LMFDB - Embedded newform 770.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [770,2,Mod(1,770)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(770, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("770.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 770 = 2 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	770.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:	$6.14848095564$
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:	$\Z[a_1, a_2, a_3]$
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	$\chi$	$=$	770.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.73205 q^{6} +1.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.73205 q^{6} +1.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} -1.00000 q^{10} +1.00000 q^{11} +2.73205 q^{12} -1.46410 q^{13} +1.00000 q^{14} -2.73205 q^{15} +1.00000 q^{16} -3.46410 q^{17} +4.46410 q^{18} +6.73205 q^{19} -1.00000 q^{20} +2.73205 q^{21} +1.00000 q^{22} -8.19615 q^{23} +2.73205 q^{24} +1.00000 q^{25} -1.46410 q^{26} +4.00000 q^{27} +1.00000 q^{28} -4.73205 q^{29} -2.73205 q^{30} +2.00000 q^{31} +1.00000 q^{32} +2.73205 q^{33} -3.46410 q^{34} -1.00000 q^{35} +4.46410 q^{36} +0.732051 q^{37} +6.73205 q^{38} -4.00000 q^{39} -1.00000 q^{40} -2.19615 q^{41} +2.73205 q^{42} +2.00000 q^{43} +1.00000 q^{44} -4.46410 q^{45} -8.19615 q^{46} +6.92820 q^{47} +2.73205 q^{48} +1.00000 q^{49} +1.00000 q^{50} -9.46410 q^{51} -1.46410 q^{52} -7.26795 q^{53} +4.00000 q^{54} -1.00000 q^{55} +1.00000 q^{56} +18.3923 q^{57} -4.73205 q^{58} +6.92820 q^{59} -2.73205 q^{60} -4.92820 q^{61} +2.00000 q^{62} +4.46410 q^{63} +1.00000 q^{64} +1.46410 q^{65} +2.73205 q^{66} -4.00000 q^{67} -3.46410 q^{68} -22.3923 q^{69} -1.00000 q^{70} +9.46410 q^{71} +4.46410 q^{72} -14.3923 q^{73} +0.732051 q^{74} +2.73205 q^{75} +6.73205 q^{76} +1.00000 q^{77} -4.00000 q^{78} -12.1962 q^{79} -1.00000 q^{80} -2.46410 q^{81} -2.19615 q^{82} -16.3923 q^{83} +2.73205 q^{84} +3.46410 q^{85} +2.00000 q^{86} -12.9282 q^{87} +1.00000 q^{88} +3.46410 q^{89} -4.46410 q^{90} -1.46410 q^{91} -8.19615 q^{92} +5.46410 q^{93} +6.92820 q^{94} -6.73205 q^{95} +2.73205 q^{96} +14.5885 q^{97} +1.00000 q^{98} +4.46410 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 2 * q^7 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} + 2 q^{12} + 4 q^{13} + 2 q^{14} - 2 q^{15} + 2 q^{16} + 2 q^{18} + 10 q^{19} - 2 q^{20} + 2 q^{21} + 2 q^{22} - 6 q^{23} + 2 q^{24} + 2 q^{25} + 4 q^{26} + 8 q^{27} + 2 q^{28} - 6 q^{29} - 2 q^{30} + 4 q^{31} + 2 q^{32} + 2 q^{33} - 2 q^{35} + 2 q^{36} - 2 q^{37} + 10 q^{38} - 8 q^{39} - 2 q^{40} + 6 q^{41} + 2 q^{42} + 4 q^{43} + 2 q^{44} - 2 q^{45} - 6 q^{46} + 2 q^{48} + 2 q^{49} + 2 q^{50} - 12 q^{51} + 4 q^{52} - 18 q^{53} + 8 q^{54} - 2 q^{55} + 2 q^{56} + 16 q^{57} - 6 q^{58} - 2 q^{60} + 4 q^{61} + 4 q^{62} + 2 q^{63} + 2 q^{64} - 4 q^{65} + 2 q^{66} - 8 q^{67} - 24 q^{69} - 2 q^{70} + 12 q^{71} + 2 q^{72} - 8 q^{73} - 2 q^{74} + 2 q^{75} + 10 q^{76} + 2 q^{77} - 8 q^{78} - 14 q^{79} - 2 q^{80} + 2 q^{81} + 6 q^{82} - 12 q^{83} + 2 q^{84} + 4 q^{86} - 12 q^{87} + 2 q^{88} - 2 q^{90} + 4 q^{91} - 6 q^{92} + 4 q^{93} - 10 q^{95} + 2 q^{96} - 2 q^{97} + 2 q^{98} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 2 * q^7 + 2 * q^8 + 2 * q^9 - 2 * q^10 + 2 * q^11 + 2 * q^12 + 4 * q^13 + 2 * q^14 - 2 * q^15 + 2 * q^16 + 2 * q^18 + 10 * q^19 - 2 * q^20 + 2 * q^21 + 2 * q^22 - 6 * q^23 + 2 * q^24 + 2 * q^25 + 4 * q^26 + 8 * q^27 + 2 * q^28 - 6 * q^29 - 2 * q^30 + 4 * q^31 + 2 * q^32 + 2 * q^33 - 2 * q^35 + 2 * q^36 - 2 * q^37 + 10 * q^38 - 8 * q^39 - 2 * q^40 + 6 * q^41 + 2 * q^42 + 4 * q^43 + 2 * q^44 - 2 * q^45 - 6 * q^46 + 2 * q^48 + 2 * q^49 + 2 * q^50 - 12 * q^51 + 4 * q^52 - 18 * q^53 + 8 * q^54 - 2 * q^55 + 2 * q^56 + 16 * q^57 - 6 * q^58 - 2 * q^60 + 4 * q^61 + 4 * q^62 + 2 * q^63 + 2 * q^64 - 4 * q^65 + 2 * q^66 - 8 * q^67 - 24 * q^69 - 2 * q^70 + 12 * q^71 + 2 * q^72 - 8 * q^73 - 2 * q^74 + 2 * q^75 + 10 * q^76 + 2 * q^77 - 8 * q^78 - 14 * q^79 - 2 * q^80 + 2 * q^81 + 6 * q^82 - 12 * q^83 + 2 * q^84 + 4 * q^86 - 12 * q^87 + 2 * q^88 - 2 * q^90 + 4 * q^91 - 6 * q^92 + 4 * q^93 - 10 * q^95 + 2 * q^96 - 2 * q^97 + 2 * q^98 + 2 * 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
$3$	2.73205	1.57735	0.788675	−	0.614810i	$-0.210767\pi$
$3$	2.73205	1.57735	0.788675	+	0.614810i	$0.210767\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.73205	1.11536
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	4.46410	1.48803
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	2.73205	0.788675
$13$	−1.46410	−0.406069	−0.203034	−	0.979172i	$-0.565080\pi$
$13$	−1.46410	−0.406069	−0.203034	+	0.979172i	$0.565080\pi$
$14$	1.00000	0.267261
$15$	−2.73205	−0.705412
$16$	1.00000	0.250000
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	4.46410	1.05220
$19$	6.73205	1.54444	0.772219	−	0.635356i	$-0.219147\pi$
$19$	6.73205	1.54444	0.772219	+	0.635356i	$0.219147\pi$
$20$	−1.00000	−0.223607
$21$	2.73205	0.596182
$22$	1.00000	0.213201
$23$	−8.19615	−1.70902	−0.854508	−	0.519438i	$-0.826141\pi$
$23$	−8.19615	−1.70902	−0.854508	+	0.519438i	$0.826141\pi$
$24$	2.73205	0.557678
$25$	1.00000	0.200000
$26$	−1.46410	−0.287134
$27$	4.00000	0.769800
$28$	1.00000	0.188982
$29$	−4.73205	−0.878720	−0.439360	−	0.898311i	$-0.644795\pi$
$29$	−4.73205	−0.878720	−0.439360	+	0.898311i	$0.644795\pi$
$30$	−2.73205	−0.498802
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000	0.176777
$33$	2.73205	0.475589
$34$	−3.46410	−0.594089
$35$	−1.00000	−0.169031
$36$	4.46410	0.744017
$37$	0.732051	0.120348	0.0601742	−	0.998188i	$-0.480834\pi$
$37$	0.732051	0.120348	0.0601742	+	0.998188i	$0.480834\pi$
$38$	6.73205	1.09208
$39$	−4.00000	−0.640513
$40$	−1.00000	−0.158114
$41$	−2.19615	−0.342981	−0.171491	−	0.985186i	$-0.554858\pi$
$41$	−2.19615	−0.342981	−0.171491	+	0.985186i	$0.554858\pi$
$42$	2.73205	0.421565
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	1.00000	0.150756
$45$	−4.46410	−0.665469
$46$	−8.19615	−1.20846
$47$	6.92820	1.01058	0.505291	−	0.862949i	$-0.331385\pi$
$47$	6.92820	1.01058	0.505291	+	0.862949i	$0.331385\pi$
$48$	2.73205	0.394338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−9.46410	−1.32524
$52$	−1.46410	−0.203034
$53$	−7.26795	−0.998330	−0.499165	−	0.866507i	$-0.666360\pi$
$53$	−7.26795	−0.998330	−0.499165	+	0.866507i	$0.666360\pi$
$54$	4.00000	0.544331
$55$	−1.00000	−0.134840
$56$	1.00000	0.133631
$57$	18.3923	2.43612
$58$	−4.73205	−0.621349
$59$	6.92820	0.901975	0.450988	−	0.892530i	$-0.351072\pi$
$59$	6.92820	0.901975	0.450988	+	0.892530i	$0.351072\pi$
$60$	−2.73205	−0.352706
$61$	−4.92820	−0.630992	−0.315496	−	0.948927i	$-0.602171\pi$
$61$	−4.92820	−0.630992	−0.315496	+	0.948927i	$0.602171\pi$
$62$	2.00000	0.254000
$63$	4.46410	0.562424
$64$	1.00000	0.125000
$65$	1.46410	0.181599
$66$	2.73205	0.336292
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−3.46410	−0.420084
$69$	−22.3923	−2.69572
$70$	−1.00000	−0.119523
$71$	9.46410	1.12318	0.561591	−	0.827415i	$-0.310189\pi$
$71$	9.46410	1.12318	0.561591	+	0.827415i	$0.310189\pi$
$72$	4.46410	0.526099
$73$	−14.3923	−1.68449	−0.842246	−	0.539093i	$-0.818767\pi$
$73$	−14.3923	−1.68449	−0.842246	+	0.539093i	$0.818767\pi$
$74$	0.732051	0.0850992
$75$	2.73205	0.315470
$76$	6.73205	0.772219
$77$	1.00000	0.113961
$78$	−4.00000	−0.452911
$79$	−12.1962	−1.37217	−0.686087	−	0.727519i	$-0.740673\pi$
$79$	−12.1962	−1.37217	−0.686087	+	0.727519i	$0.740673\pi$
$80$	−1.00000	−0.111803
$81$	−2.46410	−0.273789
$82$	−2.19615	−0.242524
$83$	−16.3923	−1.79929	−0.899645	−	0.436623i	$-0.856174\pi$
$83$	−16.3923	−1.79929	−0.899645	+	0.436623i	$0.856174\pi$
$84$	2.73205	0.298091
$85$	3.46410	0.375735
$86$	2.00000	0.215666
$87$	−12.9282	−1.38605
$88$	1.00000	0.106600
$89$	3.46410	0.367194	0.183597	−	0.983002i	$-0.441226\pi$
$89$	3.46410	0.367194	0.183597	+	0.983002i	$0.441226\pi$
$90$	−4.46410	−0.470558
$91$	−1.46410	−0.153480
$92$	−8.19615	−0.854508
$93$	5.46410	0.566601
$94$	6.92820	0.714590
$95$	−6.73205	−0.690694
$96$	2.73205	0.278839
$97$	14.5885	1.48123	0.740617	−	0.671928i	$-0.234533\pi$
$97$	14.5885	1.48123	0.740617	+	0.671928i	$0.234533\pi$
$98$	1.00000	0.101015
$99$	4.46410	0.448659
$100$	1.00000	0.100000
$101$	−7.85641	−0.781742	−0.390871	−	0.920446i	$-0.627826\pi$
$101$	−7.85641	−0.781742	−0.390871	+	0.920446i	$0.627826\pi$
$102$	−9.46410	−0.937086
$103$	12.3923	1.22105	0.610525	−	0.791997i	$-0.290958\pi$
$103$	12.3923	1.22105	0.610525	+	0.791997i	$0.290958\pi$
$104$	−1.46410	−0.143567
$105$	−2.73205	−0.266621
$106$	−7.26795	−0.705926
$107$	7.85641	0.759507	0.379754	−	0.925088i	$-0.376009\pi$
$107$	7.85641	0.759507	0.379754	+	0.925088i	$0.376009\pi$
$108$	4.00000	0.384900
$109$	−15.6603	−1.49998	−0.749990	−	0.661449i	$-0.769942\pi$
$109$	−15.6603	−1.49998	−0.749990	+	0.661449i	$0.769942\pi$
$110$	−1.00000	−0.0953463
$111$	2.00000	0.189832
$112$	1.00000	0.0944911
$113$	7.85641	0.739069	0.369534	−	0.929217i	$-0.379517\pi$
$113$	7.85641	0.739069	0.369534	+	0.929217i	$0.379517\pi$
$114$	18.3923	1.72260
$115$	8.19615	0.764295
$116$	−4.73205	−0.439360
$117$	−6.53590	−0.604244
$118$	6.92820	0.637793
$119$	−3.46410	−0.317554
$120$	−2.73205	−0.249401
$121$	1.00000	0.0909091
$122$	−4.92820	−0.446179
$123$	−6.00000	−0.541002
$124$	2.00000	0.179605
$125$	−1.00000	−0.0894427
$126$	4.46410	0.397694
$127$	1.07180	0.0951066	0.0475533	−	0.998869i	$-0.484858\pi$
$127$	1.07180	0.0951066	0.0475533	+	0.998869i	$0.484858\pi$
$128$	1.00000	0.0883883
$129$	5.46410	0.481087
$130$	1.46410	0.128410
$131$	5.66025	0.494539	0.247269	−	0.968947i	$-0.420467\pi$
$131$	5.66025	0.494539	0.247269	+	0.968947i	$0.420467\pi$
$132$	2.73205	0.237795
$133$	6.73205	0.583743
$134$	−4.00000	−0.345547
$135$	−4.00000	−0.344265
$136$	−3.46410	−0.297044
$137$	−0.928203	−0.0793018	−0.0396509	−	0.999214i	$-0.512625\pi$
$137$	−0.928203	−0.0793018	−0.0396509	+	0.999214i	$0.512625\pi$
$138$	−22.3923	−1.90616
$139$	13.6603	1.15865	0.579324	−	0.815097i	$-0.303317\pi$
$139$	13.6603	1.15865	0.579324	+	0.815097i	$0.303317\pi$
$140$	−1.00000	−0.0845154
$141$	18.9282	1.59404
$142$	9.46410	0.794210
$143$	−1.46410	−0.122434
$144$	4.46410	0.372008
$145$	4.73205	0.392975
$146$	−14.3923	−1.19112
$147$	2.73205	0.225336
$148$	0.732051	0.0601742
$149$	−7.26795	−0.595414	−0.297707	−	0.954657i	$-0.596222\pi$
$149$	−7.26795	−0.595414	−0.297707	+	0.954657i	$0.596222\pi$
$150$	2.73205	0.223071
$151$	11.1244	0.905287	0.452644	−	0.891692i	$-0.350481\pi$
$151$	11.1244	0.905287	0.452644	+	0.891692i	$0.350481\pi$
$152$	6.73205	0.546041
$153$	−15.4641	−1.25020
$154$	1.00000	0.0805823
$155$	−2.00000	−0.160644
$156$	−4.00000	−0.320256
$157$	4.53590	0.362004	0.181002	−	0.983483i	$-0.442066\pi$
$157$	4.53590	0.362004	0.181002	+	0.983483i	$0.442066\pi$
$158$	−12.1962	−0.970274
$159$	−19.8564	−1.57472
$160$	−1.00000	−0.0790569
$161$	−8.19615	−0.645947
$162$	−2.46410	−0.193598
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−2.19615	−0.171491
$165$	−2.73205	−0.212690
$166$	−16.3923	−1.27229
$167$	13.8564	1.07224	0.536120	−	0.844141i	$-0.319889\pi$
$167$	13.8564	1.07224	0.536120	+	0.844141i	$0.319889\pi$
$168$	2.73205	0.210782
$169$	−10.8564	−0.835108
$170$	3.46410	0.265684
$171$	30.0526	2.29818
$172$	2.00000	0.152499
$173$	0.928203	0.0705700	0.0352850	−	0.999377i	$-0.488766\pi$
$173$	0.928203	0.0705700	0.0352850	+	0.999377i	$0.488766\pi$
$174$	−12.9282	−0.980085
$175$	1.00000	0.0755929
$176$	1.00000	0.0753778
$177$	18.9282	1.42273
$178$	3.46410	0.259645
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	−4.46410	−0.332734
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	−1.46410	−0.108526
$183$	−13.4641	−0.995295
$184$	−8.19615	−0.604228
$185$	−0.732051	−0.0538214
$186$	5.46410	0.400647
$187$	−3.46410	−0.253320
$188$	6.92820	0.505291
$189$	4.00000	0.290957
$190$	−6.73205	−0.488394
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	2.73205	0.197169
$193$	12.3923	0.892018	0.446009	−	0.895029i	$-0.352845\pi$
$193$	12.3923	0.892018	0.446009	+	0.895029i	$0.352845\pi$
$194$	14.5885	1.04739
$195$	4.00000	0.286446
$196$	1.00000	0.0714286
$197$	−24.2487	−1.72765	−0.863825	−	0.503793i	$-0.831938\pi$
$197$	−24.2487	−1.72765	−0.863825	+	0.503793i	$0.831938\pi$
$198$	4.46410	0.317250
$199$	2.92820	0.207575	0.103787	−	0.994600i	$-0.466904\pi$
$199$	2.92820	0.207575	0.103787	+	0.994600i	$0.466904\pi$
$200$	1.00000	0.0707107
$201$	−10.9282	−0.770816
$202$	−7.85641	−0.552775
$203$	−4.73205	−0.332125
$204$	−9.46410	−0.662620
$205$	2.19615	0.153386
$206$	12.3923	0.863413
$207$	−36.5885	−2.54307
$208$	−1.46410	−0.101517
$209$	6.73205	0.465666
$210$	−2.73205	−0.188529
$211$	26.9282	1.85381	0.926907	−	0.375291i	$-0.122457\pi$
$211$	26.9282	1.85381	0.926907	+	0.375291i	$0.122457\pi$
$212$	−7.26795	−0.499165
$213$	25.8564	1.77165
$214$	7.85641	0.537053
$215$	−2.00000	−0.136399
$216$	4.00000	0.272166
$217$	2.00000	0.135769
$218$	−15.6603	−1.06065
$219$	−39.3205	−2.65703
$220$	−1.00000	−0.0674200
$221$	5.07180	0.341166
$222$	2.00000	0.134231
$223$	−25.4641	−1.70520	−0.852601	−	0.522562i	$-0.824976\pi$
$223$	−25.4641	−1.70520	−0.852601	+	0.522562i	$0.824976\pi$
$224$	1.00000	0.0668153
$225$	4.46410	0.297607
$226$	7.85641	0.522600
$227$	6.92820	0.459841	0.229920	−	0.973209i	$-0.426153\pi$
$227$	6.92820	0.459841	0.229920	+	0.973209i	$0.426153\pi$
$228$	18.3923	1.21806
$229$	24.3923	1.61189	0.805944	−	0.591991i	$-0.201658\pi$
$229$	24.3923	1.61189	0.805944	+	0.591991i	$0.201658\pi$
$230$	8.19615	0.540438
$231$	2.73205	0.179756
$232$	−4.73205	−0.310674
$233$	−7.85641	−0.514690	−0.257345	−	0.966320i	$-0.582848\pi$
$233$	−7.85641	−0.514690	−0.257345	+	0.966320i	$0.582848\pi$
$234$	−6.53590	−0.427265
$235$	−6.92820	−0.451946
$236$	6.92820	0.450988
$237$	−33.3205	−2.16440
$238$	−3.46410	−0.224544
$239$	1.26795	0.0820168	0.0410084	−	0.999159i	$-0.486943\pi$
$239$	1.26795	0.0820168	0.0410084	+	0.999159i	$0.486943\pi$
$240$	−2.73205	−0.176353
$241$	3.26795	0.210507	0.105254	−	0.994445i	$-0.466435\pi$
$241$	3.26795	0.210507	0.105254	+	0.994445i	$0.466435\pi$
$242$	1.00000	0.0642824
$243$	−18.7321	−1.20166
$244$	−4.92820	−0.315496
$245$	−1.00000	−0.0638877
$246$	−6.00000	−0.382546
$247$	−9.85641	−0.627148
$248$	2.00000	0.127000
$249$	−44.7846	−2.83811
$250$	−1.00000	−0.0632456
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	4.46410	0.281212
$253$	−8.19615	−0.515288
$254$	1.07180	0.0672505
$255$	9.46410	0.592665
$256$	1.00000	0.0625000
$257$	−23.6603	−1.47589	−0.737943	−	0.674863i	$-0.764203\pi$
$257$	−23.6603	−1.47589	−0.737943	+	0.674863i	$0.764203\pi$
$258$	5.46410	0.340180
$259$	0.732051	0.0454874
$260$	1.46410	0.0907997
$261$	−21.1244	−1.30756
$262$	5.66025	0.349692
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	2.73205	0.168146
$265$	7.26795	0.446467
$266$	6.73205	0.412769
$267$	9.46410	0.579194
$268$	−4.00000	−0.244339
$269$	28.3923	1.73111	0.865555	−	0.500814i	$-0.166966\pi$
$269$	28.3923	1.73111	0.865555	+	0.500814i	$0.166966\pi$
$270$	−4.00000	−0.243432
$271$	0.392305	0.0238308	0.0119154	−	0.999929i	$-0.496207\pi$
$271$	0.392305	0.0238308	0.0119154	+	0.999929i	$0.496207\pi$
$272$	−3.46410	−0.210042
$273$	−4.00000	−0.242091
$274$	−0.928203	−0.0560748
$275$	1.00000	0.0603023
$276$	−22.3923	−1.34786
$277$	7.07180	0.424903	0.212452	−	0.977172i	$-0.431855\pi$
$277$	7.07180	0.424903	0.212452	+	0.977172i	$0.431855\pi$
$278$	13.6603	0.819288
$279$	8.92820	0.534518
$280$	−1.00000	−0.0597614
$281$	22.3923	1.33581	0.667906	−	0.744245i	$-0.267191\pi$
$281$	22.3923	1.33581	0.667906	+	0.744245i	$0.267191\pi$
$282$	18.9282	1.12716
$283$	−31.7128	−1.88513	−0.942566	−	0.334021i	$-0.891594\pi$
$283$	−31.7128	−1.88513	−0.942566	+	0.334021i	$0.891594\pi$
$284$	9.46410	0.561591
$285$	−18.3923	−1.08947
$286$	−1.46410	−0.0865741
$287$	−2.19615	−0.129635
$288$	4.46410	0.263050
$289$	−5.00000	−0.294118
$290$	4.73205	0.277876
$291$	39.8564	2.33642
$292$	−14.3923	−0.842246
$293$	21.4641	1.25395	0.626973	−	0.779041i	$-0.284294\pi$
$293$	21.4641	1.25395	0.626973	+	0.779041i	$0.284294\pi$
$294$	2.73205	0.159336
$295$	−6.92820	−0.403376
$296$	0.732051	0.0425496
$297$	4.00000	0.232104
$298$	−7.26795	−0.421021
$299$	12.0000	0.693978
$300$	2.73205	0.157735
$301$	2.00000	0.115278
$302$	11.1244	0.640135
$303$	−21.4641	−1.23308
$304$	6.73205	0.386110
$305$	4.92820	0.282188
$306$	−15.4641	−0.884024
$307$	24.3923	1.39214	0.696071	−	0.717973i	$-0.254930\pi$
$307$	24.3923	1.39214	0.696071	+	0.717973i	$0.254930\pi$
$308$	1.00000	0.0569803
$309$	33.8564	1.92602
$310$	−2.00000	−0.113592
$311$	24.9282	1.41355	0.706774	−	0.707439i	$-0.250150\pi$
$311$	24.9282	1.41355	0.706774	+	0.707439i	$0.250150\pi$
$312$	−4.00000	−0.226455
$313$	22.1962	1.25460	0.627300	−	0.778777i	$-0.284160\pi$
$313$	22.1962	1.25460	0.627300	+	0.778777i	$0.284160\pi$
$314$	4.53590	0.255976
$315$	−4.46410	−0.251524
$316$	−12.1962	−0.686087
$317$	30.5885	1.71802	0.859009	−	0.511960i	$-0.171080\pi$
$317$	30.5885	1.71802	0.859009	+	0.511960i	$0.171080\pi$
$318$	−19.8564	−1.11349
$319$	−4.73205	−0.264944
$320$	−1.00000	−0.0559017
$321$	21.4641	1.19801
$322$	−8.19615	−0.456754
$323$	−23.3205	−1.29759
$324$	−2.46410	−0.136895
$325$	−1.46410	−0.0812137
$326$	−4.00000	−0.221540
$327$	−42.7846	−2.36599
$328$	−2.19615	−0.121262
$329$	6.92820	0.381964
$330$	−2.73205	−0.150394
$331$	−18.7846	−1.03250	−0.516248	−	0.856439i	$-0.672672\pi$
$331$	−18.7846	−1.03250	−0.516248	+	0.856439i	$0.672672\pi$
$332$	−16.3923	−0.899645
$333$	3.26795	0.179083
$334$	13.8564	0.758189
$335$	4.00000	0.218543
$336$	2.73205	0.149046
$337$	22.7846	1.24116	0.620578	−	0.784144i	$-0.286898\pi$
$337$	22.7846	1.24116	0.620578	+	0.784144i	$0.286898\pi$
$338$	−10.8564	−0.590511
$339$	21.4641	1.16577
$340$	3.46410	0.187867
$341$	2.00000	0.108306
$342$	30.0526	1.62506
$343$	1.00000	0.0539949
$344$	2.00000	0.107833
$345$	22.3923	1.20556
$346$	0.928203	0.0499005
$347$	−23.0718	−1.23856	−0.619279	−	0.785171i	$-0.712575\pi$
$347$	−23.0718	−1.23856	−0.619279	+	0.785171i	$0.712575\pi$
$348$	−12.9282	−0.693024
$349$	−5.60770	−0.300173	−0.150087	−	0.988673i	$-0.547955\pi$
$349$	−5.60770	−0.300173	−0.150087	+	0.988673i	$0.547955\pi$
$350$	1.00000	0.0534522
$351$	−5.85641	−0.312592
$352$	1.00000	0.0533002
$353$	14.1962	0.755585	0.377792	−	0.925890i	$-0.376683\pi$
$353$	14.1962	0.755585	0.377792	+	0.925890i	$0.376683\pi$
$354$	18.9282	1.00602
$355$	−9.46410	−0.502302
$356$	3.46410	0.183597
$357$	−9.46410	−0.500893
$358$	−6.00000	−0.317110
$359$	−34.0526	−1.79723	−0.898613	−	0.438743i	$-0.855424\pi$
$359$	−34.0526	−1.79723	−0.898613	+	0.438743i	$0.855424\pi$
$360$	−4.46410	−0.235279
$361$	26.3205	1.38529
$362$	14.0000	0.735824
$363$	2.73205	0.143395
$364$	−1.46410	−0.0767398
$365$	14.3923	0.753328
$366$	−13.4641	−0.703780
$367$	12.3923	0.646873	0.323437	−	0.946250i	$-0.395162\pi$
$367$	12.3923	0.646873	0.323437	+	0.946250i	$0.395162\pi$
$368$	−8.19615	−0.427254
$369$	−9.80385	−0.510368
$370$	−0.732051	−0.0380575
$371$	−7.26795	−0.377333
$372$	5.46410	0.283300
$373$	18.3923	0.952317	0.476159	−	0.879359i	$-0.342029\pi$
$373$	18.3923	0.952317	0.476159	+	0.879359i	$0.342029\pi$
$374$	−3.46410	−0.179124
$375$	−2.73205	−0.141082
$376$	6.92820	0.357295
$377$	6.92820	0.356821
$378$	4.00000	0.205738
$379$	6.14359	0.315575	0.157788	−	0.987473i	$-0.449564\pi$
$379$	6.14359	0.315575	0.157788	+	0.987473i	$0.449564\pi$
$380$	−6.73205	−0.345347
$381$	2.92820	0.150016
$382$	−12.0000	−0.613973
$383$	−33.4641	−1.70994	−0.854968	−	0.518681i	$-0.826423\pi$
$383$	−33.4641	−1.70994	−0.854968	+	0.518681i	$0.826423\pi$
$384$	2.73205	0.139419
$385$	−1.00000	−0.0509647
$386$	12.3923	0.630752
$387$	8.92820	0.453846
$388$	14.5885	0.740617
$389$	−1.60770	−0.0815134	−0.0407567	−	0.999169i	$-0.512977\pi$
$389$	−1.60770	−0.0815134	−0.0407567	+	0.999169i	$0.512977\pi$
$390$	4.00000	0.202548
$391$	28.3923	1.43586
$392$	1.00000	0.0505076
$393$	15.4641	0.780061
$394$	−24.2487	−1.22163
$395$	12.1962	0.613655
$396$	4.46410	0.224330
$397$	30.3923	1.52535	0.762673	−	0.646784i	$-0.223887\pi$
$397$	30.3923	1.52535	0.762673	+	0.646784i	$0.223887\pi$
$398$	2.92820	0.146778
$399$	18.3923	0.920767
$400$	1.00000	0.0500000
$401$	2.53590	0.126637	0.0633184	−	0.997993i	$-0.479832\pi$
$401$	2.53590	0.126637	0.0633184	+	0.997993i	$0.479832\pi$
$402$	−10.9282	−0.545049
$403$	−2.92820	−0.145864
$404$	−7.85641	−0.390871
$405$	2.46410	0.122442
$406$	−4.73205	−0.234848
$407$	0.732051	0.0362864
$408$	−9.46410	−0.468543
$409$	10.8756	0.537766	0.268883	−	0.963173i	$-0.413346\pi$
$409$	10.8756	0.537766	0.268883	+	0.963173i	$0.413346\pi$
$410$	2.19615	0.108460
$411$	−2.53590	−0.125087
$412$	12.3923	0.610525
$413$	6.92820	0.340915
$414$	−36.5885	−1.79822
$415$	16.3923	0.804667
$416$	−1.46410	−0.0717835
$417$	37.3205	1.82759
$418$	6.73205	0.329275
$419$	30.9282	1.51094	0.755471	−	0.655182i	$-0.227408\pi$
$419$	30.9282	1.51094	0.755471	+	0.655182i	$0.227408\pi$
$420$	−2.73205	−0.133310
$421$	−35.8564	−1.74753	−0.873767	−	0.486344i	$-0.838330\pi$
$421$	−35.8564	−1.74753	−0.873767	+	0.486344i	$0.838330\pi$
$422$	26.9282	1.31084
$423$	30.9282	1.50378
$424$	−7.26795	−0.352963
$425$	−3.46410	−0.168034
$426$	25.8564	1.25275
$427$	−4.92820	−0.238492
$428$	7.85641	0.379754
$429$	−4.00000	−0.193122
$430$	−2.00000	−0.0964486
$431$	3.12436	0.150495	0.0752475	−	0.997165i	$-0.476025\pi$
$431$	3.12436	0.150495	0.0752475	+	0.997165i	$0.476025\pi$
$432$	4.00000	0.192450
$433$	−18.1962	−0.874451	−0.437226	−	0.899352i	$-0.644039\pi$
$433$	−18.1962	−0.874451	−0.437226	+	0.899352i	$0.644039\pi$
$434$	2.00000	0.0960031
$435$	12.9282	0.619860
$436$	−15.6603	−0.749990
$437$	−55.1769	−2.63947
$438$	−39.3205	−1.87881
$439$	26.2487	1.25278	0.626391	−	0.779509i	$-0.284531\pi$
$439$	26.2487	1.25278	0.626391	+	0.779509i	$0.284531\pi$
$440$	−1.00000	−0.0476731
$441$	4.46410	0.212576
$442$	5.07180	0.241241
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	2.00000	0.0949158
$445$	−3.46410	−0.164214
$446$	−25.4641	−1.20576
$447$	−19.8564	−0.939176
$448$	1.00000	0.0472456
$449$	40.3923	1.90623	0.953115	−	0.302607i	$-0.0978570\pi$
$449$	40.3923	1.90623	0.953115	+	0.302607i	$0.0978570\pi$
$450$	4.46410	0.210440
$451$	−2.19615	−0.103413
$452$	7.85641	0.369534
$453$	30.3923	1.42796
$454$	6.92820	0.325157
$455$	1.46410	0.0686381
$456$	18.3923	0.861299
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	24.3923	1.13978
$459$	−13.8564	−0.646762
$460$	8.19615	0.382148
$461$	22.3923	1.04291	0.521457	−	0.853278i	$-0.325389\pi$
$461$	22.3923	1.04291	0.521457	+	0.853278i	$0.325389\pi$
$462$	2.73205	0.127107
$463$	27.5167	1.27881	0.639404	−	0.768871i	$-0.279181\pi$
$463$	27.5167	1.27881	0.639404	+	0.768871i	$0.279181\pi$
$464$	−4.73205	−0.219680
$465$	−5.46410	−0.253392
$466$	−7.85641	−0.363941
$467$	−34.0526	−1.57576	−0.787882	−	0.615826i	$-0.788822\pi$
$467$	−34.0526	−1.57576	−0.787882	+	0.615826i	$0.788822\pi$
$468$	−6.53590	−0.302122
$469$	−4.00000	−0.184703
$470$	−6.92820	−0.319574
$471$	12.3923	0.571007
$472$	6.92820	0.318896
$473$	2.00000	0.0919601
$474$	−33.3205	−1.53046
$475$	6.73205	0.308888
$476$	−3.46410	−0.158777
$477$	−32.4449	−1.48555
$478$	1.26795	0.0579946
$479$	−32.7846	−1.49797	−0.748984	−	0.662589i	$-0.769458\pi$
$479$	−32.7846	−1.49797	−0.748984	+	0.662589i	$0.769458\pi$
$480$	−2.73205	−0.124700
$481$	−1.07180	−0.0488697
$482$	3.26795	0.148851
$483$	−22.3923	−1.01889
$484$	1.00000	0.0454545
$485$	−14.5885	−0.662428
$486$	−18.7321	−0.849703
$487$	4.19615	0.190146	0.0950729	−	0.995470i	$-0.469692\pi$
$487$	4.19615	0.190146	0.0950729	+	0.995470i	$0.469692\pi$
$488$	−4.92820	−0.223089
$489$	−10.9282	−0.494190
$490$	−1.00000	−0.0451754
$491$	−27.7128	−1.25066	−0.625331	−	0.780360i	$-0.715036\pi$
$491$	−27.7128	−1.25066	−0.625331	+	0.780360i	$0.715036\pi$
$492$	−6.00000	−0.270501
$493$	16.3923	0.738272
$494$	−9.85641	−0.443461
$495$	−4.46410	−0.200646
$496$	2.00000	0.0898027
$497$	9.46410	0.424523
$498$	−44.7846	−2.00685
$499$	12.1436	0.543622	0.271811	−	0.962351i	$-0.412377\pi$
$499$	12.1436	0.543622	0.271811	+	0.962351i	$0.412377\pi$
$500$	−1.00000	−0.0447214
$501$	37.8564	1.69130
$502$	12.0000	0.535586
$503$	8.78461	0.391686	0.195843	−	0.980635i	$-0.437256\pi$
$503$	8.78461	0.391686	0.195843	+	0.980635i	$0.437256\pi$
$504$	4.46410	0.198847
$505$	7.85641	0.349605
$506$	−8.19615	−0.364363
$507$	−29.6603	−1.31726
$508$	1.07180	0.0475533
$509$	11.0718	0.490749	0.245374	−	0.969428i	$-0.421089\pi$
$509$	11.0718	0.490749	0.245374	+	0.969428i	$0.421089\pi$
$510$	9.46410	0.419077
$511$	−14.3923	−0.636678
$512$	1.00000	0.0441942
$513$	26.9282	1.18891
$514$	−23.6603	−1.04361
$515$	−12.3923	−0.546070
$516$	5.46410	0.240544
$517$	6.92820	0.304702
$518$	0.732051	0.0321645
$519$	2.53590	0.111314
$520$	1.46410	0.0642051
$521$	10.3923	0.455295	0.227648	−	0.973744i	$-0.426897\pi$
$521$	10.3923	0.455295	0.227648	+	0.973744i	$0.426897\pi$
$522$	−21.1244	−0.924588
$523$	−29.1769	−1.27582	−0.637909	−	0.770112i	$-0.720200\pi$
$523$	−29.1769	−1.27582	−0.637909	+	0.770112i	$0.720200\pi$
$524$	5.66025	0.247269
$525$	2.73205	0.119236
$526$	−24.0000	−1.04645
$527$	−6.92820	−0.301797
$528$	2.73205	0.118897
$529$	44.1769	1.92074
$530$	7.26795	0.315700
$531$	30.9282	1.34217
$532$	6.73205	0.291871
$533$	3.21539	0.139274
$534$	9.46410	0.409552
$535$	−7.85641	−0.339662
$536$	−4.00000	−0.172774
$537$	−16.3923	−0.707380
$538$	28.3923	1.22408
$539$	1.00000	0.0430730
$540$	−4.00000	−0.172133
$541$	−20.7321	−0.891340	−0.445670	−	0.895197i	$-0.647035\pi$
$541$	−20.7321	−0.891340	−0.445670	+	0.895197i	$0.647035\pi$
$542$	0.392305	0.0168509
$543$	38.2487	1.64141
$544$	−3.46410	−0.148522
$545$	15.6603	0.670812
$546$	−4.00000	−0.171184
$547$	28.7846	1.23074	0.615371	−	0.788238i	$-0.289006\pi$
$547$	28.7846	1.23074	0.615371	+	0.788238i	$0.289006\pi$
$548$	−0.928203	−0.0396509
$549$	−22.0000	−0.938937
$550$	1.00000	0.0426401
$551$	−31.8564	−1.35713
$552$	−22.3923	−0.953080
$553$	−12.1962	−0.518633
$554$	7.07180	0.300452
$555$	−2.00000	−0.0848953
$556$	13.6603	0.579324
$557$	−25.6077	−1.08503	−0.542516	−	0.840045i	$-0.682528\pi$
$557$	−25.6077	−1.08503	−0.542516	+	0.840045i	$0.682528\pi$
$558$	8.92820	0.377961
$559$	−2.92820	−0.123850
$560$	−1.00000	−0.0422577
$561$	−9.46410	−0.399575
$562$	22.3923	0.944562
$563$	5.07180	0.213751	0.106875	−	0.994272i	$-0.465915\pi$
$563$	5.07180	0.213751	0.106875	+	0.994272i	$0.465915\pi$
$564$	18.9282	0.797021
$565$	−7.85641	−0.330522
$566$	−31.7128	−1.33299
$567$	−2.46410	−0.103483
$568$	9.46410	0.397105
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	−18.3923	−0.770369
$571$	3.60770	0.150977	0.0754887	−	0.997147i	$-0.475948\pi$
$571$	3.60770	0.150977	0.0754887	+	0.997147i	$0.475948\pi$
$572$	−1.46410	−0.0612172
$573$	−32.7846	−1.36960
$574$	−2.19615	−0.0916656
$575$	−8.19615	−0.341803
$576$	4.46410	0.186004
$577$	−34.5885	−1.43994	−0.719968	−	0.694007i	$-0.755844\pi$
$577$	−34.5885	−1.43994	−0.719968	+	0.694007i	$0.755844\pi$
$578$	−5.00000	−0.207973
$579$	33.8564	1.40702
$580$	4.73205	0.196488
$581$	−16.3923	−0.680067
$582$	39.8564	1.65210
$583$	−7.26795	−0.301008
$584$	−14.3923	−0.595558
$585$	6.53590	0.270226
$586$	21.4641	0.886674
$587$	−11.4115	−0.471005	−0.235502	−	0.971874i	$-0.575674\pi$
$587$	−11.4115	−0.471005	−0.235502	+	0.971874i	$0.575674\pi$
$588$	2.73205	0.112668
$589$	13.4641	0.554779
$590$	−6.92820	−0.285230
$591$	−66.2487	−2.72511
$592$	0.732051	0.0300871
$593$	0.248711	0.0102133	0.00510667	−	0.999987i	$-0.498374\pi$
$593$	0.248711	0.0102133	0.00510667	+	0.999987i	$0.498374\pi$
$594$	4.00000	0.164122
$595$	3.46410	0.142014
$596$	−7.26795	−0.297707
$597$	8.00000	0.327418
$598$	12.0000	0.490716
$599$	37.1769	1.51901	0.759504	−	0.650503i	$-0.225442\pi$
$599$	37.1769	1.51901	0.759504	+	0.650503i	$0.225442\pi$
$600$	2.73205	0.111536
$601$	−5.51666	−0.225029	−0.112515	−	0.993650i	$-0.535891\pi$
$601$	−5.51666	−0.225029	−0.112515	+	0.993650i	$0.535891\pi$
$602$	2.00000	0.0815139
$603$	−17.8564	−0.727169
$604$	11.1244	0.452644
$605$	−1.00000	−0.0406558
$606$	−21.4641	−0.871920
$607$	20.9282	0.849450	0.424725	−	0.905323i	$-0.360371\pi$
$607$	20.9282	0.849450	0.424725	+	0.905323i	$0.360371\pi$
$608$	6.73205	0.273021
$609$	−12.9282	−0.523877
$610$	4.92820	0.199537
$611$	−10.1436	−0.410366
$612$	−15.4641	−0.625099
$613$	−35.1769	−1.42078	−0.710391	−	0.703807i	$-0.751482\pi$
$613$	−35.1769	−1.42078	−0.710391	+	0.703807i	$0.751482\pi$
$614$	24.3923	0.984393
$615$	6.00000	0.241943
$616$	1.00000	0.0402911
$617$	17.3205	0.697297	0.348649	−	0.937253i	$-0.386641\pi$
$617$	17.3205	0.697297	0.348649	+	0.937253i	$0.386641\pi$
$618$	33.8564	1.36190
$619$	28.7846	1.15695	0.578476	−	0.815700i	$-0.303648\pi$
$619$	28.7846	1.15695	0.578476	+	0.815700i	$0.303648\pi$
$620$	−2.00000	−0.0803219
$621$	−32.7846	−1.31560
$622$	24.9282	0.999530
$623$	3.46410	0.138786
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	22.1962	0.887137
$627$	18.3923	0.734518
$628$	4.53590	0.181002
$629$	−2.53590	−0.101113
$630$	−4.46410	−0.177854
$631$	−46.9282	−1.86818	−0.934091	−	0.357035i	$-0.883788\pi$
$631$	−46.9282	−1.86818	−0.934091	+	0.357035i	$0.883788\pi$
$632$	−12.1962	−0.485137
$633$	73.5692	2.92411
$634$	30.5885	1.21482
$635$	−1.07180	−0.0425330
$636$	−19.8564	−0.787358
$637$	−1.46410	−0.0580098
$638$	−4.73205	−0.187344
$639$	42.2487	1.67133
$640$	−1.00000	−0.0395285
$641$	−35.5692	−1.40490	−0.702450	−	0.711733i	$-0.747910\pi$
$641$	−35.5692	−1.40490	−0.702450	+	0.711733i	$0.747910\pi$
$642$	21.4641	0.847121
$643$	33.2679	1.31196	0.655980	−	0.754778i	$-0.272256\pi$
$643$	33.2679	1.31196	0.655980	+	0.754778i	$0.272256\pi$
$644$	−8.19615	−0.322974
$645$	−5.46410	−0.215149
$646$	−23.3205	−0.917533
$647$	−26.5359	−1.04323	−0.521617	−	0.853180i	$-0.674671\pi$
$647$	−26.5359	−1.04323	−0.521617	+	0.853180i	$0.674671\pi$
$648$	−2.46410	−0.0967991
$649$	6.92820	0.271956
$650$	−1.46410	−0.0574268
$651$	5.46410	0.214155
$652$	−4.00000	−0.156652
$653$	11.6603	0.456301	0.228151	−	0.973626i	$-0.426732\pi$
$653$	11.6603	0.456301	0.228151	+	0.973626i	$0.426732\pi$
$654$	−42.7846	−1.67301
$655$	−5.66025	−0.221164
$656$	−2.19615	−0.0857453
$657$	−64.2487	−2.50658
$658$	6.92820	0.270089
$659$	30.2487	1.17832	0.589161	−	0.808015i	$-0.299458\pi$
$659$	30.2487	1.17832	0.589161	+	0.808015i	$0.299458\pi$
$660$	−2.73205	−0.106345
$661$	−46.0000	−1.78919	−0.894596	−	0.446875i	$-0.852537\pi$
$661$	−46.0000	−1.78919	−0.894596	+	0.446875i	$0.852537\pi$
$662$	−18.7846	−0.730085
$663$	13.8564	0.538138
$664$	−16.3923	−0.636145
$665$	−6.73205	−0.261058
$666$	3.26795	0.126630
$667$	38.7846	1.50175
$668$	13.8564	0.536120
$669$	−69.5692	−2.68970
$670$	4.00000	0.154533
$671$	−4.92820	−0.190251
$672$	2.73205	0.105391
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	22.7846	0.877630
$675$	4.00000	0.153960
$676$	−10.8564	−0.417554
$677$	−4.14359	−0.159251	−0.0796256	−	0.996825i	$-0.525372\pi$
$677$	−4.14359	−0.159251	−0.0796256	+	0.996825i	$0.525372\pi$
$678$	21.4641	0.824324
$679$	14.5885	0.559854
$680$	3.46410	0.132842
$681$	18.9282	0.725330
$682$	2.00000	0.0765840
$683$	6.24871	0.239100	0.119550	−	0.992828i	$-0.461855\pi$
$683$	6.24871	0.239100	0.119550	+	0.992828i	$0.461855\pi$
$684$	30.0526	1.14909
$685$	0.928203	0.0354648
$686$	1.00000	0.0381802
$687$	66.6410	2.54251
$688$	2.00000	0.0762493
$689$	10.6410	0.405390
$690$	22.3923	0.852460
$691$	−13.4641	−0.512199	−0.256099	−	0.966650i	$-0.582437\pi$
$691$	−13.4641	−0.512199	−0.256099	+	0.966650i	$0.582437\pi$
$692$	0.928203	0.0352850
$693$	4.46410	0.169577
$694$	−23.0718	−0.875793
$695$	−13.6603	−0.518163
$696$	−12.9282	−0.490042
$697$	7.60770	0.288162
$698$	−5.60770	−0.212254
$699$	−21.4641	−0.811847
$700$	1.00000	0.0377964
$701$	41.9090	1.58288	0.791440	−	0.611247i	$-0.209332\pi$
$701$	41.9090	1.58288	0.791440	+	0.611247i	$0.209332\pi$
$702$	−5.85641	−0.221036
$703$	4.92820	0.185871
$704$	1.00000	0.0376889
$705$	−18.9282	−0.712877
$706$	14.1962	0.534279
$707$	−7.85641	−0.295471
$708$	18.9282	0.711365
$709$	25.3205	0.950932	0.475466	−	0.879734i	$-0.342280\pi$
$709$	25.3205	0.950932	0.475466	+	0.879734i	$0.342280\pi$
$710$	−9.46410	−0.355181
$711$	−54.4449	−2.04184
$712$	3.46410	0.129823
$713$	−16.3923	−0.613897
$714$	−9.46410	−0.354185
$715$	1.46410	0.0547543
$716$	−6.00000	−0.224231
$717$	3.46410	0.129369
$718$	−34.0526	−1.27083
$719$	−27.7128	−1.03351	−0.516757	−	0.856132i	$-0.672861\pi$
$719$	−27.7128	−1.03351	−0.516757	+	0.856132i	$0.672861\pi$
$720$	−4.46410	−0.166367
$721$	12.3923	0.461514
$722$	26.3205	0.979548
$723$	8.92820	0.332043
$724$	14.0000	0.520306
$725$	−4.73205	−0.175744
$726$	2.73205	0.101396
$727$	−4.00000	−0.148352	−0.0741759	−	0.997245i	$-0.523633\pi$
$727$	−4.00000	−0.148352	−0.0741759	+	0.997245i	$0.523633\pi$
$728$	−1.46410	−0.0542632
$729$	−43.7846	−1.62165
$730$	14.3923	0.532683
$731$	−6.92820	−0.256249
$732$	−13.4641	−0.497648
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	12.3923	0.457408
$735$	−2.73205	−0.100773
$736$	−8.19615	−0.302114
$737$	−4.00000	−0.147342
$738$	−9.80385	−0.360885
$739$	11.7128	0.430863	0.215431	−	0.976519i	$-0.430884\pi$
$739$	11.7128	0.430863	0.215431	+	0.976519i	$0.430884\pi$
$740$	−0.732051	−0.0269107
$741$	−26.9282	−0.989232
$742$	−7.26795	−0.266815
$743$	−32.7846	−1.20275	−0.601375	−	0.798967i	$-0.705380\pi$
$743$	−32.7846	−1.20275	−0.601375	+	0.798967i	$0.705380\pi$
$744$	5.46410	0.200324
$745$	7.26795	0.266277
$746$	18.3923	0.673390
$747$	−73.1769	−2.67740
$748$	−3.46410	−0.126660
$749$	7.85641	0.287067
$750$	−2.73205	−0.0997604
$751$	−11.6077	−0.423571	−0.211785	−	0.977316i	$-0.567928\pi$
$751$	−11.6077	−0.423571	−0.211785	+	0.977316i	$0.567928\pi$
$752$	6.92820	0.252646
$753$	32.7846	1.19474
$754$	6.92820	0.252310
$755$	−11.1244	−0.404857
$756$	4.00000	0.145479
$757$	−43.3731	−1.57642	−0.788210	−	0.615406i	$-0.788992\pi$
$757$	−43.3731	−1.57642	−0.788210	+	0.615406i	$0.788992\pi$
$758$	6.14359	0.223145
$759$	−22.3923	−0.812789
$760$	−6.73205	−0.244197
$761$	−12.3397	−0.447315	−0.223658	−	0.974668i	$-0.571800\pi$
$761$	−12.3397	−0.447315	−0.223658	+	0.974668i	$0.571800\pi$
$762$	2.92820	0.106078
$763$	−15.6603	−0.566939
$764$	−12.0000	−0.434145
$765$	15.4641	0.559106
$766$	−33.4641	−1.20911
$767$	−10.1436	−0.366264
$768$	2.73205	0.0985844
$769$	−18.1962	−0.656170	−0.328085	−	0.944648i	$-0.606403\pi$
$769$	−18.1962	−0.656170	−0.328085	+	0.944648i	$0.606403\pi$
$770$	−1.00000	−0.0360375
$771$	−64.6410	−2.32799
$772$	12.3923	0.446009
$773$	−0.928203	−0.0333851	−0.0166926	−	0.999861i	$-0.505314\pi$
$773$	−0.928203	−0.0333851	−0.0166926	+	0.999861i	$0.505314\pi$
$774$	8.92820	0.320918
$775$	2.00000	0.0718421
$776$	14.5885	0.523695
$777$	2.00000	0.0717496
$778$	−1.60770	−0.0576387
$779$	−14.7846	−0.529714
$780$	4.00000	0.143223
$781$	9.46410	0.338652
$782$	28.3923	1.01531
$783$	−18.9282	−0.676439
$784$	1.00000	0.0357143
$785$	−4.53590	−0.161893
$786$	15.4641	0.551586
$787$	18.1436	0.646749	0.323375	−	0.946271i	$-0.395183\pi$
$787$	18.1436	0.646749	0.323375	+	0.946271i	$0.395183\pi$
$788$	−24.2487	−0.863825
$789$	−65.5692	−2.33433
$790$	12.1962	0.433920
$791$	7.85641	0.279342
$792$	4.46410	0.158625
$793$	7.21539	0.256226
$794$	30.3923	1.07858
$795$	19.8564	0.704234
$796$	2.92820	0.103787
$797$	−25.6077	−0.907071	−0.453536	−	0.891238i	$-0.649837\pi$
$797$	−25.6077	−0.907071	−0.453536	+	0.891238i	$0.649837\pi$
$798$	18.3923	0.651081
$799$	−24.0000	−0.849059
$800$	1.00000	0.0353553
$801$	15.4641	0.546397
$802$	2.53590	0.0895457
$803$	−14.3923	−0.507893
$804$	−10.9282	−0.385408
$805$	8.19615	0.288876
$806$	−2.92820	−0.103142
$807$	77.5692	2.73057
$808$	−7.85641	−0.276387
$809$	−31.1769	−1.09612	−0.548061	−	0.836438i	$-0.684634\pi$
$809$	−31.1769	−1.09612	−0.548061	+	0.836438i	$0.684634\pi$
$810$	2.46410	0.0865797
$811$	−43.1244	−1.51430	−0.757150	−	0.653241i	$-0.773409\pi$
$811$	−43.1244	−1.51430	−0.757150	+	0.653241i	$0.773409\pi$
$812$	−4.73205	−0.166062
$813$	1.07180	0.0375896
$814$	0.732051	0.0256584
$815$	4.00000	0.140114
$816$	−9.46410	−0.331310
$817$	13.4641	0.471049
$818$	10.8756	0.380258
$819$	−6.53590	−0.228383
$820$	2.19615	0.0766930
$821$	17.9090	0.625027	0.312514	−	0.949913i	$-0.398829\pi$
$821$	17.9090	0.625027	0.312514	+	0.949913i	$0.398829\pi$
$822$	−2.53590	−0.0884496
$823$	−0.875644	−0.0305230	−0.0152615	−	0.999884i	$-0.504858\pi$
$823$	−0.875644	−0.0305230	−0.0152615	+	0.999884i	$0.504858\pi$
$824$	12.3923	0.431706
$825$	2.73205	0.0951178
$826$	6.92820	0.241063
$827$	−25.8564	−0.899115	−0.449558	−	0.893251i	$-0.648418\pi$
$827$	−25.8564	−0.899115	−0.449558	+	0.893251i	$0.648418\pi$
$828$	−36.5885	−1.27154
$829$	2.24871	0.0781010	0.0390505	−	0.999237i	$-0.487567\pi$
$829$	2.24871	0.0781010	0.0390505	+	0.999237i	$0.487567\pi$
$830$	16.3923	0.568985
$831$	19.3205	0.670221
$832$	−1.46410	−0.0507586
$833$	−3.46410	−0.120024
$834$	37.3205	1.29230
$835$	−13.8564	−0.479521
$836$	6.73205	0.232833
$837$	8.00000	0.276520
$838$	30.9282	1.06840
$839$	−31.8564	−1.09981	−0.549903	−	0.835229i	$-0.685335\pi$
$839$	−31.8564	−1.09981	−0.549903	+	0.835229i	$0.685335\pi$
$840$	−2.73205	−0.0942647
$841$	−6.60770	−0.227852
$842$	−35.8564	−1.23569
$843$	61.1769	2.10704
$844$	26.9282	0.926907
$845$	10.8564	0.373472
$846$	30.9282	1.06333
$847$	1.00000	0.0343604
$848$	−7.26795	−0.249582
$849$	−86.6410	−2.97351
$850$	−3.46410	−0.118818
$851$	−6.00000	−0.205677
$852$	25.8564	0.885826
$853$	−54.7846	−1.87579	−0.937895	−	0.346920i	$-0.887227\pi$
$853$	−54.7846	−1.87579	−0.937895	+	0.346920i	$0.887227\pi$
$854$	−4.92820	−0.168640
$855$	−30.0526	−1.02778
$856$	7.85641	0.268526
$857$	−27.4641	−0.938156	−0.469078	−	0.883157i	$-0.655414\pi$
$857$	−27.4641	−0.938156	−0.469078	+	0.883157i	$0.655414\pi$
$858$	−4.00000	−0.136558
$859$	−12.7846	−0.436205	−0.218103	−	0.975926i	$-0.569987\pi$
$859$	−12.7846	−0.436205	−0.218103	+	0.975926i	$0.569987\pi$
$860$	−2.00000	−0.0681994
$861$	−6.00000	−0.204479
$862$	3.12436	0.106416
$863$	−7.51666	−0.255870	−0.127935	−	0.991783i	$-0.540835\pi$
$863$	−7.51666	−0.255870	−0.127935	+	0.991783i	$0.540835\pi$
$864$	4.00000	0.136083
$865$	−0.928203	−0.0315599
$866$	−18.1962	−0.618330
$867$	−13.6603	−0.463927
$868$	2.00000	0.0678844
$869$	−12.1962	−0.413726
$870$	12.9282	0.438307
$871$	5.85641	0.198437
$872$	−15.6603	−0.530323
$873$	65.1244	2.20413
$874$	−55.1769	−1.86639
$875$	−1.00000	−0.0338062
$876$	−39.3205	−1.32852
$877$	−21.3205	−0.719942	−0.359971	−	0.932963i	$-0.617213\pi$
$877$	−21.3205	−0.719942	−0.359971	+	0.932963i	$0.617213\pi$
$878$	26.2487	0.885851
$879$	58.6410	1.97791
$880$	−1.00000	−0.0337100
$881$	11.0718	0.373018	0.186509	−	0.982453i	$-0.440283\pi$
$881$	11.0718	0.373018	0.186509	+	0.982453i	$0.440283\pi$
$882$	4.46410	0.150314
$883$	−35.6077	−1.19829	−0.599147	−	0.800639i	$-0.704494\pi$
$883$	−35.6077	−1.19829	−0.599147	+	0.800639i	$0.704494\pi$
$884$	5.07180	0.170583
$885$	−18.9282	−0.636265
$886$	0	0
$887$	13.8564	0.465253	0.232626	−	0.972566i	$-0.425268\pi$
$887$	13.8564	0.465253	0.232626	+	0.972566i	$0.425268\pi$
$888$	2.00000	0.0671156
$889$	1.07180	0.0359469
$890$	−3.46410	−0.116117
$891$	−2.46410	−0.0825505
$892$	−25.4641	−0.852601
$893$	46.6410	1.56078
$894$	−19.8564	−0.664098
$895$	6.00000	0.200558
$896$	1.00000	0.0334077
$897$	32.7846	1.09465
$898$	40.3923	1.34791
$899$	−9.46410	−0.315645
$900$	4.46410	0.148803
$901$	25.1769	0.838765
$902$	−2.19615	−0.0731239
$903$	5.46410	0.181834
$904$	7.85641	0.261300
$905$	−14.0000	−0.465376
$906$	30.3923	1.00972
$907$	−31.0333	−1.03044	−0.515222	−	0.857057i	$-0.672291\pi$
$907$	−31.0333	−1.03044	−0.515222	+	0.857057i	$0.672291\pi$
$908$	6.92820	0.229920
$909$	−35.0718	−1.16326
$910$	1.46410	0.0485345
$911$	9.46410	0.313560	0.156780	−	0.987634i	$-0.449889\pi$
$911$	9.46410	0.313560	0.156780	+	0.987634i	$0.449889\pi$
$912$	18.3923	0.609030
$913$	−16.3923	−0.542506
$914$	2.00000	0.0661541
$915$	13.4641	0.445109
$916$	24.3923	0.805944
$917$	5.66025	0.186918
$918$	−13.8564	−0.457330
$919$	−25.3731	−0.836980	−0.418490	−	0.908221i	$-0.637441\pi$
$919$	−25.3731	−0.836980	−0.418490	+	0.908221i	$0.637441\pi$
$920$	8.19615	0.270219
$921$	66.6410	2.19590
$922$	22.3923	0.737451
$923$	−13.8564	−0.456089
$924$	2.73205	0.0898779
$925$	0.732051	0.0240697
$926$	27.5167	0.904254
$927$	55.3205	1.81696
$928$	−4.73205	−0.155337
$929$	−25.6077	−0.840161	−0.420081	−	0.907487i	$-0.637998\pi$
$929$	−25.6077	−0.840161	−0.420081	+	0.907487i	$0.637998\pi$
$930$	−5.46410	−0.179175
$931$	6.73205	0.220634
$932$	−7.85641	−0.257345
$933$	68.1051	2.22966
$934$	−34.0526	−1.11423
$935$	3.46410	0.113288
$936$	−6.53590	−0.213633
$937$	−1.21539	−0.0397051	−0.0198525	−	0.999803i	$-0.506320\pi$
$937$	−1.21539	−0.0397051	−0.0198525	+	0.999803i	$0.506320\pi$
$938$	−4.00000	−0.130605
$939$	60.6410	1.97894
$940$	−6.92820	−0.225973
$941$	−14.7846	−0.481965	−0.240982	−	0.970530i	$-0.577470\pi$
$941$	−14.7846	−0.481965	−0.240982	+	0.970530i	$0.577470\pi$
$942$	12.3923	0.403763
$943$	18.0000	0.586161
$944$	6.92820	0.225494
$945$	−4.00000	−0.130120
$946$	2.00000	0.0650256
$947$	47.3205	1.53771	0.768855	−	0.639423i	$-0.220827\pi$
$947$	47.3205	1.53771	0.768855	+	0.639423i	$0.220827\pi$
$948$	−33.3205	−1.08220
$949$	21.0718	0.684019
$950$	6.73205	0.218417
$951$	83.5692	2.70992
$952$	−3.46410	−0.112272
$953$	26.5359	0.859582	0.429791	−	0.902928i	$-0.358587\pi$
$953$	26.5359	0.859582	0.429791	+	0.902928i	$0.358587\pi$
$954$	−32.4449	−1.05044
$955$	12.0000	0.388311
$956$	1.26795	0.0410084
$957$	−12.9282	−0.417909
$958$	−32.7846	−1.05922
$959$	−0.928203	−0.0299732
$960$	−2.73205	−0.0881766
$961$	−27.0000	−0.870968
$962$	−1.07180	−0.0345561
$963$	35.0718	1.13017
$964$	3.26795	0.105254
$965$	−12.3923	−0.398922
$966$	−22.3923	−0.720461
$967$	26.9282	0.865953	0.432976	−	0.901405i	$-0.357463\pi$
$967$	26.9282	0.865953	0.432976	+	0.901405i	$0.357463\pi$
$968$	1.00000	0.0321412
$969$	−63.7128	−2.04675
$970$	−14.5885	−0.468407
$971$	42.9282	1.37763	0.688816	−	0.724936i	$-0.258131\pi$
$971$	42.9282	1.37763	0.688816	+	0.724936i	$0.258131\pi$
$972$	−18.7321	−0.600831
$973$	13.6603	0.437928
$974$	4.19615	0.134453
$975$	−4.00000	−0.128103
$976$	−4.92820	−0.157748
$977$	−33.7128	−1.07857	−0.539284	−	0.842124i	$-0.681305\pi$
$977$	−33.7128	−1.07857	−0.539284	+	0.842124i	$0.681305\pi$
$978$	−10.9282	−0.349445
$979$	3.46410	0.110713
$980$	−1.00000	−0.0319438
$981$	−69.9090	−2.23202
$982$	−27.7128	−0.884351
$983$	−37.1769	−1.18576	−0.592880	−	0.805291i	$-0.702009\pi$
$983$	−37.1769	−1.18576	−0.592880	+	0.805291i	$0.702009\pi$
$984$	−6.00000	−0.191273
$985$	24.2487	0.772628
$986$	16.3923	0.522037
$987$	18.9282	0.602491
$988$	−9.85641	−0.313574
$989$	−16.3923	−0.521245
$990$	−4.46410	−0.141878
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	2.00000	0.0635001
$993$	−51.3205	−1.62861
$994$	9.46410	0.300183
$995$	−2.92820	−0.0928303
$996$	−44.7846	−1.41905
$997$	17.7128	0.560970	0.280485	−	0.959858i	$-0.409505\pi$
$997$	17.7128	0.560970	0.280485	+	0.959858i	$0.409505\pi$
$998$	12.1436	0.384399
$999$	2.92820	0.0926443

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	770.2.a.j.1.2	✓	2
3.2	odd	2		6930.2.a.bv.1.1		2
4.3	odd	2		6160.2.a.t.1.1		2
5.2	odd	4		3850.2.c.x.1849.3		4
5.3	odd	4		3850.2.c.x.1849.2		4
5.4	even	2		3850.2.a.bd.1.1		2
7.6	odd	2		5390.2.a.bs.1.1		2
11.10	odd	2		8470.2.a.br.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.j.1.2	✓	2	1.1	even	1	trivial
3850.2.a.bd.1.1		2	5.4	even	2
3850.2.c.x.1849.2		4	5.3	odd	4
3850.2.c.x.1849.3		4	5.2	odd	4
5390.2.a.bs.1.1		2	7.6	odd	2
6160.2.a.t.1.1		2	4.3	odd	2
6930.2.a.bv.1.1		2	3.2	odd	2
8470.2.a.br.1.2		2	11.10	odd	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:	\( N \)	\(=\)	\( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	770.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.73205 q^{6} +1.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.73205 q^{6} +1.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} -1.00000 q^{10} +1.00000 q^{11} +2.73205 q^{12} -1.46410 q^{13} +1.00000 q^{14} -2.73205 q^{15} +1.00000 q^{16} -3.46410 q^{17} +4.46410 q^{18} +6.73205 q^{19} -1.00000 q^{20} +2.73205 q^{21} +1.00000 q^{22} -8.19615 q^{23} +2.73205 q^{24} +1.00000 q^{25} -1.46410 q^{26} +4.00000 q^{27} +1.00000 q^{28} -4.73205 q^{29} -2.73205 q^{30} +2.00000 q^{31} +1.00000 q^{32} +2.73205 q^{33} -3.46410 q^{34} -1.00000 q^{35} +4.46410 q^{36} +0.732051 q^{37} +6.73205 q^{38} -4.00000 q^{39} -1.00000 q^{40} -2.19615 q^{41} +2.73205 q^{42} +2.00000 q^{43} +1.00000 q^{44} -4.46410 q^{45} -8.19615 q^{46} +6.92820 q^{47} +2.73205 q^{48} +1.00000 q^{49} +1.00000 q^{50} -9.46410 q^{51} -1.46410 q^{52} -7.26795 q^{53} +4.00000 q^{54} -1.00000 q^{55} +1.00000 q^{56} +18.3923 q^{57} -4.73205 q^{58} +6.92820 q^{59} -2.73205 q^{60} -4.92820 q^{61} +2.00000 q^{62} +4.46410 q^{63} +1.00000 q^{64} +1.46410 q^{65} +2.73205 q^{66} -4.00000 q^{67} -3.46410 q^{68} -22.3923 q^{69} -1.00000 q^{70} +9.46410 q^{71} +4.46410 q^{72} -14.3923 q^{73} +0.732051 q^{74} +2.73205 q^{75} +6.73205 q^{76} +1.00000 q^{77} -4.00000 q^{78} -12.1962 q^{79} -1.00000 q^{80} -2.46410 q^{81} -2.19615 q^{82} -16.3923 q^{83} +2.73205 q^{84} +3.46410 q^{85} +2.00000 q^{86} -12.9282 q^{87} +1.00000 q^{88} +3.46410 q^{89} -4.46410 q^{90} -1.46410 q^{91} -8.19615 q^{92} +5.46410 q^{93} +6.92820 q^{94} -6.73205 q^{95} +2.73205 q^{96} +14.5885 q^{97} +1.00000 q^{98} +4.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 2 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} + 2 q^{12} + 4 q^{13} + 2 q^{14} - 2 q^{15} + 2 q^{16} + 2 q^{18} + 10 q^{19} - 2 q^{20} + 2 q^{21} + 2 q^{22} - 6 q^{23} + 2 q^{24} + 2 q^{25} + 4 q^{26} + 8 q^{27} + 2 q^{28} - 6 q^{29} - 2 q^{30} + 4 q^{31} + 2 q^{32} + 2 q^{33} - 2 q^{35} + 2 q^{36} - 2 q^{37} + 10 q^{38} - 8 q^{39} - 2 q^{40} + 6 q^{41} + 2 q^{42} + 4 q^{43} + 2 q^{44} - 2 q^{45} - 6 q^{46} + 2 q^{48} + 2 q^{49} + 2 q^{50} - 12 q^{51} + 4 q^{52} - 18 q^{53} + 8 q^{54} - 2 q^{55} + 2 q^{56} + 16 q^{57} - 6 q^{58} - 2 q^{60} + 4 q^{61} + 4 q^{62} + 2 q^{63} + 2 q^{64} - 4 q^{65} + 2 q^{66} - 8 q^{67} - 24 q^{69} - 2 q^{70} + 12 q^{71} + 2 q^{72} - 8 q^{73} - 2 q^{74} + 2 q^{75} + 10 q^{76} + 2 q^{77} - 8 q^{78} - 14 q^{79} - 2 q^{80} + 2 q^{81} + 6 q^{82} - 12 q^{83} + 2 q^{84} + 4 q^{86} - 12 q^{87} + 2 q^{88} - 2 q^{90} + 4 q^{91} - 6 q^{92} + 4 q^{93} - 10 q^{95} + 2 q^{96} - 2 q^{97} + 2 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 2 * q^7 + 2 * q^8 + 2 * q^9 - 2 * q^10 + 2 * q^11 + 2 * q^12 + 4 * q^13 + 2 * q^14 - 2 * q^15 + 2 * q^16 + 2 * q^18 + 10 * q^19 - 2 * q^20 + 2 * q^21 + 2 * q^22 - 6 * q^23 + 2 * q^24 + 2 * q^25 + 4 * q^26 + 8 * q^27 + 2 * q^28 - 6 * q^29 - 2 * q^30 + 4 * q^31 + 2 * q^32 + 2 * q^33 - 2 * q^35 + 2 * q^36 - 2 * q^37 + 10 * q^38 - 8 * q^39 - 2 * q^40 + 6 * q^41 + 2 * q^42 + 4 * q^43 + 2 * q^44 - 2 * q^45 - 6 * q^46 + 2 * q^48 + 2 * q^49 + 2 * q^50 - 12 * q^51 + 4 * q^52 - 18 * q^53 + 8 * q^54 - 2 * q^55 + 2 * q^56 + 16 * q^57 - 6 * q^58 - 2 * q^60 + 4 * q^61 + 4 * q^62 + 2 * q^63 + 2 * q^64 - 4 * q^65 + 2 * q^66 - 8 * q^67 - 24 * q^69 - 2 * q^70 + 12 * q^71 + 2 * q^72 - 8 * q^73 - 2 * q^74 + 2 * q^75 + 10 * q^76 + 2 * q^77 - 8 * q^78 - 14 * q^79 - 2 * q^80 + 2 * q^81 + 6 * q^82 - 12 * q^83 + 2 * q^84 + 4 * q^86 - 12 * q^87 + 2 * q^88 - 2 * q^90 + 4 * q^91 - 6 * q^92 + 4 * q^93 - 10 * q^95 + 2 * q^96 - 2 * q^97 + 2 * q^98 + 2 * q^99

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