LMFDB - Embedded newform 1280.3.e.k.639.17

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			639.17
Character	$\chi$	$=$	1280.639
Dual form			1280.3.e.k.639.18

$q$-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	$-0.635560\pi$
$3$		−	5.46407i		−	1.82136i	0.413117	−	0.910678i	$-0.364440\pi$
$4$		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	$-0.379558\pi$
$7$	5.17181			0.738831			0.369415	+	0.929264i	$0.379558\pi$
$8$		0			0
$9$	−20.8560			−2.31734
$10$		0			0
$11$	−3.02733			−0.275212			−0.137606	−	0.990487i	$-0.543941\pi$
$11$	−3.02733			−0.275212			−0.137606	+	0.990487i	$0.543941\pi$
$12$		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	$-0.326518\pi$
$19$	19.7002			1.03685			0.518427	+	0.855122i	$0.326518\pi$
$20$		0			0
$21$		−	28.2591i		−	1.34567i
$22$		0			0
$23$	−6.31692			−0.274649			−0.137324	−	0.990526i	$-0.543850\pi$
$23$	−6.31692			−0.274649			−0.137324	+	0.990526i	$0.543850\pi$
$24$		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	$-0.899051\pi$
$31$		−	19.3348i		−	0.623704i	0.950131	−	0.311852i	$-0.100949\pi$
$32$		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	$0.0321556\pi$
$43$			8.67294i			0.201696i	−0.994902	+	0.100848i	$0.967844\pi$
$44$		0			0
$45$	77.3669	−	69.9193i	1.71927	−	1.55376i
$46$		0			0
$47$	−87.7563			−1.86715			−0.933577	−	0.358376i	$-0.883330\pi$
$47$	−87.7563			−1.86715			−0.933577	+	0.358376i	$0.883330\pi$
$48$		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	$-0.323564\pi$
$59$	62.1080			1.05268			0.526339	+	0.850275i	$0.323564\pi$
$60$		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	$0.128455\pi$
$67$			52.6204i			0.785378i	−0.919671	+	0.392689i	$0.871545\pi$
$68$		0			0
$69$			34.5161i			0.500233i
$70$		0			0
$71$			90.9146i			1.28049i	0.768172	+	0.640244i	$0.221167\pi$
$71$			90.9146i			1.28049i	−0.768172	+	0.640244i	$0.778833\pi$
$72$		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	$-0.802485\pi$
$79$		−	91.8692i		−	1.16290i	0.813582	−	0.581451i	$-0.197515\pi$
$80$		0			0
$81$	166.270			2.05271
$82$		0			0
$83$		−	42.9296i		−	0.517224i	−0.965981	−	0.258612i	$-0.916735\pi$
$83$		−	42.9296i		−	0.517224i	0.965981	−	0.258612i	$-0.0832651\pi$
$84$		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	$-0.354775\pi$
$103$	90.7585			0.881150			0.440575	+	0.897716i	$0.354775\pi$
$104$		0			0
$105$	94.7380	+	104.829i	0.902267	+	0.998374i
$106$		0			0
$107$		−	125.043i		−	1.16862i	−0.811530	−	0.584311i	$-0.801365\pi$
$107$		−	125.043i		−	1.16862i	0.811530	−	0.584311i	$-0.198635\pi$
$108$		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	$-0.306586\pi$
$127$	145.014			1.14184			0.570922	+	0.821004i	$0.306586\pi$
$128$		0			0
$129$	47.3895			0.367361
$130$		0			0
$131$	−140.500			−1.07252			−0.536258	−	0.844054i	$-0.680163\pi$
$131$	−140.500			−1.07252			−0.536258	+	0.844054i	$0.680163\pi$
$132$		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	$-0.828115\pi$
$139$	−238.444			−1.71542			−0.857712	+	0.514131i	$0.828115\pi$
$140$		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	$0.0626482\pi$
$151$			59.0552i			0.391094i	−0.980694	+	0.195547i	$0.937352\pi$
$152$		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	$0.162091\pi$
$163$			158.925i			0.974998i	−0.873124	+	0.487499i	$0.837909\pi$
$164$		0			0
$165$	−55.4550	−	61.3619i	−0.336091	−	0.371891i
$166$		0			0
$167$	−146.536			−0.877460			−0.438730	−	0.898619i	$-0.644572\pi$
$167$	−146.536			−0.877460			−0.438730	+	0.898619i	$0.644572\pi$
$168$		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	$-0.702804\pi$
$179$	−212.970			−1.18978			−0.594889	+	0.803808i	$0.702804\pi$
$180$		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	$-0.830754\pi$
$191$		−	193.675i		−	1.01400i	0.861945	−	0.507001i	$-0.169246\pi$
$192$		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	$0.281504\pi$
$199$			307.859i			1.54703i	−0.633777	+	0.773516i	$0.718496\pi$
$200$		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	$-0.437114\pi$
$211$	82.8301			0.392560			0.196280	+	0.980548i	$0.437114\pi$
$212$		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	$-0.671775\pi$
$223$	−229.170			−1.02767			−0.513834	+	0.857890i	$0.671775\pi$
$224$		0			0
$225$	−52.5954	+	518.741i	−0.233757	+	2.30552i
$226$		0			0
$227$		−	82.3448i		−	0.362752i	−0.983414	−	0.181376i	$-0.941945\pi$
$227$		−	82.3448i		−	0.362752i	0.983414	−	0.181376i	$-0.0580551\pi$
$228$		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	$0.0970682\pi$
$239$			143.517i			0.600489i	−0.953862	+	0.300244i	$0.902932\pi$
$240$		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	$-0.503935\pi$
$251$	−6.20502			−0.0247212			−0.0123606	+	0.999924i	$0.503935\pi$
$252$		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	$-0.165004\pi$
$263$	456.897			1.73725			0.868625	+	0.495470i	$0.165004\pi$
$264$		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	$0.247012\pi$
$271$			379.638i			1.40088i	−0.713713	+	0.700438i	$0.752988\pi$
$272$		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	$-0.840593\pi$
$283$		−	271.748i		−	0.960242i	0.877202	−	0.480121i	$-0.159407\pi$
$284$		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	$0.139712\pi$
$307$			260.926i			0.849922i	−0.905212	+	0.424961i	$0.860288\pi$
$308$		0			0
$309$		−	495.910i		−	1.60489i
$310$		0			0
$311$		−	87.4619i		−	0.281228i	−0.990065	−	0.140614i	$-0.955092\pi$
$311$		−	87.4619i		−	0.281228i	0.990065	−	0.140614i	$-0.0449076\pi$
$312$		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	$-0.556061\pi$
$331$	−115.989			−0.350421			−0.175210	+	0.984531i	$0.556061\pi$
$332$		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	$0.0855317\pi$
$347$			184.246i			0.530968i	−0.964115	+	0.265484i	$0.914468\pi$
$348$		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	$-0.750280\pi$
$359$		−	507.257i		−	1.41297i	0.707727	−	0.706486i	$-0.249720\pi$
$360$		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	$-0.302743\pi$
$367$	426.301			1.16158			0.580791	+	0.814053i	$0.302743\pi$
$368$		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	$-0.347880\pi$
$379$	348.615			0.919830			0.459915	+	0.887963i	$0.347880\pi$
$380$		0			0
$381$		−	792.368i		−	2.07971i
$382$		0			0
$383$	−23.1856			−0.0605368			−0.0302684	−	0.999542i	$-0.509636\pi$
$383$	−23.1856			−0.0605368			−0.0302684	+	0.999542i	$0.509636\pi$
$384$		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	$-0.430028\pi$
$419$	182.733			0.436118			0.218059	+	0.975936i	$0.430028\pi$
$420$		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	$0.302437\pi$
$431$			701.232i			1.62699i	−0.581573	+	0.813494i	$0.697563\pi$
$432$		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	$-0.742355\pi$
$439$		−	635.571i		−	1.44777i	0.689921	−	0.723884i	$-0.257645\pi$
$440$		0			0
$441$	464.096			1.05237
$442$		0			0
$443$			264.036i			0.596017i	0.954563	+	0.298009i	$0.0963224\pi$
$443$			264.036i			0.596017i	−0.954563	+	0.298009i	$0.903678\pi$
$444$		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	$-0.195292\pi$
$463$	757.117			1.63524			0.817621	+	0.575757i	$0.195292\pi$
$464$		0			0
$465$	391.904	−	354.178i	0.842805	−	0.761674i
$466$		0			0
$467$			51.8514i			0.111031i	0.998458	+	0.0555155i	$0.0176802\pi$
$467$			51.8514i			0.111031i	−0.998458	+	0.0555155i	$0.982320\pi$
$468$		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	$0.0132864\pi$
$479$			39.9757i			0.0834566i	−0.999129	+	0.0417283i	$0.986714\pi$
$480$		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	$-0.326836\pi$
$487$	504.115			1.03514			0.517572	+	0.855640i	$0.326836\pi$
$488$		0			0
$489$	868.375			1.77582
$490$		0			0
$491$	498.079			1.01442			0.507209	−	0.861823i	$-0.330677\pi$
$491$	498.079			1.01442			0.507209	+	0.861823i	$0.330677\pi$
$492$		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	$-0.168660\pi$
$499$	861.152			1.72576			0.862878	+	0.505413i	$0.168660\pi$
$500$		0			0
$501$			800.681i			1.59817i
$502$		0			0
$503$	−88.7312			−0.176404			−0.0882020	−	0.996103i	$-0.528112\pi$
$503$	−88.7312			−0.176404			−0.0882020	+	0.996103i	$0.528112\pi$
$504$		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	$0.0582832\pi$
$523$			190.456i			0.364161i	−0.983284	+	0.182081i	$0.941717\pi$
$524$		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	$-0.688640\pi$
$547$		−	907.445i		−	1.65895i	0.558545	−	0.829474i	$-0.311360\pi$
$548$		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	$0.0505323\pi$
$563$			178.005i			0.316172i	−0.987425	+	0.158086i	$0.949468\pi$
$564$		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	$-0.618606\pi$
$571$	−415.743			−0.728096			−0.364048	+	0.931380i	$0.618606\pi$
$572$		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	$0.270385\pi$
$587$			881.568i			1.50182i	−0.660404	+	0.750910i	$0.729615\pi$
$588$		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	$-0.914222\pi$
$599$		−	318.943i		−	0.532459i	0.963910	−	0.266229i	$-0.0857778\pi$
$600$		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	$-0.419773\pi$
$607$	302.748			0.498761			0.249380	+	0.968406i	$0.419773\pi$
$608$		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	$-0.519857\pi$
$619$	−77.1806			−0.124686			−0.0623429	+	0.998055i	$0.519857\pi$
$620$		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	$0.130997\pi$
$631$			504.826i			0.800041i	−0.916506	+	0.400020i	$0.869003\pi$
$632$		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	$-0.942788\pi$
$643$		−	229.899i		−	0.357541i	0.983891	−	0.178771i	$-0.0572120\pi$
$644$		0			0
$645$	−175.795	+	158.872i	−0.272550	+	0.246314i
$646$		0			0
$647$	−209.977			−0.324539			−0.162270	−	0.986746i	$-0.551881\pi$
$647$	−209.977			−0.324539			−0.162270	+	0.986746i	$0.551881\pi$
$648$		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	$-0.287914\pi$
$659$	814.620			1.23615			0.618073	+	0.786121i	$0.287914\pi$
$660$		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	$-0.989582\pi$
$683$		−	44.7000i		−	0.0654465i	0.999464	−	0.0327233i	$-0.0104180\pi$
$684$		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	$-0.297525\pi$
$691$	820.990			1.18812			0.594059	+	0.804421i	$0.297525\pi$
$692$		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	$0.189430\pi$
$719$			806.144i			1.12120i	−0.828086	+	0.560601i	$0.810570\pi$
$720$		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	$-0.532325\pi$
$727$	−147.404			−0.202756			−0.101378	+	0.994848i	$0.532325\pi$
$728$		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	$-0.347694\pi$
$739$	680.521			0.920868			0.460434	+	0.887694i	$0.347694\pi$
$740$		0			0
$741$		−	2131.36i		−	2.87632i
$742$		0			0
$743$	949.164			1.27748			0.638738	−	0.769425i	$-0.279457\pi$
$743$	949.164			1.27748			0.638738	+	0.769425i	$0.279457\pi$
$744$		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	$-0.849062\pi$
$751$		−	685.833i		−	0.913227i	0.889665	−	0.456613i	$-0.150938\pi$
$752$		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	$-0.875023\pi$
$787$		−	602.238i		−	0.765232i	0.923907	−	0.382616i	$-0.124977\pi$
$788$		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	$-0.400333\pi$
$811$	499.613			0.616046			0.308023	+	0.951379i	$0.400333\pi$
$812$		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	$-0.523068\pi$
$823$	−119.183			−0.144816			−0.0724079	+	0.997375i	$0.523068\pi$
$824$		0			0
$825$	411.429	+	41.7149i	0.498701	+	0.0505635i
$826$		0			0
$827$			29.2660i			0.0353882i	0.999843	+	0.0176941i	$0.00563249\pi$
$827$			29.2660i			0.0353882i	−0.999843	+	0.0176941i	$0.994368\pi$
$828$		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	$-0.725648\pi$
$839$		−	1273.74i		−	1.51816i	0.650996	−	0.759081i	$-0.274352\pi$
$840$		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	$-0.538141\pi$
$859$	−205.363			−0.239072			−0.119536	+	0.992830i	$0.538141\pi$
$860$		0			0
$861$			1879.85i			2.18334i
$862$		0			0
$863$	−533.186			−0.617829			−0.308914	−	0.951090i	$-0.599966\pi$
$863$	−533.186			−0.617829			−0.308914	+	0.951090i	$0.599966\pi$
$864$		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	$0.156999\pi$
$883$			836.150i			0.946943i	−0.880809	+	0.473471i	$0.843001\pi$
$884$		0			0
$885$	1137.70	+	1258.89i	1.28554	+	1.42247i
$886$		0			0
$887$	−59.0080			−0.0665253			−0.0332627	−	0.999447i	$-0.510590\pi$
$887$	−59.0080			−0.0665253			−0.0332627	+	0.999447i	$0.510590\pi$
$888$		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	$-0.747998\pi$
$907$		−	1290.73i		−	1.42308i	0.702645	−	0.711540i	$-0.252002\pi$
$908$		0			0
$909$			508.939i			0.559889i
$910$		0			0
$911$		−	1179.87i		−	1.29514i	−0.762007	−	0.647569i	$-0.775786\pi$
$911$		−	1179.87i		−	1.29514i	0.762007	−	0.647569i	$-0.224214\pi$
$912$		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	$0.303627\pi$
$919$			1499.19i			1.63133i	−0.578528	+	0.815663i	$0.696373\pi$
$920$		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	$0.322378\pi$
$947$			1606.69i			1.69661i	−0.529505	+	0.848307i	$0.677622\pi$
$948$		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	$-0.610547\pi$
$967$	−658.248			−0.680711			−0.340356	+	0.940297i	$0.610547\pi$
$968$		0			0
$969$	−3178.34			−3.28002
$970$		0			0
$971$	−1382.65			−1.42394			−0.711972	−	0.702208i	$-0.752198\pi$
$971$	−1382.65			−1.42394			−0.711972	+	0.702208i	$0.752198\pi$
$972$		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	$-0.756682\pi$
$983$	−1419.04			−1.44359			−0.721793	+	0.692109i	$0.756682\pi$
$984$		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	$-0.972866\pi$
$991$		−	168.749i		−	0.170281i	0.996369	−	0.0851405i	$-0.0271339\pi$
$992$		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.k.639.17		24
4.3	odd	2		1280.3.e.l.639.8		24
5.4	even	2	inner	1280.3.e.k.639.8		24
8.3	odd	2	inner	1280.3.e.k.639.7		24
8.5	even	2		1280.3.e.l.639.18		24
16.3	odd	4		640.3.h.b.639.1	yes	24
16.5	even	4		640.3.h.a.639.1	✓	24
16.11	odd	4		640.3.h.a.639.24	yes	24
16.13	even	4		640.3.h.b.639.24	yes	24
20.19	odd	2		1280.3.e.l.639.17		24
40.19	odd	2	inner	1280.3.e.k.639.18		24
40.29	even	2		1280.3.e.l.639.7		24
80.19	odd	4		640.3.h.b.639.23	yes	24
80.29	even	4		640.3.h.b.639.2	yes	24
80.59	odd	4		640.3.h.a.639.2	yes	24
80.69	even	4		640.3.h.a.639.23	yes	24

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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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