LMFDB - Embedded newform 800.3.m.b.593.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,3,Mod(593,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 3]))

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

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

chi := DirichletCharacter("800.593");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	800.m (of order $4$, 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:	$21.7984211488$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{18} - 3x^{16} + 11x^{14} + x^{12} - 40x^{10} + 4x^{8} + 176x^{6} - 192x^{4} - 256x^{2} + 1024 $ x^20 - x^18 - 3x^16 + 11x^14 + x^12 - 40x^10 + 4x^8 + 176x^6 - 192x^4 - 256*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{36} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.3
Root			$-1.39859 - 0.209644i$ of defining polynomial
Character	$\chi$	$=$	800.593
Dual form			800.3.m.b.657.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.52630 + 2.52630i) q^{3} +(-5.20520 + 5.20520i) q^{7} -3.76437i q^{9} +2.49236i q^{11} +(6.65988 - 6.65988i) q^{13} +(-21.9428 + 21.9428i) q^{17} -17.5567 q^{19} -26.2998i q^{21} +(20.1791 + 20.1791i) q^{23} +(-13.2268 - 13.2268i) q^{27} +1.04754 q^{29} +2.47263 q^{31} +(-6.29646 - 6.29646i) q^{33} +(-19.2786 - 19.2786i) q^{37} +33.6497i q^{39} -43.9414 q^{41} +(32.9394 - 32.9394i) q^{43} +(33.7079 - 33.7079i) q^{47} -5.18827i q^{49} -110.868i q^{51} +(36.4034 - 36.4034i) q^{53} +(44.3536 - 44.3536i) q^{57} -30.5963 q^{59} -23.8366i q^{61} +(19.5943 + 19.5943i) q^{63} +(19.9932 + 19.9932i) q^{67} -101.957 q^{69} +21.8350 q^{71} +(-32.6450 - 32.6450i) q^{73} +(-12.9733 - 12.9733i) q^{77} -16.4124i q^{79} +100.709 q^{81} +(-0.343799 + 0.343799i) q^{83} +(-2.64640 + 2.64640i) q^{87} -84.4631i q^{89} +69.3320i q^{91} +(-6.24661 + 6.24661i) q^{93} +(24.1311 - 24.1311i) q^{97} +9.38218 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{7} + 12 q^{17} - 4 q^{23} + 136 q^{31} - 32 q^{33} - 8 q^{41} + 188 q^{47} + 40 q^{57} + 228 q^{63} - 248 q^{71} + 124 q^{73} + 132 q^{81} - 488 q^{87} - 100 q^{97}+O(q^{100}) $ 20 * q - 4 * q^7 + 12 * q^17 - 4 * q^23 + 136 * q^31 - 32 * q^33 - 8 * q^41 + 188 * q^47 + 40 * q^57 + 228 * q^63 - 248 * q^71 + 124 * q^73 + 132 * q^81 - 488 * q^87 - 100 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$1$	$e\left(\frac{3}{4}\right)$

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
$3$	−2.52630	+	2.52630i	−0.842100	+	0.842100i	−0.989132	−	0.147032i	$-0.953028\pi$
$3$	−2.52630	+	2.52630i	−0.842100	+	0.842100i	0.147032	+	0.989132i	$0.453028\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−5.20520	+	5.20520i	−0.743600	+	0.743600i	−0.973269	−	0.229669i	$-0.926236\pi$
$7$	−5.20520	+	5.20520i	−0.743600	+	0.743600i	0.229669	+	0.973269i	$0.426236\pi$
$8$		0			0
$9$		−	3.76437i		−	0.418263i
$10$		0			0
$11$			2.49236i			0.226579i	0.993562	+	0.113289i	$0.0361387\pi$
$11$			2.49236i			0.226579i	−0.993562	+	0.113289i	$0.963861\pi$
$12$		0			0
$13$	6.65988	−	6.65988i	0.512298	−	0.512298i	−0.402932	−	0.915230i	$-0.632009\pi$
$13$	6.65988	−	6.65988i	0.512298	−	0.512298i	0.915230	+	0.402932i	$0.132009\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−21.9428	+	21.9428i	−1.29075	+	1.29075i	−0.356434	+	0.934320i	$0.616008\pi$
$17$	−21.9428	+	21.9428i	−1.29075	+	1.29075i	−0.934320	+	0.356434i	$0.883992\pi$
$18$		0			0
$19$	−17.5567			−0.924039			−0.462020	−	0.886870i	$-0.652875\pi$
$19$	−17.5567			−0.924039			−0.462020	+	0.886870i	$0.652875\pi$
$20$		0			0
$21$		−	26.2998i		−	1.25237i
$22$		0			0
$23$	20.1791	+	20.1791i	0.877354	+	0.877354i	0.993260	−	0.115906i	$-0.0369772\pi$
$23$	20.1791	+	20.1791i	0.877354	+	0.877354i	−0.115906	+	0.993260i	$0.536977\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−13.2268	−	13.2268i	−0.489880	−	0.489880i
$28$		0			0
$29$	1.04754			0.0361220			0.0180610	−	0.999837i	$-0.494251\pi$
$29$	1.04754			0.0361220			0.0180610	+	0.999837i	$0.494251\pi$
$30$		0			0
$31$	2.47263			0.0797623			0.0398812	−	0.999204i	$-0.487302\pi$
$31$	2.47263			0.0797623			0.0398812	+	0.999204i	$0.487302\pi$
$32$		0			0
$33$	−6.29646	−	6.29646i	−0.190802	−	0.190802i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−19.2786	−	19.2786i	−0.521043	−	0.521043i	0.396843	−	0.917886i	$-0.370106\pi$
$37$	−19.2786	−	19.2786i	−0.521043	−	0.521043i	−0.917886	+	0.396843i	$0.870106\pi$
$38$		0			0
$39$			33.6497i			0.862813i
$40$		0			0
$41$	−43.9414			−1.07174			−0.535871	−	0.844300i	$-0.680017\pi$
$41$	−43.9414			−1.07174			−0.535871	+	0.844300i	$0.680017\pi$
$42$		0			0
$43$	32.9394	−	32.9394i	0.766032	−	0.766032i	−0.211373	−	0.977405i	$-0.567794\pi$
$43$	32.9394	−	32.9394i	0.766032	−	0.766032i	0.977405	+	0.211373i	$0.0677936\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	33.7079	−	33.7079i	0.717189	−	0.717189i	−0.250840	−	0.968029i	$-0.580707\pi$
$47$	33.7079	−	33.7079i	0.717189	−	0.717189i	0.968029	+	0.250840i	$0.0807067\pi$
$48$		0			0
$49$		−	5.18827i		−	0.105883i
$50$		0			0
$51$		−	110.868i		−	2.17389i
$52$		0			0
$53$	36.4034	−	36.4034i	0.686856	−	0.686856i	−0.274679	−	0.961536i	$-0.588572\pi$
$53$	36.4034	−	36.4034i	0.686856	−	0.686856i	0.961536	+	0.274679i	$0.0885717\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	44.3536	−	44.3536i	0.778133	−	0.778133i
$58$		0			0
$59$	−30.5963			−0.518581			−0.259291	−	0.965799i	$-0.583489\pi$
$59$	−30.5963			−0.518581			−0.259291	+	0.965799i	$0.583489\pi$
$60$		0			0
$61$		−	23.8366i		−	0.390764i	−0.980727	−	0.195382i	$-0.937405\pi$
$61$		−	23.8366i		−	0.390764i	0.980727	−	0.195382i	$-0.0625947\pi$
$62$		0			0
$63$	19.5943	+	19.5943i	0.311021	+	0.311021i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	19.9932	+	19.9932i	0.298407	+	0.298407i	0.840390	−	0.541983i	$-0.182326\pi$
$67$	19.9932	+	19.9932i	0.298407	+	0.298407i	−0.541983	+	0.840390i	$0.682326\pi$
$68$		0			0
$69$	−101.957			−1.47764
$70$		0			0
$71$	21.8350			0.307535			0.153767	−	0.988107i	$-0.450859\pi$
$71$	21.8350			0.307535			0.153767	+	0.988107i	$0.450859\pi$
$72$		0			0
$73$	−32.6450	−	32.6450i	−0.447191	−	0.447191i	0.447228	−	0.894420i	$-0.352411\pi$
$73$	−32.6450	−	32.6450i	−0.447191	−	0.447191i	−0.894420	+	0.447228i	$0.852411\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.9733	−	12.9733i	−0.168484	−	0.168484i
$78$		0			0
$79$		−	16.4124i		−	0.207753i	−0.994590	−	0.103876i	$-0.966875\pi$
$79$		−	16.4124i		−	0.207753i	0.994590	−	0.103876i	$-0.0331246\pi$
$80$		0			0
$81$	100.709			1.24332
$82$		0			0
$83$	−0.343799	+	0.343799i	−0.00414216	+	0.00414216i	−0.709175	−	0.705033i	$-0.750932\pi$
$83$	−0.343799	+	0.343799i	−0.00414216	+	0.00414216i	0.705033	+	0.709175i	$0.250932\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−2.64640	+	2.64640i	−0.0304184	+	0.0304184i
$88$		0			0
$89$		−	84.4631i		−	0.949024i	−0.880249	−	0.474512i	$-0.842625\pi$
$89$		−	84.4631i		−	0.949024i	0.880249	−	0.474512i	$-0.157375\pi$
$90$		0			0
$91$			69.3320i			0.761891i
$92$		0			0
$93$	−6.24661	+	6.24661i	−0.0671678	+	0.0671678i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	24.1311	−	24.1311i	0.248774	−	0.248774i	−0.571693	−	0.820467i	$-0.693713\pi$
$97$	24.1311	−	24.1311i	0.248774	−	0.248774i	0.820467	+	0.571693i	$0.193713\pi$
$98$		0			0
$99$	9.38218			0.0947695
$100$		0			0
$101$		−	152.095i		−	1.50589i	−0.658085	−	0.752944i	$-0.728633\pi$
$101$		−	152.095i		−	1.50589i	0.658085	−	0.752944i	$-0.271367\pi$
$102$		0			0
$103$	−120.657	−	120.657i	−1.17143	−	1.17143i	−0.981870	−	0.189558i	$-0.939295\pi$
$103$	−120.657	−	120.657i	−1.17143	−	1.17143i	−0.189558	−	0.981870i	$-0.560705\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−75.2789	−	75.2789i	−0.703541	−	0.703541i	0.261628	−	0.965169i	$-0.415741\pi$
$107$	−75.2789	−	75.2789i	−0.703541	−	0.703541i	−0.965169	+	0.261628i	$0.915741\pi$
$108$		0			0
$109$	11.8614			0.108820			0.0544100	−	0.998519i	$-0.482672\pi$
$109$	11.8614			0.108820			0.0544100	+	0.998519i	$0.482672\pi$
$110$		0			0
$111$	97.4070			0.877541
$112$		0			0
$113$	82.0090	+	82.0090i	0.725743	+	0.725743i	0.969769	−	0.244025i	$-0.0784680\pi$
$113$	82.0090	+	82.0090i	0.725743	+	0.725743i	−0.244025	+	0.969769i	$0.578468\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−25.0702	−	25.0702i	−0.214276	−	0.214276i
$118$		0			0
$119$		−	228.434i		−	1.91961i
$120$		0			0
$121$	114.788			0.948662
$122$		0			0
$123$	111.009	−	111.009i	0.902513	−	0.902513i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−55.8589	+	55.8589i	−0.439834	+	0.439834i	−0.891956	−	0.452122i	$-0.850667\pi$
$127$	−55.8589	+	55.8589i	−0.439834	+	0.439834i	0.452122	+	0.891956i	$0.350667\pi$
$128$		0			0
$129$			166.429i			1.29015i
$130$		0			0
$131$		−	151.619i		−	1.15740i	−0.815541	−	0.578700i	$-0.803560\pi$
$131$		−	151.619i		−	1.15740i	0.815541	−	0.578700i	$-0.196440\pi$
$132$		0			0
$133$	91.3864	−	91.3864i	0.687116	−	0.687116i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−68.2473	+	68.2473i	−0.498156	+	0.498156i	−0.910863	−	0.412708i	$-0.864583\pi$
$137$	−68.2473	+	68.2473i	−0.498156	+	0.498156i	0.412708	+	0.910863i	$0.364583\pi$
$138$		0			0
$139$	−69.7245			−0.501615			−0.250807	−	0.968037i	$-0.580696\pi$
$139$	−69.7245			−0.501615			−0.250807	+	0.968037i	$0.580696\pi$
$140$		0			0
$141$			170.312i			1.20789i
$142$		0			0
$143$	16.5988	+	16.5988i	0.116076	+	0.116076i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	13.1071	+	13.1071i	0.0891640	+	0.0891640i
$148$		0			0
$149$	78.9012			0.529538			0.264769	−	0.964312i	$-0.414704\pi$
$149$	78.9012			0.529538			0.264769	+	0.964312i	$0.414704\pi$
$150$		0			0
$151$	−197.562			−1.30835			−0.654177	−	0.756341i	$-0.726985\pi$
$151$	−197.562			−1.30835			−0.654177	+	0.756341i	$0.726985\pi$
$152$		0			0
$153$	82.6009	+	82.6009i	0.539875	+	0.539875i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	94.1106	+	94.1106i	0.599431	+	0.599431i	0.940161	−	0.340730i	$-0.110674\pi$
$157$	94.1106	+	94.1106i	0.599431	+	0.599431i	−0.340730	+	0.940161i	$0.610674\pi$
$158$		0			0
$159$			183.932i			1.15680i
$160$		0			0
$161$	−210.073			−1.30480
$162$		0			0
$163$	−132.980	+	132.980i	−0.815827	+	0.815827i	−0.985500	−	0.169673i	$-0.945729\pi$
$163$	−132.980	+	132.980i	−0.815827	+	0.815827i	0.169673	+	0.985500i	$0.445729\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−51.4111	+	51.4111i	−0.307851	+	0.307851i	−0.844075	−	0.536225i	$-0.819850\pi$
$167$	−51.4111	+	51.4111i	−0.307851	+	0.307851i	0.536225	+	0.844075i	$0.319850\pi$
$168$		0			0
$169$			80.2920i			0.475101i
$170$		0			0
$171$			66.0900i			0.386492i
$172$		0			0
$173$	−183.164	+	183.164i	−1.05875	+	1.05875i	−0.0605880	+	0.998163i	$0.519298\pi$
$173$	−183.164	+	183.164i	−1.05875	+	1.05875i	−0.998163	+	0.0605880i	$0.980702\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	77.2954	−	77.2954i	0.436697	−	0.436697i
$178$		0			0
$179$	31.3468			0.175122			0.0875610	−	0.996159i	$-0.472093\pi$
$179$	31.3468			0.175122			0.0875610	+	0.996159i	$0.472093\pi$
$180$		0			0
$181$		−	57.4156i		−	0.317214i	−0.987342	−	0.158607i	$-0.949300\pi$
$181$		−	57.4156i		−	0.317214i	0.987342	−	0.158607i	$-0.0507002\pi$
$182$		0			0
$183$	60.2184	+	60.2184i	0.329063	+	0.329063i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−54.6895	−	54.6895i	−0.292457	−	0.292457i
$188$		0			0
$189$	137.696			0.728550
$190$		0			0
$191$	−161.923			−0.847767			−0.423884	−	0.905717i	$-0.639333\pi$
$191$	−161.923			−0.847767			−0.423884	+	0.905717i	$0.639333\pi$
$192$		0			0
$193$	−63.0037	−	63.0037i	−0.326444	−	0.326444i	0.524789	−	0.851233i	$-0.324144\pi$
$193$	−63.0037	−	63.0037i	−0.326444	−	0.326444i	−0.851233	+	0.524789i	$0.824144\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3.96178	+	3.96178i	0.0201106	+	0.0201106i	0.717091	−	0.696980i	$-0.245473\pi$
$197$	3.96178	+	3.96178i	0.0201106	+	0.0201106i	−0.696980	+	0.717091i	$0.745473\pi$
$198$		0			0
$199$			123.026i			0.618221i	0.951026	+	0.309110i	$0.100031\pi$
$199$			123.026i			0.618221i	−0.951026	+	0.309110i	$0.899969\pi$
$200$		0			0
$201$	−101.018			−0.502576
$202$		0			0
$203$	−5.45265	+	5.45265i	−0.0268604	+	0.0268604i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	75.9617	−	75.9617i	0.366965	−	0.366965i
$208$		0			0
$209$		−	43.7578i		−	0.209367i
$210$		0			0
$211$			273.854i			1.29789i	0.760837	+	0.648943i	$0.224789\pi$
$211$			273.854i			1.29789i	−0.760837	+	0.648943i	$0.775211\pi$
$212$		0			0
$213$	−55.1617	+	55.1617i	−0.258975	+	0.258975i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−12.8706	+	12.8706i	−0.0593113	+	0.0593113i
$218$		0			0
$219$	164.942			0.753159
$220$		0			0
$221$			292.273i			1.32250i
$222$		0			0
$223$	128.870	+	128.870i	0.577894	+	0.577894i	0.934323	−	0.356429i	$-0.116006\pi$
$223$	128.870	+	128.870i	0.577894	+	0.577894i	−0.356429	+	0.934323i	$0.616006\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	242.455	+	242.455i	1.06808	+	1.06808i	0.997506	+	0.0705768i	$0.0224840\pi$
$227$	242.455	+	242.455i	1.06808	+	1.06808i	0.0705768	+	0.997506i	$0.477516\pi$
$228$		0			0
$229$	−385.669			−1.68414			−0.842072	−	0.539365i	$-0.818664\pi$
$229$	−385.669			−1.68414			−0.842072	+	0.539365i	$0.818664\pi$
$230$		0			0
$231$	65.5487			0.283760
$232$		0			0
$233$	−9.90826	−	9.90826i	−0.0425247	−	0.0425247i	0.685525	−	0.728049i	$-0.259573\pi$
$233$	−9.90826	−	9.90826i	−0.0425247	−	0.0425247i	−0.728049	+	0.685525i	$0.759573\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	41.4627	+	41.4627i	0.174948	+	0.174948i
$238$		0			0
$239$			44.7391i			0.187193i	0.995610	+	0.0935965i	$0.0298364\pi$
$239$			44.7391i			0.187193i	−0.995610	+	0.0935965i	$0.970164\pi$
$240$		0			0
$241$	−21.4785			−0.0891223			−0.0445611	−	0.999007i	$-0.514189\pi$
$241$	−21.4785			−0.0891223			−0.0445611	+	0.999007i	$0.514189\pi$
$242$		0			0
$243$	−135.380	+	135.380i	−0.557118	+	0.557118i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−116.926	+	116.926i	−0.473384	+	0.473384i
$248$		0			0
$249$		−	1.73708i		−	0.00697623i
$250$		0			0
$251$			149.651i			0.596218i	0.954532	+	0.298109i	$0.0963560\pi$
$251$			149.651i			0.596218i	−0.954532	+	0.298109i	$0.903644\pi$
$252$		0			0
$253$	−50.2938	+	50.2938i	−0.198790	+	0.198790i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−244.045	+	244.045i	−0.949590	+	0.949590i	−0.998789	−	0.0491992i	$-0.984333\pi$
$257$	−244.045	+	244.045i	−0.949590	+	0.949590i	0.0491992	+	0.998789i	$0.484333\pi$
$258$		0			0
$259$	200.698			0.774896
$260$		0			0
$261$		−	3.94332i		−	0.0151085i
$262$		0			0
$263$	−241.557	−	241.557i	−0.918467	−	0.918467i	0.0784512	−	0.996918i	$-0.475003\pi$
$263$	−241.557	−	241.557i	−0.918467	−	0.918467i	−0.996918	+	0.0784512i	$0.975003\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	213.379	+	213.379i	0.799172	+	0.799172i
$268$		0			0
$269$	350.619			1.30342			0.651708	−	0.758470i	$-0.274053\pi$
$269$	350.619			1.30342			0.651708	+	0.758470i	$0.274053\pi$
$270$		0			0
$271$	298.610			1.10188			0.550940	−	0.834545i	$-0.314269\pi$
$271$	298.610			1.10188			0.550940	+	0.834545i	$0.314269\pi$
$272$		0			0
$273$	−175.153	−	175.153i	−0.641588	−	0.641588i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−275.798	−	275.798i	−0.995660	−	0.995660i	0.00433109	−	0.999991i	$-0.498621\pi$
$277$	−275.798	−	275.798i	−0.995660	−	0.995660i	−0.999991	+	0.00433109i	$0.998621\pi$
$278$		0			0
$279$		−	9.30790i		−	0.0333617i
$280$		0			0
$281$	13.2859			0.0472808			0.0236404	−	0.999721i	$-0.492474\pi$
$281$	13.2859			0.0472808			0.0236404	+	0.999721i	$0.492474\pi$
$282$		0			0
$283$	−253.189	+	253.189i	−0.894662	+	0.894662i	−0.994958	−	0.100296i	$-0.968021\pi$
$283$	−253.189	+	253.189i	−0.894662	+	0.894662i	0.100296	+	0.994958i	$0.468021\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	228.724	−	228.724i	0.796948	−	0.796948i
$288$		0			0
$289$		−	673.975i		−	2.33209i
$290$		0			0
$291$			121.925i			0.418985i
$292$		0			0
$293$	143.408	−	143.408i	0.489447	−	0.489447i	−0.418685	−	0.908132i	$-0.637509\pi$
$293$	143.408	−	143.408i	0.489447	−	0.489447i	0.908132	+	0.418685i	$0.137509\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	32.9659	−	32.9659i	0.110996	−	0.110996i
$298$		0			0
$299$	268.781			0.898934
$300$		0			0
$301$			342.912i			1.13924i
$302$		0			0
$303$	384.236	+	384.236i	1.26811	+	1.26811i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−357.761	−	357.761i	−1.16535	−	1.16535i	−0.983288	−	0.182059i	$-0.941724\pi$
$307$	−357.761	−	357.761i	−1.16535	−	1.16535i	−0.182059	−	0.983288i	$-0.558276\pi$
$308$		0			0
$309$	609.631			1.97292
$310$		0			0
$311$	380.204			1.22252			0.611261	−	0.791429i	$-0.290663\pi$
$311$	380.204			1.22252			0.611261	+	0.791429i	$0.290663\pi$
$312$		0			0
$313$	−270.798	−	270.798i	−0.865169	−	0.865169i	0.126764	−	0.991933i	$-0.459541\pi$
$313$	−270.798	−	270.798i	−0.865169	−	0.865169i	−0.991933	+	0.126764i	$0.959541\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	238.939	+	238.939i	0.753752	+	0.753752i	0.975177	−	0.221426i	$-0.0710710\pi$
$317$	238.939	+	238.939i	0.753752	+	0.753752i	−0.221426	+	0.975177i	$0.571071\pi$
$318$		0			0
$319$			2.61085i			0.00818448i
$320$		0			0
$321$	380.354			1.18490
$322$		0			0
$323$	385.245	−	385.245i	1.19271	−	1.19271i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−29.9654	+	29.9654i	−0.0916373	+	0.0916373i
$328$		0			0
$329$			350.913i			1.06660i
$330$		0			0
$331$		−	73.0725i		−	0.220763i	−0.993889	−	0.110381i	$-0.964793\pi$
$331$		−	73.0725i		−	0.220763i	0.993889	−	0.110381i	$-0.0352072\pi$
$332$		0			0
$333$	−72.5718	+	72.5718i	−0.217933	+	0.217933i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	207.932	−	207.932i	0.617008	−	0.617008i	−0.327755	−	0.944763i	$-0.606292\pi$
$337$	207.932	−	207.932i	0.617008	−	0.617008i	0.944763	+	0.327755i	$0.106292\pi$
$338$		0			0
$339$	−414.358			−1.22230
$340$		0			0
$341$			6.16270i			0.0180724i
$342$		0			0
$343$	−228.049	−	228.049i	−0.664866	−	0.664866i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	150.516	+	150.516i	0.433765	+	0.433765i	0.889907	−	0.456142i	$-0.150769\pi$
$347$	150.516	+	150.516i	0.433765	+	0.433765i	−0.456142	+	0.889907i	$0.650769\pi$
$348$		0			0
$349$	−159.797			−0.457871			−0.228935	−	0.973442i	$-0.573524\pi$
$349$	−159.797			−0.457871			−0.228935	+	0.973442i	$0.573524\pi$
$350$		0			0
$351$	−176.177			−0.501930
$352$		0			0
$353$	−31.8920	−	31.8920i	−0.0903455	−	0.0903455i	0.660490	−	0.750835i	$-0.270349\pi$
$353$	−31.8920	−	31.8920i	−0.0903455	−	0.0903455i	−0.750835	+	0.660490i	$0.770349\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	577.092	+	577.092i	1.61650	+	1.61650i
$358$		0			0
$359$		−	374.114i		−	1.04210i	−0.853526	−	0.521050i	$-0.825541\pi$
$359$		−	374.114i		−	1.04210i	0.853526	−	0.521050i	$-0.174459\pi$
$360$		0			0
$361$	−52.7608			−0.146152
$362$		0			0
$363$	−289.989	+	289.989i	−0.798868	+	0.798868i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	195.775	−	195.775i	0.533446	−	0.533446i	−0.388150	−	0.921596i	$-0.626886\pi$
$367$	195.775	−	195.775i	0.533446	−	0.533446i	0.921596	+	0.388150i	$0.126886\pi$
$368$		0			0
$369$			165.412i			0.448270i
$370$		0			0
$371$			378.974i			1.02149i
$372$		0			0
$373$	−342.423	+	342.423i	−0.918023	+	0.918023i	−0.996885	−	0.0788627i	$-0.974871\pi$
$373$	−342.423	+	342.423i	−0.918023	+	0.918023i	0.0788627	+	0.996885i	$0.474871\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6.97649	−	6.97649i	0.0185053	−	0.0185053i
$378$		0			0
$379$	−607.050			−1.60171			−0.800857	−	0.598855i	$-0.795622\pi$
$379$	−607.050			−1.60171			−0.800857	+	0.598855i	$0.795622\pi$
$380$		0			0
$381$		−	282.232i		−	0.740767i
$382$		0			0
$383$	−154.687	−	154.687i	−0.403882	−	0.403882i	0.475717	−	0.879599i	$-0.342189\pi$
$383$	−154.687	−	154.687i	−0.403882	−	0.403882i	−0.879599	+	0.475717i	$0.842189\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−123.996	−	123.996i	−0.320403	−	0.320403i
$388$		0			0
$389$	−323.730			−0.832210			−0.416105	−	0.909317i	$-0.636605\pi$
$389$	−323.730			−0.832210			−0.416105	+	0.909317i	$0.636605\pi$
$390$		0			0
$391$	−885.575			−2.26490
$392$		0			0
$393$	383.036	+	383.036i	0.974645	+	0.974645i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−376.174	−	376.174i	−0.947543	−	0.947543i	0.0511485	−	0.998691i	$-0.483712\pi$
$397$	−376.174	−	376.174i	−0.947543	−	0.947543i	−0.998691	+	0.0511485i	$0.983712\pi$
$398$		0			0
$399$			461.739i			1.15724i
$400$		0			0
$401$	−401.761			−1.00190			−0.500948	−	0.865477i	$-0.667015\pi$
$401$	−401.761			−1.00190			−0.500948	+	0.865477i	$0.667015\pi$
$402$		0			0
$403$	16.4674	−	16.4674i	0.0408621	−	0.0408621i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	48.0493	−	48.0493i	0.118057	−	0.118057i
$408$		0			0
$409$			166.880i			0.408020i	0.978969	+	0.204010i	$0.0653974\pi$
$409$			166.880i			0.408020i	−0.978969	+	0.204010i	$0.934603\pi$
$410$		0			0
$411$		−	344.826i		−	0.838993i
$412$		0			0
$413$	159.260	−	159.260i	0.385617	−	0.385617i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	176.145	−	176.145i	0.422410	−	0.422410i
$418$		0			0
$419$	533.694			1.27373			0.636866	−	0.770974i	$-0.280230\pi$
$419$	533.694			1.27373			0.636866	+	0.770974i	$0.280230\pi$
$420$		0			0
$421$			214.523i			0.509556i	0.967000	+	0.254778i	$0.0820023\pi$
$421$			214.523i			0.509556i	−0.967000	+	0.254778i	$0.917998\pi$
$422$		0			0
$423$	−126.889	−	126.889i	−0.299974	−	0.299974i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	124.074	+	124.074i	0.290573	+	0.290573i
$428$		0			0
$429$	−83.8673			−0.195495
$430$		0			0
$431$	525.371			1.21896			0.609479	−	0.792802i	$-0.291379\pi$
$431$	525.371			1.21896			0.609479	+	0.792802i	$0.291379\pi$
$432$		0			0
$433$	262.829	+	262.829i	0.606994	+	0.606994i	0.942159	−	0.335165i	$-0.108792\pi$
$433$	262.829	+	262.829i	0.606994	+	0.606994i	−0.335165	+	0.942159i	$0.608792\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−354.280	−	354.280i	−0.810709	−	0.810709i
$438$		0			0
$439$		−	419.145i		−	0.954771i	−0.878694	−	0.477386i	$-0.841584\pi$
$439$		−	419.145i		−	0.954771i	0.878694	−	0.477386i	$-0.158416\pi$
$440$		0			0
$441$	−19.5306			−0.0442870
$442$		0			0
$443$	−119.234	+	119.234i	−0.269152	+	0.269152i	−0.828758	−	0.559606i	$-0.810952\pi$
$443$	−119.234	+	119.234i	−0.269152	+	0.269152i	0.559606	+	0.828758i	$0.310952\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−199.328	+	199.328i	−0.445924	+	0.445924i
$448$		0			0
$449$			596.070i			1.32755i	0.747932	+	0.663775i	$0.231047\pi$
$449$			596.070i			1.32755i	−0.747932	+	0.663775i	$0.768953\pi$
$450$		0			0
$451$		−	109.518i		−	0.242834i
$452$		0			0
$453$	499.099	−	499.099i	1.10176	−	1.10176i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−144.279	+	144.279i	−0.315708	+	0.315708i	−0.847116	−	0.531408i	$-0.821663\pi$
$457$	−144.279	+	144.279i	−0.315708	+	0.315708i	0.531408	+	0.847116i	$0.321663\pi$
$458$		0			0
$459$	580.465			1.26463
$460$		0			0
$461$		−	743.994i		−	1.61387i	−0.590640	−	0.806935i	$-0.701125\pi$
$461$		−	743.994i		−	1.61387i	0.590640	−	0.806935i	$-0.298875\pi$
$462$		0			0
$463$	446.519	+	446.519i	0.964404	+	0.964404i	0.999388	−	0.0349842i	$-0.0111381\pi$
$463$	446.519	+	446.519i	0.964404	+	0.964404i	−0.0349842	+	0.999388i	$0.511138\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5.61916	+	5.61916i	0.0120325	+	0.0120325i	0.713097	−	0.701065i	$-0.247292\pi$
$467$	5.61916	+	5.61916i	0.0120325	+	0.0120325i	−0.701065	+	0.713097i	$0.747292\pi$
$468$		0			0
$469$	−208.138			−0.443791
$470$		0			0
$471$	−475.503			−1.00956
$472$		0			0
$473$	82.0969	+	82.0969i	0.173566	+	0.173566i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−137.036	−	137.036i	−0.287287	−	0.287287i
$478$		0			0
$479$			765.340i			1.59779i	0.601472	+	0.798894i	$0.294581\pi$
$479$			765.340i			1.59779i	−0.601472	+	0.798894i	$0.705419\pi$
$480$		0			0
$481$	−256.786			−0.533859
$482$		0			0
$483$	530.707	−	530.707i	1.09877	−	1.09877i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−611.323	+	611.323i	−1.25528	+	1.25528i	−0.301963	+	0.953320i	$0.597642\pi$
$487$	−611.323	+	611.323i	−1.25528	+	1.25528i	−0.953320	+	0.301963i	$0.902358\pi$
$488$		0			0
$489$		−	671.893i		−	1.37402i
$490$		0			0
$491$		−	899.211i		−	1.83139i	−0.401877	−	0.915694i	$-0.631642\pi$
$491$		−	899.211i		−	1.83139i	0.401877	−	0.915694i	$-0.368358\pi$
$492$		0			0
$493$	−22.9860	+	22.9860i	−0.0466247	+	0.0466247i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−113.655	+	113.655i	−0.228683	+	0.228683i
$498$		0			0
$499$	−755.350			−1.51373			−0.756864	−	0.653572i	$-0.773270\pi$
$499$	−755.350			−1.51373			−0.756864	+	0.653572i	$0.773270\pi$
$500$		0			0
$501$		−	259.760i		−	0.518482i
$502$		0			0
$503$	−505.226	−	505.226i	−1.00443	−	1.00443i	−0.999990	−	0.00443600i	$-0.998588\pi$
$503$	−505.226	−	505.226i	−1.00443	−	1.00443i	−0.00443600	−	0.999990i	$-0.501412\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−202.842	−	202.842i	−0.400082	−	0.400082i
$508$		0			0
$509$	−594.029			−1.16705			−0.583526	−	0.812095i	$-0.698327\pi$
$509$	−594.029			−1.16705			−0.583526	+	0.812095i	$0.698327\pi$
$510$		0			0
$511$	339.847			0.665063
$512$		0			0
$513$	232.219	+	232.219i	0.452668	+	0.452668i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	84.0123	+	84.0123i	0.162500	+	0.162500i
$518$		0			0
$519$		−	925.453i		−	1.78315i
$520$		0			0
$521$	871.615			1.67297			0.836483	−	0.547993i	$-0.184608\pi$
$521$	871.615			1.67297			0.836483	+	0.547993i	$0.184608\pi$
$522$		0			0
$523$	601.907	−	601.907i	1.15087	−	1.15087i	0.164497	−	0.986378i	$-0.447400\pi$
$523$	601.907	−	601.907i	1.15087	−	1.15087i	0.986378	−	0.164497i	$-0.0526001\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−54.2566	+	54.2566i	−0.102954	+	0.102954i
$528$		0			0
$529$			285.396i			0.539500i
$530$		0			0
$531$			115.176i			0.216903i
$532$		0			0
$533$	−292.645	+	292.645i	−0.549052	+	0.549052i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−79.1915	+	79.1915i	−0.147470	+	0.147470i
$538$		0			0
$539$	12.9311			0.0239908
$540$		0			0
$541$			109.548i			0.202492i	0.994861	+	0.101246i	$0.0322830\pi$
$541$			109.548i			0.202492i	−0.994861	+	0.101246i	$0.967717\pi$
$542$		0			0
$543$	145.049	+	145.049i	0.267125	+	0.267125i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−330.968	−	330.968i	−0.605060	−	0.605060i	0.336591	−	0.941651i	$-0.390726\pi$
$547$	−330.968	−	330.968i	−0.605060	−	0.605060i	−0.941651	+	0.336591i	$0.890726\pi$
$548$		0			0
$549$	−89.7299			−0.163442
$550$		0			0
$551$	−18.3914			−0.0333782
$552$		0			0
$553$	85.4301	+	85.4301i	0.154485	+	0.154485i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	62.7080	+	62.7080i	0.112582	+	0.112582i	0.761153	−	0.648572i	$-0.224633\pi$
$557$	62.7080	+	62.7080i	0.112582	+	0.112582i	−0.648572	+	0.761153i	$0.724633\pi$
$558$		0			0
$559$		−	438.745i		−	0.784874i
$560$		0			0
$561$	276.324			0.492556
$562$		0			0
$563$	−116.120	+	116.120i	−0.206252	+	0.206252i	−0.802672	−	0.596420i	$-0.796589\pi$
$563$	−116.120	+	116.120i	−0.206252	+	0.206252i	0.596420	+	0.802672i	$0.296589\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−524.210	+	524.210i	−0.924533	+	0.924533i
$568$		0			0
$569$		−	323.733i		−	0.568950i	−0.958683	−	0.284475i	$-0.908181\pi$
$569$		−	323.733i		−	0.568950i	0.958683	−	0.284475i	$-0.0918193\pi$
$570$		0			0
$571$		−	433.708i		−	0.759558i	−0.925077	−	0.379779i	$-0.876000\pi$
$571$		−	433.708i		−	0.759558i	0.925077	−	0.379779i	$-0.124000\pi$
$572$		0			0
$573$	409.067	−	409.067i	0.713904	−	0.713904i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−27.1282	+	27.1282i	−0.0470159	+	0.0470159i	−0.730224	−	0.683208i	$-0.760584\pi$
$577$	−27.1282	+	27.1282i	−0.0470159	+	0.0470159i	0.683208	+	0.730224i	$0.260584\pi$
$578$		0			0
$579$	318.332			0.549797
$580$		0			0
$581$		−	3.57909i		−	0.00616023i
$582$		0			0
$583$	90.7305	+	90.7305i	0.155627	+	0.155627i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−163.733	−	163.733i	−0.278932	−	0.278932i	0.553751	−	0.832682i	$-0.313196\pi$
$587$	−163.733	−	163.733i	−0.278932	−	0.278932i	−0.832682	+	0.553751i	$0.813196\pi$
$588$		0			0
$589$	−43.4114			−0.0737035
$590$		0			0
$591$	−20.0173			−0.0338702
$592$		0			0
$593$	−280.606	−	280.606i	−0.473198	−	0.473198i	0.429750	−	0.902948i	$-0.358602\pi$
$593$	−280.606	−	280.606i	−0.473198	−	0.473198i	−0.902948	+	0.429750i	$0.858602\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−310.800	−	310.800i	−0.520603	−	0.520603i
$598$		0			0
$599$			639.232i			1.06716i	0.845748	+	0.533582i	$0.179155\pi$
$599$			639.232i			1.06716i	−0.845748	+	0.533582i	$0.820845\pi$
$600$		0			0
$601$	−332.979			−0.554041			−0.277021	−	0.960864i	$-0.589347\pi$
$601$	−332.979			−0.554041			−0.277021	+	0.960864i	$0.589347\pi$
$602$		0			0
$603$	75.2619	−	75.2619i	0.124812	−	0.124812i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	681.745	−	681.745i	1.12314	−	1.12314i	0.131873	−	0.991267i	$-0.457901\pi$
$607$	681.745	−	681.745i	1.12314	−	1.12314i	0.991267	−	0.131873i	$-0.0420990\pi$
$608$		0			0
$609$		−	27.5501i		−	0.0452382i
$610$		0			0
$611$		−	448.981i		−	0.734830i
$612$		0			0
$613$	−378.926	+	378.926i	−0.618150	+	0.618150i	−0.945057	−	0.326907i	$-0.893994\pi$
$613$	−378.926	+	378.926i	−0.618150	+	0.618150i	0.326907	+	0.945057i	$0.393994\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	68.3041	−	68.3041i	0.110704	−	0.110704i	−0.649585	−	0.760289i	$-0.725057\pi$
$617$	68.3041	−	68.3041i	0.110704	−	0.110704i	0.760289	+	0.649585i	$0.225057\pi$
$618$		0			0
$619$	−1009.69			−1.63116			−0.815580	−	0.578645i	$-0.803582\pi$
$619$	−1009.69			−1.63116			−0.815580	+	0.578645i	$0.803582\pi$
$620$		0			0
$621$		−	533.810i		−	0.859597i
$622$		0			0
$623$	439.648	+	439.648i	0.705694	+	0.705694i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	110.545	+	110.545i	0.176308	+	0.176308i
$628$		0			0
$629$	846.054			1.34508
$630$		0			0
$631$	867.965			1.37554			0.687769	−	0.725929i	$-0.258590\pi$
$631$	867.965			1.37554			0.687769	+	0.725929i	$0.258590\pi$
$632$		0			0
$633$	−691.837	−	691.837i	−1.09295	−	1.09295i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−34.5532	−	34.5532i	−0.0542437	−	0.0542437i
$638$		0			0
$639$		−	82.1949i		−	0.128631i
$640$		0			0
$641$	116.032			0.181017			0.0905083	−	0.995896i	$-0.471151\pi$
$641$	116.032			0.181017			0.0905083	+	0.995896i	$0.471151\pi$
$642$		0			0
$643$	−429.493	+	429.493i	−0.667951	+	0.667951i	−0.957241	−	0.289290i	$-0.906581\pi$
$643$	−429.493	+	429.493i	−0.667951	+	0.667951i	0.289290	+	0.957241i	$0.406581\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	511.695	−	511.695i	0.790873	−	0.790873i	−0.190763	−	0.981636i	$-0.561096\pi$
$647$	511.695	−	511.695i	0.790873	−	0.790873i	0.981636	+	0.190763i	$0.0610963\pi$
$648$		0			0
$649$		−	76.2571i		−	0.117499i
$650$		0			0
$651$		−	65.0297i		−	0.0998920i
$652$		0			0
$653$	777.556	−	777.556i	1.19074	−	1.19074i	0.213886	−	0.976859i	$-0.431388\pi$
$653$	777.556	−	777.556i	1.19074	−	1.19074i	0.976859	−	0.213886i	$-0.0686120\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−122.888	+	122.888i	−0.187044	+	0.187044i
$658$		0			0
$659$	−734.265			−1.11421			−0.557105	−	0.830442i	$-0.688088\pi$
$659$	−734.265			−1.11421			−0.557105	+	0.830442i	$0.688088\pi$
$660$		0			0
$661$			799.237i			1.20913i	0.796555	+	0.604567i	$0.206654\pi$
$661$			799.237i			1.20913i	−0.796555	+	0.604567i	$0.793346\pi$
$662$		0			0
$663$	−738.369	−	738.369i	−1.11368	−	1.11368i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	21.1384	+	21.1384i	0.0316918	+	0.0316918i
$668$		0			0
$669$	−651.130			−0.973288
$670$		0			0
$671$	59.4096			0.0885388
$672$		0			0
$673$	185.806	+	185.806i	0.276087	+	0.276087i	0.831545	−	0.555458i	$-0.187457\pi$
$673$	185.806	+	185.806i	0.276087	+	0.276087i	−0.555458	+	0.831545i	$0.687457\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	448.763	+	448.763i	0.662870	+	0.662870i	0.956055	−	0.293186i	$-0.0947155\pi$
$677$	448.763	+	448.763i	0.662870	+	0.662870i	−0.293186	+	0.956055i	$0.594715\pi$
$678$		0			0
$679$			251.214i			0.369977i
$680$		0			0
$681$	−1225.03			−1.79886
$682$		0			0
$683$	−67.1358	+	67.1358i	−0.0982955	+	0.0982955i	−0.754544	−	0.656249i	$-0.772142\pi$
$683$	−67.1358	+	67.1358i	−0.0982955	+	0.0982955i	0.656249	+	0.754544i	$0.272142\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	974.315	−	974.315i	1.41822	−	1.41822i
$688$		0			0
$689$		−	484.884i		−	0.703751i
$690$		0			0
$691$			533.282i			0.771753i	0.922550	+	0.385877i	$0.126101\pi$
$691$			533.282i			0.771753i	−0.922550	+	0.385877i	$0.873899\pi$
$692$		0			0
$693$	−48.8361	+	48.8361i	−0.0704706	+	0.0704706i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	964.199	−	964.199i	1.38336	−	1.38336i
$698$		0			0
$699$	50.0625			0.0716201
$700$		0			0
$701$		−	1203.60i		−	1.71697i	−0.512835	−	0.858487i	$-0.671405\pi$
$701$		−	1203.60i		−	1.71697i	0.512835	−	0.858487i	$-0.328595\pi$
$702$		0			0
$703$	338.469	+	338.469i	0.481464	+	0.481464i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	791.683	+	791.683i	1.11978	+	1.11978i
$708$		0			0
$709$	−456.445			−0.643788			−0.321894	−	0.946776i	$-0.604319\pi$
$709$	−456.445			−0.643788			−0.321894	+	0.946776i	$0.604319\pi$
$710$		0			0
$711$	−61.7825			−0.0868952
$712$		0			0
$713$	49.8956	+	49.8956i	0.0699798	+	0.0699798i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−113.024	−	113.024i	−0.157635	−	0.157635i
$718$		0			0
$719$		−	63.2841i		−	0.0880168i	−0.999031	−	0.0440084i	$-0.985987\pi$
$719$		−	63.2841i		−	0.0880168i	0.999031	−	0.0440084i	$-0.0140128\pi$
$720$		0			0
$721$	1256.09			1.74215
$722$		0			0
$723$	54.2610	−	54.2610i	0.0750498	−	0.0750498i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	408.139	−	408.139i	0.561401	−	0.561401i	−0.368304	−	0.929705i	$-0.620061\pi$
$727$	408.139	−	408.139i	0.561401	−	0.561401i	0.929705	+	0.368304i	$0.120061\pi$
$728$		0			0
$729$			222.361i			0.305021i
$730$		0			0
$731$			1445.57i			1.97752i
$732$		0			0
$733$	90.0317	−	90.0317i	0.122826	−	0.122826i	−0.643022	−	0.765848i	$-0.722319\pi$
$733$	90.0317	−	90.0317i	0.122826	−	0.122826i	0.765848	+	0.643022i	$0.222319\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−49.8304	+	49.8304i	−0.0676125	+	0.0676125i
$738$		0			0
$739$	875.498			1.18471			0.592353	−	0.805679i	$-0.298199\pi$
$739$	875.498			1.18471			0.592353	+	0.805679i	$0.298199\pi$
$740$		0			0
$741$		−	590.779i		−	0.797272i
$742$		0			0
$743$	282.061	+	282.061i	0.379624	+	0.379624i	0.870967	−	0.491342i	$-0.163494\pi$
$743$	282.061	+	282.061i	0.379624	+	0.379624i	−0.491342	+	0.870967i	$0.663494\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1.29419	+	1.29419i	0.00173251	+	0.00173251i
$748$		0			0
$749$	783.683			1.04631
$750$		0			0
$751$	−606.986			−0.808237			−0.404119	−	0.914707i	$-0.632422\pi$
$751$	−606.986			−0.808237			−0.404119	+	0.914707i	$0.632422\pi$
$752$		0			0
$753$	−378.063	−	378.063i	−0.502075	−	0.502075i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	173.075	+	173.075i	0.228633	+	0.228633i	0.812121	−	0.583488i	$-0.198313\pi$
$757$	173.075	+	173.075i	0.228633	+	0.228633i	−0.583488	+	0.812121i	$0.698313\pi$
$758$		0			0
$759$		−	254.114i		−	0.334801i
$760$		0			0
$761$	100.678			0.132297			0.0661483	−	0.997810i	$-0.478929\pi$
$761$	100.678			0.132297			0.0661483	+	0.997810i	$0.478929\pi$
$762$		0			0
$763$	−61.7409	+	61.7409i	−0.0809187	+	0.0809187i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−203.768	+	203.768i	−0.265668	+	0.265668i
$768$		0			0
$769$			370.732i			0.482097i	0.970513	+	0.241048i	$0.0774913\pi$
$769$			370.732i			0.482097i	−0.970513	+	0.241048i	$0.922509\pi$
$770$		0			0
$771$		−	1233.06i		−	1.59930i
$772$		0			0
$773$	319.455	−	319.455i	0.413267	−	0.413267i	−0.469608	−	0.882875i	$-0.655605\pi$
$773$	319.455	−	319.455i	0.413267	−	0.413267i	0.882875	+	0.469608i	$0.155605\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−507.023	+	507.023i	−0.652539	+	0.652539i
$778$		0			0
$779$	771.468			0.990331
$780$		0			0
$781$			54.4207i			0.0696808i
$782$		0			0
$783$	−13.8556	−	13.8556i	−0.0176955	−	0.0176955i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−1104.16	−	1104.16i	−1.40300	−	1.40300i	−0.790372	−	0.612627i	$-0.790113\pi$
$787$	−1104.16	−	1104.16i	−1.40300	−	1.40300i	−0.612627	−	0.790372i	$-0.709887\pi$
$788$		0			0
$789$	1220.49			1.54688
$790$		0			0
$791$	−853.747			−1.07933
$792$		0			0
$793$	−158.749	−	158.749i	−0.200188	−	0.200188i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−434.605	−	434.605i	−0.545301	−	0.545301i	0.379777	−	0.925078i	$-0.376001\pi$
$797$	−434.605	−	434.605i	−0.545301	−	0.545301i	−0.925078	+	0.379777i	$0.876001\pi$
$798$		0			0
$799$			1479.29i			1.85143i
$800$		0			0
$801$	−317.950			−0.396942
$802$		0			0
$803$	81.3631	−	81.3631i	0.101324	−	0.101324i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−885.768	+	885.768i	−1.09761	+	1.09761i
$808$		0			0
$809$			164.175i			0.202935i	0.994839	+	0.101468i	$0.0323538\pi$
$809$			164.175i			0.202935i	−0.994839	+	0.101468i	$0.967646\pi$
$810$		0			0
$811$			1283.08i			1.58210i	0.611752	+	0.791049i	$0.290465\pi$
$811$			1283.08i			1.58210i	−0.611752	+	0.791049i	$0.709535\pi$
$812$		0			0
$813$	−754.377	+	754.377i	−0.927893	+	0.927893i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−578.308	+	578.308i	−0.707843	+	0.707843i
$818$		0			0
$819$	260.991			0.318671
$820$		0			0
$821$			1292.43i			1.57421i	0.616817	+	0.787106i	$0.288422\pi$
$821$			1292.43i			1.57421i	−0.616817	+	0.787106i	$0.711578\pi$
$822$		0			0
$823$	38.6290	+	38.6290i	0.0469368	+	0.0469368i	0.730186	−	0.683249i	$-0.239434\pi$
$823$	38.6290	+	38.6290i	0.0469368	+	0.0469368i	−0.683249	+	0.730186i	$0.739434\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−432.150	−	432.150i	−0.522552	−	0.522552i	0.395789	−	0.918341i	$-0.370471\pi$
$827$	−432.150	−	432.150i	−0.522552	−	0.522552i	−0.918341	+	0.395789i	$0.870471\pi$
$828$		0			0
$829$	−684.217			−0.825353			−0.412676	−	0.910878i	$-0.635406\pi$
$829$	−684.217			−0.825353			−0.412676	+	0.910878i	$0.635406\pi$
$830$		0			0
$831$	1393.49			1.67689
$832$		0			0
$833$	113.845	+	113.845i	0.136669	+	0.136669i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−32.7049	−	32.7049i	−0.0390740	−	0.0390740i
$838$		0			0
$839$		−	579.065i		−	0.690184i	−0.938569	−	0.345092i	$-0.887848\pi$
$839$		−	579.065i		−	0.690184i	0.938569	−	0.345092i	$-0.112152\pi$
$840$		0			0
$841$	−839.903			−0.998695
$842$		0			0
$843$	−33.5642	+	33.5642i	−0.0398151	+	0.0398151i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−597.495	+	597.495i	−0.705426	+	0.705426i
$848$		0			0
$849$		−	1279.26i		−	1.50679i
$850$		0			0
$851$		−	778.051i		−	0.914279i
$852$		0			0
$853$	−89.6610	+	89.6610i	−0.105113	+	0.105113i	−0.757707	−	0.652595i	$-0.773680\pi$
$853$	−89.6610	+	89.6610i	−0.105113	+	0.105113i	0.652595	+	0.757707i	$0.273680\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	409.705	−	409.705i	0.478069	−	0.478069i	−0.426445	−	0.904514i	$-0.640234\pi$
$857$	409.705	−	409.705i	0.478069	−	0.478069i	0.904514	+	0.426445i	$0.140234\pi$
$858$		0			0
$859$	−799.305			−0.930507			−0.465253	−	0.885178i	$-0.654037\pi$
$859$	−799.305			−0.930507			−0.465253	+	0.885178i	$0.654037\pi$
$860$		0			0
$861$			1155.65i			1.34222i
$862$		0			0
$863$	1116.04	+	1116.04i	1.29321	+	1.29321i	0.932793	+	0.360414i	$0.117364\pi$
$863$	1116.04	+	1116.04i	1.29321	+	1.29321i	0.360414	+	0.932793i	$0.382636\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	1702.66	+	1702.66i	1.96386	+	1.96386i
$868$		0			0
$869$	40.9058			0.0470723
$870$		0			0
$871$	266.305			0.305746
$872$		0			0
$873$	−90.8383	−	90.8383i	−0.104053	−	0.104053i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	98.3606	+	98.3606i	0.112156	+	0.112156i	0.760957	−	0.648802i	$-0.224730\pi$
$877$	98.3606	+	98.3606i	0.112156	+	0.112156i	−0.648802	+	0.760957i	$0.724730\pi$
$878$		0			0
$879$			724.582i			0.824326i
$880$		0			0
$881$	−654.962			−0.743430			−0.371715	−	0.928347i	$-0.621230\pi$
$881$	−654.962			−0.743430			−0.371715	+	0.928347i	$0.621230\pi$
$882$		0			0
$883$	−963.144	+	963.144i	−1.09076	+	1.09076i	−0.0953158	+	0.995447i	$0.530386\pi$
$883$	−963.144	+	963.144i	−1.09076	+	1.09076i	−0.995447	+	0.0953158i	$0.969614\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−588.066	+	588.066i	−0.662983	+	0.662983i	−0.956082	−	0.293099i	$-0.905313\pi$
$887$	−588.066	+	588.066i	−0.662983	+	0.662983i	0.293099	+	0.956082i	$0.405313\pi$
$888$		0			0
$889$		−	581.514i		−	0.654121i
$890$		0			0
$891$			251.003i			0.281709i
$892$		0			0
$893$	−591.801	+	591.801i	−0.662711	+	0.662711i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−679.022	+	679.022i	−0.756992	+	0.756992i
$898$		0			0
$899$	2.59018			0.00288118
$900$		0			0
$901$			1597.59i			1.77313i
$902$		0			0
$903$	−866.299	−	866.299i	−0.959356	−	0.959356i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	692.104	+	692.104i	0.763069	+	0.763069i	0.976876	−	0.213807i	$-0.0685863\pi$
$907$	692.104	+	692.104i	0.763069	+	0.763069i	−0.213807	+	0.976876i	$0.568586\pi$
$908$		0			0
$909$	−572.540			−0.629857
$910$		0			0
$911$	45.1707			0.0495836			0.0247918	−	0.999693i	$-0.492108\pi$
$911$	45.1707			0.0495836			0.0247918	+	0.999693i	$0.492108\pi$
$912$		0			0
$913$	−0.856873	−	0.856873i	−0.000938525	−	0.000938525i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	789.209	+	789.209i	0.860643	+	0.860643i
$918$		0			0
$919$		−	1598.45i		−	1.73934i	−0.493637	−	0.869668i	$-0.664333\pi$
$919$		−	1598.45i		−	1.73934i	0.493637	−	0.869668i	$-0.335667\pi$
$920$		0			0
$921$	1807.62			1.96268
$922$		0			0
$923$	145.418	−	145.418i	0.157550	−	0.157550i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−454.197	+	454.197i	−0.489965	+	0.489965i
$928$		0			0
$929$			887.585i			0.955420i	0.878518	+	0.477710i	$0.158533\pi$
$929$			887.585i			0.955420i	−0.878518	+	0.477710i	$0.841467\pi$
$930$		0			0
$931$			91.0891i			0.0978400i
$932$		0			0
$933$	−960.509	+	960.509i	−1.02948	+	1.02948i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1241.34	+	1241.34i	−1.32480	+	1.32480i	−0.414958	+	0.909841i	$0.636204\pi$
$937$	−1241.34	+	1241.34i	−1.32480	+	1.32480i	−0.909841	+	0.414958i	$0.863796\pi$
$938$		0			0
$939$	1368.23			1.45712
$940$		0			0
$941$			328.028i			0.348595i	0.984693	+	0.174298i	$0.0557654\pi$
$941$			328.028i			0.348595i	−0.984693	+	0.174298i	$0.944235\pi$
$942$		0			0
$943$	−886.700	−	886.700i	−0.940297	−	0.940297i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−228.351	−	228.351i	−0.241131	−	0.241131i	0.576187	−	0.817318i	$-0.304540\pi$
$947$	−228.351	−	228.351i	−0.241131	−	0.241131i	−0.817318	+	0.576187i	$0.804540\pi$
$948$		0			0
$949$	−434.823			−0.458191
$950$		0			0
$951$	−1207.26			−1.26947
$952$		0			0
$953$	1291.50	+	1291.50i	1.35519	+	1.35519i	0.879742	+	0.475451i	$0.157715\pi$
$953$	1291.50	+	1291.50i	1.35519	+	1.35519i	0.475451	+	0.879742i	$0.342285\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−6.59578	−	6.59578i	−0.00689215	−	0.00689215i
$958$		0			0
$959$		−	710.482i		−	0.740857i
$960$		0			0
$961$	−954.886			−0.993638
$962$		0			0
$963$	−283.377	+	283.377i	−0.294265	+	0.294265i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	486.969	−	486.969i	0.503588	−	0.503588i	−0.408963	−	0.912551i	$-0.634110\pi$
$967$	486.969	−	486.969i	0.503588	−	0.503588i	0.912551	+	0.408963i	$0.134110\pi$
$968$		0			0
$969$			1946.49i			2.00876i
$970$		0			0
$971$			361.550i			0.372348i	0.982517	+	0.186174i	$0.0596088\pi$
$971$			361.550i			0.372348i	−0.982517	+	0.186174i	$0.940391\pi$
$972$		0			0
$973$	362.930	−	362.930i	0.373001	−	0.373001i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1073.65	−	1073.65i	1.09892	−	1.09892i	0.104387	−	0.994537i	$-0.466712\pi$
$977$	1073.65	−	1073.65i	1.09892	−	1.09892i	0.994537	−	0.104387i	$-0.0332881\pi$
$978$		0			0
$979$	210.513			0.215028
$980$		0			0
$981$		−	44.6506i		−	0.0455154i
$982$		0			0
$983$	122.156	+	122.156i	0.124269	+	0.124269i	0.766506	−	0.642237i	$-0.221994\pi$
$983$	122.156	+	122.156i	0.124269	+	0.124269i	−0.642237	+	0.766506i	$0.721994\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−886.510	−	886.510i	−0.898187	−	0.898187i
$988$		0			0
$989$	1329.38			1.34416
$990$		0			0
$991$	−324.300			−0.327245			−0.163623	−	0.986523i	$-0.552318\pi$
$991$	−324.300			−0.327245			−0.163623	+	0.986523i	$0.552318\pi$
$992$		0			0
$993$	184.603	+	184.603i	0.185904	+	0.185904i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−743.909	−	743.909i	−0.746148	−	0.746148i	0.227606	−	0.973753i	$-0.426910\pi$
$997$	−743.909	−	743.909i	−0.746148	−	0.746148i	−0.973753	+	0.227606i	$0.926910\pi$
$998$		0			0
$999$			509.987i			0.510498i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.3.m.b.593.3		20
4.3	odd	2		200.3.i.b.93.9		20
5.2	odd	4	inner	800.3.m.b.657.8		20
5.3	odd	4		160.3.m.a.17.3		20
5.4	even	2		160.3.m.a.113.8		20
8.3	odd	2		200.3.i.b.93.3		20
8.5	even	2	inner	800.3.m.b.593.8		20
20.3	even	4		40.3.i.a.37.8	yes	20
20.7	even	4		200.3.i.b.157.3		20
20.19	odd	2		40.3.i.a.13.2	✓	20
40.3	even	4		40.3.i.a.37.2	yes	20
40.13	odd	4		160.3.m.a.17.8		20
40.19	odd	2		40.3.i.a.13.8	yes	20
40.27	even	4		200.3.i.b.157.9		20
40.29	even	2		160.3.m.a.113.3		20
40.37	odd	4	inner	800.3.m.b.657.3		20
60.23	odd	4		360.3.u.b.37.3		20
60.59	even	2		360.3.u.b.253.9		20
120.59	even	2		360.3.u.b.253.3		20
120.83	odd	4		360.3.u.b.37.9		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.3.i.a.13.2	✓	20	20.19	odd	2
40.3.i.a.13.8	yes	20	40.19	odd	2
40.3.i.a.37.2	yes	20	40.3	even	4
40.3.i.a.37.8	yes	20	20.3	even	4
160.3.m.a.17.3		20	5.3	odd	4
160.3.m.a.17.8		20	40.13	odd	4
160.3.m.a.113.3		20	40.29	even	2
160.3.m.a.113.8		20	5.4	even	2
200.3.i.b.93.3		20	8.3	odd	2
200.3.i.b.93.9		20	4.3	odd	2
200.3.i.b.157.3		20	20.7	even	4
200.3.i.b.157.9		20	40.27	even	4
360.3.u.b.37.3		20	60.23	odd	4
360.3.u.b.37.9		20	120.83	odd	4
360.3.u.b.253.3		20	120.59	even	2
360.3.u.b.253.9		20	60.59	even	2
800.3.m.b.593.3		20	1.1	even	1	trivial
800.3.m.b.593.8		20	8.5	even	2	inner
800.3.m.b.657.3		20	40.37	odd	4	inner
800.3.m.b.657.8		20	5.2	odd	4	inner

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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	800.m (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.52630 + 2.52630i) q^{3} +(-5.20520 + 5.20520i) q^{7} -3.76437i q^{9} +2.49236i q^{11} +(6.65988 - 6.65988i) q^{13} +(-21.9428 + 21.9428i) q^{17} -17.5567 q^{19} -26.2998i q^{21} +(20.1791 + 20.1791i) q^{23} +(-13.2268 - 13.2268i) q^{27} +1.04754 q^{29} +2.47263 q^{31} +(-6.29646 - 6.29646i) q^{33} +(-19.2786 - 19.2786i) q^{37} +33.6497i q^{39} -43.9414 q^{41} +(32.9394 - 32.9394i) q^{43} +(33.7079 - 33.7079i) q^{47} -5.18827i q^{49} -110.868i q^{51} +(36.4034 - 36.4034i) q^{53} +(44.3536 - 44.3536i) q^{57} -30.5963 q^{59} -23.8366i q^{61} +(19.5943 + 19.5943i) q^{63} +(19.9932 + 19.9932i) q^{67} -101.957 q^{69} +21.8350 q^{71} +(-32.6450 - 32.6450i) q^{73} +(-12.9733 - 12.9733i) q^{77} -16.4124i q^{79} +100.709 q^{81} +(-0.343799 + 0.343799i) q^{83} +(-2.64640 + 2.64640i) q^{87} -84.4631i q^{89} +69.3320i q^{91} +(-6.24661 + 6.24661i) q^{93} +(24.1311 - 24.1311i) q^{97} +9.38218 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{7} + 12 q^{17} - 4 q^{23} + 136 q^{31} - 32 q^{33} - 8 q^{41} + 188 q^{47} + 40 q^{57} + 228 q^{63} - 248 q^{71} + 124 q^{73} + 132 q^{81} - 488 q^{87} - 100 q^{97}+O(q^{100}) \) 20 * q - 4 * q^7 + 12 * q^17 - 4 * q^23 + 136 * q^31 - 32 * q^33 - 8 * q^41 + 188 * q^47 + 40 * q^57 + 228 * q^63 - 248 * q^71 + 124 * q^73 + 132 * q^81 - 488 * q^87 - 100 * q^97

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