LMFDB - Embedded newform 1280.3.e.l.639.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,3,Mod(639,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

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

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

chi := DirichletCharacter("1280.639");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1280.e (of order $2$, degree $1$, 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:	$34.8774738381$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 640)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			639.8
Character	$\chi$	$=$	1280.639
Dual form			1280.3.e.l.639.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.46407i q^{3} +(-3.70957 + 3.35247i) q^{5} -5.17181 q^{7} -20.8560 q^{9} +O(q^{10})$	$q+5.46407i q^{3} +(-3.70957 + 3.35247i) q^{5} -5.17181 q^{7} -20.8560 q^{9} +3.02733 q^{11} +19.8001 q^{13} +(-18.3181 - 20.2693i) q^{15} -29.5266i q^{17} -19.7002 q^{19} -28.2591i q^{21} +6.31692 q^{23} +(2.52183 - 24.8725i) q^{25} -64.7822i q^{27} -19.8111i q^{29} +19.3348i q^{31} +16.5415i q^{33} +(19.1852 - 17.3384i) q^{35} +7.92148 q^{37} +108.189i q^{39} -66.5219 q^{41} -8.67294i q^{43} +(77.3669 - 69.9193i) q^{45} +87.7563 q^{47} -22.2523 q^{49} +161.335 q^{51} +5.66765 q^{53} +(-11.2301 + 10.1490i) q^{55} -107.643i q^{57} -62.1080 q^{59} -25.2666i q^{61} +107.864 q^{63} +(-73.4500 + 66.3795i) q^{65} -52.6204i q^{67} +34.5161i q^{69} -90.9146i q^{71} +11.9724i q^{73} +(135.905 + 13.7795i) q^{75} -15.6568 q^{77} +91.8692i q^{79} +166.270 q^{81} +42.9296i q^{83} +(98.9871 + 109.531i) q^{85} +108.249 q^{87} +48.1520 q^{89} -102.403 q^{91} -105.647 q^{93} +(73.0794 - 66.0446i) q^{95} +5.13149i q^{97} -63.1380 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 72 q^{9}+O(q^{10}) $ 24 * q - 72 * q^9	$ 24 q - 72 q^{9} + 16 q^{11} - 48 q^{19} - 24 q^{25} + 112 q^{35} - 80 q^{41} + 168 q^{49} + 576 q^{51} - 496 q^{59} + 32 q^{65} - 224 q^{75} + 184 q^{81} - 144 q^{89} + 864 q^{91} - 368 q^{99}+O(q^{100}) $ 24 * q - 72 * q^9 + 16 * q^11 - 48 * q^19 - 24 * q^25 + 112 * q^35 - 80 * q^41 + 168 * q^49 + 576 * q^51 - 496 * q^59 + 32 * q^65 - 224 * q^75 + 184 * q^81 - 144 * q^89 + 864 * q^91 - 368 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$			5.46407i			1.82136i	0.413117	+	0.910678i	$0.364440\pi$
$3$			5.46407i			1.82136i	−0.413117	+	0.910678i	$0.635560\pi$
$4$		0			0
$5$	−3.70957	+	3.35247i	−0.741914	+	0.670495i
$5$	−3.70957	+	3.35247i	−0.741914	+	0.670495i
$6$		0			0
$7$	−5.17181			−0.738831			−0.369415	−	0.929264i	$-0.620442\pi$
$7$	−5.17181			−0.738831			−0.369415	+	0.929264i	$0.620442\pi$
$8$		0			0
$9$	−20.8560			−2.31734
$10$		0			0
$11$	3.02733			0.275212			0.137606	−	0.990487i	$-0.456059\pi$
$11$	3.02733			0.275212			0.137606	+	0.990487i	$0.456059\pi$
$12$		0			0
$13$	19.8001			1.52309			0.761544	−	0.648114i	$-0.224442\pi$
$13$	19.8001			1.52309			0.761544	+	0.648114i	$0.224442\pi$
$14$		0			0
$15$	−18.3181	−	20.2693i	−1.22121	−	1.35129i
$16$		0			0
$17$		−	29.5266i		−	1.73686i	−0.495814	−	0.868429i	$-0.665130\pi$
$17$		−	29.5266i		−	1.73686i	0.495814	−	0.868429i	$-0.334870\pi$
$18$		0			0
$19$	−19.7002			−1.03685			−0.518427	−	0.855122i	$-0.673482\pi$
$19$	−19.7002			−1.03685			−0.518427	+	0.855122i	$0.673482\pi$
$20$		0			0
$21$		−	28.2591i		−	1.34567i
$22$		0			0
$23$	6.31692			0.274649			0.137324	−	0.990526i	$-0.456150\pi$
$23$	6.31692			0.274649			0.137324	+	0.990526i	$0.456150\pi$
$24$		0			0
$25$	2.52183	−	24.8725i	0.100873	−	0.994899i
$26$		0			0
$27$		−	64.7822i		−	2.39934i
$28$		0			0
$29$		−	19.8111i		−	0.683142i	−0.939856	−	0.341571i	$-0.889041\pi$
$29$		−	19.8111i		−	0.683142i	0.939856	−	0.341571i	$-0.110959\pi$
$30$		0			0
$31$			19.3348i			0.623704i	0.950131	+	0.311852i	$0.100949\pi$
$31$			19.3348i			0.623704i	−0.950131	+	0.311852i	$0.899051\pi$
$32$		0			0
$33$			16.5415i			0.501258i
$34$		0			0
$35$	19.1852	−	17.3384i	0.548149	−	0.495382i
$36$		0			0
$37$	7.92148			0.214094			0.107047	−	0.994254i	$-0.465860\pi$
$37$	7.92148			0.214094			0.107047	+	0.994254i	$0.465860\pi$
$38$		0			0
$39$			108.189i			2.77408i
$40$		0			0
$41$	−66.5219			−1.62249			−0.811243	−	0.584709i	$-0.801209\pi$
$41$	−66.5219			−1.62249			−0.811243	+	0.584709i	$0.801209\pi$
$42$		0			0
$43$		−	8.67294i		−	0.201696i	−0.994902	−	0.100848i	$-0.967844\pi$
$43$		−	8.67294i		−	0.201696i	0.994902	−	0.100848i	$-0.0321556\pi$
$44$		0			0
$45$	77.3669	−	69.9193i	1.71927	−	1.55376i
$46$		0			0
$47$	87.7563			1.86715			0.933577	−	0.358376i	$-0.116670\pi$
$47$	87.7563			1.86715			0.933577	+	0.358376i	$0.116670\pi$
$48$		0			0
$49$	−22.2523			−0.454129
$50$		0			0
$51$	161.335			3.16344
$52$		0			0
$53$	5.66765			0.106937			0.0534684	−	0.998570i	$-0.482972\pi$
$53$	5.66765			0.106937			0.0534684	+	0.998570i	$0.482972\pi$
$54$		0			0
$55$	−11.2301	+	10.1490i	−0.204183	+	0.184528i
$56$		0			0
$57$		−	107.643i		−	1.88848i
$58$		0			0
$59$	−62.1080			−1.05268			−0.526339	−	0.850275i	$-0.676436\pi$
$59$	−62.1080			−1.05268			−0.526339	+	0.850275i	$0.676436\pi$
$60$		0			0
$61$		−	25.2666i		−	0.414206i	−0.978319	−	0.207103i	$-0.933596\pi$
$61$		−	25.2666i		−	0.414206i	0.978319	−	0.207103i	$-0.0664035\pi$
$62$		0			0
$63$	107.864			1.71212
$64$		0			0
$65$	−73.4500	+	66.3795i	−1.13000	+	1.02122i
$66$		0			0
$67$		−	52.6204i		−	0.785378i	−0.919671	−	0.392689i	$-0.871545\pi$
$67$		−	52.6204i		−	0.785378i	0.919671	−	0.392689i	$-0.128455\pi$
$68$		0			0
$69$			34.5161i			0.500233i
$70$		0			0
$71$		−	90.9146i		−	1.28049i	−0.768172	−	0.640244i	$-0.778833\pi$
$71$		−	90.9146i		−	1.28049i	0.768172	−	0.640244i	$-0.221167\pi$
$72$		0			0
$73$			11.9724i			0.164006i	0.996632	+	0.0820028i	$0.0261316\pi$
$73$			11.9724i			0.164006i	−0.996632	+	0.0820028i	$0.973868\pi$
$74$		0			0
$75$	135.905	+	13.7795i	1.81207	+	0.183726i
$76$		0			0
$77$	−15.6568			−0.203335
$78$		0			0
$79$			91.8692i			1.16290i	0.813582	+	0.581451i	$0.197515\pi$
$79$			91.8692i			1.16290i	−0.813582	+	0.581451i	$0.802485\pi$
$80$		0			0
$81$	166.270			2.05271
$82$		0			0
$83$			42.9296i			0.517224i	0.965981	+	0.258612i	$0.0832651\pi$
$83$			42.9296i			0.517224i	−0.965981	+	0.258612i	$0.916735\pi$
$84$		0			0
$85$	98.9871	+	109.531i	1.16455	+	1.28860i
$86$		0			0
$87$	108.249			1.24424
$88$		0			0
$89$	48.1520			0.541034			0.270517	−	0.962715i	$-0.412805\pi$
$89$	48.1520			0.541034			0.270517	+	0.962715i	$0.412805\pi$
$90$		0			0
$91$	−102.403			−1.12530
$92$		0			0
$93$	−105.647			−1.13599
$94$		0			0
$95$	73.0794	−	66.0446i	0.769257	−	0.695206i
$96$		0			0
$97$			5.13149i			0.0529020i	0.999650	+	0.0264510i	$0.00842060\pi$
$97$			5.13149i			0.0529020i	−0.999650	+	0.0264510i	$0.991579\pi$
$98$		0			0
$99$	−63.1380			−0.637758
$100$		0			0
$101$		−	24.4025i		−	0.241609i	−0.992676	−	0.120804i	$-0.961453\pi$
$101$		−	24.4025i		−	0.241609i	0.992676	−	0.120804i	$-0.0385474\pi$
$102$		0			0
$103$	−90.7585			−0.881150			−0.440575	−	0.897716i	$-0.645225\pi$
$103$	−90.7585			−0.881150			−0.440575	+	0.897716i	$0.645225\pi$
$104$		0			0
$105$	94.7380	+	104.829i	0.902267	+	0.998374i
$106$		0			0
$107$			125.043i			1.16862i	0.811530	+	0.584311i	$0.198635\pi$
$107$			125.043i			1.16862i	−0.811530	+	0.584311i	$0.801365\pi$
$108$		0			0
$109$		−	108.216i		−	0.992811i	−0.868091	−	0.496406i	$-0.834653\pi$
$109$		−	108.216i		−	0.992811i	0.868091	−	0.496406i	$-0.165347\pi$
$110$		0			0
$111$			43.2835i			0.389941i
$112$		0			0
$113$			37.0522i			0.327896i	0.986469	+	0.163948i	$0.0524229\pi$
$113$			37.0522i			0.327896i	−0.986469	+	0.163948i	$0.947577\pi$
$114$		0			0
$115$	−23.4331	+	21.1773i	−0.203766	+	0.184150i
$116$		0			0
$117$	−412.952			−3.52951
$118$		0			0
$119$			152.706i			1.28324i
$120$		0			0
$121$	−111.835			−0.924259
$122$		0			0
$123$		−	363.480i		−	2.95512i
$124$		0			0
$125$	74.0295	+	100.721i	0.592236	+	0.805765i
$126$		0			0
$127$	−145.014			−1.14184			−0.570922	−	0.821004i	$-0.693414\pi$
$127$	−145.014			−1.14184			−0.570922	+	0.821004i	$0.693414\pi$
$128$		0			0
$129$	47.3895			0.367361
$130$		0			0
$131$	140.500			1.07252			0.536258	−	0.844054i	$-0.319837\pi$
$131$	140.500			1.07252			0.536258	+	0.844054i	$0.319837\pi$
$132$		0			0
$133$	101.886			0.766060
$134$		0			0
$135$	217.181	+	240.314i	1.60875	+	1.78010i
$136$		0			0
$137$		−	216.649i		−	1.58138i	−0.612216	−	0.790690i	$-0.709722\pi$
$137$		−	216.649i		−	1.58138i	0.612216	−	0.790690i	$-0.290278\pi$
$138$		0			0
$139$	238.444			1.71542			0.857712	−	0.514131i	$-0.171885\pi$
$139$	238.444			1.71542			0.857712	+	0.514131i	$0.171885\pi$
$140$		0			0
$141$			479.506i			3.40075i
$142$		0			0
$143$	59.9415			0.419171
$144$		0			0
$145$	66.4163	+	73.4908i	0.458043	+	0.506833i
$146$		0			0
$147$		−	121.588i		−	0.827131i
$148$		0			0
$149$			122.992i			0.825447i	0.910856	+	0.412724i	$0.135422\pi$
$149$			122.992i			0.825447i	−0.910856	+	0.412724i	$0.864578\pi$
$150$		0			0
$151$		−	59.0552i		−	0.391094i	−0.980694	−	0.195547i	$-0.937352\pi$
$151$		−	59.0552i		−	0.391094i	0.980694	−	0.195547i	$-0.0626482\pi$
$152$		0			0
$153$			615.807i			4.02489i
$154$		0			0
$155$	−64.8195	−	71.7239i	−0.418191	−	0.462735i
$156$		0			0
$157$	−230.585			−1.46870			−0.734348	−	0.678774i	$-0.762512\pi$
$157$	−230.585			−1.46870			−0.734348	+	0.678774i	$0.762512\pi$
$158$		0			0
$159$			30.9684i			0.194770i
$160$		0			0
$161$	−32.6699			−0.202919
$162$		0			0
$163$		−	158.925i		−	0.974998i	−0.873124	−	0.487499i	$-0.837909\pi$
$163$		−	158.925i		−	0.974998i	0.873124	−	0.487499i	$-0.162091\pi$
$164$		0			0
$165$	−55.4550	−	61.3619i	−0.336091	−	0.371891i
$166$		0			0
$167$	146.536			0.877460			0.438730	−	0.898619i	$-0.355428\pi$
$167$	146.536			0.877460			0.438730	+	0.898619i	$0.355428\pi$
$168$		0			0
$169$	223.045			1.31980
$170$		0			0
$171$	410.869			2.40274
$172$		0			0
$173$	−40.4179			−0.233630			−0.116815	−	0.993154i	$-0.537268\pi$
$173$	−40.4179			−0.233630			−0.116815	+	0.993154i	$0.537268\pi$
$174$		0			0
$175$	−13.0424	+	128.636i	−0.0745282	+	0.735062i
$176$		0			0
$177$		−	339.362i		−	1.91730i
$178$		0			0
$179$	212.970			1.18978			0.594889	−	0.803808i	$-0.297196\pi$
$179$	212.970			1.18978			0.594889	+	0.803808i	$0.297196\pi$
$180$		0			0
$181$		−	217.576i		−	1.20208i	−0.799219	−	0.601040i	$-0.794753\pi$
$181$		−	217.576i		−	1.20208i	0.799219	−	0.601040i	$-0.205247\pi$
$182$		0			0
$183$	138.058			0.754416
$184$		0			0
$185$	−29.3853	+	26.5566i	−0.158839	+	0.143549i
$186$		0			0
$187$		−	89.3866i		−	0.478003i
$188$		0			0
$189$			335.041i			1.77271i
$190$		0			0
$191$			193.675i			1.01400i	0.861945	+	0.507001i	$0.169246\pi$
$191$			193.675i			1.01400i	−0.861945	+	0.507001i	$0.830754\pi$
$192$		0			0
$193$		−	55.1991i		−	0.286006i	−0.989722	−	0.143003i	$-0.954324\pi$
$193$		−	55.1991i		−	0.286006i	0.989722	−	0.143003i	$-0.0456758\pi$
$194$		0			0
$195$	−362.702	−	401.336i	−1.86001	−	2.05813i
$196$		0			0
$197$	−23.0489			−0.116999			−0.0584997	−	0.998287i	$-0.518632\pi$
$197$	−23.0489			−0.116999			−0.0584997	+	0.998287i	$0.518632\pi$
$198$		0			0
$199$		−	307.859i		−	1.54703i	−0.633777	−	0.773516i	$-0.718496\pi$
$199$		−	307.859i		−	1.54703i	0.633777	−	0.773516i	$-0.281504\pi$
$200$		0			0
$201$	287.521			1.43045
$202$		0			0
$203$			102.459i			0.504726i
$204$		0			0
$205$	246.768	−	223.013i	1.20375	−	1.08787i
$206$		0			0
$207$	−131.746			−0.636454
$208$		0			0
$209$	−59.6391			−0.285354
$210$		0			0
$211$	−82.8301			−0.392560			−0.196280	−	0.980548i	$-0.562886\pi$
$211$	−82.8301			−0.392560			−0.196280	+	0.980548i	$0.562886\pi$
$212$		0			0
$213$	496.764			2.33222
$214$		0			0
$215$	29.0758	+	32.1729i	0.135236	+	0.149641i
$216$		0			0
$217$		−	99.9961i		−	0.460812i
$218$		0			0
$219$	−65.4180			−0.298712
$220$		0			0
$221$		−	584.630i		−	2.64539i
$222$		0			0
$223$	229.170			1.02767			0.513834	−	0.857890i	$-0.328225\pi$
$223$	229.170			1.02767			0.513834	+	0.857890i	$0.328225\pi$
$224$		0			0
$225$	−52.5954	+	518.741i	−0.233757	+	2.30552i
$226$		0			0
$227$			82.3448i			0.362752i	0.983414	+	0.181376i	$0.0580551\pi$
$227$			82.3448i			0.362752i	−0.983414	+	0.181376i	$0.941945\pi$
$228$		0			0
$229$		−	177.615i		−	0.775613i	−0.921741	−	0.387806i	$-0.873233\pi$
$229$		−	177.615i		−	0.775613i	0.921741	−	0.387806i	$-0.126767\pi$
$230$		0			0
$231$		−	85.5496i		−	0.370345i
$232$		0			0
$233$		−	301.554i		−	1.29422i	−0.762395	−	0.647111i	$-0.775977\pi$
$233$		−	301.554i		−	1.29422i	0.762395	−	0.647111i	$-0.224023\pi$
$234$		0			0
$235$	−325.538	+	294.201i	−1.38527	+	1.25192i
$236$		0			0
$237$	−501.980			−2.11806
$238$		0			0
$239$		−	143.517i		−	0.600489i	−0.953862	−	0.300244i	$-0.902932\pi$
$239$		−	143.517i		−	0.600489i	0.953862	−	0.300244i	$-0.0970682\pi$
$240$		0			0
$241$	−171.092			−0.709926			−0.354963	−	0.934880i	$-0.615507\pi$
$241$	−171.092			−0.709926			−0.354963	+	0.934880i	$0.615507\pi$
$242$		0			0
$243$			325.470i			1.33938i
$244$		0			0
$245$	82.5466	−	74.6004i	0.336925	−	0.304491i
$246$		0			0
$247$	−390.068			−1.57922
$248$		0			0
$249$	−234.570			−0.942049
$250$		0			0
$251$	6.20502			0.0247212			0.0123606	−	0.999924i	$-0.496065\pi$
$251$	6.20502			0.0247212			0.0123606	+	0.999924i	$0.496065\pi$
$252$		0			0
$253$	19.1234			0.0755865
$254$		0			0
$255$	−598.484	+	540.872i	−2.34700	+	2.12107i
$256$		0			0
$257$		−	127.254i		−	0.495153i	−0.968868	−	0.247576i	$-0.920366\pi$
$257$		−	127.254i		−	0.495153i	0.968868	−	0.247576i	$-0.0796341\pi$
$258$		0			0
$259$	−40.9684			−0.158179
$260$		0			0
$261$			413.181i			1.58307i
$262$		0			0
$263$	−456.897			−1.73725			−0.868625	−	0.495470i	$-0.834996\pi$
$263$	−456.897			−1.73725			−0.868625	+	0.495470i	$0.834996\pi$
$264$		0			0
$265$	−21.0246	+	19.0007i	−0.0793379	+	0.0717006i
$266$		0			0
$267$			263.106i			0.985416i
$268$		0			0
$269$		−	304.065i		−	1.13035i	−0.824970	−	0.565176i	$-0.808808\pi$
$269$		−	304.065i		−	1.13035i	0.824970	−	0.565176i	$-0.191192\pi$
$270$		0			0
$271$		−	379.638i		−	1.40088i	−0.713713	−	0.700438i	$-0.752988\pi$
$271$		−	379.638i		−	1.40088i	0.713713	−	0.700438i	$-0.247012\pi$
$272$		0			0
$273$		−	559.535i		−	2.04958i
$274$		0			0
$275$	7.63440	−	75.2971i	0.0277615	−	0.273808i
$276$		0			0
$277$	179.550			0.648195			0.324098	−	0.946024i	$-0.394939\pi$
$277$	179.550			0.648195			0.324098	+	0.946024i	$0.394939\pi$
$278$		0			0
$279$		−	403.248i		−	1.44533i
$280$		0			0
$281$	−19.7475			−0.0702759			−0.0351379	−	0.999382i	$-0.511187\pi$
$281$	−19.7475			−0.0702759			−0.0351379	+	0.999382i	$0.511187\pi$
$282$		0			0
$283$			271.748i			0.960242i	0.877202	+	0.480121i	$0.159407\pi$
$283$			271.748i			0.960242i	−0.877202	+	0.480121i	$0.840593\pi$
$284$		0			0
$285$	360.872	+	399.311i	1.26622	+	1.40109i
$286$		0			0
$287$	344.039			1.19874
$288$		0			0
$289$	−582.819			−2.01667
$290$		0			0
$291$	−28.0388			−0.0963534
$292$		0			0
$293$	248.936			0.849612			0.424806	−	0.905284i	$-0.360342\pi$
$293$	248.936			0.849612			0.424806	+	0.905284i	$0.360342\pi$
$294$		0			0
$295$	230.394	−	208.215i	0.780996	−	0.705815i
$296$		0			0
$297$		−	196.117i		−	0.660326i
$298$		0			0
$299$	125.076			0.418314
$300$		0			0
$301$			44.8548i			0.149019i
$302$		0			0
$303$	133.337			0.440055
$304$		0			0
$305$	84.7055	+	93.7281i	0.277723	+	0.307305i
$306$		0			0
$307$		−	260.926i		−	0.849922i	−0.905212	−	0.424961i	$-0.860288\pi$
$307$		−	260.926i		−	0.849922i	0.905212	−	0.424961i	$-0.139712\pi$
$308$		0			0
$309$		−	495.910i		−	1.60489i
$310$		0			0
$311$			87.4619i			0.281228i	0.990065	+	0.140614i	$0.0449076\pi$
$311$			87.4619i			0.281228i	−0.990065	+	0.140614i	$0.955092\pi$
$312$		0			0
$313$		−	63.2707i		−	0.202143i	−0.994879	−	0.101071i	$-0.967773\pi$
$313$		−	63.2707i		−	0.202143i	0.994879	−	0.101071i	$-0.0322271\pi$
$314$		0			0
$315$	−400.127	+	361.610i	−1.27025	+	1.14797i
$316$		0			0
$317$	414.483			1.30752			0.653759	−	0.756703i	$-0.273191\pi$
$317$	414.483			1.30752			0.653759	+	0.756703i	$0.273191\pi$
$318$		0			0
$319$		−	59.9747i		−	0.188009i
$320$		0			0
$321$	−683.242			−2.12848
$322$		0			0
$323$			581.681i			1.80087i
$324$		0			0
$325$	49.9326	−	492.479i	0.153639	−	1.51532i
$326$		0			0
$327$	591.302			1.80826
$328$		0			0
$329$	−453.859			−1.37951
$330$		0			0
$331$	115.989			0.350421			0.175210	−	0.984531i	$-0.443939\pi$
$331$	115.989			0.350421			0.175210	+	0.984531i	$0.443939\pi$
$332$		0			0
$333$	−165.211			−0.496128
$334$		0			0
$335$	176.408	+	195.199i	0.526592	+	0.582683i
$336$		0			0
$337$		−	385.324i		−	1.14339i	−0.820465	−	0.571697i	$-0.806285\pi$
$337$		−	385.324i		−	1.14339i	0.820465	−	0.571697i	$-0.193715\pi$
$338$		0			0
$339$	−202.456			−0.597215
$340$		0			0
$341$			58.5329i			0.171651i
$342$		0			0
$343$	368.504			1.07436
$344$		0			0
$345$	−115.714	−	128.040i	−0.335404	−	0.371130i
$346$		0			0
$347$		−	184.246i		−	0.530968i	−0.964115	−	0.265484i	$-0.914468\pi$
$347$		−	184.246i		−	0.530968i	0.964115	−	0.265484i	$-0.0855317\pi$
$348$		0			0
$349$			163.692i			0.469031i	0.972112	+	0.234515i	$0.0753503\pi$
$349$			163.692i			0.469031i	−0.972112	+	0.234515i	$0.924650\pi$
$350$		0			0
$351$		−	1282.70i		−	3.65440i
$352$		0			0
$353$		−	6.35178i		−	0.0179937i	−0.999960	−	0.00899686i	$-0.997136\pi$
$353$		−	6.35178i		−	0.0179937i	0.999960	−	0.00899686i	$-0.00286383\pi$
$354$		0			0
$355$	304.789	+	337.254i	0.858560	+	0.950012i
$356$		0			0
$357$	−834.396			−2.33724
$358$		0			0
$359$			507.257i			1.41297i	0.707727	+	0.706486i	$0.249720\pi$
$359$			507.257i			1.41297i	−0.707727	+	0.706486i	$0.750280\pi$
$360$		0			0
$361$	27.0996			0.0750682
$362$		0			0
$363$		−	611.076i		−	1.68340i
$364$		0			0
$365$	−40.1372	−	44.4125i	−0.109965	−	0.121678i
$366$		0			0
$367$	−426.301			−1.16158			−0.580791	−	0.814053i	$-0.697257\pi$
$367$	−426.301			−1.16158			−0.580791	+	0.814053i	$0.697257\pi$
$368$		0			0
$369$	1387.38			3.75985
$370$		0			0
$371$	−29.3120			−0.0790082
$372$		0			0
$373$	166.211			0.445606			0.222803	−	0.974863i	$-0.428479\pi$
$373$	166.211			0.445606			0.222803	+	0.974863i	$0.428479\pi$
$374$		0			0
$375$	−550.344	+	404.502i	−1.46758	+	1.07867i
$376$		0			0
$377$		−	392.263i		−	1.04049i
$378$		0			0
$379$	−348.615			−0.919830			−0.459915	−	0.887963i	$-0.652120\pi$
$379$	−348.615			−0.919830			−0.459915	+	0.887963i	$0.652120\pi$
$380$		0			0
$381$		−	792.368i		−	2.07971i
$382$		0			0
$383$	23.1856			0.0605368			0.0302684	−	0.999542i	$-0.490364\pi$
$383$	23.1856			0.0605368			0.0302684	+	0.999542i	$0.490364\pi$
$384$		0			0
$385$	58.0799	−	52.4889i	0.150857	−	0.136335i
$386$		0			0
$387$			180.883i			0.467398i
$388$		0			0
$389$			309.175i			0.794795i	0.917647	+	0.397397i	$0.130087\pi$
$389$			309.175i			0.794795i	−0.917647	+	0.397397i	$0.869913\pi$
$390$		0			0
$391$		−	186.517i		−	0.477026i
$392$		0			0
$393$			767.700i			1.95343i
$394$		0			0
$395$	−307.989	−	340.795i	−0.779719	−	0.862773i
$396$		0			0
$397$	485.459			1.22282			0.611410	−	0.791314i	$-0.290603\pi$
$397$	485.459			1.22282			0.611410	+	0.791314i	$0.290603\pi$
$398$		0			0
$399$			556.712i			1.39527i
$400$		0			0
$401$	395.047			0.985155			0.492577	−	0.870269i	$-0.336055\pi$
$401$	395.047			0.985155			0.492577	+	0.870269i	$0.336055\pi$
$402$		0			0
$403$			382.832i			0.949956i
$404$		0			0
$405$	−616.790	+	557.416i	−1.52294	+	1.37633i
$406$		0			0
$407$	23.9809			0.0589211
$408$		0			0
$409$	−90.7677			−0.221926			−0.110963	−	0.993825i	$-0.535393\pi$
$409$	−90.7677			−0.221926			−0.110963	+	0.993825i	$0.535393\pi$
$410$		0			0
$411$	1183.79			2.88026
$412$		0			0
$413$	321.211			0.777750
$414$		0			0
$415$	−143.920	−	159.250i	−0.346796	−	0.383736i
$416$		0			0
$417$			1302.87i			3.12440i
$418$		0			0
$419$	−182.733			−0.436118			−0.218059	−	0.975936i	$-0.569972\pi$
$419$	−182.733			−0.436118			−0.218059	+	0.975936i	$0.569972\pi$
$420$		0			0
$421$			69.7252i			0.165618i	0.996565	+	0.0828091i	$0.0263892\pi$
$421$			69.7252i			0.165618i	−0.996565	+	0.0828091i	$0.973611\pi$
$422$		0			0
$423$	−1830.25			−4.32683
$424$		0			0
$425$	−734.399	−	74.4610i	−1.72800	−	0.175202i
$426$		0			0
$427$			130.674i			0.306028i
$428$		0			0
$429$			327.524i			0.763460i
$430$		0			0
$431$		−	701.232i		−	1.62699i	−0.581573	−	0.813494i	$-0.697563\pi$
$431$		−	701.232i		−	1.62699i	0.581573	−	0.813494i	$-0.302437\pi$
$432$		0			0
$433$		−	400.326i		−	0.924540i	−0.886739	−	0.462270i	$-0.847035\pi$
$433$		−	400.326i		−	0.924540i	0.886739	−	0.462270i	$-0.152965\pi$
$434$		0			0
$435$	−401.558	+	362.903i	−0.923123	+	0.834260i
$436$		0			0
$437$	−124.445			−0.284771
$438$		0			0
$439$			635.571i			1.44777i	0.689921	+	0.723884i	$0.257645\pi$
$439$			635.571i			1.44777i	−0.689921	+	0.723884i	$0.742355\pi$
$440$		0			0
$441$	464.096			1.05237
$442$		0			0
$443$		−	264.036i		−	0.596017i	−0.954563	−	0.298009i	$-0.903678\pi$
$443$		−	264.036i		−	0.596017i	0.954563	−	0.298009i	$-0.0963224\pi$
$444$		0			0
$445$	−178.623	+	161.428i	−0.401401	+	0.362761i
$446$		0			0
$447$	−672.035			−1.50343
$448$		0			0
$449$	−110.826			−0.246829			−0.123414	−	0.992355i	$-0.539384\pi$
$449$	−110.826			−0.246829			−0.123414	+	0.992355i	$0.539384\pi$
$450$		0			0
$451$	−201.384			−0.446527
$452$		0			0
$453$	322.682			0.712321
$454$		0			0
$455$	379.870	−	343.302i	0.834879	−	0.754510i
$456$		0			0
$457$		−	60.7249i		−	0.132877i	−0.997791	−	0.0664386i	$-0.978836\pi$
$457$		−	60.7249i		−	0.132877i	0.997791	−	0.0664386i	$-0.0211637\pi$
$458$		0			0
$459$	−1912.80			−4.16731
$460$		0			0
$461$			265.448i			0.575809i	0.957659	+	0.287904i	$0.0929585\pi$
$461$			265.448i			0.575809i	−0.957659	+	0.287904i	$0.907041\pi$
$462$		0			0
$463$	−757.117			−1.63524			−0.817621	−	0.575757i	$-0.804708\pi$
$463$	−757.117			−1.63524			−0.817621	+	0.575757i	$0.804708\pi$
$464$		0			0
$465$	391.904	−	354.178i	0.842805	−	0.761674i
$466$		0			0
$467$		−	51.8514i		−	0.111031i	−0.998458	−	0.0555155i	$-0.982320\pi$
$467$		−	51.8514i		−	0.111031i	0.998458	−	0.0555155i	$-0.0176802\pi$
$468$		0			0
$469$			272.143i			0.580262i
$470$		0			0
$471$		−	1259.93i		−	2.67502i
$472$		0			0
$473$		−	26.2558i		−	0.0555092i
$474$		0			0
$475$	−49.6807	+	489.994i	−0.104591	+	1.03157i
$476$		0			0
$477$	−118.205			−0.247809
$478$		0			0
$479$		−	39.9757i		−	0.0834566i	−0.999129	−	0.0417283i	$-0.986714\pi$
$479$		−	39.9757i		−	0.0834566i	0.999129	−	0.0417283i	$-0.0132864\pi$
$480$		0			0
$481$	156.846			0.326084
$482$		0			0
$483$		−	178.511i		−	0.369587i
$484$		0			0
$485$	−17.2032	−	19.0356i	−0.0354705	−	0.0392487i
$486$		0			0
$487$	−504.115			−1.03514			−0.517572	−	0.855640i	$-0.673164\pi$
$487$	−504.115			−1.03514			−0.517572	+	0.855640i	$0.673164\pi$
$488$		0			0
$489$	868.375			1.77582
$490$		0			0
$491$	−498.079			−1.01442			−0.507209	−	0.861823i	$-0.669323\pi$
$491$	−498.079			−1.01442			−0.507209	+	0.861823i	$0.669323\pi$
$492$		0			0
$493$	−584.955			−1.18652
$494$		0			0
$495$	234.215	−	211.669i	0.473162	−	0.427613i
$496$		0			0
$497$			470.193i			0.946063i
$498$		0			0
$499$	−861.152			−1.72576			−0.862878	−	0.505413i	$-0.831340\pi$
$499$	−861.152			−1.72576			−0.862878	+	0.505413i	$0.831340\pi$
$500$		0			0
$501$			800.681i			1.59817i
$502$		0			0
$503$	88.7312			0.176404			0.0882020	−	0.996103i	$-0.471888\pi$
$503$	88.7312			0.176404			0.0882020	+	0.996103i	$0.471888\pi$
$504$		0			0
$505$	81.8087	+	90.5227i	0.161997	+	0.179253i
$506$		0			0
$507$			1218.74i			2.40382i
$508$		0			0
$509$		−	522.908i		−	1.02732i	−0.857993	−	0.513662i	$-0.828289\pi$
$509$		−	522.908i		−	1.02732i	0.857993	−	0.513662i	$-0.171711\pi$
$510$		0			0
$511$		−	61.9190i		−	0.121172i
$512$		0			0
$513$			1276.22i			2.48777i
$514$		0			0
$515$	336.675	−	304.265i	0.653738	−	0.590807i
$516$		0			0
$517$	265.667			0.513862
$518$		0			0
$519$		−	220.846i		−	0.425523i
$520$		0			0
$521$	155.341			0.298160			0.149080	−	0.988825i	$-0.452369\pi$
$521$	155.341			0.298160			0.149080	+	0.988825i	$0.452369\pi$
$522$		0			0
$523$		−	190.456i		−	0.364161i	−0.983284	−	0.182081i	$-0.941717\pi$
$523$		−	190.456i		−	0.364161i	0.983284	−	0.182081i	$-0.0582832\pi$
$524$		0			0
$525$	−702.875	−	71.2648i	−1.33881	−	0.135742i
$526$		0			0
$527$	570.891			1.08329
$528$		0			0
$529$	−489.097			−0.924568
$530$		0			0
$531$	1295.33			2.43941
$532$		0			0
$533$	−1317.14			−2.47119
$534$		0			0
$535$	−419.202	−	463.855i	−0.783556	−	0.867018i
$536$		0			0
$537$			1163.68i			2.16701i
$538$		0			0
$539$	−67.3651			−0.124982
$540$		0			0
$541$			810.433i			1.49803i	0.662554	+	0.749014i	$0.269472\pi$
$541$			810.433i			1.49803i	−0.662554	+	0.749014i	$0.730528\pi$
$542$		0			0
$543$	1188.85			2.18941
$544$		0			0
$545$	362.793	+	401.436i	0.665675	+	0.736581i
$546$		0			0
$547$			907.445i			1.65895i	0.558545	+	0.829474i	$0.311360\pi$
$547$			907.445i			1.65895i	−0.558545	+	0.829474i	$0.688640\pi$
$548$		0			0
$549$			526.960i			0.959855i
$550$		0			0
$551$			390.284i			0.708319i
$552$		0			0
$553$		−	475.130i		−	0.859187i
$554$		0			0
$555$	−145.107	−	160.563i	−0.261454	−	0.289303i
$556$		0			0
$557$	820.193			1.47252			0.736260	−	0.676699i	$-0.236590\pi$
$557$	820.193			1.47252			0.736260	+	0.676699i	$0.236590\pi$
$558$		0			0
$559$		−	171.725i		−	0.307201i
$560$		0			0
$561$	488.414			0.870614
$562$		0			0
$563$		−	178.005i		−	0.316172i	−0.987425	−	0.158086i	$-0.949468\pi$
$563$		−	178.005i		−	0.316172i	0.987425	−	0.158086i	$-0.0505323\pi$
$564$		0			0
$565$	−124.217	−	137.448i	−0.219852	−	0.243270i
$566$		0			0
$567$	−859.917			−1.51661
$568$		0			0
$569$	−206.386			−0.362717			−0.181359	−	0.983417i	$-0.558049\pi$
$569$	−206.386			−0.362717			−0.181359	+	0.983417i	$0.558049\pi$
$570$		0			0
$571$	415.743			0.728096			0.364048	−	0.931380i	$-0.381394\pi$
$571$	415.743			0.728096			0.364048	+	0.931380i	$0.381394\pi$
$572$		0			0
$573$	−1058.25			−1.84686
$574$		0			0
$575$	15.9302	−	157.117i	0.0277047	−	0.273248i
$576$		0			0
$577$			103.213i			0.178878i	0.995992	+	0.0894390i	$0.0285074\pi$
$577$			103.213i			0.178878i	−0.995992	+	0.0894390i	$0.971493\pi$
$578$		0			0
$579$	301.612			0.520919
$580$		0			0
$581$		−	222.024i		−	0.382141i
$582$		0			0
$583$	17.1578			0.0294302
$584$		0			0
$585$	1531.88	−	1384.41i	2.61859	−	2.36652i
$586$		0			0
$587$		−	881.568i		−	1.50182i	−0.660404	−	0.750910i	$-0.729615\pi$
$587$		−	881.568i		−	1.50182i	0.660404	−	0.750910i	$-0.270385\pi$
$588$		0			0
$589$		−	380.901i		−	0.646691i
$590$		0			0
$591$		−	125.941i		−	0.213097i
$592$		0			0
$593$		−	718.330i		−	1.21135i	−0.795713	−	0.605674i	$-0.792903\pi$
$593$		−	718.330i		−	1.21135i	0.795713	−	0.605674i	$-0.207097\pi$
$594$		0			0
$595$	−511.943	−	566.474i	−0.860408	−	0.952056i
$596$		0			0
$597$	1682.16			2.81770
$598$		0			0
$599$			318.943i			0.532459i	0.963910	+	0.266229i	$0.0857778\pi$
$599$			318.943i			0.532459i	−0.963910	+	0.266229i	$0.914222\pi$
$600$		0			0
$601$	−693.248			−1.15349			−0.576745	−	0.816924i	$-0.695678\pi$
$601$	−693.248			−1.15349			−0.576745	+	0.816924i	$0.695678\pi$
$602$		0			0
$603$			1097.45i			1.81999i
$604$		0			0
$605$	414.861	−	374.925i	0.685721	−	0.619711i
$606$		0			0
$607$	−302.748			−0.498761			−0.249380	−	0.968406i	$-0.580227\pi$
$607$	−302.748			−0.498761			−0.249380	+	0.968406i	$0.580227\pi$
$608$		0			0
$609$	−559.845			−0.919286
$610$		0			0
$611$	1737.59			2.84384
$612$		0			0
$613$	−739.961			−1.20711			−0.603557	−	0.797320i	$-0.706250\pi$
$613$	−739.961			−1.20711			−0.603557	+	0.797320i	$0.706250\pi$
$614$		0			0
$615$	1218.56	+	1348.36i	1.98140	+	2.19245i
$616$		0			0
$617$			193.263i			0.313230i	0.987660	+	0.156615i	$0.0500582\pi$
$617$			193.263i			0.313230i	−0.987660	+	0.156615i	$0.949942\pi$
$618$		0			0
$619$	77.1806			0.124686			0.0623429	−	0.998055i	$-0.480143\pi$
$619$	77.1806			0.124686			0.0623429	+	0.998055i	$0.480143\pi$
$620$		0			0
$621$		−	409.224i		−	0.658975i
$622$		0			0
$623$	−249.033			−0.399733
$624$		0			0
$625$	−612.281	−	125.448i	−0.979649	−	0.200717i
$626$		0			0
$627$		−	325.872i		−	0.519732i
$628$		0			0
$629$		−	233.894i		−	0.371851i
$630$		0			0
$631$		−	504.826i		−	0.800041i	−0.916506	−	0.400020i	$-0.869003\pi$
$631$		−	504.826i		−	0.800041i	0.916506	−	0.400020i	$-0.130997\pi$
$632$		0			0
$633$		−	452.589i		−	0.714991i
$634$		0			0
$635$	537.941	−	486.157i	0.847151	−	0.765601i
$636$		0			0
$637$	−440.599			−0.691679
$638$		0			0
$639$			1896.12i			2.96732i
$640$		0			0
$641$	69.9672			0.109153			0.0545766	−	0.998510i	$-0.482619\pi$
$641$	69.9672			0.109153			0.0545766	+	0.998510i	$0.482619\pi$
$642$		0			0
$643$			229.899i			0.357541i	0.983891	+	0.178771i	$0.0572120\pi$
$643$			229.899i			0.357541i	−0.983891	+	0.178771i	$0.942788\pi$
$644$		0			0
$645$	−175.795	+	158.872i	−0.272550	+	0.246314i
$646$		0			0
$647$	209.977			0.324539			0.162270	−	0.986746i	$-0.448119\pi$
$647$	209.977			0.324539			0.162270	+	0.986746i	$0.448119\pi$
$648$		0			0
$649$	−188.021			−0.289709
$650$		0			0
$651$	546.386			0.839302
$652$		0			0
$653$	−62.0918			−0.0950870			−0.0475435	−	0.998869i	$-0.515139\pi$
$653$	−62.0918			−0.0950870			−0.0475435	+	0.998869i	$0.515139\pi$
$654$		0			0
$655$	−521.194	+	471.022i	−0.795715	+	0.719117i
$656$		0			0
$657$		−	249.697i		−	0.380056i
$658$		0			0
$659$	−814.620			−1.23615			−0.618073	−	0.786121i	$-0.712086\pi$
$659$	−814.620			−1.23615			−0.618073	+	0.786121i	$0.712086\pi$
$660$		0			0
$661$		−	931.804i		−	1.40969i	−0.709362	−	0.704844i	$-0.751017\pi$
$661$		−	931.804i		−	1.40969i	0.709362	−	0.704844i	$-0.248983\pi$
$662$		0			0
$663$	3194.46			4.81819
$664$		0			0
$665$	−377.953	+	341.570i	−0.568351	+	0.513639i
$666$		0			0
$667$		−	125.145i		−	0.187624i
$668$		0			0
$669$			1252.20i			1.87175i
$670$		0			0
$671$		−	76.4901i		−	0.113994i
$672$		0			0
$673$		−	717.706i		−	1.06643i	−0.845980	−	0.533214i	$-0.820984\pi$
$673$		−	717.706i		−	1.06643i	0.845980	−	0.533214i	$-0.179016\pi$
$674$		0			0
$675$	−1611.29	−	163.370i	−2.38710	−	0.242029i
$676$		0			0
$677$	917.188			1.35478			0.677391	−	0.735623i	$-0.263110\pi$
$677$	917.188			1.35478			0.677391	+	0.735623i	$0.263110\pi$
$678$		0			0
$679$		−	26.5391i		−	0.0390856i
$680$		0			0
$681$	−449.937			−0.660701
$682$		0			0
$683$			44.7000i			0.0654465i	0.999464	+	0.0327233i	$0.0104180\pi$
$683$			44.7000i			0.0654465i	−0.999464	+	0.0327233i	$0.989582\pi$
$684$		0			0
$685$	726.311	+	803.675i	1.06031	+	1.17325i
$686$		0			0
$687$	970.502			1.41267
$688$		0			0
$689$	112.220			0.162874
$690$		0			0
$691$	−820.990			−1.18812			−0.594059	−	0.804421i	$-0.702475\pi$
$691$	−820.990			−1.18812			−0.594059	+	0.804421i	$0.702475\pi$
$692$		0			0
$693$	326.538			0.471195
$694$		0			0
$695$	−884.525	+	799.377i	−1.27270	+	1.15018i
$696$		0			0
$697$			1964.17i			2.81803i
$698$		0			0
$699$	1647.71			2.35724
$700$		0			0
$701$			943.710i			1.34623i	0.739536	+	0.673117i	$0.235045\pi$
$701$			943.710i			1.34623i	−0.739536	+	0.673117i	$0.764955\pi$
$702$		0			0
$703$	−156.055			−0.221984
$704$		0			0
$705$	−1607.53	−	1778.76i	−2.28019	−	2.52307i
$706$		0			0
$707$			126.205i			0.178508i
$708$		0			0
$709$			675.064i			0.952135i	0.879409	+	0.476068i	$0.157938\pi$
$709$			675.064i			0.952135i	−0.879409	+	0.476068i	$0.842062\pi$
$710$		0			0
$711$		−	1916.03i		−	2.69483i
$712$		0			0
$713$			122.137i			0.171300i
$714$		0			0
$715$	−222.357	+	200.952i	−0.310989	+	0.281052i
$716$		0			0
$717$	784.185			1.09370
$718$		0			0
$719$		−	806.144i		−	1.12120i	−0.828086	−	0.560601i	$-0.810570\pi$
$719$		−	806.144i		−	1.12120i	0.828086	−	0.560601i	$-0.189430\pi$
$720$		0			0
$721$	469.386			0.651021
$722$		0			0
$723$		−	934.860i		−	1.29303i
$724$		0			0
$725$	−492.752	−	49.9603i	−0.679658	−	0.0689107i
$726$		0			0
$727$	147.404			0.202756			0.101378	−	0.994848i	$-0.467675\pi$
$727$	147.404			0.202756			0.101378	+	0.994848i	$0.467675\pi$
$728$		0			0
$729$	−281.963			−0.386781
$730$		0			0
$731$	−256.082			−0.350318
$732$		0			0
$733$	614.811			0.838759			0.419380	−	0.907811i	$-0.362248\pi$
$733$	614.811			0.838759			0.419380	+	0.907811i	$0.362248\pi$
$734$		0			0
$735$	407.622	+	451.040i	0.554587	+	0.613661i
$736$		0			0
$737$		−	159.299i		−	0.216145i
$738$		0			0
$739$	−680.521			−0.920868			−0.460434	−	0.887694i	$-0.652306\pi$
$739$	−680.521			−0.920868			−0.460434	+	0.887694i	$0.652306\pi$
$740$		0			0
$741$		−	2131.36i		−	2.87632i
$742$		0			0
$743$	−949.164			−1.27748			−0.638738	−	0.769425i	$-0.720543\pi$
$743$	−949.164			−1.27748			−0.638738	+	0.769425i	$0.720543\pi$
$744$		0			0
$745$	−412.326	−	456.246i	−0.553458	−	0.612411i
$746$		0			0
$747$		−	895.341i		−	1.19858i
$748$		0			0
$749$		−	646.697i		−	0.863414i
$750$		0			0
$751$			685.833i			0.913227i	0.889665	+	0.456613i	$0.150938\pi$
$751$			685.833i			0.913227i	−0.889665	+	0.456613i	$0.849062\pi$
$752$		0			0
$753$			33.9046i			0.0450261i
$754$		0			0
$755$	197.981	+	219.069i	0.262227	+	0.290158i
$756$		0			0
$757$	288.505			0.381116			0.190558	−	0.981676i	$-0.438970\pi$
$757$	288.505			0.381116			0.190558	+	0.981676i	$0.438970\pi$
$758$		0			0
$759$			104.491i			0.137670i
$760$		0			0
$761$	−1441.62			−1.89438			−0.947190	−	0.320672i	$-0.896091\pi$
$761$	−1441.62			−1.89438			−0.947190	+	0.320672i	$0.896091\pi$
$762$		0			0
$763$			559.675i			0.733519i
$764$		0			0
$765$	−2064.48	−	2284.38i	−2.69866	−	2.98612i
$766$		0			0
$767$	−1229.75			−1.60332
$768$		0			0
$769$	347.381			0.451731			0.225865	−	0.974159i	$-0.427479\pi$
$769$	347.381			0.451731			0.225865	+	0.974159i	$0.427479\pi$
$770$		0			0
$771$	695.326			0.901849
$772$		0			0
$773$	158.564			0.205128			0.102564	−	0.994726i	$-0.467295\pi$
$773$	158.564			0.205128			0.102564	+	0.994726i	$0.467295\pi$
$774$		0			0
$775$	480.905	+	48.7592i	0.620523	+	0.0629150i
$776$		0			0
$777$		−	223.854i		−	0.288101i
$778$		0			0
$779$	1310.50			1.68228
$780$		0			0
$781$		−	275.228i		−	0.352405i
$782$		0			0
$783$	−1283.41			−1.63909
$784$		0			0
$785$	855.372	−	773.031i	1.08965	−	0.984753i
$786$		0			0
$787$			602.238i			0.765232i	0.923907	+	0.382616i	$0.124977\pi$
$787$			602.238i			0.765232i	−0.923907	+	0.382616i	$0.875023\pi$
$788$		0			0
$789$		−	2496.51i		−	3.16415i
$790$		0			0
$791$		−	191.627i		−	0.242259i
$792$		0			0
$793$		−	500.281i		−	0.630872i
$794$		0			0
$795$	−103.821	−	114.880i	−0.130592	−	0.144503i
$796$		0			0
$797$	1399.64			1.75614			0.878069	−	0.478535i	$-0.158832\pi$
$797$	1399.64			1.75614			0.878069	+	0.478535i	$0.158832\pi$
$798$		0			0
$799$		−	2591.14i		−	3.24298i
$800$		0			0
$801$	−1004.26			−1.25376
$802$		0			0
$803$			36.2444i			0.0451362i
$804$		0			0
$805$	121.191	−	109.525i	0.150548	−	0.136056i
$806$		0			0
$807$	1661.43			2.05877
$808$		0			0
$809$	1235.38			1.52704			0.763522	−	0.645782i	$-0.223469\pi$
$809$	1235.38			1.52704			0.763522	+	0.645782i	$0.223469\pi$
$810$		0			0
$811$	−499.613			−0.616046			−0.308023	−	0.951379i	$-0.599667\pi$
$811$	−499.613			−0.616046			−0.308023	+	0.951379i	$0.599667\pi$
$812$		0			0
$813$	2074.37			2.55149
$814$		0			0
$815$	532.791	+	589.542i	0.653731	+	0.723365i
$816$		0			0
$817$			170.859i			0.209130i
$818$		0			0
$819$	2135.71			2.60771
$820$		0			0
$821$			341.687i			0.416184i	0.978109	+	0.208092i	$0.0667254\pi$
$821$			341.687i			0.416184i	−0.978109	+	0.208092i	$0.933275\pi$
$822$		0			0
$823$	119.183			0.144816			0.0724079	−	0.997375i	$-0.476932\pi$
$823$	119.183			0.144816			0.0724079	+	0.997375i	$0.476932\pi$
$824$		0			0
$825$	411.429	+	41.7149i	0.498701	+	0.0505635i
$826$		0			0
$827$		−	29.2660i		−	0.0353882i	−0.999843	−	0.0176941i	$-0.994368\pi$
$827$		−	29.2660i		−	0.0353882i	0.999843	−	0.0176941i	$-0.00563249\pi$
$828$		0			0
$829$		−	1187.20i		−	1.43209i	−0.698056	−	0.716043i	$-0.745952\pi$
$829$		−	1187.20i		−	1.43209i	0.698056	−	0.716043i	$-0.254048\pi$
$830$		0			0
$831$			981.074i			1.18059i
$832$		0			0
$833$			657.036i			0.788758i
$834$		0			0
$835$	−543.585	+	491.257i	−0.651000	+	0.588332i
$836$		0			0
$837$	1252.55			1.49648
$838$		0			0
$839$			1273.74i			1.51816i	0.650996	+	0.759081i	$0.274352\pi$
$839$			1273.74i			1.51816i	−0.650996	+	0.759081i	$0.725648\pi$
$840$		0			0
$841$	448.519			0.533317
$842$		0			0
$843$		−	107.902i		−	0.127997i
$844$		0			0
$845$	−827.403	+	747.754i	−0.979175	+	0.884916i
$846$		0			0
$847$	578.391			0.682870
$848$		0			0
$849$	−1484.85			−1.74894
$850$		0			0
$851$	50.0393			0.0588006
$852$		0			0
$853$	148.346			0.173911			0.0869554	−	0.996212i	$-0.472286\pi$
$853$	148.346			0.173911			0.0869554	+	0.996212i	$0.472286\pi$
$854$		0			0
$855$	−1524.15	+	1377.43i	−1.78263	+	1.61103i
$856$		0			0
$857$		−	999.389i		−	1.16615i	−0.812419	−	0.583074i	$-0.801850\pi$
$857$		−	999.389i		−	1.16615i	0.812419	−	0.583074i	$-0.198150\pi$
$858$		0			0
$859$	205.363			0.239072			0.119536	−	0.992830i	$-0.461859\pi$
$859$	205.363			0.239072			0.119536	+	0.992830i	$0.461859\pi$
$860$		0			0
$861$			1879.85i			2.18334i
$862$		0			0
$863$	533.186			0.617829			0.308914	−	0.951090i	$-0.400034\pi$
$863$	533.186			0.617829			0.308914	+	0.951090i	$0.400034\pi$
$864$		0			0
$865$	149.933	−	135.500i	0.173333	−	0.156648i
$866$		0			0
$867$		−	3184.56i		−	3.67308i
$868$		0			0
$869$			278.118i			0.320044i
$870$		0			0
$871$		−	1041.89i		−	1.19620i
$872$		0			0
$873$		−	107.023i		−	0.122592i
$874$		0			0
$875$	−382.867	−	520.908i	−0.437562	−	0.595324i
$876$		0			0
$877$	−635.936			−0.725126			−0.362563	−	0.931959i	$-0.618098\pi$
$877$	−635.936			−0.725126			−0.362563	+	0.931959i	$0.618098\pi$
$878$		0			0
$879$			1360.20i			1.54745i
$880$		0			0
$881$	−240.207			−0.272652			−0.136326	−	0.990664i	$-0.543530\pi$
$881$	−240.207			−0.272652			−0.136326	+	0.990664i	$0.543530\pi$
$882$		0			0
$883$		−	836.150i		−	0.946943i	−0.880809	−	0.473471i	$-0.843001\pi$
$883$		−	836.150i		−	0.946943i	0.880809	−	0.473471i	$-0.156999\pi$
$884$		0			0
$885$	1137.70	+	1258.89i	1.28554	+	1.42247i
$886$		0			0
$887$	59.0080			0.0665253			0.0332627	−	0.999447i	$-0.489410\pi$
$887$	59.0080			0.0665253			0.0332627	+	0.999447i	$0.489410\pi$
$888$		0			0
$889$	749.987			0.843630
$890$		0			0
$891$	503.353			0.564931
$892$		0			0
$893$	−1728.82			−1.93597
$894$		0			0
$895$	−790.029	+	713.978i	−0.882713	+	0.797740i
$896$		0			0
$897$			683.423i			0.761898i
$898$		0			0
$899$	383.045			0.426079
$900$		0			0
$901$		−	167.346i		−	0.185734i
$902$		0			0
$903$	−245.090			−0.271417
$904$		0			0
$905$	729.419	+	807.115i	0.805988	+	0.891840i
$906$		0			0
$907$			1290.73i			1.42308i	0.702645	+	0.711540i	$0.252002\pi$
$907$			1290.73i			1.42308i	−0.702645	+	0.711540i	$0.747998\pi$
$908$		0			0
$909$			508.939i			0.559889i
$910$		0			0
$911$			1179.87i			1.29514i	0.762007	+	0.647569i	$0.224214\pi$
$911$			1179.87i			1.29514i	−0.762007	+	0.647569i	$0.775786\pi$
$912$		0			0
$913$			129.962i			0.142346i
$914$		0			0
$915$	−512.137	+	462.837i	−0.559712	+	0.505832i
$916$		0			0
$917$	−726.638			−0.792408
$918$		0			0
$919$		−	1499.19i		−	1.63133i	−0.578528	−	0.815663i	$-0.696373\pi$
$919$		−	1499.19i		−	1.63133i	0.578528	−	0.815663i	$-0.303627\pi$
$920$		0			0
$921$	1425.72			1.54801
$922$		0			0
$923$		−	1800.12i		−	1.95029i
$924$		0			0
$925$	19.9766	−	197.027i	0.0215964	−	0.213002i
$926$		0			0
$927$	1892.86			2.04192
$928$		0			0
$929$	−197.581			−0.212682			−0.106341	−	0.994330i	$-0.533913\pi$
$929$	−197.581			−0.212682			−0.106341	+	0.994330i	$0.533913\pi$
$930$		0			0
$931$	438.377			0.470866
$932$		0			0
$933$	−477.898			−0.512216
$934$		0			0
$935$	299.666	+	331.586i	0.320499	+	0.354637i
$936$		0			0
$937$			1137.75i			1.21424i	0.794608	+	0.607122i	$0.207676\pi$
$937$			1137.75i			1.21424i	−0.794608	+	0.607122i	$0.792324\pi$
$938$		0			0
$939$	345.715			0.368174
$940$		0			0
$941$		−	1142.82i		−	1.21447i	−0.794522	−	0.607235i	$-0.792278\pi$
$941$		−	1142.82i		−	1.21447i	0.794522	−	0.607235i	$-0.207722\pi$
$942$		0			0
$943$	−420.214			−0.445614
$944$		0			0
$945$	−1123.22	−	1242.86i	−1.18859	−	1.31520i
$946$		0			0
$947$		−	1606.69i		−	1.69661i	−0.529505	−	0.848307i	$-0.677622\pi$
$947$		−	1606.69i		−	1.69661i	0.529505	−	0.848307i	$-0.322378\pi$
$948$		0			0
$949$			237.055i			0.249795i
$950$		0			0
$951$			2264.76i			2.38146i
$952$		0			0
$953$		−	1308.38i		−	1.37291i	−0.727174	−	0.686453i	$-0.759167\pi$
$953$		−	1308.38i		−	1.37291i	0.727174	−	0.686453i	$-0.240833\pi$
$954$		0			0
$955$	−649.289	−	718.450i	−0.679884	−	0.752303i
$956$		0			0
$957$	327.706			0.342431
$958$		0			0
$959$			1120.47i			1.16837i
$960$		0			0
$961$	587.164			0.610993
$962$		0			0
$963$		−	2607.89i		−	2.70809i
$964$		0			0
$965$	185.054	+	204.765i	0.191766	+	0.212192i
$966$		0			0
$967$	658.248			0.680711			0.340356	−	0.940297i	$-0.389453\pi$
$967$	658.248			0.680711			0.340356	+	0.940297i	$0.389453\pi$
$968$		0			0
$969$	−3178.34			−3.28002
$970$		0			0
$971$	1382.65			1.42394			0.711972	−	0.702208i	$-0.247802\pi$
$971$	1382.65			1.42394			0.711972	+	0.702208i	$0.247802\pi$
$972$		0			0
$973$	−1233.19			−1.26741
$974$		0			0
$975$	2690.94	+	272.835i	2.75993	+	0.279831i
$976$		0			0
$977$			1187.66i			1.21562i	0.794081	+	0.607812i	$0.207952\pi$
$977$			1187.66i			1.21562i	−0.794081	+	0.607812i	$0.792048\pi$
$978$		0			0
$979$	145.772			0.148899
$980$		0			0
$981$			2256.97i			2.30068i
$982$		0			0
$983$	1419.04			1.44359			0.721793	−	0.692109i	$-0.243318\pi$
$983$	1419.04			1.44359			0.721793	+	0.692109i	$0.243318\pi$
$984$		0			0
$985$	85.5014	−	77.2707i	0.0868035	−	0.0784475i
$986$		0			0
$987$		−	2479.92i		−	2.51258i
$988$		0			0
$989$		−	54.7863i		−	0.0553956i
$990$		0			0
$991$			168.749i			0.170281i	0.996369	+	0.0851405i	$0.0271339\pi$
$991$			168.749i			0.170281i	−0.996369	+	0.0851405i	$0.972866\pi$
$992$		0			0
$993$			633.774i			0.638241i
$994$		0			0
$995$	1032.09	+	1142.03i	1.03728	+	1.14776i
$996$		0			0
$997$	221.827			0.222495			0.111247	−	0.993793i	$-0.464515\pi$
$997$	221.827			0.222495			0.111247	+	0.993793i	$0.464515\pi$
$998$		0			0
$999$		−	513.171i		−	0.513684i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.3.e.l.639.8		24
4.3	odd	2		1280.3.e.k.639.17		24
5.4	even	2	inner	1280.3.e.l.639.17		24
8.3	odd	2	inner	1280.3.e.l.639.18		24
8.5	even	2		1280.3.e.k.639.7		24
16.3	odd	4		640.3.h.b.639.24	yes	24
16.5	even	4		640.3.h.a.639.24	yes	24
16.11	odd	4		640.3.h.a.639.1	✓	24
16.13	even	4		640.3.h.b.639.1	yes	24
20.19	odd	2		1280.3.e.k.639.8		24
40.19	odd	2	inner	1280.3.e.l.639.7		24
40.29	even	2		1280.3.e.k.639.18		24
80.19	odd	4		640.3.h.b.639.2	yes	24
80.29	even	4		640.3.h.b.639.23	yes	24
80.59	odd	4		640.3.h.a.639.23	yes	24
80.69	even	4		640.3.h.a.639.2	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.3.h.a.639.1	✓	24	16.11	odd	4
640.3.h.a.639.2	yes	24	80.69	even	4
640.3.h.a.639.23	yes	24	80.59	odd	4
640.3.h.a.639.24	yes	24	16.5	even	4
640.3.h.b.639.1	yes	24	16.13	even	4
640.3.h.b.639.2	yes	24	80.19	odd	4
640.3.h.b.639.23	yes	24	80.29	even	4
640.3.h.b.639.24	yes	24	16.3	odd	4
1280.3.e.k.639.7		24	8.5	even	2
1280.3.e.k.639.8		24	20.19	odd	2
1280.3.e.k.639.17		24	4.3	odd	2
1280.3.e.k.639.18		24	40.29	even	2
1280.3.e.l.639.7		24	40.19	odd	2	inner
1280.3.e.l.639.8		24	1.1	even	1	trivial
1280.3.e.l.639.17		24	5.4	even	2	inner
1280.3.e.l.639.18		24	8.3	odd	2	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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1280.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.46407i q^{3} +(-3.70957 + 3.35247i) q^{5} -5.17181 q^{7} -20.8560 q^{9} +O(q^{10})\)	\(q+5.46407i q^{3} +(-3.70957 + 3.35247i) q^{5} -5.17181 q^{7} -20.8560 q^{9} +3.02733 q^{11} +19.8001 q^{13} +(-18.3181 - 20.2693i) q^{15} -29.5266i q^{17} -19.7002 q^{19} -28.2591i q^{21} +6.31692 q^{23} +(2.52183 - 24.8725i) q^{25} -64.7822i q^{27} -19.8111i q^{29} +19.3348i q^{31} +16.5415i q^{33} +(19.1852 - 17.3384i) q^{35} +7.92148 q^{37} +108.189i q^{39} -66.5219 q^{41} -8.67294i q^{43} +(77.3669 - 69.9193i) q^{45} +87.7563 q^{47} -22.2523 q^{49} +161.335 q^{51} +5.66765 q^{53} +(-11.2301 + 10.1490i) q^{55} -107.643i q^{57} -62.1080 q^{59} -25.2666i q^{61} +107.864 q^{63} +(-73.4500 + 66.3795i) q^{65} -52.6204i q^{67} +34.5161i q^{69} -90.9146i q^{71} +11.9724i q^{73} +(135.905 + 13.7795i) q^{75} -15.6568 q^{77} +91.8692i q^{79} +166.270 q^{81} +42.9296i q^{83} +(98.9871 + 109.531i) q^{85} +108.249 q^{87} +48.1520 q^{89} -102.403 q^{91} -105.647 q^{93} +(73.0794 - 66.0446i) q^{95} +5.13149i q^{97} -63.1380 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 72 q^{9}+O(q^{10}) \) 24 * q - 72 * q^9	\( 24 q - 72 q^{9} + 16 q^{11} - 48 q^{19} - 24 q^{25} + 112 q^{35} - 80 q^{41} + 168 q^{49} + 576 q^{51} - 496 q^{59} + 32 q^{65} - 224 q^{75} + 184 q^{81} - 144 q^{89} + 864 q^{91} - 368 q^{99}+O(q^{100}) \) 24 * q - 72 * q^9 + 16 * q^11 - 48 * q^19 - 24 * q^25 + 112 * q^35 - 80 * q^41 + 168 * q^49 + 576 * q^51 - 496 * q^59 + 32 * q^65 - 224 * q^75 + 184 * q^81 - 144 * q^89 + 864 * q^91 - 368 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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