LMFDB - Embedded newform 8047.2.a.b.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8047,2,Mod(1,8047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8047, 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("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8047 = 13 \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8047.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:	$64.2556185065$
Analytic rank:	$1$
Dimension:	$142$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	8047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.52091 q^{2} -0.134089 q^{3} +4.35498 q^{4} +2.66858 q^{5} +0.338025 q^{6} -2.31114 q^{7} -5.93668 q^{8} -2.98202 q^{9} +O(q^{10})$	\(q-2.52091 q^{2} -0.134089 q^{3} +4.35498 q^{4} +2.66858 q^{5} +0.338025 q^{6} -2.31114 q^{7} -5.93668 q^{8} -2.98202 q^{9} -6.72724 q^{10} -4.30592 q^{11} -0.583953 q^{12} +1.00000 q^{13} +5.82616 q^{14} -0.357826 q^{15} +6.25588 q^{16} +4.36671 q^{17} +7.51740 q^{18} +1.70053 q^{19} +11.6216 q^{20} +0.309897 q^{21} +10.8548 q^{22} +4.64277 q^{23} +0.796042 q^{24} +2.12131 q^{25} -2.52091 q^{26} +0.802121 q^{27} -10.0649 q^{28} +2.89705 q^{29} +0.902047 q^{30} +0.593276 q^{31} -3.89712 q^{32} +0.577375 q^{33} -11.0081 q^{34} -6.16745 q^{35} -12.9866 q^{36} -8.80335 q^{37} -4.28688 q^{38} -0.134089 q^{39} -15.8425 q^{40} +4.18820 q^{41} -0.781223 q^{42} -2.50410 q^{43} -18.7522 q^{44} -7.95775 q^{45} -11.7040 q^{46} -1.06636 q^{47} -0.838842 q^{48} -1.65865 q^{49} -5.34762 q^{50} -0.585526 q^{51} +4.35498 q^{52} -11.4976 q^{53} -2.02207 q^{54} -11.4907 q^{55} +13.7205 q^{56} -0.228022 q^{57} -7.30319 q^{58} +11.6878 q^{59} -1.55832 q^{60} -13.2811 q^{61} -1.49559 q^{62} +6.89186 q^{63} -2.68746 q^{64} +2.66858 q^{65} -1.45551 q^{66} +8.24829 q^{67} +19.0169 q^{68} -0.622543 q^{69} +15.5476 q^{70} +6.16272 q^{71} +17.7033 q^{72} +8.40385 q^{73} +22.1924 q^{74} -0.284443 q^{75} +7.40578 q^{76} +9.95157 q^{77} +0.338025 q^{78} +8.86338 q^{79} +16.6943 q^{80} +8.83851 q^{81} -10.5581 q^{82} +1.30966 q^{83} +1.34960 q^{84} +11.6529 q^{85} +6.31261 q^{86} -0.388461 q^{87} +25.5629 q^{88} -16.6895 q^{89} +20.0608 q^{90} -2.31114 q^{91} +20.2192 q^{92} -0.0795516 q^{93} +2.68820 q^{94} +4.53800 q^{95} +0.522560 q^{96} +9.89727 q^{97} +4.18130 q^{98} +12.8403 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9}+O(q^{10}) $ 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9} - 25 q^{10} - 25 q^{11} - 62 q^{12} + 142 q^{13} - 57 q^{14} - 14 q^{15} + 111 q^{16} - 141 q^{17} - 29 q^{18} - 3 q^{19} - 87 q^{20} - 19 q^{21} - 24 q^{22} - 69 q^{23} - 40 q^{24} + 87 q^{25} - 13 q^{26} - 95 q^{27} - 34 q^{28} - 147 q^{29} - 2 q^{30} - 21 q^{31} - 66 q^{32} - 62 q^{33} - 6 q^{34} - 59 q^{35} + 74 q^{36} - 37 q^{37} - 76 q^{38} - 26 q^{39} - 61 q^{40} - 97 q^{41} - 29 q^{42} - 33 q^{43} - 57 q^{44} - 86 q^{45} - q^{46} - 102 q^{47} - 141 q^{48} + 70 q^{49} - 28 q^{50} - 13 q^{51} + 129 q^{52} - 137 q^{53} - 29 q^{54} - 24 q^{55} - 130 q^{56} - 65 q^{57} - 15 q^{58} - 56 q^{59} + 11 q^{60} - 77 q^{61} - 150 q^{62} - 32 q^{63} + 73 q^{64} - 37 q^{65} - 32 q^{66} - 9 q^{67} - 226 q^{68} - 113 q^{69} + 6 q^{70} - 18 q^{71} - 82 q^{72} - 117 q^{73} - 70 q^{74} - 83 q^{75} + 40 q^{76} - 214 q^{77} - 15 q^{78} - 52 q^{79} - 161 q^{80} - 10 q^{81} - 36 q^{82} - 74 q^{83} + 53 q^{84} + 2 q^{85} + 17 q^{86} - 49 q^{87} - 29 q^{88} - 171 q^{89} - 57 q^{90} - 14 q^{91} - 187 q^{92} - 39 q^{93} + 13 q^{94} - 150 q^{95} - 47 q^{96} - 126 q^{97} - 85 q^{98}+O(q^{100}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9 - 25 * q^10 - 25 * q^11 - 62 * q^12 + 142 * q^13 - 57 * q^14 - 14 * q^15 + 111 * q^16 - 141 * q^17 - 29 * q^18 - 3 * q^19 - 87 * q^20 - 19 * q^21 - 24 * q^22 - 69 * q^23 - 40 * q^24 + 87 * q^25 - 13 * q^26 - 95 * q^27 - 34 * q^28 - 147 * q^29 - 2 * q^30 - 21 * q^31 - 66 * q^32 - 62 * q^33 - 6 * q^34 - 59 * q^35 + 74 * q^36 - 37 * q^37 - 76 * q^38 - 26 * q^39 - 61 * q^40 - 97 * q^41 - 29 * q^42 - 33 * q^43 - 57 * q^44 - 86 * q^45 - q^46 - 102 * q^47 - 141 * q^48 + 70 * q^49 - 28 * q^50 - 13 * q^51 + 129 * q^52 - 137 * q^53 - 29 * q^54 - 24 * q^55 - 130 * q^56 - 65 * q^57 - 15 * q^58 - 56 * q^59 + 11 * q^60 - 77 * q^61 - 150 * q^62 - 32 * q^63 + 73 * q^64 - 37 * q^65 - 32 * q^66 - 9 * q^67 - 226 * q^68 - 113 * q^69 + 6 * q^70 - 18 * q^71 - 82 * q^72 - 117 * q^73 - 70 * q^74 - 83 * q^75 + 40 * q^76 - 214 * q^77 - 15 * q^78 - 52 * q^79 - 161 * q^80 - 10 * q^81 - 36 * q^82 - 74 * q^83 + 53 * q^84 + 2 * q^85 + 17 * q^86 - 49 * q^87 - 29 * q^88 - 171 * q^89 - 57 * q^90 - 14 * q^91 - 187 * q^92 - 39 * q^93 + 13 * q^94 - 150 * q^95 - 47 * q^96 - 126 * q^97 - 85 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.52091	−1.78255	−0.891276	−	0.453462i	$-0.850189\pi$
$2$	−2.52091	−1.78255	−0.891276	+	0.453462i	$0.850189\pi$
$3$	−0.134089	−0.0774161	−0.0387081	−	0.999251i	$-0.512324\pi$
$3$	−0.134089	−0.0774161	−0.0387081	+	0.999251i	$0.512324\pi$
$4$	4.35498	2.17749
$5$	2.66858	1.19342	0.596712	−	0.802455i	$-0.296473\pi$
$5$	2.66858	1.19342	0.596712	+	0.802455i	$0.296473\pi$
$6$	0.338025	0.137998
$7$	−2.31114	−0.873528	−0.436764	−	0.899576i	$-0.643875\pi$
$7$	−2.31114	−0.873528	−0.436764	+	0.899576i	$0.643875\pi$
$8$	−5.93668	−2.09893
$9$	−2.98202	−0.994007
$10$	−6.72724	−2.12734
$11$	−4.30592	−1.29828	−0.649142	−	0.760667i	$-0.724872\pi$
$11$	−4.30592	−1.29828	−0.649142	+	0.760667i	$0.724872\pi$
$12$	−0.583953	−0.168573
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	5.82616	1.55711
$15$	−0.357826	−0.0923903
$16$	6.25588	1.56397
$17$	4.36671	1.05908	0.529541	−	0.848284i	$-0.322364\pi$
$17$	4.36671	1.05908	0.529541	+	0.848284i	$0.322364\pi$
$18$	7.51740	1.77187
$19$	1.70053	0.390129	0.195064	−	0.980790i	$-0.437508\pi$
$19$	1.70053	0.390129	0.195064	+	0.980790i	$0.437508\pi$
$20$	11.6216	2.59867
$21$	0.309897	0.0676251
$22$	10.8548	2.31426
$23$	4.64277	0.968084	0.484042	−	0.875045i	$-0.339168\pi$
$23$	4.64277	0.968084	0.484042	+	0.875045i	$0.339168\pi$
$24$	0.796042	0.162491
$25$	2.12131	0.424261
$26$	−2.52091	−0.494391
$27$	0.802121	0.154368
$28$	−10.0649	−1.90210
$29$	2.89705	0.537968	0.268984	−	0.963145i	$-0.413312\pi$
$29$	2.89705	0.537968	0.268984	+	0.963145i	$0.413312\pi$
$30$	0.902047	0.164690
$31$	0.593276	0.106556	0.0532778	−	0.998580i	$-0.483033\pi$
$31$	0.593276	0.106556	0.0532778	+	0.998580i	$0.483033\pi$
$32$	−3.89712	−0.688921
$33$	0.577375	0.100508
$34$	−11.0081	−1.88787
$35$	−6.16745	−1.04249
$36$	−12.9866	−2.16444
$37$	−8.80335	−1.44726	−0.723631	−	0.690187i	$-0.757528\pi$
$37$	−8.80335	−1.44726	−0.723631	+	0.690187i	$0.757528\pi$
$38$	−4.28688	−0.695425
$39$	−0.134089	−0.0214714
$40$	−15.8425	−2.50492
$41$	4.18820	0.654087	0.327044	−	0.945009i	$-0.393948\pi$
$41$	4.18820	0.654087	0.327044	+	0.945009i	$0.393948\pi$
$42$	−0.781223	−0.120545
$43$	−2.50410	−0.381872	−0.190936	−	0.981602i	$-0.561152\pi$
$43$	−2.50410	−0.381872	−0.190936	+	0.981602i	$0.561152\pi$
$44$	−18.7522	−2.82700
$45$	−7.95775	−1.18627
$46$	−11.7040	−1.72566
$47$	−1.06636	−0.155545	−0.0777724	−	0.996971i	$-0.524781\pi$
$47$	−1.06636	−0.155545	−0.0777724	+	0.996971i	$0.524781\pi$
$48$	−0.838842	−0.121076
$49$	−1.65865	−0.236950
$50$	−5.34762	−0.756268
$51$	−0.585526	−0.0819901
$52$	4.35498	0.603927
$53$	−11.4976	−1.57932	−0.789658	−	0.613548i	$-0.789742\pi$
$53$	−11.4976	−1.57932	−0.789658	+	0.613548i	$0.789742\pi$
$54$	−2.02207	−0.275169
$55$	−11.4907	−1.54940
$56$	13.7205	1.83348
$57$	−0.228022	−0.0302023
$58$	−7.30319	−0.958956
$59$	11.6878	1.52162	0.760811	−	0.648973i	$-0.224801\pi$
$59$	11.6878	1.52162	0.760811	+	0.648973i	$0.224801\pi$
$60$	−1.55832	−0.201179
$61$	−13.2811	−1.70047	−0.850234	−	0.526404i	$-0.823540\pi$
$61$	−13.2811	−1.70047	−0.850234	+	0.526404i	$0.823540\pi$
$62$	−1.49559	−0.189941
$63$	6.89186	0.868292
$64$	−2.68746	−0.335933
$65$	2.66858	0.330996
$66$	−1.45551	−0.179161
$67$	8.24829	1.00769	0.503845	−	0.863794i	$-0.331918\pi$
$67$	8.24829	1.00769	0.503845	+	0.863794i	$0.331918\pi$
$68$	19.0169	2.30614
$69$	−0.622543	−0.0749454
$70$	15.5476	1.85829
$71$	6.16272	0.731381	0.365690	−	0.930737i	$-0.380833\pi$
$71$	6.16272	0.731381	0.365690	+	0.930737i	$0.380833\pi$
$72$	17.7033	2.08635
$73$	8.40385	0.983596	0.491798	−	0.870709i	$-0.336340\pi$
$73$	8.40385	0.983596	0.491798	+	0.870709i	$0.336340\pi$
$74$	22.1924	2.57982
$75$	−0.284443	−0.0328447
$76$	7.40578	0.849501
$77$	9.95157	1.13409
$78$	0.338025	0.0382738
$79$	8.86338	0.997209	0.498604	−	0.866830i	$-0.333846\pi$
$79$	8.86338	0.997209	0.498604	+	0.866830i	$0.333846\pi$
$80$	16.6943	1.86648
$81$	8.83851	0.982056
$82$	−10.5581	−1.16594
$83$	1.30966	0.143753	0.0718767	−	0.997414i	$-0.477101\pi$
$83$	1.30966	0.143753	0.0718767	+	0.997414i	$0.477101\pi$
$84$	1.34960	0.147253
$85$	11.6529	1.26394
$86$	6.31261	0.680707
$87$	−0.388461	−0.0416474
$88$	25.5629	2.72501
$89$	−16.6895	−1.76908	−0.884542	−	0.466461i	$-0.845529\pi$
$89$	−16.6895	−1.76908	−0.884542	+	0.466461i	$0.845529\pi$
$90$	20.0608	2.11459
$91$	−2.31114	−0.242273
$92$	20.2192	2.10799
$93$	−0.0795516	−0.00824912
$94$	2.68820	0.277267
$95$	4.53800	0.465589
$96$	0.522560	0.0533336
$97$	9.89727	1.00492	0.502458	−	0.864602i	$-0.332429\pi$
$97$	9.89727	1.00492	0.502458	+	0.864602i	$0.332429\pi$
$98$	4.18130	0.422375
$99$	12.8403	1.29050
$100$	9.23824	0.923824
$101$	−8.47800	−0.843593	−0.421796	−	0.906691i	$-0.638600\pi$
$101$	−8.47800	−0.843593	−0.421796	+	0.906691i	$0.638600\pi$
$102$	1.47606	0.146152
$103$	8.38448	0.826148	0.413074	−	0.910698i	$-0.364455\pi$
$103$	8.38448	0.826148	0.413074	+	0.910698i	$0.364455\pi$
$104$	−5.93668	−0.582140
$105$	0.826985	0.0807055
$106$	28.9844	2.81521
$107$	−17.1638	−1.65929	−0.829645	−	0.558292i	$-0.811457\pi$
$107$	−17.1638	−1.65929	−0.829645	+	0.558292i	$0.811457\pi$
$108$	3.49322	0.336135
$109$	7.89476	0.756181	0.378091	−	0.925769i	$-0.376581\pi$
$109$	7.89476	0.756181	0.378091	+	0.925769i	$0.376581\pi$
$110$	28.9670	2.76189
$111$	1.18043	0.112041
$112$	−14.4582	−1.36617
$113$	−5.51218	−0.518542	−0.259271	−	0.965805i	$-0.583482\pi$
$113$	−5.51218	−0.518542	−0.259271	+	0.965805i	$0.583482\pi$
$114$	0.574823	0.0538371
$115$	12.3896	1.15534
$116$	12.6166	1.17142
$117$	−2.98202	−0.275688
$118$	−29.4639	−2.71237
$119$	−10.0921	−0.925138
$120$	2.12430	0.193921
$121$	7.54094	0.685540
$122$	33.4804	3.03117
$123$	−0.561590	−0.0506369
$124$	2.58370	0.232024
$125$	−7.68202	−0.687100
$126$	−17.3737	−1.54778
$127$	−19.5045	−1.73075	−0.865373	−	0.501128i	$-0.832918\pi$
$127$	−19.5045	−1.73075	−0.865373	+	0.501128i	$0.832918\pi$
$128$	14.5691	1.28774
$129$	0.335772	0.0295631
$130$	−6.72724	−0.590018
$131$	−13.4032	−1.17105	−0.585523	−	0.810656i	$-0.699111\pi$
$131$	−13.4032	−1.17105	−0.585523	+	0.810656i	$0.699111\pi$
$132$	2.51446	0.218855
$133$	−3.93016	−0.340788
$134$	−20.7932	−1.79626
$135$	2.14052	0.184227
$136$	−25.9238	−2.22295
$137$	13.5244	1.15546	0.577732	−	0.816226i	$-0.303938\pi$
$137$	13.5244	1.15546	0.577732	+	0.816226i	$0.303938\pi$
$138$	1.56937	0.133594
$139$	−3.46956	−0.294284	−0.147142	−	0.989115i	$-0.547007\pi$
$139$	−3.46956	−0.294284	−0.147142	+	0.989115i	$0.547007\pi$
$140$	−26.8591	−2.27001
$141$	0.142987	0.0120417
$142$	−15.5357	−1.30372
$143$	−4.30592	−0.360079
$144$	−18.6552	−1.55460
$145$	7.73099	0.642024
$146$	−21.1853	−1.75331
$147$	0.222406	0.0183437
$148$	−38.3384	−3.15140
$149$	−8.84673	−0.724753	−0.362376	−	0.932032i	$-0.618034\pi$
$149$	−8.84673	−0.724753	−0.362376	+	0.932032i	$0.618034\pi$
$150$	0.717055	0.0585473
$151$	6.16787	0.501934	0.250967	−	0.967996i	$-0.419251\pi$
$151$	6.16787	0.501934	0.250967	+	0.967996i	$0.419251\pi$
$152$	−10.0955	−0.818855
$153$	−13.0216	−1.05274
$154$	−25.0870	−2.02157
$155$	1.58320	0.127166
$156$	−0.583953	−0.0467537
$157$	14.8084	1.18184	0.590920	−	0.806730i	$-0.298765\pi$
$157$	14.8084	1.18184	0.590920	+	0.806730i	$0.298765\pi$
$158$	−22.3438	−1.77758
$159$	1.54170	0.122264
$160$	−10.3998	−0.822175
$161$	−10.7301	−0.845648
$162$	−22.2811	−1.75057
$163$	14.1037	1.10469	0.552345	−	0.833615i	$-0.313733\pi$
$163$	14.1037	1.10469	0.552345	+	0.833615i	$0.313733\pi$
$164$	18.2395	1.42427
$165$	1.54077	0.119949
$166$	−3.30152	−0.256248
$167$	1.90309	0.147266	0.0736328	−	0.997285i	$-0.476541\pi$
$167$	1.90309	0.147266	0.0736328	+	0.997285i	$0.476541\pi$
$168$	−1.83976	−0.141941
$169$	1.00000	0.0769231
$170$	−29.3759	−2.25303
$171$	−5.07102	−0.387791
$172$	−10.9053	−0.831522
$173$	10.6420	0.809094	0.404547	−	0.914517i	$-0.367429\pi$
$173$	10.6420	0.809094	0.404547	+	0.914517i	$0.367429\pi$
$174$	0.979275	0.0742386
$175$	−4.90263	−0.370604
$176$	−26.9373	−2.03048
$177$	−1.56720	−0.117798
$178$	42.0727	3.15348
$179$	7.90704	0.591000	0.295500	−	0.955343i	$-0.404514\pi$
$179$	7.90704	0.591000	0.295500	+	0.955343i	$0.404514\pi$
$180$	−34.6558	−2.58309
$181$	23.7712	1.76690	0.883451	−	0.468524i	$-0.155214\pi$
$181$	23.7712	1.76690	0.883451	+	0.468524i	$0.155214\pi$
$182$	5.82616	0.431864
$183$	1.78084	0.131644
$184$	−27.5627	−2.03195
$185$	−23.4924	−1.72720
$186$	0.200542	0.0147045
$187$	−18.8027	−1.37499
$188$	−4.64398	−0.338697
$189$	−1.85381	−0.134845
$190$	−11.4399	−0.829937
$191$	8.76374	0.634122	0.317061	−	0.948405i	$-0.397304\pi$
$191$	8.76374	0.634122	0.317061	+	0.948405i	$0.397304\pi$
$192$	0.360358	0.0260066
$193$	−6.53781	−0.470602	−0.235301	−	0.971923i	$-0.575608\pi$
$193$	−6.53781	−0.470602	−0.235301	+	0.971923i	$0.575608\pi$
$194$	−24.9501	−1.79131
$195$	−0.357826	−0.0256245
$196$	−7.22337	−0.515955
$197$	22.4887	1.60225	0.801125	−	0.598497i	$-0.204235\pi$
$197$	22.4887	1.60225	0.801125	+	0.598497i	$0.204235\pi$
$198$	−32.3693	−2.30039
$199$	−19.1206	−1.35543	−0.677713	−	0.735326i	$-0.737029\pi$
$199$	−19.1206	−1.35543	−0.677713	+	0.735326i	$0.737029\pi$
$200$	−12.5935	−0.890497
$201$	−1.10600	−0.0780114
$202$	21.3723	1.50375
$203$	−6.69547	−0.469930
$204$	−2.54995	−0.178533
$205$	11.1765	0.780603
$206$	−21.1365	−1.47265
$207$	−13.8448	−0.962282
$208$	6.25588	0.433767
$209$	−7.32235	−0.506498
$210$	−2.08475	−0.143862
$211$	−19.9949	−1.37650	−0.688251	−	0.725473i	$-0.741621\pi$
$211$	−19.9949	−1.37650	−0.688251	+	0.725473i	$0.741621\pi$
$212$	−50.0717	−3.43894
$213$	−0.826351	−0.0566207
$214$	43.2684	2.95777
$215$	−6.68239	−0.455735
$216$	−4.76194	−0.324009
$217$	−1.37114	−0.0930792
$218$	−19.9020	−1.34793
$219$	−1.12686	−0.0761462
$220$	−50.0417	−3.37381
$221$	4.36671	0.293737
$222$	−2.97575	−0.199720
$223$	−9.01874	−0.603939	−0.301970	−	0.953318i	$-0.597644\pi$
$223$	−9.01874	−0.603939	−0.301970	+	0.953318i	$0.597644\pi$
$224$	9.00679	0.601791
$225$	−6.32578	−0.421719
$226$	13.8957	0.924328
$227$	−18.0066	−1.19514	−0.597570	−	0.801817i	$-0.703867\pi$
$227$	−18.0066	−1.19514	−0.597570	+	0.801817i	$0.703867\pi$
$228$	−0.993031	−0.0657651
$229$	8.27309	0.546701	0.273351	−	0.961914i	$-0.411868\pi$
$229$	8.27309	0.546701	0.273351	+	0.961914i	$0.411868\pi$
$230$	−31.2330	−2.05944
$231$	−1.33439	−0.0877966
$232$	−17.1988	−1.12916
$233$	−5.60006	−0.366872	−0.183436	−	0.983032i	$-0.558722\pi$
$233$	−5.60006	−0.366872	−0.183436	+	0.983032i	$0.558722\pi$
$234$	7.51740	0.491428
$235$	−2.84567	−0.185631
$236$	50.9001	3.31332
$237$	−1.18848	−0.0772000
$238$	25.4412	1.64911
$239$	−17.1605	−1.11002	−0.555011	−	0.831843i	$-0.687286\pi$
$239$	−17.1605	−1.11002	−0.555011	+	0.831843i	$0.687286\pi$
$240$	−2.23852	−0.144496
$241$	17.8489	1.14975	0.574876	−	0.818241i	$-0.305050\pi$
$241$	17.8489	1.14975	0.574876	+	0.818241i	$0.305050\pi$
$242$	−19.0100	−1.22201
$243$	−3.59151	−0.230395
$244$	−57.8388	−3.70275
$245$	−4.42623	−0.282781
$246$	1.41572	0.0902629
$247$	1.70053	0.108202
$248$	−3.52209	−0.223653
$249$	−0.175610	−0.0111288
$250$	19.3657	1.22479
$251$	30.5955	1.93117	0.965586	−	0.260084i	$-0.0837501\pi$
$251$	30.5955	1.93117	0.965586	+	0.260084i	$0.0837501\pi$
$252$	30.0139	1.89070
$253$	−19.9914	−1.25685
$254$	49.1691	3.08514
$255$	−1.56252	−0.0978490
$256$	−31.3524	−1.95953
$257$	−12.8894	−0.804020	−0.402010	−	0.915635i	$-0.631688\pi$
$257$	−12.8894	−0.804020	−0.402010	+	0.915635i	$0.631688\pi$
$258$	−0.846450	−0.0526977
$259$	20.3457	1.26422
$260$	11.6216	0.720741
$261$	−8.63905	−0.534744
$262$	33.7883	2.08745
$263$	0.0281087	0.00173325	0.000866627	−	1.00000i	$-0.499724\pi$
$263$	0.0281087	0.00173325	0.000866627		1.00000i	$0.499724\pi$
$264$	−3.42769	−0.210960
$265$	−30.6822	−1.88479
$266$	9.90758	0.607472
$267$	2.23787	0.136956
$268$	35.9211	2.19423
$269$	−20.3903	−1.24322	−0.621610	−	0.783327i	$-0.713521\pi$
$269$	−20.3903	−1.24322	−0.621610	+	0.783327i	$0.713521\pi$
$270$	−5.39606	−0.328394
$271$	11.4585	0.696057	0.348028	−	0.937484i	$-0.386851\pi$
$271$	11.4585	0.696057	0.348028	+	0.937484i	$0.386851\pi$
$272$	27.3176	1.65637
$273$	0.309897	0.0187558
$274$	−34.0937	−2.05968
$275$	−9.13418	−0.550812
$276$	−2.71116	−0.163193
$277$	17.0459	1.02419	0.512095	−	0.858929i	$-0.328870\pi$
$277$	17.0459	1.02419	0.512095	+	0.858929i	$0.328870\pi$
$278$	8.74644	0.524576
$279$	−1.76916	−0.105917
$280$	36.6142	2.18812
$281$	−6.55549	−0.391068	−0.195534	−	0.980697i	$-0.562644\pi$
$281$	−6.55549	−0.391068	−0.195534	+	0.980697i	$0.562644\pi$
$282$	−0.360457	−0.0214649
$283$	−4.77508	−0.283849	−0.141924	−	0.989877i	$-0.545329\pi$
$283$	−4.77508	−0.283849	−0.141924	+	0.989877i	$0.545329\pi$
$284$	26.8385	1.59257
$285$	−0.608495	−0.0360441
$286$	10.8548	0.641859
$287$	−9.67950	−0.571363
$288$	11.6213	0.684792
$289$	2.06816	0.121657
$290$	−19.4891	−1.14444
$291$	−1.32711	−0.0777967
$292$	36.5986	2.14177
$293$	−26.3525	−1.53953	−0.769765	−	0.638327i	$-0.779627\pi$
$293$	−26.3525	−1.53953	−0.769765	+	0.638327i	$0.779627\pi$
$294$	−0.560665	−0.0326986
$295$	31.1898	1.81594
$296$	52.2627	3.03771
$297$	−3.45387	−0.200414
$298$	22.3018	1.29191
$299$	4.64277	0.268498
$300$	−1.23874	−0.0715189
$301$	5.78732	0.333576
$302$	−15.5486	−0.894724
$303$	1.13680	0.0653077
$304$	10.6383	0.610149
$305$	−35.4416	−2.02938
$306$	32.8263	1.87656
$307$	−29.7198	−1.69620	−0.848100	−	0.529836i	$-0.822254\pi$
$307$	−29.7198	−1.69620	−0.848100	+	0.529836i	$0.822254\pi$
$308$	43.3389	2.46946
$309$	−1.12426	−0.0639572
$310$	−3.99111	−0.226680
$311$	11.0739	0.627941	0.313971	−	0.949433i	$-0.398341\pi$
$311$	11.0739	0.627941	0.313971	+	0.949433i	$0.398341\pi$
$312$	0.796042	0.0450670
$313$	−33.6092	−1.89970	−0.949852	−	0.312699i	$-0.898767\pi$
$313$	−33.6092	−1.89970	−0.949852	+	0.312699i	$0.898767\pi$
$314$	−37.3307	−2.10669
$315$	18.3915	1.03624
$316$	38.5998	2.17141
$317$	−34.0084	−1.91010	−0.955052	−	0.296437i	$-0.904201\pi$
$317$	−34.0084	−1.91010	−0.955052	+	0.296437i	$0.904201\pi$
$318$	−3.88648	−0.217943
$319$	−12.4744	−0.698435
$320$	−7.17170	−0.400910
$321$	2.30147	0.128456
$322$	27.0495	1.50741
$323$	7.42573	0.413179
$324$	38.4915	2.13842
$325$	2.12131	0.117669
$326$	−35.5542	−1.96917
$327$	−1.05860	−0.0585406
$328$	−24.8640	−1.37289
$329$	2.46451	0.135873
$330$	−3.88414	−0.213815
$331$	−22.4861	−1.23595	−0.617975	−	0.786198i	$-0.712047\pi$
$331$	−22.4861	−1.23595	−0.617975	+	0.786198i	$0.712047\pi$
$332$	5.70352	0.313021
$333$	26.2518	1.43859
$334$	−4.79752	−0.262509
$335$	22.0112	1.20260
$336$	1.93868	0.105764
$337$	0.973163	0.0530116	0.0265058	−	0.999649i	$-0.491562\pi$
$337$	0.973163	0.0530116	0.0265058	+	0.999649i	$0.491562\pi$
$338$	−2.52091	−0.137119
$339$	0.739121	0.0401436
$340$	50.7481	2.75220
$341$	−2.55460	−0.138339
$342$	12.7836	0.691257
$343$	20.0113	1.08051
$344$	14.8661	0.801524
$345$	−1.66130	−0.0894416
$346$	−26.8274	−1.44225
$347$	25.3284	1.35970	0.679851	−	0.733350i	$-0.262045\pi$
$347$	25.3284	1.35970	0.679851	+	0.733350i	$0.262045\pi$
$348$	−1.69174	−0.0906868
$349$	7.23725	0.387401	0.193701	−	0.981061i	$-0.437951\pi$
$349$	7.23725	0.387401	0.193701	+	0.981061i	$0.437951\pi$
$350$	12.3591	0.660621
$351$	0.802121	0.0428141
$352$	16.7807	0.894414
$353$	−26.3637	−1.40320	−0.701599	−	0.712572i	$-0.747530\pi$
$353$	−26.3637	−1.40320	−0.701599	+	0.712572i	$0.747530\pi$
$354$	3.95077	0.209981
$355$	16.4457	0.872847
$356$	−72.6824	−3.85216
$357$	1.35323	0.0716206
$358$	−19.9329	−1.05349
$359$	32.4237	1.71126	0.855629	−	0.517590i	$-0.173171\pi$
$359$	32.4237	1.71126	0.855629	+	0.517590i	$0.173171\pi$
$360$	47.2427	2.48991
$361$	−16.1082	−0.847800
$362$	−59.9251	−3.14959
$363$	−1.01115	−0.0530719
$364$	−10.0649	−0.527547
$365$	22.4263	1.17385
$366$	−4.48934	−0.234662
$367$	7.49989	0.391491	0.195746	−	0.980655i	$-0.437287\pi$
$367$	7.49989	0.391491	0.195746	+	0.980655i	$0.437287\pi$
$368$	29.0446	1.51405
$369$	−12.4893	−0.650167
$370$	59.2222	3.07882
$371$	26.5725	1.37958
$372$	−0.346446	−0.0179624
$373$	−17.8886	−0.926239	−0.463119	−	0.886296i	$-0.653270\pi$
$373$	−17.8886	−0.926239	−0.463119	+	0.886296i	$0.653270\pi$
$374$	47.3999	2.45099
$375$	1.03007	0.0531927
$376$	6.33065	0.326478
$377$	2.89705	0.149205
$378$	4.67329	0.240368
$379$	16.5101	0.848065	0.424033	−	0.905647i	$-0.360614\pi$
$379$	16.5101	0.848065	0.424033	+	0.905647i	$0.360614\pi$
$380$	19.7629	1.01382
$381$	2.61533	0.133988
$382$	−22.0926	−1.13036
$383$	−10.7627	−0.549950	−0.274975	−	0.961451i	$-0.588670\pi$
$383$	−10.7627	−0.549950	−0.274975	+	0.961451i	$0.588670\pi$
$384$	−1.95355	−0.0996917
$385$	26.5565	1.35345
$386$	16.4812	0.838872
$387$	7.46728	0.379583
$388$	43.1024	2.18819
$389$	−20.4939	−1.03908	−0.519541	−	0.854445i	$-0.673897\pi$
$389$	−20.4939	−1.03908	−0.519541	+	0.854445i	$0.673897\pi$
$390$	0.902047	0.0456769
$391$	20.2736	1.02528
$392$	9.84686	0.497342
$393$	1.79722	0.0906579
$394$	−56.6918	−2.85609
$395$	23.6526	1.19009
$396$	55.9194	2.81006
$397$	5.81577	0.291885	0.145943	−	0.989293i	$-0.453379\pi$
$397$	5.81577	0.291885	0.145943	+	0.989293i	$0.453379\pi$
$398$	48.2014	2.41612
$399$	0.526990	0.0263825
$400$	13.2706	0.663532
$401$	−37.5287	−1.87409	−0.937046	−	0.349205i	$-0.886452\pi$
$401$	−37.5287	−1.87409	−0.937046	+	0.349205i	$0.886452\pi$
$402$	2.78813	0.139059
$403$	0.593276	0.0295532
$404$	−36.9215	−1.83691
$405$	23.5862	1.17201
$406$	16.8787	0.837674
$407$	37.9065	1.87896
$408$	3.47608	0.172092
$409$	−3.41907	−0.169062	−0.0845310	−	0.996421i	$-0.526939\pi$
$409$	−3.41907	−0.169062	−0.0845310	+	0.996421i	$0.526939\pi$
$410$	−28.1750	−1.39147
$411$	−1.81346	−0.0894516
$412$	36.5142	1.79893
$413$	−27.0121	−1.32918
$414$	34.9016	1.71532
$415$	3.49492	0.171559
$416$	−3.89712	−0.191072
$417$	0.465228	0.0227823
$418$	18.4590	0.902858
$419$	0.550538	0.0268956	0.0134478	−	0.999910i	$-0.495719\pi$
$419$	0.550538	0.0268956	0.0134478	+	0.999910i	$0.495719\pi$
$420$	3.60150	0.175735
$421$	−29.2404	−1.42509	−0.712545	−	0.701626i	$-0.752458\pi$
$421$	−29.2404	−1.42509	−0.712545	+	0.701626i	$0.752458\pi$
$422$	50.4052	2.45369
$423$	3.17991	0.154613
$424$	68.2575	3.31488
$425$	9.26313	0.449328
$426$	2.08316	0.100929
$427$	30.6944	1.48541
$428$	−74.7481	−3.61308
$429$	0.577375	0.0278759
$430$	16.8457	0.812372
$431$	30.0688	1.44836	0.724181	−	0.689610i	$-0.242218\pi$
$431$	30.0688	1.44836	0.724181	+	0.689610i	$0.242218\pi$
$432$	5.01797	0.241427
$433$	−12.5009	−0.600756	−0.300378	−	0.953820i	$-0.597113\pi$
$433$	−12.5009	−0.600756	−0.300378	+	0.953820i	$0.597113\pi$
$434$	3.45652	0.165918
$435$	−1.03664	−0.0497030
$436$	34.3815	1.64658
$437$	7.89518	0.377678
$438$	2.84071	0.135734
$439$	16.2255	0.774402	0.387201	−	0.921995i	$-0.373442\pi$
$439$	16.2255	0.774402	0.387201	+	0.921995i	$0.373442\pi$
$440$	68.2165	3.25210
$441$	4.94612	0.235530
$442$	−11.0081	−0.523601
$443$	−17.1852	−0.816493	−0.408246	−	0.912872i	$-0.633860\pi$
$443$	−17.1852	−0.816493	−0.408246	+	0.912872i	$0.633860\pi$
$444$	5.14074	0.243969
$445$	−44.5372	−2.11127
$446$	22.7354	1.07655
$447$	1.18625	0.0561076
$448$	6.21109	0.293446
$449$	−17.2635	−0.814715	−0.407357	−	0.913269i	$-0.633550\pi$
$449$	−17.2635	−0.814715	−0.407357	+	0.913269i	$0.633550\pi$
$450$	15.9467	0.751735
$451$	−18.0341	−0.849190
$452$	−24.0054	−1.12912
$453$	−0.827042	−0.0388578
$454$	45.3930	2.13040
$455$	−6.16745	−0.289134
$456$	1.35369	0.0633926
$457$	20.0952	0.940013	0.470007	−	0.882663i	$-0.344252\pi$
$457$	20.0952	0.940013	0.470007	+	0.882663i	$0.344252\pi$
$458$	−20.8557	−0.974523
$459$	3.50263	0.163489
$460$	53.9564	2.51573
$461$	27.2188	1.26770	0.633852	−	0.773454i	$-0.281473\pi$
$461$	27.2188	1.26770	0.633852	+	0.773454i	$0.281473\pi$
$462$	3.36388	0.156502
$463$	−38.0902	−1.77020	−0.885100	−	0.465400i	$-0.845910\pi$
$463$	−38.0902	−1.77020	−0.885100	+	0.465400i	$0.845910\pi$
$464$	18.1236	0.841365
$465$	−0.212290	−0.00984470
$466$	14.1172	0.653969
$467$	−28.8973	−1.33721	−0.668604	−	0.743618i	$-0.733108\pi$
$467$	−28.8973	−1.33721	−0.668604	+	0.743618i	$0.733108\pi$
$468$	−12.9866	−0.600307
$469$	−19.0629	−0.880245
$470$	7.17367	0.330897
$471$	−1.98564	−0.0914936
$472$	−69.3868	−3.19379
$473$	10.7825	0.495778
$474$	2.99605	0.137613
$475$	3.60735	0.165517
$476$	−43.9507	−2.01448
$477$	34.2860	1.56985
$478$	43.2601	1.97867
$479$	−32.8177	−1.49948	−0.749739	−	0.661733i	$-0.769821\pi$
$479$	−32.8177	−1.49948	−0.749739	+	0.661733i	$0.769821\pi$
$480$	1.39449	0.0636496
$481$	−8.80335	−0.401398
$482$	−44.9955	−2.04949
$483$	1.43878	0.0654668
$484$	32.8406	1.49276
$485$	26.4116	1.19929
$486$	9.05386	0.410691
$487$	−21.8058	−0.988114	−0.494057	−	0.869430i	$-0.664487\pi$
$487$	−21.8058	−0.988114	−0.494057	+	0.869430i	$0.664487\pi$
$488$	78.8456	3.56917
$489$	−1.89115	−0.0855209
$490$	11.1581	0.504072
$491$	2.72274	0.122875	0.0614377	−	0.998111i	$-0.480431\pi$
$491$	2.72274	0.122875	0.0614377	+	0.998111i	$0.480431\pi$
$492$	−2.44571	−0.110261
$493$	12.6506	0.569753
$494$	−4.28688	−0.192876
$495$	34.2654	1.54012
$496$	3.71146	0.166650
$497$	−14.2429	−0.638881
$498$	0.442697	0.0198377
$499$	33.5633	1.50250	0.751249	−	0.660019i	$-0.229452\pi$
$499$	33.5633	1.50250	0.751249	+	0.660019i	$0.229452\pi$
$500$	−33.4550	−1.49615
$501$	−0.255183	−0.0114007
$502$	−77.1285	−3.44241
$503$	−22.8760	−1.01999	−0.509995	−	0.860177i	$-0.670353\pi$
$503$	−22.8760	−1.01999	−0.509995	+	0.860177i	$0.670353\pi$
$504$	−40.9148	−1.82249
$505$	−22.6242	−1.00676
$506$	50.3965	2.24040
$507$	−0.134089	−0.00595509
$508$	−84.9417	−3.76868
$509$	3.26178	0.144576	0.0722880	−	0.997384i	$-0.476970\pi$
$509$	3.26178	0.144576	0.0722880	+	0.997384i	$0.476970\pi$
$510$	3.93898	0.174421
$511$	−19.4224	−0.859198
$512$	49.8984	2.20522
$513$	1.36403	0.0602235
$514$	32.4931	1.43321
$515$	22.3746	0.985945
$516$	1.46228	0.0643732
$517$	4.59167	0.201941
$518$	−51.2897	−2.25354
$519$	−1.42697	−0.0626369
$520$	−15.8425	−0.694740
$521$	−39.9467	−1.75010	−0.875048	−	0.484036i	$-0.839170\pi$
$521$	−39.9467	−1.75010	−0.875048	+	0.484036i	$0.839170\pi$
$522$	21.7783	0.953208
$523$	30.2971	1.32480	0.662400	−	0.749151i	$-0.269538\pi$
$523$	30.2971	1.32480	0.662400	+	0.749151i	$0.269538\pi$
$524$	−58.3708	−2.54994
$525$	0.657387	0.0286907
$526$	−0.0708593	−0.00308961
$527$	2.59067	0.112851
$528$	3.61199	0.157192
$529$	−1.44469	−0.0628125
$530$	77.3470	3.35974
$531$	−34.8533	−1.51250
$532$	−17.1158	−0.742063
$533$	4.18820	0.181411
$534$	−5.64147	−0.244130
$535$	−45.8030	−1.98024
$536$	−48.9675	−2.11507
$537$	−1.06024	−0.0457529
$538$	51.4021	2.21610
$539$	7.14200	0.307628
$540$	9.32193	0.401152
$541$	28.9666	1.24537	0.622686	−	0.782472i	$-0.286041\pi$
$541$	28.9666	1.24537	0.622686	+	0.782472i	$0.286041\pi$
$542$	−28.8859	−1.24076
$543$	−3.18745	−0.136787
$544$	−17.0176	−0.729624
$545$	21.0678	0.902445
$546$	−0.781223	−0.0334332
$547$	22.7500	0.972719	0.486360	−	0.873759i	$-0.338324\pi$
$547$	22.7500	0.972719	0.486360	+	0.873759i	$0.338324\pi$
$548$	58.8983	2.51601
$549$	39.6045	1.69028
$550$	23.0264	0.981850
$551$	4.92652	0.209877
$552$	3.69584	0.157305
$553$	−20.4845	−0.871089
$554$	−42.9712	−1.82567
$555$	3.15007	0.133713
$556$	−15.1098	−0.640800
$557$	−17.4669	−0.740098	−0.370049	−	0.929012i	$-0.620659\pi$
$557$	−17.4669	−0.740098	−0.370049	+	0.929012i	$0.620659\pi$
$558$	4.45989	0.188802
$559$	−2.50410	−0.105912
$560$	−38.5828	−1.63042
$561$	2.52123	0.106446
$562$	16.5258	0.697098
$563$	−30.8146	−1.29868	−0.649341	−	0.760498i	$-0.724955\pi$
$563$	−30.8146	−1.29868	−0.649341	+	0.760498i	$0.724955\pi$
$564$	0.622705	0.0262206
$565$	−14.7097	−0.618841
$566$	12.0375	0.505975
$567$	−20.4270	−0.857853
$568$	−36.5861	−1.53512
$569$	0.238834	0.0100125	0.00500623	−	0.999987i	$-0.498406\pi$
$569$	0.238834	0.0100125	0.00500623	+	0.999987i	$0.498406\pi$
$570$	1.53396	0.0642505
$571$	35.0106	1.46515	0.732575	−	0.680686i	$-0.238318\pi$
$571$	35.0106	1.46515	0.732575	+	0.680686i	$0.238318\pi$
$572$	−18.7522	−0.784068
$573$	−1.17512	−0.0490913
$574$	24.4011	1.01848
$575$	9.84874	0.410721
$576$	8.01406	0.333919
$577$	−30.6537	−1.27613	−0.638064	−	0.769983i	$-0.720265\pi$
$577$	−30.6537	−1.27613	−0.638064	+	0.769983i	$0.720265\pi$
$578$	−5.21364	−0.216859
$579$	0.876647	0.0364322
$580$	33.6683	1.39800
$581$	−3.02679	−0.125573
$582$	3.34553	0.138677
$583$	49.5077	2.05040
$584$	−49.8910	−2.06450
$585$	−7.95775	−0.329013
$586$	66.4323	2.74429
$587$	−17.8569	−0.737032	−0.368516	−	0.929621i	$-0.620134\pi$
$587$	−17.8569	−0.737032	−0.368516	+	0.929621i	$0.620134\pi$
$588$	0.968573	0.0399433
$589$	1.00889	0.0415704
$590$	−78.6267	−3.23701
$591$	−3.01547	−0.124040
$592$	−55.0727	−2.26347
$593$	−20.3269	−0.834725	−0.417363	−	0.908740i	$-0.637046\pi$
$593$	−20.3269	−0.834725	−0.417363	+	0.908740i	$0.637046\pi$
$594$	8.70689	0.357248
$595$	−26.9315	−1.10408
$596$	−38.5273	−1.57814
$597$	2.56386	0.104932
$598$	−11.7040	−0.478612
$599$	−40.8749	−1.67010	−0.835052	−	0.550172i	$-0.814562\pi$
$599$	−40.8749	−1.67010	−0.835052	+	0.550172i	$0.814562\pi$
$600$	1.68865	0.0689388
$601$	11.5645	0.471726	0.235863	−	0.971786i	$-0.424208\pi$
$601$	11.5645	0.471726	0.235863	+	0.971786i	$0.424208\pi$
$602$	−14.5893	−0.594616
$603$	−24.5966	−1.00165
$604$	26.8609	1.09296
$605$	20.1236	0.818140
$606$	−2.86578	−0.116414
$607$	−35.1241	−1.42564	−0.712822	−	0.701345i	$-0.752583\pi$
$607$	−35.1241	−1.42564	−0.712822	+	0.701345i	$0.752583\pi$
$608$	−6.62718	−0.268768
$609$	0.897787	0.0363802
$610$	89.3450	3.61748
$611$	−1.06636	−0.0431404
$612$	−56.7089	−2.29232
$613$	−25.0637	−1.01231	−0.506157	−	0.862441i	$-0.668934\pi$
$613$	−25.0637	−1.01231	−0.506157	+	0.862441i	$0.668934\pi$
$614$	74.9210	3.02356
$615$	−1.49865	−0.0604313
$616$	−59.0793	−2.38037
$617$	19.5377	0.786557	0.393279	−	0.919419i	$-0.371341\pi$
$617$	19.5377	0.786557	0.393279	+	0.919419i	$0.371341\pi$
$618$	2.83417	0.114007
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	6.89482	0.276903
$621$	3.72406	0.149442
$622$	−27.9162	−1.11934
$623$	38.5717	1.54534
$624$	−0.838842	−0.0335806
$625$	−31.1066	−1.24426
$626$	84.7257	3.38632
$627$	0.981845	0.0392111
$628$	64.4904	2.57345
$629$	−38.4417	−1.53277
$630$	−46.3632	−1.84715
$631$	33.6900	1.34118	0.670588	−	0.741830i	$-0.266042\pi$
$631$	33.6900	1.34118	0.670588	+	0.741830i	$0.266042\pi$
$632$	−52.6191	−2.09308
$633$	2.68108	0.106563
$634$	85.7322	3.40486
$635$	−52.0493	−2.06551
$636$	6.71405	0.266230
$637$	−1.65865	−0.0657180
$638$	31.4469	1.24500
$639$	−18.3774	−0.726997
$640$	38.8787	1.53682
$641$	28.2358	1.11525	0.557623	−	0.830094i	$-0.311713\pi$
$641$	28.2358	1.11525	0.557623	+	0.830094i	$0.311713\pi$
$642$	−5.80180	−0.228979
$643$	−40.3508	−1.59128	−0.795641	−	0.605769i	$-0.792866\pi$
$643$	−40.3508	−1.59128	−0.795641	+	0.605769i	$0.792866\pi$
$644$	−46.7292	−1.84139
$645$	0.896033	0.0352813
$646$	−18.7196	−0.736512
$647$	−33.2397	−1.30679	−0.653393	−	0.757019i	$-0.726655\pi$
$647$	−33.2397	−1.30679	−0.653393	+	0.757019i	$0.726655\pi$
$648$	−52.4714	−2.06127
$649$	−50.3267	−1.97550
$650$	−5.34762	−0.209751
$651$	0.183855	0.00720583
$652$	61.4215	2.40545
$653$	−38.6940	−1.51421	−0.757106	−	0.653292i	$-0.773388\pi$
$653$	−38.6940	−1.51421	−0.757106	+	0.653292i	$0.773388\pi$
$654$	2.66863	0.104352
$655$	−35.7676	−1.39756
$656$	26.2009	1.02297
$657$	−25.0604	−0.977701
$658$	−6.21280	−0.242200
$659$	−16.1560	−0.629347	−0.314674	−	0.949200i	$-0.601895\pi$
$659$	−16.1560	−0.629347	−0.314674	+	0.949200i	$0.601895\pi$
$660$	6.71002	0.261187
$661$	−28.7146	−1.11687	−0.558435	−	0.829548i	$-0.688598\pi$
$661$	−28.7146	−1.11687	−0.558435	+	0.829548i	$0.688598\pi$
$662$	56.6855	2.20314
$663$	−0.585526	−0.0227400
$664$	−7.77501	−0.301729
$665$	−10.4879	−0.406705
$666$	−66.1783	−2.56436
$667$	13.4503	0.520798
$668$	8.28792	0.320669
$669$	1.20931	0.0467547
$670$	−55.4882	−2.14370
$671$	57.1873	2.20769
$672$	−1.20771	−0.0465884
$673$	−3.66181	−0.141152	−0.0705762	−	0.997506i	$-0.522484\pi$
$673$	−3.66181	−0.141152	−0.0705762	+	0.997506i	$0.522484\pi$
$674$	−2.45325	−0.0944959
$675$	1.70155	0.0654925
$676$	4.35498	0.167499
$677$	19.2590	0.740183	0.370091	−	0.928995i	$-0.379326\pi$
$677$	19.2590	0.740183	0.370091	+	0.928995i	$0.379326\pi$
$678$	−1.86326	−0.0715579
$679$	−22.8740	−0.877822
$680$	−69.1796	−2.65292
$681$	2.41448	0.0925231
$682$	6.43991	0.246597
$683$	48.0493	1.83855	0.919277	−	0.393611i	$-0.128774\pi$
$683$	48.0493	1.83855	0.919277	+	0.393611i	$0.128774\pi$
$684$	−22.0842	−0.844410
$685$	36.0908	1.37896
$686$	−50.4467	−1.92606
$687$	−1.10933	−0.0423235
$688$	−15.6654	−0.597236
$689$	−11.4976	−0.438023
$690$	4.18800	0.159434
$691$	2.45957	0.0935663	0.0467831	−	0.998905i	$-0.485103\pi$
$691$	2.45957	0.0935663	0.0467831	+	0.998905i	$0.485103\pi$
$692$	46.3455	1.76179
$693$	−29.6758	−1.12729
$694$	−63.8507	−2.42374
$695$	−9.25878	−0.351206
$696$	2.30617	0.0874152
$697$	18.2887	0.692732
$698$	−18.2444	−0.690562
$699$	0.750905	0.0284018
$700$	−21.3508	−0.806986
$701$	−51.1814	−1.93309	−0.966547	−	0.256488i	$-0.917435\pi$
$701$	−51.1814	−1.93309	−0.966547	+	0.256488i	$0.917435\pi$
$702$	−2.02207	−0.0763183
$703$	−14.9704	−0.564618
$704$	11.5720	0.436136
$705$	0.381572	0.0143708
$706$	66.4605	2.50127
$707$	19.5938	0.736902
$708$	−6.82513	−0.256504
$709$	0.507797	0.0190707	0.00953536	−	0.999955i	$-0.496965\pi$
$709$	0.507797	0.0190707	0.00953536	+	0.999955i	$0.496965\pi$
$710$	−41.4581	−1.55590
$711$	−26.4308	−0.991232
$712$	99.0803	3.71319
$713$	2.75445	0.103155
$714$	−3.41137	−0.127667
$715$	−11.4907	−0.429727
$716$	34.4350	1.28690
$717$	2.30103	0.0859337
$718$	−81.7372	−3.05041
$719$	40.8356	1.52291	0.761456	−	0.648217i	$-0.224485\pi$
$719$	40.8356	1.52291	0.761456	+	0.648217i	$0.224485\pi$
$720$	−49.7827	−1.85529
$721$	−19.3777	−0.721663
$722$	40.6073	1.51125
$723$	−2.39334	−0.0890093
$724$	103.523	3.84741
$725$	6.14552	0.228239
$726$	2.54903	0.0946033
$727$	7.64443	0.283516	0.141758	−	0.989901i	$-0.454724\pi$
$727$	7.64443	0.283516	0.141758	+	0.989901i	$0.454724\pi$
$728$	13.7205	0.508515
$729$	−26.0339	−0.964220
$730$	−56.5347	−2.09244
$731$	−10.9347	−0.404434
$732$	7.75553	0.286653
$733$	16.9914	0.627591	0.313795	−	0.949491i	$-0.398399\pi$
$733$	16.9914	0.627591	0.313795	+	0.949491i	$0.398399\pi$
$734$	−18.9065	−0.697853
$735$	0.593507	0.0218918
$736$	−18.0935	−0.666933
$737$	−35.5165	−1.30827
$738$	31.4844	1.15896
$739$	21.3547	0.785544	0.392772	−	0.919636i	$-0.371516\pi$
$739$	21.3547	0.785544	0.392772	+	0.919636i	$0.371516\pi$
$740$	−102.309	−3.76095
$741$	−0.228022	−0.00837660
$742$	−66.9868	−2.45916
$743$	−5.36281	−0.196743	−0.0983713	−	0.995150i	$-0.531363\pi$
$743$	−5.36281	−0.196743	−0.0983713	+	0.995150i	$0.531363\pi$
$744$	0.472273	0.0173144
$745$	−23.6082	−0.864937
$746$	45.0956	1.65107
$747$	−3.90542	−0.142892
$748$	−81.8854	−2.99403
$749$	39.6679	1.44943
$750$	−2.59672	−0.0948186
$751$	2.91113	0.106228	0.0531142	−	0.998588i	$-0.483085\pi$
$751$	2.91113	0.106228	0.0531142	+	0.998588i	$0.483085\pi$
$752$	−6.67103	−0.243267
$753$	−4.10251	−0.149504
$754$	−7.30319	−0.265966
$755$	16.4594	0.599020
$756$	−8.07331	−0.293623
$757$	−49.5981	−1.80268	−0.901338	−	0.433117i	$-0.857414\pi$
$757$	−49.5981	−1.80268	−0.901338	+	0.433117i	$0.857414\pi$
$758$	−41.6204	−1.51172
$759$	2.68062	0.0973003
$760$	−26.9407	−0.977241
$761$	−14.5873	−0.528791	−0.264395	−	0.964414i	$-0.585172\pi$
$761$	−14.5873	−0.528791	−0.264395	+	0.964414i	$0.585172\pi$
$762$	−6.59302	−0.238840
$763$	−18.2459	−0.660545
$764$	38.1659	1.38079
$765$	−34.7492	−1.25636
$766$	27.1319	0.980314
$767$	11.6878	0.422022
$768$	4.20400	0.151699
$769$	−22.9567	−0.827840	−0.413920	−	0.910313i	$-0.635841\pi$
$769$	−22.9567	−0.827840	−0.413920	+	0.910313i	$0.635841\pi$
$770$	−66.9466	−2.41259
$771$	1.72833	0.0622442
$772$	−28.4720	−1.02473
$773$	48.5768	1.74719	0.873594	−	0.486656i	$-0.161783\pi$
$773$	48.5768	1.74719	0.873594	+	0.486656i	$0.161783\pi$
$774$	−18.8243	−0.676627
$775$	1.25852	0.0452074
$776$	−58.7570	−2.10925
$777$	−2.72813	−0.0978712
$778$	51.6633	1.85222
$779$	7.12217	0.255178
$780$	−1.55832	−0.0557970
$781$	−26.5362	−0.949539
$782$	−51.1080	−1.82762
$783$	2.32378	0.0830452
$784$	−10.3763	−0.370582
$785$	39.5174	1.41044
$786$	−4.53063	−0.161602
$787$	37.9621	1.35320	0.676601	−	0.736350i	$-0.263452\pi$
$787$	37.9621	1.35320	0.676601	+	0.736350i	$0.263452\pi$
$788$	97.9376	3.48888
$789$	−0.00376905	−0.000134182 0
$790$	−59.6261	−2.12140
$791$	12.7394	0.452961
$792$	−76.2290	−2.70868
$793$	−13.2811	−0.471625
$794$	−14.6610	−0.520300
$795$	4.11414	0.145913
$796$	−83.2700	−2.95143
$797$	28.4925	1.00926	0.504628	−	0.863337i	$-0.331630\pi$
$797$	28.4925	1.00926	0.504628	+	0.863337i	$0.331630\pi$
$798$	−1.32849	−0.0470282
$799$	−4.65649	−0.164735
$800$	−8.26700	−0.292282
$801$	49.7684	1.75848
$802$	94.6063	3.34067
$803$	−36.1863	−1.27699
$804$	−4.81662	−0.169869
$805$	−28.6340	−1.00922
$806$	−1.49559	−0.0526801
$807$	2.73411	0.0962453
$808$	50.3312	1.77065
$809$	−49.8076	−1.75114	−0.875570	−	0.483091i	$-0.839514\pi$
$809$	−49.8076	−1.75114	−0.875570	+	0.483091i	$0.839514\pi$
$810$	−59.4587	−2.08917
$811$	3.38009	0.118691	0.0593455	−	0.998238i	$-0.481099\pi$
$811$	3.38009	0.118691	0.0593455	+	0.998238i	$0.481099\pi$
$812$	−29.1586	−1.02327
$813$	−1.53646	−0.0538860
$814$	−95.5588	−3.34933
$815$	37.6369	1.31836
$816$	−3.66298	−0.128230
$817$	−4.25831	−0.148979
$818$	8.61915	0.301362
$819$	6.89186	0.240821
$820$	48.6736	1.69976
$821$	27.2381	0.950617	0.475309	−	0.879819i	$-0.342336\pi$
$821$	27.2381	0.950617	0.475309	+	0.879819i	$0.342336\pi$
$822$	4.57158	0.159452
$823$	43.7128	1.52373	0.761866	−	0.647735i	$-0.224284\pi$
$823$	43.7128	1.52373	0.761866	+	0.647735i	$0.224284\pi$
$824$	−49.7760	−1.73403
$825$	1.22479	0.0426417
$826$	68.0951	2.36933
$827$	−36.8412	−1.28109	−0.640546	−	0.767919i	$-0.721292\pi$
$827$	−36.8412	−1.28109	−0.640546	+	0.767919i	$0.721292\pi$
$828$	−60.2939	−2.09536
$829$	5.99200	0.208111	0.104055	−	0.994572i	$-0.466818\pi$
$829$	5.99200	0.208111	0.104055	+	0.994572i	$0.466818\pi$
$830$	−8.81037	−0.305812
$831$	−2.28566	−0.0792888
$832$	−2.68746	−0.0931709
$833$	−7.24283	−0.250949
$834$	−1.17280	−0.0406107
$835$	5.07855	0.175750
$836$	−31.8887	−1.10289
$837$	0.475879	0.0164488
$838$	−1.38786	−0.0479427
$839$	−9.83153	−0.339422	−0.169711	−	0.985494i	$-0.554283\pi$
$839$	−9.83153	−0.339422	−0.169711	+	0.985494i	$0.554283\pi$
$840$	−4.90955	−0.169395
$841$	−20.6071	−0.710590
$842$	73.7124	2.54030
$843$	0.879017	0.0302749
$844$	−87.0771	−2.99732
$845$	2.66858	0.0918019
$846$	−8.01626	−0.275605
$847$	−17.4281	−0.598838
$848$	−71.9275	−2.47000
$849$	0.640284	0.0219745
$850$	−23.3515	−0.800950
$851$	−40.8719	−1.40107
$852$	−3.59874	−0.123291
$853$	−23.9072	−0.818566	−0.409283	−	0.912407i	$-0.634221\pi$
$853$	−23.9072	−0.818566	−0.409283	+	0.912407i	$0.634221\pi$
$854$	−77.3778	−2.64781
$855$	−13.5324	−0.462799
$856$	101.896	3.48274
$857$	−3.32465	−0.113568	−0.0567839	−	0.998386i	$-0.518085\pi$
$857$	−3.32465	−0.113568	−0.0567839	+	0.998386i	$0.518085\pi$
$858$	−1.45551	−0.0496903
$859$	32.0118	1.09223	0.546114	−	0.837711i	$-0.316107\pi$
$859$	32.0118	1.09223	0.546114	+	0.837711i	$0.316107\pi$
$860$	−29.1017	−0.992359
$861$	1.29791	0.0442327
$862$	−75.8006	−2.58178
$863$	7.78569	0.265028	0.132514	−	0.991181i	$-0.457695\pi$
$863$	7.78569	0.265028	0.132514	+	0.991181i	$0.457695\pi$
$864$	−3.12597	−0.106348
$865$	28.3989	0.965592
$866$	31.5137	1.07088
$867$	−0.277317	−0.00941818
$868$	−5.97129	−0.202679
$869$	−38.1650	−1.29466
$870$	2.61327	0.0885982
$871$	8.24829	0.279483
$872$	−46.8687	−1.58717
$873$	−29.5139	−0.998893
$874$	−19.9030	−0.673230
$875$	17.7542	0.600201
$876$	−4.90745	−0.165807
$877$	28.0023	0.945571	0.472786	−	0.881178i	$-0.343249\pi$
$877$	28.0023	0.945571	0.472786	+	0.881178i	$0.343249\pi$
$878$	−40.9030	−1.38041
$879$	3.53358	0.119185
$880$	−71.8843	−2.42322
$881$	−7.95159	−0.267896	−0.133948	−	0.990988i	$-0.542766\pi$
$881$	−7.95159	−0.267896	−0.133948	+	0.990988i	$0.542766\pi$
$882$	−12.4687	−0.419843
$883$	−28.8971	−0.972463	−0.486232	−	0.873830i	$-0.661629\pi$
$883$	−28.8971	−0.972463	−0.486232	+	0.873830i	$0.661629\pi$
$884$	19.0169	0.639609
$885$	−4.18220	−0.140583
$886$	43.3223	1.45544
$887$	−58.6775	−1.97020	−0.985098	−	0.171992i	$-0.944980\pi$
$887$	−58.6775	−1.97020	−0.985098	+	0.171992i	$0.944980\pi$
$888$	−7.00783	−0.235168
$889$	45.0776	1.51185
$890$	112.274	3.76344
$891$	−38.0579	−1.27499
$892$	−39.2764	−1.31507
$893$	−1.81338	−0.0606825
$894$	−2.99042	−0.100015
$895$	21.1005	0.705313
$896$	−33.6712	−1.12487
$897$	−0.622543	−0.0207861
$898$	43.5197	1.45227
$899$	1.71875	0.0573235
$900$	−27.5486	−0.918288
$901$	−50.2066	−1.67263
$902$	45.4622	1.51373
$903$	−0.776015	−0.0258241
$904$	32.7241	1.08839
$905$	63.4354	2.10866
$906$	2.08490	0.0692660
$907$	−39.7518	−1.31994	−0.659968	−	0.751294i	$-0.729430\pi$
$907$	−39.7518	−1.31994	−0.659968	+	0.751294i	$0.729430\pi$
$908$	−78.4183	−2.60240
$909$	25.2816	0.838537
$910$	15.5476	0.515397
$911$	19.2810	0.638807	0.319403	−	0.947619i	$-0.396517\pi$
$911$	19.2810	0.638807	0.319403	+	0.947619i	$0.396517\pi$
$912$	−1.42648	−0.0472354
$913$	−5.63927	−0.186633
$914$	−50.6581	−1.67562
$915$	4.75232	0.157107
$916$	36.0291	1.19044
$917$	30.9767	1.02294
$918$	−8.82981	−0.291427
$919$	26.5001	0.874159	0.437080	−	0.899423i	$-0.356013\pi$
$919$	26.5001	0.874159	0.437080	+	0.899423i	$0.356013\pi$
$920$	−73.5531	−2.42497
$921$	3.98509	0.131313
$922$	−68.6160	−2.25975
$923$	6.16272	0.202848
$924$	−5.81125	−0.191176
$925$	−18.6746	−0.614017
$926$	96.0219	3.15547
$927$	−25.0027	−0.821196
$928$	−11.2901	−0.370617
$929$	−37.6606	−1.23560	−0.617802	−	0.786333i	$-0.711977\pi$
$929$	−37.6606	−1.23560	−0.617802	+	0.786333i	$0.711977\pi$
$930$	0.535163	0.0175487
$931$	−2.82058	−0.0924409
$932$	−24.3882	−0.798860
$933$	−1.48488	−0.0486128
$934$	72.8475	2.38364
$935$	−50.1765	−1.64095
$936$	17.7033	0.578651
$937$	−29.4009	−0.960487	−0.480243	−	0.877135i	$-0.659452\pi$
$937$	−29.4009	−0.960487	−0.480243	+	0.877135i	$0.659452\pi$
$938$	48.0559	1.56908
$939$	4.50661	0.147068
$940$	−12.3928	−0.404209
$941$	16.3793	0.533950	0.266975	−	0.963703i	$-0.413976\pi$
$941$	16.3793	0.533950	0.266975	+	0.963703i	$0.413976\pi$
$942$	5.00562	0.163092
$943$	19.4449	0.633212
$944$	73.1175	2.37977
$945$	−4.94704	−0.160927
$946$	−27.1816	−0.883750
$947$	8.30810	0.269977	0.134988	−	0.990847i	$-0.456900\pi$
$947$	8.30810	0.269977	0.134988	+	0.990847i	$0.456900\pi$
$948$	−5.17580	−0.168102
$949$	8.40385	0.272800
$950$	−9.09380	−0.295042
$951$	4.56015	0.147873
$952$	59.9134	1.94180
$953$	−26.8427	−0.869522	−0.434761	−	0.900546i	$-0.643167\pi$
$953$	−26.8427	−0.869522	−0.434761	+	0.900546i	$0.643167\pi$
$954$	−86.4320	−2.79834
$955$	23.3867	0.756777
$956$	−74.7338	−2.41706
$957$	1.67268	0.0540701
$958$	82.7304	2.67290
$959$	−31.2567	−1.00933
$960$	0.961644	0.0310369
$961$	−30.6480	−0.988646
$962$	22.1924	0.715513
$963$	51.1829	1.64934
$964$	77.7318	2.50357
$965$	−17.4467	−0.561628
$966$	−3.62704	−0.116698
$967$	20.8408	0.670196	0.335098	−	0.942183i	$-0.391230\pi$
$967$	20.8408	0.670196	0.335098	+	0.942183i	$0.391230\pi$
$968$	−44.7682	−1.43890
$969$	−0.995706	−0.0319867
$970$	−66.5813	−2.13780
$971$	48.9512	1.57092	0.785459	−	0.618914i	$-0.212427\pi$
$971$	48.9512	1.57092	0.785459	+	0.618914i	$0.212427\pi$
$972$	−15.6409	−0.501683
$973$	8.01862	0.257065
$974$	54.9703	1.76136
$975$	−0.284443	−0.00910947
$976$	−83.0848	−2.65948
$977$	−10.8291	−0.346454	−0.173227	−	0.984882i	$-0.555419\pi$
$977$	−10.8291	−0.346454	−0.173227	+	0.984882i	$0.555419\pi$
$978$	4.76742	0.152445
$979$	71.8636	2.29677
$980$	−19.2761	−0.615753
$981$	−23.5423	−0.751649
$982$	−6.86377	−0.219032
$983$	−15.7468	−0.502244	−0.251122	−	0.967955i	$-0.580800\pi$
$983$	−15.7468	−0.502244	−0.251122	+	0.967955i	$0.580800\pi$
$984$	3.33398	0.106284
$985$	60.0127	1.91216
$986$	−31.8909	−1.01561
$987$	−0.330463	−0.0105187
$988$	7.40578	0.235609
$989$	−11.6260	−0.369684
$990$	−86.3800	−2.74534
$991$	−32.7034	−1.03886	−0.519428	−	0.854514i	$-0.673855\pi$
$991$	−32.7034	−1.03886	−0.519428	+	0.854514i	$0.673855\pi$
$992$	−2.31207	−0.0734083
$993$	3.01514	0.0956824
$994$	35.9050	1.13884
$995$	−51.0249	−1.61760
$996$	−0.764778	−0.0242329
$997$	−5.11255	−0.161916	−0.0809581	−	0.996718i	$-0.525798\pi$
$997$	−5.11255	−0.161916	−0.0809581	+	0.996718i	$0.525798\pi$
$998$	−84.6099	−2.67828
$999$	−7.06135	−0.223411

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.b.1.11	✓	142

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.b.1.11	✓	142	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:	\( N \)	\(=\)	\( 8047 = 13 \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.52091 q^{2} -0.134089 q^{3} +4.35498 q^{4} +2.66858 q^{5} +0.338025 q^{6} -2.31114 q^{7} -5.93668 q^{8} -2.98202 q^{9} +O(q^{10})\)	\(q-2.52091 q^{2} -0.134089 q^{3} +4.35498 q^{4} +2.66858 q^{5} +0.338025 q^{6} -2.31114 q^{7} -5.93668 q^{8} -2.98202 q^{9} -6.72724 q^{10} -4.30592 q^{11} -0.583953 q^{12} +1.00000 q^{13} +5.82616 q^{14} -0.357826 q^{15} +6.25588 q^{16} +4.36671 q^{17} +7.51740 q^{18} +1.70053 q^{19} +11.6216 q^{20} +0.309897 q^{21} +10.8548 q^{22} +4.64277 q^{23} +0.796042 q^{24} +2.12131 q^{25} -2.52091 q^{26} +0.802121 q^{27} -10.0649 q^{28} +2.89705 q^{29} +0.902047 q^{30} +0.593276 q^{31} -3.89712 q^{32} +0.577375 q^{33} -11.0081 q^{34} -6.16745 q^{35} -12.9866 q^{36} -8.80335 q^{37} -4.28688 q^{38} -0.134089 q^{39} -15.8425 q^{40} +4.18820 q^{41} -0.781223 q^{42} -2.50410 q^{43} -18.7522 q^{44} -7.95775 q^{45} -11.7040 q^{46} -1.06636 q^{47} -0.838842 q^{48} -1.65865 q^{49} -5.34762 q^{50} -0.585526 q^{51} +4.35498 q^{52} -11.4976 q^{53} -2.02207 q^{54} -11.4907 q^{55} +13.7205 q^{56} -0.228022 q^{57} -7.30319 q^{58} +11.6878 q^{59} -1.55832 q^{60} -13.2811 q^{61} -1.49559 q^{62} +6.89186 q^{63} -2.68746 q^{64} +2.66858 q^{65} -1.45551 q^{66} +8.24829 q^{67} +19.0169 q^{68} -0.622543 q^{69} +15.5476 q^{70} +6.16272 q^{71} +17.7033 q^{72} +8.40385 q^{73} +22.1924 q^{74} -0.284443 q^{75} +7.40578 q^{76} +9.95157 q^{77} +0.338025 q^{78} +8.86338 q^{79} +16.6943 q^{80} +8.83851 q^{81} -10.5581 q^{82} +1.30966 q^{83} +1.34960 q^{84} +11.6529 q^{85} +6.31261 q^{86} -0.388461 q^{87} +25.5629 q^{88} -16.6895 q^{89} +20.0608 q^{90} -2.31114 q^{91} +20.2192 q^{92} -0.0795516 q^{93} +2.68820 q^{94} +4.53800 q^{95} +0.522560 q^{96} +9.89727 q^{97} +4.18130 q^{98} +12.8403 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9}+O(q^{10}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9} - 25 q^{10} - 25 q^{11} - 62 q^{12} + 142 q^{13} - 57 q^{14} - 14 q^{15} + 111 q^{16} - 141 q^{17} - 29 q^{18} - 3 q^{19} - 87 q^{20} - 19 q^{21} - 24 q^{22} - 69 q^{23} - 40 q^{24} + 87 q^{25} - 13 q^{26} - 95 q^{27} - 34 q^{28} - 147 q^{29} - 2 q^{30} - 21 q^{31} - 66 q^{32} - 62 q^{33} - 6 q^{34} - 59 q^{35} + 74 q^{36} - 37 q^{37} - 76 q^{38} - 26 q^{39} - 61 q^{40} - 97 q^{41} - 29 q^{42} - 33 q^{43} - 57 q^{44} - 86 q^{45} - q^{46} - 102 q^{47} - 141 q^{48} + 70 q^{49} - 28 q^{50} - 13 q^{51} + 129 q^{52} - 137 q^{53} - 29 q^{54} - 24 q^{55} - 130 q^{56} - 65 q^{57} - 15 q^{58} - 56 q^{59} + 11 q^{60} - 77 q^{61} - 150 q^{62} - 32 q^{63} + 73 q^{64} - 37 q^{65} - 32 q^{66} - 9 q^{67} - 226 q^{68} - 113 q^{69} + 6 q^{70} - 18 q^{71} - 82 q^{72} - 117 q^{73} - 70 q^{74} - 83 q^{75} + 40 q^{76} - 214 q^{77} - 15 q^{78} - 52 q^{79} - 161 q^{80} - 10 q^{81} - 36 q^{82} - 74 q^{83} + 53 q^{84} + 2 q^{85} + 17 q^{86} - 49 q^{87} - 29 q^{88} - 171 q^{89} - 57 q^{90} - 14 q^{91} - 187 q^{92} - 39 q^{93} + 13 q^{94} - 150 q^{95} - 47 q^{96} - 126 q^{97} - 85 q^{98}+O(q^{100}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9 - 25 * q^10 - 25 * q^11 - 62 * q^12 + 142 * q^13 - 57 * q^14 - 14 * q^15 + 111 * q^16 - 141 * q^17 - 29 * q^18 - 3 * q^19 - 87 * q^20 - 19 * q^21 - 24 * q^22 - 69 * q^23 - 40 * q^24 + 87 * q^25 - 13 * q^26 - 95 * q^27 - 34 * q^28 - 147 * q^29 - 2 * q^30 - 21 * q^31 - 66 * q^32 - 62 * q^33 - 6 * q^34 - 59 * q^35 + 74 * q^36 - 37 * q^37 - 76 * q^38 - 26 * q^39 - 61 * q^40 - 97 * q^41 - 29 * q^42 - 33 * q^43 - 57 * q^44 - 86 * q^45 - q^46 - 102 * q^47 - 141 * q^48 + 70 * q^49 - 28 * q^50 - 13 * q^51 + 129 * q^52 - 137 * q^53 - 29 * q^54 - 24 * q^55 - 130 * q^56 - 65 * q^57 - 15 * q^58 - 56 * q^59 + 11 * q^60 - 77 * q^61 - 150 * q^62 - 32 * q^63 + 73 * q^64 - 37 * q^65 - 32 * q^66 - 9 * q^67 - 226 * q^68 - 113 * q^69 + 6 * q^70 - 18 * q^71 - 82 * q^72 - 117 * q^73 - 70 * q^74 - 83 * q^75 + 40 * q^76 - 214 * q^77 - 15 * q^78 - 52 * q^79 - 161 * q^80 - 10 * q^81 - 36 * q^82 - 74 * q^83 + 53 * q^84 + 2 * q^85 + 17 * q^86 - 49 * q^87 - 29 * q^88 - 171 * q^89 - 57 * q^90 - 14 * q^91 - 187 * q^92 - 39 * q^93 + 13 * q^94 - 150 * q^95 - 47 * q^96 - 126 * q^97 - 85 * q^98

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