LMFDB - Embedded newform 8007.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.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:	$63.9362168984$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} -0.585786 q^{5} -2.41421 q^{6} -0.585786 q^{7} -4.41421 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} -0.585786 q^{5} -2.41421 q^{6} -0.585786 q^{7} -4.41421 q^{8} +1.00000 q^{9} +1.41421 q^{10} -2.82843 q^{11} +3.82843 q^{12} +5.65685 q^{13} +1.41421 q^{14} -0.585786 q^{15} +3.00000 q^{16} -1.00000 q^{17} -2.41421 q^{18} -4.82843 q^{19} -2.24264 q^{20} -0.585786 q^{21} +6.82843 q^{22} +2.58579 q^{23} -4.41421 q^{24} -4.65685 q^{25} -13.6569 q^{26} +1.00000 q^{27} -2.24264 q^{28} -5.07107 q^{29} +1.41421 q^{30} +1.17157 q^{31} +1.58579 q^{32} -2.82843 q^{33} +2.41421 q^{34} +0.343146 q^{35} +3.82843 q^{36} +4.82843 q^{37} +11.6569 q^{38} +5.65685 q^{39} +2.58579 q^{40} +3.41421 q^{41} +1.41421 q^{42} +11.3137 q^{43} -10.8284 q^{44} -0.585786 q^{45} -6.24264 q^{46} +0.343146 q^{47} +3.00000 q^{48} -6.65685 q^{49} +11.2426 q^{50} -1.00000 q^{51} +21.6569 q^{52} -4.00000 q^{53} -2.41421 q^{54} +1.65685 q^{55} +2.58579 q^{56} -4.82843 q^{57} +12.2426 q^{58} -6.00000 q^{59} -2.24264 q^{60} +5.41421 q^{61} -2.82843 q^{62} -0.585786 q^{63} -9.82843 q^{64} -3.31371 q^{65} +6.82843 q^{66} +15.3137 q^{67} -3.82843 q^{68} +2.58579 q^{69} -0.828427 q^{70} -8.00000 q^{71} -4.41421 q^{72} -9.41421 q^{73} -11.6569 q^{74} -4.65685 q^{75} -18.4853 q^{76} +1.65685 q^{77} -13.6569 q^{78} -3.89949 q^{79} -1.75736 q^{80} +1.00000 q^{81} -8.24264 q^{82} +0.828427 q^{83} -2.24264 q^{84} +0.585786 q^{85} -27.3137 q^{86} -5.07107 q^{87} +12.4853 q^{88} -5.31371 q^{89} +1.41421 q^{90} -3.31371 q^{91} +9.89949 q^{92} +1.17157 q^{93} -0.828427 q^{94} +2.82843 q^{95} +1.58579 q^{96} +4.24264 q^{97} +16.0711 q^{98} -2.82843 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9} + 2 q^{12} - 4 q^{15} + 6 q^{16} - 2 q^{17} - 2 q^{18} - 4 q^{19} + 4 q^{20} - 4 q^{21} + 8 q^{22} + 8 q^{23} - 6 q^{24} + 2 q^{25} - 16 q^{26} + 2 q^{27} + 4 q^{28} + 4 q^{29} + 8 q^{31} + 6 q^{32} + 2 q^{34} + 12 q^{35} + 2 q^{36} + 4 q^{37} + 12 q^{38} + 8 q^{40} + 4 q^{41} - 16 q^{44} - 4 q^{45} - 4 q^{46} + 12 q^{47} + 6 q^{48} - 2 q^{49} + 14 q^{50} - 2 q^{51} + 32 q^{52} - 8 q^{53} - 2 q^{54} - 8 q^{55} + 8 q^{56} - 4 q^{57} + 16 q^{58} - 12 q^{59} + 4 q^{60} + 8 q^{61} - 4 q^{63} - 14 q^{64} + 16 q^{65} + 8 q^{66} + 8 q^{67} - 2 q^{68} + 8 q^{69} + 4 q^{70} - 16 q^{71} - 6 q^{72} - 16 q^{73} - 12 q^{74} + 2 q^{75} - 20 q^{76} - 8 q^{77} - 16 q^{78} + 12 q^{79} - 12 q^{80} + 2 q^{81} - 8 q^{82} - 4 q^{83} + 4 q^{84} + 4 q^{85} - 32 q^{86} + 4 q^{87} + 8 q^{88} + 12 q^{89} + 16 q^{91} + 8 q^{93} + 4 q^{94} + 6 q^{96} + 18 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9 + 2 * q^12 - 4 * q^15 + 6 * q^16 - 2 * q^17 - 2 * q^18 - 4 * q^19 + 4 * q^20 - 4 * q^21 + 8 * q^22 + 8 * q^23 - 6 * q^24 + 2 * q^25 - 16 * q^26 + 2 * q^27 + 4 * q^28 + 4 * q^29 + 8 * q^31 + 6 * q^32 + 2 * q^34 + 12 * q^35 + 2 * q^36 + 4 * q^37 + 12 * q^38 + 8 * q^40 + 4 * q^41 - 16 * q^44 - 4 * q^45 - 4 * q^46 + 12 * q^47 + 6 * q^48 - 2 * q^49 + 14 * q^50 - 2 * q^51 + 32 * q^52 - 8 * q^53 - 2 * q^54 - 8 * q^55 + 8 * q^56 - 4 * q^57 + 16 * q^58 - 12 * q^59 + 4 * q^60 + 8 * q^61 - 4 * q^63 - 14 * q^64 + 16 * q^65 + 8 * q^66 + 8 * q^67 - 2 * q^68 + 8 * q^69 + 4 * q^70 - 16 * q^71 - 6 * q^72 - 16 * q^73 - 12 * q^74 + 2 * q^75 - 20 * q^76 - 8 * q^77 - 16 * q^78 + 12 * q^79 - 12 * q^80 + 2 * q^81 - 8 * q^82 - 4 * q^83 + 4 * q^84 + 4 * q^85 - 32 * q^86 + 4 * q^87 + 8 * q^88 + 12 * q^89 + 16 * q^91 + 8 * q^93 + 4 * q^94 + 6 * q^96 + 18 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.82843	1.91421
$5$	−0.585786	−0.261972	−0.130986	−	0.991384i	$-0.541814\pi$
$5$	−0.585786	−0.261972	−0.130986	+	0.991384i	$0.541814\pi$
$6$	−2.41421	−0.985599
$7$	−0.585786	−0.221406	−0.110703	−	0.993854i	$-0.535310\pi$
$7$	−0.585786	−0.221406	−0.110703	+	0.993854i	$0.535310\pi$
$8$	−4.41421	−1.56066
$9$	1.00000	0.333333
$10$	1.41421	0.447214
$11$	−2.82843	−0.852803	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843	−0.852803	−0.426401	+	0.904534i	$0.640219\pi$
$12$	3.82843	1.10517
$13$	5.65685	1.56893	0.784465	−	0.620174i	$-0.212938\pi$
$13$	5.65685	1.56893	0.784465	+	0.620174i	$0.212938\pi$
$14$	1.41421	0.377964
$15$	−0.585786	−0.151249
$16$	3.00000	0.750000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−2.41421	−0.569036
$19$	−4.82843	−1.10772	−0.553859	−	0.832611i	$-0.686845\pi$
$19$	−4.82843	−1.10772	−0.553859	+	0.832611i	$0.686845\pi$
$20$	−2.24264	−0.501470
$21$	−0.585786	−0.127829
$22$	6.82843	1.45583
$23$	2.58579	0.539174	0.269587	−	0.962976i	$-0.413113\pi$
$23$	2.58579	0.539174	0.269587	+	0.962976i	$0.413113\pi$
$24$	−4.41421	−0.901048
$25$	−4.65685	−0.931371
$26$	−13.6569	−2.67833
$27$	1.00000	0.192450
$28$	−2.24264	−0.423819
$29$	−5.07107	−0.941674	−0.470837	−	0.882220i	$-0.656048\pi$
$29$	−5.07107	−0.941674	−0.470837	+	0.882220i	$0.656048\pi$
$30$	1.41421	0.258199
$31$	1.17157	0.210421	0.105210	−	0.994450i	$-0.466448\pi$
$31$	1.17157	0.210421	0.105210	+	0.994450i	$0.466448\pi$
$32$	1.58579	0.280330
$33$	−2.82843	−0.492366
$34$	2.41421	0.414034
$35$	0.343146	0.0580022
$36$	3.82843	0.638071
$37$	4.82843	0.793789	0.396894	−	0.917864i	$-0.370088\pi$
$37$	4.82843	0.793789	0.396894	+	0.917864i	$0.370088\pi$
$38$	11.6569	1.89099
$39$	5.65685	0.905822
$40$	2.58579	0.408849
$41$	3.41421	0.533211	0.266605	−	0.963806i	$-0.414098\pi$
$41$	3.41421	0.533211	0.266605	+	0.963806i	$0.414098\pi$
$42$	1.41421	0.218218
$43$	11.3137	1.72532	0.862662	−	0.505781i	$-0.168795\pi$
$43$	11.3137	1.72532	0.862662	+	0.505781i	$0.168795\pi$
$44$	−10.8284	−1.63245
$45$	−0.585786	−0.0873239
$46$	−6.24264	−0.920427
$47$	0.343146	0.0500530	0.0250265	−	0.999687i	$-0.492033\pi$
$47$	0.343146	0.0500530	0.0250265	+	0.999687i	$0.492033\pi$
$48$	3.00000	0.433013
$49$	−6.65685	−0.950979
$50$	11.2426	1.58995
$51$	−1.00000	−0.140028
$52$	21.6569	3.00327
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	−2.41421	−0.328533
$55$	1.65685	0.223410
$56$	2.58579	0.345540
$57$	−4.82843	−0.639541
$58$	12.2426	1.60754
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	−2.24264	−0.289524
$61$	5.41421	0.693219	0.346610	−	0.938010i	$-0.387333\pi$
$61$	5.41421	0.693219	0.346610	+	0.938010i	$0.387333\pi$
$62$	−2.82843	−0.359211
$63$	−0.585786	−0.0738022
$64$	−9.82843	−1.22855
$65$	−3.31371	−0.411015
$66$	6.82843	0.840521
$67$	15.3137	1.87087	0.935434	−	0.353502i	$-0.115009\pi$
$67$	15.3137	1.87087	0.935434	+	0.353502i	$0.115009\pi$
$68$	−3.82843	−0.464265
$69$	2.58579	0.311292
$70$	−0.828427	−0.0990160
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	−4.41421	−0.520220
$73$	−9.41421	−1.10185	−0.550925	−	0.834555i	$-0.685725\pi$
$73$	−9.41421	−1.10185	−0.550925	+	0.834555i	$0.685725\pi$
$74$	−11.6569	−1.35508
$75$	−4.65685	−0.537727
$76$	−18.4853	−2.12041
$77$	1.65685	0.188816
$78$	−13.6569	−1.54633
$79$	−3.89949	−0.438727	−0.219364	−	0.975643i	$-0.570398\pi$
$79$	−3.89949	−0.438727	−0.219364	+	0.975643i	$0.570398\pi$
$80$	−1.75736	−0.196479
$81$	1.00000	0.111111
$82$	−8.24264	−0.910247
$83$	0.828427	0.0909317	0.0454658	−	0.998966i	$-0.485523\pi$
$83$	0.828427	0.0909317	0.0454658	+	0.998966i	$0.485523\pi$
$84$	−2.24264	−0.244692
$85$	0.585786	0.0635375
$86$	−27.3137	−2.94531
$87$	−5.07107	−0.543676
$88$	12.4853	1.33094
$89$	−5.31371	−0.563252	−0.281626	−	0.959524i	$-0.590874\pi$
$89$	−5.31371	−0.563252	−0.281626	+	0.959524i	$0.590874\pi$
$90$	1.41421	0.149071
$91$	−3.31371	−0.347371
$92$	9.89949	1.03209
$93$	1.17157	0.121486
$94$	−0.828427	−0.0854457
$95$	2.82843	0.290191
$96$	1.58579	0.161849
$97$	4.24264	0.430775	0.215387	−	0.976529i	$-0.430899\pi$
$97$	4.24264	0.430775	0.215387	+	0.976529i	$0.430899\pi$
$98$	16.0711	1.62342
$99$	−2.82843	−0.284268
$100$	−17.8284	−1.78284
$101$	−14.1421	−1.40720	−0.703598	−	0.710599i	$-0.748424\pi$
$101$	−14.1421	−1.40720	−0.703598	+	0.710599i	$0.748424\pi$
$102$	2.41421	0.239043
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	−24.9706	−2.44857
$105$	0.343146	0.0334876
$106$	9.65685	0.937957
$107$	−0.242641	−0.0234570	−0.0117285	−	0.999931i	$-0.503733\pi$
$107$	−0.242641	−0.0234570	−0.0117285	+	0.999931i	$0.503733\pi$
$108$	3.82843	0.368391
$109$	8.14214	0.779875	0.389938	−	0.920841i	$-0.372497\pi$
$109$	8.14214	0.779875	0.389938	+	0.920841i	$0.372497\pi$
$110$	−4.00000	−0.381385
$111$	4.82843	0.458294
$112$	−1.75736	−0.166055
$113$	8.14214	0.765948	0.382974	−	0.923759i	$-0.374900\pi$
$113$	8.14214	0.765948	0.382974	+	0.923759i	$0.374900\pi$
$114$	11.6569	1.09176
$115$	−1.51472	−0.141248
$116$	−19.4142	−1.80256
$117$	5.65685	0.522976
$118$	14.4853	1.33348
$119$	0.585786	0.0536990
$120$	2.58579	0.236049
$121$	−3.00000	−0.272727
$122$	−13.0711	−1.18340
$123$	3.41421	0.307849
$124$	4.48528	0.402790
$125$	5.65685	0.505964
$126$	1.41421	0.125988
$127$	8.14214	0.722498	0.361249	−	0.932469i	$-0.382350\pi$
$127$	8.14214	0.722498	0.361249	+	0.932469i	$0.382350\pi$
$128$	20.5563	1.81694
$129$	11.3137	0.996116
$130$	8.00000	0.701646
$131$	−16.7279	−1.46153	−0.730763	−	0.682632i	$-0.760835\pi$
$131$	−16.7279	−1.46153	−0.730763	+	0.682632i	$0.760835\pi$
$132$	−10.8284	−0.942494
$133$	2.82843	0.245256
$134$	−36.9706	−3.19377
$135$	−0.585786	−0.0504165
$136$	4.41421	0.378516
$137$	5.65685	0.483298	0.241649	−	0.970364i	$-0.422312\pi$
$137$	5.65685	0.483298	0.241649	+	0.970364i	$0.422312\pi$
$138$	−6.24264	−0.531409
$139$	−1.07107	−0.0908468	−0.0454234	−	0.998968i	$-0.514464\pi$
$139$	−1.07107	−0.0908468	−0.0454234	+	0.998968i	$0.514464\pi$
$140$	1.31371	0.111029
$141$	0.343146	0.0288981
$142$	19.3137	1.62077
$143$	−16.0000	−1.33799
$144$	3.00000	0.250000
$145$	2.97056	0.246692
$146$	22.7279	1.88098
$147$	−6.65685	−0.549048
$148$	18.4853	1.51948
$149$	6.82843	0.559407	0.279703	−	0.960086i	$-0.409764\pi$
$149$	6.82843	0.559407	0.279703	+	0.960086i	$0.409764\pi$
$150$	11.2426	0.917958
$151$	−13.6569	−1.11138	−0.555690	−	0.831390i	$-0.687546\pi$
$151$	−13.6569	−1.11138	−0.555690	+	0.831390i	$0.687546\pi$
$152$	21.3137	1.72877
$153$	−1.00000	−0.0808452
$154$	−4.00000	−0.322329
$155$	−0.686292	−0.0551243
$156$	21.6569	1.73394
$157$	1.00000	0.0798087
$157$	1.00000	0.0798087
$158$	9.41421	0.748955
$159$	−4.00000	−0.317221
$160$	−0.928932	−0.0734385
$161$	−1.51472	−0.119377
$162$	−2.41421	−0.189679
$163$	−9.07107	−0.710501	−0.355250	−	0.934771i	$-0.615604\pi$
$163$	−9.07107	−0.710501	−0.355250	+	0.934771i	$0.615604\pi$
$164$	13.0711	1.02068
$165$	1.65685	0.128986
$166$	−2.00000	−0.155230
$167$	−9.65685	−0.747270	−0.373635	−	0.927576i	$-0.621889\pi$
$167$	−9.65685	−0.747270	−0.373635	+	0.927576i	$0.621889\pi$
$168$	2.58579	0.199498
$169$	19.0000	1.46154
$170$	−1.41421	−0.108465
$171$	−4.82843	−0.369239
$172$	43.3137	3.30264
$173$	5.31371	0.403994	0.201997	−	0.979386i	$-0.435257\pi$
$173$	5.31371	0.403994	0.201997	+	0.979386i	$0.435257\pi$
$174$	12.2426	0.928112
$175$	2.72792	0.206212
$176$	−8.48528	−0.639602
$177$	−6.00000	−0.450988
$178$	12.8284	0.961531
$179$	−11.6569	−0.871274	−0.435637	−	0.900122i	$-0.643477\pi$
$179$	−11.6569	−0.871274	−0.435637	+	0.900122i	$0.643477\pi$
$180$	−2.24264	−0.167157
$181$	0.242641	0.0180353	0.00901767	−	0.999959i	$-0.497130\pi$
$181$	0.242641	0.0180353	0.00901767	+	0.999959i	$0.497130\pi$
$182$	8.00000	0.592999
$183$	5.41421	0.400230
$184$	−11.4142	−0.841467
$185$	−2.82843	−0.207950
$186$	−2.82843	−0.207390
$187$	2.82843	0.206835
$188$	1.31371	0.0958120
$189$	−0.585786	−0.0426097
$190$	−6.82843	−0.495386
$191$	10.9706	0.793802	0.396901	−	0.917861i	$-0.370086\pi$
$191$	10.9706	0.793802	0.396901	+	0.917861i	$0.370086\pi$
$192$	−9.82843	−0.709306
$193$	−3.65685	−0.263226	−0.131613	−	0.991301i	$-0.542016\pi$
$193$	−3.65685	−0.263226	−0.131613	+	0.991301i	$0.542016\pi$
$194$	−10.2426	−0.735379
$195$	−3.31371	−0.237300
$196$	−25.4853	−1.82038
$197$	−13.7990	−0.983137	−0.491569	−	0.870839i	$-0.663576\pi$
$197$	−13.7990	−0.983137	−0.491569	+	0.870839i	$0.663576\pi$
$198$	6.82843	0.485275
$199$	−13.1716	−0.933708	−0.466854	−	0.884334i	$-0.654613\pi$
$199$	−13.1716	−0.933708	−0.466854	+	0.884334i	$0.654613\pi$
$200$	20.5563	1.45355
$201$	15.3137	1.08015
$202$	34.1421	2.40223
$203$	2.97056	0.208493
$204$	−3.82843	−0.268044
$205$	−2.00000	−0.139686
$206$	−9.65685	−0.672825
$207$	2.58579	0.179725
$208$	16.9706	1.17670
$209$	13.6569	0.944664
$210$	−0.828427	−0.0571669
$211$	−9.55635	−0.657886	−0.328943	−	0.944350i	$-0.606692\pi$
$211$	−9.55635	−0.657886	−0.328943	+	0.944350i	$0.606692\pi$
$212$	−15.3137	−1.05175
$213$	−8.00000	−0.548151
$214$	0.585786	0.0400435
$215$	−6.62742	−0.451986
$216$	−4.41421	−0.300349
$217$	−0.686292	−0.0465885
$218$	−19.6569	−1.33133
$219$	−9.41421	−0.636154
$220$	6.34315	0.427655
$221$	−5.65685	−0.380521
$222$	−11.6569	−0.782357
$223$	−14.1421	−0.947027	−0.473514	−	0.880786i	$-0.657015\pi$
$223$	−14.1421	−0.947027	−0.473514	+	0.880786i	$0.657015\pi$
$224$	−0.928932	−0.0620669
$225$	−4.65685	−0.310457
$226$	−19.6569	−1.30755
$227$	−7.75736	−0.514874	−0.257437	−	0.966295i	$-0.582878\pi$
$227$	−7.75736	−0.514874	−0.257437	+	0.966295i	$0.582878\pi$
$228$	−18.4853	−1.22422
$229$	0.343146	0.0226757	0.0113379	−	0.999936i	$-0.496391\pi$
$229$	0.343146	0.0226757	0.0113379	+	0.999936i	$0.496391\pi$
$230$	3.65685	0.241126
$231$	1.65685	0.109013
$232$	22.3848	1.46963
$233$	−25.7990	−1.69015	−0.845074	−	0.534649i	$-0.820444\pi$
$233$	−25.7990	−1.69015	−0.845074	+	0.534649i	$0.820444\pi$
$234$	−13.6569	−0.892776
$235$	−0.201010	−0.0131125
$236$	−22.9706	−1.49526
$237$	−3.89949	−0.253299
$238$	−1.41421	−0.0916698
$239$	6.68629	0.432500	0.216250	−	0.976338i	$-0.430617\pi$
$239$	6.68629	0.432500	0.216250	+	0.976338i	$0.430617\pi$
$240$	−1.75736	−0.113437
$241$	19.5563	1.25974	0.629868	−	0.776703i	$-0.283109\pi$
$241$	19.5563	1.25974	0.629868	+	0.776703i	$0.283109\pi$
$242$	7.24264	0.465575
$243$	1.00000	0.0641500
$244$	20.7279	1.32697
$245$	3.89949	0.249130
$246$	−8.24264	−0.525532
$247$	−27.3137	−1.73793
$248$	−5.17157	−0.328395
$249$	0.828427	0.0524994
$250$	−13.6569	−0.863735
$251$	14.9706	0.944934	0.472467	−	0.881348i	$-0.343364\pi$
$251$	14.9706	0.944934	0.472467	+	0.881348i	$0.343364\pi$
$252$	−2.24264	−0.141273
$253$	−7.31371	−0.459809
$254$	−19.6569	−1.23338
$255$	0.585786	0.0366834
$256$	−29.9706	−1.87316
$257$	−13.1716	−0.821620	−0.410810	−	0.911721i	$-0.634754\pi$
$257$	−13.1716	−0.821620	−0.410810	+	0.911721i	$0.634754\pi$
$258$	−27.3137	−1.70048
$259$	−2.82843	−0.175750
$260$	−12.6863	−0.786770
$261$	−5.07107	−0.313891
$262$	40.3848	2.49498
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	12.4853	0.768416
$265$	2.34315	0.143938
$266$	−6.82843	−0.418678
$267$	−5.31371	−0.325194
$268$	58.6274	3.58124
$269$	−13.5563	−0.826545	−0.413273	−	0.910607i	$-0.635614\pi$
$269$	−13.5563	−0.826545	−0.413273	+	0.910607i	$0.635614\pi$
$270$	1.41421	0.0860663
$271$	18.1421	1.10206	0.551028	−	0.834487i	$-0.314236\pi$
$271$	18.1421	1.10206	0.551028	+	0.834487i	$0.314236\pi$
$272$	−3.00000	−0.181902
$273$	−3.31371	−0.200555
$274$	−13.6569	−0.825041
$275$	13.1716	0.794276
$276$	9.89949	0.595880
$277$	−20.8284	−1.25146	−0.625729	−	0.780040i	$-0.715199\pi$
$277$	−20.8284	−1.25146	−0.625729	+	0.780040i	$0.715199\pi$
$278$	2.58579	0.155085
$279$	1.17157	0.0701402
$280$	−1.51472	−0.0905218
$281$	21.4558	1.27995	0.639974	−	0.768396i	$-0.278945\pi$
$281$	21.4558	1.27995	0.639974	+	0.768396i	$0.278945\pi$
$282$	−0.828427	−0.0493321
$283$	−6.34315	−0.377061	−0.188530	−	0.982067i	$-0.560372\pi$
$283$	−6.34315	−0.377061	−0.188530	+	0.982067i	$0.560372\pi$
$284$	−30.6274	−1.81740
$285$	2.82843	0.167542
$286$	38.6274	2.28409
$287$	−2.00000	−0.118056
$288$	1.58579	0.0934434
$289$	1.00000	0.0588235
$290$	−7.17157	−0.421129
$291$	4.24264	0.248708
$292$	−36.0416	−2.10918
$293$	−25.4558	−1.48715	−0.743573	−	0.668655i	$-0.766870\pi$
$293$	−25.4558	−1.48715	−0.743573	+	0.668655i	$0.766870\pi$
$294$	16.0711	0.937284
$295$	3.51472	0.204635
$296$	−21.3137	−1.23883
$297$	−2.82843	−0.164122
$298$	−16.4853	−0.954967
$299$	14.6274	0.845925
$300$	−17.8284	−1.02932
$301$	−6.62742	−0.381998
$302$	32.9706	1.89724
$303$	−14.1421	−0.812444
$304$	−14.4853	−0.830788
$305$	−3.17157	−0.181604
$306$	2.41421	0.138011
$307$	14.1421	0.807134	0.403567	−	0.914950i	$-0.367770\pi$
$307$	14.1421	0.807134	0.403567	+	0.914950i	$0.367770\pi$
$308$	6.34315	0.361434
$309$	4.00000	0.227552
$310$	1.65685	0.0941030
$311$	−11.7990	−0.669059	−0.334530	−	0.942385i	$-0.608577\pi$
$311$	−11.7990	−0.669059	−0.334530	+	0.942385i	$0.608577\pi$
$312$	−24.9706	−1.41368
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	−2.41421	−0.136242
$315$	0.343146	0.0193341
$316$	−14.9289	−0.839818
$317$	−20.6274	−1.15855	−0.579276	−	0.815132i	$-0.696664\pi$
$317$	−20.6274	−1.15855	−0.579276	+	0.815132i	$0.696664\pi$
$318$	9.65685	0.541529
$319$	14.3431	0.803062
$320$	5.75736	0.321846
$321$	−0.242641	−0.0135429
$322$	3.65685	0.203789
$323$	4.82843	0.268661
$324$	3.82843	0.212690
$325$	−26.3431	−1.46125
$326$	21.8995	1.21290
$327$	8.14214	0.450261
$328$	−15.0711	−0.832161
$329$	−0.201010	−0.0110820
$330$	−4.00000	−0.220193
$331$	7.17157	0.394185	0.197093	−	0.980385i	$-0.436850\pi$
$331$	7.17157	0.394185	0.197093	+	0.980385i	$0.436850\pi$
$332$	3.17157	0.174063
$333$	4.82843	0.264596
$334$	23.3137	1.27567
$335$	−8.97056	−0.490114
$336$	−1.75736	−0.0958718
$337$	13.8995	0.757154	0.378577	−	0.925570i	$-0.376414\pi$
$337$	13.8995	0.757154	0.378577	+	0.925570i	$0.376414\pi$
$338$	−45.8701	−2.49500
$339$	8.14214	0.442220
$340$	2.24264	0.121624
$341$	−3.31371	−0.179447
$342$	11.6569	0.630330
$343$	8.00000	0.431959
$344$	−49.9411	−2.69265
$345$	−1.51472	−0.0815497
$346$	−12.8284	−0.689661
$347$	13.1716	0.707087	0.353544	−	0.935418i	$-0.384977\pi$
$347$	13.1716	0.707087	0.353544	+	0.935418i	$0.384977\pi$
$348$	−19.4142	−1.04071
$349$	16.3431	0.874829	0.437414	−	0.899260i	$-0.355894\pi$
$349$	16.3431	0.874829	0.437414	+	0.899260i	$0.355894\pi$
$350$	−6.58579	−0.352025
$351$	5.65685	0.301941
$352$	−4.48528	−0.239066
$353$	24.4853	1.30322	0.651610	−	0.758554i	$-0.274094\pi$
$353$	24.4853	1.30322	0.651610	+	0.758554i	$0.274094\pi$
$354$	14.4853	0.769884
$355$	4.68629	0.248723
$356$	−20.3431	−1.07818
$357$	0.585786	0.0310031
$358$	28.1421	1.48736
$359$	−20.1421	−1.06306	−0.531531	−	0.847039i	$-0.678383\pi$
$359$	−20.1421	−1.06306	−0.531531	+	0.847039i	$0.678383\pi$
$360$	2.58579	0.136283
$361$	4.31371	0.227037
$362$	−0.585786	−0.0307883
$363$	−3.00000	−0.157459
$364$	−12.6863	−0.664942
$365$	5.51472	0.288654
$366$	−13.0711	−0.683236
$367$	1.07107	0.0559093	0.0279546	−	0.999609i	$-0.491101\pi$
$367$	1.07107	0.0559093	0.0279546	+	0.999609i	$0.491101\pi$
$368$	7.75736	0.404380
$369$	3.41421	0.177737
$370$	6.82843	0.354993
$371$	2.34315	0.121650
$372$	4.48528	0.232551
$373$	−29.7990	−1.54293	−0.771467	−	0.636270i	$-0.780477\pi$
$373$	−29.7990	−1.54293	−0.771467	+	0.636270i	$0.780477\pi$
$374$	−6.82843	−0.353090
$375$	5.65685	0.292119
$376$	−1.51472	−0.0781156
$377$	−28.6863	−1.47742
$378$	1.41421	0.0727393
$379$	−35.8995	−1.84403	−0.922017	−	0.387150i	$-0.873459\pi$
$379$	−35.8995	−1.84403	−0.922017	+	0.387150i	$0.873459\pi$
$380$	10.8284	0.555487
$381$	8.14214	0.417134
$382$	−26.4853	−1.35510
$383$	23.4558	1.19854	0.599269	−	0.800548i	$-0.295458\pi$
$383$	23.4558	1.19854	0.599269	+	0.800548i	$0.295458\pi$
$384$	20.5563	1.04901
$385$	−0.970563	−0.0494645
$386$	8.82843	0.449355
$387$	11.3137	0.575108
$388$	16.2426	0.824595
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	8.00000	0.405096
$391$	−2.58579	−0.130769
$392$	29.3848	1.48416
$393$	−16.7279	−0.843812
$394$	33.3137	1.67832
$395$	2.28427	0.114934
$396$	−10.8284	−0.544149
$397$	−14.3848	−0.721951	−0.360976	−	0.932575i	$-0.617556\pi$
$397$	−14.3848	−0.721951	−0.360976	+	0.932575i	$0.617556\pi$
$398$	31.7990	1.59394
$399$	2.82843	0.141598
$400$	−13.9706	−0.698528
$401$	−5.27208	−0.263275	−0.131638	−	0.991298i	$-0.542023\pi$
$401$	−5.27208	−0.263275	−0.131638	+	0.991298i	$0.542023\pi$
$402$	−36.9706	−1.84392
$403$	6.62742	0.330135
$404$	−54.1421	−2.69367
$405$	−0.585786	−0.0291080
$406$	−7.17157	−0.355919
$407$	−13.6569	−0.676945
$408$	4.41421	0.218536
$409$	−11.6569	−0.576394	−0.288197	−	0.957571i	$-0.593056\pi$
$409$	−11.6569	−0.576394	−0.288197	+	0.957571i	$0.593056\pi$
$410$	4.82843	0.238459
$411$	5.65685	0.279032
$412$	15.3137	0.754452
$413$	3.51472	0.172948
$414$	−6.24264	−0.306809
$415$	−0.485281	−0.0238215
$416$	8.97056	0.439818
$417$	−1.07107	−0.0524504
$418$	−32.9706	−1.61264
$419$	25.4558	1.24360	0.621800	−	0.783176i	$-0.286402\pi$
$419$	25.4558	1.24360	0.621800	+	0.783176i	$0.286402\pi$
$420$	1.31371	0.0641024
$421$	−30.4853	−1.48576	−0.742881	−	0.669424i	$-0.766541\pi$
$421$	−30.4853	−1.48576	−0.742881	+	0.669424i	$0.766541\pi$
$422$	23.0711	1.12308
$423$	0.343146	0.0166843
$424$	17.6569	0.857493
$425$	4.65685	0.225891
$426$	19.3137	0.935752
$427$	−3.17157	−0.153483
$428$	−0.928932	−0.0449016
$429$	−16.0000	−0.772487
$430$	16.0000	0.771589
$431$	7.79899	0.375664	0.187832	−	0.982201i	$-0.439854\pi$
$431$	7.79899	0.375664	0.187832	+	0.982201i	$0.439854\pi$
$432$	3.00000	0.144338
$433$	−9.31371	−0.447588	−0.223794	−	0.974636i	$-0.571844\pi$
$433$	−9.31371	−0.447588	−0.223794	+	0.974636i	$0.571844\pi$
$434$	1.65685	0.0795315
$435$	2.97056	0.142428
$436$	31.1716	1.49285
$437$	−12.4853	−0.597252
$438$	22.7279	1.08598
$439$	−3.89949	−0.186113	−0.0930564	−	0.995661i	$-0.529664\pi$
$439$	−3.89949	−0.186113	−0.0930564	+	0.995661i	$0.529664\pi$
$440$	−7.31371	−0.348667
$441$	−6.65685	−0.316993
$442$	13.6569	0.649590
$443$	−4.82843	−0.229405	−0.114703	−	0.993400i	$-0.536592\pi$
$443$	−4.82843	−0.229405	−0.114703	+	0.993400i	$0.536592\pi$
$444$	18.4853	0.877273
$445$	3.11270	0.147556
$446$	34.1421	1.61668
$447$	6.82843	0.322974
$448$	5.75736	0.272010
$449$	19.4142	0.916213	0.458107	−	0.888897i	$-0.348528\pi$
$449$	19.4142	0.916213	0.458107	+	0.888897i	$0.348528\pi$
$450$	11.2426	0.529983
$451$	−9.65685	−0.454724
$452$	31.1716	1.46619
$453$	−13.6569	−0.641655
$454$	18.7279	0.878945
$455$	1.94113	0.0910014
$456$	21.3137	0.998106
$457$	14.0000	0.654892	0.327446	−	0.944870i	$-0.393812\pi$
$457$	14.0000	0.654892	0.327446	+	0.944870i	$0.393812\pi$
$458$	−0.828427	−0.0387099
$459$	−1.00000	−0.0466760
$460$	−5.79899	−0.270379
$461$	−1.85786	−0.0865294	−0.0432647	−	0.999064i	$-0.513776\pi$
$461$	−1.85786	−0.0865294	−0.0432647	+	0.999064i	$0.513776\pi$
$462$	−4.00000	−0.186097
$463$	−4.68629	−0.217790	−0.108895	−	0.994053i	$-0.534731\pi$
$463$	−4.68629	−0.217790	−0.108895	+	0.994053i	$0.534731\pi$
$464$	−15.2132	−0.706255
$465$	−0.686292	−0.0318260
$466$	62.2843	2.88526
$467$	24.2843	1.12374	0.561871	−	0.827225i	$-0.310082\pi$
$467$	24.2843	1.12374	0.561871	+	0.827225i	$0.310082\pi$
$468$	21.6569	1.00109
$469$	−8.97056	−0.414222
$470$	0.485281	0.0223844
$471$	1.00000	0.0460776
$472$	26.4853	1.21908
$473$	−32.0000	−1.47136
$474$	9.41421	0.432409
$475$	22.4853	1.03170
$476$	2.24264	0.102791
$477$	−4.00000	−0.183147
$478$	−16.1421	−0.738324
$479$	13.8995	0.635084	0.317542	−	0.948244i	$-0.397143\pi$
$479$	13.8995	0.635084	0.317542	+	0.948244i	$0.397143\pi$
$480$	−0.928932	−0.0423998
$481$	27.3137	1.24540
$482$	−47.2132	−2.15050
$483$	−1.51472	−0.0689221
$484$	−11.4853	−0.522058
$485$	−2.48528	−0.112851
$486$	−2.41421	−0.109511
$487$	28.4853	1.29079	0.645396	−	0.763848i	$-0.276693\pi$
$487$	28.4853	1.29079	0.645396	+	0.763848i	$0.276693\pi$
$488$	−23.8995	−1.08188
$489$	−9.07107	−0.410208
$490$	−9.41421	−0.425291
$491$	−2.48528	−0.112159	−0.0560796	−	0.998426i	$-0.517860\pi$
$491$	−2.48528	−0.112159	−0.0560796	+	0.998426i	$0.517860\pi$
$492$	13.0711	0.589289
$493$	5.07107	0.228389
$494$	65.9411	2.96683
$495$	1.65685	0.0744701
$496$	3.51472	0.157816
$497$	4.68629	0.210209
$498$	−2.00000	−0.0896221
$499$	−24.3848	−1.09161	−0.545806	−	0.837911i	$-0.683777\pi$
$499$	−24.3848	−1.09161	−0.545806	+	0.837911i	$0.683777\pi$
$500$	21.6569	0.968524
$501$	−9.65685	−0.431436
$502$	−36.1421	−1.61310
$503$	−10.5858	−0.471997	−0.235998	−	0.971753i	$-0.575836\pi$
$503$	−10.5858	−0.471997	−0.235998	+	0.971753i	$0.575836\pi$
$504$	2.58579	0.115180
$505$	8.28427	0.368645
$506$	17.6569	0.784943
$507$	19.0000	0.843820
$508$	31.1716	1.38301
$509$	−31.7990	−1.40947	−0.704733	−	0.709473i	$-0.748933\pi$
$509$	−31.7990	−1.40947	−0.704733	+	0.709473i	$0.748933\pi$
$510$	−1.41421	−0.0626224
$511$	5.51472	0.243957
$512$	31.2426	1.38074
$513$	−4.82843	−0.213180
$514$	31.7990	1.40259
$515$	−2.34315	−0.103251
$516$	43.3137	1.90678
$517$	−0.970563	−0.0426853
$518$	6.82843	0.300024
$519$	5.31371	0.233246
$520$	14.6274	0.641455
$521$	11.4142	0.500066	0.250033	−	0.968237i	$-0.419559\pi$
$521$	11.4142	0.500066	0.250033	+	0.968237i	$0.419559\pi$
$522$	12.2426	0.535846
$523$	15.3137	0.669622	0.334811	−	0.942285i	$-0.391328\pi$
$523$	15.3137	0.669622	0.334811	+	0.942285i	$0.391328\pi$
$524$	−64.0416	−2.79767
$525$	2.72792	0.119056
$526$	19.3137	0.842118
$527$	−1.17157	−0.0510345
$528$	−8.48528	−0.369274
$529$	−16.3137	−0.709292
$530$	−5.65685	−0.245718
$531$	−6.00000	−0.260378
$532$	10.8284	0.469472
$533$	19.3137	0.836570
$534$	12.8284	0.555140
$535$	0.142136	0.00614506
$536$	−67.5980	−2.91979
$537$	−11.6569	−0.503030
$538$	32.7279	1.41100
$539$	18.8284	0.810998
$540$	−2.24264	−0.0965079
$541$	−6.38478	−0.274503	−0.137251	−	0.990536i	$-0.543827\pi$
$541$	−6.38478	−0.274503	−0.137251	+	0.990536i	$0.543827\pi$
$542$	−43.7990	−1.88133
$543$	0.242641	0.0104127
$544$	−1.58579	−0.0679900
$545$	−4.76955	−0.204305
$546$	8.00000	0.342368
$547$	−1.65685	−0.0708420	−0.0354210	−	0.999372i	$-0.511277\pi$
$547$	−1.65685	−0.0708420	−0.0354210	+	0.999372i	$0.511277\pi$
$548$	21.6569	0.925135
$549$	5.41421	0.231073
$550$	−31.7990	−1.35591
$551$	24.4853	1.04311
$552$	−11.4142	−0.485821
$553$	2.28427	0.0971371
$554$	50.2843	2.13637
$555$	−2.82843	−0.120060
$556$	−4.10051	−0.173900
$557$	30.2843	1.28319	0.641593	−	0.767045i	$-0.278274\pi$
$557$	30.2843	1.28319	0.641593	+	0.767045i	$0.278274\pi$
$558$	−2.82843	−0.119737
$559$	64.0000	2.70691
$560$	1.02944	0.0435017
$561$	2.82843	0.119416
$562$	−51.7990	−2.18501
$563$	−28.8284	−1.21497	−0.607487	−	0.794330i	$-0.707822\pi$
$563$	−28.8284	−1.21497	−0.607487	+	0.794330i	$0.707822\pi$
$564$	1.31371	0.0553171
$565$	−4.76955	−0.200657
$566$	15.3137	0.643683
$567$	−0.585786	−0.0246007
$568$	35.3137	1.48173
$569$	−9.65685	−0.404836	−0.202418	−	0.979299i	$-0.564880\pi$
$569$	−9.65685	−0.404836	−0.202418	+	0.979299i	$0.564880\pi$
$570$	−6.82843	−0.286011
$571$	16.9706	0.710196	0.355098	−	0.934829i	$-0.384448\pi$
$571$	16.9706	0.710196	0.355098	+	0.934829i	$0.384448\pi$
$572$	−61.2548	−2.56119
$573$	10.9706	0.458302
$574$	4.82843	0.201535
$575$	−12.0416	−0.502171
$576$	−9.82843	−0.409518
$577$	−30.0000	−1.24892	−0.624458	−	0.781058i	$-0.714680\pi$
$577$	−30.0000	−1.24892	−0.624458	+	0.781058i	$0.714680\pi$
$578$	−2.41421	−0.100418
$579$	−3.65685	−0.151974
$580$	11.3726	0.472221
$581$	−0.485281	−0.0201329
$582$	−10.2426	−0.424571
$583$	11.3137	0.468566
$584$	41.5563	1.71961
$585$	−3.31371	−0.137005
$586$	61.4558	2.53872
$587$	−7.17157	−0.296002	−0.148001	−	0.988987i	$-0.547284\pi$
$587$	−7.17157	−0.296002	−0.148001	+	0.988987i	$0.547284\pi$
$588$	−25.4853	−1.05100
$589$	−5.65685	−0.233087
$590$	−8.48528	−0.349334
$591$	−13.7990	−0.567615
$592$	14.4853	0.595341
$593$	8.34315	0.342612	0.171306	−	0.985218i	$-0.445201\pi$
$593$	8.34315	0.342612	0.171306	+	0.985218i	$0.445201\pi$
$594$	6.82843	0.280174
$595$	−0.343146	−0.0140676
$596$	26.1421	1.07082
$597$	−13.1716	−0.539077
$598$	−35.3137	−1.44408
$599$	−47.6569	−1.94721	−0.973603	−	0.228248i	$-0.926700\pi$
$599$	−47.6569	−1.94721	−0.973603	+	0.228248i	$0.926700\pi$
$600$	20.5563	0.839209
$601$	9.31371	0.379914	0.189957	−	0.981792i	$-0.439165\pi$
$601$	9.31371	0.379914	0.189957	+	0.981792i	$0.439165\pi$
$602$	16.0000	0.652111
$603$	15.3137	0.623622
$604$	−52.2843	−2.12742
$605$	1.75736	0.0714468
$606$	34.1421	1.38693
$607$	28.8701	1.17180	0.585900	−	0.810384i	$-0.300741\pi$
$607$	28.8701	1.17180	0.585900	+	0.810384i	$0.300741\pi$
$608$	−7.65685	−0.310526
$609$	2.97056	0.120373
$610$	7.65685	0.310017
$611$	1.94113	0.0785295
$612$	−3.82843	−0.154755
$613$	−31.6569	−1.27861	−0.639304	−	0.768954i	$-0.720778\pi$
$613$	−31.6569	−1.27861	−0.639304	+	0.768954i	$0.720778\pi$
$614$	−34.1421	−1.37786
$615$	−2.00000	−0.0806478
$616$	−7.31371	−0.294678
$617$	2.97056	0.119590	0.0597952	−	0.998211i	$-0.480955\pi$
$617$	2.97056	0.119590	0.0597952	+	0.998211i	$0.480955\pi$
$618$	−9.65685	−0.388456
$619$	−16.2843	−0.654520	−0.327260	−	0.944934i	$-0.606125\pi$
$619$	−16.2843	−0.654520	−0.327260	+	0.944934i	$0.606125\pi$
$620$	−2.62742	−0.105520
$621$	2.58579	0.103764
$622$	28.4853	1.14216
$623$	3.11270	0.124708
$624$	16.9706	0.679366
$625$	19.9706	0.798823
$626$	−4.82843	−0.192983
$627$	13.6569	0.545402
$628$	3.82843	0.152771
$629$	−4.82843	−0.192522
$630$	−0.828427	−0.0330053
$631$	−25.5147	−1.01572	−0.507862	−	0.861438i	$-0.669564\pi$
$631$	−25.5147	−1.01572	−0.507862	+	0.861438i	$0.669564\pi$
$632$	17.2132	0.684704
$633$	−9.55635	−0.379831
$634$	49.7990	1.97777
$635$	−4.76955	−0.189274
$636$	−15.3137	−0.607228
$637$	−37.6569	−1.49202
$638$	−34.6274	−1.37091
$639$	−8.00000	−0.316475
$640$	−12.0416	−0.475987
$641$	−11.6569	−0.460418	−0.230209	−	0.973141i	$-0.573941\pi$
$641$	−11.6569	−0.460418	−0.230209	+	0.973141i	$0.573941\pi$
$642$	0.585786	0.0231191
$643$	4.58579	0.180846	0.0904229	−	0.995903i	$-0.471178\pi$
$643$	4.58579	0.180846	0.0904229	+	0.995903i	$0.471178\pi$
$644$	−5.79899	−0.228512
$645$	−6.62742	−0.260954
$646$	−11.6569	−0.458633
$647$	−11.0294	−0.433612	−0.216806	−	0.976215i	$-0.569564\pi$
$647$	−11.0294	−0.433612	−0.216806	+	0.976215i	$0.569564\pi$
$648$	−4.41421	−0.173407
$649$	16.9706	0.666153
$650$	63.5980	2.49452
$651$	−0.686292	−0.0268979
$652$	−34.7279	−1.36005
$653$	−8.14214	−0.318626	−0.159313	−	0.987228i	$-0.550928\pi$
$653$	−8.14214	−0.318626	−0.159313	+	0.987228i	$0.550928\pi$
$654$	−19.6569	−0.768644
$655$	9.79899	0.382878
$656$	10.2426	0.399908
$657$	−9.41421	−0.367283
$658$	0.485281	0.0189182
$659$	−22.6863	−0.883732	−0.441866	−	0.897081i	$-0.645683\pi$
$659$	−22.6863	−0.883732	−0.441866	+	0.897081i	$0.645683\pi$
$660$	6.34315	0.246907
$661$	−16.6274	−0.646732	−0.323366	−	0.946274i	$-0.604814\pi$
$661$	−16.6274	−0.646732	−0.323366	+	0.946274i	$0.604814\pi$
$662$	−17.3137	−0.672916
$663$	−5.65685	−0.219694
$664$	−3.65685	−0.141913
$665$	−1.65685	−0.0642501
$666$	−11.6569	−0.451694
$667$	−13.1127	−0.507726
$668$	−36.9706	−1.43043
$669$	−14.1421	−0.546767
$670$	21.6569	0.836677
$671$	−15.3137	−0.591179
$672$	−0.928932	−0.0358343
$673$	−10.5858	−0.408052	−0.204026	−	0.978965i	$-0.565403\pi$
$673$	−10.5858	−0.408052	−0.204026	+	0.978965i	$0.565403\pi$
$674$	−33.5563	−1.29254
$675$	−4.65685	−0.179242
$676$	72.7401	2.79770
$677$	−12.1421	−0.466660	−0.233330	−	0.972398i	$-0.574962\pi$
$677$	−12.1421	−0.466660	−0.233330	+	0.972398i	$0.574962\pi$
$678$	−19.6569	−0.754917
$679$	−2.48528	−0.0953763
$680$	−2.58579	−0.0991604
$681$	−7.75736	−0.297263
$682$	8.00000	0.306336
$683$	34.5858	1.32339	0.661694	−	0.749774i	$-0.269838\pi$
$683$	34.5858	1.32339	0.661694	+	0.749774i	$0.269838\pi$
$684$	−18.4853	−0.706802
$685$	−3.31371	−0.126610
$686$	−19.3137	−0.737401
$687$	0.343146	0.0130918
$688$	33.9411	1.29399
$689$	−22.6274	−0.862036
$690$	3.65685	0.139214
$691$	−35.4142	−1.34722	−0.673610	−	0.739087i	$-0.735257\pi$
$691$	−35.4142	−1.34722	−0.673610	+	0.739087i	$0.735257\pi$
$692$	20.3431	0.773330
$693$	1.65685	0.0629387
$694$	−31.7990	−1.20707
$695$	0.627417	0.0237993
$696$	22.3848	0.848493
$697$	−3.41421	−0.129323
$698$	−39.4558	−1.49343
$699$	−25.7990	−0.975807
$700$	10.4437	0.394733
$701$	−24.2843	−0.917204	−0.458602	−	0.888642i	$-0.651650\pi$
$701$	−24.2843	−0.917204	−0.458602	+	0.888642i	$0.651650\pi$
$702$	−13.6569	−0.515445
$703$	−23.3137	−0.879293
$704$	27.7990	1.04771
$705$	−0.201010	−0.00757048
$706$	−59.1127	−2.22474
$707$	8.28427	0.311562
$708$	−22.9706	−0.863287
$709$	12.3431	0.463557	0.231778	−	0.972769i	$-0.425546\pi$
$709$	12.3431	0.463557	0.231778	+	0.972769i	$0.425546\pi$
$710$	−11.3137	−0.424596
$711$	−3.89949	−0.146242
$712$	23.4558	0.879045
$713$	3.02944	0.113453
$714$	−1.41421	−0.0529256
$715$	9.37258	0.350515
$716$	−44.6274	−1.66780
$717$	6.68629	0.249704
$718$	48.6274	1.81476
$719$	0.727922	0.0271469	0.0135735	−	0.999908i	$-0.495679\pi$
$719$	0.727922	0.0271469	0.0135735	+	0.999908i	$0.495679\pi$
$720$	−1.75736	−0.0654929
$721$	−2.34315	−0.0872633
$722$	−10.4142	−0.387577
$723$	19.5563	0.727308
$724$	0.928932	0.0345235
$725$	23.6152	0.877047
$726$	7.24264	0.268800
$727$	−5.79899	−0.215073	−0.107536	−	0.994201i	$-0.534296\pi$
$727$	−5.79899	−0.215073	−0.107536	+	0.994201i	$0.534296\pi$
$728$	14.6274	0.542128
$729$	1.00000	0.0370370
$730$	−13.3137	−0.492762
$731$	−11.3137	−0.418453
$732$	20.7279	0.766126
$733$	−10.9706	−0.405207	−0.202603	−	0.979261i	$-0.564940\pi$
$733$	−10.9706	−0.405207	−0.202603	+	0.979261i	$0.564940\pi$
$734$	−2.58579	−0.0954431
$735$	3.89949	0.143835
$736$	4.10051	0.151147
$737$	−43.3137	−1.59548
$738$	−8.24264	−0.303416
$739$	−4.97056	−0.182845	−0.0914226	−	0.995812i	$-0.529141\pi$
$739$	−4.97056	−0.182845	−0.0914226	+	0.995812i	$0.529141\pi$
$740$	−10.8284	−0.398061
$741$	−27.3137	−1.00339
$742$	−5.65685	−0.207670
$743$	3.51472	0.128943	0.0644713	−	0.997920i	$-0.479464\pi$
$743$	3.51472	0.128943	0.0644713	+	0.997920i	$0.479464\pi$
$744$	−5.17157	−0.189599
$745$	−4.00000	−0.146549
$746$	71.9411	2.63395
$747$	0.828427	0.0303106
$748$	10.8284	0.395927
$749$	0.142136	0.00519352
$750$	−13.6569	−0.498678
$751$	41.8406	1.52679	0.763393	−	0.645934i	$-0.223532\pi$
$751$	41.8406	1.52679	0.763393	+	0.645934i	$0.223532\pi$
$752$	1.02944	0.0375397
$753$	14.9706	0.545558
$754$	69.2548	2.52211
$755$	8.00000	0.291150
$756$	−2.24264	−0.0815641
$757$	12.6274	0.458951	0.229476	−	0.973314i	$-0.426299\pi$
$757$	12.6274	0.458951	0.229476	+	0.973314i	$0.426299\pi$
$758$	86.6690	3.14796
$759$	−7.31371	−0.265471
$760$	−12.4853	−0.452889
$761$	45.2548	1.64049	0.820243	−	0.572015i	$-0.193838\pi$
$761$	45.2548	1.64049	0.820243	+	0.572015i	$0.193838\pi$
$762$	−19.6569	−0.712093
$763$	−4.76955	−0.172669
$764$	42.0000	1.51951
$765$	0.585786	0.0211792
$766$	−56.6274	−2.04603
$767$	−33.9411	−1.22554
$768$	−29.9706	−1.08147
$769$	0.686292	0.0247483	0.0123742	−	0.999923i	$-0.496061\pi$
$769$	0.686292	0.0247483	0.0123742	+	0.999923i	$0.496061\pi$
$770$	2.34315	0.0844411
$771$	−13.1716	−0.474363
$772$	−14.0000	−0.503871
$773$	13.0294	0.468636	0.234318	−	0.972160i	$-0.424714\pi$
$773$	13.0294	0.468636	0.234318	+	0.972160i	$0.424714\pi$
$774$	−27.3137	−0.981771
$775$	−5.45584	−0.195980
$776$	−18.7279	−0.672293
$777$	−2.82843	−0.101469
$778$	24.1421	0.865537
$779$	−16.4853	−0.590647
$780$	−12.6863	−0.454242
$781$	22.6274	0.809673
$782$	6.24264	0.223236
$783$	−5.07107	−0.181225
$784$	−19.9706	−0.713234
$785$	−0.585786	−0.0209076
$786$	40.3848	1.44048
$787$	−3.89949	−0.139002	−0.0695010	−	0.997582i	$-0.522141\pi$
$787$	−3.89949	−0.139002	−0.0695010	+	0.997582i	$0.522141\pi$
$788$	−52.8284	−1.88193
$789$	−8.00000	−0.284808
$790$	−5.51472	−0.196205
$791$	−4.76955	−0.169586
$792$	12.4853	0.443645
$793$	30.6274	1.08761
$794$	34.7279	1.23245
$795$	2.34315	0.0831028
$796$	−50.4264	−1.78732
$797$	14.1421	0.500940	0.250470	−	0.968124i	$-0.419415\pi$
$797$	14.1421	0.500940	0.250470	+	0.968124i	$0.419415\pi$
$798$	−6.82843	−0.241724
$799$	−0.343146	−0.0121396
$800$	−7.38478	−0.261091
$801$	−5.31371	−0.187751
$802$	12.7279	0.449439
$803$	26.6274	0.939661
$804$	58.6274	2.06763
$805$	0.887302	0.0312733
$806$	−16.0000	−0.563576
$807$	−13.5563	−0.477206
$808$	62.4264	2.19615
$809$	−50.2426	−1.76644	−0.883219	−	0.468962i	$-0.844628\pi$
$809$	−50.2426	−1.76644	−0.883219	+	0.468962i	$0.844628\pi$
$810$	1.41421	0.0496904
$811$	−17.2721	−0.606505	−0.303252	−	0.952910i	$-0.598073\pi$
$811$	−17.2721	−0.606505	−0.303252	+	0.952910i	$0.598073\pi$
$812$	11.3726	0.399099
$813$	18.1421	0.636272
$814$	32.9706	1.15562
$815$	5.31371	0.186131
$816$	−3.00000	−0.105021
$817$	−54.6274	−1.91117
$818$	28.1421	0.983967
$819$	−3.31371	−0.115790
$820$	−7.65685	−0.267389
$821$	13.7990	0.481588	0.240794	−	0.970576i	$-0.422592\pi$
$821$	13.7990	0.481588	0.240794	+	0.970576i	$0.422592\pi$
$822$	−13.6569	−0.476337
$823$	3.21320	0.112005	0.0560026	−	0.998431i	$-0.482164\pi$
$823$	3.21320	0.112005	0.0560026	+	0.998431i	$0.482164\pi$
$824$	−17.6569	−0.615106
$825$	13.1716	0.458575
$826$	−8.48528	−0.295241
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	9.89949	0.344031
$829$	−54.6274	−1.89729	−0.948644	−	0.316345i	$-0.897544\pi$
$829$	−54.6274	−1.89729	−0.948644	+	0.316345i	$0.897544\pi$
$830$	1.17157	0.0406659
$831$	−20.8284	−0.722530
$832$	−55.5980	−1.92751
$833$	6.65685	0.230646
$834$	2.58579	0.0895385
$835$	5.65685	0.195764
$836$	52.2843	1.80829
$837$	1.17157	0.0404955
$838$	−61.4558	−2.12296
$839$	−16.9289	−0.584452	−0.292226	−	0.956349i	$-0.594396\pi$
$839$	−16.9289	−0.584452	−0.292226	+	0.956349i	$0.594396\pi$
$840$	−1.51472	−0.0522628
$841$	−3.28427	−0.113251
$842$	73.5980	2.53635
$843$	21.4558	0.738979
$844$	−36.5858	−1.25933
$845$	−11.1299	−0.382882
$846$	−0.828427	−0.0284819
$847$	1.75736	0.0603836
$848$	−12.0000	−0.412082
$849$	−6.34315	−0.217696
$850$	−11.2426	−0.385619
$851$	12.4853	0.427990
$852$	−30.6274	−1.04928
$853$	40.6274	1.39106	0.695528	−	0.718499i	$-0.255170\pi$
$853$	40.6274	1.39106	0.695528	+	0.718499i	$0.255170\pi$
$854$	7.65685	0.262012
$855$	2.82843	0.0967302
$856$	1.07107	0.0366083
$857$	−18.9289	−0.646600	−0.323300	−	0.946297i	$-0.604792\pi$
$857$	−18.9289	−0.646600	−0.323300	+	0.946297i	$0.604792\pi$
$858$	38.6274	1.31872
$859$	20.6863	0.705807	0.352904	−	0.935660i	$-0.385194\pi$
$859$	20.6863	0.705807	0.352904	+	0.935660i	$0.385194\pi$
$860$	−25.3726	−0.865198
$861$	−2.00000	−0.0681598
$862$	−18.8284	−0.641299
$863$	−8.14214	−0.277162	−0.138581	−	0.990351i	$-0.544254\pi$
$863$	−8.14214	−0.277162	−0.138581	+	0.990351i	$0.544254\pi$
$864$	1.58579	0.0539496
$865$	−3.11270	−0.105835
$866$	22.4853	0.764081
$867$	1.00000	0.0339618
$868$	−2.62742	−0.0891803
$869$	11.0294	0.374148
$870$	−7.17157	−0.243139
$871$	86.6274	2.93526
$872$	−35.9411	−1.21712
$873$	4.24264	0.143592
$874$	30.1421	1.01957
$875$	−3.31371	−0.112024
$876$	−36.0416	−1.21773
$877$	−21.6152	−0.729894	−0.364947	−	0.931028i	$-0.618913\pi$
$877$	−21.6152	−0.729894	−0.364947	+	0.931028i	$0.618913\pi$
$878$	9.41421	0.317714
$879$	−25.4558	−0.858604
$880$	4.97056	0.167558
$881$	9.75736	0.328734	0.164367	−	0.986399i	$-0.447442\pi$
$881$	9.75736	0.328734	0.164367	+	0.986399i	$0.447442\pi$
$882$	16.0711	0.541141
$883$	−2.14214	−0.0720886	−0.0360443	−	0.999350i	$-0.511476\pi$
$883$	−2.14214	−0.0720886	−0.0360443	+	0.999350i	$0.511476\pi$
$884$	−21.6569	−0.728399
$885$	3.51472	0.118146
$886$	11.6569	0.391620
$887$	−1.61522	−0.0542339	−0.0271170	−	0.999632i	$-0.508633\pi$
$887$	−1.61522	−0.0542339	−0.0271170	+	0.999632i	$0.508633\pi$
$888$	−21.3137	−0.715241
$889$	−4.76955	−0.159966
$890$	−7.51472	−0.251894
$891$	−2.82843	−0.0947559
$892$	−54.1421	−1.81281
$893$	−1.65685	−0.0554445
$894$	−16.4853	−0.551350
$895$	6.82843	0.228249
$896$	−12.0416	−0.402283
$897$	14.6274	0.488395
$898$	−46.8701	−1.56407
$899$	−5.94113	−0.198148
$900$	−17.8284	−0.594281
$901$	4.00000	0.133259
$902$	23.3137	0.776262
$903$	−6.62742	−0.220547
$904$	−35.9411	−1.19538
$905$	−0.142136	−0.00472475
$906$	32.9706	1.09537
$907$	−7.79899	−0.258961	−0.129481	−	0.991582i	$-0.541331\pi$
$907$	−7.79899	−0.258961	−0.129481	+	0.991582i	$0.541331\pi$
$908$	−29.6985	−0.985579
$909$	−14.1421	−0.469065
$910$	−4.68629	−0.155349
$911$	−25.8579	−0.856709	−0.428355	−	0.903611i	$-0.640907\pi$
$911$	−25.8579	−0.856709	−0.428355	+	0.903611i	$0.640907\pi$
$912$	−14.4853	−0.479656
$913$	−2.34315	−0.0775468
$914$	−33.7990	−1.11797
$915$	−3.17157	−0.104849
$916$	1.31371	0.0434062
$917$	9.79899	0.323591
$918$	2.41421	0.0796809
$919$	21.6569	0.714394	0.357197	−	0.934029i	$-0.383733\pi$
$919$	21.6569	0.714394	0.357197	+	0.934029i	$0.383733\pi$
$920$	6.68629	0.220441
$921$	14.1421	0.465999
$922$	4.48528	0.147715
$923$	−45.2548	−1.48958
$924$	6.34315	0.208674
$925$	−22.4853	−0.739311
$926$	11.3137	0.371792
$927$	4.00000	0.131377
$928$	−8.04163	−0.263979
$929$	−8.62742	−0.283056	−0.141528	−	0.989934i	$-0.545202\pi$
$929$	−8.62742	−0.283056	−0.141528	+	0.989934i	$0.545202\pi$
$930$	1.65685	0.0543304
$931$	32.1421	1.05342
$932$	−98.7696	−3.23530
$933$	−11.7990	−0.386282
$934$	−58.6274	−1.91835
$935$	−1.65685	−0.0541849
$936$	−24.9706	−0.816188
$937$	−39.1716	−1.27968	−0.639840	−	0.768508i	$-0.720999\pi$
$937$	−39.1716	−1.27968	−0.639840	+	0.768508i	$0.720999\pi$
$938$	21.6569	0.707121
$939$	2.00000	0.0652675
$940$	−0.769553	−0.0251000
$941$	13.0294	0.424748	0.212374	−	0.977189i	$-0.431881\pi$
$941$	13.0294	0.424748	0.212374	+	0.977189i	$0.431881\pi$
$942$	−2.41421	−0.0786593
$943$	8.82843	0.287493
$944$	−18.0000	−0.585850
$945$	0.343146	0.0111625
$946$	77.2548	2.51177
$947$	−29.8995	−0.971603	−0.485802	−	0.874069i	$-0.661472\pi$
$947$	−29.8995	−0.971603	−0.485802	+	0.874069i	$0.661472\pi$
$948$	−14.9289	−0.484869
$949$	−53.2548	−1.72873
$950$	−54.2843	−1.76121
$951$	−20.6274	−0.668890
$952$	−2.58579	−0.0838058
$953$	−18.1421	−0.587681	−0.293841	−	0.955854i	$-0.594934\pi$
$953$	−18.1421	−0.587681	−0.293841	+	0.955854i	$0.594934\pi$
$954$	9.65685	0.312652
$955$	−6.42641	−0.207954
$956$	25.5980	0.827898
$957$	14.3431	0.463648
$958$	−33.5563	−1.08416
$959$	−3.31371	−0.107005
$960$	5.75736	0.185818
$961$	−29.6274	−0.955723
$962$	−65.9411	−2.12603
$963$	−0.242641	−0.00781899
$964$	74.8701	2.41140
$965$	2.14214	0.0689578
$966$	3.65685	0.117657
$967$	0.970563	0.0312112	0.0156056	−	0.999878i	$-0.495032\pi$
$967$	0.970563	0.0312112	0.0156056	+	0.999878i	$0.495032\pi$
$968$	13.2426	0.425635
$969$	4.82843	0.155111
$970$	6.00000	0.192648
$971$	−41.3137	−1.32582	−0.662910	−	0.748699i	$-0.730679\pi$
$971$	−41.3137	−1.32582	−0.662910	+	0.748699i	$0.730679\pi$
$972$	3.82843	0.122797
$973$	0.627417	0.0201141
$974$	−68.7696	−2.20352
$975$	−26.3431	−0.843656
$976$	16.2426	0.519914
$977$	−20.6274	−0.659930	−0.329965	−	0.943993i	$-0.607037\pi$
$977$	−20.6274	−0.659930	−0.329965	+	0.943993i	$0.607037\pi$
$978$	21.8995	0.700269
$979$	15.0294	0.480343
$980$	14.9289	0.476887
$981$	8.14214	0.259958
$982$	6.00000	0.191468
$983$	−9.89949	−0.315745	−0.157872	−	0.987460i	$-0.550463\pi$
$983$	−9.89949	−0.315745	−0.157872	+	0.987460i	$0.550463\pi$
$984$	−15.0711	−0.480448
$985$	8.08326	0.257554
$986$	−12.2426	−0.389885
$987$	−0.201010	−0.00639822
$988$	−104.569	−3.32677
$989$	29.2548	0.930250
$990$	−4.00000	−0.127128
$991$	−16.7696	−0.532702	−0.266351	−	0.963876i	$-0.585818\pi$
$991$	−16.7696	−0.532702	−0.266351	+	0.963876i	$0.585818\pi$
$992$	1.85786	0.0589873
$993$	7.17157	0.227583
$994$	−11.3137	−0.358849
$995$	7.71573	0.244605
$996$	3.17157	0.100495
$997$	41.4975	1.31424	0.657119	−	0.753787i	$-0.271775\pi$
$997$	41.4975	1.31424	0.657119	+	0.753787i	$0.271775\pi$
$998$	58.8701	1.86350
$999$	4.82843	0.152765

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.b.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.b.1.1	✓	2	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} -0.585786 q^{5} -2.41421 q^{6} -0.585786 q^{7} -4.41421 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} -0.585786 q^{5} -2.41421 q^{6} -0.585786 q^{7} -4.41421 q^{8} +1.00000 q^{9} +1.41421 q^{10} -2.82843 q^{11} +3.82843 q^{12} +5.65685 q^{13} +1.41421 q^{14} -0.585786 q^{15} +3.00000 q^{16} -1.00000 q^{17} -2.41421 q^{18} -4.82843 q^{19} -2.24264 q^{20} -0.585786 q^{21} +6.82843 q^{22} +2.58579 q^{23} -4.41421 q^{24} -4.65685 q^{25} -13.6569 q^{26} +1.00000 q^{27} -2.24264 q^{28} -5.07107 q^{29} +1.41421 q^{30} +1.17157 q^{31} +1.58579 q^{32} -2.82843 q^{33} +2.41421 q^{34} +0.343146 q^{35} +3.82843 q^{36} +4.82843 q^{37} +11.6569 q^{38} +5.65685 q^{39} +2.58579 q^{40} +3.41421 q^{41} +1.41421 q^{42} +11.3137 q^{43} -10.8284 q^{44} -0.585786 q^{45} -6.24264 q^{46} +0.343146 q^{47} +3.00000 q^{48} -6.65685 q^{49} +11.2426 q^{50} -1.00000 q^{51} +21.6569 q^{52} -4.00000 q^{53} -2.41421 q^{54} +1.65685 q^{55} +2.58579 q^{56} -4.82843 q^{57} +12.2426 q^{58} -6.00000 q^{59} -2.24264 q^{60} +5.41421 q^{61} -2.82843 q^{62} -0.585786 q^{63} -9.82843 q^{64} -3.31371 q^{65} +6.82843 q^{66} +15.3137 q^{67} -3.82843 q^{68} +2.58579 q^{69} -0.828427 q^{70} -8.00000 q^{71} -4.41421 q^{72} -9.41421 q^{73} -11.6569 q^{74} -4.65685 q^{75} -18.4853 q^{76} +1.65685 q^{77} -13.6569 q^{78} -3.89949 q^{79} -1.75736 q^{80} +1.00000 q^{81} -8.24264 q^{82} +0.828427 q^{83} -2.24264 q^{84} +0.585786 q^{85} -27.3137 q^{86} -5.07107 q^{87} +12.4853 q^{88} -5.31371 q^{89} +1.41421 q^{90} -3.31371 q^{91} +9.89949 q^{92} +1.17157 q^{93} -0.828427 q^{94} +2.82843 q^{95} +1.58579 q^{96} +4.24264 q^{97} +16.0711 q^{98} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9} + 2 q^{12} - 4 q^{15} + 6 q^{16} - 2 q^{17} - 2 q^{18} - 4 q^{19} + 4 q^{20} - 4 q^{21} + 8 q^{22} + 8 q^{23} - 6 q^{24} + 2 q^{25} - 16 q^{26} + 2 q^{27} + 4 q^{28} + 4 q^{29} + 8 q^{31} + 6 q^{32} + 2 q^{34} + 12 q^{35} + 2 q^{36} + 4 q^{37} + 12 q^{38} + 8 q^{40} + 4 q^{41} - 16 q^{44} - 4 q^{45} - 4 q^{46} + 12 q^{47} + 6 q^{48} - 2 q^{49} + 14 q^{50} - 2 q^{51} + 32 q^{52} - 8 q^{53} - 2 q^{54} - 8 q^{55} + 8 q^{56} - 4 q^{57} + 16 q^{58} - 12 q^{59} + 4 q^{60} + 8 q^{61} - 4 q^{63} - 14 q^{64} + 16 q^{65} + 8 q^{66} + 8 q^{67} - 2 q^{68} + 8 q^{69} + 4 q^{70} - 16 q^{71} - 6 q^{72} - 16 q^{73} - 12 q^{74} + 2 q^{75} - 20 q^{76} - 8 q^{77} - 16 q^{78} + 12 q^{79} - 12 q^{80} + 2 q^{81} - 8 q^{82} - 4 q^{83} + 4 q^{84} + 4 q^{85} - 32 q^{86} + 4 q^{87} + 8 q^{88} + 12 q^{89} + 16 q^{91} + 8 q^{93} + 4 q^{94} + 6 q^{96} + 18 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9 + 2 * q^12 - 4 * q^15 + 6 * q^16 - 2 * q^17 - 2 * q^18 - 4 * q^19 + 4 * q^20 - 4 * q^21 + 8 * q^22 + 8 * q^23 - 6 * q^24 + 2 * q^25 - 16 * q^26 + 2 * q^27 + 4 * q^28 + 4 * q^29 + 8 * q^31 + 6 * q^32 + 2 * q^34 + 12 * q^35 + 2 * q^36 + 4 * q^37 + 12 * q^38 + 8 * q^40 + 4 * q^41 - 16 * q^44 - 4 * q^45 - 4 * q^46 + 12 * q^47 + 6 * q^48 - 2 * q^49 + 14 * q^50 - 2 * q^51 + 32 * q^52 - 8 * q^53 - 2 * q^54 - 8 * q^55 + 8 * q^56 - 4 * q^57 + 16 * q^58 - 12 * q^59 + 4 * q^60 + 8 * q^61 - 4 * q^63 - 14 * q^64 + 16 * q^65 + 8 * q^66 + 8 * q^67 - 2 * q^68 + 8 * q^69 + 4 * q^70 - 16 * q^71 - 6 * q^72 - 16 * q^73 - 12 * q^74 + 2 * q^75 - 20 * q^76 - 8 * q^77 - 16 * q^78 + 12 * q^79 - 12 * q^80 + 2 * q^81 - 8 * q^82 - 4 * q^83 + 4 * q^84 + 4 * q^85 - 32 * q^86 + 4 * q^87 + 8 * q^88 + 12 * q^89 + 16 * q^91 + 8 * q^93 + 4 * q^94 + 6 * q^96 + 18 * q^98

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