LMFDB - Embedded newform 819.2.o.b.757.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [819,2,Mod(568,819)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(819, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("819.568");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 819 = 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	819.o (of order $3$, degree $2$, minimal)

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:	no
Analytic conductor:	$6.53974792554$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 91)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			757.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	819.757
Dual form			819.2.o.b.568.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 + 1.50000i) q^{2} +(-0.500000 + 0.866025i) q^{4} -1.73205 q^{5} +(-0.500000 + 0.866025i) q^{7} +1.73205 q^{8} +O(q^{10})$	\(q+(0.866025 + 1.50000i) q^{2} +(-0.500000 + 0.866025i) q^{4} -1.73205 q^{5} +(-0.500000 + 0.866025i) q^{7} +1.73205 q^{8} +(-1.50000 - 2.59808i) q^{10} +(0.633975 + 1.09808i) q^{11} +(3.59808 + 0.232051i) q^{13} -1.73205 q^{14} +(2.50000 + 4.33013i) q^{16} +(-3.86603 + 6.69615i) q^{17} +(-1.00000 + 1.73205i) q^{19} +(0.866025 - 1.50000i) q^{20} +(-1.09808 + 1.90192i) q^{22} +(2.36603 + 4.09808i) q^{23} -2.00000 q^{25} +(2.76795 + 5.59808i) q^{26} +(-0.500000 - 0.866025i) q^{28} +(-1.50000 - 2.59808i) q^{29} +4.19615 q^{31} +(-2.59808 + 4.50000i) q^{32} -13.3923 q^{34} +(0.866025 - 1.50000i) q^{35} +(3.50000 + 6.06218i) q^{37} -3.46410 q^{38} -3.00000 q^{40} +(-2.59808 - 4.50000i) q^{41} +(0.0980762 - 0.169873i) q^{43} -1.26795 q^{44} +(-4.09808 + 7.09808i) q^{46} -12.9282 q^{47} +(-0.500000 - 0.866025i) q^{49} +(-1.73205 - 3.00000i) q^{50} +(-2.00000 + 3.00000i) q^{52} +9.92820 q^{53} +(-1.09808 - 1.90192i) q^{55} +(-0.866025 + 1.50000i) q^{56} +(2.59808 - 4.50000i) q^{58} +(3.63397 - 6.29423i) q^{59} +(2.40192 - 4.16025i) q^{61} +(3.63397 + 6.29423i) q^{62} +1.00000 q^{64} +(-6.23205 - 0.401924i) q^{65} +(3.09808 + 5.36603i) q^{67} +(-3.86603 - 6.69615i) q^{68} +3.00000 q^{70} +(3.00000 - 5.19615i) q^{71} -3.19615 q^{73} +(-6.06218 + 10.5000i) q^{74} +(-1.00000 - 1.73205i) q^{76} -1.26795 q^{77} +16.1962 q^{79} +(-4.33013 - 7.50000i) q^{80} +(4.50000 - 7.79423i) q^{82} +2.19615 q^{83} +(6.69615 - 11.5981i) q^{85} +0.339746 q^{86} +(1.09808 + 1.90192i) q^{88} +(6.46410 + 11.1962i) q^{89} +(-2.00000 + 3.00000i) q^{91} -4.73205 q^{92} +(-11.1962 - 19.3923i) q^{94} +(1.73205 - 3.00000i) q^{95} +(-3.19615 + 5.53590i) q^{97} +(0.866025 - 1.50000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{4} - 2 q^{7}+O(q^{10}) $ 4 * q - 2 * q^4 - 2 * q^7	$ 4 q - 2 q^{4} - 2 q^{7} - 6 q^{10} + 6 q^{11} + 4 q^{13} + 10 q^{16} - 12 q^{17} - 4 q^{19} + 6 q^{22} + 6 q^{23} - 8 q^{25} + 18 q^{26} - 2 q^{28} - 6 q^{29} - 4 q^{31} - 12 q^{34} + 14 q^{37} - 12 q^{40} - 10 q^{43} - 12 q^{44} - 6 q^{46} - 24 q^{47} - 2 q^{49} - 8 q^{52} + 12 q^{53} + 6 q^{55} + 18 q^{59} + 20 q^{61} + 18 q^{62} + 4 q^{64} - 18 q^{65} + 2 q^{67} - 12 q^{68} + 12 q^{70} + 12 q^{71} + 8 q^{73} - 4 q^{76} - 12 q^{77} + 44 q^{79} + 18 q^{82} - 12 q^{83} + 6 q^{85} + 36 q^{86} - 6 q^{88} + 12 q^{89} - 8 q^{91} - 12 q^{92} - 24 q^{94} + 8 q^{97}+O(q^{100}) $ 4 * q - 2 * q^4 - 2 * q^7 - 6 * q^10 + 6 * q^11 + 4 * q^13 + 10 * q^16 - 12 * q^17 - 4 * q^19 + 6 * q^22 + 6 * q^23 - 8 * q^25 + 18 * q^26 - 2 * q^28 - 6 * q^29 - 4 * q^31 - 12 * q^34 + 14 * q^37 - 12 * q^40 - 10 * q^43 - 12 * q^44 - 6 * q^46 - 24 * q^47 - 2 * q^49 - 8 * q^52 + 12 * q^53 + 6 * q^55 + 18 * q^59 + 20 * q^61 + 18 * q^62 + 4 * q^64 - 18 * q^65 + 2 * q^67 - 12 * q^68 + 12 * q^70 + 12 * q^71 + 8 * q^73 - 4 * q^76 - 12 * q^77 + 44 * q^79 + 18 * q^82 - 12 * q^83 + 6 * q^85 + 36 * q^86 - 6 * q^88 + 12 * q^89 - 8 * q^91 - 12 * q^92 - 24 * q^94 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/819\mathbb{Z}\right)^\times$.

$n$	$92$	$379$	$703$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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 $

$2$	0.866025	+	1.50000i	0.612372	+	1.06066i	0.990839	+	0.135045i	$0.0431180\pi$
$2$	0.866025	+	1.50000i	0.612372	+	1.06066i	−0.378467	+	0.925615i	$0.623549\pi$
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$	−1.73205			−0.774597			−0.387298	−	0.921954i	$-0.626592\pi$
$5$	−1.73205			−0.774597			−0.387298	+	0.921954i	$0.626592\pi$
$6$		0			0
$7$	−0.500000	+	0.866025i	−0.188982	+	0.327327i
$7$	−0.500000	+	0.866025i	−0.188982	+	0.327327i
$8$	1.73205			0.612372
$9$		0			0
$10$	−1.50000	−	2.59808i	−0.474342	−	0.821584i
$11$	0.633975	+	1.09808i	0.191151	+	0.331082i	0.945632	−	0.325239i	$-0.105445\pi$
$11$	0.633975	+	1.09808i	0.191151	+	0.331082i	−0.754481	+	0.656322i	$0.772111\pi$
$12$		0			0
$13$	3.59808	+	0.232051i	0.997927	+	0.0643593i
$13$	3.59808	+	0.232051i	0.997927	+	0.0643593i
$14$	−1.73205			−0.462910
$15$		0			0
$16$	2.50000	+	4.33013i	0.625000	+	1.08253i
$17$	−3.86603	+	6.69615i	−0.937649	+	1.62406i	−0.167808	+	0.985820i	$0.553669\pi$
$17$	−3.86603	+	6.69615i	−0.937649	+	1.62406i	−0.769841	+	0.638236i	$0.779664\pi$
$18$		0			0
$19$	−1.00000	+	1.73205i	−0.229416	+	0.397360i	−0.957635	−	0.287984i	$-0.907015\pi$
$19$	−1.00000	+	1.73205i	−0.229416	+	0.397360i	0.728219	+	0.685344i	$0.240348\pi$
$20$	0.866025	−	1.50000i	0.193649	−	0.335410i
$21$		0			0
$22$	−1.09808	+	1.90192i	−0.234111	+	0.405492i
$23$	2.36603	+	4.09808i	0.493350	+	0.854508i	0.999971	−	0.00766135i	$-0.00243871\pi$
$23$	2.36603	+	4.09808i	0.493350	+	0.854508i	−0.506620	+	0.862169i	$0.669105\pi$
$24$		0			0
$25$	−2.00000			−0.400000
$26$	2.76795	+	5.59808i	0.542839	+	1.09787i
$27$		0			0
$28$	−0.500000	−	0.866025i	−0.0944911	−	0.163663i
$29$	−1.50000	−	2.59808i	−0.278543	−	0.482451i	0.692480	−	0.721437i	$-0.256518\pi$
$29$	−1.50000	−	2.59808i	−0.278543	−	0.482451i	−0.971023	+	0.238987i	$0.923185\pi$
$30$		0			0
$31$	4.19615			0.753651			0.376826	−	0.926284i	$-0.377016\pi$
$31$	4.19615			0.753651			0.376826	+	0.926284i	$0.377016\pi$
$32$	−2.59808	+	4.50000i	−0.459279	+	0.795495i
$33$		0			0
$34$	−13.3923			−2.29676
$35$	0.866025	−	1.50000i	0.146385	−	0.253546i
$36$		0			0
$37$	3.50000	+	6.06218i	0.575396	+	0.996616i	0.995998	+	0.0893706i	$0.0284856\pi$
$37$	3.50000	+	6.06218i	0.575396	+	0.996616i	−0.420602	+	0.907245i	$0.638181\pi$
$38$	−3.46410			−0.561951
$39$		0			0
$40$	−3.00000			−0.474342
$41$	−2.59808	−	4.50000i	−0.405751	−	0.702782i	0.588657	−	0.808383i	$-0.299657\pi$
$41$	−2.59808	−	4.50000i	−0.405751	−	0.702782i	−0.994409	+	0.105601i	$0.966323\pi$
$42$		0			0
$43$	0.0980762	−	0.169873i	0.0149565	−	0.0259054i	−0.858450	−	0.512897i	$-0.828572\pi$
$43$	0.0980762	−	0.169873i	0.0149565	−	0.0259054i	0.873407	+	0.486991i	$0.161906\pi$
$44$	−1.26795			−0.191151
$45$		0			0
$46$	−4.09808	+	7.09808i	−0.604228	+	1.04655i
$47$	−12.9282			−1.88577			−0.942886	−	0.333115i	$-0.891900\pi$
$47$	−12.9282			−1.88577			−0.942886	+	0.333115i	$0.891900\pi$
$48$		0			0
$49$	−0.500000	−	0.866025i	−0.0714286	−	0.123718i
$50$	−1.73205	−	3.00000i	−0.244949	−	0.424264i
$51$		0			0
$52$	−2.00000	+	3.00000i	−0.277350	+	0.416025i
$53$	9.92820			1.36374			0.681872	−	0.731472i	$-0.261166\pi$
$53$	9.92820			1.36374			0.681872	+	0.731472i	$0.261166\pi$
$54$		0			0
$55$	−1.09808	−	1.90192i	−0.148065	−	0.256455i
$56$	−0.866025	+	1.50000i	−0.115728	+	0.200446i
$57$		0			0
$58$	2.59808	−	4.50000i	0.341144	−	0.590879i
$59$	3.63397	−	6.29423i	0.473103	−	0.819439i	−0.526423	−	0.850223i	$-0.676467\pi$
$59$	3.63397	−	6.29423i	0.473103	−	0.819439i	0.999526	+	0.0307841i	$0.00980044\pi$
$60$		0			0
$61$	2.40192	−	4.16025i	0.307535	−	0.532666i	−0.670288	−	0.742101i	$-0.733829\pi$
$61$	2.40192	−	4.16025i	0.307535	−	0.532666i	0.977822	+	0.209435i	$0.0671626\pi$
$62$	3.63397	+	6.29423i	0.461515	+	0.799368i
$63$		0			0
$64$	1.00000			0.125000
$65$	−6.23205	−	0.401924i	−0.772991	−	0.0498525i
$66$		0			0
$67$	3.09808	+	5.36603i	0.378490	+	0.655564i	0.990843	−	0.135020i	$-0.0431100\pi$
$67$	3.09808	+	5.36603i	0.378490	+	0.655564i	−0.612353	+	0.790585i	$0.709777\pi$
$68$	−3.86603	−	6.69615i	−0.468824	−	0.812028i
$69$		0			0
$70$	3.00000			0.358569
$71$	3.00000	−	5.19615i	0.356034	−	0.616670i	−0.631260	−	0.775571i	$-0.717462\pi$
$71$	3.00000	−	5.19615i	0.356034	−	0.616670i	0.987294	+	0.158901i	$0.0507952\pi$
$72$		0			0
$73$	−3.19615			−0.374081			−0.187041	−	0.982352i	$-0.559890\pi$
$73$	−3.19615			−0.374081			−0.187041	+	0.982352i	$0.559890\pi$
$74$	−6.06218	+	10.5000i	−0.704714	+	1.22060i
$75$		0			0
$76$	−1.00000	−	1.73205i	−0.114708	−	0.198680i
$77$	−1.26795			−0.144496
$78$		0			0
$79$	16.1962			1.82221			0.911105	−	0.412175i	$-0.135231\pi$
$79$	16.1962			1.82221			0.911105	+	0.412175i	$0.135231\pi$
$80$	−4.33013	−	7.50000i	−0.484123	−	0.838525i
$81$		0			0
$82$	4.50000	−	7.79423i	0.496942	−	0.860729i
$83$	2.19615			0.241059			0.120530	−	0.992710i	$-0.461541\pi$
$83$	2.19615			0.241059			0.120530	+	0.992710i	$0.461541\pi$
$84$		0			0
$85$	6.69615	−	11.5981i	0.726300	−	1.25799i
$86$	0.339746			0.0366357
$87$		0			0
$88$	1.09808	+	1.90192i	0.117055	+	0.202746i
$89$	6.46410	+	11.1962i	0.685193	+	1.18679i	0.973376	+	0.229214i	$0.0736157\pi$
$89$	6.46410	+	11.1962i	0.685193	+	1.18679i	−0.288183	+	0.957575i	$0.593051\pi$
$90$		0			0
$91$	−2.00000	+	3.00000i	−0.209657	+	0.314485i
$92$	−4.73205			−0.493350
$93$		0			0
$94$	−11.1962	−	19.3923i	−1.15479	−	2.00016i
$95$	1.73205	−	3.00000i	0.177705	−	0.307794i
$96$		0			0
$97$	−3.19615	+	5.53590i	−0.324520	+	0.562085i	−0.981415	−	0.191897i	$-0.938536\pi$
$97$	−3.19615	+	5.53590i	−0.324520	+	0.562085i	0.656895	+	0.753982i	$0.271869\pi$
$98$	0.866025	−	1.50000i	0.0874818	−	0.151523i
$99$		0			0
$100$	1.00000	−	1.73205i	0.100000	−	0.173205i
$101$	−3.86603	−	6.69615i	−0.384684	−	0.666292i	0.607041	−	0.794670i	$-0.292356\pi$
$101$	−3.86603	−	6.69615i	−0.384684	−	0.666292i	−0.991725	+	0.128378i	$0.959023\pi$
$102$		0			0
$103$	−14.3923			−1.41812			−0.709058	−	0.705150i	$-0.750880\pi$
$103$	−14.3923			−1.41812			−0.709058	+	0.705150i	$0.750880\pi$
$104$	6.23205	+	0.401924i	0.611103	+	0.0394119i
$105$		0			0
$106$	8.59808	+	14.8923i	0.835119	+	1.44647i
$107$	−3.92820	−	6.80385i	−0.379754	−	0.657753i	0.611273	−	0.791420i	$-0.290658\pi$
$107$	−3.92820	−	6.80385i	−0.379754	−	0.657753i	−0.991026	+	0.133667i	$0.957325\pi$
$108$		0			0
$109$	−8.39230			−0.803837			−0.401919	−	0.915675i	$-0.631656\pi$
$109$	−8.39230			−0.803837			−0.401919	+	0.915675i	$0.631656\pi$
$110$	1.90192	−	3.29423i	0.181341	−	0.314092i
$111$		0			0
$112$	−5.00000			−0.472456
$113$	6.69615	−	11.5981i	0.629921	−	1.09106i	−0.357646	−	0.933857i	$-0.616421\pi$
$113$	6.69615	−	11.5981i	0.629921	−	1.09106i	0.987567	−	0.157198i	$-0.0502461\pi$
$114$		0			0
$115$	−4.09808	−	7.09808i	−0.382148	−	0.661899i
$116$	3.00000			0.278543
$117$		0			0
$118$	12.5885			1.15886
$119$	−3.86603	−	6.69615i	−0.354398	−	0.613835i
$120$		0			0
$121$	4.69615	−	8.13397i	0.426923	−	0.739452i
$122$	8.32051			0.753303
$123$		0			0
$124$	−2.09808	+	3.63397i	−0.188413	+	0.326341i
$125$	12.1244			1.08444
$126$		0			0
$127$	−9.19615	−	15.9282i	−0.816027	−	1.41340i	−0.908588	−	0.417693i	$-0.862839\pi$
$127$	−9.19615	−	15.9282i	−0.816027	−	1.41340i	0.0925619	−	0.995707i	$-0.470494\pi$
$128$	6.06218	+	10.5000i	0.535826	+	0.928078i
$129$		0			0
$130$	−4.79423	−	9.69615i	−0.420482	−	0.850409i
$131$	3.46410			0.302660			0.151330	−	0.988483i	$-0.451644\pi$
$131$	3.46410			0.302660			0.151330	+	0.988483i	$0.451644\pi$
$132$		0			0
$133$	−1.00000	−	1.73205i	−0.0867110	−	0.150188i
$134$	−5.36603	+	9.29423i	−0.463554	+	0.802899i
$135$		0			0
$136$	−6.69615	+	11.5981i	−0.574190	+	0.994527i
$137$	4.03590	−	6.99038i	0.344810	−	0.597229i	−0.640509	−	0.767951i	$-0.721277\pi$
$137$	4.03590	−	6.99038i	0.344810	−	0.597229i	0.985319	+	0.170722i	$0.0546100\pi$
$138$		0			0
$139$	5.29423	−	9.16987i	0.449051	−	0.777778i	−0.549274	−	0.835642i	$-0.685096\pi$
$139$	5.29423	−	9.16987i	0.449051	−	0.777778i	0.998324	+	0.0578639i	$0.0184290\pi$
$140$	0.866025	+	1.50000i	0.0731925	+	0.126773i
$141$		0			0
$142$	10.3923			0.872103
$143$	2.02628	+	4.09808i	0.169446	+	0.342698i
$144$		0			0
$145$	2.59808	+	4.50000i	0.215758	+	0.373705i
$146$	−2.76795	−	4.79423i	−0.229077	−	0.396773i
$147$		0			0
$148$	−7.00000			−0.575396
$149$	−3.23205	+	5.59808i	−0.264780	+	0.458612i	−0.967506	−	0.252848i	$-0.918633\pi$
$149$	−3.23205	+	5.59808i	−0.264780	+	0.458612i	0.702726	+	0.711461i	$0.251966\pi$
$150$		0			0
$151$	2.00000			0.162758			0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000			0.162758			0.0813788	+	0.996683i	$0.474068\pi$
$152$	−1.73205	+	3.00000i	−0.140488	+	0.243332i
$153$		0			0
$154$	−1.09808	−	1.90192i	−0.0884855	−	0.153261i
$155$	−7.26795			−0.583776
$156$		0			0
$157$	1.19615			0.0954634			0.0477317	−	0.998860i	$-0.484801\pi$
$157$	1.19615			0.0954634			0.0477317	+	0.998860i	$0.484801\pi$
$158$	14.0263	+	24.2942i	1.11587	+	1.93275i
$159$		0			0
$160$	4.50000	−	7.79423i	0.355756	−	0.616188i
$161$	−4.73205			−0.372938
$162$		0			0
$163$	−8.09808	+	14.0263i	−0.634290	+	1.09862i	0.352375	+	0.935859i	$0.385374\pi$
$163$	−8.09808	+	14.0263i	−0.634290	+	1.09862i	−0.986665	+	0.162764i	$0.947959\pi$
$164$	5.19615			0.405751
$165$		0			0
$166$	1.90192	+	3.29423i	0.147618	+	0.255682i
$167$	−3.29423	−	5.70577i	−0.254915	−	0.441526i	0.709957	−	0.704245i	$-0.248714\pi$
$167$	−3.29423	−	5.70577i	−0.254915	−	0.441526i	−0.964872	+	0.262719i	$0.915381\pi$
$168$		0			0
$169$	12.8923	+	1.66987i	0.991716	+	0.128452i
$170$	23.1962			1.77906
$171$		0			0
$172$	0.0980762	+	0.169873i	0.00747824	+	0.0129527i
$173$	4.26795	−	7.39230i	0.324486	−	0.562027i	−0.656922	−	0.753958i	$-0.728142\pi$
$173$	4.26795	−	7.39230i	0.324486	−	0.562027i	0.981408	+	0.191932i	$0.0614753\pi$
$174$		0			0
$175$	1.00000	−	1.73205i	0.0755929	−	0.130931i
$176$	−3.16987	+	5.49038i	−0.238938	+	0.413853i
$177$		0			0
$178$	−11.1962	+	19.3923i	−0.839187	+	1.45351i
$179$	−3.46410	−	6.00000i	−0.258919	−	0.448461i	0.707034	−	0.707180i	$-0.250033\pi$
$179$	−3.46410	−	6.00000i	−0.258919	−	0.448461i	−0.965953	+	0.258719i	$0.916700\pi$
$180$		0			0
$181$	5.58846			0.415387			0.207693	−	0.978194i	$-0.433404\pi$
$181$	5.58846			0.415387			0.207693	+	0.978194i	$0.433404\pi$
$182$	−6.23205	−	0.401924i	−0.461950	−	0.0297926i
$183$		0			0
$184$	4.09808	+	7.09808i	0.302114	+	0.523277i
$185$	−6.06218	−	10.5000i	−0.445700	−	0.771975i
$186$		0			0
$187$	−9.80385			−0.716928
$188$	6.46410	−	11.1962i	0.471443	−	0.816563i
$189$		0			0
$190$	6.00000			0.435286
$191$	2.36603	−	4.09808i	0.171200	−	0.296526i	−0.767640	−	0.640881i	$-0.778569\pi$
$191$	2.36603	−	4.09808i	0.171200	−	0.296526i	0.938840	+	0.344355i	$0.111902\pi$
$192$		0			0
$193$	−2.50000	−	4.33013i	−0.179954	−	0.311689i	0.761911	−	0.647682i	$-0.224262\pi$
$193$	−2.50000	−	4.33013i	−0.179954	−	0.311689i	−0.941865	+	0.335993i	$0.890928\pi$
$194$	−11.0718			−0.794909
$195$		0			0
$196$	1.00000			0.0714286
$197$	6.00000	+	10.3923i	0.427482	+	0.740421i	0.996649	−	0.0818013i	$-0.0260673\pi$
$197$	6.00000	+	10.3923i	0.427482	+	0.740421i	−0.569166	+	0.822222i	$0.692734\pi$
$198$		0			0
$199$	−1.00000	+	1.73205i	−0.0708881	+	0.122782i	−0.899291	−	0.437351i	$-0.855917\pi$
$199$	−1.00000	+	1.73205i	−0.0708881	+	0.122782i	0.828403	+	0.560133i	$0.189250\pi$
$200$	−3.46410			−0.244949
$201$		0			0
$202$	6.69615	−	11.5981i	0.471140	−	0.816038i
$203$	3.00000			0.210559
$204$		0			0
$205$	4.50000	+	7.79423i	0.314294	+	0.544373i
$206$	−12.4641	−	21.5885i	−0.868415	−	1.50414i
$207$		0			0
$208$	7.99038	+	16.1603i	0.554033	+	1.12051i
$209$	−2.53590			−0.175412
$210$		0			0
$211$	0.901924	+	1.56218i	0.0620910	+	0.107545i	0.895400	−	0.445263i	$-0.146890\pi$
$211$	0.901924	+	1.56218i	0.0620910	+	0.107545i	−0.833309	+	0.552808i	$0.813556\pi$
$212$	−4.96410	+	8.59808i	−0.340936	+	0.590518i
$213$		0			0
$214$	6.80385	−	11.7846i	0.465101	−	0.805579i
$215$	−0.169873	+	0.294229i	−0.0115852	+	0.0200662i
$216$		0			0
$217$	−2.09808	+	3.63397i	−0.142427	+	0.246690i
$218$	−7.26795	−	12.5885i	−0.492248	−	0.852598i
$219$		0			0
$220$	2.19615			0.148065
$221$	−15.4641	+	23.1962i	−1.04023	+	1.56034i
$222$		0			0
$223$	5.00000	+	8.66025i	0.334825	+	0.579934i	0.983451	−	0.181173i	$-0.0579895\pi$
$223$	5.00000	+	8.66025i	0.334825	+	0.579934i	−0.648626	+	0.761107i	$0.724656\pi$
$224$	−2.59808	−	4.50000i	−0.173591	−	0.300669i
$225$		0			0
$226$	23.1962			1.54299
$227$	2.83013	−	4.90192i	0.187842	−	0.325352i	−0.756688	−	0.653776i	$-0.773184\pi$
$227$	2.83013	−	4.90192i	0.187842	−	0.325352i	0.944531	+	0.328424i	$0.106517\pi$
$228$		0			0
$229$	−14.3923			−0.951070			−0.475535	−	0.879697i	$-0.657746\pi$
$229$	−14.3923			−0.951070			−0.475535	+	0.879697i	$0.657746\pi$
$230$	7.09808	−	12.2942i	0.468033	−	0.810657i
$231$		0			0
$232$	−2.59808	−	4.50000i	−0.170572	−	0.295439i
$233$	1.85641			0.121617			0.0608086	−	0.998149i	$-0.480632\pi$
$233$	1.85641			0.121617			0.0608086	+	0.998149i	$0.480632\pi$
$234$		0			0
$235$	22.3923			1.46071
$236$	3.63397	+	6.29423i	0.236552	+	0.409719i
$237$		0			0
$238$	6.69615	−	11.5981i	0.434047	−	0.751792i
$239$	15.8038			1.02227			0.511133	−	0.859502i	$-0.329226\pi$
$239$	15.8038			1.02227			0.511133	+	0.859502i	$0.329226\pi$
$240$		0			0
$241$	10.5981	−	18.3564i	0.682682	−	1.18244i	−0.291477	−	0.956578i	$-0.594147\pi$
$241$	10.5981	−	18.3564i	0.682682	−	1.18244i	0.974159	−	0.225862i	$-0.0725199\pi$
$242$	16.2679			1.04574
$243$		0			0
$244$	2.40192	+	4.16025i	0.153767	+	0.266333i
$245$	0.866025	+	1.50000i	0.0553283	+	0.0958315i
$246$		0			0
$247$	−4.00000	+	6.00000i	−0.254514	+	0.381771i
$248$	7.26795			0.461515
$249$		0			0
$250$	10.5000	+	18.1865i	0.664078	+	1.15022i
$251$	−0.803848	+	1.39230i	−0.0507384	+	0.0878815i	−0.890279	−	0.455415i	$-0.849491\pi$
$251$	−0.803848	+	1.39230i	−0.0507384	+	0.0878815i	0.839541	+	0.543297i	$0.182824\pi$
$252$		0			0
$253$	−3.00000	+	5.19615i	−0.188608	+	0.326679i
$254$	15.9282	−	27.5885i	0.999424	−	1.73105i
$255$		0			0
$256$	−9.50000	+	16.4545i	−0.593750	+	1.02841i
$257$	3.06218	+	5.30385i	0.191013	+	0.330845i	0.945586	−	0.325371i	$-0.105489\pi$
$257$	3.06218	+	5.30385i	0.191013	+	0.330845i	−0.754573	+	0.656216i	$0.772156\pi$
$258$		0			0
$259$	−7.00000			−0.434959
$260$	3.46410	−	5.19615i	0.214834	−	0.322252i
$261$		0			0
$262$	3.00000	+	5.19615i	0.185341	+	0.321019i
$263$	0.633975	+	1.09808i	0.0390925	+	0.0677103i	0.884910	−	0.465763i	$-0.154220\pi$
$263$	0.633975	+	1.09808i	0.0390925	+	0.0677103i	−0.845817	+	0.533473i	$0.820887\pi$
$264$		0			0
$265$	−17.1962			−1.05635
$266$	1.73205	−	3.00000i	0.106199	−	0.183942i
$267$		0			0
$268$	−6.19615			−0.378490
$269$	2.53590	−	4.39230i	0.154616	−	0.267804i	−0.778303	−	0.627889i	$-0.783919\pi$
$269$	2.53590	−	4.39230i	0.154616	−	0.267804i	0.932919	+	0.360086i	$0.117252\pi$
$270$		0			0
$271$	−2.90192	−	5.02628i	−0.176279	−	0.305325i	0.764324	−	0.644832i	$-0.223073\pi$
$271$	−2.90192	−	5.02628i	−0.176279	−	0.305325i	−0.940603	+	0.339508i	$0.889740\pi$
$272$	−38.6603			−2.34412
$273$		0			0
$274$	13.9808			0.844609
$275$	−1.26795	−	2.19615i	−0.0764602	−	0.132433i
$276$		0			0
$277$	−8.50000	+	14.7224i	−0.510716	+	0.884585i	0.489207	+	0.872167i	$0.337286\pi$
$277$	−8.50000	+	14.7224i	−0.510716	+	0.884585i	−0.999923	+	0.0124177i	$0.996047\pi$
$278$	18.3397			1.09994
$279$		0			0
$280$	1.50000	−	2.59808i	0.0896421	−	0.155265i
$281$	−13.3923			−0.798918			−0.399459	−	0.916751i	$-0.630802\pi$
$281$	−13.3923			−0.798918			−0.399459	+	0.916751i	$0.630802\pi$
$282$		0			0
$283$	−5.09808	−	8.83013i	−0.303049	−	0.524897i	0.673776	−	0.738936i	$-0.264671\pi$
$283$	−5.09808	−	8.83013i	−0.303049	−	0.524897i	−0.976825	+	0.214039i	$0.931338\pi$
$284$	3.00000	+	5.19615i	0.178017	+	0.308335i
$285$		0			0
$286$	−4.39230	+	6.58846i	−0.259722	+	0.389584i
$287$	5.19615			0.306719
$288$		0			0
$289$	−21.3923	−	37.0526i	−1.25837	−	2.17956i
$290$	−4.50000	+	7.79423i	−0.264249	+	0.457693i
$291$		0			0
$292$	1.59808	−	2.76795i	0.0935203	−	0.161982i
$293$	−0.401924	+	0.696152i	−0.0234806	+	0.0406697i	−0.877527	−	0.479527i	$-0.840808\pi$
$293$	−0.401924	+	0.696152i	−0.0234806	+	0.0406697i	0.854046	+	0.520197i	$0.174141\pi$
$294$		0			0
$295$	−6.29423	+	10.9019i	−0.366464	+	0.634735i
$296$	6.06218	+	10.5000i	0.352357	+	0.610300i
$297$		0			0
$298$	−11.1962			−0.648576
$299$	7.56218	+	15.2942i	0.437332	+	0.884488i
$300$		0			0
$301$	0.0980762	+	0.169873i	0.00565302	+	0.00979132i
$302$	1.73205	+	3.00000i	0.0996683	+	0.172631i
$303$		0			0
$304$	−10.0000			−0.573539
$305$	−4.16025	+	7.20577i	−0.238215	+	0.412601i
$306$		0			0
$307$	−4.58846			−0.261877			−0.130939	−	0.991390i	$-0.541799\pi$
$307$	−4.58846			−0.261877			−0.130939	+	0.991390i	$0.541799\pi$
$308$	0.633975	−	1.09808i	0.0361241	−	0.0625687i
$309$		0			0
$310$	−6.29423	−	10.9019i	−0.357488	−	0.619188i
$311$	−1.26795			−0.0718988			−0.0359494	−	0.999354i	$-0.511446\pi$
$311$	−1.26795			−0.0718988			−0.0359494	+	0.999354i	$0.511446\pi$
$312$		0			0
$313$	28.7846			1.62700			0.813501	−	0.581563i	$-0.197559\pi$
$313$	28.7846			1.62700			0.813501	+	0.581563i	$0.197559\pi$
$314$	1.03590	+	1.79423i	0.0584591	+	0.101254i
$315$		0			0
$316$	−8.09808	+	14.0263i	−0.455552	+	0.789040i
$317$	6.46410			0.363060			0.181530	−	0.983385i	$-0.441895\pi$
$317$	6.46410			0.363060			0.181530	+	0.983385i	$0.441895\pi$
$318$		0			0
$319$	1.90192	−	3.29423i	0.106487	−	0.184441i
$320$	−1.73205			−0.0968246
$321$		0			0
$322$	−4.09808	−	7.09808i	−0.228377	−	0.395560i
$323$	−7.73205	−	13.3923i	−0.430223	−	0.745168i
$324$		0			0
$325$	−7.19615	−	0.464102i	−0.399171	−	0.0257437i
$326$	−28.0526			−1.55369
$327$		0			0
$328$	−4.50000	−	7.79423i	−0.248471	−	0.430364i
$329$	6.46410	−	11.1962i	0.356377	−	0.617264i
$330$		0			0
$331$	−12.4904	+	21.6340i	−0.686533	+	1.18911i	0.286419	+	0.958105i	$0.407535\pi$
$331$	−12.4904	+	21.6340i	−0.686533	+	1.18911i	−0.972952	+	0.231006i	$0.925798\pi$
$332$	−1.09808	+	1.90192i	−0.0602648	+	0.104382i
$333$		0			0
$334$	5.70577	−	9.88269i	0.312206	−	0.540757i
$335$	−5.36603	−	9.29423i	−0.293177	−	0.507798i
$336$		0			0
$337$	11.0000			0.599208			0.299604	−	0.954064i	$-0.403145\pi$
$337$	11.0000			0.599208			0.299604	+	0.954064i	$0.403145\pi$
$338$	8.66025	+	20.7846i	0.471056	+	1.13053i
$339$		0			0
$340$	6.69615	+	11.5981i	0.363150	+	0.628994i
$341$	2.66025	+	4.60770i	0.144061	+	0.249521i
$342$		0			0
$343$	1.00000			0.0539949
$344$	0.169873	−	0.294229i	0.00915894	−	0.0158637i
$345$		0			0
$346$	14.7846			0.794826
$347$	3.63397	−	6.29423i	0.195082	−	0.337892i	−0.751845	−	0.659339i	$-0.770836\pi$
$347$	3.63397	−	6.29423i	0.195082	−	0.337892i	0.946927	+	0.321448i	$0.104169\pi$
$348$		0			0
$349$	12.3923	+	21.4641i	0.663345	+	1.14895i	0.979731	+	0.200317i	$0.0641971\pi$
$349$	12.3923	+	21.4641i	0.663345	+	1.14895i	−0.316386	+	0.948630i	$0.602470\pi$
$350$	3.46410			0.185164
$351$		0			0
$352$	−6.58846			−0.351166
$353$	10.3301	+	17.8923i	0.549817	+	0.952311i	0.998287	+	0.0585131i	$0.0186360\pi$
$353$	10.3301	+	17.8923i	0.549817	+	0.952311i	−0.448469	+	0.893798i	$0.648031\pi$
$354$		0			0
$355$	−5.19615	+	9.00000i	−0.275783	+	0.477670i
$356$	−12.9282			−0.685193
$357$		0			0
$358$	6.00000	−	10.3923i	0.317110	−	0.549250i
$359$	−18.9282			−0.998992			−0.499496	−	0.866316i	$-0.666482\pi$
$359$	−18.9282			−0.998992			−0.499496	+	0.866316i	$0.666482\pi$
$360$		0			0
$361$	7.50000	+	12.9904i	0.394737	+	0.683704i
$362$	4.83975	+	8.38269i	0.254371	+	0.440584i
$363$		0			0
$364$	−1.59808	−	3.23205i	−0.0837620	−	0.169405i
$365$	5.53590			0.289762
$366$		0			0
$367$	−2.09808	−	3.63397i	−0.109519	−	0.189692i	0.806057	−	0.591838i	$-0.201598\pi$
$367$	−2.09808	−	3.63397i	−0.109519	−	0.189692i	−0.915575	+	0.402146i	$0.868264\pi$
$368$	−11.8301	+	20.4904i	−0.616688	+	1.06813i
$369$		0			0
$370$	10.5000	−	18.1865i	0.545869	−	0.945473i
$371$	−4.96410	+	8.59808i	−0.257723	+	0.446390i
$372$		0			0
$373$	5.69615	−	9.86603i	0.294936	−	0.510843i	−0.680034	−	0.733180i	$-0.738035\pi$
$373$	5.69615	−	9.86603i	0.294936	−	0.510843i	0.974970	+	0.222337i	$0.0713685\pi$
$374$	−8.49038	−	14.7058i	−0.439027	−	0.760417i
$375$		0			0
$376$	−22.3923			−1.15479
$377$	−4.79423	−	9.69615i	−0.246915	−	0.499377i
$378$		0			0
$379$	−13.2942	−	23.0263i	−0.682879	−	1.18278i	−0.974098	−	0.226124i	$-0.927394\pi$
$379$	−13.2942	−	23.0263i	−0.682879	−	1.18278i	0.291220	−	0.956656i	$-0.405939\pi$
$380$	1.73205	+	3.00000i	0.0888523	+	0.153897i
$381$		0			0
$382$	8.19615			0.419352
$383$	−5.83013	+	10.0981i	−0.297906	+	0.515988i	−0.975657	−	0.219304i	$-0.929621\pi$
$383$	−5.83013	+	10.0981i	−0.297906	+	0.515988i	0.677751	+	0.735291i	$0.262955\pi$
$384$		0			0
$385$	2.19615			0.111926
$386$	4.33013	−	7.50000i	0.220398	−	0.381740i
$387$		0			0
$388$	−3.19615	−	5.53590i	−0.162260	−	0.281043i
$389$	23.5359			1.19332			0.596659	−	0.802495i	$-0.296495\pi$
$389$	23.5359			1.19332			0.596659	+	0.802495i	$0.296495\pi$
$390$		0			0
$391$	−36.5885			−1.85036
$392$	−0.866025	−	1.50000i	−0.0437409	−	0.0757614i
$393$		0			0
$394$	−10.3923	+	18.0000i	−0.523557	+	0.906827i
$395$	−28.0526			−1.41148
$396$		0			0
$397$	9.39230	−	16.2679i	0.471386	−	0.816465i	−0.528078	−	0.849196i	$-0.677087\pi$
$397$	9.39230	−	16.2679i	0.471386	−	0.816465i	0.999464	+	0.0327309i	$0.0104204\pi$
$398$	−3.46410			−0.173640
$399$		0			0
$400$	−5.00000	−	8.66025i	−0.250000	−	0.433013i
$401$	−5.42820	−	9.40192i	−0.271072	−	0.469510i	0.698065	−	0.716034i	$-0.254045\pi$
$401$	−5.42820	−	9.40192i	−0.271072	−	0.469510i	−0.969136	+	0.246525i	$0.920711\pi$
$402$		0			0
$403$	15.0981	+	0.973721i	0.752089	+	0.0485045i
$404$	7.73205			0.384684
$405$		0			0
$406$	2.59808	+	4.50000i	0.128940	+	0.223331i
$407$	−4.43782	+	7.68653i	−0.219975	+	0.381007i
$408$		0			0
$409$	8.40192	−	14.5526i	0.415448	−	0.719578i	−0.580027	−	0.814597i	$-0.696958\pi$
$409$	8.40192	−	14.5526i	0.415448	−	0.719578i	0.995475	+	0.0950195i	$0.0302913\pi$
$410$	−7.79423	+	13.5000i	−0.384930	+	0.666717i
$411$		0			0
$412$	7.19615	−	12.4641i	0.354529	−	0.614062i
$413$	3.63397	+	6.29423i	0.178816	+	0.309719i
$414$		0			0
$415$	−3.80385			−0.186724
$416$	−10.3923	+	15.5885i	−0.509525	+	0.764287i
$417$		0			0
$418$	−2.19615	−	3.80385i	−0.107417	−	0.186052i
$419$	−16.0981	−	27.8827i	−0.786442	−	1.36216i	−0.928134	−	0.372247i	$-0.878587\pi$
$419$	−16.0981	−	27.8827i	−0.786442	−	1.36216i	0.141691	−	0.989911i	$-0.454746\pi$
$420$		0			0
$421$	−32.1769			−1.56821			−0.784103	−	0.620630i	$-0.786877\pi$
$421$	−32.1769			−1.56821			−0.784103	+	0.620630i	$0.786877\pi$
$422$	−1.56218	+	2.70577i	−0.0760456	+	0.131715i
$423$		0			0
$424$	17.1962			0.835119
$425$	7.73205	−	13.3923i	0.375060	−	0.649622i
$426$		0			0
$427$	2.40192	+	4.16025i	0.116237	+	0.201329i
$428$	7.85641			0.379754
$429$		0			0
$430$	−0.588457			−0.0283779
$431$	0.339746	+	0.588457i	0.0163650	+	0.0283450i	0.874092	−	0.485761i	$-0.161457\pi$
$431$	0.339746	+	0.588457i	0.0163650	+	0.0283450i	−0.857727	+	0.514106i	$0.828124\pi$
$432$		0			0
$433$	6.79423	−	11.7679i	0.326510	−	0.565532i	−0.655307	−	0.755363i	$-0.727461\pi$
$433$	6.79423	−	11.7679i	0.326510	−	0.565532i	0.981817	+	0.189831i	$0.0607941\pi$
$434$	−7.26795			−0.348873
$435$		0			0
$436$	4.19615	−	7.26795i	0.200959	−	0.348072i
$437$	−9.46410			−0.452729
$438$		0			0
$439$	−7.29423	−	12.6340i	−0.348135	−	0.602987i	0.637784	−	0.770216i	$-0.279851\pi$
$439$	−7.29423	−	12.6340i	−0.348135	−	0.602987i	−0.985918	+	0.167229i	$0.946518\pi$
$440$	−1.90192	−	3.29423i	−0.0906707	−	0.157046i
$441$		0			0
$442$	−48.1865	−	3.10770i	−2.29200	−	0.147818i
$443$	−23.3205			−1.10799			−0.553995	−	0.832520i	$-0.686897\pi$
$443$	−23.3205			−1.10799			−0.553995	+	0.832520i	$0.686897\pi$
$444$		0			0
$445$	−11.1962	−	19.3923i	−0.530749	−	0.919283i
$446$	−8.66025	+	15.0000i	−0.410075	+	0.710271i
$447$		0			0
$448$	−0.500000	+	0.866025i	−0.0236228	+	0.0409159i
$449$	−6.00000	+	10.3923i	−0.283158	+	0.490443i	−0.972161	−	0.234315i	$-0.924715\pi$
$449$	−6.00000	+	10.3923i	−0.283158	+	0.490443i	0.689003	+	0.724758i	$0.258049\pi$
$450$		0			0
$451$	3.29423	−	5.70577i	0.155119	−	0.268674i
$452$	6.69615	+	11.5981i	0.314961	+	0.545528i
$453$		0			0
$454$	9.80385			0.460117
$455$	3.46410	−	5.19615i	0.162400	−	0.243599i
$456$		0			0
$457$	−5.50000	−	9.52628i	−0.257279	−	0.445621i	0.708233	−	0.705979i	$-0.249493\pi$
$457$	−5.50000	−	9.52628i	−0.257279	−	0.445621i	−0.965512	+	0.260358i	$0.916159\pi$
$458$	−12.4641	−	21.5885i	−0.582409	−	1.00876i
$459$		0			0
$460$	8.19615			0.382148
$461$	−7.79423	+	13.5000i	−0.363013	+	0.628758i	−0.988455	−	0.151513i	$-0.951585\pi$
$461$	−7.79423	+	13.5000i	−0.363013	+	0.628758i	0.625442	+	0.780271i	$0.284919\pi$
$462$		0			0
$463$	−4.58846			−0.213244			−0.106622	−	0.994300i	$-0.534003\pi$
$463$	−4.58846			−0.213244			−0.106622	+	0.994300i	$0.534003\pi$
$464$	7.50000	−	12.9904i	0.348179	−	0.603063i
$465$		0			0
$466$	1.60770	+	2.78461i	0.0744750	+	0.128995i
$467$	−25.5167			−1.18077			−0.590385	−	0.807122i	$-0.701024\pi$
$467$	−25.5167			−1.18077			−0.590385	+	0.807122i	$0.701024\pi$
$468$		0			0
$469$	−6.19615			−0.286112
$470$	19.3923	+	33.5885i	0.894500	+	1.54932i
$471$		0			0
$472$	6.29423	−	10.9019i	0.289715	−	0.501802i
$473$	0.248711			0.0114358
$474$		0			0
$475$	2.00000	−	3.46410i	0.0917663	−	0.158944i
$476$	7.73205			0.354398
$477$		0			0
$478$	13.6865	+	23.7058i	0.626007	+	1.08428i
$479$	0.633975	+	1.09808i	0.0289670	+	0.0501724i	0.880146	−	0.474704i	$-0.157445\pi$
$479$	0.633975	+	1.09808i	0.0289670	+	0.0501724i	−0.851178	+	0.524876i	$0.824112\pi$
$480$		0			0
$481$	11.1865	+	22.6244i	0.510062	+	1.03158i
$482$	36.7128			1.67222
$483$		0			0
$484$	4.69615	+	8.13397i	0.213461	+	0.369726i
$485$	5.53590	−	9.58846i	0.251372	−	0.435389i
$486$		0			0
$487$	−20.3923	+	35.3205i	−0.924064	+	1.60052i	−0.131003	+	0.991382i	$0.541820\pi$
$487$	−20.3923	+	35.3205i	−0.924064	+	1.60052i	−0.793060	+	0.609143i	$0.791514\pi$
$488$	4.16025	−	7.20577i	0.188326	−	0.326190i
$489$		0			0
$490$	−1.50000	+	2.59808i	−0.0677631	+	0.117369i
$491$	3.80385	+	6.58846i	0.171665	+	0.297333i	0.939002	−	0.343911i	$-0.111752\pi$
$491$	3.80385	+	6.58846i	0.171665	+	0.297333i	−0.767337	+	0.641244i	$0.778419\pi$
$492$		0			0
$493$	23.1962			1.04470
$494$	−12.4641	−	0.803848i	−0.560786	−	0.0361668i
$495$		0			0
$496$	10.4904	+	18.1699i	0.471032	+	0.815851i
$497$	3.00000	+	5.19615i	0.134568	+	0.233079i
$498$		0			0
$499$	−38.9808			−1.74502			−0.872509	−	0.488598i	$-0.837509\pi$
$499$	−38.9808			−1.74502			−0.872509	+	0.488598i	$0.837509\pi$
$500$	−6.06218	+	10.5000i	−0.271109	+	0.469574i
$501$		0			0
$502$	−2.78461			−0.124283
$503$	−9.29423	+	16.0981i	−0.414409	+	0.717778i	−0.995366	−	0.0961565i	$-0.969345\pi$
$503$	−9.29423	+	16.0981i	−0.414409	+	0.717778i	0.580957	+	0.813934i	$0.302678\pi$
$504$		0			0
$505$	6.69615	+	11.5981i	0.297975	+	0.516108i
$506$	−10.3923			−0.461994
$507$		0			0
$508$	18.3923			0.816027
$509$	6.86603	+	11.8923i	0.304331	+	0.527117i	0.977112	−	0.212725i	$-0.0682337\pi$
$509$	6.86603	+	11.8923i	0.304331	+	0.527117i	−0.672781	+	0.739842i	$0.734900\pi$
$510$		0			0
$511$	1.59808	−	2.76795i	0.0706947	−	0.122447i
$512$	−8.66025			−0.382733
$513$		0			0
$514$	−5.30385	+	9.18653i	−0.233943	+	0.405201i
$515$	24.9282			1.09847
$516$		0			0
$517$	−8.19615	−	14.1962i	−0.360466	−	0.624346i
$518$	−6.06218	−	10.5000i	−0.266357	−	0.461344i
$519$		0			0
$520$	−10.7942	−	0.696152i	−0.473358	−	0.0305283i
$521$	24.1244			1.05691			0.528454	−	0.848962i	$-0.322772\pi$
$521$	24.1244			1.05691			0.528454	+	0.848962i	$0.322772\pi$
$522$		0			0
$523$	14.5885	+	25.2679i	0.637909	+	1.10489i	0.985891	+	0.167389i	$0.0535336\pi$
$523$	14.5885	+	25.2679i	0.637909	+	1.10489i	−0.347982	+	0.937501i	$0.613133\pi$
$524$	−1.73205	+	3.00000i	−0.0756650	+	0.131056i
$525$		0			0
$526$	−1.09808	+	1.90192i	−0.0478784	+	0.0829278i
$527$	−16.2224	+	28.0981i	−0.706660	+	1.22397i
$528$		0			0
$529$	0.303848	−	0.526279i	0.0132108	−	0.0228817i
$530$	−14.8923	−	25.7942i	−0.646880	−	1.12043i
$531$		0			0
$532$	2.00000			0.0867110
$533$	−8.30385	−	16.7942i	−0.359680	−	0.727439i
$534$		0			0
$535$	6.80385	+	11.7846i	0.294156	+	0.509493i
$536$	5.36603	+	9.29423i	0.231777	+	0.401450i
$537$		0			0
$538$	8.78461			0.378731
$539$	0.633975	−	1.09808i	0.0273072	−	0.0472975i
$540$		0			0
$541$	−14.6077			−0.628034			−0.314017	−	0.949417i	$-0.601675\pi$
$541$	−14.6077			−0.628034			−0.314017	+	0.949417i	$0.601675\pi$
$542$	5.02628	−	8.70577i	0.215897	−	0.373945i
$543$		0			0
$544$	−20.0885	−	34.7942i	−0.861285	−	1.49179i
$545$	14.5359			0.622649
$546$		0			0
$547$	17.8038			0.761238			0.380619	−	0.924732i	$-0.375711\pi$
$547$	17.8038			0.761238			0.380619	+	0.924732i	$0.375711\pi$
$548$	4.03590	+	6.99038i	0.172405	+	0.298614i
$549$		0			0
$550$	2.19615	−	3.80385i	0.0936443	−	0.162197i
$551$	6.00000			0.255609
$552$		0			0
$553$	−8.09808	+	14.0263i	−0.344365	+	0.596458i
$554$	−29.4449			−1.25099
$555$		0			0
$556$	5.29423	+	9.16987i	0.224525	+	0.388889i
$557$	21.8205	+	37.7942i	0.924565	+	1.60139i	0.792260	+	0.610184i	$0.208904\pi$
$557$	21.8205	+	37.7942i	0.924565	+	1.60139i	0.132305	+	0.991209i	$0.457762\pi$
$558$		0			0
$559$	0.392305	−	0.588457i	0.0165927	−	0.0248891i
$560$	8.66025			0.365963
$561$		0			0
$562$	−11.5981	−	20.0885i	−0.489235	−	0.847380i
$563$	14.0263	−	24.2942i	0.591137	−	1.02388i	−0.402942	−	0.915225i	$-0.632013\pi$
$563$	14.0263	−	24.2942i	0.591137	−	1.02388i	0.994080	−	0.108654i	$-0.0346542\pi$
$564$		0			0
$565$	−11.5981	+	20.0885i	−0.487935	+	0.845128i
$566$	8.83013	−	15.2942i	0.371158	−	0.642864i
$567$		0			0
$568$	5.19615	−	9.00000i	0.218026	−	0.377632i
$569$	21.4641	+	37.1769i	0.899822	+	1.55854i	0.827721	+	0.561140i	$0.189637\pi$
$569$	21.4641	+	37.1769i	0.899822	+	1.55854i	0.0721010	+	0.997397i	$0.477030\pi$
$570$		0			0
$571$	16.7846			0.702414			0.351207	−	0.936298i	$-0.385771\pi$
$571$	16.7846			0.702414			0.351207	+	0.936298i	$0.385771\pi$
$572$	−4.56218	−	0.294229i	−0.190754	−	0.0123023i
$573$		0			0
$574$	4.50000	+	7.79423i	0.187826	+	0.325325i
$575$	−4.73205	−	8.19615i	−0.197340	−	0.341803i
$576$		0			0
$577$	43.1962			1.79828			0.899140	−	0.437662i	$-0.144193\pi$
$577$	43.1962			1.79828			0.899140	+	0.437662i	$0.144193\pi$
$578$	37.0526	−	64.1769i	1.54118	−	2.66941i
$579$		0			0
$580$	−5.19615			−0.215758
$581$	−1.09808	+	1.90192i	−0.0455559	+	0.0789051i
$582$		0			0
$583$	6.29423	+	10.9019i	0.260680	+	0.451512i
$584$	−5.53590			−0.229077
$585$		0			0
$586$	−1.39230			−0.0575156
$587$	8.19615	+	14.1962i	0.338291	+	0.585938i	0.984111	−	0.177552i	$-0.0568177\pi$
$587$	8.19615	+	14.1962i	0.338291	+	0.585938i	−0.645820	+	0.763490i	$0.723484\pi$
$588$		0			0
$589$	−4.19615	+	7.26795i	−0.172899	+	0.299471i
$590$	−21.8038			−0.897650
$591$		0			0
$592$	−17.5000	+	30.3109i	−0.719246	+	1.24577i
$593$	17.4449			0.716375			0.358187	−	0.933650i	$-0.383395\pi$
$593$	17.4449			0.716375			0.358187	+	0.933650i	$0.383395\pi$
$594$		0			0
$595$	6.69615	+	11.5981i	0.274515	+	0.475475i
$596$	−3.23205	−	5.59808i	−0.132390	−	0.229306i
$597$		0			0
$598$	−16.3923	+	24.5885i	−0.670331	+	1.00550i
$599$	−43.8564			−1.79192			−0.895962	−	0.444131i	$-0.853513\pi$
$599$	−43.8564			−1.79192			−0.895962	+	0.444131i	$0.853513\pi$
$600$		0			0
$601$	14.9904	+	25.9641i	0.611470	+	1.05910i	0.990993	+	0.133915i	$0.0427550\pi$
$601$	14.9904	+	25.9641i	0.611470	+	1.05910i	−0.379522	+	0.925183i	$0.623912\pi$
$602$	−0.169873	+	0.294229i	−0.00692351	+	0.0119919i
$603$		0			0
$604$	−1.00000	+	1.73205i	−0.0406894	+	0.0704761i
$605$	−8.13397	+	14.0885i	−0.330693	+	0.572777i
$606$		0			0
$607$	7.19615	−	12.4641i	0.292083	−	0.505902i	−0.682219	−	0.731148i	$-0.738985\pi$
$607$	7.19615	−	12.4641i	0.292083	−	0.505902i	0.974302	+	0.225245i	$0.0723184\pi$
$608$	−5.19615	−	9.00000i	−0.210732	−	0.364998i
$609$		0			0
$610$	−14.4115			−0.583506
$611$	−46.5167	−	3.00000i	−1.88186	−	0.121367i
$612$		0			0
$613$	−1.69615	−	2.93782i	−0.0685070	−	0.118658i	0.829737	−	0.558154i	$-0.188490\pi$
$613$	−1.69615	−	2.93782i	−0.0685070	−	0.118658i	−0.898244	+	0.439497i	$0.855157\pi$
$614$	−3.97372	−	6.88269i	−0.160366	−	0.277763i
$615$		0			0
$616$	−2.19615			−0.0884855
$617$	24.6962	−	42.7750i	0.994230	−	1.72206i	0.404214	−	0.914664i	$-0.367545\pi$
$617$	24.6962	−	42.7750i	0.994230	−	1.72206i	0.590016	−	0.807392i	$-0.299122\pi$
$618$		0			0
$619$	35.3731			1.42176			0.710882	−	0.703311i	$-0.248296\pi$
$619$	35.3731			1.42176			0.710882	+	0.703311i	$0.248296\pi$
$620$	3.63397	−	6.29423i	0.145944	−	0.252782i
$621$		0			0
$622$	−1.09808	−	1.90192i	−0.0440288	−	0.0762602i
$623$	−12.9282			−0.517958
$624$		0			0
$625$	−11.0000			−0.440000
$626$	24.9282	+	43.1769i	0.996331	+	1.72570i
$627$		0			0
$628$	−0.598076	+	1.03590i	−0.0238658	+	0.0413368i
$629$	−54.1244			−2.15808
$630$		0			0
$631$	6.39230	−	11.0718i	0.254474	−	0.440761i	−0.710279	−	0.703920i	$-0.751431\pi$
$631$	6.39230	−	11.0718i	0.254474	−	0.440761i	0.964752	+	0.263159i	$0.0847645\pi$
$632$	28.0526			1.11587
$633$		0			0
$634$	5.59808	+	9.69615i	0.222328	+	0.385083i
$635$	15.9282	+	27.5885i	0.632091	+	1.09481i
$636$		0			0
$637$	−1.59808	−	3.23205i	−0.0633181	−	0.128059i
$638$	6.58846			0.260840
$639$		0			0
$640$	−10.5000	−	18.1865i	−0.415049	−	0.718886i
$641$	14.4282	−	24.9904i	0.569880	−	0.987061i	−0.426698	−	0.904394i	$-0.640323\pi$
$641$	14.4282	−	24.9904i	0.569880	−	0.987061i	0.996577	−	0.0826663i	$-0.0263436\pi$
$642$		0			0
$643$	0.392305	−	0.679492i	0.0154710	−	0.0267965i	−0.858186	−	0.513338i	$-0.828409\pi$
$643$	0.392305	−	0.679492i	0.0154710	−	0.0267965i	0.873657	+	0.486542i	$0.161742\pi$
$644$	2.36603	−	4.09808i	0.0932345	−	0.161487i
$645$		0			0
$646$	13.3923	−	23.1962i	0.526913	−	0.912640i
$647$	22.5167	+	39.0000i	0.885221	+	1.53325i	0.845460	+	0.534039i	$0.179326\pi$
$647$	22.5167	+	39.0000i	0.885221	+	1.53325i	0.0397614	+	0.999209i	$0.487340\pi$
$648$		0			0
$649$	9.21539			0.361736
$650$	−5.53590	−	11.1962i	−0.217136	−	0.439149i
$651$		0			0
$652$	−8.09808	−	14.0263i	−0.317145	−	0.549311i
$653$	−18.9282	−	32.7846i	−0.740718	−	1.28296i	−0.952169	−	0.305572i	$-0.901152\pi$
$653$	−18.9282	−	32.7846i	−0.740718	−	1.28296i	0.211451	−	0.977389i	$-0.432181\pi$
$654$		0			0
$655$	−6.00000			−0.234439
$656$	12.9904	−	22.5000i	0.507189	−	0.878477i
$657$		0			0
$658$	22.3923			0.872943
$659$	14.1962	−	24.5885i	0.553004	−	0.957830i	−0.445052	−	0.895505i	$-0.646815\pi$
$659$	14.1962	−	24.5885i	0.553004	−	0.957830i	0.998056	−	0.0623257i	$-0.0198518\pi$
$660$		0			0
$661$	16.5981	+	28.7487i	0.645590	+	1.11820i	0.984165	+	0.177256i	$0.0567220\pi$
$661$	16.5981	+	28.7487i	0.645590	+	1.11820i	−0.338574	+	0.940940i	$0.609945\pi$
$662$	−43.2679			−1.68166
$663$		0			0
$664$	3.80385			0.147618
$665$	1.73205	+	3.00000i	0.0671660	+	0.116335i
$666$		0			0
$667$	7.09808	−	12.2942i	0.274839	−	0.476034i
$668$	6.58846			0.254915
$669$		0			0
$670$	9.29423	−	16.0981i	0.359067	−	0.621923i
$671$	6.09103			0.235142
$672$		0			0
$673$	22.0885	+	38.2583i	0.851447	+	1.47475i	0.879902	+	0.475155i	$0.157608\pi$
$673$	22.0885	+	38.2583i	0.851447	+	1.47475i	−0.0284546	+	0.999595i	$0.509059\pi$
$674$	9.52628	+	16.5000i	0.366939	+	0.635556i
$675$		0			0
$676$	−7.89230	+	10.3301i	−0.303550	+	0.397313i
$677$	23.0718			0.886721			0.443361	−	0.896343i	$-0.353786\pi$
$677$	23.0718			0.886721			0.443361	+	0.896343i	$0.353786\pi$
$678$		0			0
$679$	−3.19615	−	5.53590i	−0.122657	−	0.212448i
$680$	11.5981	−	20.0885i	0.444766	−	0.770357i
$681$		0			0
$682$	−4.60770	+	7.98076i	−0.176438	+	0.305599i
$683$	7.73205	−	13.3923i	0.295859	−	0.512442i	−0.679326	−	0.733837i	$-0.737728\pi$
$683$	7.73205	−	13.3923i	0.295859	−	0.512442i	0.975184	+	0.221395i	$0.0710609\pi$
$684$		0			0
$685$	−6.99038	+	12.1077i	−0.267089	+	0.462611i
$686$	0.866025	+	1.50000i	0.0330650	+	0.0572703i
$687$		0			0
$688$	0.980762			0.0373912
$689$	35.7224	+	2.30385i	1.36092	+	0.0877696i
$690$		0			0
$691$	−0.196152	−	0.339746i	−0.00746199	−	0.0129245i	0.862270	−	0.506448i	$-0.169042\pi$
$691$	−0.196152	−	0.339746i	−0.00746199	−	0.0129245i	−0.869732	+	0.493524i	$0.835709\pi$
$692$	4.26795	+	7.39230i	0.162243	+	0.281013i
$693$		0			0
$694$	12.5885			0.477851
$695$	−9.16987	+	15.8827i	−0.347833	+	0.602465i
$696$		0			0
$697$	40.1769			1.52181
$698$	−21.4641	+	37.1769i	−0.812428	+	1.40717i
$699$		0			0
$700$	1.00000	+	1.73205i	0.0377964	+	0.0654654i
$701$	−20.7846			−0.785024			−0.392512	−	0.919747i	$-0.628394\pi$
$701$	−20.7846			−0.785024			−0.392512	+	0.919747i	$0.628394\pi$
$702$		0			0
$703$	−14.0000			−0.528020
$704$	0.633975	+	1.09808i	0.0238938	+	0.0413853i
$705$		0			0
$706$	−17.8923	+	30.9904i	−0.673386	+	1.16634i
$707$	7.73205			0.290794
$708$		0			0
$709$	−15.0885	+	26.1340i	−0.566659	+	0.981482i	0.430234	+	0.902717i	$0.358431\pi$
$709$	−15.0885	+	26.1340i	−0.566659	+	0.981482i	−0.996893	+	0.0787648i	$0.974902\pi$
$710$	−18.0000			−0.675528
$711$		0			0
$712$	11.1962	+	19.3923i	0.419594	+	0.726757i
$713$	9.92820	+	17.1962i	0.371814	+	0.644001i
$714$		0			0
$715$	−3.50962	−	7.09808i	−0.131252	−	0.265453i
$716$	6.92820			0.258919
$717$		0			0
$718$	−16.3923	−	28.3923i	−0.611755	−	1.05959i
$719$	−3.63397	+	6.29423i	−0.135524	+	0.234735i	−0.925798	−	0.378019i	$-0.876605\pi$
$719$	−3.63397	+	6.29423i	−0.135524	+	0.234735i	0.790273	+	0.612755i	$0.209939\pi$
$720$		0			0
$721$	7.19615	−	12.4641i	0.267999	−	0.464187i
$722$	−12.9904	+	22.5000i	−0.483452	+	0.837363i
$723$		0			0
$724$	−2.79423	+	4.83975i	−0.103847	+	0.179868i
$725$	3.00000	+	5.19615i	0.111417	+	0.192980i
$726$		0			0
$727$	−41.1769			−1.52717			−0.763584	−	0.645709i	$-0.776562\pi$
$727$	−41.1769			−1.52717			−0.763584	+	0.645709i	$0.776562\pi$
$728$	−3.46410	+	5.19615i	−0.128388	+	0.192582i
$729$		0			0
$730$	4.79423	+	8.30385i	0.177442	+	0.307339i
$731$	0.758330	+	1.31347i	0.0280479	+	0.0485803i
$732$		0			0
$733$	23.5885			0.871260			0.435630	−	0.900126i	$-0.356526\pi$
$733$	23.5885			0.871260			0.435630	+	0.900126i	$0.356526\pi$
$734$	3.63397	−	6.29423i	0.134132	−	0.232324i
$735$		0			0
$736$	−24.5885			−0.906343
$737$	−3.92820	+	6.80385i	−0.144697	+	0.250623i
$738$		0			0
$739$	−20.3923	−	35.3205i	−0.750143	−	1.29929i	−0.947753	−	0.319005i	$-0.896651\pi$
$739$	−20.3923	−	35.3205i	−0.750143	−	1.29929i	0.197610	−	0.980281i	$-0.436682\pi$
$740$	12.1244			0.445700
$741$		0			0
$742$	−17.1962			−0.631291
$743$	−3.80385	−	6.58846i	−0.139550	−	0.241707i	0.787777	−	0.615961i	$-0.211232\pi$
$743$	−3.80385	−	6.58846i	−0.139550	−	0.241707i	−0.927326	+	0.374254i	$0.877899\pi$
$744$		0			0
$745$	5.59808	−	9.69615i	0.205098	−	0.355240i
$746$	19.7321			0.722442
$747$		0			0
$748$	4.90192	−	8.49038i	0.179232	−	0.310439i
$749$	7.85641			0.287067
$750$		0			0
$751$	−17.9019	−	31.0070i	−0.653250	−	1.13146i	−0.982329	−	0.187161i	$-0.940072\pi$
$751$	−17.9019	−	31.0070i	−0.653250	−	1.13146i	0.329079	−	0.944302i	$-0.393262\pi$
$752$	−32.3205	−	55.9808i	−1.17861	−	2.04141i
$753$		0			0
$754$	10.3923	−	15.5885i	0.378465	−	0.567698i
$755$	−3.46410			−0.126072
$756$		0			0
$757$	8.00000	+	13.8564i	0.290765	+	0.503620i	0.973991	−	0.226587i	$-0.0727569\pi$
$757$	8.00000	+	13.8564i	0.290765	+	0.503620i	−0.683226	+	0.730207i	$0.739424\pi$
$758$	23.0263	−	39.8827i	0.836352	−	1.44860i
$759$		0			0
$760$	3.00000	−	5.19615i	0.108821	−	0.188484i
$761$	20.6603	−	35.7846i	0.748934	−	1.29719i	−0.199401	−	0.979918i	$-0.563900\pi$
$761$	20.6603	−	35.7846i	0.748934	−	1.29719i	0.948334	−	0.317273i	$-0.102767\pi$
$762$		0			0
$763$	4.19615	−	7.26795i	0.151911	−	0.263117i
$764$	2.36603	+	4.09808i	0.0855998	+	0.148263i
$765$		0			0
$766$	−20.1962			−0.729717
$767$	14.5359	−	21.8038i	0.524861	−	0.787291i
$768$		0			0
$769$	−7.58846	−	13.1436i	−0.273647	−	0.473970i	0.696146	−	0.717900i	$-0.254897\pi$
$769$	−7.58846	−	13.1436i	−0.273647	−	0.473970i	−0.969793	+	0.243930i	$0.921563\pi$
$770$	1.90192	+	3.29423i	0.0685406	+	0.118716i
$771$		0			0
$772$	5.00000			0.179954
$773$	−6.46410	+	11.1962i	−0.232498	+	0.402698i	−0.958542	−	0.284950i	$-0.908023\pi$
$773$	−6.46410	+	11.1962i	−0.232498	+	0.402698i	0.726045	+	0.687647i	$0.241356\pi$
$774$		0			0
$775$	−8.39230			−0.301460
$776$	−5.53590	+	9.58846i	−0.198727	+	0.344206i
$777$		0			0
$778$	20.3827	+	35.3038i	0.730755	+	1.26570i
$779$	10.3923			0.372343
$780$		0			0
$781$	7.60770			0.272225
$782$	−31.6865	−	54.8827i	−1.13311	−	1.96260i
$783$		0			0
$784$	2.50000	−	4.33013i	0.0892857	−	0.1

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:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.o (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 + 1.50000i) q^{2} +(-0.500000 + 0.866025i) q^{4} -1.73205 q^{5} +(-0.500000 + 0.866025i) q^{7} +1.73205 q^{8} +O(q^{10})\)	\(q+(0.866025 + 1.50000i) q^{2} +(-0.500000 + 0.866025i) q^{4} -1.73205 q^{5} +(-0.500000 + 0.866025i) q^{7} +1.73205 q^{8} +(-1.50000 - 2.59808i) q^{10} +(0.633975 + 1.09808i) q^{11} +(3.59808 + 0.232051i) q^{13} -1.73205 q^{14} +(2.50000 + 4.33013i) q^{16} +(-3.86603 + 6.69615i) q^{17} +(-1.00000 + 1.73205i) q^{19} +(0.866025 - 1.50000i) q^{20} +(-1.09808 + 1.90192i) q^{22} +(2.36603 + 4.09808i) q^{23} -2.00000 q^{25} +(2.76795 + 5.59808i) q^{26} +(-0.500000 - 0.866025i) q^{28} +(-1.50000 - 2.59808i) q^{29} +4.19615 q^{31} +(-2.59808 + 4.50000i) q^{32} -13.3923 q^{34} +(0.866025 - 1.50000i) q^{35} +(3.50000 + 6.06218i) q^{37} -3.46410 q^{38} -3.00000 q^{40} +(-2.59808 - 4.50000i) q^{41} +(0.0980762 - 0.169873i) q^{43} -1.26795 q^{44} +(-4.09808 + 7.09808i) q^{46} -12.9282 q^{47} +(-0.500000 - 0.866025i) q^{49} +(-1.73205 - 3.00000i) q^{50} +(-2.00000 + 3.00000i) q^{52} +9.92820 q^{53} +(-1.09808 - 1.90192i) q^{55} +(-0.866025 + 1.50000i) q^{56} +(2.59808 - 4.50000i) q^{58} +(3.63397 - 6.29423i) q^{59} +(2.40192 - 4.16025i) q^{61} +(3.63397 + 6.29423i) q^{62} +1.00000 q^{64} +(-6.23205 - 0.401924i) q^{65} +(3.09808 + 5.36603i) q^{67} +(-3.86603 - 6.69615i) q^{68} +3.00000 q^{70} +(3.00000 - 5.19615i) q^{71} -3.19615 q^{73} +(-6.06218 + 10.5000i) q^{74} +(-1.00000 - 1.73205i) q^{76} -1.26795 q^{77} +16.1962 q^{79} +(-4.33013 - 7.50000i) q^{80} +(4.50000 - 7.79423i) q^{82} +2.19615 q^{83} +(6.69615 - 11.5981i) q^{85} +0.339746 q^{86} +(1.09808 + 1.90192i) q^{88} +(6.46410 + 11.1962i) q^{89} +(-2.00000 + 3.00000i) q^{91} -4.73205 q^{92} +(-11.1962 - 19.3923i) q^{94} +(1.73205 - 3.00000i) q^{95} +(-3.19615 + 5.53590i) q^{97} +(0.866025 - 1.50000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{4} - 2 q^{7}+O(q^{10}) \) 4 * q - 2 * q^4 - 2 * q^7	\( 4 q - 2 q^{4} - 2 q^{7} - 6 q^{10} + 6 q^{11} + 4 q^{13} + 10 q^{16} - 12 q^{17} - 4 q^{19} + 6 q^{22} + 6 q^{23} - 8 q^{25} + 18 q^{26} - 2 q^{28} - 6 q^{29} - 4 q^{31} - 12 q^{34} + 14 q^{37} - 12 q^{40} - 10 q^{43} - 12 q^{44} - 6 q^{46} - 24 q^{47} - 2 q^{49} - 8 q^{52} + 12 q^{53} + 6 q^{55} + 18 q^{59} + 20 q^{61} + 18 q^{62} + 4 q^{64} - 18 q^{65} + 2 q^{67} - 12 q^{68} + 12 q^{70} + 12 q^{71} + 8 q^{73} - 4 q^{76} - 12 q^{77} + 44 q^{79} + 18 q^{82} - 12 q^{83} + 6 q^{85} + 36 q^{86} - 6 q^{88} + 12 q^{89} - 8 q^{91} - 12 q^{92} - 24 q^{94} + 8 q^{97}+O(q^{100}) \) 4 * q - 2 * q^4 - 2 * q^7 - 6 * q^10 + 6 * q^11 + 4 * q^13 + 10 * q^16 - 12 * q^17 - 4 * q^19 + 6 * q^22 + 6 * q^23 - 8 * q^25 + 18 * q^26 - 2 * q^28 - 6 * q^29 - 4 * q^31 - 12 * q^34 + 14 * q^37 - 12 * q^40 - 10 * q^43 - 12 * q^44 - 6 * q^46 - 24 * q^47 - 2 * q^49 - 8 * q^52 + 12 * q^53 + 6 * q^55 + 18 * q^59 + 20 * q^61 + 18 * q^62 + 4 * q^64 - 18 * q^65 + 2 * q^67 - 12 * q^68 + 12 * q^70 + 12 * q^71 + 8 * q^73 - 4 * q^76 - 12 * q^77 + 44 * q^79 + 18 * q^82 - 12 * q^83 + 6 * q^85 + 36 * q^86 - 6 * q^88 + 12 * q^89 - 8 * q^91 - 12 * q^92 - 24 * q^94 + 8 * q^97

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