LMFDB - Embedded newform 4009.2.a.d.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, 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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$75$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.58872 q^{2} +0.285133 q^{3} +4.70149 q^{4} -3.09212 q^{5} -0.738130 q^{6} +5.20222 q^{7} -6.99340 q^{8} -2.91870 q^{9} +O(q^{10})$	\(q-2.58872 q^{2} +0.285133 q^{3} +4.70149 q^{4} -3.09212 q^{5} -0.738130 q^{6} +5.20222 q^{7} -6.99340 q^{8} -2.91870 q^{9} +8.00463 q^{10} +2.28232 q^{11} +1.34055 q^{12} +1.50725 q^{13} -13.4671 q^{14} -0.881664 q^{15} +8.70100 q^{16} -1.74516 q^{17} +7.55570 q^{18} -1.00000 q^{19} -14.5375 q^{20} +1.48332 q^{21} -5.90829 q^{22} -1.32846 q^{23} -1.99405 q^{24} +4.56118 q^{25} -3.90185 q^{26} -1.68762 q^{27} +24.4581 q^{28} +4.11892 q^{29} +2.28238 q^{30} -1.59007 q^{31} -8.53767 q^{32} +0.650765 q^{33} +4.51774 q^{34} -16.0859 q^{35} -13.7222 q^{36} +4.74428 q^{37} +2.58872 q^{38} +0.429767 q^{39} +21.6244 q^{40} -7.87681 q^{41} -3.83991 q^{42} -6.35192 q^{43} +10.7303 q^{44} +9.02496 q^{45} +3.43902 q^{46} -10.9350 q^{47} +2.48094 q^{48} +20.0630 q^{49} -11.8076 q^{50} -0.497603 q^{51} +7.08631 q^{52} -1.36775 q^{53} +4.36877 q^{54} -7.05720 q^{55} -36.3812 q^{56} -0.285133 q^{57} -10.6628 q^{58} +2.27962 q^{59} -4.14513 q^{60} -11.5708 q^{61} +4.11626 q^{62} -15.1837 q^{63} +4.69968 q^{64} -4.66059 q^{65} -1.68465 q^{66} -1.39455 q^{67} -8.20486 q^{68} -0.378788 q^{69} +41.6418 q^{70} +3.32433 q^{71} +20.4116 q^{72} +11.0187 q^{73} -12.2816 q^{74} +1.30054 q^{75} -4.70149 q^{76} +11.8731 q^{77} -1.11255 q^{78} -9.38562 q^{79} -26.9045 q^{80} +8.27490 q^{81} +20.3909 q^{82} -6.42097 q^{83} +6.97382 q^{84} +5.39624 q^{85} +16.4434 q^{86} +1.17444 q^{87} -15.9612 q^{88} +5.31513 q^{89} -23.3631 q^{90} +7.84104 q^{91} -6.24574 q^{92} -0.453382 q^{93} +28.3076 q^{94} +3.09212 q^{95} -2.43437 q^{96} -10.5397 q^{97} -51.9377 q^{98} -6.66140 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9}+O(q^{10}) $ 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9} - 48 q^{11} - 14 q^{12} - 3 q^{13} - 4 q^{14} - 39 q^{15} + 59 q^{16} - 23 q^{17} - 24 q^{18} - 75 q^{19} - 62 q^{20} - 3 q^{21} - 6 q^{22} - 73 q^{23} - 64 q^{24} + 57 q^{25} - 46 q^{26} - 22 q^{27} - 26 q^{28} - 39 q^{29} - 14 q^{30} - 44 q^{31} - 71 q^{32} - 3 q^{33} - 9 q^{34} - 49 q^{35} + 20 q^{36} - 12 q^{37} + 11 q^{38} - 90 q^{39} - 8 q^{40} - 42 q^{41} - 45 q^{42} - 24 q^{43} - 120 q^{44} - 63 q^{45} - 39 q^{46} - 59 q^{47} - 4 q^{48} + 48 q^{49} - 100 q^{50} - 55 q^{51} + 2 q^{52} + 13 q^{53} - 87 q^{54} - 36 q^{55} - 12 q^{56} + 4 q^{57} - 17 q^{58} - 47 q^{59} - 45 q^{60} - 35 q^{61} - 40 q^{62} - 69 q^{63} + 26 q^{64} - 44 q^{65} + 33 q^{66} - 39 q^{67} - 63 q^{68} + 42 q^{69} + 40 q^{70} - 154 q^{71} - 51 q^{72} - 29 q^{73} - 95 q^{74} + 37 q^{75} - 67 q^{76} - 24 q^{77} - 19 q^{78} - 95 q^{79} - 146 q^{80} + 23 q^{81} + 7 q^{82} - 52 q^{83} - 72 q^{84} - 36 q^{85} - 44 q^{86} - 103 q^{87} + 67 q^{88} + q^{89} - 2 q^{90} - 64 q^{91} - 183 q^{92} - 49 q^{93} + 5 q^{94} + 18 q^{95} - 69 q^{96} - 7 q^{97} - 23 q^{98} - 100 q^{99}+O(q^{100}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9 - 48 * q^11 - 14 * q^12 - 3 * q^13 - 4 * q^14 - 39 * q^15 + 59 * q^16 - 23 * q^17 - 24 * q^18 - 75 * q^19 - 62 * q^20 - 3 * q^21 - 6 * q^22 - 73 * q^23 - 64 * q^24 + 57 * q^25 - 46 * q^26 - 22 * q^27 - 26 * q^28 - 39 * q^29 - 14 * q^30 - 44 * q^31 - 71 * q^32 - 3 * q^33 - 9 * q^34 - 49 * q^35 + 20 * q^36 - 12 * q^37 + 11 * q^38 - 90 * q^39 - 8 * q^40 - 42 * q^41 - 45 * q^42 - 24 * q^43 - 120 * q^44 - 63 * q^45 - 39 * q^46 - 59 * q^47 - 4 * q^48 + 48 * q^49 - 100 * q^50 - 55 * q^51 + 2 * q^52 + 13 * q^53 - 87 * q^54 - 36 * q^55 - 12 * q^56 + 4 * q^57 - 17 * q^58 - 47 * q^59 - 45 * q^60 - 35 * q^61 - 40 * q^62 - 69 * q^63 + 26 * q^64 - 44 * q^65 + 33 * q^66 - 39 * q^67 - 63 * q^68 + 42 * q^69 + 40 * q^70 - 154 * q^71 - 51 * q^72 - 29 * q^73 - 95 * q^74 + 37 * q^75 - 67 * q^76 - 24 * q^77 - 19 * q^78 - 95 * q^79 - 146 * q^80 + 23 * q^81 + 7 * q^82 - 52 * q^83 - 72 * q^84 - 36 * q^85 - 44 * q^86 - 103 * q^87 + 67 * q^88 + q^89 - 2 * q^90 - 64 * q^91 - 183 * q^92 - 49 * q^93 + 5 * q^94 + 18 * q^95 - 69 * q^96 - 7 * q^97 - 23 * q^98 - 100 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.58872	−1.83050	−0.915252	−	0.402882i	$-0.868008\pi$
$2$	−2.58872	−1.83050	−0.915252	+	0.402882i	$0.868008\pi$
$3$	0.285133	0.164622	0.0823108	−	0.996607i	$-0.473770\pi$
$3$	0.285133	0.164622	0.0823108	+	0.996607i	$0.473770\pi$
$4$	4.70149	2.35074
$5$	−3.09212	−1.38284	−0.691418	−	0.722455i	$-0.743014\pi$
$5$	−3.09212	−1.38284	−0.691418	+	0.722455i	$0.743014\pi$
$6$	−0.738130	−0.301340
$7$	5.20222	1.96625	0.983126	−	0.182928i	$-0.0585574\pi$
$7$	5.20222	1.96625	0.983126	+	0.182928i	$0.0585574\pi$
$8$	−6.99340	−2.47254
$9$	−2.91870	−0.972900
$10$	8.00463	2.53129
$11$	2.28232	0.688145	0.344073	−	0.938943i	$-0.388193\pi$
$11$	2.28232	0.688145	0.344073	+	0.938943i	$0.388193\pi$
$12$	1.34055	0.386983
$13$	1.50725	0.418036	0.209018	−	0.977912i	$-0.432973\pi$
$13$	1.50725	0.418036	0.209018	+	0.977912i	$0.432973\pi$
$14$	−13.4671	−3.59923
$15$	−0.881664	−0.227645
$16$	8.70100	2.17525
$17$	−1.74516	−0.423264	−0.211632	−	0.977349i	$-0.567878\pi$
$17$	−1.74516	−0.423264	−0.211632	+	0.977349i	$0.567878\pi$
$18$	7.55570	1.78090
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−14.5375	−3.25069
$21$	1.48332	0.323688
$22$	−5.90829	−1.25965
$23$	−1.32846	−0.277003	−0.138502	−	0.990362i	$-0.544229\pi$
$23$	−1.32846	−0.277003	−0.138502	+	0.990362i	$0.544229\pi$
$24$	−1.99405	−0.407033
$25$	4.56118	0.912236
$26$	−3.90185	−0.765216
$27$	−1.68762	−0.324782
$28$	24.4581	4.62215
$29$	4.11892	0.764865	0.382433	−	0.923983i	$-0.375086\pi$
$29$	4.11892	0.764865	0.382433	+	0.923983i	$0.375086\pi$
$30$	2.28238	0.416704
$31$	−1.59007	−0.285586	−0.142793	−	0.989753i	$-0.545608\pi$
$31$	−1.59007	−0.285586	−0.142793	+	0.989753i	$0.545608\pi$
$32$	−8.53767	−1.50926
$33$	0.650765	0.113284
$34$	4.51774	0.774786
$35$	−16.0859	−2.71901
$36$	−13.7222	−2.28704
$37$	4.74428	0.779956	0.389978	−	0.920824i	$-0.372483\pi$
$37$	4.74428	0.779956	0.389978	+	0.920824i	$0.372483\pi$
$38$	2.58872	0.419946
$39$	0.429767	0.0688177
$40$	21.6244	3.41912
$41$	−7.87681	−1.23015	−0.615075	−	0.788468i	$-0.710874\pi$
$41$	−7.87681	−1.23015	−0.615075	+	0.788468i	$0.710874\pi$
$42$	−3.83991	−0.592511
$43$	−6.35192	−0.968658	−0.484329	−	0.874886i	$-0.660936\pi$
$43$	−6.35192	−0.968658	−0.484329	+	0.874886i	$0.660936\pi$
$44$	10.7303	1.61765
$45$	9.02496	1.34536
$46$	3.43902	0.507055
$47$	−10.9350	−1.59503	−0.797515	−	0.603299i	$-0.793852\pi$
$47$	−10.9350	−1.59503	−0.797515	+	0.603299i	$0.793852\pi$
$48$	2.48094	0.358093
$49$	20.0630	2.86615
$50$	−11.8076	−1.66985
$51$	−0.497603	−0.0696784
$52$	7.08631	0.982695
$53$	−1.36775	−0.187875	−0.0939377	−	0.995578i	$-0.529945\pi$
$53$	−1.36775	−0.187875	−0.0939377	+	0.995578i	$0.529945\pi$
$54$	4.36877	0.594514
$55$	−7.05720	−0.951592
$56$	−36.3812	−4.86164
$57$	−0.285133	−0.0377668
$58$	−10.6628	−1.40009
$59$	2.27962	0.296781	0.148391	−	0.988929i	$-0.452591\pi$
$59$	2.27962	0.296781	0.148391	+	0.988929i	$0.452591\pi$
$60$	−4.14513	−0.535134
$61$	−11.5708	−1.48148	−0.740742	−	0.671789i	$-0.765526\pi$
$61$	−11.5708	−1.48148	−0.740742	+	0.671789i	$0.765526\pi$
$62$	4.11626	0.522766
$63$	−15.1837	−1.91297
$64$	4.69968	0.587460
$65$	−4.66059	−0.578075
$66$	−1.68465	−0.207366
$67$	−1.39455	−0.170371	−0.0851856	−	0.996365i	$-0.527148\pi$
$67$	−1.39455	−0.170371	−0.0851856	+	0.996365i	$0.527148\pi$
$68$	−8.20486	−0.994985
$69$	−0.378788	−0.0456007
$70$	41.6418	4.97715
$71$	3.32433	0.394526	0.197263	−	0.980351i	$-0.436795\pi$
$71$	3.32433	0.394526	0.197263	+	0.980351i	$0.436795\pi$
$72$	20.4116	2.40553
$73$	11.0187	1.28965	0.644823	−	0.764332i	$-0.276931\pi$
$73$	11.0187	1.28965	0.644823	+	0.764332i	$0.276931\pi$
$74$	−12.2816	−1.42771
$75$	1.30054	0.150174
$76$	−4.70149	−0.539297
$77$	11.8731	1.35307
$78$	−1.11255	−0.125971
$79$	−9.38562	−1.05597	−0.527983	−	0.849255i	$-0.677051\pi$
$79$	−9.38562	−1.05597	−0.527983	+	0.849255i	$0.677051\pi$
$80$	−26.9045	−3.00801
$81$	8.27490	0.919434
$82$	20.3909	2.25180
$83$	−6.42097	−0.704793	−0.352397	−	0.935851i	$-0.614633\pi$
$83$	−6.42097	−0.704793	−0.352397	+	0.935851i	$0.614633\pi$
$84$	6.97382	0.760906
$85$	5.39624	0.585305
$86$	16.4434	1.77313
$87$	1.17444	0.125913
$88$	−15.9612	−1.70147
$89$	5.31513	0.563402	0.281701	−	0.959502i	$-0.409101\pi$
$89$	5.31513	0.563402	0.281701	+	0.959502i	$0.409101\pi$
$90$	−23.3631	−2.46269
$91$	7.84104	0.821964
$92$	−6.24574	−0.651163
$93$	−0.453382	−0.0470136
$94$	28.3076	2.91971
$95$	3.09212	0.317244
$96$	−2.43437	−0.248457
$97$	−10.5397	−1.07014	−0.535072	−	0.844806i	$-0.679716\pi$
$97$	−10.5397	−1.07014	−0.535072	+	0.844806i	$0.679716\pi$
$98$	−51.9377	−5.24650
$99$	−6.66140	−0.669496
$100$	21.4443	2.14443
$101$	9.12416	0.907888	0.453944	−	0.891030i	$-0.350017\pi$
$101$	9.12416	0.907888	0.453944	+	0.891030i	$0.350017\pi$
$102$	1.28816	0.127547
$103$	−8.83939	−0.870971	−0.435485	−	0.900196i	$-0.643423\pi$
$103$	−8.83939	−0.870971	−0.435485	+	0.900196i	$0.643423\pi$
$104$	−10.5408	−1.03361
$105$	−4.58661	−0.447607
$106$	3.54074	0.343907
$107$	−2.60534	−0.251868	−0.125934	−	0.992039i	$-0.540193\pi$
$107$	−2.60534	−0.251868	−0.125934	+	0.992039i	$0.540193\pi$
$108$	−7.93430	−0.763479
$109$	−5.91816	−0.566857	−0.283429	−	0.958993i	$-0.591472\pi$
$109$	−5.91816	−0.566857	−0.283429	+	0.958993i	$0.591472\pi$
$110$	18.2691	1.74189
$111$	1.35275	0.128398
$112$	45.2645	4.27709
$113$	4.34956	0.409172	0.204586	−	0.978849i	$-0.434415\pi$
$113$	4.34956	0.409172	0.204586	+	0.978849i	$0.434415\pi$
$114$	0.738130	0.0691322
$115$	4.10776	0.383050
$116$	19.3651	1.79800
$117$	−4.39921	−0.406707
$118$	−5.90130	−0.543259
$119$	−9.07871	−0.832244
$120$	6.16583	0.562861
$121$	−5.79102	−0.526456
$122$	29.9535	2.71186
$123$	−2.24594	−0.202509
$124$	−7.47571	−0.671338
$125$	1.35689	0.121364
$126$	39.3064	3.50169
$127$	−0.281993	−0.0250228	−0.0125114	−	0.999922i	$-0.503983\pi$
$127$	−0.281993	−0.0250228	−0.0125114	+	0.999922i	$0.503983\pi$
$128$	4.90919	0.433915
$129$	−1.81114	−0.159462
$130$	12.0650	1.05817
$131$	9.33760	0.815830	0.407915	−	0.913020i	$-0.366256\pi$
$131$	9.33760	0.815830	0.407915	+	0.913020i	$0.366256\pi$
$132$	3.05956	0.266301
$133$	−5.20222	−0.451089
$134$	3.61010	0.311865
$135$	5.21830	0.449120
$136$	12.2046	1.04654
$137$	−2.64302	−0.225808	−0.112904	−	0.993606i	$-0.536015\pi$
$137$	−2.64302	−0.225808	−0.112904	+	0.993606i	$0.536015\pi$
$138$	0.980577	0.0834723
$139$	−0.0983001	−0.00833770	−0.00416885	−	0.999991i	$-0.501327\pi$
$139$	−0.0983001	−0.00833770	−0.00416885	+	0.999991i	$0.501327\pi$
$140$	−75.6274	−6.39168
$141$	−3.11792	−0.262576
$142$	−8.60577	−0.722180
$143$	3.44003	0.287669
$144$	−25.3956	−2.11630
$145$	−12.7362	−1.05768
$146$	−28.5245	−2.36070
$147$	5.72064	0.471830
$148$	22.3052	1.83348
$149$	−15.4532	−1.26598	−0.632988	−	0.774162i	$-0.718172\pi$
$149$	−15.4532	−1.26598	−0.632988	+	0.774162i	$0.718172\pi$
$150$	−3.36674	−0.274894
$151$	13.3891	1.08959	0.544796	−	0.838569i	$-0.316607\pi$
$151$	13.3891	1.08959	0.544796	+	0.838569i	$0.316607\pi$
$152$	6.99340	0.567239
$153$	5.09360	0.411793
$154$	−30.7362	−2.47679
$155$	4.91669	0.394918
$156$	2.02054	0.161773
$157$	−19.3622	−1.54527	−0.772635	−	0.634850i	$-0.781062\pi$
$157$	−19.3622	−1.54527	−0.772635	+	0.634850i	$0.781062\pi$
$158$	24.2968	1.93295
$159$	−0.389992	−0.0309284
$160$	26.3995	2.08706
$161$	−6.91094	−0.544658
$162$	−21.4214	−1.68303
$163$	−9.75999	−0.764462	−0.382231	−	0.924067i	$-0.624844\pi$
$163$	−9.75999	−0.764462	−0.382231	+	0.924067i	$0.624844\pi$
$164$	−37.0327	−2.89177
$165$	−2.01224	−0.156653
$166$	16.6221	1.29013
$167$	21.8572	1.69136	0.845682	−	0.533687i	$-0.179194\pi$
$167$	21.8572	1.69136	0.845682	+	0.533687i	$0.179194\pi$
$168$	−10.3735	−0.800331
$169$	−10.7282	−0.825246
$170$	−13.9694	−1.07140
$171$	2.91870	0.223199
$172$	−29.8634	−2.27707
$173$	19.7860	1.50430	0.752152	−	0.658989i	$-0.229016\pi$
$173$	19.7860	1.50430	0.752152	+	0.658989i	$0.229016\pi$
$174$	−3.04030	−0.230485
$175$	23.7282	1.79369
$176$	19.8585	1.49689
$177$	0.649995	0.0488566
$178$	−13.7594	−1.03131
$179$	−2.14581	−0.160386	−0.0801928	−	0.996779i	$-0.525554\pi$
$179$	−2.14581	−0.160386	−0.0801928	+	0.996779i	$0.525554\pi$
$180$	42.4307	3.16260
$181$	−7.72058	−0.573866	−0.286933	−	0.957951i	$-0.592636\pi$
$181$	−7.72058	−0.573866	−0.286933	+	0.957951i	$0.592636\pi$
$182$	−20.2983	−1.50461
$183$	−3.29921	−0.243884
$184$	9.29046	0.684902
$185$	−14.6699	−1.07855
$186$	1.17368	0.0860585
$187$	−3.98302	−0.291267
$188$	−51.4106	−3.74950
$189$	−8.77934	−0.638603
$190$	−8.00463	−0.580717
$191$	4.38425	0.317233	0.158617	−	0.987340i	$-0.449297\pi$
$191$	4.38425	0.317233	0.158617	+	0.987340i	$0.449297\pi$
$192$	1.34003	0.0967085
$193$	−0.727728	−0.0523830	−0.0261915	−	0.999657i	$-0.508338\pi$
$193$	−0.727728	−0.0523830	−0.0261915	+	0.999657i	$0.508338\pi$
$194$	27.2844	1.95890
$195$	−1.32889	−0.0951637
$196$	94.3261	6.73758
$197$	−11.0519	−0.787414	−0.393707	−	0.919236i	$-0.628807\pi$
$197$	−11.0519	−0.787414	−0.393707	+	0.919236i	$0.628807\pi$
$198$	17.2445	1.22552
$199$	12.7585	0.904428	0.452214	−	0.891910i	$-0.350634\pi$
$199$	12.7585	0.904428	0.452214	+	0.891910i	$0.350634\pi$
$200$	−31.8981	−2.25554
$201$	−0.397632	−0.0280468
$202$	−23.6199	−1.66189
$203$	21.4275	1.50392
$204$	−2.33948	−0.163796
$205$	24.3560	1.70110
$206$	22.8827	1.59432
$207$	3.87738	0.269496
$208$	13.1146	0.909332
$209$	−2.28232	−0.157871
$210$	11.8735	0.819346
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	−6.43048	−0.441647
$213$	0.947876	0.0649474
$214$	6.74450	0.461045
$215$	19.6409	1.33950
$216$	11.8022	0.803036
$217$	−8.27191	−0.561534
$218$	15.3205	1.03763
$219$	3.14181	0.212304
$220$	−33.1793	−2.23695
$221$	−2.63040	−0.176940
$222$	−3.50190	−0.235032
$223$	11.9137	0.797802	0.398901	−	0.916994i	$-0.369392\pi$
$223$	11.9137	0.797802	0.398901	+	0.916994i	$0.369392\pi$
$224$	−44.4148	−2.96759
$225$	−13.3127	−0.887514
$226$	−11.2598	−0.748991
$227$	1.58313	0.105076	0.0525381	−	0.998619i	$-0.483269\pi$
$227$	1.58313	0.105076	0.0525381	+	0.998619i	$0.483269\pi$
$228$	−1.34055	−0.0887800
$229$	9.68534	0.640025	0.320012	−	0.947413i	$-0.396313\pi$
$229$	9.68534	0.640025	0.320012	+	0.947413i	$0.396313\pi$
$230$	−10.6338	−0.701175
$231$	3.38542	0.222744
$232$	−28.8053	−1.89116
$233$	−7.84501	−0.513944	−0.256972	−	0.966419i	$-0.582725\pi$
$233$	−7.84501	−0.513944	−0.256972	+	0.966419i	$0.582725\pi$
$234$	11.3883	0.744479
$235$	33.8122	2.20566
$236$	10.7176	0.697656
$237$	−2.67615	−0.173835
$238$	23.5023	1.52343
$239$	−11.4499	−0.740630	−0.370315	−	0.928906i	$-0.620750\pi$
$239$	−11.4499	−0.740630	−0.370315	+	0.928906i	$0.620750\pi$
$240$	−7.67136	−0.495184
$241$	30.0337	1.93464	0.967321	−	0.253557i	$-0.0816004\pi$
$241$	30.0337	1.93464	0.967321	+	0.253557i	$0.0816004\pi$
$242$	14.9913	0.963680
$243$	7.42230	0.476141
$244$	−54.3998	−3.48259
$245$	−62.0373	−3.96342
$246$	5.81411	0.370694
$247$	−1.50725	−0.0959040
$248$	11.1200	0.706122
$249$	−1.83083	−0.116024
$250$	−3.51260	−0.222156
$251$	−27.8907	−1.76044	−0.880222	−	0.474563i	$-0.842606\pi$
$251$	−27.8907	−1.76044	−0.880222	+	0.474563i	$0.842606\pi$
$252$	−71.3860	−4.49689
$253$	−3.03197	−0.190618
$254$	0.730002	0.0458044
$255$	1.53865	0.0963538
$256$	−22.1079	−1.38174
$257$	−23.6386	−1.47453	−0.737267	−	0.675601i	$-0.763884\pi$
$257$	−23.6386	−1.47453	−0.737267	+	0.675601i	$0.763884\pi$
$258$	4.68854	0.291896
$259$	24.6808	1.53359
$260$	−21.9117	−1.35891
$261$	−12.0219	−0.744137
$262$	−24.1725	−1.49338
$263$	6.07426	0.374555	0.187277	−	0.982307i	$-0.440034\pi$
$263$	6.07426	0.374555	0.187277	+	0.982307i	$0.440034\pi$
$264$	−4.55106	−0.280098
$265$	4.22925	0.259801
$266$	13.4671	0.825721
$267$	1.51552	0.0927482
$268$	−6.55645	−0.400499
$269$	−7.93816	−0.483998	−0.241999	−	0.970277i	$-0.577803\pi$
$269$	−7.93816	−0.483998	−0.241999	+	0.970277i	$0.577803\pi$
$270$	−13.5087	−0.822116
$271$	−5.42487	−0.329538	−0.164769	−	0.986332i	$-0.552688\pi$
$271$	−5.42487	−0.329538	−0.164769	+	0.986332i	$0.552688\pi$
$272$	−15.1847	−0.920705
$273$	2.23574	0.135313
$274$	6.84204	0.413343
$275$	10.4101	0.627751
$276$	−1.78087	−0.107196
$277$	1.88755	0.113412	0.0567060	−	0.998391i	$-0.481940\pi$
$277$	1.88755	0.113412	0.0567060	+	0.998391i	$0.481940\pi$
$278$	0.254472	0.0152622
$279$	4.64095	0.277846
$280$	112.495	6.72285
$281$	−20.5225	−1.22427	−0.612136	−	0.790753i	$-0.709689\pi$
$281$	−20.5225	−1.22427	−0.612136	+	0.790753i	$0.709689\pi$
$282$	8.07143	0.480647
$283$	−17.3747	−1.03282	−0.516409	−	0.856342i	$-0.672731\pi$
$283$	−17.3747	−1.03282	−0.516409	+	0.856342i	$0.672731\pi$
$284$	15.6293	0.927428
$285$	0.881664	0.0522253
$286$	−8.90528	−0.526580
$287$	−40.9769	−2.41879
$288$	24.9189	1.46836
$289$	−13.9544	−0.820848
$290$	32.9705	1.93609
$291$	−3.00522	−0.176169
$292$	51.8045	3.03163
$293$	25.4450	1.48651	0.743255	−	0.669008i	$-0.233281\pi$
$293$	25.4450	1.48651	0.743255	+	0.669008i	$0.233281\pi$
$294$	−14.8091	−0.863687
$295$	−7.04885	−0.410400
$296$	−33.1787	−1.92847
$297$	−3.85168	−0.223497
$298$	40.0041	2.31737
$299$	−2.00232	−0.115797
$300$	6.11448	0.353020
$301$	−33.0440	−1.90463
$302$	−34.6608	−1.99450
$303$	2.60160	0.149458
$304$	−8.70100	−0.499036
$305$	35.7781	2.04865
$306$	−13.1859	−0.753789
$307$	17.6874	1.00948	0.504738	−	0.863273i	$-0.331589\pi$
$307$	17.6874	1.00948	0.504738	+	0.863273i	$0.331589\pi$
$308$	55.8213	3.18071
$309$	−2.52040	−0.143381
$310$	−12.7280	−0.722899
$311$	−9.70024	−0.550050	−0.275025	−	0.961437i	$-0.588686\pi$
$311$	−9.70024	−0.550050	−0.275025	+	0.961437i	$0.588686\pi$
$312$	−3.00553	−0.170155
$313$	−1.57027	−0.0887569	−0.0443784	−	0.999015i	$-0.514131\pi$
$313$	−1.57027	−0.0887569	−0.0443784	+	0.999015i	$0.514131\pi$
$314$	50.1233	2.82862
$315$	46.9498	2.64532
$316$	−44.1264	−2.48230
$317$	22.3805	1.25701	0.628507	−	0.777804i	$-0.283666\pi$
$317$	22.3805	1.25701	0.628507	+	0.777804i	$0.283666\pi$
$318$	1.00958	0.0566145
$319$	9.40070	0.526338
$320$	−14.5319	−0.812360
$321$	−0.742868	−0.0414629
$322$	17.8905	0.996999
$323$	1.74516	0.0971034
$324$	38.9043	2.16135
$325$	6.87484	0.381347
$326$	25.2659	1.39935
$327$	−1.68746	−0.0933169
$328$	55.0857	3.04160
$329$	−56.8861	−3.13623
$330$	5.20913	0.286753
$331$	4.89056	0.268810	0.134405	−	0.990927i	$-0.457088\pi$
$331$	4.89056	0.268810	0.134405	+	0.990927i	$0.457088\pi$
$332$	−30.1881	−1.65679
$333$	−13.8471	−0.758819
$334$	−56.5824	−3.09605
$335$	4.31210	0.235595
$336$	12.9064	0.704101
$337$	−0.674748	−0.0367558	−0.0183779	−	0.999831i	$-0.505850\pi$
$337$	−0.674748	−0.0367558	−0.0183779	+	0.999831i	$0.505850\pi$
$338$	27.7723	1.51062
$339$	1.24020	0.0673586
$340$	25.3704	1.37590
$341$	−3.62906	−0.196524
$342$	−7.55570	−0.408566
$343$	67.9568	3.66932
$344$	44.4215	2.39505
$345$	1.17126	0.0630583
$346$	−51.2206	−2.75364
$347$	−11.0976	−0.595751	−0.297875	−	0.954605i	$-0.596278\pi$
$347$	−11.0976	−0.595751	−0.297875	+	0.954605i	$0.596278\pi$
$348$	5.52162	0.295990
$349$	−18.2566	−0.977254	−0.488627	−	0.872493i	$-0.662502\pi$
$349$	−18.2566	−0.977254	−0.488627	+	0.872493i	$0.662502\pi$
$350$	−61.4258	−3.28335
$351$	−2.54366	−0.135771
$352$	−19.4857	−1.03859
$353$	−5.67471	−0.302034	−0.151017	−	0.988531i	$-0.548255\pi$
$353$	−5.67471	−0.302034	−0.151017	+	0.988531i	$0.548255\pi$
$354$	−1.68266	−0.0894322
$355$	−10.2792	−0.545564
$356$	24.9890	1.32441
$357$	−2.58864	−0.137005
$358$	5.55491	0.293586
$359$	−10.5218	−0.555319	−0.277659	−	0.960680i	$-0.589559\pi$
$359$	−10.5218	−0.555319	−0.277659	+	0.960680i	$0.589559\pi$
$360$	−63.1151	−3.32646
$361$	1.00000	0.0526316
$362$	19.9865	1.05046
$363$	−1.65121	−0.0866660
$364$	36.8645	1.93223
$365$	−34.0712	−1.78337
$366$	8.54073	0.446431
$367$	10.7357	0.560401	0.280201	−	0.959941i	$-0.409599\pi$
$367$	10.7357	0.560401	0.280201	+	0.959941i	$0.409599\pi$
$368$	−11.5589	−0.602551
$369$	22.9900	1.19681
$370$	37.9762	1.97429
$371$	−7.11535	−0.369411
$372$	−2.13157	−0.110517
$373$	−15.4161	−0.798217	−0.399108	−	0.916904i	$-0.630680\pi$
$373$	−15.4161	−0.798217	−0.399108	+	0.916904i	$0.630680\pi$
$374$	10.3109	0.533166
$375$	0.386893	0.0199791
$376$	76.4726	3.94377
$377$	6.20825	0.319741
$378$	22.7273	1.16897
$379$	−36.7046	−1.88539	−0.942694	−	0.333658i	$-0.891717\pi$
$379$	−36.7046	−1.88539	−0.942694	+	0.333658i	$0.891717\pi$
$380$	14.5375	0.745760
$381$	−0.0804055	−0.00411930
$382$	−11.3496	−0.580696
$383$	−8.75116	−0.447163	−0.223582	−	0.974685i	$-0.571775\pi$
$383$	−8.75116	−0.447163	−0.223582	+	0.974685i	$0.571775\pi$
$384$	1.39977	0.0714318
$385$	−36.7131	−1.87107
$386$	1.88389	0.0958873
$387$	18.5393	0.942407
$388$	−49.5523	−2.51563
$389$	38.8360	1.96906	0.984532	−	0.175205i	$-0.0560590\pi$
$389$	38.8360	1.96906	0.984532	+	0.175205i	$0.0560590\pi$
$390$	3.44012	0.174197
$391$	2.31838	0.117246
$392$	−140.309	−7.08667
$393$	2.66246	0.134303
$394$	28.6102	1.44136
$395$	29.0214	1.46023
$396$	−31.3185	−1.57381
$397$	6.13998	0.308157	0.154078	−	0.988059i	$-0.450759\pi$
$397$	6.13998	0.308157	0.154078	+	0.988059i	$0.450759\pi$
$398$	−33.0283	−1.65556
$399$	−1.48332	−0.0742590
$400$	39.6868	1.98434
$401$	−26.4372	−1.32021	−0.660106	−	0.751172i	$-0.729489\pi$
$401$	−26.4372	−1.32021	−0.660106	+	0.751172i	$0.729489\pi$
$402$	1.02936	0.0513397
$403$	−2.39664	−0.119385
$404$	42.8971	2.13421
$405$	−25.5870	−1.27143
$406$	−55.4699	−2.75293
$407$	10.8280	0.536723
$408$	3.47994	0.172283
$409$	18.4908	0.914310	0.457155	−	0.889387i	$-0.348868\pi$
$409$	18.4908	0.914310	0.457155	+	0.889387i	$0.348868\pi$
$410$	−63.0509	−3.11386
$411$	−0.753611	−0.0371729
$412$	−41.5583	−2.04743
$413$	11.8591	0.583547
$414$	−10.0375	−0.493314
$415$	19.8544	0.974614
$416$	−12.8684	−0.630926
$417$	−0.0280286	−0.00137257
$418$	5.90829	0.288984
$419$	−16.2017	−0.791506	−0.395753	−	0.918357i	$-0.629516\pi$
$419$	−16.2017	−0.791506	−0.395753	+	0.918357i	$0.629516\pi$
$420$	−21.5639	−1.05221
$421$	−8.28470	−0.403771	−0.201886	−	0.979409i	$-0.564707\pi$
$421$	−8.28470	−0.403771	−0.201886	+	0.979409i	$0.564707\pi$
$422$	2.58872	0.126017
$423$	31.9159	1.55180
$424$	9.56525	0.464530
$425$	−7.96000	−0.386117
$426$	−2.45379	−0.118886
$427$	−60.1936	−2.91297
$428$	−12.2490	−0.592076
$429$	0.980865	0.0473566
$430$	−50.8448	−2.45195
$431$	−28.2796	−1.36218	−0.681091	−	0.732199i	$-0.738494\pi$
$431$	−28.2796	−1.36218	−0.681091	+	0.732199i	$0.738494\pi$
$432$	−14.6839	−0.706482
$433$	−10.3302	−0.496437	−0.248218	−	0.968704i	$-0.579845\pi$
$433$	−10.3302	−0.496437	−0.248218	+	0.968704i	$0.579845\pi$
$434$	21.4137	1.02789
$435$	−3.63151	−0.174117
$436$	−27.8242	−1.33254
$437$	1.32846	0.0635489
$438$	−8.13327	−0.388623
$439$	−23.7352	−1.13282	−0.566410	−	0.824124i	$-0.691668\pi$
$439$	−23.7352	−1.13282	−0.566410	+	0.824124i	$0.691668\pi$
$440$	49.3538	2.35285
$441$	−58.5580	−2.78848
$442$	6.80937	0.323889
$443$	−10.1458	−0.482039	−0.241020	−	0.970520i	$-0.577482\pi$
$443$	−10.1458	−0.482039	−0.241020	+	0.970520i	$0.577482\pi$
$444$	6.35994	0.301830
$445$	−16.4350	−0.779093
$446$	−30.8413	−1.46038
$447$	−4.40622	−0.208407
$448$	24.4487	1.15509
$449$	4.15163	0.195928	0.0979639	−	0.995190i	$-0.468767\pi$
$449$	4.15163	0.195928	0.0979639	+	0.995190i	$0.468767\pi$
$450$	34.4629	1.62460
$451$	−17.9774	−0.846523
$452$	20.4494	0.961858
$453$	3.81768	0.179370
$454$	−4.09829	−0.192343
$455$	−24.2454	−1.13664
$456$	1.99405	0.0933799
$457$	−11.2849	−0.527887	−0.263944	−	0.964538i	$-0.585023\pi$
$457$	−11.2849	−0.527887	−0.263944	+	0.964538i	$0.585023\pi$
$458$	−25.0726	−1.17157
$459$	2.94516	0.137469
$460$	19.3126	0.900452
$461$	−27.1816	−1.26597	−0.632986	−	0.774163i	$-0.718171\pi$
$461$	−27.1816	−1.26597	−0.632986	+	0.774163i	$0.718171\pi$
$462$	−8.76391	−0.407734
$463$	−37.4274	−1.73940	−0.869698	−	0.493583i	$-0.835687\pi$
$463$	−37.4274	−1.73940	−0.869698	+	0.493583i	$0.835687\pi$
$464$	35.8388	1.66377
$465$	1.40191	0.0650121
$466$	20.3086	0.940776
$467$	19.9662	0.923927	0.461963	−	0.886899i	$-0.347145\pi$
$467$	19.9662	0.923927	0.461963	+	0.886899i	$0.347145\pi$
$468$	−20.6828	−0.956064
$469$	−7.25474	−0.334993
$470$	−87.5304	−4.03748
$471$	−5.52080	−0.254385
$472$	−15.9423	−0.733803
$473$	−14.4971	−0.666578
$474$	6.92781	0.318205
$475$	−4.56118	−0.209281
$476$	−42.6834	−1.95639
$477$	3.99206	0.182784
$478$	29.6405	1.35573
$479$	−39.9624	−1.82593	−0.912964	−	0.408041i	$-0.866212\pi$
$479$	−39.9624	−1.82593	−0.912964	+	0.408041i	$0.866212\pi$
$480$	7.52736	0.343575
$481$	7.15082	0.326050
$482$	−77.7489	−3.54137
$483$	−1.97054	−0.0896625
$484$	−27.2264	−1.23756
$485$	32.5900	1.47983
$486$	−19.2143	−0.871577
$487$	−37.3887	−1.69424	−0.847122	−	0.531399i	$-0.821667\pi$
$487$	−37.3887	−1.69424	−0.847122	+	0.531399i	$0.821667\pi$
$488$	80.9189	3.66303
$489$	−2.78290	−0.125847
$490$	160.597	7.25505
$491$	25.1058	1.13301	0.566504	−	0.824059i	$-0.308296\pi$
$491$	25.1058	1.13301	0.566504	+	0.824059i	$0.308296\pi$
$492$	−10.5592	−0.476048
$493$	−7.18819	−0.323740
$494$	3.90185	0.175553
$495$	20.5978	0.925804
$496$	−13.8352	−0.621220
$497$	17.2939	0.775737
$498$	4.73951	0.212383
$499$	−36.0951	−1.61584	−0.807918	−	0.589294i	$-0.799406\pi$
$499$	−36.0951	−1.61584	−0.807918	+	0.589294i	$0.799406\pi$
$500$	6.37938	0.285294
$501$	6.23222	0.278435
$502$	72.2012	3.22250
$503$	−10.0340	−0.447394	−0.223697	−	0.974659i	$-0.571813\pi$
$503$	−10.0340	−0.447394	−0.223697	+	0.974659i	$0.571813\pi$
$504$	106.186	4.72989
$505$	−28.2130	−1.25546
$506$	7.84894	0.348928
$507$	−3.05896	−0.135853
$508$	−1.32579	−0.0588223
$509$	−30.9989	−1.37400	−0.687001	−	0.726657i	$-0.741073\pi$
$509$	−30.9989	−1.37400	−0.687001	+	0.726657i	$0.741073\pi$
$510$	−3.98313	−0.176376
$511$	57.3219	2.53577
$512$	47.4128	2.09537
$513$	1.68762	0.0745101
$514$	61.1938	2.69914
$515$	27.3324	1.20441
$516$	−8.51505	−0.374854
$517$	−24.9571	−1.09761
$518$	−63.8917	−2.80724
$519$	5.64165	0.247641
$520$	32.5934	1.42931
$521$	−34.4026	−1.50720	−0.753602	−	0.657331i	$-0.771685\pi$
$521$	−34.4026	−1.50720	−0.753602	+	0.657331i	$0.771685\pi$
$522$	31.1214	1.36215
$523$	−23.0352	−1.00726	−0.503629	−	0.863920i	$-0.668002\pi$
$523$	−23.0352	−1.00726	−0.503629	+	0.863920i	$0.668002\pi$
$524$	43.9006	1.91781
$525$	6.76570	0.295279
$526$	−15.7246	−0.685624
$527$	2.77494	0.120878
$528$	5.66230	0.246420
$529$	−21.2352	−0.923269
$530$	−10.9484	−0.475567
$531$	−6.65352	−0.288738
$532$	−24.4581	−1.06040
$533$	−11.8723	−0.514247
$534$	−3.92326	−0.169776
$535$	8.05601	0.348292
$536$	9.75263	0.421249
$537$	−0.611842	−0.0264029
$538$	20.5497	0.885960
$539$	45.7903	1.97233
$540$	24.5338	1.05577
$541$	−40.9267	−1.75958	−0.879788	−	0.475365i	$-0.842316\pi$
$541$	−40.9267	−1.75958	−0.879788	+	0.475365i	$0.842316\pi$
$542$	14.0435	0.603220
$543$	−2.20139	−0.0944708
$544$	14.8996	0.638816
$545$	18.2996	0.783870
$546$	−5.78771	−0.247691
$547$	−13.0531	−0.558109	−0.279054	−	0.960275i	$-0.590021\pi$
$547$	−13.0531	−0.558109	−0.279054	+	0.960275i	$0.590021\pi$
$548$	−12.4261	−0.530817
$549$	33.7716	1.44134
$550$	−26.9488	−1.14910
$551$	−4.11892	−0.175472
$552$	2.64902	0.112750
$553$	−48.8260	−2.07629
$554$	−4.88635	−0.207601
$555$	−4.18287	−0.177553
$556$	−0.462156	−0.0195998
$557$	−34.7855	−1.47391	−0.736954	−	0.675943i	$-0.763737\pi$
$557$	−34.7855	−1.47391	−0.736954	+	0.675943i	$0.763737\pi$
$558$	−12.0141	−0.508598
$559$	−9.57393	−0.404934
$560$	−139.963	−5.91451
$561$	−1.13569	−0.0479489
$562$	53.1272	2.24103
$563$	23.3903	0.985786	0.492893	−	0.870090i	$-0.335939\pi$
$563$	23.3903	0.985786	0.492893	+	0.870090i	$0.335939\pi$
$564$	−14.6589	−0.617249
$565$	−13.4493	−0.565818
$566$	44.9782	1.89058
$567$	43.0478	1.80784
$568$	−23.2484	−0.975480
$569$	3.53186	0.148063	0.0740315	−	0.997256i	$-0.476413\pi$
$569$	3.53186	0.148063	0.0740315	+	0.997256i	$0.476413\pi$
$570$	−2.28238	−0.0955985
$571$	26.5079	1.10932	0.554660	−	0.832077i	$-0.312848\pi$
$571$	26.5079	1.10932	0.554660	+	0.832077i	$0.312848\pi$
$572$	16.1732	0.676237
$573$	1.25009	0.0522234
$574$	106.078	4.42760
$575$	−6.05935	−0.252692
$576$	−13.7169	−0.571539
$577$	−25.2543	−1.05135	−0.525674	−	0.850686i	$-0.676187\pi$
$577$	−25.2543	−1.05135	−0.525674	+	0.850686i	$0.676187\pi$
$578$	36.1241	1.50256
$579$	−0.207499	−0.00862337
$580$	−59.8790	−2.48634
$581$	−33.4033	−1.38580
$582$	7.77967	0.322478
$583$	−3.12165	−0.129286
$584$	−77.0585	−3.18870
$585$	13.6029	0.562409
$586$	−65.8699	−2.72106
$587$	−44.9328	−1.85458	−0.927288	−	0.374349i	$-0.877866\pi$
$587$	−44.9328	−1.85458	−0.927288	+	0.374349i	$0.877866\pi$
$588$	26.8955	1.10915
$589$	1.59007	0.0655178
$590$	18.2475	0.751238
$591$	−3.15125	−0.129625
$592$	41.2800	1.69660
$593$	2.29097	0.0940787	0.0470394	−	0.998893i	$-0.485021\pi$
$593$	2.29097	0.0940787	0.0470394	+	0.998893i	$0.485021\pi$
$594$	9.97093	0.409112
$595$	28.0724	1.15086
$596$	−72.6530	−2.97598
$597$	3.63788	0.148888
$598$	5.18346	0.211967
$599$	29.6130	1.20995	0.604977	−	0.796243i	$-0.293182\pi$
$599$	29.6130	1.20995	0.604977	+	0.796243i	$0.293182\pi$
$600$	−9.09521	−0.371310
$601$	24.4349	0.996721	0.498360	−	0.866970i	$-0.333936\pi$
$601$	24.4349	0.996721	0.498360	+	0.866970i	$0.333936\pi$
$602$	85.5419	3.48643
$603$	4.07027	0.165754
$604$	62.9488	2.56135
$605$	17.9065	0.728002
$606$	−6.73482	−0.273583
$607$	19.1755	0.778310	0.389155	−	0.921172i	$-0.372767\pi$
$607$	19.1755	0.778310	0.389155	+	0.921172i	$0.372767\pi$
$608$	8.53767	0.346248
$609$	6.10970	0.247577
$610$	−92.6197	−3.75006
$611$	−16.4817	−0.666780
$612$	23.9475	0.968021
$613$	22.9010	0.924964	0.462482	−	0.886629i	$-0.346959\pi$
$613$	22.9010	0.924964	0.462482	+	0.886629i	$0.346959\pi$
$614$	−45.7879	−1.84785
$615$	6.94470	0.280037
$616$	−83.0335	−3.34551
$617$	7.36581	0.296536	0.148268	−	0.988947i	$-0.452630\pi$
$617$	7.36581	0.296536	0.148268	+	0.988947i	$0.452630\pi$
$618$	6.52462	0.262459
$619$	20.2575	0.814216	0.407108	−	0.913380i	$-0.366537\pi$
$619$	20.2575	0.814216	0.407108	+	0.913380i	$0.366537\pi$
$620$	23.1158	0.928351
$621$	2.24193	0.0899657
$622$	25.1112	1.00687
$623$	27.6504	1.10779
$624$	3.73940	0.149696
$625$	−27.0015	−1.08006
$626$	4.06499	0.162470
$627$	−0.650765	−0.0259890
$628$	−91.0310	−3.63253
$629$	−8.27955	−0.330127
$630$	−121.540	−4.84227
$631$	2.53545	0.100935	0.0504673	−	0.998726i	$-0.483929\pi$
$631$	2.53545	0.100935	0.0504673	+	0.998726i	$0.483929\pi$
$632$	65.6374	2.61092
$633$	−0.285133	−0.0113330
$634$	−57.9369	−2.30097
$635$	0.871955	0.0346025
$636$	−1.83354	−0.0727046
$637$	30.2400	1.19815
$638$	−24.3358	−0.963464
$639$	−9.70272	−0.383834
$640$	−15.1798	−0.600033
$641$	−29.6333	−1.17045	−0.585223	−	0.810872i	$-0.698993\pi$
$641$	−29.6333	−1.17045	−0.585223	+	0.810872i	$0.698993\pi$
$642$	1.92308	0.0758979
$643$	16.5580	0.652985	0.326492	−	0.945200i	$-0.394133\pi$
$643$	16.5580	0.652985	0.326492	+	0.945200i	$0.394133\pi$
$644$	−32.4917	−1.28035
$645$	5.60026	0.220510
$646$	−4.51774	−0.177748
$647$	−19.4641	−0.765213	−0.382606	−	0.923911i	$-0.624973\pi$
$647$	−19.4641	−0.765213	−0.382606	+	0.923911i	$0.624973\pi$
$648$	−57.8697	−2.27334
$649$	5.20282	0.204229
$650$	−17.7970	−0.698058
$651$	−2.35859	−0.0924406
$652$	−45.8865	−1.79705
$653$	30.5047	1.19374	0.596871	−	0.802337i	$-0.296410\pi$
$653$	30.5047	1.19374	0.596871	+	0.802337i	$0.296410\pi$
$654$	4.36837	0.170817
$655$	−28.8729	−1.12816
$656$	−68.5361	−2.67589
$657$	−32.1604	−1.25470
$658$	147.262	5.74088
$659$	−26.6670	−1.03880	−0.519399	−	0.854532i	$-0.673844\pi$
$659$	−26.6670	−1.03880	−0.519399	+	0.854532i	$0.673844\pi$
$660$	−9.46052	−0.368250
$661$	17.6805	0.687692	0.343846	−	0.939026i	$-0.388270\pi$
$661$	17.6805	0.687692	0.343846	+	0.939026i	$0.388270\pi$
$662$	−12.6603	−0.492057
$663$	−0.750013	−0.0291281
$664$	44.9044	1.74263
$665$	16.0859	0.623783
$666$	35.8464	1.38902
$667$	−5.47183	−0.211870
$668$	102.762	3.97596
$669$	3.39699	0.131335
$670$	−11.1628	−0.431258
$671$	−26.4082	−1.01948
$672$	−12.6641	−0.488529
$673$	21.4655	0.827435	0.413718	−	0.910405i	$-0.364230\pi$
$673$	21.4655	0.827435	0.413718	+	0.910405i	$0.364230\pi$
$674$	1.74673	0.0672817
$675$	−7.69752	−0.296278
$676$	−50.4385	−1.93994
$677$	37.1612	1.42822	0.714110	−	0.700033i	$-0.246831\pi$
$677$	37.1612	1.42822	0.714110	+	0.700033i	$0.246831\pi$
$678$	−3.21054	−0.123300
$679$	−54.8298	−2.10417
$680$	−37.7381	−1.44719
$681$	0.451404	0.0172978
$682$	9.39462	0.359739
$683$	−35.9439	−1.37536	−0.687678	−	0.726015i	$-0.741370\pi$
$683$	−35.9439	−1.37536	−0.687678	+	0.726015i	$0.741370\pi$
$684$	13.7222	0.524682
$685$	8.17251	0.312256
$686$	−175.921	−6.71671
$687$	2.76161	0.105362
$688$	−55.2680	−2.10707
$689$	−2.06155	−0.0785387
$690$	−3.03206	−0.115428
$691$	26.4526	1.00630	0.503151	−	0.864198i	$-0.332174\pi$
$691$	26.4526	1.00630	0.503151	+	0.864198i	$0.332174\pi$
$692$	93.0238	3.53623
$693$	−34.6541	−1.31640
$694$	28.7286	1.09052
$695$	0.303955	0.0115297
$696$	−8.21334	−0.311326
$697$	13.7463	0.520679
$698$	47.2613	1.78887
$699$	−2.23687	−0.0846062
$700$	111.558	4.21649
$701$	12.2389	0.462258	0.231129	−	0.972923i	$-0.425758\pi$
$701$	12.2389	0.462258	0.231129	+	0.972923i	$0.425758\pi$
$702$	6.58483	0.248528
$703$	−4.74428	−0.178934
$704$	10.7262	0.404258
$705$	9.64097	0.363100
$706$	14.6903	0.552875
$707$	47.4659	1.78514
$708$	3.05594	0.114849
$709$	−3.99073	−0.149875	−0.0749375	−	0.997188i	$-0.523876\pi$
$709$	−3.99073	−0.149875	−0.0749375	+	0.997188i	$0.523876\pi$
$710$	26.6100	0.998657
$711$	27.3938	1.02735
$712$	−37.1708	−1.39303
$713$	2.11235	0.0791082
$714$	6.70127	0.250789
$715$	−10.6370	−0.397800
$716$	−10.0885	−0.377025
$717$	−3.26473	−0.121924
$718$	27.2380	1.01651
$719$	−44.0201	−1.64167	−0.820836	−	0.571163i	$-0.806492\pi$
$719$	−44.0201	−1.64167	−0.820836	+	0.571163i	$0.806492\pi$
$720$	78.5261	2.92650
$721$	−45.9844	−1.71255
$722$	−2.58872	−0.0963423
$723$	8.56360	0.318484
$724$	−36.2982	−1.34901
$725$	18.7872	0.697737
$726$	4.27452	0.158642
$727$	53.1012	1.96942	0.984708	−	0.174214i	$-0.0557385\pi$
$727$	53.1012	1.96942	0.984708	+	0.174214i	$0.0557385\pi$
$728$	−54.8355	−2.03234
$729$	−22.7084	−0.841051
$730$	88.2010	3.26447
$731$	11.0851	0.409998
$732$	−15.5112	−0.573309
$733$	22.8335	0.843373	0.421686	−	0.906742i	$-0.361438\pi$
$733$	22.8335	0.843373	0.421686	+	0.906742i	$0.361438\pi$
$734$	−27.7919	−1.02582
$735$	−17.6889	−0.652464
$736$	11.3420	0.418070
$737$	−3.18280	−0.117240
$738$	−59.5148	−2.19077
$739$	−16.9908	−0.625015	−0.312507	−	0.949915i	$-0.601169\pi$
$739$	−16.9908	−0.625015	−0.312507	+	0.949915i	$0.601169\pi$
$740$	−68.9702	−2.53540
$741$	−0.429767	−0.0157879
$742$	18.4197	0.676207
$743$	43.8881	1.61010	0.805050	−	0.593207i	$-0.202138\pi$
$743$	43.8881	1.61010	0.805050	+	0.593207i	$0.202138\pi$
$744$	3.17068	0.116243
$745$	47.7831	1.75064
$746$	39.9081	1.46114
$747$	18.7409	0.685693
$748$	−18.7261	−0.684694
$749$	−13.5535	−0.495236
$750$	−1.00156	−0.0365717
$751$	−38.0621	−1.38891	−0.694453	−	0.719538i	$-0.744354\pi$
$751$	−38.0621	−1.38891	−0.694453	+	0.719538i	$0.744354\pi$
$752$	−95.1452	−3.46959
$753$	−7.95255	−0.289807
$754$	−16.0714	−0.585287
$755$	−41.4008	−1.50673
$756$	−41.2760	−1.50119
$757$	−20.4248	−0.742353	−0.371176	−	0.928562i	$-0.621045\pi$
$757$	−20.4248	−0.742353	−0.371176	+	0.928562i	$0.621045\pi$
$758$	95.0180	3.45121
$759$	−0.864515	−0.0313799
$760$	−21.6244	−0.784399
$761$	2.85115	0.103354	0.0516770	−	0.998664i	$-0.483543\pi$
$761$	2.85115	0.103354	0.0516770	+	0.998664i	$0.483543\pi$
$762$	0.208148	0.00754039
$763$	−30.7876	−1.11458
$764$	20.6125	0.745733
$765$	−15.7500	−0.569443
$766$	22.6543	0.818534
$767$	3.43596	0.124065
$768$	−6.30368	−0.227465
$769$	−32.0754	−1.15667	−0.578334	−	0.815800i	$-0.696297\pi$
$769$	−32.0754	−1.15667	−0.578334	+	0.815800i	$0.696297\pi$
$770$	95.0399	3.42500
$771$	−6.74014	−0.242740
$772$	−3.42140	−0.123139
$773$	46.6440	1.67767	0.838834	−	0.544388i	$-0.183238\pi$
$773$	46.6440	1.67767	0.838834	+	0.544388i	$0.183238\pi$
$774$	−47.9932	−1.72508
$775$	−7.25261	−0.260521
$776$	73.7083	2.64597
$777$	7.03731	0.252462
$778$	−100.536	−3.60438
$779$	7.87681	0.282216
$780$	−6.24775	−0.223705
$781$	7.58719	0.271491
$782$	−6.00164	−0.214618
$783$	−6.95116	−0.248414
$784$	174.569	6.23459
$785$	59.8701	2.13686
$786$	−6.89237	−0.245843
$787$	−22.0944	−0.787581	−0.393790	−	0.919200i	$-0.628836\pi$
$787$	−22.0944	−0.787581	−0.393790	+	0.919200i	$0.628836\pi$
$788$	−51.9602	−1.85101
$789$	1.73197	0.0616598
$790$	−75.1285	−2.67295
$791$	22.6273	0.804536
$792$	46.5859	1.65536
$793$	−17.4400	−0.619314
$794$	−15.8947	−0.564082
$795$	1.20590	0.0427689
$796$	59.9840	2.12608
$797$	18.3667	0.650581	0.325291	−	0.945614i	$-0.394538\pi$
$797$	18.3667	0.650581	0.325291	+	0.945614i	$0.394538\pi$
$798$	3.83991	0.135931
$799$	19.0833	0.675119
$800$	−38.9419	−1.37680
$801$	−15.5133	−0.548134
$802$	68.4387	2.41665
$803$	25.1483	0.887464
$804$	−1.86946	−0.0659307
$805$	21.3694	0.753173
$806$	6.20423	0.218535
$807$	−2.26343	−0.0796765
$808$	−63.8089	−2.24479
$809$	10.7308	0.377273	0.188637	−	0.982047i	$-0.439593\pi$
$809$	10.7308	0.377273	0.188637	+	0.982047i	$0.439593\pi$
$810$	66.2375	2.32735
$811$	41.2174	1.44734	0.723670	−	0.690146i	$-0.242454\pi$
$811$	41.2174	1.44734	0.723670	+	0.690146i	$0.242454\pi$
$812$	100.741	3.53532
$813$	−1.54681	−0.0542490
$814$	−28.0306	−0.982473
$815$	30.1790	1.05712
$816$	−4.32965	−0.151568
$817$	6.35192	0.222225
$818$	−47.8675	−1.67365
$819$	−22.8856	−0.799689
$820$	114.509	3.99884
$821$	−31.1642	−1.08764	−0.543819	−	0.839202i	$-0.683022\pi$
$821$	−31.1642	−1.08764	−0.543819	+	0.839202i	$0.683022\pi$
$822$	1.95089	0.0680451
$823$	48.1677	1.67902	0.839510	−	0.543344i	$-0.182842\pi$
$823$	48.1677	1.67902	0.839510	+	0.543344i	$0.182842\pi$
$824$	61.8174	2.15351
$825$	2.96825	0.103341
$826$	−30.6999	−1.06818
$827$	−38.2048	−1.32851	−0.664256	−	0.747506i	$-0.731251\pi$
$827$	−38.2048	−1.32851	−0.664256	+	0.747506i	$0.731251\pi$
$828$	18.2294	0.633517
$829$	−36.8263	−1.27903	−0.639515	−	0.768778i	$-0.720865\pi$
$829$	−36.8263	−1.27903	−0.639515	+	0.768778i	$0.720865\pi$
$830$	−51.3975	−1.78403
$831$	0.538203	0.0186701
$832$	7.08359	0.245579
$833$	−35.0133	−1.21314
$834$	0.0725583	0.00251249
$835$	−67.5851	−2.33888
$836$	−10.7303	−0.371115
$837$	2.68343	0.0927530
$838$	41.9418	1.44885
$839$	28.5113	0.984318	0.492159	−	0.870505i	$-0.336208\pi$
$839$	28.5113	0.984318	0.492159	+	0.870505i	$0.336208\pi$
$840$	32.0760	1.10673
$841$	−12.0345	−0.414981
$842$	21.4468	0.739105
$843$	−5.85165	−0.201542
$844$	−4.70149	−0.161832
$845$	33.1728	1.14118
$846$	−82.6214	−2.84058
$847$	−30.1261	−1.03515
$848$	−11.9008	−0.408676
$849$	−4.95409	−0.170024
$850$	20.6062	0.706788
$851$	−6.30260	−0.216050
$852$	4.45643	0.152675
$853$	54.8693	1.87869	0.939344	−	0.342977i	$-0.111435\pi$
$853$	54.8693	1.87869	0.939344	+	0.342977i	$0.111435\pi$
$854$	155.825	5.33221
$855$	−9.02496	−0.308647
$856$	18.2202	0.622753
$857$	−52.8488	−1.80528	−0.902641	−	0.430394i	$-0.858375\pi$
$857$	−52.8488	−1.80528	−0.902641	+	0.430394i	$0.858375\pi$
$858$	−2.53919	−0.0866864
$859$	12.2086	0.416550	0.208275	−	0.978070i	$-0.433215\pi$
$859$	12.2086	0.416550	0.208275	+	0.978070i	$0.433215\pi$
$860$	92.3412	3.14881
$861$	−11.6839	−0.398185
$862$	73.2081	2.49348
$863$	−32.7683	−1.11545	−0.557723	−	0.830027i	$-0.688325\pi$
$863$	−32.7683	−1.11545	−0.557723	+	0.830027i	$0.688325\pi$
$864$	14.4083	0.490181
$865$	−61.1807	−2.08021
$866$	26.7420	0.908730
$867$	−3.97886	−0.135129
$868$	−38.8902	−1.32002
$869$	−21.4210	−0.726658
$870$	9.40097	0.318723
$871$	−2.10193	−0.0712213
$872$	41.3881	1.40158
$873$	30.7622	1.04114
$874$	−3.43902	−0.116326
$875$	7.05881	0.238631
$876$	14.7712	0.499072
$877$	52.4494	1.77109	0.885546	−	0.464552i	$-0.153785\pi$
$877$	52.4494	1.77109	0.885546	+	0.464552i	$0.153785\pi$
$878$	61.4439	2.07363
$879$	7.25520	0.244712
$880$	−61.4046	−2.06995
$881$	42.7191	1.43924	0.719621	−	0.694367i	$-0.244316\pi$
$881$	42.7191	1.43924	0.719621	+	0.694367i	$0.244316\pi$
$882$	151.590	5.10432
$883$	40.7874	1.37260	0.686302	−	0.727317i	$-0.259233\pi$
$883$	40.7874	1.37260	0.686302	+	0.727317i	$0.259233\pi$
$884$	−12.3668	−0.415940
$885$	−2.00986	−0.0675607
$886$	26.2645	0.882375
$887$	36.8567	1.23753	0.618764	−	0.785577i	$-0.287634\pi$
$887$	36.8567	1.23753	0.618764	+	0.785577i	$0.287634\pi$
$888$	−9.46033	−0.317468
$889$	−1.46699	−0.0492012
$890$	42.5456	1.42613
$891$	18.8860	0.632704
$892$	56.0122	1.87543
$893$	10.9350	0.365925
$894$	11.4065	0.381490
$895$	6.63510	0.221787
$896$	25.5386	0.853186
$897$	−0.570928	−0.0190627
$898$	−10.7474	−0.358646
$899$	−6.54939	−0.218434
$900$	−62.5895	−2.08632
$901$	2.38695	0.0795209
$902$	46.5385	1.54956
$903$	−9.42195	−0.313543
$904$	−30.4182	−1.01169
$905$	23.8729	0.793563
$906$	−9.88292	−0.328338
$907$	−12.2932	−0.408188	−0.204094	−	0.978951i	$-0.565425\pi$
$907$	−12.2932	−0.408188	−0.204094	+	0.978951i	$0.565425\pi$
$908$	7.44308	0.247007
$909$	−26.6307	−0.883284
$910$	62.7646	2.08063
$911$	−10.9007	−0.361155	−0.180577	−	0.983561i	$-0.557797\pi$
$911$	−10.9007	−0.361155	−0.180577	+	0.983561i	$0.557797\pi$
$912$	−2.48094	−0.0821522
$913$	−14.6547	−0.485000
$914$	29.2136	0.966300
$915$	10.2015	0.337252
$916$	45.5355	1.50453
$917$	48.5762	1.60413
$918$	−7.62421	−0.251637
$919$	−5.11134	−0.168608	−0.0843038	−	0.996440i	$-0.526867\pi$
$919$	−5.11134	−0.168608	−0.0843038	+	0.996440i	$0.526867\pi$
$920$	−28.7272	−0.947107
$921$	5.04327	0.166181
$922$	70.3656	2.31737
$923$	5.01060	0.164926
$924$	15.9165	0.523614
$925$	21.6395	0.711503
$926$	96.8891	3.18397
$927$	25.7995	0.847367
$928$	−35.1660	−1.15438
$929$	−3.21095	−0.105348	−0.0526739	−	0.998612i	$-0.516774\pi$
$929$	−3.21095	−0.105348	−0.0526739	+	0.998612i	$0.516774\pi$
$930$	−3.62916	−0.119005
$931$	−20.0630	−0.657540
$932$	−36.8832	−1.20815
$933$	−2.76586	−0.0905501
$934$	−51.6870	−1.69125
$935$	12.3160	0.402775
$936$	30.7654	1.00560
$937$	−19.9106	−0.650452	−0.325226	−	0.945636i	$-0.605440\pi$
$937$	−19.9106	−0.650452	−0.325226	+	0.945636i	$0.605440\pi$
$938$	18.7805	0.613205
$939$	−0.447736	−0.0146113
$940$	158.968	5.18495
$941$	57.4210	1.87187	0.935936	−	0.352170i	$-0.114556\pi$
$941$	57.4210	1.87187	0.935936	+	0.352170i	$0.114556\pi$
$942$	14.2918	0.465652
$943$	10.4640	0.340756
$944$	19.8350	0.645573
$945$	27.1467	0.883084
$946$	37.5290	1.22017
$947$	37.4411	1.21667	0.608335	−	0.793680i	$-0.291838\pi$
$947$	37.4411	1.21667	0.608335	+	0.793680i	$0.291838\pi$
$948$	−12.5819	−0.408641
$949$	16.6080	0.539119
$950$	11.8076	0.383090
$951$	6.38142	0.206932
$952$	63.4910	2.05776
$953$	−8.89143	−0.288022	−0.144011	−	0.989576i	$-0.546000\pi$
$953$	−8.89143	−0.288022	−0.144011	+	0.989576i	$0.546000\pi$
$954$	−10.3343	−0.334587
$955$	−13.5566	−0.438681
$956$	−53.8314	−1.74103
$957$	2.68045	0.0866467
$958$	103.451	3.34237
$959$	−13.7495	−0.443996
$960$	−4.14354	−0.133732
$961$	−28.4717	−0.918441
$962$	−18.5115	−0.596835
$963$	7.60420	0.245042
$964$	141.203	4.54784
$965$	2.25022	0.0724371
$966$	5.10117	0.164128
$967$	−8.88560	−0.285742	−0.142871	−	0.989741i	$-0.545633\pi$
$967$	−8.88560	−0.285742	−0.142871	+	0.989741i	$0.545633\pi$
$968$	40.4989	1.30168
$969$	0.497603	0.0159853
$970$	−84.3664	−2.70884
$971$	40.8626	1.31134	0.655671	−	0.755047i	$-0.272386\pi$
$971$	40.8626	1.31134	0.655671	+	0.755047i	$0.272386\pi$
$972$	34.8958	1.11928
$973$	−0.511378	−0.0163940
$974$	96.7890	3.10132
$975$	1.96024	0.0627780
$976$	−100.677	−3.22260
$977$	15.9410	0.509999	0.255000	−	0.966941i	$-0.417925\pi$
$977$	15.9410	0.509999	0.255000	+	0.966941i	$0.417925\pi$
$978$	7.20414	0.230363
$979$	12.1308	0.387703
$980$	−291.667	−9.31697
$981$	17.2733	0.551495
$982$	−64.9919	−2.07397
$983$	−51.1045	−1.62998	−0.814991	−	0.579474i	$-0.803258\pi$
$983$	−51.1045	−1.62998	−0.814991	+	0.579474i	$0.803258\pi$
$984$	15.7067	0.500713
$985$	34.1737	1.08886
$986$	18.6082	0.592607
$987$	−16.2201	−0.516291
$988$	−7.08631	−0.225446
$989$	8.43827	0.268322
$990$	−53.3221	−1.69469
$991$	−7.12344	−0.226283	−0.113142	−	0.993579i	$-0.536091\pi$
$991$	−7.12344	−0.226283	−0.113142	+	0.993579i	$0.536091\pi$
$992$	13.5755	0.431023
$993$	1.39446	0.0442519
$994$	−44.7691	−1.41999
$995$	−39.4508	−1.25068
$996$	−8.60763	−0.272743
$997$	−43.5066	−1.37787	−0.688934	−	0.724824i	$-0.741921\pi$
$997$	−43.5066	−1.37787	−0.688934	+	0.724824i	$0.741921\pi$
$998$	93.4401	2.95779
$999$	−8.00653	−0.253315

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.d.1.6	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.d.1.6	✓	75	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.58872 q^{2} +0.285133 q^{3} +4.70149 q^{4} -3.09212 q^{5} -0.738130 q^{6} +5.20222 q^{7} -6.99340 q^{8} -2.91870 q^{9} +O(q^{10})\)	\(q-2.58872 q^{2} +0.285133 q^{3} +4.70149 q^{4} -3.09212 q^{5} -0.738130 q^{6} +5.20222 q^{7} -6.99340 q^{8} -2.91870 q^{9} +8.00463 q^{10} +2.28232 q^{11} +1.34055 q^{12} +1.50725 q^{13} -13.4671 q^{14} -0.881664 q^{15} +8.70100 q^{16} -1.74516 q^{17} +7.55570 q^{18} -1.00000 q^{19} -14.5375 q^{20} +1.48332 q^{21} -5.90829 q^{22} -1.32846 q^{23} -1.99405 q^{24} +4.56118 q^{25} -3.90185 q^{26} -1.68762 q^{27} +24.4581 q^{28} +4.11892 q^{29} +2.28238 q^{30} -1.59007 q^{31} -8.53767 q^{32} +0.650765 q^{33} +4.51774 q^{34} -16.0859 q^{35} -13.7222 q^{36} +4.74428 q^{37} +2.58872 q^{38} +0.429767 q^{39} +21.6244 q^{40} -7.87681 q^{41} -3.83991 q^{42} -6.35192 q^{43} +10.7303 q^{44} +9.02496 q^{45} +3.43902 q^{46} -10.9350 q^{47} +2.48094 q^{48} +20.0630 q^{49} -11.8076 q^{50} -0.497603 q^{51} +7.08631 q^{52} -1.36775 q^{53} +4.36877 q^{54} -7.05720 q^{55} -36.3812 q^{56} -0.285133 q^{57} -10.6628 q^{58} +2.27962 q^{59} -4.14513 q^{60} -11.5708 q^{61} +4.11626 q^{62} -15.1837 q^{63} +4.69968 q^{64} -4.66059 q^{65} -1.68465 q^{66} -1.39455 q^{67} -8.20486 q^{68} -0.378788 q^{69} +41.6418 q^{70} +3.32433 q^{71} +20.4116 q^{72} +11.0187 q^{73} -12.2816 q^{74} +1.30054 q^{75} -4.70149 q^{76} +11.8731 q^{77} -1.11255 q^{78} -9.38562 q^{79} -26.9045 q^{80} +8.27490 q^{81} +20.3909 q^{82} -6.42097 q^{83} +6.97382 q^{84} +5.39624 q^{85} +16.4434 q^{86} +1.17444 q^{87} -15.9612 q^{88} +5.31513 q^{89} -23.3631 q^{90} +7.84104 q^{91} -6.24574 q^{92} -0.453382 q^{93} +28.3076 q^{94} +3.09212 q^{95} -2.43437 q^{96} -10.5397 q^{97} -51.9377 q^{98} -6.66140 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9}+O(q^{10}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9} - 48 q^{11} - 14 q^{12} - 3 q^{13} - 4 q^{14} - 39 q^{15} + 59 q^{16} - 23 q^{17} - 24 q^{18} - 75 q^{19} - 62 q^{20} - 3 q^{21} - 6 q^{22} - 73 q^{23} - 64 q^{24} + 57 q^{25} - 46 q^{26} - 22 q^{27} - 26 q^{28} - 39 q^{29} - 14 q^{30} - 44 q^{31} - 71 q^{32} - 3 q^{33} - 9 q^{34} - 49 q^{35} + 20 q^{36} - 12 q^{37} + 11 q^{38} - 90 q^{39} - 8 q^{40} - 42 q^{41} - 45 q^{42} - 24 q^{43} - 120 q^{44} - 63 q^{45} - 39 q^{46} - 59 q^{47} - 4 q^{48} + 48 q^{49} - 100 q^{50} - 55 q^{51} + 2 q^{52} + 13 q^{53} - 87 q^{54} - 36 q^{55} - 12 q^{56} + 4 q^{57} - 17 q^{58} - 47 q^{59} - 45 q^{60} - 35 q^{61} - 40 q^{62} - 69 q^{63} + 26 q^{64} - 44 q^{65} + 33 q^{66} - 39 q^{67} - 63 q^{68} + 42 q^{69} + 40 q^{70} - 154 q^{71} - 51 q^{72} - 29 q^{73} - 95 q^{74} + 37 q^{75} - 67 q^{76} - 24 q^{77} - 19 q^{78} - 95 q^{79} - 146 q^{80} + 23 q^{81} + 7 q^{82} - 52 q^{83} - 72 q^{84} - 36 q^{85} - 44 q^{86} - 103 q^{87} + 67 q^{88} + q^{89} - 2 q^{90} - 64 q^{91} - 183 q^{92} - 49 q^{93} + 5 q^{94} + 18 q^{95} - 69 q^{96} - 7 q^{97} - 23 q^{98} - 100 q^{99}+O(q^{100}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9 - 48 * q^11 - 14 * q^12 - 3 * q^13 - 4 * q^14 - 39 * q^15 + 59 * q^16 - 23 * q^17 - 24 * q^18 - 75 * q^19 - 62 * q^20 - 3 * q^21 - 6 * q^22 - 73 * q^23 - 64 * q^24 + 57 * q^25 - 46 * q^26 - 22 * q^27 - 26 * q^28 - 39 * q^29 - 14 * q^30 - 44 * q^31 - 71 * q^32 - 3 * q^33 - 9 * q^34 - 49 * q^35 + 20 * q^36 - 12 * q^37 + 11 * q^38 - 90 * q^39 - 8 * q^40 - 42 * q^41 - 45 * q^42 - 24 * q^43 - 120 * q^44 - 63 * q^45 - 39 * q^46 - 59 * q^47 - 4 * q^48 + 48 * q^49 - 100 * q^50 - 55 * q^51 + 2 * q^52 + 13 * q^53 - 87 * q^54 - 36 * q^55 - 12 * q^56 + 4 * q^57 - 17 * q^58 - 47 * q^59 - 45 * q^60 - 35 * q^61 - 40 * q^62 - 69 * q^63 + 26 * q^64 - 44 * q^65 + 33 * q^66 - 39 * q^67 - 63 * q^68 + 42 * q^69 + 40 * q^70 - 154 * q^71 - 51 * q^72 - 29 * q^73 - 95 * q^74 + 37 * q^75 - 67 * q^76 - 24 * q^77 - 19 * q^78 - 95 * q^79 - 146 * q^80 + 23 * q^81 + 7 * q^82 - 52 * q^83 - 72 * q^84 - 36 * q^85 - 44 * q^86 - 103 * q^87 + 67 * q^88 + q^89 - 2 * q^90 - 64 * q^91 - 183 * q^92 - 49 * q^93 + 5 * q^94 + 18 * q^95 - 69 * q^96 - 7 * q^97 - 23 * q^98 - 100 * q^99

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