LMFDB - Embedded newform 8047.2.a.d.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:	$0$
Dimension:	$156$
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)

$f(q)$	$=$	$q-2.43558 q^{2} -1.59767 q^{3} +3.93207 q^{4} -2.50698 q^{5} +3.89125 q^{6} -1.32806 q^{7} -4.70573 q^{8} -0.447465 q^{9} +O(q^{10})$	\(q-2.43558 q^{2} -1.59767 q^{3} +3.93207 q^{4} -2.50698 q^{5} +3.89125 q^{6} -1.32806 q^{7} -4.70573 q^{8} -0.447465 q^{9} +6.10595 q^{10} -0.0818645 q^{11} -6.28214 q^{12} -1.00000 q^{13} +3.23460 q^{14} +4.00531 q^{15} +3.59705 q^{16} -3.03467 q^{17} +1.08984 q^{18} +1.84620 q^{19} -9.85762 q^{20} +2.12179 q^{21} +0.199388 q^{22} +1.52399 q^{23} +7.51818 q^{24} +1.28493 q^{25} +2.43558 q^{26} +5.50790 q^{27} -5.22203 q^{28} -4.78664 q^{29} -9.75527 q^{30} -8.23735 q^{31} +0.650528 q^{32} +0.130792 q^{33} +7.39119 q^{34} +3.32941 q^{35} -1.75946 q^{36} -5.92421 q^{37} -4.49658 q^{38} +1.59767 q^{39} +11.7972 q^{40} +4.58663 q^{41} -5.16781 q^{42} +3.55579 q^{43} -0.321897 q^{44} +1.12178 q^{45} -3.71180 q^{46} +8.20409 q^{47} -5.74689 q^{48} -5.23626 q^{49} -3.12956 q^{50} +4.84838 q^{51} -3.93207 q^{52} -3.16709 q^{53} -13.4149 q^{54} +0.205232 q^{55} +6.24948 q^{56} -2.94961 q^{57} +11.6583 q^{58} -3.56473 q^{59} +15.7492 q^{60} -9.20735 q^{61} +20.0628 q^{62} +0.594260 q^{63} -8.77852 q^{64} +2.50698 q^{65} -0.318555 q^{66} -0.445217 q^{67} -11.9325 q^{68} -2.43482 q^{69} -8.10907 q^{70} +4.51008 q^{71} +2.10565 q^{72} -8.94648 q^{73} +14.4289 q^{74} -2.05289 q^{75} +7.25939 q^{76} +0.108721 q^{77} -3.89125 q^{78} +2.48568 q^{79} -9.01773 q^{80} -7.45738 q^{81} -11.1711 q^{82} +5.08585 q^{83} +8.34305 q^{84} +7.60784 q^{85} -8.66043 q^{86} +7.64744 q^{87} +0.385232 q^{88} -8.03176 q^{89} -2.73220 q^{90} +1.32806 q^{91} +5.99243 q^{92} +13.1605 q^{93} -19.9817 q^{94} -4.62838 q^{95} -1.03933 q^{96} -8.27256 q^{97} +12.7534 q^{98} +0.0366315 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9}+O(q^{10}) $ 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9} + 11 q^{10} + 23 q^{11} + 57 q^{12} - 156 q^{13} + 18 q^{14} + 32 q^{15} + 159 q^{16} + 119 q^{17} + 36 q^{18} + 35 q^{19} + 109 q^{20} + 33 q^{21} + 11 q^{22} + 55 q^{23} + 63 q^{24} + 189 q^{25} - 13 q^{26} + 89 q^{27} + 54 q^{28} - 55 q^{29} + 47 q^{31} + 112 q^{32} + 109 q^{33} + 51 q^{34} + 25 q^{35} + 162 q^{36} + 53 q^{37} + 37 q^{38} - 23 q^{39} + 25 q^{40} + 113 q^{41} + 26 q^{42} + 31 q^{43} + 86 q^{44} + 144 q^{45} + 37 q^{46} + 115 q^{47} + 129 q^{48} + 189 q^{49} + 72 q^{50} - 4 q^{51} - 161 q^{52} + 51 q^{53} + 108 q^{54} + 22 q^{55} + 39 q^{56} + 102 q^{57} + 31 q^{58} + 75 q^{59} + 97 q^{60} + 7 q^{61} + 77 q^{62} + 94 q^{63} + 158 q^{64} - 39 q^{65} + 48 q^{66} + 37 q^{67} + 235 q^{68} + 27 q^{69} + 38 q^{70} + 70 q^{71} + 152 q^{72} + 155 q^{73} - 18 q^{74} + 80 q^{75} + 21 q^{76} + 101 q^{77} - 25 q^{78} + 10 q^{79} + 211 q^{80} + 220 q^{81} + 45 q^{82} + 132 q^{83} + 86 q^{84} + 74 q^{85} + 35 q^{86} + 53 q^{87} + 51 q^{88} + 190 q^{89} - 27 q^{90} - 19 q^{91} + 125 q^{92} + 96 q^{93} - 19 q^{94} + 72 q^{95} + 146 q^{96} + 155 q^{97} + 135 q^{98} + 89 q^{99}+O(q^{100}) \) 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9 + 11 * q^10 + 23 * q^11 + 57 * q^12 - 156 * q^13 + 18 * q^14 + 32 * q^15 + 159 * q^16 + 119 * q^17 + 36 * q^18 + 35 * q^19 + 109 * q^20 + 33 * q^21 + 11 * q^22 + 55 * q^23 + 63 * q^24 + 189 * q^25 - 13 * q^26 + 89 * q^27 + 54 * q^28 - 55 * q^29 + 47 * q^31 + 112 * q^32 + 109 * q^33 + 51 * q^34 + 25 * q^35 + 162 * q^36 + 53 * q^37 + 37 * q^38 - 23 * q^39 + 25 * q^40 + 113 * q^41 + 26 * q^42 + 31 * q^43 + 86 * q^44 + 144 * q^45 + 37 * q^46 + 115 * q^47 + 129 * q^48 + 189 * q^49 + 72 * q^50 - 4 * q^51 - 161 * q^52 + 51 * q^53 + 108 * q^54 + 22 * q^55 + 39 * q^56 + 102 * q^57 + 31 * q^58 + 75 * q^59 + 97 * q^60 + 7 * q^61 + 77 * q^62 + 94 * q^63 + 158 * q^64 - 39 * q^65 + 48 * q^66 + 37 * q^67 + 235 * q^68 + 27 * q^69 + 38 * q^70 + 70 * q^71 + 152 * q^72 + 155 * q^73 - 18 * q^74 + 80 * q^75 + 21 * q^76 + 101 * q^77 - 25 * q^78 + 10 * q^79 + 211 * q^80 + 220 * q^81 + 45 * q^82 + 132 * q^83 + 86 * q^84 + 74 * q^85 + 35 * q^86 + 53 * q^87 + 51 * q^88 + 190 * q^89 - 27 * q^90 - 19 * q^91 + 125 * q^92 + 96 * q^93 - 19 * q^94 + 72 * q^95 + 146 * q^96 + 155 * q^97 + 135 * q^98 + 89 * 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 $

$2$	−2.43558	−1.72222	−0.861109	−	0.508420i	$-0.830230\pi$
$2$	−2.43558	−1.72222	−0.861109	+	0.508420i	$0.830230\pi$
$3$	−1.59767	−0.922413	−0.461206	−	0.887293i	$-0.652583\pi$
$3$	−1.59767	−0.922413	−0.461206	+	0.887293i	$0.652583\pi$
$4$	3.93207	1.96604
$5$	−2.50698	−1.12115	−0.560577	−	0.828102i	$-0.689421\pi$
$5$	−2.50698	−1.12115	−0.560577	+	0.828102i	$0.689421\pi$
$6$	3.89125	1.58860
$7$	−1.32806	−0.501959	−0.250980	−	0.967992i	$-0.580753\pi$
$7$	−1.32806	−0.501959	−0.250980	+	0.967992i	$0.580753\pi$
$8$	−4.70573	−1.66373
$9$	−0.447465	−0.149155
$10$	6.10595	1.93087
$11$	−0.0818645	−0.0246831	−0.0123415	−	0.999924i	$-0.503929\pi$
$11$	−0.0818645	−0.0246831	−0.0123415	+	0.999924i	$0.503929\pi$
$12$	−6.28214	−1.81350
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	3.23460	0.864483
$15$	4.00531	1.03417
$16$	3.59705	0.899263
$17$	−3.03467	−0.736015	−0.368007	−	0.929823i	$-0.619960\pi$
$17$	−3.03467	−0.736015	−0.368007	+	0.929823i	$0.619960\pi$
$18$	1.08984	0.256877
$19$	1.84620	0.423547	0.211774	−	0.977319i	$-0.432076\pi$
$19$	1.84620	0.423547	0.211774	+	0.977319i	$0.432076\pi$
$20$	−9.85762	−2.20423
$21$	2.12179	0.463013
$22$	0.199388	0.0425096
$23$	1.52399	0.317773	0.158887	−	0.987297i	$-0.449210\pi$
$23$	1.52399	0.317773	0.158887	+	0.987297i	$0.449210\pi$
$24$	7.51818	1.53464
$25$	1.28493	0.256987
$26$	2.43558	0.477657
$27$	5.50790	1.06000
$28$	−5.22203	−0.986870
$29$	−4.78664	−0.888856	−0.444428	−	0.895815i	$-0.646593\pi$
$29$	−4.78664	−0.888856	−0.444428	+	0.895815i	$0.646593\pi$
$30$	−9.75527	−1.78106
$31$	−8.23735	−1.47947	−0.739735	−	0.672898i	$-0.765049\pi$
$31$	−8.23735	−1.47947	−0.739735	+	0.672898i	$0.765049\pi$
$32$	0.650528	0.114998
$33$	0.130792	0.0227680
$34$	7.39119	1.26758
$35$	3.32941	0.562774
$36$	−1.75946	−0.293244
$37$	−5.92421	−0.973934	−0.486967	−	0.873420i	$-0.661897\pi$
$37$	−5.92421	−0.973934	−0.486967	+	0.873420i	$0.661897\pi$
$38$	−4.49658	−0.729441
$39$	1.59767	0.255831
$40$	11.7972	1.86529
$41$	4.58663	0.716311	0.358156	−	0.933662i	$-0.383406\pi$
$41$	4.58663	0.716311	0.358156	+	0.933662i	$0.383406\pi$
$42$	−5.16781	−0.797410
$43$	3.55579	0.542253	0.271126	−	0.962544i	$-0.412604\pi$
$43$	3.55579	0.542253	0.271126	+	0.962544i	$0.412604\pi$
$44$	−0.321897	−0.0485278
$45$	1.12178	0.167226
$46$	−3.71180	−0.547275
$47$	8.20409	1.19669	0.598345	−	0.801239i	$-0.295825\pi$
$47$	8.20409	1.19669	0.598345	+	0.801239i	$0.295825\pi$
$48$	−5.74689	−0.829492
$49$	−5.23626	−0.748037
$50$	−3.12956	−0.442587
$51$	4.84838	0.678909
$52$	−3.93207	−0.545280
$53$	−3.16709	−0.435034	−0.217517	−	0.976057i	$-0.569796\pi$
$53$	−3.16709	−0.435034	−0.217517	+	0.976057i	$0.569796\pi$
$54$	−13.4149	−1.82554
$55$	0.205232	0.0276735
$56$	6.24948	0.835123
$57$	−2.94961	−0.390685
$58$	11.6583	1.53080
$59$	−3.56473	−0.464088	−0.232044	−	0.972705i	$-0.574541\pi$
$59$	−3.56473	−0.464088	−0.232044	+	0.972705i	$0.574541\pi$
$60$	15.7492	2.03321
$61$	−9.20735	−1.17888	−0.589440	−	0.807812i	$-0.700652\pi$
$61$	−9.20735	−1.17888	−0.589440	+	0.807812i	$0.700652\pi$
$62$	20.0628	2.54797
$63$	0.594260	0.0748697
$64$	−8.77852	−1.09732
$65$	2.50698	0.310952
$66$	−0.318555	−0.0392114
$67$	−0.445217	−0.0543919	−0.0271960	−	0.999630i	$-0.508658\pi$
$67$	−0.445217	−0.0543919	−0.0271960	+	0.999630i	$0.508658\pi$
$68$	−11.9325	−1.44703
$69$	−2.43482	−0.293118
$70$	−8.10907	−0.969219
$71$	4.51008	0.535248	0.267624	−	0.963523i	$-0.413761\pi$
$71$	4.51008	0.535248	0.267624	+	0.963523i	$0.413761\pi$
$72$	2.10565	0.248153
$73$	−8.94648	−1.04711	−0.523553	−	0.851993i	$-0.675394\pi$
$73$	−8.94648	−1.04711	−0.523553	+	0.851993i	$0.675394\pi$
$74$	14.4289	1.67733
$75$	−2.05289	−0.237048
$76$	7.25939	0.832709
$77$	0.108721	0.0123899
$78$	−3.89125	−0.440597
$79$	2.48568	0.279661	0.139831	−	0.990175i	$-0.455344\pi$
$79$	2.48568	0.279661	0.139831	+	0.990175i	$0.455344\pi$
$80$	−9.01773	−1.00821
$81$	−7.45738	−0.828598
$82$	−11.1711	−1.23364
$83$	5.08585	0.558245	0.279122	−	0.960256i	$-0.409956\pi$
$83$	5.08585	0.558245	0.279122	+	0.960256i	$0.409956\pi$
$84$	8.34305	0.910301
$85$	7.60784	0.825186
$86$	−8.66043	−0.933878
$87$	7.64744	0.819892
$88$	0.385232	0.0410659
$89$	−8.03176	−0.851365	−0.425683	−	0.904873i	$-0.639966\pi$
$89$	−8.03176	−0.851365	−0.425683	+	0.904873i	$0.639966\pi$
$90$	−2.73220	−0.287999
$91$	1.32806	0.139218
$92$	5.99243	0.624754
$93$	13.1605	1.36468
$94$	−19.9817	−2.06096
$95$	−4.62838	−0.474862
$96$	−1.03933	−0.106076
$97$	−8.27256	−0.839951	−0.419975	−	0.907535i	$-0.637961\pi$
$97$	−8.27256	−0.839951	−0.419975	+	0.907535i	$0.637961\pi$
$98$	12.7534	1.28828
$99$	0.0366315	0.00368160
$100$	5.05245	0.505245
$101$	11.3815	1.13250	0.566251	−	0.824233i	$-0.308393\pi$
$101$	11.3815	1.13250	0.566251	+	0.824233i	$0.308393\pi$
$102$	−11.8086	−1.16923
$103$	−10.4231	−1.02701	−0.513507	−	0.858085i	$-0.671654\pi$
$103$	−10.4231	−1.02701	−0.513507	+	0.858085i	$0.671654\pi$
$104$	4.70573	0.461435
$105$	−5.31929	−0.519110
$106$	7.71373	0.749223
$107$	0.472089	0.0456386	0.0228193	−	0.999740i	$-0.492736\pi$
$107$	0.472089	0.0456386	0.0228193	+	0.999740i	$0.492736\pi$
$108$	21.6574	2.08399
$109$	−14.7733	−1.41503	−0.707513	−	0.706700i	$-0.750183\pi$
$109$	−14.7733	−1.41503	−0.707513	+	0.706700i	$0.750183\pi$
$110$	−0.499861	−0.0476599
$111$	9.46490	0.898369
$112$	−4.77710	−0.451393
$113$	−17.4137	−1.63814	−0.819072	−	0.573690i	$-0.805511\pi$
$113$	−17.4137	−1.63814	−0.819072	+	0.573690i	$0.805511\pi$
$114$	7.18402	0.672845
$115$	−3.82060	−0.356273
$116$	−18.8214	−1.74752
$117$	0.447465	0.0413681
$118$	8.68221	0.799262
$119$	4.03022	0.369449
$120$	−18.8479	−1.72057
$121$	−10.9933	−0.999391
$122$	22.4253	2.03029
$123$	−7.32790	−0.660735
$124$	−32.3898	−2.90869
$125$	9.31359	0.833032
$126$	−1.44737	−0.128942
$127$	8.28930	0.735556	0.367778	−	0.929914i	$-0.380119\pi$
$127$	8.28930	0.735556	0.367778	+	0.929914i	$0.380119\pi$
$128$	20.0798	1.77482
$129$	−5.68096	−0.500181
$130$	−6.10595	−0.535528
$131$	6.50705	0.568524	0.284262	−	0.958747i	$-0.408251\pi$
$131$	6.50705	0.568524	0.284262	+	0.958747i	$0.408251\pi$
$132$	0.514284	0.0447627
$133$	−2.45186	−0.212603
$134$	1.08436	0.0936748
$135$	−13.8082	−1.18842
$136$	14.2803	1.22453
$137$	−10.7246	−0.916260	−0.458130	−	0.888885i	$-0.651481\pi$
$137$	−10.7246	−0.916260	−0.458130	+	0.888885i	$0.651481\pi$
$138$	5.93022	0.504813
$139$	−7.13823	−0.605457	−0.302728	−	0.953077i	$-0.597898\pi$
$139$	−7.13823	−0.605457	−0.302728	+	0.953077i	$0.597898\pi$
$140$	13.0915	1.10643
$141$	−13.1074	−1.10384
$142$	−10.9847	−0.921814
$143$	0.0818645	0.00684585
$144$	−1.60955	−0.134130
$145$	12.0000	0.996545
$146$	21.7899	1.80335
$147$	8.36579	0.689999
$148$	−23.2944	−1.91479
$149$	−14.9138	−1.22178	−0.610892	−	0.791714i	$-0.709189\pi$
$149$	−14.9138	−1.22178	−0.610892	+	0.791714i	$0.709189\pi$
$150$	5.00000	0.408248
$151$	2.82900	0.230221	0.115110	−	0.993353i	$-0.463278\pi$
$151$	2.82900	0.230221	0.115110	+	0.993353i	$0.463278\pi$
$152$	−8.68771	−0.704667
$153$	1.35791	0.109780
$154$	−0.264799	−0.0213381
$155$	20.6508	1.65871
$156$	6.28214	0.502974
$157$	−11.1318	−0.888416	−0.444208	−	0.895924i	$-0.646515\pi$
$157$	−11.1318	−0.888416	−0.444208	+	0.895924i	$0.646515\pi$
$158$	−6.05409	−0.481638
$159$	5.05996	0.401281
$160$	−1.63086	−0.128931
$161$	−2.02395	−0.159509
$162$	18.1631	1.42703
$163$	18.5586	1.45362	0.726810	−	0.686838i	$-0.241002\pi$
$163$	18.5586	1.45362	0.726810	+	0.686838i	$0.241002\pi$
$164$	18.0350	1.40829
$165$	−0.327893	−0.0255264
$166$	−12.3870	−0.961420
$167$	−5.34058	−0.413267	−0.206633	−	0.978418i	$-0.566251\pi$
$167$	−5.34058	−0.413267	−0.206633	+	0.978418i	$0.566251\pi$
$168$	−9.98459	−0.770328
$169$	1.00000	0.0769231
$170$	−18.5295	−1.42115
$171$	−0.826109	−0.0631742
$172$	13.9816	1.06609
$173$	−3.16590	−0.240699	−0.120350	−	0.992732i	$-0.538402\pi$
$173$	−3.16590	−0.240699	−0.120350	+	0.992732i	$0.538402\pi$
$174$	−18.6260	−1.41203
$175$	−1.70647	−0.128997
$176$	−0.294471	−0.0221966
$177$	5.69525	0.428081
$178$	19.5620	1.46624
$179$	−10.0078	−0.748021	−0.374011	−	0.927424i	$-0.622018\pi$
$179$	−10.0078	−0.748021	−0.374011	+	0.927424i	$0.622018\pi$
$180$	4.41094	0.328772
$181$	−16.7932	−1.24823	−0.624114	−	0.781333i	$-0.714540\pi$
$181$	−16.7932	−1.24823	−0.624114	+	0.781333i	$0.714540\pi$
$182$	−3.23460	−0.239765
$183$	14.7103	1.08741
$184$	−7.17147	−0.528688
$185$	14.8519	1.09193
$186$	−32.0536	−2.35028
$187$	0.248431	0.0181671
$188$	32.2591	2.35273
$189$	−7.31481	−0.532074
$190$	11.2728	0.817816
$191$	−19.9150	−1.44100	−0.720499	−	0.693456i	$-0.756087\pi$
$191$	−19.9150	−1.44100	−0.720499	+	0.693456i	$0.756087\pi$
$192$	14.0251	1.01218
$193$	−9.22002	−0.663672	−0.331836	−	0.943337i	$-0.607668\pi$
$193$	−9.22002	−0.663672	−0.331836	+	0.943337i	$0.607668\pi$
$194$	20.1485	1.44658
$195$	−4.00531	−0.286826
$196$	−20.5894	−1.47067
$197$	3.25996	0.232262	0.116131	−	0.993234i	$-0.462951\pi$
$197$	3.25996	0.232262	0.116131	+	0.993234i	$0.462951\pi$
$198$	−0.0892191	−0.00634052
$199$	−0.0915695	−0.00649119	−0.00324559	−	0.999995i	$-0.501033\pi$
$199$	−0.0915695	−0.00649119	−0.00324559	+	0.999995i	$0.501033\pi$
$200$	−6.04655	−0.427555
$201$	0.711308	0.0501718
$202$	−27.7206	−1.95042
$203$	6.35694	0.446169
$204$	19.0642	1.33476
$205$	−11.4986	−0.803095
$206$	25.3863	1.76874
$207$	−0.681931	−0.0473975
$208$	−3.59705	−0.249411
$209$	−0.151138	−0.0104544
$210$	12.9556	0.894020
$211$	−27.5262	−1.89498	−0.947492	−	0.319779i	$-0.896391\pi$
$211$	−27.5262	−1.89498	−0.947492	+	0.319779i	$0.896391\pi$
$212$	−12.4532	−0.855292
$213$	−7.20560	−0.493720
$214$	−1.14981	−0.0785996
$215$	−8.91428	−0.607949
$216$	−25.9187	−1.76354
$217$	10.9397	0.742634
$218$	35.9816	2.43698
$219$	14.2935	0.965864
$220$	0.806989	0.0544072
$221$	3.03467	0.204134
$222$	−23.0526	−1.54719
$223$	−20.7384	−1.38874	−0.694372	−	0.719616i	$-0.744318\pi$
$223$	−20.7384	−1.38874	−0.694372	+	0.719616i	$0.744318\pi$
$224$	−0.863940	−0.0577244
$225$	−0.574963	−0.0383308
$226$	42.4126	2.82124
$227$	−2.74059	−0.181899	−0.0909497	−	0.995855i	$-0.528990\pi$
$227$	−2.74059	−0.181899	−0.0909497	+	0.995855i	$0.528990\pi$
$228$	−11.5981	−0.768102
$229$	20.1930	1.33439	0.667196	−	0.744882i	$-0.267494\pi$
$229$	20.1930	1.33439	0.667196	+	0.744882i	$0.267494\pi$
$230$	9.30540	0.613580
$231$	−0.173700	−0.0114286
$232$	22.5246	1.47881
$233$	23.2831	1.52533	0.762665	−	0.646794i	$-0.223891\pi$
$233$	23.2831	1.52533	0.762665	+	0.646794i	$0.223891\pi$
$234$	−1.08984	−0.0712450
$235$	−20.5675	−1.34167
$236$	−14.0168	−0.912415
$237$	−3.97129	−0.257963
$238$	−9.81593	−0.636272
$239$	−0.494455	−0.0319836	−0.0159918	−	0.999872i	$-0.505091\pi$
$239$	−0.494455	−0.0319836	−0.0159918	+	0.999872i	$0.505091\pi$
$240$	14.4073	0.929988
$241$	25.7368	1.65786	0.828928	−	0.559355i	$-0.188951\pi$
$241$	25.7368	1.65786	0.828928	+	0.559355i	$0.188951\pi$
$242$	26.7751	1.72117
$243$	−4.60929	−0.295686
$244$	−36.2040	−2.31772
$245$	13.1272	0.838665
$246$	17.8477	1.13793
$247$	−1.84620	−0.117471
$248$	38.7627	2.46143
$249$	−8.12549	−0.514932
$250$	−22.6840	−1.43466
$251$	−11.7881	−0.744057	−0.372029	−	0.928221i	$-0.621338\pi$
$251$	−11.7881	−0.744057	−0.372029	+	0.928221i	$0.621338\pi$
$252$	2.33667	0.147197
$253$	−0.124760	−0.00784362
$254$	−20.1893	−1.26679
$255$	−12.1548	−0.761162
$256$	−31.3490	−1.95931
$257$	−1.52470	−0.0951081	−0.0475541	−	0.998869i	$-0.515143\pi$
$257$	−1.52470	−0.0951081	−0.0475541	+	0.998869i	$0.515143\pi$
$258$	13.8365	0.861421
$259$	7.86770	0.488875
$260$	9.85762	0.611343
$261$	2.14185	0.132577
$262$	−15.8485	−0.979122
$263$	−5.65818	−0.348898	−0.174449	−	0.984666i	$-0.555814\pi$
$263$	−5.65818	−0.348898	−0.174449	+	0.984666i	$0.555814\pi$
$264$	−0.615472	−0.0378797
$265$	7.93983	0.487740
$266$	5.97172	0.366150
$267$	12.8321	0.785310
$268$	−1.75063	−0.106937
$269$	−13.1069	−0.799142	−0.399571	−	0.916702i	$-0.630841\pi$
$269$	−13.1069	−0.799142	−0.399571	+	0.916702i	$0.630841\pi$
$270$	33.6310	2.04672
$271$	−20.1402	−1.22343	−0.611716	−	0.791078i	$-0.709520\pi$
$271$	−20.1402	−1.22343	−0.611716	+	0.791078i	$0.709520\pi$
$272$	−10.9159	−0.661871
$273$	−2.12179	−0.128417
$274$	26.1206	1.57800
$275$	−0.105190	−0.00634322
$276$	−9.57390	−0.576281
$277$	8.49900	0.510655	0.255328	−	0.966855i	$-0.417817\pi$
$277$	8.49900	0.510655	0.255328	+	0.966855i	$0.417817\pi$
$278$	17.3858	1.04273
$279$	3.68592	0.220670
$280$	−15.6673	−0.936301
$281$	−7.97738	−0.475891	−0.237945	−	0.971279i	$-0.576474\pi$
$281$	−7.97738	−0.475891	−0.237945	+	0.971279i	$0.576474\pi$
$282$	31.9242	1.90106
$283$	−12.1424	−0.721792	−0.360896	−	0.932606i	$-0.617529\pi$
$283$	−12.1424	−0.721792	−0.360896	+	0.932606i	$0.617529\pi$
$284$	17.7340	1.05232
$285$	7.39460	0.438018
$286$	−0.199388	−0.0117901
$287$	−6.09132	−0.359559
$288$	−0.291089	−0.0171526
$289$	−7.79080	−0.458283
$290$	−29.2270	−1.71627
$291$	13.2168	0.774781
$292$	−35.1782	−2.05865
$293$	11.9372	0.697376	0.348688	−	0.937239i	$-0.386627\pi$
$293$	11.9372	0.697376	0.348688	+	0.937239i	$0.386627\pi$
$294$	−20.3756	−1.18833
$295$	8.93670	0.520315
$296$	27.8777	1.62036
$297$	−0.450901	−0.0261639
$298$	36.3237	2.10418
$299$	−1.52399	−0.0881345
$300$	−8.07213	−0.466045
$301$	−4.72230	−0.272189
$302$	−6.89027	−0.396490
$303$	−18.1838	−1.04463
$304$	6.64088	0.380880
$305$	23.0826	1.32171
$306$	−3.30730	−0.189065
$307$	−15.9848	−0.912303	−0.456152	−	0.889902i	$-0.650773\pi$
$307$	−15.9848	−0.912303	−0.456152	+	0.889902i	$0.650773\pi$
$308$	0.427499	0.0243590
$309$	16.6526	0.947332
$310$	−50.2969	−2.85667
$311$	−6.98418	−0.396037	−0.198018	−	0.980198i	$-0.563451\pi$
$311$	−6.98418	−0.396037	−0.198018	+	0.980198i	$0.563451\pi$
$312$	−7.51818	−0.425633
$313$	−30.3023	−1.71279	−0.856394	−	0.516323i	$-0.827300\pi$
$313$	−30.3023	−1.71279	−0.856394	+	0.516323i	$0.827300\pi$
$314$	27.1125	1.53005
$315$	−1.48980	−0.0839405
$316$	9.77389	0.549824
$317$	−20.2025	−1.13468	−0.567342	−	0.823482i	$-0.692028\pi$
$317$	−20.2025	−1.13468	−0.567342	+	0.823482i	$0.692028\pi$
$318$	−12.3240	−0.691093
$319$	0.391856	0.0219397
$320$	22.0076	1.23026
$321$	−0.754240	−0.0420976
$322$	4.92949	0.274710
$323$	−5.60260	−0.311737
$324$	−29.3230	−1.62905
$325$	−1.28493	−0.0712753
$326$	−45.2010	−2.50345
$327$	23.6028	1.30524
$328$	−21.5834	−1.19175
$329$	−10.8955	−0.600689
$330$	0.798611	0.0439621
$331$	−5.94385	−0.326703	−0.163352	−	0.986568i	$-0.552231\pi$
$331$	−5.94385	−0.326703	−0.163352	+	0.986568i	$0.552231\pi$
$332$	19.9979	1.09753
$333$	2.65087	0.145267
$334$	13.0074	0.711736
$335$	1.11615	0.0609817
$336$	7.63221	0.416371
$337$	25.5594	1.39231	0.696155	−	0.717891i	$-0.254892\pi$
$337$	25.5594	1.39231	0.696155	+	0.717891i	$0.254892\pi$
$338$	−2.43558	−0.132478
$339$	27.8213	1.51105
$340$	29.9146	1.62235
$341$	0.674346	0.0365179
$342$	2.01206	0.108800
$343$	16.2505	0.877443
$344$	−16.7326	−0.902160
$345$	6.10404	0.328631
$346$	7.71082	0.414536
$347$	−10.3442	−0.555304	−0.277652	−	0.960682i	$-0.589556\pi$
$347$	−10.3442	−0.555304	−0.277652	+	0.960682i	$0.589556\pi$
$348$	30.0703	1.61194
$349$	−35.0327	−1.87525	−0.937627	−	0.347642i	$-0.886983\pi$
$349$	−35.0327	−1.87525	−0.937627	+	0.347642i	$0.886983\pi$
$350$	4.15625	0.222161
$351$	−5.50790	−0.293990
$352$	−0.0532552	−0.00283851
$353$	−8.36434	−0.445189	−0.222594	−	0.974911i	$-0.571453\pi$
$353$	−8.36434	−0.445189	−0.222594	+	0.974911i	$0.571453\pi$
$354$	−13.8713	−0.737249
$355$	−11.3067	−0.600096
$356$	−31.5815	−1.67382
$357$	−6.43894	−0.340785
$358$	24.3750	1.28826
$359$	5.65616	0.298521	0.149260	−	0.988798i	$-0.452311\pi$
$359$	5.65616	0.298521	0.149260	+	0.988798i	$0.452311\pi$
$360$	−5.27881	−0.278218
$361$	−15.5915	−0.820608
$362$	40.9013	2.14972
$363$	17.5636	0.921851
$364$	5.22203	0.273709
$365$	22.4286	1.17397
$366$	−35.8281	−1.87276
$367$	−10.4385	−0.544885	−0.272442	−	0.962172i	$-0.587831\pi$
$367$	−10.4385	−0.544885	−0.272442	+	0.962172i	$0.587831\pi$
$368$	5.48186	0.285762
$369$	−2.05236	−0.106841
$370$	−36.1730	−1.88054
$371$	4.20609	0.218369
$372$	51.7481	2.68302
$373$	−1.50418	−0.0778833	−0.0389417	−	0.999241i	$-0.512399\pi$
$373$	−1.50418	−0.0778833	−0.0389417	+	0.999241i	$0.512399\pi$
$374$	−0.605076	−0.0312877
$375$	−14.8800	−0.768400
$376$	−38.6062	−1.99096
$377$	4.78664	0.246524
$378$	17.8158	0.916348
$379$	−14.0096	−0.719625	−0.359813	−	0.933025i	$-0.617159\pi$
$379$	−14.0096	−0.719625	−0.359813	+	0.933025i	$0.617159\pi$
$380$	−18.1991	−0.933596
$381$	−13.2435	−0.678487
$382$	48.5046	2.48171
$383$	18.7117	0.956124	0.478062	−	0.878326i	$-0.341339\pi$
$383$	18.7117	0.956124	0.478062	+	0.878326i	$0.341339\pi$
$384$	−32.0808	−1.63712
$385$	−0.272561	−0.0138910
$386$	22.4561	1.14299
$387$	−1.59109	−0.0808797
$388$	−32.5283	−1.65137
$389$	−24.2764	−1.23086	−0.615431	−	0.788191i	$-0.711018\pi$
$389$	−24.2764	−1.23086	−0.615431	+	0.788191i	$0.711018\pi$
$390$	9.75527	0.493977
$391$	−4.62479	−0.233886
$392$	24.6404	1.24453
$393$	−10.3961	−0.524414
$394$	−7.93990	−0.400006
$395$	−6.23155	−0.313543
$396$	0.144038	0.00723817
$397$	−0.951470	−0.0477529	−0.0238764	−	0.999715i	$-0.507601\pi$
$397$	−0.951470	−0.0477529	−0.0238764	+	0.999715i	$0.507601\pi$
$398$	0.223025	0.0111792
$399$	3.91726	0.196108
$400$	4.62197	0.231099
$401$	4.08644	0.204067	0.102033	−	0.994781i	$-0.467465\pi$
$401$	4.08644	0.204067	0.102033	+	0.994781i	$0.467465\pi$
$402$	−1.73245	−0.0864068
$403$	8.23735	0.410331
$404$	44.7529	2.22654
$405$	18.6955	0.928986
$406$	−15.4829	−0.768401
$407$	0.484982	0.0240397
$408$	−22.8152	−1.12952
$409$	−17.8488	−0.882565	−0.441282	−	0.897368i	$-0.645476\pi$
$409$	−17.8488	−0.882565	−0.441282	+	0.897368i	$0.645476\pi$
$410$	28.0058	1.38311
$411$	17.1342	0.845170
$412$	−40.9842	−2.01915
$413$	4.73417	0.232953
$414$	1.66090	0.0816288
$415$	−12.7501	−0.625879
$416$	−0.650528	−0.0318948
$417$	11.4045	0.558481
$418$	0.368110	0.0180048
$419$	−15.1391	−0.739594	−0.369797	−	0.929112i	$-0.620573\pi$
$419$	−15.1391	−0.739594	−0.369797	+	0.929112i	$0.620573\pi$
$420$	−20.9158	−1.02059
$421$	9.99427	0.487091	0.243545	−	0.969889i	$-0.421690\pi$
$421$	9.99427	0.487091	0.243545	+	0.969889i	$0.421690\pi$
$422$	67.0425	3.26358
$423$	−3.67104	−0.178492
$424$	14.9035	0.723777
$425$	−3.89934	−0.189146
$426$	17.5498	0.850293
$427$	12.2279	0.591750
$428$	1.85629	0.0897271
$429$	−0.130792	−0.00631470
$430$	21.7115	1.04702
$431$	−38.6975	−1.86399	−0.931996	−	0.362468i	$-0.881934\pi$
$431$	−38.6975	−1.86399	−0.931996	+	0.362468i	$0.881934\pi$
$432$	19.8122	0.953215
$433$	−8.70363	−0.418270	−0.209135	−	0.977887i	$-0.567065\pi$
$433$	−8.70363	−0.418270	−0.209135	+	0.977887i	$0.567065\pi$
$434$	−26.6445	−1.27898
$435$	−19.1720	−0.919226
$436$	−58.0897	−2.78199
$437$	2.81358	0.134592
$438$	−34.8130	−1.66343
$439$	7.77716	0.371184	0.185592	−	0.982627i	$-0.440580\pi$
$439$	7.77716	0.371184	0.185592	+	0.982627i	$0.440580\pi$
$440$	−0.965768	−0.0460412
$441$	2.34304	0.111573
$442$	−7.39119	−0.351563
$443$	2.82133	0.134046	0.0670228	−	0.997751i	$-0.478650\pi$
$443$	2.82133	0.134046	0.0670228	+	0.997751i	$0.478650\pi$
$444$	37.2167	1.76623
$445$	20.1354	0.954512
$446$	50.5100	2.39172
$447$	23.8272	1.12699
$448$	11.6584	0.550808
$449$	−24.9130	−1.17572	−0.587860	−	0.808963i	$-0.700029\pi$
$449$	−24.9130	−1.17572	−0.587860	+	0.808963i	$0.700029\pi$
$450$	1.40037	0.0660141
$451$	−0.375482	−0.0176808
$452$	−68.4720	−3.22065
$453$	−4.51979	−0.212358
$454$	6.67494	0.313270
$455$	−3.32941	−0.156085
$456$	13.8801	0.649993
$457$	27.1922	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$457$	27.1922	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$458$	−49.1818	−2.29811
$459$	−16.7146	−0.780172
$460$	−15.0229	−0.700445
$461$	−23.4038	−1.09003	−0.545013	−	0.838428i	$-0.683475\pi$
$461$	−23.4038	−1.09003	−0.545013	+	0.838428i	$0.683475\pi$
$462$	0.423060	0.0196825
$463$	34.8865	1.62131	0.810657	−	0.585521i	$-0.199110\pi$
$463$	34.8865	1.62131	0.810657	+	0.585521i	$0.199110\pi$
$464$	−17.2178	−0.799316
$465$	−32.9931	−1.53002
$466$	−56.7081	−2.62695
$467$	19.9207	0.921819	0.460910	−	0.887447i	$-0.347523\pi$
$467$	19.9207	0.921819	0.460910	+	0.887447i	$0.347523\pi$
$468$	1.75946	0.0813313
$469$	0.591275	0.0273025
$470$	50.0938	2.31065
$471$	17.7849	0.819486
$472$	16.7747	0.772116
$473$	−0.291093	−0.0133845
$474$	9.67241	0.444269
$475$	2.37224	0.108846
$476$	15.8471	0.726351
$477$	1.41716	0.0648874
$478$	1.20429	0.0550828
$479$	3.70213	0.169155	0.0845774	−	0.996417i	$-0.473046\pi$
$479$	3.70213	0.169155	0.0845774	+	0.996417i	$0.473046\pi$
$480$	2.60557	0.118927
$481$	5.92421	0.270121
$482$	−62.6843	−2.85519
$483$	3.23359	0.147133
$484$	−43.2265	−1.96484
$485$	20.7391	0.941714
$486$	11.2263	0.509236
$487$	6.57823	0.298088	0.149044	−	0.988831i	$-0.452380\pi$
$487$	6.57823	0.298088	0.149044	+	0.988831i	$0.452380\pi$
$488$	43.3273	1.96133
$489$	−29.6504	−1.34084
$490$	−31.9724	−1.44436
$491$	−28.0197	−1.26451	−0.632255	−	0.774761i	$-0.717870\pi$
$491$	−28.0197	−1.26451	−0.632255	+	0.774761i	$0.717870\pi$
$492$	−28.8138	−1.29903
$493$	14.5258	0.654211
$494$	4.49658	0.202311
$495$	−0.0918343	−0.00412764
$496$	−29.6302	−1.33043
$497$	−5.98965	−0.268673
$498$	19.7903	0.886826
$499$	33.5776	1.50314	0.751571	−	0.659652i	$-0.229296\pi$
$499$	33.5776	1.50314	0.751571	+	0.659652i	$0.229296\pi$
$500$	36.6217	1.63777
$501$	8.53247	0.381202
$502$	28.7109	1.28143
$503$	4.23301	0.188741	0.0943703	−	0.995537i	$-0.469916\pi$
$503$	4.23301	0.188741	0.0943703	+	0.995537i	$0.469916\pi$
$504$	−2.79642	−0.124563
$505$	−28.5332	−1.26971
$506$	0.303865	0.0135084
$507$	−1.59767	−0.0709548
$508$	32.5941	1.44613
$509$	12.5482	0.556191	0.278095	−	0.960553i	$-0.410297\pi$
$509$	12.5482	0.556191	0.278095	+	0.960553i	$0.410297\pi$
$510$	29.6040	1.31089
$511$	11.8815	0.525605
$512$	36.1935	1.59954
$513$	10.1687	0.448958
$514$	3.71353	0.163797
$515$	26.1304	1.15144
$516$	−22.3380	−0.983374
$517$	−0.671623	−0.0295380
$518$	−19.1624	−0.841950
$519$	5.05805	0.222024
$520$	−11.7972	−0.517339
$521$	−21.7258	−0.951827	−0.475913	−	0.879492i	$-0.657882\pi$
$521$	−21.7258	−0.951827	−0.475913	+	0.879492i	$0.657882\pi$
$522$	−5.21666	−0.228327
$523$	−44.3723	−1.94026	−0.970132	−	0.242577i	$-0.922007\pi$
$523$	−44.3723	−1.94026	−0.970132	+	0.242577i	$0.922007\pi$
$524$	25.5862	1.11774
$525$	2.72636	0.118988
$526$	13.7810	0.600879
$527$	24.9976	1.08891
$528$	0.470466	0.0204744
$529$	−20.6775	−0.899020
$530$	−19.3381	−0.839995
$531$	1.59509	0.0692211
$532$	−9.64090	−0.417986
$533$	−4.58663	−0.198669
$534$	−31.2536	−1.35248
$535$	−1.18352	−0.0511679
$536$	2.09507	0.0904933
$537$	15.9892	0.689984
$538$	31.9230	1.37630
$539$	0.428664	0.0184639
$540$	−54.2947	−2.33647
$541$	8.11716	0.348984	0.174492	−	0.984659i	$-0.444172\pi$
$541$	8.11716	0.348984	0.174492	+	0.984659i	$0.444172\pi$
$542$	49.0532	2.10702
$543$	26.8299	1.15138
$544$	−1.97414	−0.0846404
$545$	37.0363	1.58646
$546$	5.16781	0.221162
$547$	27.2581	1.16547	0.582735	−	0.812662i	$-0.301982\pi$
$547$	27.2581	1.16547	0.582735	+	0.812662i	$0.301982\pi$
$548$	−42.1697	−1.80140
$549$	4.11996	0.175836
$550$	0.256200	0.0109244
$551$	−8.83709	−0.376473
$552$	11.4576	0.487668
$553$	−3.30113	−0.140379
$554$	−20.7000	−0.879460
$555$	−23.7283	−1.00721
$556$	−28.0680	−1.19035
$557$	−1.12080	−0.0474897	−0.0237448	−	0.999718i	$-0.507559\pi$
$557$	−1.12080	−0.0474897	−0.0237448	+	0.999718i	$0.507559\pi$
$558$	−8.97738	−0.380043
$559$	−3.55579	−0.150394
$560$	11.9761	0.506082
$561$	−0.396910	−0.0167576
$562$	19.4296	0.819588
$563$	6.69766	0.282273	0.141136	−	0.989990i	$-0.454924\pi$
$563$	6.69766	0.282273	0.141136	+	0.989990i	$0.454924\pi$
$564$	−51.5392	−2.17019
$565$	43.6558	1.83661
$566$	29.5739	1.24308
$567$	9.90384	0.415922
$568$	−21.2232	−0.890506
$569$	21.4844	0.900672	0.450336	−	0.892859i	$-0.351304\pi$
$569$	21.4844	0.900672	0.450336	+	0.892859i	$0.351304\pi$
$570$	−18.0102	−0.754364
$571$	−13.9764	−0.584895	−0.292448	−	0.956282i	$-0.594470\pi$
$571$	−13.9764	−0.584895	−0.292448	+	0.956282i	$0.594470\pi$
$572$	0.321897	0.0134592
$573$	31.8175	1.32919
$574$	14.8359	0.619239
$575$	1.95822	0.0816635
$576$	3.92808	0.163670
$577$	−10.1516	−0.422618	−0.211309	−	0.977419i	$-0.567773\pi$
$577$	−10.1516	−0.422618	−0.211309	+	0.977419i	$0.567773\pi$
$578$	18.9752	0.789263
$579$	14.7305	0.612179
$580$	47.1848	1.95924
$581$	−6.75431	−0.280216
$582$	−32.1906	−1.33434
$583$	0.259273	0.0107380
$584$	42.0997	1.74210
$585$	−1.12178	−0.0463801
$586$	−29.0739	−1.20103
$587$	39.4332	1.62758	0.813791	−	0.581157i	$-0.197400\pi$
$587$	39.4332	1.62758	0.813791	+	0.581157i	$0.197400\pi$
$588$	32.8949	1.35656
$589$	−15.2078	−0.626626
$590$	−21.7661	−0.896096
$591$	−5.20832	−0.214242
$592$	−21.3097	−0.875823
$593$	−34.5303	−1.41799	−0.708995	−	0.705214i	$-0.750851\pi$
$593$	−34.5303	−1.41799	−0.708995	+	0.705214i	$0.750851\pi$
$594$	1.09821	0.0450600
$595$	−10.1037	−0.414210
$596$	−58.6420	−2.40207
$597$	0.146297	0.00598756
$598$	3.71180	0.151787
$599$	1.88807	0.0771446	0.0385723	−	0.999256i	$-0.487719\pi$
$599$	1.88807	0.0771446	0.0385723	+	0.999256i	$0.487719\pi$
$600$	9.66036	0.394383
$601$	14.1595	0.577576	0.288788	−	0.957393i	$-0.406748\pi$
$601$	14.1595	0.577576	0.288788	+	0.957393i	$0.406748\pi$
$602$	11.5016	0.468768
$603$	0.199219	0.00811283
$604$	11.1238	0.452622
$605$	27.5599	1.12047
$606$	44.2883	1.79909
$607$	19.7947	0.803443	0.401721	−	0.915762i	$-0.368412\pi$
$607$	19.7947	0.803443	0.401721	+	0.915762i	$0.368412\pi$
$608$	1.20101	0.0487072
$609$	−10.1563	−0.411552
$610$	−56.2196	−2.27627
$611$	−8.20409	−0.331902
$612$	5.33939	0.215832
$613$	−9.61688	−0.388422	−0.194211	−	0.980960i	$-0.562215\pi$
$613$	−9.61688	−0.388422	−0.194211	+	0.980960i	$0.562215\pi$
$614$	38.9324	1.57119
$615$	18.3709	0.740785
$616$	−0.511611	−0.0206134
$617$	27.3971	1.10297	0.551484	−	0.834186i	$-0.314062\pi$
$617$	27.3971	1.10297	0.551484	+	0.834186i	$0.314062\pi$
$618$	−40.5587	−1.63151
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	81.2006	3.26109
$621$	8.39396	0.336838
$622$	17.0106	0.682062
$623$	10.6667	0.427351
$624$	5.74689	0.230060
$625$	−29.7736	−1.19094
$626$	73.8038	2.94979
$627$	0.241468	0.00964332
$628$	−43.7711	−1.74666
$629$	17.9780	0.716830
$630$	3.62852	0.144564
$631$	7.45598	0.296818	0.148409	−	0.988926i	$-0.452585\pi$
$631$	7.45598	0.296818	0.148409	+	0.988926i	$0.452585\pi$
$632$	−11.6969	−0.465280
$633$	43.9777	1.74796
$634$	49.2048	1.95417
$635$	−20.7811	−0.824672
$636$	19.8961	0.788933
$637$	5.23626	0.207468
$638$	−0.954398	−0.0377850
$639$	−2.01810	−0.0798349
$640$	−50.3395	−1.98985
$641$	−25.8869	−1.02247	−0.511236	−	0.859441i	$-0.670812\pi$
$641$	−25.8869	−1.02247	−0.511236	+	0.859441i	$0.670812\pi$
$642$	1.83702	0.0725012
$643$	42.1205	1.66107	0.830536	−	0.556965i	$-0.188034\pi$
$643$	42.1205	1.66107	0.830536	+	0.556965i	$0.188034\pi$
$644$	−7.95830	−0.313601
$645$	14.2420	0.560780
$646$	13.6456	0.536879
$647$	−1.28092	−0.0503582	−0.0251791	−	0.999683i	$-0.508016\pi$
$647$	−1.28092	−0.0503582	−0.0251791	+	0.999683i	$0.508016\pi$
$648$	35.0924	1.37856
$649$	0.291825	0.0114551
$650$	3.12956	0.122752
$651$	−17.4780	−0.685015
$652$	72.9737	2.85787
$653$	26.3148	1.02978	0.514888	−	0.857257i	$-0.327834\pi$
$653$	26.3148	1.02978	0.514888	+	0.857257i	$0.327834\pi$
$654$	−57.4866	−2.24790
$655$	−16.3130	−0.637403
$656$	16.4984	0.644152
$657$	4.00323	0.156181
$658$	26.5369	1.03452
$659$	6.72659	0.262031	0.131015	−	0.991380i	$-0.458176\pi$
$659$	6.72659	0.262031	0.131015	+	0.991380i	$0.458176\pi$
$660$	−1.28930	−0.0501859
$661$	−19.5946	−0.762142	−0.381071	−	0.924546i	$-0.624445\pi$
$661$	−19.5946	−0.762142	−0.381071	+	0.924546i	$0.624445\pi$
$662$	14.4767	0.562654
$663$	−4.84838	−0.188296
$664$	−23.9326	−0.928767
$665$	6.14676	0.238361
$666$	−6.45643	−0.250182
$667$	−7.29477	−0.282455
$668$	−20.9996	−0.812498
$669$	33.1330	1.28099
$670$	−2.71848	−0.105024
$671$	0.753755	0.0290984
$672$	1.38029	0.0532458
$673$	−2.18594	−0.0842617	−0.0421308	−	0.999112i	$-0.513415\pi$
$673$	−2.18594	−0.0842617	−0.0421308	+	0.999112i	$0.513415\pi$
$674$	−62.2521	−2.39786
$675$	7.07728	0.272405
$676$	3.93207	0.151234
$677$	−0.112317	−0.00431668	−0.00215834	−	0.999998i	$-0.500687\pi$
$677$	−0.112317	−0.00431668	−0.00215834	+	0.999998i	$0.500687\pi$
$678$	−67.7611	−2.60235
$679$	10.9864	0.421621
$680$	−35.8004	−1.37288
$681$	4.37855	0.167786
$682$	−1.64243	−0.0628918
$683$	14.4955	0.554656	0.277328	−	0.960775i	$-0.410551\pi$
$683$	14.4955	0.554656	0.277328	+	0.960775i	$0.410551\pi$
$684$	−3.24832	−0.124203
$685$	26.8862	1.02727
$686$	−39.5794	−1.51115
$687$	−32.2617	−1.23086
$688$	12.7904	0.487628
$689$	3.16709	0.120657
$690$	−14.8669	−0.565974
$691$	28.7993	1.09558	0.547788	−	0.836617i	$-0.315470\pi$
$691$	28.7993	1.09558	0.547788	+	0.836617i	$0.315470\pi$
$692$	−12.4486	−0.473223
$693$	−0.0486488	−0.00184801
$694$	25.1941	0.956354
$695$	17.8954	0.678810
$696$	−35.9868	−1.36408
$697$	−13.9189	−0.527216
$698$	85.3250	3.22960
$699$	−37.1987	−1.40698
$700$	−6.70996	−0.253613
$701$	−12.3663	−0.467069	−0.233535	−	0.972349i	$-0.575029\pi$
$701$	−12.3663	−0.467069	−0.233535	+	0.972349i	$0.575029\pi$
$702$	13.4149	0.506315
$703$	−10.9373	−0.412507
$704$	0.718649	0.0270851
$705$	32.8599	1.23758
$706$	20.3721	0.766712
$707$	−15.1153	−0.568470
$708$	22.3941	0.841623
$709$	−19.0132	−0.714056	−0.357028	−	0.934094i	$-0.616210\pi$
$709$	−19.0132	−0.714056	−0.357028	+	0.934094i	$0.616210\pi$
$710$	27.5383	1.03350
$711$	−1.11226	−0.0417128
$712$	37.7953	1.41644
$713$	−12.5536	−0.470136
$714$	15.6826	0.586906
$715$	−0.205232	−0.00767526
$716$	−39.3516	−1.47064
$717$	0.789973	0.0295021
$718$	−13.7761	−0.514118
$719$	34.8903	1.30119	0.650594	−	0.759426i	$-0.274520\pi$
$719$	34.8903	1.30119	0.650594	+	0.759426i	$0.274520\pi$
$720$	4.03512	0.150380
$721$	13.8424	0.515520
$722$	37.9745	1.41327
$723$	−41.1189	−1.52923
$724$	−66.0321	−2.45406
$725$	−6.15051	−0.228424
$726$	−42.7777	−1.58763
$727$	−46.6534	−1.73028	−0.865139	−	0.501532i	$-0.832770\pi$
$727$	−46.6534	−1.73028	−0.865139	+	0.501532i	$0.832770\pi$
$728$	−6.24948	−0.231621
$729$	29.7362	1.10134
$730$	−54.6268	−2.02183
$731$	−10.7906	−0.399106
$732$	57.8418	2.13790
$733$	−6.29810	−0.232626	−0.116313	−	0.993213i	$-0.537108\pi$
$733$	−6.29810	−0.232626	−0.116313	+	0.993213i	$0.537108\pi$
$734$	25.4238	0.938411
$735$	−20.9728	−0.773595
$736$	0.991397	0.0365434
$737$	0.0364475	0.00134256
$738$	4.99869	0.184004
$739$	−2.76728	−0.101796	−0.0508979	−	0.998704i	$-0.516208\pi$
$739$	−2.76728	−0.101796	−0.0508979	+	0.998704i	$0.516208\pi$
$740$	58.3986	2.14677
$741$	2.94961	0.108357
$742$	−10.2443	−0.376080
$743$	−15.0759	−0.553083	−0.276541	−	0.961002i	$-0.589188\pi$
$743$	−15.0759	−0.553083	−0.276541	+	0.961002i	$0.589188\pi$
$744$	−61.9298	−2.27046
$745$	37.3885	1.36981
$746$	3.66355	0.134132
$747$	−2.27574	−0.0832650
$748$	0.976850	0.0357172
$749$	−0.626962	−0.0229087
$750$	36.2415	1.32335
$751$	30.6701	1.11917	0.559584	−	0.828774i	$-0.310961\pi$
$751$	30.6701	1.11917	0.559584	+	0.828774i	$0.310961\pi$
$752$	29.5105	1.07614
$753$	18.8334	0.686328
$754$	−11.6583	−0.424569
$755$	−7.09223	−0.258113
$756$	−28.7624	−1.04608
$757$	−38.8986	−1.41379	−0.706897	−	0.707316i	$-0.749906\pi$
$757$	−38.8986	−1.41379	−0.706897	+	0.707316i	$0.749906\pi$
$758$	34.1216	1.23935
$759$	0.199325	0.00723506
$760$	21.7799	0.790040
$761$	38.1964	1.38462	0.692309	−	0.721601i	$-0.256593\pi$
$761$	38.1964	1.38462	0.692309	+	0.721601i	$0.256593\pi$
$762$	32.2557	1.16850
$763$	19.6198	0.710285
$764$	−78.3072	−2.83305
$765$	−3.40424	−0.123081
$766$	−45.5740	−1.64665
$767$	3.56473	0.128715
$768$	50.0852	1.80729
$769$	26.9390	0.971444	0.485722	−	0.874113i	$-0.338557\pi$
$769$	26.9390	0.971444	0.485722	+	0.874113i	$0.338557\pi$
$770$	0.663845	0.0239233
$771$	2.43596	0.0877289
$772$	−36.2538	−1.30480
$773$	−9.06703	−0.326119	−0.163059	−	0.986616i	$-0.552136\pi$
$773$	−9.06703	−0.326119	−0.163059	+	0.986616i	$0.552136\pi$
$774$	3.87524	0.139292
$775$	−10.5844	−0.380204
$776$	38.9284	1.39745
$777$	−12.5700	−0.450945
$778$	59.1272	2.11981
$779$	8.46783	0.303392
$780$	−15.7492	−0.563911
$781$	−0.369215	−0.0132116
$782$	11.2641	0.402802
$783$	−26.3643	−0.942183
$784$	−18.8351	−0.672682
$785$	27.9072	0.996051
$786$	25.3206	0.903155
$787$	21.5905	0.769619	0.384809	−	0.922996i	$-0.374267\pi$
$787$	21.5905	0.769619	0.384809	+	0.922996i	$0.374267\pi$
$788$	12.8184	0.456636
$789$	9.03988	0.321828
$790$	15.1775	0.539990
$791$	23.1264	0.822282
$792$	−0.172378	−0.00612518
$793$	9.20735	0.326962
$794$	2.31738	0.0822409
$795$	−12.6852	−0.449898
$796$	−0.360058	−0.0127619
$797$	−36.6348	−1.29767	−0.648836	−	0.760929i	$-0.724744\pi$
$797$	−36.6348	−1.29767	−0.648836	+	0.760929i	$0.724744\pi$
$798$	−9.54081	−0.337741
$799$	−24.8967	−0.880781
$800$	0.835886	0.0295530
$801$	3.59393	0.126985
$802$	−9.95286	−0.351448
$803$	0.732399	0.0258458
$804$	2.79692	0.0986396
$805$	5.07398	0.178834
$806$	−20.0628	−0.706680
$807$	20.9405	0.737139
$808$	−53.5583	−1.88417
$809$	26.4326	0.929320	0.464660	−	0.885489i	$-0.346177\pi$
$809$	26.4326	0.929320	0.464660	+	0.885489i	$0.346177\pi$
$810$	−45.5344	−1.59992
$811$	33.1354	1.16354	0.581770	−	0.813353i	$-0.302360\pi$
$811$	33.1354	1.16354	0.581770	+	0.813353i	$0.302360\pi$
$812$	24.9959	0.877186
$813$	32.1773	1.12851
$814$	−1.18122	−0.0414016
$815$	−46.5260	−1.62973
$816$	17.4399	0.610518
$817$	6.56470	0.229670
$818$	43.4722	1.51997
$819$	−0.594260	−0.0207651
$820$	−45.2132	−1.57891
$821$	−27.6169	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$821$	−27.6169	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$822$	−41.7319	−1.45557
$823$	−49.5308	−1.72653	−0.863267	−	0.504748i	$-0.831585\pi$
$823$	−49.5308	−1.72653	−0.863267	+	0.504748i	$0.831585\pi$
$824$	49.0481	1.70867
$825$	0.168059	0.00585107
$826$	−11.5305	−0.401197
$827$	−11.1783	−0.388708	−0.194354	−	0.980931i	$-0.562261\pi$
$827$	−11.1783	−0.388708	−0.194354	+	0.980931i	$0.562261\pi$
$828$	−2.68140	−0.0931851
$829$	5.36437	0.186312	0.0931562	−	0.995652i	$-0.470304\pi$
$829$	5.36437	0.186312	0.0931562	+	0.995652i	$0.470304\pi$
$830$	31.0540	1.07790
$831$	−13.5786	−0.471035
$832$	8.77852	0.304341
$833$	15.8903	0.550566
$834$	−27.7766	−0.961826
$835$	13.3887	0.463336
$836$	−0.594286	−0.0205538
$837$	−45.3704	−1.56823
$838$	36.8726	1.27374
$839$	17.6587	0.609645	0.304823	−	0.952409i	$-0.401403\pi$
$839$	17.6587	0.609645	0.304823	+	0.952409i	$0.401403\pi$
$840$	25.0311	0.863656
$841$	−6.08811	−0.209935
$842$	−24.3419	−0.838877
$843$	12.7452	0.438968
$844$	−108.235	−3.72561
$845$	−2.50698	−0.0862426
$846$	8.94113	0.307402
$847$	14.5997	0.501653
$848$	−11.3922	−0.391210
$849$	19.3995	0.665790
$850$	9.49718	0.325751
$851$	−9.02842	−0.309490
$852$	−28.3329	−0.970671
$853$	−17.4016	−0.595819	−0.297910	−	0.954594i	$-0.596289\pi$
$853$	−17.4016	−0.595819	−0.297910	+	0.954594i	$0.596289\pi$
$854$	−29.7821	−1.01912
$855$	2.07104	0.0708280
$856$	−2.22152	−0.0759301
$857$	23.8492	0.814675	0.407337	−	0.913278i	$-0.366457\pi$
$857$	23.8492	0.814675	0.407337	+	0.913278i	$0.366457\pi$
$858$	0.318555	0.0108753
$859$	35.2878	1.20400	0.602002	−	0.798495i	$-0.294370\pi$
$859$	35.2878	1.20400	0.602002	+	0.798495i	$0.294370\pi$
$860$	−35.0516	−1.19525
$861$	9.73189	0.331662
$862$	94.2510	3.21020
$863$	−23.6797	−0.806065	−0.403033	−	0.915186i	$-0.632044\pi$
$863$	−23.6797	−0.806065	−0.403033	+	0.915186i	$0.632044\pi$
$864$	3.58304	0.121898
$865$	7.93684	0.269861
$866$	21.1984	0.720352
$867$	12.4471	0.422726
$868$	43.0156	1.46005
$869$	−0.203489	−0.00690290
$870$	46.6950	1.58311
$871$	0.445217	0.0150856
$872$	69.5192	2.35422
$873$	3.70168	0.125283
$874$	−6.85272	−0.231797
$875$	−12.3690	−0.418148
$876$	56.2030	1.89892
$877$	39.4084	1.33073	0.665364	−	0.746519i	$-0.268276\pi$
$877$	39.4084	1.33073	0.665364	+	0.746519i	$0.268276\pi$
$878$	−18.9419	−0.639259
$879$	−19.0716	−0.643268
$880$	0.738232	0.0248858
$881$	18.9999	0.640123	0.320061	−	0.947397i	$-0.396297\pi$
$881$	18.9999	0.640123	0.320061	+	0.947397i	$0.396297\pi$
$882$	−5.70668	−0.192154
$883$	6.24984	0.210324	0.105162	−	0.994455i	$-0.466464\pi$
$883$	6.24984	0.210324	0.105162	+	0.994455i	$0.466464\pi$
$884$	11.9325	0.401334
$885$	−14.2779	−0.479945
$886$	−6.87160	−0.230856
$887$	3.52826	0.118467	0.0592337	−	0.998244i	$-0.481134\pi$
$887$	3.52826	0.118467	0.0592337	+	0.998244i	$0.481134\pi$
$888$	−44.5393	−1.49464
$889$	−11.0087	−0.369219
$890$	−49.0416	−1.64388
$891$	0.610495	0.0204523
$892$	−81.5448	−2.73032
$893$	15.1464	0.506854
$894$	−58.0332	−1.94092
$895$	25.0894	0.838647
$896$	−26.6671	−0.890886
$897$	2.43482	0.0812963
$898$	60.6778	2.02485
$899$	39.4292	1.31504
$900$	−2.26079	−0.0753598
$901$	9.61107	0.320191
$902$	0.914519	0.0304501
$903$	7.54465	0.251070
$904$	81.9442	2.72542
$905$	42.1002	1.39946
$906$	11.0083	0.365728
$907$	−14.4197	−0.478797	−0.239399	−	0.970921i	$-0.576950\pi$
$907$	−14.4197	−0.478797	−0.239399	+	0.970921i	$0.576950\pi$
$908$	−10.7762	−0.357621
$909$	−5.09282	−0.168918
$910$	8.10907	0.268813
$911$	18.5078	0.613191	0.306596	−	0.951840i	$-0.400810\pi$
$911$	18.5078	0.613191	0.306596	+	0.951840i	$0.400810\pi$
$912$	−10.6099	−0.351329
$913$	−0.416351	−0.0137792
$914$	−66.2290	−2.19066
$915$	−36.8783	−1.21916
$916$	79.4004	2.62346
$917$	−8.64175	−0.285376
$918$	40.7099	1.34363
$919$	6.62327	0.218482	0.109241	−	0.994015i	$-0.465158\pi$
$919$	6.62327	0.218482	0.109241	+	0.994015i	$0.465158\pi$
$920$	17.9787	0.592740
$921$	25.5384	0.841520
$922$	57.0020	1.87726
$923$	−4.51008	−0.148451
$924$	−0.683000	−0.0224690
$925$	−7.61221	−0.250288
$926$	−84.9691	−2.79226
$927$	4.66395	0.153184
$928$	−3.11384	−0.102217
$929$	1.45448	0.0477200	0.0238600	−	0.999715i	$-0.492404\pi$
$929$	1.45448	0.0477200	0.0238600	+	0.999715i	$0.492404\pi$
$930$	80.3576	2.63503
$931$	−9.66718	−0.316829
$932$	91.5510	2.99885
$933$	11.1584	0.365309
$934$	−48.5185	−1.58757
$935$	−0.622812	−0.0203681
$936$	−2.10565	−0.0688252
$937$	50.4817	1.64916	0.824582	−	0.565742i	$-0.191410\pi$
$937$	50.4817	1.64916	0.824582	+	0.565742i	$0.191410\pi$
$938$	−1.44010	−0.0470209
$939$	48.4130	1.57990
$940$	−80.8727	−2.63778
$941$	15.0323	0.490040	0.245020	−	0.969518i	$-0.421206\pi$
$941$	15.0323	0.490040	0.245020	+	0.969518i	$0.421206\pi$
$942$	−43.3167	−1.41133
$943$	6.98997	0.227625
$944$	−12.8225	−0.417338
$945$	18.3381	0.596537
$946$	0.708981	0.0230510
$947$	25.9346	0.842760	0.421380	−	0.906884i	$-0.361546\pi$
$947$	25.9346	0.842760	0.421380	+	0.906884i	$0.361546\pi$
$948$	−15.6154	−0.507165
$949$	8.94648	0.290415
$950$	−5.77780	−0.187457
$951$	32.2768	1.04665
$952$	−18.9651	−0.614662
$953$	14.7463	0.477678	0.238839	−	0.971059i	$-0.423233\pi$
$953$	14.7463	0.477678	0.238839	+	0.971059i	$0.423233\pi$
$954$	−3.45162	−0.111750
$955$	49.9264	1.61558
$956$	−1.94423	−0.0628809
$957$	−0.626054	−0.0202375
$958$	−9.01686	−0.291321
$959$	14.2428	0.459925
$960$	−35.1607	−1.13481
$961$	36.8539	1.18883
$962$	−14.4289	−0.465207
$963$	−0.211243	−0.00680722
$964$	101.199	3.25941
$965$	23.1144	0.744078
$966$	−7.87568	−0.253396
$967$	11.5228	0.370547	0.185274	−	0.982687i	$-0.440683\pi$
$967$	11.5228	0.370547	0.185274	+	0.982687i	$0.440683\pi$
$968$	51.7315	1.66271
$969$	8.95108	0.287550
$970$	−50.5119	−1.62184
$971$	−8.33017	−0.267328	−0.133664	−	0.991027i	$-0.542674\pi$
$971$	−8.33017	−0.267328	−0.133664	+	0.991027i	$0.542674\pi$
$972$	−18.1241	−0.581329
$973$	9.47999	0.303915
$974$	−16.0218	−0.513373
$975$	2.05289	0.0657452
$976$	−33.1193	−1.06012
$977$	51.3905	1.64413	0.822064	−	0.569395i	$-0.192823\pi$
$977$	51.3905	1.64413	0.822064	+	0.569395i	$0.192823\pi$
$978$	72.2161	2.30922
$979$	0.657516	0.0210143
$980$	51.6170	1.64885
$981$	6.61053	0.211058
$982$	68.2442	2.17776
$983$	59.5944	1.90077	0.950383	−	0.311082i	$-0.100691\pi$
$983$	59.5944	1.90077	0.950383	+	0.311082i	$0.100691\pi$
$984$	34.4831	1.09928
$985$	−8.17264	−0.260402
$986$	−35.3789	−1.12669
$987$	17.4074	0.554083
$988$	−7.25939	−0.230952
$989$	5.41898	0.172313
$990$	0.223670	0.00710870
$991$	−32.5170	−1.03294	−0.516468	−	0.856307i	$-0.672753\pi$
$991$	−32.5170	−1.03294	−0.516468	+	0.856307i	$0.672753\pi$
$992$	−5.35863	−0.170137
$993$	9.49628	0.301355
$994$	14.5883	0.462713
$995$	0.229563	0.00727762
$996$	−31.9500	−1.01238
$997$	42.5266	1.34683	0.673416	−	0.739264i	$-0.264826\pi$
$997$	42.5266	1.34683	0.673416	+	0.739264i	$0.264826\pi$
$998$	−81.7812	−2.58874
$999$	−32.6299	−1.03237

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.d.1.11	✓	156

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.d.1.11	✓	156	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)

\(f(q)\)	\(=\)	\(q-2.43558 q^{2} -1.59767 q^{3} +3.93207 q^{4} -2.50698 q^{5} +3.89125 q^{6} -1.32806 q^{7} -4.70573 q^{8} -0.447465 q^{9} +O(q^{10})\)	\(q-2.43558 q^{2} -1.59767 q^{3} +3.93207 q^{4} -2.50698 q^{5} +3.89125 q^{6} -1.32806 q^{7} -4.70573 q^{8} -0.447465 q^{9} +6.10595 q^{10} -0.0818645 q^{11} -6.28214 q^{12} -1.00000 q^{13} +3.23460 q^{14} +4.00531 q^{15} +3.59705 q^{16} -3.03467 q^{17} +1.08984 q^{18} +1.84620 q^{19} -9.85762 q^{20} +2.12179 q^{21} +0.199388 q^{22} +1.52399 q^{23} +7.51818 q^{24} +1.28493 q^{25} +2.43558 q^{26} +5.50790 q^{27} -5.22203 q^{28} -4.78664 q^{29} -9.75527 q^{30} -8.23735 q^{31} +0.650528 q^{32} +0.130792 q^{33} +7.39119 q^{34} +3.32941 q^{35} -1.75946 q^{36} -5.92421 q^{37} -4.49658 q^{38} +1.59767 q^{39} +11.7972 q^{40} +4.58663 q^{41} -5.16781 q^{42} +3.55579 q^{43} -0.321897 q^{44} +1.12178 q^{45} -3.71180 q^{46} +8.20409 q^{47} -5.74689 q^{48} -5.23626 q^{49} -3.12956 q^{50} +4.84838 q^{51} -3.93207 q^{52} -3.16709 q^{53} -13.4149 q^{54} +0.205232 q^{55} +6.24948 q^{56} -2.94961 q^{57} +11.6583 q^{58} -3.56473 q^{59} +15.7492 q^{60} -9.20735 q^{61} +20.0628 q^{62} +0.594260 q^{63} -8.77852 q^{64} +2.50698 q^{65} -0.318555 q^{66} -0.445217 q^{67} -11.9325 q^{68} -2.43482 q^{69} -8.10907 q^{70} +4.51008 q^{71} +2.10565 q^{72} -8.94648 q^{73} +14.4289 q^{74} -2.05289 q^{75} +7.25939 q^{76} +0.108721 q^{77} -3.89125 q^{78} +2.48568 q^{79} -9.01773 q^{80} -7.45738 q^{81} -11.1711 q^{82} +5.08585 q^{83} +8.34305 q^{84} +7.60784 q^{85} -8.66043 q^{86} +7.64744 q^{87} +0.385232 q^{88} -8.03176 q^{89} -2.73220 q^{90} +1.32806 q^{91} +5.99243 q^{92} +13.1605 q^{93} -19.9817 q^{94} -4.62838 q^{95} -1.03933 q^{96} -8.27256 q^{97} +12.7534 q^{98} +0.0366315 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9}+O(q^{10}) \) 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9} + 11 q^{10} + 23 q^{11} + 57 q^{12} - 156 q^{13} + 18 q^{14} + 32 q^{15} + 159 q^{16} + 119 q^{17} + 36 q^{18} + 35 q^{19} + 109 q^{20} + 33 q^{21} + 11 q^{22} + 55 q^{23} + 63 q^{24} + 189 q^{25} - 13 q^{26} + 89 q^{27} + 54 q^{28} - 55 q^{29} + 47 q^{31} + 112 q^{32} + 109 q^{33} + 51 q^{34} + 25 q^{35} + 162 q^{36} + 53 q^{37} + 37 q^{38} - 23 q^{39} + 25 q^{40} + 113 q^{41} + 26 q^{42} + 31 q^{43} + 86 q^{44} + 144 q^{45} + 37 q^{46} + 115 q^{47} + 129 q^{48} + 189 q^{49} + 72 q^{50} - 4 q^{51} - 161 q^{52} + 51 q^{53} + 108 q^{54} + 22 q^{55} + 39 q^{56} + 102 q^{57} + 31 q^{58} + 75 q^{59} + 97 q^{60} + 7 q^{61} + 77 q^{62} + 94 q^{63} + 158 q^{64} - 39 q^{65} + 48 q^{66} + 37 q^{67} + 235 q^{68} + 27 q^{69} + 38 q^{70} + 70 q^{71} + 152 q^{72} + 155 q^{73} - 18 q^{74} + 80 q^{75} + 21 q^{76} + 101 q^{77} - 25 q^{78} + 10 q^{79} + 211 q^{80} + 220 q^{81} + 45 q^{82} + 132 q^{83} + 86 q^{84} + 74 q^{85} + 35 q^{86} + 53 q^{87} + 51 q^{88} + 190 q^{89} - 27 q^{90} - 19 q^{91} + 125 q^{92} + 96 q^{93} - 19 q^{94} + 72 q^{95} + 146 q^{96} + 155 q^{97} + 135 q^{98} + 89 q^{99}+O(q^{100}) \) 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9 + 11 * q^10 + 23 * q^11 + 57 * q^12 - 156 * q^13 + 18 * q^14 + 32 * q^15 + 159 * q^16 + 119 * q^17 + 36 * q^18 + 35 * q^19 + 109 * q^20 + 33 * q^21 + 11 * q^22 + 55 * q^23 + 63 * q^24 + 189 * q^25 - 13 * q^26 + 89 * q^27 + 54 * q^28 - 55 * q^29 + 47 * q^31 + 112 * q^32 + 109 * q^33 + 51 * q^34 + 25 * q^35 + 162 * q^36 + 53 * q^37 + 37 * q^38 - 23 * q^39 + 25 * q^40 + 113 * q^41 + 26 * q^42 + 31 * q^43 + 86 * q^44 + 144 * q^45 + 37 * q^46 + 115 * q^47 + 129 * q^48 + 189 * q^49 + 72 * q^50 - 4 * q^51 - 161 * q^52 + 51 * q^53 + 108 * q^54 + 22 * q^55 + 39 * q^56 + 102 * q^57 + 31 * q^58 + 75 * q^59 + 97 * q^60 + 7 * q^61 + 77 * q^62 + 94 * q^63 + 158 * q^64 - 39 * q^65 + 48 * q^66 + 37 * q^67 + 235 * q^68 + 27 * q^69 + 38 * q^70 + 70 * q^71 + 152 * q^72 + 155 * q^73 - 18 * q^74 + 80 * q^75 + 21 * q^76 + 101 * q^77 - 25 * q^78 + 10 * q^79 + 211 * q^80 + 220 * q^81 + 45 * q^82 + 132 * q^83 + 86 * q^84 + 74 * q^85 + 35 * q^86 + 53 * q^87 + 51 * q^88 + 190 * q^89 - 27 * q^90 - 19 * q^91 + 125 * q^92 + 96 * q^93 - 19 * q^94 + 72 * q^95 + 146 * q^96 + 155 * q^97 + 135 * q^98 + 89 * q^99

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