LMFDB - Embedded newform 2640.2.k.e.1871.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2640,2,Mod(1871,2640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2640, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2640.1871");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2640.k (of order $2$ , degree $1$ , minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$21.0805061336$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.14786560000.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} + 43x^{4} + 361$ x^8 + 43*x^4 + 361
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{3}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1871.8
Root			$1.67601 - 1.67601i$ of defining polynomial
Character	$\chi$	$=$	2640.1871
Dual form			2640.2.k.e.1871.5

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.61803 + 0.618034i) q^{3} +1.00000i q^{5} +2.07167i q^{7} +(2.23607 + 2.00000i) q^{9} +1.00000 q^{11} +6.58809 q^{13} +(-0.618034 + 1.61803i) q^{15} +4.07167i q^{17} -4.18762i q^{19} +(-1.28036 + 3.35202i) q^{21} +6.70405 q^{23} -1.00000 q^{25} +(2.38197 + 4.61803i) q^{27} -2.18762i q^{29} -8.70405i q^{31} +(1.61803 + 0.618034i) q^{33} -2.07167 q^{35} -9.61131 q^{37} +(10.6598 + 4.07167i) q^{39} -6.18762i q^{41} +11.2478i q^{43} +(-2.00000 + 2.23607i) q^{45} -9.94012 q^{47} +2.70820 q^{49} +(-2.51643 + 6.58809i) q^{51} +5.94012i q^{53} +1.00000i q^{55} +(2.58809 - 6.77571i) q^{57} +0.532019 q^{59} +0.704049 q^{61} +(-4.14333 + 4.63238i) q^{63} +6.58809i q^{65} -4.14333i q^{67} +(10.8474 + 4.14333i) q^{69} +5.37940 q^{71} -4.25929 q^{73} +(-1.61803 - 0.618034i) q^{75} +2.07167i q^{77} -11.9557i q^{79} +(1.00000 + 8.94427i) q^{81} -2.25929 q^{83} -4.07167 q^{85} +(1.35202 - 3.53965i) q^{87} +17.5050i q^{89} +13.6483i q^{91} +(5.37940 - 14.0834i) q^{93} +4.18762 q^{95} +7.85153 q^{97} +(2.23607 + 2.00000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 4 q^{3} + 8 q^{11} + 8 q^{13} + 4 q^{15} - 8 q^{25} + 28 q^{27} + 4 q^{33} - 8 q^{37} + 24 q^{39} - 16 q^{45} - 8 q^{47} - 32 q^{49} + 8 q^{51} - 24 q^{57} + 40 q^{59} - 48 q^{61} - 8 q^{71} + 8 q^{73}+ \cdots - 24 q^{97}+O(q^{100})$ 8 * q + 4 * q^3 + 8 * q^11 + 8 * q^13 + 4 * q^15 - 8 * q^25 + 28 * q^27 + 4 * q^33 - 8 * q^37 + 24 * q^39 - 16 * q^45 - 8 * q^47 - 32 * q^49 + 8 * q^51 - 24 * q^57 + 40 * q^59 - 48 * q^61 - 8 * q^71 + 8 * q^73 - 4 * q^75 + 8 * q^81 + 24 * q^83 - 16 * q^85 - 16 * q^87 - 8 * q^93 + 8 * q^95 - 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2640\mathbb{Z}\right)^\times$ .

$n$	$661$	$881$	$991$	$1057$	$1201$
$\chi(n)$	$1$	$-1$	$-1$	$1$	$1$

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
$3$	1.61803	+	0.618034i	0.934172	+	0.356822i
$3$	1.61803	+	0.618034i	0.934172	+	0.356822i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$			2.07167i			0.783016i	0.920175	+	0.391508i	$0.128046\pi$
$7$			2.07167i			0.783016i	−0.920175	+	0.391508i	$0.871954\pi$
$8$		0			0
$9$	2.23607	+	2.00000i	0.745356	+	0.666667i
$10$		0			0
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$		0			0
$13$	6.58809			1.82721			0.913604	−	0.406605i	$-0.133287\pi$
$13$	6.58809			1.82721			0.913604	+	0.406605i	$0.133287\pi$
$14$		0			0
$15$	−0.618034	+	1.61803i	−0.159576	+	0.417775i
$16$		0			0
$17$			4.07167i			0.987524i	0.869597	+	0.493762i	$0.164379\pi$
$17$			4.07167i			0.987524i	−0.869597	+	0.493762i	$0.835621\pi$
$18$		0			0
$19$		−	4.18762i		−	0.960706i	−0.877075	−	0.480353i	$-0.840509\pi$
$19$		−	4.18762i		−	0.960706i	0.877075	−	0.480353i	$-0.159491\pi$
$20$		0			0
$21$	−1.28036	+	3.35202i	−0.279397	+	0.731472i
$22$		0			0
$23$	6.70405			1.39789			0.698945	−	0.715175i	$-0.253653\pi$
$23$	6.70405			1.39789			0.698945	+	0.715175i	$0.253653\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	2.38197	+	4.61803i	0.458410	+	0.888741i
$28$		0			0
$29$		−	2.18762i		−	0.406231i	−0.979155	−	0.203116i	$-0.934893\pi$
$29$		−	2.18762i		−	0.406231i	0.979155	−	0.203116i	$-0.0651067\pi$
$30$		0			0
$31$		−	8.70405i		−	1.56329i	−0.623721	−	0.781647i	$-0.714380\pi$
$31$		−	8.70405i		−	1.56329i	0.623721	−	0.781647i	$-0.285620\pi$
$32$		0			0
$33$	1.61803	+	0.618034i	0.281664	+	0.107586i
$34$		0			0
$35$	−2.07167			−0.350175
$36$		0			0
$37$	−9.61131			−1.58009			−0.790045	−	0.613049i	$-0.789943\pi$
$37$	−9.61131			−1.58009			−0.790045	+	0.613049i	$0.789943\pi$
$38$		0			0
$39$	10.6598	+	4.07167i	1.70693	+	0.651988i
$40$		0			0
$41$		−	6.18762i		−	0.966344i	−0.875525	−	0.483172i	$-0.839485\pi$
$41$		−	6.18762i		−	0.966344i	0.875525	−	0.483172i	$-0.160515\pi$
$42$		0			0
$43$			11.2478i			1.71528i	0.514250	+	0.857641i	$0.328070\pi$
$43$			11.2478i			1.71528i	−0.514250	+	0.857641i	$0.671930\pi$
$44$		0			0
$45$	−2.00000	+	2.23607i	−0.298142	+	0.333333i
$46$		0			0
$47$	−9.94012			−1.44992			−0.724958	−	0.688794i	$-0.758141\pi$
$47$	−9.94012			−1.44992			−0.724958	+	0.688794i	$0.758141\pi$
$48$		0			0
$49$	2.70820			0.386886
$50$		0			0
$51$	−2.51643	+	6.58809i	−0.352370	+	0.922517i
$52$		0			0
$53$			5.94012i			0.815938i	0.912996	+	0.407969i	$0.133763\pi$
$53$			5.94012i			0.815938i	−0.912996	+	0.407969i	$0.866237\pi$
$54$		0			0
$55$			1.00000i			0.134840i
$56$		0			0
$57$	2.58809	−	6.77571i	0.342801	−	0.897465i
$58$		0			0
$59$	0.532019			0.0692630			0.0346315	−	0.999400i	$-0.488974\pi$
$59$	0.532019			0.0692630			0.0346315	+	0.999400i	$0.488974\pi$
$60$		0			0
$61$	0.704049			0.0901442			0.0450721	−	0.998984i	$-0.485648\pi$
$61$	0.704049			0.0901442			0.0450721	+	0.998984i	$0.485648\pi$
$62$		0			0
$63$	−4.14333	+	4.63238i	−0.522011	+	0.583626i
$64$		0			0
$65$			6.58809i			0.817152i
$66$		0			0
$67$		−	4.14333i		−	0.506188i	−0.967442	−	0.253094i	$-0.918552\pi$
$67$		−	4.14333i		−	0.506188i	0.967442	−	0.253094i	$-0.0814482\pi$
$68$		0			0
$69$	10.8474	+	4.14333i	1.30587	+	0.498798i
$70$		0			0
$71$	5.37940			0.638417			0.319209	−	0.947684i	$-0.396583\pi$
$71$	5.37940			0.638417			0.319209	+	0.947684i	$0.396583\pi$
$72$		0			0
$73$	−4.25929			−0.498512			−0.249256	−	0.968438i	$-0.580186\pi$
$73$	−4.25929			−0.498512			−0.249256	+	0.968438i	$0.580186\pi$
$74$		0			0
$75$	−1.61803	−	0.618034i	−0.186834	−	0.0713644i
$76$		0			0
$77$			2.07167i			0.236088i
$78$		0			0
$79$		−	11.9557i		−	1.34512i	−0.740042	−	0.672561i	$-0.765194\pi$
$79$		−	11.9557i		−	1.34512i	0.740042	−	0.672561i	$-0.234806\pi$
$80$		0			0
$81$	1.00000	+	8.94427i	0.111111	+	0.993808i
$82$		0			0
$83$	−2.25929			−0.247989			−0.123994	−	0.992283i	$-0.539571\pi$
$83$	−2.25929			−0.247989			−0.123994	+	0.992283i	$0.539571\pi$
$84$		0			0
$85$	−4.07167			−0.441634
$86$		0			0
$87$	1.35202	−	3.53965i	0.144952	−	0.379490i
$88$		0			0
$89$			17.5050i			1.85553i	0.373171	+	0.927763i	$0.378271\pi$
$89$			17.5050i			1.85553i	−0.373171	+	0.927763i	$0.621729\pi$
$90$		0			0
$91$			13.6483i			1.43073i
$92$		0			0
$93$	5.37940	−	14.0834i	0.557818	−	1.46039i
$94$		0			0
$95$	4.18762			0.429641
$96$		0			0
$97$	7.85153			0.797202			0.398601	−	0.917124i	$-0.369496\pi$
$97$	7.85153			0.797202			0.398601	+	0.917124i	$0.369496\pi$
$98$		0			0
$99$	2.23607	+	2.00000i	0.224733	+	0.201008i
$100$		0			0
$101$			12.5712i			1.25088i	0.780273	+	0.625439i	$0.215080\pi$
$101$			12.5712i			1.25088i	−0.780273	+	0.625439i	$0.784920\pi$
$102$		0			0
$103$		−	8.94427i		−	0.881305i	−0.897678	−	0.440653i	$-0.854747\pi$
$103$		−	8.94427i		−	0.881305i	0.897678	−	0.440653i	$-0.145253\pi$
$104$		0			0
$105$	−3.35202	−	1.28036i	−0.327124	−	0.124950i
$106$		0			0
$107$	−2.68499			−0.259567			−0.129784	−	0.991542i	$-0.541428\pi$
$107$	−2.68499			−0.259567			−0.129784	+	0.991542i	$0.541428\pi$
$108$		0			0
$109$	−10.3752			−0.993768			−0.496884	−	0.867817i	$-0.665523\pi$
$109$	−10.3752			−0.993768			−0.496884	+	0.867817i	$0.665523\pi$
$110$		0			0
$111$	−15.5514	−	5.94012i	−1.47608	−	0.563811i
$112$		0			0
$113$			13.6196i			1.28123i	0.767864	+	0.640613i	$0.221320\pi$
$113$			13.6196i			1.28123i	−0.767864	+	0.640613i	$0.778680\pi$
$114$		0			0
$115$			6.70405i			0.625156i
$116$		0			0
$117$	14.7314	+	13.1762i	1.36192	+	1.21814i
$118$		0			0
$119$	−8.43513			−0.773247
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$	3.82416	−	10.0118i	0.344813	−	0.902732i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			10.0717i			0.893716i	0.894605	+	0.446858i	$0.147457\pi$
$127$			10.0717i			0.893716i	−0.894605	+	0.446858i	$0.852543\pi$
$128$		0			0
$129$	−6.95155	+	18.1994i	−0.612050	+	1.60237i
$130$		0			0
$131$	2.61547			0.228514			0.114257	−	0.993451i	$-0.463551\pi$
$131$	2.61547			0.228514			0.114257	+	0.993451i	$0.463551\pi$
$132$		0			0
$133$	8.67535			0.752248
$134$		0			0
$135$	−4.61803	+	2.38197i	−0.397457	+	0.205007i
$136$		0			0
$137$			19.4081i			1.65815i	0.559141	+	0.829073i	$0.311131\pi$
$137$			19.4081i			1.65815i	−0.559141	+	0.829073i	$0.688869\pi$
$138$		0			0
$139$		−	8.04429i		−	0.682308i	−0.940007	−	0.341154i	$-0.889182\pi$
$139$		−	8.04429i		−	0.682308i	0.940007	−	0.341154i	$-0.110818\pi$
$140$		0			0
$141$	−16.0834	−	6.14333i	−1.35447	−	0.517362i
$142$		0			0
$143$	6.58809			0.550924
$144$		0			0
$145$	2.18762			0.181672
$146$		0			0
$147$	4.38197	+	1.67376i	0.361418	+	0.138050i
$148$		0			0
$149$		−	17.9793i		−	1.47292i	−0.676481	−	0.736460i	$-0.736496\pi$
$149$		−	17.9793i		−	1.47292i	0.676481	−	0.736460i	$-0.263504\pi$
$150$		0			0
$151$		−	12.9886i		−	1.05699i	−0.848935	−	0.528497i	$-0.822756\pi$
$151$		−	12.9886i		−	1.05699i	0.848935	−	0.528497i	$-0.177244\pi$
$152$		0			0
$153$	−8.14333	+	9.10452i	−0.658349	+	0.736057i
$154$		0			0
$155$	8.70405			0.699126
$156$		0			0
$157$	21.6483			1.72772			0.863862	−	0.503728i	$-0.168039\pi$
$157$	21.6483			1.72772			0.863862	+	0.503728i	$0.168039\pi$
$158$		0			0
$159$	−3.67119	+	9.61131i	−0.291145	+	0.762227i
$160$		0			0
$161$			13.8885i			1.09457i
$162$		0			0
$163$		−	12.0000i		−	0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$		−	12.0000i		−	0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$		0			0
$165$	−0.618034	+	1.61803i	−0.0481139	+	0.125964i
$166$		0			0
$167$	−25.5324			−1.97575			−0.987877	−	0.155240i	$-0.950385\pi$
$167$	−25.5324			−1.97575			−0.987877	+	0.155240i	$0.950385\pi$
$168$		0			0
$169$	30.4030			2.33869
$170$		0			0
$171$	8.37524	−	9.36381i	0.640471	−	0.716068i
$172$		0			0
$173$		−	11.5767i		−	0.880157i	−0.897959	−	0.440078i	$-0.854951\pi$
$173$		−	11.5767i		−	0.880157i	0.897959	−	0.440078i	$-0.145049\pi$
$174$		0			0
$175$		−	2.07167i		−	0.156603i
$176$		0			0
$177$	0.860825	+	0.328806i	0.0647036	+	0.0247146i
$178$		0			0
$179$	13.7917			1.03084			0.515418	−	0.856939i	$-0.327637\pi$
$179$	13.7917			1.03084			0.515418	+	0.856939i	$0.327637\pi$
$180$		0			0
$181$	−23.9949			−1.78352			−0.891762	−	0.452505i	$-0.850531\pi$
$181$	−23.9949			−1.78352			−0.891762	+	0.452505i	$0.850531\pi$
$182$		0			0
$183$	1.13918	+	0.435126i	0.0842102	+	0.0321655i
$184$		0			0
$185$		−	9.61131i		−	0.706638i
$186$		0			0
$187$			4.07167i			0.297750i
$188$		0			0
$189$	−9.56702	+	4.93464i	−0.695898	+	0.358942i
$190$		0			0
$191$	0.560719			0.0405722			0.0202861	−	0.999794i	$-0.493542\pi$
$191$	0.560719			0.0405722			0.0202861	+	0.999794i	$0.493542\pi$
$192$		0			0
$193$	−5.26858			−0.379241			−0.189620	−	0.981858i	$-0.560726\pi$
$193$	−5.26858			−0.379241			−0.189620	+	0.981858i	$0.560726\pi$
$194$		0			0
$195$	−4.07167	+	10.6598i	−0.291578	+	0.763361i
$196$		0			0
$197$		−	22.3355i		−	1.59134i	−0.605734	−	0.795668i	$-0.707120\pi$
$197$		−	22.3355i		−	1.59134i	0.605734	−	0.795668i	$-0.292880\pi$
$198$		0			0
$199$			0.375243i			0.0266003i	0.999912	+	0.0133001i	$0.00423369\pi$
$199$			0.375243i			0.0266003i	−0.999912	+	0.0133001i	$0.995766\pi$
$200$		0			0
$201$	2.56072	−	6.70405i	0.180619	−	0.472867i
$202$		0			0
$203$	4.53202			0.318085
$204$		0			0
$205$	6.18762			0.432162
$206$		0			0
$207$	14.9907	+	13.4081i	1.04193	+	0.931927i
$208$		0			0
$209$		−	4.18762i		−	0.289664i
$210$		0			0
$211$		−	22.1647i		−	1.52588i	−0.646467	−	0.762942i	$-0.723754\pi$
$211$		−	22.1647i		−	1.52588i	0.646467	−	0.762942i	$-0.276246\pi$
$212$		0			0
$213$	8.70405	+	3.32465i	0.596392	+	0.227801i
$214$		0			0
$215$	−11.2478			−0.767097
$216$		0			0
$217$	18.0319			1.22408
$218$		0			0
$219$	−6.89167	−	2.63238i	−0.465696	−	0.177880i
$220$		0			0
$221$			26.8245i			1.80441i
$222$		0			0
$223$		−	13.8432i		−	0.927011i	−0.886094	−	0.463505i	$-0.846591\pi$
$223$		−	13.8432i		−	0.927011i	0.886094	−	0.463505i	$-0.153409\pi$
$224$		0			0
$225$	−2.23607	−	2.00000i	−0.149071	−	0.133333i
$226$		0			0
$227$	−25.8190			−1.71367			−0.856834	−	0.515592i	$-0.827572\pi$
$227$	−25.8190			−1.71367			−0.856834	+	0.515592i	$0.827572\pi$
$228$		0			0
$229$	−7.88023			−0.520741			−0.260370	−	0.965509i	$-0.583845\pi$
$229$	−7.88023			−0.520741			−0.260370	+	0.965509i	$0.583845\pi$
$230$		0			0
$231$	−1.28036	+	3.35202i	−0.0842415	+	0.220547i
$232$		0			0
$233$		−	4.96119i		−	0.325018i	−0.986707	−	0.162509i	$-0.948041\pi$
$233$		−	4.96119i		−	0.325018i	0.986707	−	0.162509i	$-0.0519587\pi$
$234$		0			0
$235$		−	9.94012i		−	0.648422i
$236$		0			0
$237$	7.38903	−	19.3447i	0.479969	−	1.25658i
$238$		0			0
$239$	0.143330			0.00927125			0.00463563	−	0.999989i	$-0.498524\pi$
$239$	0.143330			0.00927125			0.00463563	+	0.999989i	$0.498524\pi$
$240$		0			0
$241$	−10.8895			−0.701456			−0.350728	−	0.936477i	$-0.614066\pi$
$241$	−10.8895			−0.701456			−0.350728	+	0.936477i	$0.614066\pi$
$242$		0			0
$243$	−3.90983	+	15.0902i	−0.250816	+	0.968035i
$244$		0			0
$245$			2.70820i			0.173021i
$246$		0			0
$247$		−	27.5884i		−	1.75541i
$248$		0			0
$249$	−3.65560	−	1.39632i	−0.231664	−	0.0884879i
$250$		0			0
$251$	−11.1939			−0.706554			−0.353277	−	0.935519i	$-0.614933\pi$
$251$	−11.1939			−0.706554			−0.353277	+	0.935519i	$0.614933\pi$
$252$		0			0
$253$	6.70405			0.421480
$254$		0			0
$255$	−6.58809	−	2.51643i	−0.412562	−	0.157585i
$256$		0			0
$257$		−	3.26477i		−	0.203651i	−0.994802	−	0.101825i	$-0.967532\pi$
$257$		−	3.26477i		−	0.203651i	0.994802	−	0.101825i	$-0.0324683\pi$
$258$		0			0
$259$		−	19.9114i		−	1.23724i
$260$		0			0
$261$	4.37524	−	4.89167i	0.270821	−	0.302787i
$262$		0			0
$263$	−11.7872			−0.726827			−0.363413	−	0.931628i	$-0.618389\pi$
$263$	−11.7872			−0.726827			−0.363413	+	0.931628i	$0.618389\pi$
$264$		0			0
$265$	−5.94012			−0.364898
$266$		0			0
$267$	−10.8187	+	28.3237i	−0.662092	+	1.73338i
$268$		0			0
$269$		−	1.06404i		−	0.0648755i	−0.999474	−	0.0324378i	$-0.989673\pi$
$269$		−	1.06404i		−	0.0648755i	0.999474	−	0.0324378i	$-0.0103271\pi$
$270$		0			0
$271$		−	7.63521i		−	0.463806i	−0.972739	−	0.231903i	$-0.925505\pi$
$271$		−	7.63521i		−	0.463806i	0.972739	−	0.231903i	$-0.0744952\pi$
$272$		0			0
$273$	−8.43513	+	22.0834i	−0.510517	+	1.33655i
$274$		0			0
$275$	−1.00000			−0.0603023
$276$		0			0
$277$	16.4912			0.990860			0.495430	−	0.868648i	$-0.335010\pi$
$277$	16.4912			0.990860			0.495430	+	0.868648i	$0.335010\pi$
$278$		0			0
$279$	17.4081	−	19.4628i	1.04220	−	1.16521i
$280$		0			0
$281$			22.2112i			1.32501i	0.749058	+	0.662504i	$0.230506\pi$
$281$			22.2112i			1.32501i	−0.749058	+	0.662504i	$0.769494\pi$
$282$		0			0
$283$		−	9.78500i		−	0.581658i	−0.956775	−	0.290829i	$-0.906069\pi$
$283$		−	9.78500i		−	0.581658i	0.956775	−	0.290829i	$-0.0939311\pi$
$284$		0			0
$285$	6.77571	+	2.58809i	0.401359	+	0.153305i
$286$		0			0
$287$	12.8187			0.756663
$288$		0			0
$289$	0.421544			0.0247967
$290$		0			0
$291$	12.7040	+	4.85251i	0.744725	+	0.284459i
$292$		0			0
$293$			19.1510i			1.11881i	0.828894	+	0.559405i	$0.188970\pi$
$293$			19.1510i			1.11881i	−0.828894	+	0.559405i	$0.811030\pi$
$294$		0			0
$295$			0.532019i			0.0309753i
$296$		0			0
$297$	2.38197	+	4.61803i	0.138216	+	0.267966i
$298$		0			0
$299$	44.1669			2.55424
$300$		0			0
$301$	−23.3018			−1.34309
$302$		0			0
$303$	−7.76941	+	20.3406i	−0.446341	+	1.16854i
$304$		0			0
$305$			0.704049i			0.0403137i
$306$		0			0
$307$			13.7200i			0.783041i	0.920169	+	0.391520i	$0.128051\pi$
$307$			13.7200i			0.783041i	−0.920169	+	0.391520i	$0.871949\pi$
$308$		0			0
$309$	5.52786	−	14.4721i	0.314469	−	0.823291i
$310$		0			0
$311$	13.3565			0.757379			0.378690	−	0.925524i	$-0.376375\pi$
$311$	13.3565			0.757379			0.378690	+	0.925524i	$0.376375\pi$
$312$		0			0
$313$	−14.7327			−0.832744			−0.416372	−	0.909194i	$-0.636699\pi$
$313$	−14.7327			−0.832744			−0.416372	+	0.909194i	$0.636699\pi$
$314$		0			0
$315$	−4.63238	−	4.14333i	−0.261005	−	0.233450i
$316$		0			0
$317$		−	3.14749i		−	0.176780i	−0.996086	−	0.0883902i	$-0.971828\pi$
$317$		−	3.14749i		−	0.176780i	0.996086	−	0.0883902i	$-0.0281722\pi$
$318$		0			0
$319$		−	2.18762i		−	0.122483i
$320$		0			0
$321$	−4.34440	−	1.65941i	−0.242481	−	0.0926194i
$322$		0			0
$323$	17.0506			0.948720
$324$		0			0
$325$	−6.58809			−0.365442
$326$		0			0
$327$	−16.7875	−	6.41225i	−0.928351	−	0.354598i
$328$		0			0
$329$		−	20.5926i		−	1.13531i
$330$		0			0
$331$			7.08760i			0.389570i	0.980846	+	0.194785i	$0.0624009\pi$
$331$			7.08760i			0.389570i	−0.980846	+	0.194785i	$0.937599\pi$
$332$		0			0
$333$	−21.4915	−	19.2226i	−1.17773	−	1.05339i
$334$		0			0
$335$	4.14333			0.226374
$336$		0			0
$337$	23.5324			1.28189			0.640945	−	0.767587i	$-0.278543\pi$
$337$	23.5324			1.28189			0.640945	+	0.767587i	$0.278543\pi$
$338$		0			0
$339$	−8.41739	+	22.0370i	−0.457170	+	1.19689i
$340$		0			0
$341$		−	8.70405i		−	0.471351i
$342$		0			0
$343$			20.1121i			1.08595i
$344$		0			0
$345$	−4.14333	+	10.8474i	−0.223069	+	0.584003i
$346$		0			0
$347$	−7.46665			−0.400831			−0.200416	−	0.979711i	$-0.564229\pi$
$347$	−7.46665			−0.400831			−0.200416	+	0.979711i	$0.564229\pi$
$348$		0			0
$349$	0.135020			0.00722746			0.00361373	−	0.999993i	$-0.498850\pi$
$349$	0.135020			0.00722746			0.00361373	+	0.999993i	$0.498850\pi$
$350$		0			0
$351$	15.6926	+	30.4240i	0.837610	+	1.62391i
$352$		0			0
$353$			6.13390i			0.326475i	0.986587	+	0.163237i	$0.0521936\pi$
$353$			6.13390i			0.326475i	−0.986587	+	0.163237i	$0.947806\pi$
$354$		0			0
$355$			5.37940i			0.285509i
$356$		0			0
$357$	−13.6483	−	5.21319i	−0.722346	−	0.275912i
$358$		0			0
$359$	−19.9350			−1.05213			−0.526064	−	0.850445i	$-0.676333\pi$
$359$	−19.9350			−1.05213			−0.526064	+	0.850445i	$0.676333\pi$
$360$		0			0
$361$	1.46383			0.0770435
$362$		0			0
$363$	1.61803	+	0.618034i	0.0849248	+	0.0324384i
$364$		0			0
$365$		−	4.25929i		−	0.222941i
$366$		0			0
$367$			0.0286998i			0.00149812i	1.00000		0.000749058i	$0.000238433\pi$
$367$			0.0286998i			0.00149812i	−1.00000		0.000749058i	$0.999762\pi$
$368$		0			0
$369$	12.3752	−	13.8359i	0.644229	−	0.720270i
$370$		0			0
$371$	−12.3059			−0.638892
$372$		0			0
$373$	7.70953			0.399184			0.199592	−	0.979879i	$-0.436038\pi$
$373$	7.70953			0.399184			0.199592	+	0.979879i	$0.436038\pi$
$374$		0			0
$375$	0.618034	−	1.61803i	0.0319151	−	0.0835549i
$376$		0			0
$377$		−	14.4123i		−	0.742269i
$378$		0			0
$379$		−	15.8802i		−	0.815713i	−0.913046	−	0.407856i	$-0.866276\pi$
$379$		−	15.8802i		−	0.815713i	0.913046	−	0.407856i	$-0.133724\pi$
$380$		0			0
$381$	−6.22463	+	16.2963i	−0.318898	+	0.834885i
$382$		0			0
$383$	−3.88023			−0.198271			−0.0991353	−	0.995074i	$-0.531608\pi$
$383$	−3.88023			−0.198271			−0.0991353	+	0.995074i	$0.531608\pi$
$384$		0			0
$385$	−2.07167			−0.105582
$386$		0			0
$387$	−22.4957	+	25.1510i	−1.14352	+	1.27849i
$388$		0			0
$389$		−	14.3836i		−	0.729275i	−0.931150	−	0.364638i	$-0.881193\pi$
$389$		−	14.3836i		−	0.729275i	0.931150	−	0.364638i	$-0.118807\pi$
$390$		0			0
$391$			27.2966i			1.38045i
$392$		0			0
$393$	4.23191	+	1.61645i	0.213472	+	0.0815390i
$394$		0			0
$395$	11.9557			0.601557
$396$		0			0
$397$	13.0927			0.657106			0.328553	−	0.944486i	$-0.393439\pi$
$397$	13.0927			0.657106			0.328553	+	0.944486i	$0.393439\pi$
$398$		0			0
$399$	14.0370	+	5.36166i	0.702729	+	0.268419i
$400$		0			0
$401$		−	11.8145i		−	0.589989i	−0.955499	−	0.294995i	$-0.904682\pi$
$401$		−	11.8145i		−	0.589989i	0.955499	−	0.294995i	$-0.0953179\pi$
$402$		0			0
$403$		−	57.3431i		−	2.85646i
$404$		0			0
$405$	−8.94427	+	1.00000i	−0.444444	+	0.0496904i
$406$		0			0
$407$	−9.61131			−0.476415
$408$		0			0
$409$	23.8885			1.18121			0.590606	−	0.806960i	$-0.298889\pi$
$409$	23.8885			1.18121			0.590606	+	0.806960i	$0.298889\pi$
$410$		0			0
$411$	−11.9949	+	31.4030i	−0.591663	+	1.54899i
$412$		0			0
$413$			1.10217i			0.0542340i
$414$		0			0
$415$		−	2.25929i		−	0.110904i
$416$		0			0
$417$	4.97165	−	13.0159i	0.243463	−	0.637393i
$418$		0			0
$419$	−28.7276			−1.40344			−0.701718	−	0.712455i	$-0.747583\pi$
$419$	−28.7276			−1.40344			−0.701718	+	0.712455i	$0.747583\pi$
$420$		0			0
$421$	−17.5967			−0.857613			−0.428807	−	0.903396i	$-0.641066\pi$
$421$	−17.5967			−0.857613			−0.428807	+	0.903396i	$0.641066\pi$
$422$		0			0
$423$	−22.2268	−	19.8802i	−1.08070	−	0.966610i
$424$		0			0
$425$		−	4.07167i		−	0.197505i
$426$		0			0
$427$			1.45855i			0.0705844i
$428$		0			0
$429$	10.6598	+	4.07167i	0.514658	+	0.196582i
$430$		0			0
$431$	−4.37524			−0.210748			−0.105374	−	0.994433i	$-0.533604\pi$
$431$	−4.37524			−0.210748			−0.105374	+	0.994433i	$0.533604\pi$
$432$		0			0
$433$	6.21418			0.298634			0.149317	−	0.988789i	$-0.452292\pi$
$433$	6.21418			0.298634			0.149317	+	0.988789i	$0.452292\pi$
$434$		0			0
$435$	3.53965	+	1.35202i	0.169713	+	0.0648246i
$436$		0			0
$437$		−	28.0740i		−	1.34296i
$438$		0			0
$439$		−	11.4753i		−	0.547684i	−0.961775	−	0.273842i	$-0.911705\pi$
$439$		−	11.4753i		−	0.547684i	0.961775	−	0.273842i	$-0.0882946\pi$
$440$		0			0
$441$	6.05573	+	5.41641i	0.288368	+	0.257924i
$442$		0			0
$443$	35.9897			1.70992			0.854962	−	0.518691i	$-0.173580\pi$
$443$	35.9897			1.70992			0.854962	+	0.518691i	$0.173580\pi$
$444$		0			0
$445$	−17.5050			−0.829816
$446$		0			0
$447$	11.1118	−	29.0911i	0.525570	−	1.37596i
$448$		0			0
$449$		−	15.9771i		−	0.754007i	−0.926212	−	0.377004i	$-0.876954\pi$
$449$		−	15.9771i		−	0.754007i	0.926212	−	0.377004i	$-0.123046\pi$
$450$		0			0
$451$		−	6.18762i		−	0.291364i
$452$		0			0
$453$	8.02737	−	21.0159i	0.377159	−	0.987415i
$454$		0			0
$455$	−13.6483			−0.639843
$456$		0			0
$457$	−31.4819			−1.47266			−0.736331	−	0.676622i	$-0.763443\pi$
$457$	−31.4819			−1.47266			−0.736331	+	0.676622i	$0.763443\pi$
$458$		0			0
$459$	−18.8031	+	9.69857i	−0.877653	+	0.452690i
$460$		0			0
$461$		−	31.1007i		−	1.44850i	−0.689535	−	0.724252i	$-0.742185\pi$
$461$		−	31.1007i		−	1.44850i	0.689535	−	0.724252i	$-0.257815\pi$
$462$		0			0
$463$			23.8515i			1.10847i	0.832359	+	0.554237i	$0.186990\pi$
$463$			23.8515i			1.10847i	−0.832359	+	0.554237i	$0.813010\pi$
$464$		0			0
$465$	14.0834	+	5.37940i	0.653104	+	0.249464i
$466$		0			0
$467$	30.5843			1.41527			0.707636	−	0.706577i	$-0.249762\pi$
$467$	30.5843			1.41527			0.707636	+	0.706577i	$0.249762\pi$
$468$		0			0
$469$	8.58359			0.396353
$470$		0			0
$471$	35.0277	+	13.3794i	1.61399	+	0.616490i
$472$		0			0
$473$			11.2478i			0.517177i
$474$		0			0
$475$			4.18762i			0.192141i
$476$		0			0
$477$	−11.8802	+	13.2825i	−0.543959	+	0.608164i
$478$		0			0
$479$	12.8895			0.588937			0.294469	−	0.955661i	$-0.404857\pi$
$479$	12.8895			0.588937			0.294469	+	0.955661i	$0.404857\pi$
$480$		0			0
$481$	−63.3202			−2.88715
$482$		0			0
$483$	−8.58359	+	22.4721i	−0.390567	+	1.02252i
$484$		0			0
$485$			7.85153i			0.356520i
$486$		0			0
$487$			23.5597i			1.06759i	0.845613	+	0.533797i	$0.179235\pi$
$487$			23.5597i			1.06759i	−0.845613	+	0.533797i	$0.820765\pi$
$488$		0			0
$489$	7.41641	−	19.4164i	0.335382	−	0.878040i
$490$		0			0
$491$	6.93596			0.313016			0.156508	−	0.987677i	$-0.449976\pi$
$491$	6.93596			0.313016			0.156508	+	0.987677i	$0.449976\pi$
$492$		0			0
$493$	8.90726			0.401163
$494$		0			0
$495$	−2.00000	+	2.23607i	−0.0898933	+	0.100504i
$496$		0			0
$497$			11.1443i			0.499891i
$498$		0			0
$499$		−	0.282369i		−	0.0126406i	−0.999980	−	0.00632028i	$-0.997988\pi$
$499$		−	0.282369i		−	0.0126406i	0.999980	−	0.00632028i	$-0.00201182\pi$
$500$		0			0
$501$	−41.3122	−	15.7799i	−1.84569	−	0.704993i
$502$		0			0
$503$	41.1952			1.83681			0.918403	−	0.395647i	$-0.129480\pi$
$503$	41.1952			1.83681			0.918403	+	0.395647i	$0.129480\pi$
$504$		0			0
$505$	−12.5712			−0.559410
$506$		0			0
$507$	49.1930	+	18.7901i	2.18474	+	0.834496i
$508$		0			0
$509$			26.8162i			1.18861i	0.804241	+	0.594303i	$0.202572\pi$
$509$			26.8162i			1.18861i	−0.804241	+	0.594303i	$0.797428\pi$
$510$		0			0
$511$		−	8.82382i		−	0.390343i
$512$		0			0
$513$	19.3386	−	9.97477i	0.853819	−	0.440397i
$514$		0			0
$515$	8.94427			0.394132
$516$		0			0
$517$	−9.94012			−0.437166
$518$		0			0
$519$	7.15477	−	18.7314i	0.314059	−	0.822218i
$520$		0			0
$521$		−	17.3998i		−	0.762299i	−0.924514	−	0.381149i	$-0.875528\pi$
$521$		−	17.3998i		−	0.762299i	0.924514	−	0.381149i	$-0.124472\pi$
$522$		0			0
$523$		−	30.1838i		−	1.31985i	−0.751333	−	0.659923i	$-0.770589\pi$
$523$		−	30.1838i		−	1.31985i	0.751333	−	0.659923i	$-0.229411\pi$
$524$		0			0
$525$	1.28036	−	3.35202i	0.0558795	−	0.146294i
$526$		0			0
$527$	35.4400			1.54379
$528$		0			0
$529$	21.9443			0.954099
$530$		0			0
$531$	1.18963	+	1.06404i	0.0516256	+	0.0461753i
$532$		0			0
$533$		−	40.7646i		−	1.76571i
$534$		0			0
$535$		−	2.68499i		−	0.116082i
$536$		0			0
$537$	22.3154	+	8.52371i	0.962979	+	0.367825i
$538$		0			0
$539$	2.70820			0.116651
$540$		0			0
$541$	−9.79165			−0.420976			−0.210488	−	0.977596i	$-0.567505\pi$
$541$	−9.79165			−0.420976			−0.210488	+	0.977596i	$0.567505\pi$
$542$		0			0
$543$	−38.8245	−	14.8296i	−1.66612	−	0.636401i
$544$		0			0
$545$		−	10.3752i		−	0.444427i
$546$		0			0
$547$		−	20.1686i		−	0.862345i	−0.902269	−	0.431173i	$-0.858100\pi$
$547$		−	20.1686i		−	0.862345i	0.902269	−	0.431173i	$-0.141900\pi$
$548$		0			0
$549$	1.57430	+	1.40810i	0.0671895	+	0.0600962i
$550$		0			0
$551$	−9.16093			−0.390269
$552$		0			0
$553$	24.7682			1.05325
$554$		0			0
$555$	5.94012	−	15.5514i	0.252144	−	0.660121i
$556$		0			0
$557$		−	31.5262i		−	1.33581i	−0.744247	−	0.667904i	$-0.767192\pi$
$557$		−	31.5262i		−	1.33581i	0.744247	−	0.667904i	$-0.232808\pi$
$558$		0			0
$559$			74.1019i			3.13418i
$560$		0			0
$561$	−2.51643	+	6.58809i	−0.106244	+	0.278149i
$562$		0			0
$563$	14.1160			0.594917			0.297458	−	0.954735i	$-0.403861\pi$
$563$	14.1160			0.594917			0.297458	+	0.954735i	$0.403861\pi$
$564$		0			0
$565$	−13.6196			−0.572982
$566$		0			0
$567$	−18.5295	+	2.07167i	−0.778167	+	0.0870018i
$568$		0			0
$569$		−	16.2195i		−	0.679957i	−0.940433	−	0.339978i	$-0.889580\pi$
$569$		−	16.2195i		−	0.679957i	0.940433	−	0.339978i	$-0.110420\pi$
$570$		0			0
$571$		−	1.50714i		−	0.0630717i	−0.999503	−	0.0315358i	$-0.989960\pi$
$571$		−	1.50714i		−	0.0630717i	0.999503	−	0.0315358i	$-0.0100398\pi$
$572$		0			0
$573$	0.907262	+	0.346543i	0.0379014	+	0.0144771i
$574$		0			0
$575$	−6.70405			−0.279578
$576$		0			0
$577$	−0.646649			−0.0269204			−0.0134602	−	0.999909i	$-0.504285\pi$
$577$	−0.646649			−0.0269204			−0.0134602	+	0.999909i	$0.504285\pi$
$578$		0			0
$579$	−8.52474	−	3.25616i	−0.354276	−	0.135321i
$580$		0			0
$581$		−	4.68049i		−	0.194179i
$582$		0			0
$583$			5.94012i			0.246015i
$584$		0			0
$585$	−13.1762	+	14.7314i	−0.544768	+	0.609069i
$586$		0			0
$587$	14.9907			0.618733			0.309366	−	0.950943i	$-0.399883\pi$
$587$	14.9907			0.618733			0.309366	+	0.950943i	$0.399883\pi$
$588$		0			0
$589$	−36.4493			−1.50187
$590$		0			0
$591$	13.8041	−	36.1395i	0.567824	−	1.48658i
$592$		0			0
$593$			23.4715i			0.963857i	0.876211	+	0.481929i	$0.160064\pi$
$593$			23.4715i			0.963857i	−0.876211	+	0.481929i	$0.839936\pi$
$594$		0			0
$595$		−	8.43513i		−	0.345806i
$596$		0			0
$597$	−0.231913	+	0.607156i	−0.00949157	+	0.0248492i
$598$		0			0
$599$	−20.0000			−0.817178			−0.408589	−	0.912719i	$-0.633979\pi$
$599$	−20.0000			−0.817178			−0.408589	+	0.912719i	$0.633979\pi$
$600$		0			0
$601$	−17.5597			−0.716277			−0.358138	−	0.933669i	$-0.616588\pi$
$601$	−17.5597			−0.716277			−0.358138	+	0.933669i	$0.616588\pi$
$602$		0			0
$603$	8.28666	−	9.26477i	0.337459	−	0.377291i
$604$		0			0
$605$			1.00000i			0.0406558i
$606$		0			0
$607$		−	7.19310i		−	0.291959i	−0.989288	−	0.145980i	$-0.953367\pi$
$607$		−	7.19310i		−	0.291959i	0.989288	−	0.145980i	$-0.0466334\pi$
$608$		0			0
$609$	7.33296	+	2.80094i	0.297147	+	0.113500i
$610$		0			0
$611$	−65.4864			−2.64930
$612$		0			0
$613$	−25.2583			−1.02017			−0.510087	−	0.860123i	$-0.670387\pi$
$613$	−25.2583			−1.02017			−0.510087	+	0.860123i	$0.670387\pi$
$614$		0			0
$615$	10.0118	+	3.82416i	0.403714	+	0.154205i
$616$		0			0
$617$			30.8339i			1.24133i	0.784077	+	0.620664i	$0.213137\pi$
$617$			30.8339i			1.24133i	−0.784077	+	0.620664i	$0.786863\pi$
$618$		0			0
$619$			35.4483i			1.42479i	0.701780	+	0.712393i	$0.252389\pi$
$619$			35.4483i			1.42479i	−0.701780	+	0.712393i	$0.747611\pi$
$620$		0			0
$621$	15.9688	+	30.9595i	0.640807	+	1.24236i
$622$		0			0
$623$	−36.2645			−1.45291
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	2.58809	−	6.77571i	0.103358	−	0.270596i
$628$		0			0
$629$		−	39.1340i		−	1.56038i
$630$		0			0
$631$		−	44.2555i		−	1.76178i	−0.473319	−	0.880891i	$-0.656944\pi$
$631$		−	44.2555i		−	1.76178i	0.473319	−	0.880891i	$-0.343056\pi$
$632$		0			0
$633$	13.6986	−	35.8633i	0.544469	−	1.42544i
$634$		0			0
$635$	−10.0717			−0.399682
$636$		0			0
$637$	17.8419			0.706922
$638$		0			0
$639$	12.0287	+	10.7588i	0.475848	+	0.425611i
$640$		0			0
$641$			38.6162i			1.52525i	0.646843	+	0.762623i	$0.276089\pi$
$641$			38.6162i			1.52525i	−0.646843	+	0.762623i	$0.723911\pi$
$642$		0			0
$643$		−	3.53617i		−	0.139453i	−0.997566	−	0.0697265i	$-0.977787\pi$
$643$		−	3.53617i		−	0.139453i	0.997566	−	0.0697265i	$-0.0222127\pi$
$644$		0			0
$645$	−18.1994	−	6.95155i	−0.716601	−	0.273717i
$646$		0			0
$647$	32.8245			1.29046			0.645232	−	0.763986i	$-0.276761\pi$
$647$	32.8245			1.29046			0.645232	+	0.763986i	$0.276761\pi$
$648$		0			0
$649$	0.532019			0.0208836
$650$		0			0
$651$	29.1762	+	11.1443i	1.14350	+	0.436780i
$652$		0			0
$653$		−	12.1516i		−	0.475530i	−0.971323	−	0.237765i	$-0.923585\pi$
$653$		−	12.1516i		−	0.475530i	0.971323	−	0.237765i	$-0.0764149\pi$
$654$		0			0
$655$			2.61547i			0.102195i
$656$		0			0
$657$	−9.52405	−	8.51857i	−0.371569	−	0.332341i
$658$		0			0
$659$	15.5936			0.607439			0.303720	−	0.952761i	$-0.401771\pi$
$659$	15.5936			0.607439			0.303720	+	0.952761i	$0.401771\pi$
$660$		0			0
$661$	−9.35901			−0.364023			−0.182012	−	0.983296i	$-0.558261\pi$
$661$	−9.35901			−0.364023			−0.182012	+	0.983296i	$0.558261\pi$
$662$		0			0
$663$	−16.5785	+	43.4030i	−0.643854	+	1.68563i
$664$		0			0
$665$			8.67535i			0.336416i
$666$		0			0
$667$		−	14.6659i		−	0.567867i
$668$		0			0
$669$	8.55558	−	22.3988i	0.330778	−	0.865988i
$670$		0			0
$671$	0.704049			0.0271795
$672$		0			0
$673$	−34.0426			−1.31225			−0.656123	−	0.754654i	$-0.727805\pi$
$673$	−34.0426			−1.31225			−0.656123	+	0.754654i	$0.727805\pi$
$674$		0			0
$675$	−2.38197	−	4.61803i	−0.0916819	−	0.177748i
$676$		0			0
$677$		−	33.1447i		−	1.27385i	−0.770924	−	0.636927i	$-0.780205\pi$
$677$		−	33.1447i		−	1.27385i	0.770924	−	0.636927i	$-0.219795\pi$
$678$		0			0
$679$			16.2657i			0.624222i
$680$		0			0
$681$	−41.7761	−	15.9570i	−1.60086	−	0.611475i
$682$		0			0
$683$	−34.0330			−1.30224			−0.651118	−	0.758976i	$-0.725700\pi$
$683$	−34.0330			−1.30224			−0.651118	+	0.758976i	$0.725700\pi$
$684$		0			0
$685$	−19.4081			−0.741545
$686$		0			0
$687$	−12.7505	−	4.87025i	−0.486461	−	0.185812i
$688$		0			0
$689$			39.1340i			1.49089i
$690$		0			0
$691$			15.2226i			0.579096i	0.957164	+	0.289548i	$0.0935050\pi$
$691$			15.2226i			0.579096i	−0.957164	+	0.289548i	$0.906495\pi$
$692$		0			0
$693$	−4.14333	+	4.63238i	−0.157392	+	0.175970i
$694$		0			0
$695$	8.04429			0.305137
$696$		0			0
$697$	25.1939			0.954288
$698$		0			0
$699$	3.06618	−	8.02737i	0.115974	−	0.303623i
$700$		0			0
$701$		−	9.90096i		−	0.373954i	−0.982364	−	0.186977i	$-0.940131\pi$
$701$		−	9.90096i		−	0.373954i	0.982364	−	0.186977i	$-0.0598690\pi$
$702$		0			0
$703$			40.2485i			1.51800i
$704$		0			0
$705$	6.14333	−	16.0834i	0.231371	−	0.605738i
$706$		0			0
$707$	−26.0433			−0.979458
$708$		0			0
$709$	41.4687			1.55739			0.778694	−	0.627403i	$-0.215882\pi$
$709$	41.4687			1.55739			0.778694	+	0.627403i	$0.215882\pi$
$710$		0			0
$711$	23.9114	−	26.7338i	0.896748	−	1.00259i
$712$		0			0
$713$		−	58.3524i		−	2.18531i
$714$		0			0
$715$			6.58809i			0.246381i
$716$		0			0
$717$	0.231913	+	0.0885829i	0.00866095	+	0.00330819i
$718$		0			0
$719$	−18.1626			−0.677351			−0.338675	−	0.940903i	$-0.609979\pi$
$719$	−18.1626			−0.677351			−0.338675	+	0.940903i	$0.609979\pi$
$720$		0			0
$721$	18.5295			0.690076
$722$		0			0
$723$	−17.6196	−	6.73010i	−0.655281	−	0.250295i
$724$		0			0
$725$			2.18762i			0.0812462i
$726$		0			0
$727$		−	45.4030i		−	1.68390i	−0.539553	−	0.841951i	$-0.681407\pi$
$727$		−	45.4030i		−	1.68390i	0.539553	−	0.841951i	$-0.318593\pi$
$728$		0			0
$729$	−15.6525	+	22.0000i	−0.579721	+	0.814815i
$730$		0			0
$731$	−45.7975			−1.69388
$732$		0			0
$733$	14.7964			0.546519			0.273260	−	0.961940i	$-0.411898\pi$
$733$	14.7964			0.546519			0.273260	+	0.961940i	$0.411898\pi$
$734$		0			0
$735$	−1.67376	+	4.38197i	−0.0617376	+	0.161631i
$736$		0			0
$737$		−	4.14333i		−	0.152622i
$738$		0			0
$739$		−	47.3071i		−	1.74022i	−0.492858	−	0.870110i	$-0.664048\pi$
$739$		−	47.3071i		−	1.74022i	0.492858	−	0.870110i	$-0.335952\pi$
$740$		0			0
$741$	17.0506	−	44.6390i	0.626369	−	1.63986i
$742$		0			0
$743$	34.1083			1.25131			0.625657	−	0.780098i	$-0.284831\pi$
$743$	34.1083			1.25131			0.625657	+	0.780098i	$0.284831\pi$
$744$		0			0
$745$	17.9793			0.658710
$746$		0			0
$747$	−5.05192	−	4.51857i	−0.184840	−	0.165326i
$748$		0			0
$749$		−	5.56239i		−	0.203245i
$750$		0			0
$751$			37.9771i			1.38581i	0.721031	+	0.692903i	$0.243668\pi$
$751$			37.9771i			1.38581i	−0.721031	+	0.692903i	$0.756332\pi$
$752$		0			0
$753$	−18.1121	−	6.91822i	−0.660043	−	0.252114i
$754$		0			0
$755$	12.9886			0.472702
$756$		0			0
$757$	−11.4628			−0.416624			−0.208312	−	0.978062i	$-0.566797\pi$
$757$	−11.4628			−0.416624			−0.208312	+	0.978062i	$0.566797\pi$
$758$		0			0
$759$	10.8474	+	4.14333i	0.393735	+	0.150393i
$760$		0			0
$761$			7.16573i			0.259757i	0.991530	+	0.129879i	$0.0414588\pi$
$761$			7.16573i			0.259757i	−0.991530	+	0.129879i	$0.958541\pi$
$762$		0			0
$763$		−	21.4940i		−	0.778136i
$764$		0			0
$765$	−9.10452	−	8.14333i	−0.329175	−	0.294423i
$766$		0			0
$767$	3.50499			0.126558
$768$		0			0
$769$	9.18449			0.331201			0.165601	−	0.986193i	$-0.447044\pi$
$769$	9.18449			0.331201			0.165601	+	0.986193i	$0.447044\pi$
$770$		0			0
$771$	2.01774	−	5.28251i	0.0726670	−	0.190245i
$772$		0			0
$773$		−	31.9121i		−	1.14780i	−0.818926	−	0.573899i	$-0.805430\pi$
$773$		−	31.9121i		−	1.14780i	0.818926	−	0.573899i	$-0.194570\pi$
$774$		0			0
$775$			8.70405i			0.312659i
$776$		0			0
$777$	12.3059	−	32.2173i	0.441473	−	1.15579i
$778$		0			0
$779$	−25.9114			−0.928373
$780$		0			0
$781$	5.37940			0.192490
$782$		0			0
$783$	10.1025	−	5.21084i	0.361034	−	0.186220i
$784$		0			0
$785$			21.6483i			0.772662i
$786$		0			0
$787$		−	32.9931i		−	1.17608i	−0.808833	−	0.588038i	$-0.799901\pi$
$787$		−	32.9931i		−	1.17608i	0.808833	−	0.588038i	$-0.200099\pi$
$788$		0			0
$789$	−19.0720	−	7.28486i	−0.678982	−	0.259348i
$790$		0			0
$791$	−28.2153			−1.00322
$792$		0			0
$793$	4.63834			0.164712
$794$		0			0
$795$	−9.61131	−	3.67119i	−0.340878	−	0.130204i
$796$		0			0
$797$		−	35.8522i		−	1.26995i	−0.772532	−	0.634975i	$-0.781010\pi$
$797$		−	35.8522i		−	1.26995i	0.772532	−	0.634975i	$-0.218990\pi$
$798$		0			0
$799$		−	40.4728i		−	1.43183i
$800$		0			0
$801$	−35.0100	+	39.1423i	−1.23702	+	1.38303i
$802$		0			0
$803$	−4.25929			−0.150307
$804$		0			0
$805$	−13.8885			−0.489507
$806$		0			0
$807$	0.657612	−	1.72165i	0.0231490	−	0.0606049i
$808$		0			0
$809$			13.3319i			0.468726i	0.972149	+	0.234363i	$0.0753004\pi$
$809$			13.3319i			0.468726i	−0.972149	+	0.234363i	$0.924700\pi$
$810$		0			0
$811$		−	20.6681i		−	0.725754i	−0.931837	−	0.362877i	$-0.881795\pi$
$811$		−	20.6681i		−	0.725754i	0.931837	−	0.362877i	$-0.118205\pi$
$812$		0			0
$813$	4.71882	−	12.3540i	0.165496	−	0.433275i
$814$		0			0
$815$	12.0000			0.420342
$816$		0			0
$817$	47.1017			1.64788
$818$		0			0
$819$	−27.2966	+	30.5186i	−0.953822	+	1.06641i
$820$		0			0
$821$		−	18.1876i		−	0.634752i	−0.948300	−	0.317376i	$-0.897198\pi$
$821$		−	18.1876i		−	0.634752i	0.948300	−	0.317376i	$-0.102802\pi$
$822$		0			0
$823$			16.4638i			0.573893i	0.957947	+	0.286946i	$0.0926402\pi$
$823$			16.4638i			0.573893i	−0.957947	+	0.286946i	$0.907360\pi$
$824$		0			0
$825$	−1.61803	−	0.618034i	−0.0563327	−	0.0215172i
$826$		0			0
$827$	−50.6435			−1.76105			−0.880524	−	0.474001i	$-0.842809\pi$
$827$	−50.6435			−1.76105			−0.880524	+	0.474001i	$0.842809\pi$
$828$		0			0
$829$	−35.9866			−1.24986			−0.624932	−	0.780679i	$-0.714873\pi$
$829$	−35.9866			−1.24986			−0.624932	+	0.780679i	$0.714873\pi$
$830$		0			0
$831$	26.6833	+	10.1921i	0.925634	+	0.353561i
$832$		0			0
$833$			11.0269i			0.382059i
$834$		0			0
$835$		−	25.5324i		−	0.883584i
$836$		0			0
$837$	40.1956	−	20.7327i	1.38936	−	0.716629i
$838$		0			0
$839$	44.4020			1.53293			0.766463	−	0.642288i	$-0.222015\pi$
$839$	44.4020			1.53293			0.766463	+	0.642288i	$0.222015\pi$
$840$		0			0
$841$	24.2143			0.834976
$842$		0			0
$843$	−13.7273	+	35.9385i	−0.472792	+	1.23779i
$844$		0			0
$845$			30.4030i			1.04589i
$846$		0			0
$847$			2.07167i			0.0711833i
$848$		0			0
$849$	6.04747	−	15.8325i	0.207548	−	0.543369i
$850$		0			0
$851$	−64.4347			−2.20879
$852$		0			0
$853$	−30.4136			−1.04134			−0.520671	−	0.853758i	$-0.674318\pi$
$853$	−30.4136			−1.04134			−0.520671	+	0.853758i	$0.674318\pi$
$854$		0			0
$855$	9.36381	+	8.37524i	0.320235	+	0.286427i
$856$		0			0
$857$			17.7976i			0.607955i	0.952679	+	0.303977i	$0.0983147\pi$
$857$			17.7976i			0.607955i	−0.952679	+	0.303977i	$0.901685\pi$
$858$		0			0
$859$		−	26.3102i		−	0.897693i	−0.893609	−	0.448847i	$-0.851835\pi$
$859$		−	26.3102i		−	0.897693i	0.893609	−	0.448847i	$-0.148165\pi$
$860$		0			0
$861$	20.7411	+	7.92238i	0.706853	+	0.269994i
$862$		0			0
$863$	31.6864			1.07862			0.539310	−	0.842108i	$-0.318685\pi$
$863$	31.6864			1.07862			0.539310	+	0.842108i	$0.318685\pi$
$864$		0			0
$865$	11.5767			0.393618
$866$		0			0
$867$	0.682072	+	0.260528i	0.0231644	+	0.00884801i
$868$		0			0
$869$		−	11.9557i		−	0.405570i
$870$		0			0
$871$		−	27.2966i		−	0.924911i
$872$		0			0
$873$	17.5566	+	15.7031i	0.594200	+	0.531468i
$874$		0			0
$875$	2.07167			0.0700351
$876$		0			0
$877$	52.9259			1.78718			0.893590	−	0.448883i	$-0.148178\pi$
$877$	52.9259			1.78718			0.893590	+	0.448883i	$0.148178\pi$
$878$		0			0
$879$	−11.8359	+	30.9869i	−0.399216	+	1.04516i
$880$		0			0
$881$			11.0412i			0.371986i	0.982551	+	0.185993i	$0.0595502\pi$
$881$			11.0412i			0.371986i	−0.982551	+	0.185993i	$0.940450\pi$
$882$		0			0
$883$			21.7739i			0.732751i	0.930467	+	0.366375i	$0.119401\pi$
$883$			21.7739i			0.732751i	−0.930467	+	0.366375i	$0.880599\pi$
$884$		0			0
$885$	−0.328806	+	0.860825i	−0.0110527	+	0.0289363i
$886$		0			0
$887$	−7.71784			−0.259140			−0.129570	−	0.991570i	$-0.541360\pi$
$887$	−7.71784			−0.259140			−0.129570	+	0.991570i	$0.541360\pi$
$888$		0			0
$889$	−20.8651			−0.699794
$890$		0			0
$891$	1.00000	+	8.94427i	0.0335013	+	0.299644i
$892$		0			0
$893$			41.6254i			1.39294i
$894$		0			0
$895$			13.7917i			0.461004i
$896$		0			0
$897$	71.4635	+	27.2966i	2.38610	+	0.911408i
$898$		0			0
$899$	−19.0412			−0.635058
$900$		0			0
$901$	−24.1862			−0.805758
$902$		0			0
$903$	−37.7031	−	14.4013i	−1.25468	−	0.479245i
$904$		0			0
$905$		−	23.9949i		−	0.797616i
$906$		0			0
$907$			15.2513i			0.506412i	0.967412	+	0.253206i	$0.0814850\pi$
$907$			15.2513i			0.506412i	−0.967412	+	0.253206i	$0.918515\pi$
$908$		0			0
$909$	−25.1423	+	28.1100i	−0.833919	+	0.932350i
$910$		0			0
$911$	24.7442			0.819813			0.409907	−	0.912128i	$-0.365561\pi$
$911$	24.7442			0.819813			0.409907	+	0.912128i	$0.365561\pi$
$912$		0			0
$913$	−2.25929			−0.0747715
$914$		0			0
$915$	−0.435126	+	1.13918i	−0.0143848	+	0.0376600i
$916$		0			0
$917$			5.41837i			0.178930i
$918$		0			0
$919$		−	20.0367i		−	0.660949i	−0.943815	−	0.330474i	$-0.892791\pi$
$919$		−	20.0367i		−	0.660949i	0.943815	−	0.330474i	$-0.107209\pi$
$920$		0			0
$921$	−8.47942	+	22.1994i	−0.279406	+	0.731495i
$922$		0			0
$923$	35.4400			1.16652
$924$		0			0
$925$	9.61131			0.316018
$926$		0			0
$927$	17.8885	−	20.0000i	0.587537	−	0.656886i
$928$		0			0
$929$		−	19.2183i		−	0.630533i	−0.949003	−	0.315266i	$-0.897906\pi$
$929$		−	19.2183i		−	0.630533i	0.949003	−	0.315266i	$-0.102094\pi$
$930$		0			0
$931$		−	11.3409i		−	0.371684i
$932$		0			0
$933$	21.6113	+	8.25479i	0.707523	+	0.270250i
$934$		0			0
$935$	−4.07167			−0.133158
$936$		0			0
$937$	−49.9083			−1.63043			−0.815217	−	0.579156i	$-0.803382\pi$
$937$	−49.9083			−1.63043			−0.815217	+	0.579156i	$0.803382\pi$
$938$		0			0
$939$	−23.8381	−	9.10534i	−0.777927	−	0.297142i
$940$		0			0
$941$			4.08644i			0.133214i	0.997779	+	0.0666070i	$0.0212174\pi$
$941$			4.08644i			0.133214i	−0.997779	+	0.0666070i	$0.978783\pi$
$942$		0			0
$943$		−	41.4821i		−	1.35084i
$944$		0			0
$945$	−4.93464	−	9.56702i	−0.160524	−	0.311215i
$946$		0			0
$947$	16.3091			0.529975			0.264987	−	0.964252i	$-0.414632\pi$
$947$	16.3091			0.529975			0.264987	+	0.964252i	$0.414632\pi$
$948$		0			0
$949$	−28.0606			−0.910885
$950$		0			0
$951$	1.94525	−	5.09274i	0.0630791	−	0.165143i
$952$		0			0
$953$			12.6407i			0.409472i	0.978817	+	0.204736i	$0.0656336\pi$
$953$			12.6407i			0.409472i	−0.978817	+	0.204736i	$0.934366\pi$
$954$		0			0
$955$			0.560719i			0.0181444i
$956$		0			0
$957$	1.35202	−	3.53965i	0.0437047	−	0.114421i
$958$		0			0
$959$	−40.2071			−1.29835
$960$		0			0
$961$	−44.7605			−1.44389
$962$		0			0
$963$	−6.00381	−	5.36997i	−0.193470	−	0.173045i
$964$		0			0
$965$		−	5.26858i		−	0.169602i
$966$		0			0
$967$			61.7290i			1.98507i	0.121959	+	0.992535i	$0.461082\pi$
$967$			61.7290i			1.98507i	−0.121959	+	0.992535i	$0.538918\pi$
$968$		0			0
$969$	27.5884	+	10.5378i	0.886268	+	0.338524i
$970$		0			0
$971$	26.9257			0.864087			0.432043	−	0.901853i	$-0.357793\pi$
$971$	26.9257			0.864087			0.432043	+	0.901853i	$0.357793\pi$
$972$		0			0
$973$	16.6651			0.534258
$974$		0			0
$975$	−10.6598	−	4.07167i	−0.341385	−	0.130398i
$976$		0			0
$977$		−	35.0017i		−	1.11980i	−0.828559	−	0.559901i	$-0.810839\pi$
$977$		−	35.0017i		−	1.11980i	0.828559	−	0.559901i	$-0.189161\pi$
$978$		0			0
$979$			17.5050i			0.559462i
$980$		0			0
$981$	−23.1997	−	20.7505i	−0.740711	−	0.662512i
$982$		0			0
$983$	−16.6473			−0.530968			−0.265484	−	0.964115i	$-0.585532\pi$
$983$	−16.6473			−0.530968			−0.265484	+	0.964115i	$0.585532\pi$
$984$		0			0
$985$	22.3355			0.711667
$986$		0			0
$987$	12.7269	−	33.3195i	0.405102	−	1.06057i
$988$		0			0
$989$			75.4061i			2.39778i
$990$		0			0
$991$			15.5826i			0.494998i	0.968888	+	0.247499i	$0.0796087\pi$
$991$			15.5826i			0.494998i	−0.968888	+	0.247499i	$0.920391\pi$
$992$		0			0
$993$	−4.38038	+	11.4680i	−0.139007	+	0.363925i
$994$		0			0
$995$	−0.375243			−0.0118960
$996$		0			0
$997$	−5.13219			−0.162538			−0.0812691	−	0.996692i	$-0.525897\pi$
$997$	−5.13219			−0.162538			−0.0812691	+	0.996692i	$0.525897\pi$
$998$		0			0
$999$	−22.8938	−	44.3854i	−0.724328	−	1.40429i

Display

a_p

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p

up to: 50 250 1000 (See

a_n

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a_n

instead) (See

a_n

instead) Display

a_n

with

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up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2640.2.k.e.1871.8	yes	8
3.2	odd	2		2640.2.k.c.1871.4	yes	8
4.3	odd	2		2640.2.k.c.1871.1	✓	8
12.11	even	2	inner	2640.2.k.e.1871.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2640.2.k.c.1871.1	✓	8	4.3	odd	2
2640.2.k.c.1871.4	yes	8	3.2	odd	2
2640.2.k.e.1871.5	yes	8	12.11	even	2	inner
2640.2.k.e.1871.8	yes	8	1.1	even	1	trivial

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2640.2.k.e.1871.8

Level	$2640$
Weight	$2$
Character	2640.1871
Analytic conductor	$21.081$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$