LMFDB - Embedded newform 819.2.o.b.757.1

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			757.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	819.757
Dual form			819.2.o.b.568.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 - 1.50000i) q^{2} +(-0.500000 + 0.866025i) q^{4} +1.73205 q^{5} +(-0.500000 + 0.866025i) q^{7} -1.73205 q^{8} +O(q^{10})$	\(q+(-0.866025 - 1.50000i) q^{2} +(-0.500000 + 0.866025i) q^{4} +1.73205 q^{5} +(-0.500000 + 0.866025i) q^{7} -1.73205 q^{8} +(-1.50000 - 2.59808i) q^{10} +(2.36603 + 4.09808i) q^{11} +(-1.59808 + 3.23205i) q^{13} +1.73205 q^{14} +(2.50000 + 4.33013i) q^{16} +(-2.13397 + 3.69615i) q^{17} +(-1.00000 + 1.73205i) q^{19} +(-0.866025 + 1.50000i) q^{20} +(4.09808 - 7.09808i) q^{22} +(0.633975 + 1.09808i) q^{23} -2.00000 q^{25} +(6.23205 - 0.401924i) q^{26} +(-0.500000 - 0.866025i) q^{28} +(-1.50000 - 2.59808i) q^{29} -6.19615 q^{31} +(2.59808 - 4.50000i) q^{32} +7.39230 q^{34} +(-0.866025 + 1.50000i) q^{35} +(3.50000 + 6.06218i) q^{37} +3.46410 q^{38} -3.00000 q^{40} +(2.59808 + 4.50000i) q^{41} +(-5.09808 + 8.83013i) q^{43} -4.73205 q^{44} +(1.09808 - 1.90192i) q^{46} +0.928203 q^{47} +(-0.500000 - 0.866025i) q^{49} +(1.73205 + 3.00000i) q^{50} +(-2.00000 - 3.00000i) q^{52} -3.92820 q^{53} +(4.09808 + 7.09808i) q^{55} +(0.866025 - 1.50000i) q^{56} +(-2.59808 + 4.50000i) q^{58} +(5.36603 - 9.29423i) q^{59} +(7.59808 - 13.1603i) q^{61} +(5.36603 + 9.29423i) q^{62} +1.00000 q^{64} +(-2.76795 + 5.59808i) q^{65} +(-2.09808 - 3.63397i) q^{67} +(-2.13397 - 3.69615i) q^{68} +3.00000 q^{70} +(3.00000 - 5.19615i) q^{71} +7.19615 q^{73} +(6.06218 - 10.5000i) q^{74} +(-1.00000 - 1.73205i) q^{76} -4.73205 q^{77} +5.80385 q^{79} +(4.33013 + 7.50000i) q^{80} +(4.50000 - 7.79423i) q^{82} -8.19615 q^{83} +(-3.69615 + 6.40192i) q^{85} +17.6603 q^{86} +(-4.09808 - 7.09808i) q^{88} +(-0.464102 - 0.803848i) q^{89} +(-2.00000 - 3.00000i) q^{91} -1.26795 q^{92} +(-0.803848 - 1.39230i) q^{94} +(-1.73205 + 3.00000i) q^{95} +(7.19615 - 12.4641i) q^{97} +(-0.866025 + 1.50000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.956882\pi$
$2$	−0.866025	−	1.50000i	−0.612372	−	1.06066i	0.378467	−	0.925615i	$-0.376451\pi$
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$	1.73205			0.774597			0.387298	−	0.921954i	$-0.373408\pi$
$5$	1.73205			0.774597			0.387298	+	0.921954i	$0.373408\pi$
$6$		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
$11$	2.36603	+	4.09808i	0.713384	+	1.23562i	0.963580	+	0.267421i	$0.0861715\pi$
$11$	2.36603	+	4.09808i	0.713384	+	1.23562i	−0.250196	+	0.968195i	$0.580495\pi$
$12$		0			0
$13$	−1.59808	+	3.23205i	−0.443227	+	0.896410i
$13$	−1.59808	+	3.23205i	−0.443227	+	0.896410i
$14$	1.73205			0.462910
$15$		0			0
$16$	2.50000	+	4.33013i	0.625000	+	1.08253i
$17$	−2.13397	+	3.69615i	−0.517565	+	0.896449i	0.482227	+	0.876046i	$0.339828\pi$
$17$	−2.13397	+	3.69615i	−0.517565	+	0.896449i	−0.999792	+	0.0204023i	$0.993505\pi$
$18$		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
$22$	4.09808	−	7.09808i	0.873713	−	1.51331i
$23$	0.633975	+	1.09808i	0.132193	+	0.228965i	0.924522	−	0.381130i	$-0.124465\pi$
$23$	0.633975	+	1.09808i	0.132193	+	0.228965i	−0.792329	+	0.610094i	$0.791132\pi$
$24$		0			0
$25$	−2.00000			−0.400000
$26$	6.23205	−	0.401924i	1.22221	−	0.0788237i
$27$		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
$31$	−6.19615			−1.11286			−0.556431	−	0.830894i	$-0.687830\pi$
$31$	−6.19615			−1.11286			−0.556431	+	0.830894i	$0.687830\pi$
$32$	2.59808	−	4.50000i	0.459279	−	0.795495i
$33$		0			0
$34$	7.39230			1.26777
$35$	−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	0.994409	−	0.105601i	$-0.0336766\pi$
$41$	2.59808	+	4.50000i	0.405751	+	0.702782i	−0.588657	+	0.808383i	$0.700343\pi$
$42$		0			0
$43$	−5.09808	+	8.83013i	−0.777449	+	1.34658i	0.155958	+	0.987764i	$0.450153\pi$
$43$	−5.09808	+	8.83013i	−0.777449	+	1.34658i	−0.933408	+	0.358818i	$0.883180\pi$
$44$	−4.73205			−0.713384
$45$		0			0
$46$	1.09808	−	1.90192i	0.161903	−	0.280423i
$47$	0.928203			0.135392			0.0676962	−	0.997706i	$-0.478435\pi$
$47$	0.928203			0.135392			0.0676962	+	0.997706i	$0.478435\pi$
$48$		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
$53$	−3.92820			−0.539580			−0.269790	−	0.962919i	$-0.586954\pi$
$53$	−3.92820			−0.539580			−0.269790	+	0.962919i	$0.586954\pi$
$54$		0			0
$55$	4.09808	+	7.09808i	0.552584	+	0.957104i
$56$	0.866025	−	1.50000i	0.115728	−	0.200446i
$57$		0			0
$58$	−2.59808	+	4.50000i	−0.341144	+	0.590879i
$59$	5.36603	−	9.29423i	0.698597	−	1.21001i	−0.270356	−	0.962760i	$-0.587141\pi$
$59$	5.36603	−	9.29423i	0.698597	−	1.21001i	0.968953	−	0.247245i	$-0.0795253\pi$
$60$		0			0
$61$	7.59808	−	13.1603i	0.972834	−	1.68500i	0.285929	−	0.958251i	$-0.407698\pi$
$61$	7.59808	−	13.1603i	0.972834	−	1.68500i	0.686905	−	0.726747i	$-0.258969\pi$
$62$	5.36603	+	9.29423i	0.681486	+	1.18037i
$63$		0			0
$64$	1.00000			0.125000
$65$	−2.76795	+	5.59808i	−0.343322	+	0.694356i
$66$		0			0
$67$	−2.09808	−	3.63397i	−0.256321	−	0.443961i	0.708933	−	0.705276i	$-0.249177\pi$
$67$	−2.09808	−	3.63397i	−0.256321	−	0.443961i	−0.965253	+	0.261316i	$0.915844\pi$
$68$	−2.13397	−	3.69615i	−0.258782	−	0.448224i
$69$		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
$73$	7.19615			0.842246			0.421123	−	0.907004i	$-0.361636\pi$
$73$	7.19615			0.842246			0.421123	+	0.907004i	$0.361636\pi$
$74$	6.06218	−	10.5000i	0.704714	−	1.22060i
$75$		0			0
$76$	−1.00000	−	1.73205i	−0.114708	−	0.198680i
$77$	−4.73205			−0.539267
$78$		0			0
$79$	5.80385			0.652984			0.326492	−	0.945200i	$-0.394133\pi$
$79$	5.80385			0.652984			0.326492	+	0.945200i	$0.394133\pi$
$80$	4.33013	+	7.50000i	0.484123	+	0.838525i
$81$		0			0
$82$	4.50000	−	7.79423i	0.496942	−	0.860729i
$83$	−8.19615			−0.899645			−0.449822	−	0.893118i	$-0.648513\pi$
$83$	−8.19615			−0.899645			−0.449822	+	0.893118i	$0.648513\pi$
$84$		0			0
$85$	−3.69615	+	6.40192i	−0.400904	+	0.694386i
$86$	17.6603			1.90435
$87$		0			0
$88$	−4.09808	−	7.09808i	−0.436856	−	0.756657i
$89$	−0.464102	−	0.803848i	−0.0491947	−	0.0852077i	0.840379	−	0.541998i	$-0.182332\pi$
$89$	−0.464102	−	0.803848i	−0.0491947	−	0.0852077i	−0.889574	+	0.456791i	$0.848999\pi$
$90$		0			0
$91$	−2.00000	−	3.00000i	−0.209657	−	0.314485i
$92$	−1.26795			−0.132193
$93$		0			0
$94$	−0.803848	−	1.39230i	−0.0829105	−	0.143605i
$95$	−1.73205	+	3.00000i	−0.177705	+	0.307794i
$96$		0			0
$97$	7.19615	−	12.4641i	0.730659	−	1.26554i	−0.225944	−	0.974140i	$-0.572546\pi$
$97$	7.19615	−	12.4641i	0.730659	−	1.26554i	0.956602	−	0.291397i	$-0.0941202\pi$
$98$	−0.866025	+	1.50000i	−0.0874818	+	0.151523i
$99$		0			0
$100$	1.00000	−	1.73205i	0.100000	−	0.173205i
$101$	−2.13397	−	3.69615i	−0.212338	−	0.367781i	0.740108	−	0.672489i	$-0.234775\pi$
$101$	−2.13397	−	3.69615i	−0.212338	−	0.367781i	−0.952446	+	0.304708i	$0.901441\pi$
$102$		0			0
$103$	6.39230			0.629853			0.314926	−	0.949116i	$-0.398020\pi$
$103$	6.39230			0.629853			0.314926	+	0.949116i	$0.398020\pi$
$104$	2.76795	−	5.59808i	0.271420	−	0.548937i
$105$		0			0
$106$	3.40192	+	5.89230i	0.330424	+	0.572311i
$107$	9.92820	+	17.1962i	0.959796	+	1.66241i	0.722991	+	0.690858i	$0.242767\pi$
$107$	9.92820	+	17.1962i	0.959796	+	1.66241i	0.236805	+	0.971557i	$0.423900\pi$
$108$		0			0
$109$	12.3923			1.18697			0.593484	−	0.804846i	$-0.297752\pi$
$109$	12.3923			1.18697			0.593484	+	0.804846i	$0.297752\pi$
$110$	7.09808	−	12.2942i	0.676775	−	1.17221i
$111$		0			0
$112$	−5.00000			−0.472456
$113$	−3.69615	+	6.40192i	−0.347705	+	0.602242i	−0.985841	−	0.167681i	$-0.946372\pi$
$113$	−3.69615	+	6.40192i	−0.347705	+	0.602242i	0.638137	+	0.769923i	$0.279706\pi$
$114$		0			0
$115$	1.09808	+	1.90192i	0.102396	+	0.177355i
$116$	3.00000			0.278543
$117$		0			0
$118$	−18.5885			−1.71121
$119$	−2.13397	−	3.69615i	−0.195621	−	0.338826i
$120$		0			0
$121$	−5.69615	+	9.86603i	−0.517832	+	0.896911i
$122$	−26.3205			−2.38295
$123$		0			0
$124$	3.09808	−	5.36603i	0.278215	−	0.481883i
$125$	−12.1244			−1.08444
$126$		0			0
$127$	1.19615	+	2.07180i	0.106141	+	0.183842i	0.914204	−	0.405254i	$-0.132817\pi$
$127$	1.19615	+	2.07180i	0.106141	+	0.183842i	−0.808063	+	0.589097i	$0.799484\pi$
$128$	−6.06218	−	10.5000i	−0.535826	−	0.928078i
$129$		0			0
$130$	10.7942	−	0.696152i	0.946716	−	0.0610566i
$131$	−3.46410			−0.302660			−0.151330	−	0.988483i	$-0.548356\pi$
$131$	−3.46410			−0.302660			−0.151330	+	0.988483i	$0.548356\pi$
$132$		0			0
$133$	−1.00000	−	1.73205i	−0.0867110	−	0.150188i
$134$	−3.63397	+	6.29423i	−0.313928	+	0.543739i
$135$		0			0
$136$	3.69615	−	6.40192i	0.316942	−	0.548960i
$137$	10.9641	−	18.9904i	0.936726	−	1.62246i	0.165200	−	0.986260i	$-0.447173\pi$
$137$	10.9641	−	18.9904i	0.936726	−	1.62246i	0.771526	−	0.636198i	$-0.219494\pi$
$138$		0			0
$139$	−10.2942	+	17.8301i	−0.873145	+	1.51233i	−0.0144194	+	0.999896i	$0.504590\pi$
$139$	−10.2942	+	17.8301i	−0.873145	+	1.51233i	−0.858726	+	0.512436i	$0.828743\pi$
$140$	−0.866025	−	1.50000i	−0.0731925	−	0.126773i
$141$		0			0
$142$	−10.3923			−0.872103
$143$	−17.0263	+	1.09808i	−1.42381	+	0.0918257i
$144$		0			0
$145$	−2.59808	−	4.50000i	−0.215758	−	0.373705i
$146$	−6.23205	−	10.7942i	−0.515768	−	0.893337i
$147$		0			0
$148$	−7.00000			−0.575396
$149$	0.232051	−	0.401924i	0.0190103	−	0.0329269i	−0.856364	−	0.516373i	$-0.827282\pi$
$149$	0.232051	−	0.401924i	0.0190103	−	0.0329269i	0.875374	+	0.483446i	$0.160615\pi$
$150$		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
$154$	4.09808	+	7.09808i	0.330232	+	0.571979i
$155$	−10.7321			−0.862019
$156$		0			0
$157$	−9.19615			−0.733933			−0.366966	−	0.930234i	$-0.619604\pi$
$157$	−9.19615			−0.733933			−0.366966	+	0.930234i	$0.619604\pi$
$158$	−5.02628	−	8.70577i	−0.399869	−	0.692594i
$159$		0			0
$160$	4.50000	−	7.79423i	0.355756	−	0.616188i
$161$	−1.26795			−0.0999284
$162$		0			0
$163$	−2.90192	+	5.02628i	−0.227296	+	0.393689i	−0.957006	−	0.290069i	$-0.906322\pi$
$163$	−2.90192	+	5.02628i	−0.227296	+	0.393689i	0.729710	+	0.683757i	$0.239655\pi$
$164$	−5.19615			−0.405751
$165$		0			0
$166$	7.09808	+	12.2942i	0.550918	+	0.954217i
$167$	12.2942	+	21.2942i	0.951356	+	1.64780i	0.742495	+	0.669852i	$0.233642\pi$
$167$	12.2942	+	21.2942i	0.951356	+	1.64780i	0.208861	+	0.977945i	$0.433024\pi$
$168$		0			0
$169$	−7.89230	−	10.3301i	−0.607100	−	0.794625i
$170$	12.8038			0.982010
$171$		0			0
$172$	−5.09808	−	8.83013i	−0.388725	−	0.673291i
$173$	7.73205	−	13.3923i	0.587857	−	1.01820i	−0.406656	−	0.913581i	$-0.633305\pi$
$173$	7.73205	−	13.3923i	0.587857	−	1.01820i	0.994513	−	0.104617i	$-0.0333615\pi$
$174$		0			0
$175$	1.00000	−	1.73205i	0.0755929	−	0.130931i
$176$	−11.8301	+	20.4904i	−0.891729	+	1.54452i
$177$		0			0
$178$	−0.803848	+	1.39230i	−0.0602509	+	0.104358i
$179$	3.46410	+	6.00000i	0.258919	+	0.448461i	0.965953	−	0.258719i	$-0.0833004\pi$
$179$	3.46410	+	6.00000i	0.258919	+	0.448461i	−0.707034	+	0.707180i	$0.749967\pi$
$180$		0			0
$181$	−25.5885			−1.90198			−0.950988	−	0.309229i	$-0.899929\pi$
$181$	−25.5885			−1.90198			−0.950988	+	0.309229i	$0.899929\pi$
$182$	−2.76795	+	5.59808i	−0.205174	+	0.414957i
$183$		0			0
$184$	−1.09808	−	1.90192i	−0.0809513	−	0.140212i
$185$	6.06218	+	10.5000i	0.445700	+	0.771975i
$186$		0			0
$187$	−20.1962			−1.47689
$188$	−0.464102	+	0.803848i	−0.0338481	+	0.0586266i
$189$		0			0
$190$	6.00000			0.435286
$191$	0.633975	−	1.09808i	0.0458728	−	0.0794540i	−0.842177	−	0.539201i	$-0.818726\pi$
$191$	0.633975	−	1.09808i	0.0458728	−	0.0794540i	0.888050	+	0.459747i	$0.152060\pi$
$192$		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	$0.890928\pi$
$194$	−24.9282			−1.78974
$195$		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
$202$	−3.69615	+	6.40192i	−0.260060	+	0.450438i
$203$	3.00000			0.210559
$204$		0			0
$205$	4.50000	+	7.79423i	0.314294	+	0.544373i
$206$	−5.53590	−	9.58846i	−0.385704	−	0.668059i
$207$		0			0
$208$	−17.9904	+	1.16025i	−1.24741	+	0.0804491i
$209$	−9.46410			−0.654646
$210$		0			0
$211$	6.09808	+	10.5622i	0.419809	+	0.727130i	0.995920	−	0.0902411i	$-0.0287638\pi$
$211$	6.09808	+	10.5622i	0.419809	+	0.727130i	−0.576111	+	0.817371i	$0.695430\pi$
$212$	1.96410	−	3.40192i	0.134895	−	0.233645i
$213$		0			0
$214$	17.1962	−	29.7846i	1.17550	−	2.03603i
$215$	−8.83013	+	15.2942i	−0.602210	+	1.04306i
$216$		0			0
$217$	3.09808	−	5.36603i	0.210311	−	0.364270i
$218$	−10.7321	−	18.5885i	−0.726866	−	1.25897i
$219$		0			0
$220$	−8.19615			−0.552584
$221$	−8.53590	−	12.8038i	−0.574187	−	0.861280i
$222$		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
$226$	12.8038			0.851699
$227$	−5.83013	+	10.0981i	−0.386959	+	0.670233i	−0.992039	−	0.125932i	$-0.959808\pi$
$227$	−5.83013	+	10.0981i	−0.386959	+	0.670233i	0.605080	+	0.796165i	$0.293141\pi$
$228$		0			0
$229$	6.39230			0.422415			0.211208	−	0.977441i	$-0.432260\pi$
$229$	6.39230			0.422415			0.211208	+	0.977441i	$0.432260\pi$
$230$	1.90192	−	3.29423i	0.125409	−	0.217215i
$231$		0			0
$232$	2.59808	+	4.50000i	0.170572	+	0.295439i
$233$	−25.8564			−1.69391			−0.846955	−	0.531665i	$-0.821567\pi$
$233$	−25.8564			−1.69391			−0.846955	+	0.531665i	$0.821567\pi$
$234$		0			0
$235$	1.60770			0.104874
$236$	5.36603	+	9.29423i	0.349299	+	0.605003i
$237$		0			0
$238$	−3.69615	+	6.40192i	−0.239586	+	0.414975i
$239$	26.1962			1.69449			0.847244	−	0.531204i	$-0.178260\pi$
$239$	26.1962			1.69449			0.847244	+	0.531204i	$0.178260\pi$
$240$		0			0
$241$	5.40192	−	9.35641i	0.347969	−	0.602699i	−0.637920	−	0.770103i	$-0.720205\pi$
$241$	5.40192	−	9.35641i	0.347969	−	0.602699i	0.985888	+	0.167404i	$0.0535383\pi$
$242$	19.7321			1.26842
$243$		0			0
$244$	7.59808	+	13.1603i	0.486417	+	0.842499i
$245$	−0.866025	−	1.50000i	−0.0553283	−	0.0958315i
$246$		0			0
$247$	−4.00000	−	6.00000i	−0.254514	−	0.381771i
$248$	10.7321			0.681486
$249$		0			0
$250$	10.5000	+	18.1865i	0.664078	+	1.15022i
$251$	−11.1962	+	19.3923i	−0.706695	+	1.22403i	0.259382	+	0.965775i	$0.416481\pi$
$251$	−11.1962	+	19.3923i	−0.706695	+	1.22403i	−0.966076	+	0.258256i	$0.916852\pi$
$252$		0			0
$253$	−3.00000	+	5.19615i	−0.188608	+	0.326679i
$254$	2.07180	−	3.58846i	0.129996	−	0.225160i
$255$		0			0
$256$	−9.50000	+	16.4545i	−0.593750	+	1.02841i
$257$	−9.06218	−	15.6962i	−0.565283	−	0.979099i	−0.997023	−	0.0771011i	$-0.975434\pi$
$257$	−9.06218	−	15.6962i	−0.565283	−	0.979099i	0.431740	−	0.901998i	$-0.357900\pi$
$258$		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
$263$	2.36603	+	4.09808i	0.145895	+	0.252698i	0.929707	−	0.368301i	$-0.120060\pi$
$263$	2.36603	+	4.09808i	0.145895	+	0.252698i	−0.783811	+	0.620999i	$0.786727\pi$
$264$		0			0
$265$	−6.80385			−0.417957
$266$	−1.73205	+	3.00000i	−0.106199	+	0.183942i
$267$		0			0
$268$	4.19615			0.256321
$269$	9.46410	−	16.3923i	0.577036	−	0.999456i	−0.418781	−	0.908087i	$-0.637542\pi$
$269$	9.46410	−	16.3923i	0.577036	−	0.999456i	0.995817	−	0.0913690i	$-0.0291243\pi$
$270$		0			0
$271$	−8.09808	−	14.0263i	−0.491923	−	0.852036i	0.508034	−	0.861337i	$-0.330373\pi$
$271$	−8.09808	−	14.0263i	−0.491923	−	0.852036i	−0.999957	+	0.00930143i	$0.997039\pi$
$272$	−21.3397			−1.29391
$273$		0			0
$274$	−37.9808			−2.29450
$275$	−4.73205	−	8.19615i	−0.285353	−	0.494247i
$276$		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	$0.996047\pi$
$278$	35.6603			2.13876
$279$		0			0
$280$	1.50000	−	2.59808i	0.0896421	−	0.155265i
$281$	7.39230			0.440988			0.220494	−	0.975388i	$-0.429233\pi$
$281$	7.39230			0.440988			0.220494	+	0.975388i	$0.429233\pi$
$282$		0			0
$283$	0.0980762	+	0.169873i	0.00583003	+	0.0100979i	0.868926	−	0.494943i	$-0.164811\pi$
$283$	0.0980762	+	0.169873i	0.00583003	+	0.0100979i	−0.863096	+	0.505040i	$0.831478\pi$
$284$	3.00000	+	5.19615i	0.178017	+	0.308335i
$285$		0			0
$286$	16.3923	+	24.5885i	0.969297	+	1.45395i
$287$	−5.19615			−0.306719
$288$		0			0
$289$	−0.607695	−	1.05256i	−0.0357468	−	0.0619152i
$290$	−4.50000	+	7.79423i	−0.264249	+	0.457693i
$291$		0			0
$292$	−3.59808	+	6.23205i	−0.210561	+	0.364703i
$293$	−5.59808	+	9.69615i	−0.327043	+	0.566455i	−0.981924	−	0.189277i	$-0.939386\pi$
$293$	−5.59808	+	9.69615i	−0.327043	+	0.566455i	0.654881	+	0.755732i	$0.272719\pi$
$294$		0			0
$295$	9.29423	−	16.0981i	0.541131	−	0.937266i
$296$	−6.06218	−	10.5000i	−0.352357	−	0.610300i
$297$		0			0
$298$	−0.803848			−0.0465656
$299$	−4.56218	+	0.294229i	−0.263838	+	0.0170157i
$300$		0			0
$301$	−5.09808	−	8.83013i	−0.293848	−	0.508960i
$302$	−1.73205	−	3.00000i	−0.0996683	−	0.172631i
$303$		0			0
$304$	−10.0000			−0.573539
$305$	13.1603	−	22.7942i	0.753554	−	1.30519i
$306$		0			0
$307$	26.5885			1.51748			0.758742	−	0.651392i	$-0.225814\pi$
$307$	26.5885			1.51748			0.758742	+	0.651392i	$0.225814\pi$
$308$	2.36603	−	4.09808i	0.134817	−	0.233510i
$309$		0			0
$310$	9.29423	+	16.0981i	0.527877	+	0.914309i
$311$	−4.73205			−0.268330			−0.134165	−	0.990959i	$-0.542835\pi$
$311$	−4.73205			−0.268330			−0.134165	+	0.990959i	$0.542835\pi$
$312$		0			0
$313$	−12.7846			−0.722629			−0.361314	−	0.932444i	$-0.617672\pi$
$313$	−12.7846			−0.722629			−0.361314	+	0.932444i	$0.617672\pi$
$314$	7.96410	+	13.7942i	0.449440	+	0.778453i
$315$		0			0
$316$	−2.90192	+	5.02628i	−0.163246	+	0.282750i
$317$	−0.464102			−0.0260665			−0.0130333	−	0.999915i	$-0.504149\pi$
$317$	−0.464102			−0.0260665			−0.0130333	+	0.999915i	$0.504149\pi$
$318$		0			0
$319$	7.09808	−	12.2942i	0.397416	−	0.688345i
$320$	1.73205			0.0968246
$321$		0			0
$322$	1.09808	+	1.90192i	0.0611934	+	0.105990i
$323$	−4.26795	−	7.39230i	−0.237475	−	0.411319i
$324$		0			0
$325$	3.19615	−	6.46410i	0.177291	−	0.358564i
$326$	10.0526			0.556760
$327$		0			0
$328$	−4.50000	−	7.79423i	−0.248471	−	0.430364i
$329$	−0.464102	+	0.803848i	−0.0255868	+	0.0443176i
$330$		0			0
$331$	13.4904	−	23.3660i	0.741498	−	1.28431i	−0.210315	−	0.977634i	$-0.567449\pi$
$331$	13.4904	−	23.3660i	0.741498	−	1.28431i	0.951813	−	0.306679i	$-0.0992179\pi$
$332$	4.09808	−	7.09808i	0.224911	−	0.389558i
$333$		0			0
$334$	21.2942	−	36.8827i	1.16517	−	2.01813i
$335$	−3.63397	−	6.29423i	−0.198545	−	0.343890i
$336$		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
$340$	−3.69615	−	6.40192i	−0.200452	−	0.347193i
$341$	−14.6603	−	25.3923i	−0.793897	−	1.37507i
$342$		0			0
$343$	1.00000			0.0539949
$344$	8.83013	−	15.2942i	0.476089	−	0.824610i
$345$		0			0
$346$	−26.7846			−1.43995
$347$	5.36603	−	9.29423i	0.288063	−	0.498940i	−0.685284	−	0.728276i	$-0.740322\pi$
$347$	5.36603	−	9.29423i	0.288063	−	0.498940i	0.973347	+	0.229336i	$0.0736553\pi$
$348$		0			0
$349$	−8.39230	−	14.5359i	−0.449230	−	0.778089i	0.549106	−	0.835753i	$-0.314968\pi$
$349$	−8.39230	−	14.5359i	−0.449230	−	0.778089i	−0.998336	+	0.0576637i	$0.981635\pi$
$350$	−3.46410			−0.185164
$351$		0			0
$352$	24.5885			1.31057
$353$	1.66987	+	2.89230i	0.0888784	+	0.153942i	0.907037	−	0.421050i	$-0.138338\pi$
$353$	1.66987	+	2.89230i	0.0888784	+	0.153942i	−0.818159	+	0.574992i	$0.805005\pi$
$354$		0			0
$355$	5.19615	−	9.00000i	0.275783	−	0.477670i
$356$	0.928203			0.0491947
$357$		0			0
$358$	6.00000	−	10.3923i	0.317110	−	0.549250i
$359$	−5.07180			−0.267679			−0.133840	−	0.991003i	$-0.542731\pi$
$359$	−5.07180			−0.267679			−0.133840	+	0.991003i	$0.542731\pi$
$360$		0			0
$361$	7.50000	+	12.9904i	0.394737	+	0.683704i
$362$	22.1603	+	38.3827i	1.16472	+	2.01735i
$363$		0			0
$364$	3.59808	−	0.232051i	0.188590	−	0.0121628i
$365$	12.4641			0.652401
$366$		0			0
$367$	3.09808	+	5.36603i	0.161718	+	0.280104i	0.935485	−	0.353366i	$-0.114963\pi$
$367$	3.09808	+	5.36603i	0.161718	+	0.280104i	−0.773767	+	0.633471i	$0.781630\pi$
$368$	−3.16987	+	5.49038i	−0.165241	+	0.286206i
$369$		0			0
$370$	10.5000	−	18.1865i	0.545869	−	0.945473i
$371$	1.96410	−	3.40192i	0.101971	−	0.176619i
$372$		0			0
$373$	−4.69615	+	8.13397i	−0.243158	+	0.421161i	−0.961612	−	0.274413i	$-0.911517\pi$
$373$	−4.69615	+	8.13397i	−0.243158	+	0.421161i	0.718454	+	0.695574i	$0.244850\pi$
$374$	17.4904	+	30.2942i	0.904406	+	1.56648i
$375$		0			0
$376$	−1.60770			−0.0829105
$377$	10.7942	−	0.696152i	0.555931	−	0.0358537i
$378$		0			0
$379$	2.29423	+	3.97372i	0.117847	+	0.204116i	0.918914	−	0.394458i	$-0.129068\pi$
$379$	2.29423	+	3.97372i	0.117847	+	0.204116i	−0.801067	+	0.598574i	$0.795734\pi$
$380$	−1.73205	−	3.00000i	−0.0888523	−	0.153897i
$381$		0			0
$382$	−2.19615			−0.112365
$383$	2.83013	−	4.90192i	0.144613	−	0.250477i	−0.784616	−	0.619982i	$-0.787140\pi$
$383$	2.83013	−	4.90192i	0.144613	−	0.250477i	0.929228	+	0.369506i	$0.120473\pi$
$384$		0			0
$385$	−8.19615			−0.417715
$386$	−4.33013	+	7.50000i	−0.220398	+	0.381740i
$387$		0			0
$388$	7.19615	+	12.4641i	0.365329	+	0.632769i
$389$	30.4641			1.54459			0.772296	−	0.635263i	$-0.219108\pi$
$389$	30.4641			1.54459			0.772296	+	0.635263i	$0.219108\pi$
$390$		0			0
$391$	−5.41154			−0.273673
$392$	0.866025	+	1.50000i	0.0437409	+	0.0757614i
$393$		0			0
$394$	10.3923	−	18.0000i	0.523557	−	0.906827i
$395$	10.0526			0.505799
$396$		0			0
$397$	−11.3923	+	19.7321i	−0.571763	+	0.990323i	0.424622	+	0.905371i	$0.360407\pi$
$397$	−11.3923	+	19.7321i	−0.571763	+	0.990323i	−0.996385	+	0.0849523i	$0.972926\pi$
$398$	3.46410			0.173640
$399$		0			0
$400$	−5.00000	−	8.66025i	−0.250000	−	0.433013i
$401$	8.42820	+	14.5981i	0.420884	+	0.728993i	0.996026	−	0.0890606i	$-0.0283865\pi$
$401$	8.42820	+	14.5981i	0.420884	+	0.728993i	−0.575142	+	0.818054i	$0.695053\pi$
$402$		0			0
$403$	9.90192	−	20.0263i	0.493250	−	0.997580i
$404$	4.26795			0.212338
$405$		0			0
$406$	−2.59808	−	4.50000i	−0.128940	−	0.223331i
$407$	−16.5622	+	28.6865i	−0.820957	+	1.42194i
$408$		0			0
$409$	13.5981	−	23.5526i	0.672382	−	1.16460i	−0.304845	−	0.952402i	$-0.598605\pi$
$409$	13.5981	−	23.5526i	0.672382	−	1.16460i	0.977227	−	0.212197i	$-0.0680619\pi$
$410$	7.79423	−	13.5000i	0.384930	−	0.666717i
$411$		0			0
$412$	−3.19615	+	5.53590i	−0.157463	+	0.272734i
$413$	5.36603	+	9.29423i	0.264045	+	0.457339i
$414$		0			0
$415$	−14.1962			−0.696862
$416$	10.3923	+	15.5885i	0.509525	+	0.764287i
$417$		0			0
$418$	8.19615	+	14.1962i	0.400887	+	0.694357i
$419$	−10.9019	−	18.8827i	−0.532594	−	0.922480i	−0.999276	−	0.0380543i	$-0.987884\pi$
$419$	−10.9019	−	18.8827i	−0.532594	−	0.922480i	0.466682	−	0.884425i	$-0.345449\pi$
$420$		0			0
$421$	30.1769			1.47073			0.735366	−	0.677670i	$-0.237010\pi$
$421$	30.1769			1.47073			0.735366	+	0.677670i	$0.237010\pi$
$422$	10.5622	−	18.2942i	0.514159	−	0.890549i
$423$		0			0
$424$	6.80385			0.330424
$425$	4.26795	−	7.39230i	0.207026	−	0.358579i
$426$		0			0
$427$	7.59808	+	13.1603i	0.367697	+	0.636869i
$428$	−19.8564			−0.959796
$429$		0			0
$430$	30.5885			1.47511
$431$	17.6603	+	30.5885i	0.850665	+	1.47339i	0.880610	+	0.473843i	$0.157133\pi$
$431$	17.6603	+	30.5885i	0.850665	+	1.47339i	−0.0299451	+	0.999552i	$0.509533\pi$
$432$		0			0
$433$	−8.79423	+	15.2321i	−0.422624	+	0.732006i	−0.996195	−	0.0871498i	$-0.972224\pi$
$433$	−8.79423	+	15.2321i	−0.422624	+	0.732006i	0.573572	+	0.819155i	$0.305557\pi$
$434$	−10.7321			−0.515155
$435$		0			0
$436$	−6.19615	+	10.7321i	−0.296742	+	0.513972i
$437$	−2.53590			−0.121308
$438$		0			0
$439$	8.29423	+	14.3660i	0.395862	+	0.685653i	0.993211	−	0.116329i	$-0.0371125\pi$
$439$	8.29423	+	14.3660i	0.395862	+	0.685653i	−0.597349	+	0.801982i	$0.703779\pi$
$440$	−7.09808	−	12.2942i	−0.338388	−	0.586104i
$441$		0			0
$442$	−11.8135	+	23.8923i	−0.561909	+	1.13644i
$443$	11.3205			0.537854			0.268927	−	0.963161i	$-0.413331\pi$
$443$	11.3205			0.537854			0.268927	+	0.963161i	$0.413331\pi$
$444$		0			0
$445$	−0.803848	−	1.39230i	−0.0381060	−	0.0660016i
$446$	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
$451$	−12.2942	+	21.2942i	−0.578913	+	1.00271i
$452$	−3.69615	−	6.40192i	−0.173852	−	0.301121i
$453$		0			0
$454$	20.1962			0.947852
$455$	−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	$0.916159\pi$
$458$	−5.53590	−	9.58846i	−0.258676	−	0.448039i
$459$		0			0
$460$	−2.19615			−0.102396
$461$	7.79423	−	13.5000i	0.363013	−	0.628758i	−0.625442	−	0.780271i	$-0.715081\pi$
$461$	7.79423	−	13.5000i	0.363013	−	0.628758i	0.988455	+	0.151513i	$0.0484146\pi$
$462$		0			0
$463$	26.5885			1.23567			0.617835	−	0.786308i	$-0.288010\pi$
$463$	26.5885			1.23567			0.617835	+	0.786308i	$0.288010\pi$
$464$	7.50000	−	12.9904i	0.348179	−	0.603063i
$465$		0			0
$466$	22.3923	+	38.7846i	1.03730	+	1.79666i
$467$	19.5167			0.903123			0.451562	−	0.892240i	$-0.350867\pi$
$467$	19.5167			0.903123			0.451562	+	0.892240i	$0.350867\pi$
$468$		0			0
$469$	4.19615			0.193760
$470$	−1.39230	−	2.41154i	−0.0642222	−	0.111236i
$471$		0			0
$472$	−9.29423	+	16.0981i	−0.427802	+	0.740974i
$473$	−48.2487			−2.21848
$474$		0			0
$475$	2.00000	−	3.46410i	0.0917663	−	0.158944i
$476$	4.26795			0.195621
$477$		0			0
$478$	−22.6865	−	39.2942i	−1.03766	−	1.79728i
$479$	2.36603	+	4.09808i	0.108106	+	0.187246i	0.915003	−	0.403447i	$-0.132188\pi$
$479$	2.36603	+	4.09808i	0.108106	+	0.187246i	−0.806897	+	0.590693i	$0.798855\pi$
$480$		0			0
$481$	−25.1865	+	1.62436i	−1.14841	+	0.0740642i
$482$	−18.7128			−0.852345
$483$		0			0
$484$	−5.69615	−	9.86603i	−0.258916	−	0.448456i
$485$	12.4641	−	21.5885i	0.565966	−	0.980281i
$486$		0			0
$487$	0.392305	−	0.679492i	0.0177770	−	0.0307907i	−0.857000	−	0.515316i	$-0.827674\pi$
$487$	0.392305	−	0.679492i	0.0177770	−	0.0307907i	0.874777	+	0.484526i	$0.161008\pi$
$488$	−13.1603	+	22.7942i	−0.595737	+	1.03185i
$489$		0			0
$490$	−1.50000	+	2.59808i	−0.0677631	+	0.117369i
$491$	14.1962	+	24.5885i	0.640663	+	1.10966i	0.985285	+	0.170920i	$0.0546739\pi$
$491$	14.1962	+	24.5885i	0.640663	+	1.10966i	−0.344622	+	0.938742i	$0.611993\pi$
$492$		0			0
$493$	12.8038			0.576656
$494$	−5.53590	+	11.1962i	−0.249072	+	0.503739i
$495$		0			0
$496$	−15.4904	−	26.8301i	−0.695539	−	1.20471i
$497$	3.00000	+	5.19615i	0.134568	+	0.233079i
$498$		0			0
$499$	12.9808			0.581099			0.290549	−	0.956860i	$-0.406162\pi$
$499$	12.9808			0.581099			0.290549	+	0.956860i	$0.406162\pi$
$500$	6.06218	−	10.5000i	0.271109	−	0.469574i
$501$		0			0
$502$	38.7846			1.73104
$503$	6.29423	−	10.9019i	0.280646	−	0.486093i	−0.690898	−	0.722952i	$-0.742785\pi$
$503$	6.29423	−	10.9019i	0.280646	−	0.486093i	0.971544	+	0.236859i	$0.0761181\pi$
$504$		0			0
$505$	−3.69615	−	6.40192i	−0.164477	−	0.284882i
$506$	10.3923			0.461994
$507$		0			0
$508$	−2.39230			−0.106141
$509$	5.13397	+	8.89230i	0.227559	+	0.394144i	0.957084	−	0.289810i	$-0.0935921\pi$
$509$	5.13397	+	8.89230i	0.227559	+	0.394144i	−0.729525	+	0.683954i	$0.760259\pi$
$510$		0			0
$511$	−3.59808	+	6.23205i	−0.159170	+	0.275690i
$512$	8.66025			0.382733
$513$		0			0
$514$	−15.6962	+	27.1865i	−0.692328	+	1.19915i
$515$	11.0718			0.487882
$516$		0			0
$517$	2.19615	+	3.80385i	0.0965867	+	0.167293i
$518$	6.06218	+	10.5000i	0.266357	+	0.461344i
$519$		0			0
$520$	4.79423	−	9.69615i	0.210241	−	0.425204i
$521$	−0.124356			−0.00544812			−0.00272406	−	0.999996i	$-0.500867\pi$
$521$	−0.124356			−0.00544812			−0.00272406	+	0.999996i	$0.500867\pi$
$522$		0			0
$523$	−16.5885	−	28.7321i	−0.725363	−	1.25636i	−0.958825	−	0.283999i	$-0.908339\pi$
$523$	−16.5885	−	28.7321i	−0.725363	−	1.25636i	0.233462	−	0.972366i	$-0.424995\pi$
$524$	1.73205	−	3.00000i	0.0756650	−	0.131056i
$525$		0			0
$526$	4.09808	−	7.09808i	0.178685	−	0.309491i
$527$	13.2224	−	22.9019i	0.575978	−	0.997623i
$528$		0			0
$529$	10.6962	−	18.5263i	0.465050	−	0.805490i
$530$	5.89230	+	10.2058i	0.255945	+	0.443310i
$531$		0			0
$532$	2.00000			0.0867110
$533$	−18.6962	+	1.20577i	−0.809820	+	0.0522278i
$534$		0			0
$535$	17.1962	+	29.7846i	0.743455	+	1.28770i
$536$	3.63397	+	6.29423i	0.156964	+	0.271869i
$537$		0			0
$538$	−32.7846			−1.41344
$539$	2.36603	−	4.09808i	0.101912	−	0.176517i
$540$		0			0
$541$	−35.3923			−1.52163			−0.760817	−	0.648966i	$-0.775202\pi$
$541$	−35.3923			−1.52163			−0.760817	+	0.648966i	$0.775202\pi$
$542$	−14.0263	+	24.2942i	−0.602480	+	1.04353i
$543$		0			0
$544$	11.0885	+	19.2058i	0.475414	+	0.823441i
$545$	21.4641			0.919421
$546$		0			0
$547$	28.1962			1.20558			0.602790	−	0.797900i	$-0.294056\pi$
$547$	28.1962			1.20558			0.602790	+	0.797900i	$0.294056\pi$
$548$	10.9641	+	18.9904i	0.468363	+	0.811229i
$549$		0			0
$550$	−8.19615	+	14.1962i	−0.349485	+	0.605326i
$551$	6.00000			0.255609
$552$		0			0
$553$	−2.90192	+	5.02628i	−0.123402	+	0.213739i
$554$	29.4449			1.25099
$555$		0			0
$556$	−10.2942	−	17.8301i	−0.436573	−	0.756166i
$557$	−12.8205	−	22.2058i	−0.543222	−	0.940889i	−0.998716	−	0.0506499i	$-0.983871\pi$
$557$	−12.8205	−	22.2058i	−0.543222	−	0.940889i	0.455494	−	0.890239i	$-0.349463\pi$
$558$		0			0
$559$	−20.3923	−	30.5885i	−0.862503	−	1.29375i
$560$	−8.66025			−0.365963
$561$		0			0
$562$	−6.40192	−	11.0885i	−0.270049	−	0.467738i
$563$	−5.02628	+	8.70577i	−0.211832	+	0.366905i	−0.952288	−	0.305201i	$-0.901276\pi$
$563$	−5.02628	+	8.70577i	−0.211832	+	0.366905i	0.740456	+	0.672105i	$0.234610\pi$
$564$		0			0
$565$	−6.40192	+	11.0885i	−0.269331	+	0.466495i
$566$	0.169873	−	0.294229i	0.00714029	−	0.0123674i
$567$		0			0
$568$	−5.19615	+	9.00000i	−0.218026	+	0.377632i
$569$	14.5359	+	25.1769i	0.609377	+	1.05547i	0.991343	+	0.131295i	$0.0419135\pi$
$569$	14.5359	+	25.1769i	0.609377	+	1.05547i	−0.381967	+	0.924176i	$0.624753\pi$
$570$		0			0
$571$	−24.7846			−1.03720			−0.518602	−	0.855016i	$-0.673547\pi$
$571$	−24.7846			−1.03720			−0.518602	+	0.855016i	$0.673547\pi$
$572$	7.56218	−	15.2942i	0.316191	−	0.639484i
$573$		0			0
$574$	4.50000	+	7.79423i	0.187826	+	0.325325i
$575$	−1.26795	−	2.19615i	−0.0528771	−	0.0915859i
$576$		0			0
$577$	32.8038			1.36564			0.682821	−	0.730586i	$-0.260753\pi$
$577$	32.8038			1.36564			0.682821	+	0.730586i	$0.260753\pi$
$578$	−1.05256	+	1.82309i	−0.0437807	+	0.0758304i
$579$		0			0
$580$	5.19615			0.215758
$581$	4.09808	−	7.09808i	0.170017	−	0.294478i
$582$		0			0
$583$	−9.29423	−	16.0981i	−0.384928	−	0.666714i
$584$	−12.4641			−0.515768
$585$		0			0
$586$	19.3923			0.801089
$587$	−2.19615	−	3.80385i	−0.0906449	−	0.157002i	0.817138	−	0.576442i	$-0.195559\pi$
$587$	−2.19615	−	3.80385i	−0.0906449	−	0.157002i	−0.907783	+	0.419441i	$0.862226\pi$
$588$		0			0
$589$	6.19615	−	10.7321i	0.255308	−	0.442206i
$590$	−32.1962			−1.32549
$591$		0			0
$592$	−17.5000	+	30.3109i	−0.719246	+	1.24577i
$593$	−41.4449			−1.70194			−0.850968	−	0.525217i	$-0.823984\pi$
$593$	−41.4449			−1.70194			−0.850968	+	0.525217i	$0.823984\pi$
$594$		0			0
$595$	−3.69615	−	6.40192i	−0.151527	−	0.262453i
$596$	0.232051	+	0.401924i	0.00950517	+	0.0164634i
$597$		0			0
$598$	4.39230	+	6.58846i	0.179615	+	0.269422i
$599$	−16.1436			−0.659609			−0.329805	−	0.944049i	$-0.606983\pi$
$599$	−16.1436			−0.659609			−0.329805	+	0.944049i	$0.606983\pi$
$600$		0			0
$601$	−10.9904	−	19.0359i	−0.448307	−	0.776490i	0.549969	−	0.835185i	$-0.314640\pi$
$601$	−10.9904	−	19.0359i	−0.448307	−	0.776490i	−0.998276	+	0.0586946i	$0.981306\pi$
$602$	−8.83013	+	15.2942i	−0.359889	+	0.623346i
$603$		0			0
$604$	−1.00000	+	1.73205i	−0.0406894	+	0.0704761i
$605$	−9.86603	+	17.0885i	−0.401111	+	0.694745i
$606$		0			0
$607$	−3.19615	+	5.53590i	−0.129728	+	0.224695i	−0.923571	−	0.383427i	$-0.874744\pi$
$607$	−3.19615	+	5.53590i	−0.129728	+	0.224695i	0.793843	+	0.608122i	$0.208077\pi$
$608$	5.19615	+	9.00000i	0.210732	+	0.364998i
$609$		0			0
$610$	−45.5885			−1.84582
$611$	−1.48334	+	3.00000i	−0.0600095	+	0.121367i
$612$		0			0
$613$	8.69615	+	15.0622i	0.351234	+	0.608356i	0.986466	−	0.163966i	$-0.0524287\pi$
$613$	8.69615	+	15.0622i	0.351234	+	0.608356i	−0.635232	+	0.772322i	$0.719095\pi$
$614$	−23.0263	−	39.8827i	−0.929265	−	1.60953i
$615$		0			0
$616$	8.19615			0.330232
$617$	14.3038	−	24.7750i	0.575851	−	0.997404i	−0.420097	−	0.907479i	$-0.638004\pi$
$617$	14.3038	−	24.7750i	0.575851	−	0.997404i	0.995949	−	0.0899245i	$-0.0286626\pi$
$618$		0			0
$619$	−37.3731			−1.50215			−0.751075	−	0.660217i	$-0.770464\pi$
$619$	−37.3731			−1.50215			−0.751075	+	0.660217i	$0.770464\pi$
$620$	5.36603	−	9.29423i	0.215505	−	0.373265i
$621$		0			0
$622$	4.09808	+	7.09808i	0.164318	+	0.284607i
$623$	0.928203			0.0371877
$624$		0			0
$625$	−11.0000			−0.440000
$626$	11.0718	+	19.1769i	0.442518	+	0.766464i
$627$		0			0
$628$	4.59808	−	7.96410i	0.183483	−	0.317802i
$629$	−29.8756			−1.19122
$630$		0			0
$631$	−14.3923	+	24.9282i	−0.572949	+	0.992376i	0.423313	+	0.905984i	$0.360867\pi$
$631$	−14.3923	+	24.9282i	−0.572949	+	0.992376i	−0.996261	+	0.0863924i	$0.972466\pi$
$632$	−10.0526			−0.399869
$633$		0			0
$634$	0.401924	+	0.696152i	0.0159624	+	0.0276477i
$635$	2.07180	+	3.58846i	0.0822167	+	0.142404i
$636$		0			0
$637$	3.59808	−	0.232051i	0.142561	−	0.00919419i
$638$	−24.5885			−0.973466
$639$		0			0
$640$	−10.5000	−	18.1865i	−0.415049	−	0.718886i
$641$	0.571797	−	0.990381i	0.0225846	−	0.0391177i	−0.854512	−	0.519431i	$-0.826144\pi$
$641$	0.571797	−	0.990381i	0.0225846	−	0.0391177i	0.877097	+	0.480314i	$0.159477\pi$
$642$		0			0
$643$	−20.3923	+	35.3205i	−0.804194	+	1.39290i	0.112640	+	0.993636i	$0.464069\pi$
$643$	−20.3923	+	35.3205i	−0.804194	+	1.39290i	−0.916834	+	0.399269i	$0.869264\pi$
$644$	0.633975	−	1.09808i	0.0249821	−	0.0432703i
$645$		0			0
$646$	−7.39230	+	12.8038i	−0.290846	+	0.503761i
$647$	−22.5167	−	39.0000i	−0.885221	−	1.53325i	−0.845460	−	0.534039i	$-0.820674\pi$
$647$	−22.5167	−	39.0000i	−0.885221	−	1.53325i	−0.0397614	−	0.999209i	$-0.512660\pi$
$648$		0			0
$649$	50.7846			1.99347
$650$	−12.4641	+	0.803848i	−0.488882	+	0.0315295i
$651$		0			0
$652$	−2.90192	−	5.02628i	−0.113648	−	0.196844i
$653$	−5.07180	−	8.78461i	−0.198475	−	0.343768i	0.749559	−	0.661937i	$-0.230265\pi$
$653$	−5.07180	−	8.78461i	−0.198475	−	0.343768i	−0.948034	+	0.318169i	$0.896932\pi$
$654$		0			0
$655$	−6.00000			−0.234439
$656$	−12.9904	+	22.5000i	−0.507189	+	0.878477i
$657$		0			0
$658$	1.60770			0.0626745
$659$	3.80385	−	6.58846i	0.148177	−	0.256650i	−0.782377	−	0.622805i	$-0.785993\pi$
$659$	3.80385	−	6.58846i	0.148177	−	0.256650i	0.930554	+	0.366156i	$0.119326\pi$
$660$		0			0
$661$	11.4019	+	19.7487i	0.443483	+	0.768136i	0.997945	−	0.0640734i	$-0.0204092\pi$
$661$	11.4019	+	19.7487i	0.443483	+	0.768136i	−0.554462	+	0.832209i	$0.687076\pi$
$662$	−46.7321			−1.81629
$663$		0			0
$664$	14.1962			0.550918
$665$	−1.73205	−	3.00000i	−0.0671660	−	0.116335i
$666$		0			0
$667$	1.90192	−	3.29423i	0.0736428	−	0.127553i
$668$	−24.5885			−0.951356
$669$		0			0
$670$	−6.29423	+	10.9019i	−0.243167	+	0.421178i
$671$	71.9090			2.77601
$672$		0			0
$673$	−9.08846	−	15.7417i	−0.350334	−	0.606797i	0.635974	−	0.771711i	$-0.280599\pi$
$673$	−9.08846	−	15.7417i	−0.350334	−	0.606797i	−0.986308	+	0.164914i	$0.947265\pi$
$674$	−9.52628	−	16.5000i	−0.366939	−	0.635556i
$675$		0			0
$676$	12.8923	−	1.66987i	0.495858	−	0.0642259i
$677$	36.9282			1.41927			0.709633	−	0.704571i	$-0.248861\pi$
$677$	36.9282			1.41927			0.709633	+	0.704571i	$0.248861\pi$
$678$		0			0
$679$	7.19615	+	12.4641i	0.276163	+	0.478328i
$680$	6.40192	−	11.0885i	0.245503	−	0.425223i
$681$		0			0
$682$	−25.3923	+	43.9808i	−0.972322	+	1.68411i
$683$	4.26795	−	7.39230i	0.163309	−	0.282859i	−0.772745	−	0.634717i	$-0.781117\pi$
$683$	4.26795	−	7.39230i	0.163309	−	0.282859i	0.936053	+	0.351858i	$0.114450\pi$
$684$		0			0
$685$	18.9904	−	32.8923i	0.725585	−	1.25675i
$686$	−0.866025	−	1.50000i	−0.0330650	−	0.0572703i
$687$		0			0
$688$	−50.9808			−1.94362
$689$	6.27757	−	12.6962i	0.239156	−	0.483685i
$690$		0			0
$691$	10.1962	+	17.6603i	0.387880	+	0.671828i	0.992164	−	0.124941i	$-0.0398742\pi$
$691$	10.1962	+	17.6603i	0.387880	+	0.671828i	−0.604284	+	0.796769i	$0.706541\pi$
$692$	7.73205	+	13.3923i	0.293928	+	0.509099i
$693$		0			0
$694$	−18.5885			−0.705608
$695$	−17.8301	+	30.8827i	−0.676335	+	1.17145i
$696$		0			0
$697$	−22.1769			−0.840011
$698$	−14.5359	+	25.1769i	−0.550192	+	0.952960i
$699$		0			0
$700$	1.00000	+	1.73205i	0.0377964	+	0.0654654i
$701$	20.7846			0.785024			0.392512	−	0.919747i	$-0.371606\pi$
$701$	20.7846			0.785024			0.392512	+	0.919747i	$0.371606\pi$
$702$		0			0
$703$	−14.0000			−0.528020
$704$	2.36603	+	4.09808i	0.0891729	+	0.154452i
$705$		0			0
$706$	2.89230	−	5.00962i	0.108853	−	0.188539i
$707$	4.26795			0.160513
$708$		0			0
$709$	16.0885	−	27.8660i	0.604215	−	1.04653i	−0.387960	−	0.921676i	$-0.626820\pi$
$709$	16.0885	−	27.8660i	0.604215	−	1.04653i	0.992175	−	0.124854i	$-0.0398464\pi$
$710$	−18.0000			−0.675528
$711$		0			0
$712$	0.803848	+	1.39230i	0.0301255	+	0.0521788i
$713$	−3.92820	−	6.80385i	−0.147112	−	0.254806i
$714$		0			0
$715$	−29.4904	+	1.90192i	−1.10288	+	0.0711279i
$716$	−6.92820			−0.258919
$717$		0			0
$718$	4.39230	+	7.60770i	0.163919	+	0.283917i
$719$	−5.36603	+	9.29423i	−0.200119	+	0.346616i	−0.948567	−	0.316578i	$-0.897466\pi$
$719$	−5.36603	+	9.29423i	−0.200119	+	0.346616i	0.748448	+	0.663194i	$0.230800\pi$
$720$		0			0
$721$	−3.19615	+	5.53590i	−0.119031	+	0.206168i
$722$	12.9904	−	22.5000i	0.483452	−	0.837363i
$723$		0			0
$724$	12.7942	−	22.1603i	0.475494	−	0.823579i
$725$	3.00000	+	5.19615i	0.111417	+	0.192980i
$726$		0			0
$727$	21.1769			0.785408			0.392704	−	0.919665i	$-0.371540\pi$
$727$	21.1769			0.785408			0.392704	+	0.919665i	$0.371540\pi$
$728$	3.46410	+	5.19615i	0.128388	+	0.192582i
$729$		0			0
$730$	−10.7942	−	18.6962i	−0.399512	−	0.691976i
$731$	−21.7583	−	37.6865i	−0.804761	−	1.39389i
$732$		0			0
$733$	−7.58846			−0.280286			−0.140143	−	0.990131i	$-0.544756\pi$
$733$	−7.58846			−0.280286			−0.140143	+	0.990131i	$0.544756\pi$
$734$	5.36603	−	9.29423i	0.198064	−	0.343056i
$735$		0			0
$736$	6.58846			0.242854
$737$	9.92820	−	17.1962i	0.365710	−	0.633428i
$738$		0			0
$739$	0.392305	+	0.679492i	0.0144312	+	0.0249955i	0.873151	−	0.487450i	$-0.162073\pi$
$739$	0.392305	+	0.679492i	0.0144312	+	0.0249955i	−0.858720	+	0.512446i	$0.828740\pi$
$740$	−12.1244			−0.445700
$741$		0			0
$742$	−6.80385			−0.249777
$743$	−14.1962	−	24.5885i	−0.520806	−	0.902063i	−0.999707	−	0.0241941i	$-0.992298\pi$
$743$	−14.1962	−	24.5885i	−0.520806	−	0.902063i	0.478901	−	0.877869i	$-0.341035\pi$
$744$		0			0
$745$	0.401924	−	0.696152i	0.0147253	−	0.0255051i
$746$	16.2679			0.595612
$747$		0			0
$748$	10.0981	−	17.4904i	0.369222	−	0.639512i
$749$	−19.8564			−0.725537
$750$		0			0
$751$	−23.0981	−	40.0070i	−0.842861	−	1.45988i	−0.887466	−	0.460873i	$-0.847536\pi$
$751$	−23.0981	−	40.0070i	−0.842861	−	1.45988i	0.0446053	−	0.999005i	$-0.485797\pi$
$752$	2.32051	+	4.01924i	0.0846202	+	0.146567i
$753$		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	$0.739424\pi$
$758$	3.97372	−	6.88269i	0.144332	−	0.249990i
$759$		0			0
$760$	3.00000	−	5.19615i	0.108821	−	0.188484i
$761$	3.33975	−	5.78461i	0.121066	−	0.209692i	−0.799123	−	0.601168i	$-0.794702\pi$
$761$	3.33975	−	5.78461i	0.121066	−	0.209692i	0.920188	+	0.391476i	$0.128035\pi$
$762$		0			0
$763$	−6.19615	+	10.7321i	−0.224316	+	0.388526i
$764$	0.633975	+	1.09808i	0.0229364	+	0.0397270i
$765$		0			0
$766$	−9.80385			−0.354227
$767$	21.4641	+	32.1962i	0.775024	+	1.16254i
$768$		0			0
$769$	23.5885	+	40.8564i	0.850622	+	1.47332i	0.880648	+	0.473771i	$0.157107\pi$
$769$	23.5885	+	40.8564i	0.850622	+	1.47332i	−0.0300268	+	0.999549i	$0.509559\pi$
$770$	7.09808	+	12.2942i	0.255797	+	0.443053i
$771$		0			0
$772$	5.00000			0.179954
$773$	0.464102	−	0.803848i	0.0166926	−	0.0289124i	−0.857558	−	0.514387i	$-0.828020\pi$
$773$	0.464102	−	0.803848i	0.0166926	−	0.0289124i	0.874251	+	0.485474i	$0.161353\pi$
$774$		0			0
$775$	12.3923			0.445145
$776$	−12.4641	+	21.5885i	−0.447435	+	0.774980i
$777$		0			0
$778$	−26.3827	−	45.6962i	−0.945865	−	1.63829i
$779$	−10.3923			−0.372343
$780$		0			0
$781$	28.3923			1.01596
$782$	4.68653	+	8.11731i	0.167590	+	0.290275i
$783$		0			0
$784$	2.50000	−	4.33013i	0.0892857

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)

\(f(q)\)	\(=\)	\(q+(-0.866025 - 1.50000i) q^{2} +(-0.500000 + 0.866025i) q^{4} +1.73205 q^{5} +(-0.500000 + 0.866025i) q^{7} -1.73205 q^{8} +O(q^{10})\)	\(q+(-0.866025 - 1.50000i) q^{2} +(-0.500000 + 0.866025i) q^{4} +1.73205 q^{5} +(-0.500000 + 0.866025i) q^{7} -1.73205 q^{8} +(-1.50000 - 2.59808i) q^{10} +(2.36603 + 4.09808i) q^{11} +(-1.59808 + 3.23205i) q^{13} +1.73205 q^{14} +(2.50000 + 4.33013i) q^{16} +(-2.13397 + 3.69615i) q^{17} +(-1.00000 + 1.73205i) q^{19} +(-0.866025 + 1.50000i) q^{20} +(4.09808 - 7.09808i) q^{22} +(0.633975 + 1.09808i) q^{23} -2.00000 q^{25} +(6.23205 - 0.401924i) q^{26} +(-0.500000 - 0.866025i) q^{28} +(-1.50000 - 2.59808i) q^{29} -6.19615 q^{31} +(2.59808 - 4.50000i) q^{32} +7.39230 q^{34} +(-0.866025 + 1.50000i) q^{35} +(3.50000 + 6.06218i) q^{37} +3.46410 q^{38} -3.00000 q^{40} +(2.59808 + 4.50000i) q^{41} +(-5.09808 + 8.83013i) q^{43} -4.73205 q^{44} +(1.09808 - 1.90192i) q^{46} +0.928203 q^{47} +(-0.500000 - 0.866025i) q^{49} +(1.73205 + 3.00000i) q^{50} +(-2.00000 - 3.00000i) q^{52} -3.92820 q^{53} +(4.09808 + 7.09808i) q^{55} +(0.866025 - 1.50000i) q^{56} +(-2.59808 + 4.50000i) q^{58} +(5.36603 - 9.29423i) q^{59} +(7.59808 - 13.1603i) q^{61} +(5.36603 + 9.29423i) q^{62} +1.00000 q^{64} +(-2.76795 + 5.59808i) q^{65} +(-2.09808 - 3.63397i) q^{67} +(-2.13397 - 3.69615i) q^{68} +3.00000 q^{70} +(3.00000 - 5.19615i) q^{71} +7.19615 q^{73} +(6.06218 - 10.5000i) q^{74} +(-1.00000 - 1.73205i) q^{76} -4.73205 q^{77} +5.80385 q^{79} +(4.33013 + 7.50000i) q^{80} +(4.50000 - 7.79423i) q^{82} -8.19615 q^{83} +(-3.69615 + 6.40192i) q^{85} +17.6603 q^{86} +(-4.09808 - 7.09808i) q^{88} +(-0.464102 - 0.803848i) q^{89} +(-2.00000 - 3.00000i) q^{91} -1.26795 q^{92} +(-0.803848 - 1.39230i) q^{94} +(-1.73205 + 3.00000i) q^{95} +(7.19615 - 12.4641i) q^{97} +(-0.866025 + 1.50000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{4} - 2 q^{7}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more