LMFDB - Embedded newform 810.3.j.c.539.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,3,Mod(269,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([1, 3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("810.269");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	810.j (of order $6$, degree $2$, not 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:	$22.0709014132$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.443364212736.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 16x^{6} + 175x^{4} - 1296x^{2} + 6561 $ x^8 - 16x^6 + 175x^4 - 1296*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			539.1
Root			$-2.17132 + 2.07011i$ of defining polynomial
Character	$\chi$	$=$	810.539
Dual form			810.3.j.c.269.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 1.22474i) q^{2} +(-1.00000 + 1.73205i) q^{4} +(-2.15650 - 4.51104i) q^{5} +(5.04975 - 2.91548i) q^{7} +2.82843 q^{8} +O(q^{10})$	\(q+(-0.707107 - 1.22474i) q^{2} +(-1.00000 + 1.73205i) q^{4} +(-2.15650 - 4.51104i) q^{5} +(5.04975 - 2.91548i) q^{7} +2.82843 q^{8} +(-4.00000 + 5.83095i) q^{10} +(-14.2829 + 8.24621i) q^{11} +(-7.14143 - 4.12311i) q^{14} +(-2.00000 - 3.46410i) q^{16} -11.3137 q^{17} +12.0000 q^{19} +(9.96986 + 0.775874i) q^{20} +(20.1990 + 11.6619i) q^{22} +(-12.0208 + 20.8207i) q^{23} +(-15.6990 + 19.4561i) q^{25} +11.6619i q^{28} +(16.0000 - 27.7128i) q^{31} +(-2.82843 + 4.89898i) q^{32} +(8.00000 + 13.8564i) q^{34} +(-24.0416 - 16.4924i) q^{35} +23.3238i q^{37} +(-8.48528 - 14.6969i) q^{38} +(-6.09950 - 12.7592i) q^{40} +(-49.9900 - 28.8617i) q^{41} +(-35.3483 + 20.4083i) q^{43} -32.9848i q^{44} +34.0000 q^{46} +(17.6777 + 30.6186i) q^{47} +(-7.50000 + 12.9904i) q^{49} +(34.9297 + 5.46972i) q^{50} +67.8823 q^{53} +(68.0000 + 46.6476i) q^{55} +(14.2829 - 8.24621i) q^{56} +(14.2829 + 8.24621i) q^{59} +(8.00000 + 13.8564i) q^{61} -45.2548 q^{62} +8.00000 q^{64} +(5.04975 + 2.91548i) q^{67} +(11.3137 - 19.5959i) q^{68} +(-3.19901 + 41.1068i) q^{70} +116.619i q^{73} +(28.5657 - 16.4924i) q^{74} +(-12.0000 + 20.7846i) q^{76} +(-48.0833 + 83.2827i) q^{77} +(36.0000 + 62.3538i) q^{79} +(-11.3137 + 16.4924i) q^{80} +81.6333i q^{82} +(21.9203 + 37.9671i) q^{83} +(24.3980 + 51.0366i) q^{85} +(49.9900 + 28.8617i) q^{86} +(-40.3980 + 23.3238i) q^{88} +65.9697i q^{89} +(-24.0416 - 41.6413i) q^{92} +(25.0000 - 43.3013i) q^{94} +(-25.8780 - 54.1325i) q^{95} +(-141.393 + 81.6333i) q^{97} +21.2132 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} - 32 q^{10} - 16 q^{16} + 96 q^{19} + 36 q^{25} + 128 q^{31} + 64 q^{34} + 32 q^{40} + 272 q^{46} - 60 q^{49} + 544 q^{55} + 64 q^{61} + 64 q^{64} + 136 q^{70} - 96 q^{76} + 288 q^{79} - 128 q^{85} + 200 q^{94}+O(q^{100}) $ 8 * q - 8 * q^4 - 32 * q^10 - 16 * q^16 + 96 * q^19 + 36 * q^25 + 128 * q^31 + 64 * q^34 + 32 * q^40 + 272 * q^46 - 60 * q^49 + 544 * q^55 + 64 * q^61 + 64 * q^64 + 136 * q^70 - 96 * q^76 + 288 * q^79 - 128 * q^85 + 200 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$

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 $

$2$	−0.707107	−	1.22474i	−0.353553	−	0.612372i
$2$	−0.707107	−	1.22474i	−0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−1.00000	+	1.73205i	−0.250000	+	0.433013i
$5$	−2.15650	−	4.51104i	−0.431300	−	0.902209i
$5$	−2.15650	−	4.51104i	−0.431300	−	0.902209i
$6$		0			0
$7$	5.04975	−	2.91548i	0.721393	−	0.416497i	−0.0938720	−	0.995584i	$-0.529924\pi$
$7$	5.04975	−	2.91548i	0.721393	−	0.416497i	0.815265	+	0.579088i	$0.196591\pi$
$8$	2.82843			0.353553
$9$		0			0
$10$	−4.00000	+	5.83095i	−0.400000	+	0.583095i
$11$	−14.2829	+	8.24621i	−1.29844	+	0.749656i	−0.980135	−	0.198332i	$-0.936448\pi$
$11$	−14.2829	+	8.24621i	−1.29844	+	0.749656i	−0.318307	+	0.947988i	$0.603114\pi$
$12$		0			0
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$	−7.14143	−	4.12311i	−0.510102	−	0.294508i
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.125000	−	0.216506i
$17$	−11.3137			−0.665512			−0.332756	−	0.943013i	$-0.607979\pi$
$17$	−11.3137			−0.665512			−0.332756	+	0.943013i	$0.607979\pi$
$18$		0			0
$19$	12.0000			0.631579			0.315789	−	0.948829i	$-0.397731\pi$
$19$	12.0000			0.631579			0.315789	+	0.948829i	$0.397731\pi$
$20$	9.96986	+	0.775874i	0.498493	+	0.0387937i
$21$		0			0
$22$	20.1990	+	11.6619i	0.918137	+	0.530087i
$23$	−12.0208	+	20.8207i	−0.522644	+	0.905246i	0.477009	+	0.878899i	$0.341721\pi$
$23$	−12.0208	+	20.8207i	−0.522644	+	0.905246i	−0.999653	+	0.0263476i	$0.991612\pi$
$24$		0			0
$25$	−15.6990	+	19.4561i	−0.627960	+	0.778245i
$26$		0			0
$27$		0			0
$28$			11.6619i			0.416497i
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$	16.0000	−	27.7128i	0.516129	−	0.893962i	−0.483696	−	0.875236i	$-0.660706\pi$
$31$	16.0000	−	27.7128i	0.516129	−	0.893962i	0.999825	−	0.0187254i	$-0.00596084\pi$
$32$	−2.82843	+	4.89898i	−0.0883883	+	0.153093i
$33$		0			0
$34$	8.00000	+	13.8564i	0.235294	+	0.407541i
$35$	−24.0416	−	16.4924i	−0.686904	−	0.471212i
$36$		0			0
$37$			23.3238i			0.630373i	0.949030	+	0.315187i	$0.102067\pi$
$37$			23.3238i			0.630373i	−0.949030	+	0.315187i	$0.897933\pi$
$38$	−8.48528	−	14.6969i	−0.223297	−	0.386762i
$39$		0			0
$40$	−6.09950	−	12.7592i	−0.152488	−	0.318979i
$41$	−49.9900	−	28.8617i	−1.21927	−	0.703945i	−0.254507	−	0.967071i	$-0.581913\pi$
$41$	−49.9900	−	28.8617i	−1.21927	−	0.703945i	−0.964761	+	0.263126i	$0.915246\pi$
$42$		0			0
$43$	−35.3483	+	20.4083i	−0.822053	+	0.474612i	−0.851124	−	0.524965i	$-0.824078\pi$
$43$	−35.3483	+	20.4083i	−0.822053	+	0.474612i	0.0290711	+	0.999577i	$0.490745\pi$
$44$		−	32.9848i		−	0.749656i
$45$		0			0
$46$	34.0000			0.739130
$47$	17.6777	+	30.6186i	0.376121	+	0.651460i	0.990494	−	0.137555i	$-0.0439245\pi$
$47$	17.6777	+	30.6186i	0.376121	+	0.651460i	−0.614373	+	0.789015i	$0.710591\pi$
$48$		0			0
$49$	−7.50000	+	12.9904i	−0.153061	+	0.265110i
$50$	34.9297	+	5.46972i	0.698593	+	0.109394i
$51$		0			0
$52$		0			0
$53$	67.8823			1.28080			0.640399	−	0.768043i	$-0.278769\pi$
$53$	67.8823			1.28080			0.640399	+	0.768043i	$0.278769\pi$
$54$		0			0
$55$	68.0000	+	46.6476i	1.23636	+	0.848138i
$56$	14.2829	−	8.24621i	0.255051	−	0.147254i
$57$		0			0
$58$		0			0
$59$	14.2829	+	8.24621i	0.242082	+	0.139766i	0.616133	−	0.787642i	$-0.288698\pi$
$59$	14.2829	+	8.24621i	0.242082	+	0.139766i	−0.374051	+	0.927408i	$0.622032\pi$
$60$		0			0
$61$	8.00000	+	13.8564i	0.131148	+	0.227154i	0.924119	−	0.382104i	$-0.124801\pi$
$61$	8.00000	+	13.8564i	0.131148	+	0.227154i	−0.792972	+	0.609259i	$0.791467\pi$
$62$	−45.2548			−0.729917
$63$		0			0
$64$	8.00000			0.125000
$65$		0			0
$66$		0			0
$67$	5.04975	+	2.91548i	0.0753694	+	0.0435146i	0.537211	−	0.843448i	$-0.319478\pi$
$67$	5.04975	+	2.91548i	0.0753694	+	0.0435146i	−0.461842	+	0.886962i	$0.652811\pi$
$68$	11.3137	−	19.5959i	0.166378	−	0.288175i
$69$		0			0
$70$	−3.19901	+	41.1068i	−0.0457001	+	0.587240i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$			116.619i			1.59752i	0.601649	+	0.798761i	$0.294511\pi$
$73$			116.619i			1.59752i	−0.601649	+	0.798761i	$0.705489\pi$
$74$	28.5657	−	16.4924i	0.386023	−	0.222871i
$75$		0			0
$76$	−12.0000	+	20.7846i	−0.157895	+	0.273482i
$77$	−48.0833	+	83.2827i	−0.624458	+	1.08159i
$78$		0			0
$79$	36.0000	+	62.3538i	0.455696	+	0.789289i	0.998728	−	0.0504232i	$-0.0160570\pi$
$79$	36.0000	+	62.3538i	0.455696	+	0.789289i	−0.543032	+	0.839712i	$0.682724\pi$
$80$	−11.3137	+	16.4924i	−0.141421	+	0.206155i
$81$		0			0
$82$			81.6333i			0.995528i
$83$	21.9203	+	37.9671i	0.264100	+	0.457435i	0.967327	−	0.253530i	$-0.0815917\pi$
$83$	21.9203	+	37.9671i	0.264100	+	0.457435i	−0.703227	+	0.710965i	$0.748258\pi$
$84$		0			0
$85$	24.3980	+	51.0366i	0.287036	+	0.600431i
$86$	49.9900	+	28.8617i	0.581279	+	0.335602i
$87$		0			0
$88$	−40.3980	+	23.3238i	−0.459068	+	0.265043i
$89$			65.9697i			0.741232i	0.928786	+	0.370616i	$0.120853\pi$
$89$			65.9697i			0.741232i	−0.928786	+	0.370616i	$0.879147\pi$
$90$		0			0
$91$		0			0
$92$	−24.0416	−	41.6413i	−0.261322	−	0.452623i
$93$		0			0
$94$	25.0000	−	43.3013i	0.265957	−	0.460652i
$95$	−25.8780	−	54.1325i	−0.272400	−	0.569816i
$96$		0			0
$97$	−141.393	+	81.6333i	−1.45766	+	0.841581i	−0.998896	−	0.0469772i	$-0.985041\pi$
$97$	−141.393	+	81.6333i	−1.45766	+	0.841581i	−0.458765	+	0.888558i	$0.651708\pi$
$98$	21.2132			0.216461
$99$		0			0
$100$	−18.0000	−	46.6476i	−0.180000	−	0.466476i
$101$	−114.263	+	65.9697i	−1.13132	+	0.653165i	−0.944265	−	0.329185i	$-0.893226\pi$
$101$	−114.263	+	65.9697i	−1.13132	+	0.653165i	−0.187050	+	0.982350i	$0.559893\pi$
$102$		0			0
$103$	85.8458	+	49.5631i	0.833454	+	0.481195i	0.855034	−	0.518572i	$-0.173536\pi$
$103$	85.8458	+	49.5631i	0.833454	+	0.481195i	−0.0215796	+	0.999767i	$0.506870\pi$
$104$		0			0
$105$		0			0
$106$	−48.0000	−	83.1384i	−0.452830	−	0.784325i
$107$	−55.1543			−0.515461			−0.257731	−	0.966217i	$-0.582975\pi$
$107$	−55.1543			−0.515461			−0.257731	+	0.966217i	$0.582975\pi$
$108$		0			0
$109$	80.0000			0.733945			0.366972	−	0.930232i	$-0.380394\pi$
$109$	80.0000			0.733945			0.366972	+	0.930232i	$0.380394\pi$
$110$	9.04817	−	116.267i	0.0822561	−	1.05698i
$111$		0			0
$112$	−20.1990	−	11.6619i	−0.180348	−	0.104124i
$113$	76.3675	−	132.272i	0.675819	−	1.17055i	−0.300410	−	0.953810i	$-0.597123\pi$
$113$	76.3675	−	132.272i	0.675819	−	1.17055i	0.976229	−	0.216742i	$-0.0695432\pi$
$114$		0			0
$115$	119.846	+	9.32664i	1.04214	+	0.0811012i
$116$		0			0
$117$		0			0
$118$		−	23.3238i		−	0.197659i
$119$	−57.1314	+	32.9848i	−0.480096	+	0.277184i
$120$		0			0
$121$	75.5000	−	130.770i	0.623967	−	1.08074i
$122$	11.3137	−	19.5959i	0.0927353	−	0.160622i
$123$		0			0
$124$	32.0000	+	55.4256i	0.258065	+	0.446981i
$125$	121.622	+	28.8617i	0.972979	+	0.230894i
$126$		0			0
$127$			40.8167i			0.321391i	0.987004	+	0.160696i	$0.0513737\pi$
$127$			40.8167i			0.321391i	−0.987004	+	0.160696i	$0.948626\pi$
$128$	−5.65685	−	9.79796i	−0.0441942	−	0.0765466i
$129$		0			0
$130$		0			0
$131$	42.8486	+	24.7386i	0.327088	+	0.188845i	0.654548	−	0.756021i	$-0.272859\pi$
$131$	42.8486	+	24.7386i	0.327088	+	0.188845i	−0.327459	+	0.944865i	$0.606192\pi$
$132$		0			0
$133$	60.5970	−	34.9857i	0.455617	−	0.263050i
$134$		−	8.24621i		−	0.0615389i
$135$		0			0
$136$	−32.0000			−0.235294
$137$	25.4558	+	44.0908i	0.185809	+	0.321831i	0.943849	−	0.330378i	$-0.107176\pi$
$137$	25.4558	+	44.0908i	0.185809	+	0.321831i	−0.758040	+	0.652208i	$0.773843\pi$
$138$		0			0
$139$	−22.0000	+	38.1051i	−0.158273	+	0.274138i	−0.934246	−	0.356629i	$-0.883926\pi$
$139$	−22.0000	+	38.1051i	−0.158273	+	0.274138i	0.775973	+	0.630766i	$0.217259\pi$
$140$	52.6073	−	25.1489i	0.375767	−	0.179635i
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$	142.829	−	82.4621i	0.978278	−	0.564809i
$147$		0			0
$148$	−40.3980	−	23.3238i	−0.272960	−	0.157593i
$149$	−7.14143	−	4.12311i	−0.0479290	−	0.0276718i	0.475844	−	0.879530i	$-0.342143\pi$
$149$	−7.14143	−	4.12311i	−0.0479290	−	0.0276718i	−0.523773	+	0.851858i	$0.675476\pi$
$150$		0			0
$151$	−68.0000	−	117.779i	−0.450331	−	0.779996i	0.548075	−	0.836429i	$-0.315361\pi$
$151$	−68.0000	−	117.779i	−0.450331	−	0.779996i	−0.998406	+	0.0564326i	$0.982027\pi$
$152$	33.9411			0.223297
$153$		0			0
$154$	136.000			0.883117
$155$	−159.518	−	12.4140i	−1.02915	−	0.0800902i
$156$		0			0
$157$	−100.995	−	58.3095i	−0.643281	−	0.371398i	0.142597	−	0.989781i	$-0.454455\pi$
$157$	−100.995	−	58.3095i	−0.643281	−	0.371398i	−0.785877	+	0.618383i	$0.787788\pi$
$158$	50.9117	−	88.1816i	0.322226	−	0.558112i
$159$		0			0
$160$	28.1990	+	2.19450i	0.176244	+	0.0137156i
$161$			140.186i			0.870718i
$162$		0			0
$163$		−	99.1262i		−	0.608136i	−0.952650	−	0.304068i	$-0.901655\pi$
$163$		−	99.1262i		−	0.608136i	0.952650	−	0.304068i	$-0.0983450\pi$
$164$	99.9800	−	57.7235i	0.609634	−	0.351972i
$165$		0			0
$166$	31.0000	−	53.6936i	0.186747	−	0.323455i
$167$	−146.371	+	253.522i	−0.876474	+	1.51810i	−0.0212891	+	0.999773i	$0.506777\pi$
$167$	−146.371	+	253.522i	−0.876474	+	1.51810i	−0.855185	+	0.518324i	$0.826556\pi$
$168$		0			0
$169$	−84.5000	−	146.358i	−0.500000	−	0.866025i
$170$	45.2548	−	65.9697i	0.266205	−	0.388057i
$171$		0			0
$172$		−	81.6333i		−	0.474612i
$173$	82.0244	+	142.070i	0.474129	+	0.821216i	0.999561	−	0.0296195i	$-0.00942955\pi$
$173$	82.0244	+	142.070i	0.474129	+	0.821216i	−0.525432	+	0.850836i	$0.676096\pi$
$174$		0			0
$175$	−22.5522	+	144.019i	−0.128870	+	0.822964i
$176$	57.1314	+	32.9848i	0.324610	+	0.187414i
$177$		0			0
$178$	80.7960	−	46.6476i	0.453910	−	0.262065i
$179$		−	16.4924i		−	0.0921364i	−0.998938	−	0.0460682i	$-0.985331\pi$
$179$		−	16.4924i		−	0.0921364i	0.998938	−	0.0460682i	$-0.0146692\pi$
$180$		0			0
$181$	−82.0000			−0.453039			−0.226519	−	0.974007i	$-0.572735\pi$
$181$	−82.0000			−0.453039			−0.226519	+	0.974007i	$0.572735\pi$
$182$		0			0
$183$		0			0
$184$	−34.0000	+	58.8897i	−0.184783	+	0.320053i
$185$	105.215	−	50.2978i	0.568728	−	0.271880i
$186$		0			0
$187$	161.592	−	93.2952i	0.864129	−	0.498905i
$188$	−70.7107			−0.376121
$189$		0			0
$190$	−48.0000	+	69.9714i	−0.252632	+	0.368271i
$191$	257.091	−	148.432i	1.34603	−	0.777130i	0.358344	−	0.933590i	$-0.383341\pi$
$191$	257.091	−	148.432i	1.34603	−	0.777130i	0.987684	+	0.156460i	$0.0500081\pi$
$192$		0			0
$193$	−100.995	−	58.3095i	−0.523290	−	0.302122i	0.214989	−	0.976616i	$-0.431028\pi$
$193$	−100.995	−	58.3095i	−0.523290	−	0.302122i	−0.738280	+	0.674495i	$0.764362\pi$
$194$	199.960	+	115.447i	1.03072	+	0.595087i
$195$		0			0
$196$	−15.0000	−	25.9808i	−0.0765306	−	0.132555i
$197$	−192.333			−0.976310			−0.488155	−	0.872757i	$-0.662330\pi$
$197$	−192.333			−0.976310			−0.488155	+	0.872757i	$0.662330\pi$
$198$		0			0
$199$	−312.000			−1.56784			−0.783920	−	0.620862i	$-0.786783\pi$
$199$	−312.000			−1.56784			−0.783920	+	0.620862i	$0.786783\pi$
$200$	−44.4035	+	55.0303i	−0.222018	+	0.275151i
$201$		0			0
$202$	161.592	+	93.2952i	0.799961	+	0.461858i
$203$		0			0
$204$		0			0
$205$	−22.3931	+	287.747i	−0.109234	+	1.40365i
$206$		−	140.186i		−	0.680513i
$207$		0			0
$208$		0			0
$209$	−171.394	+	98.9545i	−0.820068	+	0.473467i
$210$		0			0
$211$	6.00000	−	10.3923i	0.0284360	−	0.0492526i	−0.851457	−	0.524424i	$-0.824281\pi$
$211$	6.00000	−	10.3923i	0.0284360	−	0.0492526i	0.879893	+	0.475171i	$0.157614\pi$
$212$	−67.8823	+	117.576i	−0.320199	+	0.554601i
$213$		0			0
$214$	39.0000	+	67.5500i	0.182243	+	0.315654i
$215$	168.291	+	115.447i	0.782751	+	0.536963i
$216$		0			0
$217$		−	186.590i		−	0.859864i
$218$	−56.5685	−	97.9796i	−0.259489	−	0.449448i
$219$		0			0
$220$	−148.796	+	71.1318i	−0.676346	+	0.323327i
$221$		0			0
$222$		0			0
$223$	−35.3483	+	20.4083i	−0.158512	+	0.0915172i	−0.577158	−	0.816633i	$-0.695838\pi$
$223$	−35.3483	+	20.4083i	−0.158512	+	0.0915172i	0.418646	+	0.908150i	$0.362505\pi$
$224$			32.9848i			0.147254i
$225$		0			0
$226$	−216.000			−0.955752
$227$	79.9031	+	138.396i	0.351996	+	0.609675i	0.986599	−	0.163163i	$-0.0521698\pi$
$227$	79.9031	+	138.396i	0.351996	+	0.609675i	−0.634603	+	0.772838i	$0.718836\pi$
$228$		0			0
$229$	−41.0000	+	71.0141i	−0.179039	+	0.310105i	−0.941552	−	0.336868i	$-0.890632\pi$
$229$	−41.0000	+	71.0141i	−0.179039	+	0.310105i	0.762512	+	0.646974i	$0.223966\pi$
$230$	−73.3210	−	153.375i	−0.318787	−	0.666850i
$231$		0			0
$232$		0			0
$233$	192.333			0.825464			0.412732	−	0.910853i	$-0.364575\pi$
$233$	192.333			0.825464			0.412732	+	0.910853i	$0.364575\pi$
$234$		0			0
$235$	100.000	−	145.774i	0.425532	−	0.620314i
$236$	−28.5657	+	16.4924i	−0.121041	+	0.0698831i
$237$		0			0
$238$	80.7960	+	46.6476i	0.339479	+	0.195998i
$239$	−399.920	−	230.894i	−1.67331	−	0.966083i	−0.965768	−	0.259407i	$-0.916473\pi$
$239$	−399.920	−	230.894i	−1.67331	−	0.966083i	−0.707537	−	0.706676i	$-0.750194\pi$
$240$		0			0
$241$	−152.000	−	263.272i	−0.630705	−	1.09241i	−0.987408	−	0.158196i	$-0.949432\pi$
$241$	−152.000	−	263.272i	−0.630705	−	1.09241i	0.356702	−	0.934218i	$-0.383901\pi$
$242$	−213.546			−0.882423
$243$		0			0
$244$	−32.0000			−0.131148
$245$	74.7739	+	5.81905i	0.305200	+	0.0237512i
$246$		0			0
$247$		0			0
$248$	45.2548	−	78.3837i	0.182479	−	0.316063i
$249$		0			0
$250$	−50.6517	−	169.365i	−0.202607	−	0.677459i
$251$		−	346.341i		−	1.37984i	−0.723884	−	0.689922i	$-0.757645\pi$
$251$		−	346.341i		−	1.37984i	0.723884	−	0.689922i	$-0.242355\pi$
$252$		0			0
$253$		−	396.505i		−	1.56721i
$254$	49.9900	−	28.8617i	0.196811	−	0.113629i
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.0312500	+	0.0541266i
$257$	−195.161	+	338.030i	−0.759383	+	1.31529i	0.183782	+	0.982967i	$0.441166\pi$
$257$	−195.161	+	338.030i	−0.759383	+	1.31529i	−0.943166	+	0.332323i	$0.892168\pi$
$258$		0			0
$259$	68.0000	+	117.779i	0.262548	+	0.454747i
$260$		0			0
$261$		0			0
$262$		−	69.9714i		−	0.267066i
$263$	−147.785	−	255.972i	−0.561921	−	0.973276i	−0.997329	−	0.0730428i	$-0.976729\pi$
$263$	−147.785	−	255.972i	−0.561921	−	0.973276i	0.435407	−	0.900234i	$-0.356604\pi$
$264$		0			0
$265$	−146.388	−	306.220i	−0.552408	−	1.15555i
$266$	−85.6971	−	49.4773i	−0.322170	−	0.186005i
$267$		0			0
$268$	−10.0995	+	5.83095i	−0.0376847	+	0.0217573i
$269$		−	74.2159i		−	0.275896i	−0.990440	−	0.137948i	$-0.955949\pi$
$269$		−	74.2159i		−	0.275896i	0.990440	−	0.137948i	$-0.0440506\pi$
$270$		0			0
$271$	40.0000			0.147601			0.0738007	−	0.997273i	$-0.476487\pi$
$271$	40.0000			0.147601			0.0738007	+	0.997273i	$0.476487\pi$
$272$	22.6274	+	39.1918i	0.0831890	+	0.144088i
$273$		0			0
$274$	36.0000	−	62.3538i	0.131387	−	0.227569i
$275$	63.7873	−	407.347i	0.231954	−	1.48126i
$276$		0			0
$277$	383.781	−	221.576i	1.38549	−	0.799914i	0.392688	−	0.919672i	$-0.371545\pi$
$277$	383.781	−	221.576i	1.38549	−	0.799914i	0.992803	+	0.119758i	$0.0382117\pi$
$278$	62.2254			0.223832
$279$		0			0
$280$	−68.0000	−	46.6476i	−0.242857	−	0.166599i
$281$	−449.910	+	259.756i	−1.60110	+	0.924397i	−0.609836	+	0.792528i	$0.708765\pi$
$281$	−449.910	+	259.756i	−1.60110	+	0.924397i	−0.991267	+	0.131870i	$0.957902\pi$
$282$		0			0
$283$	−277.736	−	160.351i	−0.981401	−	0.566612i	−0.0787079	−	0.996898i	$-0.525079\pi$
$283$	−277.736	−	160.351i	−0.981401	−	0.566612i	−0.902693	+	0.430286i	$0.858413\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−336.583			−1.17276
$288$		0			0
$289$	−161.000			−0.557093
$290$		0			0
$291$		0			0
$292$	−201.990	−	116.619i	−0.691747	−	0.399380i
$293$	−42.4264	+	73.4847i	−0.144800	+	0.250801i	−0.929298	−	0.369330i	$-0.879587\pi$
$293$	−42.4264	+	73.4847i	−0.144800	+	0.250801i	0.784498	+	0.620131i	$0.212921\pi$
$294$		0			0
$295$	6.39802	−	82.2135i	0.0216882	−	0.278690i
$296$			65.9697i			0.222871i
$297$		0			0
$298$			11.6619i			0.0391339i
$299$		0			0
$300$		0			0
$301$	−119.000	+	206.114i	−0.395349	+	0.684764i
$302$	−96.1665	+	166.565i	−0.318432	+	0.551541i
$303$		0			0
$304$	−24.0000	−	41.5692i	−0.0789474	−	0.136741i
$305$	45.2548	−	65.9697i	0.148377	−	0.216294i
$306$		0			0
$307$		−	367.350i		−	1.19658i	−0.801280	−	0.598290i	$-0.795847\pi$
$307$		−	367.350i		−	1.19658i	0.801280	−	0.598290i	$-0.204153\pi$
$308$	−96.1665	−	166.565i	−0.312229	−	0.540796i
$309$		0			0
$310$	97.5921	+	204.146i	0.314813	+	0.658537i
$311$	−85.6971	−	49.4773i	−0.275554	−	0.159091i	0.355855	−	0.934541i	$-0.384190\pi$
$311$	−85.6971	−	49.4773i	−0.275554	−	0.159091i	−0.631409	+	0.775450i	$0.717523\pi$
$312$		0			0
$313$	161.592	−	93.2952i	0.516269	−	0.298068i	−0.219138	−	0.975694i	$-0.570325\pi$
$313$	161.592	−	93.2952i	0.516269	−	0.298068i	0.735407	+	0.677626i	$0.236991\pi$
$314$			164.924i			0.525236i
$315$		0			0
$316$	−144.000			−0.455696
$317$	260.215	+	450.706i	0.820868	+	1.42179i	0.905036	+	0.425334i	$0.139843\pi$
$317$	260.215	+	450.706i	0.820868	+	1.42179i	−0.0841679	+	0.996452i	$0.526823\pi$
$318$		0			0
$319$		0			0
$320$	−17.2520	−	36.0883i	−0.0539125	−	0.112776i
$321$		0			0
$322$	171.692	−	99.1262i	0.533204	−	0.307845i
$323$	−135.765			−0.420324
$324$		0			0
$325$		0			0
$326$	−121.404	+	70.0928i	−0.372406	+	0.215009i
$327$		0			0
$328$	−141.393	−	81.6333i	−0.431076	−	0.248882i
$329$	178.536	+	103.078i	0.542662	+	0.313306i
$330$		0			0
$331$	−146.000	−	252.879i	−0.441088	−	0.763986i	0.556683	−	0.830725i	$-0.312074\pi$
$331$	−146.000	−	252.879i	−0.441088	−	0.763986i	−0.997770	+	0.0667389i	$0.978741\pi$
$332$	−87.6812			−0.264100
$333$		0			0
$334$	414.000			1.23952
$335$	2.26204	−	29.0669i	0.00675236	−	0.0867668i
$336$		0			0
$337$	282.786	+	163.267i	0.839128	+	0.484471i	0.856968	−	0.515370i	$-0.172346\pi$
$337$	282.786	+	163.267i	0.839128	+	0.484471i	−0.0178397	+	0.999841i	$0.505679\pi$
$338$	−119.501	+	206.982i	−0.353553	+	0.612372i
$339$		0			0
$340$	−112.796	−	8.77801i	−0.331753	−	0.0258177i
$341$			527.758i			1.54768i
$342$		0			0
$343$			373.181i			1.08799i
$344$	−99.9800	+	57.7235i	−0.290640	+	0.167801i
$345$		0			0
$346$	116.000	−	200.918i	0.335260	−	0.580688i
$347$	−197.283	+	341.704i	−0.568538	+	0.984737i	0.428173	+	0.903697i	$0.359158\pi$
$347$	−197.283	+	341.704i	−0.568538	+	0.984737i	−0.996711	+	0.0810402i	$0.974176\pi$
$348$		0			0
$349$	−127.000	−	219.970i	−0.363897	−	0.630288i	0.624702	−	0.780864i	$-0.285221\pi$
$349$	−127.000	−	219.970i	−0.363897	−	0.630288i	−0.988598	+	0.150576i	$0.951887\pi$
$350$	192.333	−	74.2159i	0.549523	−	0.212045i
$351$		0			0
$352$		−	93.2952i		−	0.265043i
$353$	−172.534	−	298.838i	−0.488765	−	0.846566i	0.511151	−	0.859491i	$-0.329219\pi$
$353$	−172.534	−	298.838i	−0.488765	−	0.846566i	−0.999916	+	0.0129248i	$0.995886\pi$
$354$		0			0
$355$		0			0
$356$	−114.263	−	65.9697i	−0.320963	−	0.185308i
$357$		0			0
$358$	−20.1990	+	11.6619i	−0.0564218	+	0.0325752i
$359$		−	395.818i		−	1.10256i	−0.834321	−	0.551279i	$-0.814140\pi$
$359$		−	395.818i		−	1.10256i	0.834321	−	0.551279i	$-0.185860\pi$
$360$		0			0
$361$	−217.000			−0.601108
$362$	57.9828	+	100.429i	0.160173	+	0.277428i
$363$		0			0
$364$		0			0
$365$	526.073	−	251.489i	1.44130	−	0.689011i
$366$		0			0
$367$	−358.532	+	206.999i	−0.976928	+	0.564029i	−0.901341	−	0.433110i	$-0.857416\pi$
$367$	−358.532	+	206.999i	−0.976928	+	0.564029i	−0.0755864	+	0.997139i	$0.524083\pi$
$368$	96.1665			0.261322
$369$		0			0
$370$	−136.000	−	93.2952i	−0.367568	−	0.252149i
$371$	342.789	−	197.909i	0.923958	−	0.533448i
$372$		0			0
$373$	545.373	+	314.871i	1.46213	+	0.844159i	0.999110	−	0.0421901i	$-0.0134335\pi$
$373$	545.373	+	314.871i	1.46213	+	0.844159i	0.463017	+	0.886349i	$0.346767\pi$
$374$	−228.526	−	131.939i	−0.611031	−	0.352779i
$375$		0			0
$376$	50.0000	+	86.6025i	0.132979	+	0.230326i
$377$		0			0
$378$		0			0
$379$	−572.000			−1.50923			−0.754617	−	0.656165i	$-0.772178\pi$
$379$	−572.000			−1.50923			−0.754617	+	0.656165i	$0.772178\pi$
$380$	119.638	+	9.31049i	0.314838	+	0.0245013i
$381$		0			0
$382$	−363.582	−	209.914i	−0.951786	−	0.549514i
$383$	96.8736	−	167.790i	0.252934	−	0.438094i	−0.711398	−	0.702789i	$-0.751938\pi$
$383$	96.8736	−	167.790i	0.252934	−	0.438094i	0.964332	+	0.264695i	$0.0852712\pi$
$384$		0			0
$385$	479.383	+	37.3065i	1.24515	+	0.0969001i
$386$			164.924i			0.427265i
$387$		0			0
$388$		−	326.533i		−	0.841581i
$389$	335.647	−	193.786i	0.862846	−	0.498164i	−0.00211824	−	0.999998i	$-0.500674\pi$
$389$	335.647	−	193.786i	0.862846	−	0.498164i	0.864964	+	0.501833i	$0.167341\pi$
$390$		0			0
$391$	136.000	−	235.559i	0.347826	−	0.602452i
$392$	−21.2132	+	36.7423i	−0.0541153	+	0.0937305i
$393$		0			0
$394$	136.000	+	235.559i	0.345178	+	0.597865i
$395$	203.647	−	296.864i	0.515561	−	0.751553i
$396$		0			0
$397$			513.124i			1.29250i	0.763124	+	0.646252i	$0.223664\pi$
$397$			513.124i			1.29250i	−0.763124	+	0.646252i	$0.776336\pi$
$398$	220.617	+	382.120i	0.554315	+	0.960102i
$399$		0			0
$400$	98.7960	+	15.4707i	0.246990	+	0.0386768i
$401$	−57.1314	−	32.9848i	−0.142472	−	0.0822565i	0.427069	−	0.904219i	$-0.359546\pi$
$401$	−57.1314	−	32.9848i	−0.142472	−	0.0822565i	−0.569542	+	0.821962i	$0.692879\pi$
$402$		0			0
$403$		0			0
$404$		−	263.879i		−	0.653165i
$405$		0			0
$406$		0			0
$407$	−192.333	−	333.131i	−0.472563	−	0.818503i
$408$		0			0
$409$	−320.000	+	554.256i	−0.782396	+	1.35515i	0.148146	+	0.988965i	$0.452669\pi$
$409$	−320.000	+	554.256i	−0.782396	+	1.35515i	−0.930542	+	0.366184i	$0.880664\pi$
$410$	368.251	−	176.042i	0.898174	−	0.429372i
$411$		0			0
$412$	−171.692	+	99.1262i	−0.416727	+	0.240598i
$413$	96.1665			0.232849
$414$		0			0
$415$	124.000	−	180.760i	0.298795	−	0.435565i
$416$		0			0
$417$		0			0
$418$	242.388	+	139.943i	0.579876	+	0.334791i
$419$	499.900	+	288.617i	1.19308	+	0.688824i	0.959004	−	0.283394i	$-0.0914605\pi$
$419$	499.900	+	288.617i	1.19308	+	0.688824i	0.234075	+	0.972218i	$0.424794\pi$
$420$		0			0
$421$	328.000	+	568.113i	0.779097	+	1.34944i	0.932463	+	0.361267i	$0.117656\pi$
$421$	328.000	+	568.113i	0.779097	+	1.34944i	−0.153365	+	0.988170i	$0.549011\pi$
$422$	−16.9706			−0.0402146
$423$		0			0
$424$	192.000			0.452830
$425$	177.614	−	220.121i	0.417915	−	0.517932i
$426$		0			0
$427$	80.7960	+	46.6476i	0.189218	+	0.109245i
$428$	55.1543	−	95.5301i	0.128865	−	0.223201i
$429$		0			0
$430$	22.3931	−	287.747i	0.0520769	−	0.669180i
$431$		−	362.833i		−	0.841841i	−0.907098	−	0.420920i	$-0.861707\pi$
$431$		−	362.833i		−	0.841841i	0.907098	−	0.420920i	$-0.138293\pi$
$432$		0			0
$433$		−	163.267i		−	0.377059i	−0.982068	−	0.188530i	$-0.939628\pi$
$433$		−	163.267i		−	0.377059i	0.982068	−	0.188530i	$-0.0603722\pi$
$434$	−228.526	+	131.939i	−0.526557	+	0.304008i
$435$		0			0
$436$	−80.0000	+	138.564i	−0.183486	+	0.317807i
$437$	−144.250	+	249.848i	−0.330091	+	0.571734i
$438$		0			0
$439$	−216.000	−	374.123i	−0.492027	−	0.852216i	0.507931	−	0.861398i	$-0.330411\pi$
$439$	−216.000	−	374.123i	−0.492027	−	0.852216i	−0.999958	+	0.00918170i	$0.997077\pi$
$440$	192.333	+	131.939i	0.437121	+	0.299862i
$441$		0			0
$442$		0			0
$443$	−61.5183	−	106.553i	−0.138867	−	0.240526i	0.788201	−	0.615418i	$-0.211013\pi$
$443$	−61.5183	−	106.553i	−0.138867	−	0.240526i	−0.927068	+	0.374893i	$0.877680\pi$
$444$		0			0
$445$	297.592	−	142.264i	0.668746	−	0.319694i
$446$	49.9900	+	28.8617i	0.112085	+	0.0647124i
$447$		0			0
$448$	40.3980	−	23.3238i	0.0901742	−	0.0520621i
$449$			865.852i			1.92840i	0.265174	+	0.964201i	$0.414571\pi$
$449$			865.852i			1.92840i	−0.265174	+	0.964201i	$0.585429\pi$
$450$		0			0
$451$	952.000			2.11086
$452$	152.735	+	264.545i	0.337909	+	0.585276i
$453$		0			0
$454$	113.000	−	195.722i	0.248899	−	0.431105i
$455$		0			0
$456$		0			0
$457$	−403.980	+	233.238i	−0.883983	+	0.510368i	−0.871970	−	0.489560i	$-0.837157\pi$
$457$	−403.980	+	233.238i	−0.883983	+	0.510368i	−0.0120133	+	0.999928i	$0.503824\pi$
$458$	115.966			0.253200
$459$		0			0
$460$	−136.000	+	198.252i	−0.295652	+	0.430983i
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$	−530.224	−	306.125i	−1.14519	−	0.661177i	−0.197481	−	0.980307i	$-0.563276\pi$
$463$	−530.224	−	306.125i	−1.14519	−	0.661177i	−0.947711	+	0.319130i	$0.896609\pi$
$464$		0			0
$465$		0			0
$466$	−136.000	−	235.559i	−0.291845	−	0.505491i
$467$	−767.918			−1.64436			−0.822182	−	0.569225i	$-0.807243\pi$
$467$	−767.918			−1.64436			−0.822182	+	0.569225i	$0.807243\pi$
$468$		0			0
$469$	34.0000			0.0724947
$470$	−249.246	−	19.3968i	−0.530311	−	0.0412699i
$471$		0			0
$472$	40.3980	+	23.3238i	0.0855890	+	0.0494148i
$473$	336.583	−	582.979i	0.711592	−	1.23251i
$474$		0			0
$475$	−188.388	+	233.474i	−0.396607	+	0.491523i
$476$		−	131.939i		−	0.277184i
$477$		0			0
$478$			653.067i			1.36625i
$479$	485.617	−	280.371i	1.01381	−	0.585326i	0.101508	−	0.994835i	$-0.467633\pi$
$479$	485.617	−	280.371i	1.01381	−	0.585326i	0.912306	+	0.409509i	$0.134300\pi$
$480$		0			0
$481$		0			0
$482$	−214.960	+	372.322i	−0.445976	+	0.772453i
$483$		0			0
$484$	151.000	+	261.540i	0.311983	+	0.540371i
$485$	673.166	+	461.788i	1.38797	+	0.952140i
$486$		0			0
$487$			647.236i			1.32903i	0.747277	+	0.664513i	$0.231361\pi$
$487$			647.236i			1.32903i	−0.747277	+	0.664513i	$0.768639\pi$
$488$	22.6274	+	39.1918i	0.0463677	+	0.0803111i
$489$		0			0
$490$	−45.7463	−	95.6937i	−0.0933598	−	0.195293i
$491$	299.940	+	173.170i	0.610876	+	0.352689i	0.773308	−	0.634030i	$-0.218601\pi$
$491$	299.940	+	173.170i	0.610876	+	0.352689i	−0.162432	+	0.986720i	$0.551934\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−128.000			−0.258065
$497$		0			0
$498$		0			0
$499$	330.000	−	571.577i	0.661323	−	1.14544i	−0.318946	−	0.947773i	$-0.603329\pi$
$499$	330.000	−	571.577i	0.661323	−	1.14544i	0.980268	−	0.197671i	$-0.0633379\pi$
$500$	−171.612	+	181.794i	−0.343225	+	0.363589i
$501$		0			0
$502$	−424.179	+	244.900i	−0.844979	+	0.487849i
$503$	−182.434			−0.362691			−0.181345	−	0.983419i	$-0.558045\pi$
$503$	−182.434			−0.362691			−0.181345	+	0.983419i	$0.558045\pi$
$504$		0			0
$505$	544.000	+	373.181i	1.07723	+	0.738972i
$506$	−485.617	+	280.371i	−0.959718	+	0.554093i
$507$		0			0
$508$	−70.6965	−	40.8167i	−0.139166	−	0.0803478i
$509$	342.789	+	197.909i	0.673455	+	0.388819i	0.797384	−	0.603472i	$-0.206216\pi$
$509$	342.789	+	197.909i	0.673455	+	0.388819i	−0.123930	+	0.992291i	$0.539550\pi$
$510$		0			0
$511$	340.000	+	588.897i	0.665362	+	1.15244i
$512$	22.6274			0.0441942
$513$		0			0
$514$	552.000			1.07393
$515$	38.4547	−	494.137i	0.0746693	−	0.959489i
$516$		0			0
$517$	−504.975	−	291.548i	−0.976741	−	0.563922i
$518$	96.1665	−	166.565i	0.185650	−	0.321555i
$519$		0			0
$520$		0			0
$521$		−	131.939i		−	0.253243i	−0.991951	−	0.126621i	$-0.959587\pi$
$521$		−	131.939i		−	0.253243i	0.991951	−	0.126621i	$-0.0404133\pi$
$522$		0			0
$523$			145.774i			0.278726i	0.990241	+	0.139363i	$0.0445055\pi$
$523$			145.774i			0.278726i	−0.990241	+	0.139363i	$0.955494\pi$
$524$	−85.6971	+	49.4773i	−0.163544	+	0.0944223i
$525$		0			0
$526$	−209.000	+	361.999i	−0.397338	+	0.688210i
$527$	−181.019	+	313.535i	−0.343490	+	0.594942i
$528$		0			0
$529$	−24.5000	−	42.4352i	−0.0463138	−	0.0802179i
$530$	−271.529	+	395.818i	−0.512319	+	0.746827i
$531$		0			0
$532$			139.943i			0.263050i
$533$		0			0
$534$		0			0
$535$	118.940	+	248.804i	0.222318	+	0.465053i
$536$	14.2829	+	8.24621i	0.0266471	+	0.0153847i
$537$		0			0
$538$	−90.8955	+	52.4786i	−0.168951	+	0.0975438i
$539$		−	247.386i		−	0.458973i
$540$		0			0
$541$	418.000			0.772643			0.386322	−	0.922364i	$-0.373745\pi$
$541$	418.000			0.772643			0.386322	+	0.922364i	$0.373745\pi$
$542$	−28.2843	−	48.9898i	−0.0521850	−	0.0903871i
$543$		0			0
$544$	32.0000	−	55.4256i	0.0588235	−	0.101885i
$545$	−172.520	−	360.883i	−0.316551	−	0.662171i
$546$		0			0
$547$	247.438	−	142.858i	0.452354	−	0.261167i	−0.256470	−	0.966552i	$-0.582559\pi$
$547$	247.438	−	142.858i	0.452354	−	0.261167i	0.708824	+	0.705385i	$0.249226\pi$
$548$	−101.823			−0.185809
$549$		0			0
$550$	−544.000	+	209.914i	−0.989091	+	0.381662i
$551$		0			0
$552$		0			0
$553$	363.582	+	209.914i	0.657472	+	0.379592i
$554$	−542.749	−	313.356i	−0.979691	−	0.565625i
$555$		0			0
$556$	−44.0000	−	76.2102i	−0.0791367	−	0.137069i
$557$	424.264			0.761695			0.380847	−	0.924638i	$-0.375632\pi$
$557$	424.264			0.761695			0.380847	+	0.924638i	$0.375632\pi$
$558$		0			0
$559$		0			0
$560$	−9.04817	+	116.267i	−0.0161574	+	0.207621i
$561$		0			0
$562$	636.269	+	367.350i	1.13215	+	0.653648i
$563$	406.586	−	704.228i	0.722178	−	1.25085i	−0.237947	−	0.971278i	$-0.576474\pi$
$563$	406.586	−	704.228i	0.722178	−	1.25085i	0.960125	−	0.279571i	$-0.0901923\pi$
$564$		0			0
$565$	−761.373	−	59.2516i	−1.34756	−	0.104870i
$566$			453.542i			0.801310i
$567$		0			0
$568$		0			0
$569$	−392.779	+	226.771i	−0.690296	+	0.398543i	−0.803723	−	0.595004i	$-0.797151\pi$
$569$	−392.779	+	226.771i	−0.690296	+	0.398543i	0.113427	+	0.993546i	$0.463817\pi$
$570$		0			0
$571$	−110.000	+	190.526i	−0.192644	+	0.333670i	−0.946126	−	0.323799i	$-0.895040\pi$
$571$	−110.000	+	190.526i	−0.192644	+	0.333670i	0.753481	+	0.657469i	$0.228373\pi$
$572$		0			0
$573$		0			0
$574$	238.000	+	412.228i	0.414634	+	0.718167i
$575$	−216.375	−	560.742i	−0.376304	−	0.975204i
$576$		0			0
$577$		−	46.6476i		−	0.0808451i	−0.999183	−	0.0404225i	$-0.987130\pi$
$577$		−	46.6476i		−	0.0808451i	0.999183	−	0.0404225i	$-0.0128704\pi$
$578$	113.844	+	197.184i	0.196962	+	0.341149i
$579$		0			0
$580$		0			0
$581$	221.384	+	127.816i	0.381040	+	0.219994i
$582$		0			0
$583$	−969.552	+	559.771i	−1.66304	+	0.960157i
$584$			329.848i			0.564809i
$585$		0			0
$586$	120.000			0.204778
$587$	−27.5772	−	47.7650i	−0.0469798	−	0.0813715i	0.841579	−	0.540134i	$-0.181626\pi$
$587$	−27.5772	−	47.7650i	−0.0469798	−	0.0813715i	−0.888559	+	0.458762i	$0.848293\pi$
$588$		0			0
$589$	192.000	−	332.554i	0.325976	−	0.564607i
$590$	−105.215	+	50.2978i	−0.178330	+	0.0852505i
$591$		0			0
$592$	80.7960	−	46.6476i	0.136480	−	0.0787966i
$593$	390.323			0.658217			0.329109	−	0.944292i	$-0.393252\pi$
$593$	390.323			0.658217			0.329109	+	0.944292i	$0.393252\pi$
$594$		0			0
$595$	272.000	+	186.590i	0.457143	+	0.313597i
$596$	14.2829	−	8.24621i	0.0239645	−	0.0138359i
$597$		0			0
$598$		0			0
$599$	−85.6971	−	49.4773i	−0.143067	−	0.0825998i	0.426758	−	0.904366i	$-0.359656\pi$
$599$	−85.6971	−	49.4773i	−0.143067	−	0.0825998i	−0.569825	+	0.821766i	$0.692989\pi$
$600$		0			0
$601$	440.000	+	762.102i	0.732113	+	1.26806i	0.955978	+	0.293437i	$0.0947991\pi$
$601$	440.000	+	762.102i	0.732113	+	1.26806i	−0.223865	+	0.974620i	$0.571868\pi$
$602$	336.583			0.559108
$603$		0			0
$604$	272.000			0.450331
$605$	−752.724	−	58.5785i	−1.24417	−	0.0968239i
$606$		0			0
$607$	368.632	+	212.830i	0.607301	+	0.350626i	0.771909	−	0.635734i	$-0.219302\pi$
$607$	368.632	+	212.830i	0.607301	+	0.350626i	−0.164607	+	0.986359i	$0.552636\pi$
$608$	−33.9411	+	58.7878i	−0.0558242	+	0.0966904i
$609$		0			0
$610$	−112.796	−	8.77801i	−0.184912	−	0.0143902i
$611$		0			0
$612$		0			0
$613$			606.419i			0.989264i	0.869102	+	0.494632i	$0.164697\pi$
$613$			606.419i			0.989264i	−0.869102	+	0.494632i	$0.835303\pi$
$614$	−449.910	+	259.756i	−0.732752	+	0.423055i
$615$		0			0
$616$	−136.000	+	235.559i	−0.220779	+	0.382401i
$617$	56.5685	−	97.9796i	0.0916832	−	0.158800i	−0.816536	−	0.577294i	$-0.804109\pi$
$617$	56.5685	−	97.9796i	0.0916832	−	0.158800i	0.908220	+	0.418494i	$0.137442\pi$
$618$		0			0
$619$	−26.0000	−	45.0333i	−0.0420032	−	0.0727517i	0.844259	−	0.535935i	$-0.180041\pi$
$619$	−26.0000	−	45.0333i	−0.0420032	−	0.0727517i	−0.886263	+	0.463183i	$0.846707\pi$
$620$	181.019	−	263.879i	0.291967	−	0.425611i
$621$		0			0
$622$			139.943i			0.224988i
$623$	192.333	+	333.131i	0.308721	+	0.534720i
$624$		0			0
$625$	−132.082	−	610.884i	−0.211331	−	0.977414i
$626$	−228.526	−	131.939i	−0.365057	−	0.210766i
$627$		0			0
$628$	201.990	−	116.619i	0.321640	−	0.185699i
$629$		−	263.879i		−	0.419521i
$630$		0			0
$631$	−544.000			−0.862124			−0.431062	−	0.902322i	$-0.641861\pi$
$631$	−544.000			−0.862124			−0.431062	+	0.902322i	$0.641861\pi$
$632$	101.823	+	176.363i	0.161113	+	0.279056i
$633$		0			0
$634$	368.000	−	637.395i	0.580442	−	1.00535i
$635$	184.126	−	88.0212i	0.289962	−	0.138616i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−32.0000	+	46.6476i	−0.0500000	+	0.0728869i
$641$	735.567	−	424.680i	1.14753	−	0.662527i	0.199246	−	0.979949i	$-0.436151\pi$
$641$	735.567	−	424.680i	1.14753	−	0.662527i	0.948284	+	0.317422i	$0.102817\pi$
$642$		0			0
$643$	318.134	+	183.675i	0.494766	+	0.285653i	0.726549	−	0.687114i	$-0.241123\pi$
$643$	318.134	+	183.675i	0.494766	+	0.285653i	−0.231784	+	0.972767i	$0.574456\pi$
$644$	−242.809	−	140.186i	−0.377032	−	0.217679i
$645$		0			0
$646$	96.0000	+	166.277i	0.148607	+	0.257395i
$647$	−971.565			−1.50165			−0.750823	−	0.660504i	$-0.770343\pi$
$647$	−971.565			−1.50165			−0.750823	+	0.660504i	$0.770343\pi$
$648$		0			0
$649$	−272.000			−0.419106
$650$		0			0
$651$		0			0
$652$	171.692	+	99.1262i	0.263331	+	0.152034i
$653$	−175.362	+	303.737i	−0.268549	+	0.465140i	−0.968487	−	0.249063i	$-0.919877\pi$
$653$	−175.362	+	303.737i	−0.268549	+	0.465140i	0.699938	+	0.714203i	$0.253211\pi$
$654$		0			0
$655$	19.1941	−	246.641i	0.0293039	−	0.376551i
$656$			230.894i			0.351972i
$657$		0			0
$658$		−	291.548i		−	0.443081i
$659$	499.900	−	288.617i	0.758574	−	0.437963i	−0.0702098	−	0.997532i	$-0.522367\pi$
$659$	499.900	−	288.617i	0.758574	−	0.437963i	0.828783	+	0.559570i	$0.189034\pi$
$660$		0			0
$661$	−40.0000	+	69.2820i	−0.0605144	+	0.104814i	−0.894695	−	0.446677i	$-0.852607\pi$
$661$	−40.0000	+	69.2820i	−0.0605144	+	0.104814i	0.834181	+	0.551491i	$0.185941\pi$
$662$	−206.475	+	357.626i	−0.311896	+	0.540220i
$663$		0			0
$664$	62.0000	+	107.387i	0.0933735	+	0.161728i
$665$	−288.500	−	197.909i	−0.433834	−	0.297608i
$666$		0			0
$667$		0			0
$668$	−292.742	−	507.044i	−0.438237	−	0.759048i
$669$		0			0
$670$	−37.1990	+	17.7830i	−0.0555209	+	0.0265417i
$671$	−228.526	−	131.939i	−0.340575	−	0.196631i
$672$		0			0
$673$	424.179	−	244.900i	0.630281	−	0.363893i	−0.150580	−	0.988598i	$-0.548114\pi$
$673$	424.179	−	244.900i	0.630281	−	0.363893i	0.780861	+	0.624705i	$0.214781\pi$
$674$		−	461.788i		−	0.685145i
$675$		0			0
$676$	338.000			0.500000
$677$	−96.1665	−	166.565i	−0.142048	−	0.246034i	0.786220	−	0.617947i	$-0.212035\pi$
$677$	−96.1665	−	166.565i	−0.142048	−	0.246034i	−0.928268	+	0.371913i	$0.878702\pi$
$678$		0			0
$679$	−476.000	+	824.456i	−0.701031	+	1.21422i
$680$	69.0080	+	144.353i	0.101482	+	0.212284i
$681$		0			0
$682$	646.368	−	373.181i	0.947754	−	0.547186i
$683$	236.174			0.345789			0.172894	−	0.984940i	$-0.444688\pi$
$683$	236.174			0.345789			0.172894	+	0.984940i	$0.444688\pi$
$684$		0			0
$685$	144.000	−	209.914i	0.210219	−	0.306444i
$686$	457.051	−	263.879i	0.666256	−	0.384663i
$687$		0			0
$688$	141.393	+	81.6333i	0.205513	+	0.118653i
$689$		0			0
$690$		0			0
$691$	274.000	+	474.582i	0.396527	+	0.686805i	0.993295	−	0.115609i	$-0.0368821\pi$
$691$	274.000	+	474.582i	0.396527	+	0.686805i	−0.596768	+	0.802414i	$0.703549\pi$
$692$	−328.098			−0.474129
$693$		0			0
$694$	558.000			0.804035
$695$	219.337	+	17.0692i	0.315593	+	0.0245600i
$696$		0			0
$697$	565.572	+	326.533i	0.811438	+	0.468484i
$698$	−179.605	+	311.085i	−0.257314	+	0.445681i
$699$		0			0
$700$	−226.896	−	183.080i	−0.324136	−	0.261543i
$701$			57.7235i			0.0823445i	0.999152	+	0.0411722i	$0.0131092\pi$
$701$			57.7235i			0.0823445i	−0.999152	+	0.0411722i	$0.986891\pi$
$702$		0			0
$703$			279.886i			0.398130i
$704$	−114.263	+	65.9697i	−0.162305	+	0.0937069i
$705$		0			0
$706$	−244.000	+	422.620i	−0.345609	+	0.598612i
$707$	−384.666	+	666.261i	−0.544082	+	0.942378i
$708$		0			0
$709$	−615.000	−	1065.21i	−0.867419	−	1.50241i	−0.864625	−	0.502418i	$-0.832444\pi$
$709$	−615.000	−	1065.21i	−0.867419	−	1.50241i	−0.00279375	−	0.999996i	$-0.500889\pi$
$710$		0			0
$711$		0			0
$712$			186.590i			0.262065i
$713$	384.666	+	666.261i	0.539504	+	0.934448i
$714$		0			0
$715$		0			0
$716$	28.5657	+	16.4924i	0.0398962	+	0.0230341i
$717$		0			0
$718$	−484.776	+	279.886i	−0.675176	+	0.389813i
$719$		−	626.712i		−	0.871644i	−0.900033	−	0.435822i	$-0.856458\pi$
$719$		−	626.712i		−	0.871644i	0.900033	−	0.435822i	$-0.143542\pi$
$720$		0			0
$721$	578.000			0.801664
$722$	153.442	+	265.770i	0.212524	+	0.368102i
$723$		0			0
$724$	82.0000	−	142.028i	0.113260	−	0.196172i
$725$		0			0
$726$		0			0
$727$	318.134	−	183.675i	0.437599	−	0.252648i	−0.264980	−	0.964254i	$-0.585365\pi$
$727$	318.134	−	183.675i	0.437599	−	0.252648i	0.702579	+	0.711606i	$0.252032\pi$
$728$		0			0
$729$		0			0
$730$	−680.000	−	466.476i	−0.931507	−	0.639008i
$731$	399.920	−	230.894i	0.547086	−	0.315860i
$732$		0			0
$733$	−868.557	−	501.462i	−1.18494	−	0.684123i	−0.227784	−	0.973712i	$-0.573148\pi$
$733$	−868.557	−	501.462i	−1.18494	−	0.684123i	−0.957151	+	0.289589i	$0.906481\pi$
$734$	507.041	+	292.740i	0.690792	+	0.398829i
$735$		0			0
$736$	−68.0000	−	117.779i	−0.0923913	−	0.160026i
$737$	−96.1665			−0.130484
$738$		0			0
$739$	−340.000			−0.460081			−0.230041	−	0.973181i	$-0.573886\pi$
$739$	−340.000			−0.460081			−0.230041	+	0.973181i	$0.573886\pi$
$740$	−18.0963	+	232.535i	−0.0244545	+	0.314236i
$741$		0			0
$742$	−484.776	−	279.886i	−0.653337	−	0.377204i
$743$	−621.547	+	1076.55i	−0.836537	+	1.44892i	0.0562362	+	0.998417i	$0.482090\pi$
$743$	−621.547	+	1076.55i	−0.836537	+	1.44892i	−0.892773	+	0.450507i	$0.851243\pi$
$744$		0			0
$745$	−3.19901	+	41.1068i	−0.00429397	+	0.0551769i
$746$		−	890.591i		−	1.19382i
$747$		0			0
$748$			373.181i			0.498905i
$749$	−278.516	+	160.801i	−0.371850	+	0.214688i
$750$		0			0
$751$	260.000	−	450.333i	0.346205	−	0.599645i	−0.639367	−	0.768902i	$-0.720803\pi$
$751$	260.000	−	450.333i	0.346205	−	0.599645i	0.985572	+	0.169257i	$0.0541368\pi$
$752$	70.7107	−	122.474i	0.0940302	−	0.162865i
$753$		0			0
$754$		0			0
$755$	−384.666	+	560.742i	−0.509492	+	0.742705i
$756$		0			0
$757$		−	816.333i		−	1.07838i	−0.842184	−	0.539190i	$-0.818731\pi$
$757$		−	816.333i		−	1.07838i	0.842184	−	0.539190i	$-0.181269\pi$
$758$	404.465	+	700.554i	0.533595	+	0.924214i
$759$		0			0
$760$	−73.1941	−	153.110i	−0.0963080	−	0.201460i
$761$	342.789	+	197.909i	0.450445	+	0.260064i	0.708018	−	0.706194i	$-0.249590\pi$
$761$	342.789	+	197.909i	0.450445	+	0.260064i	−0.257573	+	0.966259i	$0.582923\pi$
$762$		0			0
$763$	403.980	−	233.238i	0.529463	−	0.305686i
$764$			593.727i			0.777130i
$765$		0			0
$766$	−274.000			−0.357702
$767$		0			0
$768$		0			0
$769$	153.000	−	265.004i	0.198960	−	0.344608i	−0.749232	−	0.662308i	$-0.769577\pi$
$769$	153.000	−	265.004i	0.198960	−	0.344608i	0.948191	+	0.317700i	$0.102910\pi$
$770$	−293.284	−	613.502i	−0.380888	−	0.796756i
$771$		0			0
$772$	201.990	−	116.619i	0.261645	−	0.151061i
$773$	305.470			0.395175			0.197587	−	0.980285i	$-0.436689\pi$
$773$	305.470			0.395175			0.197587	+	0.980285i	$0.436689\pi$
$774$		0			0
$775$	288.000	+	746.362i	0.371613	+	0.963048i
$776$	−399.920	+	230.894i	−0.515361	+	0.297544i
$777$		0			0
$778$	−474.677	−	274.055i	−0.610124	−	0.352255i
$779$	−599.880	−	346.341i	−0.770064	−	0.444597i
$780$		0			0
$781$		0			0
$782$	−384.666			−0.491900
$783$		0			0
$784$	60.0000			0.0765306
$785$	−45.2408	+	581.337i	−0.0576316	+	0.740557i
$786$		0			0

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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	810.j (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 1.22474i) q^{2} +(-1.00000 + 1.73205i) q^{4} +(-2.15650 - 4.51104i) q^{5} +(5.04975 - 2.91548i) q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+(-0.707107 - 1.22474i) q^{2} +(-1.00000 + 1.73205i) q^{4} +(-2.15650 - 4.51104i) q^{5} +(5.04975 - 2.91548i) q^{7} +2.82843 q^{8} +(-4.00000 + 5.83095i) q^{10} +(-14.2829 + 8.24621i) q^{11} +(-7.14143 - 4.12311i) q^{14} +(-2.00000 - 3.46410i) q^{16} -11.3137 q^{17} +12.0000 q^{19} +(9.96986 + 0.775874i) q^{20} +(20.1990 + 11.6619i) q^{22} +(-12.0208 + 20.8207i) q^{23} +(-15.6990 + 19.4561i) q^{25} +11.6619i q^{28} +(16.0000 - 27.7128i) q^{31} +(-2.82843 + 4.89898i) q^{32} +(8.00000 + 13.8564i) q^{34} +(-24.0416 - 16.4924i) q^{35} +23.3238i q^{37} +(-8.48528 - 14.6969i) q^{38} +(-6.09950 - 12.7592i) q^{40} +(-49.9900 - 28.8617i) q^{41} +(-35.3483 + 20.4083i) q^{43} -32.9848i q^{44} +34.0000 q^{46} +(17.6777 + 30.6186i) q^{47} +(-7.50000 + 12.9904i) q^{49} +(34.9297 + 5.46972i) q^{50} +67.8823 q^{53} +(68.0000 + 46.6476i) q^{55} +(14.2829 - 8.24621i) q^{56} +(14.2829 + 8.24621i) q^{59} +(8.00000 + 13.8564i) q^{61} -45.2548 q^{62} +8.00000 q^{64} +(5.04975 + 2.91548i) q^{67} +(11.3137 - 19.5959i) q^{68} +(-3.19901 + 41.1068i) q^{70} +116.619i q^{73} +(28.5657 - 16.4924i) q^{74} +(-12.0000 + 20.7846i) q^{76} +(-48.0833 + 83.2827i) q^{77} +(36.0000 + 62.3538i) q^{79} +(-11.3137 + 16.4924i) q^{80} +81.6333i q^{82} +(21.9203 + 37.9671i) q^{83} +(24.3980 + 51.0366i) q^{85} +(49.9900 + 28.8617i) q^{86} +(-40.3980 + 23.3238i) q^{88} +65.9697i q^{89} +(-24.0416 - 41.6413i) q^{92} +(25.0000 - 43.3013i) q^{94} +(-25.8780 - 54.1325i) q^{95} +(-141.393 + 81.6333i) q^{97} +21.2132 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} - 32 q^{10} - 16 q^{16} + 96 q^{19} + 36 q^{25} + 128 q^{31} + 64 q^{34} + 32 q^{40} + 272 q^{46} - 60 q^{49} + 544 q^{55} + 64 q^{61} + 64 q^{64} + 136 q^{70} - 96 q^{76} + 288 q^{79} - 128 q^{85} + 200 q^{94}+O(q^{100}) \) 8 * q - 8 * q^4 - 32 * q^10 - 16 * q^16 + 96 * q^19 + 36 * q^25 + 128 * q^31 + 64 * q^34 + 32 * q^40 + 272 * q^46 - 60 * q^49 + 544 * q^55 + 64 * q^61 + 64 * q^64 + 136 * q^70 - 96 * q^76 + 288 * q^79 - 128 * q^85 + 200 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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