LMFDB - Embedded newform 768.3.h.d.641.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [768,3,Mod(641,768)] 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("768.641"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	768.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,24,0,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$20.9264843029$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			641.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	768.641
Dual form			768.3.h.d.641.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.82843 - 1.00000i) q^{3} -5.65685 q^{5} +6.00000 q^{7} +(7.00000 + 5.65685i) q^{9} -5.65685 q^{11} -10.0000i q^{13} +(16.0000 + 5.65685i) q^{15} -22.6274i q^{17} -2.00000i q^{19} +(-16.9706 - 6.00000i) q^{21} +11.3137i q^{23} +7.00000 q^{25} +(-14.1421 - 23.0000i) q^{27} +16.9706 q^{29} -22.0000 q^{31} +(16.0000 + 5.65685i) q^{33} -33.9411 q^{35} -6.00000i q^{37} +(-10.0000 + 28.2843i) q^{39} -33.9411i q^{41} +82.0000i q^{43} +(-39.5980 - 32.0000i) q^{45} +67.8823i q^{47} -13.0000 q^{49} +(-22.6274 + 64.0000i) q^{51} +62.2254 q^{53} +32.0000 q^{55} +(-2.00000 + 5.65685i) q^{57} -73.5391 q^{59} +86.0000i q^{61} +(42.0000 + 33.9411i) q^{63} +56.5685i q^{65} -2.00000i q^{67} +(11.3137 - 32.0000i) q^{69} +124.451i q^{71} -82.0000 q^{73} +(-19.7990 - 7.00000i) q^{75} -33.9411 q^{77} +10.0000 q^{79} +(17.0000 + 79.1960i) q^{81} -73.5391 q^{83} +128.000i q^{85} +(-48.0000 - 16.9706i) q^{87} +33.9411i q^{89} -60.0000i q^{91} +(62.2254 + 22.0000i) q^{93} +11.3137i q^{95} -94.0000 q^{97} +(-39.5980 - 32.0000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 24 q^{7} + 28 q^{9} + 64 q^{15} + 28 q^{25} - 88 q^{31} + 64 q^{33} - 40 q^{39} - 52 q^{49} + 128 q^{55} - 8 q^{57} + 168 q^{63} - 328 q^{73} + 40 q^{79} + 68 q^{81} - 192 q^{87} - 376 q^{97}+O(q^{100}) $ 4 * q + 24 * q^7 + 28 * q^9 + 64 * q^15 + 28 * q^25 - 88 * q^31 + 64 * q^33 - 40 * q^39 - 52 * q^49 + 128 * q^55 - 8 * q^57 + 168 * q^63 - 328 * q^73 + 40 * q^79 + 68 * q^81 - 192 * q^87 - 376 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−2.82843	−	1.00000i	−0.942809	−	0.333333i
$3$	−2.82843	−	1.00000i	−0.942809	−	0.333333i
$4$		0			0
$5$	−5.65685			−1.13137			−0.565685	−	0.824621i	$-0.691388\pi$
$5$	−5.65685			−1.13137			−0.565685	+	0.824621i	$0.691388\pi$
$6$		0			0
$7$	6.00000			0.857143			0.428571	−	0.903508i	$-0.359017\pi$
$7$	6.00000			0.857143			0.428571	+	0.903508i	$0.359017\pi$
$8$		0			0
$9$	7.00000	+	5.65685i	0.777778	+	0.628539i
$10$		0			0
$11$	−5.65685			−0.514259			−0.257130	−	0.966377i	$-0.582777\pi$
$11$	−5.65685			−0.514259			−0.257130	+	0.966377i	$0.582777\pi$
$12$		0			0
$13$		−	10.0000i		−	0.769231i	−0.923077	−	0.384615i	$-0.874334\pi$
$13$		−	10.0000i		−	0.769231i	0.923077	−	0.384615i	$-0.125666\pi$
$14$		0			0
$15$	16.0000	+	5.65685i	1.06667	+	0.377124i
$16$		0			0
$17$		−	22.6274i		−	1.33102i	−0.746387	−	0.665512i	$-0.768213\pi$
$17$		−	22.6274i		−	1.33102i	0.746387	−	0.665512i	$-0.231787\pi$
$18$		0			0
$19$		−	2.00000i		−	0.105263i	−0.998614	−	0.0526316i	$-0.983239\pi$
$19$		−	2.00000i		−	0.105263i	0.998614	−	0.0526316i	$-0.0167609\pi$
$20$		0			0
$21$	−16.9706	−	6.00000i	−0.808122	−	0.285714i
$22$		0			0
$23$			11.3137i			0.491900i	0.969282	+	0.245950i	$0.0791000\pi$
$23$			11.3137i			0.491900i	−0.969282	+	0.245950i	$0.920900\pi$
$24$		0			0
$25$	7.00000			0.280000
$26$		0			0
$27$	−14.1421	−	23.0000i	−0.523783	−	0.851852i
$28$		0			0
$29$	16.9706			0.585192			0.292596	−	0.956236i	$-0.405481\pi$
$29$	16.9706			0.585192			0.292596	+	0.956236i	$0.405481\pi$
$30$		0			0
$31$	−22.0000			−0.709677			−0.354839	−	0.934928i	$-0.615464\pi$
$31$	−22.0000			−0.709677			−0.354839	+	0.934928i	$0.615464\pi$
$32$		0			0
$33$	16.0000	+	5.65685i	0.484848	+	0.171420i
$34$		0			0
$35$	−33.9411			−0.969746
$36$		0			0
$37$		−	6.00000i		−	0.162162i	−0.996708	−	0.0810811i	$-0.974163\pi$
$37$		−	6.00000i		−	0.162162i	0.996708	−	0.0810811i	$-0.0258373\pi$
$38$		0			0
$39$	−10.0000	+	28.2843i	−0.256410	+	0.725238i
$40$		0			0
$41$		−	33.9411i		−	0.827832i	−0.910315	−	0.413916i	$-0.864161\pi$
$41$		−	33.9411i		−	0.827832i	0.910315	−	0.413916i	$-0.135839\pi$
$42$		0			0
$43$			82.0000i			1.90698i	0.301430	+	0.953488i	$0.402536\pi$
$43$			82.0000i			1.90698i	−0.301430	+	0.953488i	$0.597464\pi$
$44$		0			0
$45$	−39.5980	−	32.0000i	−0.879955	−	0.711111i
$46$		0			0
$47$			67.8823i			1.44430i	0.691735	+	0.722152i	$0.256847\pi$
$47$			67.8823i			1.44430i	−0.691735	+	0.722152i	$0.743153\pi$
$48$		0			0
$49$	−13.0000			−0.265306
$50$		0			0
$51$	−22.6274	+	64.0000i	−0.443675	+	1.25490i
$52$		0			0
$53$	62.2254			1.17406			0.587032	−	0.809564i	$-0.300296\pi$
$53$	62.2254			1.17406			0.587032	+	0.809564i	$0.300296\pi$
$54$		0			0
$55$	32.0000			0.581818
$56$		0			0
$57$	−2.00000	+	5.65685i	−0.0350877	+	0.0992431i
$58$		0			0
$59$	−73.5391			−1.24643			−0.623213	−	0.782052i	$-0.714173\pi$
$59$	−73.5391			−1.24643			−0.623213	+	0.782052i	$0.714173\pi$
$60$		0			0
$61$			86.0000i			1.40984i	0.709289	+	0.704918i	$0.249016\pi$
$61$			86.0000i			1.40984i	−0.709289	+	0.704918i	$0.750984\pi$
$62$		0			0
$63$	42.0000	+	33.9411i	0.666667	+	0.538748i
$64$		0			0
$65$			56.5685i			0.870285i
$66$		0			0
$67$		−	2.00000i		−	0.0298507i	−0.999889	−	0.0149254i	$-0.995249\pi$
$67$		−	2.00000i		−	0.0298507i	0.999889	−	0.0149254i	$-0.00475107\pi$
$68$		0			0
$69$	11.3137	−	32.0000i	0.163967	−	0.463768i
$70$		0			0
$71$			124.451i			1.75283i	0.481558	+	0.876414i	$0.340071\pi$
$71$			124.451i			1.75283i	−0.481558	+	0.876414i	$0.659929\pi$
$72$		0			0
$73$	−82.0000			−1.12329			−0.561644	−	0.827379i	$-0.689831\pi$
$73$	−82.0000			−1.12329			−0.561644	+	0.827379i	$0.689831\pi$
$74$		0			0
$75$	−19.7990	−	7.00000i	−0.263987	−	0.0933333i
$76$		0			0
$77$	−33.9411			−0.440794
$78$		0			0
$79$	10.0000			0.126582			0.0632911	−	0.997995i	$-0.479840\pi$
$79$	10.0000			0.126582			0.0632911	+	0.997995i	$0.479840\pi$
$80$		0			0
$81$	17.0000	+	79.1960i	0.209877	+	0.977728i
$82$		0			0
$83$	−73.5391			−0.886013			−0.443007	−	0.896518i	$-0.646088\pi$
$83$	−73.5391			−0.886013			−0.443007	+	0.896518i	$0.646088\pi$
$84$		0			0
$85$			128.000i			1.50588i
$86$		0			0
$87$	−48.0000	−	16.9706i	−0.551724	−	0.195064i
$88$		0			0
$89$			33.9411i			0.381361i	0.981652	+	0.190680i	$0.0610694\pi$
$89$			33.9411i			0.381361i	−0.981652	+	0.190680i	$0.938931\pi$
$90$		0			0
$91$		−	60.0000i		−	0.659341i
$92$		0			0
$93$	62.2254	+	22.0000i	0.669090	+	0.236559i
$94$		0			0
$95$			11.3137i			0.119092i
$96$		0			0
$97$	−94.0000			−0.969072			−0.484536	−	0.874771i	$-0.661012\pi$
$97$	−94.0000			−0.969072			−0.484536	+	0.874771i	$0.661012\pi$
$98$		0			0
$99$	−39.5980	−	32.0000i	−0.399980	−	0.323232i
$100$		0			0
$101$	−50.9117			−0.504076			−0.252038	−	0.967717i	$-0.581101\pi$
$101$	−50.9117			−0.504076			−0.252038	+	0.967717i	$0.581101\pi$
$102$		0			0
$103$	134.000			1.30097			0.650485	−	0.759519i	$-0.274565\pi$
$103$	134.000			1.30097			0.650485	+	0.759519i	$0.274565\pi$
$104$		0			0
$105$	96.0000	+	33.9411i	0.914286	+	0.323249i
$106$		0			0
$107$	−50.9117			−0.475810			−0.237905	−	0.971288i	$-0.576461\pi$
$107$	−50.9117			−0.475810			−0.237905	+	0.971288i	$0.576461\pi$
$108$		0			0
$109$		−	10.0000i		−	0.0917431i	−0.998947	−	0.0458716i	$-0.985394\pi$
$109$		−	10.0000i		−	0.0917431i	0.998947	−	0.0458716i	$-0.0146065\pi$
$110$		0			0
$111$	−6.00000	+	16.9706i	−0.0540541	+	0.152888i
$112$		0			0
$113$		−	67.8823i		−	0.600728i	−0.953825	−	0.300364i	$-0.902892\pi$
$113$		−	67.8823i		−	0.600728i	0.953825	−	0.300364i	$-0.0971081\pi$
$114$		0			0
$115$		−	64.0000i		−	0.556522i
$116$		0			0
$117$	56.5685	−	70.0000i	0.483492	−	0.598291i
$118$		0			0
$119$		−	135.765i		−	1.14088i
$120$		0			0
$121$	−89.0000			−0.735537
$122$		0			0
$123$	−33.9411	+	96.0000i	−0.275944	+	0.780488i
$124$		0			0
$125$	101.823			0.814587
$126$		0			0
$127$	106.000			0.834646			0.417323	−	0.908758i	$-0.362968\pi$
$127$	106.000			0.834646			0.417323	+	0.908758i	$0.362968\pi$
$128$		0			0
$129$	82.0000	−	231.931i	0.635659	−	1.79791i
$130$		0			0
$131$	−5.65685			−0.0431821			−0.0215910	−	0.999767i	$-0.506873\pi$
$131$	−5.65685			−0.0431821			−0.0215910	+	0.999767i	$0.506873\pi$
$132$		0			0
$133$		−	12.0000i		−	0.0902256i
$134$		0			0
$135$	80.0000	+	130.108i	0.592593	+	0.963760i
$136$		0			0
$137$			101.823i			0.743236i	0.928386	+	0.371618i	$0.121197\pi$
$137$			101.823i			0.743236i	−0.928386	+	0.371618i	$0.878803\pi$
$138$		0			0
$139$		−	78.0000i		−	0.561151i	−0.959832	−	0.280576i	$-0.909475\pi$
$139$		−	78.0000i		−	0.561151i	0.959832	−	0.280576i	$-0.0905253\pi$
$140$		0			0
$141$	67.8823	−	192.000i	0.481434	−	1.36170i
$142$		0			0
$143$			56.5685i			0.395584i
$144$		0			0
$145$	−96.0000			−0.662069
$146$		0			0
$147$	36.7696	+	13.0000i	0.250133	+	0.0884354i
$148$		0			0
$149$	−164.049			−1.10100			−0.550499	−	0.834836i	$-0.685563\pi$
$149$	−164.049			−1.10100			−0.550499	+	0.834836i	$0.685563\pi$
$150$		0			0
$151$	−218.000			−1.44371			−0.721854	−	0.692045i	$-0.756710\pi$
$151$	−218.000			−1.44371			−0.721854	+	0.692045i	$0.756710\pi$
$152$		0			0
$153$	128.000	−	158.392i	0.836601	−	1.03524i
$154$		0			0
$155$	124.451			0.802908
$156$		0			0
$157$			86.0000i			0.547771i	0.961762	+	0.273885i	$0.0883089\pi$
$157$			86.0000i			0.547771i	−0.961762	+	0.273885i	$0.911691\pi$
$158$		0			0
$159$	−176.000	−	62.2254i	−1.10692	−	0.391355i
$160$		0			0
$161$			67.8823i			0.421629i
$162$		0			0
$163$			222.000i			1.36196i	0.732301	+	0.680982i	$0.238447\pi$
$163$			222.000i			1.36196i	−0.732301	+	0.680982i	$0.761553\pi$
$164$		0			0
$165$	−90.5097	−	32.0000i	−0.548543	−	0.193939i
$166$		0			0
$167$			169.706i			1.01620i	0.861298	+	0.508101i	$0.169652\pi$
$167$			169.706i			1.01620i	−0.861298	+	0.508101i	$0.830348\pi$
$168$		0			0
$169$	69.0000			0.408284
$170$		0			0
$171$	11.3137	−	14.0000i	0.0661620	−	0.0818713i
$172$		0			0
$173$	−186.676			−1.07905			−0.539527	−	0.841969i	$-0.681397\pi$
$173$	−186.676			−1.07905			−0.539527	+	0.841969i	$0.681397\pi$
$174$		0			0
$175$	42.0000			0.240000
$176$		0			0
$177$	208.000	+	73.5391i	1.17514	+	0.415475i
$178$		0			0
$179$	152.735			0.853269			0.426634	−	0.904424i	$-0.359699\pi$
$179$	152.735			0.853269			0.426634	+	0.904424i	$0.359699\pi$
$180$		0			0
$181$			90.0000i			0.497238i	0.968601	+	0.248619i	$0.0799766\pi$
$181$			90.0000i			0.497238i	−0.968601	+	0.248619i	$0.920023\pi$
$182$		0			0
$183$	86.0000	−	243.245i	0.469945	−	1.32921i
$184$		0			0
$185$			33.9411i			0.183466i
$186$		0			0
$187$			128.000i			0.684492i
$188$		0			0
$189$	−84.8528	−	138.000i	−0.448957	−	0.730159i
$190$		0			0
$191$		−	271.529i		−	1.42162i	−0.703385	−	0.710809i	$-0.748329\pi$
$191$		−	271.529i		−	1.42162i	0.703385	−	0.710809i	$-0.251671\pi$
$192$		0			0
$193$	2.00000			0.0103627			0.00518135	−	0.999987i	$-0.498351\pi$
$193$	2.00000			0.0103627			0.00518135	+	0.999987i	$0.498351\pi$
$194$		0			0
$195$	56.5685	−	160.000i	0.290095	−	0.820513i
$196$		0			0
$197$	84.8528			0.430725			0.215362	−	0.976534i	$-0.430907\pi$
$197$	84.8528			0.430725			0.215362	+	0.976534i	$0.430907\pi$
$198$		0			0
$199$	−250.000			−1.25628			−0.628141	−	0.778100i	$-0.716184\pi$
$199$	−250.000			−1.25628			−0.628141	+	0.778100i	$0.716184\pi$
$200$		0			0
$201$	−2.00000	+	5.65685i	−0.00995025	+	0.0281436i
$202$		0			0
$203$	101.823			0.501593
$204$		0			0
$205$			192.000i			0.936585i
$206$		0			0
$207$	−64.0000	+	79.1960i	−0.309179	+	0.382589i
$208$		0			0
$209$			11.3137i			0.0541326i
$210$		0			0
$211$		−	34.0000i		−	0.161137i	−0.996749	−	0.0805687i	$-0.974326\pi$
$211$		−	34.0000i		−	0.161137i	0.996749	−	0.0805687i	$-0.0256736\pi$
$212$		0			0
$213$	124.451	−	352.000i	0.584276	−	1.65258i
$214$		0			0
$215$		−	463.862i		−	2.15750i
$216$		0			0
$217$	−132.000			−0.608295
$218$		0			0
$219$	231.931	+	82.0000i	1.05905	+	0.374429i
$220$		0			0
$221$	−226.274			−1.02387
$222$		0			0
$223$	−278.000			−1.24664			−0.623318	−	0.781968i	$-0.714216\pi$
$223$	−278.000			−1.24664			−0.623318	+	0.781968i	$0.714216\pi$
$224$		0			0
$225$	49.0000	+	39.5980i	0.217778	+	0.175991i
$226$		0			0
$227$	220.617			0.971882			0.485941	−	0.873991i	$-0.338477\pi$
$227$	220.617			0.971882			0.485941	+	0.873991i	$0.338477\pi$
$228$		0			0
$229$			58.0000i			0.253275i	0.991949	+	0.126638i	$0.0404185\pi$
$229$			58.0000i			0.253275i	−0.991949	+	0.126638i	$0.959581\pi$
$230$		0			0
$231$	96.0000	+	33.9411i	0.415584	+	0.146931i
$232$		0			0
$233$		−	395.980i		−	1.69948i	−0.527199	−	0.849742i	$-0.676758\pi$
$233$		−	395.980i		−	1.69948i	0.527199	−	0.849742i	$-0.323242\pi$
$234$		0			0
$235$		−	384.000i		−	1.63404i
$236$		0			0
$237$	−28.2843	−	10.0000i	−0.119343	−	0.0421941i
$238$		0			0
$239$		−	22.6274i		−	0.0946754i	−0.998879	−	0.0473377i	$-0.984926\pi$
$239$		−	22.6274i		−	0.0946754i	0.998879	−	0.0473377i	$-0.0150737\pi$
$240$		0			0
$241$	−30.0000			−0.124481			−0.0622407	−	0.998061i	$-0.519825\pi$
$241$	−30.0000			−0.124481			−0.0622407	+	0.998061i	$0.519825\pi$
$242$		0			0
$243$	31.1127	−	241.000i	0.128036	−	0.991770i
$244$		0			0
$245$	73.5391			0.300160
$246$		0			0
$247$	−20.0000			−0.0809717
$248$		0			0
$249$	208.000	+	73.5391i	0.835341	+	0.295338i
$250$		0			0
$251$	107.480			0.428208			0.214104	−	0.976811i	$-0.431317\pi$
$251$	107.480			0.428208			0.214104	+	0.976811i	$0.431317\pi$
$252$		0			0
$253$		−	64.0000i		−	0.252964i
$254$		0			0
$255$	128.000	−	362.039i	0.501961	−	1.41976i
$256$		0			0
$257$			181.019i			0.704355i	0.935933	+	0.352178i	$0.114559\pi$
$257$			181.019i			0.704355i	−0.935933	+	0.352178i	$0.885441\pi$
$258$		0			0
$259$		−	36.0000i		−	0.138996i
$260$		0			0
$261$	118.794	+	96.0000i	0.455149	+	0.367816i
$262$		0			0
$263$			214.960i			0.817340i	0.912682	+	0.408670i	$0.134007\pi$
$263$			214.960i			0.817340i	−0.912682	+	0.408670i	$0.865993\pi$
$264$		0			0
$265$	−352.000			−1.32830
$266$		0			0
$267$	33.9411	−	96.0000i	0.127120	−	0.359551i
$268$		0			0
$269$	401.637			1.49307			0.746537	−	0.665344i	$-0.231715\pi$
$269$	401.637			1.49307			0.746537	+	0.665344i	$0.231715\pi$
$270$		0			0
$271$	266.000			0.981550			0.490775	−	0.871286i	$-0.336714\pi$
$271$	266.000			0.981550			0.490775	+	0.871286i	$0.336714\pi$
$272$		0			0
$273$	−60.0000	+	169.706i	−0.219780	+	0.621632i
$274$		0			0
$275$	−39.5980			−0.143993
$276$		0			0
$277$			346.000i			1.24910i	0.780986	+	0.624549i	$0.214717\pi$
$277$			346.000i			1.24910i	−0.780986	+	0.624549i	$0.785283\pi$
$278$		0			0
$279$	−154.000	−	124.451i	−0.551971	−	0.446060i
$280$		0			0
$281$			124.451i			0.442885i	0.975173	+	0.221443i	$0.0710766\pi$
$281$			124.451i			0.442885i	−0.975173	+	0.221443i	$0.928923\pi$
$282$		0			0
$283$		−	46.0000i		−	0.162544i	−0.996692	−	0.0812721i	$-0.974102\pi$
$283$		−	46.0000i		−	0.162544i	0.996692	−	0.0812721i	$-0.0258983\pi$
$284$		0			0
$285$	11.3137	−	32.0000i	0.0396972	−	0.112281i
$286$		0			0
$287$		−	203.647i		−	0.709571i
$288$		0			0
$289$	−223.000			−0.771626
$290$		0			0
$291$	265.872	+	94.0000i	0.913650	+	0.323024i
$292$		0			0
$293$	220.617			0.752960			0.376480	−	0.926425i	$-0.377134\pi$
$293$	220.617			0.752960			0.376480	+	0.926425i	$0.377134\pi$
$294$		0			0
$295$	416.000			1.41017
$296$		0			0
$297$	80.0000	+	130.108i	0.269360	+	0.438073i
$298$		0			0
$299$	113.137			0.378385
$300$		0			0
$301$			492.000i			1.63455i
$302$		0			0
$303$	144.000	+	50.9117i	0.475248	+	0.168025i
$304$		0			0
$305$		−	486.489i		−	1.59505i
$306$		0			0
$307$			30.0000i			0.0977199i	0.998806	+	0.0488599i	$0.0155588\pi$
$307$			30.0000i			0.0977199i	−0.998806	+	0.0488599i	$0.984441\pi$
$308$		0			0
$309$	−379.009	−	134.000i	−1.22657	−	0.433657i
$310$		0			0
$311$		−	576.999i		−	1.85530i	−0.373447	−	0.927651i	$-0.621824\pi$
$311$		−	576.999i		−	1.85530i	0.373447	−	0.927651i	$-0.378176\pi$
$312$		0			0
$313$	−210.000			−0.670927			−0.335463	−	0.942053i	$-0.608893\pi$
$313$	−210.000			−0.670927			−0.335463	+	0.942053i	$0.608893\pi$
$314$		0			0
$315$	−237.588	−	192.000i	−0.754247	−	0.609524i
$316$		0			0
$317$	152.735			0.481814			0.240907	−	0.970548i	$-0.422555\pi$
$317$	152.735			0.481814			0.240907	+	0.970548i	$0.422555\pi$
$318$		0			0
$319$	−96.0000			−0.300940
$320$		0			0
$321$	144.000	+	50.9117i	0.448598	+	0.158603i
$322$		0			0
$323$	−45.2548			−0.140108
$324$		0			0
$325$		−	70.0000i		−	0.215385i
$326$		0			0
$327$	−10.0000	+	28.2843i	−0.0305810	+	0.0864962i
$328$		0			0
$329$			407.294i			1.23797i
$330$		0			0
$331$			434.000i			1.31118i	0.755118	+	0.655589i	$0.227580\pi$
$331$			434.000i			1.31118i	−0.755118	+	0.655589i	$0.772420\pi$
$332$		0			0
$333$	33.9411	−	42.0000i	0.101925	−	0.126126i
$334$		0			0
$335$			11.3137i			0.0337723i
$336$		0			0
$337$	−510.000			−1.51335			−0.756677	−	0.653789i	$-0.773178\pi$
$337$	−510.000			−1.51335			−0.756677	+	0.653789i	$0.773178\pi$
$338$		0			0
$339$	−67.8823	+	192.000i	−0.200243	+	0.566372i
$340$		0			0
$341$	124.451			0.364958
$342$		0			0
$343$	−372.000			−1.08455
$344$		0			0
$345$	−64.0000	+	181.019i	−0.185507	+	0.524694i
$346$		0			0
$347$	152.735			0.440159			0.220079	−	0.975482i	$-0.429368\pi$
$347$	152.735			0.440159			0.220079	+	0.975482i	$0.429368\pi$
$348$		0			0
$349$		−	426.000i		−	1.22063i	−0.792159	−	0.610315i	$-0.791043\pi$
$349$		−	426.000i		−	1.22063i	0.792159	−	0.610315i	$-0.208957\pi$
$350$		0			0
$351$	−230.000	+	141.421i	−0.655271	+	0.402910i
$352$		0			0
$353$			45.2548i			0.128201i	0.997943	+	0.0641003i	$0.0204178\pi$
$353$			45.2548i			0.128201i	−0.997943	+	0.0641003i	$0.979582\pi$
$354$		0			0
$355$		−	704.000i		−	1.98310i
$356$		0			0
$357$	−135.765	+	384.000i	−0.380293	+	1.07563i
$358$		0			0
$359$			441.235i			1.22907i	0.788891	+	0.614533i	$0.210655\pi$
$359$			441.235i			1.22907i	−0.788891	+	0.614533i	$0.789345\pi$
$360$		0			0
$361$	357.000			0.988920
$362$		0			0
$363$	251.730	+	89.0000i	0.693471	+	0.245179i
$364$		0			0
$365$	463.862			1.27085
$366$		0			0
$367$	−566.000			−1.54223			−0.771117	−	0.636693i	$-0.780302\pi$
$367$	−566.000			−1.54223			−0.771117	+	0.636693i	$0.780302\pi$
$368$		0			0
$369$	192.000	−	237.588i	0.520325	−	0.643870i
$370$		0			0
$371$	373.352			1.00634
$372$		0			0
$373$			218.000i			0.584450i	0.956350	+	0.292225i	$0.0943957\pi$
$373$			218.000i			0.584450i	−0.956350	+	0.292225i	$0.905604\pi$
$374$		0			0
$375$	−288.000	−	101.823i	−0.768000	−	0.271529i
$376$		0			0
$377$		−	169.706i		−	0.450148i
$378$		0			0
$379$		−	142.000i		−	0.374670i	−0.982296	−	0.187335i	$-0.940015\pi$
$379$		−	142.000i		−	0.374670i	0.982296	−	0.187335i	$-0.0599850\pi$
$380$		0			0
$381$	−299.813	−	106.000i	−0.786911	−	0.278215i
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$	192.000			0.498701
$386$		0			0
$387$	−463.862	+	574.000i	−1.19861	+	1.48320i
$388$		0			0
$389$	−548.715			−1.41058			−0.705289	−	0.708920i	$-0.749183\pi$
$389$	−548.715			−1.41058			−0.705289	+	0.708920i	$0.749183\pi$
$390$		0			0
$391$	256.000			0.654731
$392$		0			0
$393$	16.0000	+	5.65685i	0.0407125	+	0.0143940i
$394$		0			0
$395$	−56.5685			−0.143211
$396$		0			0
$397$			310.000i			0.780856i	0.920633	+	0.390428i	$0.127673\pi$
$397$			310.000i			0.780856i	−0.920633	+	0.390428i	$0.872327\pi$
$398$		0			0
$399$	−12.0000	+	33.9411i	−0.0300752	+	0.0850655i
$400$		0			0
$401$			339.411i			0.846412i	0.906033	+	0.423206i	$0.139095\pi$
$401$			339.411i			0.846412i	−0.906033	+	0.423206i	$0.860905\pi$
$402$		0			0
$403$			220.000i			0.545906i
$404$		0			0
$405$	−96.1665	−	448.000i	−0.237448	−	1.10617i
$406$		0			0
$407$			33.9411i			0.0833934i
$408$		0			0
$409$	270.000			0.660147			0.330073	−	0.943955i	$-0.392927\pi$
$409$	270.000			0.660147			0.330073	+	0.943955i	$0.392927\pi$
$410$		0			0
$411$	101.823	−	288.000i	0.247745	−	0.700730i
$412$		0			0
$413$	−441.235			−1.06836
$414$		0			0
$415$	416.000			1.00241
$416$		0			0
$417$	−78.0000	+	220.617i	−0.187050	+	0.529058i
$418$		0			0
$419$	−50.9117			−0.121508			−0.0607538	−	0.998153i	$-0.519350\pi$
$419$	−50.9117			−0.121508			−0.0607538	+	0.998153i	$0.519350\pi$
$420$		0			0
$421$		−	454.000i		−	1.07838i	−0.842183	−	0.539192i	$-0.818730\pi$
$421$		−	454.000i		−	1.07838i	0.842183	−	0.539192i	$-0.181270\pi$
$422$		0			0
$423$	−384.000	+	475.176i	−0.907801	+	1.12335i
$424$		0			0
$425$		−	158.392i		−	0.372687i
$426$		0			0
$427$			516.000i			1.20843i
$428$		0			0
$429$	56.5685	−	160.000i	0.131861	−	0.372960i
$430$		0			0
$431$			248.902i			0.577498i	0.957405	+	0.288749i	$0.0932393\pi$
$431$			248.902i			0.577498i	−0.957405	+	0.288749i	$0.906761\pi$
$432$		0			0
$433$	706.000			1.63048			0.815242	−	0.579120i	$-0.196604\pi$
$433$	706.000			1.63048			0.815242	+	0.579120i	$0.196604\pi$
$434$		0			0
$435$	271.529	+	96.0000i	0.624205	+	0.220690i
$436$		0			0
$437$	22.6274			0.0517790
$438$		0			0
$439$	486.000			1.10706			0.553531	−	0.832829i	$-0.313280\pi$
$439$	486.000			1.10706			0.553531	+	0.832829i	$0.313280\pi$
$440$		0			0
$441$	−91.0000	−	73.5391i	−0.206349	−	0.166755i
$442$		0			0
$443$	−707.107			−1.59618			−0.798089	−	0.602540i	$-0.794156\pi$
$443$	−707.107			−1.59618			−0.798089	+	0.602540i	$0.794156\pi$
$444$		0			0
$445$		−	192.000i		−	0.431461i
$446$		0			0
$447$	464.000	+	164.049i	1.03803	+	0.366999i
$448$		0			0
$449$		−	724.077i		−	1.61264i	−0.591477	−	0.806322i	$-0.701455\pi$
$449$		−	724.077i		−	1.61264i	0.591477	−	0.806322i	$-0.298545\pi$
$450$		0			0
$451$			192.000i			0.425721i
$452$		0			0
$453$	616.597	+	218.000i	1.36114	+	0.481236i
$454$		0			0
$455$			339.411i			0.745959i
$456$		0			0
$457$	−338.000			−0.739606			−0.369803	−	0.929110i	$-0.620575\pi$
$457$	−338.000			−0.739606			−0.369803	+	0.929110i	$0.620575\pi$
$458$		0			0
$459$	−520.431	+	320.000i	−1.13384	+	0.697168i
$460$		0			0
$461$	−774.989			−1.68110			−0.840552	−	0.541731i	$-0.817769\pi$
$461$	−774.989			−1.68110			−0.840552	+	0.541731i	$0.817769\pi$
$462$		0			0
$463$	74.0000			0.159827			0.0799136	−	0.996802i	$-0.474536\pi$
$463$	74.0000			0.159827			0.0799136	+	0.996802i	$0.474536\pi$
$464$		0			0
$465$	−352.000	−	124.451i	−0.756989	−	0.267636i
$466$		0			0
$467$	−797.616			−1.70796			−0.853979	−	0.520307i	$-0.825817\pi$
$467$	−797.616			−1.70796			−0.853979	+	0.520307i	$0.825817\pi$
$468$		0			0
$469$		−	12.0000i		−	0.0255864i
$470$		0			0
$471$	86.0000	−	243.245i	0.182590	−	0.516443i
$472$		0			0
$473$		−	463.862i		−	0.980681i
$474$		0			0
$475$		−	14.0000i		−	0.0294737i
$476$		0			0
$477$	435.578	+	352.000i	0.913161	+	0.737945i
$478$		0			0
$479$			316.784i			0.661344i	0.943746	+	0.330672i	$0.107275\pi$
$479$			316.784i			0.661344i	−0.943746	+	0.330672i	$0.892725\pi$
$480$		0			0
$481$	−60.0000			−0.124740
$482$		0			0
$483$	67.8823	−	192.000i	0.140543	−	0.397516i
$484$		0			0
$485$	531.744			1.09638
$486$		0			0
$487$	134.000			0.275154			0.137577	−	0.990491i	$-0.456069\pi$
$487$	134.000			0.275154			0.137577	+	0.990491i	$0.456069\pi$
$488$		0			0
$489$	222.000	−	627.911i	0.453988	−	1.28407i
$490$		0			0
$491$	−50.9117			−0.103690			−0.0518449	−	0.998655i	$-0.516510\pi$
$491$	−50.9117			−0.103690			−0.0518449	+	0.998655i	$0.516510\pi$
$492$		0			0
$493$		−	384.000i		−	0.778905i
$494$		0			0
$495$	224.000	+	181.019i	0.452525	+	0.365696i
$496$		0			0
$497$			746.705i			1.50242i
$498$		0			0
$499$			30.0000i			0.0601202i	0.999548	+	0.0300601i	$0.00956988\pi$
$499$			30.0000i			0.0601202i	−0.999548	+	0.0300601i	$0.990430\pi$
$500$		0			0
$501$	169.706	−	480.000i	0.338734	−	0.958084i
$502$		0			0
$503$			237.588i			0.472342i	0.971712	+	0.236171i	$0.0758925\pi$
$503$			237.588i			0.472342i	−0.971712	+	0.236171i	$0.924107\pi$
$504$		0			0
$505$	288.000			0.570297
$506$		0			0
$507$	−195.161	−	69.0000i	−0.384934	−	0.136095i
$508$		0			0
$509$	−118.794			−0.233387			−0.116693	−	0.993168i	$-0.537230\pi$
$509$	−118.794			−0.233387			−0.116693	+	0.993168i	$0.537230\pi$
$510$		0			0
$511$	−492.000			−0.962818
$512$		0			0
$513$	−46.0000	+	28.2843i	−0.0896686	+	0.0551350i
$514$		0			0
$515$	−758.018			−1.47188
$516$		0			0
$517$		−	384.000i		−	0.742747i
$518$		0			0
$519$	528.000	+	186.676i	1.01734	+	0.359684i
$520$		0			0
$521$		−	79.1960i		−	0.152008i	−0.997108	−	0.0760038i	$-0.975784\pi$
$521$		−	79.1960i		−	0.152008i	0.997108	−	0.0760038i	$-0.0242161\pi$
$522$		0			0
$523$		−	494.000i		−	0.944551i	−0.881451	−	0.472275i	$-0.843433\pi$
$523$		−	494.000i		−	0.944551i	0.881451	−	0.472275i	$-0.156567\pi$
$524$		0			0
$525$	−118.794	−	42.0000i	−0.226274	−	0.0800000i
$526$		0			0
$527$			497.803i			0.944598i
$528$		0			0
$529$	401.000			0.758034
$530$		0			0
$531$	−514.774	−	416.000i	−0.969442	−	0.783427i
$532$		0			0
$533$	−339.411			−0.636794
$534$		0			0
$535$	288.000			0.538318
$536$		0			0
$537$	−432.000	−	152.735i	−0.804469	−	0.284423i
$538$		0			0
$539$	73.5391			0.136436
$540$		0			0
$541$		−	234.000i		−	0.432532i	−0.976334	−	0.216266i	$-0.930612\pi$
$541$		−	234.000i		−	0.432532i	0.976334	−	0.216266i	$-0.0693879\pi$
$542$		0			0
$543$	90.0000	−	254.558i	0.165746	−	0.468800i
$544$		0			0
$545$			56.5685i			0.103795i
$546$		0			0
$547$		−	290.000i		−	0.530165i	−0.964226	−	0.265082i	$-0.914601\pi$
$547$		−	290.000i		−	0.530165i	0.964226	−	0.265082i	$-0.0853991\pi$
$548$		0			0
$549$	−486.489	+	602.000i	−0.886137	+	1.09654i
$550$		0			0
$551$		−	33.9411i		−	0.0615991i
$552$		0			0
$553$	60.0000			0.108499
$554$		0			0
$555$	33.9411	−	96.0000i	0.0611552	−	0.172973i
$556$		0			0
$557$	175.362			0.314834			0.157417	−	0.987532i	$-0.449683\pi$
$557$	175.362			0.314834			0.157417	+	0.987532i	$0.449683\pi$
$558$		0			0
$559$	820.000			1.46691
$560$		0			0
$561$	128.000	−	362.039i	0.228164	−	0.645345i
$562$		0			0
$563$	243.245			0.432051			0.216026	−	0.976388i	$-0.430691\pi$
$563$	243.245			0.432051			0.216026	+	0.976388i	$0.430691\pi$
$564$		0			0
$565$			384.000i			0.679646i
$566$		0			0
$567$	102.000	+	475.176i	0.179894	+	0.838052i
$568$		0			0
$569$			622.254i			1.09359i	0.837266	+	0.546796i	$0.184153\pi$
$569$			622.254i			1.09359i	−0.837266	+	0.546796i	$0.815847\pi$
$570$		0			0
$571$			402.000i			0.704028i	0.935995	+	0.352014i	$0.114503\pi$
$571$			402.000i			0.704028i	−0.935995	+	0.352014i	$0.885497\pi$
$572$		0			0
$573$	−271.529	+	768.000i	−0.473873	+	1.34031i
$574$		0			0
$575$			79.1960i			0.137732i
$576$		0			0
$577$	98.0000			0.169844			0.0849220	−	0.996388i	$-0.472936\pi$
$577$	98.0000			0.169844			0.0849220	+	0.996388i	$0.472936\pi$
$578$		0			0
$579$	−5.65685	−	2.00000i	−0.00977004	−	0.00345423i
$580$		0			0
$581$	−441.235			−0.759440
$582$		0			0
$583$	−352.000			−0.603774
$584$		0			0
$585$	−320.000	+	395.980i	−0.547009	+	0.676889i
$586$		0			0
$587$	−548.715			−0.934778			−0.467389	−	0.884052i	$-0.654805\pi$
$587$	−548.715			−0.934778			−0.467389	+	0.884052i	$0.654805\pi$
$588$		0			0
$589$			44.0000i			0.0747029i
$590$		0			0
$591$	−240.000	−	84.8528i	−0.406091	−	0.143575i
$592$		0			0
$593$			701.450i			1.18288i	0.806348	+	0.591442i	$0.201441\pi$
$593$			701.450i			1.18288i	−0.806348	+	0.591442i	$0.798559\pi$
$594$		0			0
$595$			768.000i			1.29076i
$596$		0			0
$597$	707.107	+	250.000i	1.18443	+	0.418760i
$598$		0			0
$599$			644.881i			1.07660i	0.842754	+	0.538298i	$0.180933\pi$
$599$			644.881i			1.07660i	−0.842754	+	0.538298i	$0.819067\pi$
$600$		0			0
$601$	398.000			0.662230			0.331115	−	0.943590i	$-0.392575\pi$
$601$	398.000			0.662230			0.331115	+	0.943590i	$0.392575\pi$
$602$		0			0
$603$	11.3137	−	14.0000i	0.0187624	−	0.0232172i
$604$		0			0
$605$	503.460			0.832165
$606$		0			0
$607$	170.000			0.280066			0.140033	−	0.990147i	$-0.455279\pi$
$607$	170.000			0.280066			0.140033	+	0.990147i	$0.455279\pi$
$608$		0			0
$609$	−288.000	−	101.823i	−0.472906	−	0.167198i
$610$		0			0
$611$	678.823			1.11100
$612$		0			0
$613$		−	1030.00i		−	1.68026i	−0.542384	−	0.840131i	$-0.682478\pi$
$613$		−	1030.00i		−	1.68026i	0.542384	−	0.840131i	$-0.317522\pi$
$614$		0			0
$615$	192.000	−	543.058i	0.312195	−	0.883021i
$616$		0			0
$617$			1052.17i			1.70531i	0.522476	+	0.852654i	$0.325008\pi$
$617$			1052.17i			1.70531i	−0.522476	+	0.852654i	$0.674992\pi$
$618$		0			0
$619$		−	14.0000i		−	0.0226171i	−0.999936	−	0.0113086i	$-0.996400\pi$
$619$		−	14.0000i		−	0.0226171i	0.999936	−	0.0113086i	$-0.00359970\pi$
$620$		0			0
$621$	260.215	−	160.000i	0.419026	−	0.257649i
$622$		0			0
$623$			203.647i			0.326881i
$624$		0			0
$625$	−751.000			−1.20160
$626$		0			0
$627$	11.3137	−	32.0000i	0.0180442	−	0.0510367i
$628$		0			0
$629$	−135.765			−0.215842
$630$		0			0
$631$	−1114.00			−1.76545			−0.882726	−	0.469888i	$-0.844294\pi$
$631$	−1114.00			−1.76545			−0.882726	+	0.469888i	$0.844294\pi$
$632$		0			0
$633$	−34.0000	+	96.1665i	−0.0537125	+	0.151922i
$634$		0			0
$635$	−599.627			−0.944294
$636$		0			0
$637$			130.000i			0.204082i
$638$		0			0
$639$	−704.000	+	871.156i	−1.10172	+	1.36331i
$640$		0			0
$641$		−	452.548i		−	0.706004i	−0.935623	−	0.353002i	$-0.885161\pi$
$641$		−	452.548i		−	0.706004i	0.935623	−	0.353002i	$-0.114839\pi$
$642$		0			0
$643$			798.000i			1.24106i	0.784184	+	0.620529i	$0.213082\pi$
$643$			798.000i			1.24106i	−0.784184	+	0.620529i	$0.786918\pi$
$644$		0			0
$645$	−463.862	+	1312.00i	−0.719166	+	2.03411i
$646$		0			0
$647$		−	147.078i		−	0.227323i	−0.993520	−	0.113662i	$-0.963742\pi$
$647$		−	147.078i		−	0.227323i	0.993520	−	0.113662i	$-0.0362580\pi$
$648$		0			0
$649$	416.000			0.640986
$650$		0			0
$651$	373.352	+	132.000i	0.573506	+	0.202765i
$652$		0			0
$653$	−322.441			−0.493784			−0.246892	−	0.969043i	$-0.579409\pi$
$653$	−322.441			−0.493784			−0.246892	+	0.969043i	$0.579409\pi$
$654$		0			0
$655$	32.0000			0.0488550
$656$		0			0
$657$	−574.000	−	463.862i	−0.873668	−	0.706031i
$658$		0			0
$659$	−797.616			−1.21034			−0.605172	−	0.796095i	$-0.706896\pi$
$659$	−797.616			−1.21034			−0.605172	+	0.796095i	$0.706896\pi$
$660$		0			0
$661$			986.000i			1.49168i	0.666126	+	0.745840i	$0.267951\pi$
$661$			986.000i			1.49168i	−0.666126	+	0.745840i	$0.732049\pi$
$662$		0			0
$663$	640.000	+	226.274i	0.965309	+	0.341288i
$664$		0			0
$665$			67.8823i			0.102079i
$666$		0			0
$667$			192.000i			0.287856i
$668$		0			0
$669$	786.303	+	278.000i	1.17534	+	0.415546i
$670$		0			0
$671$		−	486.489i		−	0.725022i
$672$		0			0
$673$	34.0000			0.0505201			0.0252600	−	0.999681i	$-0.491959\pi$
$673$	34.0000			0.0505201			0.0252600	+	0.999681i	$0.491959\pi$
$674$		0			0
$675$	−98.9949	−	161.000i	−0.146659	−	0.238519i
$676$		0			0
$677$	401.637			0.593259			0.296630	−	0.954993i	$-0.404137\pi$
$677$	401.637			0.593259			0.296630	+	0.954993i	$0.404137\pi$
$678$		0			0
$679$	−564.000			−0.830633
$680$		0			0
$681$	−624.000	−	220.617i	−0.916300	−	0.323961i
$682$		0			0
$683$	130.108			0.190494			0.0952472	−	0.995454i	$-0.469636\pi$
$683$	130.108			0.190494			0.0952472	+	0.995454i	$0.469636\pi$
$684$		0			0
$685$		−	576.000i		−	0.840876i
$686$		0			0
$687$	58.0000	−	164.049i	0.0844250	−	0.238790i
$688$		0			0
$689$		−	622.254i		−	0.903126i
$690$		0			0
$691$		−	578.000i		−	0.836469i	−0.908339	−	0.418234i	$-0.862649\pi$
$691$		−	578.000i		−	0.836469i	0.908339	−	0.418234i	$-0.137351\pi$
$692$		0			0
$693$	−237.588	−	192.000i	−0.342840	−	0.277056i
$694$		0			0
$695$			441.235i			0.634870i
$696$		0			0
$697$	−768.000			−1.10187
$698$		0			0
$699$	−395.980	+	1120.00i	−0.566495	+	1.60229i
$700$		0			0
$701$	1238.85			1.76726			0.883631	−	0.468183i	$-0.155091\pi$
$701$	1238.85			1.76726			0.883631	+	0.468183i	$0.155091\pi$
$702$		0			0
$703$	−12.0000			−0.0170697
$704$		0			0
$705$	−384.000	+	1086.12i	−0.544681	+	1.54059i
$706$		0			0
$707$	−305.470			−0.432065
$708$		0			0
$709$		−	1222.00i		−	1.72355i	−0.507287	−	0.861777i	$-0.669352\pi$
$709$		−	1222.00i		−	1.72355i	0.507287	−	0.861777i	$-0.330648\pi$
$710$		0			0
$711$	70.0000	+	56.5685i	0.0984529	+	0.0795619i
$712$		0			0
$713$		−	248.902i		−	0.349091i
$714$		0			0
$715$		−	320.000i		−	0.447552i
$716$		0			0
$717$	−22.6274	+	64.0000i	−0.0315585	+	0.0892608i
$718$		0			0
$719$		−	248.902i		−	0.346177i	−0.984906	−	0.173089i	$-0.944625\pi$
$719$		−	248.902i		−	0.346177i	0.984906	−	0.173089i	$-0.0553747\pi$
$720$		0			0
$721$	804.000			1.11512
$722$		0			0
$723$	84.8528	+	30.0000i	0.117362	+	0.0414938i
$724$		0			0
$725$	118.794			0.163854
$726$		0			0
$727$	870.000			1.19670			0.598349	−	0.801235i	$-0.295823\pi$
$727$	870.000			1.19670			0.598349	+	0.801235i	$0.295823\pi$
$728$		0			0
$729$	−329.000	+	650.538i	−0.451303	+	0.892371i
$730$		0			0
$731$	1855.45			2.53823
$732$		0			0
$733$			214.000i			0.291951i	0.989288	+	0.145975i	$0.0466321\pi$
$733$			214.000i			0.291951i	−0.989288	+	0.145975i	$0.953368\pi$
$734$		0			0
$735$	−208.000	−	73.5391i	−0.282993	−	0.100053i
$736$		0			0
$737$			11.3137i			0.0153510i
$738$		0			0
$739$			958.000i			1.29635i	0.761493	+	0.648173i	$0.224467\pi$
$739$			958.000i			1.29635i	−0.761493	+	0.648173i	$0.775533\pi$
$740$		0			0
$741$	56.5685	+	20.0000i	0.0763408	+	0.0269906i
$742$		0			0
$743$		−	1006.92i		−	1.35521i	−0.735427	−	0.677604i	$-0.763018\pi$
$743$		−	1006.92i		−	1.35521i	0.735427	−	0.677604i	$-0.236982\pi$
$744$		0			0
$745$	928.000			1.24564
$746$		0			0
$747$	−514.774	−	416.000i	−0.689121	−	0.556894i
$748$		0			0
$749$	−305.470			−0.407837
$750$		0			0
$751$	−630.000			−0.838881			−0.419441	−	0.907783i	$-0.637774\pi$
$751$	−630.000			−0.838881			−0.419441	+	0.907783i	$0.637774\pi$
$752$		0			0
$753$	−304.000	−	107.480i	−0.403718	−	0.142736i
$754$		0			0
$755$	1233.19			1.63337
$756$		0			0
$757$			602.000i			0.795244i	0.917549	+	0.397622i	$0.130165\pi$
$757$			602.000i			0.795244i	−0.917549	+	0.397622i	$0.869835\pi$
$758$		0			0
$759$	−64.0000	+	181.019i	−0.0843215	+	0.238497i
$760$		0			0
$761$		−	1097.43i		−	1.44209i	−0.692889	−	0.721044i	$-0.743662\pi$
$761$		−	1097.43i		−	1.44209i	0.692889	−	0.721044i	$-0.256338\pi$
$762$		0			0
$763$		−	60.0000i		−	0.0786370i
$764$		0			0
$765$	−724.077	+	896.000i	−0.946506	+	1.17124i
$766$		0			0
$767$			735.391i			0.958789i
$768$		0			0
$769$	770.000			1.00130			0.500650	−	0.865650i	$-0.333094\pi$
$769$	770.000			1.00130			0.500650	+	0.865650i	$0.333094\pi$
$770$		0			0
$771$	181.019	−	512.000i	0.234785	−	0.664073i
$772$		0			0
$773$	−186.676			−0.241496			−0.120748	−	0.992683i	$-0.538529\pi$
$773$	−186.676			−0.241496			−0.120748	+	0.992683i	$0.538529\pi$
$774$		0			0
$775$	−154.000			−0.198710
$776$		0			0
$777$	−36.0000	+	101.823i	−0.0463320	+	0.131047i
$778$		0			0
$779$	−67.8823			−0.0871402
$780$		0			0
$781$		−	704.000i		−	0.901408i
$782$		0			0
$783$	−240.000	−	390.323i	−0.306513	−	0.498497i
$784$		0			0
$785$		−	486.489i		−	0.619732i
$786$		0			0
$787$		−	514.000i		−	0.653113i	−0.945178	−	0.326557i	$-0.894112\pi$
$787$		−	514.000i		−	0.653113i	0.945178	−	0.326557i	$-0.105888\pi$
$788$		0			0
$789$	214.960	−	608.000i	0.272447	−	0.770596i
$790$		0			0
$791$		−	407.294i		−	0.514910i
$792$		0			0
$793$	860.000			1.08449
$794$		0			0
$795$	995.606	+	352.000i	1.25234	+	0.442767i
$796$		0			0
$797$	−707.107			−0.887211			−0.443605	−	0.896222i	$-0.646301\pi$
$797$	−707.107			−0.887211			−0.443605	+	0.896222i	$0.646301\pi$
$798$		0			0
$799$	1536.00			1.92240
$800$		0			0
$801$	−192.000	+	237.588i	−0.239700	+	0.296614i
$802$		0			0
$803$	463.862			0.577661
$804$		0			0
$805$		−	384.000i		−	0.477019i
$806$		0			0
$807$	−1136.00	−	401.637i	−1.40768	−	0.497691i
$808$		0			0
$809$		−	758.018i		−	0.936982i	−0.883468	−	0.468491i	$-0.844798\pi$
$809$		−	758.018i		−	0.936982i	0.883468	−	0.468491i	$-0.155202\pi$
$810$		0			0
$811$		−	1454.00i		−	1.79285i	−0.443197	−	0.896424i	$-0.646156\pi$
$811$		−	1454.00i		−	1.79285i	0.443197	−	0.896424i	$-0.353844\pi$
$812$		0			0
$813$	−752.362	−	266.000i	−0.925414	−	0.327183i
$814$		0			0
$815$		−	1255.82i		−	1.54089i
$816$		0			0
$817$	164.000			0.200734
$818$		0			0
$819$	339.411	−	420.000i	0.414422	−	0.512821i
$820$		0			0
$821$	967.322			1.17822			0.589112	−	0.808051i	$-0.299478\pi$
$821$	967.322			1.17822			0.589112	+	0.808051i	$0.299478\pi$
$822$		0			0
$823$	166.000			0.201701			0.100851	−	0.994902i	$-0.467844\pi$
$823$	166.000			0.201701			0.100851	+	0.994902i	$0.467844\pi$
$824$		0			0
$825$	112.000	+	39.5980i	0.135758	+	0.0479976i
$826$		0			0
$827$	−978.636			−1.18336			−0.591678	−	0.806174i	$-0.701534\pi$
$827$	−978.636			−1.18336			−0.591678	+	0.806174i	$0.701534\pi$
$828$		0			0
$829$		−	1258.00i		−	1.51749i	−0.651387	−	0.758745i	$-0.725813\pi$
$829$		−	1258.00i		−	1.51749i	0.651387	−	0.758745i	$-0.274187\pi$
$830$		0			0
$831$	346.000	−	978.636i	0.416366	−	1.17766i
$832$		0			0
$833$			294.156i			0.353129i
$834$		0			0
$835$		−	960.000i		−	1.14970i
$836$		0			0
$837$	311.127	+	506.000i	0.371717	+	0.604540i
$838$		0			0
$839$		−	1323.70i		−	1.57772i	−0.614575	−	0.788858i	$-0.710673\pi$
$839$		−	1323.70i		−	1.57772i	0.614575	−	0.788858i	$-0.289327\pi$
$840$		0			0
$841$	−553.000			−0.657551
$842$		0			0
$843$	124.451	−	352.000i	0.147628	−	0.417556i
$844$		0			0
$845$	−390.323			−0.461921
$846$		0			0
$847$	−534.000			−0.630460
$848$		0			0
$849$	−46.0000	+	130.108i	−0.0541814	+	0.153248i
$850$		0			0
$851$	67.8823			0.0797676
$852$		0			0
$853$		−	742.000i		−	0.869871i	−0.900462	−	0.434936i	$-0.856771\pi$
$853$		−	742.000i		−	0.869871i	0.900462	−	0.434936i	$-0.143229\pi$
$854$		0			0
$855$	−64.0000	+	79.1960i	−0.0748538	+	0.0926269i
$856$		0			0
$857$		−	237.588i		−	0.277232i	−0.990346	−	0.138616i	$-0.955735\pi$
$857$		−	237.588i		−	0.277232i	0.990346	−	0.138616i	$-0.0442654\pi$
$858$		0			0
$859$		−	1230.00i		−	1.43190i	−0.698153	−	0.715949i	$-0.745994\pi$
$859$		−	1230.00i		−	1.43190i	0.698153	−	0.715949i	$-0.254006\pi$
$860$		0			0
$861$	−203.647	+	576.000i	−0.236524	+	0.668990i
$862$		0			0
$863$		−	45.2548i		−	0.0524390i	−0.999656	−	0.0262195i	$-0.991653\pi$
$863$		−	45.2548i		−	0.0524390i	0.999656	−	0.0262195i	$-0.00834688\pi$
$864$		0			0
$865$	1056.00			1.22081
$866$		0			0
$867$	630.739	+	223.000i	0.727496	+	0.257209i
$868$		0			0
$869$	−56.5685			−0.0650961
$870$		0			0
$871$	−20.0000			−0.0229621
$872$		0			0
$873$	−658.000	−	531.744i	−0.753723	−	0.609100i
$874$		0			0
$875$	610.940			0.698217
$876$		0			0
$877$			822.000i			0.937286i	0.883388	+	0.468643i	$0.155257\pi$
$877$			822.000i			0.937286i	−0.883388	+	0.468643i	$0.844743\pi$
$878$		0			0
$879$	−624.000	−	220.617i	−0.709898	−	0.250987i
$880$		0			0
$881$			656.195i			0.744830i	0.928066	+	0.372415i	$0.121470\pi$
$881$			656.195i			0.744830i	−0.928066	+	0.372415i	$0.878530\pi$
$882$		0			0
$883$		−	962.000i		−	1.08947i	−0.838609	−	0.544734i	$-0.816631\pi$
$883$		−	962.000i		−	1.08947i	0.838609	−	0.544734i	$-0.183369\pi$
$884$		0			0
$885$	−1176.63	−	416.000i	−1.32952	−	0.470056i
$886$		0			0
$887$			1142.68i			1.28826i	0.764917	+	0.644129i	$0.222780\pi$
$887$			1142.68i			1.28826i	−0.764917	+	0.644129i	$0.777220\pi$
$888$		0			0
$889$	636.000			0.715411
$890$		0			0
$891$	−96.1665	−	448.000i	−0.107931	−	0.502806i
$892$		0			0
$893$	135.765			0.152032
$894$		0			0
$895$	−864.000			−0.965363
$896$		0			0
$897$	−320.000	−	113.137i	−0.356745	−	0.126128i
$898$		0			0
$899$	−373.352			−0.415297
$900$		0			0
$901$		−	1408.00i		−	1.56271i
$902$		0			0
$903$	492.000	−	1391.59i	0.544850	−	1.54107i
$904$		0			0
$905$		−	509.117i		−	0.562560i
$906$		0			0
$907$			1042.00i			1.14884i	0.818560	+	0.574421i	$0.194773\pi$
$907$			1042.00i			1.14884i	−0.818560	+	0.574421i	$0.805227\pi$
$908$		0			0
$909$	−356.382	−	288.000i	−0.392059	−	0.316832i
$910$		0			0
$911$		−	1606.55i		−	1.76350i	−0.471719	−	0.881749i	$-0.656366\pi$
$911$		−	1606.55i		−	1.76350i	0.471719	−	0.881749i	$-0.343634\pi$
$912$		0			0
$913$	416.000			0.455641
$914$		0			0
$915$	−486.489	+	1376.00i	−0.531682	+	1.50383i
$916$		0			0
$917$	−33.9411			−0.0370132
$918$		0			0
$919$	614.000			0.668118			0.334059	−	0.942552i	$-0.391582\pi$
$919$	614.000			0.668118			0.334059	+	0.942552i	$0.391582\pi$
$920$		0			0
$921$	30.0000	−	84.8528i	0.0325733	−	0.0921312i
$922$		0			0
$923$	1244.51			1.34833
$924$		0			0
$925$		−	42.0000i		−	0.0454054i
$926$		0			0
$927$	938.000	+	758.018i	1.01187	+	0.817711i
$928$		0			0
$929$			45.2548i			0.0487135i	0.999703	+	0.0243567i	$0.00775376\pi$
$929$			45.2548i			0.0487135i	−0.999703	+	0.0243567i	$0.992246\pi$
$930$		0			0
$931$			26.0000i			0.0279270i
$932$		0			0
$933$	−576.999	+	1632.00i	−0.618434	+	1.74920i
$934$		0			0
$935$		−	724.077i		−	0.774414i
$936$		0			0
$937$	462.000			0.493063			0.246531	−	0.969135i	$-0.420709\pi$
$937$	462.000			0.493063			0.246531	+	0.969135i	$0.420709\pi$
$938$		0			0
$939$	593.970	+	210.000i	0.632556	+	0.223642i
$940$		0			0
$941$	356.382			0.378727			0.189363	−	0.981907i	$-0.439358\pi$
$941$	356.382			0.378727			0.189363	+	0.981907i	$0.439358\pi$
$942$		0			0
$943$	384.000			0.407211
$944$		0			0
$945$	480.000	+	780.646i	0.507937	+	0.826080i
$946$		0			0
$947$	695.793			0.734734			0.367367	−	0.930076i	$-0.380259\pi$
$947$	695.793			0.734734			0.367367	+	0.930076i	$0.380259\pi$
$948$		0			0
$949$			820.000i			0.864067i
$950$		0			0
$951$	−432.000	−	152.735i	−0.454259	−	0.160605i
$952$		0			0
$953$			1527.35i			1.60268i	0.598212	+	0.801338i	$0.295878\pi$
$953$			1527.35i			1.60268i	−0.598212	+	0.801338i	$0.704122\pi$
$954$		0			0
$955$			1536.00i			1.60838i
$956$		0			0
$957$	271.529	+	96.0000i	0.283729	+	0.100313i
$958$		0			0
$959$			610.940i			0.637060i
$960$		0			0
$961$	−477.000			−0.496358
$962$		0			0
$963$	−356.382	−	288.000i	−0.370075	−	0.299065i
$964$		0			0
$965$	−11.3137			−0.0117241
$966$		0			0
$967$	70.0000			0.0723888			0.0361944	−	0.999345i	$-0.488476\pi$
$967$	70.0000			0.0723888			0.0361944	+	0.999345i	$0.488476\pi$
$968$		0			0
$969$	128.000	+	45.2548i	0.132095	+	0.0467026i
$970$		0			0
$971$	627.911			0.646664			0.323332	−	0.946286i	$-0.395197\pi$
$971$	627.911			0.646664			0.323332	+	0.946286i	$0.395197\pi$
$972$		0			0
$973$		−	468.000i		−	0.480987i
$974$		0			0
$975$	−70.0000	+	197.990i	−0.0717949	+	0.203067i
$976$		0			0
$977$		−	1018.23i		−	1.04220i	−0.853494	−	0.521102i	$-0.825521\pi$
$977$		−	1018.23i		−	1.04220i	0.853494	−	0.521102i	$-0.174479\pi$
$978$		0			0
$979$		−	192.000i		−	0.196118i
$980$		0			0
$981$	56.5685	−	70.0000i	0.0576642	−	0.0713558i
$982$		0			0
$983$			644.881i			0.656034i	0.944672	+	0.328017i	$0.106380\pi$
$983$			644.881i			0.656034i	−0.944672	+	0.328017i	$0.893620\pi$
$984$		0			0
$985$	−480.000			−0.487310
$986$		0			0
$987$	407.294	−	1152.00i	0.412658	−	1.16717i
$988$		0			0
$989$	−927.724			−0.938043
$990$		0			0
$991$	−854.000			−0.861756			−0.430878	−	0.902410i	$-0.641796\pi$
$991$	−854.000			−0.861756			−0.430878	+	0.902410i	$0.641796\pi$
$992$		0			0
$993$	434.000	−	1227.54i	0.437059	−	1.23619i
$994$		0			0
$995$	1414.21			1.42132
$996$		0			0
$997$		−	518.000i		−	0.519559i	−0.965668	−	0.259779i	$-0.916350\pi$
$997$		−	518.000i		−	0.519559i	0.965668	−	0.259779i	$-0.0836498\pi$
$998$		0			0
$999$	−138.000	+	84.8528i	−0.138138	+	0.0849378i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.3.h.d.641.1		4
3.2	odd	2	inner	768.3.h.d.641.3		4
4.3	odd	2		768.3.h.c.641.4		4
8.3	odd	2		768.3.h.c.641.1		4
8.5	even	2	inner	768.3.h.d.641.4		4
12.11	even	2		768.3.h.c.641.2		4
16.3	odd	4		48.3.e.b.17.2		2
16.5	even	4		192.3.e.c.65.2		2
16.11	odd	4		192.3.e.d.65.1		2
16.13	even	4		24.3.e.a.17.1	✓	2
24.5	odd	2	inner	768.3.h.d.641.2		4
24.11	even	2		768.3.h.c.641.3		4
48.5	odd	4		192.3.e.c.65.1		2
48.11	even	4		192.3.e.d.65.2		2
48.29	odd	4		24.3.e.a.17.2	yes	2
48.35	even	4		48.3.e.b.17.1		2
80.3	even	4		1200.3.c.i.449.1		4
80.13	odd	4		600.3.c.a.449.4		4
80.19	odd	4		1200.3.l.n.401.1		2
80.29	even	4		600.3.l.b.401.2		2
80.67	even	4		1200.3.c.i.449.4		4
80.77	odd	4		600.3.c.a.449.1		4
112.13	odd	4		1176.3.d.a.785.2		2
144.13	even	12		648.3.m.d.593.2		4
144.29	odd	12		648.3.m.d.377.2		4
144.61	even	12		648.3.m.d.377.1		4
144.67	odd	12		1296.3.q.e.593.2		4
144.77	odd	12		648.3.m.d.593.1		4
144.83	even	12		1296.3.q.e.1025.2		4
144.115	odd	12		1296.3.q.e.1025.1		4
144.131	even	12		1296.3.q.e.593.1		4
240.29	odd	4		600.3.l.b.401.1		2
240.77	even	4		600.3.c.a.449.3		4
240.83	odd	4		1200.3.c.i.449.3		4
240.173	even	4		600.3.c.a.449.2		4
240.179	even	4		1200.3.l.n.401.2		2
240.227	odd	4		1200.3.c.i.449.2		4
336.125	even	4		1176.3.d.a.785.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.3.e.a.17.1	✓	2	16.13	even	4
24.3.e.a.17.2	yes	2	48.29	odd	4
48.3.e.b.17.1		2	48.35	even	4
48.3.e.b.17.2		2	16.3	odd	4
192.3.e.c.65.1		2	48.5	odd	4
192.3.e.c.65.2		2	16.5	even	4
192.3.e.d.65.1		2	16.11	odd	4
192.3.e.d.65.2		2	48.11	even	4
600.3.c.a.449.1		4	80.77	odd	4
600.3.c.a.449.2		4	240.173	even	4
600.3.c.a.449.3		4	240.77	even	4
600.3.c.a.449.4		4	80.13	odd	4
600.3.l.b.401.1		2	240.29	odd	4
600.3.l.b.401.2		2	80.29	even	4
648.3.m.d.377.1		4	144.61	even	12
648.3.m.d.377.2		4	144.29	odd	12
648.3.m.d.593.1		4	144.77	odd	12
648.3.m.d.593.2		4	144.13	even	12
768.3.h.c.641.1		4	8.3	odd	2
768.3.h.c.641.2		4	12.11	even	2
768.3.h.c.641.3		4	24.11	even	2
768.3.h.c.641.4		4	4.3	odd	2
768.3.h.d.641.1		4	1.1	even	1	trivial
768.3.h.d.641.2		4	24.5	odd	2	inner
768.3.h.d.641.3		4	3.2	odd	2	inner
768.3.h.d.641.4		4	8.5	even	2	inner
1176.3.d.a.785.1		2	336.125	even	4
1176.3.d.a.785.2		2	112.13	odd	4
1200.3.c.i.449.1		4	80.3	even	4
1200.3.c.i.449.2		4	240.227	odd	4
1200.3.c.i.449.3		4	240.83	odd	4
1200.3.c.i.449.4		4	80.67	even	4
1200.3.l.n.401.1		2	80.19	odd	4
1200.3.l.n.401.2		2	240.179	even	4
1296.3.q.e.593.1		4	144.131	even	12
1296.3.q.e.593.2		4	144.67	odd	12
1296.3.q.e.1025.1		4	144.115	odd	12
1296.3.q.e.1025.2		4	144.83	even	12

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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	768.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.82843 - 1.00000i) q^{3} -5.65685 q^{5} +6.00000 q^{7} +(7.00000 + 5.65685i) q^{9} -5.65685 q^{11} -10.0000i q^{13} +(16.0000 + 5.65685i) q^{15} -22.6274i q^{17} -2.00000i q^{19} +(-16.9706 - 6.00000i) q^{21} +11.3137i q^{23} +7.00000 q^{25} +(-14.1421 - 23.0000i) q^{27} +16.9706 q^{29} -22.0000 q^{31} +(16.0000 + 5.65685i) q^{33} -33.9411 q^{35} -6.00000i q^{37} +(-10.0000 + 28.2843i) q^{39} -33.9411i q^{41} +82.0000i q^{43} +(-39.5980 - 32.0000i) q^{45} +67.8823i q^{47} -13.0000 q^{49} +(-22.6274 + 64.0000i) q^{51} +62.2254 q^{53} +32.0000 q^{55} +(-2.00000 + 5.65685i) q^{57} -73.5391 q^{59} +86.0000i q^{61} +(42.0000 + 33.9411i) q^{63} +56.5685i q^{65} -2.00000i q^{67} +(11.3137 - 32.0000i) q^{69} +124.451i q^{71} -82.0000 q^{73} +(-19.7990 - 7.00000i) q^{75} -33.9411 q^{77} +10.0000 q^{79} +(17.0000 + 79.1960i) q^{81} -73.5391 q^{83} +128.000i q^{85} +(-48.0000 - 16.9706i) q^{87} +33.9411i q^{89} -60.0000i q^{91} +(62.2254 + 22.0000i) q^{93} +11.3137i q^{95} -94.0000 q^{97} +(-39.5980 - 32.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 24 q^{7} + 28 q^{9} + 64 q^{15} + 28 q^{25} - 88 q^{31} + 64 q^{33} - 40 q^{39} - 52 q^{49} + 128 q^{55} - 8 q^{57} + 168 q^{63} - 328 q^{73} + 40 q^{79} + 68 q^{81} - 192 q^{87} - 376 q^{97}+O(q^{100}) \) 4 * q + 24 * q^7 + 28 * q^9 + 64 * q^15 + 28 * q^25 - 88 * q^31 + 64 * q^33 - 40 * q^39 - 52 * q^49 + 128 * q^55 - 8 * q^57 + 168 * q^63 - 328 * q^73 + 40 * q^79 + 68 * q^81 - 192 * q^87 - 376 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more