LMFDB - Embedded newform 8005.2.a.g.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8005,2,Mod(1,8005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8005, 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("8005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8005 = 5 \cdot 1601 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8005.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:	$63.9202468180$
Analytic rank:	$0$
Dimension:	$137$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.53925 q^{2} -0.755801 q^{3} +4.44781 q^{4} -1.00000 q^{5} +1.91917 q^{6} -1.64604 q^{7} -6.21562 q^{8} -2.42876 q^{9} +O(q^{10})$	\(q-2.53925 q^{2} -0.755801 q^{3} +4.44781 q^{4} -1.00000 q^{5} +1.91917 q^{6} -1.64604 q^{7} -6.21562 q^{8} -2.42876 q^{9} +2.53925 q^{10} +4.50239 q^{11} -3.36166 q^{12} -1.34071 q^{13} +4.17970 q^{14} +0.755801 q^{15} +6.88742 q^{16} -6.70357 q^{17} +6.16725 q^{18} -0.667519 q^{19} -4.44781 q^{20} +1.24408 q^{21} -11.4327 q^{22} +2.39279 q^{23} +4.69777 q^{24} +1.00000 q^{25} +3.40440 q^{26} +4.10307 q^{27} -7.32126 q^{28} -2.34112 q^{29} -1.91917 q^{30} +0.170375 q^{31} -5.05766 q^{32} -3.40291 q^{33} +17.0221 q^{34} +1.64604 q^{35} -10.8027 q^{36} +1.92937 q^{37} +1.69500 q^{38} +1.01331 q^{39} +6.21562 q^{40} -0.393932 q^{41} -3.15902 q^{42} -9.82719 q^{43} +20.0258 q^{44} +2.42876 q^{45} -6.07591 q^{46} -2.79875 q^{47} -5.20552 q^{48} -4.29057 q^{49} -2.53925 q^{50} +5.06657 q^{51} -5.96322 q^{52} -3.67538 q^{53} -10.4187 q^{54} -4.50239 q^{55} +10.2311 q^{56} +0.504512 q^{57} +5.94469 q^{58} +1.95017 q^{59} +3.36166 q^{60} +7.12777 q^{61} -0.432627 q^{62} +3.99783 q^{63} -0.932145 q^{64} +1.34071 q^{65} +8.64086 q^{66} +4.45132 q^{67} -29.8162 q^{68} -1.80848 q^{69} -4.17970 q^{70} +7.41128 q^{71} +15.0963 q^{72} -6.17523 q^{73} -4.89915 q^{74} -0.755801 q^{75} -2.96900 q^{76} -7.41109 q^{77} -2.57305 q^{78} -11.3247 q^{79} -6.88742 q^{80} +4.18519 q^{81} +1.00029 q^{82} -8.75600 q^{83} +5.53342 q^{84} +6.70357 q^{85} +24.9537 q^{86} +1.76942 q^{87} -27.9851 q^{88} +3.77493 q^{89} -6.16725 q^{90} +2.20685 q^{91} +10.6427 q^{92} -0.128770 q^{93} +7.10675 q^{94} +0.667519 q^{95} +3.82259 q^{96} -0.682798 q^{97} +10.8948 q^{98} -10.9352 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 137 q + 4 q^{2} + 20 q^{3} + 152 q^{4} - 137 q^{5} + 14 q^{6} - 30 q^{7} + 24 q^{8} + 163 q^{9}+O(q^{10}) $ 137 * q + 4 * q^2 + 20 * q^3 + 152 * q^4 - 137 * q^5 + 14 * q^6 - 30 * q^7 + 24 * q^8 + 163 * q^9	\( 137 q + 4 q^{2} + 20 q^{3} + 152 q^{4} - 137 q^{5} + 14 q^{6} - 30 q^{7} + 24 q^{8} + 163 q^{9} - 4 q^{10} + 51 q^{11} + 32 q^{12} - 6 q^{13} + 49 q^{14} - 20 q^{15} + 170 q^{16} + 46 q^{17} + 4 q^{18} + 47 q^{19} - 152 q^{20} + 28 q^{21} - 19 q^{22} + 29 q^{23} + 39 q^{24} + 137 q^{25} + 67 q^{26} + 77 q^{27} - 58 q^{28} + 27 q^{29} - 14 q^{30} + 23 q^{31} + 42 q^{32} + 40 q^{33} + 38 q^{34} + 30 q^{35} + 222 q^{36} - 56 q^{37} + 87 q^{38} + 44 q^{39} - 24 q^{40} + 66 q^{41} + 34 q^{42} + 15 q^{43} + 87 q^{44} - 163 q^{45} + 37 q^{46} + 52 q^{47} + 56 q^{48} + 195 q^{49} + 4 q^{50} + 106 q^{51} - 31 q^{52} + 45 q^{53} + 83 q^{54} - 51 q^{55} + 148 q^{56} + 4 q^{57} - 101 q^{58} + 239 q^{59} - 32 q^{60} + 46 q^{61} + 63 q^{62} - 59 q^{63} + 200 q^{64} + 6 q^{65} + 108 q^{66} - 18 q^{67} + 152 q^{68} + 63 q^{69} - 49 q^{70} + 110 q^{71} + 6 q^{72} - 19 q^{73} + 81 q^{74} + 20 q^{75} + 94 q^{76} + 43 q^{77} - 3 q^{78} + 40 q^{79} - 170 q^{80} + 229 q^{81} + 3 q^{82} + 235 q^{83} + 94 q^{84} - 46 q^{85} + 110 q^{86} + 31 q^{87} - 105 q^{88} + 150 q^{89} - 4 q^{90} + 110 q^{91} + 76 q^{92} + 11 q^{93} + 56 q^{94} - 47 q^{95} + 146 q^{96} + 17 q^{97} + 75 q^{98} + 125 q^{99}+O(q^{100}) \) 137 * q + 4 * q^2 + 20 * q^3 + 152 * q^4 - 137 * q^5 + 14 * q^6 - 30 * q^7 + 24 * q^8 + 163 * q^9 - 4 * q^10 + 51 * q^11 + 32 * q^12 - 6 * q^13 + 49 * q^14 - 20 * q^15 + 170 * q^16 + 46 * q^17 + 4 * q^18 + 47 * q^19 - 152 * q^20 + 28 * q^21 - 19 * q^22 + 29 * q^23 + 39 * q^24 + 137 * q^25 + 67 * q^26 + 77 * q^27 - 58 * q^28 + 27 * q^29 - 14 * q^30 + 23 * q^31 + 42 * q^32 + 40 * q^33 + 38 * q^34 + 30 * q^35 + 222 * q^36 - 56 * q^37 + 87 * q^38 + 44 * q^39 - 24 * q^40 + 66 * q^41 + 34 * q^42 + 15 * q^43 + 87 * q^44 - 163 * q^45 + 37 * q^46 + 52 * q^47 + 56 * q^48 + 195 * q^49 + 4 * q^50 + 106 * q^51 - 31 * q^52 + 45 * q^53 + 83 * q^54 - 51 * q^55 + 148 * q^56 + 4 * q^57 - 101 * q^58 + 239 * q^59 - 32 * q^60 + 46 * q^61 + 63 * q^62 - 59 * q^63 + 200 * q^64 + 6 * q^65 + 108 * q^66 - 18 * q^67 + 152 * q^68 + 63 * q^69 - 49 * q^70 + 110 * q^71 + 6 * q^72 - 19 * q^73 + 81 * q^74 + 20 * q^75 + 94 * q^76 + 43 * q^77 - 3 * q^78 + 40 * q^79 - 170 * q^80 + 229 * q^81 + 3 * q^82 + 235 * q^83 + 94 * q^84 - 46 * q^85 + 110 * q^86 + 31 * q^87 - 105 * q^88 + 150 * q^89 - 4 * q^90 + 110 * q^91 + 76 * q^92 + 11 * q^93 + 56 * q^94 - 47 * q^95 + 146 * q^96 + 17 * q^97 + 75 * q^98 + 125 * 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 $

$2$	−2.53925	−1.79552	−0.897762	−	0.440481i	$-0.854808\pi$
$2$	−2.53925	−1.79552	−0.897762	+	0.440481i	$0.854808\pi$
$3$	−0.755801	−0.436362	−0.218181	−	0.975908i	$-0.570012\pi$
$3$	−0.755801	−0.436362	−0.218181	+	0.975908i	$0.570012\pi$
$4$	4.44781	2.22391
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.91917	0.783499
$7$	−1.64604	−0.622143	−0.311071	−	0.950387i	$-0.600688\pi$
$7$	−1.64604	−0.622143	−0.311071	+	0.950387i	$0.600688\pi$
$8$	−6.21562	−2.19755
$9$	−2.42876	−0.809588
$10$	2.53925	0.802983
$11$	4.50239	1.35752	0.678761	−	0.734360i	$-0.262517\pi$
$11$	4.50239	1.35752	0.678761	+	0.734360i	$0.262517\pi$
$12$	−3.36166	−0.970429
$13$	−1.34071	−0.371846	−0.185923	−	0.982564i	$-0.559527\pi$
$13$	−1.34071	−0.371846	−0.185923	+	0.982564i	$0.559527\pi$
$14$	4.17970	1.11707
$15$	0.755801	0.195147
$16$	6.88742	1.72185
$17$	−6.70357	−1.62586	−0.812928	−	0.582365i	$-0.802128\pi$
$17$	−6.70357	−1.62586	−0.812928	+	0.582365i	$0.802128\pi$
$18$	6.16725	1.45363
$19$	−0.667519	−0.153139	−0.0765697	−	0.997064i	$-0.524397\pi$
$19$	−0.667519	−0.153139	−0.0765697	+	0.997064i	$0.524397\pi$
$20$	−4.44781	−0.994561
$21$	1.24408	0.271480
$22$	−11.4327	−2.43746
$23$	2.39279	0.498932	0.249466	−	0.968384i	$-0.419745\pi$
$23$	2.39279	0.498932	0.249466	+	0.968384i	$0.419745\pi$
$24$	4.69777	0.958929
$25$	1.00000	0.200000
$26$	3.40440	0.667658
$27$	4.10307	0.789636
$28$	−7.32126	−1.38359
$29$	−2.34112	−0.434735	−0.217367	−	0.976090i	$-0.569747\pi$
$29$	−2.34112	−0.434735	−0.217367	+	0.976090i	$0.569747\pi$
$30$	−1.91917	−0.350391
$31$	0.170375	0.0306003	0.0153002	−	0.999883i	$-0.495130\pi$
$31$	0.170375	0.0306003	0.0153002	+	0.999883i	$0.495130\pi$
$32$	−5.05766	−0.894077
$33$	−3.40291	−0.592371
$34$	17.0221	2.91926
$35$	1.64604	0.278231
$36$	−10.8027	−1.80045
$37$	1.92937	0.317186	0.158593	−	0.987344i	$-0.449304\pi$
$37$	1.92937	0.317186	0.158593	+	0.987344i	$0.449304\pi$
$38$	1.69500	0.274965
$39$	1.01331	0.162259
$40$	6.21562	0.982776
$41$	−0.393932	−0.0615219	−0.0307610	−	0.999527i	$-0.509793\pi$
$41$	−0.393932	−0.0615219	−0.0307610	+	0.999527i	$0.509793\pi$
$42$	−3.15902	−0.487448
$43$	−9.82719	−1.49863	−0.749316	−	0.662212i	$-0.769618\pi$
$43$	−9.82719	−1.49863	−0.749316	+	0.662212i	$0.769618\pi$
$44$	20.0258	3.01900
$45$	2.42876	0.362059
$46$	−6.07591	−0.895844
$47$	−2.79875	−0.408240	−0.204120	−	0.978946i	$-0.565433\pi$
$47$	−2.79875	−0.408240	−0.204120	+	0.978946i	$0.565433\pi$
$48$	−5.20552	−0.751352
$49$	−4.29057	−0.612938
$50$	−2.53925	−0.359105
$51$	5.06657	0.709462
$52$	−5.96322	−0.826950
$53$	−3.67538	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$53$	−3.67538	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$54$	−10.4187	−1.41781
$55$	−4.50239	−0.607102
$56$	10.2311	1.36719
$57$	0.504512	0.0668242
$58$	5.94469	0.780576
$59$	1.95017	0.253891	0.126945	−	0.991910i	$-0.459483\pi$
$59$	1.95017	0.253891	0.126945	+	0.991910i	$0.459483\pi$
$60$	3.36166	0.433989
$61$	7.12777	0.912617	0.456309	−	0.889822i	$-0.349171\pi$
$61$	7.12777	0.912617	0.456309	+	0.889822i	$0.349171\pi$
$62$	−0.432627	−0.0549436
$63$	3.99783	0.503679
$64$	−0.932145	−0.116518
$65$	1.34071	0.166295
$66$	8.64086	1.06362
$67$	4.45132	0.543815	0.271907	−	0.962323i	$-0.412346\pi$
$67$	4.45132	0.543815	0.271907	+	0.962323i	$0.412346\pi$
$68$	−29.8162	−3.61575
$69$	−1.80848	−0.217715
$70$	−4.17970	−0.499570
$71$	7.41128	0.879557	0.439779	−	0.898106i	$-0.355057\pi$
$71$	7.41128	0.879557	0.439779	+	0.898106i	$0.355057\pi$
$72$	15.0963	1.77911
$73$	−6.17523	−0.722756	−0.361378	−	0.932419i	$-0.617694\pi$
$73$	−6.17523	−0.722756	−0.361378	+	0.932419i	$0.617694\pi$
$74$	−4.89915	−0.569515
$75$	−0.755801	−0.0872724
$76$	−2.96900	−0.340568
$77$	−7.41109	−0.844572
$78$	−2.57305	−0.291341
$79$	−11.3247	−1.27413	−0.637066	−	0.770809i	$-0.719852\pi$
$79$	−11.3247	−1.27413	−0.637066	+	0.770809i	$0.719852\pi$
$80$	−6.88742	−0.770037
$81$	4.18519	0.465021
$82$	1.00029	0.110464
$83$	−8.75600	−0.961096	−0.480548	−	0.876968i	$-0.659562\pi$
$83$	−8.75600	−0.961096	−0.480548	+	0.876968i	$0.659562\pi$
$84$	5.53342	0.603745
$85$	6.70357	0.727105
$86$	24.9537	2.69083
$87$	1.76942	0.189702
$88$	−27.9851	−2.98323
$89$	3.77493	0.400142	0.200071	−	0.979781i	$-0.435883\pi$
$89$	3.77493	0.400142	0.200071	+	0.979781i	$0.435883\pi$
$90$	−6.16725	−0.650085
$91$	2.20685	0.231341
$92$	10.6427	1.10958
$93$	−0.128770	−0.0133528
$94$	7.10675	0.733005
$95$	0.667519	0.0684860
$96$	3.82259	0.390141
$97$	−0.682798	−0.0693277	−0.0346638	−	0.999399i	$-0.511036\pi$
$97$	−0.682798	−0.0693277	−0.0346638	+	0.999399i	$0.511036\pi$
$98$	10.8948	1.10055
$99$	−10.9352	−1.09903
$100$	4.44781	0.444781
$101$	10.9930	1.09385	0.546923	−	0.837183i	$-0.315799\pi$
$101$	10.9930	1.09385	0.546923	+	0.837183i	$0.315799\pi$
$102$	−12.8653	−1.27386
$103$	6.61615	0.651909	0.325954	−	0.945386i	$-0.394314\pi$
$103$	6.61615	0.651909	0.325954	+	0.945386i	$0.394314\pi$
$104$	8.33334	0.817151
$105$	−1.24408	−0.121409
$106$	9.33273	0.906475
$107$	−15.7508	−1.52269	−0.761346	−	0.648346i	$-0.775461\pi$
$107$	−15.7508	−1.52269	−0.761346	+	0.648346i	$0.775461\pi$
$108$	18.2497	1.75608
$109$	−7.83901	−0.750841	−0.375421	−	0.926855i	$-0.622502\pi$
$109$	−7.83901	−0.750841	−0.375421	+	0.926855i	$0.622502\pi$
$110$	11.4327	1.09007
$111$	−1.45822	−0.138408
$112$	−11.3369	−1.07124
$113$	−9.16226	−0.861913	−0.430956	−	0.902373i	$-0.641824\pi$
$113$	−9.16226	−0.861913	−0.430956	+	0.902373i	$0.641824\pi$
$114$	−1.28108	−0.119984
$115$	−2.39279	−0.223129
$116$	−10.4129	−0.966809
$117$	3.25627	0.301042
$118$	−4.95198	−0.455867
$119$	11.0343	1.01151
$120$	−4.69777	−0.428846
$121$	9.27151	0.842864
$122$	−18.0992	−1.63863
$123$	0.297735	0.0268458
$124$	0.757798	0.0680523
$125$	−1.00000	−0.0894427
$126$	−10.1515	−0.904369
$127$	−20.4293	−1.81281	−0.906404	−	0.422412i	$-0.861184\pi$
$127$	−20.4293	−1.81281	−0.906404	+	0.422412i	$0.861184\pi$
$128$	12.4823	1.10329
$129$	7.42741	0.653947
$130$	−3.40440	−0.298586
$131$	−10.7316	−0.937620	−0.468810	−	0.883299i	$-0.655317\pi$
$131$	−10.7316	−0.937620	−0.468810	+	0.883299i	$0.655317\pi$
$132$	−15.1355	−1.31738
$133$	1.09876	0.0952745
$134$	−11.3030	−0.976433
$135$	−4.10307	−0.353136
$136$	41.6669	3.57290
$137$	3.22957	0.275921	0.137961	−	0.990438i	$-0.455945\pi$
$137$	3.22957	0.275921	0.137961	+	0.990438i	$0.455945\pi$
$138$	4.59218	0.390912
$139$	−6.59952	−0.559764	−0.279882	−	0.960034i	$-0.590295\pi$
$139$	−6.59952	−0.559764	−0.279882	+	0.960034i	$0.590295\pi$
$140$	7.32126	0.618759
$141$	2.11530	0.178141
$142$	−18.8191	−1.57927
$143$	−6.03640	−0.504789
$144$	−16.7279	−1.39399
$145$	2.34112	0.194419
$146$	15.6805	1.29773
$147$	3.24282	0.267463
$148$	8.58146	0.705392
$149$	−5.24604	−0.429772	−0.214886	−	0.976639i	$-0.568938\pi$
$149$	−5.24604	−0.429772	−0.214886	+	0.976639i	$0.568938\pi$
$150$	1.91917	0.156700
$151$	−13.4648	−1.09575	−0.547876	−	0.836560i	$-0.684563\pi$
$151$	−13.4648	−1.09575	−0.547876	+	0.836560i	$0.684563\pi$
$152$	4.14904	0.336532
$153$	16.2814	1.31627
$154$	18.8186	1.51645
$155$	−0.170375	−0.0136849
$156$	4.50701	0.360850
$157$	23.1422	1.84695	0.923475	−	0.383659i	$-0.125336\pi$
$157$	23.1422	1.84695	0.923475	+	0.383659i	$0.125336\pi$
$158$	28.7564	2.28774
$159$	2.77786	0.220299
$160$	5.05766	0.399843
$161$	−3.93862	−0.310407
$162$	−10.6273	−0.834956
$163$	2.06408	0.161671	0.0808356	−	0.996727i	$-0.474241\pi$
$163$	2.06408	0.161671	0.0808356	+	0.996727i	$0.474241\pi$
$164$	−1.75214	−0.136819
$165$	3.40291	0.264916
$166$	22.2337	1.72567
$167$	8.63830	0.668452	0.334226	−	0.942493i	$-0.391525\pi$
$167$	8.63830	0.668452	0.334226	+	0.942493i	$0.391525\pi$
$168$	−7.73270	−0.596591
$169$	−11.2025	−0.861731
$170$	−17.0221	−1.30553
$171$	1.62125	0.123980
$172$	−43.7095	−3.33282
$173$	−0.750029	−0.0570236	−0.0285118	−	0.999593i	$-0.509077\pi$
$173$	−0.750029	−0.0570236	−0.0285118	+	0.999593i	$0.509077\pi$
$174$	−4.49301	−0.340614
$175$	−1.64604	−0.124429
$176$	31.0098	2.33745
$177$	−1.47394	−0.110788
$178$	−9.58552	−0.718465
$179$	2.00606	0.149940	0.0749700	−	0.997186i	$-0.476114\pi$
$179$	2.00606	0.149940	0.0749700	+	0.997186i	$0.476114\pi$
$180$	10.8027	0.805185
$181$	5.12389	0.380855	0.190428	−	0.981701i	$-0.439013\pi$
$181$	5.12389	0.380855	0.190428	+	0.981701i	$0.439013\pi$
$182$	−5.60377	−0.415379
$183$	−5.38718	−0.398231
$184$	−14.8727	−1.09643
$185$	−1.92937	−0.141850
$186$	0.326980	0.0239753
$187$	−30.1821	−2.20713
$188$	−12.4483	−0.907888
$189$	−6.75379	−0.491266
$190$	−1.69500	−0.122968
$191$	17.2724	1.24979	0.624893	−	0.780710i	$-0.285143\pi$
$191$	17.2724	1.24979	0.624893	+	0.780710i	$0.285143\pi$
$192$	0.704516	0.0508441
$193$	−11.4927	−0.827260	−0.413630	−	0.910445i	$-0.635739\pi$
$193$	−11.4927	−0.827260	−0.413630	+	0.910445i	$0.635739\pi$
$194$	1.73380	0.124479
$195$	−1.01331	−0.0725646
$196$	−19.0836	−1.36312
$197$	13.5280	0.963833	0.481916	−	0.876217i	$-0.339941\pi$
$197$	13.5280	0.963833	0.481916	+	0.876217i	$0.339941\pi$
$198$	27.7674	1.97334
$199$	−4.44823	−0.315326	−0.157663	−	0.987493i	$-0.550396\pi$
$199$	−4.44823	−0.315326	−0.157663	+	0.987493i	$0.550396\pi$
$200$	−6.21562	−0.439511
$201$	−3.36431	−0.237300
$202$	−27.9140	−1.96403
$203$	3.85356	0.270467
$204$	22.5352	1.57778
$205$	0.393932	0.0275134
$206$	−16.8001	−1.17052
$207$	−5.81153	−0.403929
$208$	−9.23402	−0.640264
$209$	−3.00543	−0.207890
$210$	3.15902	0.217993
$211$	−13.3985	−0.922391	−0.461196	−	0.887298i	$-0.652579\pi$
$211$	−13.3985	−0.922391	−0.461196	+	0.887298i	$0.652579\pi$
$212$	−16.3474	−1.12275
$213$	−5.60146	−0.383805
$214$	39.9954	2.73403
$215$	9.82719	0.670209
$216$	−25.5031	−1.73527
$217$	−0.280444	−0.0190378
$218$	19.9053	1.34815
$219$	4.66725	0.315384
$220$	−20.0258	−1.35014
$221$	8.98754	0.604568
$222$	3.70279	0.248515
$223$	−8.16267	−0.546613	−0.273306	−	0.961927i	$-0.588117\pi$
$223$	−8.16267	−0.546613	−0.273306	+	0.961927i	$0.588117\pi$
$224$	8.32509	0.556243
$225$	−2.42876	−0.161918
$226$	23.2653	1.54758
$227$	27.9553	1.85546	0.927730	−	0.373253i	$-0.121758\pi$
$227$	27.9553	1.85546	0.927730	+	0.373253i	$0.121758\pi$
$228$	2.24397	0.148611
$229$	13.8832	0.917426	0.458713	−	0.888585i	$-0.348311\pi$
$229$	13.8832	0.917426	0.458713	+	0.888585i	$0.348311\pi$
$230$	6.07591	0.400633
$231$	5.60131	0.368539
$232$	14.5515	0.955353
$233$	−8.57569	−0.561812	−0.280906	−	0.959735i	$-0.590635\pi$
$233$	−8.57569	−0.561812	−0.280906	+	0.959735i	$0.590635\pi$
$234$	−8.26849	−0.540528
$235$	2.79875	0.182571
$236$	8.67400	0.564629
$237$	8.55925	0.555983
$238$	−28.0189	−1.81620
$239$	22.4807	1.45416	0.727078	−	0.686555i	$-0.240878\pi$
$239$	22.4807	1.45416	0.727078	+	0.686555i	$0.240878\pi$
$240$	5.20552	0.336015
$241$	11.4197	0.735605	0.367803	−	0.929904i	$-0.380110\pi$
$241$	11.4197	0.735605	0.367803	+	0.929904i	$0.380110\pi$
$242$	−23.5427	−1.51338
$243$	−15.4724	−0.992553
$244$	31.7030	2.02957
$245$	4.29057	0.274114
$246$	−0.756024	−0.0482023
$247$	0.894949	0.0569442
$248$	−1.05899	−0.0672459
$249$	6.61780	0.419386
$250$	2.53925	0.160597
$251$	−27.3164	−1.72420	−0.862098	−	0.506742i	$-0.830850\pi$
$251$	−27.3164	−1.72420	−0.862098	+	0.506742i	$0.830850\pi$
$252$	17.7816	1.12014
$253$	10.7733	0.677310
$254$	51.8752	3.25494
$255$	−5.06657	−0.317281
$256$	−29.8314	−1.86446
$257$	−14.4720	−0.902741	−0.451371	−	0.892337i	$-0.649065\pi$
$257$	−14.4720	−0.902741	−0.451371	+	0.892337i	$0.649065\pi$
$258$	−18.8601	−1.17418
$259$	−3.17580	−0.197335
$260$	5.96322	0.369823
$261$	5.68602	0.351956
$262$	27.2501	1.68352
$263$	−3.15571	−0.194589	−0.0972947	−	0.995256i	$-0.531019\pi$
$263$	−3.15571	−0.194589	−0.0972947	+	0.995256i	$0.531019\pi$
$264$	21.1512	1.30177
$265$	3.67538	0.225777
$266$	−2.79003	−0.171068
$267$	−2.85310	−0.174607
$268$	19.7986	1.20939
$269$	1.23715	0.0754306	0.0377153	−	0.999289i	$-0.487992\pi$
$269$	1.23715	0.0754306	0.0377153	+	0.999289i	$0.487992\pi$
$270$	10.4187	0.634064
$271$	3.02192	0.183569	0.0917844	−	0.995779i	$-0.470743\pi$
$271$	3.02192	0.183569	0.0917844	+	0.995779i	$0.470743\pi$
$272$	−46.1703	−2.79949
$273$	−1.66794	−0.100949
$274$	−8.20071	−0.495423
$275$	4.50239	0.271504
$276$	−8.04376	−0.484177
$277$	−0.600981	−0.0361094	−0.0180547	−	0.999837i	$-0.505747\pi$
$277$	−0.600981	−0.0361094	−0.0180547	+	0.999837i	$0.505747\pi$
$278$	16.7579	1.00507
$279$	−0.413802	−0.0247737
$280$	−10.2311	−0.611427
$281$	14.7516	0.880006	0.440003	−	0.897996i	$-0.354977\pi$
$281$	14.7516	0.880006	0.440003	+	0.897996i	$0.354977\pi$
$282$	−5.37129	−0.319856
$283$	−31.4013	−1.86661	−0.933307	−	0.359078i	$-0.883091\pi$
$283$	−31.4013	−1.86661	−0.933307	+	0.359078i	$0.883091\pi$
$284$	32.9640	1.95605
$285$	−0.504512	−0.0298847
$286$	15.3279	0.906360
$287$	0.648427	0.0382754
$288$	12.2839	0.723834
$289$	27.9379	1.64341
$290$	−5.94469	−0.349084
$291$	0.516060	0.0302520
$292$	−27.4663	−1.60734
$293$	−2.74232	−0.160208	−0.0801040	−	0.996787i	$-0.525525\pi$
$293$	−2.74232	−0.160208	−0.0801040	+	0.996787i	$0.525525\pi$
$294$	−8.23434	−0.480236
$295$	−1.95017	−0.113543
$296$	−11.9922	−0.697033
$297$	18.4736	1.07195
$298$	13.3210	0.771667
$299$	−3.20804	−0.185526
$300$	−3.36166	−0.194086
$301$	16.1759	0.932364
$302$	34.1906	1.96745
$303$	−8.30853	−0.477313
$304$	−4.59748	−0.263684
$305$	−7.12777	−0.408135
$306$	−41.3426	−2.36340
$307$	−9.84297	−0.561768	−0.280884	−	0.959742i	$-0.590628\pi$
$307$	−9.84297	−0.561768	−0.280884	+	0.959742i	$0.590628\pi$
$308$	−32.9631	−1.87825
$309$	−5.00050	−0.284468
$310$	0.432627	0.0245715
$311$	30.0269	1.70267	0.851336	−	0.524621i	$-0.175793\pi$
$311$	30.0269	1.70267	0.851336	+	0.524621i	$0.175793\pi$
$312$	−6.29835	−0.356574
$313$	2.31667	0.130946	0.0654730	−	0.997854i	$-0.479144\pi$
$313$	2.31667	0.130946	0.0654730	+	0.997854i	$0.479144\pi$
$314$	−58.7640	−3.31624
$315$	−3.99783	−0.225252
$316$	−50.3703	−2.83355
$317$	−2.37173	−0.133210	−0.0666048	−	0.997779i	$-0.521217\pi$
$317$	−2.37173	−0.133210	−0.0666048	+	0.997779i	$0.521217\pi$
$318$	−7.05369	−0.395551
$319$	−10.5406	−0.590161
$320$	0.932145	0.0521085
$321$	11.9045	0.664445
$322$	10.0012	0.557343
$323$	4.47476	0.248982
$324$	18.6149	1.03416
$325$	−1.34071	−0.0743692
$326$	−5.24123	−0.290285
$327$	5.92474	0.327639
$328$	2.44853	0.135198
$329$	4.60685	0.253984
$330$	−8.64086	−0.475664
$331$	−7.91616	−0.435111	−0.217556	−	0.976048i	$-0.569808\pi$
$331$	−7.91616	−0.435111	−0.217556	+	0.976048i	$0.569808\pi$
$332$	−38.9451	−2.13739
$333$	−4.68598	−0.256790
$334$	−21.9348	−1.20022
$335$	−4.45132	−0.243201
$336$	8.56847	0.467448
$337$	9.06642	0.493880	0.246940	−	0.969031i	$-0.420575\pi$
$337$	9.06642	0.493880	0.246940	+	0.969031i	$0.420575\pi$
$338$	28.4460	1.54726
$339$	6.92485	0.376106
$340$	29.8162	1.61701
$341$	0.767097	0.0415406
$342$	−4.11676	−0.222609
$343$	18.5847	1.00348
$344$	61.0821	3.29333
$345$	1.80848	0.0973650
$346$	1.90451	0.102387
$347$	11.9916	0.643742	0.321871	−	0.946783i	$-0.395688\pi$
$347$	11.9916	0.643742	0.321871	+	0.946783i	$0.395688\pi$
$348$	7.87005	0.421879
$349$	−24.4525	−1.30891	−0.654457	−	0.756099i	$-0.727103\pi$
$349$	−24.4525	−1.30891	−0.654457	+	0.756099i	$0.727103\pi$
$350$	4.17970	0.223414
$351$	−5.50102	−0.293623
$352$	−22.7716	−1.21373
$353$	−4.93048	−0.262423	−0.131212	−	0.991354i	$-0.541887\pi$
$353$	−4.93048	−0.262423	−0.131212	+	0.991354i	$0.541887\pi$
$354$	3.74271	0.198923
$355$	−7.41128	−0.393350
$356$	16.7902	0.889879
$357$	−8.33975	−0.441387
$358$	−5.09390	−0.269221
$359$	12.4273	0.655888	0.327944	−	0.944697i	$-0.393644\pi$
$359$	12.4273	0.655888	0.327944	+	0.944697i	$0.393644\pi$
$360$	−15.0963	−0.795644
$361$	−18.5544	−0.976548
$362$	−13.0109	−0.683835
$363$	−7.00742	−0.367794
$364$	9.81568	0.514481
$365$	6.17523	0.323227
$366$	13.6794	0.715034
$367$	−33.7950	−1.76408	−0.882041	−	0.471172i	$-0.843831\pi$
$367$	−33.7950	−1.76408	−0.882041	+	0.471172i	$0.843831\pi$
$368$	16.4802	0.859087
$369$	0.956769	0.0498074
$370$	4.89915	0.254695
$371$	6.04981	0.314090
$372$	−0.572745	−0.0296954
$373$	−2.61275	−0.135283	−0.0676416	−	0.997710i	$-0.521547\pi$
$373$	−2.61275	−0.135283	−0.0676416	+	0.997710i	$0.521547\pi$
$374$	76.6400	3.96296
$375$	0.755801	0.0390294
$376$	17.3960	0.897130
$377$	3.13876	0.161654
$378$	17.1496	0.882080
$379$	4.61157	0.236880	0.118440	−	0.992961i	$-0.462211\pi$
$379$	4.61157	0.236880	0.118440	+	0.992961i	$0.462211\pi$
$380$	2.96900	0.152306
$381$	15.4405	0.791041
$382$	−43.8590	−2.24402
$383$	−18.9752	−0.969589	−0.484795	−	0.874628i	$-0.661106\pi$
$383$	−18.9752	−0.969589	−0.484795	+	0.874628i	$0.661106\pi$
$384$	−9.43412	−0.481433
$385$	7.41109	0.377704
$386$	29.1828	1.48537
$387$	23.8679	1.21328
$388$	−3.03696	−0.154178
$389$	−12.4790	−0.632712	−0.316356	−	0.948640i	$-0.602459\pi$
$389$	−12.4790	−0.632712	−0.316356	+	0.948640i	$0.602459\pi$
$390$	2.57305	0.130292
$391$	−16.0403	−0.811191
$392$	26.6685	1.34696
$393$	8.11092	0.409142
$394$	−34.3511	−1.73058
$395$	11.3247	0.569809
$396$	−48.6379	−2.44415
$397$	−9.78152	−0.490920	−0.245460	−	0.969407i	$-0.578939\pi$
$397$	−9.78152	−0.490920	−0.245460	+	0.969407i	$0.578939\pi$
$398$	11.2952	0.566176
$399$	−0.830444	−0.0415742
$400$	6.88742	0.344371
$401$	23.8093	1.18898	0.594490	−	0.804103i	$-0.297354\pi$
$401$	23.8093	1.18898	0.594490	+	0.804103i	$0.297354\pi$
$402$	8.54284	0.426078
$403$	−0.228424	−0.0113786
$404$	48.8948	2.43261
$405$	−4.18519	−0.207964
$406$	−9.78517	−0.485630
$407$	8.68676	0.430587
$408$	−31.4919	−1.55908
$409$	−0.905986	−0.0447981	−0.0223991	−	0.999749i	$-0.507130\pi$
$409$	−0.905986	−0.0447981	−0.0223991	+	0.999749i	$0.507130\pi$
$410$	−1.00029	−0.0494010
$411$	−2.44092	−0.120402
$412$	29.4274	1.44978
$413$	−3.21005	−0.157956
$414$	14.7569	0.725264
$415$	8.75600	0.429815
$416$	6.78085	0.332459
$417$	4.98793	0.244260
$418$	7.63155	0.373271
$419$	−40.4272	−1.97500	−0.987498	−	0.157630i	$-0.949615\pi$
$419$	−40.4272	−1.97500	−0.987498	+	0.157630i	$0.949615\pi$
$420$	−5.53342	−0.270003
$421$	−26.5407	−1.29351	−0.646757	−	0.762696i	$-0.723875\pi$
$421$	−26.5407	−1.29351	−0.646757	+	0.762696i	$0.723875\pi$
$422$	34.0222	1.65618
$423$	6.79751	0.330506
$424$	22.8448	1.10944
$425$	−6.70357	−0.325171
$426$	14.2235	0.689132
$427$	−11.7326	−0.567778
$428$	−70.0568	−3.38632
$429$	4.56232	0.220271
$430$	−24.9537	−1.20338
$431$	−1.14318	−0.0550649	−0.0275325	−	0.999621i	$-0.508765\pi$
$431$	−1.14318	−0.0550649	−0.0275325	+	0.999621i	$0.508765\pi$
$432$	28.2595	1.35964
$433$	6.57821	0.316129	0.158064	−	0.987429i	$-0.449475\pi$
$433$	6.57821	0.316129	0.158064	+	0.987429i	$0.449475\pi$
$434$	0.712119	0.0341828
$435$	−1.76942	−0.0848372
$436$	−34.8665	−1.66980
$437$	−1.59723	−0.0764060
$438$	−11.8513	−0.566279
$439$	−36.4532	−1.73982	−0.869908	−	0.493215i	$-0.835822\pi$
$439$	−36.4532	−1.73982	−0.869908	+	0.493215i	$0.835822\pi$
$440$	27.9851	1.33414
$441$	10.4208	0.496228
$442$	−22.8217	−1.08552
$443$	32.8236	1.55950	0.779748	−	0.626094i	$-0.215347\pi$
$443$	32.8236	1.55950	0.779748	+	0.626094i	$0.215347\pi$
$444$	−6.48588	−0.307806
$445$	−3.77493	−0.178949
$446$	20.7271	0.981456
$447$	3.96496	0.187536
$448$	1.53434	0.0724909
$449$	−14.6277	−0.690324	−0.345162	−	0.938543i	$-0.612176\pi$
$449$	−14.6277	−0.690324	−0.345162	+	0.938543i	$0.612176\pi$
$450$	6.16725	0.290727
$451$	−1.77364	−0.0835173
$452$	−40.7520	−1.91681
$453$	10.1767	0.478144
$454$	−70.9856	−3.33152
$455$	−2.20685	−0.103459
$456$	−3.13585	−0.146850
$457$	14.0199	0.655823	0.327911	−	0.944708i	$-0.393655\pi$
$457$	14.0199	0.655823	0.327911	+	0.944708i	$0.393655\pi$
$458$	−35.2529	−1.64726
$459$	−27.5052	−1.28383
$460$	−10.6427	−0.496218
$461$	−20.8220	−0.969779	−0.484890	−	0.874575i	$-0.661140\pi$
$461$	−20.8220	−0.969779	−0.484890	+	0.874575i	$0.661140\pi$
$462$	−14.2232	−0.661721
$463$	14.2222	0.660960	0.330480	−	0.943813i	$-0.392789\pi$
$463$	14.2222	0.660960	0.330480	+	0.943813i	$0.392789\pi$
$464$	−16.1242	−0.748549
$465$	0.128770	0.00597156
$466$	21.7759	1.00875
$467$	13.3879	0.619517	0.309759	−	0.950815i	$-0.399752\pi$
$467$	13.3879	0.619517	0.309759	+	0.950815i	$0.399752\pi$
$468$	14.4833	0.669489
$469$	−7.32702	−0.338331
$470$	−7.10675	−0.327810
$471$	−17.4909	−0.805939
$472$	−12.1215	−0.557938
$473$	−44.2458	−2.03443
$474$	−21.7341	−0.998281
$475$	−0.667519	−0.0306279
$476$	49.0786	2.24951
$477$	8.92664	0.408723
$478$	−57.0843	−2.61097
$479$	−20.3235	−0.928603	−0.464302	−	0.885677i	$-0.653695\pi$
$479$	−20.3235	−0.928603	−0.464302	+	0.885677i	$0.653695\pi$
$480$	−3.82259	−0.174476
$481$	−2.58672	−0.117944
$482$	−28.9974	−1.32080
$483$	2.97681	0.135450
$484$	41.2379	1.87445
$485$	0.682798	0.0310043
$486$	39.2883	1.78215
$487$	25.4781	1.15452	0.577262	−	0.816559i	$-0.304121\pi$
$487$	25.4781	1.15452	0.577262	+	0.816559i	$0.304121\pi$
$488$	−44.3035	−2.00552
$489$	−1.56003	−0.0705472
$490$	−10.8948	−0.492179
$491$	27.1340	1.22454	0.612271	−	0.790648i	$-0.290256\pi$
$491$	27.1340	1.22454	0.612271	+	0.790648i	$0.290256\pi$
$492$	1.32427	0.0597026
$493$	15.6939	0.706816
$494$	−2.27250	−0.102245
$495$	10.9352	0.491503
$496$	1.17345	0.0526893
$497$	−12.1992	−0.547210
$498$	−16.8043	−0.753018
$499$	−10.1250	−0.453256	−0.226628	−	0.973981i	$-0.572770\pi$
$499$	−10.1250	−0.453256	−0.226628	+	0.973981i	$0.572770\pi$
$500$	−4.44781	−0.198912
$501$	−6.52884	−0.291687
$502$	69.3633	3.09583
$503$	−31.7939	−1.41762	−0.708811	−	0.705399i	$-0.750768\pi$
$503$	−31.7939	−1.41762	−0.708811	+	0.705399i	$0.750768\pi$
$504$	−24.8490	−1.10686
$505$	−10.9930	−0.489182
$506$	−27.3561	−1.21613
$507$	8.46686	0.376027
$508$	−90.8657	−4.03152
$509$	17.4669	0.774207	0.387104	−	0.922036i	$-0.373476\pi$
$509$	17.4669	0.774207	0.387104	+	0.922036i	$0.373476\pi$
$510$	12.8653	0.569686
$511$	10.1647	0.449658
$512$	50.7849	2.24440
$513$	−2.73887	−0.120924
$514$	36.7482	1.62089
$515$	−6.61615	−0.291542
$516$	33.0357	1.45432
$517$	−12.6011	−0.554195
$518$	8.06418	0.354320
$519$	0.566873	0.0248830
$520$	−8.33334	−0.365441
$521$	38.7327	1.69691	0.848454	−	0.529269i	$-0.177534\pi$
$521$	38.7327	1.69691	0.848454	+	0.529269i	$0.177534\pi$
$522$	−14.4383	−0.631945
$523$	0.169845	0.00742679	0.00371339	−	0.999993i	$-0.498818\pi$
$523$	0.169845	0.00742679	0.00371339	+	0.999993i	$0.498818\pi$
$524$	−47.7319	−2.08518
$525$	1.24408	0.0542959
$526$	8.01315	0.349390
$527$	−1.14212	−0.0497517
$528$	−23.4373	−1.01998
$529$	−17.2745	−0.751067
$530$	−9.33273	−0.405388
$531$	−4.73651	−0.205547
$532$	4.88708	0.211882
$533$	0.528149	0.0228767
$534$	7.24475	0.313511
$535$	15.7508	0.680968
$536$	−27.6677	−1.19506
$537$	−1.51618	−0.0654282
$538$	−3.14145	−0.135438
$539$	−19.3178	−0.832077
$540$	−18.2497	−0.785341
$541$	−5.48832	−0.235961	−0.117981	−	0.993016i	$-0.537642\pi$
$541$	−5.48832	−0.235961	−0.117981	+	0.993016i	$0.537642\pi$
$542$	−7.67343	−0.329602
$543$	−3.87264	−0.166191
$544$	33.9044	1.45364
$545$	7.83901	0.335786
$546$	4.23533	0.181256
$547$	4.85117	0.207421	0.103711	−	0.994608i	$-0.466928\pi$
$547$	4.85117	0.207421	0.103711	+	0.994608i	$0.466928\pi$
$548$	14.3645	0.613623
$549$	−17.3117	−0.738844
$550$	−11.4327	−0.487492
$551$	1.56274	0.0665749
$552$	11.2408	0.478440
$553$	18.6409	0.792693
$554$	1.52604	0.0648353
$555$	1.45822	0.0618979
$556$	−29.3534	−1.24486
$557$	8.84925	0.374955	0.187477	−	0.982269i	$-0.439969\pi$
$557$	8.84925	0.374955	0.187477	+	0.982269i	$0.439969\pi$
$558$	1.05075	0.0444817
$559$	13.1754	0.557260
$560$	11.3369	0.479073
$561$	22.8117	0.963109
$562$	−37.4580	−1.58007
$563$	−18.5233	−0.780662	−0.390331	−	0.920675i	$-0.627639\pi$
$563$	−18.5233	−0.780662	−0.390331	+	0.920675i	$0.627639\pi$
$564$	9.40847	0.396168
$565$	9.16226	0.385459
$566$	79.7360	3.35155
$567$	−6.88897	−0.289309
$568$	−46.0657	−1.93287
$569$	13.1758	0.552357	0.276179	−	0.961106i	$-0.410932\pi$
$569$	13.1758	0.552357	0.276179	+	0.961106i	$0.410932\pi$
$570$	1.28108	0.0536587
$571$	32.5139	1.36067	0.680333	−	0.732903i	$-0.261835\pi$
$571$	32.5139	1.36067	0.680333	+	0.732903i	$0.261835\pi$
$572$	−26.8488	−1.12260
$573$	−13.0545	−0.545359
$574$	−1.64652	−0.0687244
$575$	2.39279	0.0997863
$576$	2.26396	0.0943317
$577$	18.9744	0.789916	0.394958	−	0.918699i	$-0.370759\pi$
$577$	18.9744	0.789916	0.394958	+	0.918699i	$0.370759\pi$
$578$	−70.9414	−2.95077
$579$	8.68617	0.360985
$580$	10.4129	0.432370
$581$	14.4127	0.597939
$582$	−1.31041	−0.0543181
$583$	−16.5480	−0.685348
$584$	38.3829	1.58830
$585$	−3.25627	−0.134630
$586$	6.96345	0.287657
$587$	11.5861	0.478211	0.239106	−	0.970994i	$-0.423146\pi$
$587$	11.5861	0.478211	0.239106	+	0.970994i	$0.423146\pi$
$588$	14.4234	0.594813
$589$	−0.113729	−0.00468611
$590$	4.95198	0.203870
$591$	−10.2245	−0.420580
$592$	13.2883	0.546148
$593$	1.70111	0.0698564	0.0349282	−	0.999390i	$-0.488880\pi$
$593$	1.70111	0.0698564	0.0349282	+	0.999390i	$0.488880\pi$
$594$	−46.9092	−1.92471
$595$	−11.0343	−0.452363
$596$	−23.3334	−0.955774
$597$	3.36198	0.137597
$598$	8.14602	0.333116
$599$	27.9410	1.14164	0.570820	−	0.821075i	$-0.306625\pi$
$599$	27.9410	1.14164	0.570820	+	0.821075i	$0.306625\pi$
$600$	4.69777	0.191786
$601$	−7.32044	−0.298607	−0.149303	−	0.988791i	$-0.547703\pi$
$601$	−7.32044	−0.298607	−0.149303	+	0.988791i	$0.547703\pi$
$602$	−41.0747	−1.67408
$603$	−10.8112	−0.440266
$604$	−59.8890	−2.43685
$605$	−9.27151	−0.376940
$606$	21.0975	0.857026
$607$	18.4448	0.748653	0.374327	−	0.927297i	$-0.377874\pi$
$607$	18.4448	0.748653	0.374327	+	0.927297i	$0.377874\pi$
$608$	3.37608	0.136918
$609$	−2.91253	−0.118022
$610$	18.0992	0.732816
$611$	3.75232	0.151802
$612$	72.4166	2.92727
$613$	−33.9776	−1.37234	−0.686171	−	0.727440i	$-0.740710\pi$
$613$	−33.9776	−1.37234	−0.686171	+	0.727440i	$0.740710\pi$
$614$	24.9938	1.00867
$615$	−0.297735	−0.0120058
$616$	46.0645	1.85599
$617$	24.9727	1.00536	0.502681	−	0.864472i	$-0.332347\pi$
$617$	24.9727	1.00536	0.502681	+	0.864472i	$0.332347\pi$
$618$	12.6975	0.510770
$619$	−17.9139	−0.720021	−0.360010	−	0.932948i	$-0.617227\pi$
$619$	−17.9139	−0.720021	−0.360010	+	0.932948i	$0.617227\pi$
$620$	−0.757798	−0.0304339
$621$	9.81779	0.393974
$622$	−76.2461	−3.05719
$623$	−6.21367	−0.248946
$624$	6.97909	0.279387
$625$	1.00000	0.0400000
$626$	−5.88262	−0.235117
$627$	2.27151	0.0907153
$628$	102.932	4.10744
$629$	−12.9336	−0.515698
$630$	10.1515	0.404446
$631$	11.6494	0.463755	0.231878	−	0.972745i	$-0.425513\pi$
$631$	11.6494	0.463755	0.231878	+	0.972745i	$0.425513\pi$
$632$	70.3903	2.79998
$633$	10.1266	0.402497
$634$	6.02242	0.239181
$635$	20.4293	0.810712
$636$	12.3554	0.489924
$637$	5.75240	0.227919
$638$	26.7653	1.05965
$639$	−18.0003	−0.712079
$640$	−12.4823	−0.493405
$641$	−42.2729	−1.66968	−0.834839	−	0.550494i	$-0.814439\pi$
$641$	−42.2729	−1.66968	−0.834839	+	0.550494i	$0.814439\pi$
$642$	−30.2286	−1.19303
$643$	26.8653	1.05946	0.529732	−	0.848165i	$-0.322292\pi$
$643$	26.8653	1.05946	0.529732	+	0.848165i	$0.322292\pi$
$644$	−17.5182	−0.690315
$645$	−7.42741	−0.292454
$646$	−11.3626	−0.447054
$647$	11.2448	0.442078	0.221039	−	0.975265i	$-0.429055\pi$
$647$	11.2448	0.442078	0.221039	+	0.975265i	$0.429055\pi$
$648$	−26.0135	−1.02191
$649$	8.78043	0.344662
$650$	3.40440	0.133532
$651$	0.211960	0.00830736
$652$	9.18065	0.359542
$653$	28.5302	1.11647	0.558237	−	0.829681i	$-0.311478\pi$
$653$	28.5302	1.11647	0.558237	+	0.829681i	$0.311478\pi$
$654$	−15.0444	−0.588283
$655$	10.7316	0.419317
$656$	−2.71318	−0.105932
$657$	14.9982	0.585135
$658$	−11.6980	−0.456034
$659$	40.9033	1.59337	0.796684	−	0.604396i	$-0.206586\pi$
$659$	40.9033	1.59337	0.796684	+	0.604396i	$0.206586\pi$
$660$	15.1355	0.589149
$661$	11.6701	0.453912	0.226956	−	0.973905i	$-0.427123\pi$
$661$	11.6701	0.453912	0.226956	+	0.973905i	$0.427123\pi$
$662$	20.1011	0.781253
$663$	−6.79280	−0.263810
$664$	54.4240	2.11206
$665$	−1.09876	−0.0426081
$666$	11.8989	0.461072
$667$	−5.60181	−0.216903
$668$	38.4215	1.48657
$669$	6.16935	0.238521
$670$	11.3030	0.436674
$671$	32.0920	1.23890
$672$	−6.29211	−0.242724
$673$	20.2625	0.781063	0.390531	−	0.920590i	$-0.372291\pi$
$673$	20.2625	0.781063	0.390531	+	0.920590i	$0.372291\pi$
$674$	−23.0220	−0.886773
$675$	4.10307	0.157927
$676$	−49.8266	−1.91641
$677$	−16.1889	−0.622190	−0.311095	−	0.950379i	$-0.600696\pi$
$677$	−16.1889	−0.622190	−0.311095	+	0.950379i	$0.600696\pi$
$678$	−17.5839	−0.675307
$679$	1.12391	0.0431317
$680$	−41.6669	−1.59785
$681$	−21.1287	−0.809652
$682$	−1.94785	−0.0745872
$683$	17.1758	0.657213	0.328606	−	0.944467i	$-0.393421\pi$
$683$	17.1758	0.657213	0.328606	+	0.944467i	$0.393421\pi$
$684$	7.21100	0.275719
$685$	−3.22957	−0.123396
$686$	−47.1912	−1.80177
$687$	−10.4929	−0.400330
$688$	−67.6840	−2.58043
$689$	4.92762	0.187727
$690$	−4.59218	−0.174821
$691$	−19.6597	−0.747891	−0.373946	−	0.927451i	$-0.621995\pi$
$691$	−19.6597	−0.747891	−0.373946	+	0.927451i	$0.621995\pi$
$692$	−3.33599	−0.126815
$693$	17.9998	0.683756
$694$	−30.4497	−1.15586
$695$	6.59952	0.250334
$696$	−10.9980	−0.416880
$697$	2.64075	0.100026
$698$	62.0912	2.35019
$699$	6.48152	0.245153
$700$	−7.32126	−0.276718
$701$	−27.0728	−1.02253	−0.511263	−	0.859425i	$-0.670822\pi$
$701$	−27.0728	−1.02253	−0.511263	+	0.859425i	$0.670822\pi$
$702$	13.9685	0.527207
$703$	−1.28789	−0.0485736
$704$	−4.19688	−0.158176
$705$	−2.11530	−0.0796669
$706$	12.5198	0.471187
$707$	−18.0949	−0.680528
$708$	−6.55582	−0.246383
$709$	23.7342	0.891357	0.445679	−	0.895193i	$-0.352962\pi$
$709$	23.7342	0.891357	0.445679	+	0.895193i	$0.352962\pi$
$710$	18.8191	0.706269
$711$	27.5051	1.03152
$712$	−23.4636	−0.879334
$713$	0.407673	0.0152675
$714$	21.1768	0.792520
$715$	6.03640	0.225748
$716$	8.92259	0.333453
$717$	−16.9910	−0.634539
$718$	−31.5561	−1.17766
$719$	30.4368	1.13510	0.567551	−	0.823338i	$-0.307891\pi$
$719$	30.4368	1.13510	0.567551	+	0.823338i	$0.307891\pi$
$720$	16.7279	0.623412
$721$	−10.8904	−0.405580
$722$	47.1144	1.75342
$723$	−8.63100	−0.320990
$724$	22.7901	0.846987
$725$	−2.34112	−0.0869469
$726$	17.7936	0.660383
$727$	30.9812	1.14903	0.574514	−	0.818495i	$-0.305191\pi$
$727$	30.9812	1.14903	0.574514	+	0.818495i	$0.305191\pi$
$728$	−13.7170	−0.508385
$729$	−0.861526	−0.0319084
$730$	−15.6805	−0.580361
$731$	65.8773	2.43656
$732$	−23.9612	−0.885630
$733$	31.8108	1.17496	0.587480	−	0.809239i	$-0.300120\pi$
$733$	31.8108	1.17496	0.587480	+	0.809239i	$0.300120\pi$
$734$	85.8140	3.16745
$735$	−3.24282	−0.119613
$736$	−12.1019	−0.446083
$737$	20.0416	0.738240
$738$	−2.42948	−0.0894304
$739$	−24.7199	−0.909336	−0.454668	−	0.890661i	$-0.650242\pi$
$739$	−24.7199	−0.909336	−0.454668	+	0.890661i	$0.650242\pi$
$740$	−8.58146	−0.315461
$741$	−0.676403	−0.0248483
$742$	−15.3620	−0.563957
$743$	26.4070	0.968778	0.484389	−	0.874853i	$-0.339042\pi$
$743$	26.4070	0.968778	0.484389	+	0.874853i	$0.339042\pi$
$744$	0.800385	0.0293436
$745$	5.24604	0.192200
$746$	6.63444	0.242904
$747$	21.2663	0.778092
$748$	−134.244	−4.90846
$749$	25.9264	0.947331
$750$	−1.91917	−0.0700783
$751$	12.9089	0.471052	0.235526	−	0.971868i	$-0.424319\pi$
$751$	12.9089	0.471052	0.235526	+	0.971868i	$0.424319\pi$
$752$	−19.2762	−0.702930
$753$	20.6458	0.752374
$754$	−7.97010	−0.290254
$755$	13.4648	0.490035
$756$	−30.0396	−1.09253
$757$	4.96353	0.180402	0.0902012	−	0.995924i	$-0.471249\pi$
$757$	4.96353	0.180402	0.0902012	+	0.995924i	$0.471249\pi$
$758$	−11.7100	−0.425325
$759$	−8.14246	−0.295553
$760$	−4.14904	−0.150502
$761$	41.8718	1.51785	0.758927	−	0.651176i	$-0.225724\pi$
$761$	41.8718	1.51785	0.758927	+	0.651176i	$0.225724\pi$
$762$	−39.2074	−1.42033
$763$	12.9033	0.467131
$764$	76.8243	2.77941
$765$	−16.2814	−0.588655
$766$	48.1830	1.74092
$767$	−2.61461	−0.0944082
$768$	22.5466	0.813580
$769$	24.0920	0.868781	0.434390	−	0.900725i	$-0.356964\pi$
$769$	24.0920	0.868781	0.434390	+	0.900725i	$0.356964\pi$
$770$	−18.8186	−0.678177
$771$	10.9380	0.393922
$772$	−51.1172	−1.83975
$773$	−7.02343	−0.252615	−0.126308	−	0.991991i	$-0.540313\pi$
$773$	−7.02343	−0.252615	−0.126308	+	0.991991i	$0.540313\pi$
$774$	−60.6068	−2.17847
$775$	0.170375	0.00612007
$776$	4.24402	0.152351
$777$	2.40028	0.0861095
$778$	31.6875	1.13605
$779$	0.262957	0.00942142
$780$	−4.50701	−0.161377
$781$	33.3685	1.19402
$782$	40.7303	1.45651
$783$	−9.60576	−0.343282
$784$	−29.5509	−1.05539
$785$	−23.1422	−0.825981
$786$	−20.5957	−0.734624
$787$	30.9973	1.10494	0.552468	−	0.833534i	$-0.313686\pi$
$787$	30.9973	1.10494	0.552468	+	0.833534i	$0.313686\pi$
$788$	60.1702	2.14347
$789$	2.38509	0.0849115
$790$	−28.7564	−1.02311
$791$	15.0814	0.536233
$792$	67.9693	2.41518
$793$	−9.55626	−0.339353
$794$	24.8378	0.881459
$795$	−2.77786	−0.0985205
$796$	−19.7849	−0.701257
$797$	7.32646	0.259517	0.129758	−	0.991546i	$-0.458580\pi$
$797$	7.32646	0.259517	0.129758	+	0.991546i	$0.458580\pi$
$798$	2.10871	0.0746475
$799$	18.7617	0.663740
$800$	−5.05766	−0.178815
$801$	−9.16842	−0.323950
$802$	−60.4579	−2.13484
$803$	−27.8033	−0.981157
$804$	−14.9638	−0.527734
$805$	3.93862	0.138818
$806$	0.580026	0.0204306
$807$	−0.935043	−0.0329151
$808$	−68.3284	−2.40378
$809$	14.9053	0.524044	0.262022	−	0.965062i	$-0.415611\pi$
$809$	14.9053	0.524044	0.262022	+	0.965062i	$0.415611\pi$
$810$	10.6273	0.373404
$811$	31.3362	1.10036	0.550181	−	0.835045i	$-0.314559\pi$
$811$	31.3362	1.10036	0.550181	+	0.835045i	$0.314559\pi$
$812$	17.1399	0.601493
$813$	−2.28397	−0.0801025
$814$	−22.0579	−0.773129
$815$	−2.06408	−0.0723016
$816$	34.8956	1.22159
$817$	6.55984	0.229500
$818$	2.30053	0.0804361
$819$	−5.35993	−0.187291
$820$	1.75214	0.0611873
$821$	−19.5347	−0.681764	−0.340882	−	0.940106i	$-0.610726\pi$
$821$	−19.5347	−0.681764	−0.340882	+	0.940106i	$0.610726\pi$
$822$	6.19811	0.216184
$823$	−7.12209	−0.248260	−0.124130	−	0.992266i	$-0.539614\pi$
$823$	−7.12209	−0.248260	−0.124130	+	0.992266i	$0.539614\pi$
$824$	−41.1235	−1.43260
$825$	−3.40291	−0.118474
$826$	8.15113	0.283614
$827$	−31.3615	−1.09055	−0.545273	−	0.838258i	$-0.683574\pi$
$827$	−31.3615	−1.09055	−0.545273	+	0.838258i	$0.683574\pi$
$828$	−25.8486	−0.898300
$829$	28.3035	0.983022	0.491511	−	0.870871i	$-0.336445\pi$
$829$	28.3035	0.983022	0.491511	+	0.870871i	$0.336445\pi$
$830$	−22.2337	−0.771744
$831$	0.454222	0.0157568
$832$	1.24974	0.0433268
$833$	28.7621	0.996549
$834$	−12.6656	−0.438574
$835$	−8.63830	−0.298941
$836$	−13.3676	−0.462328
$837$	0.699062	0.0241631
$838$	102.655	3.54615
$839$	26.9373	0.929979	0.464989	−	0.885316i	$-0.346058\pi$
$839$	26.9373	0.929979	0.464989	+	0.885316i	$0.346058\pi$
$840$	7.73270	0.266804
$841$	−23.5192	−0.811006
$842$	67.3936	2.32254
$843$	−11.1493	−0.384001
$844$	−59.5941	−2.05131
$845$	11.2025	0.385378
$846$	−17.2606	−0.593432
$847$	−15.2612	−0.524382
$848$	−25.3139	−0.869283
$849$	23.7332	0.814520
$850$	17.0221	0.583853
$851$	4.61657	0.158254
$852$	−24.9142	−0.853548
$853$	41.4142	1.41800	0.708998	−	0.705210i	$-0.249147\pi$
$853$	41.4142	1.41800	0.708998	+	0.705210i	$0.249147\pi$
$854$	29.7919	1.01946
$855$	−1.62125	−0.0554454
$856$	97.9013	3.34620
$857$	42.0078	1.43496	0.717479	−	0.696580i	$-0.245296\pi$
$857$	42.0078	1.43496	0.717479	+	0.696580i	$0.245296\pi$
$858$	−11.5849	−0.395501
$859$	24.1657	0.824525	0.412262	−	0.911065i	$-0.364739\pi$
$859$	24.1657	0.824525	0.412262	+	0.911065i	$0.364739\pi$
$860$	43.7095	1.49048
$861$	−0.490082	−0.0167019
$862$	2.90282	0.0988704
$863$	−14.3112	−0.487158	−0.243579	−	0.969881i	$-0.578321\pi$
$863$	−14.3112	−0.487158	−0.243579	+	0.969881i	$0.578321\pi$
$864$	−20.7519	−0.705995
$865$	0.750029	0.0255017
$866$	−16.7037	−0.567616
$867$	−21.1155	−0.717120
$868$	−1.24736	−0.0423382
$869$	−50.9884	−1.72966
$870$	4.49301	0.152327
$871$	−5.96792	−0.202215
$872$	48.7243	1.65001
$873$	1.65836	0.0561269
$874$	4.05578	0.137189
$875$	1.64604	0.0556461
$876$	20.7591	0.701384
$877$	8.11146	0.273905	0.136952	−	0.990578i	$-0.456269\pi$
$877$	8.11146	0.273905	0.136952	+	0.990578i	$0.456269\pi$
$878$	92.5639	3.12388
$879$	2.07265	0.0699087
$880$	−31.0098	−1.04534
$881$	19.5630	0.659094	0.329547	−	0.944139i	$-0.393104\pi$
$881$	19.5630	0.659094	0.329547	+	0.944139i	$0.393104\pi$
$882$	−26.4610	−0.890989
$883$	47.1927	1.58816	0.794080	−	0.607814i	$-0.207953\pi$
$883$	47.1927	1.58816	0.794080	+	0.607814i	$0.207953\pi$
$884$	39.9749	1.34450
$885$	1.47394	0.0495460
$886$	−83.3474	−2.80011
$887$	−42.3220	−1.42103	−0.710517	−	0.703680i	$-0.751539\pi$
$887$	−42.3220	−1.42103	−0.710517	+	0.703680i	$0.751539\pi$
$888$	9.06373	0.304159
$889$	33.6274	1.12783
$890$	9.58552	0.321307
$891$	18.8433	0.631276
$892$	−36.3060	−1.21562
$893$	1.86822	0.0625176
$894$	−10.0681	−0.336726
$895$	−2.00606	−0.0670552
$896$	−20.5463	−0.686402
$897$	2.42464	0.0809564
$898$	37.1435	1.23949
$899$	−0.398869	−0.0133030
$900$	−10.8027	−0.360090
$901$	24.6382	0.820818
$902$	4.50372	0.149957
$903$	−12.2258	−0.406848
$904$	56.9491	1.89410
$905$	−5.12389	−0.170324
$906$	−25.8413	−0.858520
$907$	−33.8769	−1.12486	−0.562432	−	0.826844i	$-0.690134\pi$
$907$	−33.8769	−1.12486	−0.562432	+	0.826844i	$0.690134\pi$
$908$	124.340	4.12637
$909$	−26.6994	−0.885564
$910$	5.60377	0.185763
$911$	18.6920	0.619292	0.309646	−	0.950852i	$-0.399789\pi$
$911$	18.6920	0.619292	0.309646	+	0.950852i	$0.399789\pi$
$912$	3.47478	0.115062
$913$	−39.4229	−1.30471
$914$	−35.6001	−1.17755
$915$	5.38718	0.178095
$916$	61.7497	2.04027
$917$	17.6645	0.583334
$918$	69.8427	2.30515
$919$	42.4763	1.40117	0.700583	−	0.713571i	$-0.252924\pi$
$919$	42.4763	1.40117	0.700583	+	0.713571i	$0.252924\pi$
$920$	14.8727	0.490338
$921$	7.43933	0.245134
$922$	52.8725	1.74126
$923$	−9.93637	−0.327060
$924$	24.9136	0.819597
$925$	1.92937	0.0634372
$926$	−36.1137	−1.18677
$927$	−16.0691	−0.527778
$928$	11.8406	0.388686
$929$	27.9027	0.915459	0.457730	−	0.889092i	$-0.348663\pi$
$929$	27.9027	0.915459	0.457730	+	0.889092i	$0.348663\pi$
$930$	−0.326980	−0.0107221
$931$	2.86403	0.0938650
$932$	−38.1431	−1.24942
$933$	−22.6944	−0.742981
$934$	−33.9952	−1.11236
$935$	30.1821	0.987060
$936$	−20.2397	−0.661556
$937$	−6.17831	−0.201836	−0.100918	−	0.994895i	$-0.532178\pi$
$937$	−6.17831	−0.201836	−0.100918	+	0.994895i	$0.532178\pi$
$938$	18.6052	0.607481
$939$	−1.75094	−0.0571399
$940$	12.4483	0.406020
$941$	46.1485	1.50440	0.752199	−	0.658936i	$-0.228993\pi$
$941$	46.1485	1.50440	0.752199	+	0.658936i	$0.228993\pi$
$942$	44.4139	1.44708
$943$	−0.942598	−0.0306952
$944$	13.4316	0.437163
$945$	6.75379	0.219701
$946$	112.351	3.65286
$947$	25.4435	0.826802	0.413401	−	0.910549i	$-0.364341\pi$
$947$	25.4435	0.826802	0.413401	+	0.910549i	$0.364341\pi$
$948$	38.0700	1.23645
$949$	8.27919	0.268754
$950$	1.69500	0.0549931
$951$	1.79256	0.0581276
$952$	−68.5851	−2.22286
$953$	21.6447	0.701142	0.350571	−	0.936536i	$-0.385988\pi$
$953$	21.6447	0.701142	0.350571	+	0.936536i	$0.385988\pi$
$954$	−22.6670	−0.733872
$955$	−17.2724	−0.558921
$956$	99.9900	3.23391
$957$	7.96662	0.257524
$958$	51.6065	1.66733
$959$	−5.31599	−0.171662
$960$	−0.704516	−0.0227382
$961$	−30.9710	−0.999064
$962$	6.56834	0.211772
$963$	38.2551	1.23275
$964$	50.7925	1.63592
$965$	11.4927	0.369962
$966$	−7.55889	−0.243203
$967$	12.3766	0.398004	0.199002	−	0.979999i	$-0.436230\pi$
$967$	12.3766	0.398004	0.199002	+	0.979999i	$0.436230\pi$
$968$	−57.6282	−1.85224
$969$	−3.38203	−0.108646
$970$	−1.73380	−0.0556689
$971$	−24.4572	−0.784868	−0.392434	−	0.919780i	$-0.628367\pi$
$971$	−24.4572	−0.784868	−0.392434	+	0.919780i	$0.628367\pi$
$972$	−68.8182	−2.20735
$973$	10.8630	0.348253
$974$	−64.6954	−2.07297
$975$	1.01331	0.0324519
$976$	49.0919	1.57139
$977$	59.4988	1.90354	0.951768	−	0.306818i	$-0.0992644\pi$
$977$	59.4988	1.90354	0.951768	+	0.306818i	$0.0992644\pi$
$978$	3.96133	0.126669
$979$	16.9962	0.543201
$980$	19.0836	0.609605
$981$	19.0391	0.607872
$982$	−68.9002	−2.19869
$983$	8.16975	0.260574	0.130287	−	0.991476i	$-0.458410\pi$
$983$	8.16975	0.260574	0.130287	+	0.991476i	$0.458410\pi$
$984$	−1.85061	−0.0589952
$985$	−13.5280	−0.431039
$986$	−39.8507	−1.26910
$987$	−3.48186	−0.110829
$988$	3.98056	0.126639
$989$	−23.5144	−0.747715
$990$	−27.7674	−0.882505
$991$	−43.7362	−1.38933	−0.694663	−	0.719336i	$-0.744446\pi$
$991$	−43.7362	−1.38933	−0.694663	+	0.719336i	$0.744446\pi$
$992$	−0.861701	−0.0273590
$993$	5.98304	0.189866
$994$	30.9769	0.982529
$995$	4.44823	0.141018
$996$	29.4347	0.932675
$997$	15.6438	0.495445	0.247722	−	0.968831i	$-0.420318\pi$
$997$	15.6438	0.495445	0.247722	+	0.968831i	$0.420318\pi$
$998$	25.7099	0.813832
$999$	7.91632	0.250461

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8005.2.a.g.1.8	✓	137

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8005.2.a.g.1.8	✓	137	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:	\( N \)	\(=\)	\( 8005 = 5 \cdot 1601 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8005.a (trivial)

\(f(q)\)	\(=\)	\(q-2.53925 q^{2} -0.755801 q^{3} +4.44781 q^{4} -1.00000 q^{5} +1.91917 q^{6} -1.64604 q^{7} -6.21562 q^{8} -2.42876 q^{9} +O(q^{10})\)	\(q-2.53925 q^{2} -0.755801 q^{3} +4.44781 q^{4} -1.00000 q^{5} +1.91917 q^{6} -1.64604 q^{7} -6.21562 q^{8} -2.42876 q^{9} +2.53925 q^{10} +4.50239 q^{11} -3.36166 q^{12} -1.34071 q^{13} +4.17970 q^{14} +0.755801 q^{15} +6.88742 q^{16} -6.70357 q^{17} +6.16725 q^{18} -0.667519 q^{19} -4.44781 q^{20} +1.24408 q^{21} -11.4327 q^{22} +2.39279 q^{23} +4.69777 q^{24} +1.00000 q^{25} +3.40440 q^{26} +4.10307 q^{27} -7.32126 q^{28} -2.34112 q^{29} -1.91917 q^{30} +0.170375 q^{31} -5.05766 q^{32} -3.40291 q^{33} +17.0221 q^{34} +1.64604 q^{35} -10.8027 q^{36} +1.92937 q^{37} +1.69500 q^{38} +1.01331 q^{39} +6.21562 q^{40} -0.393932 q^{41} -3.15902 q^{42} -9.82719 q^{43} +20.0258 q^{44} +2.42876 q^{45} -6.07591 q^{46} -2.79875 q^{47} -5.20552 q^{48} -4.29057 q^{49} -2.53925 q^{50} +5.06657 q^{51} -5.96322 q^{52} -3.67538 q^{53} -10.4187 q^{54} -4.50239 q^{55} +10.2311 q^{56} +0.504512 q^{57} +5.94469 q^{58} +1.95017 q^{59} +3.36166 q^{60} +7.12777 q^{61} -0.432627 q^{62} +3.99783 q^{63} -0.932145 q^{64} +1.34071 q^{65} +8.64086 q^{66} +4.45132 q^{67} -29.8162 q^{68} -1.80848 q^{69} -4.17970 q^{70} +7.41128 q^{71} +15.0963 q^{72} -6.17523 q^{73} -4.89915 q^{74} -0.755801 q^{75} -2.96900 q^{76} -7.41109 q^{77} -2.57305 q^{78} -11.3247 q^{79} -6.88742 q^{80} +4.18519 q^{81} +1.00029 q^{82} -8.75600 q^{83} +5.53342 q^{84} +6.70357 q^{85} +24.9537 q^{86} +1.76942 q^{87} -27.9851 q^{88} +3.77493 q^{89} -6.16725 q^{90} +2.20685 q^{91} +10.6427 q^{92} -0.128770 q^{93} +7.10675 q^{94} +0.667519 q^{95} +3.82259 q^{96} -0.682798 q^{97} +10.8948 q^{98} -10.9352 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 137 q + 4 q^{2} + 20 q^{3} + 152 q^{4} - 137 q^{5} + 14 q^{6} - 30 q^{7} + 24 q^{8} + 163 q^{9}+O(q^{10}) \) 137 * q + 4 * q^2 + 20 * q^3 + 152 * q^4 - 137 * q^5 + 14 * q^6 - 30 * q^7 + 24 * q^8 + 163 * q^9	\( 137 q + 4 q^{2} + 20 q^{3} + 152 q^{4} - 137 q^{5} + 14 q^{6} - 30 q^{7} + 24 q^{8} + 163 q^{9} - 4 q^{10} + 51 q^{11} + 32 q^{12} - 6 q^{13} + 49 q^{14} - 20 q^{15} + 170 q^{16} + 46 q^{17} + 4 q^{18} + 47 q^{19} - 152 q^{20} + 28 q^{21} - 19 q^{22} + 29 q^{23} + 39 q^{24} + 137 q^{25} + 67 q^{26} + 77 q^{27} - 58 q^{28} + 27 q^{29} - 14 q^{30} + 23 q^{31} + 42 q^{32} + 40 q^{33} + 38 q^{34} + 30 q^{35} + 222 q^{36} - 56 q^{37} + 87 q^{38} + 44 q^{39} - 24 q^{40} + 66 q^{41} + 34 q^{42} + 15 q^{43} + 87 q^{44} - 163 q^{45} + 37 q^{46} + 52 q^{47} + 56 q^{48} + 195 q^{49} + 4 q^{50} + 106 q^{51} - 31 q^{52} + 45 q^{53} + 83 q^{54} - 51 q^{55} + 148 q^{56} + 4 q^{57} - 101 q^{58} + 239 q^{59} - 32 q^{60} + 46 q^{61} + 63 q^{62} - 59 q^{63} + 200 q^{64} + 6 q^{65} + 108 q^{66} - 18 q^{67} + 152 q^{68} + 63 q^{69} - 49 q^{70} + 110 q^{71} + 6 q^{72} - 19 q^{73} + 81 q^{74} + 20 q^{75} + 94 q^{76} + 43 q^{77} - 3 q^{78} + 40 q^{79} - 170 q^{80} + 229 q^{81} + 3 q^{82} + 235 q^{83} + 94 q^{84} - 46 q^{85} + 110 q^{86} + 31 q^{87} - 105 q^{88} + 150 q^{89} - 4 q^{90} + 110 q^{91} + 76 q^{92} + 11 q^{93} + 56 q^{94} - 47 q^{95} + 146 q^{96} + 17 q^{97} + 75 q^{98} + 125 q^{99}+O(q^{100}) \) 137 * q + 4 * q^2 + 20 * q^3 + 152 * q^4 - 137 * q^5 + 14 * q^6 - 30 * q^7 + 24 * q^8 + 163 * q^9 - 4 * q^10 + 51 * q^11 + 32 * q^12 - 6 * q^13 + 49 * q^14 - 20 * q^15 + 170 * q^16 + 46 * q^17 + 4 * q^18 + 47 * q^19 - 152 * q^20 + 28 * q^21 - 19 * q^22 + 29 * q^23 + 39 * q^24 + 137 * q^25 + 67 * q^26 + 77 * q^27 - 58 * q^28 + 27 * q^29 - 14 * q^30 + 23 * q^31 + 42 * q^32 + 40 * q^33 + 38 * q^34 + 30 * q^35 + 222 * q^36 - 56 * q^37 + 87 * q^38 + 44 * q^39 - 24 * q^40 + 66 * q^41 + 34 * q^42 + 15 * q^43 + 87 * q^44 - 163 * q^45 + 37 * q^46 + 52 * q^47 + 56 * q^48 + 195 * q^49 + 4 * q^50 + 106 * q^51 - 31 * q^52 + 45 * q^53 + 83 * q^54 - 51 * q^55 + 148 * q^56 + 4 * q^57 - 101 * q^58 + 239 * q^59 - 32 * q^60 + 46 * q^61 + 63 * q^62 - 59 * q^63 + 200 * q^64 + 6 * q^65 + 108 * q^66 - 18 * q^67 + 152 * q^68 + 63 * q^69 - 49 * q^70 + 110 * q^71 + 6 * q^72 - 19 * q^73 + 81 * q^74 + 20 * q^75 + 94 * q^76 + 43 * q^77 - 3 * q^78 + 40 * q^79 - 170 * q^80 + 229 * q^81 + 3 * q^82 + 235 * q^83 + 94 * q^84 - 46 * q^85 + 110 * q^86 + 31 * q^87 - 105 * q^88 + 150 * q^89 - 4 * q^90 + 110 * q^91 + 76 * q^92 + 11 * q^93 + 56 * q^94 - 47 * q^95 + 146 * q^96 + 17 * q^97 + 75 * q^98 + 125 * q^99

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