LMFDB - Embedded newform 576.3.m.c.559.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [576,3,Mod(271,576)] 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("576.271"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(576, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	576.m (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$15.6948632272$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + \cdots + 65536 $ x^16 - 6x^14 - 4x^13 + 10x^12 + 56x^11 + 88x^10 - 128x^9 - 496x^8 - 512x^7 + 1408x^6 + 3584x^5 + 2560x^4 - 4096x^3 - 24576*x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			559.2
Root			$0.125358 - 1.99607i$ of defining polynomial
Character	$\chi$	$=$	576.559
Dual form			576.3.m.c.271.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.32679 + 3.32679i) q^{5} +4.04088 q^{7} +(6.82458 + 6.82458i) q^{11} +(4.29091 + 4.29091i) q^{13} -30.1192 q^{17} +(19.7548 - 19.7548i) q^{19} -28.2345 q^{23} +2.86488i q^{25} +(21.3607 + 21.3607i) q^{29} +38.0396i q^{31} +(-13.4432 + 13.4432i) q^{35} +(-42.8916 + 42.8916i) q^{37} +48.2343i q^{41} +(-32.6765 - 32.6765i) q^{43} +15.8305i q^{47} -32.6713 q^{49} +(0.476870 - 0.476870i) q^{53} -45.4079 q^{55} +(9.97719 + 9.97719i) q^{59} +(37.9455 + 37.9455i) q^{61} -28.5500 q^{65} +(-20.0705 + 20.0705i) q^{67} +40.0818 q^{71} -30.8095i q^{73} +(27.5773 + 27.5773i) q^{77} +130.125i q^{79} +(-2.26155 + 2.26155i) q^{83} +(100.200 - 100.200i) q^{85} -72.2232i q^{89} +(17.3391 + 17.3391i) q^{91} +131.441i q^{95} -112.343 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{11} + 32 q^{19} - 128 q^{23} - 32 q^{29} + 96 q^{35} - 96 q^{37} - 160 q^{43} + 112 q^{49} + 160 q^{53} + 256 q^{55} - 128 q^{59} - 32 q^{61} + 32 q^{65} - 320 q^{67} + 512 q^{71} - 224 q^{77}+ \cdots + 480 q^{91}+O(q^{100}) $ 16 * q + 32 * q^11 + 32 * q^19 - 128 * q^23 - 32 * q^29 + 96 * q^35 - 96 * q^37 - 160 * q^43 + 112 * q^49 + 160 * q^53 + 256 * q^55 - 128 * q^59 - 32 * q^61 + 32 * q^65 - 320 * q^67 + 512 * q^71 - 224 * q^77 - 160 * q^83 + 160 * q^85 + 480 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$5$	−3.32679	+	3.32679i	−0.665359	+	0.665359i	−0.956638	−	0.291279i	$-0.905919\pi$
$5$	−3.32679	+	3.32679i	−0.665359	+	0.665359i	0.291279	+	0.956638i	$0.405919\pi$
$6$		0			0
$7$	4.04088			0.577269			0.288635	−	0.957439i	$-0.406799\pi$
$7$	4.04088			0.577269			0.288635	+	0.957439i	$0.406799\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	6.82458	+	6.82458i	0.620416	+	0.620416i	0.945638	−	0.325222i	$-0.105439\pi$
$11$	6.82458	+	6.82458i	0.620416	+	0.620416i	−0.325222	+	0.945638i	$0.605439\pi$
$12$		0			0
$13$	4.29091	+	4.29091i	0.330070	+	0.330070i	0.852613	−	0.522543i	$-0.175017\pi$
$13$	4.29091	+	4.29091i	0.330070	+	0.330070i	−0.522543	+	0.852613i	$0.675017\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−30.1192			−1.77172			−0.885859	−	0.463954i	$-0.846430\pi$
$17$	−30.1192			−1.77172			−0.885859	+	0.463954i	$0.846430\pi$
$18$		0			0
$19$	19.7548	−	19.7548i	1.03973	−	1.03973i	0.0405505	−	0.999177i	$-0.487089\pi$
$19$	19.7548	−	19.7548i	1.03973	−	1.03973i	0.999177	−	0.0405505i	$-0.0129112\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−28.2345			−1.22759			−0.613794	−	0.789466i	$-0.710358\pi$
$23$	−28.2345			−1.22759			−0.613794	+	0.789466i	$0.710358\pi$
$24$		0			0
$25$			2.86488i			0.114595i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	21.3607	+	21.3607i	0.736575	+	0.736575i	0.971914	−	0.235338i	$-0.0756198\pi$
$29$	21.3607	+	21.3607i	0.736575	+	0.736575i	−0.235338	+	0.971914i	$0.575620\pi$
$30$		0			0
$31$			38.0396i			1.22708i	0.789662	+	0.613541i	$0.210256\pi$
$31$			38.0396i			1.22708i	−0.789662	+	0.613541i	$0.789744\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−13.4432	+	13.4432i	−0.384091	+	0.384091i
$36$		0			0
$37$	−42.8916	+	42.8916i	−1.15923	+	1.15923i	−0.174590	+	0.984641i	$0.555860\pi$
$37$	−42.8916	+	42.8916i	−1.15923	+	1.15923i	−0.984641	+	0.174590i	$0.944140\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			48.2343i			1.17645i	0.808699	+	0.588223i	$0.200172\pi$
$41$			48.2343i			1.17645i	−0.808699	+	0.588223i	$0.799828\pi$
$42$		0			0
$43$	−32.6765	−	32.6765i	−0.759918	−	0.759918i	0.216389	−	0.976307i	$-0.430572\pi$
$43$	−32.6765	−	32.6765i	−0.759918	−	0.759918i	−0.976307	+	0.216389i	$0.930572\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			15.8305i			0.336818i	0.985717	+	0.168409i	$0.0538630\pi$
$47$			15.8305i			0.336818i	−0.985717	+	0.168409i	$0.946137\pi$
$48$		0			0
$49$	−32.6713			−0.666760
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.476870	−	0.476870i	0.00899755	−	0.00899755i	−0.702594	−	0.711591i	$-0.747975\pi$
$53$	0.476870	−	0.476870i	0.00899755	−	0.00899755i	0.711591	+	0.702594i	$0.247975\pi$
$54$		0			0
$55$	−45.4079			−0.825599
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.97719	+	9.97719i	0.169105	+	0.169105i	0.786586	−	0.617481i	$-0.211847\pi$
$59$	9.97719	+	9.97719i	0.169105	+	0.169105i	−0.617481	+	0.786586i	$0.711847\pi$
$60$		0			0
$61$	37.9455	+	37.9455i	0.622057	+	0.622057i	0.946057	−	0.324000i	$-0.105028\pi$
$61$	37.9455	+	37.9455i	0.622057	+	0.622057i	−0.324000	+	0.946057i	$0.605028\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−28.5500			−0.439230
$66$		0			0
$67$	−20.0705	+	20.0705i	−0.299559	+	0.299559i	−0.840841	−	0.541282i	$-0.817939\pi$
$67$	−20.0705	+	20.0705i	−0.299559	+	0.299559i	0.541282	+	0.840841i	$0.317939\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	40.0818			0.564532			0.282266	−	0.959336i	$-0.408914\pi$
$71$	40.0818			0.564532			0.282266	+	0.959336i	$0.408914\pi$
$72$		0			0
$73$		−	30.8095i		−	0.422049i	−0.977481	−	0.211024i	$-0.932320\pi$
$73$		−	30.8095i		−	0.422049i	0.977481	−	0.211024i	$-0.0676799\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	27.5773	+	27.5773i	0.358147	+	0.358147i
$78$		0			0
$79$			130.125i			1.64716i	0.567203	+	0.823578i	$0.308025\pi$
$79$			130.125i			1.64716i	−0.567203	+	0.823578i	$0.691975\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.26155	+	2.26155i	−0.0272476	+	0.0272476i	−0.720599	−	0.693352i	$-0.756133\pi$
$83$	−2.26155	+	2.26155i	−0.0272476	+	0.0272476i	0.693352	+	0.720599i	$0.256133\pi$
$84$		0			0
$85$	100.200	−	100.200i	1.17883	−	1.17883i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	72.2232i		−	0.811496i	−0.913985	−	0.405748i	$-0.867011\pi$
$89$		−	72.2232i		−	0.811496i	0.913985	−	0.405748i	$-0.132989\pi$
$90$		0			0
$91$	17.3391	+	17.3391i	0.190539	+	0.190539i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			131.441i			1.38358i
$96$		0			0
$97$	−112.343			−1.15817			−0.579085	−	0.815267i	$-0.696590\pi$
$97$	−112.343			−1.15817			−0.579085	+	0.815267i	$0.696590\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.61933	−	1.61933i	0.0160330	−	0.0160330i	−0.699045	−	0.715078i	$-0.746391\pi$
$101$	1.61933	−	1.61933i	0.0160330	−	0.0160330i	0.715078	+	0.699045i	$0.246391\pi$
$102$		0			0
$103$	−27.9974			−0.271819			−0.135910	−	0.990721i	$-0.543396\pi$
$103$	−27.9974			−0.271819			−0.135910	+	0.990721i	$0.543396\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−40.3835	−	40.3835i	−0.377416	−	0.377416i	0.492753	−	0.870169i	$-0.335990\pi$
$107$	−40.3835	−	40.3835i	−0.377416	−	0.377416i	−0.870169	+	0.492753i	$0.835990\pi$
$108$		0			0
$109$	36.8336	+	36.8336i	0.337923	+	0.337923i	0.855585	−	0.517662i	$-0.173198\pi$
$109$	36.8336	+	36.8336i	0.337923	+	0.337923i	−0.517662	+	0.855585i	$0.673198\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	55.5952			0.491993			0.245997	−	0.969271i	$-0.420885\pi$
$113$	55.5952			0.491993			0.245997	+	0.969271i	$0.420885\pi$
$114$		0			0
$115$	93.9305	−	93.9305i	0.816787	−	0.816787i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−121.708			−1.02276
$120$		0			0
$121$		−	27.8503i		−	0.230167i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−92.7007	−	92.7007i	−0.741606	−	0.741606i
$126$		0			0
$127$		−	109.927i		−	0.865569i	−0.901497	−	0.432785i	$-0.857531\pi$
$127$		−	109.927i		−	0.865569i	0.901497	−	0.432785i	$-0.142469\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	75.6795	−	75.6795i	0.577706	−	0.577706i	−0.356565	−	0.934271i	$-0.616052\pi$
$131$	75.6795	−	75.6795i	0.577706	−	0.577706i	0.934271	+	0.356565i	$0.116052\pi$
$132$		0			0
$133$	79.8270	−	79.8270i	0.600203	−	0.600203i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	2.14751i		−	0.0156752i	−0.999969	−	0.00783762i	$-0.997505\pi$
$137$		−	2.14751i		−	0.0156752i	0.999969	−	0.00783762i	$-0.00249482\pi$
$138$		0			0
$139$	109.246	+	109.246i	0.785941	+	0.785941i	0.980826	−	0.194885i	$-0.0624334\pi$
$139$	109.246	+	109.246i	0.785941	+	0.785941i	−0.194885	+	0.980826i	$0.562433\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			58.5673i			0.409562i
$144$		0			0
$145$	−142.125			−0.980174
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−79.6950	+	79.6950i	−0.534866	+	0.534866i	−0.922016	−	0.387151i	$-0.873459\pi$
$149$	−79.6950	+	79.6950i	−0.534866	+	0.534866i	0.387151	+	0.922016i	$0.373459\pi$
$150$		0			0
$151$	−105.546			−0.698982			−0.349491	−	0.936940i	$-0.613645\pi$
$151$	−105.546			−0.698982			−0.349491	+	0.936940i	$0.613645\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−126.550	−	126.550i	−0.816451	−	0.816451i
$156$		0			0
$157$	190.060	+	190.060i	1.21057	+	1.21057i	0.970839	+	0.239733i	$0.0770598\pi$
$157$	190.060	+	190.060i	1.21057	+	1.21057i	0.239733	+	0.970839i	$0.422940\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−114.092			−0.708649
$162$		0			0
$163$	59.4130	−	59.4130i	0.364497	−	0.364497i	−0.500969	−	0.865465i	$-0.667023\pi$
$163$	59.4130	−	59.4130i	0.364497	−	0.364497i	0.865465	+	0.500969i	$0.167023\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	65.3894			0.391553			0.195777	−	0.980649i	$-0.437277\pi$
$167$	65.3894			0.391553			0.195777	+	0.980649i	$0.437277\pi$
$168$		0			0
$169$		−	132.176i		−	0.782107i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	212.939	+	212.939i	1.23086	+	1.23086i	0.963633	+	0.267228i	$0.0861077\pi$
$173$	212.939	+	212.939i	1.23086	+	1.23086i	0.267228	+	0.963633i	$0.413892\pi$
$174$		0			0
$175$			11.5766i			0.0661522i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	196.852	−	196.852i	1.09973	−	1.09973i	0.105289	−	0.994442i	$-0.466423\pi$
$179$	196.852	−	196.852i	1.09973	−	1.09973i	0.994442	−	0.105289i	$-0.0335768\pi$
$180$		0			0
$181$	−27.4330	+	27.4330i	−0.151564	+	0.151564i	−0.778816	−	0.627252i	$-0.784179\pi$
$181$	−27.4330	+	27.4330i	−0.151564	+	0.151564i	0.627252	+	0.778816i	$0.284179\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	285.383i		−	1.54261i
$186$		0			0
$187$	−205.551	−	205.551i	−1.09920	−	1.09920i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	244.409i		−	1.27963i	−0.768530	−	0.639814i	$-0.779011\pi$
$191$		−	244.409i		−	1.27963i	0.768530	−	0.639814i	$-0.220989\pi$
$192$		0			0
$193$	255.040			1.32145			0.660726	−	0.750627i	$-0.270249\pi$
$193$	255.040			1.32145			0.660726	+	0.750627i	$0.270249\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	194.229	−	194.229i	0.985936	−	0.985936i	−0.0139666	−	0.999902i	$-0.504446\pi$
$197$	194.229	−	194.229i	0.985936	−	0.985936i	0.999902	+	0.0139666i	$0.00444586\pi$
$198$		0			0
$199$	169.797			0.853252			0.426626	−	0.904428i	$-0.359702\pi$
$199$	169.797			0.853252			0.426626	+	0.904428i	$0.359702\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	86.3160	+	86.3160i	0.425202	+	0.425202i
$204$		0			0
$205$	−160.466	−	160.466i	−0.782759	−	0.782759i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	269.637			1.29013
$210$		0			0
$211$	132.691	−	132.691i	0.628868	−	0.628868i	−0.318915	−	0.947783i	$-0.603319\pi$
$211$	132.691	−	132.691i	0.628868	−	0.628868i	0.947783	+	0.318915i	$0.103319\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	217.416			1.01124
$216$		0			0
$217$			153.713i			0.708357i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−129.239	−	129.239i	−0.584791	−	0.584791i
$222$		0			0
$223$		−	26.3436i		−	0.118133i	−0.998254	−	0.0590664i	$-0.981188\pi$
$223$		−	26.3436i		−	0.118133i	0.998254	−	0.0590664i	$-0.0188124\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−70.3362	+	70.3362i	−0.309851	+	0.309851i	−0.844852	−	0.535001i	$-0.820311\pi$
$227$	−70.3362	+	70.3362i	−0.309851	+	0.309851i	0.535001	+	0.844852i	$0.320311\pi$
$228$		0			0
$229$	−215.607	+	215.607i	−0.941516	+	0.941516i	−0.998382	−	0.0568658i	$-0.981889\pi$
$229$	−215.607	+	215.607i	−0.941516	+	0.941516i	0.0568658	+	0.998382i	$0.481889\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			183.853i			0.789069i	0.918881	+	0.394534i	$0.129094\pi$
$233$			183.853i			0.789069i	−0.918881	+	0.394534i	$0.870906\pi$
$234$		0			0
$235$	−52.6647	−	52.6647i	−0.224105	−	0.224105i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	315.183i		−	1.31876i	−0.751811	−	0.659379i	$-0.770819\pi$
$239$		−	315.183i		−	1.31876i	0.751811	−	0.659379i	$-0.229181\pi$
$240$		0			0
$241$	−327.804			−1.36018			−0.680090	−	0.733128i	$-0.738059\pi$
$241$	−327.804			−1.36018			−0.680090	+	0.733128i	$0.738059\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	108.691	−	108.691i	0.443635	−	0.443635i
$246$		0			0
$247$	169.532			0.686366
$248$		0			0
$249$		0			0
$250$		0			0
$251$	219.813	+	219.813i	0.875747	+	0.875747i	0.993091	−	0.117344i	$-0.0374381\pi$
$251$	219.813	+	219.813i	0.875747	+	0.875747i	−0.117344	+	0.993091i	$0.537438\pi$
$252$		0			0
$253$	−192.689	−	192.689i	−0.761616	−	0.761616i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−150.042			−0.583823			−0.291911	−	0.956445i	$-0.594291\pi$
$257$	−150.042			−0.583823			−0.291911	+	0.956445i	$0.594291\pi$
$258$		0			0
$259$	−173.320	+	173.320i	−0.669188	+	0.669188i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	14.8922			0.0566242			0.0283121	−	0.999599i	$-0.490987\pi$
$263$	14.8922			0.0566242			0.0283121	+	0.999599i	$0.490987\pi$
$264$		0			0
$265$			3.17290i			0.0119732i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−95.4169	−	95.4169i	−0.354710	−	0.354710i	0.507149	−	0.861858i	$-0.330699\pi$
$269$	−95.4169	−	95.4169i	−0.354710	−	0.354710i	−0.861858	+	0.507149i	$0.830699\pi$
$270$		0			0
$271$		−	46.4991i		−	0.171583i	−0.996313	−	0.0857917i	$-0.972658\pi$
$271$		−	46.4991i		−	0.171583i	0.996313	−	0.0857917i	$-0.0273420\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−19.5516	+	19.5516i	−0.0710967	+	0.0710967i
$276$		0			0
$277$	30.5071	−	30.5071i	0.110134	−	0.110134i	−0.649892	−	0.760026i	$-0.725186\pi$
$277$	30.5071	−	30.5071i	0.110134	−	0.110134i	0.760026	+	0.649892i	$0.225186\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			217.239i			0.773093i	0.922270	+	0.386547i	$0.126332\pi$
$281$			217.239i			0.773093i	−0.922270	+	0.386547i	$0.873668\pi$
$282$		0			0
$283$	136.055	+	136.055i	0.480760	+	0.480760i	0.905374	−	0.424614i	$-0.139590\pi$
$283$	136.055	+	136.055i	0.480760	+	0.480760i	−0.424614	+	0.905374i	$0.639590\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			194.909i			0.679126i
$288$		0			0
$289$	618.167			2.13898
$290$		0			0
$291$		0			0
$292$		0			0
$293$	56.8362	−	56.8362i	0.193980	−	0.193980i	−0.603433	−	0.797414i	$-0.706201\pi$
$293$	56.8362	−	56.8362i	0.193980	−	0.193980i	0.797414	+	0.603433i	$0.206201\pi$
$294$		0			0
$295$	−66.3841			−0.225031
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−121.152	−	121.152i	−0.405190	−	0.405190i
$300$		0			0
$301$	−132.042	−	132.042i	−0.438677	−	0.438677i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−252.474			−0.827782
$306$		0			0
$307$	−245.927	+	245.927i	−0.801067	+	0.801067i	−0.983262	−	0.182196i	$-0.941680\pi$
$307$	−245.927	+	245.927i	−0.801067	+	0.801067i	0.182196	+	0.983262i	$0.441680\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−359.964			−1.15744			−0.578721	−	0.815526i	$-0.696448\pi$
$311$	−359.964			−1.15744			−0.578721	+	0.815526i	$0.696448\pi$
$312$		0			0
$313$			131.023i			0.418605i	0.977851	+	0.209303i	$0.0671194\pi$
$313$			131.023i			0.418605i	−0.977851	+	0.209303i	$0.932881\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−89.0470	−	89.0470i	−0.280905	−	0.280905i	0.552565	−	0.833470i	$-0.313649\pi$
$317$	−89.0470	−	89.0470i	−0.280905	−	0.280905i	−0.833470	+	0.552565i	$0.813649\pi$
$318$		0			0
$319$			291.555i			0.913966i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−595.000	+	595.000i	−1.84210	+	1.84210i
$324$		0			0
$325$	−12.2929	+	12.2929i	−0.0378244	+	0.0378244i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			63.9690i			0.194435i
$330$		0			0
$331$	−95.5992	−	95.5992i	−0.288819	−	0.288819i	0.547794	−	0.836613i	$-0.315468\pi$
$331$	−95.5992	−	95.5992i	−0.288819	−	0.288819i	−0.836613	+	0.547794i	$0.815468\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	133.541i		−	0.398629i
$336$		0			0
$337$	583.717			1.73210			0.866050	−	0.499958i	$-0.166651\pi$
$337$	583.717			1.73210			0.866050	+	0.499958i	$0.166651\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−259.604	+	259.604i	−0.761302	+	0.761302i
$342$		0			0
$343$	−330.024			−0.962169
$344$		0			0
$345$		0			0
$346$		0			0
$347$	191.655	+	191.655i	0.552320	+	0.552320i	0.927110	−	0.374790i	$-0.122285\pi$
$347$	191.655	+	191.655i	0.552320	+	0.552320i	−0.374790	+	0.927110i	$0.622285\pi$
$348$		0			0
$349$	−19.4781	−	19.4781i	−0.0558112	−	0.0558112i	0.678650	−	0.734462i	$-0.262565\pi$
$349$	−19.4781	−	19.4781i	−0.0558112	−	0.0558112i	−0.734462	+	0.678650i	$0.762565\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	82.9610			0.235017			0.117509	−	0.993072i	$-0.462509\pi$
$353$	82.9610			0.235017			0.117509	+	0.993072i	$0.462509\pi$
$354$		0			0
$355$	−133.344	+	133.344i	−0.375616	+	0.375616i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	357.792			0.996634			0.498317	−	0.866995i	$-0.333952\pi$
$359$	357.792			0.996634			0.498317	+	0.866995i	$0.333952\pi$
$360$		0			0
$361$		−	419.507i		−	1.16207i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	102.497	+	102.497i	0.280814	+	0.280814i
$366$		0			0
$367$		−	651.729i		−	1.77583i	−0.460010	−	0.887914i	$-0.652154\pi$
$367$		−	651.729i		−	1.77583i	0.460010	−	0.887914i	$-0.347846\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.92698	−	1.92698i	0.00519401	−	0.00519401i
$372$		0			0
$373$	199.720	−	199.720i	0.535442	−	0.535442i	−0.386745	−	0.922187i	$-0.626401\pi$
$373$	199.720	−	199.720i	0.535442	−	0.535442i	0.922187	+	0.386745i	$0.126401\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			183.314i			0.486243i
$378$		0			0
$379$	330.204	+	330.204i	0.871251	+	0.871251i	0.992609	−	0.121358i	$-0.0387248\pi$
$379$	330.204	+	330.204i	0.871251	+	0.871251i	−0.121358	+	0.992609i	$0.538725\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			174.284i			0.455049i	0.973772	+	0.227524i	$0.0730631\pi$
$383$			174.284i			0.455049i	−0.973772	+	0.227524i	$0.926937\pi$
$384$		0			0
$385$	−183.488			−0.476593
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−207.835	+	207.835i	−0.534279	+	0.534279i	−0.921843	−	0.387564i	$-0.873317\pi$
$389$	−207.835	+	207.835i	−0.534279	+	0.534279i	0.387564	+	0.921843i	$0.373317\pi$
$390$		0			0
$391$	850.402			2.17494
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−432.900	−	432.900i	−1.09595	−	1.09595i
$396$		0			0
$397$	−37.2994	−	37.2994i	−0.0939533	−	0.0939533i	0.658568	−	0.752521i	$-0.271162\pi$
$397$	−37.2994	−	37.2994i	−0.0939533	−	0.0939533i	−0.752521	+	0.658568i	$0.771162\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	524.704			1.30849			0.654244	−	0.756284i	$-0.272987\pi$
$401$	524.704			1.30849			0.654244	+	0.756284i	$0.272987\pi$
$402$		0			0
$403$	−163.224	+	163.224i	−0.405023	+	0.405023i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−585.434			−1.43841
$408$		0			0
$409$			787.357i			1.92508i	0.271141	+	0.962540i	$0.412599\pi$
$409$			787.357i			1.92508i	−0.271141	+	0.962540i	$0.587401\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	40.3166	+	40.3166i	0.0976190	+	0.0976190i
$414$		0			0
$415$		−	15.0475i		−	0.0362589i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	30.1767	−	30.1767i	0.0720209	−	0.0720209i	−0.670179	−	0.742200i	$-0.733783\pi$
$419$	30.1767	−	30.1767i	0.0720209	−	0.0720209i	0.742200	+	0.670179i	$0.233783\pi$
$420$		0			0
$421$	−261.021	+	261.021i	−0.620003	+	0.620003i	−0.945532	−	0.325529i	$-0.894458\pi$
$421$	−261.021	+	261.021i	−0.620003	+	0.620003i	0.325529	+	0.945532i	$0.394458\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	86.2878i		−	0.203030i
$426$		0			0
$427$	153.333	+	153.333i	0.359094	+	0.359094i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			459.989i			1.06726i	0.845718	+	0.533630i	$0.179172\pi$
$431$			459.989i			1.06726i	−0.845718	+	0.533630i	$0.820828\pi$
$432$		0			0
$433$	−445.246			−1.02828			−0.514140	−	0.857706i	$-0.671889\pi$
$433$	−445.246			−1.02828			−0.514140	+	0.857706i	$0.671889\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−557.768	+	557.768i	−1.27636	+	1.27636i
$438$		0			0
$439$	−356.467			−0.811998			−0.405999	−	0.913874i	$-0.633076\pi$
$439$	−356.467			−0.811998			−0.405999	+	0.913874i	$0.633076\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	358.752	+	358.752i	0.809824	+	0.809824i	0.984607	−	0.174783i	$-0.0559224\pi$
$443$	358.752	+	358.752i	0.809824	+	0.809824i	−0.174783	+	0.984607i	$0.555922\pi$
$444$		0			0
$445$	240.272	+	240.272i	0.539936	+	0.539936i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	44.6564			0.0994576			0.0497288	−	0.998763i	$-0.484164\pi$
$449$	44.6564			0.0994576			0.0497288	+	0.998763i	$0.484164\pi$
$450$		0			0
$451$	−329.179	+	329.179i	−0.729886	+	0.729886i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−115.367			−0.253554
$456$		0			0
$457$			84.2332i			0.184318i	0.995744	+	0.0921589i	$0.0293768\pi$
$457$			84.2332i			0.184318i	−0.995744	+	0.0921589i	$0.970623\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−205.347	−	205.347i	−0.445438	−	0.445438i	0.448397	−	0.893835i	$-0.351995\pi$
$461$	−205.347	−	205.347i	−0.445438	−	0.445438i	−0.893835	+	0.448397i	$0.851995\pi$
$462$		0			0
$463$		−	270.647i		−	0.584550i	−0.956334	−	0.292275i	$-0.905588\pi$
$463$		−	270.647i		−	0.584550i	0.956334	−	0.292275i	$-0.0944123\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	230.389	−	230.389i	0.493338	−	0.493338i	−0.416018	−	0.909356i	$-0.636575\pi$
$467$	230.389	−	230.389i	0.493338	−	0.493338i	0.909356	+	0.416018i	$0.136575\pi$
$468$		0			0
$469$	−81.1024	+	81.1024i	−0.172926	+	0.172926i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	446.006i		−	0.942931i
$474$		0			0
$475$	56.5952	+	56.5952i	0.119148	+	0.119148i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			575.911i			1.20232i	0.799129	+	0.601159i	$0.205294\pi$
$479$			575.911i			1.20232i	−0.799129	+	0.601159i	$0.794706\pi$
$480$		0			0
$481$	−368.088			−0.765255
$482$		0			0
$483$		0			0
$484$		0			0
$485$	373.740	−	373.740i	0.770599	−	0.770599i
$486$		0			0
$487$	−600.355			−1.23276			−0.616381	−	0.787448i	$-0.711402\pi$
$487$	−600.355			−1.23276			−0.616381	+	0.787448i	$0.711402\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	79.7182	+	79.7182i	0.162359	+	0.162359i	0.783611	−	0.621252i	$-0.213376\pi$
$491$	79.7182	+	79.7182i	0.162359	+	0.162359i	−0.621252	+	0.783611i	$0.713376\pi$
$492$		0			0
$493$	−643.367	−	643.367i	−1.30500	−	1.30500i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	161.966			0.325887
$498$		0			0
$499$	13.4912	−	13.4912i	0.0270365	−	0.0270365i	−0.693459	−	0.720496i	$-0.743914\pi$
$499$	13.4912	−	13.4912i	0.0270365	−	0.0270365i	0.720496	+	0.693459i	$0.243914\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	892.196			1.77375			0.886875	−	0.462009i	$-0.152871\pi$
$503$	892.196			1.77375			0.886875	+	0.462009i	$0.152871\pi$
$504$		0			0
$505$			10.7744i			0.0213354i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	44.9128	+	44.9128i	0.0882374	+	0.0882374i	0.749848	−	0.661610i	$-0.230127\pi$
$509$	44.9128	+	44.9128i	0.0882374	+	0.0882374i	−0.661610	+	0.749848i	$0.730127\pi$
$510$		0			0
$511$		−	124.498i		−	0.243636i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	93.1416	−	93.1416i	0.180857	−	0.180857i
$516$		0			0
$517$	−108.036	+	108.036i	−0.208967	+	0.208967i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			866.038i			1.66226i	0.556078	+	0.831130i	$0.312306\pi$
$521$			866.038i			1.66226i	−0.556078	+	0.831130i	$0.687694\pi$
$522$		0			0
$523$	−359.579	−	359.579i	−0.687531	−	0.687531i	0.274155	−	0.961686i	$-0.411602\pi$
$523$	−359.579	−	359.579i	−0.687531	−	0.687531i	−0.961686	+	0.274155i	$0.911602\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	1145.72i		−	2.17405i
$528$		0			0
$529$	268.189			0.506973
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−206.969	+	206.969i	−0.388310	+	0.388310i
$534$		0			0
$535$	268.695			0.502234
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−222.968	−	222.968i	−0.413669	−	0.413669i
$540$		0			0
$541$	−9.41176	−	9.41176i	−0.0173970	−	0.0173970i	0.698355	−	0.715752i	$-0.253916\pi$
$541$	−9.41176	−	9.41176i	−0.0173970	−	0.0173970i	−0.715752	+	0.698355i	$0.753916\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−245.076			−0.449680
$546$		0			0
$547$	37.6377	−	37.6377i	0.0688075	−	0.0688075i	−0.671866	−	0.740673i	$-0.734507\pi$
$547$	37.6377	−	37.6377i	0.0688075	−	0.0688075i	0.740673	+	0.671866i	$0.234507\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	843.953			1.53168
$552$		0			0
$553$			525.821i			0.950852i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−369.172	−	369.172i	−0.662786	−	0.662786i	0.293250	−	0.956036i	$-0.405263\pi$
$557$	−369.172	−	369.172i	−0.662786	−	0.662786i	−0.956036	+	0.293250i	$0.905263\pi$
$558$		0			0
$559$		−	280.424i		−	0.501652i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−141.210	+	141.210i	−0.250817	+	0.250817i	−0.821306	−	0.570489i	$-0.806754\pi$
$563$	−141.210	+	141.210i	−0.250817	+	0.250817i	0.570489	+	0.821306i	$0.306754\pi$
$564$		0			0
$565$	−184.954	+	184.954i	−0.327352	+	0.327352i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	134.928i		−	0.237131i	−0.992946	−	0.118566i	$-0.962170\pi$
$569$		−	134.928i		−	0.237131i	0.992946	−	0.118566i	$-0.0378296\pi$
$570$		0			0
$571$	486.485	+	486.485i	0.851988	+	0.851988i	0.990378	−	0.138390i	$-0.0441926\pi$
$571$	486.485	+	486.485i	0.851988	+	0.851988i	−0.138390	+	0.990378i	$0.544193\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	80.8885i		−	0.140676i
$576$		0			0
$577$	−310.050			−0.537349			−0.268674	−	0.963231i	$-0.586586\pi$
$577$	−310.050			−0.537349			−0.268674	+	0.963231i	$0.586586\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−9.13868	+	9.13868i	−0.0157292	+	0.0157292i
$582$		0			0
$583$	6.50888			0.0111645
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−301.021	−	301.021i	−0.512812	−	0.512812i	0.402575	−	0.915387i	$-0.368115\pi$
$587$	−301.021	−	301.021i	−0.512812	−	0.512812i	−0.915387	+	0.402575i	$0.868115\pi$
$588$		0			0
$589$	751.465	+	751.465i	1.27583	+	1.27583i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−6.08782			−0.0102661			−0.00513307	−	0.999987i	$-0.501634\pi$
$593$	−6.08782			−0.0102661			−0.00513307	+	0.999987i	$0.501634\pi$
$594$		0			0
$595$	404.898	−	404.898i	0.680501	−	0.680501i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−756.472			−1.26289			−0.631446	−	0.775420i	$-0.717538\pi$
$599$	−756.472			−1.26289			−0.631446	+	0.775420i	$0.717538\pi$
$600$		0			0
$601$		−	753.072i		−	1.25303i	−0.779409	−	0.626516i	$-0.784480\pi$
$601$		−	753.072i		−	1.25303i	0.779409	−	0.626516i	$-0.215520\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	92.6521	+	92.6521i	0.153144	+	0.153144i
$606$		0			0
$607$		−	47.1200i		−	0.0776277i	−0.999246	−	0.0388139i	$-0.987642\pi$
$607$		−	47.1200i		−	0.0776277i	0.999246	−	0.0388139i	$-0.0123579\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−67.9271	+	67.9271i	−0.111174	+	0.111174i
$612$		0			0
$613$	637.192	−	637.192i	1.03947	−	1.03947i	0.0402769	−	0.999189i	$-0.487176\pi$
$613$	637.192	−	637.192i	1.03947	−	1.03947i	0.999189	−	0.0402769i	$-0.0128240\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	514.635i		−	0.834092i	−0.908885	−	0.417046i	$-0.863065\pi$
$617$		−	514.635i		−	0.834092i	0.908885	−	0.417046i	$-0.136935\pi$
$618$		0			0
$619$	−313.704	−	313.704i	−0.506791	−	0.506791i	0.406749	−	0.913540i	$-0.366662\pi$
$619$	−313.704	−	313.704i	−0.506791	−	0.506791i	−0.913540	+	0.406749i	$0.866662\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	291.845i		−	0.468452i
$624$		0			0
$625$	545.171			0.872273
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1291.86	−	1291.86i	2.05383	−	2.05383i
$630$		0			0
$631$	1226.20			1.94326			0.971631	−	0.236502i	$-0.0760009\pi$
$631$	1226.20			1.94326			0.971631	+	0.236502i	$0.0760009\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	365.706	+	365.706i	0.575914	+	0.575914i
$636$		0			0
$637$	−140.189	−	140.189i	−0.220078	−	0.220078i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−241.218			−0.376314			−0.188157	−	0.982139i	$-0.560251\pi$
$641$	−241.218			−0.376314			−0.188157	+	0.982139i	$0.560251\pi$
$642$		0			0
$643$	736.141	−	736.141i	1.14485	−	1.14485i	0.157304	−	0.987550i	$-0.449720\pi$
$643$	736.141	−	736.141i	1.14485	−	1.14485i	0.987550	−	0.157304i	$-0.0502803\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−680.082			−1.05113			−0.525565	−	0.850753i	$-0.676146\pi$
$647$	−680.082			−1.05113			−0.525565	+	0.850753i	$0.676146\pi$
$648$		0			0
$649$			136.180i			0.209831i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	716.929	+	716.929i	1.09790	+	1.09790i	0.994656	+	0.103244i	$0.0329224\pi$
$653$	716.929	+	716.929i	1.09790	+	1.09790i	0.103244	+	0.994656i	$0.467078\pi$
$654$		0			0
$655$			503.540i			0.768763i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	276.868	−	276.868i	0.420133	−	0.420133i	−0.465116	−	0.885250i	$-0.653988\pi$
$659$	276.868	−	276.868i	0.420133	−	0.420133i	0.885250	+	0.465116i	$0.153988\pi$
$660$		0			0
$661$	251.780	−	251.780i	0.380907	−	0.380907i	−0.490522	−	0.871429i	$-0.663194\pi$
$661$	251.780	−	251.780i	0.380907	−	0.380907i	0.871429	+	0.490522i	$0.163194\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			531.136i			0.798700i
$666$		0			0
$667$	−603.109	−	603.109i	−0.904211	−	0.904211i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			517.924i			0.771869i
$672$		0			0
$673$	674.332			1.00198			0.500990	−	0.865453i	$-0.332969\pi$
$673$	674.332			1.00198			0.500990	+	0.865453i	$0.332969\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−109.048	+	109.048i	−0.161075	+	0.161075i	−0.783043	−	0.621968i	$-0.786334\pi$
$677$	−109.048	+	109.048i	−0.161075	+	0.161075i	0.621968	+	0.783043i	$0.286334\pi$
$678$		0			0
$679$	−453.963			−0.668576
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−784.278	−	784.278i	−1.14828	−	1.14828i	−0.986890	−	0.161394i	$-0.948401\pi$
$683$	−784.278	−	784.278i	−1.14828	−	1.14828i	−0.161394	−	0.986890i	$-0.551599\pi$
$684$		0			0
$685$	7.14432	+	7.14432i	0.0104297	+	0.0104297i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4.09241			0.00593964
$690$		0			0
$691$	99.4915	−	99.4915i	0.143982	−	0.143982i	−0.631442	−	0.775423i	$-0.717536\pi$
$691$	99.4915	−	99.4915i	0.143982	−	0.143982i	0.775423	+	0.631442i	$0.217536\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−726.877			−1.04587
$696$		0			0
$697$		−	1452.78i		−	2.08433i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−177.909	−	177.909i	−0.253794	−	0.253794i	0.568730	−	0.822524i	$-0.307435\pi$
$701$	−177.909	−	177.909i	−0.253794	−	0.253794i	−0.822524	+	0.568730i	$0.807435\pi$
$702$		0			0
$703$			1694.63i			2.41057i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.54353	−	6.54353i	0.00925535	−	0.00925535i
$708$		0			0
$709$	−208.080	+	208.080i	−0.293484	+	0.293484i	−0.838455	−	0.544971i	$-0.816541\pi$
$709$	−208.080	+	208.080i	−0.293484	+	0.293484i	0.544971	+	0.838455i	$0.316541\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1074.03i		−	1.50635i
$714$		0			0
$715$	−194.841	−	194.841i	−0.272506	−	0.272506i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1013.84i		−	1.41007i	−0.709171	−	0.705036i	$-0.750931\pi$
$719$		−	1013.84i		−	1.41007i	0.709171	−	0.705036i	$-0.249069\pi$
$720$		0			0
$721$	−113.134			−0.156913
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−61.1957	+	61.1957i	−0.0844079	+	0.0844079i
$726$		0			0
$727$	−697.156			−0.958949			−0.479474	−	0.877556i	$-0.659173\pi$
$727$	−697.156			−0.958949			−0.479474	+	0.877556i	$0.659173\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	984.189	+	984.189i	1.34636	+	1.34636i
$732$		0			0
$733$	−39.9608	−	39.9608i	−0.0545168	−	0.0545168i	0.679323	−	0.733840i	$-0.262274\pi$
$733$	−39.9608	−	39.9608i	−0.0545168	−	0.0545168i	−0.733840	+	0.679323i	$0.762274\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−273.945			−0.371703
$738$		0			0
$739$	236.377	−	236.377i	0.319860	−	0.319860i	−0.528853	−	0.848713i	$-0.677378\pi$
$739$	236.377	−	236.377i	0.319860	−	0.319860i	0.848713	+	0.528853i	$0.177378\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	804.248			1.08243			0.541217	−	0.840883i	$-0.317964\pi$
$743$	804.248			1.08243			0.541217	+	0.840883i	$0.317964\pi$
$744$		0			0
$745$		−	530.258i		−	0.711755i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−163.185	−	163.185i	−0.217870	−	0.217870i
$750$		0			0
$751$			607.492i			0.808911i	0.914558	+	0.404456i	$0.132539\pi$
$751$			607.492i			0.808911i	−0.914558	+	0.404456i	$0.867461\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	351.131	−	351.131i	0.465074	−	0.465074i
$756$		0			0
$757$	−11.6797	+	11.6797i	−0.0154289	+	0.0154289i	−0.714779	−	0.699350i	$-0.753473\pi$
$757$	−11.6797	+	11.6797i	−0.0154289	+	0.0154289i	0.699350	+	0.714779i	$0.253473\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	659.125i		−	0.866130i	−0.901363	−	0.433065i	$-0.857432\pi$
$761$		−	659.125i		−	0.866130i	0.901363	−	0.433065i	$-0.142568\pi$
$762$		0			0
$763$	148.840	+	148.840i	0.195073	+	0.195073i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			85.6224i			0.111633i
$768$		0			0
$769$	−178.802			−0.232512			−0.116256	−	0.993219i	$-0.537089\pi$
$769$	−178.802			−0.232512			−0.116256	+	0.993219i	$0.537089\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	91.8171	−	91.8171i	0.118780	−	0.118780i	−0.645218	−	0.763998i	$-0.723234\pi$
$773$	91.8171	−	91.8171i	0.118780	−	0.118780i	0.763998	+	0.645218i	$0.223234\pi$
$774$		0			0
$775$	−108.979			−0.140618
$776$		0			0
$777$		0			0
$778$		0			0
$779$	952.860	+	952.860i	1.22318	+	1.22318i
$780$		0			0
$781$	273.541	+	273.541i	0.350245	+	0.350245i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1264.58			−1.61093
$786$		0			0
$787$	−214.856	+	214.856i	−0.273006	+	0.273006i	−0.830309	−	0.557303i	$-0.811836\pi$
$787$	−214.856	+	214.856i	−0.273006	+	0.273006i	0.557303	+	0.830309i	$0.311836\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	224.654			0.284012
$792$		0			0
$793$			325.641i			0.410645i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	332.028	+	332.028i	0.416598	+	0.416598i	0.884029	−	0.467432i	$-0.154821\pi$
$797$	332.028	+	332.028i	0.416598	+	0.416598i	−0.467432	+	0.884029i	$0.654821\pi$
$798$		0			0
$799$		−	476.801i		−	0.596747i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	210.262	−	210.262i	0.261846	−	0.261846i
$804$		0			0
$805$	379.562	−	379.562i	0.471506	−	0.471506i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	421.989i		−	0.521618i	−0.965390	−	0.260809i	$-0.916011\pi$
$809$		−	421.989i		−	0.521618i	0.965390	−	0.260809i	$-0.0839893\pi$
$810$		0			0
$811$	−532.822	−	532.822i	−0.656994	−	0.656994i	0.297674	−	0.954668i	$-0.403789\pi$
$811$	−532.822	−	532.822i	−0.656994	−	0.656994i	−0.954668	+	0.297674i	$0.903789\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			395.310i			0.485042i
$816$		0			0
$817$	−1291.04			−1.58022
$818$		0			0
$819$		0			0
$820$		0			0
$821$	797.754	−	797.754i	0.971686	−	0.971686i	−0.0279241	−	0.999610i	$-0.508890\pi$
$821$	797.754	−	797.754i	0.971686	−	0.971686i	0.999610	+	0.0279241i	$0.00888968\pi$
$822$		0			0
$823$	−863.777			−1.04955			−0.524773	−	0.851242i	$-0.675850\pi$
$823$	−863.777			−1.04955			−0.524773	+	0.851242i	$0.675850\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1154.08	−	1154.08i	−1.39550	−	1.39550i	−0.812402	−	0.583098i	$-0.801840\pi$
$827$	−1154.08	−	1154.08i	−1.39550	−	1.39550i	−0.583098	−	0.812402i	$-0.698160\pi$
$828$		0			0
$829$	473.183	+	473.183i	0.570788	+	0.570788i	0.932349	−	0.361561i	$-0.117756\pi$
$829$	473.183	+	473.183i	0.570788	+	0.570788i	−0.361561	+	0.932349i	$0.617756\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	984.033			1.18131
$834$		0			0
$835$	−217.537	+	217.537i	−0.260523	+	0.260523i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	561.776			0.669578			0.334789	−	0.942293i	$-0.391335\pi$
$839$	561.776			0.669578			0.334789	+	0.942293i	$0.391335\pi$
$840$		0			0
$841$			71.5574i			0.0850861i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	439.723	+	439.723i	0.520382	+	0.520382i
$846$		0			0
$847$		−	112.540i		−	0.132869i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1211.02	−	1211.02i	1.42306	−	1.42306i
$852$		0			0
$853$	−431.517	+	431.517i	−0.505881	+	0.505881i	−0.913259	−	0.407378i	$-0.866443\pi$
$853$	−431.517	+	431.517i	−0.505881	+	0.505881i	0.407378	+	0.913259i	$0.366443\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			448.237i			0.523030i	0.965199	+	0.261515i	$0.0842221\pi$
$857$			448.237i			0.523030i	−0.965199	+	0.261515i	$0.915778\pi$
$858$		0			0
$859$	617.299	+	617.299i	0.718625	+	0.718625i	0.968324	−	0.249699i	$-0.0803316\pi$
$859$	617.299	+	617.299i	0.718625	+	0.718625i	−0.249699	+	0.968324i	$0.580332\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1588.33i			1.84047i	0.391363	+	0.920236i	$0.372004\pi$
$863$			1588.33i			1.84047i	−0.391363	+	0.920236i	$0.627996\pi$
$864$		0			0
$865$	−1416.81			−1.63793
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−888.050	+	888.050i	−1.02192	+	1.02192i
$870$		0			0
$871$	−172.241			−0.197751
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−374.593	−	374.593i	−0.428106	−	0.428106i
$876$		0			0
$877$	535.285	+	535.285i	0.610359	+	0.610359i	0.943040	−	0.332680i	$-0.107953\pi$
$877$	535.285	+	535.285i	0.610359	+	0.610359i	−0.332680	+	0.943040i	$0.607953\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−517.437			−0.587330			−0.293665	−	0.955908i	$-0.594875\pi$
$881$	−517.437			−0.587330			−0.293665	+	0.955908i	$0.594875\pi$
$882$		0			0
$883$	1192.91	−	1192.91i	1.35097	−	1.35097i	0.466399	−	0.884574i	$-0.345551\pi$
$883$	1192.91	−	1192.91i	1.35097	−	1.35097i	0.884574	−	0.466399i	$-0.154449\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1750.36			−1.97335			−0.986675	−	0.162704i	$-0.947978\pi$
$887$	−1750.36			−1.97335			−0.986675	+	0.162704i	$0.947978\pi$
$888$		0			0
$889$		−	444.203i		−	0.499666i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	312.728	+	312.728i	0.350199	+	0.350199i
$894$		0			0
$895$			1309.77i			1.46343i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−812.551	+	812.551i	−0.903839	+	0.903839i
$900$		0			0
$901$	−14.3630	+	14.3630i	−0.0159411	+	0.0159411i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	182.528i		−	0.201689i
$906$		0			0
$907$	−86.0833	−	86.0833i	−0.0949099	−	0.0949099i	0.658058	−	0.752968i	$-0.271378\pi$
$907$	−86.0833	−	86.0833i	−0.0949099	−	0.0949099i	−0.752968	+	0.658058i	$0.771378\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			44.7701i			0.0491439i	0.999698	+	0.0245719i	$0.00782228\pi$
$911$			44.7701i			0.0491439i	−0.999698	+	0.0245719i	$0.992178\pi$
$912$		0			0
$913$	−30.8683			−0.0338098
$914$		0			0
$915$		0			0
$916$		0			0
$917$	305.812	−	305.812i	0.333492	−	0.333492i
$918$		0			0
$919$	−498.982			−0.542962			−0.271481	−	0.962444i	$-0.587513\pi$
$919$	−498.982			−0.542962			−0.271481	+	0.962444i	$0.587513\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	171.987	+	171.987i	0.186335	+	0.186335i
$924$		0			0
$925$	−122.879	−	122.879i	−0.132842	−	0.132842i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1460.97			1.57262			0.786311	−	0.617831i	$-0.211988\pi$
$929$	1460.97			1.57262			0.786311	+	0.617831i	$0.211988\pi$
$930$		0			0
$931$	−645.415	+	645.415i	−0.693250	+	0.693250i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1367.65			1.46273
$936$		0			0
$937$			594.005i			0.633943i	0.948435	+	0.316972i	$0.102666\pi$
$937$			594.005i			0.633943i	−0.948435	+	0.316972i	$0.897334\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−767.764	−	767.764i	−0.815902	−	0.815902i	0.169610	−	0.985511i	$-0.445749\pi$
$941$	−767.764	−	767.764i	−0.815902	−	0.815902i	−0.985511	+	0.169610i	$0.945749\pi$
$942$		0			0
$943$		−	1361.87i		−	1.44419i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−205.644	+	205.644i	−0.217153	+	0.217153i	−0.807297	−	0.590145i	$-0.799071\pi$
$947$	−205.644	+	205.644i	−0.217153	+	0.217153i	0.590145	+	0.807297i	$0.299071\pi$
$948$		0			0
$949$	132.201	−	132.201i	0.139306	−	0.139306i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1714.17i			1.79871i	0.437215	+	0.899357i	$0.355965\pi$
$953$			1714.17i			1.79871i	−0.437215	+	0.899357i	$0.644035\pi$
$954$		0			0
$955$	813.098	+	813.098i	0.851412	+	0.851412i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	8.67783i		−	0.00904883i
$960$		0			0
$961$	−486.009			−0.505733
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−848.466	+	848.466i	−0.879240	+	0.879240i
$966$		0			0
$967$	1375.76			1.42271			0.711356	−	0.702832i	$-0.248081\pi$
$967$	1375.76			1.42271			0.711356	+	0.702832i	$0.248081\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1081.41	−	1081.41i	−1.11371	−	1.11371i	−0.992645	−	0.121062i	$-0.961370\pi$
$971$	−1081.41	−	1081.41i	−1.11371	−	1.11371i	−0.121062	−	0.992645i	$-0.538630\pi$
$972$		0			0
$973$	441.449	+	441.449i	0.453699	+	0.453699i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	18.0081			0.0184320			0.00921601	−	0.999958i	$-0.497066\pi$
$977$	18.0081			0.0184320			0.00921601	+	0.999958i	$0.497066\pi$
$978$		0			0
$979$	492.893	−	492.893i	0.503465	−	0.503465i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	579.164			0.589180			0.294590	−	0.955624i	$-0.404817\pi$
$983$	579.164			0.589180			0.294590	+	0.955624i	$0.404817\pi$
$984$		0			0
$985$			1292.32i			1.31200i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	922.605	+	922.605i	0.932866	+	0.932866i
$990$		0			0
$991$		−	1389.22i		−	1.40184i	−0.713242	−	0.700918i	$-0.752774\pi$
$991$		−	1389.22i		−	1.40184i	0.713242	−	0.700918i	$-0.247226\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−564.880	+	564.880i	−0.567719	+	0.567719i
$996$		0			0
$997$	867.073	−	867.073i	0.869682	−	0.869682i	−0.122755	−	0.992437i	$-0.539173\pi$
$997$	867.073	−	867.073i	0.869682	−	0.869682i	0.992437	+	0.122755i	$0.0391730\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.3.m.c.559.2		16
3.2	odd	2		192.3.l.a.175.4		16
4.3	odd	2		144.3.m.c.19.5		16
8.3	odd	2		1152.3.m.f.991.7		16
8.5	even	2		1152.3.m.c.991.7		16
12.11	even	2		48.3.l.a.19.4	✓	16
16.3	odd	4		1152.3.m.c.415.7		16
16.5	even	4		144.3.m.c.91.5		16
16.11	odd	4	inner	576.3.m.c.271.2		16
16.13	even	4		1152.3.m.f.415.7		16
24.5	odd	2		384.3.l.b.223.5		16
24.11	even	2		384.3.l.a.223.1		16
48.5	odd	4		48.3.l.a.43.4	yes	16
48.11	even	4		192.3.l.a.79.4		16
48.29	odd	4		384.3.l.a.31.1		16
48.35	even	4		384.3.l.b.31.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.l.a.19.4	✓	16	12.11	even	2
48.3.l.a.43.4	yes	16	48.5	odd	4
144.3.m.c.19.5		16	4.3	odd	2
144.3.m.c.91.5		16	16.5	even	4
192.3.l.a.79.4		16	48.11	even	4
192.3.l.a.175.4		16	3.2	odd	2
384.3.l.a.31.1		16	48.29	odd	4
384.3.l.a.223.1		16	24.11	even	2
384.3.l.b.31.5		16	48.35	even	4
384.3.l.b.223.5		16	24.5	odd	2
576.3.m.c.271.2		16	16.11	odd	4	inner
576.3.m.c.559.2		16	1.1	even	1	trivial
1152.3.m.c.415.7		16	16.3	odd	4
1152.3.m.c.991.7		16	8.5	even	2
1152.3.m.f.415.7		16	16.13	even	4
1152.3.m.f.991.7		16	8.3	odd	2

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:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	576.m (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.32679 + 3.32679i) q^{5} +4.04088 q^{7} +(6.82458 + 6.82458i) q^{11} +(4.29091 + 4.29091i) q^{13} -30.1192 q^{17} +(19.7548 - 19.7548i) q^{19} -28.2345 q^{23} +2.86488i q^{25} +(21.3607 + 21.3607i) q^{29} +38.0396i q^{31} +(-13.4432 + 13.4432i) q^{35} +(-42.8916 + 42.8916i) q^{37} +48.2343i q^{41} +(-32.6765 - 32.6765i) q^{43} +15.8305i q^{47} -32.6713 q^{49} +(0.476870 - 0.476870i) q^{53} -45.4079 q^{55} +(9.97719 + 9.97719i) q^{59} +(37.9455 + 37.9455i) q^{61} -28.5500 q^{65} +(-20.0705 + 20.0705i) q^{67} +40.0818 q^{71} -30.8095i q^{73} +(27.5773 + 27.5773i) q^{77} +130.125i q^{79} +(-2.26155 + 2.26155i) q^{83} +(100.200 - 100.200i) q^{85} -72.2232i q^{89} +(17.3391 + 17.3391i) q^{91} +131.441i q^{95} -112.343 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{11} + 32 q^{19} - 128 q^{23} - 32 q^{29} + 96 q^{35} - 96 q^{37} - 160 q^{43} + 112 q^{49} + 160 q^{53} + 256 q^{55} - 128 q^{59} - 32 q^{61} + 32 q^{65} - 320 q^{67} + 512 q^{71} - 224 q^{77}+ \cdots + 480 q^{91}+O(q^{100}) \) 16 * q + 32 * q^11 + 32 * q^19 - 128 * q^23 - 32 * q^29 + 96 * q^35 - 96 * q^37 - 160 * q^43 + 112 * q^49 + 160 * q^53 + 256 * q^55 - 128 * q^59 - 32 * q^61 + 32 * q^65 - 320 * q^67 + 512 * q^71 - 224 * q^77 - 160 * q^83 + 160 * q^85 + 480 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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