LMFDB - Embedded newform 8006.2.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8006,2,Mod(1,8006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8006, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8006 = 2 \cdot 4003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8006.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:	$63.9282318582$
Analytic rank:	$1$
Dimension:	$75$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.99270 q^{3} +1.00000 q^{4} -0.0677361 q^{5} +2.99270 q^{6} -2.03477 q^{7} -1.00000 q^{8} +5.95625 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.99270 q^{3} +1.00000 q^{4} -0.0677361 q^{5} +2.99270 q^{6} -2.03477 q^{7} -1.00000 q^{8} +5.95625 q^{9} +0.0677361 q^{10} +2.06793 q^{11} -2.99270 q^{12} +4.05312 q^{13} +2.03477 q^{14} +0.202714 q^{15} +1.00000 q^{16} +2.14596 q^{17} -5.95625 q^{18} +6.54839 q^{19} -0.0677361 q^{20} +6.08944 q^{21} -2.06793 q^{22} +7.28115 q^{23} +2.99270 q^{24} -4.99541 q^{25} -4.05312 q^{26} -8.84715 q^{27} -2.03477 q^{28} -9.58367 q^{29} -0.202714 q^{30} -7.07241 q^{31} -1.00000 q^{32} -6.18870 q^{33} -2.14596 q^{34} +0.137827 q^{35} +5.95625 q^{36} +4.07241 q^{37} -6.54839 q^{38} -12.1298 q^{39} +0.0677361 q^{40} -4.26718 q^{41} -6.08944 q^{42} -6.92155 q^{43} +2.06793 q^{44} -0.403453 q^{45} -7.28115 q^{46} -4.25664 q^{47} -2.99270 q^{48} -2.85973 q^{49} +4.99541 q^{50} -6.42222 q^{51} +4.05312 q^{52} -5.40084 q^{53} +8.84715 q^{54} -0.140074 q^{55} +2.03477 q^{56} -19.5974 q^{57} +9.58367 q^{58} +5.13928 q^{59} +0.202714 q^{60} -7.58843 q^{61} +7.07241 q^{62} -12.1196 q^{63} +1.00000 q^{64} -0.274542 q^{65} +6.18870 q^{66} +1.02118 q^{67} +2.14596 q^{68} -21.7903 q^{69} -0.137827 q^{70} +11.1012 q^{71} -5.95625 q^{72} +12.7617 q^{73} -4.07241 q^{74} +14.9498 q^{75} +6.54839 q^{76} -4.20776 q^{77} +12.1298 q^{78} +0.115909 q^{79} -0.0677361 q^{80} +8.60812 q^{81} +4.26718 q^{82} +2.46156 q^{83} +6.08944 q^{84} -0.145359 q^{85} +6.92155 q^{86} +28.6810 q^{87} -2.06793 q^{88} -15.0457 q^{89} +0.403453 q^{90} -8.24714 q^{91} +7.28115 q^{92} +21.1656 q^{93} +4.25664 q^{94} -0.443562 q^{95} +2.99270 q^{96} -4.48255 q^{97} +2.85973 q^{98} +12.3171 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 75 q - 75 q^{2} + q^{3} + 75 q^{4} - 9 q^{5} - q^{6} - 8 q^{7} - 75 q^{8} + 66 q^{9}+O(q^{10}) $ 75 * q - 75 * q^2 + q^3 + 75 * q^4 - 9 * q^5 - q^6 - 8 * q^7 - 75 * q^8 + 66 * q^9	\( 75 q - 75 q^{2} + q^{3} + 75 q^{4} - 9 q^{5} - q^{6} - 8 q^{7} - 75 q^{8} + 66 q^{9} + 9 q^{10} - 5 q^{11} + q^{12} - 35 q^{13} + 8 q^{14} - 21 q^{15} + 75 q^{16} + 4 q^{17} - 66 q^{18} - 59 q^{19} - 9 q^{20} - 62 q^{21} + 5 q^{22} + 43 q^{23} - q^{24} + 44 q^{25} + 35 q^{26} + 4 q^{27} - 8 q^{28} - 38 q^{29} + 21 q^{30} - 51 q^{31} - 75 q^{32} - 19 q^{33} - 4 q^{34} + 14 q^{35} + 66 q^{36} - 63 q^{37} + 59 q^{38} - 34 q^{39} + 9 q^{40} - 27 q^{41} + 62 q^{42} - 39 q^{43} - 5 q^{44} - 52 q^{45} - 43 q^{46} + 40 q^{47} + q^{48} + 29 q^{49} - 44 q^{50} - 34 q^{51} - 35 q^{52} - 39 q^{53} - 4 q^{54} - 48 q^{55} + 8 q^{56} - 28 q^{57} + 38 q^{58} + 5 q^{59} - 21 q^{60} - 98 q^{61} + 51 q^{62} + 2 q^{63} + 75 q^{64} - q^{65} + 19 q^{66} - 59 q^{67} + 4 q^{68} - 69 q^{69} - 14 q^{70} - 9 q^{71} - 66 q^{72} - 51 q^{73} + 63 q^{74} - q^{75} - 59 q^{76} - 25 q^{77} + 34 q^{78} - 139 q^{79} - 9 q^{80} + 23 q^{81} + 27 q^{82} + 31 q^{83} - 62 q^{84} - 149 q^{85} + 39 q^{86} + q^{87} + 5 q^{88} - 39 q^{89} + 52 q^{90} - 93 q^{91} + 43 q^{92} - 83 q^{93} - 40 q^{94} + 2 q^{95} - q^{96} - 70 q^{97} - 29 q^{98} - 61 q^{99}+O(q^{100}) \) 75 * q - 75 * q^2 + q^3 + 75 * q^4 - 9 * q^5 - q^6 - 8 * q^7 - 75 * q^8 + 66 * q^9 + 9 * q^10 - 5 * q^11 + q^12 - 35 * q^13 + 8 * q^14 - 21 * q^15 + 75 * q^16 + 4 * q^17 - 66 * q^18 - 59 * q^19 - 9 * q^20 - 62 * q^21 + 5 * q^22 + 43 * q^23 - q^24 + 44 * q^25 + 35 * q^26 + 4 * q^27 - 8 * q^28 - 38 * q^29 + 21 * q^30 - 51 * q^31 - 75 * q^32 - 19 * q^33 - 4 * q^34 + 14 * q^35 + 66 * q^36 - 63 * q^37 + 59 * q^38 - 34 * q^39 + 9 * q^40 - 27 * q^41 + 62 * q^42 - 39 * q^43 - 5 * q^44 - 52 * q^45 - 43 * q^46 + 40 * q^47 + q^48 + 29 * q^49 - 44 * q^50 - 34 * q^51 - 35 * q^52 - 39 * q^53 - 4 * q^54 - 48 * q^55 + 8 * q^56 - 28 * q^57 + 38 * q^58 + 5 * q^59 - 21 * q^60 - 98 * q^61 + 51 * q^62 + 2 * q^63 + 75 * q^64 - q^65 + 19 * q^66 - 59 * q^67 + 4 * q^68 - 69 * q^69 - 14 * q^70 - 9 * q^71 - 66 * q^72 - 51 * q^73 + 63 * q^74 - q^75 - 59 * q^76 - 25 * q^77 + 34 * q^78 - 139 * q^79 - 9 * q^80 + 23 * q^81 + 27 * q^82 + 31 * q^83 - 62 * q^84 - 149 * q^85 + 39 * q^86 + q^87 + 5 * q^88 - 39 * q^89 + 52 * q^90 - 93 * q^91 + 43 * q^92 - 83 * q^93 - 40 * q^94 + 2 * q^95 - q^96 - 70 * q^97 - 29 * q^98 - 61 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.99270	−1.72784	−0.863918	−	0.503633i	$-0.831996\pi$
$3$	−2.99270	−1.72784	−0.863918	+	0.503633i	$0.831996\pi$
$4$	1.00000	0.500000
$5$	−0.0677361	−0.0302925	−0.0151462	−	0.999885i	$-0.504821\pi$
$5$	−0.0677361	−0.0302925	−0.0151462	+	0.999885i	$0.504821\pi$
$6$	2.99270	1.22176
$7$	−2.03477	−0.769069	−0.384535	−	0.923111i	$-0.625638\pi$
$7$	−2.03477	−0.769069	−0.384535	+	0.923111i	$0.625638\pi$
$8$	−1.00000	−0.353553
$9$	5.95625	1.98542
$10$	0.0677361	0.0214200
$11$	2.06793	0.623505	0.311753	−	0.950163i	$-0.399084\pi$
$11$	2.06793	0.623505	0.311753	+	0.950163i	$0.399084\pi$
$12$	−2.99270	−0.863918
$13$	4.05312	1.12413	0.562066	−	0.827092i	$-0.310007\pi$
$13$	4.05312	1.12413	0.562066	+	0.827092i	$0.310007\pi$
$14$	2.03477	0.543814
$15$	0.202714	0.0523404
$16$	1.00000	0.250000
$17$	2.14596	0.520473	0.260236	−	0.965545i	$-0.416200\pi$
$17$	2.14596	0.520473	0.260236	+	0.965545i	$0.416200\pi$
$18$	−5.95625	−1.40390
$19$	6.54839	1.50230	0.751152	−	0.660129i	$-0.229499\pi$
$19$	6.54839	1.50230	0.751152	+	0.660129i	$0.229499\pi$
$20$	−0.0677361	−0.0151462
$21$	6.08944	1.32882
$22$	−2.06793	−0.440885
$23$	7.28115	1.51823	0.759113	−	0.650959i	$-0.225633\pi$
$23$	7.28115	1.51823	0.759113	+	0.650959i	$0.225633\pi$
$24$	2.99270	0.610882
$25$	−4.99541	−0.999082
$26$	−4.05312	−0.794881
$27$	−8.84715	−1.70264
$28$	−2.03477	−0.384535
$29$	−9.58367	−1.77964	−0.889821	−	0.456309i	$-0.849171\pi$
$29$	−9.58367	−1.77964	−0.889821	+	0.456309i	$0.849171\pi$
$30$	−0.202714	−0.0370103
$31$	−7.07241	−1.27024	−0.635121	−	0.772413i	$-0.719050\pi$
$31$	−7.07241	−1.27024	−0.635121	+	0.772413i	$0.719050\pi$
$32$	−1.00000	−0.176777
$33$	−6.18870	−1.07731
$34$	−2.14596	−0.368030
$35$	0.137827	0.0232970
$36$	5.95625	0.992708
$37$	4.07241	0.669500	0.334750	−	0.942307i	$-0.391348\pi$
$37$	4.07241	0.669500	0.334750	+	0.942307i	$0.391348\pi$
$38$	−6.54839	−1.06229
$39$	−12.1298	−1.94231
$40$	0.0677361	0.0107100
$41$	−4.26718	−0.666421	−0.333210	−	0.942852i	$-0.608132\pi$
$41$	−4.26718	−0.666421	−0.333210	+	0.942852i	$0.608132\pi$
$42$	−6.08944	−0.939621
$43$	−6.92155	−1.05553	−0.527763	−	0.849392i	$-0.676969\pi$
$43$	−6.92155	−1.05553	−0.527763	+	0.849392i	$0.676969\pi$
$44$	2.06793	0.311753
$45$	−0.403453	−0.0601432
$46$	−7.28115	−1.07355
$47$	−4.25664	−0.620895	−0.310447	−	0.950591i	$-0.600479\pi$
$47$	−4.25664	−0.620895	−0.310447	+	0.950591i	$0.600479\pi$
$48$	−2.99270	−0.431959
$49$	−2.85973	−0.408533
$50$	4.99541	0.706458
$51$	−6.42222	−0.899291
$52$	4.05312	0.562066
$53$	−5.40084	−0.741862	−0.370931	−	0.928660i	$-0.620961\pi$
$53$	−5.40084	−0.741862	−0.370931	+	0.928660i	$0.620961\pi$
$54$	8.84715	1.20394
$55$	−0.140074	−0.0188875
$56$	2.03477	0.271907
$57$	−19.5974	−2.59573
$58$	9.58367	1.25840
$59$	5.13928	0.669077	0.334538	−	0.942382i	$-0.391420\pi$
$59$	5.13928	0.669077	0.334538	+	0.942382i	$0.391420\pi$
$60$	0.202714	0.0261702
$61$	−7.58843	−0.971599	−0.485800	−	0.874070i	$-0.661472\pi$
$61$	−7.58843	−0.971599	−0.485800	+	0.874070i	$0.661472\pi$
$62$	7.07241	0.898197
$63$	−12.1196	−1.52692
$64$	1.00000	0.125000
$65$	−0.274542	−0.0340527
$66$	6.18870	0.761776
$67$	1.02118	0.124758	0.0623788	−	0.998053i	$-0.480131\pi$
$67$	1.02118	0.124758	0.0623788	+	0.998053i	$0.480131\pi$
$68$	2.14596	0.260236
$69$	−21.7903	−2.62324
$70$	−0.137827	−0.0164735
$71$	11.1012	1.31748	0.658738	−	0.752373i	$-0.271091\pi$
$71$	11.1012	1.31748	0.658738	+	0.752373i	$0.271091\pi$
$72$	−5.95625	−0.701950
$73$	12.7617	1.49365	0.746825	−	0.665021i	$-0.231577\pi$
$73$	12.7617	1.49365	0.746825	+	0.665021i	$0.231577\pi$
$74$	−4.07241	−0.473408
$75$	14.9498	1.72625
$76$	6.54839	0.751152
$77$	−4.20776	−0.479519
$78$	12.1298	1.37342
$79$	0.115909	0.0130408	0.00652040	−	0.999979i	$-0.497924\pi$
$79$	0.115909	0.0130408	0.00652040	+	0.999979i	$0.497924\pi$
$80$	−0.0677361	−0.00757312
$81$	8.60812	0.956458
$82$	4.26718	0.471231
$83$	2.46156	0.270191	0.135096	−	0.990833i	$-0.456866\pi$
$83$	2.46156	0.270191	0.135096	+	0.990833i	$0.456866\pi$
$84$	6.08944	0.664412
$85$	−0.145359	−0.0157664
$86$	6.92155	0.746370
$87$	28.6810	3.07493
$88$	−2.06793	−0.220442
$89$	−15.0457	−1.59485	−0.797423	−	0.603421i	$-0.793804\pi$
$89$	−15.0457	−1.59485	−0.797423	+	0.603421i	$0.793804\pi$
$90$	0.403453	0.0425276
$91$	−8.24714	−0.864535
$92$	7.28115	0.759113
$93$	21.1656	2.19477
$94$	4.25664	0.439039
$95$	−0.443562	−0.0455085
$96$	2.99270	0.305441
$97$	−4.48255	−0.455134	−0.227567	−	0.973762i	$-0.573077\pi$
$97$	−4.48255	−0.455134	−0.227567	+	0.973762i	$0.573077\pi$
$98$	2.85973	0.288876
$99$	12.3171	1.23792
$100$	−4.99541	−0.499541
$101$	4.98904	0.496428	0.248214	−	0.968705i	$-0.420156\pi$
$101$	4.98904	0.496428	0.248214	+	0.968705i	$0.420156\pi$
$102$	6.42222	0.635895
$103$	9.98597	0.983947	0.491973	−	0.870610i	$-0.336276\pi$
$103$	9.98597	0.983947	0.491973	+	0.870610i	$0.336276\pi$
$104$	−4.05312	−0.397441
$105$	−0.412475	−0.0402534
$106$	5.40084	0.524576
$107$	8.16003	0.788859	0.394430	−	0.918926i	$-0.370942\pi$
$107$	8.16003	0.788859	0.394430	+	0.918926i	$0.370942\pi$
$108$	−8.84715	−0.851318
$109$	10.3712	0.993384	0.496692	−	0.867927i	$-0.334548\pi$
$109$	10.3712	0.993384	0.496692	+	0.867927i	$0.334548\pi$
$110$	0.140074	0.0133555
$111$	−12.1875	−1.15679
$112$	−2.03477	−0.192267
$113$	−7.46512	−0.702259	−0.351130	−	0.936327i	$-0.614202\pi$
$113$	−7.46512	−0.702259	−0.351130	+	0.936327i	$0.614202\pi$
$114$	19.5974	1.83546
$115$	−0.493197	−0.0459908
$116$	−9.58367	−0.889821
$117$	24.1413	2.23187
$118$	−5.13928	−0.473109
$119$	−4.36653	−0.400279
$120$	−0.202714	−0.0185051
$121$	−6.72365	−0.611241
$122$	7.58843	0.687025
$123$	12.7704	1.15147
$124$	−7.07241	−0.635121
$125$	0.677050	0.0605572
$126$	12.1196	1.07970
$127$	−12.6262	−1.12039	−0.560197	−	0.828359i	$-0.689275\pi$
$127$	−12.6262	−1.12039	−0.560197	+	0.828359i	$0.689275\pi$
$128$	−1.00000	−0.0883883
$129$	20.7141	1.82378
$130$	0.274542	0.0240789
$131$	−16.5516	−1.44612	−0.723059	−	0.690787i	$-0.757264\pi$
$131$	−16.5516	−1.44612	−0.723059	+	0.690787i	$0.757264\pi$
$132$	−6.18870	−0.538657
$133$	−13.3244	−1.15538
$134$	−1.02118	−0.0882169
$135$	0.599271	0.0515770
$136$	−2.14596	−0.184015
$137$	4.91138	0.419607	0.209804	−	0.977744i	$-0.432718\pi$
$137$	4.91138	0.419607	0.209804	+	0.977744i	$0.432718\pi$
$138$	21.7903	1.85491
$139$	−3.70390	−0.314160	−0.157080	−	0.987586i	$-0.550208\pi$
$139$	−3.70390	−0.314160	−0.157080	+	0.987586i	$0.550208\pi$
$140$	0.137827	0.0116485
$141$	12.7388	1.07280
$142$	−11.1012	−0.931596
$143$	8.38157	0.700902
$144$	5.95625	0.496354
$145$	0.649160	0.0539098
$146$	−12.7617	−1.05617
$147$	8.55831	0.705877
$148$	4.07241	0.334750
$149$	12.9062	1.05731	0.528657	−	0.848835i	$-0.322696\pi$
$149$	12.9062	1.05731	0.528657	+	0.848835i	$0.322696\pi$
$150$	−14.9498	−1.22064
$151$	−12.6540	−1.02976	−0.514882	−	0.857261i	$-0.672164\pi$
$151$	−12.6540	−1.02976	−0.514882	+	0.857261i	$0.672164\pi$
$152$	−6.54839	−0.531145
$153$	12.7819	1.03335
$154$	4.20776	0.339071
$155$	0.479057	0.0384788
$156$	−12.1298	−0.971157
$157$	13.4859	1.07629	0.538145	−	0.842852i	$-0.319125\pi$
$157$	13.4859	1.07629	0.538145	+	0.842852i	$0.319125\pi$
$158$	−0.115909	−0.00922123
$159$	16.1631	1.28182
$160$	0.0677361	0.00535501
$161$	−14.8154	−1.16762
$162$	−8.60812	−0.676318
$163$	−1.76629	−0.138347	−0.0691733	−	0.997605i	$-0.522036\pi$
$163$	−1.76629	−0.138347	−0.0691733	+	0.997605i	$0.522036\pi$
$164$	−4.26718	−0.333210
$165$	0.419198	0.0326345
$166$	−2.46156	−0.191054
$167$	8.18746	0.633565	0.316782	−	0.948498i	$-0.397397\pi$
$167$	8.18746	0.633565	0.316782	+	0.948498i	$0.397397\pi$
$168$	−6.08944	−0.469810
$169$	3.42774	0.263672
$170$	0.145359	0.0111485
$171$	39.0038	2.98270
$172$	−6.92155	−0.527763
$173$	−11.1436	−0.847235	−0.423617	−	0.905841i	$-0.639240\pi$
$173$	−11.1436	−0.847235	−0.423617	+	0.905841i	$0.639240\pi$
$174$	−28.6810	−2.17430
$175$	10.1645	0.768363
$176$	2.06793	0.155876
$177$	−15.3803	−1.15605
$178$	15.0457	1.12773
$179$	−13.9831	−1.04515	−0.522573	−	0.852594i	$-0.675028\pi$
$179$	−13.9831	−1.04515	−0.522573	+	0.852594i	$0.675028\pi$
$180$	−0.403453	−0.0300716
$181$	−1.18370	−0.0879838	−0.0439919	−	0.999032i	$-0.514008\pi$
$181$	−1.18370	−0.0879838	−0.0439919	+	0.999032i	$0.514008\pi$
$182$	8.24714	0.611319
$183$	22.7099	1.67876
$184$	−7.28115	−0.536774
$185$	−0.275849	−0.0202808
$186$	−21.1656	−1.55194
$187$	4.43771	0.324517
$188$	−4.25664	−0.310447
$189$	18.0019	1.30944
$190$	0.443562	0.0321794
$191$	5.60255	0.405386	0.202693	−	0.979242i	$-0.435031\pi$
$191$	5.60255	0.405386	0.202693	+	0.979242i	$0.435031\pi$
$192$	−2.99270	−0.215979
$193$	3.32006	0.238983	0.119492	−	0.992835i	$-0.461874\pi$
$193$	3.32006	0.238983	0.119492	+	0.992835i	$0.461874\pi$
$194$	4.48255	0.321828
$195$	0.821622	0.0588375
$196$	−2.85973	−0.204266
$197$	−18.9372	−1.34922	−0.674608	−	0.738176i	$-0.735687\pi$
$197$	−18.9372	−1.34922	−0.674608	+	0.738176i	$0.735687\pi$
$198$	−12.3171	−0.875339
$199$	13.3701	0.947780	0.473890	−	0.880584i	$-0.342849\pi$
$199$	13.3701	0.947780	0.473890	+	0.880584i	$0.342849\pi$
$200$	4.99541	0.353229
$201$	−3.05610	−0.215560
$202$	−4.98904	−0.351028
$203$	19.5005	1.36867
$204$	−6.42222	−0.449645
$205$	0.289042	0.0201875
$206$	−9.98597	−0.695755
$207$	43.3683	3.01431
$208$	4.05312	0.281033
$209$	13.5416	0.936694
$210$	0.412475	0.0284635
$211$	4.63518	0.319099	0.159550	−	0.987190i	$-0.448996\pi$
$211$	4.63518	0.319099	0.159550	+	0.987190i	$0.448996\pi$
$212$	−5.40084	−0.370931
$213$	−33.2227	−2.27638
$214$	−8.16003	−0.557808
$215$	0.468838	0.0319745
$216$	8.84715	0.601972
$217$	14.3907	0.976904
$218$	−10.3712	−0.702428
$219$	−38.1921	−2.58078
$220$	−0.140074	−0.00944376
$221$	8.69784	0.585080
$222$	12.1875	0.817971
$223$	−9.88158	−0.661719	−0.330860	−	0.943680i	$-0.607339\pi$
$223$	−9.88158	−0.661719	−0.330860	+	0.943680i	$0.607339\pi$
$224$	2.03477	0.135953
$225$	−29.7539	−1.98359
$226$	7.46512	0.496572
$227$	18.0229	1.19622	0.598112	−	0.801412i	$-0.295918\pi$
$227$	18.0229	1.19622	0.598112	+	0.801412i	$0.295918\pi$
$228$	−19.5974	−1.29787
$229$	−29.5601	−1.95339	−0.976693	−	0.214640i	$-0.931142\pi$
$229$	−29.5601	−1.95339	−0.976693	+	0.214640i	$0.931142\pi$
$230$	0.493197	0.0325204
$231$	12.5926	0.828529
$232$	9.58367	0.629199
$233$	−5.74634	−0.376456	−0.188228	−	0.982125i	$-0.560274\pi$
$233$	−5.74634	−0.376456	−0.188228	+	0.982125i	$0.560274\pi$
$234$	−24.1413	−1.57817
$235$	0.288328	0.0188084
$236$	5.13928	0.334538
$237$	−0.346881	−0.0225323
$238$	4.36653	0.283040
$239$	20.2666	1.31094	0.655469	−	0.755222i	$-0.272471\pi$
$239$	20.2666	1.31094	0.655469	+	0.755222i	$0.272471\pi$
$240$	0.202714	0.0130851
$241$	−29.6342	−1.90891	−0.954453	−	0.298363i	$-0.903560\pi$
$241$	−29.6342	−1.90891	−0.954453	+	0.298363i	$0.903560\pi$
$242$	6.72365	0.432213
$243$	0.779939	0.0500331
$244$	−7.58843	−0.485800
$245$	0.193707	0.0123755
$246$	−12.7704	−0.814209
$247$	26.5414	1.68879
$248$	7.07241	0.449098
$249$	−7.36671	−0.466846
$250$	−0.677050	−0.0428204
$251$	−11.4335	−0.721676	−0.360838	−	0.932629i	$-0.617509\pi$
$251$	−11.4335	−0.721676	−0.360838	+	0.932629i	$0.617509\pi$
$252$	−12.1196	−0.763461
$253$	15.0569	0.946622
$254$	12.6262	0.792238
$255$	0.435016	0.0272418
$256$	1.00000	0.0625000
$257$	8.34716	0.520682	0.260341	−	0.965517i	$-0.416165\pi$
$257$	8.34716	0.520682	0.260341	+	0.965517i	$0.416165\pi$
$258$	−20.7141	−1.28960
$259$	−8.28640	−0.514892
$260$	−0.274542	−0.0170264
$261$	−57.0827	−3.53333
$262$	16.5516	1.02256
$263$	4.08590	0.251948	0.125974	−	0.992034i	$-0.459794\pi$
$263$	4.08590	0.251948	0.125974	+	0.992034i	$0.459794\pi$
$264$	6.18870	0.380888
$265$	0.365832	0.0224729
$266$	13.3244	0.816974
$267$	45.0274	2.75563
$268$	1.02118	0.0623788
$269$	−3.02764	−0.184598	−0.0922991	−	0.995731i	$-0.529422\pi$
$269$	−3.02764	−0.184598	−0.0922991	+	0.995731i	$0.529422\pi$
$270$	−0.599271	−0.0364705
$271$	−31.0409	−1.88560	−0.942801	−	0.333357i	$-0.891818\pi$
$271$	−31.0409	−1.88560	−0.942801	+	0.333357i	$0.891818\pi$
$272$	2.14596	0.130118
$273$	24.6812	1.49377
$274$	−4.91138	−0.296707
$275$	−10.3302	−0.622933
$276$	−21.7903	−1.31162
$277$	14.0195	0.842351	0.421176	−	0.906979i	$-0.361618\pi$
$277$	14.0195	0.842351	0.421176	+	0.906979i	$0.361618\pi$
$278$	3.70390	0.222145
$279$	−42.1250	−2.52196
$280$	−0.137827	−0.00823674
$281$	0.243472	0.0145243	0.00726215	−	0.999974i	$-0.497688\pi$
$281$	0.243472	0.0145243	0.00726215	+	0.999974i	$0.497688\pi$
$282$	−12.7388	−0.758587
$283$	15.4261	0.916984	0.458492	−	0.888699i	$-0.348390\pi$
$283$	15.4261	0.916984	0.458492	+	0.888699i	$0.348390\pi$
$284$	11.1012	0.658738
$285$	1.32745	0.0786312
$286$	−8.38157	−0.495613
$287$	8.68270	0.512524
$288$	−5.95625	−0.350975
$289$	−12.3948	−0.729108
$290$	−0.649160	−0.0381200
$291$	13.4149	0.786396
$292$	12.7617	0.746825
$293$	−25.5963	−1.49535	−0.747676	−	0.664064i	$-0.768830\pi$
$293$	−25.5963	−1.49535	−0.747676	+	0.664064i	$0.768830\pi$
$294$	−8.55831	−0.499131
$295$	−0.348114	−0.0202680
$296$	−4.07241	−0.236704
$297$	−18.2953	−1.06160
$298$	−12.9062	−0.747634
$299$	29.5114	1.70669
$300$	14.9498	0.863125
$301$	14.0837	0.811772
$302$	12.6540	0.728153
$303$	−14.9307	−0.857746
$304$	6.54839	0.375576
$305$	0.514011	0.0294322
$306$	−12.7819	−0.730692
$307$	−8.26620	−0.471777	−0.235889	−	0.971780i	$-0.575800\pi$
$307$	−8.26620	−0.471777	−0.235889	+	0.971780i	$0.575800\pi$
$308$	−4.20776	−0.239759
$309$	−29.8850	−1.70010
$310$	−0.479057	−0.0272086
$311$	13.3408	0.756489	0.378244	−	0.925706i	$-0.376528\pi$
$311$	13.3408	0.756489	0.378244	+	0.925706i	$0.376528\pi$
$312$	12.1298	0.686712
$313$	4.75811	0.268944	0.134472	−	0.990917i	$-0.457066\pi$
$313$	4.75811	0.268944	0.134472	+	0.990917i	$0.457066\pi$
$314$	−13.4859	−0.761052
$315$	0.820931	0.0462542
$316$	0.115909	0.00652040
$317$	−0.642703	−0.0360978	−0.0180489	−	0.999837i	$-0.505745\pi$
$317$	−0.642703	−0.0360978	−0.0180489	+	0.999837i	$0.505745\pi$
$318$	−16.1631	−0.906381
$319$	−19.8184	−1.10962
$320$	−0.0677361	−0.00378656
$321$	−24.4205	−1.36302
$322$	14.8154	0.825632
$323$	14.0526	0.781908
$324$	8.60812	0.478229
$325$	−20.2470	−1.12310
$326$	1.76629	0.0978258
$327$	−31.0380	−1.71640
$328$	4.26718	0.235615
$329$	8.66126	0.477511
$330$	−0.419198	−0.0230761
$331$	15.8514	0.871272	0.435636	−	0.900123i	$-0.356523\pi$
$331$	15.8514	0.871272	0.435636	+	0.900123i	$0.356523\pi$
$332$	2.46156	0.135096
$333$	24.2563	1.32924
$334$	−8.18746	−0.447998
$335$	−0.0691710	−0.00377922
$336$	6.08944	0.332206
$337$	1.27879	0.0696601	0.0348300	−	0.999393i	$-0.488911\pi$
$337$	1.27879	0.0696601	0.0348300	+	0.999393i	$0.488911\pi$
$338$	−3.42774	−0.186445
$339$	22.3408	1.21339
$340$	−0.145359	−0.00788320
$341$	−14.6253	−0.792003
$342$	−39.0038	−2.10908
$343$	20.0622	1.08326
$344$	6.92155	0.373185
$345$	1.47599	0.0794646
$346$	11.1436	0.599086
$347$	33.7201	1.81019	0.905094	−	0.425212i	$-0.139800\pi$
$347$	33.7201	1.81019	0.905094	+	0.425212i	$0.139800\pi$
$348$	28.6810	1.53746
$349$	20.2075	1.08168	0.540842	−	0.841124i	$-0.318106\pi$
$349$	20.2075	1.08168	0.540842	+	0.841124i	$0.318106\pi$
$350$	−10.1645	−0.543315
$351$	−35.8585	−1.91399
$352$	−2.06793	−0.110221
$353$	7.43432	0.395689	0.197844	−	0.980233i	$-0.436606\pi$
$353$	7.43432	0.395689	0.197844	+	0.980233i	$0.436606\pi$
$354$	15.3803	0.817454
$355$	−0.751955	−0.0399096
$356$	−15.0457	−0.797423
$357$	13.0677	0.691617
$358$	13.9831	0.739030
$359$	27.4887	1.45080	0.725400	−	0.688328i	$-0.241655\pi$
$359$	27.4887	1.45080	0.725400	+	0.688328i	$0.241655\pi$
$360$	0.403453	0.0212638
$361$	23.8814	1.25692
$362$	1.18370	0.0622139
$363$	20.1219	1.05612
$364$	−8.24714	−0.432268
$365$	−0.864430	−0.0452464
$366$	−22.7099	−1.18707
$367$	−28.4350	−1.48430	−0.742148	−	0.670236i	$-0.766193\pi$
$367$	−28.4350	−1.48430	−0.742148	+	0.670236i	$0.766193\pi$
$368$	7.28115	0.379556
$369$	−25.4163	−1.32312
$370$	0.275849	0.0143407
$371$	10.9894	0.570543
$372$	21.1656	1.09738
$373$	12.6934	0.657242	0.328621	−	0.944462i	$-0.393416\pi$
$373$	12.6934	0.657242	0.328621	+	0.944462i	$0.393416\pi$
$374$	−4.43771	−0.229468
$375$	−2.02621	−0.104633
$376$	4.25664	0.219519
$377$	−38.8437	−2.00055
$378$	−18.0019	−0.925917
$379$	9.97790	0.512531	0.256265	−	0.966606i	$-0.417508\pi$
$379$	9.97790	0.512531	0.256265	+	0.966606i	$0.417508\pi$
$380$	−0.443562	−0.0227543
$381$	37.7864	1.93586
$382$	−5.60255	−0.286651
$383$	−16.9324	−0.865204	−0.432602	−	0.901585i	$-0.642404\pi$
$383$	−16.9324	−0.865204	−0.432602	+	0.901585i	$0.642404\pi$
$384$	2.99270	0.152721
$385$	0.285017	0.0145258
$386$	−3.32006	−0.168987
$387$	−41.2264	−2.09566
$388$	−4.48255	−0.227567
$389$	−14.4597	−0.733134	−0.366567	−	0.930392i	$-0.619467\pi$
$389$	−14.4597	−0.733134	−0.366567	+	0.930392i	$0.619467\pi$
$390$	−0.821622	−0.0416044
$391$	15.6251	0.790195
$392$	2.85973	0.144438
$393$	49.5339	2.49865
$394$	18.9372	0.954040
$395$	−0.00785123	−0.000395038 0
$396$	12.3171	0.618958
$397$	−26.1886	−1.31437	−0.657184	−	0.753730i	$-0.728253\pi$
$397$	−26.1886	−1.31437	−0.657184	+	0.753730i	$0.728253\pi$
$398$	−13.3701	−0.670182
$399$	39.8760	1.99630
$400$	−4.99541	−0.249771
$401$	4.47741	0.223591	0.111796	−	0.993731i	$-0.464340\pi$
$401$	4.47741	0.223591	0.111796	+	0.993731i	$0.464340\pi$
$402$	3.05610	0.152424
$403$	−28.6653	−1.42792
$404$	4.98904	0.248214
$405$	−0.583080	−0.0289735
$406$	−19.5005	−0.967794
$407$	8.42147	0.417437
$408$	6.42222	0.317947
$409$	−15.9218	−0.787282	−0.393641	−	0.919264i	$-0.628785\pi$
$409$	−15.9218	−0.787282	−0.393641	+	0.919264i	$0.628785\pi$
$410$	−0.289042	−0.0142747
$411$	−14.6983	−0.725013
$412$	9.98597	0.491973
$413$	−10.4572	−0.514566
$414$	−43.3683	−2.13144
$415$	−0.166736	−0.00818477
$416$	−4.05312	−0.198720
$417$	11.0846	0.542818
$418$	−13.5416	−0.662343
$419$	−28.3243	−1.38373	−0.691867	−	0.722025i	$-0.743212\pi$
$419$	−28.3243	−1.38373	−0.691867	+	0.722025i	$0.743212\pi$
$420$	−0.412475	−0.0201267
$421$	31.6110	1.54063	0.770314	−	0.637665i	$-0.220100\pi$
$421$	31.6110	1.54063	0.770314	+	0.637665i	$0.220100\pi$
$422$	−4.63518	−0.225637
$423$	−25.3536	−1.23273
$424$	5.40084	0.262288
$425$	−10.7200	−0.519995
$426$	33.2227	1.60964
$427$	15.4407	0.747227
$428$	8.16003	0.394430
$429$	−25.0835	−1.21104
$430$	−0.468838	−0.0226094
$431$	−31.6842	−1.52618	−0.763088	−	0.646295i	$-0.776318\pi$
$431$	−31.6842	−1.52618	−0.763088	+	0.646295i	$0.776318\pi$
$432$	−8.84715	−0.425659
$433$	−36.6970	−1.76355	−0.881773	−	0.471674i	$-0.843650\pi$
$433$	−36.6970	−1.76355	−0.881773	+	0.471674i	$0.843650\pi$
$434$	−14.3907	−0.690775
$435$	−1.94274	−0.0931472
$436$	10.3712	0.496692
$437$	47.6798	2.28084
$438$	38.1921	1.82489
$439$	8.72882	0.416604	0.208302	−	0.978065i	$-0.433206\pi$
$439$	8.72882	0.416604	0.208302	+	0.978065i	$0.433206\pi$
$440$	0.140074	0.00667775
$441$	−17.0333	−0.811107
$442$	−8.69784	−0.413714
$443$	−5.01358	−0.238202	−0.119101	−	0.992882i	$-0.538001\pi$
$443$	−5.01358	−0.238202	−0.119101	+	0.992882i	$0.538001\pi$
$444$	−12.1875	−0.578393
$445$	1.01914	0.0483118
$446$	9.88158	0.467906
$447$	−38.6243	−1.82687
$448$	−2.03477	−0.0961336
$449$	−17.5458	−0.828036	−0.414018	−	0.910269i	$-0.635875\pi$
$449$	−17.5458	−0.828036	−0.414018	+	0.910269i	$0.635875\pi$
$450$	29.7539	1.40261
$451$	−8.82423	−0.415517
$452$	−7.46512	−0.351130
$453$	37.8695	1.77926
$454$	−18.0229	−0.845858
$455$	0.558629	0.0261889
$456$	19.5974	0.917730
$457$	−20.1369	−0.941966	−0.470983	−	0.882142i	$-0.656101\pi$
$457$	−20.1369	−0.941966	−0.470983	+	0.882142i	$0.656101\pi$
$458$	29.5601	1.38125
$459$	−18.9857	−0.886175
$460$	−0.493197	−0.0229954
$461$	16.1037	0.750026	0.375013	−	0.927020i	$-0.377638\pi$
$461$	16.1037	0.750026	0.375013	+	0.927020i	$0.377638\pi$
$462$	−12.5926	−0.585859
$463$	30.5546	1.41999	0.709996	−	0.704206i	$-0.248697\pi$
$463$	30.5546	1.41999	0.709996	+	0.704206i	$0.248697\pi$
$464$	−9.58367	−0.444911
$465$	−1.43367	−0.0664850
$466$	5.74634	0.266194
$467$	−14.9703	−0.692741	−0.346370	−	0.938098i	$-0.612586\pi$
$467$	−14.9703	−0.692741	−0.346370	+	0.938098i	$0.612586\pi$
$468$	24.1413	1.11593
$469$	−2.07787	−0.0959472
$470$	−0.288328	−0.0132996
$471$	−40.3592	−1.85965
$472$	−5.13928	−0.236554
$473$	−14.3133	−0.658126
$474$	0.346881	0.0159328
$475$	−32.7119	−1.50093
$476$	−4.36653	−0.200140
$477$	−32.1687	−1.47290
$478$	−20.2666	−0.926973
$479$	31.4681	1.43781	0.718907	−	0.695106i	$-0.244643\pi$
$479$	31.4681	1.43781	0.718907	+	0.695106i	$0.244643\pi$
$480$	−0.202714	−0.00925257
$481$	16.5059	0.752606
$482$	29.6342	1.34980
$483$	44.3382	2.01746
$484$	−6.72365	−0.305621
$485$	0.303630	0.0137871
$486$	−0.779939	−0.0353788
$487$	−14.8080	−0.671015	−0.335508	−	0.942037i	$-0.608908\pi$
$487$	−14.8080	−0.671015	−0.335508	+	0.942037i	$0.608908\pi$
$488$	7.58843	0.343512
$489$	5.28597	0.239040
$490$	−0.193707	−0.00875078
$491$	−29.9936	−1.35359	−0.676796	−	0.736170i	$-0.736632\pi$
$491$	−29.9936	−1.35359	−0.676796	+	0.736170i	$0.736632\pi$
$492$	12.7704	0.575733
$493$	−20.5662	−0.926255
$494$	−26.5414	−1.19415
$495$	−0.834313	−0.0374996
$496$	−7.07241	−0.317561
$497$	−22.5884	−1.01323
$498$	7.36671	0.330110
$499$	−22.8285	−1.02194	−0.510971	−	0.859598i	$-0.670714\pi$
$499$	−22.8285	−1.02194	−0.510971	+	0.859598i	$0.670714\pi$
$500$	0.677050	0.0302786
$501$	−24.5026	−1.09470
$502$	11.4335	0.510302
$503$	25.8604	1.15306	0.576528	−	0.817077i	$-0.304407\pi$
$503$	25.8604	1.15306	0.576528	+	0.817077i	$0.304407\pi$
$504$	12.1196	0.539848
$505$	−0.337938	−0.0150380
$506$	−15.0569	−0.669363
$507$	−10.2582	−0.455583
$508$	−12.6262	−0.560197
$509$	0.666585	0.0295459	0.0147729	−	0.999891i	$-0.495297\pi$
$509$	0.666585	0.0295459	0.0147729	+	0.999891i	$0.495297\pi$
$510$	−0.435016	−0.0192628
$511$	−25.9672	−1.14872
$512$	−1.00000	−0.0441942
$513$	−57.9346	−2.55787
$514$	−8.34716	−0.368178
$515$	−0.676410	−0.0298062
$516$	20.7141	0.911888
$517$	−8.80245	−0.387131
$518$	8.28640	0.364083
$519$	33.3495	1.46388
$520$	0.274542	0.0120395
$521$	31.2015	1.36696	0.683481	−	0.729968i	$-0.260465\pi$
$521$	31.2015	1.36696	0.683481	+	0.729968i	$0.260465\pi$
$522$	57.0827	2.49844
$523$	−13.5286	−0.591565	−0.295782	−	0.955255i	$-0.595580\pi$
$523$	−13.5286	−0.591565	−0.295782	+	0.955255i	$0.595580\pi$
$524$	−16.5516	−0.723059
$525$	−30.4193	−1.32761
$526$	−4.08590	−0.178154
$527$	−15.1771	−0.661126
$528$	−6.18870	−0.269329
$529$	30.0152	1.30501
$530$	−0.365832	−0.0158907
$531$	30.6108	1.32840
$532$	−13.3244	−0.577688
$533$	−17.2954	−0.749145
$534$	−45.0274	−1.94853
$535$	−0.552728	−0.0238965
$536$	−1.02118	−0.0441084
$537$	41.8472	1.80584
$538$	3.02764	0.130531
$539$	−5.91373	−0.254722
$540$	0.599271	0.0257885
$541$	−24.5845	−1.05697	−0.528486	−	0.848942i	$-0.677240\pi$
$541$	−24.5845	−1.05697	−0.528486	+	0.848942i	$0.677240\pi$
$542$	31.0409	1.33332
$543$	3.54246	0.152022
$544$	−2.14596	−0.0920074
$545$	−0.702506	−0.0300921
$546$	−24.6812	−1.05626
$547$	−16.0205	−0.684985	−0.342493	−	0.939521i	$-0.611271\pi$
$547$	−16.0205	−0.684985	−0.342493	+	0.939521i	$0.611271\pi$
$548$	4.91138	0.209804
$549$	−45.1986	−1.92903
$550$	10.3302	0.440480
$551$	−62.7576	−2.67356
$552$	21.7903	0.927457
$553$	−0.235848	−0.0100293
$554$	−14.0195	−0.595632
$555$	0.825533	0.0350419
$556$	−3.70390	−0.157080
$557$	−14.6979	−0.622772	−0.311386	−	0.950284i	$-0.600793\pi$
$557$	−14.6979	−0.622772	−0.311386	+	0.950284i	$0.600793\pi$
$558$	42.1250	1.78329
$559$	−28.0538	−1.18655
$560$	0.137827	0.00582425
$561$	−13.2807	−0.560713
$562$	−0.243472	−0.0102702
$563$	5.94684	0.250629	0.125315	−	0.992117i	$-0.460006\pi$
$563$	5.94684	0.250629	0.125315	+	0.992117i	$0.460006\pi$
$564$	12.7388	0.536402
$565$	0.505658	0.0212732
$566$	−15.4261	−0.648406
$567$	−17.5155	−0.735582
$568$	−11.1012	−0.465798
$569$	−8.54371	−0.358171	−0.179086	−	0.983834i	$-0.557314\pi$
$569$	−8.54371	−0.358171	−0.179086	+	0.983834i	$0.557314\pi$
$570$	−1.32745	−0.0556007
$571$	−1.92265	−0.0804605	−0.0402303	−	0.999190i	$-0.512809\pi$
$571$	−1.92265	−0.0804605	−0.0402303	+	0.999190i	$0.512809\pi$
$572$	8.38157	0.350451
$573$	−16.7667	−0.700441
$574$	−8.68270	−0.362409
$575$	−36.3724	−1.51683
$576$	5.95625	0.248177
$577$	31.8711	1.32681	0.663405	−	0.748261i	$-0.269111\pi$
$577$	31.8711	1.32681	0.663405	+	0.748261i	$0.269111\pi$
$578$	12.3948	0.515557
$579$	−9.93595	−0.412924
$580$	0.649160	0.0269549
$581$	−5.00870	−0.207796
$582$	−13.4149	−0.556066
$583$	−11.1686	−0.462555
$584$	−12.7617	−0.528085
$585$	−1.63524	−0.0676088
$586$	25.5963	1.05737
$587$	43.2906	1.78679	0.893396	−	0.449269i	$-0.148316\pi$
$587$	43.2906	1.78679	0.893396	+	0.449269i	$0.148316\pi$
$588$	8.55831	0.352939
$589$	−46.3129	−1.90829
$590$	0.348114	0.0143316
$591$	56.6732	2.33122
$592$	4.07241	0.167375
$593$	29.8241	1.22473	0.612364	−	0.790576i	$-0.290219\pi$
$593$	29.8241	1.22473	0.612364	+	0.790576i	$0.290219\pi$
$594$	18.2953	0.750666
$595$	0.295772	0.0121255
$596$	12.9062	0.528657
$597$	−40.0126	−1.63761
$598$	−29.5114	−1.20681
$599$	−7.91073	−0.323223	−0.161612	−	0.986854i	$-0.551669\pi$
$599$	−7.91073	−0.323223	−0.161612	+	0.986854i	$0.551669\pi$
$600$	−14.9498	−0.610321
$601$	13.9827	0.570364	0.285182	−	0.958473i	$-0.407946\pi$
$601$	13.9827	0.570364	0.285182	+	0.958473i	$0.407946\pi$
$602$	−14.0837	−0.574010
$603$	6.08242	0.247695
$604$	−12.6540	−0.514882
$605$	0.455434	0.0185160
$606$	14.9307	0.606518
$607$	5.37606	0.218207	0.109104	−	0.994030i	$-0.465202\pi$
$607$	5.37606	0.218207	0.109104	+	0.994030i	$0.465202\pi$
$608$	−6.54839	−0.265572
$609$	−58.3592	−2.36483
$610$	−0.514011	−0.0208117
$611$	−17.2527	−0.697968
$612$	12.7819	0.516677
$613$	−45.6770	−1.84487	−0.922437	−	0.386146i	$-0.873806\pi$
$613$	−45.6770	−1.84487	−0.922437	+	0.386146i	$0.873806\pi$
$614$	8.26620	0.333597
$615$	−0.865015	−0.0348808
$616$	4.20776	0.169535
$617$	21.4929	0.865272	0.432636	−	0.901569i	$-0.357584\pi$
$617$	21.4929	0.865272	0.432636	+	0.901569i	$0.357584\pi$
$618$	29.8850	1.20215
$619$	−13.2377	−0.532069	−0.266035	−	0.963963i	$-0.585714\pi$
$619$	−13.2377	−0.532069	−0.266035	+	0.963963i	$0.585714\pi$
$620$	0.479057	0.0192394
$621$	−64.4175	−2.58498
$622$	−13.3408	−0.534918
$623$	30.6146	1.22655
$624$	−12.1298	−0.485579
$625$	24.9312	0.997248
$626$	−4.75811	−0.190172
$627$	−40.5260	−1.61845
$628$	13.4859	0.538145
$629$	8.73924	0.348456
$630$	−0.820931	−0.0327067
$631$	−1.98666	−0.0790876	−0.0395438	−	0.999218i	$-0.512590\pi$
$631$	−1.98666	−0.0790876	−0.0395438	+	0.999218i	$0.512590\pi$
$632$	−0.115909	−0.00461062
$633$	−13.8717	−0.551351
$634$	0.642703	0.0255250
$635$	0.855249	0.0339395
$636$	16.1631	0.640908
$637$	−11.5908	−0.459245
$638$	19.8184	0.784617
$639$	66.1218	2.61574
$640$	0.0677361	0.00267750
$641$	−36.8438	−1.45524	−0.727621	−	0.685980i	$-0.759374\pi$
$641$	−36.8438	−1.45524	−0.727621	+	0.685980i	$0.759374\pi$
$642$	24.4205	0.963800
$643$	−10.2737	−0.405156	−0.202578	−	0.979266i	$-0.564932\pi$
$643$	−10.2737	−0.405156	−0.202578	+	0.979266i	$0.564932\pi$
$644$	−14.8154	−0.583810
$645$	−1.40309	−0.0552467
$646$	−14.0526	−0.552892
$647$	−19.7168	−0.775146	−0.387573	−	0.921839i	$-0.626687\pi$
$647$	−19.7168	−0.775146	−0.387573	+	0.921839i	$0.626687\pi$
$648$	−8.60812	−0.338159
$649$	10.6277	0.417173
$650$	20.2470	0.794152
$651$	−43.0670	−1.68793
$652$	−1.76629	−0.0691733
$653$	−9.10604	−0.356347	−0.178173	−	0.983999i	$-0.557019\pi$
$653$	−9.10604	−0.356347	−0.178173	+	0.983999i	$0.557019\pi$
$654$	31.0380	1.21368
$655$	1.12114	0.0438065
$656$	−4.26718	−0.166605
$657$	76.0121	2.96551
$658$	−8.66126	−0.337651
$659$	24.6397	0.959825	0.479912	−	0.877316i	$-0.340668\pi$
$659$	24.6397	0.959825	0.479912	+	0.877316i	$0.340668\pi$
$660$	0.419198	0.0163173
$661$	−34.1497	−1.32827	−0.664134	−	0.747613i	$-0.731200\pi$
$661$	−34.1497	−1.32827	−0.664134	+	0.747613i	$0.731200\pi$
$662$	−15.8514	−0.616082
$663$	−26.0300	−1.01092
$664$	−2.46156	−0.0955271
$665$	0.902545	0.0349992
$666$	−24.2563	−0.939911
$667$	−69.7802	−2.70190
$668$	8.18746	0.316782
$669$	29.5726	1.14334
$670$	0.0691710	0.00267231
$671$	−15.6924	−0.605797
$672$	−6.08944	−0.234905
$673$	−38.8059	−1.49586	−0.747930	−	0.663778i	$-0.768952\pi$
$673$	−38.8059	−1.49586	−0.747930	+	0.663778i	$0.768952\pi$
$674$	−1.27879	−0.0492571
$675$	44.1952	1.70107
$676$	3.42774	0.131836
$677$	0.0582366	0.00223821	0.00111911	−	0.999999i	$-0.499644\pi$
$677$	0.0582366	0.00223821	0.00111911	+	0.999999i	$0.499644\pi$
$678$	−22.3408	−0.857995
$679$	9.12094	0.350029
$680$	0.145359	0.00557427
$681$	−53.9372	−2.06688
$682$	14.6253	0.560030
$683$	26.2945	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$683$	26.2945	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$684$	39.0038	1.49135
$685$	−0.332677	−0.0127110
$686$	−20.0622	−0.765980
$687$	88.4645	3.37513
$688$	−6.92155	−0.263882
$689$	−21.8902	−0.833951
$690$	−1.47599	−0.0561899
$691$	−11.7417	−0.446675	−0.223338	−	0.974741i	$-0.571695\pi$
$691$	−11.7417	−0.446675	−0.223338	+	0.974741i	$0.571695\pi$
$692$	−11.1436	−0.423617
$693$	−25.0624	−0.952043
$694$	−33.7201	−1.28000
$695$	0.250887	0.00951670
$696$	−28.6810	−1.08715
$697$	−9.15720	−0.346854
$698$	−20.2075	−0.764866
$699$	17.1971	0.650453
$700$	10.1645	0.384182
$701$	−30.6049	−1.15593	−0.577966	−	0.816061i	$-0.696153\pi$
$701$	−30.6049	−1.15593	−0.577966	+	0.816061i	$0.696153\pi$
$702$	35.8585	1.35339
$703$	26.6677	1.00579
$704$	2.06793	0.0779382
$705$	−0.862879	−0.0324979
$706$	−7.43432	−0.279794
$707$	−10.1515	−0.381787
$708$	−15.3803	−0.578027
$709$	19.5777	0.735256	0.367628	−	0.929973i	$-0.380170\pi$
$709$	19.5777	0.735256	0.367628	+	0.929973i	$0.380170\pi$
$710$	0.751955	0.0282204
$711$	0.690383	0.0258914
$712$	15.0457	0.563863
$713$	−51.4953	−1.92851
$714$	−13.0677	−0.489047
$715$	−0.567735	−0.0212321
$716$	−13.9831	−0.522573
$717$	−60.6518	−2.26508
$718$	−27.4887	−1.02587
$719$	1.36823	0.0510264	0.0255132	−	0.999674i	$-0.491878\pi$
$719$	1.36823	0.0510264	0.0255132	+	0.999674i	$0.491878\pi$
$720$	−0.403453	−0.0150358
$721$	−20.3191	−0.756723
$722$	−23.8814	−0.888774
$723$	88.6862	3.29827
$724$	−1.18370	−0.0439919
$725$	47.8744	1.77801
$726$	−20.1219	−0.746793
$727$	30.8679	1.14483	0.572413	−	0.819965i	$-0.306007\pi$
$727$	30.8679	1.14483	0.572413	+	0.819965i	$0.306007\pi$
$728$	8.24714	0.305659
$729$	−28.1585	−1.04291
$730$	0.864430	0.0319940
$731$	−14.8534	−0.549372
$732$	22.7099	0.839382
$733$	−18.0409	−0.666357	−0.333179	−	0.942864i	$-0.608121\pi$
$733$	−18.0409	−0.666357	−0.333179	+	0.942864i	$0.608121\pi$
$734$	28.4350	1.04956
$735$	−0.579706	−0.0213828
$736$	−7.28115	−0.268387
$737$	2.11174	0.0777870
$738$	25.4163	0.935589
$739$	−41.4612	−1.52518	−0.762588	−	0.646884i	$-0.776072\pi$
$739$	−41.4612	−1.52518	−0.762588	+	0.646884i	$0.776072\pi$
$740$	−0.275849	−0.0101404
$741$	−79.4303	−2.91795
$742$	−10.9894	−0.403435
$743$	39.3306	1.44290	0.721450	−	0.692466i	$-0.243476\pi$
$743$	39.3306	1.44290	0.721450	+	0.692466i	$0.243476\pi$
$744$	−21.1656	−0.775968
$745$	−0.874213	−0.0320287
$746$	−12.6934	−0.464740
$747$	14.6617	0.536442
$748$	4.43771	0.162259
$749$	−16.6037	−0.606687
$750$	2.02621	0.0739866
$751$	−10.0799	−0.367820	−0.183910	−	0.982943i	$-0.558875\pi$
$751$	−10.0799	−0.367820	−0.183910	+	0.982943i	$0.558875\pi$
$752$	−4.25664	−0.155224
$753$	34.2170	1.24694
$754$	38.8437	1.41460
$755$	0.857129	0.0311941
$756$	18.0019	0.654722
$757$	15.9540	0.579858	0.289929	−	0.957048i	$-0.406368\pi$
$757$	15.9540	0.579858	0.289929	+	0.957048i	$0.406368\pi$
$758$	−9.97790	−0.362414
$759$	−45.0609	−1.63561
$760$	0.443562	0.0160897
$761$	−19.1936	−0.695768	−0.347884	−	0.937538i	$-0.613100\pi$
$761$	−19.1936	−0.695768	−0.347884	+	0.937538i	$0.613100\pi$
$762$	−37.7864	−1.36886
$763$	−21.1030	−0.763981
$764$	5.60255	0.202693
$765$	−0.865794	−0.0313029
$766$	16.9324	0.611791
$767$	20.8301	0.752131
$768$	−2.99270	−0.107990
$769$	17.9026	0.645585	0.322793	−	0.946470i	$-0.395378\pi$
$769$	17.9026	0.645585	0.322793	+	0.946470i	$0.395378\pi$
$770$	−0.285017	−0.0102713
$771$	−24.9805	−0.899652
$772$	3.32006	0.119492
$773$	19.2907	0.693840	0.346920	−	0.937895i	$-0.387228\pi$
$773$	19.2907	0.693840	0.346920	+	0.937895i	$0.387228\pi$
$774$	41.2264	1.48185
$775$	35.3296	1.26908
$776$	4.48255	0.160914
$777$	24.7987	0.889648
$778$	14.4597	0.518404
$779$	−27.9431	−1.00117
$780$	0.821622	0.0294188
$781$	22.9566	0.821453
$782$	−15.6251	−0.558752
$783$	84.7882	3.03008
$784$	−2.85973	−0.102133
$785$	−0.913481	−0.0326035
$786$	−49.5339	−1.76681
$787$	25.6091	0.912865	0.456433	−	0.889758i	$-0.349127\pi$
$787$	25.6091	0.912865	0.456433	+	0.889758i	$0.349127\pi$
$788$	−18.9372	−0.674608
$789$	−12.2279	−0.435324
$790$	0.00785123	0.000279334 0
$791$	15.1898	0.540086
$792$	−12.3171	−0.437670
$793$	−30.7568	−1.09221
$794$	26.1886	0.929399
$795$	−1.09482	−0.0388294
$796$	13.3701	0.473890
$797$	−27.6356	−0.978903	−0.489452	−	0.872030i	$-0.662803\pi$
$797$	−27.6356	−0.978903	−0.489452	+	0.872030i	$0.662803\pi$
$798$	−39.8760	−1.41160
$799$	−9.13459	−0.323159
$800$	4.99541	0.176614
$801$	−89.6161	−3.16643
$802$	−4.47741	−0.158103
$803$	26.3904	0.931298
$804$	−3.05610	−0.107780
$805$	1.00354	0.0353701
$806$	28.6653	1.00969
$807$	9.06080	0.318955
$808$	−4.98904	−0.175514
$809$	−25.8265	−0.908012	−0.454006	−	0.890999i	$-0.650006\pi$
$809$	−25.8265	−0.908012	−0.454006	+	0.890999i	$0.650006\pi$
$810$	0.583080	0.0204874
$811$	39.5546	1.38895	0.694475	−	0.719517i	$-0.255637\pi$
$811$	39.5546	1.38895	0.694475	+	0.719517i	$0.255637\pi$
$812$	19.5005	0.684334
$813$	92.8961	3.25801
$814$	−8.42147	−0.295172
$815$	0.119642	0.00419086
$816$	−6.42222	−0.224823
$817$	−45.3250	−1.58572
$818$	15.9218	0.556692
$819$	−49.1220	−1.71646
$820$	0.289042	0.0100938
$821$	−13.7607	−0.480251	−0.240126	−	0.970742i	$-0.577189\pi$
$821$	−13.7607	−0.480251	−0.240126	+	0.970742i	$0.577189\pi$
$822$	14.6983	0.512661
$823$	18.3103	0.638257	0.319129	−	0.947711i	$-0.396610\pi$
$823$	18.3103	0.638257	0.319129	+	0.947711i	$0.396610\pi$
$824$	−9.98597	−0.347878
$825$	30.9151	1.07633
$826$	10.4572	0.363853
$827$	5.00769	0.174135	0.0870673	−	0.996202i	$-0.472250\pi$
$827$	5.00769	0.174135	0.0870673	+	0.996202i	$0.472250\pi$
$828$	43.3683	1.50715
$829$	1.92666	0.0669158	0.0334579	−	0.999440i	$-0.489348\pi$
$829$	1.92666	0.0669158	0.0334579	+	0.999440i	$0.489348\pi$
$830$	0.166736	0.00578751
$831$	−41.9562	−1.45544
$832$	4.05312	0.140516
$833$	−6.13688	−0.212630
$834$	−11.0846	−0.383830
$835$	−0.554586	−0.0191923
$836$	13.5416	0.468347
$837$	62.5707	2.16276
$838$	28.3243	0.978448
$839$	0.256649	0.00886050	0.00443025	−	0.999990i	$-0.498590\pi$
$839$	0.256649	0.00886050	0.00443025	+	0.999990i	$0.498590\pi$
$840$	0.412475	0.0142317
$841$	62.8467	2.16713
$842$	−31.6110	−1.08939
$843$	−0.728638	−0.0250956
$844$	4.63518	0.159550
$845$	−0.232182	−0.00798730
$846$	25.3536	0.871674
$847$	13.6811	0.470087
$848$	−5.40084	−0.185466
$849$	−46.1655	−1.58440
$850$	10.7200	0.367692
$851$	29.6518	1.01645
$852$	−33.2227	−1.13819
$853$	39.3371	1.34688	0.673439	−	0.739243i	$-0.264816\pi$
$853$	39.3371	1.34688	0.673439	+	0.739243i	$0.264816\pi$
$854$	−15.4407	−0.528369
$855$	−2.64196	−0.0903533
$856$	−8.16003	−0.278904
$857$	−7.40116	−0.252819	−0.126409	−	0.991978i	$-0.540345\pi$
$857$	−7.40116	−0.252819	−0.126409	+	0.991978i	$0.540345\pi$
$858$	25.0835	0.856337
$859$	27.1748	0.927192	0.463596	−	0.886047i	$-0.346559\pi$
$859$	27.1748	0.927192	0.463596	+	0.886047i	$0.346559\pi$
$860$	0.468838	0.0159873
$861$	−25.9847	−0.885556
$862$	31.6842	1.07917
$863$	31.4617	1.07097	0.535485	−	0.844545i	$-0.320129\pi$
$863$	31.4617	1.07097	0.535485	+	0.844545i	$0.320129\pi$
$864$	8.84715	0.300986
$865$	0.754826	0.0256649
$866$	36.6970	1.24702
$867$	37.0940	1.25978
$868$	14.3907	0.488452
$869$	0.239692	0.00813100
$870$	1.94274	0.0658650
$871$	4.13898	0.140244
$872$	−10.3712	−0.351214
$873$	−26.6992	−0.903630
$874$	−47.6798	−1.61279
$875$	−1.37764	−0.0465726
$876$	−38.1921	−1.29039
$877$	−19.5279	−0.659409	−0.329705	−	0.944084i	$-0.606949\pi$
$877$	−19.5279	−0.659409	−0.329705	+	0.944084i	$0.606949\pi$
$878$	−8.72882	−0.294583
$879$	76.6020	2.58372
$880$	−0.140074	−0.00472188
$881$	−22.4676	−0.756953	−0.378476	−	0.925611i	$-0.623552\pi$
$881$	−22.4676	−0.756953	−0.378476	+	0.925611i	$0.623552\pi$
$882$	17.0333	0.573539
$883$	−37.4440	−1.26009	−0.630045	−	0.776558i	$-0.716964\pi$
$883$	−37.4440	−1.26009	−0.630045	+	0.776558i	$0.716964\pi$
$884$	8.69784	0.292540
$885$	1.04180	0.0350198
$886$	5.01358	0.168435
$887$	−30.6229	−1.02822	−0.514109	−	0.857725i	$-0.671877\pi$
$887$	−30.6229	−1.02822	−0.514109	+	0.857725i	$0.671877\pi$
$888$	12.1875	0.408986
$889$	25.6914	0.861661
$890$	−1.01914	−0.0341616
$891$	17.8010	0.596357
$892$	−9.88158	−0.330860
$893$	−27.8741	−0.932772
$894$	38.6243	1.29179
$895$	0.947161	0.0316601
$896$	2.03477	0.0679767
$897$	−88.3186	−2.94887
$898$	17.5458	0.585510
$899$	67.7796	2.26058
$900$	−29.7539	−0.991797
$901$	−11.5900	−0.386119
$902$	8.82423	0.293815
$903$	−42.1483	−1.40261
$904$	7.46512	0.248286
$905$	0.0801792	0.00266525
$906$	−37.8695	−1.25813
$907$	−51.1953	−1.69991	−0.849956	−	0.526854i	$-0.823372\pi$
$907$	−51.1953	−1.69991	−0.849956	+	0.526854i	$0.823372\pi$
$908$	18.0229	0.598112
$909$	29.7159	0.985616
$910$	−0.558629	−0.0185184
$911$	−2.58794	−0.0857422	−0.0428711	−	0.999081i	$-0.513650\pi$
$911$	−2.58794	−0.0857422	−0.0428711	+	0.999081i	$0.513650\pi$
$912$	−19.5974	−0.648933
$913$	5.09034	0.168466
$914$	20.1369	0.666070
$915$	−1.53828	−0.0508539
$916$	−29.5601	−0.976693
$917$	33.6786	1.11216
$918$	18.9857	0.626620
$919$	−45.5641	−1.50302	−0.751510	−	0.659722i	$-0.770674\pi$
$919$	−45.5641	−1.50302	−0.751510	+	0.659722i	$0.770674\pi$
$920$	0.493197	0.0162602
$921$	24.7383	0.815153
$922$	−16.1037	−0.530348
$923$	44.9946	1.48102
$924$	12.5926	0.414265
$925$	−20.3434	−0.668886
$926$	−30.5546	−1.00409
$927$	59.4789	1.95354
$928$	9.58367	0.314599
$929$	33.8250	1.10976	0.554881	−	0.831929i	$-0.312764\pi$
$929$	33.8250	1.10976	0.554881	+	0.831929i	$0.312764\pi$
$930$	1.43367	0.0470120
$931$	−18.7266	−0.613740
$932$	−5.74634	−0.188228
$933$	−39.9251	−1.30709
$934$	14.9703	0.489842
$935$	−0.300593	−0.00983044
$936$	−24.1413	−0.789085
$937$	25.1201	0.820639	0.410319	−	0.911942i	$-0.365417\pi$
$937$	25.1201	0.820639	0.410319	+	0.911942i	$0.365417\pi$
$938$	2.07787	0.0678449
$939$	−14.2396	−0.464691
$940$	0.288328	0.00940422
$941$	14.2579	0.464793	0.232397	−	0.972621i	$-0.425343\pi$
$941$	14.2579	0.464793	0.232397	+	0.972621i	$0.425343\pi$
$942$	40.3592	1.31497
$943$	−31.0700	−1.01178
$944$	5.13928	0.167269
$945$	−1.21938	−0.0396663
$946$	14.3133	0.465365
$947$	16.7193	0.543303	0.271652	−	0.962396i	$-0.412430\pi$
$947$	16.7193	0.543303	0.271652	+	0.962396i	$0.412430\pi$
$948$	−0.346881	−0.0112662
$949$	51.7248	1.67906
$950$	32.7119	1.06131
$951$	1.92342	0.0623710
$952$	4.36653	0.141520
$953$	0.717637	0.0232466	0.0116233	−	0.999932i	$-0.496300\pi$
$953$	0.717637	0.0232466	0.0116233	+	0.999932i	$0.496300\pi$
$954$	32.1687	1.04150
$955$	−0.379495	−0.0122802
$956$	20.2666	0.655469
$957$	59.3104	1.91723
$958$	−31.4681	−1.01669
$959$	−9.99350	−0.322707
$960$	0.202714	0.00654255
$961$	19.0190	0.613515
$962$	−16.5059	−0.532173
$963$	48.6031	1.56621
$964$	−29.6342	−0.954453
$965$	−0.224888	−0.00723940
$966$	−44.3382	−1.42656
$967$	−35.7878	−1.15086	−0.575430	−	0.817851i	$-0.695165\pi$
$967$	−35.7878	−1.15086	−0.575430	+	0.817851i	$0.695165\pi$
$968$	6.72365	0.216106
$969$	−42.0552	−1.35101
$970$	−0.303630	−0.00974898
$971$	−48.6581	−1.56151	−0.780756	−	0.624836i	$-0.785166\pi$
$971$	−48.6581	−1.56151	−0.780756	+	0.624836i	$0.785166\pi$
$972$	0.779939	0.0250166
$973$	7.53656	0.241611
$974$	14.8080	0.474480
$975$	60.5931	1.94053
$976$	−7.58843	−0.242900
$977$	30.1605	0.964921	0.482460	−	0.875918i	$-0.339743\pi$
$977$	30.1605	0.964921	0.482460	+	0.875918i	$0.339743\pi$
$978$	−5.28597	−0.169027
$979$	−31.1136	−0.994395
$980$	0.193707	0.00618774
$981$	61.7736	1.97228
$982$	29.9936	0.957135
$983$	−49.3234	−1.57317	−0.786586	−	0.617481i	$-0.788153\pi$
$983$	−49.3234	−1.57317	−0.786586	+	0.617481i	$0.788153\pi$
$984$	−12.7704	−0.407105
$985$	1.28273	0.0408711
$986$	20.5662	0.654961
$987$	−25.9206	−0.825060
$988$	26.5414	0.844394
$989$	−50.3969	−1.60253
$990$	0.834313	0.0265162
$991$	−6.39300	−0.203080	−0.101540	−	0.994831i	$-0.532377\pi$
$991$	−6.39300	−0.203080	−0.101540	+	0.994831i	$0.532377\pi$
$992$	7.07241	0.224549
$993$	−47.4385	−1.50541
$994$	22.5884	0.716462
$995$	−0.905637	−0.0287106
$996$	−7.36671	−0.233423
$997$	−40.5833	−1.28528	−0.642642	−	0.766166i	$-0.722162\pi$
$997$	−40.5833	−1.28528	−0.642642	+	0.766166i	$0.722162\pi$
$998$	22.8285	0.722623
$999$	−36.0292	−1.13991

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8006.2.a.b.1.3	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8006.2.a.b.1.3	✓	75	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8006 = 2 \cdot 4003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8006.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.99270 q^{3} +1.00000 q^{4} -0.0677361 q^{5} +2.99270 q^{6} -2.03477 q^{7} -1.00000 q^{8} +5.95625 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.99270 q^{3} +1.00000 q^{4} -0.0677361 q^{5} +2.99270 q^{6} -2.03477 q^{7} -1.00000 q^{8} +5.95625 q^{9} +0.0677361 q^{10} +2.06793 q^{11} -2.99270 q^{12} +4.05312 q^{13} +2.03477 q^{14} +0.202714 q^{15} +1.00000 q^{16} +2.14596 q^{17} -5.95625 q^{18} +6.54839 q^{19} -0.0677361 q^{20} +6.08944 q^{21} -2.06793 q^{22} +7.28115 q^{23} +2.99270 q^{24} -4.99541 q^{25} -4.05312 q^{26} -8.84715 q^{27} -2.03477 q^{28} -9.58367 q^{29} -0.202714 q^{30} -7.07241 q^{31} -1.00000 q^{32} -6.18870 q^{33} -2.14596 q^{34} +0.137827 q^{35} +5.95625 q^{36} +4.07241 q^{37} -6.54839 q^{38} -12.1298 q^{39} +0.0677361 q^{40} -4.26718 q^{41} -6.08944 q^{42} -6.92155 q^{43} +2.06793 q^{44} -0.403453 q^{45} -7.28115 q^{46} -4.25664 q^{47} -2.99270 q^{48} -2.85973 q^{49} +4.99541 q^{50} -6.42222 q^{51} +4.05312 q^{52} -5.40084 q^{53} +8.84715 q^{54} -0.140074 q^{55} +2.03477 q^{56} -19.5974 q^{57} +9.58367 q^{58} +5.13928 q^{59} +0.202714 q^{60} -7.58843 q^{61} +7.07241 q^{62} -12.1196 q^{63} +1.00000 q^{64} -0.274542 q^{65} +6.18870 q^{66} +1.02118 q^{67} +2.14596 q^{68} -21.7903 q^{69} -0.137827 q^{70} +11.1012 q^{71} -5.95625 q^{72} +12.7617 q^{73} -4.07241 q^{74} +14.9498 q^{75} +6.54839 q^{76} -4.20776 q^{77} +12.1298 q^{78} +0.115909 q^{79} -0.0677361 q^{80} +8.60812 q^{81} +4.26718 q^{82} +2.46156 q^{83} +6.08944 q^{84} -0.145359 q^{85} +6.92155 q^{86} +28.6810 q^{87} -2.06793 q^{88} -15.0457 q^{89} +0.403453 q^{90} -8.24714 q^{91} +7.28115 q^{92} +21.1656 q^{93} +4.25664 q^{94} -0.443562 q^{95} +2.99270 q^{96} -4.48255 q^{97} +2.85973 q^{98} +12.3171 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q - 75 q^{2} + q^{3} + 75 q^{4} - 9 q^{5} - q^{6} - 8 q^{7} - 75 q^{8} + 66 q^{9}+O(q^{10}) \) 75 * q - 75 * q^2 + q^3 + 75 * q^4 - 9 * q^5 - q^6 - 8 * q^7 - 75 * q^8 + 66 * q^9	\( 75 q - 75 q^{2} + q^{3} + 75 q^{4} - 9 q^{5} - q^{6} - 8 q^{7} - 75 q^{8} + 66 q^{9} + 9 q^{10} - 5 q^{11} + q^{12} - 35 q^{13} + 8 q^{14} - 21 q^{15} + 75 q^{16} + 4 q^{17} - 66 q^{18} - 59 q^{19} - 9 q^{20} - 62 q^{21} + 5 q^{22} + 43 q^{23} - q^{24} + 44 q^{25} + 35 q^{26} + 4 q^{27} - 8 q^{28} - 38 q^{29} + 21 q^{30} - 51 q^{31} - 75 q^{32} - 19 q^{33} - 4 q^{34} + 14 q^{35} + 66 q^{36} - 63 q^{37} + 59 q^{38} - 34 q^{39} + 9 q^{40} - 27 q^{41} + 62 q^{42} - 39 q^{43} - 5 q^{44} - 52 q^{45} - 43 q^{46} + 40 q^{47} + q^{48} + 29 q^{49} - 44 q^{50} - 34 q^{51} - 35 q^{52} - 39 q^{53} - 4 q^{54} - 48 q^{55} + 8 q^{56} - 28 q^{57} + 38 q^{58} + 5 q^{59} - 21 q^{60} - 98 q^{61} + 51 q^{62} + 2 q^{63} + 75 q^{64} - q^{65} + 19 q^{66} - 59 q^{67} + 4 q^{68} - 69 q^{69} - 14 q^{70} - 9 q^{71} - 66 q^{72} - 51 q^{73} + 63 q^{74} - q^{75} - 59 q^{76} - 25 q^{77} + 34 q^{78} - 139 q^{79} - 9 q^{80} + 23 q^{81} + 27 q^{82} + 31 q^{83} - 62 q^{84} - 149 q^{85} + 39 q^{86} + q^{87} + 5 q^{88} - 39 q^{89} + 52 q^{90} - 93 q^{91} + 43 q^{92} - 83 q^{93} - 40 q^{94} + 2 q^{95} - q^{96} - 70 q^{97} - 29 q^{98} - 61 q^{99}+O(q^{100}) \) 75 * q - 75 * q^2 + q^3 + 75 * q^4 - 9 * q^5 - q^6 - 8 * q^7 - 75 * q^8 + 66 * q^9 + 9 * q^10 - 5 * q^11 + q^12 - 35 * q^13 + 8 * q^14 - 21 * q^15 + 75 * q^16 + 4 * q^17 - 66 * q^18 - 59 * q^19 - 9 * q^20 - 62 * q^21 + 5 * q^22 + 43 * q^23 - q^24 + 44 * q^25 + 35 * q^26 + 4 * q^27 - 8 * q^28 - 38 * q^29 + 21 * q^30 - 51 * q^31 - 75 * q^32 - 19 * q^33 - 4 * q^34 + 14 * q^35 + 66 * q^36 - 63 * q^37 + 59 * q^38 - 34 * q^39 + 9 * q^40 - 27 * q^41 + 62 * q^42 - 39 * q^43 - 5 * q^44 - 52 * q^45 - 43 * q^46 + 40 * q^47 + q^48 + 29 * q^49 - 44 * q^50 - 34 * q^51 - 35 * q^52 - 39 * q^53 - 4 * q^54 - 48 * q^55 + 8 * q^56 - 28 * q^57 + 38 * q^58 + 5 * q^59 - 21 * q^60 - 98 * q^61 + 51 * q^62 + 2 * q^63 + 75 * q^64 - q^65 + 19 * q^66 - 59 * q^67 + 4 * q^68 - 69 * q^69 - 14 * q^70 - 9 * q^71 - 66 * q^72 - 51 * q^73 + 63 * q^74 - q^75 - 59 * q^76 - 25 * q^77 + 34 * q^78 - 139 * q^79 - 9 * q^80 + 23 * q^81 + 27 * q^82 + 31 * q^83 - 62 * q^84 - 149 * q^85 + 39 * q^86 + q^87 + 5 * q^88 - 39 * q^89 + 52 * q^90 - 93 * q^91 + 43 * q^92 - 83 * q^93 - 40 * q^94 + 2 * q^95 - q^96 - 70 * q^97 - 29 * q^98 - 61 * q^99

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