LMFDB - Embedded newform 85.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [85,2,Mod(1,85)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(85, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("85.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 85 = 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	85.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:	$0.678728417181$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
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			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	85.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.73205 q^{2} -0.732051 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.26795 q^{6} +0.732051 q^{7} -1.73205 q^{8} -2.46410 q^{9} +1.73205 q^{10} +1.26795 q^{11} -0.732051 q^{12} -4.00000 q^{13} +1.26795 q^{14} -0.732051 q^{15} -5.00000 q^{16} -1.00000 q^{17} -4.26795 q^{18} +5.46410 q^{19} +1.00000 q^{20} -0.535898 q^{21} +2.19615 q^{22} +2.19615 q^{23} +1.26795 q^{24} +1.00000 q^{25} -6.92820 q^{26} +4.00000 q^{27} +0.732051 q^{28} +3.46410 q^{29} -1.26795 q^{30} +6.73205 q^{31} -5.19615 q^{32} -0.928203 q^{33} -1.73205 q^{34} +0.732051 q^{35} -2.46410 q^{36} -7.46410 q^{37} +9.46410 q^{38} +2.92820 q^{39} -1.73205 q^{40} +3.46410 q^{41} -0.928203 q^{42} -7.46410 q^{43} +1.26795 q^{44} -2.46410 q^{45} +3.80385 q^{46} -0.928203 q^{47} +3.66025 q^{48} -6.46410 q^{49} +1.73205 q^{50} +0.732051 q^{51} -4.00000 q^{52} +6.00000 q^{53} +6.92820 q^{54} +1.26795 q^{55} -1.26795 q^{56} -4.00000 q^{57} +6.00000 q^{58} +9.46410 q^{59} -0.732051 q^{60} +8.92820 q^{61} +11.6603 q^{62} -1.80385 q^{63} +1.00000 q^{64} -4.00000 q^{65} -1.60770 q^{66} -10.0000 q^{67} -1.00000 q^{68} -1.60770 q^{69} +1.26795 q^{70} -5.66025 q^{71} +4.26795 q^{72} -14.3923 q^{73} -12.9282 q^{74} -0.732051 q^{75} +5.46410 q^{76} +0.928203 q^{77} +5.07180 q^{78} -16.5885 q^{79} -5.00000 q^{80} +4.46410 q^{81} +6.00000 q^{82} +15.4641 q^{83} -0.535898 q^{84} -1.00000 q^{85} -12.9282 q^{86} -2.53590 q^{87} -2.19615 q^{88} -16.3923 q^{89} -4.26795 q^{90} -2.92820 q^{91} +2.19615 q^{92} -4.92820 q^{93} -1.60770 q^{94} +5.46410 q^{95} +3.80385 q^{96} +8.92820 q^{97} -11.1962 q^{98} -3.12436 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{4} + 2 q^{5} - 6 q^{6} - 2 q^{7} + 2 q^{9} + 6 q^{11} + 2 q^{12} - 8 q^{13} + 6 q^{14} + 2 q^{15} - 10 q^{16} - 2 q^{17} - 12 q^{18} + 4 q^{19} + 2 q^{20} - 8 q^{21} - 6 q^{22} - 6 q^{23}+ \cdots + 18 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^4 + 2 * q^5 - 6 * q^6 - 2 * q^7 + 2 * q^9 + 6 * q^11 + 2 * q^12 - 8 * q^13 + 6 * q^14 + 2 * q^15 - 10 * q^16 - 2 * q^17 - 12 * q^18 + 4 * q^19 + 2 * q^20 - 8 * q^21 - 6 * q^22 - 6 * q^23 + 6 * q^24 + 2 * q^25 + 8 * q^27 - 2 * q^28 - 6 * q^30 + 10 * q^31 + 12 * q^33 - 2 * q^35 + 2 * q^36 - 8 * q^37 + 12 * q^38 - 8 * q^39 + 12 * q^42 - 8 * q^43 + 6 * q^44 + 2 * q^45 + 18 * q^46 + 12 * q^47 - 10 * q^48 - 6 * q^49 - 2 * q^51 - 8 * q^52 + 12 * q^53 + 6 * q^55 - 6 * q^56 - 8 * q^57 + 12 * q^58 + 12 * q^59 + 2 * q^60 + 4 * q^61 + 6 * q^62 - 14 * q^63 + 2 * q^64 - 8 * q^65 - 24 * q^66 - 20 * q^67 - 2 * q^68 - 24 * q^69 + 6 * q^70 + 6 * q^71 + 12 * q^72 - 8 * q^73 - 12 * q^74 + 2 * q^75 + 4 * q^76 - 12 * q^77 + 24 * q^78 - 2 * q^79 - 10 * q^80 + 2 * q^81 + 12 * q^82 + 24 * q^83 - 8 * q^84 - 2 * q^85 - 12 * q^86 - 12 * q^87 + 6 * q^88 - 12 * q^89 - 12 * q^90 + 8 * q^91 - 6 * q^92 + 4 * q^93 - 24 * q^94 + 4 * q^95 + 18 * q^96 + 4 * q^97 - 12 * q^98 + 18 * q^99

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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	−0.732051	−0.422650	−0.211325	−	0.977416i	$-0.567778\pi$
$3$	−0.732051	−0.422650	−0.211325	+	0.977416i	$0.567778\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.26795	−0.517638
$7$	0.732051	0.276689	0.138345	−	0.990384i	$-0.455822\pi$
$7$	0.732051	0.276689	0.138345	+	0.990384i	$0.455822\pi$
$8$	−1.73205	−0.612372
$9$	−2.46410	−0.821367
$10$	1.73205	0.547723
$11$	1.26795	0.382301	0.191151	−	0.981561i	$-0.438778\pi$
$11$	1.26795	0.382301	0.191151	+	0.981561i	$0.438778\pi$
$12$	−0.732051	−0.211325
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	1.26795	0.338874
$15$	−0.732051	−0.189015
$16$	−5.00000	−1.25000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−4.26795	−1.00597
$19$	5.46410	1.25355	0.626775	−	0.779200i	$-0.284374\pi$
$19$	5.46410	1.25355	0.626775	+	0.779200i	$0.284374\pi$
$20$	1.00000	0.223607
$21$	−0.535898	−0.116943
$22$	2.19615	0.468221
$23$	2.19615	0.457929	0.228965	−	0.973435i	$-0.426466\pi$
$23$	2.19615	0.457929	0.228965	+	0.973435i	$0.426466\pi$
$24$	1.26795	0.258819
$25$	1.00000	0.200000
$26$	−6.92820	−1.35873
$27$	4.00000	0.769800
$28$	0.732051	0.138345
$29$	3.46410	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	−1.26795	−0.231495
$31$	6.73205	1.20911	0.604556	−	0.796563i	$-0.293351\pi$
$31$	6.73205	1.20911	0.604556	+	0.796563i	$0.293351\pi$
$32$	−5.19615	−0.918559
$33$	−0.928203	−0.161579
$34$	−1.73205	−0.297044
$35$	0.732051	0.123739
$36$	−2.46410	−0.410684
$37$	−7.46410	−1.22709	−0.613545	−	0.789659i	$-0.710257\pi$
$37$	−7.46410	−1.22709	−0.613545	+	0.789659i	$0.710257\pi$
$38$	9.46410	1.53528
$39$	2.92820	0.468888
$40$	−1.73205	−0.273861
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	−0.928203	−0.143225
$43$	−7.46410	−1.13826	−0.569132	−	0.822246i	$-0.692721\pi$
$43$	−7.46410	−1.13826	−0.569132	+	0.822246i	$0.692721\pi$
$44$	1.26795	0.191151
$45$	−2.46410	−0.367327
$46$	3.80385	0.560847
$47$	−0.928203	−0.135392	−0.0676962	−	0.997706i	$-0.521565\pi$
$47$	−0.928203	−0.135392	−0.0676962	+	0.997706i	$0.521565\pi$
$48$	3.66025	0.528312
$49$	−6.46410	−0.923443
$50$	1.73205	0.244949
$51$	0.732051	0.102508
$52$	−4.00000	−0.554700
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	6.92820	0.942809
$55$	1.26795	0.170970
$56$	−1.26795	−0.169437
$57$	−4.00000	−0.529813
$58$	6.00000	0.787839
$59$	9.46410	1.23212	0.616061	−	0.787699i	$-0.288728\pi$
$59$	9.46410	1.23212	0.616061	+	0.787699i	$0.288728\pi$
$60$	−0.732051	−0.0945074
$61$	8.92820	1.14314	0.571570	−	0.820554i	$-0.306335\pi$
$61$	8.92820	1.14314	0.571570	+	0.820554i	$0.306335\pi$
$62$	11.6603	1.48085
$63$	−1.80385	−0.227263
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	−1.60770	−0.197894
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	−1.00000	−0.121268
$69$	−1.60770	−0.193544
$70$	1.26795	0.151549
$71$	−5.66025	−0.671749	−0.335874	−	0.941907i	$-0.609032\pi$
$71$	−5.66025	−0.671749	−0.335874	+	0.941907i	$0.609032\pi$
$72$	4.26795	0.502983
$73$	−14.3923	−1.68449	−0.842246	−	0.539093i	$-0.818767\pi$
$73$	−14.3923	−1.68449	−0.842246	+	0.539093i	$0.818767\pi$
$74$	−12.9282	−1.50287
$75$	−0.732051	−0.0845299
$76$	5.46410	0.626775
$77$	0.928203	0.105779
$78$	5.07180	0.574268
$79$	−16.5885	−1.86635	−0.933174	−	0.359426i	$-0.882973\pi$
$79$	−16.5885	−1.86635	−0.933174	+	0.359426i	$0.882973\pi$
$80$	−5.00000	−0.559017
$81$	4.46410	0.496011
$82$	6.00000	0.662589
$83$	15.4641	1.69741	0.848703	−	0.528870i	$-0.177384\pi$
$83$	15.4641	1.69741	0.848703	+	0.528870i	$0.177384\pi$
$84$	−0.535898	−0.0584713
$85$	−1.00000	−0.108465
$86$	−12.9282	−1.39408
$87$	−2.53590	−0.271877
$88$	−2.19615	−0.234111
$89$	−16.3923	−1.73758	−0.868790	−	0.495180i	$-0.835102\pi$
$89$	−16.3923	−1.73758	−0.868790	+	0.495180i	$0.835102\pi$
$90$	−4.26795	−0.449881
$91$	−2.92820	−0.306959
$92$	2.19615	0.228965
$93$	−4.92820	−0.511031
$94$	−1.60770	−0.165821
$95$	5.46410	0.560605
$96$	3.80385	0.388229
$97$	8.92820	0.906522	0.453261	−	0.891378i	$-0.350261\pi$
$97$	8.92820	0.906522	0.453261	+	0.891378i	$0.350261\pi$
$98$	−11.1962	−1.13098
$99$	−3.12436	−0.314010
$100$	1.00000	0.100000
$101$	−2.53590	−0.252331	−0.126166	−	0.992009i	$-0.540267\pi$
$101$	−2.53590	−0.252331	−0.126166	+	0.992009i	$0.540267\pi$
$102$	1.26795	0.125546
$103$	−4.92820	−0.485590	−0.242795	−	0.970078i	$-0.578064\pi$
$103$	−4.92820	−0.485590	−0.242795	+	0.970078i	$0.578064\pi$
$104$	6.92820	0.679366
$105$	−0.535898	−0.0522983
$106$	10.3923	1.00939
$107$	0.339746	0.0328445	0.0164222	−	0.999865i	$-0.494772\pi$
$107$	0.339746	0.0328445	0.0164222	+	0.999865i	$0.494772\pi$
$108$	4.00000	0.384900
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	2.19615	0.209395
$111$	5.46410	0.518630
$112$	−3.66025	−0.345861
$113$	17.3205	1.62938	0.814688	−	0.579899i	$-0.196908\pi$
$113$	17.3205	1.62938	0.814688	+	0.579899i	$0.196908\pi$
$114$	−6.92820	−0.648886
$115$	2.19615	0.204792
$116$	3.46410	0.321634
$117$	9.85641	0.911225
$118$	16.3923	1.50903
$119$	−0.732051	−0.0671070
$120$	1.26795	0.115747
$121$	−9.39230	−0.853846
$122$	15.4641	1.40005
$123$	−2.53590	−0.228654
$124$	6.73205	0.604556
$125$	1.00000	0.0894427
$126$	−3.12436	−0.278340
$127$	6.39230	0.567225	0.283613	−	0.958939i	$-0.408467\pi$
$127$	6.39230	0.567225	0.283613	+	0.958939i	$0.408467\pi$
$128$	12.1244	1.07165
$129$	5.46410	0.481087
$130$	−6.92820	−0.607644
$131$	8.19615	0.716101	0.358051	−	0.933702i	$-0.383442\pi$
$131$	8.19615	0.716101	0.358051	+	0.933702i	$0.383442\pi$
$132$	−0.928203	−0.0807897
$133$	4.00000	0.346844
$134$	−17.3205	−1.49626
$135$	4.00000	0.344265
$136$	1.73205	0.148522
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	−2.78461	−0.237042
$139$	13.6603	1.15865	0.579324	−	0.815097i	$-0.303317\pi$
$139$	13.6603	1.15865	0.579324	+	0.815097i	$0.303317\pi$
$140$	0.732051	0.0618696
$141$	0.679492	0.0572235
$142$	−9.80385	−0.822721
$143$	−5.07180	−0.424125
$144$	12.3205	1.02671
$145$	3.46410	0.287678
$146$	−24.9282	−2.06307
$147$	4.73205	0.390293
$148$	−7.46410	−0.613545
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	−1.26795	−0.103528
$151$	5.46410	0.444662	0.222331	−	0.974971i	$-0.428633\pi$
$151$	5.46410	0.444662	0.222331	+	0.974971i	$0.428633\pi$
$152$	−9.46410	−0.767640
$153$	2.46410	0.199211
$154$	1.60770	0.129552
$155$	6.73205	0.540731
$156$	2.92820	0.234444
$157$	−4.92820	−0.393313	−0.196657	−	0.980472i	$-0.563008\pi$
$157$	−4.92820	−0.393313	−0.196657	+	0.980472i	$0.563008\pi$
$158$	−28.7321	−2.28580
$159$	−4.39230	−0.348332
$160$	−5.19615	−0.410792
$161$	1.60770	0.126704
$162$	7.73205	0.607487
$163$	10.1962	0.798624	0.399312	−	0.916815i	$-0.369249\pi$
$163$	10.1962	0.798624	0.399312	+	0.916815i	$0.369249\pi$
$164$	3.46410	0.270501
$165$	−0.928203	−0.0722605
$166$	26.7846	2.07889
$167$	−18.5885	−1.43842	−0.719209	−	0.694794i	$-0.755496\pi$
$167$	−18.5885	−1.43842	−0.719209	+	0.694794i	$0.755496\pi$
$168$	0.928203	0.0716124
$169$	3.00000	0.230769
$170$	−1.73205	−0.132842
$171$	−13.4641	−1.02963
$172$	−7.46410	−0.569132
$173$	−3.46410	−0.263371	−0.131685	−	0.991292i	$-0.542039\pi$
$173$	−3.46410	−0.263371	−0.131685	+	0.991292i	$0.542039\pi$
$174$	−4.39230	−0.332980
$175$	0.732051	0.0553378
$176$	−6.33975	−0.477876
$177$	−6.92820	−0.520756
$178$	−28.3923	−2.12809
$179$	23.3205	1.74306	0.871528	−	0.490345i	$-0.163129\pi$
$179$	23.3205	1.74306	0.871528	+	0.490345i	$0.163129\pi$
$180$	−2.46410	−0.183663
$181$	18.3923	1.36709	0.683545	−	0.729909i	$-0.260437\pi$
$181$	18.3923	1.36709	0.683545	+	0.729909i	$0.260437\pi$
$182$	−5.07180	−0.375947
$183$	−6.53590	−0.483148
$184$	−3.80385	−0.280423
$185$	−7.46410	−0.548772
$186$	−8.53590	−0.625882
$187$	−1.26795	−0.0927216
$188$	−0.928203	−0.0676962
$189$	2.92820	0.212995
$190$	9.46410	0.686598
$191$	−25.8564	−1.87090	−0.935452	−	0.353454i	$-0.885007\pi$
$191$	−25.8564	−1.87090	−0.935452	+	0.353454i	$0.885007\pi$
$192$	−0.732051	−0.0528312
$193$	23.4641	1.68898	0.844491	−	0.535569i	$-0.179903\pi$
$193$	23.4641	1.68898	0.844491	+	0.535569i	$0.179903\pi$
$194$	15.4641	1.11026
$195$	2.92820	0.209693
$196$	−6.46410	−0.461722
$197$	−17.3205	−1.23404	−0.617018	−	0.786949i	$-0.711659\pi$
$197$	−17.3205	−1.23404	−0.617018	+	0.786949i	$0.711659\pi$
$198$	−5.41154	−0.384582
$199$	−0.196152	−0.0139049	−0.00695244	−	0.999976i	$-0.502213\pi$
$199$	−0.196152	−0.0139049	−0.00695244	+	0.999976i	$0.502213\pi$
$200$	−1.73205	−0.122474
$201$	7.32051	0.516349
$202$	−4.39230	−0.309041
$203$	2.53590	0.177985
$204$	0.732051	0.0512538
$205$	3.46410	0.241943
$206$	−8.53590	−0.594724
$207$	−5.41154	−0.376128
$208$	20.0000	1.38675
$209$	6.92820	0.479234
$210$	−0.928203	−0.0640521
$211$	−0.196152	−0.0135037	−0.00675184	−	0.999977i	$-0.502149\pi$
$211$	−0.196152	−0.0135037	−0.00675184	+	0.999977i	$0.502149\pi$
$212$	6.00000	0.412082
$213$	4.14359	0.283914
$214$	0.588457	0.0402261
$215$	−7.46410	−0.509048
$216$	−6.92820	−0.471405
$217$	4.92820	0.334548
$218$	−17.3205	−1.17309
$219$	10.5359	0.711950
$220$	1.26795	0.0854851
$221$	4.00000	0.269069
$222$	9.46410	0.635189
$223$	−5.60770	−0.375519	−0.187760	−	0.982215i	$-0.560123\pi$
$223$	−5.60770	−0.375519	−0.187760	+	0.982215i	$0.560123\pi$
$224$	−3.80385	−0.254155
$225$	−2.46410	−0.164273
$226$	30.0000	1.99557
$227$	19.2679	1.27886	0.639429	−	0.768850i	$-0.279171\pi$
$227$	19.2679	1.27886	0.639429	+	0.768850i	$0.279171\pi$
$228$	−4.00000	−0.264906
$229$	12.3923	0.818907	0.409453	−	0.912331i	$-0.365719\pi$
$229$	12.3923	0.818907	0.409453	+	0.912331i	$0.365719\pi$
$230$	3.80385	0.250818
$231$	−0.679492	−0.0447073
$232$	−6.00000	−0.393919
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	17.0718	1.11602
$235$	−0.928203	−0.0605493
$236$	9.46410	0.616061
$237$	12.1436	0.788811
$238$	−1.26795	−0.0821889
$239$	20.7846	1.34444	0.672222	−	0.740349i	$-0.265340\pi$
$239$	20.7846	1.34444	0.672222	+	0.740349i	$0.265340\pi$
$240$	3.66025	0.236268
$241$	−26.3923	−1.70008	−0.850039	−	0.526720i	$-0.823422\pi$
$241$	−26.3923	−1.70008	−0.850039	+	0.526720i	$0.823422\pi$
$242$	−16.2679	−1.04574
$243$	−15.2679	−0.979439
$244$	8.92820	0.571570
$245$	−6.46410	−0.412976
$246$	−4.39230	−0.280043
$247$	−21.8564	−1.39069
$248$	−11.6603	−0.740427
$249$	−11.3205	−0.717408
$250$	1.73205	0.109545
$251$	−6.92820	−0.437304	−0.218652	−	0.975803i	$-0.570166\pi$
$251$	−6.92820	−0.437304	−0.218652	+	0.975803i	$0.570166\pi$
$252$	−1.80385	−0.113632
$253$	2.78461	0.175067
$254$	11.0718	0.694706
$255$	0.732051	0.0458428
$256$	19.0000	1.18750
$257$	6.92820	0.432169	0.216085	−	0.976375i	$-0.430671\pi$
$257$	6.92820	0.432169	0.216085	+	0.976375i	$0.430671\pi$
$258$	9.46410	0.589209
$259$	−5.46410	−0.339523
$260$	−4.00000	−0.248069
$261$	−8.53590	−0.528359
$262$	14.1962	0.877041
$263$	−22.3923	−1.38077	−0.690384	−	0.723443i	$-0.742559\pi$
$263$	−22.3923	−1.38077	−0.690384	+	0.723443i	$0.742559\pi$
$264$	1.60770	0.0989468
$265$	6.00000	0.368577
$266$	6.92820	0.424795
$267$	12.0000	0.734388
$268$	−10.0000	−0.610847
$269$	−12.9282	−0.788246	−0.394123	−	0.919058i	$-0.628952\pi$
$269$	−12.9282	−0.788246	−0.394123	+	0.919058i	$0.628952\pi$
$270$	6.92820	0.421637
$271$	−10.9282	−0.663841	−0.331921	−	0.943307i	$-0.607697\pi$
$271$	−10.9282	−0.663841	−0.331921	+	0.943307i	$0.607697\pi$
$272$	5.00000	0.303170
$273$	2.14359	0.129736
$274$	0	0
$275$	1.26795	0.0764602
$276$	−1.60770	−0.0967719
$277$	7.07180	0.424903	0.212452	−	0.977172i	$-0.431855\pi$
$277$	7.07180	0.424903	0.212452	+	0.977172i	$0.431855\pi$
$278$	23.6603	1.41905
$279$	−16.5885	−0.993125
$280$	−1.26795	−0.0757745
$281$	0.928203	0.0553720	0.0276860	−	0.999617i	$-0.491186\pi$
$281$	0.928203	0.0553720	0.0276860	+	0.999617i	$0.491186\pi$
$282$	1.17691	0.0700842
$283$	−8.73205	−0.519067	−0.259533	−	0.965734i	$-0.583569\pi$
$283$	−8.73205	−0.519067	−0.259533	+	0.965734i	$0.583569\pi$
$284$	−5.66025	−0.335874
$285$	−4.00000	−0.236940
$286$	−8.78461	−0.519445
$287$	2.53590	0.149689
$288$	12.8038	0.754474
$289$	1.00000	0.0588235
$290$	6.00000	0.352332
$291$	−6.53590	−0.383141
$292$	−14.3923	−0.842246
$293$	12.9282	0.755274	0.377637	−	0.925954i	$-0.376737\pi$
$293$	12.9282	0.755274	0.377637	+	0.925954i	$0.376737\pi$
$294$	8.19615	0.478009
$295$	9.46410	0.551021
$296$	12.9282	0.751437
$297$	5.07180	0.294295
$298$	−10.3923	−0.602010
$299$	−8.78461	−0.508027
$300$	−0.732051	−0.0422650
$301$	−5.46410	−0.314946
$302$	9.46410	0.544598
$303$	1.85641	0.106648
$304$	−27.3205	−1.56694
$305$	8.92820	0.511227
$306$	4.26795	0.243982
$307$	−10.0000	−0.570730	−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000	−0.570730	−0.285365	+	0.958419i	$0.592115\pi$
$308$	0.928203	0.0528893
$309$	3.60770	0.205235
$310$	11.6603	0.662258
$311$	22.0526	1.25049	0.625243	−	0.780430i	$-0.285000\pi$
$311$	22.0526	1.25049	0.625243	+	0.780430i	$0.285000\pi$
$312$	−5.07180	−0.287134
$313$	−5.60770	−0.316966	−0.158483	−	0.987362i	$-0.550660\pi$
$313$	−5.60770	−0.316966	−0.158483	+	0.987362i	$0.550660\pi$
$314$	−8.53590	−0.481709
$315$	−1.80385	−0.101635
$316$	−16.5885	−0.933174
$317$	−11.0718	−0.621854	−0.310927	−	0.950434i	$-0.600639\pi$
$317$	−11.0718	−0.621854	−0.310927	+	0.950434i	$0.600639\pi$
$318$	−7.60770	−0.426618
$319$	4.39230	0.245922
$320$	1.00000	0.0559017
$321$	−0.248711	−0.0138817
$322$	2.78461	0.155180
$323$	−5.46410	−0.304031
$324$	4.46410	0.248006
$325$	−4.00000	−0.221880
$326$	17.6603	0.978111
$327$	7.32051	0.404825
$328$	−6.00000	−0.331295
$329$	−0.679492	−0.0374616
$330$	−1.60770	−0.0885007
$331$	−13.4641	−0.740054	−0.370027	−	0.929021i	$-0.620652\pi$
$331$	−13.4641	−0.740054	−0.370027	+	0.929021i	$0.620652\pi$
$332$	15.4641	0.848703
$333$	18.3923	1.00789
$334$	−32.1962	−1.76170
$335$	−10.0000	−0.546358
$336$	2.67949	0.146178
$337$	34.7846	1.89484	0.947419	−	0.319995i	$-0.103681\pi$
$337$	34.7846	1.89484	0.947419	+	0.319995i	$0.103681\pi$
$338$	5.19615	0.282633
$339$	−12.6795	−0.688655
$340$	−1.00000	−0.0542326
$341$	8.53590	0.462245
$342$	−23.3205	−1.26103
$343$	−9.85641	−0.532196
$344$	12.9282	0.697042
$345$	−1.60770	−0.0865554
$346$	−6.00000	−0.322562
$347$	14.1962	0.762089	0.381045	−	0.924557i	$-0.375564\pi$
$347$	14.1962	0.762089	0.381045	+	0.924557i	$0.375564\pi$
$348$	−2.53590	−0.135938
$349$	−30.7846	−1.64786	−0.823931	−	0.566690i	$-0.808224\pi$
$349$	−30.7846	−1.64786	−0.823931	+	0.566690i	$0.808224\pi$
$350$	1.26795	0.0677747
$351$	−16.0000	−0.854017
$352$	−6.58846	−0.351166
$353$	−14.7846	−0.786905	−0.393453	−	0.919345i	$-0.628719\pi$
$353$	−14.7846	−0.786905	−0.393453	+	0.919345i	$0.628719\pi$
$354$	−12.0000	−0.637793
$355$	−5.66025	−0.300415
$356$	−16.3923	−0.868790
$357$	0.535898	0.0283628
$358$	40.3923	2.13480
$359$	−14.5359	−0.767175	−0.383588	−	0.923504i	$-0.625312\pi$
$359$	−14.5359	−0.767175	−0.383588	+	0.923504i	$0.625312\pi$
$360$	4.26795	0.224941
$361$	10.8564	0.571390
$362$	31.8564	1.67434
$363$	6.87564	0.360878
$364$	−2.92820	−0.153480
$365$	−14.3923	−0.753328
$366$	−11.3205	−0.591732
$367$	−18.1962	−0.949831	−0.474916	−	0.880031i	$-0.657521\pi$
$367$	−18.1962	−0.949831	−0.474916	+	0.880031i	$0.657521\pi$
$368$	−10.9808	−0.572412
$369$	−8.53590	−0.444361
$370$	−12.9282	−0.672105
$371$	4.39230	0.228037
$372$	−4.92820	−0.255515
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	−2.19615	−0.113560
$375$	−0.732051	−0.0378029
$376$	1.60770	0.0829105
$377$	−13.8564	−0.713641
$378$	5.07180	0.260865
$379$	28.1962	1.44834	0.724170	−	0.689622i	$-0.242223\pi$
$379$	28.1962	1.44834	0.724170	+	0.689622i	$0.242223\pi$
$380$	5.46410	0.280302
$381$	−4.67949	−0.239738
$382$	−44.7846	−2.29138
$383$	8.53590	0.436164	0.218082	−	0.975930i	$-0.430020\pi$
$383$	8.53590	0.436164	0.218082	+	0.975930i	$0.430020\pi$
$384$	−8.87564	−0.452933
$385$	0.928203	0.0473056
$386$	40.6410	2.06857
$387$	18.3923	0.934933
$388$	8.92820	0.453261
$389$	16.3923	0.831123	0.415561	−	0.909565i	$-0.363585\pi$
$389$	16.3923	0.831123	0.415561	+	0.909565i	$0.363585\pi$
$390$	5.07180	0.256820
$391$	−2.19615	−0.111064
$392$	11.1962	0.565491
$393$	−6.00000	−0.302660
$394$	−30.0000	−1.51138
$395$	−16.5885	−0.834656
$396$	−3.12436	−0.157005
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	−0.339746	−0.0170299
$399$	−2.92820	−0.146594
$400$	−5.00000	−0.250000
$401$	−0.928203	−0.0463523	−0.0231761	−	0.999731i	$-0.507378\pi$
$401$	−0.928203	−0.0463523	−0.0231761	+	0.999731i	$0.507378\pi$
$402$	12.6795	0.632396
$403$	−26.9282	−1.34139
$404$	−2.53590	−0.126166
$405$	4.46410	0.221823
$406$	4.39230	0.217986
$407$	−9.46410	−0.469118
$408$	−1.26795	−0.0627728
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	6.00000	0.296319
$411$	0	0
$412$	−4.92820	−0.242795
$413$	6.92820	0.340915
$414$	−9.37307	−0.460661
$415$	15.4641	0.759103
$416$	20.7846	1.01905
$417$	−10.0000	−0.489702
$418$	12.0000	0.586939
$419$	27.8038	1.35831	0.679153	−	0.733996i	$-0.262347\pi$
$419$	27.8038	1.35831	0.679153	+	0.733996i	$0.262347\pi$
$420$	−0.535898	−0.0261492
$421$	−1.46410	−0.0713559	−0.0356780	−	0.999363i	$-0.511359\pi$
$421$	−1.46410	−0.0713559	−0.0356780	+	0.999363i	$0.511359\pi$
$422$	−0.339746	−0.0165386
$423$	2.28719	0.111207
$424$	−10.3923	−0.504695
$425$	−1.00000	−0.0485071
$426$	7.17691	0.347723
$427$	6.53590	0.316294
$428$	0.339746	0.0164222
$429$	3.71281	0.179256
$430$	−12.9282	−0.623453
$431$	20.1962	0.972814	0.486407	−	0.873732i	$-0.338307\pi$
$431$	20.1962	0.972814	0.486407	+	0.873732i	$0.338307\pi$
$432$	−20.0000	−0.962250
$433$	9.85641	0.473669	0.236834	−	0.971550i	$-0.423890\pi$
$433$	9.85641	0.473669	0.236834	+	0.971550i	$0.423890\pi$
$434$	8.53590	0.409736
$435$	−2.53590	−0.121587
$436$	−10.0000	−0.478913
$437$	12.0000	0.574038
$438$	18.2487	0.871957
$439$	18.0526	0.861602	0.430801	−	0.902447i	$-0.358231\pi$
$439$	18.0526	0.861602	0.430801	+	0.902447i	$0.358231\pi$
$440$	−2.19615	−0.104697
$441$	15.9282	0.758486
$442$	6.92820	0.329541
$443$	0.928203	0.0441003	0.0220501	−	0.999757i	$-0.492981\pi$
$443$	0.928203	0.0441003	0.0220501	+	0.999757i	$0.492981\pi$
$444$	5.46410	0.259315
$445$	−16.3923	−0.777070
$446$	−9.71281	−0.459915
$447$	4.39230	0.207749
$448$	0.732051	0.0345861
$449$	−13.6077	−0.642187	−0.321093	−	0.947048i	$-0.604050\pi$
$449$	−13.6077	−0.642187	−0.321093	+	0.947048i	$0.604050\pi$
$450$	−4.26795	−0.201193
$451$	4.39230	0.206826
$452$	17.3205	0.814688
$453$	−4.00000	−0.187936
$454$	33.3731	1.56628
$455$	−2.92820	−0.137276
$456$	6.92820	0.324443
$457$	4.78461	0.223815	0.111907	−	0.993719i	$-0.464304\pi$
$457$	4.78461	0.223815	0.111907	+	0.993719i	$0.464304\pi$
$458$	21.4641	1.00295
$459$	−4.00000	−0.186704
$460$	2.19615	0.102396
$461$	−11.0718	−0.515665	−0.257832	−	0.966190i	$-0.583008\pi$
$461$	−11.0718	−0.515665	−0.257832	+	0.966190i	$0.583008\pi$
$462$	−1.17691	−0.0547550
$463$	3.85641	0.179222	0.0896112	−	0.995977i	$-0.471438\pi$
$463$	3.85641	0.179222	0.0896112	+	0.995977i	$0.471438\pi$
$464$	−17.3205	−0.804084
$465$	−4.92820	−0.228540
$466$	10.3923	0.481414
$467$	22.3923	1.03619	0.518096	−	0.855322i	$-0.326641\pi$
$467$	22.3923	1.03619	0.518096	+	0.855322i	$0.326641\pi$
$468$	9.85641	0.455613
$469$	−7.32051	−0.338030
$470$	−1.60770	−0.0741574
$471$	3.60770	0.166234
$472$	−16.3923	−0.754517
$473$	−9.46410	−0.435160
$474$	21.0333	0.966092
$475$	5.46410	0.250710
$476$	−0.732051	−0.0335535
$477$	−14.7846	−0.676941
$478$	36.0000	1.64660
$479$	5.66025	0.258624	0.129312	−	0.991604i	$-0.458723\pi$
$479$	5.66025	0.258624	0.129312	+	0.991604i	$0.458723\pi$
$480$	3.80385	0.173621
$481$	29.8564	1.36133
$482$	−45.7128	−2.08216
$483$	−1.17691	−0.0535515
$484$	−9.39230	−0.426923
$485$	8.92820	0.405409
$486$	−26.4449	−1.19956
$487$	−26.9808	−1.22262	−0.611308	−	0.791393i	$-0.709356\pi$
$487$	−26.9808	−1.22262	−0.611308	+	0.791393i	$0.709356\pi$
$488$	−15.4641	−0.700027
$489$	−7.46410	−0.337538
$490$	−11.1962	−0.505791
$491$	−40.3923	−1.82288	−0.911440	−	0.411434i	$-0.865028\pi$
$491$	−40.3923	−1.82288	−0.911440	+	0.411434i	$0.865028\pi$
$492$	−2.53590	−0.114327
$493$	−3.46410	−0.156015
$494$	−37.8564	−1.70324
$495$	−3.12436	−0.140429
$496$	−33.6603	−1.51139
$497$	−4.14359	−0.185866
$498$	−19.6077	−0.878642
$499$	1.66025	0.0743232	0.0371616	−	0.999309i	$-0.488168\pi$
$499$	1.66025	0.0743232	0.0371616	+	0.999309i	$0.488168\pi$
$500$	1.00000	0.0447214
$501$	13.6077	0.607947
$502$	−12.0000	−0.535586
$503$	−9.12436	−0.406835	−0.203417	−	0.979092i	$-0.565205\pi$
$503$	−9.12436	−0.406835	−0.203417	+	0.979092i	$0.565205\pi$
$504$	3.12436	0.139170
$505$	−2.53590	−0.112846
$506$	4.82309	0.214412
$507$	−2.19615	−0.0975346
$508$	6.39230	0.283613
$509$	−7.85641	−0.348229	−0.174115	−	0.984725i	$-0.555706\pi$
$509$	−7.85641	−0.348229	−0.174115	+	0.984725i	$0.555706\pi$
$510$	1.26795	0.0561457
$511$	−10.5359	−0.466081
$512$	8.66025	0.382733
$513$	21.8564	0.964984
$514$	12.0000	0.529297
$515$	−4.92820	−0.217163
$516$	5.46410	0.240544
$517$	−1.17691	−0.0517606
$518$	−9.46410	−0.415829
$519$	2.53590	0.111314
$520$	6.92820	0.303822
$521$	31.8564	1.39565	0.697827	−	0.716266i	$-0.254150\pi$
$521$	31.8564	1.39565	0.697827	+	0.716266i	$0.254150\pi$
$522$	−14.7846	−0.647105
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	8.19615	0.358051
$525$	−0.535898	−0.0233885
$526$	−38.7846	−1.69109
$527$	−6.73205	−0.293253
$528$	4.64102	0.201974
$529$	−18.1769	−0.790301
$530$	10.3923	0.451413
$531$	−23.3205	−1.01202
$532$	4.00000	0.173422
$533$	−13.8564	−0.600188
$534$	20.7846	0.899438
$535$	0.339746	0.0146885
$536$	17.3205	0.748132
$537$	−17.0718	−0.736702
$538$	−22.3923	−0.965401
$539$	−8.19615	−0.353033
$540$	4.00000	0.172133
$541$	−23.1769	−0.996453	−0.498227	−	0.867047i	$-0.666015\pi$
$541$	−23.1769	−0.996453	−0.498227	+	0.867047i	$0.666015\pi$
$542$	−18.9282	−0.813036
$543$	−13.4641	−0.577800
$544$	5.19615	0.222783
$545$	−10.0000	−0.428353
$546$	3.71281	0.158894
$547$	25.9090	1.10779	0.553894	−	0.832587i	$-0.313141\pi$
$547$	25.9090	1.10779	0.553894	+	0.832587i	$0.313141\pi$
$548$	0	0
$549$	−22.0000	−0.938937
$550$	2.19615	0.0936443
$551$	18.9282	0.806369
$552$	2.78461	0.118521
$553$	−12.1436	−0.516398
$554$	12.2487	0.520398
$555$	5.46410	0.231938
$556$	13.6603	0.579324
$557$	6.92820	0.293557	0.146779	−	0.989169i	$-0.453109\pi$
$557$	6.92820	0.293557	0.146779	+	0.989169i	$0.453109\pi$
$558$	−28.7321	−1.21632
$559$	29.8564	1.26279
$560$	−3.66025	−0.154674
$561$	0.928203	0.0391888
$562$	1.60770	0.0678165
$563$	20.5359	0.865485	0.432742	−	0.901518i	$-0.357546\pi$
$563$	20.5359	0.865485	0.432742	+	0.901518i	$0.357546\pi$
$564$	0.679492	0.0286118
$565$	17.3205	0.728679
$566$	−15.1244	−0.635724
$567$	3.26795	0.137241
$568$	9.80385	0.411360
$569$	−28.6410	−1.20069	−0.600347	−	0.799740i	$-0.704971\pi$
$569$	−28.6410	−1.20069	−0.600347	+	0.799740i	$0.704971\pi$
$570$	−6.92820	−0.290191
$571$	22.4449	0.939288	0.469644	−	0.882856i	$-0.344382\pi$
$571$	22.4449	0.939288	0.469644	+	0.882856i	$0.344382\pi$
$572$	−5.07180	−0.212062
$573$	18.9282	0.790737
$574$	4.39230	0.183331
$575$	2.19615	0.0915859
$576$	−2.46410	−0.102671
$577$	30.6410	1.27560	0.637801	−	0.770201i	$-0.279844\pi$
$577$	30.6410	1.27560	0.637801	+	0.770201i	$0.279844\pi$
$578$	1.73205	0.0720438
$579$	−17.1769	−0.713848
$580$	3.46410	0.143839
$581$	11.3205	0.469654
$582$	−11.3205	−0.469250
$583$	7.60770	0.315079
$584$	24.9282	1.03154
$585$	9.85641	0.407512
$586$	22.3923	0.925018
$587$	25.6077	1.05694	0.528471	−	0.848951i	$-0.322765\pi$
$587$	25.6077	1.05694	0.528471	+	0.848951i	$0.322765\pi$
$588$	4.73205	0.195146
$589$	36.7846	1.51568
$590$	16.3923	0.674861
$591$	12.6795	0.521565
$592$	37.3205	1.53386
$593$	−7.85641	−0.322624	−0.161312	−	0.986903i	$-0.551573\pi$
$593$	−7.85641	−0.322624	−0.161312	+	0.986903i	$0.551573\pi$
$594$	8.78461	0.360437
$595$	−0.732051	−0.0300112
$596$	−6.00000	−0.245770
$597$	0.143594	0.00587689
$598$	−15.2154	−0.622204
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	1.26795	0.0517638
$601$	−40.2487	−1.64178	−0.820890	−	0.571087i	$-0.806522\pi$
$601$	−40.2487	−1.64178	−0.820890	+	0.571087i	$0.806522\pi$
$602$	−9.46410	−0.385728
$603$	24.6410	1.00346
$604$	5.46410	0.222331
$605$	−9.39230	−0.381851
$606$	3.21539	0.130616
$607$	−4.33975	−0.176145	−0.0880724	−	0.996114i	$-0.528071\pi$
$607$	−4.33975	−0.176145	−0.0880724	+	0.996114i	$0.528071\pi$
$608$	−28.3923	−1.15146
$609$	−1.85641	−0.0752254
$610$	15.4641	0.626123
$611$	3.71281	0.150204
$612$	2.46410	0.0996054
$613$	−11.8564	−0.478876	−0.239438	−	0.970912i	$-0.576963\pi$
$613$	−11.8564	−0.478876	−0.239438	+	0.970912i	$0.576963\pi$
$614$	−17.3205	−0.698999
$615$	−2.53590	−0.102257
$616$	−1.60770	−0.0647759
$617$	−20.5359	−0.826744	−0.413372	−	0.910562i	$-0.635649\pi$
$617$	−20.5359	−0.826744	−0.413372	+	0.910562i	$0.635649\pi$
$618$	6.24871	0.251360
$619$	7.41154	0.297895	0.148948	−	0.988845i	$-0.452411\pi$
$619$	7.41154	0.297895	0.148948	+	0.988845i	$0.452411\pi$
$620$	6.73205	0.270366
$621$	8.78461	0.352514
$622$	38.1962	1.53153
$623$	−12.0000	−0.480770
$624$	−14.6410	−0.586110
$625$	1.00000	0.0400000
$626$	−9.71281	−0.388202
$627$	−5.07180	−0.202548
$628$	−4.92820	−0.196657
$629$	7.46410	0.297613
$630$	−3.12436	−0.124477
$631$	−11.6077	−0.462095	−0.231048	−	0.972942i	$-0.574215\pi$
$631$	−11.6077	−0.462095	−0.231048	+	0.972942i	$0.574215\pi$
$632$	28.7321	1.14290
$633$	0.143594	0.00570733
$634$	−19.1769	−0.761613
$635$	6.39230	0.253671
$636$	−4.39230	−0.174166
$637$	25.8564	1.02447
$638$	7.60770	0.301192
$639$	13.9474	0.551752
$640$	12.1244	0.479257
$641$	−31.1769	−1.23141	−0.615707	−	0.787975i	$-0.711130\pi$
$641$	−31.1769	−1.23141	−0.615707	+	0.787975i	$0.711130\pi$
$642$	−0.430781	−0.0170016
$643$	−13.8038	−0.544371	−0.272185	−	0.962245i	$-0.587746\pi$
$643$	−13.8038	−0.544371	−0.272185	+	0.962245i	$0.587746\pi$
$644$	1.60770	0.0633521
$645$	5.46410	0.215149
$646$	−9.46410	−0.372360
$647$	2.78461	0.109474	0.0547372	−	0.998501i	$-0.482568\pi$
$647$	2.78461	0.109474	0.0547372	+	0.998501i	$0.482568\pi$
$648$	−7.73205	−0.303744
$649$	12.0000	0.471041
$650$	−6.92820	−0.271746
$651$	−3.60770	−0.141397
$652$	10.1962	0.399312
$653$	−46.3923	−1.81547	−0.907736	−	0.419543i	$-0.862190\pi$
$653$	−46.3923	−1.81547	−0.907736	+	0.419543i	$0.862190\pi$
$654$	12.6795	0.495807
$655$	8.19615	0.320250
$656$	−17.3205	−0.676252
$657$	35.4641	1.38359
$658$	−1.17691	−0.0458809
$659$	8.78461	0.342200	0.171100	−	0.985254i	$-0.445268\pi$
$659$	8.78461	0.342200	0.171100	+	0.985254i	$0.445268\pi$
$660$	−0.928203	−0.0361303
$661$	−35.8564	−1.39465	−0.697326	−	0.716754i	$-0.745627\pi$
$661$	−35.8564	−1.39465	−0.697326	+	0.716754i	$0.745627\pi$
$662$	−23.3205	−0.906377
$663$	−2.92820	−0.113722
$664$	−26.7846	−1.03944
$665$	4.00000	0.155113
$666$	31.8564	1.23441
$667$	7.60770	0.294571
$668$	−18.5885	−0.719209
$669$	4.10512	0.158713
$670$	−17.3205	−0.669150
$671$	11.3205	0.437023
$672$	2.78461	0.107419
$673$	16.5359	0.637412	0.318706	−	0.947854i	$-0.396752\pi$
$673$	16.5359	0.637412	0.318706	+	0.947854i	$0.396752\pi$
$674$	60.2487	2.32069
$675$	4.00000	0.153960
$676$	3.00000	0.115385
$677$	−38.7846	−1.49061	−0.745307	−	0.666722i	$-0.767697\pi$
$677$	−38.7846	−1.49061	−0.745307	+	0.666722i	$0.767697\pi$
$678$	−21.9615	−0.843427
$679$	6.53590	0.250825
$680$	1.73205	0.0664211
$681$	−14.1051	−0.540509
$682$	14.7846	0.566132
$683$	48.8372	1.86870	0.934351	−	0.356354i	$-0.115980\pi$
$683$	48.8372	1.86870	0.934351	+	0.356354i	$0.115980\pi$
$684$	−13.4641	−0.514813
$685$	0	0
$686$	−17.0718	−0.651804
$687$	−9.07180	−0.346111
$688$	37.3205	1.42283
$689$	−24.0000	−0.914327
$690$	−2.78461	−0.106008
$691$	12.9808	0.493811	0.246906	−	0.969040i	$-0.420586\pi$
$691$	12.9808	0.493811	0.246906	+	0.969040i	$0.420586\pi$
$692$	−3.46410	−0.131685
$693$	−2.28719	−0.0868831
$694$	24.5885	0.933365
$695$	13.6603	0.518163
$696$	4.39230	0.166490
$697$	−3.46410	−0.131212
$698$	−53.3205	−2.01821
$699$	−4.39230	−0.166132
$700$	0.732051	0.0276689
$701$	−23.3205	−0.880803	−0.440402	−	0.897801i	$-0.645164\pi$
$701$	−23.3205	−0.880803	−0.440402	+	0.897801i	$0.645164\pi$
$702$	−27.7128	−1.04595
$703$	−40.7846	−1.53822
$704$	1.26795	0.0477876
$705$	0.679492	0.0255911
$706$	−25.6077	−0.963758
$707$	−1.85641	−0.0698174
$708$	−6.92820	−0.260378
$709$	11.4641	0.430543	0.215272	−	0.976554i	$-0.430936\pi$
$709$	11.4641	0.430543	0.215272	+	0.976554i	$0.430936\pi$
$710$	−9.80385	−0.367932
$711$	40.8756	1.53296
$712$	28.3923	1.06405
$713$	14.7846	0.553688
$714$	0.928203	0.0347371
$715$	−5.07180	−0.189674
$716$	23.3205	0.871528
$717$	−15.2154	−0.568229
$718$	−25.1769	−0.939594
$719$	−36.5885	−1.36452	−0.682260	−	0.731110i	$-0.739003\pi$
$719$	−36.5885	−1.36452	−0.682260	+	0.731110i	$0.739003\pi$
$720$	12.3205	0.459158
$721$	−3.60770	−0.134358
$722$	18.8038	0.699807
$723$	19.3205	0.718537
$724$	18.3923	0.683545
$725$	3.46410	0.128654
$726$	11.9090	0.441983
$727$	27.8564	1.03314	0.516568	−	0.856246i	$-0.327209\pi$
$727$	27.8564	1.03314	0.516568	+	0.856246i	$0.327209\pi$
$728$	5.07180	0.187973
$729$	−2.21539	−0.0820515
$730$	−24.9282	−0.922634
$731$	7.46410	0.276070
$732$	−6.53590	−0.241574
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	−31.5167	−1.16330
$735$	4.73205	0.174544
$736$	−11.4115	−0.420635
$737$	−12.6795	−0.467055
$738$	−14.7846	−0.544229
$739$	−32.3923	−1.19157	−0.595785	−	0.803144i	$-0.703159\pi$
$739$	−32.3923	−1.19157	−0.595785	+	0.803144i	$0.703159\pi$
$740$	−7.46410	−0.274386
$741$	16.0000	0.587775
$742$	7.60770	0.279287
$743$	−19.2679	−0.706872	−0.353436	−	0.935459i	$-0.614987\pi$
$743$	−19.2679	−0.706872	−0.353436	+	0.935459i	$0.614987\pi$
$744$	8.53590	0.312941
$745$	−6.00000	−0.219823
$746$	34.6410	1.26830
$747$	−38.1051	−1.39419
$748$	−1.26795	−0.0463608
$749$	0.248711	0.00908771
$750$	−1.26795	−0.0462990
$751$	26.3397	0.961151	0.480575	−	0.876953i	$-0.340428\pi$
$751$	26.3397	0.961151	0.480575	+	0.876953i	$0.340428\pi$
$752$	4.64102	0.169240
$753$	5.07180	0.184827
$754$	−24.0000	−0.874028
$755$	5.46410	0.198859
$756$	2.92820	0.106498
$757$	−38.6410	−1.40443	−0.702216	−	0.711964i	$-0.747806\pi$
$757$	−38.6410	−1.40443	−0.702216	+	0.711964i	$0.747806\pi$
$758$	48.8372	1.77385
$759$	−2.03848	−0.0739920
$760$	−9.46410	−0.343299
$761$	−7.60770	−0.275779	−0.137889	−	0.990448i	$-0.544032\pi$
$761$	−7.60770	−0.275779	−0.137889	+	0.990448i	$0.544032\pi$
$762$	−8.10512	−0.293617
$763$	−7.32051	−0.265020
$764$	−25.8564	−0.935452
$765$	2.46410	0.0890898
$766$	14.7846	0.534190
$767$	−37.8564	−1.36692
$768$	−13.9090	−0.501897
$769$	15.6077	0.562828	0.281414	−	0.959586i	$-0.409197\pi$
$769$	15.6077	0.562828	0.281414	+	0.959586i	$0.409197\pi$
$770$	1.60770	0.0579373
$771$	−5.07180	−0.182656
$772$	23.4641	0.844491
$773$	22.6410	0.814341	0.407170	−	0.913352i	$-0.366516\pi$
$773$	22.6410	0.814341	0.407170	+	0.913352i	$0.366516\pi$
$774$	31.8564	1.14505
$775$	6.73205	0.241822
$776$	−15.4641	−0.555129
$777$	4.00000	0.143499
$778$	28.3923	1.01791
$779$	18.9282	0.678173
$780$	2.92820	0.104846
$781$	−7.17691	−0.256810
$782$	−3.80385	−0.136025
$783$	13.8564	0.495188
$784$	32.3205	1.15430
$785$	−4.92820	−0.175895
$786$	−10.3923	−0.370681
$787$	−21.9090	−0.780970	−0.390485	−	0.920609i	$-0.627693\pi$
$787$	−21.9090	−0.780970	−0.390485	+	0.920609i	$0.627693\pi$
$788$	−17.3205	−0.617018
$789$	16.3923	0.583582
$790$	−28.7321	−1.02224
$791$	12.6795	0.450831
$792$	5.41154	0.192291
$793$	−35.7128	−1.26820
$794$	24.2487	0.860555
$795$	−4.39230	−0.155779
$796$	−0.196152	−0.00695244
$797$	−11.0718	−0.392183	−0.196092	−	0.980586i	$-0.562825\pi$
$797$	−11.0718	−0.392183	−0.196092	+	0.980586i	$0.562825\pi$
$798$	−5.07180	−0.179540
$799$	0.928203	0.0328375
$800$	−5.19615	−0.183712
$801$	40.3923	1.42719
$802$	−1.60770	−0.0567697
$803$	−18.2487	−0.643983
$804$	7.32051	0.258174
$805$	1.60770	0.0566638
$806$	−46.6410	−1.64286
$807$	9.46410	0.333152
$808$	4.39230	0.154521
$809$	−28.1436	−0.989476	−0.494738	−	0.869042i	$-0.664736\pi$
$809$	−28.1436	−0.989476	−0.494738	+	0.869042i	$0.664736\pi$
$810$	7.73205	0.271677
$811$	−34.8372	−1.22330	−0.611649	−	0.791129i	$-0.709494\pi$
$811$	−34.8372	−1.22330	−0.611649	+	0.791129i	$0.709494\pi$
$812$	2.53590	0.0889926
$813$	8.00000	0.280572
$814$	−16.3923	−0.574550
$815$	10.1962	0.357156
$816$	−3.66025	−0.128135
$817$	−40.7846	−1.42687
$818$	45.0333	1.57455
$819$	7.21539	0.252126
$820$	3.46410	0.120972
$821$	−11.0718	−0.386408	−0.193204	−	0.981159i	$-0.561888\pi$
$821$	−11.0718	−0.386408	−0.193204	+	0.981159i	$0.561888\pi$
$822$	0	0
$823$	42.9808	1.49822	0.749108	−	0.662448i	$-0.230483\pi$
$823$	42.9808	1.49822	0.749108	+	0.662448i	$0.230483\pi$
$824$	8.53590	0.297362
$825$	−0.928203	−0.0323159
$826$	12.0000	0.417533
$827$	−26.1962	−0.910929	−0.455465	−	0.890254i	$-0.650527\pi$
$827$	−26.1962	−0.910929	−0.455465	+	0.890254i	$0.650527\pi$
$828$	−5.41154	−0.188064
$829$	−37.7128	−1.30982	−0.654910	−	0.755707i	$-0.727294\pi$
$829$	−37.7128	−1.30982	−0.654910	+	0.755707i	$0.727294\pi$
$830$	26.7846	0.929707
$831$	−5.17691	−0.179585
$832$	−4.00000	−0.138675
$833$	6.46410	0.223968
$834$	−17.3205	−0.599760
$835$	−18.5885	−0.643280
$836$	6.92820	0.239617
$837$	26.9282	0.930775
$838$	48.1577	1.66358
$839$	22.7321	0.784798	0.392399	−	0.919795i	$-0.371645\pi$
$839$	22.7321	0.784798	0.392399	+	0.919795i	$0.371645\pi$
$840$	0.928203	0.0320261
$841$	−17.0000	−0.586207
$842$	−2.53590	−0.0873928
$843$	−0.679492	−0.0234029
$844$	−0.196152	−0.00675184
$845$	3.00000	0.103203
$846$	3.96152	0.136200
$847$	−6.87564	−0.236250
$848$	−30.0000	−1.03020
$849$	6.39230	0.219383
$850$	−1.73205	−0.0594089
$851$	−16.3923	−0.561921
$852$	4.14359	0.141957
$853$	39.1769	1.34139	0.670696	−	0.741732i	$-0.265996\pi$
$853$	39.1769	1.34139	0.670696	+	0.741732i	$0.265996\pi$
$854$	11.3205	0.387380
$855$	−13.4641	−0.460463
$856$	−0.588457	−0.0201131
$857$	−31.1769	−1.06498	−0.532492	−	0.846435i	$-0.678744\pi$
$857$	−31.1769	−1.06498	−0.532492	+	0.846435i	$0.678744\pi$
$858$	6.43078	0.219543
$859$	−18.5359	−0.632437	−0.316218	−	0.948686i	$-0.602413\pi$
$859$	−18.5359	−0.632437	−0.316218	+	0.948686i	$0.602413\pi$
$860$	−7.46410	−0.254524
$861$	−1.85641	−0.0632662
$862$	34.9808	1.19145
$863$	36.9282	1.25705	0.628525	−	0.777789i	$-0.283659\pi$
$863$	36.9282	1.25705	0.628525	+	0.777789i	$0.283659\pi$
$864$	−20.7846	−0.707107
$865$	−3.46410	−0.117783
$866$	17.0718	0.580123
$867$	−0.732051	−0.0248617
$868$	4.92820	0.167274
$869$	−21.0333	−0.713507
$870$	−4.39230	−0.148913
$871$	40.0000	1.35535
$872$	17.3205	0.586546
$873$	−22.0000	−0.744587
$874$	20.7846	0.703050
$875$	0.732051	0.0247478
$876$	10.5359	0.355975
$877$	−42.7846	−1.44473	−0.722367	−	0.691510i	$-0.756946\pi$
$877$	−42.7846	−1.44473	−0.722367	+	0.691510i	$0.756946\pi$
$878$	31.2679	1.05524
$879$	−9.46410	−0.319216
$880$	−6.33975	−0.213713
$881$	−6.67949	−0.225038	−0.112519	−	0.993650i	$-0.535892\pi$
$881$	−6.67949	−0.225038	−0.112519	+	0.993650i	$0.535892\pi$
$882$	27.5885	0.928952
$883$	−10.0000	−0.336527	−0.168263	−	0.985742i	$-0.553816\pi$
$883$	−10.0000	−0.336527	−0.168263	+	0.985742i	$0.553816\pi$
$884$	4.00000	0.134535
$885$	−6.92820	−0.232889
$886$	1.60770	0.0540116
$887$	−21.1244	−0.709286	−0.354643	−	0.935002i	$-0.615398\pi$
$887$	−21.1244	−0.709286	−0.354643	+	0.935002i	$0.615398\pi$
$888$	−9.46410	−0.317594
$889$	4.67949	0.156945
$890$	−28.3923	−0.951712
$891$	5.66025	0.189626
$892$	−5.60770	−0.187760
$893$	−5.07180	−0.169721
$894$	7.60770	0.254439
$895$	23.3205	0.779519
$896$	8.87564	0.296514
$897$	6.43078	0.214718
$898$	−23.5692	−0.786515
$899$	23.3205	0.777782
$900$	−2.46410	−0.0821367
$901$	−6.00000	−0.199889
$902$	7.60770	0.253309
$903$	4.00000	0.133112
$904$	−30.0000	−0.997785
$905$	18.3923	0.611381
$906$	−6.92820	−0.230174
$907$	27.2679	0.905417	0.452709	−	0.891658i	$-0.350458\pi$
$907$	27.2679	0.905417	0.452709	+	0.891658i	$0.350458\pi$
$908$	19.2679	0.639429
$909$	6.24871	0.207257
$910$	−5.07180	−0.168128
$911$	53.6603	1.77784	0.888922	−	0.458059i	$-0.151455\pi$
$911$	53.6603	1.77784	0.888922	+	0.458059i	$0.151455\pi$
$912$	20.0000	0.662266
$913$	19.6077	0.648920
$914$	8.28719	0.274116
$915$	−6.53590	−0.216070
$916$	12.3923	0.409453
$917$	6.00000	0.198137
$918$	−6.92820	−0.228665
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	−3.80385	−0.125409
$921$	7.32051	0.241219
$922$	−19.1769	−0.631558
$923$	22.6410	0.745238
$924$	−0.679492	−0.0223536
$925$	−7.46410	−0.245418
$926$	6.67949	0.219502
$927$	12.1436	0.398848
$928$	−18.0000	−0.590879
$929$	3.46410	0.113653	0.0568267	−	0.998384i	$-0.481902\pi$
$929$	3.46410	0.113653	0.0568267	+	0.998384i	$0.481902\pi$
$930$	−8.53590	−0.279903
$931$	−35.3205	−1.15758
$932$	6.00000	0.196537
$933$	−16.1436	−0.528518
$934$	38.7846	1.26907
$935$	−1.26795	−0.0414664
$936$	−17.0718	−0.558009
$937$	36.6410	1.19701	0.598505	−	0.801119i	$-0.295762\pi$
$937$	36.6410	1.19701	0.598505	+	0.801119i	$0.295762\pi$
$938$	−12.6795	−0.414000
$939$	4.10512	0.133965
$940$	−0.928203	−0.0302747
$941$	−0.248711	−0.00810776	−0.00405388	−	0.999992i	$-0.501290\pi$
$941$	−0.248711	−0.00810776	−0.00405388	+	0.999992i	$0.501290\pi$
$942$	6.24871	0.203594
$943$	7.60770	0.247741
$944$	−47.3205	−1.54015
$945$	2.92820	0.0952545
$946$	−16.3923	−0.532960
$947$	−2.19615	−0.0713654	−0.0356827	−	0.999363i	$-0.511361\pi$
$947$	−2.19615	−0.0713654	−0.0356827	+	0.999363i	$0.511361\pi$
$948$	12.1436	0.394406
$949$	57.5692	1.86878
$950$	9.46410	0.307056
$951$	8.10512	0.262826
$952$	1.26795	0.0410945
$953$	32.7846	1.06200	0.530999	−	0.847373i	$-0.321817\pi$
$953$	32.7846	1.06200	0.530999	+	0.847373i	$0.321817\pi$
$954$	−25.6077	−0.829080
$955$	−25.8564	−0.836694
$956$	20.7846	0.672222
$957$	−3.21539	−0.103939
$958$	9.80385	0.316748
$959$	0	0
$960$	−0.732051	−0.0236268
$961$	14.3205	0.461952
$962$	51.7128	1.66729
$963$	−0.837169	−0.0269774
$964$	−26.3923	−0.850039
$965$	23.4641	0.755336
$966$	−2.03848	−0.0655869
$967$	−23.1769	−0.745319	−0.372660	−	0.927968i	$-0.621554\pi$
$967$	−23.1769	−0.745319	−0.372660	+	0.927968i	$0.621554\pi$
$968$	16.2679	0.522872
$969$	4.00000	0.128499
$970$	15.4641	0.496522
$971$	5.75129	0.184568	0.0922838	−	0.995733i	$-0.470583\pi$
$971$	5.75129	0.184568	0.0922838	+	0.995733i	$0.470583\pi$
$972$	−15.2679	−0.489720
$973$	10.0000	0.320585
$974$	−46.7321	−1.49739
$975$	2.92820	0.0937776
$976$	−44.6410	−1.42892
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	−12.9282	−0.413398
$979$	−20.7846	−0.664279
$980$	−6.46410	−0.206488
$981$	24.6410	0.786727
$982$	−69.9615	−2.23256
$983$	55.2679	1.76277	0.881387	−	0.472395i	$-0.156610\pi$
$983$	55.2679	1.76277	0.881387	+	0.472395i	$0.156610\pi$
$984$	4.39230	0.140022
$985$	−17.3205	−0.551877
$986$	−6.00000	−0.191079
$987$	0.497423	0.0158331
$988$	−21.8564	−0.695345
$989$	−16.3923	−0.521245
$990$	−5.41154	−0.171990
$991$	48.9808	1.55593	0.777963	−	0.628311i	$-0.216253\pi$
$991$	48.9808	1.55593	0.777963	+	0.628311i	$0.216253\pi$
$992$	−34.9808	−1.11064
$993$	9.85641	0.312784
$994$	−7.17691	−0.227638
$995$	−0.196152	−0.00621845
$996$	−11.3205	−0.358704
$997$	18.3923	0.582490	0.291245	−	0.956648i	$-0.405930\pi$
$997$	18.3923	0.582490	0.291245	+	0.956648i	$0.405930\pi$
$998$	2.87564	0.0910269
$999$	−29.8564	−0.944615

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	85.2.a.c.1.2	✓	2
3.2	odd	2		765.2.a.g.1.1		2
4.3	odd	2		1360.2.a.k.1.2		2
5.2	odd	4		425.2.b.d.324.3		4
5.3	odd	4		425.2.b.d.324.2		4
5.4	even	2		425.2.a.e.1.1		2
7.6	odd	2		4165.2.a.t.1.2		2
8.3	odd	2		5440.2.a.bl.1.1		2
8.5	even	2		5440.2.a.bb.1.2		2
15.14	odd	2		3825.2.a.v.1.2		2
17.4	even	4		1445.2.d.e.866.2		4
17.13	even	4		1445.2.d.e.866.1		4
17.16	even	2		1445.2.a.g.1.2		2
20.19	odd	2		6800.2.a.bg.1.1		2
85.84	even	2		7225.2.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.a.c.1.2	✓	2	1.1	even	1	trivial
425.2.a.e.1.1		2	5.4	even	2
425.2.b.d.324.2		4	5.3	odd	4
425.2.b.d.324.3		4	5.2	odd	4
765.2.a.g.1.1		2	3.2	odd	2
1360.2.a.k.1.2		2	4.3	odd	2
1445.2.a.g.1.2		2	17.16	even	2
1445.2.d.e.866.1		4	17.13	even	4
1445.2.d.e.866.2		4	17.4	even	4
3825.2.a.v.1.2		2	15.14	odd	2
4165.2.a.t.1.2		2	7.6	odd	2
5440.2.a.bb.1.2		2	8.5	even	2
5440.2.a.bl.1.1		2	8.3	odd	2
6800.2.a.bg.1.1		2	20.19	odd	2
7225.2.a.l.1.1		2	85.84	even	2

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:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	85.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} -0.732051 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.26795 q^{6} +0.732051 q^{7} -1.73205 q^{8} -2.46410 q^{9} +1.73205 q^{10} +1.26795 q^{11} -0.732051 q^{12} -4.00000 q^{13} +1.26795 q^{14} -0.732051 q^{15} -5.00000 q^{16} -1.00000 q^{17} -4.26795 q^{18} +5.46410 q^{19} +1.00000 q^{20} -0.535898 q^{21} +2.19615 q^{22} +2.19615 q^{23} +1.26795 q^{24} +1.00000 q^{25} -6.92820 q^{26} +4.00000 q^{27} +0.732051 q^{28} +3.46410 q^{29} -1.26795 q^{30} +6.73205 q^{31} -5.19615 q^{32} -0.928203 q^{33} -1.73205 q^{34} +0.732051 q^{35} -2.46410 q^{36} -7.46410 q^{37} +9.46410 q^{38} +2.92820 q^{39} -1.73205 q^{40} +3.46410 q^{41} -0.928203 q^{42} -7.46410 q^{43} +1.26795 q^{44} -2.46410 q^{45} +3.80385 q^{46} -0.928203 q^{47} +3.66025 q^{48} -6.46410 q^{49} +1.73205 q^{50} +0.732051 q^{51} -4.00000 q^{52} +6.00000 q^{53} +6.92820 q^{54} +1.26795 q^{55} -1.26795 q^{56} -4.00000 q^{57} +6.00000 q^{58} +9.46410 q^{59} -0.732051 q^{60} +8.92820 q^{61} +11.6603 q^{62} -1.80385 q^{63} +1.00000 q^{64} -4.00000 q^{65} -1.60770 q^{66} -10.0000 q^{67} -1.00000 q^{68} -1.60770 q^{69} +1.26795 q^{70} -5.66025 q^{71} +4.26795 q^{72} -14.3923 q^{73} -12.9282 q^{74} -0.732051 q^{75} +5.46410 q^{76} +0.928203 q^{77} +5.07180 q^{78} -16.5885 q^{79} -5.00000 q^{80} +4.46410 q^{81} +6.00000 q^{82} +15.4641 q^{83} -0.535898 q^{84} -1.00000 q^{85} -12.9282 q^{86} -2.53590 q^{87} -2.19615 q^{88} -16.3923 q^{89} -4.26795 q^{90} -2.92820 q^{91} +2.19615 q^{92} -4.92820 q^{93} -1.60770 q^{94} +5.46410 q^{95} +3.80385 q^{96} +8.92820 q^{97} -11.1962 q^{98} -3.12436 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{4} + 2 q^{5} - 6 q^{6} - 2 q^{7} + 2 q^{9} + 6 q^{11} + 2 q^{12} - 8 q^{13} + 6 q^{14} + 2 q^{15} - 10 q^{16} - 2 q^{17} - 12 q^{18} + 4 q^{19} + 2 q^{20} - 8 q^{21} - 6 q^{22} - 6 q^{23}+ \cdots + 18 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^4 + 2 * q^5 - 6 * q^6 - 2 * q^7 + 2 * q^9 + 6 * q^11 + 2 * q^12 - 8 * q^13 + 6 * q^14 + 2 * q^15 - 10 * q^16 - 2 * q^17 - 12 * q^18 + 4 * q^19 + 2 * q^20 - 8 * q^21 - 6 * q^22 - 6 * q^23 + 6 * q^24 + 2 * q^25 + 8 * q^27 - 2 * q^28 - 6 * q^30 + 10 * q^31 + 12 * q^33 - 2 * q^35 + 2 * q^36 - 8 * q^37 + 12 * q^38 - 8 * q^39 + 12 * q^42 - 8 * q^43 + 6 * q^44 + 2 * q^45 + 18 * q^46 + 12 * q^47 - 10 * q^48 - 6 * q^49 - 2 * q^51 - 8 * q^52 + 12 * q^53 + 6 * q^55 - 6 * q^56 - 8 * q^57 + 12 * q^58 + 12 * q^59 + 2 * q^60 + 4 * q^61 + 6 * q^62 - 14 * q^63 + 2 * q^64 - 8 * q^65 - 24 * q^66 - 20 * q^67 - 2 * q^68 - 24 * q^69 + 6 * q^70 + 6 * q^71 + 12 * q^72 - 8 * q^73 - 12 * q^74 + 2 * q^75 + 4 * q^76 - 12 * q^77 + 24 * q^78 - 2 * q^79 - 10 * q^80 + 2 * q^81 + 12 * q^82 + 24 * q^83 - 8 * q^84 - 2 * q^85 - 12 * q^86 - 12 * q^87 + 6 * q^88 - 12 * q^89 - 12 * q^90 + 8 * q^91 - 6 * q^92 + 4 * q^93 - 24 * q^94 + 4 * q^95 + 18 * q^96 + 4 * q^97 - 12 * q^98 + 18 * q^99

Introduction

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