LMFDB - Embedded newform 1728.3.q.l.1601.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(449,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

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

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

chi := DirichletCharacter("1728.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.q (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:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1601.6
Character	$\chi$	$=$	1728.1601
Dual form			1728.3.q.l.449.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.439631 - 0.253821i) q^{5} +(-6.44757 + 11.1675i) q^{7} +O(q^{10})$	$q+(0.439631 - 0.253821i) q^{5} +(-6.44757 + 11.1675i) q^{7} +(-13.3339 - 7.69831i) q^{11} +(8.16955 + 14.1501i) q^{13} +15.2050i q^{17} +5.84378 q^{19} +(21.4189 - 12.3662i) q^{23} +(-12.3711 + 21.4275i) q^{25} +(34.5464 + 19.9454i) q^{29} +(-18.1841 - 31.4958i) q^{31} +6.54611i q^{35} -6.15309 q^{37} +(-33.5076 + 19.3456i) q^{41} +(-2.89558 + 5.01529i) q^{43} +(-45.8699 - 26.4830i) q^{47} +(-58.6424 - 101.572i) q^{49} -17.2825i q^{53} -7.81597 q^{55} +(31.3548 - 18.1027i) q^{59} +(2.36999 - 4.10494i) q^{61} +(7.18317 + 4.14720i) q^{65} +(-21.0649 - 36.4854i) q^{67} +16.6118i q^{71} -134.364 q^{73} +(171.942 - 99.2708i) q^{77} +(-27.4107 + 47.4767i) q^{79} +(-92.1440 - 53.1994i) q^{83} +(3.85934 + 6.68458i) q^{85} +6.93873i q^{89} -210.695 q^{91} +(2.56910 - 1.48327i) q^{95} +(56.0607 - 97.0999i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 60 q^{25} + 72 q^{29} + 36 q^{41} - 132 q^{49} - 96 q^{61} - 576 q^{65} + 24 q^{73} + 432 q^{77} + 96 q^{85} + 252 q^{97}+O(q^{100}) $ 24 * q + 60 * q^25 + 72 * q^29 + 36 * q^41 - 132 * q^49 - 96 * q^61 - 576 * q^65 + 24 * q^73 + 432 * q^77 + 96 * q^85 + 252 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.439631	−	0.253821i	0.0879261	−	0.0507642i	−0.455392	−	0.890291i	$-0.650501\pi$
$5$	0.439631	−	0.253821i	0.0879261	−	0.0507642i	0.543318	+	0.839527i	$0.317168\pi$
$6$		0			0
$7$	−6.44757	+	11.1675i	−0.921082	+	1.59536i	−0.123337	+	0.992365i	$0.539360\pi$
$7$	−6.44757	+	11.1675i	−0.921082	+	1.59536i	−0.797744	+	0.602996i	$0.793974\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−13.3339	−	7.69831i	−1.21217	−	0.699846i	−0.248938	−	0.968519i	$-0.580082\pi$
$11$	−13.3339	−	7.69831i	−1.21217	−	0.699846i	−0.963231	+	0.268673i	$0.913415\pi$
$12$		0			0
$13$	8.16955	+	14.1501i	0.628427	+	1.08847i	0.987868	+	0.155299i	$0.0496341\pi$
$13$	8.16955	+	14.1501i	0.628427	+	1.08847i	−0.359441	+	0.933168i	$0.617033\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			15.2050i			0.894411i	0.894431	+	0.447206i	$0.147581\pi$
$17$			15.2050i			0.894411i	−0.894431	+	0.447206i	$0.852419\pi$
$18$		0			0
$19$	5.84378			0.307567			0.153784	−	0.988105i	$-0.450854\pi$
$19$	5.84378			0.307567			0.153784	+	0.988105i	$0.450854\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	21.4189	−	12.3662i	0.931255	−	0.537661i	0.0440470	−	0.999029i	$-0.485975\pi$
$23$	21.4189	−	12.3662i	0.931255	−	0.537661i	0.887208	+	0.461369i	$0.152642\pi$
$24$		0			0
$25$	−12.3711	+	21.4275i	−0.494846	+	0.857098i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	34.5464	+	19.9454i	1.19126	+	0.687772i	0.958591	−	0.284786i	$-0.0919224\pi$
$29$	34.5464	+	19.9454i	1.19126	+	0.687772i	0.232664	+	0.972557i	$0.425256\pi$
$30$		0			0
$31$	−18.1841	−	31.4958i	−0.586584	−	1.01599i	−0.994676	−	0.103053i	$-0.967139\pi$
$31$	−18.1841	−	31.4958i	−0.586584	−	1.01599i	0.408092	−	0.912941i	$-0.366194\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			6.54611i			0.187032i
$36$		0			0
$37$	−6.15309			−0.166300			−0.0831498	−	0.996537i	$-0.526498\pi$
$37$	−6.15309			−0.166300			−0.0831498	+	0.996537i	$0.526498\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−33.5076	+	19.3456i	−0.817258	+	0.471844i	−0.849470	−	0.527637i	$-0.823078\pi$
$41$	−33.5076	+	19.3456i	−0.817258	+	0.471844i	0.0322119	+	0.999481i	$0.489745\pi$
$42$		0			0
$43$	−2.89558	+	5.01529i	−0.0673390	+	0.116635i	−0.897729	−	0.440548i	$-0.854784\pi$
$43$	−2.89558	+	5.01529i	−0.0673390	+	0.116635i	0.830390	+	0.557182i	$0.188118\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−45.8699	−	26.4830i	−0.975955	−	0.563468i	−0.0749083	−	0.997190i	$-0.523866\pi$
$47$	−45.8699	−	26.4830i	−0.975955	−	0.563468i	−0.901046	+	0.433723i	$0.857200\pi$
$48$		0			0
$49$	−58.6424	−	101.572i	−1.19678	−	2.07289i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	17.2825i		−	0.326085i	−0.986619	−	0.163042i	$-0.947869\pi$
$53$		−	17.2825i		−	0.326085i	0.986619	−	0.163042i	$-0.0521308\pi$
$54$		0			0
$55$	−7.81597			−0.142108
$56$		0			0
$57$		0			0
$58$		0			0
$59$	31.3548	−	18.1027i	0.531437	−	0.306825i	−0.210164	−	0.977666i	$-0.567400\pi$
$59$	31.3548	−	18.1027i	0.531437	−	0.306825i	0.741602	+	0.670841i	$0.234067\pi$
$60$		0			0
$61$	2.36999	−	4.10494i	0.0388523	−	0.0672941i	−0.845945	−	0.533270i	$-0.820963\pi$
$61$	2.36999	−	4.10494i	0.0388523	−	0.0672941i	0.884798	+	0.465975i	$0.154296\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	7.18317	+	4.14720i	0.110510	+	0.0638031i
$66$		0			0
$67$	−21.0649	−	36.4854i	−0.314401	−	0.544559i	0.664909	−	0.746925i	$-0.268470\pi$
$67$	−21.0649	−	36.4854i	−0.314401	−	0.544559i	−0.979310	+	0.202366i	$0.935137\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			16.6118i			0.233969i	0.993134	+	0.116985i	$0.0373228\pi$
$71$			16.6118i			0.233969i	−0.993134	+	0.116985i	$0.962677\pi$
$72$		0			0
$73$	−134.364			−1.84060			−0.920300	−	0.391213i	$-0.872056\pi$
$73$	−134.364			−1.84060			−0.920300	+	0.391213i	$0.872056\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	171.942	−	99.2708i	2.23301	−	1.28923i
$78$		0			0
$79$	−27.4107	+	47.4767i	−0.346971	+	0.600971i	−0.985710	−	0.168453i	$-0.946123\pi$
$79$	−27.4107	+	47.4767i	−0.346971	+	0.600971i	0.638739	+	0.769423i	$0.279456\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−92.1440	−	53.1994i	−1.11017	−	0.640956i	−0.171296	−	0.985220i	$-0.554795\pi$
$83$	−92.1440	−	53.1994i	−1.11017	−	0.640956i	−0.938873	+	0.344263i	$0.888129\pi$
$84$		0			0
$85$	3.85934	+	6.68458i	0.0454041	+	0.0786421i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			6.93873i			0.0779633i	0.999240	+	0.0389816i	$0.0124114\pi$
$89$			6.93873i			0.0779633i	−0.999240	+	0.0389816i	$0.987589\pi$
$90$		0			0
$91$	−210.695			−2.31533
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.56910	−	1.48327i	0.0270432	−	0.0156134i
$96$		0			0
$97$	56.0607	−	97.0999i	0.577945	−	1.00103i	−0.417770	−	0.908553i	$-0.637188\pi$
$97$	56.0607	−	97.0999i	0.577945	−	1.00103i	0.995715	−	0.0924770i	$-0.0294785\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	17.9551	+	10.3664i	0.177774	+	0.102638i	0.586246	−	0.810133i	$-0.300605\pi$
$101$	17.9551	+	10.3664i	0.177774	+	0.102638i	−0.408473	+	0.912771i	$0.633938\pi$
$102$		0			0
$103$	−31.0601	−	53.7977i	−0.301554	−	0.522307i	0.674934	−	0.737878i	$-0.264172\pi$
$103$	−31.0601	−	53.7977i	−0.301554	−	0.522307i	−0.976488	+	0.215571i	$0.930839\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	74.3234i		−	0.694611i	−0.937752	−	0.347305i	$-0.887097\pi$
$107$		−	74.3234i		−	0.694611i	0.937752	−	0.347305i	$-0.112903\pi$
$108$		0			0
$109$	131.093			1.20269			0.601346	−	0.798989i	$-0.294631\pi$
$109$	131.093			1.20269			0.601346	+	0.798989i	$0.294631\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	26.5455	−	15.3261i	0.234916	−	0.135629i	−0.377922	−	0.925837i	$-0.623361\pi$
$113$	26.5455	−	15.3261i	0.234916	−	0.135629i	0.612838	+	0.790209i	$0.290028\pi$
$114$		0			0
$115$	6.27760	−	10.8731i	0.0545878	−	0.0945488i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−169.802	−	98.0353i	−1.42691	−	0.823826i
$120$		0			0
$121$	58.0280	+	100.507i	0.479570	+	0.830640i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			25.2513i			0.202010i
$126$		0			0
$127$	−13.1521			−0.103560			−0.0517800	−	0.998659i	$-0.516489\pi$
$127$	−13.1521			−0.103560			−0.0517800	+	0.998659i	$0.516489\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	119.041	−	68.7285i	0.908711	−	0.524645i	0.0286951	−	0.999588i	$-0.490865\pi$
$131$	119.041	−	68.7285i	0.908711	−	0.524645i	0.880016	+	0.474943i	$0.157531\pi$
$132$		0			0
$133$	−37.6782	+	65.2605i	−0.283295	+	0.490681i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−216.146	−	124.792i	−1.57771	−	0.910892i	−0.995178	−	0.0980849i	$-0.968728\pi$
$137$	−216.146	−	124.792i	−1.57771	−	0.910892i	−0.582533	−	0.812807i	$-0.697938\pi$
$138$		0			0
$139$	96.4347	+	167.030i	0.693775	+	1.20165i	0.970592	+	0.240731i	$0.0773870\pi$
$139$	96.4347	+	167.030i	0.693775	+	1.20165i	−0.276817	+	0.960923i	$0.589280\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	251.567i		−	1.75921i
$144$		0			0
$145$	20.2502			0.139657
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−81.7318	+	47.1879i	−0.548535	+	0.316697i	−0.748531	−	0.663100i	$-0.769241\pi$
$149$	−81.7318	+	47.1879i	−0.548535	+	0.316697i	0.199996	+	0.979797i	$0.435907\pi$
$150$		0			0
$151$	−80.1510	+	138.826i	−0.530801	+	0.919375i	0.468553	+	0.883436i	$0.344776\pi$
$151$	−80.1510	+	138.826i	−0.530801	+	0.919375i	−0.999354	+	0.0359393i	$0.988558\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−15.9886	−	9.23101i	−0.103152	−	0.0595549i
$156$		0			0
$157$	60.6643	+	105.074i	0.386397	+	0.669259i	0.991962	−	0.126537i	$-0.0403861\pi$
$157$	60.6643	+	105.074i	0.386397	+	0.669259i	−0.605565	+	0.795796i	$0.707053\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			318.928i			1.98092i
$162$		0			0
$163$	−251.871			−1.54522			−0.772609	−	0.634882i	$-0.781049\pi$
$163$	−251.871			−1.54522			−0.772609	+	0.634882i	$0.781049\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	37.8670	−	21.8625i	0.226749	−	0.130913i	−0.382323	−	0.924029i	$-0.624876\pi$
$167$	37.8670	−	21.8625i	0.226749	−	0.130913i	0.609071	+	0.793116i	$0.291542\pi$
$168$		0			0
$169$	−48.9830	+	84.8410i	−0.289840	+	0.502018i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−278.211	−	160.625i	−1.60816	−	0.928470i	−0.989782	−	0.142587i	$-0.954458\pi$
$173$	−278.211	−	160.625i	−1.60816	−	0.928470i	−0.618375	−	0.785883i	$-0.712209\pi$
$174$		0			0
$175$	−159.528	−	276.310i	−0.911587	−	1.57892i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	201.116i		−	1.12355i	−0.827290	−	0.561776i	$-0.810118\pi$
$179$		−	201.116i		−	1.12355i	0.827290	−	0.561776i	$-0.189882\pi$
$180$		0			0
$181$	77.5050			0.428204			0.214102	−	0.976811i	$-0.431317\pi$
$181$	77.5050			0.428204			0.214102	+	0.976811i	$0.431317\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.70508	+	1.56178i	−0.0146221	+	0.00844206i
$186$		0			0
$187$	117.053	−	202.741i	0.625951	−	1.08418i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	212.320	+	122.583i	1.11162	+	0.641795i	0.939249	−	0.343236i	$-0.111523\pi$
$191$	212.320	+	122.583i	1.11162	+	0.641795i	0.172373	+	0.985032i	$0.444857\pi$
$192$		0			0
$193$	51.8435	+	89.7955i	0.268619	+	0.465262i	0.968505	−	0.248992i	$-0.0800994\pi$
$193$	51.8435	+	89.7955i	0.268619	+	0.465262i	−0.699886	+	0.714254i	$0.746766\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			258.993i			1.31469i	0.753591	+	0.657344i	$0.228320\pi$
$197$			258.993i			1.31469i	−0.753591	+	0.657344i	$0.771680\pi$
$198$		0			0
$199$	−177.365			−0.891283			−0.445641	−	0.895212i	$-0.647024\pi$
$199$	−177.365			−0.891283			−0.445641	+	0.895212i	$0.647024\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−445.481	+	257.199i	−2.19449	+	1.26699i
$204$		0			0
$205$	−9.82064	+	17.0098i	−0.0479056	+	0.0829749i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−77.9202	−	44.9872i	−0.372824	−	0.215250i
$210$		0			0
$211$	−69.5887	−	120.531i	−0.329804	−	0.571238i	0.652669	−	0.757644i	$-0.273649\pi$
$211$	−69.5887	−	120.531i	−0.329804	−	0.571238i	−0.982473	+	0.186406i	$0.940316\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			2.93983i			0.0136736i
$216$		0			0
$217$	468.974			2.16117
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−215.152	+	124.218i	−0.973537	+	0.562072i
$222$		0			0
$223$	−113.174	+	196.024i	−0.507508	+	0.879029i	0.492454	+	0.870338i	$0.336100\pi$
$223$	−113.174	+	196.024i	−0.507508	+	0.879029i	−0.999962	+	0.00869103i	$0.997234\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−168.223	−	97.1238i	−0.741072	−	0.427858i	0.0813870	−	0.996683i	$-0.474065\pi$
$227$	−168.223	−	97.1238i	−0.741072	−	0.427858i	−0.822459	+	0.568825i	$0.807398\pi$
$228$		0			0
$229$	−94.0612	−	162.919i	−0.410748	−	0.711436i	0.584224	−	0.811592i	$-0.301399\pi$
$229$	−94.0612	−	162.919i	−0.410748	−	0.711436i	−0.994972	+	0.100157i	$0.968066\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	422.067i		−	1.81144i	−0.423871	−	0.905722i	$-0.639329\pi$
$233$		−	422.067i		−	1.81144i	0.423871	−	0.905722i	$-0.360671\pi$
$234$		0			0
$235$	−26.8877			−0.114416
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−127.081	+	73.3705i	−0.531721	+	0.306989i	−0.741717	−	0.670713i	$-0.765988\pi$
$239$	−127.081	+	73.3705i	−0.531721	+	0.306989i	0.209996	+	0.977702i	$0.432655\pi$
$240$		0			0
$241$	−79.2527	+	137.270i	−0.328849	+	0.569584i	−0.982284	−	0.187399i	$-0.939994\pi$
$241$	−79.2527	+	137.270i	−0.328849	+	0.569584i	0.653434	+	0.756983i	$0.273328\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−51.5620	−	29.7693i	−0.210457	−	0.121507i
$246$		0			0
$247$	47.7410	+	82.6899i	0.193283	+	0.334777i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	173.536i		−	0.691377i	−0.938349	−	0.345689i	$-0.887645\pi$
$251$		−	173.536i		−	0.691377i	0.938349	−	0.345689i	$-0.112355\pi$
$252$		0			0
$253$	−380.795			−1.50512
$254$		0			0
$255$		0			0
$256$		0			0
$257$	296.784	−	171.348i	1.15480	−	0.666725i	0.204748	−	0.978815i	$-0.434362\pi$
$257$	296.784	−	171.348i	1.15480	−	0.666725i	0.950053	+	0.312090i	$0.101029\pi$
$258$		0			0
$259$	39.6725	−	68.7147i	0.153176	−	0.265308i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−114.275	−	65.9769i	−0.434507	−	0.250863i	0.266758	−	0.963764i	$-0.414048\pi$
$263$	−114.275	−	65.9769i	−0.434507	−	0.250863i	−0.701265	+	0.712901i	$0.747381\pi$
$264$		0			0
$265$	−4.38666	−	7.59792i	−0.0165534	−	0.0286714i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	206.392i		−	0.767258i	−0.923487	−	0.383629i	$-0.874674\pi$
$269$		−	206.392i		−	0.767258i	0.923487	−	0.383629i	$-0.125326\pi$
$270$		0			0
$271$	124.308			0.458702			0.229351	−	0.973344i	$-0.426340\pi$
$271$	124.308			0.458702			0.229351	+	0.973344i	$0.426340\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	329.910	−	190.474i	1.19967	−	0.692632i
$276$		0			0
$277$	241.493	−	418.279i	0.871817	−	1.51003i	0.0117017	−	0.999932i	$-0.496275\pi$
$277$	241.493	−	418.279i	0.871817	−	1.51003i	0.860115	−	0.510100i	$-0.170392\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	410.951	+	237.263i	1.46246	+	0.844351i	0.999125	−	0.0418320i	$-0.0133194\pi$
$281$	410.951	+	237.263i	1.46246	+	0.844351i	0.463335	+	0.886183i	$0.346653\pi$
$282$		0			0
$283$	137.428	+	238.032i	0.485611	+	0.841103i	0.999863	−	0.0165361i	$-0.00526383\pi$
$283$	137.428	+	238.032i	0.485611	+	0.841103i	−0.514252	+	0.857639i	$0.671930\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	498.929i		−	1.73843i
$288$		0			0
$289$	57.8081			0.200028
$290$		0			0
$291$		0			0
$292$		0			0
$293$	341.288	−	197.043i	1.16481	−	0.672502i	0.212356	−	0.977192i	$-0.431886\pi$
$293$	341.288	−	197.043i	1.16481	−	0.672502i	0.952451	+	0.304691i	$0.0985532\pi$
$294$		0			0
$295$	9.18969	−	15.9170i	0.0311515	−	0.0539560i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	349.965	+	202.052i	1.17045	+	0.675760i
$300$		0			0
$301$	−37.3389	−	64.6729i	−0.124049	−	0.214860i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	2.40621i		−	0.00788921i
$306$		0			0
$307$	−360.968			−1.17579			−0.587895	−	0.808937i	$-0.700043\pi$
$307$	−360.968			−1.17579			−0.587895	+	0.808937i	$0.700043\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−228.619	+	131.993i	−0.735108	+	0.424415i	−0.820288	−	0.571951i	$-0.806187\pi$
$311$	−228.619	+	131.993i	−0.735108	+	0.424415i	0.0851797	+	0.996366i	$0.472854\pi$
$312$		0			0
$313$	93.7598	−	162.397i	0.299552	−	0.518840i	−0.676481	−	0.736460i	$-0.736496\pi$
$313$	93.7598	−	162.397i	0.299552	−	0.518840i	0.976034	+	0.217620i	$0.0698294\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−221.667	−	127.980i	−0.699266	−	0.403721i	0.107808	−	0.994172i	$-0.465617\pi$
$317$	−221.667	−	127.980i	−0.699266	−	0.403721i	−0.807074	+	0.590451i	$0.798950\pi$
$318$		0			0
$319$	−307.091	−	531.898i	−0.962669	−	1.66739i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			88.8546i			0.275092i
$324$		0			0
$325$	−404.267			−1.24390
$326$		0			0
$327$		0			0
$328$		0			0
$329$	591.499	−	341.502i	1.79787	−	1.03800i
$330$		0			0
$331$	−174.081	+	301.517i	−0.525924	+	0.910928i	0.473620	+	0.880730i	$0.342947\pi$
$331$	−174.081	+	301.517i	−0.525924	+	0.910928i	−0.999544	+	0.0301982i	$0.990386\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−18.5215	−	10.6934i	−0.0552882	−	0.0319206i
$336$		0			0
$337$	22.1297	+	38.3298i	0.0656668	+	0.113738i	0.896990	−	0.442052i	$-0.145749\pi$
$337$	22.1297	+	38.3298i	0.0656668	+	0.113738i	−0.831323	+	0.555790i	$0.812416\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			559.948i			1.64208i
$342$		0			0
$343$	880.542			2.56718
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−72.5052	+	41.8609i	−0.208949	+	0.120637i	−0.600823	−	0.799382i	$-0.705160\pi$
$347$	−72.5052	+	41.8609i	−0.208949	+	0.120637i	0.391874	+	0.920019i	$0.371827\pi$
$348$		0			0
$349$	−170.829	+	295.885i	−0.489482	+	0.847808i	−0.999927	−	0.0121024i	$-0.996148\pi$
$349$	−170.829	+	295.885i	−0.489482	+	0.847808i	0.510444	+	0.859911i	$0.329481\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−290.206	−	167.550i	−0.822113	−	0.474647i	0.0290315	−	0.999578i	$-0.490758\pi$
$353$	−290.206	−	167.550i	−0.822113	−	0.474647i	−0.851145	+	0.524931i	$0.824091\pi$
$354$		0			0
$355$	4.21643	+	7.30307i	0.0118773	+	0.0205720i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			680.296i			1.89497i	0.319793	+	0.947487i	$0.396387\pi$
$359$			680.296i			1.89497i	−0.319793	+	0.947487i	$0.603613\pi$
$360$		0			0
$361$	−326.850			−0.905402
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−59.0705	+	34.1043i	−0.161837	+	0.0934366i
$366$		0			0
$367$	−107.245	+	185.753i	−0.292220	+	0.506139i	−0.974334	−	0.225106i	$-0.927727\pi$
$367$	−107.245	+	185.753i	−0.292220	+	0.506139i	0.682115	+	0.731245i	$0.261061\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	193.003	+	111.430i	0.520223	+	0.300351i
$372$		0			0
$373$	−222.543	−	385.455i	−0.596629	−	1.03339i	−0.993315	−	0.115437i	$-0.963173\pi$
$373$	−222.543	−	385.455i	−0.596629	−	1.03339i	0.396686	−	0.917954i	$-0.370160\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			651.779i			1.72886i
$378$		0			0
$379$	129.508			0.341710			0.170855	−	0.985296i	$-0.445347\pi$
$379$	129.508			0.341710			0.170855	+	0.985296i	$0.445347\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	328.642	−	189.741i	0.858072	−	0.495408i	−0.00529429	−	0.999986i	$-0.501685\pi$
$383$	328.642	−	189.741i	0.858072	−	0.495408i	0.863366	+	0.504578i	$0.168352\pi$
$384$		0			0
$385$	50.3940	−	87.2850i	0.130894	−	0.226714i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−181.454	−	104.763i	−0.466463	−	0.269312i	0.248295	−	0.968684i	$-0.420130\pi$
$389$	−181.454	−	104.763i	−0.466463	−	0.269312i	−0.714758	+	0.699372i	$0.753463\pi$
$390$		0			0
$391$	188.028	+	325.674i	0.480890	+	0.832925i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			27.8296i			0.0704547i
$396$		0			0
$397$	61.5407			0.155014			0.0775072	−	0.996992i	$-0.475304\pi$
$397$	61.5407			0.155014			0.0775072	+	0.996992i	$0.475304\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−65.5977	+	37.8728i	−0.163585	+	0.0944460i	−0.579558	−	0.814931i	$-0.696775\pi$
$401$	−65.5977	+	37.8728i	−0.163585	+	0.0944460i	0.415972	+	0.909377i	$0.363441\pi$
$402$		0			0
$403$	297.112	−	514.613i	0.737250	−	1.27696i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	82.0444	+	47.3684i	0.201583	+	0.116384i
$408$		0			0
$409$	156.578	+	271.201i	0.382831	+	0.663082i	0.991466	−	0.130368i	$-0.0416160\pi$
$409$	156.578	+	271.201i	0.382831	+	0.663082i	−0.608635	+	0.793450i	$0.708283\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			466.874i			1.13045i
$414$		0			0
$415$	−54.0125			−0.130151
$416$		0			0
$417$		0			0
$418$		0			0
$419$	115.127	−	66.4688i	0.274767	−	0.158637i	−0.356285	−	0.934377i	$-0.615957\pi$
$419$	115.127	−	66.4688i	0.274767	−	0.158637i	0.631052	+	0.775741i	$0.282623\pi$
$420$		0			0
$421$	−39.8633	+	69.0452i	−0.0946871	+	0.164003i	−0.909478	−	0.415752i	$-0.863518\pi$
$421$	−39.8633	+	69.0452i	−0.0946871	+	0.164003i	0.814791	+	0.579755i	$0.196852\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−325.804	−	188.103i	−0.766599	−	0.442596i
$426$		0			0
$427$	30.5613	+	52.9338i	0.0715722	+	0.123967i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			95.0522i			0.220539i	0.993902	+	0.110269i	$0.0351714\pi$
$431$			95.0522i			0.220539i	−0.993902	+	0.110269i	$0.964829\pi$
$432$		0			0
$433$	−456.089			−1.05332			−0.526661	−	0.850075i	$-0.676556\pi$
$433$	−456.089			−1.05332			−0.526661	+	0.850075i	$0.676556\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	125.167	−	72.2653i	0.286424	−	0.165367i
$438$		0			0
$439$	−242.535	+	420.083i	−0.552472	+	0.956909i	0.445624	+	0.895220i	$0.352982\pi$
$439$	−242.535	+	420.083i	−0.552472	+	0.956909i	−0.998095	+	0.0616886i	$0.980351\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−397.022	−	229.220i	−0.896211	−	0.517428i	−0.0202421	−	0.999795i	$-0.506444\pi$
$443$	−397.022	−	229.220i	−0.896211	−	0.517428i	−0.875969	+	0.482367i	$0.839777\pi$
$444$		0			0
$445$	1.76119	+	3.05048i	0.00395774	+	0.00685501i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			554.694i			1.23540i	0.786414	+	0.617699i	$0.211935\pi$
$449$			554.694i			1.23540i	−0.786414	+	0.617699i	$0.788065\pi$
$450$		0			0
$451$	595.714			1.32087
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−92.6280	+	53.4788i	−0.203578	+	0.117536i
$456$		0			0
$457$	−73.0010	+	126.441i	−0.159740	+	0.276677i	−0.934775	−	0.355241i	$-0.884399\pi$
$457$	−73.0010	+	126.441i	−0.159740	+	0.276677i	0.775035	+	0.631918i	$0.217732\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	430.283	+	248.424i	0.933368	+	0.538880i	0.887875	−	0.460084i	$-0.152181\pi$
$461$	430.283	+	248.424i	0.933368	+	0.538880i	0.0454929	+	0.998965i	$0.485514\pi$
$462$		0			0
$463$	192.295	+	333.065i	0.415324	+	0.719363i	0.995462	−	0.0951553i	$-0.0303348\pi$
$463$	192.295	+	333.065i	0.415324	+	0.719363i	−0.580138	+	0.814518i	$0.697001\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			430.997i			0.922907i	0.887164	+	0.461453i	$0.152672\pi$
$467$			430.997i			0.922907i	−0.887164	+	0.461453i	$0.847328\pi$
$468$		0			0
$469$	543.269			1.15836
$470$		0			0
$471$		0			0
$472$		0			0
$473$	77.2185	−	44.5821i	0.163253	−	0.0942539i
$474$		0			0
$475$	−72.2943	+	125.217i	−0.152198	+	0.263615i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	185.048	+	106.838i	0.386322	+	0.223043i	0.680565	−	0.732688i	$-0.261734\pi$
$479$	185.048	+	106.838i	0.386322	+	0.223043i	−0.294243	+	0.955730i	$0.595068\pi$
$480$		0			0
$481$	−50.2679	−	87.0666i	−0.104507	−	0.181012i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	56.9175i		−	0.117356i
$486$		0			0
$487$	−312.876			−0.642456			−0.321228	−	0.947002i	$-0.604096\pi$
$487$	−312.876			−0.642456			−0.321228	+	0.947002i	$0.604096\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−232.474	+	134.219i	−0.473471	+	0.273358i	−0.717691	−	0.696361i	$-0.754801\pi$
$491$	−232.474	+	134.219i	−0.473471	+	0.273358i	0.244221	+	0.969720i	$0.421468\pi$
$492$		0			0
$493$	−303.269	+	525.278i	−0.615151	+	1.06547i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−185.513	−	107.106i	−0.373266	−	0.215505i
$498$		0			0
$499$	42.3336	+	73.3240i	0.0848369	+	0.146942i	0.905322	−	0.424727i	$-0.139630\pi$
$499$	42.3336	+	73.3240i	0.0848369	+	0.146942i	−0.820485	+	0.571668i	$0.806296\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			618.121i			1.22887i	0.788968	+	0.614434i	$0.210616\pi$
$503$			618.121i			1.22887i	−0.788968	+	0.614434i	$0.789384\pi$
$504$		0			0
$505$	10.5248			0.0208413
$506$		0			0
$507$		0			0
$508$		0			0
$509$	543.018	−	313.512i	1.06683	−	0.615936i	0.139519	−	0.990219i	$-0.455445\pi$
$509$	543.018	−	313.512i	1.06683	−	0.615936i	0.927314	+	0.374283i	$0.122111\pi$
$510$		0			0
$511$	866.321	−	1500.51i	1.69534	−	2.93642i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−27.3099	−	15.7674i	−0.0530290	−	0.0306163i
$516$		0			0
$517$	407.748	+	706.241i	0.788682	+	1.36604i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			426.201i			0.818044i	0.912524	+	0.409022i	$0.134130\pi$
$521$			426.201i			0.818044i	−0.912524	+	0.409022i	$0.865870\pi$
$522$		0			0
$523$	535.721			1.02432			0.512162	−	0.858889i	$-0.328845\pi$
$523$	535.721			1.02432			0.512162	+	0.858889i	$0.328845\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	478.894	−	276.489i	0.908716	−	0.524648i
$528$		0			0
$529$	41.3455	−	71.6124i	0.0781578	−	0.135373i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−547.484	−	316.090i	−1.02717	−	0.593039i
$534$		0			0
$535$	−18.8648	−	32.6748i	−0.0352613	−	0.0610744i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			1805.79i			3.35026i
$540$		0			0
$541$	740.501			1.36876			0.684382	−	0.729124i	$-0.260072\pi$
$541$	740.501			1.36876			0.684382	+	0.729124i	$0.260072\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	57.6327	−	33.2742i	0.105748	−	0.0610537i
$546$		0			0
$547$	190.999	−	330.820i	0.349175	−	0.604789i	−0.636928	−	0.770923i	$-0.719795\pi$
$547$	190.999	−	330.820i	0.349175	−	0.604789i	0.986103	+	0.166134i	$0.0531284\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	201.882	+	116.556i	0.366391	+	0.211536i
$552$		0			0
$553$	−353.465	−	612.219i	−0.639177	−	1.10709i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	66.7107i		−	0.119768i	−0.998205	−	0.0598839i	$-0.980927\pi$
$557$		−	66.7107i		−	0.119768i	0.998205	−	0.0598839i	$-0.0190730\pi$
$558$		0			0
$559$	−94.6222			−0.169271
$560$		0			0
$561$		0			0
$562$		0			0
$563$	511.625	−	295.387i	0.908748	−	0.524666i	0.0287199	−	0.999587i	$-0.490857\pi$
$563$	511.625	−	295.387i	0.908748	−	0.524666i	0.880028	+	0.474922i	$0.157524\pi$
$564$		0			0
$565$	7.78014	−	13.4756i	0.0137702	−	0.0238506i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−355.334	−	205.152i	−0.624489	−	0.360549i	0.154126	−	0.988051i	$-0.450744\pi$
$569$	−355.334	−	205.152i	−0.624489	−	0.360549i	−0.778615	+	0.627502i	$0.784077\pi$
$570$		0			0
$571$	502.284	+	869.982i	0.879657	+	1.52361i	0.851718	+	0.524001i	$0.175561\pi$
$571$	502.284	+	869.982i	0.879657	+	1.52361i	0.0279391	+	0.999610i	$0.491106\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			611.936i			1.06424i
$576$		0			0
$577$	238.644			0.413595			0.206798	−	0.978384i	$-0.433696\pi$
$577$	238.644			0.413595			0.206798	+	0.978384i	$0.433696\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1188.21	−	686.014i	2.04511	−	1.18075i
$582$		0			0
$583$	−133.046	+	230.443i	−0.228209	+	0.395270i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	560.752	+	323.751i	0.955285	+	0.551534i	0.894719	−	0.446630i	$-0.147376\pi$
$587$	560.752	+	323.751i	0.955285	+	0.551534i	0.0605664	+	0.998164i	$0.480709\pi$
$588$		0			0
$589$	−106.264	−	184.055i	−0.180414	−	0.312486i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	669.903i		−	1.12969i	−0.825199	−	0.564843i	$-0.808937\pi$
$593$		−	669.903i		−	1.12969i	0.825199	−	0.564843i	$-0.191063\pi$
$594$		0			0
$595$	−99.5336			−0.167283
$596$		0			0
$597$		0			0
$598$		0			0
$599$	123.532	−	71.3213i	0.206231	−	0.119067i	−0.393328	−	0.919398i	$-0.628676\pi$
$599$	123.532	−	71.3213i	0.206231	−	0.119067i	0.599558	+	0.800331i	$0.295343\pi$
$600$		0			0
$601$	50.4375	−	87.3604i	0.0839227	−	0.145358i	−0.821009	−	0.570915i	$-0.806588\pi$
$601$	50.4375	−	87.3604i	0.0839227	−	0.145358i	0.904932	+	0.425557i	$0.139922\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	51.0217	+	29.4574i	0.0843335	+	0.0486899i
$606$		0			0
$607$	292.988	+	507.470i	0.482682	+	0.836029i	0.999802	−	0.0198835i	$-0.00632952\pi$
$607$	292.988	+	507.470i	0.482682	+	0.836029i	−0.517121	+	0.855912i	$0.672996\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	865.416i		−	1.41639i
$612$		0			0
$613$	−444.730			−0.725497			−0.362749	−	0.931887i	$-0.618162\pi$
$613$	−444.730			−0.725497			−0.362749	+	0.931887i	$0.618162\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−737.916	+	426.036i	−1.19597	+	0.690496i	−0.959655	−	0.281180i	$-0.909274\pi$
$617$	−737.916	+	426.036i	−1.19597	+	0.690496i	−0.236318	+	0.971676i	$0.575941\pi$
$618$		0			0
$619$	118.567	−	205.365i	0.191546	−	0.331768i	−0.754216	−	0.656626i	$-0.771983\pi$
$619$	118.567	−	205.365i	0.191546	−	0.331768i	0.945763	+	0.324858i	$0.105316\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−77.4884	−	44.7380i	−0.124380	−	0.0718105i
$624$		0			0
$625$	−302.869	−	524.585i	−0.484591	−	0.839336i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	93.5576i		−	0.148740i
$630$		0			0
$631$	386.236			0.612102			0.306051	−	0.952015i	$-0.400992\pi$
$631$	386.236			0.612102			0.306051	+	0.952015i	$0.400992\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5.78207	+	3.33828i	−0.00910563	+	0.00525714i
$636$		0			0
$637$	958.163	−	1659.59i	1.50418	−	2.60532i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−344.762	−	199.048i	−0.537850	−	0.310528i	0.206357	−	0.978477i	$-0.433839\pi$
$641$	−344.762	−	199.048i	−0.537850	−	0.310528i	−0.744207	+	0.667949i	$0.767172\pi$
$642$		0			0
$643$	−453.394	−	785.301i	−0.705122	−	1.22131i	−0.966647	−	0.256111i	$-0.917559\pi$
$643$	−453.394	−	785.301i	−0.705122	−	1.22131i	0.261525	−	0.965197i	$-0.415775\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	146.621i		−	0.226617i	−0.993560	−	0.113308i	$-0.963855\pi$
$647$		−	146.621i		−	0.226617i	0.993560	−	0.113308i	$-0.0361448\pi$
$648$		0			0
$649$	−557.441			−0.858923
$650$		0			0
$651$		0			0
$652$		0			0
$653$	208.775	−	120.536i	0.319716	−	0.184588i	−0.331550	−	0.943438i	$-0.607571\pi$
$653$	208.775	−	120.536i	0.319716	−	0.184588i	0.651266	+	0.758849i	$0.274238\pi$
$654$		0			0
$655$	34.8894	−	60.4303i	0.0532663	−	0.0922600i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−161.540	−	93.2650i	−0.245129	−	0.141525i	0.372403	−	0.928071i	$-0.378534\pi$
$659$	−161.540	−	93.2650i	−0.245129	−	0.141525i	−0.617532	+	0.786546i	$0.711867\pi$
$660$		0			0
$661$	569.443	+	986.303i	0.861486	+	1.49214i	0.870494	+	0.492179i	$0.163799\pi$
$661$	569.443	+	986.303i	0.861486	+	1.49214i	−0.00900763	+	0.999959i	$0.502867\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			38.2540i			0.0575249i
$666$		0			0
$667$	986.594			1.47915
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−63.2022	+	36.4898i	−0.0941910	+	0.0543812i
$672$		0			0
$673$	−123.699	+	214.254i	−0.183803	+	0.318356i	−0.943173	−	0.332303i	$-0.892174\pi$
$673$	−123.699	+	214.254i	−0.183803	+	0.318356i	0.759369	+	0.650660i	$0.225508\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−728.122	−	420.382i	−1.07551	−	0.620948i	−0.145831	−	0.989310i	$-0.546585\pi$
$677$	−728.122	−	420.382i	−1.07551	−	0.620948i	−0.929683	+	0.368362i	$0.879919\pi$
$678$		0			0
$679$	722.910	+	1252.12i	1.06467	+	1.84406i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	487.692i		−	0.714043i	−0.934096	−	0.357022i	$-0.883792\pi$
$683$		−	487.692i		−	0.714043i	0.934096	−	0.357022i	$-0.116208\pi$
$684$		0			0
$685$	−126.699			−0.184963
$686$		0			0
$687$		0			0
$688$		0			0
$689$	244.549	−	141.190i	0.354933	−	0.204920i
$690$		0			0
$691$	−200.461	+	347.209i	−0.290103	+	0.502473i	−0.973834	−	0.227261i	$-0.927023\pi$
$691$	−200.461	+	347.209i	−0.290103	+	0.502473i	0.683731	+	0.729734i	$0.260356\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	84.7913	+	48.9543i	0.122002	+	0.0704378i
$696$		0			0
$697$	−294.150	−	509.483i	−0.422023	−	0.730965i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	194.336i		−	0.277227i	−0.990347	−	0.138613i	$-0.955735\pi$
$701$		−	194.336i		−	0.277227i	0.990347	−	0.138613i	$-0.0442646\pi$
$702$		0			0
$703$	−35.9573			−0.0511483
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−231.534	+	133.676i	−0.327488	+	0.189075i
$708$		0			0
$709$	−580.616	+	1005.66i	−0.818922	+	1.41841i	0.0875552	+	0.996160i	$0.472095\pi$
$709$	−580.616	+	1005.66i	−0.818922	+	1.41841i	−0.906477	+	0.422255i	$0.861239\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−778.966	−	449.736i	−1.09252	−	0.630766i
$714$		0			0
$715$	−63.8529	−	110.596i	−0.0893048	−	0.154680i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	284.601i		−	0.395829i	−0.980219	−	0.197915i	$-0.936583\pi$
$719$		−	284.601i		−	0.395829i	0.980219	−	0.197915i	$-0.0634169\pi$
$720$		0			0
$721$	801.049			1.11102
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−854.758	+	493.494i	−1.17898	+	0.680682i
$726$		0			0
$727$	98.7275	−	171.001i	0.135801	−	0.235215i	−0.790102	−	0.612975i	$-0.789972\pi$
$727$	98.7275	−	171.001i	0.135801	−	0.235215i	0.925903	+	0.377761i	$0.123306\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−76.2574	−	44.0272i	−0.104319	−	0.0602288i
$732$		0			0
$733$	−4.46828	−	7.73928i	−0.00609588	−	0.0105584i	0.862961	−	0.505270i	$-0.168607\pi$
$733$	−4.46828	−	7.73928i	−0.00609588	−	0.0105584i	−0.869057	+	0.494712i	$0.835274\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			648.656i			0.880130i
$738$		0			0
$739$	−545.359			−0.737968			−0.368984	−	0.929436i	$-0.620294\pi$
$739$	−545.359			−0.737968			−0.368984	+	0.929436i	$0.620294\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1102.50	+	636.530i	−1.48385	+	0.856702i	−0.999832	−	0.0183549i	$-0.994157\pi$
$743$	−1102.50	+	636.530i	−1.48385	+	0.856702i	−0.484020	+	0.875057i	$0.660824\pi$
$744$		0			0
$745$	−23.9545	+	41.4905i	−0.0321537	+	0.0556919i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	830.008	+	479.205i	1.10815	+	0.639793i
$750$		0			0
$751$	−330.544	−	572.519i	−0.440138	−	0.762342i	0.557561	−	0.830136i	$-0.311737\pi$
$751$	−330.544	−	572.519i	−0.440138	−	0.762342i	−0.997699	+	0.0677940i	$0.978404\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			81.3760i			0.107783i
$756$		0			0
$757$	441.298			0.582956			0.291478	−	0.956577i	$-0.405853\pi$
$757$	441.298			0.582956			0.291478	+	0.956577i	$0.405853\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−943.045	+	544.467i	−1.23922	+	0.715463i	−0.968934	−	0.247318i	$-0.920451\pi$
$761$	−943.045	+	544.467i	−1.23922	+	0.715463i	−0.270283	+	0.962781i	$0.587117\pi$
$762$		0			0
$763$	−845.234	+	1463.99i	−1.10778	+	1.91873i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	512.309	+	295.782i	0.667939	+	0.385635i
$768$		0			0
$769$	457.795	+	792.923i	0.595312	+	1.03111i	0.993503	+	0.113807i	$0.0363047\pi$
$769$	457.795	+	792.923i	0.595312	+	1.03111i	−0.398191	+	0.917302i	$0.630362\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	162.264i		−	0.209915i	−0.994477	−	0.104957i	$-0.966529\pi$
$773$		−	162.264i		−	0.209915i	0.994477	−	0.104957i	$-0.0334706\pi$
$774$		0			0
$775$	899.833			1.16108
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−195.811	+	113.051i	−0.251362	+	0.145124i
$780$		0			0
$781$	127.883	−	221.500i	0.163743	−	0.283611i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	53.3398	+	30.7958i	0.0679488	+	0.0392303i
$786$		0			0
$787$	−244.801	−	424.007i	−0.311055	−	0.538764i	0.667536	−	0.744578i	$-0.267349\pi$
$787$	−244.801	−	424.007i	−0.311055	−	0.538764i	−0.978591	+	0.205814i	$0.934016\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			395.263i			0.499701i
$792$		0			0
$793$	77.4469			0.0976632
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−769.799	+	444.444i	−0.965871	+	0.557646i	−0.897975	−	0.440047i	$-0.854962\pi$
$797$	−769.799	+	444.444i	−0.965871	+	0.557646i	−0.0678959	+	0.997692i	$0.521629\pi$
$798$		0			0
$799$	402.674	−	697.451i	0.503972	−	0.872905i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1791.59	+	1034.37i	2.23112	+	1.28814i
$804$		0			0
$805$	80.9505	+	140.210i	0.100560	+	0.174174i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			25.3522i			0.0313377i	0.999877	+	0.0156689i	$0.00498776\pi$
$809$			25.3522i			0.0313377i	−0.999877	+	0.0156689i	$0.995012\pi$
$810$		0			0
$811$	700.162			0.863332			0.431666	−	0.902034i	$-0.357926\pi$
$811$	700.162			0.863332			0.431666	+	0.902034i	$0.357926\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−110.730	+	63.9300i	−0.135865	+	0.0784417i
$816$		0			0
$817$	−16.9211	+	29.3082i	−0.0207113	+	0.0358730i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−375.552	−	216.825i	−0.457432	−	0.264099i	0.253532	−	0.967327i	$-0.418408\pi$
$821$	−375.552	−	216.825i	−0.457432	−	0.264099i	−0.710964	+	0.703228i	$0.751741\pi$
$822$		0			0
$823$	−173.900	−	301.204i	−0.211300	−	0.365983i	0.740821	−	0.671702i	$-0.234437\pi$
$823$	−173.900	−	301.204i	−0.211300	−	0.365983i	−0.952122	+	0.305719i	$0.901103\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			941.259i			1.13816i	0.822282	+	0.569080i	$0.192701\pi$
$827$			941.259i			1.13816i	−0.822282	+	0.569080i	$0.807299\pi$
$828$		0			0
$829$	496.147			0.598489			0.299244	−	0.954177i	$-0.403265\pi$
$829$	496.147			0.598489			0.299244	+	0.954177i	$0.403265\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1544.40	−	891.657i	1.85402	−	1.07042i
$834$		0			0
$835$	11.0983	−	19.2229i	0.0132914	−	0.0230214i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	120.646	+	69.6548i	0.143797	+	0.0830212i	0.570172	−	0.821525i	$-0.306876\pi$
$839$	120.646	+	69.6548i	0.143797	+	0.0830212i	−0.426375	+	0.904546i	$0.640210\pi$
$840$		0			0
$841$	375.136	+	649.755i	0.446060	+	0.772598i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			49.7316i			0.0588540i
$846$		0			0
$847$	−1496.56			−1.76689
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−131.792	+	76.0902i	−0.154867	+	0.0894127i
$852$		0			0
$853$	−483.535	+	837.507i	−0.566864	+	0.981837i	0.430009	+	0.902824i	$0.358510\pi$
$853$	−483.535	+	837.507i	−0.566864	+	0.981837i	−0.996874	+	0.0790130i	$0.974823\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	936.914	+	540.927i	1.09325	+	0.631187i	0.934439	−	0.356122i	$-0.115901\pi$
$857$	936.914	+	540.927i	1.09325	+	0.631187i	0.158809	+	0.987309i	$0.449235\pi$
$858$		0			0
$859$	−630.719	−	1092.44i	−0.734248	−	1.27175i	−0.955053	−	0.296436i	$-0.904202\pi$
$859$	−630.719	−	1092.44i	−0.734248	−	1.27175i	0.220805	−	0.975318i	$-0.429132\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1160.44i		−	1.34466i	−0.740252	−	0.672330i	$-0.765294\pi$
$863$		−	1160.44i		−	1.34466i	0.740252	−	0.672330i	$-0.234706\pi$
$864$		0			0
$865$	−163.080			−0.188532
$866$		0			0
$867$		0			0
$868$		0			0
$869$	730.981	−	422.032i	0.841174	−	0.485652i
$870$		0			0
$871$	344.181	−	596.139i	0.395156	−	0.684431i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−281.994	−	162.809i	−0.322279	−	0.186068i
$876$		0			0
$877$	−8.24639	−	14.2832i	−0.00940295	−	0.0162864i	0.861286	−	0.508121i	$-0.169660\pi$
$877$	−8.24639	−	14.2832i	−0.00940295	−	0.0162864i	−0.870689	+	0.491835i	$0.836326\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			858.235i			0.974160i	0.873357	+	0.487080i	$0.161938\pi$
$881$			858.235i			0.974160i	−0.873357	+	0.487080i	$0.838062\pi$
$882$		0			0
$883$	1484.58			1.68130			0.840648	−	0.541583i	$-0.182175\pi$
$883$	1484.58			1.68130			0.840648	+	0.541583i	$0.182175\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−926.033	+	534.645i	−1.04401	+	0.602757i	−0.920965	−	0.389645i	$-0.872598\pi$
$887$	−926.033	+	534.645i	−1.04401	+	0.602757i	−0.123040	+	0.992402i	$0.539264\pi$
$888$		0			0
$889$	84.7992	−	146.877i	0.0953872	−	0.165215i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−268.053	−	154.761i	−0.300172	−	0.173304i
$894$		0			0
$895$	−51.0474	−	88.4166i	−0.0570362	−	0.0987895i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	1450.76i		−	1.61374i
$900$		0			0
$901$	262.780			0.291654
$902$		0			0
$903$		0			0
$904$		0			0
$905$	34.0736	−	19.6724i	0.0376504	−	0.0217374i
$906$		0			0
$907$	−243.206	+	421.246i	−0.268144	+	0.464438i	−0.968382	−	0.249470i	$-0.919743\pi$
$907$	−243.206	+	421.246i	−0.268144	+	0.464438i	0.700239	+	0.713909i	$0.253077\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−753.527	−	435.049i	−0.827142	−	0.477551i	0.0257309	−	0.999669i	$-0.491809\pi$
$911$	−753.527	−	435.049i	−0.827142	−	0.477551i	−0.852873	+	0.522118i	$0.825142\pi$
$912$		0			0
$913$	819.091	+	1418.71i	0.897142	+	1.55390i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			1772.53i			1.93296i
$918$		0			0
$919$	−681.374			−0.741429			−0.370715	−	0.928747i	$-0.620887\pi$
$919$	−681.374			−0.741429			−0.370715	+	0.928747i	$0.620887\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−235.059	+	135.711i	−0.254668	+	0.147033i
$924$		0			0
$925$	76.1207	−	131.845i	0.0822927	−	0.142535i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−923.850	−	533.385i	−0.994456	−	0.574150i	−0.0878528	−	0.996133i	$-0.528001\pi$
$929$	−923.850	−	533.385i	−0.994456	−	0.574150i	−0.906603	+	0.421984i	$0.861334\pi$
$930$		0			0
$931$	−342.693	−	593.562i	−0.368091	−	0.637553i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	118.842i		−	0.127103i
$936$		0			0
$937$	774.576			0.826655			0.413328	−	0.910582i	$-0.364366\pi$
$937$	774.576			0.826655			0.413328	+	0.910582i	$0.364366\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−954.461	+	551.058i	−1.01431	+	0.585609i	−0.912449	−	0.409190i	$-0.865811\pi$
$941$	−954.461	+	551.058i	−1.01431	+	0.585609i	−0.101856	+	0.994799i	$0.532478\pi$
$942$		0			0
$943$	−478.463	+	828.722i	−0.507384	+	0.878815i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−922.431	−	532.566i	−0.974056	−	0.562372i	−0.0735857	−	0.997289i	$-0.523444\pi$
$947$	−922.431	−	532.566i	−0.974056	−	0.562372i	−0.900470	+	0.434917i	$0.856778\pi$
$948$		0			0
$949$	−1097.69	−	1901.26i	−1.15668	−	2.00343i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1376.41i		−	1.44430i	−0.691739	−	0.722148i	$-0.743155\pi$
$953$		−	1376.41i		−	1.44430i	0.691739	−	0.722148i	$-0.256845\pi$
$954$		0			0
$955$	124.456			0.130321
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2787.24	−	1609.21i	2.90640	−	1.67801i
$960$		0			0
$961$	−180.824	+	313.196i	−0.188162	+	0.325906i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	45.5839	+	26.3179i	0.0472372	+	0.0272724i
$966$		0			0
$967$	668.134	+	1157.24i	0.690935	+	1.19673i	0.971532	+	0.236909i	$0.0761344\pi$
$967$	668.134	+	1157.24i	0.690935	+	1.19673i	−0.280597	+	0.959826i	$0.590532\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	512.682i		−	0.527994i	−0.964524	−	0.263997i	$-0.914959\pi$
$971$		−	512.682i		−	0.527994i	0.964524	−	0.263997i	$-0.0850409\pi$
$972$		0			0
$973$	−2487.08			−2.55609
$974$		0			0
$975$		0			0
$976$		0			0
$977$	229.612	−	132.567i	0.235017	−	0.135687i	−0.377867	−	0.925860i	$-0.623342\pi$
$977$	229.612	−	132.567i	0.235017	−	0.135687i	0.612885	+	0.790172i	$0.290009\pi$
$978$		0			0
$979$	53.4165	−	92.5201i	0.0545623	−	0.0945047i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1459.14	+	842.433i	1.48437	+	0.857002i	0.999842	−	0.0177726i	$-0.00565750\pi$
$983$	1459.14	+	842.433i	1.48437	+	0.857002i	0.484529	+	0.874775i	$0.338991\pi$
$984$		0			0
$985$	65.7379	+	113.861i	0.0667390	+	0.115595i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			143.229i			0.144822i
$990$		0			0
$991$	−842.730			−0.850383			−0.425192	−	0.905103i	$-0.639793\pi$
$991$	−842.730			−0.850383			−0.425192	+	0.905103i	$0.639793\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−77.9752	+	45.0190i	−0.0783671	+	0.0452452i
$996$		0			0
$997$	55.4961	−	96.1221i	0.0556631	−	0.0964114i	−0.836851	−	0.547430i	$-0.815606\pi$
$997$	55.4961	−	96.1221i	0.0556631	−	0.0964114i	0.892514	+	0.451019i	$0.148939\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.q.l.1601.6		24
3.2	odd	2		576.3.q.k.65.5		24
4.3	odd	2	inner	1728.3.q.l.1601.5		24
8.3	odd	2		864.3.q.b.737.8		24
8.5	even	2		864.3.q.b.737.7		24
9.4	even	3		576.3.q.k.257.5		24
9.5	odd	6	inner	1728.3.q.l.449.6		24
12.11	even	2		576.3.q.k.65.8		24
24.5	odd	2		288.3.q.a.65.8	yes	24
24.11	even	2		288.3.q.a.65.5	✓	24
36.23	even	6	inner	1728.3.q.l.449.5		24
36.31	odd	6		576.3.q.k.257.8		24
72.5	odd	6		864.3.q.b.449.7		24
72.11	even	6		2592.3.e.j.161.4		24
72.13	even	6		288.3.q.a.257.8	yes	24
72.29	odd	6		2592.3.e.j.161.22		24
72.43	odd	6		2592.3.e.j.161.3		24
72.59	even	6		864.3.q.b.449.8		24
72.61	even	6		2592.3.e.j.161.21		24
72.67	odd	6		288.3.q.a.257.5	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.3.q.a.65.5	✓	24	24.11	even	2
288.3.q.a.65.8	yes	24	24.5	odd	2
288.3.q.a.257.5	yes	24	72.67	odd	6
288.3.q.a.257.8	yes	24	72.13	even	6
576.3.q.k.65.5		24	3.2	odd	2
576.3.q.k.65.8		24	12.11	even	2
576.3.q.k.257.5		24	9.4	even	3
576.3.q.k.257.8		24	36.31	odd	6
864.3.q.b.449.7		24	72.5	odd	6
864.3.q.b.449.8		24	72.59	even	6
864.3.q.b.737.7		24	8.5	even	2
864.3.q.b.737.8		24	8.3	odd	2
1728.3.q.l.449.5		24	36.23	even	6	inner
1728.3.q.l.449.6		24	9.5	odd	6	inner
1728.3.q.l.1601.5		24	4.3	odd	2	inner
1728.3.q.l.1601.6		24	1.1	even	1	trivial
2592.3.e.j.161.3		24	72.43	odd	6
2592.3.e.j.161.4		24	72.11	even	6
2592.3.e.j.161.21		24	72.61	even	6
2592.3.e.j.161.22		24	72.29	odd	6

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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.439631 - 0.253821i) q^{5} +(-6.44757 + 11.1675i) q^{7} +O(q^{10})\)	\(q+(0.439631 - 0.253821i) q^{5} +(-6.44757 + 11.1675i) q^{7} +(-13.3339 - 7.69831i) q^{11} +(8.16955 + 14.1501i) q^{13} +15.2050i q^{17} +5.84378 q^{19} +(21.4189 - 12.3662i) q^{23} +(-12.3711 + 21.4275i) q^{25} +(34.5464 + 19.9454i) q^{29} +(-18.1841 - 31.4958i) q^{31} +6.54611i q^{35} -6.15309 q^{37} +(-33.5076 + 19.3456i) q^{41} +(-2.89558 + 5.01529i) q^{43} +(-45.8699 - 26.4830i) q^{47} +(-58.6424 - 101.572i) q^{49} -17.2825i q^{53} -7.81597 q^{55} +(31.3548 - 18.1027i) q^{59} +(2.36999 - 4.10494i) q^{61} +(7.18317 + 4.14720i) q^{65} +(-21.0649 - 36.4854i) q^{67} +16.6118i q^{71} -134.364 q^{73} +(171.942 - 99.2708i) q^{77} +(-27.4107 + 47.4767i) q^{79} +(-92.1440 - 53.1994i) q^{83} +(3.85934 + 6.68458i) q^{85} +6.93873i q^{89} -210.695 q^{91} +(2.56910 - 1.48327i) q^{95} +(56.0607 - 97.0999i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 60 q^{25} + 72 q^{29} + 36 q^{41} - 132 q^{49} - 96 q^{61} - 576 q^{65} + 24 q^{73} + 432 q^{77} + 96 q^{85} + 252 q^{97}+O(q^{100}) \) 24 * q + 60 * q^25 + 72 * q^29 + 36 * q^41 - 132 * q^49 - 96 * q^61 - 576 * q^65 + 24 * q^73 + 432 * q^77 + 96 * q^85 + 252 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more