LMFDB - Embedded newform 1728.3.b.f.1567.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1728,3,Mod(1567,1728)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1728.1567"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,-44,0,0,0,0, 0,0,0,0,0,-120,0,0,0,0,0,0,0,0,0,0,0,0,0,96] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(49)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.4
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1728.1567
Dual form			1728.3.b.f.1567.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+6.00000i q^{5} +5.00000i q^{7} +6.00000 q^{11} +5.19615i q^{13} -31.1769 q^{17} -25.9808 q^{19} +31.1769i q^{23} -11.0000 q^{25} -36.0000i q^{29} -26.0000i q^{31} -30.0000 q^{35} -25.9808i q^{37} -20.7846 q^{43} +31.1769i q^{47} +24.0000 q^{49} +60.0000i q^{53} +36.0000i q^{55} +18.0000 q^{59} -77.9423i q^{61} -31.1769 q^{65} -77.9423 q^{67} +25.0000 q^{73} +30.0000i q^{77} +31.0000i q^{79} +120.000 q^{83} -187.061i q^{85} +155.885 q^{89} -25.9808 q^{91} -155.885i q^{95} -85.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 24 q^{11} - 44 q^{25} - 120 q^{35} + 96 q^{49} + 72 q^{59} + 100 q^{73} + 480 q^{83} - 340 q^{97}+O(q^{100}) $ 4 * q + 24 * q^11 - 44 * q^25 - 120 * q^35 + 96 * q^49 + 72 * q^59 + 100 * q^73 + 480 * q^83 - 340 * q^97

Character values

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

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	6.00000i	1.20000i	0.800000	+	0.600000i	$0.204833\pi$
$5$	6.00000i	1.20000i	−0.800000	+	0.600000i	$0.795167\pi$
$6$	0	0
$7$	5.00000i	0.714286i	0.934050	+	0.357143i	$0.116249\pi$
$7$	5.00000i	0.714286i	−0.934050	+	0.357143i	$0.883751\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.00000	0.545455	0.272727	−	0.962091i	$-0.412074\pi$
$11$	6.00000	0.545455	0.272727	+	0.962091i	$0.412074\pi$
$12$	0	0
$13$	5.19615i	0.399704i	0.979826	+	0.199852i	$0.0640461\pi$
$13$	5.19615i	0.399704i	−0.979826	+	0.199852i	$0.935954\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−31.1769	−1.83394	−0.916968	−	0.398961i	$-0.869371\pi$
$17$	−31.1769	−1.83394	−0.916968	+	0.398961i	$0.869371\pi$
$18$	0	0
$19$	−25.9808	−1.36741	−0.683704	−	0.729759i	$-0.739632\pi$
$19$	−25.9808	−1.36741	−0.683704	+	0.729759i	$0.739632\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	31.1769i	1.35552i	0.735284	+	0.677759i	$0.237049\pi$
$23$	31.1769i	1.35552i	−0.735284	+	0.677759i	$0.762951\pi$
$24$	0	0
$25$	−11.0000	−0.440000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 36.0000i	− 1.24138i	−0.784056	−	0.620690i	$-0.786853\pi$
$29$	− 36.0000i	− 1.24138i	0.784056	−	0.620690i	$-0.213147\pi$
$30$	0	0
$31$	− 26.0000i	− 0.838710i	−0.907822	−	0.419355i	$-0.862256\pi$
$31$	− 26.0000i	− 0.838710i	0.907822	−	0.419355i	$-0.137744\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−30.0000	−0.857143
$36$	0	0
$37$	− 25.9808i	− 0.702183i	−0.936341	−	0.351091i	$-0.885811\pi$
$37$	− 25.9808i	− 0.702183i	0.936341	−	0.351091i	$-0.114189\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−20.7846	−0.483363	−0.241682	−	0.970356i	$-0.577699\pi$
$43$	−20.7846	−0.483363	−0.241682	+	0.970356i	$0.577699\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	31.1769i	0.663339i	0.943396	+	0.331669i	$0.107612\pi$
$47$	31.1769i	0.663339i	−0.943396	+	0.331669i	$0.892388\pi$
$48$	0	0
$49$	24.0000	0.489796
$50$	0	0
$51$	0	0
$52$	0	0
$53$	60.0000i	1.13208i	0.824379	+	0.566038i	$0.191524\pi$
$53$	60.0000i	1.13208i	−0.824379	+	0.566038i	$0.808476\pi$
$54$	0	0
$55$	36.0000i	0.654545i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	18.0000	0.305085	0.152542	−	0.988297i	$-0.451254\pi$
$59$	18.0000	0.305085	0.152542	+	0.988297i	$0.451254\pi$
$60$	0	0
$61$	− 77.9423i	− 1.27774i	−0.769314	−	0.638871i	$-0.779402\pi$
$61$	− 77.9423i	− 1.27774i	0.769314	−	0.638871i	$-0.220598\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−31.1769	−0.479645
$66$	0	0
$67$	−77.9423	−1.16332	−0.581659	−	0.813433i	$-0.697596\pi$
$67$	−77.9423	−1.16332	−0.581659	+	0.813433i	$0.697596\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	25.0000	0.342466	0.171233	−	0.985231i	$-0.445225\pi$
$73$	25.0000	0.342466	0.171233	+	0.985231i	$0.445225\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	30.0000i	0.389610i
$78$	0	0
$79$	31.0000i	0.392405i	0.980563	+	0.196203i	$0.0628610\pi$
$79$	31.0000i	0.392405i	−0.980563	+	0.196203i	$0.937139\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	120.000	1.44578	0.722892	−	0.690961i	$-0.242813\pi$
$83$	120.000	1.44578	0.722892	+	0.690961i	$0.242813\pi$
$84$	0	0
$85$	− 187.061i	− 2.20072i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	155.885	1.75151	0.875756	−	0.482754i	$-0.160363\pi$
$89$	155.885	1.75151	0.875756	+	0.482754i	$0.160363\pi$
$90$	0	0
$91$	−25.9808	−0.285503
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 155.885i	− 1.64089i
$96$	0	0
$97$	−85.0000	−0.876289	−0.438144	−	0.898905i	$-0.644364\pi$
$97$	−85.0000	−0.876289	−0.438144	+	0.898905i	$0.644364\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 156.000i	− 1.54455i	−0.635286	−	0.772277i	$-0.719118\pi$
$101$	− 156.000i	− 1.54455i	0.635286	−	0.772277i	$-0.280882\pi$
$102$	0	0
$103$	− 125.000i	− 1.21359i	−0.794858	−	0.606796i	$-0.792454\pi$
$103$	− 125.000i	− 1.21359i	0.794858	−	0.606796i	$-0.207546\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−150.000	−1.40187	−0.700935	−	0.713226i	$-0.747234\pi$
$107$	−150.000	−1.40187	−0.700935	+	0.713226i	$0.747234\pi$
$108$	0	0
$109$	− 103.923i	− 0.953422i	−0.879060	−	0.476711i	$-0.841829\pi$
$109$	− 103.923i	− 0.953422i	0.879060	−	0.476711i	$-0.158171\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−31.1769	−0.275902	−0.137951	−	0.990439i	$-0.544052\pi$
$113$	−31.1769	−0.275902	−0.137951	+	0.990439i	$0.544052\pi$
$114$	0	0
$115$	−187.061	−1.62662
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 155.885i	− 1.30995i
$120$	0	0
$121$	−85.0000	−0.702479
$122$	0	0
$123$	0	0
$124$	0	0
$125$	84.0000i	0.672000i
$126$	0	0
$127$	110.000i	0.866142i	0.901360	+	0.433071i	$0.142570\pi$
$127$	110.000i	0.866142i	−0.901360	+	0.433071i	$0.857430\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−156.000	−1.19084	−0.595420	−	0.803415i	$-0.703014\pi$
$131$	−156.000	−1.19084	−0.595420	+	0.803415i	$0.703014\pi$
$132$	0	0
$133$	− 129.904i	− 0.976720i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	155.885	1.13784	0.568922	−	0.822392i	$-0.307361\pi$
$137$	155.885	1.13784	0.568922	+	0.822392i	$0.307361\pi$
$138$	0	0
$139$	−181.865	−1.30838	−0.654192	−	0.756329i	$-0.726991\pi$
$139$	−181.865	−1.30838	−0.654192	+	0.756329i	$0.726991\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	31.1769i	0.218020i
$144$	0	0
$145$	216.000	1.48966
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 24.0000i	− 0.161074i	−0.996752	−	0.0805369i	$-0.974337\pi$
$149$	− 24.0000i	− 0.161074i	0.996752	−	0.0805369i	$-0.0256635\pi$
$150$	0	0
$151$	− 31.0000i	− 0.205298i	−0.994718	−	0.102649i	$-0.967268\pi$
$151$	− 31.0000i	− 0.205298i	0.994718	−	0.102649i	$-0.0327318\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	156.000	1.00645
$156$	0	0
$157$	20.7846i	0.132386i	0.997807	+	0.0661930i	$0.0210853\pi$
$157$	20.7846i	0.132386i	−0.997807	+	0.0661930i	$0.978915\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−155.885	−0.968227
$162$	0	0
$163$	−306.573	−1.88082	−0.940408	−	0.340048i	$-0.889557\pi$
$163$	−306.573	−1.88082	−0.940408	+	0.340048i	$0.889557\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	155.885i	0.933441i	0.884405	+	0.466720i	$0.154565\pi$
$167$	155.885i	0.933441i	−0.884405	+	0.466720i	$0.845435\pi$
$168$	0	0
$169$	142.000	0.840237
$170$	0	0
$171$	0	0
$172$	0	0
$173$	120.000i	0.693642i	0.937931	+	0.346821i	$0.112739\pi$
$173$	120.000i	0.693642i	−0.937931	+	0.346821i	$0.887261\pi$
$174$	0	0
$175$	− 55.0000i	− 0.314286i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	192.000	1.07263	0.536313	−	0.844019i	$-0.319817\pi$
$179$	192.000	1.07263	0.536313	+	0.844019i	$0.319817\pi$
$180$	0	0
$181$	− 337.750i	− 1.86602i	−0.359848	−	0.933011i	$-0.617172\pi$
$181$	− 337.750i	− 1.86602i	0.359848	−	0.933011i	$-0.382828\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	155.885	0.842619
$186$	0	0
$187$	−187.061	−1.00033
$188$	0	0
$189$	0	0
$190$	0	0
$191$	155.885i	0.816150i	0.912948	+	0.408075i	$0.133800\pi$
$191$	155.885i	0.816150i	−0.912948	+	0.408075i	$0.866200\pi$
$192$	0	0
$193$	−325.000	−1.68394	−0.841969	−	0.539526i	$-0.818603\pi$
$193$	−325.000	−1.68394	−0.841969	+	0.539526i	$0.818603\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	30.0000i	0.152284i	0.997097	+	0.0761421i	$0.0242603\pi$
$197$	30.0000i	0.152284i	−0.997097	+	0.0761421i	$0.975740\pi$
$198$	0	0
$199$	259.000i	1.30151i	0.759289	+	0.650754i	$0.225547\pi$
$199$	259.000i	1.30151i	−0.759289	+	0.650754i	$0.774453\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	180.000	0.886700
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−155.885	−0.745859
$210$	0	0
$211$	−25.9808	−0.123132	−0.0615658	−	0.998103i	$-0.519609\pi$
$211$	−25.9808	−0.123132	−0.0615658	+	0.998103i	$0.519609\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 124.708i	− 0.580036i
$216$	0	0
$217$	130.000	0.599078
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 162.000i	− 0.733032i
$222$	0	0
$223$	− 310.000i	− 1.39013i	−0.718945	−	0.695067i	$-0.755375\pi$
$223$	− 310.000i	− 1.39013i	0.718945	−	0.695067i	$-0.244625\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−360.000	−1.58590	−0.792952	−	0.609285i	$-0.791457\pi$
$227$	−360.000	−1.58590	−0.792952	+	0.609285i	$0.791457\pi$
$228$	0	0
$229$	− 207.846i	− 0.907625i	−0.891097	−	0.453812i	$-0.850064\pi$
$229$	− 207.846i	− 0.907625i	0.891097	−	0.453812i	$-0.149936\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−311.769	−1.33807	−0.669033	−	0.743233i	$-0.733291\pi$
$233$	−311.769	−1.33807	−0.669033	+	0.743233i	$0.733291\pi$
$234$	0	0
$235$	−187.061	−0.796006
$236$	0	0
$237$	0	0
$238$	0	0
$239$	311.769i	1.30447i	0.758015	+	0.652237i	$0.226169\pi$
$239$	311.769i	1.30447i	−0.758015	+	0.652237i	$0.773831\pi$
$240$	0	0
$241$	−131.000	−0.543568	−0.271784	−	0.962358i	$-0.587614\pi$
$241$	−131.000	−0.543568	−0.271784	+	0.962358i	$0.587614\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	144.000i	0.587755i
$246$	0	0
$247$	− 135.000i	− 0.546559i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−216.000	−0.860558	−0.430279	−	0.902696i	$-0.641585\pi$
$251$	−216.000	−0.860558	−0.430279	+	0.902696i	$0.641585\pi$
$252$	0	0
$253$	187.061i	0.739373i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−187.061	−0.727866	−0.363933	−	0.931425i	$-0.618566\pi$
$257$	−187.061	−0.727866	−0.363933	+	0.931425i	$0.618566\pi$
$258$	0	0
$259$	129.904	0.501559
$260$	0	0
$261$	0	0
$262$	0	0
$263$	311.769i	1.18543i	0.805411	+	0.592717i	$0.201945\pi$
$263$	311.769i	1.18543i	−0.805411	+	0.592717i	$0.798055\pi$
$264$	0	0
$265$	−360.000	−1.35849
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 414.000i	− 1.53903i	−0.638627	−	0.769517i	$-0.720497\pi$
$269$	− 414.000i	− 1.53903i	0.638627	−	0.769517i	$-0.279503\pi$
$270$	0	0
$271$	317.000i	1.16974i	0.811126	+	0.584871i	$0.198855\pi$
$271$	317.000i	1.16974i	−0.811126	+	0.584871i	$0.801145\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−66.0000	−0.240000
$276$	0	0
$277$	− 519.615i	− 1.87587i	−0.346815	−	0.937934i	$-0.612737\pi$
$277$	− 519.615i	− 1.87587i	0.346815	−	0.937934i	$-0.387263\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−311.769	−1.10950	−0.554749	−	0.832018i	$-0.687186\pi$
$281$	−311.769	−1.10950	−0.554749	+	0.832018i	$0.687186\pi$
$282$	0	0
$283$	−103.923	−0.367219	−0.183610	−	0.982999i	$-0.558778\pi$
$283$	−103.923	−0.367219	−0.183610	+	0.982999i	$0.558778\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	683.000	2.36332
$290$	0	0
$291$	0	0
$292$	0	0
$293$	270.000i	0.921502i	0.887530	+	0.460751i	$0.152420\pi$
$293$	270.000i	0.921502i	−0.887530	+	0.460751i	$0.847580\pi$
$294$	0	0
$295$	108.000i	0.366102i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−162.000	−0.541806
$300$	0	0
$301$	− 103.923i	− 0.345259i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	467.654	1.53329
$306$	0	0
$307$	436.477	1.42175	0.710874	−	0.703319i	$-0.248299\pi$
$307$	436.477	1.42175	0.710874	+	0.703319i	$0.248299\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 467.654i	− 1.50371i	−0.659329	−	0.751855i	$-0.729159\pi$
$311$	− 467.654i	− 1.50371i	0.659329	−	0.751855i	$-0.270841\pi$
$312$	0	0
$313$	85.0000	0.271565	0.135783	−	0.990739i	$-0.456645\pi$
$313$	85.0000	0.271565	0.135783	+	0.990739i	$0.456645\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	120.000i	0.378549i	0.981924	+	0.189274i	$0.0606136\pi$
$317$	120.000i	0.378549i	−0.981924	+	0.189274i	$0.939386\pi$
$318$	0	0
$319$	− 216.000i	− 0.677116i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	810.000	2.50774
$324$	0	0
$325$	− 57.1577i	− 0.175870i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−155.885	−0.473813
$330$	0	0
$331$	25.9808	0.0784917	0.0392459	−	0.999230i	$-0.487504\pi$
$331$	25.9808	0.0784917	0.0392459	+	0.999230i	$0.487504\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 467.654i	− 1.39598i
$336$	0	0
$337$	−325.000	−0.964392	−0.482196	−	0.876063i	$-0.660161\pi$
$337$	−325.000	−0.964392	−0.482196	+	0.876063i	$0.660161\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 156.000i	− 0.457478i
$342$	0	0
$343$	365.000i	1.06414i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	300.000	0.864553	0.432277	−	0.901741i	$-0.357710\pi$
$347$	300.000	0.864553	0.432277	+	0.901741i	$0.357710\pi$
$348$	0	0
$349$	597.558i	1.71220i	0.516811	+	0.856100i	$0.327119\pi$
$349$	597.558i	1.71220i	−0.516811	+	0.856100i	$0.672881\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	436.477	1.23648	0.618239	−	0.785990i	$-0.287846\pi$
$353$	436.477	1.23648	0.618239	+	0.785990i	$0.287846\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	467.654i	1.30266i	0.758796	+	0.651328i	$0.225788\pi$
$359$	467.654i	1.30266i	−0.758796	+	0.651328i	$0.774212\pi$
$360$	0	0
$361$	314.000	0.869806
$362$	0	0
$363$	0	0
$364$	0	0
$365$	150.000i	0.410959i
$366$	0	0
$367$	− 185.000i	− 0.504087i	−0.967716	−	0.252044i	$-0.918897\pi$
$367$	− 185.000i	− 0.504087i	0.967716	−	0.252044i	$-0.0811026\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−300.000	−0.808625
$372$	0	0
$373$	57.1577i	0.153238i	0.997060	+	0.0766189i	$0.0244125\pi$
$373$	57.1577i	0.153238i	−0.997060	+	0.0766189i	$0.975588\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	187.061	0.496184
$378$	0	0
$379$	129.904	0.342754	0.171377	−	0.985206i	$-0.445178\pi$
$379$	129.904	0.342754	0.171377	+	0.985206i	$0.445178\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 187.061i	− 0.488411i	−0.969723	−	0.244206i	$-0.921473\pi$
$383$	− 187.061i	− 0.488411i	0.969723	−	0.244206i	$-0.0785272\pi$
$384$	0	0
$385$	−180.000	−0.467532
$386$	0	0
$387$	0	0
$388$	0	0
$389$	354.000i	0.910026i	0.890485	+	0.455013i	$0.150365\pi$
$389$	354.000i	0.910026i	−0.890485	+	0.455013i	$0.849635\pi$
$390$	0	0
$391$	− 972.000i	− 2.48593i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−186.000	−0.470886
$396$	0	0
$397$	644.323i	1.62298i	0.584367	+	0.811490i	$0.301343\pi$
$397$	644.323i	1.62298i	−0.584367	+	0.811490i	$0.698657\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	311.769	0.777479	0.388740	−	0.921348i	$-0.372911\pi$
$401$	311.769	0.777479	0.388740	+	0.921348i	$0.372911\pi$
$402$	0	0
$403$	135.100	0.335236
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 155.885i	− 0.383009i
$408$	0	0
$409$	−539.000	−1.31785	−0.658924	−	0.752209i	$-0.728988\pi$
$409$	−539.000	−1.31785	−0.658924	+	0.752209i	$0.728988\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	90.0000i	0.217918i
$414$	0	0
$415$	720.000i	1.73494i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−306.000	−0.730310	−0.365155	−	0.930947i	$-0.618984\pi$
$419$	−306.000	−0.730310	−0.365155	+	0.930947i	$0.618984\pi$
$420$	0	0
$421$	25.9808i	0.0617120i	0.999524	+	0.0308560i	$0.00982333\pi$
$421$	25.9808i	0.0617120i	−0.999524	+	0.0308560i	$0.990177\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	342.946	0.806932
$426$	0	0
$427$	389.711	0.912673
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 155.885i	− 0.361681i	−0.983512	−	0.180841i	$-0.942118\pi$
$431$	− 155.885i	− 0.361681i	0.983512	−	0.180841i	$-0.0578818\pi$
$432$	0	0
$433$	250.000	0.577367	0.288684	−	0.957425i	$-0.406782\pi$
$433$	250.000	0.577367	0.288684	+	0.957425i	$0.406782\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 810.000i	− 1.85355i
$438$	0	0
$439$	22.0000i	0.0501139i	0.999686	+	0.0250569i	$0.00797671\pi$
$439$	22.0000i	0.0501139i	−0.999686	+	0.0250569i	$0.992023\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−540.000	−1.21896	−0.609481	−	0.792801i	$-0.708622\pi$
$443$	−540.000	−1.21896	−0.609481	+	0.792801i	$0.708622\pi$
$444$	0	0
$445$	935.307i	2.10181i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	155.885	0.347182	0.173591	−	0.984818i	$-0.444463\pi$
$449$	155.885	0.347182	0.173591	+	0.984818i	$0.444463\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 155.885i	− 0.342603i
$456$	0	0
$457$	−230.000	−0.503282	−0.251641	−	0.967821i	$-0.580970\pi$
$457$	−230.000	−0.503282	−0.251641	+	0.967821i	$0.580970\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	78.0000i	0.169197i	0.996415	+	0.0845987i	$0.0269608\pi$
$461$	78.0000i	0.169197i	−0.996415	+	0.0845987i	$0.973039\pi$
$462$	0	0
$463$	− 55.0000i	− 0.118790i	−0.998235	−	0.0593952i	$-0.981083\pi$
$463$	− 55.0000i	− 0.118790i	0.998235	−	0.0593952i	$-0.0189172\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−150.000	−0.321199	−0.160600	−	0.987020i	$-0.551343\pi$
$467$	−150.000	−0.321199	−0.160600	+	0.987020i	$0.551343\pi$
$468$	0	0
$469$	− 389.711i	− 0.830941i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−124.708	−0.263653
$474$	0	0
$475$	285.788	0.601660
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 311.769i	− 0.650875i	−0.945564	−	0.325438i	$-0.894488\pi$
$479$	− 311.769i	− 0.650875i	0.945564	−	0.325438i	$-0.105512\pi$
$480$	0	0
$481$	135.000	0.280665
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 510.000i	− 1.05155i
$486$	0	0
$487$	785.000i	1.61191i	0.591977	+	0.805955i	$0.298348\pi$
$487$	785.000i	1.61191i	−0.591977	+	0.805955i	$0.701652\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	618.000	1.25866	0.629328	−	0.777140i	$-0.283330\pi$
$491$	618.000	1.25866	0.629328	+	0.777140i	$0.283330\pi$
$492$	0	0
$493$	1122.37i	2.27661i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	623.538	1.24958	0.624788	−	0.780795i	$-0.285185\pi$
$499$	623.538	1.24958	0.624788	+	0.780795i	$0.285185\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 654.715i	− 1.30162i	−0.759240	−	0.650810i	$-0.774429\pi$
$503$	− 654.715i	− 1.30162i	0.759240	−	0.650810i	$-0.225571\pi$
$504$	0	0
$505$	936.000	1.85347
$506$	0	0
$507$	0	0
$508$	0	0
$509$	366.000i	0.719057i	0.933134	+	0.359528i	$0.117062\pi$
$509$	366.000i	0.719057i	−0.933134	+	0.359528i	$0.882938\pi$
$510$	0	0
$511$	125.000i	0.244618i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	750.000	1.45631
$516$	0	0
$517$	187.061i	0.361821i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−779.423	−1.49601	−0.748007	−	0.663691i	$-0.768989\pi$
$521$	−779.423	−1.49601	−0.748007	+	0.663691i	$0.768989\pi$
$522$	0	0
$523$	109.119	0.208641	0.104320	−	0.994544i	$-0.466733\pi$
$523$	109.119	0.208641	0.104320	+	0.994544i	$0.466733\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	810.600i	1.53814i
$528$	0	0
$529$	−443.000	−0.837429
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 900.000i	− 1.68224i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	144.000	0.267161
$540$	0	0
$541$	649.519i	1.20059i	0.799779	+	0.600295i	$0.204950\pi$
$541$	649.519i	1.20059i	−0.799779	+	0.600295i	$0.795050\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	623.538	1.14411
$546$	0	0
$547$	420.888	0.769449	0.384724	−	0.923032i	$-0.374297\pi$
$547$	420.888	0.769449	0.384724	+	0.923032i	$0.374297\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	935.307i	1.69747i
$552$	0	0
$553$	−155.000	−0.280289
$554$	0	0
$555$	0	0
$556$	0	0
$557$	690.000i	1.23878i	0.785084	+	0.619390i	$0.212620\pi$
$557$	690.000i	1.23878i	−0.785084	+	0.619390i	$0.787380\pi$
$558$	0	0
$559$	− 108.000i	− 0.193202i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−840.000	−1.49201	−0.746004	−	0.665942i	$-0.768030\pi$
$563$	−840.000	−1.49201	−0.746004	+	0.665942i	$0.768030\pi$
$564$	0	0
$565$	− 187.061i	− 0.331082i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1091.19	−1.91774	−0.958868	−	0.283852i	$-0.908388\pi$
$569$	−1091.19	−1.91774	−0.958868	+	0.283852i	$0.908388\pi$
$570$	0	0
$571$	805.404	1.41051	0.705257	−	0.708952i	$-0.250832\pi$
$571$	805.404	1.41051	0.705257	+	0.708952i	$0.250832\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 342.946i	− 0.596428i
$576$	0	0
$577$	385.000	0.667244	0.333622	−	0.942707i	$-0.391729\pi$
$577$	385.000	0.667244	0.333622	+	0.942707i	$0.391729\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	600.000i	1.03270i
$582$	0	0
$583$	360.000i	0.617496i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	930.000	1.58433	0.792164	−	0.610309i	$-0.208955\pi$
$587$	930.000	1.58433	0.792164	+	0.610309i	$0.208955\pi$
$588$	0	0
$589$	675.500i	1.14686i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−498.831	−0.841198	−0.420599	−	0.907247i	$-0.638180\pi$
$593$	−498.831	−0.841198	−0.420599	+	0.907247i	$0.638180\pi$
$594$	0	0
$595$	935.307	1.57195
$596$	0	0
$597$	0	0
$598$	0	0
$599$	311.769i	0.520483i	0.965544	+	0.260241i	$0.0838021\pi$
$599$	311.769i	0.520483i	−0.965544	+	0.260241i	$0.916198\pi$
$600$	0	0
$601$	−346.000	−0.575707	−0.287854	−	0.957674i	$-0.592942\pi$
$601$	−346.000	−0.575707	−0.287854	+	0.957674i	$0.592942\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 510.000i	− 0.842975i
$606$	0	0
$607$	− 595.000i	− 0.980231i	−0.871658	−	0.490115i	$-0.836955\pi$
$607$	− 595.000i	− 0.980231i	0.871658	−	0.490115i	$-0.163045\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−162.000	−0.265139
$612$	0	0
$613$	109.119i	0.178008i	0.996031	+	0.0890042i	$0.0283685\pi$
$613$	109.119i	0.178008i	−0.996031	+	0.0890042i	$0.971632\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−592.361	−0.960067	−0.480034	−	0.877250i	$-0.659376\pi$
$617$	−592.361	−0.960067	−0.480034	+	0.877250i	$0.659376\pi$
$618$	0	0
$619$	701.481	1.13325	0.566624	−	0.823976i	$-0.308249\pi$
$619$	701.481	1.13325	0.566624	+	0.823976i	$0.308249\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	779.423i	1.25108i
$624$	0	0
$625$	−779.000	−1.24640
$626$	0	0
$627$	0	0
$628$	0	0
$629$	810.000i	1.28776i
$630$	0	0
$631$	713.000i	1.12995i	0.825107	+	0.564976i	$0.191115\pi$
$631$	713.000i	1.12995i	−0.825107	+	0.564976i	$0.808885\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−660.000	−1.03937
$636$	0	0
$637$	124.708i	0.195773i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	−436.477	−0.678813	−0.339407	−	0.940640i	$-0.610226\pi$
$643$	−436.477	−0.678813	−0.339407	+	0.940640i	$0.610226\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	623.538i	0.963738i	0.876243	+	0.481869i	$0.160042\pi$
$647$	623.538i	0.963738i	−0.876243	+	0.481869i	$0.839958\pi$
$648$	0	0
$649$	108.000	0.166410
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 960.000i	− 1.47014i	−0.677992	−	0.735069i	$-0.737150\pi$
$653$	− 960.000i	− 1.47014i	0.677992	−	0.735069i	$-0.262850\pi$
$654$	0	0
$655$	− 936.000i	− 1.42901i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−132.000	−0.200303	−0.100152	−	0.994972i	$-0.531933\pi$
$659$	−132.000	−0.200303	−0.100152	+	0.994972i	$0.531933\pi$
$660$	0	0
$661$	− 1013.25i	− 1.53290i	−0.642301	−	0.766452i	$-0.722020\pi$
$661$	− 1013.25i	− 1.53290i	0.642301	−	0.766452i	$-0.277980\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	779.423	1.17206
$666$	0	0
$667$	1122.37	1.68271
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 467.654i	− 0.696950i
$672$	0	0
$673$	445.000	0.661218	0.330609	−	0.943768i	$-0.392746\pi$
$673$	445.000	0.661218	0.330609	+	0.943768i	$0.392746\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 990.000i	− 1.46233i	−0.682199	−	0.731167i	$-0.738976\pi$
$677$	− 990.000i	− 1.46233i	0.682199	−	0.731167i	$-0.261024\pi$
$678$	0	0
$679$	− 425.000i	− 0.625920i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1260.00	−1.84480	−0.922401	−	0.386233i	$-0.873776\pi$
$683$	−1260.00	−1.84480	−0.922401	+	0.386233i	$0.873776\pi$
$684$	0	0
$685$	935.307i	1.36541i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−311.769	−0.452495
$690$	0	0
$691$	103.923	0.150395	0.0751976	−	0.997169i	$-0.476041\pi$
$691$	103.923	0.150395	0.0751976	+	0.997169i	$0.476041\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1091.19i	− 1.57006i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 246.000i	− 0.350927i	−0.984486	−	0.175464i	$-0.943858\pi$
$701$	− 246.000i	− 0.350927i	0.984486	−	0.175464i	$-0.0561424\pi$
$702$	0	0
$703$	675.000i	0.960171i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	780.000	1.10325
$708$	0	0
$709$	129.904i	0.183221i	0.995795	+	0.0916106i	$0.0292015\pi$
$709$	129.904i	0.183221i	−0.995795	+	0.0916106i	$0.970799\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	810.600	1.13689
$714$	0	0
$715$	−187.061	−0.261624
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1247.08i	1.73446i	0.497908	+	0.867230i	$0.334102\pi$
$719$	1247.08i	1.73446i	−0.497908	+	0.867230i	$0.665898\pi$
$720$	0	0
$721$	625.000	0.866852
$722$	0	0
$723$	0	0
$724$	0	0
$725$	396.000i	0.546207i
$726$	0	0
$727$	770.000i	1.05915i	0.848264	+	0.529574i	$0.177648\pi$
$727$	770.000i	1.05915i	−0.848264	+	0.529574i	$0.822352\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	648.000	0.886457
$732$	0	0
$733$	1143.15i	1.55955i	0.626057	+	0.779777i	$0.284668\pi$
$733$	1143.15i	1.55955i	−0.626057	+	0.779777i	$0.715332\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−467.654	−0.634537
$738$	0	0
$739$	−727.461	−0.984386	−0.492193	−	0.870486i	$-0.663805\pi$
$739$	−727.461	−0.984386	−0.492193	+	0.870486i	$0.663805\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	592.361i	0.797256i	0.917113	+	0.398628i	$0.130514\pi$
$743$	592.361i	0.797256i	−0.917113	+	0.398628i	$0.869486\pi$
$744$	0	0
$745$	144.000	0.193289
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 750.000i	− 1.00134i
$750$	0	0
$751$	1099.00i	1.46338i	0.681636	+	0.731691i	$0.261269\pi$
$751$	1099.00i	1.46338i	−0.681636	+	0.731691i	$0.738731\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	186.000	0.246358
$756$	0	0
$757$	420.888i	0.555995i	0.960582	+	0.277998i	$0.0896707\pi$
$757$	420.888i	0.555995i	−0.960582	+	0.277998i	$0.910329\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−155.885	−0.204842	−0.102421	−	0.994741i	$-0.532659\pi$
$761$	−155.885	−0.204842	−0.102421	+	0.994741i	$0.532659\pi$
$762$	0	0
$763$	519.615	0.681016
$764$	0	0
$765$	0	0
$766$	0	0
$767$	93.5307i	0.121944i
$768$	0	0
$769$	−1009.00	−1.31209	−0.656047	−	0.754720i	$-0.727773\pi$
$769$	−1009.00	−1.31209	−0.656047	+	0.754720i	$0.727773\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	360.000i	0.465718i	0.972511	+	0.232859i	$0.0748081\pi$
$773$	360.000i	0.465718i	−0.972511	+	0.232859i	$0.925192\pi$
$774$	0	0
$775$	286.000i	0.369032i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−124.708	−0.158863
$786$	0	0
$787$	514.419	0.653646	0.326823	−	0.945086i	$-0.394022\pi$
$787$	514.419	0.653646	0.326823	+	0.945086i	$0.394022\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 155.885i	− 0.197073i
$792$	0	0
$793$	405.000	0.510719
$794$	0	0
$795$	0	0
$796$	0	0
$797$	660.000i	0.828105i	0.910253	+	0.414053i	$0.135887\pi$
$797$	660.000i	0.828105i	−0.910253	+	0.414053i	$0.864113\pi$
$798$	0	0
$799$	− 972.000i	− 1.21652i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	150.000	0.186800
$804$	0	0
$805$	− 935.307i	− 1.16187i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1247.08	1.54150	0.770752	−	0.637135i	$-0.219881\pi$
$809$	1247.08	1.54150	0.770752	+	0.637135i	$0.219881\pi$
$810$	0	0
$811$	−519.615	−0.640709	−0.320355	−	0.947298i	$-0.603802\pi$
$811$	−519.615	−0.640709	−0.320355	+	0.947298i	$0.603802\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1839.44i	− 2.25698i
$816$	0	0
$817$	540.000	0.660955
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1242.00i	1.51279i	0.654116	+	0.756395i	$0.273041\pi$
$821$	1242.00i	1.51279i	−0.654116	+	0.756395i	$0.726959\pi$
$822$	0	0
$823$	− 955.000i	− 1.16039i	−0.814478	−	0.580194i	$-0.802977\pi$
$823$	− 955.000i	− 1.16039i	0.814478	−	0.580194i	$-0.197023\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−90.0000	−0.108827	−0.0544135	−	0.998518i	$-0.517329\pi$
$827$	−90.0000	−0.108827	−0.0544135	+	0.998518i	$0.517329\pi$
$828$	0	0
$829$	181.865i	0.219379i	0.993966	+	0.109690i	$0.0349857\pi$
$829$	181.865i	0.219379i	−0.993966	+	0.109690i	$0.965014\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−748.246	−0.898254
$834$	0	0
$835$	−935.307	−1.12013
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1558.85i	− 1.85798i	−0.370104	−	0.928990i	$-0.620678\pi$
$839$	− 1558.85i	− 1.85798i	0.370104	−	0.928990i	$-0.379322\pi$
$840$	0	0
$841$	−455.000	−0.541023
$842$	0	0
$843$	0	0
$844$	0	0
$845$	852.000i	1.00828i
$846$	0	0
$847$	− 425.000i	− 0.501771i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	810.000	0.951821
$852$	0	0
$853$	− 46.7654i	− 0.0548246i	−0.999624	−	0.0274123i	$-0.991273\pi$
$853$	− 46.7654i	− 0.0548246i	0.999624	−	0.0274123i	$-0.00872670\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	623.538	0.727583	0.363791	−	0.931480i	$-0.381482\pi$
$857$	623.538	0.727583	0.363791	+	0.931480i	$0.381482\pi$
$858$	0	0
$859$	−129.904	−0.151227	−0.0756134	−	0.997137i	$-0.524091\pi$
$859$	−129.904	−0.151227	−0.0756134	+	0.997137i	$0.524091\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 342.946i	− 0.397388i	−0.980062	−	0.198694i	$-0.936330\pi$
$863$	− 342.946i	− 0.397388i	0.980062	−	0.198694i	$-0.0636700\pi$
$864$	0	0
$865$	−720.000	−0.832370
$866$	0	0
$867$	0	0
$868$	0	0
$869$	186.000i	0.214039i
$870$	0	0
$871$	− 405.000i	− 0.464983i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−420.000	−0.480000
$876$	0	0
$877$	1034.03i	1.17906i	0.807747	+	0.589529i	$0.200687\pi$
$877$	1034.03i	1.17906i	−0.807747	+	0.589529i	$0.799313\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	467.654	0.530821	0.265411	−	0.964135i	$-0.414492\pi$
$881$	467.654	0.530821	0.265411	+	0.964135i	$0.414492\pi$
$882$	0	0
$883$	−285.788	−0.323656	−0.161828	−	0.986819i	$-0.551739\pi$
$883$	−285.788	−0.323656	−0.161828	+	0.986819i	$0.551739\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 498.831i	− 0.562380i	−0.959652	−	0.281190i	$-0.909271\pi$
$887$	− 498.831i	− 0.562380i	0.959652	−	0.281190i	$-0.0907290\pi$
$888$	0	0
$889$	−550.000	−0.618673
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 810.000i	− 0.907055i
$894$	0	0
$895$	1152.00i	1.28715i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−936.000	−1.04116
$900$	0	0
$901$	− 1870.61i	− 2.07615i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2026.50	2.23923
$906$	0	0
$907$	−1096.39	−1.20881	−0.604404	−	0.796678i	$-0.706589\pi$
$907$	−1096.39	−1.20881	−0.604404	+	0.796678i	$0.706589\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	623.538i	0.684455i	0.939617	+	0.342227i	$0.111181\pi$
$911$	623.538i	0.684455i	−0.939617	+	0.342227i	$0.888819\pi$
$912$	0	0
$913$	720.000	0.788609
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 780.000i	− 0.850600i
$918$	0	0
$919$	− 1006.00i	− 1.09467i	−0.836914	−	0.547334i	$-0.815643\pi$
$919$	− 1006.00i	− 1.09467i	0.836914	−	0.547334i	$-0.184357\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	285.788i	0.308960i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−311.769	−0.335596	−0.167798	−	0.985821i	$-0.553666\pi$
$929$	−311.769	−0.335596	−0.167798	+	0.985821i	$0.553666\pi$
$930$	0	0
$931$	−623.538	−0.669751
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1122.37i	− 1.20039i
$936$	0	0
$937$	565.000	0.602988	0.301494	−	0.953468i	$-0.402515\pi$
$937$	565.000	0.602988	0.301494	+	0.953468i	$0.402515\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	294.000i	0.312434i	0.987723	+	0.156217i	$0.0499298\pi$
$941$	294.000i	0.312434i	−0.987723	+	0.156217i	$0.950070\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.0000	0.0316790	0.0158395	−	0.999875i	$-0.494958\pi$
$947$	30.0000	0.0316790	0.0158395	+	0.999875i	$0.494958\pi$
$948$	0	0
$949$	129.904i	0.136885i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−904.131	−0.948720	−0.474360	−	0.880331i	$-0.657321\pi$
$953$	−904.131	−0.948720	−0.474360	+	0.880331i	$0.657321\pi$
$954$	0	0
$955$	−935.307	−0.979380
$956$	0	0
$957$	0	0
$958$	0	0
$959$	779.423i	0.812745i
$960$	0	0
$961$	285.000	0.296566
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1950.00i	− 2.02073i
$966$	0	0
$967$	− 305.000i	− 0.315408i	−0.987486	−	0.157704i	$-0.949591\pi$
$967$	− 305.000i	− 0.315408i	0.987486	−	0.157704i	$-0.0504093\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1194.00	1.22966	0.614830	−	0.788660i	$-0.289225\pi$
$971$	1194.00	1.22966	0.614830	+	0.788660i	$0.289225\pi$
$972$	0	0
$973$	− 909.327i	− 0.934560i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−124.708	−0.127643	−0.0638217	−	0.997961i	$-0.520329\pi$
$977$	−124.708	−0.127643	−0.0638217	+	0.997961i	$0.520329\pi$
$978$	0	0
$979$	935.307	0.955370
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1278.25i	1.30036i	0.759780	+	0.650180i	$0.225306\pi$
$983$	1278.25i	1.30036i	−0.759780	+	0.650180i	$0.774694\pi$
$984$	0	0
$985$	−180.000	−0.182741
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 648.000i	− 0.655207i
$990$	0	0
$991$	511.000i	0.515641i	0.966193	+	0.257820i	$0.0830043\pi$
$991$	511.000i	0.515641i	−0.966193	+	0.257820i	$0.916996\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1554.00	−1.56181
$996$	0	0
$997$	− 1060.02i	− 1.06320i	−0.846994	−	0.531602i	$-0.821590\pi$
$997$	− 1060.02i	− 1.06320i	0.846994	−	0.531602i	$-0.178410\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.b.f.1567.4	yes	4
3.2	odd	2		1728.3.b.a.1567.2	yes	4
4.3	odd	2		1728.3.b.a.1567.4	yes	4
8.3	odd	2	inner	1728.3.b.f.1567.1	yes	4
8.5	even	2		1728.3.b.a.1567.1	✓	4
12.11	even	2	inner	1728.3.b.f.1567.2	yes	4
24.5	odd	2	inner	1728.3.b.f.1567.3	yes	4
24.11	even	2		1728.3.b.a.1567.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.3.b.a.1567.1	✓	4	8.5	even	2
1728.3.b.a.1567.2	yes	4	3.2	odd	2
1728.3.b.a.1567.3	yes	4	24.11	even	2
1728.3.b.a.1567.4	yes	4	4.3	odd	2
1728.3.b.f.1567.1	yes	4	8.3	odd	2	inner
1728.3.b.f.1567.2	yes	4	12.11	even	2	inner
1728.3.b.f.1567.3	yes	4	24.5	odd	2	inner
1728.3.b.f.1567.4	yes	4	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+6.00000i q^{5} +5.00000i q^{7} +6.00000 q^{11} +5.19615i q^{13} -31.1769 q^{17} -25.9808 q^{19} +31.1769i q^{23} -11.0000 q^{25} -36.0000i q^{29} -26.0000i q^{31} -30.0000 q^{35} -25.9808i q^{37} -20.7846 q^{43} +31.1769i q^{47} +24.0000 q^{49} +60.0000i q^{53} +36.0000i q^{55} +18.0000 q^{59} -77.9423i q^{61} -31.1769 q^{65} -77.9423 q^{67} +25.0000 q^{73} +30.0000i q^{77} +31.0000i q^{79} +120.000 q^{83} -187.061i q^{85} +155.885 q^{89} -25.9808 q^{91} -155.885i q^{95} -85.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 24 q^{11} - 44 q^{25} - 120 q^{35} + 96 q^{49} + 72 q^{59} + 100 q^{73} + 480 q^{83} - 340 q^{97}+O(q^{100}) \) 4 * q + 24 * q^11 - 44 * q^25 - 120 * q^35 + 96 * q^49 + 72 * q^59 + 100 * q^73 + 480 * q^83 - 340 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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