LMFDB - Embedded newform 1280.3.e.l.639.3

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			639.3
Character	$\chi$	$=$	1280.639
Dual form			1280.3.e.l.639.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73799i q^{3} +(3.35996 - 3.70279i) q^{5} -7.32638 q^{7} +5.97938 q^{9} +O(q^{10})$	$q-1.73799i q^{3} +(3.35996 - 3.70279i) q^{5} -7.32638 q^{7} +5.97938 q^{9} -13.9852 q^{11} +12.4106 q^{13} +(-6.43544 - 5.83959i) q^{15} +4.07874i q^{17} +14.9804 q^{19} +12.7332i q^{21} -24.8788 q^{23} +(-2.42138 - 24.8825i) q^{25} -26.0341i q^{27} -42.6447i q^{29} -27.9864i q^{31} +24.3062i q^{33} +(-24.6163 + 27.1281i) q^{35} +12.4664 q^{37} -21.5695i q^{39} -75.4673 q^{41} +20.5971i q^{43} +(20.0904 - 22.1404i) q^{45} +31.8709 q^{47} +4.67586 q^{49} +7.08883 q^{51} -88.3667 q^{53} +(-46.9896 + 51.7843i) q^{55} -26.0358i q^{57} -64.3508 q^{59} +44.1435i q^{61} -43.8072 q^{63} +(41.6990 - 45.9538i) q^{65} +85.0952i q^{67} +43.2391i q^{69} -44.9608i q^{71} -51.8688i q^{73} +(-43.2456 + 4.20834i) q^{75} +102.461 q^{77} -145.130i q^{79} +8.56734 q^{81} +100.687i q^{83} +(15.1028 + 13.7044i) q^{85} -74.1163 q^{87} -15.4980 q^{89} -90.9245 q^{91} -48.6402 q^{93} +(50.3333 - 55.4692i) q^{95} -0.492654i q^{97} -83.6227 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	1.73799i		−	0.579331i	−0.957128	−	0.289666i	$-0.906456\pi$
$3$		−	1.73799i		−	0.579331i	0.957128	−	0.289666i	$-0.0935441\pi$
$4$		0			0
$5$	3.35996	−	3.70279i	0.671991	−	0.740559i
$5$	3.35996	−	3.70279i	0.671991	−	0.740559i
$6$		0			0
$7$	−7.32638			−1.04663			−0.523313	−	0.852141i	$-0.675304\pi$
$7$	−7.32638			−1.04663			−0.523313	+	0.852141i	$0.675304\pi$
$8$		0			0
$9$	5.97938			0.664375
$10$		0			0
$11$	−13.9852			−1.27138			−0.635690	−	0.771944i	$-0.719284\pi$
$11$	−13.9852			−1.27138			−0.635690	+	0.771944i	$0.719284\pi$
$12$		0			0
$13$	12.4106			0.954659			0.477329	−	0.878724i	$-0.341605\pi$
$13$	12.4106			0.954659			0.477329	+	0.878724i	$0.341605\pi$
$14$		0			0
$15$	−6.43544	−	5.83959i	−0.429029	−	0.389306i
$16$		0			0
$17$			4.07874i			0.239926i	0.992778	+	0.119963i	$0.0382776\pi$
$17$			4.07874i			0.239926i	−0.992778	+	0.119963i	$0.961722\pi$
$18$		0			0
$19$	14.9804			0.788440			0.394220	−	0.919016i	$-0.371015\pi$
$19$	14.9804			0.788440			0.394220	+	0.919016i	$0.371015\pi$
$20$		0			0
$21$			12.7332i			0.606343i
$22$		0			0
$23$	−24.8788			−1.08168			−0.540842	−	0.841124i	$-0.681894\pi$
$23$	−24.8788			−1.08168			−0.540842	+	0.841124i	$0.681894\pi$
$24$		0			0
$25$	−2.42138	−	24.8825i	−0.0968551	−	0.995298i
$26$		0			0
$27$		−	26.0341i		−	0.964225i
$28$		0			0
$29$		−	42.6447i		−	1.47051i	−0.677792	−	0.735254i	$-0.737063\pi$
$29$		−	42.6447i		−	1.47051i	0.677792	−	0.735254i	$-0.262937\pi$
$30$		0			0
$31$		−	27.9864i		−	0.902787i	−0.892325	−	0.451393i	$-0.850927\pi$
$31$		−	27.9864i		−	0.902787i	0.892325	−	0.451393i	$-0.149073\pi$
$32$		0			0
$33$			24.3062i			0.736551i
$34$		0			0
$35$	−24.6163	+	27.1281i	−0.703324	+	0.775088i
$36$		0			0
$37$	12.4664			0.336929			0.168465	−	0.985708i	$-0.446119\pi$
$37$	12.4664			0.336929			0.168465	+	0.985708i	$0.446119\pi$
$38$		0			0
$39$		−	21.5695i		−	0.553064i
$40$		0			0
$41$	−75.4673			−1.84067			−0.920333	−	0.391136i	$-0.872082\pi$
$41$	−75.4673			−1.84067			−0.920333	+	0.391136i	$0.872082\pi$
$42$		0			0
$43$			20.5971i			0.479002i	0.970896	+	0.239501i	$0.0769839\pi$
$43$			20.5971i			0.479002i	−0.970896	+	0.239501i	$0.923016\pi$
$44$		0			0
$45$	20.0904	−	22.1404i	0.446454	−	0.492009i
$46$		0			0
$47$	31.8709			0.678104			0.339052	−	0.940768i	$-0.389894\pi$
$47$	31.8709			0.678104			0.339052	+	0.940768i	$0.389894\pi$
$48$		0			0
$49$	4.67586			0.0954256
$50$		0			0
$51$	7.08883			0.138997
$52$		0			0
$53$	−88.3667			−1.66730			−0.833648	−	0.552296i	$-0.813752\pi$
$53$	−88.3667			−1.66730			−0.833648	+	0.552296i	$0.813752\pi$
$54$		0			0
$55$	−46.9896	+	51.7843i	−0.854357	+	0.941532i
$56$		0			0
$57$		−	26.0358i		−	0.456768i
$58$		0			0
$59$	−64.3508			−1.09069			−0.545345	−	0.838211i	$-0.683602\pi$
$59$	−64.3508			−1.09069			−0.545345	+	0.838211i	$0.683602\pi$
$60$		0			0
$61$			44.1435i			0.723664i	0.932243	+	0.361832i	$0.117849\pi$
$61$			44.1435i			0.723664i	−0.932243	+	0.361832i	$0.882151\pi$
$62$		0			0
$63$	−43.8072			−0.695352
$64$		0			0
$65$	41.6990	−	45.9538i	0.641523	−	0.706981i
$66$		0			0
$67$			85.0952i			1.27008i	0.772481	+	0.635038i	$0.219016\pi$
$67$			85.0952i			1.27008i	−0.772481	+	0.635038i	$0.780984\pi$
$68$		0			0
$69$			43.2391i			0.626654i
$70$		0			0
$71$		−	44.9608i		−	0.633250i	−0.948551	−	0.316625i	$-0.897450\pi$
$71$		−	44.9608i		−	0.633250i	0.948551	−	0.316625i	$-0.102550\pi$
$72$		0			0
$73$		−	51.8688i		−	0.710531i	−0.934765	−	0.355265i	$-0.884390\pi$
$73$		−	51.8688i		−	0.710531i	0.934765	−	0.355265i	$-0.115610\pi$
$74$		0			0
$75$	−43.2456	+	4.20834i	−0.576608	+	0.0561112i
$76$		0			0
$77$	102.461			1.33066
$78$		0			0
$79$		−	145.130i		−	1.83709i	−0.395316	−	0.918545i	$-0.629365\pi$
$79$		−	145.130i		−	1.83709i	0.395316	−	0.918545i	$-0.370635\pi$
$80$		0			0
$81$	8.56734			0.105770
$82$		0			0
$83$			100.687i			1.21309i	0.795048	+	0.606547i	$0.207446\pi$
$83$			100.687i			1.21309i	−0.795048	+	0.606547i	$0.792554\pi$
$84$		0			0
$85$	15.1028	+	13.7044i	0.177679	+	0.161228i
$86$		0			0
$87$	−74.1163			−0.851911
$88$		0			0
$89$	−15.4980			−0.174135			−0.0870674	−	0.996202i	$-0.527750\pi$
$89$	−15.4980			−0.174135			−0.0870674	+	0.996202i	$0.527750\pi$
$90$		0			0
$91$	−90.9245			−0.999171
$92$		0			0
$93$	−48.6402			−0.523013
$94$		0			0
$95$	50.3333	−	55.4692i	0.529825	−	0.583886i
$96$		0			0
$97$		−	0.492654i		−	0.00507891i	−0.999997	−	0.00253946i	$-0.999192\pi$
$97$		−	0.492654i		−	0.00507891i	0.999997	−	0.00253946i	$-0.000808335\pi$
$98$		0			0
$99$	−83.6227			−0.844674
$100$		0			0
$101$		−	24.5991i		−	0.243556i	−0.992557	−	0.121778i	$-0.961140\pi$
$101$		−	24.5991i		−	0.243556i	0.992557	−	0.121778i	$-0.0388595\pi$
$102$		0			0
$103$	45.2049			0.438882			0.219441	−	0.975626i	$-0.429577\pi$
$103$	45.2049			0.438882			0.219441	+	0.975626i	$0.429577\pi$
$104$		0			0
$105$	47.1484	+	42.7830i	0.449033	+	0.407457i
$106$		0			0
$107$			113.659i			1.06223i	0.847299	+	0.531116i	$0.178227\pi$
$107$			113.659i			1.06223i	−0.847299	+	0.531116i	$0.821773\pi$
$108$		0			0
$109$			33.0086i			0.302831i	0.988470	+	0.151416i	$0.0483832\pi$
$109$			33.0086i			0.302831i	−0.988470	+	0.151416i	$0.951617\pi$
$110$		0			0
$111$		−	21.6665i		−	0.195194i
$112$		0			0
$113$			215.991i			1.91142i	0.294303	+	0.955712i	$0.404912\pi$
$113$			215.991i			1.91142i	−0.294303	+	0.955712i	$0.595088\pi$
$114$		0			0
$115$	−83.5915	+	92.1209i	−0.726883	+	0.801051i
$116$		0			0
$117$	74.2075			0.634252
$118$		0			0
$119$		−	29.8824i		−	0.251113i
$120$		0			0
$121$	74.5855			0.616409
$122$		0			0
$123$			131.162i			1.06636i
$124$		0			0
$125$	−100.270	−	74.6381i	−0.802163	−	0.597105i
$126$		0			0
$127$	−113.998			−0.897619			−0.448810	−	0.893627i	$-0.648152\pi$
$127$	−113.998			−0.897619			−0.448810	+	0.893627i	$0.648152\pi$
$128$		0			0
$129$	35.7976			0.277501
$130$		0			0
$131$	35.7036			0.272547			0.136273	−	0.990671i	$-0.456487\pi$
$131$	35.7036			0.272547			0.136273	+	0.990671i	$0.456487\pi$
$132$		0			0
$133$	−109.752			−0.825201
$134$		0			0
$135$	−96.3988	−	87.4733i	−0.714065	−	0.647951i
$136$		0			0
$137$			157.165i			1.14719i	0.819139	+	0.573595i	$0.194452\pi$
$137$			157.165i			1.14719i	−0.819139	+	0.573595i	$0.805548\pi$
$138$		0			0
$139$	38.7762			0.278965			0.139483	−	0.990225i	$-0.455456\pi$
$139$	38.7762			0.278965			0.139483	+	0.990225i	$0.455456\pi$
$140$		0			0
$141$		−	55.3914i		−	0.392847i
$142$		0			0
$143$	−173.564			−1.21373
$144$		0			0
$145$	−157.905	−	143.284i	−1.08900	−	0.988168i
$146$		0			0
$147$		−	8.12661i		−	0.0552831i
$148$		0			0
$149$		−	50.5027i		−	0.338944i	−0.985535	−	0.169472i	$-0.945794\pi$
$149$		−	50.5027i		−	0.338944i	0.985535	−	0.169472i	$-0.0542063\pi$
$150$		0			0
$151$		−	280.148i		−	1.85529i	−0.373467	−	0.927643i	$-0.621831\pi$
$151$		−	280.148i		−	1.85529i	0.373467	−	0.927643i	$-0.378169\pi$
$152$		0			0
$153$			24.3883i			0.159401i
$154$		0			0
$155$	−103.628	−	94.0331i	−0.668567	−	0.606665i
$156$		0			0
$157$	−109.217			−0.695650			−0.347825	−	0.937560i	$-0.613080\pi$
$157$	−109.217			−0.695650			−0.347825	+	0.937560i	$0.613080\pi$
$158$		0			0
$159$			153.581i			0.965917i
$160$		0			0
$161$	182.271			1.13212
$162$		0			0
$163$		−	312.727i		−	1.91857i	−0.282438	−	0.959286i	$-0.591143\pi$
$163$		−	312.727i		−	1.91857i	0.282438	−	0.959286i	$-0.408857\pi$
$164$		0			0
$165$	90.0008	+	81.6677i	0.545459	+	0.494956i
$166$		0			0
$167$	−163.812			−0.980913			−0.490456	−	0.871466i	$-0.663170\pi$
$167$	−163.812			−0.980913			−0.490456	+	0.871466i	$0.663170\pi$
$168$		0			0
$169$	−14.9778			−0.0886262
$170$		0			0
$171$	89.5732			0.523820
$172$		0			0
$173$	120.423			0.696087			0.348043	−	0.937478i	$-0.386846\pi$
$173$	120.423			0.696087			0.348043	+	0.937478i	$0.386846\pi$
$174$		0			0
$175$	17.7399	+	182.298i	0.101371	+	1.04171i
$176$		0			0
$177$			111.841i			0.631871i
$178$		0			0
$179$	−234.996			−1.31283			−0.656413	−	0.754401i	$-0.727927\pi$
$179$	−234.996			−1.31283			−0.656413	+	0.754401i	$0.727927\pi$
$180$		0			0
$181$		−	51.8066i		−	0.286224i	−0.989706	−	0.143112i	$-0.954289\pi$
$181$		−	51.8066i		−	0.286224i	0.989706	−	0.143112i	$-0.0457109\pi$
$182$		0			0
$183$	76.7211			0.419241
$184$		0			0
$185$	41.8865	−	46.1604i	0.226413	−	0.249516i
$186$		0			0
$187$		−	57.0420i		−	0.305037i
$188$		0			0
$189$			190.735i			1.00918i
$190$		0			0
$191$		−	259.662i		−	1.35949i	−0.733450	−	0.679744i	$-0.762091\pi$
$191$		−	259.662i		−	1.35949i	0.733450	−	0.679744i	$-0.237909\pi$
$192$		0			0
$193$		−	246.349i		−	1.27642i	−0.769861	−	0.638211i	$-0.779675\pi$
$193$		−	246.349i		−	1.27642i	0.769861	−	0.638211i	$-0.220325\pi$
$194$		0			0
$195$	−79.8674	−	72.4726i	−0.409576	−	0.371654i
$196$		0			0
$197$	176.610			0.896499			0.448249	−	0.893908i	$-0.352048\pi$
$197$	176.610			0.896499			0.448249	+	0.893908i	$0.352048\pi$
$198$		0			0
$199$			133.272i			0.669711i	0.942270	+	0.334855i	$0.108687\pi$
$199$			133.272i			0.669711i	−0.942270	+	0.334855i	$0.891313\pi$
$200$		0			0
$201$	147.895			0.735795
$202$		0			0
$203$			312.431i			1.53907i
$204$		0			0
$205$	−253.567	+	279.440i	−1.23691	+	1.36312i
$206$		0			0
$207$	−148.759			−0.718645
$208$		0			0
$209$	−209.503			−1.00241
$210$		0			0
$211$	93.5098			0.443174			0.221587	−	0.975141i	$-0.428876\pi$
$211$	93.5098			0.443174			0.221587	+	0.975141i	$0.428876\pi$
$212$		0			0
$213$	−78.1416			−0.366862
$214$		0			0
$215$	76.2668	+	69.2054i	0.354729	+	0.321885i
$216$		0			0
$217$			205.039i			0.944880i
$218$		0			0
$219$	−90.1476			−0.411633
$220$		0			0
$221$			50.6195i			0.229048i
$222$		0			0
$223$	−275.006			−1.23321			−0.616606	−	0.787272i	$-0.711493\pi$
$223$	−275.006			−1.23321			−0.616606	+	0.787272i	$0.711493\pi$
$224$		0			0
$225$	−14.4783	−	148.782i	−0.0643482	−	0.661252i
$226$		0			0
$227$			279.388i			1.23078i	0.788222	+	0.615391i	$0.211002\pi$
$227$			279.388i			1.23078i	−0.788222	+	0.615391i	$0.788998\pi$
$228$		0			0
$229$		−	74.4214i		−	0.324984i	−0.986710	−	0.162492i	$-0.948047\pi$
$229$		−	74.4214i		−	0.324984i	0.986710	−	0.162492i	$-0.0519532\pi$
$230$		0			0
$231$		−	178.076i		−	0.770893i
$232$		0			0
$233$		−	345.559i		−	1.48308i	−0.670907	−	0.741542i	$-0.734095\pi$
$233$		−	345.559i		−	1.48308i	0.670907	−	0.741542i	$-0.265905\pi$
$234$		0			0
$235$	107.085	−	118.011i	0.455680	−	0.502176i
$236$		0			0
$237$	−252.235			−1.06428
$238$		0			0
$239$			3.25176i			0.0136057i	0.999977	+	0.00680284i	$0.00216543\pi$
$239$			3.25176i			0.0136057i	−0.999977	+	0.00680284i	$0.997835\pi$
$240$		0			0
$241$	−395.249			−1.64004			−0.820018	−	0.572337i	$-0.806037\pi$
$241$	−395.249			−1.64004			−0.820018	+	0.572337i	$0.806037\pi$
$242$		0			0
$243$		−	249.197i		−	1.02550i
$244$		0			0
$245$	15.7107	−	17.3137i	0.0641252	−	0.0706683i
$246$		0			0
$247$	185.915			0.752691
$248$		0			0
$249$	174.993			0.702783
$250$		0			0
$251$	366.962			1.46200			0.730999	−	0.682378i	$-0.239054\pi$
$251$	366.962			1.46200			0.730999	+	0.682378i	$0.239054\pi$
$252$		0			0
$253$	347.934			1.37523
$254$		0			0
$255$	23.8182	−	26.2485i	0.0934046	−	0.102935i
$256$		0			0
$257$		−	354.526i		−	1.37948i	−0.724058	−	0.689739i	$-0.757725\pi$
$257$		−	354.526i		−	1.37948i	0.724058	−	0.689739i	$-0.242275\pi$
$258$		0			0
$259$	−91.3334			−0.352639
$260$		0			0
$261$		−	254.989i		−	0.976969i
$262$		0			0
$263$	380.361			1.44624			0.723120	−	0.690722i	$-0.242707\pi$
$263$	380.361			1.44624			0.723120	+	0.690722i	$0.242707\pi$
$264$		0			0
$265$	−296.908	+	327.204i	−1.12041	+	1.23473i
$266$		0			0
$267$			26.9354i			0.100882i
$268$		0			0
$269$		−	130.893i		−	0.486591i	−0.969952	−	0.243295i	$-0.921772\pi$
$269$		−	130.893i		−	0.486591i	0.969952	−	0.243295i	$-0.0782284\pi$
$270$		0			0
$271$			29.1591i			0.107598i	0.998552	+	0.0537991i	$0.0171330\pi$
$271$			29.1591i			0.107598i	−0.998552	+	0.0537991i	$0.982867\pi$
$272$		0			0
$273$			158.026i			0.578851i
$274$		0			0
$275$	33.8634	+	347.986i	0.123140	+	1.26540i
$276$		0			0
$277$	239.772			0.865601			0.432801	−	0.901490i	$-0.357525\pi$
$277$	239.772			0.865601			0.432801	+	0.901490i	$0.357525\pi$
$278$		0			0
$279$		−	167.341i		−	0.599789i
$280$		0			0
$281$	−38.7771			−0.137997			−0.0689984	−	0.997617i	$-0.521980\pi$
$281$	−38.7771			−0.137997			−0.0689984	+	0.997617i	$0.521980\pi$
$282$		0			0
$283$			105.503i			0.372804i	0.982474	+	0.186402i	$0.0596826\pi$
$283$			105.503i			0.372804i	−0.982474	+	0.186402i	$0.940317\pi$
$284$		0			0
$285$	−96.4051	−	87.4790i	−0.338263	−	0.306944i
$286$		0			0
$287$	552.902			1.92649
$288$		0			0
$289$	272.364			0.942435
$290$		0			0
$291$	−0.856230			−0.00294237
$292$		0			0
$293$	−116.858			−0.398832			−0.199416	−	0.979915i	$-0.563904\pi$
$293$	−116.858			−0.398832			−0.199416	+	0.979915i	$0.563904\pi$
$294$		0			0
$295$	−216.216	+	238.278i	−0.732935	+	0.807721i
$296$		0			0
$297$			364.091i			1.22590i
$298$		0			0
$299$	−308.759			−1.03264
$300$		0			0
$301$		−	150.902i		−	0.501336i
$302$		0			0
$303$	−42.7531			−0.141099
$304$		0			0
$305$	163.454	+	148.320i	0.535916	+	0.486296i
$306$		0			0
$307$		−	440.306i		−	1.43422i	−0.696960	−	0.717110i	$-0.745464\pi$
$307$		−	440.306i		−	1.43422i	0.696960	−	0.717110i	$-0.254536\pi$
$308$		0			0
$309$		−	78.5658i		−	0.254258i
$310$		0			0
$311$			160.201i			0.515115i	0.966263	+	0.257558i	$0.0829178\pi$
$311$			160.201i			0.515115i	−0.966263	+	0.257558i	$0.917082\pi$
$312$		0			0
$313$		−	529.200i		−	1.69073i	−0.534186	−	0.845367i	$-0.679382\pi$
$313$		−	529.200i		−	1.69073i	0.534186	−	0.845367i	$-0.320618\pi$
$314$		0			0
$315$	−147.190	+	162.209i	−0.467271	+	0.514949i
$316$		0			0
$317$	428.110			1.35051			0.675253	−	0.737586i	$-0.264035\pi$
$317$	428.110			1.35051			0.675253	+	0.737586i	$0.264035\pi$
$318$		0			0
$319$			596.394i			1.86957i
$320$		0			0
$321$	197.538			0.615385
$322$		0			0
$323$			61.1010i			0.189167i
$324$		0			0
$325$	−30.0507	−	308.805i	−0.0924636	−	0.950171i
$326$		0			0
$327$	57.3688			0.175440
$328$		0			0
$329$	−233.498			−0.709721
$330$		0			0
$331$	649.786			1.96310			0.981550	−	0.191204i	$-0.0612391\pi$
$331$	649.786			1.96310			0.981550	+	0.191204i	$0.0612391\pi$
$332$		0			0
$333$	74.5412			0.223847
$334$		0			0
$335$	315.090	+	285.916i	0.940567	+	0.853481i
$336$		0			0
$337$			296.274i			0.879152i	0.898205	+	0.439576i	$0.144871\pi$
$337$			296.274i			0.879152i	−0.898205	+	0.439576i	$0.855129\pi$
$338$		0			0
$339$	375.391			1.10735
$340$		0			0
$341$			391.395i			1.14779i
$342$		0			0
$343$	324.736			0.946751
$344$		0			0
$345$	160.106	+	145.282i	0.464074	+	0.421106i
$346$		0			0
$347$		−	510.003i		−	1.46975i	−0.678203	−	0.734874i	$-0.737241\pi$
$347$		−	510.003i		−	1.46975i	0.678203	−	0.734874i	$-0.262759\pi$
$348$		0			0
$349$		−	380.141i		−	1.08923i	−0.838686	−	0.544615i	$-0.816676\pi$
$349$		−	380.141i		−	1.08923i	0.838686	−	0.544615i	$-0.183324\pi$
$350$		0			0
$351$		−	323.098i		−	0.920506i
$352$		0			0
$353$			171.150i			0.484844i	0.970171	+	0.242422i	$0.0779419\pi$
$353$			171.150i			0.484844i	−0.970171	+	0.242422i	$0.922058\pi$
$354$		0			0
$355$	−166.481	−	151.066i	−0.468959	−	0.425539i
$356$		0			0
$357$	−51.9355			−0.145478
$358$		0			0
$359$			418.429i			1.16554i	0.812637	+	0.582770i	$0.198031\pi$
$359$			418.429i			1.16554i	−0.812637	+	0.582770i	$0.801969\pi$
$360$		0			0
$361$	−136.589			−0.378363
$362$		0			0
$363$		−	129.629i		−	0.357105i
$364$		0			0
$365$	−192.059	−	174.277i	−0.526190	−	0.477471i
$366$		0			0
$367$	304.286			0.829118			0.414559	−	0.910023i	$-0.363936\pi$
$367$	304.286			0.829118			0.414559	+	0.910023i	$0.363936\pi$
$368$		0			0
$369$	−451.247			−1.22289
$370$		0			0
$371$	647.408			1.74504
$372$		0			0
$373$	184.208			0.493856			0.246928	−	0.969034i	$-0.420579\pi$
$373$	184.208			0.493856			0.246928	+	0.969034i	$0.420579\pi$
$374$		0			0
$375$	−129.721	+	174.269i	−0.345922	+	0.464718i
$376$		0			0
$377$		−	529.245i		−	1.40383i
$378$		0			0
$379$	180.181			0.475410			0.237705	−	0.971337i	$-0.423605\pi$
$379$	180.181			0.475410			0.237705	+	0.971337i	$0.423605\pi$
$380$		0			0
$381$			198.127i			0.520019i
$382$		0			0
$383$	−67.9402			−0.177390			−0.0886948	−	0.996059i	$-0.528270\pi$
$383$	−67.9402			−0.177390			−0.0886948	+	0.996059i	$0.528270\pi$
$384$		0			0
$385$	344.264	−	379.391i	0.894192	−	0.985432i
$386$		0			0
$387$			123.158i			0.318237i
$388$		0			0
$389$			547.634i			1.40780i	0.710299	+	0.703900i	$0.248560\pi$
$389$			547.634i			1.40780i	−0.710299	+	0.703900i	$0.751440\pi$
$390$		0			0
$391$		−	101.474i		−	0.259524i
$392$		0			0
$393$		−	62.0527i		−	0.157895i
$394$		0			0
$395$	−537.387	−	487.631i	−1.36047	−	1.23451i
$396$		0			0
$397$	−102.144			−0.257289			−0.128644	−	0.991691i	$-0.541063\pi$
$397$	−102.144			−0.257289			−0.128644	+	0.991691i	$0.541063\pi$
$398$		0			0
$399$			190.748i			0.478065i
$400$		0			0
$401$	−260.718			−0.650170			−0.325085	−	0.945685i	$-0.605393\pi$
$401$	−260.718			−0.650170			−0.325085	+	0.945685i	$0.605393\pi$
$402$		0			0
$403$		−	347.327i		−	0.861853i
$404$		0			0
$405$	28.7859	−	31.7231i	0.0710763	−	0.0783286i
$406$		0			0
$407$	−174.345			−0.428365
$408$		0			0
$409$	237.138			0.579800			0.289900	−	0.957057i	$-0.406378\pi$
$409$	237.138			0.579800			0.289900	+	0.957057i	$0.406378\pi$
$410$		0			0
$411$	273.152			0.664604
$412$		0			0
$413$	471.458			1.14155
$414$		0			0
$415$	372.823	+	338.303i	0.898368	+	0.815189i
$416$		0			0
$417$		−	67.3928i		−	0.161613i
$418$		0			0
$419$	378.980			0.904487			0.452244	−	0.891894i	$-0.350624\pi$
$419$	378.980			0.904487			0.452244	+	0.891894i	$0.350624\pi$
$420$		0			0
$421$			514.293i			1.22160i	0.791786	+	0.610799i	$0.209152\pi$
$421$			514.293i			1.22160i	−0.791786	+	0.610799i	$0.790848\pi$
$422$		0			0
$423$	190.568			0.450515
$424$		0			0
$425$	101.489	−	9.87618i	0.238798	−	0.0232381i
$426$		0			0
$427$		−	323.412i		−	0.757405i
$428$		0			0
$429$			301.653i			0.703155i
$430$		0			0
$431$		−	747.351i		−	1.73399i	−0.498315	−	0.866996i	$-0.666048\pi$
$431$		−	747.351i		−	1.73399i	0.498315	−	0.866996i	$-0.333952\pi$
$432$		0			0
$433$			604.587i			1.39628i	0.715963	+	0.698138i	$0.245988\pi$
$433$			604.587i			1.39628i	−0.715963	+	0.698138i	$0.754012\pi$
$434$		0			0
$435$	−249.027	+	274.437i	−0.572477	+	0.630890i
$436$		0			0
$437$	−372.692			−0.852843
$438$		0			0
$439$			560.191i			1.27606i	0.770010	+	0.638031i	$0.220251\pi$
$439$			560.191i			1.27606i	−0.770010	+	0.638031i	$0.779749\pi$
$440$		0			0
$441$	27.9587			0.0633984
$442$		0			0
$443$		−	63.4566i		−	0.143243i	−0.997432	−	0.0716215i	$-0.977183\pi$
$443$		−	63.4566i		−	0.143243i	0.997432	−	0.0716215i	$-0.0228174\pi$
$444$		0			0
$445$	−52.0726	+	57.3859i	−0.117017	+	0.128957i
$446$		0			0
$447$	−87.7734			−0.196361
$448$		0			0
$449$	684.142			1.52370			0.761851	−	0.647753i	$-0.224291\pi$
$449$	684.142			1.52370			0.761851	+	0.647753i	$0.224291\pi$
$450$		0			0
$451$	1055.42			2.34019
$452$		0			0
$453$	−486.896			−1.07483
$454$		0			0
$455$	−305.503	+	336.675i	−0.671434	+	0.739945i
$456$		0			0
$457$		−	98.0585i		−	0.214570i	−0.994228	−	0.107285i	$-0.965784\pi$
$457$		−	98.0585i		−	0.214570i	0.994228	−	0.107285i	$-0.0342157\pi$
$458$		0			0
$459$	106.186			0.231343
$460$		0			0
$461$		−	261.156i		−	0.566499i	−0.959046	−	0.283249i	$-0.908588\pi$
$461$		−	261.156i		−	0.566499i	0.959046	−	0.283249i	$-0.0914124\pi$
$462$		0			0
$463$	485.876			1.04941			0.524704	−	0.851285i	$-0.324176\pi$
$463$	485.876			1.04941			0.524704	+	0.851285i	$0.324176\pi$
$464$		0			0
$465$	−163.429	+	180.105i	−0.351460	+	0.387322i
$466$		0			0
$467$		−	95.7017i		−	0.204929i	−0.994737	−	0.102464i	$-0.967327\pi$
$467$		−	95.7017i		−	0.204929i	0.994737	−	0.102464i	$-0.0326727\pi$
$468$		0			0
$469$		−	623.440i		−	1.32930i
$470$		0			0
$471$			189.818i			0.403012i
$472$		0			0
$473$		−	288.054i		−	0.608994i
$474$		0			0
$475$	−36.2731	−	372.748i	−0.0763644	−	0.784733i
$476$		0			0
$477$	−528.378			−1.10771
$478$		0			0
$479$		−	521.584i		−	1.08890i	−0.838793	−	0.544451i	$-0.816738\pi$
$479$		−	521.584i		−	1.08890i	0.838793	−	0.544451i	$-0.183262\pi$
$480$		0			0
$481$	154.715			0.321652
$482$		0			0
$483$		−	316.786i		−	0.655872i
$484$		0			0
$485$	−1.82420	−	1.65530i	−0.00376123	−	0.00341298i
$486$		0			0
$487$	176.883			0.363209			0.181604	−	0.983372i	$-0.441871\pi$
$487$	176.883			0.363209			0.181604	+	0.983372i	$0.441871\pi$
$488$		0			0
$489$	−543.518			−1.11149
$490$		0			0
$491$	−196.467			−0.400137			−0.200068	−	0.979782i	$-0.564116\pi$
$491$	−196.467			−0.400137			−0.200068	+	0.979782i	$0.564116\pi$
$492$		0			0
$493$	173.937			0.352813
$494$		0			0
$495$	−280.969	+	309.638i	−0.567614	+	0.625531i
$496$		0			0
$497$			329.400i			0.662776i
$498$		0			0
$499$	−220.261			−0.441406			−0.220703	−	0.975341i	$-0.570835\pi$
$499$	−220.261			−0.441406			−0.220703	+	0.975341i	$0.570835\pi$
$500$		0			0
$501$			284.705i			0.568274i
$502$		0			0
$503$	−939.069			−1.86694			−0.933468	−	0.358661i	$-0.883233\pi$
$503$	−939.069			−1.86694			−0.933468	+	0.358661i	$0.883233\pi$
$504$		0			0
$505$	−91.0855	−	82.6520i	−0.180367	−	0.163667i
$506$		0			0
$507$			26.0314i			0.0513440i
$508$		0			0
$509$			288.731i			0.567251i	0.958935	+	0.283625i	$0.0915372\pi$
$509$			288.731i			0.567251i	−0.958935	+	0.283625i	$0.908463\pi$
$510$		0			0
$511$			380.010i			0.743660i
$512$		0			0
$513$		−	390.000i		−	0.760233i
$514$		0			0
$515$	151.886	−	167.384i	0.294925	−	0.325018i
$516$		0			0
$517$	−445.720			−0.862128
$518$		0			0
$519$		−	209.294i		−	0.403265i
$520$		0			0
$521$	−674.737			−1.29508			−0.647541	−	0.762031i	$-0.724202\pi$
$521$	−674.737			−1.29508			−0.647541	+	0.762031i	$0.724202\pi$
$522$		0			0
$523$		−	447.061i		−	0.854801i	−0.904063	−	0.427400i	$-0.859430\pi$
$523$		−	447.061i		−	0.854801i	0.904063	−	0.427400i	$-0.140570\pi$
$524$		0			0
$525$	316.834	−	30.8319i	0.603492	−	0.0587275i
$526$		0			0
$527$	114.149			0.216602
$528$		0			0
$529$	89.9523			0.170042
$530$		0			0
$531$	−384.777			−0.724628
$532$		0			0
$533$	−936.592			−1.75721
$534$		0			0
$535$	420.856	+	381.889i	0.786646	+	0.713811i
$536$		0			0
$537$			408.422i			0.760562i
$538$		0			0
$539$	−65.3927			−0.121322
$540$		0			0
$541$		−	229.726i		−	0.424632i	−0.977201	−	0.212316i	$-0.931899\pi$
$541$		−	229.726i		−	0.424632i	0.977201	−	0.212316i	$-0.0681006\pi$
$542$		0			0
$543$	−90.0395			−0.165819
$544$		0			0
$545$	122.224	+	110.908i	0.224264	+	0.203500i
$546$		0			0
$547$			218.904i			0.400190i	0.979776	+	0.200095i	$0.0641251\pi$
$547$			218.904i			0.400190i	−0.979776	+	0.200095i	$0.935875\pi$
$548$		0			0
$549$			263.951i			0.480784i
$550$		0			0
$551$		−	638.833i		−	1.15941i
$552$		0			0
$553$			1063.28i			1.92275i
$554$		0			0
$555$	−80.2266	−	72.7985i	−0.144552	−	0.131168i
$556$		0			0
$557$	193.072			0.346628			0.173314	−	0.984867i	$-0.444552\pi$
$557$	193.072			0.346628			0.173314	+	0.984867i	$0.444552\pi$
$558$		0			0
$559$			255.622i			0.457284i
$560$		0			0
$561$	−99.1387			−0.176718
$562$		0			0
$563$			184.168i			0.327119i	0.986533	+	0.163559i	$0.0522975\pi$
$563$			184.168i			0.327119i	−0.986533	+	0.163559i	$0.947702\pi$
$564$		0			0
$565$	799.770	+	725.720i	1.41552	+	1.28446i
$566$		0			0
$567$	−62.7676			−0.110701
$568$		0			0
$569$	−80.6773			−0.141788			−0.0708940	−	0.997484i	$-0.522585\pi$
$569$	−80.6773			−0.141788			−0.0708940	+	0.997484i	$0.522585\pi$
$570$		0			0
$571$	905.670			1.58611			0.793056	−	0.609148i	$-0.208489\pi$
$571$	905.670			1.58611			0.793056	+	0.609148i	$0.208489\pi$
$572$		0			0
$573$	−451.291			−0.787594
$574$		0			0
$575$	60.2409	+	619.045i	0.104767	+	1.07660i
$576$		0			0
$577$		−	557.704i		−	0.966557i	−0.875467	−	0.483279i	$-0.839446\pi$
$577$		−	557.704i		−	0.966557i	0.875467	−	0.483279i	$-0.160554\pi$
$578$		0			0
$579$	−428.154			−0.739471
$580$		0			0
$581$		−	737.670i		−	1.26966i
$582$		0			0
$583$	1235.82			2.11977
$584$		0			0
$585$	249.334	−	274.775i	0.426212	−	0.469701i
$586$		0			0
$587$		−	188.682i		−	0.321435i	−0.987000	−	0.160717i	$-0.948619\pi$
$587$		−	188.682i		−	0.321435i	0.987000	−	0.160717i	$-0.0513808\pi$
$588$		0			0
$589$		−	419.246i		−	0.711793i
$590$		0			0
$591$		−	306.948i		−	0.519370i
$592$		0			0
$593$			676.550i			1.14089i	0.821335	+	0.570446i	$0.193230\pi$
$593$			676.550i			1.14089i	−0.821335	+	0.570446i	$0.806770\pi$
$594$		0			0
$595$	−110.649	−	100.404i	−0.185964	−	0.168746i
$596$		0			0
$597$	231.627			0.387985
$598$		0			0
$599$			565.309i			0.943755i	0.881664	+	0.471877i	$0.156424\pi$
$599$			565.309i			0.943755i	−0.881664	+	0.471877i	$0.843576\pi$
$600$		0			0
$601$	891.353			1.48312			0.741559	−	0.670888i	$-0.234087\pi$
$601$	891.353			1.48312			0.741559	+	0.670888i	$0.234087\pi$
$602$		0			0
$603$			508.816i			0.843808i
$604$		0			0
$605$	250.604	−	276.175i	0.414221	−	0.456487i
$606$		0			0
$607$	−112.539			−0.185402			−0.0927011	−	0.995694i	$-0.529550\pi$
$607$	−112.539			−0.185402			−0.0927011	+	0.995694i	$0.529550\pi$
$608$		0			0
$609$	543.004			0.891632
$610$		0			0
$611$	395.536			0.647358
$612$		0			0
$613$	609.554			0.994379			0.497190	−	0.867642i	$-0.334365\pi$
$613$	609.554			0.994379			0.497190	+	0.867642i	$0.334365\pi$
$614$		0			0
$615$	485.665	+	440.698i	0.789699	+	0.716582i
$616$		0			0
$617$		−	599.080i		−	0.970957i	−0.874249	−	0.485479i	$-0.838645\pi$
$617$		−	599.080i		−	0.970957i	0.874249	−	0.485479i	$-0.161355\pi$
$618$		0			0
$619$	358.707			0.579494			0.289747	−	0.957103i	$-0.406429\pi$
$619$	358.707			0.579494			0.289747	+	0.957103i	$0.406429\pi$
$620$		0			0
$621$			647.695i			1.04299i
$622$		0			0
$623$	113.544			0.182254
$624$		0			0
$625$	−613.274	+	120.500i	−0.981238	+	0.192800i
$626$		0			0
$627$			364.115i			0.580726i
$628$		0			0
$629$			50.8472i			0.0808381i
$630$		0			0
$631$		−	827.851i		−	1.31197i	−0.754776	−	0.655983i	$-0.772254\pi$
$631$		−	827.851i		−	1.31197i	0.754776	−	0.655983i	$-0.227746\pi$
$632$		0			0
$633$		−	162.519i		−	0.256745i
$634$		0			0
$635$	−383.027	+	422.110i	−0.603193	+	0.664740i
$636$		0			0
$637$	58.0300			0.0910989
$638$		0			0
$639$		−	268.837i		−	0.420716i
$640$		0			0
$641$	747.305			1.16584			0.582921	−	0.812529i	$-0.301910\pi$
$641$	747.305			1.16584			0.582921	+	0.812529i	$0.301910\pi$
$642$		0			0
$643$		−	209.992i		−	0.326582i	−0.986578	−	0.163291i	$-0.947789\pi$
$643$		−	209.992i		−	0.326582i	0.986578	−	0.163291i	$-0.0522110\pi$
$644$		0			0
$645$	120.278	−	132.551i	0.186478	−	0.205506i
$646$		0			0
$647$	−703.401			−1.08717			−0.543586	−	0.839353i	$-0.682934\pi$
$647$	−703.401			−1.08717			−0.543586	+	0.839353i	$0.682934\pi$
$648$		0			0
$649$	899.957			1.38668
$650$		0			0
$651$	356.356			0.547399
$652$		0			0
$653$	−235.011			−0.359894			−0.179947	−	0.983676i	$-0.557593\pi$
$653$	−235.011			−0.359894			−0.179947	+	0.983676i	$0.557593\pi$
$654$		0			0
$655$	119.963	−	132.203i	0.183149	−	0.201837i
$656$		0			0
$657$		−	310.143i		−	0.472059i
$658$		0			0
$659$	−478.498			−0.726097			−0.363049	−	0.931770i	$-0.618264\pi$
$659$	−478.498			−0.726097			−0.363049	+	0.931770i	$0.618264\pi$
$660$		0			0
$661$		−	714.347i		−	1.08071i	−0.841438	−	0.540354i	$-0.818291\pi$
$661$		−	714.347i		−	1.08071i	0.841438	−	0.540354i	$-0.181709\pi$
$662$		0			0
$663$	87.9764			0.132694
$664$		0			0
$665$	−368.761	+	406.388i	−0.554528	+	0.611110i
$666$		0			0
$667$			1060.95i			1.59063i
$668$		0			0
$669$			477.959i			0.714439i
$670$		0			0
$671$		−	617.355i		−	0.920052i
$672$		0			0
$673$		−	54.3728i		−	0.0807917i	−0.999184	−	0.0403958i	$-0.987138\pi$
$673$		−	54.3728i		−	0.0807917i	0.999184	−	0.0403958i	$-0.0128619\pi$
$674$		0			0
$675$	−647.792	+	63.0383i	−0.959691	+	0.0933901i
$676$		0			0
$677$	1035.13			1.52899			0.764495	−	0.644629i	$-0.222988\pi$
$677$	1035.13			1.52899			0.764495	+	0.644629i	$0.222988\pi$
$678$		0			0
$679$			3.60937i			0.00531572i
$680$		0			0
$681$	485.574			0.713031
$682$		0			0
$683$			99.8208i			0.146151i	0.997326	+	0.0730753i	$0.0232813\pi$
$683$			99.8208i			0.146151i	−0.997326	+	0.0730753i	$0.976719\pi$
$684$		0			0
$685$	581.950	+	528.068i	0.849562	+	0.770902i
$686$		0			0
$687$	−129.344			−0.188274
$688$		0			0
$689$	−1096.68			−1.59170
$690$		0			0
$691$	1201.63			1.73898			0.869489	−	0.493952i	$-0.164448\pi$
$691$	1201.63			1.73898			0.869489	+	0.493952i	$0.164448\pi$
$692$		0			0
$693$	612.652			0.884057
$694$		0			0
$695$	130.286	−	143.580i	0.187462	−	0.206590i
$696$		0			0
$697$		−	307.812i		−	0.441624i
$698$		0			0
$699$	−600.579			−0.859197
$700$		0			0
$701$		−	357.520i		−	0.510014i	−0.966939	−	0.255007i	$-0.917922\pi$
$701$		−	357.520i		−	0.510014i	0.966939	−	0.255007i	$-0.0820777\pi$
$702$		0			0
$703$	186.751			0.265648
$704$		0			0
$705$	−205.103	−	186.113i	−0.290926	−	0.263990i
$706$		0			0
$707$			180.222i			0.254912i
$708$		0			0
$709$		−	199.220i		−	0.280988i	−0.990082	−	0.140494i	$-0.955131\pi$
$709$		−	199.220i		−	0.280988i	0.990082	−	0.140494i	$-0.0448690\pi$
$710$		0			0
$711$		−	867.788i		−	1.22052i
$712$		0			0
$713$			696.266i			0.976531i
$714$		0			0
$715$	−583.168	+	642.672i	−0.815619	+	0.898842i
$716$		0			0
$717$	5.65154			0.00788220
$718$		0			0
$719$			1102.66i			1.53360i	0.641888	+	0.766799i	$0.278152\pi$
$719$			1102.66i			1.53360i	−0.641888	+	0.766799i	$0.721848\pi$
$720$		0			0
$721$	−331.188			−0.459345
$722$		0			0
$723$			686.940i			0.950125i
$724$		0			0
$725$	−1061.11	+	103.259i	−1.46359	+	0.142426i
$726$		0			0
$727$	1126.09			1.54895			0.774476	−	0.632603i	$-0.218014\pi$
$727$	1126.09			1.54895			0.774476	+	0.632603i	$0.218014\pi$
$728$		0			0
$729$	−355.996			−0.488335
$730$		0			0
$731$	−84.0103			−0.114925
$732$		0			0
$733$	−1351.90			−1.84434			−0.922171	−	0.386784i	$-0.873586\pi$
$733$	−1351.90			−1.84434			−0.922171	+	0.386784i	$0.873586\pi$
$734$		0			0
$735$	−30.0912	−	27.3051i	−0.0409404	−	0.0371497i
$736$		0			0
$737$		−	1190.07i		−	1.61475i
$738$		0			0
$739$	−167.536			−0.226706			−0.113353	−	0.993555i	$-0.536159\pi$
$739$	−167.536			−0.226706			−0.113353	+	0.993555i	$0.536159\pi$
$740$		0			0
$741$		−	323.119i		−	0.436057i
$742$		0			0
$743$	−948.386			−1.27643			−0.638214	−	0.769859i	$-0.720326\pi$
$743$	−948.386			−1.27643			−0.638214	+	0.769859i	$0.720326\pi$
$744$		0			0
$745$	−187.001	−	169.687i	−0.251008	−	0.227768i
$746$		0			0
$747$			602.044i			0.805950i
$748$		0			0
$749$		−	832.708i		−	1.11176i
$750$		0			0
$751$			52.4990i			0.0699055i	0.999389	+	0.0349528i	$0.0111281\pi$
$751$			52.4990i			0.0699055i	−0.999389	+	0.0349528i	$0.988872\pi$
$752$		0			0
$753$		−	637.777i		−	0.846981i
$754$		0			0
$755$	−1037.33	−	941.286i	−1.37395	−	1.24674i
$756$		0			0
$757$	−654.674			−0.864827			−0.432414	−	0.901675i	$-0.642338\pi$
$757$	−654.674			−0.864827			−0.432414	+	0.901675i	$0.642338\pi$
$758$		0			0
$759$		−	604.707i		−	0.796716i
$760$		0			0
$761$	−88.5344			−0.116340			−0.0581698	−	0.998307i	$-0.518526\pi$
$761$	−88.5344			−0.116340			−0.0581698	+	0.998307i	$0.518526\pi$
$762$		0			0
$763$		−	241.834i		−	0.316951i
$764$		0			0
$765$	90.3050	+	81.9438i	0.118046	+	0.107116i
$766$		0			0
$767$	−798.629			−1.04124
$768$		0			0
$769$	−1308.40			−1.70144			−0.850718	−	0.525623i	$-0.823832\pi$
$769$	−1308.40			−1.70144			−0.850718	+	0.525623i	$0.823832\pi$
$770$		0			0
$771$	−616.164			−0.799175
$772$		0			0
$773$	604.591			0.782136			0.391068	−	0.920362i	$-0.372106\pi$
$773$	604.591			0.782136			0.391068	+	0.920362i	$0.372106\pi$
$774$		0			0
$775$	−696.370	+	67.7656i	−0.898542	+	0.0874395i
$776$		0			0
$777$			158.737i			0.204295i
$778$		0			0
$779$	−1130.53			−1.45125
$780$		0			0
$781$			628.785i			0.805102i
$782$		0			0
$783$	−1110.22			−1.41790
$784$		0			0
$785$	−366.964	+	404.408i	−0.467471	+	0.515170i
$786$		0			0
$787$		−	1375.23i		−	1.74743i	−0.486434	−	0.873717i	$-0.661703\pi$
$787$		−	1375.23i		−	1.74743i	0.486434	−	0.873717i	$-0.338297\pi$
$788$		0			0
$789$		−	661.066i		−	0.837852i
$790$		0			0
$791$		−	1582.43i		−	2.00055i
$792$		0			0
$793$			547.846i			0.690852i
$794$		0			0
$795$	568.678	+	516.025i	0.715318	+	0.649088i
$796$		0			0
$797$	−924.699			−1.16022			−0.580112	−	0.814537i	$-0.696991\pi$
$797$	−924.699			−1.16022			−0.580112	+	0.814537i	$0.696991\pi$
$798$		0			0
$799$			129.993i			0.162695i
$800$		0			0
$801$	−92.6683			−0.115691
$802$		0			0
$803$			725.394i			0.903355i
$804$		0			0
$805$	612.423	−	674.913i	0.760774	−	0.838401i
$806$		0			0
$807$	−227.491			−0.281897
$808$		0			0
$809$	−241.708			−0.298774			−0.149387	−	0.988779i	$-0.547730\pi$
$809$	−241.708			−0.298774			−0.149387	+	0.988779i	$0.547730\pi$
$810$		0			0
$811$	61.5266			0.0758651			0.0379325	−	0.999280i	$-0.487923\pi$
$811$	61.5266			0.0758651			0.0379325	+	0.999280i	$0.487923\pi$
$812$		0			0
$813$	50.6783			0.0623350
$814$		0			0
$815$	−1157.96	−	1050.75i	−1.42082	−	1.28926i
$816$		0			0
$817$			308.552i			0.377664i
$818$		0			0
$819$	−543.672			−0.663824
$820$		0			0
$821$			99.4438i			0.121125i	0.998164	+	0.0605626i	$0.0192895\pi$
$821$			99.4438i			0.121125i	−0.998164	+	0.0605626i	$0.980711\pi$
$822$		0			0
$823$	−972.951			−1.18220			−0.591100	−	0.806598i	$-0.701306\pi$
$823$	−972.951			−1.18220			−0.591100	+	0.806598i	$0.701306\pi$
$824$		0			0
$825$	604.797	−	58.8544i	0.733088	−	0.0713387i
$826$		0			0
$827$		−	364.016i		−	0.440164i	−0.975481	−	0.220082i	$-0.929368\pi$
$827$		−	364.016i		−	0.440164i	0.975481	−	0.220082i	$-0.0706325\pi$
$828$		0			0
$829$			1013.84i			1.22296i	0.791258	+	0.611482i	$0.209426\pi$
$829$			1013.84i			1.22296i	−0.791258	+	0.611482i	$0.790574\pi$
$830$		0			0
$831$		−	416.722i		−	0.501470i
$832$		0			0
$833$			19.0716i			0.0228951i
$834$		0			0
$835$	−550.403	+	606.564i	−0.659165	+	0.726424i
$836$		0			0
$837$	−728.599			−0.870489
$838$		0			0
$839$		−	377.536i		−	0.449984i	−0.974361	−	0.224992i	$-0.927764\pi$
$839$		−	377.536i		−	0.449984i	0.974361	−	0.224992i	$-0.0722355\pi$
$840$		0			0
$841$	−977.571			−1.16239
$842$		0			0
$843$			67.3944i			0.0799459i
$844$		0			0
$845$	−50.3249	+	55.4598i	−0.0595561	+	0.0656330i
$846$		0			0
$847$	−546.442			−0.645149
$848$		0			0
$849$	183.364			0.215977
$850$		0			0
$851$	−310.148			−0.364451
$852$		0			0
$853$	884.533			1.03697			0.518483	−	0.855088i	$-0.326497\pi$
$853$	884.533			1.03697			0.518483	+	0.855088i	$0.326497\pi$
$854$		0			0
$855$	300.962	−	331.671i	0.352002	−	0.387919i
$856$		0			0
$857$		−	1036.74i		−	1.20974i	−0.796326	−	0.604868i	$-0.793226\pi$
$857$		−	1036.74i		−	1.20974i	0.796326	−	0.604868i	$-0.206774\pi$
$858$		0			0
$859$	658.011			0.766019			0.383010	−	0.923744i	$-0.374888\pi$
$859$	658.011			0.766019			0.383010	+	0.923744i	$0.374888\pi$
$860$		0			0
$861$		−	960.941i		−	1.11607i
$862$		0			0
$863$	−496.875			−0.575753			−0.287877	−	0.957668i	$-0.592949\pi$
$863$	−496.875			−0.575753			−0.287877	+	0.957668i	$0.592949\pi$
$864$		0			0
$865$	404.616	−	445.902i	0.467764	−	0.515493i
$866$		0			0
$867$		−	473.367i		−	0.545982i
$868$		0			0
$869$			2029.67i			2.33564i
$870$		0			0
$871$			1056.08i			1.21249i
$872$		0			0
$873$		−	2.94577i		−	0.00337430i
$874$		0			0
$875$	734.619	+	546.827i	0.839565	+	0.624946i
$876$		0			0
$877$	288.006			0.328399			0.164199	−	0.986427i	$-0.447496\pi$
$877$	288.006			0.328399			0.164199	+	0.986427i	$0.447496\pi$
$878$		0			0
$879$			203.098i			0.231056i
$880$		0			0
$881$	411.220			0.466765			0.233382	−	0.972385i	$-0.425021\pi$
$881$	411.220			0.466765			0.233382	+	0.972385i	$0.425021\pi$
$882$		0			0
$883$			382.459i			0.433136i	0.976268	+	0.216568i	$0.0694863\pi$
$883$			382.459i			0.433136i	−0.976268	+	0.216568i	$0.930514\pi$
$884$		0			0
$885$	414.125	+	375.782i	0.467938	+	0.424612i
$886$		0			0
$887$	−20.7959			−0.0234452			−0.0117226	−	0.999931i	$-0.503732\pi$
$887$	−20.7959			−0.0234452			−0.0117226	+	0.999931i	$0.503732\pi$
$888$		0			0
$889$	835.190			0.939472
$890$		0			0
$891$	−119.816			−0.134473
$892$		0			0
$893$	477.437			0.534644
$894$		0			0
$895$	−789.576	+	870.142i	−0.882208	+	0.972226i
$896$		0			0
$897$			536.622i			0.598241i
$898$		0			0
$899$	−1193.47			−1.32755
$900$		0			0
$901$		−	360.425i		−	0.400028i
$902$		0			0
$903$	−262.267			−0.290440
$904$		0			0
$905$	−191.829	−	174.068i	−0.211966	−	0.192340i
$906$		0			0
$907$			376.924i			0.415572i	0.978174	+	0.207786i	$0.0666258\pi$
$907$			376.924i			0.415572i	−0.978174	+	0.207786i	$0.933374\pi$
$908$		0			0
$909$		−	147.087i		−	0.161812i
$910$		0			0
$911$		−	1143.44i		−	1.25515i	−0.778555	−	0.627577i	$-0.784047\pi$
$911$		−	1143.44i		−	1.25515i	0.778555	−	0.627577i	$-0.215953\pi$
$912$		0			0
$913$		−	1408.12i		−	1.54230i
$914$		0			0
$915$	257.780	−	284.083i	0.281726	−	0.310473i
$916$		0			0
$917$	−261.578			−0.285254
$918$		0			0
$919$		−	47.1893i		−	0.0513485i	−0.999670	−	0.0256743i	$-0.991827\pi$
$919$		−	47.1893i		−	0.0513485i	0.999670	−	0.0256743i	$-0.00817327\pi$
$920$		0			0
$921$	−765.248			−0.830889
$922$		0			0
$923$		−	557.989i		−	0.604538i
$924$		0			0
$925$	−30.1858	−	310.194i	−0.0326333	−	0.335345i
$926$		0			0
$927$	270.297			0.291582
$928$		0			0
$929$	226.732			0.244061			0.122030	−	0.992526i	$-0.461059\pi$
$929$	226.732			0.244061			0.122030	+	0.992526i	$0.461059\pi$
$930$		0			0
$931$	70.0460			0.0752374
$932$		0			0
$933$	278.428			0.298423
$934$		0			0
$935$	−211.215	−	191.659i	−0.225898	−	0.204983i
$936$		0			0
$937$			54.3501i			0.0580044i	0.999579	+	0.0290022i	$0.00923299\pi$
$937$			54.3501i			0.0580044i	−0.999579	+	0.0290022i	$0.990767\pi$
$938$		0			0
$939$	−919.746			−0.979495
$940$		0			0
$941$			787.840i			0.837237i	0.908162	+	0.418619i	$0.137486\pi$
$941$			787.840i			0.837237i	−0.908162	+	0.418619i	$0.862514\pi$
$942$		0			0
$943$	1877.53			1.99102
$944$		0			0
$945$	706.254	+	640.863i	0.747359	+	0.678162i
$946$		0			0
$947$			1718.24i			1.81440i	0.420701	+	0.907199i	$0.361784\pi$
$947$			1718.24i			1.81440i	−0.420701	+	0.907199i	$0.638216\pi$
$948$		0			0
$949$		−	643.721i		−	0.678315i
$950$		0			0
$951$		−	744.053i		−	0.782390i
$952$		0			0
$953$			968.692i			1.01647i	0.861220	+	0.508233i	$0.169701\pi$
$953$			968.692i			1.01647i	−0.861220	+	0.508233i	$0.830299\pi$
$954$		0			0
$955$	−961.476	−	872.454i	−1.00678	−	0.913564i
$956$		0			0
$957$	1036.53			1.08310
$958$		0			0
$959$		−	1151.45i		−	1.20068i
$960$		0			0
$961$	177.762			0.184976
$962$		0			0
$963$			679.609i			0.705721i
$964$		0			0
$965$	−912.181	−	827.724i	−0.945266	−	0.857745i
$966$		0			0
$967$	527.058			0.545044			0.272522	−	0.962150i	$-0.412142\pi$
$967$	527.058			0.545044			0.272522	+	0.962150i	$0.412142\pi$
$968$		0			0
$969$	106.193			0.109591
$970$		0			0
$971$	591.749			0.609422			0.304711	−	0.952445i	$-0.401440\pi$
$971$	591.749			0.609422			0.304711	+	0.952445i	$0.401440\pi$
$972$		0			0
$973$	−284.089			−0.291972
$974$		0			0
$975$	−536.702	+	52.2279i	−0.550464	+	0.0535671i
$976$		0			0
$977$			590.566i			0.604469i	0.953234	+	0.302234i	$0.0977325\pi$
$977$			590.566i			0.604469i	−0.953234	+	0.302234i	$0.902267\pi$
$978$		0			0
$979$	216.742			0.221392
$980$		0			0
$981$			197.371i			0.201194i
$982$		0			0
$983$	−1128.26			−1.14777			−0.573885	−	0.818936i	$-0.694564\pi$
$983$	−1128.26			−1.14777			−0.573885	+	0.818936i	$0.694564\pi$
$984$		0			0
$985$	593.403	−	653.952i	0.602440	−	0.663910i
$986$		0			0
$987$			405.819i			0.411164i
$988$		0			0
$989$		−	512.430i		−	0.518129i
$990$		0			0
$991$		−	504.911i		−	0.509496i	−0.967007	−	0.254748i	$-0.918007\pi$
$991$		−	504.911i		−	0.509496i	0.967007	−	0.254748i	$-0.0819926\pi$
$992$		0			0
$993$		−	1129.32i		−	1.13729i
$994$		0			0
$995$	493.481	+	447.790i	0.495960	+	0.450040i
$996$		0			0
$997$	−1166.48			−1.16999			−0.584996	−	0.811036i	$-0.698904\pi$
$997$	−1166.48			−1.16999			−0.584996	+	0.811036i	$0.698904\pi$
$998$		0			0
$999$		−	324.551i		−	0.324875i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.3.e.l.639.3		24
4.3	odd	2		1280.3.e.k.639.22		24
5.4	even	2	inner	1280.3.e.l.639.22		24
8.3	odd	2	inner	1280.3.e.l.639.21		24
8.5	even	2		1280.3.e.k.639.4		24
16.3	odd	4		640.3.h.b.639.10	yes	24
16.5	even	4		640.3.h.a.639.10	yes	24
16.11	odd	4		640.3.h.a.639.15	yes	24
16.13	even	4		640.3.h.b.639.15	yes	24
20.19	odd	2		1280.3.e.k.639.3		24
40.19	odd	2	inner	1280.3.e.l.639.4		24
40.29	even	2		1280.3.e.k.639.21		24
80.19	odd	4		640.3.h.b.639.16	yes	24
80.29	even	4		640.3.h.b.639.9	yes	24
80.59	odd	4		640.3.h.a.639.9	✓	24
80.69	even	4		640.3.h.a.639.16	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.3.h.a.639.9	✓	24	80.59	odd	4
640.3.h.a.639.10	yes	24	16.5	even	4
640.3.h.a.639.15	yes	24	16.11	odd	4
640.3.h.a.639.16	yes	24	80.69	even	4
640.3.h.b.639.9	yes	24	80.29	even	4
640.3.h.b.639.10	yes	24	16.3	odd	4
640.3.h.b.639.15	yes	24	16.13	even	4
640.3.h.b.639.16	yes	24	80.19	odd	4
1280.3.e.k.639.3		24	20.19	odd	2
1280.3.e.k.639.4		24	8.5	even	2
1280.3.e.k.639.21		24	40.29	even	2
1280.3.e.k.639.22		24	4.3	odd	2
1280.3.e.l.639.3		24	1.1	even	1	trivial
1280.3.e.l.639.4		24	40.19	odd	2	inner
1280.3.e.l.639.21		24	8.3	odd	2	inner
1280.3.e.l.639.22		24	5.4	even	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)

\(f(q)\)	\(=\)	\(q-1.73799i q^{3} +(3.35996 - 3.70279i) q^{5} -7.32638 q^{7} +5.97938 q^{9} +O(q^{10})\)	\(q-1.73799i q^{3} +(3.35996 - 3.70279i) q^{5} -7.32638 q^{7} +5.97938 q^{9} -13.9852 q^{11} +12.4106 q^{13} +(-6.43544 - 5.83959i) q^{15} +4.07874i q^{17} +14.9804 q^{19} +12.7332i q^{21} -24.8788 q^{23} +(-2.42138 - 24.8825i) q^{25} -26.0341i q^{27} -42.6447i q^{29} -27.9864i q^{31} +24.3062i q^{33} +(-24.6163 + 27.1281i) q^{35} +12.4664 q^{37} -21.5695i q^{39} -75.4673 q^{41} +20.5971i q^{43} +(20.0904 - 22.1404i) q^{45} +31.8709 q^{47} +4.67586 q^{49} +7.08883 q^{51} -88.3667 q^{53} +(-46.9896 + 51.7843i) q^{55} -26.0358i q^{57} -64.3508 q^{59} +44.1435i q^{61} -43.8072 q^{63} +(41.6990 - 45.9538i) q^{65} +85.0952i q^{67} +43.2391i q^{69} -44.9608i q^{71} -51.8688i q^{73} +(-43.2456 + 4.20834i) q^{75} +102.461 q^{77} -145.130i q^{79} +8.56734 q^{81} +100.687i q^{83} +(15.1028 + 13.7044i) q^{85} -74.1163 q^{87} -15.4980 q^{89} -90.9245 q^{91} -48.6402 q^{93} +(50.3333 - 55.4692i) q^{95} -0.492654i q^{97} -83.6227 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 72 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more