^{8} -2.61803 q^{9} +O(q^{10})$	\(q+0.777484 q^{2} +0.618034 q^{3} -1.39552 q^{4} -0.222516 q^{5} +0.480512 q^{6} +1.00000 q^{7} -2.63996 q^{8} -2.61803 q^{9} -0.173002 q^{10} -0.862478 q^{12} -6.52954 q^{13} +0.777484 q^{14} -0.137522 q^{15} +0.738508 q^{16} +4.33461 q^{17} -2.03548 q^{18} -2.91501 q^{19} +0.310525 q^{20} +0.618034 q^{21} -3.89796 q^{23} -1.63159 q^{24} -4.95049 q^{25} -5.07662 q^{26} -3.47214 q^{27} -1.39552 q^{28} +3.77399 q^{29} -0.106921 q^{30} -6.88958 q^{31} +5.85410 q^{32} +3.37009 q^{34} -0.222516 q^{35} +3.65351 q^{36} -5.65351 q^{37} -2.26637 q^{38} -4.03548 q^{39} +0.587433 q^{40} -1.33811 q^{41} +0.480512 q^{42} +4.70820 q^{43} +0.582554 q^{45} -3.03060 q^{46} +6.04554 q^{47} +0.456423 q^{48} +1.00000 q^{49} -3.84893 q^{50} +2.67894 q^{51} +9.11210 q^{52} -1.71792 q^{53} -2.69953 q^{54} -2.63996 q^{56} -1.80157 q^{57} +2.93422 q^{58} -9.53304 q^{59} +0.191915 q^{60} +9.62943 q^{61} -5.35654 q^{62} -2.61803 q^{63} +3.07446 q^{64} +1.45293 q^{65} +1.27155 q^{67} -6.04903 q^{68} -2.40907 q^{69} -0.173002 q^{70} -9.30919 q^{71} +6.91151 q^{72} -5.58911 q^{73} -4.39552 q^{74} -3.05957 q^{75} +4.06794 q^{76} -3.13752 q^{78} +4.52954 q^{79} -0.164330 q^{80} +5.70820 q^{81} -1.04036 q^{82} -11.2838 q^{83} -0.862478 q^{84} -0.964520 q^{85} +3.66055 q^{86} +2.33245 q^{87} +7.92157 q^{89} +0.452926 q^{90} -6.52954 q^{91} +5.43967 q^{92} -4.25800 q^{93} +4.70031 q^{94} +0.648635 q^{95} +3.61803 q^{96} -9.05391 q^{97} +0.777484 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{5} + q^{6} + 4 q^{7} - 9 q^{8} - 6 q^{9}+O(q^{10}) $ 4 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 6 * q^5 + q^6 + 4 * q^7 - 9 * q^8 - 6 * q^9	$ 4 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{5} + q^{6} + 4 q^{7} - 9 q^{8} - 6 q^{9} + 14 q^{10} - 7 q^{12} - 2 q^{14} + 3 q^{15} - 4 q^{16} + 3 q^{17} + 3 q^{18} - 3 q^{19} - 17 q^{20} - 2 q^{21} - 8 q^{23} + 12 q^{24} - 12 q^{26} + 4 q^{27} + 4 q^{28} - 3 q^{29} - 12 q^{30} - 3 q^{31} + 10 q^{32} - 12 q^{34} - 6 q^{35} - q^{36} - 7 q^{37} - 20 q^{38} - 5 q^{39} + 13 q^{40} - 4 q^{41} + q^{42} - 8 q^{43} + 9 q^{45} + 3 q^{46} - 14 q^{47} - 3 q^{48} + 4 q^{49} - 33 q^{50} + 11 q^{51} + 17 q^{52} - 9 q^{53} - 2 q^{54} - 9 q^{56} - 6 q^{57} + 3 q^{58} - 25 q^{59} + 21 q^{60} + 19 q^{61} - 10 q^{62} - 6 q^{63} + 3 q^{64} - 12 q^{65} - 15 q^{67} + q^{68} + 14 q^{69} + 14 q^{70} - 7 q^{71} + 6 q^{72} + 11 q^{73} - 8 q^{74} - 5 q^{75} + 26 q^{76} - 9 q^{78} - 8 q^{79} - 4 q^{80} - 4 q^{81} + 3 q^{82} + q^{83} - 7 q^{84} - 15 q^{85} + 4 q^{86} - 6 q^{87} - 17 q^{89} - 16 q^{90} - 17 q^{92} - 11 q^{93} + 20 q^{94} - 17 q^{95} + 10 q^{96} - 15 q^{97} - 2 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 6 * q^5 + q^6 + 4 * q^7 - 9 * q^8 - 6 * q^9 + 14 * q^10 - 7 * q^12 - 2 * q^14 + 3 * q^15 - 4 * q^16 + 3 * q^17 + 3 * q^18 - 3 * q^19 - 17 * q^20 - 2 * q^21 - 8 * q^23 + 12 * q^24 - 12 * q^26 + 4 * q^27 + 4 * q^28 - 3 * q^29 - 12 * q^30 - 3 * q^31 + 10 * q^32 - 12 * q^34 - 6 * q^35 - q^36 - 7 * q^37 - 20 * q^38 - 5 * q^39 + 13 * q^40 - 4 * q^41 + q^42 - 8 * q^43 + 9 * q^45 + 3 * q^46 - 14 * q^47 - 3 * q^48 + 4 * q^49 - 33 * q^50 + 11 * q^51 + 17 * q^52 - 9 * q^53 - 2 * q^54 - 9 * q^56 - 6 * q^57 + 3 * q^58 - 25 * q^59 + 21 * q^60 + 19 * q^61 - 10 * q^62 - 6 * q^63 + 3 * q^64 - 12 * q^65 - 15 * q^67 + q^68 + 14 * q^69 + 14 * q^70 - 7 * q^71 + 6 * q^72 + 11 * q^73 - 8 * q^74 - 5 * q^75 + 26 * q^76 - 9 * q^78 - 8 * q^79 - 4 * q^80 - 4 * q^81 + 3 * q^82 + q^83 - 7 * q^84 - 15 * q^85 + 4 * q^86 - 6 * q^87 - 17 * q^89 - 16 * q^90 - 17 * q^92 - 11 * q^93 + 20 * q^94 - 17 * q^95 + 10 * q^96 - 15 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.777484	0.549764	0.274882	−	0.961478i	$-0.411361\pi$
$2$	0.777484	0.549764	0.274882	+	0.961478i	$0.411361\pi$
$3$	0.618034	0.356822	0.178411	−	0.983956i	$-0.442904\pi$
$3$	0.618034	0.356822	0.178411	+	0.983956i	$0.442904\pi$
$4$	−1.39552	−0.697759
$5$	−0.222516	−0.0995121	−0.0497560	−	0.998761i	$-0.515844\pi$
$5$	−0.222516	−0.0995121	−0.0497560	+	0.998761i	$0.515844\pi$
$6$	0.480512	0.196168
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.63996	−0.933368
$9$	−2.61803	−0.872678
$10$	−0.173002	−0.0547082
$11$	0	0
$11$	0	0
$12$	−0.862478	−0.248976
$13$	−6.52954	−1.81097	−0.905485	−	0.424379i	$-0.860493\pi$
$13$	−6.52954	−1.81097	−0.905485	+	0.424379i	$0.860493\pi$
$14$	0.777484	0.207791
$15$	−0.137522	−0.0355081
$16$	0.738508	0.184627
$17$	4.33461	1.05130	0.525649	−	0.850701i	$-0.323822\pi$
$17$	4.33461	1.05130	0.525649	+	0.850701i	$0.323822\pi$
$18$	−2.03548	−0.479767
$19$	−2.91501	−0.668748	−0.334374	−	0.942440i	$-0.608525\pi$
$19$	−2.91501	−0.668748	−0.334374	+	0.942440i	$0.608525\pi$
$20$	0.310525	0.0694355
$21$	0.618034	0.134866
$22$	0	0
$23$	−3.89796	−0.812780	−0.406390	−	0.913700i	$-0.633213\pi$
$23$	−3.89796	−0.812780	−0.406390	+	0.913700i	$0.633213\pi$
$24$	−1.63159	−0.333046
$25$	−4.95049	−0.990097
$26$	−5.07662	−0.995607
$27$	−3.47214	−0.668213
$28$	−1.39552	−0.263728
$29$	3.77399	0.700812	0.350406	−	0.936598i	$-0.386044\pi$
$29$	3.77399	0.700812	0.350406	+	0.936598i	$0.386044\pi$
$30$	−0.106921	−0.0195211
$31$	−6.88958	−1.23741	−0.618703	−	0.785625i	$-0.712341\pi$
$31$	−6.88958	−1.23741	−0.618703	+	0.785625i	$0.712341\pi$
$32$	5.85410	1.03487
$33$	0	0
$34$	3.37009	0.577966
$35$	−0.222516	−0.0376120
$36$	3.65351	0.608919
$37$	−5.65351	−0.929432	−0.464716	−	0.885460i	$-0.653844\pi$
$37$	−5.65351	−0.929432	−0.464716	+	0.885460i	$0.653844\pi$
$38$	−2.26637	−0.367654
$39$	−4.03548	−0.646194
$40$	0.587433	0.0928813
$41$	−1.33811	−0.208978	−0.104489	−	0.994526i	$-0.533321\pi$
$41$	−1.33811	−0.208978	−0.104489	+	0.994526i	$0.533321\pi$
$42$	0.480512	0.0741446
$43$	4.70820	0.717994	0.358997	−	0.933339i	$-0.383119\pi$
$43$	4.70820	0.717994	0.358997	+	0.933339i	$0.383119\pi$
$44$	0	0
$45$	0.582554	0.0868420
$46$	−3.03060	−0.446838
$47$	6.04554	0.881832	0.440916	−	0.897548i	$-0.354654\pi$
$47$	6.04554	0.881832	0.440916	+	0.897548i	$0.354654\pi$
$48$	0.456423	0.0658790
$49$	1.00000	0.142857
$50$	−3.84893	−0.544320
$51$	2.67894	0.375126
$52$	9.11210	1.26362
$53$	−1.71792	−0.235974	−0.117987	−	0.993015i	$-0.537644\pi$
$53$	−1.71792	−0.235974	−0.117987	+	0.993015i	$0.537644\pi$
$54$	−2.69953	−0.367360
$55$	0	0
$56$	−2.63996	−0.352780
$57$	−1.80157	−0.238624
$58$	2.93422	0.385281
$59$	−9.53304	−1.24110	−0.620548	−	0.784168i	$-0.713090\pi$
$59$	−9.53304	−1.24110	−0.620548	+	0.784168i	$0.713090\pi$
$60$	0.191915	0.0247761
$61$	9.62943	1.23292	0.616461	−	0.787386i	$-0.288566\pi$
$61$	9.62943	1.23292	0.616461	+	0.787386i	$0.288566\pi$
$62$	−5.35654	−0.680281
$63$	−2.61803	−0.329841
$64$	3.07446	0.384307
$65$	1.45293	0.180213
$66$	0	0
$67$	1.27155	0.155344	0.0776722	−	0.996979i	$-0.475251\pi$
$67$	1.27155	0.155344	0.0776722	+	0.996979i	$0.475251\pi$
$68$	−6.04903	−0.733553
$69$	−2.40907	−0.290018
$70$	−0.173002	−0.0206778
$71$	−9.30919	−1.10480	−0.552399	−	0.833580i	$-0.686287\pi$
$71$	−9.30919	−1.10480	−0.552399	+	0.833580i	$0.686287\pi$
$72$	6.91151	0.814529
$73$	−5.58911	−0.654156	−0.327078	−	0.944997i	$-0.606064\pi$
$73$	−5.58911	−0.654156	−0.327078	+	0.944997i	$0.606064\pi$
$74$	−4.39552	−0.510969
$75$	−3.05957	−0.353289
$76$	4.06794	0.466625
$77$	0	0
$78$	−3.13752	−0.355254
$79$	4.52954	0.509614	0.254807	−	0.966992i	$-0.417988\pi$
$79$	4.52954	0.509614	0.254807	+	0.966992i	$0.417988\pi$
$80$	−0.164330	−0.0183726
$81$	5.70820	0.634245
$82$	−1.04036	−0.114888
$83$	−11.2838	−1.23855	−0.619277	−	0.785173i	$-0.712574\pi$
$83$	−11.2838	−1.23855	−0.619277	+	0.785173i	$0.712574\pi$
$84$	−0.862478	−0.0941040
$85$	−0.964520	−0.104617
$86$	3.66055	0.394728
$87$	2.33245	0.250065
$88$	0	0
$89$	7.92157	0.839684	0.419842	−	0.907597i	$-0.362085\pi$
$89$	7.92157	0.839684	0.419842	+	0.907597i	$0.362085\pi$
$90$	0.452926	0.0477426
$91$	−6.52954	−0.684482
$92$	5.43967	0.567125
$93$	−4.25800	−0.441534
$94$	4.70031	0.484800
$95$	0.648635	0.0665485
$96$	3.61803	0.369264
$97$	−9.05391	−0.919285	−0.459643	−	0.888104i	$-0.652023\pi$
$97$	−9.05391	−0.919285	−0.459643	+	0.888104i	$0.652023\pi$
$98$	0.777484	0.0785378
$99$	0	0
$100$	6.90849	0.690849
$101$	19.2102	1.91148	0.955741	−	0.294208i	$-0.0950559\pi$
$101$	19.2102	1.91148	0.955741	+	0.294208i	$0.0950559\pi$
$102$	2.08283	0.206231
$103$	16.2383	1.60001	0.800004	−	0.599995i	$-0.204831\pi$
$103$	16.2383	1.60001	0.800004	+	0.599995i	$0.204831\pi$
$104$	17.2377	1.69030
$105$	−0.137522	−0.0134208
$106$	−1.33565	−0.129730
$107$	5.39336	0.521396	0.260698	−	0.965420i	$-0.416047\pi$
$107$	5.39336	0.521396	0.260698	+	0.965420i	$0.416047\pi$
$108$	4.84543	0.466252
$109$	−5.39901	−0.517132	−0.258566	−	0.965994i	$-0.583250\pi$
$109$	−5.39901	−0.517132	−0.258566	+	0.965994i	$0.583250\pi$
$110$	0	0
$111$	−3.49406	−0.331642
$112$	0.738508	0.0697824
$113$	−16.4962	−1.55183	−0.775917	−	0.630835i	$-0.782713\pi$
$113$	−16.4962	−1.55183	−0.775917	+	0.630835i	$0.782713\pi$
$114$	−1.40069	−0.131187
$115$	0.867357	0.0808815
$116$	−5.26667	−0.488998
$117$	17.0946	1.58039
$118$	−7.41179	−0.682310
$119$	4.33461	0.397353
$120$	0.363054	0.0331421
$121$	0	0
$122$	7.48673	0.677816
$123$	−0.826998	−0.0745679
$124$	9.61454	0.863411
$125$	2.21414	0.198039
$126$	−2.03548	−0.181335
$127$	−1.99728	−0.177230	−0.0886150	−	0.996066i	$-0.528244\pi$
$127$	−1.99728	−0.177230	−0.0886150	+	0.996066i	$0.528244\pi$
$128$	−9.31786	−0.823590
$129$	2.90983	0.256196
$130$	1.12963	0.0990749
$131$	−1.37009	−0.119706	−0.0598528	−	0.998207i	$-0.519063\pi$
$131$	−1.37009	−0.119706	−0.0598528	+	0.998207i	$0.519063\pi$
$132$	0	0
$133$	−2.91501	−0.252763
$134$	0.988609	0.0854028
$135$	0.772605	0.0664952
$136$	−11.4432	−0.981248
$137$	2.49924	0.213525	0.106762	−	0.994285i	$-0.465952\pi$
$137$	2.49924	0.213525	0.106762	+	0.994285i	$0.465952\pi$
$138$	−1.87301	−0.159442
$139$	4.76260	0.403958	0.201979	−	0.979390i	$-0.435263\pi$
$139$	4.76260	0.403958	0.201979	+	0.979390i	$0.435263\pi$
$140$	0.310525	0.0262441
$141$	3.73635	0.314657
$142$	−7.23775	−0.607378
$143$	0	0
$144$	−1.93344	−0.161120
$145$	−0.839772	−0.0697392
$146$	−4.34545	−0.359632
$147$	0.618034	0.0509746
$148$	7.88958	0.648520
$149$	7.22985	0.592293	0.296146	−	0.955143i	$-0.404298\pi$
$149$	7.22985	0.592293	0.296146	+	0.955143i	$0.404298\pi$
$150$	−2.37877	−0.194225
$151$	−8.13968	−0.662398	−0.331199	−	0.943561i	$-0.607453\pi$
$151$	−8.13968	−0.662398	−0.331199	+	0.943561i	$0.607453\pi$
$152$	7.69551	0.624188
$153$	−11.3482	−0.917445
$154$	0	0
$155$	1.53304	0.123137
$156$	5.63159	0.450888
$157$	20.1514	1.60826	0.804129	−	0.594455i	$-0.202632\pi$
$157$	20.1514	1.60826	0.804129	+	0.594455i	$0.202632\pi$
$158$	3.52165	0.280167
$159$	−1.06173	−0.0842007
$160$	−1.30263	−0.102982
$161$	−3.89796	−0.307202
$162$	4.43804	0.348685
$163$	7.43449	0.582315	0.291157	−	0.956675i	$-0.405960\pi$
$163$	7.43449	0.582315	0.291157	+	0.956675i	$0.405960\pi$
$164$	1.86736	0.145816
$165$	0	0
$166$	−8.77295	−0.680913
$167$	2.21112	0.171102	0.0855510	−	0.996334i	$-0.472735\pi$
$167$	2.21112	0.171102	0.0855510	+	0.996334i	$0.472735\pi$
$168$	−1.63159	−0.125880
$169$	29.6349	2.27961
$170$	−0.749899	−0.0575146
$171$	7.63159	0.583602
$172$	−6.57038	−0.500987
$173$	10.3622	0.787823	0.393912	−	0.919148i	$-0.371122\pi$
$173$	10.3622	0.787823	0.393912	+	0.919148i	$0.371122\pi$
$174$	1.81345	0.137477
$175$	−4.95049	−0.374222
$176$	0	0
$177$	−5.89174	−0.442851
$178$	6.15889	0.461629
$179$	−23.4094	−1.74970	−0.874851	−	0.484392i	$-0.839041\pi$
$179$	−23.4094	−1.74970	−0.874851	+	0.484392i	$0.839041\pi$
$180$	−0.812964	−0.0605948
$181$	−13.7161	−1.01951	−0.509755	−	0.860320i	$-0.670264\pi$
$181$	−13.7161	−1.01951	−0.509755	+	0.860320i	$0.670264\pi$
$182$	−5.07662	−0.376304
$183$	5.95131	0.439934
$184$	10.2905	0.758623
$185$	1.25800	0.0924897
$186$	−3.31052	−0.242739
$187$	0	0
$188$	−8.43666	−0.615306
$189$	−3.47214	−0.252561
$190$	0.504303	0.0365860
$191$	−12.0249	−0.870094	−0.435047	−	0.900408i	$-0.643268\pi$
$191$	−12.0249	−0.870094	−0.435047	+	0.900408i	$0.643268\pi$
$192$	1.90012	0.137129
$193$	−15.8839	−1.14335	−0.571675	−	0.820480i	$-0.693706\pi$
$193$	−15.8839	−1.14335	−0.571675	+	0.820480i	$0.693706\pi$
$194$	−7.03927	−0.505390
$195$	0.897958	0.0643041
$196$	−1.39552	−0.0996799
$197$	−12.3035	−0.876590	−0.438295	−	0.898831i	$-0.644418\pi$
$197$	−12.3035	−0.876590	−0.438295	+	0.898831i	$0.644418\pi$
$198$	0	0
$199$	15.2615	1.08186	0.540929	−	0.841068i	$-0.318073\pi$
$199$	15.2615	1.08186	0.540929	+	0.841068i	$0.318073\pi$
$200$	13.0691	0.924125
$201$	0.785860	0.0554303
$202$	14.9356	1.05087
$203$	3.77399	0.264882
$204$	−3.73851	−0.261748
$205$	0.297751	0.0207958
$206$	12.6250	0.879627
$207$	10.2050	0.709296
$208$	−4.82212	−0.334354
$209$	0	0
$210$	−0.106921	−0.00737828
$211$	−8.93344	−0.615003	−0.307502	−	0.951548i	$-0.599493\pi$
$211$	−8.93344	−0.615003	−0.307502	+	0.951548i	$0.599493\pi$
$212$	2.39738	0.164653
$213$	−5.75340	−0.394216
$214$	4.19325	0.286645
$215$	−1.04765	−0.0714491
$216$	9.16631	0.623688
$217$	−6.88958	−0.467695
$218$	−4.19765	−0.284301
$219$	−3.45426	−0.233417
$220$	0	0
$221$	−28.3031	−1.90387
$222$	−2.71658	−0.182325
$223$	0.715244	0.0478963	0.0239481	−	0.999713i	$-0.492376\pi$
$223$	0.715244	0.0478963	0.0239481	+	0.999713i	$0.492376\pi$
$224$	5.85410	0.391144
$225$	12.9605	0.864036
$226$	−12.8256	−0.853143
$227$	5.32162	0.353208	0.176604	−	0.984282i	$-0.443489\pi$
$227$	5.32162	0.353208	0.176604	+	0.984282i	$0.443489\pi$
$228$	2.51413	0.166502
$229$	−6.60832	−0.436690	−0.218345	−	0.975872i	$-0.570066\pi$
$229$	−6.60832	−0.436690	−0.218345	+	0.975872i	$0.570066\pi$
$230$	0.674356	0.0444657
$231$	0	0
$232$	−9.96318	−0.654115
$233$	9.55713	0.626108	0.313054	−	0.949735i	$-0.398648\pi$
$233$	9.55713	0.626108	0.313054	+	0.949735i	$0.398648\pi$
$234$	13.2908	0.868844
$235$	−1.34523	−0.0877529
$236$	13.3035	0.865986
$237$	2.79941	0.181841
$238$	3.37009	0.218451
$239$	−26.8466	−1.73656	−0.868282	−	0.496071i	$-0.834776\pi$
$239$	−26.8466	−1.73656	−0.868282	+	0.496071i	$0.834776\pi$
$240$	−0.101561	−0.00655575
$241$	−18.8663	−1.21529	−0.607643	−	0.794210i	$-0.707885\pi$
$241$	−18.8663	−1.21529	−0.607643	+	0.794210i	$0.707885\pi$
$242$	0	0
$243$	13.9443	0.894525
$244$	−13.4380	−0.860282
$245$	−0.222516	−0.0142160
$246$	−0.642978	−0.0409948
$247$	19.0337	1.21108
$248$	18.1882	1.15495
$249$	−6.97375	−0.441943
$250$	1.72146	0.108875
$251$	29.3732	1.85402	0.927009	−	0.375040i	$-0.122371\pi$
$251$	29.3732	1.85402	0.927009	+	0.375040i	$0.122371\pi$
$252$	3.65351	0.230150
$253$	0	0
$254$	−1.55285	−0.0974348
$255$	−0.596106	−0.0373296
$256$	−13.3934	−0.837088
$257$	−16.9164	−1.05522	−0.527608	−	0.849488i	$-0.676911\pi$
$257$	−16.9164	−1.05522	−0.527608	+	0.849488i	$0.676911\pi$
$258$	2.26235	0.140848
$259$	−5.65351	−0.351292
$260$	−2.02759	−0.125746
$261$	−9.88043	−0.611583
$262$	−1.06523	−0.0658099
$263$	8.18034	0.504421	0.252211	−	0.967672i	$-0.418842\pi$
$263$	8.18034	0.504421	0.252211	+	0.967672i	$0.418842\pi$
$264$	0	0
$265$	0.382263	0.0234822
$266$	−2.26637	−0.138960
$267$	4.89580	0.299618
$268$	−1.77447	−0.108393
$269$	−6.24716	−0.380896	−0.190448	−	0.981697i	$-0.560994\pi$
$269$	−6.24716	−0.380896	−0.190448	+	0.981697i	$0.560994\pi$
$270$	0.600688	0.0365567
$271$	−7.86688	−0.477879	−0.238939	−	0.971034i	$-0.576800\pi$
$271$	−7.86688	−0.477879	−0.238939	+	0.971034i	$0.576800\pi$
$272$	3.20115	0.194098
$273$	−4.03548	−0.244238
$274$	1.94312	0.117388
$275$	0	0
$276$	3.36190	0.202363
$277$	−12.0476	−0.723873	−0.361937	−	0.932203i	$-0.617884\pi$
$277$	−12.0476	−0.723873	−0.361937	+	0.932203i	$0.617884\pi$
$278$	3.70284	0.222082
$279$	18.0372	1.07986
$280$	0.587433	0.0351058
$281$	−21.1549	−1.26200	−0.630998	−	0.775784i	$-0.717354\pi$
$281$	−21.1549	−1.26200	−0.630998	+	0.775784i	$0.717354\pi$
$282$	2.90495	0.172987
$283$	25.0034	1.48630	0.743148	−	0.669127i	$-0.233332\pi$
$283$	25.0034	1.48630	0.743148	+	0.669127i	$0.233332\pi$
$284$	12.9911	0.770883
$285$	0.400878	0.0237460
$286$	0	0
$287$	−1.33811	−0.0789861
$288$	−15.3262	−0.903107
$289$	1.78888	0.105228
$290$	−0.652909	−0.0383402
$291$	−5.59563	−0.328021
$292$	7.79971	0.456443
$293$	0.455087	0.0265865	0.0132932	−	0.999912i	$-0.495769\pi$
$293$	0.455087	0.0265865	0.0132932	+	0.999912i	$0.495769\pi$
$294$	0.480512	0.0280240
$295$	2.12125	0.123504
$296$	14.9251	0.867502
$297$	0	0
$298$	5.62110	0.325621
$299$	25.4519	1.47192
$300$	4.26968	0.246510
$301$	4.70820	0.271376
$302$	−6.32848	−0.364163
$303$	11.8725	0.682059
$304$	−2.15275	−0.123469
$305$	−2.14270	−0.122691
$306$	−8.82302	−0.504378
$307$	8.03578	0.458626	0.229313	−	0.973353i	$-0.426352\pi$
$307$	8.03578	0.458626	0.229313	+	0.973353i	$0.426352\pi$
$308$	0	0
$309$	10.0358	0.570918
$310$	1.19191	0.0676962
$311$	0.136965	0.00776656	0.00388328	−	0.999992i	$-0.498764\pi$
$311$	0.136965	0.00776656	0.00388328	+	0.999992i	$0.498764\pi$
$312$	10.6535	0.603136
$313$	−15.2899	−0.864238	−0.432119	−	0.901817i	$-0.642234\pi$
$313$	−15.2899	−0.864238	−0.432119	+	0.901817i	$0.642234\pi$
$314$	15.6674	0.884163
$315$	0.582554	0.0328232
$316$	−6.32106	−0.355587
$317$	−32.9925	−1.85304	−0.926522	−	0.376239i	$-0.877217\pi$
$317$	−32.9925	−1.85304	−0.926522	+	0.376239i	$0.877217\pi$
$318$	−0.825478	−0.0462905
$319$	0	0
$320$	−0.684115	−0.0382432
$321$	3.33328	0.186045
$322$	−3.03060	−0.168889
$323$	−12.6354	−0.703054
$324$	−7.96590	−0.442550
$325$	32.3244	1.79304
$326$	5.78020	0.320136
$327$	−3.33677	−0.184524
$328$	3.53256	0.195053
$329$	6.04554	0.333301
$330$	0	0
$331$	29.5335	1.62331	0.811653	−	0.584140i	$-0.198568\pi$
$331$	29.5335	1.62331	0.811653	+	0.584140i	$0.198568\pi$
$332$	15.7467	0.864212
$333$	14.8011	0.811095
$334$	1.71911	0.0940658
$335$	−0.282939	−0.0154586
$336$	0.456423	0.0248999
$337$	−5.63025	−0.306699	−0.153350	−	0.988172i	$-0.549006\pi$
$337$	−5.63025	−0.306699	−0.153350	+	0.988172i	$0.549006\pi$
$338$	23.0407	1.25325
$339$	−10.1952	−0.553729
$340$	1.34600	0.0729974
$341$	0	0
$342$	5.93344	0.320844
$343$	1.00000	0.0539949
$344$	−12.4295	−0.670153
$345$	0.536056	0.0288603
$346$	8.05645	0.433117
$347$	−8.73061	−0.468684	−0.234342	−	0.972154i	$-0.575294\pi$
$347$	−8.73061	−0.468684	−0.234342	+	0.972154i	$0.575294\pi$
$348$	−3.25498	−0.174485
$349$	−18.4670	−0.988514	−0.494257	−	0.869316i	$-0.664560\pi$
$349$	−18.4670	−0.988514	−0.494257	+	0.869316i	$0.664560\pi$
$350$	−3.84893	−0.205734
$351$	22.6715	1.21011
$352$	0	0
$353$	−23.4857	−1.25002	−0.625009	−	0.780618i	$-0.714905\pi$
$353$	−23.4857	−1.25002	−0.625009	+	0.780618i	$0.714905\pi$
$354$	−4.58074	−0.243463
$355$	2.07144	0.109941
$356$	−11.0547	−0.585897
$357$	2.67894	0.141784
$358$	−18.2005	−0.961924
$359$	−14.8812	−0.785400	−0.392700	−	0.919667i	$-0.628459\pi$
$359$	−14.8812	−0.785400	−0.392700	+	0.919667i	$0.628459\pi$
$360$	−1.53792	−0.0810555
$361$	−10.5027	−0.552776
$362$	−10.6641	−0.560490
$363$	0	0
$364$	9.11210	0.477604
$365$	1.24367	0.0650964
$366$	4.62705	0.241860
$367$	−34.2259	−1.78658	−0.893289	−	0.449482i	$-0.851609\pi$
$367$	−34.2259	−1.78658	−0.893289	+	0.449482i	$0.851609\pi$
$368$	−2.87867	−0.150061
$369$	3.50322	0.182370
$370$	0.978072	0.0508475
$371$	−1.71792	−0.0891897
$372$	5.94211	0.308084
$373$	−3.01739	−0.156235	−0.0781173	−	0.996944i	$-0.524891\pi$
$373$	−3.01739	−0.156235	−0.0781173	+	0.996944i	$0.524891\pi$
$374$	0	0
$375$	1.36841	0.0706646
$376$	−15.9600	−0.823073
$377$	−24.6424	−1.26915
$378$	−2.69953	−0.138849
$379$	−6.12431	−0.314585	−0.157292	−	0.987552i	$-0.550277\pi$
$379$	−6.12431	−0.314585	−0.157292	+	0.987552i	$0.550277\pi$
$380$	−0.905182	−0.0464348
$381$	−1.23439	−0.0632396
$382$	−9.34920	−0.478347
$383$	4.39098	0.224369	0.112184	−	0.993687i	$-0.464215\pi$
$383$	4.39098	0.224369	0.112184	+	0.993687i	$0.464215\pi$
$384$	−5.75876	−0.293875
$385$	0	0
$386$	−12.3495	−0.628573
$387$	−12.3262	−0.626578
$388$	12.6349	0.641440
$389$	10.4960	0.532169	0.266084	−	0.963950i	$-0.414270\pi$
$389$	10.4960	0.532169	0.266084	+	0.963950i	$0.414270\pi$
$390$	0.698148	0.0353521
$391$	−16.8961	−0.854475
$392$	−2.63996	−0.133338
$393$	−0.846765	−0.0427136
$394$	−9.56580	−0.481918
$395$	−1.00789	−0.0507127
$396$	0	0
$397$	11.3888	0.571589	0.285794	−	0.958291i	$-0.407743\pi$
$397$	11.3888	0.571589	0.285794	+	0.958291i	$0.407743\pi$
$398$	11.8656	0.594767
$399$	−1.80157	−0.0901915
$400$	−3.65597	−0.182799
$401$	−4.72496	−0.235953	−0.117977	−	0.993016i	$-0.537641\pi$
$401$	−4.72496	−0.235953	−0.117977	+	0.993016i	$0.537641\pi$
$402$	0.610994	0.0304736
$403$	44.9858	2.24090
$404$	−26.8081	−1.33375
$405$	−1.27017	−0.0631150
$406$	2.93422	0.145623
$407$	0	0
$408$	−7.07230	−0.350131
$409$	−2.46544	−0.121908	−0.0609541	−	0.998141i	$-0.519414\pi$
$409$	−2.46544	−0.121908	−0.0609541	+	0.998141i	$0.519414\pi$
$410$	0.231496	0.0114328
$411$	1.54462	0.0761903
$412$	−22.6609	−1.11642
$413$	−9.53304	−0.469090
$414$	7.93422	0.389945
$415$	2.51082	0.123251
$416$	−38.2246	−1.87412
$417$	2.94345	0.144141
$418$	0	0
$419$	−14.3399	−0.700548	−0.350274	−	0.936647i	$-0.613912\pi$
$419$	−14.3399	−0.700548	−0.350274	+	0.936647i	$0.613912\pi$
$420$	0.191915	0.00936449
$421$	−17.3157	−0.843918	−0.421959	−	0.906615i	$-0.638657\pi$
$421$	−17.3157	−0.843918	−0.421959	+	0.906615i	$0.638657\pi$
$422$	−6.94561	−0.338107
$423$	−15.8274	−0.769555
$424$	4.53523	0.220250
$425$	−21.4584	−1.04089
$426$	−4.47317	−0.216726
$427$	9.62943	0.466001
$428$	−7.52653	−0.363808
$429$	0	0
$430$	−0.814531	−0.0392802
$431$	−27.8923	−1.34352	−0.671762	−	0.740767i	$-0.734462\pi$
$431$	−27.8923	−1.34352	−0.671762	+	0.740767i	$0.734462\pi$
$432$	−2.56420	−0.123370
$433$	−29.5470	−1.41994	−0.709969	−	0.704232i	$-0.751291\pi$
$433$	−29.5470	−1.41994	−0.709969	+	0.704232i	$0.751291\pi$
$434$	−5.35654	−0.257122
$435$	−0.519007	−0.0248845
$436$	7.53442	0.360833
$437$	11.3626	0.543546
$438$	−2.68563	−0.128325
$439$	33.6655	1.60677	0.803384	−	0.595461i	$-0.203030\pi$
$439$	33.6655	1.60677	0.803384	+	0.595461i	$0.203030\pi$
$440$	0	0
$441$	−2.61803	−0.124668
$442$	−22.0052	−1.04668
$443$	10.3267	0.490637	0.245319	−	0.969443i	$-0.421107\pi$
$443$	10.3267	0.490637	0.245319	+	0.969443i	$0.421107\pi$
$444$	4.87603	0.231406
$445$	−1.76267	−0.0835587
$446$	0.556091	0.0263317
$447$	4.46829	0.211343
$448$	3.07446	0.145254
$449$	−2.75785	−0.130151	−0.0650756	−	0.997880i	$-0.520729\pi$
$449$	−2.75785	−0.130151	−0.0650756	+	0.997880i	$0.520729\pi$
$450$	10.0766	0.475016
$451$	0	0
$452$	23.0208	1.08281
$453$	−5.03060	−0.236358
$454$	4.13747	0.194181
$455$	1.45293	0.0681142
$456$	4.75608	0.222724
$457$	40.3305	1.88658	0.943291	−	0.331968i	$-0.107713\pi$
$457$	40.3305	1.88658	0.943291	+	0.331968i	$0.107713\pi$
$458$	−5.13787	−0.240077
$459$	−15.0504	−0.702491
$460$	−1.21041	−0.0564358
$461$	34.2251	1.59402	0.797011	−	0.603965i	$-0.206413\pi$
$461$	34.2251	1.59402	0.797011	+	0.603965i	$0.206413\pi$
$462$	0	0
$463$	0.707349	0.0328733	0.0164367	−	0.999865i	$-0.494768\pi$
$463$	0.707349	0.0328733	0.0164367	+	0.999865i	$0.494768\pi$
$464$	2.78712	0.129389
$465$	0.947471	0.0439379
$466$	7.43052	0.344212
$467$	−28.5924	−1.32310	−0.661550	−	0.749901i	$-0.730101\pi$
$467$	−28.5924	−1.32310	−0.661550	+	0.749901i	$0.730101\pi$
$468$	−23.8558	−1.10273
$469$	1.27155	0.0587146
$470$	−1.04589	−0.0482434
$471$	12.4543	0.573862
$472$	25.1669	1.15840
$473$	0	0
$474$	2.17650	0.0999699
$475$	14.4307	0.662126
$476$	−6.04903	−0.277257
$477$	4.49756	0.205929
$478$	−20.8728	−0.954701
$479$	5.09716	0.232895	0.116448	−	0.993197i	$-0.462849\pi$
$479$	5.09716	0.232895	0.116448	+	0.993197i	$0.462849\pi$
$480$	−0.805070	−0.0367462
$481$	36.9149	1.68317
$482$	−14.6683	−0.668121
$483$	−2.40907	−0.109617
$484$	0	0
$485$	2.01464	0.0914800
$486$	10.8415	0.491778
$487$	29.1883	1.32265	0.661324	−	0.750101i	$-0.269995\pi$
$487$	29.1883	1.32265	0.661324	+	0.750101i	$0.269995\pi$
$488$	−25.4213	−1.15077
$489$	4.59477	0.207783
$490$	−0.173002	−0.00781546
$491$	−30.1439	−1.36038	−0.680189	−	0.733037i	$-0.738102\pi$
$491$	−30.1439	−1.36038	−0.680189	+	0.733037i	$0.738102\pi$
$492$	1.15409	0.0520304
$493$	16.3588	0.736762
$494$	14.7984	0.665810
$495$	0	0
$496$	−5.08801	−0.228458
$497$	−9.30919	−0.417574
$498$	−5.42198	−0.242965
$499$	5.95104	0.266405	0.133203	−	0.991089i	$-0.457474\pi$
$499$	5.95104	0.266405	0.133203	+	0.991089i	$0.457474\pi$
$500$	−3.08987	−0.138183
$501$	1.36655	0.0610530
$502$	22.8372	1.01927
$503$	4.97855	0.221983	0.110991	−	0.993821i	$-0.464597\pi$
$503$	4.97855	0.221983	0.110991	+	0.993821i	$0.464597\pi$
$504$	6.91151	0.307863
$505$	−4.27456	−0.190216
$506$	0	0
$507$	18.3154	0.813416
$508$	2.78724	0.123664
$509$	−21.3285	−0.945370	−0.472685	−	0.881231i	$-0.656715\pi$
$509$	−21.3285	−0.945370	−0.472685	+	0.881231i	$0.656715\pi$
$510$	−0.463463	−0.0205225
$511$	−5.58911	−0.247248
$512$	8.22256	0.363389
$513$	10.1213	0.446866
$514$	−13.1522	−0.580120
$515$	−3.61328	−0.159220
$516$	−4.06072	−0.178763
$517$	0	0
$518$	−4.39552	−0.193128
$519$	6.40419	0.281113
$520$	−3.83567	−0.168205
$521$	−22.1383	−0.969899	−0.484949	−	0.874542i	$-0.661162\pi$
$521$	−22.1383	−0.969899	−0.484949	+	0.874542i	$0.661162\pi$
$522$	−7.68188	−0.336227
$523$	−4.91146	−0.214763	−0.107382	−	0.994218i	$-0.534247\pi$
$523$	−4.91146	−0.214763	−0.107382	+	0.994218i	$0.534247\pi$
$524$	1.91199	0.0835257
$525$	−3.05957	−0.133531
$526$	6.36009	0.277313
$527$	−29.8637	−1.30088
$528$	0	0
$529$	−7.80592	−0.339388
$530$	0.297204	0.0129097
$531$	24.9578	1.08308
$532$	4.06794	0.176368
$533$	8.73725	0.378452
$534$	3.80641	0.164719
$535$	−1.20011	−0.0518851
$536$	−3.35684	−0.144993
$537$	−14.4678	−0.624332
$538$	−4.85707	−0.209403
$539$	0	0
$540$	−1.07818	−0.0463977
$541$	−19.3845	−0.833404	−0.416702	−	0.909043i	$-0.636814\pi$
$541$	−19.3845	−0.833404	−0.416702	+	0.909043i	$0.636814\pi$
$542$	−6.11637	−0.262721
$543$	−8.47702	−0.363784
$544$	25.3753	1.08796
$545$	1.20137	0.0514609
$546$	−3.13752	−0.134274
$547$	−14.4501	−0.617840	−0.308920	−	0.951088i	$-0.599968\pi$
$547$	−14.4501	−0.617840	−0.308920	+	0.951088i	$0.599968\pi$
$548$	−3.48774	−0.148989
$549$	−25.2102	−1.07594
$550$	0	0
$551$	−11.0012	−0.468667
$552$	6.35985	0.270693
$553$	4.52954	0.192616
$554$	−9.36686	−0.397960
$555$	0.777484	0.0330024
$556$	−6.64629	−0.281865
$557$	−18.4708	−0.782634	−0.391317	−	0.920256i	$-0.627980\pi$
$557$	−18.4708	−0.782634	−0.391317	+	0.920256i	$0.627980\pi$
$558$	14.0236	0.593667
$559$	−30.7424	−1.30027
$560$	−0.164330	−0.00694419
$561$	0	0
$562$	−16.4476	−0.693801
$563$	−31.2838	−1.31846	−0.659228	−	0.751943i	$-0.729117\pi$
$563$	−31.2838	−1.31846	−0.659228	+	0.751943i	$0.729117\pi$
$564$	−5.21414	−0.219555
$565$	3.67067	0.154426
$566$	19.4397	0.817112
$567$	5.70820	0.239722
$568$	24.5759	1.03118
$569$	−1.73605	−0.0727790	−0.0363895	−	0.999338i	$-0.511586\pi$
$569$	−1.73605	−0.0727790	−0.0363895	+	0.999338i	$0.511586\pi$
$570$	0.311677	0.0130547
$571$	−12.5309	−0.524403	−0.262201	−	0.965013i	$-0.584449\pi$
$571$	−12.5309	−0.524403	−0.262201	+	0.965013i	$0.584449\pi$
$572$	0	0
$573$	−7.43182	−0.310469
$574$	−1.04036	−0.0434238
$575$	19.2968	0.804732
$576$	−8.04903	−0.335376
$577$	20.1896	0.840505	0.420252	−	0.907407i	$-0.361942\pi$
$577$	20.1896	0.840505	0.420252	+	0.907407i	$0.361942\pi$
$578$	1.39082	0.0578506
$579$	−9.81681	−0.407973
$580$	1.17192	0.0486612
$581$	−11.2838	−0.468129
$582$	−4.35051	−0.180334
$583$	0	0
$584$	14.7550	0.610568
$585$	−3.80381	−0.157268
$586$	0.353823	0.0146163
$587$	−0.00837574	−0.000345704 0	−0.000172852	−	1.00000i	$-0.500055\pi$
$587$	−0.00837574	−0.000345704 0	−0.000172852		1.00000i	$0.500055\pi$
$588$	−0.862478	−0.0355680
$589$	20.0832	0.827513
$590$	1.64924	0.0678981
$591$	−7.60400	−0.312787
$592$	−4.17516	−0.171598
$593$	0.439298	0.0180398	0.00901989	−	0.999959i	$-0.497129\pi$
$593$	0.439298	0.0180398	0.00901989	+	0.999959i	$0.497129\pi$
$594$	0	0
$595$	−0.964520	−0.0395415
$596$	−10.0894	−0.413278
$597$	9.43212	0.386031
$598$	19.7884	0.809210
$599$	45.1664	1.84545	0.922724	−	0.385462i	$-0.125958\pi$
$599$	45.1664	1.84545	0.922724	+	0.385462i	$0.125958\pi$
$600$	8.07715	0.329748
$601$	−11.1437	−0.454559	−0.227280	−	0.973830i	$-0.572983\pi$
$601$	−11.1437	−0.454559	−0.227280	+	0.973830i	$0.572983\pi$
$602$	3.66055	0.149193
$603$	−3.32896	−0.135566
$604$	11.3591	0.462194
$605$	0	0
$606$	9.23071	0.374972
$607$	35.6998	1.44901	0.724504	−	0.689270i	$-0.242069\pi$
$607$	35.6998	1.44901	0.724504	+	0.689270i	$0.242069\pi$
$608$	−17.0647	−0.692067
$609$	2.33245	0.0945158
$610$	−1.66591	−0.0674509
$611$	−39.4746	−1.59697
$612$	15.8366	0.640156
$613$	−17.1980	−0.694620	−0.347310	−	0.937750i	$-0.612905\pi$
$613$	−17.1980	−0.694620	−0.347310	+	0.937750i	$0.612905\pi$
$614$	6.24769	0.252136
$615$	0.184020	0.00742040
$616$	0	0
$617$	−16.8852	−0.679774	−0.339887	−	0.940466i	$-0.610389\pi$
$617$	−16.8852	−0.679774	−0.339887	+	0.940466i	$0.610389\pi$
$618$	7.80270	0.313870
$619$	31.0692	1.24878	0.624389	−	0.781114i	$-0.285348\pi$
$619$	31.0692	1.24878	0.624389	+	0.781114i	$0.285348\pi$
$620$	−2.13939	−0.0859198
$621$	13.5342	0.543110
$622$	0.106488	0.00426978
$623$	7.92157	0.317371
$624$	−2.98023	−0.119305
$625$	24.2598	0.970390
$626$	−11.8877	−0.475127
$627$	0	0
$628$	−28.1217	−1.12218
$629$	−24.5058	−0.977110
$630$	0.452926	0.0180450
$631$	7.25364	0.288763	0.144382	−	0.989522i	$-0.453881\pi$
$631$	7.25364	0.288763	0.144382	+	0.989522i	$0.453881\pi$
$632$	−11.9578	−0.475657
$633$	−5.52117	−0.219447
$634$	−25.6512	−1.01874
$635$	0.444427	0.0176365
$636$	1.48166	0.0587518
$637$	−6.52954	−0.258710
$638$	0	0
$639$	24.3718	0.964132
$640$	2.07337	0.0819572
$641$	−20.7499	−0.819572	−0.409786	−	0.912182i	$-0.634397\pi$
$641$	−20.7499	−0.819572	−0.409786	+	0.912182i	$0.634397\pi$
$642$	2.59157	0.102281
$643$	7.11127	0.280441	0.140221	−	0.990120i	$-0.455219\pi$
$643$	7.11127	0.280441	0.140221	+	0.990120i	$0.455219\pi$
$644$	5.43967	0.214353
$645$	−0.647483	−0.0254946
$646$	−9.82385	−0.386514
$647$	27.5789	1.08424	0.542119	−	0.840302i	$-0.317622\pi$
$647$	27.5789	1.08424	0.542119	+	0.840302i	$0.317622\pi$
$648$	−15.0694	−0.591984
$649$	0	0
$650$	25.1317	0.985748
$651$	−4.25800	−0.166884
$652$	−10.3750	−0.406315
$653$	16.1541	0.632160	0.316080	−	0.948732i	$-0.397633\pi$
$653$	16.1541	0.632160	0.316080	+	0.948732i	$0.397633\pi$
$654$	−2.59429	−0.101445
$655$	0.304867	0.0119122
$656$	−0.988205	−0.0385829
$657$	14.6325	0.570868
$658$	4.70031	0.183237
$659$	13.2085	0.514531	0.257266	−	0.966341i	$-0.417178\pi$
$659$	13.2085	0.514531	0.257266	+	0.966341i	$0.417178\pi$
$660$	0	0
$661$	−4.90660	−0.190845	−0.0954223	−	0.995437i	$-0.530420\pi$
$661$	−4.90660	−0.190845	−0.0954223	+	0.995437i	$0.530420\pi$
$662$	22.9618	0.892436
$663$	−17.4922	−0.679343
$664$	29.7887	1.15603
$665$	0.648635	0.0251530
$666$	11.5076	0.445911
$667$	−14.7108	−0.569606
$668$	−3.08566	−0.119388
$669$	0.442045	0.0170905
$670$	−0.219981	−0.00849861
$671$	0	0
$672$	3.61803	0.139569
$673$	29.9872	1.15592	0.577960	−	0.816065i	$-0.303849\pi$
$673$	29.9872	1.15592	0.577960	+	0.816065i	$0.303849\pi$
$674$	−4.37743	−0.168612
$675$	17.1888	0.661596
$676$	−41.3561	−1.59062
$677$	−12.5889	−0.483832	−0.241916	−	0.970297i	$-0.577776\pi$
$677$	−12.5889	−0.483832	−0.241916	+	0.970297i	$0.577776\pi$
$678$	−7.92663	−0.304420
$679$	−9.05391	−0.347457
$680$	2.54630	0.0976460
$681$	3.28894	0.126033
$682$	0	0
$683$	−28.5342	−1.09183	−0.545916	−	0.837840i	$-0.683818\pi$
$683$	−28.5342	−1.09183	−0.545916	+	0.837840i	$0.683818\pi$
$684$	−10.6500	−0.407214
$685$	−0.556120	−0.0212483
$686$	0.777484	0.0296845
$687$	−4.08417	−0.155821
$688$	3.47704	0.132561
$689$	11.2172	0.427341
$690$	0.416775	0.0158664
$691$	25.3742	0.965281	0.482641	−	0.875818i	$-0.339678\pi$
$691$	25.3742	0.965281	0.482641	+	0.875818i	$0.339678\pi$
$692$	−14.4606	−0.549711
$693$	0	0
$694$	−6.78791	−0.257666
$695$	−1.05975	−0.0401987
$696$	−6.15759	−0.233403
$697$	−5.80019	−0.219698
$698$	−14.3578	−0.543450
$699$	5.90663	0.223409
$700$	6.90849	0.261117
$701$	45.7021	1.72615	0.863073	−	0.505079i	$-0.168537\pi$
$701$	45.7021	1.72615	0.863073	+	0.505079i	$0.168537\pi$
$702$	17.6267	0.665277
$703$	16.4800	0.621556
$704$	0	0
$705$	−0.831396	−0.0313122
$706$	−18.2598	−0.687215
$707$	19.2102	0.722473
$708$	8.22203	0.309003
$709$	38.5304	1.44704	0.723520	−	0.690304i	$-0.242523\pi$
$709$	38.5304	1.44704	0.723520	+	0.690304i	$0.242523\pi$
$710$	1.61051	0.0604415
$711$	−11.8585	−0.444729
$712$	−20.9126	−0.783734
$713$	26.8553	1.00574
$714$	2.08283	0.0779480
$715$	0	0
$716$	32.6683	1.22087
$717$	−16.5921	−0.619644
$718$	−11.5699	−0.431785
$719$	5.70213	0.212653	0.106327	−	0.994331i	$-0.466091\pi$
$719$	5.70213	0.212653	0.106327	+	0.994331i	$0.466091\pi$
$720$	0.430220	0.0160334
$721$	16.2383	0.604746
$722$	−8.16571	−0.303896
$723$	−11.6600	−0.433641
$724$	19.1411	0.711372
$725$	−18.6831	−0.693872
$726$	0	0
$727$	11.8221	0.438458	0.219229	−	0.975673i	$-0.429646\pi$
$727$	11.8221	0.438458	0.219229	+	0.975673i	$0.429646\pi$
$728$	17.2377	0.638873
$729$	−8.50658	−0.315058
$730$	0.966931	0.0357877
$731$	20.4082	0.754826
$732$	−8.30516	−0.306968
$733$	−8.81193	−0.325476	−0.162738	−	0.986669i	$-0.552033\pi$
$733$	−8.81193	−0.325476	−0.162738	+	0.986669i	$0.552033\pi$
$734$	−26.6101	−0.982197
$735$	−0.137522	−0.00507259
$736$	−22.8190	−0.841121
$737$	0	0
$738$	2.72370	0.100261
$739$	−0.480809	−0.0176868	−0.00884342	−	0.999961i	$-0.502815\pi$
$739$	−0.480809	−0.0176868	−0.00884342	+	0.999961i	$0.502815\pi$
$740$	−1.75556	−0.0645355
$741$	11.7635	0.432141
$742$	−1.33565	−0.0490333
$743$	−33.2443	−1.21961	−0.609807	−	0.792550i	$-0.708753\pi$
$743$	−33.2443	−1.21961	−0.609807	+	0.792550i	$0.708753\pi$
$744$	11.2409	0.412113
$745$	−1.60876	−0.0589403
$746$	−2.34598	−0.0858923
$747$	29.5413	1.08086
$748$	0	0
$749$	5.39336	0.197069
$750$	1.06392	0.0388489
$751$	28.0066	1.02198	0.510988	−	0.859588i	$-0.329280\pi$
$751$	28.0066	1.02198	0.510988	+	0.859588i	$0.329280\pi$
$752$	4.46467	0.162810
$753$	18.1536	0.661554
$754$	−19.1591	−0.697733
$755$	1.81121	0.0659166
$756$	4.84543	0.176227
$757$	−22.2468	−0.808575	−0.404288	−	0.914632i	$-0.632481\pi$
$757$	−22.2468	−0.808575	−0.404288	+	0.914632i	$0.632481\pi$
$758$	−4.76156	−0.172948
$759$	0	0
$760$	−1.71237	−0.0621142
$761$	48.2254	1.74817	0.874085	−	0.485774i	$-0.161462\pi$
$761$	48.2254	1.74817	0.874085	+	0.485774i	$0.161462\pi$
$762$	−0.959717	−0.0347669
$763$	−5.39901	−0.195457
$764$	16.7810	0.607116
$765$	2.52515	0.0912968
$766$	3.41392	0.123350
$767$	62.2464	2.24759
$768$	−8.27758	−0.298691
$769$	−43.6883	−1.57544	−0.787721	−	0.616032i	$-0.788739\pi$
$769$	−43.6883	−1.57544	−0.787721	+	0.616032i	$0.788739\pi$
$770$	0	0
$771$	−10.4549	−0.376524
$772$	22.1663	0.797783
$773$	−0.585569	−0.0210615	−0.0105307	−	0.999945i	$-0.503352\pi$
$773$	−0.585569	−0.0210615	−0.0105307	+	0.999945i	$0.503352\pi$
$774$	−9.58346	−0.344470
$775$	34.1068	1.22515
$776$	23.9020	0.858031
$777$	−3.49406	−0.125349
$778$	8.16048	0.292567
$779$	3.90060	0.139753
$780$	−1.25312	−0.0448688
$781$	0	0
$782$	−13.1365	−0.469760
$783$	−13.1038	−0.468292
$784$	0.738508	0.0263753
$785$	−4.48401	−0.160041
$786$	−0.658346	−0.0234824
$787$	−30.7166	−1.09493	−0.547464	−	0.836829i	$-0.684407\pi$
$787$	−30.7166	−1.09493	−0.547464	+	0.836829i	$0.684407\pi$
$788$	17.1698	0.611649
$789$	5.05573	0.179989
$790$	−0.783622	−0.0278800
$791$	−16.4962	−0.586538
$792$	0	0
$793$	−62.8758	−2.23278
$794$	8.85463	0.314239
$795$	0.236252	0.00837898
$796$	−21.2977	−0.754877
$797$	−10.8333	−0.383735	−0.191867	−	0.981421i	$-0.561454\pi$
$797$	−10.8333	−0.383735	−0.191867	+	0.981421i	$0.561454\pi$
$798$	−1.40069	−0.0495841
$799$	26.2051	0.927068
$800$	−28.9807	−1.02462
$801$	−20.7389	−0.732774
$802$	−3.67358	−0.129719
$803$	0	0
$804$	−1.09668	−0.0386770
$805$	0.867357	0.0305703
$806$	34.9758	1.23197
$807$	−3.86096	−0.135912
$808$	−50.7141	−1.78412
$809$	38.5901	1.35675	0.678377	−	0.734714i	$-0.262684\pi$
$809$	38.5901	1.35675	0.678377	+	0.734714i	$0.262684\pi$
$810$	−0.987533	−0.0346984
$811$	50.8935	1.78711	0.893556	−	0.448951i	$-0.148202\pi$
$811$	50.8935	1.78711	0.893556	+	0.448951i	$0.148202\pi$
$812$	−5.26667	−0.184824
$813$	−4.86200	−0.170518
$814$	0	0
$815$	−1.65429	−0.0579473
$816$	1.97842	0.0692584
$817$	−13.7244	−0.480158
$818$	−1.91684	−0.0670208
$819$	17.0946	0.597333
$820$	−0.415516	−0.0145105
$821$	40.3094	1.40681	0.703404	−	0.710790i	$-0.251662\pi$
$821$	40.3094	1.40681	0.703404	+	0.710790i	$0.251662\pi$
$822$	1.20091	0.0418867
$823$	25.6817	0.895209	0.447605	−	0.894232i	$-0.352277\pi$
$823$	25.6817	0.895209	0.447605	+	0.894232i	$0.352277\pi$
$824$	−42.8685	−1.49340
$825$	0	0
$826$	−7.41179	−0.257889
$827$	31.9557	1.11121	0.555604	−	0.831447i	$-0.312487\pi$
$827$	31.9557	1.11121	0.555604	+	0.831447i	$0.312487\pi$
$828$	−14.2412	−0.494917
$829$	48.4002	1.68101	0.840504	−	0.541805i	$-0.182259\pi$
$829$	48.4002	1.68101	0.840504	+	0.541805i	$0.182259\pi$
$830$	1.95212	0.0677591
$831$	−7.44586	−0.258294
$832$	−20.0748	−0.695969
$833$	4.33461	0.150185
$834$	2.28848	0.0792437
$835$	−0.492010	−0.0170267
$836$	0	0
$837$	23.9216	0.826850
$838$	−11.1490	−0.385137
$839$	−37.7165	−1.30212	−0.651059	−	0.759027i	$-0.725675\pi$
$839$	−37.7165	−1.30212	−0.651059	+	0.759027i	$0.725675\pi$
$840$	0.363054	0.0125265
$841$	−14.7570	−0.508863
$842$	−13.4627	−0.463956
$843$	−13.0745	−0.450308
$844$	12.4668	0.429124
$845$	−6.59424	−0.226849
$846$	−12.3056	−0.423074
$847$	0	0
$848$	−1.26869	−0.0435671
$849$	15.4529	0.530343
$850$	−16.6836	−0.572243
$851$	22.0372	0.755424
$852$	8.02897	0.275068
$853$	−9.28864	−0.318037	−0.159019	−	0.987276i	$-0.550833\pi$
$853$	−9.28864	−0.318037	−0.159019	+	0.987276i	$0.550833\pi$
$854$	7.48673	0.256191
$855$	−1.69815	−0.0580754
$856$	−14.2383	−0.486654
$857$	29.7644	1.01673	0.508365	−	0.861141i	$-0.330250\pi$
$857$	29.7644	1.01673	0.508365	+	0.861141i	$0.330250\pi$
$858$	0	0
$859$	33.2611	1.13485	0.567427	−	0.823424i	$-0.307939\pi$
$859$	33.2611	1.13485	0.567427	+	0.823424i	$0.307939\pi$
$860$	1.46201	0.0498543
$861$	−0.826998	−0.0281840
$862$	−21.6858	−0.738622
$863$	−18.5677	−0.632051	−0.316025	−	0.948751i	$-0.602348\pi$
$863$	−18.5677	−0.632051	−0.316025	+	0.948751i	$0.602348\pi$
$864$	−20.3262	−0.691513
$865$	−2.30575	−0.0783979
$866$	−22.9723	−0.780632
$867$	1.10559	0.0375477
$868$	9.61454	0.326339
$869$	0	0
$870$	−0.403520	−0.0136806
$871$	−8.30263	−0.281324
$872$	14.2532	0.482674
$873$	23.7034	0.802240
$874$	8.83422	0.298822
$875$	2.21414	0.0748516
$876$	4.82049	0.162869
$877$	22.4921	0.759505	0.379752	−	0.925088i	$-0.376009\pi$
$877$	22.4921	0.759505	0.379752	+	0.925088i	$0.376009\pi$
$878$	26.1744	0.883344
$879$	0.281259	0.00948664
$880$	0	0
$881$	7.06565	0.238048	0.119024	−	0.992891i	$-0.462023\pi$
$881$	7.06565	0.238048	0.119024	+	0.992891i	$0.462023\pi$
$882$	−2.03548	−0.0685382
$883$	−19.9382	−0.670974	−0.335487	−	0.942045i	$-0.608901\pi$
$883$	−19.9382	−0.670974	−0.335487	+	0.942045i	$0.608901\pi$
$884$	39.4974	1.32844
$885$	1.31101	0.0440690
$886$	8.02886	0.269735
$887$	−6.95481	−0.233520	−0.116760	−	0.993160i	$-0.537251\pi$
$887$	−6.95481	−0.233520	−0.116760	+	0.993160i	$0.537251\pi$
$888$	9.22420	0.309544
$889$	−1.99728	−0.0669867
$890$	−1.37045	−0.0459376
$891$	0	0
$892$	−0.998136	−0.0334201
$893$	−17.6228	−0.589724
$894$	3.47403	0.116189
$895$	5.20896	0.174116
$896$	−9.31786	−0.311288
$897$	15.7301	0.525214
$898$	−2.14419	−0.0715525
$899$	−26.0012	−0.867189
$900$	−18.0867	−0.602889
$901$	−7.44650	−0.248079
$902$	0	0
$903$	2.90983	0.0968331
$904$	43.5494	1.44843
$905$	3.05205	0.101454
$906$	−3.91121	−0.129941
$907$	−23.4401	−0.778317	−0.389159	−	0.921171i	$-0.627234\pi$
$907$	−23.4401	−0.778317	−0.389159	+	0.921171i	$0.627234\pi$
$908$	−7.42642	−0.246454
$909$	−50.2929	−1.66811
$910$	1.12963	0.0374468
$911$	46.8096	1.55087	0.775436	−	0.631426i	$-0.217530\pi$
$911$	46.8096	1.55087	0.775436	+	0.631426i	$0.217530\pi$
$912$	−1.33048	−0.0440564
$913$	0	0
$914$	31.3563	1.03718
$915$	−1.32426	−0.0437787
$916$	9.22203	0.304705
$917$	−1.37009	−0.0452445
$918$	−11.7014	−0.386205
$919$	−10.6273	−0.350563	−0.175281	−	0.984518i	$-0.556084\pi$
$919$	−10.6273	−0.350563	−0.175281	+	0.984518i	$0.556084\pi$
$920$	−2.28979	−0.0754921
$921$	4.96638	0.163648
$922$	26.6095	0.876336
$923$	60.7848	2.00075
$924$	0	0
$925$	27.9876	0.920228
$926$	0.549953	0.0180726
$927$	−42.5124	−1.39629
$928$	22.0933	0.725248
$929$	−36.6710	−1.20313	−0.601567	−	0.798822i	$-0.705457\pi$
$929$	−36.6710	−1.20313	−0.601567	+	0.798822i	$0.705457\pi$
$930$	0.736644	0.0241555
$931$	−2.91501	−0.0955355
$932$	−13.3371	−0.436873
$933$	0.0846489	0.00277128
$934$	−22.2302	−0.727393
$935$	0	0
$936$	−45.1290	−1.47509
$937$	40.2174	1.31384	0.656922	−	0.753959i	$-0.271858\pi$
$937$	40.2174	1.31384	0.656922	+	0.753959i	$0.271858\pi$
$938$	0.988609	0.0322792
$939$	−9.44970	−0.308379
$940$	1.87729	0.0612304
$941$	−24.9097	−0.812032	−0.406016	−	0.913866i	$-0.633082\pi$
$941$	−24.9097	−0.812032	−0.406016	+	0.913866i	$0.633082\pi$
$942$	9.68299	0.315489
$943$	5.21590	0.169853
$944$	−7.04022	−0.229140
$945$	0.772605	0.0251328
$946$	0	0
$947$	−32.2061	−1.04656	−0.523279	−	0.852161i	$-0.675292\pi$
$947$	−32.2061	−1.04656	−0.523279	+	0.852161i	$0.675292\pi$
$948$	−3.90663	−0.126881
$949$	36.4944	1.18466
$950$	11.2196	0.364013
$951$	−20.3905	−0.661207
$952$	−11.4432	−0.370877
$953$	−45.0611	−1.45967	−0.729837	−	0.683621i	$-0.760404\pi$
$953$	−45.0611	−1.45967	−0.729837	+	0.683621i	$0.760404\pi$
$954$	3.49678	0.113213
$955$	2.67574	0.0865849
$956$	37.4650	1.21170
$957$	0	0
$958$	3.96296	0.128038
$959$	2.49924	0.0807047
$960$	−0.422806	−0.0136460
$961$	16.4663	0.531172
$962$	28.7007	0.925349
$963$	−14.1200	−0.455010
$964$	26.3283	0.847977
$965$	3.53442	0.113777
$966$	−1.87301	−0.0602632
$967$	−1.81387	−0.0583300	−0.0291650	−	0.999575i	$-0.509285\pi$
$967$	−1.81387	−0.0583300	−0.0291650	+	0.999575i	$0.509285\pi$
$968$	0	0
$969$	−7.80912	−0.250865
$970$	1.56635	0.0502924
$971$	23.0539	0.739835	0.369918	−	0.929065i	$-0.379386\pi$
$971$	23.0539	0.739835	0.369918	+	0.929065i	$0.379386\pi$
$972$	−19.4595	−0.624163
$973$	4.76260	0.152682
$974$	22.6934	0.727144
$975$	19.9776	0.639795
$976$	7.11140	0.227630
$977$	−7.09449	−0.226973	−0.113486	−	0.993540i	$-0.536202\pi$
$977$	−7.09449	−0.226973	−0.113486	+	0.993540i	$0.536202\pi$
$978$	3.57236	0.114232
$979$	0	0
$980$	0.310525	0.00991935
$981$	14.1348	0.451290
$982$	−23.4364	−0.747887
$983$	38.3502	1.22318	0.611591	−	0.791174i	$-0.290530\pi$
$983$	38.3502	1.22318	0.611591	+	0.791174i	$0.290530\pi$
$984$	2.18324	0.0695992
$985$	2.73773	0.0872313
$986$	12.7187	0.405046
$987$	3.73635	0.118929
$988$	−26.5618	−0.845044
$989$	−18.3524	−0.583572
$990$	0	0
$991$	20.2722	0.643967	0.321984	−	0.946745i	$-0.395650\pi$
$991$	20.2722	0.643967	0.321984	+	0.946745i	$0.395650\pi$
$992$	−40.3323	−1.28055
$993$	18.2527	0.579231
$994$	−7.23775	−0.229567
$995$	−3.39592	−0.107658
$996$	9.73200	0.308370
$997$	−15.4107	−0.488062	−0.244031	−	0.969767i	$-0.578470\pi$
$997$	−15.4107	−0.488062	−0.244031	+	0.969767i	$0.578470\pi$
$998$	4.62684	0.146460
$999$	19.6298	0.621058

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.a.k.1.3		4
3.2	odd	2		7623.2.a.co.1.2		4
7.6	odd	2		5929.2.a.bb.1.3		4
11.2	odd	10		847.2.f.p.323.2		8
11.3	even	5		847.2.f.q.372.2		8
11.4	even	5		847.2.f.q.148.2		8
11.5	even	5		847.2.f.s.729.1		8
11.6	odd	10		847.2.f.p.729.2		8
11.7	odd	10		77.2.f.a.71.1	yes	8
11.8	odd	10		77.2.f.a.64.1	✓	8
11.9	even	5		847.2.f.s.323.1		8
11.10	odd	2		847.2.a.l.1.2		4
33.8	even	10		693.2.m.g.64.2		8
33.29	even	10		693.2.m.g.379.2		8
33.32	even	2		7623.2.a.ch.1.3		4
77.18	odd	30		539.2.q.c.324.2		16
77.19	even	30		539.2.q.b.361.2		16
77.30	odd	30		539.2.q.c.361.2		16
77.40	even	30		539.2.q.b.214.1		16
77.41	even	10		539.2.f.d.295.1		8
77.51	odd	30		539.2.q.c.214.1		16
77.52	even	30		539.2.q.b.471.1		16
77.62	even	10		539.2.f.d.148.1		8
77.73	even	30		539.2.q.b.324.2		16
77.74	odd	30		539.2.q.c.471.1		16
77.76	even	2		5929.2.a.bi.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.a.64.1	✓	8	11.8	odd	10
77.2.f.a.71.1	yes	8	11.7	odd	10
539.2.f.d.148.1		8	77.62	even	10
539.2.f.d.295.1		8	77.41	even	10
539.2.q.b.214.1		16	77.40	even	30
539.2.q.b.324.2		16	77.73	even	30
539.2.q.b.361.2		16	77.19	even	30
539.2.q.b.471.1		16	77.52	even	30
539.2.q.c.214.1		16	77.51	odd	30
539.2.q.c.324.2		16	77.18	odd	30
539.2.q.c.361.2		16	77.30	odd	30
539.2.q.c.471.1		16	77.74	odd	30
693.2.m.g.64.2		8	33.8	even	10
693.2.m.g.379.2		8	33.29	even	10
847.2.a.k.1.3		4	1.1	even	1	trivial
847.2.a.l.1.2		4	11.10	odd	2
847.2.f.p.323.2		8	11.2	odd	10
847.2.f.p.729.2		8	11.6	odd	10
847.2.f.q.148.2		8	11.4	even	5
847.2.f.q.372.2		8	11.3	even	5
847.2.f.s.323.1		8	11.9	even	5
847.2.f.s.729.1		8	11.5	even	5
5929.2.a.bb.1.3		4	7.6	odd	2
5929.2.a.bi.1.2		4	77.76	even	2
7623.2.a.ch.1.3		4	33.32	even	2
7623.2.a.co.1.2		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.777484 q^{2} +0.618034 q^{3} -1.39552 q^{4} -0.222516 q^{5} +0.480512 q^{6} +1.00000 q^{7} -2.63996 q^{8} -2.61803 q^{9} +O(q^{10})\)	\(q+0.777484 q^{2} +0.618034 q^{3} -1.39552 q^{4} -0.222516 q^{5} +0.480512 q^{6} +1.00000 q^{7} -2.63996 q^{8} -2.61803 q^{9} -0.173002 q^{10} -0.862478 q^{12} -6.52954 q^{13} +0.777484 q^{14} -0.137522 q^{15} +0.738508 q^{16} +4.33461 q^{17} -2.03548 q^{18} -2.91501 q^{19} +0.310525 q^{20} +0.618034 q^{21} -3.89796 q^{23} -1.63159 q^{24} -4.95049 q^{25} -5.07662 q^{26} -3.47214 q^{27} -1.39552 q^{28} +3.77399 q^{29} -0.106921 q^{30} -6.88958 q^{31} +5.85410 q^{32} +3.37009 q^{34} -0.222516 q^{35} +3.65351 q^{36} -5.65351 q^{37} -2.26637 q^{38} -4.03548 q^{39} +0.587433 q^{40} -1.33811 q^{41} +0.480512 q^{42} +4.70820 q^{43} +0.582554 q^{45} -3.03060 q^{46} +6.04554 q^{47} +0.456423 q^{48} +1.00000 q^{49} -3.84893 q^{50} +2.67894 q^{51} +9.11210 q^{52} -1.71792 q^{53} -2.69953 q^{54} -2.63996 q^{56} -1.80157 q^{57} +2.93422 q^{58} -9.53304 q^{59} +0.191915 q^{60} +9.62943 q^{61} -5.35654 q^{62} -2.61803 q^{63} +3.07446 q^{64} +1.45293 q^{65} +1.27155 q^{67} -6.04903 q^{68} -2.40907 q^{69} -0.173002 q^{70} -9.30919 q^{71} +6.91151 q^{72} -5.58911 q^{73} -4.39552 q^{74} -3.05957 q^{75} +4.06794 q^{76} -3.13752 q^{78} +4.52954 q^{79} -0.164330 q^{80} +5.70820 q^{81} -1.04036 q^{82} -11.2838 q^{83} -0.862478 q^{84} -0.964520 q^{85} +3.66055 q^{86} +2.33245 q^{87} +7.92157 q^{89} +0.452926 q^{90} -6.52954 q^{91} +5.43967 q^{92} -4.25800 q^{93} +4.70031 q^{94} +0.648635 q^{95} +3.61803 q^{96} -9.05391 q^{97} +0.777484 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{5} + q^{6} + 4 q^{7} - 9 q^{8} - 6 q^{9}+O(q^{10}) \) 4 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 6 * q^5 + q^6 + 4 * q^7 - 9 * q^8 - 6 * q^9	\( 4 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{5} + q^{6} + 4 q^{7} - 9 q^{8} - 6 q^{9} + 14 q^{10} - 7 q^{12} - 2 q^{14} + 3 q^{15} - 4 q^{16} + 3 q^{17} + 3 q^{18} - 3 q^{19} - 17 q^{20} - 2 q^{21} - 8 q^{23} + 12 q^{24} - 12 q^{26} + 4 q^{27} + 4 q^{28} - 3 q^{29} - 12 q^{30} - 3 q^{31} + 10 q^{32} - 12 q^{34} - 6 q^{35} - q^{36} - 7 q^{37} - 20 q^{38} - 5 q^{39} + 13 q^{40} - 4 q^{41} + q^{42} - 8 q^{43} + 9 q^{45} + 3 q^{46} - 14 q^{47} - 3 q^{48} + 4 q^{49} - 33 q^{50} + 11 q^{51} + 17 q^{52} - 9 q^{53} - 2 q^{54} - 9 q^{56} - 6 q^{57} + 3 q^{58} - 25 q^{59} + 21 q^{60} + 19 q^{61} - 10 q^{62} - 6 q^{63} + 3 q^{64} - 12 q^{65} - 15 q^{67} + q^{68} + 14 q^{69} + 14 q^{70} - 7 q^{71} + 6 q^{72} + 11 q^{73} - 8 q^{74} - 5 q^{75} + 26 q^{76} - 9 q^{78} - 8 q^{79} - 4 q^{80} - 4 q^{81} + 3 q^{82} + q^{83} - 7 q^{84} - 15 q^{85} + 4 q^{86} - 6 q^{87} - 17 q^{89} - 16 q^{90} - 17 q^{92} - 11 q^{93} + 20 q^{94} - 17 q^{95} + 10 q^{96} - 15 q^{97} - 2 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 6 * q^5 + q^6 + 4 * q^7 - 9 * q^8 - 6 * q^9 + 14 * q^10 - 7 * q^12 - 2 * q^14 + 3 * q^15 - 4 * q^16 + 3 * q^17 + 3 * q^18 - 3 * q^19 - 17 * q^20 - 2 * q^21 - 8 * q^23 + 12 * q^24 - 12 * q^26 + 4 * q^27 + 4 * q^28 - 3 * q^29 - 12 * q^30 - 3 * q^31 + 10 * q^32 - 12 * q^34 - 6 * q^35 - q^36 - 7 * q^37 - 20 * q^38 - 5 * q^39 + 13 * q^40 - 4 * q^41 + q^42 - 8 * q^43 + 9 * q^45 + 3 * q^46 - 14 * q^47 - 3 * q^48 + 4 * q^49 - 33 * q^50 + 11 * q^51 + 17 * q^52 - 9 * q^53 - 2 * q^54 - 9 * q^56 - 6 * q^57 + 3 * q^58 - 25 * q^59 + 21 * q^60 + 19 * q^61 - 10 * q^62 - 6 * q^63 + 3 * q^64 - 12 * q^65 - 15 * q^67 + q^68 + 14 * q^69 + 14 * q^70 - 7 * q^71 + 6 * q^72 + 11 * q^73 - 8 * q^74 - 5 * q^75 + 26 * q^76 - 9 * q^78 - 8 * q^79 - 4 * q^80 - 4 * q^81 + 3 * q^82 + q^83 - 7 * q^84 - 15 * q^85 + 4 * q^86 - 6 * q^87 - 17 * q^89 - 16 * q^90 - 17 * q^92 - 11 * q^93 + 20 * q^94 - 17 * q^95 + 10 * q^96 - 15 * q^97 - 2 * q^98

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