LMFDB - Embedded newform 961.2.a.l.1.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [961,2,Mod(1,961)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("961.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(961, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 961 = 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	961.a (trivial)

Newform invariants

comment:select newform

sage:traces = [16,-8,0,8,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.67362363425$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 24x^{14} + 220x^{12} - 992x^{10} + 2366x^{8} - 2944x^{6} + 1688x^{4} - 288x^{2} + 4 $ x^16 - 24x^14 + 220x^12 - 992x^10 + 2366x^8 - 2944x^6 + 1688x^4 - 288*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-1.47925$ of defining polynomial
Character	$\chi$	$=$	961.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.124175 q^{2} -1.47925 q^{3} -1.98458 q^{4} -1.22944 q^{5} -0.183686 q^{6} +2.66703 q^{7} -0.494784 q^{8} -0.811809 q^{9} -0.152666 q^{10} +5.63192 q^{11} +2.93570 q^{12} +0.0134654 q^{13} +0.331177 q^{14} +1.81866 q^{15} +3.90772 q^{16} -3.54485 q^{17} -0.100806 q^{18} -5.96385 q^{19} +2.43993 q^{20} -3.94521 q^{21} +0.699342 q^{22} +4.08969 q^{23} +0.731911 q^{24} -3.48847 q^{25} +0.00167206 q^{26} +5.63863 q^{27} -5.29293 q^{28} +2.96497 q^{29} +0.225832 q^{30} +1.47481 q^{32} -8.33104 q^{33} -0.440181 q^{34} -3.27896 q^{35} +1.61110 q^{36} -10.6352 q^{37} -0.740559 q^{38} -0.0199187 q^{39} +0.608309 q^{40} -7.14473 q^{41} -0.489895 q^{42} +3.17563 q^{43} -11.1770 q^{44} +0.998075 q^{45} +0.507836 q^{46} -9.81029 q^{47} -5.78051 q^{48} +0.113032 q^{49} -0.433179 q^{50} +5.24373 q^{51} -0.0267232 q^{52} -7.90828 q^{53} +0.700175 q^{54} -6.92413 q^{55} -1.31960 q^{56} +8.82205 q^{57} +0.368174 q^{58} -10.3909 q^{59} -3.60928 q^{60} -3.70951 q^{61} -2.16512 q^{63} -7.63231 q^{64} -0.0165550 q^{65} -1.03450 q^{66} +3.56973 q^{67} +7.03504 q^{68} -6.04969 q^{69} -0.407164 q^{70} +3.76169 q^{71} +0.401670 q^{72} +3.55021 q^{73} -1.32063 q^{74} +5.16032 q^{75} +11.8357 q^{76} +15.0205 q^{77} -0.00247340 q^{78} +3.25987 q^{79} -4.80433 q^{80} -5.90554 q^{81} -0.887194 q^{82} -2.07024 q^{83} +7.82959 q^{84} +4.35820 q^{85} +0.394333 q^{86} -4.38594 q^{87} -2.78658 q^{88} -8.80431 q^{89} +0.123936 q^{90} +0.0359126 q^{91} -8.11632 q^{92} -1.21819 q^{94} +7.33223 q^{95} -2.18161 q^{96} +3.38749 q^{97} +0.0140358 q^{98} -4.57204 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{2} + 8 q^{4} - 16 q^{5} - 16 q^{7} + 8 q^{10} - 8 q^{14} - 8 q^{16} - 24 q^{18} - 32 q^{19} - 24 q^{20} - 8 q^{28} - 8 q^{32} - 32 q^{33} - 16 q^{35} - 40 q^{36} - 24 q^{38} - 32 q^{39} - 32 q^{41}+ \cdots + 16 q^{98}+O(q^{100}) $ 16 * q - 8 * q^2 + 8 * q^4 - 16 * q^5 - 16 * q^7 + 8 * q^10 - 8 * q^14 - 8 * q^16 - 24 * q^18 - 32 * q^19 - 24 * q^20 - 8 * q^28 - 8 * q^32 - 32 * q^33 - 16 * q^35 - 40 * q^36 - 24 * q^38 - 32 * q^39 - 32 * q^41 - 16 * q^45 - 32 * q^47 - 16 * q^49 - 32 * q^50 - 32 * q^51 - 48 * q^56 - 64 * q^59 - 16 * q^63 - 16 * q^64 + 16 * q^67 - 32 * q^69 + 88 * q^70 - 48 * q^71 + 48 * q^72 + 40 * q^76 + 32 * q^78 - 40 * q^80 - 16 * q^81 + 88 * q^82 - 32 * q^87 + 40 * q^90 - 32 * q^94 - 48 * q^95 + 16 * q^98

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 $

$2$	0.124175	0.0878047	0.0439024	−	0.999036i	$-0.486021\pi$
$2$	0.124175	0.0878047	0.0439024	+	0.999036i	$0.486021\pi$
$3$	−1.47925	−0.854047	−0.427024	−	0.904240i	$-0.640438\pi$
$3$	−1.47925	−0.854047	−0.427024	+	0.904240i	$0.640438\pi$
$4$	−1.98458	−0.992290
$5$	−1.22944	−0.549824	−0.274912	−	0.961469i	$-0.588649\pi$
$5$	−1.22944	−0.549824	−0.274912	+	0.961469i	$0.588649\pi$
$6$	−0.183686	−0.0749894
$7$	2.66703	1.00804	0.504021	−	0.863692i	$-0.331854\pi$
$7$	2.66703	1.00804	0.504021	+	0.863692i	$0.331854\pi$
$8$	−0.494784	−0.174933
$9$	−0.811809	−0.270603
$10$	−0.152666	−0.0482772
$11$	5.63192	1.69809	0.849044	−	0.528322i	$-0.177179\pi$
$11$	5.63192	1.69809	0.849044	+	0.528322i	$0.177179\pi$
$12$	2.93570	0.847463
$13$	0.0134654	0.00373463	0.00186731	−	0.999998i	$-0.499406\pi$
$13$	0.0134654	0.00373463	0.00186731	+	0.999998i	$0.499406\pi$
$14$	0.331177	0.0885108
$15$	1.81866	0.469576
$16$	3.90772	0.976930
$17$	−3.54485	−0.859753	−0.429876	−	0.902888i	$-0.641443\pi$
$17$	−3.54485	−0.859753	−0.429876	+	0.902888i	$0.641443\pi$
$18$	−0.100806	−0.0237602
$19$	−5.96385	−1.36820	−0.684101	−	0.729388i	$-0.739805\pi$
$19$	−5.96385	−1.36820	−0.684101	+	0.729388i	$0.739805\pi$
$20$	2.43993	0.545585
$21$	−3.94521	−0.860915
$22$	0.699342	0.149100
$23$	4.08969	0.852759	0.426380	−	0.904544i	$-0.359789\pi$
$23$	4.08969	0.852759	0.426380	+	0.904544i	$0.359789\pi$
$24$	0.731911	0.149401
$25$	−3.48847	−0.697693
$26$	0.00167206	0.000327918 0
$27$	5.63863	1.08516
$28$	−5.29293	−1.00027
$29$	2.96497	0.550581	0.275290	−	0.961361i	$-0.411226\pi$
$29$	2.96497	0.550581	0.275290	+	0.961361i	$0.411226\pi$
$30$	0.225832	0.0412310
$31$	0	0
$31$	0	0
$32$	1.47481	0.260712
$33$	−8.33104	−1.45025
$34$	−0.440181	−0.0754903
$35$	−3.27896	−0.554246
$36$	1.61110	0.268517
$37$	−10.6352	−1.74842	−0.874211	−	0.485546i	$-0.838621\pi$
$37$	−10.6352	−1.74842	−0.874211	+	0.485546i	$0.838621\pi$
$38$	−0.740559	−0.120135
$39$	−0.0199187	−0.00318955
$40$	0.608309	0.0961822
$41$	−7.14473	−1.11582	−0.557910	−	0.829902i	$-0.688396\pi$
$41$	−7.14473	−1.11582	−0.557910	+	0.829902i	$0.688396\pi$
$42$	−0.489895	−0.0755924
$43$	3.17563	0.484279	0.242139	−	0.970241i	$-0.422151\pi$
$43$	3.17563	0.484279	0.242139	+	0.970241i	$0.422151\pi$
$44$	−11.1770	−1.68500
$45$	0.998075	0.148784
$46$	0.507836	0.0748763
$47$	−9.81029	−1.43098	−0.715489	−	0.698624i	$-0.753796\pi$
$47$	−9.81029	−1.43098	−0.715489	+	0.698624i	$0.753796\pi$
$48$	−5.78051	−0.834345
$49$	0.113032	0.0161475
$50$	−0.433179	−0.0612608
$51$	5.24373	0.734269
$52$	−0.0267232	−0.00370584
$53$	−7.90828	−1.08629	−0.543143	−	0.839640i	$-0.682766\pi$
$53$	−7.90828	−1.08629	−0.543143	+	0.839640i	$0.682766\pi$
$54$	0.700175	0.0952818
$55$	−6.92413	−0.933650
$56$	−1.31960	−0.176339
$57$	8.82205	1.16851
$58$	0.368174	0.0483436
$59$	−10.3909	−1.35278	−0.676392	−	0.736542i	$-0.736457\pi$
$59$	−10.3909	−1.35278	−0.676392	+	0.736542i	$0.736457\pi$
$60$	−3.60928	−0.465956
$61$	−3.70951	−0.474954	−0.237477	−	0.971393i	$-0.576320\pi$
$61$	−3.70951	−0.474954	−0.237477	+	0.971393i	$0.576320\pi$
$62$	0	0
$63$	−2.16512	−0.272779
$64$	−7.63231	−0.954039
$65$	−0.0165550	−0.00205339
$66$	−1.03450	−0.127339
$67$	3.56973	0.436112	0.218056	−	0.975936i	$-0.430028\pi$
$67$	3.56973	0.436112	0.218056	+	0.975936i	$0.430028\pi$
$68$	7.03504	0.853124
$69$	−6.04969	−0.728297
$70$	−0.407164	−0.0486654
$71$	3.76169	0.446430	0.223215	−	0.974769i	$-0.428345\pi$
$71$	3.76169	0.446430	0.223215	+	0.974769i	$0.428345\pi$
$72$	0.401670	0.0473373
$73$	3.55021	0.415520	0.207760	−	0.978180i	$-0.433383\pi$
$73$	3.55021	0.415520	0.207760	+	0.978180i	$0.433383\pi$
$74$	−1.32063	−0.153520
$75$	5.16032	0.595863
$76$	11.8357	1.35765
$77$	15.0205	1.71174
$78$	−0.00247340	−0.000280058 0
$79$	3.25987	0.366765	0.183382	−	0.983042i	$-0.441295\pi$
$79$	3.25987	0.366765	0.183382	+	0.983042i	$0.441295\pi$
$80$	−4.80433	−0.537140
$81$	−5.90554	−0.656171
$82$	−0.887194	−0.0979742
$83$	−2.07024	−0.227238	−0.113619	−	0.993524i	$-0.536244\pi$
$83$	−2.07024	−0.227238	−0.113619	+	0.993524i	$0.536244\pi$
$84$	7.82959	0.854278
$85$	4.35820	0.472713
$86$	0.394333	0.0425220
$87$	−4.38594	−0.470222
$88$	−2.78658	−0.297051
$89$	−8.80431	−0.933255	−0.466627	−	0.884454i	$-0.654531\pi$
$89$	−8.80431	−0.933255	−0.466627	+	0.884454i	$0.654531\pi$
$90$	0.123936	0.0130640
$91$	0.0359126	0.00376466
$92$	−8.11632	−0.846185
$93$	0	0
$94$	−1.21819	−0.125647
$95$	7.33223	0.752271
$96$	−2.18161	−0.222660
$97$	3.38749	0.343948	0.171974	−	0.985101i	$-0.444986\pi$
$97$	3.38749	0.343948	0.171974	+	0.985101i	$0.444986\pi$
$98$	0.0140358	0.00141783
$99$	−4.57204	−0.459508
$100$	6.92314	0.692314
$101$	−8.66837	−0.862535	−0.431267	−	0.902224i	$-0.641933\pi$
$101$	−8.66837	−0.862535	−0.431267	+	0.902224i	$0.641933\pi$
$102$	0.651139	0.0644723
$103$	2.68096	0.264163	0.132081	−	0.991239i	$-0.457834\pi$
$103$	2.68096	0.264163	0.132081	+	0.991239i	$0.457834\pi$
$104$	−0.00666246	−0.000653308 0
$105$	4.85042	0.473352
$106$	−0.982008	−0.0953811
$107$	−7.97570	−0.771040	−0.385520	−	0.922700i	$-0.625978\pi$
$107$	−7.97570	−0.771040	−0.385520	+	0.922700i	$0.625978\pi$
$108$	−11.1903	−1.07679
$109$	6.29895	0.603330	0.301665	−	0.953414i	$-0.402458\pi$
$109$	6.29895	0.603330	0.301665	+	0.953414i	$0.402458\pi$
$110$	−0.859802	−0.0819789
$111$	15.7322	1.49324
$112$	10.4220	0.984786
$113$	−2.74728	−0.258442	−0.129221	−	0.991616i	$-0.541248\pi$
$113$	−2.74728	−0.258442	−0.129221	+	0.991616i	$0.541248\pi$
$114$	1.09547	0.102601
$115$	−5.02805	−0.468868
$116$	−5.88422	−0.546336
$117$	−0.0109313	−0.00101060
$118$	−1.29029	−0.118781
$119$	−9.45421	−0.866666
$120$	−0.899844	−0.0821441
$121$	20.7185	1.88350
$122$	−0.460627	−0.0417032
$123$	10.5689	0.952962
$124$	0	0
$125$	10.4361	0.933433
$126$	−0.268853	−0.0239513
$127$	−1.73534	−0.153986	−0.0769931	−	0.997032i	$-0.524532\pi$
$127$	−1.73534	−0.153986	−0.0769931	+	0.997032i	$0.524532\pi$
$128$	−3.89735	−0.344481
$129$	−4.69756	−0.413597
$130$	−0.00205571	−0.000180297 0
$131$	7.06939	0.617656	0.308828	−	0.951118i	$-0.400063\pi$
$131$	7.06939	0.617656	0.308828	+	0.951118i	$0.400063\pi$
$132$	16.5336	1.43907
$133$	−15.9058	−1.37920
$134$	0.443270	0.0382927
$135$	−6.93239	−0.596645
$136$	1.75394	0.150399
$137$	9.53710	0.814809	0.407405	−	0.913248i	$-0.366434\pi$
$137$	9.53710	0.814809	0.407405	+	0.913248i	$0.366434\pi$
$138$	−0.751218	−0.0639479
$139$	−13.3484	−1.13220	−0.566098	−	0.824338i	$-0.691547\pi$
$139$	−13.3484	−1.13220	−0.566098	+	0.824338i	$0.691547\pi$
$140$	6.50737	0.549973
$141$	14.5119	1.22212
$142$	0.467106	0.0391987
$143$	0.0758360	0.00634173
$144$	−3.17232	−0.264360
$145$	−3.64526	−0.302723
$146$	0.440846	0.0364846
$147$	−0.167204	−0.0137907
$148$	21.1065	1.73494
$149$	18.1985	1.49088	0.745440	−	0.666573i	$-0.232239\pi$
$149$	18.1985	1.49088	0.745440	+	0.666573i	$0.232239\pi$
$150$	0.640781	0.0523196
$151$	6.59561	0.536743	0.268371	−	0.963316i	$-0.413515\pi$
$151$	6.59561	0.536743	0.268371	+	0.963316i	$0.413515\pi$
$152$	2.95082	0.239343
$153$	2.87774	0.232652
$154$	1.86516	0.150299
$155$	0	0
$156$	0.0395303	0.00316496
$157$	15.7251	1.25500	0.627502	−	0.778615i	$-0.284078\pi$
$157$	15.7251	1.25500	0.627502	+	0.778615i	$0.284078\pi$
$158$	0.404794	0.0322037
$159$	11.6984	0.927740
$160$	−1.81319	−0.143346
$161$	10.9073	0.859617
$162$	−0.733318	−0.0576149
$163$	−7.19636	−0.563663	−0.281831	−	0.959464i	$-0.590942\pi$
$163$	−7.19636	−0.563663	−0.281831	+	0.959464i	$0.590942\pi$
$164$	14.1793	1.10722
$165$	10.2425	0.797381
$166$	−0.257071	−0.0199526
$167$	−20.4703	−1.58404	−0.792018	−	0.610498i	$-0.790969\pi$
$167$	−20.4703	−1.58404	−0.792018	+	0.610498i	$0.790969\pi$
$168$	1.95203	0.150602
$169$	−12.9998	−0.999986
$170$	0.541178	0.0415064
$171$	4.84151	0.370240
$172$	−6.30229	−0.480545
$173$	−21.0674	−1.60173	−0.800863	−	0.598848i	$-0.795625\pi$
$173$	−21.0674	−1.60173	−0.800863	+	0.598848i	$0.795625\pi$
$174$	−0.544622	−0.0412877
$175$	−9.30383	−0.703304
$176$	22.0080	1.65891
$177$	15.3708	1.15534
$178$	−1.09327	−0.0819442
$179$	−10.4789	−0.783231	−0.391615	−	0.920129i	$-0.628084\pi$
$179$	−10.4789	−0.783231	−0.391615	+	0.920129i	$0.628084\pi$
$180$	−1.98076	−0.147637
$181$	−9.53710	−0.708887	−0.354443	−	0.935077i	$-0.615330\pi$
$181$	−9.53710	−0.708887	−0.354443	+	0.935077i	$0.615330\pi$
$182$	0.00445943	0.000330555 0
$183$	5.48730	0.405633
$184$	−2.02351	−0.149175
$185$	13.0754	0.961325
$186$	0	0
$187$	−19.9643	−1.45994
$188$	19.4693	1.41994
$189$	15.0384	1.09388
$190$	0.910477	0.0660529
$191$	−10.0046	−0.723905	−0.361952	−	0.932197i	$-0.617890\pi$
$191$	−10.0046	−0.723905	−0.361952	+	0.932197i	$0.617890\pi$
$192$	11.2901	0.814794
$193$	−20.6806	−1.48862	−0.744310	−	0.667835i	$-0.767221\pi$
$193$	−20.6806	−1.48862	−0.744310	+	0.667835i	$0.767221\pi$
$194$	0.420641	0.0302003
$195$	0.0244890	0.00175369
$196$	−0.224322	−0.0160230
$197$	−0.753285	−0.0536693	−0.0268347	−	0.999640i	$-0.508543\pi$
$197$	−0.753285	−0.0536693	−0.0268347	+	0.999640i	$0.508543\pi$
$198$	−0.567732	−0.0403470
$199$	9.89063	0.701128	0.350564	−	0.936539i	$-0.385990\pi$
$199$	9.89063	0.701128	0.350564	+	0.936539i	$0.385990\pi$
$200$	1.72604	0.122049
$201$	−5.28053	−0.372460
$202$	−1.07639	−0.0757346
$203$	7.90765	0.555008
$204$	−10.4066	−0.728609
$205$	8.78405	0.613505
$206$	0.332907	0.0231947
$207$	−3.32005	−0.230759
$208$	0.0526190	0.00364847
$209$	−33.5879	−2.32333
$210$	0.602299	0.0415626
$211$	14.0629	0.968128	0.484064	−	0.875033i	$-0.339160\pi$
$211$	14.0629	0.968128	0.484064	+	0.875033i	$0.339160\pi$
$212$	15.6946	1.07791
$213$	−5.56449	−0.381273
$214$	−0.990379	−0.0677009
$215$	−3.90426	−0.266268
$216$	−2.78990	−0.189829
$217$	0	0
$218$	0.782170	0.0529753
$219$	−5.25165	−0.354874
$220$	13.7415	0.926452
$221$	−0.0477328	−0.00321086
$222$	1.95354	0.131113
$223$	9.89557	0.662656	0.331328	−	0.943516i	$-0.392503\pi$
$223$	9.89557	0.662656	0.331328	+	0.943516i	$0.392503\pi$
$224$	3.93335	0.262808
$225$	2.83197	0.188798
$226$	−0.341142	−0.0226924
$227$	9.40457	0.624204	0.312102	−	0.950049i	$-0.398967\pi$
$227$	9.40457	0.624204	0.312102	+	0.950049i	$0.398967\pi$
$228$	−17.5081	−1.15950
$229$	−14.7546	−0.975011	−0.487506	−	0.873120i	$-0.662093\pi$
$229$	−14.7546	−0.975011	−0.487506	+	0.873120i	$0.662093\pi$
$230$	−0.624356	−0.0411688
$231$	−22.2191	−1.46191
$232$	−1.46702	−0.0963145
$233$	−28.5860	−1.87273	−0.936365	−	0.351028i	$-0.885832\pi$
$233$	−28.5860	−1.87273	−0.936365	+	0.351028i	$0.885832\pi$
$234$	−0.00135739	−8.87356e−5 0
$235$	12.0612	0.786786
$236$	20.6216	1.34235
$237$	−4.82218	−0.313234
$238$	−1.17397	−0.0760974
$239$	−9.00876	−0.582729	−0.291364	−	0.956612i	$-0.594109\pi$
$239$	−9.00876	−0.582729	−0.291364	+	0.956612i	$0.594109\pi$
$240$	7.10682	0.458743
$241$	0.788163	0.0507700	0.0253850	−	0.999678i	$-0.491919\pi$
$241$	0.788163	0.0507700	0.0253850	+	0.999678i	$0.491919\pi$
$242$	2.57271	0.165380
$243$	−8.18011	−0.524754
$244$	7.36182	0.471292
$245$	−0.138967	−0.00887829
$246$	1.31238	0.0836746
$247$	−0.0803056	−0.00510972
$248$	0	0
$249$	3.06241	0.194072
$250$	1.29590	0.0819598
$251$	−19.7718	−1.24798	−0.623992	−	0.781430i	$-0.714490\pi$
$251$	−19.7718	−1.24798	−0.623992	+	0.781430i	$0.714490\pi$
$252$	4.29685	0.270676
$253$	23.0328	1.44806
$254$	−0.215485	−0.0135207
$255$	−6.44688	−0.403719
$256$	14.7807	0.923792
$257$	−9.02462	−0.562940	−0.281470	−	0.959570i	$-0.590822\pi$
$257$	−9.02462	−0.562940	−0.281470	+	0.959570i	$0.590822\pi$
$258$	−0.583318	−0.0363158
$259$	−28.3645	−1.76248
$260$	0.0328546	0.00203756
$261$	−2.40699	−0.148989
$262$	0.877840	0.0542331
$263$	−3.12800	−0.192881	−0.0964403	−	0.995339i	$-0.530746\pi$
$263$	−3.12800	−0.192881	−0.0964403	+	0.995339i	$0.530746\pi$
$264$	4.12206	0.253695
$265$	9.72280	0.597267
$266$	−1.97509	−0.121101
$267$	13.0238	0.797044
$268$	−7.08441	−0.432749
$269$	13.9776	0.852231	0.426115	−	0.904669i	$-0.359882\pi$
$269$	13.9776	0.852231	0.426115	+	0.904669i	$0.359882\pi$
$270$	−0.860827	−0.0523882
$271$	14.0587	0.854008	0.427004	−	0.904250i	$-0.359569\pi$
$271$	14.0587	0.854008	0.427004	+	0.904250i	$0.359569\pi$
$272$	−13.8523	−0.839918
$273$	−0.0531238	−0.00321520
$274$	1.18427	0.0715441
$275$	−19.6468	−1.18474
$276$	12.0061	0.722682
$277$	22.0218	1.32316	0.661581	−	0.749874i	$-0.269886\pi$
$277$	22.0218	1.32316	0.661581	+	0.749874i	$0.269886\pi$
$278$	−1.65753	−0.0994121
$279$	0	0
$280$	1.62238	0.0969556
$281$	2.32625	0.138772	0.0693862	−	0.997590i	$-0.477896\pi$
$281$	2.32625	0.138772	0.0693862	+	0.997590i	$0.477896\pi$
$282$	1.80201	0.107308
$283$	13.9295	0.828021	0.414010	−	0.910272i	$-0.364128\pi$
$283$	13.9295	0.828021	0.414010	+	0.910272i	$0.364128\pi$
$284$	−7.46537	−0.442988
$285$	−10.8462	−0.642475
$286$	0.00941691	0.000556834 0
$287$	−19.0552	−1.12479
$288$	−1.19726	−0.0705494
$289$	−4.43403	−0.260825
$290$	−0.452649	−0.0265805
$291$	−5.01096	−0.293748
$292$	−7.04567	−0.412317
$293$	25.8604	1.51078	0.755390	−	0.655276i	$-0.227448\pi$
$293$	25.8604	1.51078	0.755390	+	0.655276i	$0.227448\pi$
$294$	−0.0207625	−0.00121089
$295$	12.7751	0.743794
$296$	5.26214	0.305856
$297$	31.7563	1.84269
$298$	2.25979	0.130906
$299$	0.0550693	0.00318474
$300$	−10.2411	−0.591269
$301$	8.46949	0.488173
$302$	0.819007	0.0471286
$303$	12.8227	0.736645
$304$	−23.3051	−1.33664
$305$	4.56064	0.261141
$306$	0.357343	0.0204279
$307$	−0.655663	−0.0374206	−0.0187103	−	0.999825i	$-0.505956\pi$
$307$	−0.655663	−0.0374206	−0.0187103	+	0.999825i	$0.505956\pi$
$308$	−29.8094	−1.69855
$309$	−3.96582	−0.225608
$310$	0	0
$311$	−22.0394	−1.24974	−0.624869	−	0.780730i	$-0.714848\pi$
$311$	−22.0394	−1.24974	−0.624869	+	0.780730i	$0.714848\pi$
$312$	0.00985547	0.000557956 0
$313$	9.51336	0.537727	0.268863	−	0.963178i	$-0.413352\pi$
$313$	9.51336	0.537727	0.268863	+	0.963178i	$0.413352\pi$
$314$	1.95266	0.110195
$315$	2.66189	0.149981
$316$	−6.46948	−0.363937
$317$	−2.09503	−0.117669	−0.0588343	−	0.998268i	$-0.518738\pi$
$317$	−2.09503	−0.117669	−0.0588343	+	0.998268i	$0.518738\pi$
$318$	1.45264	0.0814599
$319$	16.6985	0.934934
$320$	9.38350	0.524554
$321$	11.7981	0.658505
$322$	1.35441	0.0754784
$323$	21.1410	1.17631
$324$	11.7200	0.651112
$325$	−0.0469736	−0.00260562
$326$	−0.893606	−0.0494923
$327$	−9.31775	−0.515273
$328$	3.53510	0.195193
$329$	−26.1643	−1.44248
$330$	1.27186	0.0700139
$331$	2.50739	0.137819	0.0689093	−	0.997623i	$-0.478048\pi$
$331$	2.50739	0.137819	0.0689093	+	0.997623i	$0.478048\pi$
$332$	4.10855	0.225486
$333$	8.63378	0.473128
$334$	−2.54189	−0.139086
$335$	−4.38878	−0.239785
$336$	−15.4168	−0.841054
$337$	−27.2137	−1.48242	−0.741212	−	0.671271i	$-0.765749\pi$
$337$	−27.2137	−1.48242	−0.741212	+	0.671271i	$0.765749\pi$
$338$	−1.61425	−0.0878035
$339$	4.06392	0.220722
$340$	−8.64920	−0.469069
$341$	0	0
$342$	0.601193	0.0325088
$343$	−18.3677	−0.991764
$344$	−1.57125	−0.0847161
$345$	7.43776	0.400435
$346$	−2.61604	−0.140639
$347$	24.3284	1.30602	0.653010	−	0.757350i	$-0.273506\pi$
$347$	24.3284	1.30602	0.653010	+	0.757350i	$0.273506\pi$
$348$	8.70425	0.466597
$349$	12.1656	0.651211	0.325605	−	0.945506i	$-0.394432\pi$
$349$	12.1656	0.651211	0.325605	+	0.945506i	$0.394432\pi$
$350$	−1.15530	−0.0617534
$351$	0.0759264	0.00405265
$352$	8.30600	0.442711
$353$	28.0154	1.49111	0.745556	−	0.666443i	$-0.232184\pi$
$353$	28.0154	1.49111	0.745556	+	0.666443i	$0.232184\pi$
$354$	1.90867	0.101444
$355$	−4.62479	−0.245458
$356$	17.4729	0.926060
$357$	13.9852	0.740174
$358$	−1.30122	−0.0687714
$359$	−24.3366	−1.28444	−0.642219	−	0.766521i	$-0.721986\pi$
$359$	−24.3366	−1.28444	−0.642219	+	0.766521i	$0.721986\pi$
$360$	−0.493831	−0.0260272
$361$	16.5675	0.871975
$362$	−1.18427	−0.0622436
$363$	−30.6479	−1.60860
$364$	−0.0712714	−0.00373564
$365$	−4.36478	−0.228463
$366$	0.681384	0.0356165
$367$	−1.68920	−0.0881753	−0.0440876	−	0.999028i	$-0.514038\pi$
$367$	−1.68920	−0.0881753	−0.0440876	+	0.999028i	$0.514038\pi$
$368$	15.9814	0.833087
$369$	5.80016	0.301944
$370$	1.62364	0.0844089
$371$	−21.0916	−1.09502
$372$	0	0
$373$	24.2842	1.25739	0.628695	−	0.777652i	$-0.283589\pi$
$373$	24.2842	1.25739	0.628695	+	0.777652i	$0.283589\pi$
$374$	−2.47906	−0.128189
$375$	−15.4376	−0.797196
$376$	4.85397	0.250324
$377$	0.0399245	0.00205621
$378$	1.86739	0.0960480
$379$	13.3445	0.685462	0.342731	−	0.939433i	$-0.388648\pi$
$379$	13.3445	0.685462	0.342731	+	0.939433i	$0.388648\pi$
$380$	−14.5514	−0.746471
$381$	2.56700	0.131512
$382$	−1.24231	−0.0635622
$383$	25.0445	1.27971	0.639857	−	0.768494i	$-0.278994\pi$
$383$	25.0445	1.27971	0.639857	+	0.768494i	$0.278994\pi$
$384$	5.76518	0.294203
$385$	−18.4669	−0.941158
$386$	−2.56800	−0.130708
$387$	−2.57800	−0.131047
$388$	−6.72276	−0.341296
$389$	21.9366	1.11223	0.556115	−	0.831105i	$-0.312291\pi$
$389$	21.9366	1.11223	0.556115	+	0.831105i	$0.312291\pi$
$390$	0.00304091	0.000153982 0
$391$	−14.4973	−0.733162
$392$	−0.0559267	−0.00282472
$393$	−10.4574	−0.527507
$394$	−0.0935389	−0.00471242
$395$	−4.00784	−0.201656
$396$	9.07359	0.455965
$397$	2.21630	0.111233	0.0556165	−	0.998452i	$-0.482288\pi$
$397$	2.21630	0.111233	0.0556165	+	0.998452i	$0.482288\pi$
$398$	1.22817	0.0615624
$399$	23.5286	1.17791
$400$	−13.6320	−0.681598
$401$	−26.6696	−1.33181	−0.665907	−	0.746035i	$-0.731955\pi$
$401$	−26.6696	−1.33181	−0.665907	+	0.746035i	$0.731955\pi$
$402$	−0.655708	−0.0327038
$403$	0	0
$404$	17.2031	0.855885
$405$	7.26053	0.360779
$406$	0.981930	0.0487323
$407$	−59.8968	−2.96897
$408$	−2.59451	−0.128448
$409$	−37.8777	−1.87293	−0.936465	−	0.350760i	$-0.885923\pi$
$409$	−37.8777	−1.87293	−0.936465	+	0.350760i	$0.885923\pi$
$410$	1.09076	0.0538686
$411$	−14.1078	−0.695886
$412$	−5.32058	−0.262126
$413$	−27.7129	−1.36366
$414$	−0.412266	−0.0202618
$415$	2.54524	0.124941
$416$	0.0198589	0.000973661 0
$417$	19.7456	0.966948
$418$	−4.17077	−0.203999
$419$	−9.16702	−0.447838	−0.223919	−	0.974608i	$-0.571885\pi$
$419$	−9.16702	−0.447838	−0.223919	+	0.974608i	$0.571885\pi$
$420$	−9.62604	−0.469703
$421$	−3.26808	−0.159277	−0.0796383	−	0.996824i	$-0.525377\pi$
$421$	−3.26808	−0.159277	−0.0796383	+	0.996824i	$0.525377\pi$
$422$	1.74625	0.0850062
$423$	7.96408	0.387227
$424$	3.91289	0.190027
$425$	12.3661	0.599843
$426$	−0.690969	−0.0334775
$427$	−9.89336	−0.478773
$428$	15.8284	0.765095
$429$	−0.112181	−0.00541613
$430$	−0.484810	−0.0233796
$431$	−23.4706	−1.13054	−0.565269	−	0.824907i	$-0.691228\pi$
$431$	−23.4706	−1.13054	−0.565269	+	0.824907i	$0.691228\pi$
$432$	22.0342	1.06012
$433$	27.3421	1.31398	0.656990	−	0.753900i	$-0.271829\pi$
$433$	27.3421	1.31398	0.656990	+	0.753900i	$0.271829\pi$
$434$	0	0
$435$	5.39227	0.258540
$436$	−12.5008	−0.598679
$437$	−24.3903	−1.16675
$438$	−0.652122	−0.0311596
$439$	−33.0439	−1.57710	−0.788551	−	0.614970i	$-0.789168\pi$
$439$	−33.0439	−1.57710	−0.788551	+	0.614970i	$0.789168\pi$
$440$	3.42595	0.163326
$441$	−0.0917608	−0.00436956
$442$	−0.00592720	−0.000281928 0
$443$	−13.3265	−0.633162	−0.316581	−	0.948566i	$-0.602535\pi$
$443$	−13.3265	−0.633162	−0.316581	+	0.948566i	$0.602535\pi$
$444$	−31.2218	−1.48172
$445$	10.8244	0.513126
$446$	1.22878	0.0581844
$447$	−26.9202	−1.27328
$448$	−20.3556	−0.961711
$449$	11.2943	0.533012	0.266506	−	0.963833i	$-0.414131\pi$
$449$	11.2943	0.533012	0.266506	+	0.963833i	$0.414131\pi$
$450$	0.351659	0.0165773
$451$	−40.2385	−1.89476
$452$	5.45219	0.256449
$453$	−9.75658	−0.458404
$454$	1.16781	0.0548080
$455$	−0.0441525	−0.00206990
$456$	−4.36501	−0.204410
$457$	4.55068	0.212872	0.106436	−	0.994320i	$-0.466056\pi$
$457$	4.55068	0.212872	0.106436	+	0.994320i	$0.466056\pi$
$458$	−1.83215	−0.0856106
$459$	−19.9881	−0.932965
$460$	9.97857	0.465253
$461$	28.7961	1.34117	0.670585	−	0.741833i	$-0.266043\pi$
$461$	28.7961	1.34117	0.670585	+	0.741833i	$0.266043\pi$
$462$	−2.75905	−0.128363
$463$	12.0515	0.560082	0.280041	−	0.959988i	$-0.409652\pi$
$463$	12.0515	0.560082	0.280041	+	0.959988i	$0.409652\pi$
$464$	11.5863	0.537879
$465$	0	0
$466$	−3.54965	−0.164435
$467$	−19.1300	−0.885230	−0.442615	−	0.896712i	$-0.645949\pi$
$467$	−19.1300	−0.885230	−0.442615	+	0.896712i	$0.645949\pi$
$468$	0.0216941	0.00100281
$469$	9.52056	0.439619
$470$	1.49770	0.0690836
$471$	−23.2615	−1.07183
$472$	5.14126	0.236646
$473$	17.8849	0.822348
$474$	−0.598793	−0.0275035
$475$	20.8047	0.954585
$476$	18.7626	0.859985
$477$	6.42002	0.293952
$478$	−1.11866	−0.0511663
$479$	−12.2911	−0.561593	−0.280797	−	0.959767i	$-0.590599\pi$
$479$	−12.2911	−0.561593	−0.280797	+	0.959767i	$0.590599\pi$
$480$	2.68217	0.122424
$481$	−0.143208	−0.00652971
$482$	0.0978698	0.00445785
$483$	−16.1347	−0.734154
$484$	−41.1176	−1.86898
$485$	−4.16474	−0.189111
$486$	−1.01576	−0.0460759
$487$	15.1267	0.685456	0.342728	−	0.939435i	$-0.388649\pi$
$487$	15.1267	0.685456	0.342728	+	0.939435i	$0.388649\pi$
$488$	1.83540	0.0830849
$489$	10.6452	0.481395
$490$	−0.0172562	−0.000779556 0
$491$	30.5340	1.37798	0.688990	−	0.724771i	$-0.258054\pi$
$491$	30.5340	1.37798	0.688990	+	0.724771i	$0.258054\pi$
$492$	−20.9748	−0.945615
$493$	−10.5104	−0.473363
$494$	−0.00997192	−0.000448658 0
$495$	5.62108	0.252649
$496$	0	0
$497$	10.0325	0.450020
$498$	0.380273	0.0170404
$499$	28.8272	1.29048	0.645240	−	0.763980i	$-0.276757\pi$
$499$	28.8272	1.29048	0.645240	+	0.763980i	$0.276757\pi$
$500$	−20.7113	−0.926237
$501$	30.2807	1.35284
$502$	−2.45516	−0.109579
$503$	−3.11786	−0.139018	−0.0695092	−	0.997581i	$-0.522143\pi$
$503$	−3.11786	−0.139018	−0.0695092	+	0.997581i	$0.522143\pi$
$504$	1.07126	0.0477179
$505$	10.6573	0.474243
$506$	2.86009	0.127147
$507$	19.2300	0.854035
$508$	3.44392	0.152799
$509$	−13.2957	−0.589323	−0.294661	−	0.955602i	$-0.595207\pi$
$509$	−13.2957	−0.589323	−0.294661	+	0.955602i	$0.595207\pi$
$510$	−0.800539	−0.0354485
$511$	9.46849	0.418862
$512$	9.63009	0.425594
$513$	−33.6280	−1.48471
$514$	−1.12063	−0.0494288
$515$	−3.29609	−0.145243
$516$	9.32269	0.410408
$517$	−55.2507	−2.42992
$518$	−3.52215	−0.154754
$519$	31.1640	1.36795
$520$	0.00819113	0.000359205 0
$521$	8.44298	0.369894	0.184947	−	0.982749i	$-0.440789\pi$
$521$	8.44298	0.369894	0.184947	+	0.982749i	$0.440789\pi$
$522$	−0.298887	−0.0130819
$523$	39.6642	1.73440	0.867198	−	0.497964i	$-0.165919\pi$
$523$	39.6642	1.73440	0.867198	+	0.497964i	$0.165919\pi$
$524$	−14.0298	−0.612894
$525$	13.7627	0.600655
$526$	−0.388418	−0.0169358
$527$	0	0
$528$	−32.5554	−1.41679
$529$	−6.27443	−0.272801
$530$	1.20732	0.0524428
$531$	8.43545	0.366067
$532$	31.5662	1.36857
$533$	−0.0962066	−0.00416717
$534$	1.61723	0.0699842
$535$	9.80568	0.423937
$536$	−1.76624	−0.0762901
$537$	15.5010	0.668916
$538$	1.73567	0.0748299
$539$	0.636590	0.0274199
$540$	13.7579	0.592045
$541$	21.8292	0.938509	0.469255	−	0.883063i	$-0.344523\pi$
$541$	21.8292	0.938509	0.469255	+	0.883063i	$0.344523\pi$
$542$	1.74574	0.0749859
$543$	14.1078	0.605423
$544$	−5.22797	−0.224148
$545$	−7.74422	−0.331726
$546$	−0.00659663	−0.000282310 0
$547$	27.3401	1.16898	0.584490	−	0.811401i	$-0.301295\pi$
$547$	27.3401	1.16898	0.584490	+	0.811401i	$0.301295\pi$
$548$	−18.9271	−0.808527
$549$	3.01141	0.128524
$550$	−2.43963	−0.104026
$551$	−17.6826	−0.753305
$552$	2.99329	0.127403
$553$	8.69417	0.369714
$554$	2.73455	0.116180
$555$	−19.3419	−0.821017
$556$	26.4909	1.12347
$557$	−27.9171	−1.18289	−0.591443	−	0.806347i	$-0.701441\pi$
$557$	−27.9171	−1.18289	−0.591443	+	0.806347i	$0.701441\pi$
$558$	0	0
$559$	0.0427611	0.00180860
$560$	−12.8133	−0.541460
$561$	29.5323	1.24685
$562$	0.288861	0.0121849
$563$	−25.7662	−1.08591	−0.542957	−	0.839760i	$-0.682695\pi$
$563$	−25.7662	−1.08591	−0.542957	+	0.839760i	$0.682695\pi$
$564$	−28.8000	−1.21270
$565$	3.37762	0.142098
$566$	1.72969	0.0727042
$567$	−15.7502	−0.661447
$568$	−1.86122	−0.0780952
$569$	16.2134	0.679701	0.339851	−	0.940479i	$-0.389623\pi$
$569$	16.2134	0.679701	0.339851	+	0.940479i	$0.389623\pi$
$570$	−1.34683	−0.0564123
$571$	−9.37945	−0.392518	−0.196259	−	0.980552i	$-0.562879\pi$
$571$	−9.37945	−0.392518	−0.196259	+	0.980552i	$0.562879\pi$
$572$	−0.150503	−0.00629283
$573$	14.7993	0.618249
$574$	−2.36617	−0.0987620
$575$	−14.2667	−0.594964
$576$	6.19598	0.258166
$577$	31.6764	1.31870	0.659352	−	0.751834i	$-0.270831\pi$
$577$	31.6764	1.31870	0.659352	+	0.751834i	$0.270831\pi$
$578$	−0.550594	−0.0229017
$579$	30.5918	1.27135
$580$	7.23432	0.300389
$581$	−5.52138	−0.229065
$582$	−0.622234	−0.0257924
$583$	−44.5388	−1.84461
$584$	−1.75658	−0.0726880
$585$	0.0134395	0.000555654 0
$586$	3.21120	0.132654
$587$	−0.186802	−0.00771014	−0.00385507	−	0.999993i	$-0.501227\pi$
$587$	−0.186802	−0.00771014	−0.00385507	+	0.999993i	$0.501227\pi$
$588$	0.331829	0.0136844
$589$	0	0
$590$	1.58634	0.0653086
$591$	1.11430	0.0458362
$592$	−41.5596	−1.70809
$593$	−13.4668	−0.553015	−0.276508	−	0.961012i	$-0.589177\pi$
$593$	−13.4668	−0.553015	−0.276508	+	0.961012i	$0.589177\pi$
$594$	3.94333	0.161797
$595$	11.6234	0.476514
$596$	−36.1164	−1.47939
$597$	−14.6307	−0.598797
$598$	0.00683821	0.000279635 0
$599$	−18.4914	−0.755540	−0.377770	−	0.925899i	$-0.623309\pi$
$599$	−18.4914	−0.755540	−0.377770	+	0.925899i	$0.623309\pi$
$600$	−2.55325	−0.104236
$601$	17.8606	0.728548	0.364274	−	0.931292i	$-0.381317\pi$
$601$	17.8606	0.728548	0.364274	+	0.931292i	$0.381317\pi$
$602$	1.05170	0.0428639
$603$	−2.89794	−0.118013
$604$	−13.0895	−0.532605
$605$	−25.4723	−1.03560
$606$	1.59226	0.0646809
$607$	28.9604	1.17547	0.587734	−	0.809054i	$-0.300020\pi$
$607$	28.9604	1.17547	0.587734	+	0.809054i	$0.300020\pi$
$608$	−8.79553	−0.356706
$609$	−11.6974	−0.474003
$610$	0.566315	0.0229294
$611$	−0.132099	−0.00534417
$612$	−5.71111	−0.230858
$613$	20.6444	0.833821	0.416910	−	0.908948i	$-0.363113\pi$
$613$	20.6444	0.833821	0.416910	+	0.908948i	$0.363113\pi$
$614$	−0.0814167	−0.00328571
$615$	−12.9938	−0.523962
$616$	−7.43189	−0.299439
$617$	0.697562	0.0280828	0.0140414	−	0.999901i	$-0.495530\pi$
$617$	0.697562	0.0280828	0.0140414	+	0.999901i	$0.495530\pi$
$618$	−0.492454	−0.0198094
$619$	2.41091	0.0969026	0.0484513	−	0.998826i	$-0.484571\pi$
$619$	2.41091	0.0969026	0.0484513	+	0.998826i	$0.484571\pi$
$620$	0	0
$621$	23.0603	0.925376
$622$	−2.73673	−0.109733
$623$	−23.4813	−0.940760
$624$	−0.0778369	−0.00311597
$625$	4.61172	0.184469
$626$	1.18132	0.0472150
$627$	49.6851	1.98423
$628$	−31.2078	−1.24533
$629$	37.7003	1.50321
$630$	0.330539	0.0131690
$631$	−36.4136	−1.44960	−0.724801	−	0.688959i	$-0.758068\pi$
$631$	−36.4136	−1.44960	−0.724801	+	0.688959i	$0.758068\pi$
$632$	−1.61293	−0.0641590
$633$	−20.8025	−0.826827
$634$	−0.260150	−0.0103319
$635$	2.13350	0.0846654
$636$	−23.2163	−0.920587
$637$	0.00152203	6.03049e−5 0
$638$	2.07353	0.0820917
$639$	−3.05377	−0.120805
$640$	4.79158	0.189404
$641$	−2.14971	−0.0849085	−0.0424542	−	0.999098i	$-0.513518\pi$
$641$	−2.14971	−0.0849085	−0.0424542	+	0.999098i	$0.513518\pi$
$642$	1.46502	0.0578198
$643$	17.7453	0.699806	0.349903	−	0.936786i	$-0.386215\pi$
$643$	17.7453	0.699806	0.349903	+	0.936786i	$0.386215\pi$
$644$	−21.6464	−0.852989
$645$	5.77539	0.227406
$646$	2.62517	0.103286
$647$	−40.1703	−1.57926	−0.789629	−	0.613584i	$-0.789727\pi$
$647$	−40.1703	−1.57926	−0.789629	+	0.613584i	$0.789727\pi$
$648$	2.92197	0.114786
$649$	−58.5209	−2.29715
$650$	−0.00583292	−0.000228786 0
$651$	0	0
$652$	14.2818	0.559317
$653$	−2.41626	−0.0945556	−0.0472778	−	0.998882i	$-0.515055\pi$
$653$	−2.41626	−0.0945556	−0.0472778	+	0.998882i	$0.515055\pi$
$654$	−1.15703	−0.0452434
$655$	−8.69143	−0.339602
$656$	−27.9196	−1.09008
$657$	−2.88209	−0.112441
$658$	−3.24894	−0.126657
$659$	26.3680	1.02715	0.513576	−	0.858044i	$-0.328321\pi$
$659$	26.3680	1.02715	0.513576	+	0.858044i	$0.328321\pi$
$660$	−20.3272	−0.791234
$661$	−37.8137	−1.47078	−0.735392	−	0.677642i	$-0.763002\pi$
$661$	−37.8137	−1.47078	−0.735392	+	0.677642i	$0.763002\pi$
$662$	0.311354	0.0121011
$663$	0.0706089	0.00274222
$664$	1.02432	0.0397513
$665$	19.5552	0.758320
$666$	1.07210	0.0415429
$667$	12.1258	0.469513
$668$	40.6249	1.57182
$669$	−14.6381	−0.565940
$670$	−0.544976	−0.0210542
$671$	−20.8916	−0.806513
$672$	−5.81842	−0.224451
$673$	−25.8793	−0.997575	−0.498788	−	0.866724i	$-0.666221\pi$
$673$	−25.8793	−0.997575	−0.498788	+	0.866724i	$0.666221\pi$
$674$	−3.37925	−0.130164
$675$	−19.6702	−0.757105
$676$	25.7992	0.992276
$677$	19.9464	0.766602	0.383301	−	0.923623i	$-0.374787\pi$
$677$	19.9464	0.766602	0.383301	+	0.923623i	$0.374787\pi$
$678$	0.504635	0.0193804
$679$	9.03454	0.346714
$680$	−2.15637	−0.0826929
$681$	−13.9117	−0.533099
$682$	0	0
$683$	−10.3960	−0.397793	−0.198897	−	0.980020i	$-0.563736\pi$
$683$	−10.3960	−0.397793	−0.198897	+	0.980020i	$0.563736\pi$
$684$	−9.60837	−0.367385
$685$	−11.7253	−0.448002
$686$	−2.28081	−0.0870816
$687$	21.8258	0.832706
$688$	12.4095	0.473107
$689$	−0.106488	−0.00405687
$690$	0.923581	0.0351601
$691$	−13.4503	−0.511675	−0.255837	−	0.966720i	$-0.582351\pi$
$691$	−13.4503	−0.511675	−0.255837	+	0.966720i	$0.582351\pi$
$692$	41.8100	1.58938
$693$	−12.1938	−0.463203
$694$	3.02098	0.114675
$695$	16.4111	0.622509
$696$	2.17009	0.0822571
$697$	25.3270	0.959328
$698$	1.51066	0.0571794
$699$	42.2859	1.59940
$700$	18.4642	0.697881
$701$	−1.52309	−0.0575264	−0.0287632	−	0.999586i	$-0.509157\pi$
$701$	−1.52309	−0.0575264	−0.0287632	+	0.999586i	$0.509157\pi$
$702$	0.00942813	0.000355842 0
$703$	63.4270	2.39219
$704$	−42.9846	−1.62004
$705$	−17.8416	−0.671953
$706$	3.47881	0.130927
$707$	−23.1188	−0.869471
$708$	−30.5046	−1.14643
$709$	−1.54507	−0.0580263	−0.0290131	−	0.999579i	$-0.509236\pi$
$709$	−1.54507	−0.0580263	−0.0290131	+	0.999579i	$0.509236\pi$
$710$	−0.574281	−0.0215524
$711$	−2.64640	−0.0992476
$712$	4.35623	0.163257
$713$	0	0
$714$	1.73660	0.0649908
$715$	−0.0932362	−0.00348684
$716$	20.7962	0.777192
$717$	13.3262	0.497678
$718$	−3.02199	−0.112780
$719$	36.4715	1.36016	0.680079	−	0.733139i	$-0.261946\pi$
$719$	36.4715	1.36016	0.680079	+	0.733139i	$0.261946\pi$
$720$	3.90020	0.145352
$721$	7.15019	0.266287
$722$	2.05727	0.0765635
$723$	−1.16589	−0.0433600
$724$	18.9271	0.703422
$725$	−10.3432	−0.384136
$726$	−3.80570	−0.141243
$727$	11.3252	0.420027	0.210013	−	0.977698i	$-0.432649\pi$
$727$	11.3252	0.420027	0.210013	+	0.977698i	$0.432649\pi$
$728$	−0.0177690	−0.000658561 0
$729$	29.8171	1.10434
$730$	−0.541995	−0.0200601
$731$	−11.2571	−0.416360
$732$	−10.8900	−0.402506
$733$	1.82359	0.0673559	0.0336779	−	0.999433i	$-0.489278\pi$
$733$	1.82359	0.0673559	0.0336779	+	0.999433i	$0.489278\pi$
$734$	−0.209755	−0.00774221
$735$	0.205568	0.00758248
$736$	6.03151	0.222324
$737$	20.1044	0.740556
$738$	0.720232	0.0265121
$739$	−35.2840	−1.29794	−0.648971	−	0.760813i	$-0.724800\pi$
$739$	−35.2840	−1.29794	−0.648971	+	0.760813i	$0.724800\pi$
$740$	−25.9493	−0.953914
$741$	0.118792	0.00436395
$742$	−2.61904	−0.0961481
$743$	−26.9859	−0.990018	−0.495009	−	0.868888i	$-0.664835\pi$
$743$	−26.9859	−0.990018	−0.495009	+	0.868888i	$0.664835\pi$
$744$	0	0
$745$	−22.3741	−0.819722
$746$	3.01549	0.110405
$747$	1.68064	0.0614913
$748$	39.6208	1.44868
$749$	−21.2714	−0.777240
$750$	−1.91696	−0.0699976
$751$	−21.1606	−0.772161	−0.386080	−	0.922465i	$-0.626171\pi$
$751$	−21.1606	−0.772161	−0.386080	+	0.922465i	$0.626171\pi$
$752$	−38.3359	−1.39797
$753$	29.2475	1.06584
$754$	0.00495761	0.000180545 0
$755$	−8.10894	−0.295114
$756$	−29.8449	−1.08545
$757$	53.6180	1.94878	0.974389	−	0.224869i	$-0.0721956\pi$
$757$	53.6180	1.94878	0.974389	+	0.224869i	$0.0721956\pi$
$758$	1.65705	0.0601868
$759$	−34.0714	−1.23671
$760$	−3.62787	−0.131597
$761$	−8.24779	−0.298982	−0.149491	−	0.988763i	$-0.547764\pi$
$761$	−8.24779	−0.298982	−0.149491	+	0.988763i	$0.547764\pi$
$762$	0.318757	0.0115473
$763$	16.7995	0.608182
$764$	19.8549	0.718323
$765$	−3.53803	−0.127918
$766$	3.10989	0.112365
$767$	−0.139918	−0.00505214
$768$	−21.8644	−0.788962
$769$	26.3326	0.949577	0.474788	−	0.880100i	$-0.342525\pi$
$769$	26.3326	0.949577	0.474788	+	0.880100i	$0.342525\pi$
$770$	−2.29311	−0.0826381
$771$	13.3497	0.480778
$772$	41.0422	1.47714
$773$	−11.8440	−0.426000	−0.213000	−	0.977052i	$-0.568323\pi$
$773$	−11.8440	−0.426000	−0.213000	+	0.977052i	$0.568323\pi$
$774$	−0.320123	−0.0115066
$775$	0	0
$776$	−1.67608	−0.0601677
$777$	41.9582	1.50524
$778$	2.72397	0.0976591
$779$	42.6101	1.52667
$780$	−0.0486004	−0.00174017
$781$	21.1855	0.758078
$782$	−1.80020	−0.0643751
$783$	16.7184	0.597466
$784$	0.441700	0.0157750
$785$	−19.3332	−0.690032
$786$	−1.29855	−0.0463176
$787$	38.8768	1.38581	0.692904	−	0.721030i	$-0.256331\pi$
$787$	38.8768	1.38581	0.692904	+	0.721030i	$0.256331\pi$
$788$	1.49496	0.0532556
$789$	4.62710	0.164729
$790$	−0.497672	−0.0177064
$791$	−7.32706	−0.260520
$792$	2.26217	0.0803828
$793$	−0.0499500	−0.00177378
$794$	0.275208	0.00976678
$795$	−14.3825	−0.510094
$796$	−19.6288	−0.695723
$797$	52.0663	1.84428	0.922141	−	0.386854i	$-0.126438\pi$
$797$	52.0663	1.84428	0.922141	+	0.386854i	$0.126438\pi$
$798$	2.92166	0.103426
$799$	34.7760	1.23029
$800$	−5.14482	−0.181897
$801$	7.14742	0.252542
$802$	−3.31168	−0.116940
$803$	19.9945	0.705590
$804$	10.4796	0.369589
$805$	−13.4099	−0.472638
$806$	0	0
$807$	−20.6764	−0.727845
$808$	4.28897	0.150885
$809$	31.2082	1.09722	0.548612	−	0.836077i	$-0.315157\pi$
$809$	31.2082	1.09722	0.548612	+	0.836077i	$0.315157\pi$
$810$	0.901574	0.0316781
$811$	16.4241	0.576729	0.288364	−	0.957521i	$-0.406889\pi$
$811$	16.4241	0.576729	0.288364	+	0.957521i	$0.406889\pi$
$812$	−15.6934	−0.550729
$813$	−20.7964	−0.729363
$814$	−7.43766	−0.260690
$815$	8.84753	0.309916
$816$	20.4910	0.717330
$817$	−18.9390	−0.662591
$818$	−4.70345	−0.164452
$819$	−0.0291542	−0.00101873
$820$	−17.4327	−0.608775
$821$	−40.2675	−1.40534	−0.702672	−	0.711514i	$-0.748010\pi$
$821$	−40.2675	−1.40534	−0.702672	+	0.711514i	$0.748010\pi$
$822$	−1.75183	−0.0611020
$823$	23.3005	0.812205	0.406102	−	0.913828i	$-0.366888\pi$
$823$	23.3005	0.812205	0.406102	+	0.913828i	$0.366888\pi$
$824$	−1.32650	−0.0462107
$825$	29.0625	1.01183
$826$	−3.44124	−0.119736
$827$	24.1368	0.839319	0.419660	−	0.907682i	$-0.362149\pi$
$827$	24.1368	0.839319	0.419660	+	0.907682i	$0.362149\pi$
$828$	6.58890	0.228980
$829$	−22.7194	−0.789076	−0.394538	−	0.918880i	$-0.629095\pi$
$829$	−22.7194	−0.789076	−0.394538	+	0.918880i	$0.629095\pi$
$830$	0.316055	0.0109704
$831$	−32.5758	−1.13004
$832$	−0.102772	−0.00356298
$833$	−0.400683	−0.0138829
$834$	2.45191	0.0849026
$835$	25.1670	0.870941
$836$	66.6580	2.30541
$837$	0	0
$838$	−1.13831	−0.0393223
$839$	−6.01460	−0.207647	−0.103823	−	0.994596i	$-0.533108\pi$
$839$	−6.01460	−0.207647	−0.103823	+	0.994596i	$0.533108\pi$
$840$	−2.39991	−0.0828047
$841$	−20.2090	−0.696861
$842$	−0.405813	−0.0139852
$843$	−3.44111	−0.118518
$844$	−27.9089	−0.960664
$845$	15.9826	0.549817
$846$	0.988937	0.0340003
$847$	55.2568	1.89865
$848$	−30.9034	−1.06123
$849$	−20.6052	−0.707169
$850$	1.53555	0.0526691
$851$	−43.4948	−1.49098
$852$	11.0432	0.378333
$853$	10.8138	0.370258	0.185129	−	0.982714i	$-0.440730\pi$
$853$	10.8138	0.370258	0.185129	+	0.982714i	$0.440730\pi$
$854$	−1.22850	−0.0420385
$855$	−5.95237	−0.203567
$856$	3.94625	0.134880
$857$	−34.0840	−1.16429	−0.582144	−	0.813086i	$-0.697786\pi$
$857$	−34.0840	−1.16429	−0.582144	+	0.813086i	$0.697786\pi$
$858$	−0.0139300	−0.000475562 0
$859$	−7.51793	−0.256508	−0.128254	−	0.991741i	$-0.540937\pi$
$859$	−7.51793	−0.256508	−0.128254	+	0.991741i	$0.540937\pi$
$860$	7.74832	0.264215
$861$	28.1874	0.960626
$862$	−2.91445	−0.0992666
$863$	−22.2748	−0.758243	−0.379121	−	0.925347i	$-0.623774\pi$
$863$	−22.2748	−0.758243	−0.379121	+	0.925347i	$0.623774\pi$
$864$	8.31590	0.282913
$865$	25.9012	0.880668
$866$	3.39520	0.115374
$867$	6.55906	0.222757
$868$	0	0
$869$	18.3594	0.622798
$870$	0.669583	0.0227010
$871$	0.0480678	0.00162872
$872$	−3.11662	−0.105542
$873$	−2.75000	−0.0930734
$874$	−3.02866	−0.102446
$875$	27.8334	0.940939
$876$	10.4223	0.352138
$877$	−27.1655	−0.917313	−0.458656	−	0.888614i	$-0.651669\pi$
$877$	−27.1655	−0.917313	−0.458656	+	0.888614i	$0.651669\pi$
$878$	−4.10322	−0.138477
$879$	−38.2541	−1.29028
$880$	−27.0576	−0.912111
$881$	−22.7696	−0.767126	−0.383563	−	0.923515i	$-0.625303\pi$
$881$	−22.7696	−0.767126	−0.383563	+	0.923515i	$0.625303\pi$
$882$	−0.0113944	−0.000383668 0
$883$	−20.5333	−0.691000	−0.345500	−	0.938419i	$-0.612291\pi$
$883$	−20.5333	−0.691000	−0.345500	+	0.938419i	$0.612291\pi$
$884$	0.0947296	0.00318610
$885$	−18.8976	−0.635235
$886$	−1.65481	−0.0555946
$887$	53.7079	1.80333	0.901667	−	0.432431i	$-0.142344\pi$
$887$	53.7079	1.80333	0.901667	+	0.432431i	$0.142344\pi$
$888$	−7.78405	−0.261215
$889$	−4.62819	−0.155224
$890$	1.34412	0.0450549
$891$	−33.2595	−1.11424
$892$	−19.6386	−0.657547
$893$	58.5071	1.95787
$894$	−3.34281	−0.111800
$895$	12.8832	0.430639
$896$	−10.3943	−0.347251
$897$	−0.0814614	−0.00271992
$898$	1.40247	0.0468010
$899$	0	0
$900$	−5.62027	−0.187342
$901$	28.0337	0.933937
$902$	−4.99661	−0.166369
$903$	−12.5285	−0.416923
$904$	1.35931	0.0452099
$905$	11.7253	0.389763
$906$	−1.21152	−0.0402500
$907$	53.3488	1.77142	0.885709	−	0.464241i	$-0.153673\pi$
$907$	53.3488	1.77142	0.885709	+	0.464241i	$0.153673\pi$
$908$	−18.6641	−0.619391
$909$	7.03706	0.233405
$910$	−0.00548262	−0.000181747 0
$911$	−44.5464	−1.47589	−0.737944	−	0.674861i	$-0.764203\pi$
$911$	−44.5464	−1.47589	−0.737944	+	0.674861i	$0.764203\pi$
$912$	34.4741	1.14155
$913$	−11.6594	−0.385870
$914$	0.565079	0.0186912
$915$	−6.74634	−0.223027
$916$	29.2817	0.967494
$917$	18.8543	0.622623
$918$	−2.48202	−0.0819187
$919$	−11.6589	−0.384593	−0.192297	−	0.981337i	$-0.561594\pi$
$919$	−11.6589	−0.384593	−0.192297	+	0.981337i	$0.561594\pi$
$920$	2.48780	0.0820203
$921$	0.969891	0.0319590
$922$	3.57575	0.117761
$923$	0.0506526	0.00166725
$924$	44.0956	1.45064
$925$	37.1007	1.21986
$926$	1.49649	0.0491779
$927$	−2.17643	−0.0714833
$928$	4.37276	0.143543
$929$	16.1137	0.528675	0.264337	−	0.964430i	$-0.414847\pi$
$929$	16.1137	0.528675	0.264337	+	0.964430i	$0.414847\pi$
$930$	0	0
$931$	−0.674109	−0.0220930
$932$	56.7312	1.85829
$933$	32.6018	1.06734
$934$	−2.37546	−0.0777274
$935$	24.5450	0.802708
$936$	0.00540865	0.000176787 0
$937$	45.2884	1.47951	0.739754	−	0.672877i	$-0.234942\pi$
$937$	45.2884	1.47951	0.739754	+	0.672877i	$0.234942\pi$
$938$	1.18221	0.0386006
$939$	−14.0727	−0.459244
$940$	−23.9364	−0.780720
$941$	−24.5088	−0.798964	−0.399482	−	0.916741i	$-0.630810\pi$
$941$	−24.5088	−0.798964	−0.399482	+	0.916741i	$0.630810\pi$
$942$	−2.88849	−0.0941120
$943$	−29.2197	−0.951525
$944$	−40.6049	−1.32158
$945$	−18.4889	−0.601443
$946$	2.22085	0.0722060
$947$	−44.7537	−1.45430	−0.727150	−	0.686478i	$-0.759156\pi$
$947$	−44.7537	−1.45430	−0.727150	+	0.686478i	$0.759156\pi$
$948$	9.57001	0.310819
$949$	0.0478049	0.00155181
$950$	2.58341	0.0838170
$951$	3.09908	0.100495
$952$	4.67779	0.151608
$953$	−30.8112	−0.998071	−0.499036	−	0.866581i	$-0.666312\pi$
$953$	−30.8112	−0.998071	−0.499036	+	0.866581i	$0.666312\pi$
$954$	0.797203	0.0258104
$955$	12.3001	0.398020
$956$	17.8786	0.578236
$957$	−24.7013	−0.798478
$958$	−1.52624	−0.0493105
$959$	25.4357	0.821361
$960$	−13.8806	−0.447994
$961$	0	0
$962$	−0.0177828	−0.000573339 0
$963$	6.47474	0.208646
$964$	−1.56417	−0.0503786
$965$	25.4256	0.818479
$966$	−2.00352	−0.0644622
$967$	44.5447	1.43246	0.716230	−	0.697864i	$-0.245866\pi$
$967$	44.5447	1.43246	0.716230	+	0.697864i	$0.245866\pi$
$968$	−10.2512	−0.329486
$969$	−31.2728	−1.00463
$970$	−0.517155	−0.0166048
$971$	6.35732	0.204016	0.102008	−	0.994784i	$-0.467473\pi$
$971$	6.35732	0.204016	0.102008	+	0.994784i	$0.467473\pi$
$972$	16.2341	0.520708
$973$	−35.6005	−1.14130
$974$	1.87835	0.0601863
$975$	0.0694858	0.00222533
$976$	−14.4957	−0.463997
$977$	−13.4329	−0.429758	−0.214879	−	0.976641i	$-0.568936\pi$
$977$	−13.4329	−0.429758	−0.214879	+	0.976641i	$0.568936\pi$
$978$	1.32187	0.0422687
$979$	−49.5852	−1.58475
$980$	0.275792	0.00880984
$981$	−5.11355	−0.163263
$982$	3.79155	0.120993
$983$	42.1777	1.34526	0.672629	−	0.739980i	$-0.265165\pi$
$983$	42.1777	1.34526	0.672629	+	0.739980i	$0.265165\pi$
$984$	−5.22930	−0.166704
$985$	0.926123	0.0295087
$986$	−1.30512	−0.0415635
$987$	38.7036	1.23195
$988$	0.159373	0.00507033
$989$	12.9873	0.412973
$990$	0.697995	0.0221837
$991$	40.1209	1.27448	0.637241	−	0.770665i	$-0.280076\pi$
$991$	40.1209	1.27448	0.637241	+	0.770665i	$0.280076\pi$
$992$	0	0
$993$	−3.70907	−0.117704
$994$	1.24579	0.0395139
$995$	−12.1600	−0.385497
$996$	−6.07759	−0.192576
$997$	−40.6053	−1.28598	−0.642991	−	0.765874i	$-0.722307\pi$
$997$	−40.6053	−1.28598	−0.642991	+	0.765874i	$0.722307\pi$
$998$	3.57960	0.113310
$999$	−59.9682	−1.89731

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	961.2.a.l.1.11	✓	16
3.2	odd	2		8649.2.a.bs.1.5		16
31.2	even	5		961.2.d.s.531.11		64
31.3	odd	30		961.2.g.w.846.11		128
31.4	even	5		961.2.d.s.388.6		64
31.5	even	3		961.2.c.l.521.12		32
31.6	odd	6		961.2.c.l.439.11		32
31.7	even	15		961.2.g.w.235.5		128
31.8	even	5		961.2.d.s.374.6		64
31.9	even	15		961.2.g.w.732.5		128
31.10	even	15		961.2.g.w.844.12		128
31.11	odd	30		961.2.g.w.338.5		128
31.12	odd	30		961.2.g.w.547.12		128
31.13	odd	30		961.2.g.w.448.12		128
31.14	even	15		961.2.g.w.816.6		128
31.15	odd	10		961.2.d.s.628.12		64
31.16	even	5		961.2.d.s.628.11		64
31.17	odd	30		961.2.g.w.816.5		128
31.18	even	15		961.2.g.w.448.11		128
31.19	even	15		961.2.g.w.547.11		128
31.20	even	15		961.2.g.w.338.6		128
31.21	odd	30		961.2.g.w.844.11		128
31.22	odd	30		961.2.g.w.732.6		128
31.23	odd	10		961.2.d.s.374.5		64
31.24	odd	30		961.2.g.w.235.6		128
31.25	even	3		961.2.c.l.439.12		32
31.26	odd	6		961.2.c.l.521.11		32
31.27	odd	10		961.2.d.s.388.5		64
31.28	even	15		961.2.g.w.846.12		128
31.29	odd	10		961.2.d.s.531.12		64
31.30	odd	2	inner	961.2.a.l.1.12	yes	16
93.92	even	2		8649.2.a.bs.1.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
961.2.a.l.1.11	✓	16	1.1	even	1	trivial
961.2.a.l.1.12	yes	16	31.30	odd	2	inner
961.2.c.l.439.11		32	31.6	odd	6
961.2.c.l.439.12		32	31.25	even	3
961.2.c.l.521.11		32	31.26	odd	6
961.2.c.l.521.12		32	31.5	even	3
961.2.d.s.374.5		64	31.23	odd	10
961.2.d.s.374.6		64	31.8	even	5
961.2.d.s.388.5		64	31.27	odd	10
961.2.d.s.388.6		64	31.4	even	5
961.2.d.s.531.11		64	31.2	even	5
961.2.d.s.531.12		64	31.29	odd	10
961.2.d.s.628.11		64	31.16	even	5
961.2.d.s.628.12		64	31.15	odd	10
961.2.g.w.235.5		128	31.7	even	15
961.2.g.w.235.6		128	31.24	odd	30
961.2.g.w.338.5		128	31.11	odd	30
961.2.g.w.338.6		128	31.20	even	15
961.2.g.w.448.11		128	31.18	even	15
961.2.g.w.448.12		128	31.13	odd	30
961.2.g.w.547.11		128	31.19	even	15
961.2.g.w.547.12		128	31.12	odd	30
961.2.g.w.732.5		128	31.9	even	15
961.2.g.w.732.6		128	31.22	odd	30
961.2.g.w.816.5		128	31.17	odd	30
961.2.g.w.816.6		128	31.14	even	15
961.2.g.w.844.11		128	31.21	odd	30
961.2.g.w.844.12		128	31.10	even	15
961.2.g.w.846.11		128	31.3	odd	30
961.2.g.w.846.12		128	31.28	even	15
8649.2.a.bs.1.5		16	3.2	odd	2
8649.2.a.bs.1.6		16	93.92	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 961 = 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	961.a (trivial)

\(f(q)\)	\(=\)	\(q+0.124175 q^{2} -1.47925 q^{3} -1.98458 q^{4} -1.22944 q^{5} -0.183686 q^{6} +2.66703 q^{7} -0.494784 q^{8} -0.811809 q^{9} -0.152666 q^{10} +5.63192 q^{11} +2.93570 q^{12} +0.0134654 q^{13} +0.331177 q^{14} +1.81866 q^{15} +3.90772 q^{16} -3.54485 q^{17} -0.100806 q^{18} -5.96385 q^{19} +2.43993 q^{20} -3.94521 q^{21} +0.699342 q^{22} +4.08969 q^{23} +0.731911 q^{24} -3.48847 q^{25} +0.00167206 q^{26} +5.63863 q^{27} -5.29293 q^{28} +2.96497 q^{29} +0.225832 q^{30} +1.47481 q^{32} -8.33104 q^{33} -0.440181 q^{34} -3.27896 q^{35} +1.61110 q^{36} -10.6352 q^{37} -0.740559 q^{38} -0.0199187 q^{39} +0.608309 q^{40} -7.14473 q^{41} -0.489895 q^{42} +3.17563 q^{43} -11.1770 q^{44} +0.998075 q^{45} +0.507836 q^{46} -9.81029 q^{47} -5.78051 q^{48} +0.113032 q^{49} -0.433179 q^{50} +5.24373 q^{51} -0.0267232 q^{52} -7.90828 q^{53} +0.700175 q^{54} -6.92413 q^{55} -1.31960 q^{56} +8.82205 q^{57} +0.368174 q^{58} -10.3909 q^{59} -3.60928 q^{60} -3.70951 q^{61} -2.16512 q^{63} -7.63231 q^{64} -0.0165550 q^{65} -1.03450 q^{66} +3.56973 q^{67} +7.03504 q^{68} -6.04969 q^{69} -0.407164 q^{70} +3.76169 q^{71} +0.401670 q^{72} +3.55021 q^{73} -1.32063 q^{74} +5.16032 q^{75} +11.8357 q^{76} +15.0205 q^{77} -0.00247340 q^{78} +3.25987 q^{79} -4.80433 q^{80} -5.90554 q^{81} -0.887194 q^{82} -2.07024 q^{83} +7.82959 q^{84} +4.35820 q^{85} +0.394333 q^{86} -4.38594 q^{87} -2.78658 q^{88} -8.80431 q^{89} +0.123936 q^{90} +0.0359126 q^{91} -8.11632 q^{92} -1.21819 q^{94} +7.33223 q^{95} -2.18161 q^{96} +3.38749 q^{97} +0.0140358 q^{98} -4.57204 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{2} + 8 q^{4} - 16 q^{5} - 16 q^{7} + 8 q^{10} - 8 q^{14} - 8 q^{16} - 24 q^{18} - 32 q^{19} - 24 q^{20} - 8 q^{28} - 8 q^{32} - 32 q^{33} - 16 q^{35} - 40 q^{36} - 24 q^{38} - 32 q^{39} - 32 q^{41}+ \cdots + 16 q^{98}+O(q^{100}) \) 16 * q - 8 * q^2 + 8 * q^4 - 16 * q^5 - 16 * q^7 + 8 * q^10 - 8 * q^14 - 8 * q^16 - 24 * q^18 - 32 * q^19 - 24 * q^20 - 8 * q^28 - 8 * q^32 - 32 * q^33 - 16 * q^35 - 40 * q^36 - 24 * q^38 - 32 * q^39 - 32 * q^41 - 16 * q^45 - 32 * q^47 - 16 * q^49 - 32 * q^50 - 32 * q^51 - 48 * q^56 - 64 * q^59 - 16 * q^63 - 16 * q^64 + 16 * q^67 - 32 * q^69 + 88 * q^70 - 48 * q^71 + 48 * q^72 + 40 * q^76 + 32 * q^78 - 40 * q^80 - 16 * q^81 + 88 * q^82 - 32 * q^87 + 40 * q^90 - 32 * q^94 - 48 * q^95 + 16 * q^98

Introduction

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