LMFDB - Embedded newform 1560.2.l.e.1249.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1560,2,Mod(1249,1560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1560, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1560.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1560.l (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:	$12.4566627153$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1698758656.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 $ x^8 + 18x^6 + 97x^4 + 176*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.5
Root			$-3.16053i$ of defining polynomial
Character	$\chi$	$=$	1560.1249
Dual form			1560.2.l.e.1249.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-1.52773 + 1.63280i) q^{5} -2.82843i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(-1.52773 + 1.63280i) q^{5} -2.82843i q^{7} -1.00000 q^{9} +1.05545 q^{11} -1.00000i q^{13} +(-1.63280 - 1.52773i) q^{15} +7.14949i q^{17} +6.61827 q^{19} +2.82843 q^{21} +3.49264i q^{23} +(-0.332104 - 4.98896i) q^{25} -1.00000i q^{27} -4.82843 q^{29} -9.65685 q^{31} +1.05545i q^{33} +(4.61827 + 4.32106i) q^{35} +8.11091i q^{37} +1.00000 q^{39} -3.26561 q^{41} +7.44670i q^{43} +(1.52773 - 1.63280i) q^{45} +11.3765i q^{47} -1.00000 q^{49} -7.14949 q^{51} -4.53122i q^{53} +(-1.61244 + 1.72335i) q^{55} +6.61827i q^{57} -1.05545 q^{59} -7.97792 q^{61} +2.82843i q^{63} +(1.63280 + 1.52773i) q^{65} +1.46878i q^{67} -3.49264 q^{69} +0.845296 q^{71} -9.28248i q^{73} +(4.98896 - 0.332104i) q^{75} -2.98527i q^{77} -6.85228 q^{79} +1.00000 q^{81} -7.97093i q^{83} +(-11.6737 - 10.9225i) q^{85} -4.82843i q^{87} -0.139976 q^{89} -2.82843 q^{91} -9.65685i q^{93} +(-10.1109 + 10.8063i) q^{95} -6.93933i q^{97} -1.05545 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} - 16 q^{11} - 4 q^{15} + 24 q^{19} - 4 q^{25} - 16 q^{29} - 32 q^{31} + 8 q^{35} + 8 q^{39} - 8 q^{41} - 8 q^{49} + 8 q^{51} - 36 q^{55} + 16 q^{59} + 24 q^{61} + 4 q^{65} - 8 q^{69} - 24 q^{71} - 4 q^{75} + 24 q^{79} + 8 q^{81} - 40 q^{85} + 8 q^{89} - 32 q^{95} + 16 q^{99}+O(q^{100}) $ 8 * q - 8 * q^9 - 16 * q^11 - 4 * q^15 + 24 * q^19 - 4 * q^25 - 16 * q^29 - 32 * q^31 + 8 * q^35 + 8 * q^39 - 8 * q^41 - 8 * q^49 + 8 * q^51 - 36 * q^55 + 16 * q^59 + 24 * q^61 + 4 * q^65 - 8 * q^69 - 24 * q^71 - 4 * q^75 + 24 * q^79 + 8 * q^81 - 40 * q^85 + 8 * q^89 - 32 * q^95 + 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times$.

$n$	$391$	$521$	$781$	$937$	$1081$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−1.52773	+	1.63280i	−0.683220	+	0.730213i
$5$	−1.52773	+	1.63280i	−0.683220	+	0.730213i
$6$		0			0
$7$		−	2.82843i		−	1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$		−	2.82843i		−	1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	1.05545			0.318231			0.159115	−	0.987260i	$-0.449136\pi$
$11$	1.05545			0.318231			0.159115	+	0.987260i	$0.449136\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$		0			0
$15$	−1.63280	−	1.52773i	−0.421588	−	0.394457i
$16$		0			0
$17$			7.14949i			1.73401i	0.498303	+	0.867003i	$0.333957\pi$
$17$			7.14949i			1.73401i	−0.498303	+	0.867003i	$0.666043\pi$
$18$		0			0
$19$	6.61827			1.51834			0.759168	−	0.650895i	$-0.225606\pi$
$19$	6.61827			1.51834			0.759168	+	0.650895i	$0.225606\pi$
$20$		0			0
$21$	2.82843			0.617213
$22$		0			0
$23$			3.49264i			0.728265i	0.931347	+	0.364132i	$0.118634\pi$
$23$			3.49264i			0.728265i	−0.931347	+	0.364132i	$0.881366\pi$
$24$		0			0
$25$	−0.332104	−	4.98896i	−0.0664208	−	0.997792i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−4.82843			−0.896616			−0.448308	−	0.893879i	$-0.647973\pi$
$29$	−4.82843			−0.896616			−0.448308	+	0.893879i	$0.647973\pi$
$30$		0			0
$31$	−9.65685			−1.73442			−0.867211	−	0.497941i	$-0.834090\pi$
$31$	−9.65685			−1.73442			−0.867211	+	0.497941i	$0.834090\pi$
$32$		0			0
$33$			1.05545i			0.183731i
$34$		0			0
$35$	4.61827	+	4.32106i	0.780630	+	0.730393i
$36$		0			0
$37$			8.11091i			1.33342i	0.745315	+	0.666712i	$0.232299\pi$
$37$			8.11091i			1.33342i	−0.745315	+	0.666712i	$0.767701\pi$
$38$		0			0
$39$	1.00000			0.160128
$40$		0			0
$41$	−3.26561			−0.510003			−0.255001	−	0.966941i	$-0.582076\pi$
$41$	−3.26561			−0.510003			−0.255001	+	0.966941i	$0.582076\pi$
$42$		0			0
$43$			7.44670i			1.13561i	0.823163	+	0.567805i	$0.192207\pi$
$43$			7.44670i			1.13561i	−0.823163	+	0.567805i	$0.807793\pi$
$44$		0			0
$45$	1.52773	−	1.63280i	0.227740	−	0.243404i
$46$		0			0
$47$			11.3765i			1.65944i	0.558183	+	0.829718i	$0.311499\pi$
$47$			11.3765i			1.65944i	−0.558183	+	0.829718i	$0.688501\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−7.14949			−1.00113
$52$		0			0
$53$		−	4.53122i		−	0.622411i	−0.950343	−	0.311205i	$-0.899267\pi$
$53$		−	4.53122i		−	0.622411i	0.950343	−	0.311205i	$-0.100733\pi$
$54$		0			0
$55$	−1.61244	+	1.72335i	−0.217422	+	0.232376i
$56$		0			0
$57$			6.61827i			0.876611i
$58$		0			0
$59$	−1.05545			−0.137408			−0.0687041	−	0.997637i	$-0.521886\pi$
$59$	−1.05545			−0.137408			−0.0687041	+	0.997637i	$0.521886\pi$
$60$		0			0
$61$	−7.97792			−1.02147			−0.510734	−	0.859739i	$-0.670626\pi$
$61$	−7.97792			−1.02147			−0.510734	+	0.859739i	$0.670626\pi$
$62$		0			0
$63$			2.82843i			0.356348i
$64$		0			0
$65$	1.63280	+	1.52773i	0.202525	+	0.189491i
$66$		0			0
$67$			1.46878i			0.179440i	0.995967	+	0.0897200i	$0.0285972\pi$
$67$			1.46878i			0.179440i	−0.995967	+	0.0897200i	$0.971403\pi$
$68$		0			0
$69$	−3.49264			−0.420464
$70$		0			0
$71$	0.845296			0.100318			0.0501591	−	0.998741i	$-0.484027\pi$
$71$	0.845296			0.100318			0.0501591	+	0.998741i	$0.484027\pi$
$72$		0			0
$73$		−	9.28248i		−	1.08643i	−0.839593	−	0.543216i	$-0.817206\pi$
$73$		−	9.28248i		−	1.08643i	0.839593	−	0.543216i	$-0.182794\pi$
$74$		0			0
$75$	4.98896	−	0.332104i	0.576075	−	0.0383481i
$76$		0			0
$77$		−	2.98527i		−	0.340203i
$78$		0			0
$79$	−6.85228			−0.770942			−0.385471	−	0.922720i	$-0.625961\pi$
$79$	−6.85228			−0.770942			−0.385471	+	0.922720i	$0.625961\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	7.97093i		−	0.874923i	−0.899237	−	0.437462i	$-0.855878\pi$
$83$		−	7.97093i		−	0.874923i	0.899237	−	0.437462i	$-0.144122\pi$
$84$		0			0
$85$	−11.6737	−	10.9225i	−1.26619	−	1.18471i
$86$		0			0
$87$		−	4.82843i		−	0.517662i
$88$		0			0
$89$	−0.139976			−0.0148374			−0.00741870	−	0.999972i	$-0.502361\pi$
$89$	−0.139976			−0.0148374			−0.00741870	+	0.999972i	$0.502361\pi$
$90$		0			0
$91$	−2.82843			−0.296500
$92$		0			0
$93$		−	9.65685i		−	1.00137i
$94$		0			0
$95$	−10.1109	+	10.8063i	−1.03736	+	1.10871i
$96$		0			0
$97$		−	6.93933i		−	0.704582i	−0.935890	−	0.352291i	$-0.885403\pi$
$97$		−	6.93933i		−	0.704582i	0.935890	−	0.352291i	$-0.114597\pi$
$98$		0			0
$99$	−1.05545			−0.106077
$100$		0			0
$101$	−16.1421			−1.60620			−0.803101	−	0.595843i	$-0.796818\pi$
$101$	−16.1421			−1.60620			−0.803101	+	0.595843i	$0.796818\pi$
$102$		0			0
$103$			16.7530i			1.65073i	0.564603	+	0.825363i	$0.309029\pi$
$103$			16.7530i			1.65073i	−0.564603	+	0.825363i	$0.690971\pi$
$104$		0			0
$105$	−4.32106	+	4.61827i	−0.421693	+	0.450697i
$106$		0			0
$107$			4.87437i			0.471223i	0.971847	+	0.235611i	$0.0757092\pi$
$107$			4.87437i			0.471223i	−0.971847	+	0.235611i	$0.924291\pi$
$108$		0			0
$109$	7.14949			0.684797			0.342398	−	0.939555i	$-0.388761\pi$
$109$	7.14949			0.684797			0.342398	+	0.939555i	$0.388761\pi$
$110$		0			0
$111$	−8.11091			−0.769853
$112$		0			0
$113$			13.2604i			1.24743i	0.781651	+	0.623717i	$0.214378\pi$
$113$			13.2604i			1.24743i	−0.781651	+	0.623717i	$0.785622\pi$
$114$		0			0
$115$	−5.70279	−	5.33579i	−0.531788	−	0.497565i
$116$		0			0
$117$			1.00000i			0.0924500i
$118$		0			0
$119$	20.2218			1.85373
$120$		0			0
$121$	−9.88602			−0.898729
$122$		0			0
$123$		−	3.26561i		−	0.294450i
$124$		0			0
$125$	8.65336	+	7.07950i	0.773980	+	0.633210i
$126$		0			0
$127$			13.3063i			1.18075i	0.807130	+	0.590373i	$0.201019\pi$
$127$			13.3063i			1.18075i	−0.807130	+	0.590373i	$0.798981\pi$
$128$		0			0
$129$	−7.44670			−0.655645
$130$		0			0
$131$	8.06497			0.704639			0.352320	−	0.935880i	$-0.385393\pi$
$131$	8.06497			0.704639			0.352320	+	0.935880i	$0.385393\pi$
$132$		0			0
$133$		−	18.7193i		−	1.62317i
$134$		0			0
$135$	1.63280	+	1.52773i	0.140529	+	0.131486i
$136$		0			0
$137$			10.8453i			0.926576i	0.886208	+	0.463288i	$0.153330\pi$
$137$			10.8453i			0.926576i	−0.886208	+	0.463288i	$0.846670\pi$
$138$		0			0
$139$	−9.23654			−0.783433			−0.391717	−	0.920086i	$-0.628119\pi$
$139$	−9.23654			−0.783433			−0.391717	+	0.920086i	$0.628119\pi$
$140$		0			0
$141$	−11.3765			−0.958075
$142$		0			0
$143$		−	1.05545i		−	0.0882614i
$144$		0			0
$145$	7.37652	−	7.88388i	0.612586	−	0.654721i
$146$		0			0
$147$		−	1.00000i		−	0.0824786i
$148$		0			0
$149$	6.60140			0.540808			0.270404	−	0.962747i	$-0.412843\pi$
$149$	6.60140			0.540808			0.270404	+	0.962747i	$0.412843\pi$
$150$		0			0
$151$	14.8934			1.21201			0.606004	−	0.795462i	$-0.292772\pi$
$151$	14.8934			1.21201			0.606004	+	0.795462i	$0.292772\pi$
$152$		0			0
$153$		−	7.14949i		−	0.578002i
$154$		0			0
$155$	14.7530	−	15.7678i	1.18499	−	1.26650i
$156$		0			0
$157$		−	11.3835i		−	0.908502i	−0.890874	−	0.454251i	$-0.849907\pi$
$157$		−	11.3835i		−	0.908502i	0.890874	−	0.454251i	$-0.150093\pi$
$158$		0			0
$159$	4.53122			0.359349
$160$		0			0
$161$	9.87867			0.778548
$162$		0			0
$163$			7.31371i			0.572854i	0.958102	+	0.286427i	$0.0924676\pi$
$163$			7.31371i			0.572854i	−0.958102	+	0.286427i	$0.907532\pi$
$164$		0			0
$165$	−1.72335	−	1.61244i	−0.134163	−	0.125529i
$166$		0			0
$167$			15.3428i			1.18726i	0.804738	+	0.593630i	$0.202306\pi$
$167$			15.3428i			1.18726i	−0.804738	+	0.593630i	$0.797694\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$	−6.61827			−0.506112
$172$		0			0
$173$		−	18.9706i		−	1.44231i	−0.692776	−	0.721153i	$-0.743613\pi$
$173$		−	18.9706i		−	1.44231i	0.692776	−	0.721153i	$-0.256387\pi$
$174$		0			0
$175$	−14.1109	+	0.939333i	−1.06668	+	0.0710069i
$176$		0			0
$177$		−	1.05545i		−	0.0793327i
$178$		0			0
$179$	15.9246			1.19026			0.595130	−	0.803629i	$-0.297100\pi$
$179$	15.9246			1.19026			0.595130	+	0.803629i	$0.297100\pi$
$180$		0			0
$181$	16.0772			1.19501			0.597503	−	0.801866i	$-0.296159\pi$
$181$	16.0772			1.19501			0.597503	+	0.801866i	$0.296159\pi$
$182$		0			0
$183$		−	7.97792i		−	0.589745i
$184$		0			0
$185$	−13.2435	−	12.3912i	−0.973683	−	0.911022i
$186$		0			0
$187$			7.54595i			0.551814i
$188$		0			0
$189$	−2.82843			−0.205738
$190$		0			0
$191$	6.29898			0.455778			0.227889	−	0.973687i	$-0.426818\pi$
$191$	6.29898			0.455778			0.227889	+	0.973687i	$0.426818\pi$
$192$		0			0
$193$			23.7695i			1.71097i	0.517829	+	0.855484i	$0.326740\pi$
$193$			23.7695i			1.71097i	−0.517829	+	0.855484i	$0.673260\pi$
$194$		0			0
$195$	−1.52773	+	1.63280i	−0.109403	+	0.116928i
$196$		0			0
$197$		−	7.62778i		−	0.543457i	−0.962374	−	0.271729i	$-0.912405\pi$
$197$		−	7.62778i		−	0.543457i	0.962374	−	0.271729i	$-0.0875953\pi$
$198$		0			0
$199$	−6.85228			−0.485745			−0.242873	−	0.970058i	$-0.578090\pi$
$199$	−6.85228			−0.485745			−0.242873	+	0.970058i	$0.578090\pi$
$200$		0			0
$201$	−1.46878			−0.103600
$202$		0			0
$203$			13.6569i			0.958523i
$204$		0			0
$205$	4.98896	−	5.33210i	0.348444	−	0.372410i
$206$		0			0
$207$		−	3.49264i		−	0.242755i
$208$		0			0
$209$	6.98527			0.483181
$210$		0			0
$211$	15.8670			1.09233			0.546165	−	0.837678i	$-0.316087\pi$
$211$	15.8670			1.09233			0.546165	+	0.837678i	$0.316087\pi$
$212$		0			0
$213$			0.845296i			0.0579187i
$214$		0			0
$215$	−12.1590	−	11.3765i	−0.829237	−	0.775872i
$216$		0			0
$217$			27.3137i			1.85418i
$218$		0			0
$219$	9.28248			0.627252
$220$		0			0
$221$	7.14949			0.480927
$222$		0			0
$223$			9.95406i			0.666573i	0.942826	+	0.333287i	$0.108158\pi$
$223$			9.95406i			0.666573i	−0.942826	+	0.333287i	$0.891842\pi$
$224$		0			0
$225$	0.332104	+	4.98896i	0.0221403	+	0.332597i
$226$		0			0
$227$			11.1547i			0.740364i	0.928959	+	0.370182i	$0.120705\pi$
$227$			11.1547i			0.740364i	−0.928959	+	0.370182i	$0.879295\pi$
$228$		0			0
$229$	−23.9657			−1.58370			−0.791850	−	0.610716i	$-0.790882\pi$
$229$	−23.9657			−1.58370			−0.791850	+	0.610716i	$0.790882\pi$
$230$		0			0
$231$	2.98527			0.196416
$232$		0			0
$233$			15.3376i			1.00480i	0.864636	+	0.502399i	$0.167549\pi$
$233$			15.3376i			1.00480i	−0.864636	+	0.502399i	$0.832451\pi$
$234$		0			0
$235$	−18.5756	−	17.3802i	−1.21174	−	1.13376i
$236$		0			0
$237$		−	6.85228i		−	0.445104i
$238$		0			0
$239$	14.7344			0.953088			0.476544	−	0.879150i	$-0.341889\pi$
$239$	14.7344			0.953088			0.476544	+	0.879150i	$0.341889\pi$
$240$		0			0
$241$	10.1881			0.656272			0.328136	−	0.944631i	$-0.393580\pi$
$241$	10.1881			0.656272			0.328136	+	0.944631i	$0.393580\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	1.52773	−	1.63280i	0.0976029	−	0.104316i
$246$		0			0
$247$		−	6.61827i		−	0.421110i
$248$		0			0
$249$	7.97093			0.505137
$250$		0			0
$251$	15.8909			1.00302			0.501511	−	0.865151i	$-0.332778\pi$
$251$	15.8909			1.00302			0.501511	+	0.865151i	$0.332778\pi$
$252$		0			0
$253$			3.68631i			0.231756i
$254$		0			0
$255$	10.9225	−	11.6737i	0.683991	−	0.731037i
$256$		0			0
$257$			20.0239i			1.24905i	0.781003	+	0.624527i	$0.214708\pi$
$257$			20.0239i			1.24905i	−0.781003	+	0.624527i	$0.785292\pi$
$258$		0			0
$259$	22.9411			1.42549
$260$		0			0
$261$	4.82843			0.298872
$262$		0			0
$263$		−	27.0282i		−	1.66663i	−0.552800	−	0.833314i	$-0.686441\pi$
$263$		−	27.0282i		−	1.66663i	0.552800	−	0.833314i	$-0.313559\pi$
$264$		0			0
$265$	7.39860	+	6.92246i	0.454492	+	0.425244i
$266$		0			0
$267$		−	0.139976i		−	0.00856637i
$268$		0			0
$269$	−31.1274			−1.89787			−0.948936	−	0.315469i	$-0.897838\pi$
$269$	−31.1274			−1.89787			−0.948936	+	0.315469i	$0.897838\pi$
$270$		0			0
$271$	21.0043			1.27592			0.637960	−	0.770069i	$-0.279778\pi$
$271$	21.0043			1.27592			0.637960	+	0.770069i	$0.279778\pi$
$272$		0			0
$273$		−	2.82843i		−	0.171184i
$274$		0			0
$275$	−0.350520	−	5.26561i	−0.0211372	−	0.317528i
$276$		0			0
$277$			13.2145i			0.793980i	0.917823	+	0.396990i	$0.129945\pi$
$277$			13.2145i			0.793980i	−0.917823	+	0.396990i	$0.870055\pi$
$278$		0			0
$279$	9.65685			0.578141
$280$		0			0
$281$	1.83057			0.109202			0.0546012	−	0.998508i	$-0.482611\pi$
$281$	1.83057			0.109202			0.0546012	+	0.998508i	$0.482611\pi$
$282$		0			0
$283$		−	27.5355i		−	1.63682i	−0.574637	−	0.818408i	$-0.694857\pi$
$283$		−	27.5355i		−	1.63682i	0.574637	−	0.818408i	$-0.305143\pi$
$284$		0			0
$285$	−10.8063	−	10.1109i	−0.640113	−	0.598918i
$286$		0			0
$287$			9.23654i			0.545216i
$288$		0			0
$289$	−34.1152			−2.00678
$290$		0			0
$291$	6.93933			0.406791
$292$		0			0
$293$		−	3.95190i		−	0.230873i	−0.993315	−	0.115436i	$-0.963173\pi$
$293$		−	3.95190i		−	0.230873i	0.993315	−	0.115436i	$-0.0368266\pi$
$294$		0			0
$295$	1.61244	−	1.72335i	0.0938801	−	0.100337i
$296$		0			0
$297$		−	1.05545i		−	0.0612436i
$298$		0			0
$299$	3.49264			0.201984
$300$		0			0
$301$	21.0624			1.21402
$302$		0			0
$303$		−	16.1421i		−	0.927341i
$304$		0			0
$305$	12.1881	−	13.0264i	0.697887	−	0.745888i
$306$		0			0
$307$		−	14.9271i		−	0.851936i	−0.904738	−	0.425968i	$-0.859934\pi$
$307$		−	14.9271i		−	0.851936i	0.904738	−	0.425968i	$-0.140066\pi$
$308$		0			0
$309$	−16.7530			−0.953047
$310$		0			0
$311$	4.87437			0.276400			0.138200	−	0.990404i	$-0.455868\pi$
$311$	4.87437			0.276400			0.138200	+	0.990404i	$0.455868\pi$
$312$		0			0
$313$		−	27.2145i		−	1.53825i	−0.639097	−	0.769126i	$-0.720692\pi$
$313$		−	27.2145i		−	1.53825i	0.639097	−	0.769126i	$-0.279308\pi$
$314$		0			0
$315$	−4.61827	−	4.32106i	−0.260210	−	0.243464i
$316$		0			0
$317$			3.68592i			0.207022i	0.994628	+	0.103511i	$0.0330077\pi$
$317$			3.68592i			0.207022i	−0.994628	+	0.103511i	$0.966992\pi$
$318$		0			0
$319$	−5.09618			−0.285331
$320$		0			0
$321$	−4.87437			−0.272061
$322$		0			0
$323$			47.3173i			2.63280i
$324$		0			0
$325$	−4.98896	+	0.332104i	−0.276738	+	0.0184218i
$326$		0			0
$327$			7.14949i			0.395368i
$328$		0			0
$329$	32.1776			1.77401
$330$		0			0
$331$	−24.6850			−1.35681			−0.678405	−	0.734688i	$-0.737329\pi$
$331$	−24.6850			−1.35681			−0.678405	+	0.734688i	$0.737329\pi$
$332$		0			0
$333$		−	8.11091i		−	0.444475i
$334$		0			0
$335$	−2.39823	−	2.24389i	−0.131029	−	0.122597i
$336$		0			0
$337$		−	30.5282i		−	1.66298i	−0.555543	−	0.831488i	$-0.687490\pi$
$337$		−	30.5282i		−	1.66298i	0.555543	−	0.831488i	$-0.312510\pi$
$338$		0			0
$339$	−13.2604			−0.720206
$340$		0			0
$341$	−10.1924			−0.551947
$342$		0			0
$343$		−	16.9706i		−	0.916324i
$344$		0			0
$345$	5.33579	−	5.70279i	0.287269	−	0.307028i
$346$		0			0
$347$		−	8.32844i		−	0.447094i	−0.974693	−	0.223547i	$-0.928236\pi$
$347$		−	8.32844i		−	0.447094i	0.974693	−	0.223547i	$-0.0717636\pi$
$348$		0			0
$349$	8.57410			0.458961			0.229481	−	0.973313i	$-0.426297\pi$
$349$	8.57410			0.458961			0.229481	+	0.973313i	$0.426297\pi$
$350$		0			0
$351$	−1.00000			−0.0533761
$352$		0			0
$353$		−	2.42498i		−	0.129069i	−0.997915	−	0.0645344i	$-0.979444\pi$
$353$		−	2.42498i		−	0.129069i	0.997915	−	0.0645344i	$-0.0205562\pi$
$354$		0			0
$355$	−1.29138	+	1.38020i	−0.0685393	+	0.0732536i
$356$		0			0
$357$			20.2218i			1.07025i
$358$		0			0
$359$	18.5499			0.979024			0.489512	−	0.871997i	$-0.337175\pi$
$359$	18.5499			0.979024			0.489512	+	0.871997i	$0.337175\pi$
$360$		0			0
$361$	24.8015			1.30534
$362$		0			0
$363$		−	9.88602i		−	0.518881i
$364$		0			0
$365$	15.1565	+	14.1811i	0.793326	+	0.742272i
$366$		0			0
$367$			5.29973i			0.276644i	0.990387	+	0.138322i	$0.0441709\pi$
$367$			5.29973i			0.276644i	−0.990387	+	0.138322i	$0.955829\pi$
$368$		0			0
$369$	3.26561			0.170001
$370$		0			0
$371$	−12.8162			−0.665385
$372$		0			0
$373$		−	29.2071i		−	1.51229i	−0.654407	−	0.756143i	$-0.727082\pi$
$373$		−	29.2071i		−	1.51229i	0.654407	−	0.756143i	$-0.272918\pi$
$374$		0			0
$375$	−7.07950	+	8.65336i	−0.365584	+	0.446858i
$376$		0			0
$377$			4.82843i			0.248677i
$378$		0			0
$379$	−10.3523			−0.531762			−0.265881	−	0.964006i	$-0.585663\pi$
$379$	−10.3523			−0.531762			−0.265881	+	0.964006i	$0.585663\pi$
$380$		0			0
$381$	−13.3063			−0.681704
$382$		0			0
$383$		−	30.0818i		−	1.53711i	−0.639784	−	0.768555i	$-0.720976\pi$
$383$		−	30.0818i		−	1.53711i	0.639784	−	0.768555i	$-0.279024\pi$
$384$		0			0
$385$	4.87437	+	4.56068i	0.248421	+	0.232434i
$386$		0			0
$387$		−	7.44670i		−	0.378537i
$388$		0			0
$389$	−33.2383			−1.68525			−0.842625	−	0.538501i	$-0.818991\pi$
$389$	−33.2383			−1.68525			−0.842625	+	0.538501i	$0.818991\pi$
$390$		0			0
$391$	−24.9706			−1.26282
$392$		0			0
$393$			8.06497i			0.406824i
$394$		0			0
$395$	10.4684	−	11.1884i	0.526723	−	0.562952i
$396$		0			0
$397$		−	13.5018i		−	0.677635i	−0.940852	−	0.338818i	$-0.889973\pi$
$397$		−	13.5018i		−	0.677635i	0.940852	−	0.338818i	$-0.110027\pi$
$398$		0			0
$399$	18.7193			0.937137
$400$		0			0
$401$	3.53159			0.176359			0.0881795	−	0.996105i	$-0.471895\pi$
$401$	3.53159			0.176359			0.0881795	+	0.996105i	$0.471895\pi$
$402$		0			0
$403$			9.65685i			0.481042i
$404$		0			0
$405$	−1.52773	+	1.63280i	−0.0759133	+	0.0811347i
$406$		0			0
$407$			8.56068i			0.424337i
$408$		0			0
$409$	24.7530			1.22396			0.611979	−	0.790874i	$-0.290374\pi$
$409$	24.7530			1.22396			0.611979	+	0.790874i	$0.290374\pi$
$410$		0			0
$411$	−10.8453			−0.534959
$412$		0			0
$413$			2.98527i			0.146896i
$414$		0			0
$415$	13.0150	+	12.1774i	0.638880	+	0.597765i
$416$		0			0
$417$		−	9.23654i		−	0.452315i
$418$		0			0
$419$	−5.39338			−0.263484			−0.131742	−	0.991284i	$-0.542057\pi$
$419$	−5.39338			−0.263484			−0.131742	+	0.991284i	$0.542057\pi$
$420$		0			0
$421$	32.4337			1.58072			0.790362	−	0.612640i	$-0.209893\pi$
$421$	32.4337			1.58072			0.790362	+	0.612640i	$0.209893\pi$
$422$		0			0
$423$		−	11.3765i		−	0.553145i
$424$		0			0
$425$	35.6685	−	2.37438i	1.73018	−	0.115174i
$426$		0			0
$427$			22.5650i			1.09199i
$428$		0			0
$429$	1.05545			0.0509577
$430$		0			0
$431$	−31.5983			−1.52204			−0.761019	−	0.648730i	$-0.775301\pi$
$431$	−31.5983			−1.52204			−0.761019	+	0.648730i	$0.775301\pi$
$432$		0			0
$433$		−	27.8566i		−	1.33870i	−0.742946	−	0.669351i	$-0.766572\pi$
$433$		−	27.8566i		−	1.33870i	0.742946	−	0.669351i	$-0.233428\pi$
$434$		0			0
$435$	7.88388	+	7.37652i	0.378003	+	0.353677i
$436$		0			0
$437$			23.1152i			1.10575i
$438$		0			0
$439$	−7.09190			−0.338478			−0.169239	−	0.985575i	$-0.554131\pi$
$439$	−7.09190			−0.338478			−0.169239	+	0.985575i	$0.554131\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$		−	0.386578i		−	0.0183669i	−0.999958	−	0.00918343i	$-0.997077\pi$
$443$		−	0.386578i		−	0.0183669i	0.999958	−	0.00918343i	$-0.00292322\pi$
$444$		0			0
$445$	0.213845	−	0.228553i	0.0101372	−	0.0108344i
$446$		0			0
$447$			6.60140i			0.312236i
$448$		0			0
$449$	−1.18844			−0.0560860			−0.0280430	−	0.999607i	$-0.508928\pi$
$449$	−1.18844			−0.0560860			−0.0280430	+	0.999607i	$0.508928\pi$
$450$		0			0
$451$	−3.44670			−0.162299
$452$		0			0
$453$			14.8934i			0.699753i
$454$		0			0
$455$	4.32106	−	4.61827i	0.202575	−	0.216508i
$456$		0			0
$457$			10.5434i			0.493200i	0.969117	+	0.246600i	$0.0793135\pi$
$457$			10.5434i			0.493200i	−0.969117	+	0.246600i	$0.920687\pi$
$458$		0			0
$459$	7.14949			0.333710
$460$		0			0
$461$	11.8961			0.554056			0.277028	−	0.960862i	$-0.410651\pi$
$461$	11.8961			0.554056			0.277028	+	0.960862i	$0.410651\pi$
$462$		0			0
$463$		−	1.92032i		−	0.0892451i	−0.999004	−	0.0446225i	$-0.985791\pi$
$463$		−	1.92032i		−	0.0892451i	0.999004	−	0.0446225i	$-0.0142085\pi$
$464$		0			0
$465$	15.7678	+	14.7530i	0.731212	+	0.684155i
$466$		0			0
$467$			19.1256i			0.885029i	0.896761	+	0.442514i	$0.145913\pi$
$467$			19.1256i			0.885029i	−0.896761	+	0.442514i	$0.854087\pi$
$468$		0			0
$469$	4.15434			0.191829
$470$		0			0
$471$	11.3835			0.524524
$472$		0			0
$473$			7.85964i			0.361386i
$474$		0			0
$475$	−2.19796	−	33.0183i	−0.100849	−	1.51498i
$476$		0			0
$477$			4.53122i			0.207470i
$478$		0			0
$479$	−12.5793			−0.574764			−0.287382	−	0.957816i	$-0.592785\pi$
$479$	−12.5793			−0.574764			−0.287382	+	0.957816i	$0.592785\pi$
$480$		0			0
$481$	8.11091			0.369825
$482$		0			0
$483$			9.87867i			0.449495i
$484$		0			0
$485$	11.3306	+	10.6014i	0.514495	+	0.481385i
$486$		0			0
$487$			22.8180i			1.03398i	0.855991	+	0.516991i	$0.172948\pi$
$487$			22.8180i			1.03398i	−0.855991	+	0.516991i	$0.827052\pi$
$488$		0			0
$489$	−7.31371			−0.330737
$490$		0			0
$491$	38.6924			1.74616			0.873081	−	0.487574i	$-0.162118\pi$
$491$	38.6924			1.74616			0.873081	+	0.487574i	$0.162118\pi$
$492$		0			0
$493$		−	34.5208i		−	1.55474i
$494$		0			0
$495$	1.61244	−	1.72335i	0.0724739	−	0.0774588i
$496$		0			0
$497$		−	2.39086i		−	0.107245i
$498$		0			0
$499$	26.5404			1.18811			0.594055	−	0.804424i	$-0.297526\pi$
$499$	26.5404			1.18811			0.594055	+	0.804424i	$0.297526\pi$
$500$		0			0
$501$	−15.3428			−0.685465
$502$		0			0
$503$			15.4008i			0.686686i	0.939210	+	0.343343i	$0.111559\pi$
$503$			15.4008i			0.686686i	−0.939210	+	0.343343i	$0.888441\pi$
$504$		0			0
$505$	24.6608	−	26.3570i	1.09739	−	1.17287i
$506$		0			0
$507$		−	1.00000i		−	0.0444116i
$508$		0			0
$509$	30.1846			1.33791			0.668955	−	0.743303i	$-0.266742\pi$
$509$	30.1846			1.33791			0.668955	+	0.743303i	$0.266742\pi$
$510$		0			0
$511$	−26.2548			−1.16144
$512$		0			0
$513$		−	6.61827i		−	0.292204i
$514$		0			0
$515$	−27.3544	−	25.5940i	−1.20538	−	1.12781i
$516$		0			0
$517$			12.0074i			0.528084i
$518$		0			0
$519$	18.9706			0.832715
$520$		0			0
$521$	−26.7825			−1.17336			−0.586681	−	0.809818i	$-0.699566\pi$
$521$	−26.7825			−1.17336			−0.586681	+	0.809818i	$0.699566\pi$
$522$		0			0
$523$		−	27.8670i		−	1.21854i	−0.792963	−	0.609270i	$-0.791463\pi$
$523$		−	27.8670i		−	1.21854i	0.792963	−	0.609270i	$-0.208537\pi$
$524$		0			0
$525$	−0.939333	−	14.1109i	−0.0409958	−	0.615850i
$526$		0			0
$527$		−	69.0416i		−	3.00750i
$528$		0			0
$529$	10.8015			0.469630
$530$		0			0
$531$	1.05545			0.0458027
$532$		0			0
$533$			3.26561i			0.141449i
$534$		0			0
$535$	−7.95889	−	7.44670i	−0.344093	−	0.321949i
$536$		0			0
$537$			15.9246i			0.687197i
$538$		0			0
$539$	−1.05545			−0.0454616
$540$		0			0
$541$	20.8982			0.898485			0.449242	−	0.893410i	$-0.351694\pi$
$541$	20.8982			0.898485			0.449242	+	0.893410i	$0.351694\pi$
$542$		0			0
$543$			16.0772i			0.689937i
$544$		0			0
$545$	−10.9225	+	11.6737i	−0.467867	+	0.500047i
$546$		0			0
$547$			24.3106i			1.03945i	0.854334	+	0.519724i	$0.173965\pi$
$547$			24.3106i			1.03945i	−0.854334	+	0.519724i	$0.826035\pi$
$548$		0			0
$549$	7.97792			0.340489
$550$		0			0
$551$	−31.9558			−1.36136
$552$		0			0
$553$			19.3812i			0.824172i
$554$		0			0
$555$	12.3912	−	13.2435i	0.525979	−	0.562156i
$556$		0			0
$557$		−	10.7197i		−	0.454207i	−0.973871	−	0.227104i	$-0.927074\pi$
$557$		−	10.7197i		−	0.454207i	0.973871	−	0.227104i	$-0.0729256\pi$
$558$		0			0
$559$	7.44670			0.314962
$560$		0			0
$561$	−7.54595			−0.318590
$562$		0			0
$563$		−	24.6898i		−	1.04055i	−0.853998	−	0.520276i	$-0.825829\pi$
$563$		−	24.6898i		−	1.04055i	0.853998	−	0.520276i	$-0.174171\pi$
$564$		0			0
$565$	−21.6516	−	20.2583i	−0.910891	−	0.852271i
$566$		0			0
$567$		−	2.82843i		−	0.118783i
$568$		0			0
$569$	35.6023			1.49252			0.746262	−	0.665652i	$-0.231847\pi$
$569$	35.6023			1.49252			0.746262	+	0.665652i	$0.231847\pi$
$570$		0			0
$571$	−13.3254			−0.557649			−0.278825	−	0.960342i	$-0.589945\pi$
$571$	−13.3254			−0.557649			−0.278825	+	0.960342i	$0.589945\pi$
$572$		0			0
$573$			6.29898i			0.263144i
$574$		0			0
$575$	17.4246	−	1.15992i	0.726657	−	0.0483720i
$576$		0			0
$577$			26.3112i			1.09535i	0.836692	+	0.547674i	$0.184487\pi$
$577$			26.3112i			1.09535i	−0.836692	+	0.547674i	$0.815513\pi$
$578$		0			0
$579$	−23.7695			−0.987828
$580$		0			0
$581$	−22.5452			−0.935332
$582$		0			0
$583$		−	4.78249i		−	0.198070i
$584$		0			0
$585$	−1.63280	−	1.52773i	−0.0675082	−	0.0631637i
$586$		0			0
$587$			28.3808i			1.17140i	0.810527	+	0.585701i	$0.199181\pi$
$587$			28.3808i			1.17140i	−0.810527	+	0.585701i	$0.800819\pi$
$588$		0			0
$589$	−63.9117			−2.63343
$590$		0			0
$591$	7.62778			0.313765
$592$		0			0
$593$			6.81156i			0.279717i	0.990171	+	0.139859i	$0.0446648\pi$
$593$			6.81156i			0.279717i	−0.990171	+	0.139859i	$0.955335\pi$
$594$		0			0
$595$	−30.8934	+	33.0183i	−1.26651	+	1.35362i
$596$		0			0
$597$		−	6.85228i		−	0.280445i
$598$		0			0
$599$	−38.9307			−1.59066			−0.795332	−	0.606174i	$-0.792704\pi$
$599$	−38.9307			−1.59066			−0.795332	+	0.606174i	$0.792704\pi$
$600$		0			0
$601$	−2.87131			−0.117123			−0.0585616	−	0.998284i	$-0.518651\pi$
$601$	−2.87131			−0.117123			−0.0585616	+	0.998284i	$0.518651\pi$
$602$		0			0
$603$		−	1.46878i		−	0.0598133i
$604$		0			0
$605$	15.1031	−	16.1419i	0.614030	−	0.656263i
$606$		0			0
$607$		−	10.9221i		−	0.443313i	−0.975125	−	0.221657i	$-0.928854\pi$
$607$		−	10.9221i		−	0.443313i	0.975125	−	0.221657i	$-0.0711464\pi$
$608$		0			0
$609$	−13.6569			−0.553404
$610$		0			0
$611$	11.3765			0.460245
$612$		0			0
$613$			38.8779i			1.57026i	0.619328	+	0.785132i	$0.287405\pi$
$613$			38.8779i			1.57026i	−0.619328	+	0.785132i	$0.712595\pi$
$614$		0			0
$615$	5.33210	+	4.98896i	0.215011	+	0.201174i
$616$		0			0
$617$		−	1.09656i		−	0.0441460i	−0.999756	−	0.0220730i	$-0.992973\pi$
$617$		−	1.09656i		−	0.0441460i	0.999756	−	0.0220730i	$-0.00702663\pi$
$618$		0			0
$619$	5.52209			0.221952			0.110976	−	0.993823i	$-0.464602\pi$
$619$	5.52209			0.221952			0.110976	+	0.993823i	$0.464602\pi$
$620$		0			0
$621$	3.49264			0.140155
$622$		0			0
$623$			0.395911i			0.0158618i
$624$		0			0
$625$	−24.7794	+	3.31371i	−0.991177	+	0.132548i
$626$		0			0
$627$			6.98527i			0.278965i
$628$		0			0
$629$	−57.9888			−2.31217
$630$		0			0
$631$	−1.74873			−0.0696159			−0.0348079	−	0.999394i	$-0.511082\pi$
$631$	−1.74873			−0.0696159			−0.0348079	+	0.999394i	$0.511082\pi$
$632$		0			0
$633$			15.8670i			0.630657i
$634$		0			0
$635$	−21.7266	−	20.3284i	−0.862196	−	0.806709i
$636$		0			0
$637$			1.00000i			0.0396214i
$638$		0			0
$639$	−0.845296			−0.0334394
$640$		0			0
$641$	2.73400			0.107987			0.0539933	−	0.998541i	$-0.482805\pi$
$641$	2.73400			0.107987			0.0539933	+	0.998541i	$0.482805\pi$
$642$		0			0
$643$			36.3180i			1.43224i	0.697976	+	0.716121i	$0.254084\pi$
$643$			36.3180i			1.43224i	−0.697976	+	0.716121i	$0.745916\pi$
$644$		0			0
$645$	11.3765	−	12.1590i	0.447950	−	0.478760i
$646$		0			0
$647$			28.4295i			1.11768i	0.829276	+	0.558839i	$0.188753\pi$
$647$			28.4295i			1.11768i	−0.829276	+	0.558839i	$0.811247\pi$
$648$		0			0
$649$	−1.11398			−0.0437276
$650$		0			0
$651$	−27.3137			−1.07051
$652$		0			0
$653$		−	7.48275i		−	0.292823i	−0.989224	−	0.146411i	$-0.953228\pi$
$653$		−	7.48275i		−	0.292823i	0.989224	−	0.146411i	$-0.0467723\pi$
$654$		0			0
$655$	−12.3211	+	13.1685i	−0.481424	+	0.514537i
$656$		0			0
$657$			9.28248i			0.362144i
$658$		0			0
$659$	1.90635			0.0742609			0.0371304	−	0.999310i	$-0.488178\pi$
$659$	1.90635			0.0742609			0.0371304	+	0.999310i	$0.488178\pi$
$660$		0			0
$661$	23.9761			0.932564			0.466282	−	0.884636i	$-0.345593\pi$
$661$	23.9761			0.932564			0.466282	+	0.884636i	$0.345593\pi$
$662$		0			0
$663$			7.14949i			0.277663i
$664$		0			0
$665$	30.5650	+	28.5980i	1.18526	+	1.10898i
$666$		0			0
$667$		−	16.8639i		−	0.652974i
$668$		0			0
$669$	−9.95406			−0.384846
$670$		0			0
$671$	−8.42031			−0.325063
$672$		0			0
$673$			18.6127i			0.717466i	0.933440	+	0.358733i	$0.116791\pi$
$673$			18.6127i			0.717466i	−0.933440	+	0.358733i	$0.883209\pi$
$674$		0			0
$675$	−4.98896	+	0.332104i	−0.192025	+	0.0127827i
$676$		0			0
$677$		−	32.7530i		−	1.25880i	−0.777081	−	0.629401i	$-0.783300\pi$
$677$		−	32.7530i		−	1.25880i	0.777081	−	0.629401i	$-0.216700\pi$
$678$		0			0
$679$	−19.6274			−0.753230
$680$		0			0
$681$	−11.1547			−0.427449
$682$		0			0
$683$		−	34.5976i		−	1.32384i	−0.749575	−	0.661920i	$-0.769742\pi$
$683$		−	34.5976i		−	1.32384i	0.749575	−	0.661920i	$-0.230258\pi$
$684$		0			0
$685$	−17.7083	−	16.5686i	−0.676598	−	0.633055i
$686$		0			0
$687$		−	23.9657i		−	0.914349i
$688$		0			0
$689$	−4.53122			−0.172626
$690$		0			0
$691$	3.99517			0.151984			0.0759918	−	0.997108i	$-0.475788\pi$
$691$	3.99517			0.151984			0.0759918	+	0.997108i	$0.475788\pi$
$692$		0			0
$693$			2.98527i			0.113401i
$694$		0			0
$695$	14.1109	−	15.0815i	0.535257	−	0.572073i
$696$		0			0
$697$		−	23.3474i		−	0.884348i
$698$		0			0
$699$	−15.3376			−0.580120
$700$		0			0
$701$	−0.267750			−0.0101128			−0.00505638	−	0.999987i	$-0.501610\pi$
$701$	−0.267750			−0.0101128			−0.00505638	+	0.999987i	$0.501610\pi$
$702$		0			0
$703$			53.6802i			2.02459i
$704$		0			0
$705$	17.3802	−	18.5756i	0.654576	−	0.699599i
$706$		0			0
$707$			45.6569i			1.71710i
$708$		0			0
$709$	1.13478			0.0426176			0.0213088	−	0.999773i	$-0.493217\pi$
$709$	1.13478			0.0426176			0.0213088	+	0.999773i	$0.493217\pi$
$710$		0			0
$711$	6.85228			0.256981
$712$		0			0
$713$		−	33.7279i		−	1.26312i
$714$		0			0
$715$	1.72335	+	1.61244i	0.0644496	+	0.0603019i
$716$		0			0
$717$			14.7344i			0.550266i
$718$		0			0
$719$	14.0434			0.523732			0.261866	−	0.965104i	$-0.415662\pi$
$719$	14.0434			0.523732			0.261866	+	0.965104i	$0.415662\pi$
$720$		0			0
$721$	47.3847			1.76470
$722$		0			0
$723$			10.1881i			0.378899i
$724$		0			0
$725$	1.60354	+	24.0888i	0.0595540	+	0.894636i
$726$		0			0
$727$		−	23.5099i		−	0.871934i	−0.899963	−	0.435967i	$-0.856407\pi$
$727$		−	23.5099i		−	0.871934i	0.899963	−	0.435967i	$-0.143593\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−53.2401			−1.96916
$732$		0			0
$733$			21.6274i			0.798826i	0.916771	+	0.399413i	$0.130786\pi$
$733$			21.6274i			0.798826i	−0.916771	+	0.399413i	$0.869214\pi$
$734$		0			0
$735$	1.63280	+	1.52773i	0.0602269	+	0.0563510i
$736$		0			0
$737$			1.55023i			0.0571034i
$738$		0			0
$739$	26.8738			0.988569			0.494285	−	0.869300i	$-0.335430\pi$
$739$	26.8738			0.988569			0.494285	+	0.869300i	$0.335430\pi$
$740$		0			0
$741$	6.61827			0.243128
$742$		0			0
$743$		−	11.5506i		−	0.423751i	−0.977297	−	0.211875i	$-0.932043\pi$
$743$		−	11.5506i		−	0.423751i	0.977297	−	0.211875i	$-0.0679571\pi$
$744$		0			0
$745$	−10.0851	+	10.7788i	−0.369491	+	0.394905i
$746$		0			0
$747$			7.97093i			0.291641i
$748$		0			0
$749$	13.7868			0.503758
$750$		0			0
$751$	52.5980			1.91933			0.959663	−	0.281151i	$-0.0907163\pi$
$751$	52.5980			1.91933			0.959663	+	0.281151i	$0.0907163\pi$
$752$		0			0
$753$			15.8909i			0.579095i
$754$		0			0
$755$	−22.7530	+	24.3180i	−0.828068	+	0.885023i
$756$		0			0
$757$			18.8406i			0.684774i	0.939559	+	0.342387i	$0.111235\pi$
$757$			18.8406i			0.684774i	−0.939559	+	0.342387i	$0.888765\pi$
$758$		0			0
$759$	−3.68631			−0.133805
$760$		0			0
$761$	−49.3323			−1.78830			−0.894148	−	0.447771i	$-0.852218\pi$
$761$	−49.3323			−1.78830			−0.894148	+	0.447771i	$0.852218\pi$
$762$		0			0
$763$		−	20.2218i		−	0.732079i
$764$		0			0
$765$	11.6737	+	10.9225i	0.422064	+	0.394903i
$766$		0			0
$767$			1.05545i			0.0381102i
$768$		0			0
$769$	−32.6274			−1.17657			−0.588287	−	0.808652i	$-0.700198\pi$
$769$	−32.6274			−1.17657			−0.588287	+	0.808652i	$0.700198\pi$
$770$		0			0
$771$	−20.0239			−0.721142
$772$		0			0
$773$			19.0229i			0.684208i	0.939662	+	0.342104i	$0.111139\pi$
$773$			19.0229i			0.684208i	−0.939662	+	0.342104i	$0.888861\pi$
$774$		0			0
$775$	3.20708	+	48.1776i	0.115202	+	1.73059i
$776$		0			0
$777$			22.9411i			0.823008i
$778$		0			0
$779$	−21.6127			−0.774355
$780$		0			0
$781$	0.892170			0.0319243
$782$		0			0
$783$			4.82843i			0.172554i
$784$		0			0
$785$	18.5870	+	17.3909i	0.663400	+	0.620707i
$786$		0			0
$787$		−	3.79216i		−	0.135176i	−0.997713	−	0.0675880i	$-0.978470\pi$
$787$		−	3.79216i		−	0.135176i	0.997713	−	0.0675880i	$-0.0215303\pi$
$788$		0			0
$789$	27.0282			0.962228
$790$		0			0
$791$	37.5061			1.33356
$792$		0			0
$793$			7.97792i			0.283304i
$794$		0			0
$795$	−6.92246	+	7.39860i	−0.245515	+	0.262401i
$796$		0			0
$797$		−	31.0287i		−	1.09909i	−0.835463	−	0.549547i	$-0.814801\pi$
$797$		−	31.0287i		−	1.09909i	0.835463	−	0.549547i	$-0.185199\pi$
$798$		0			0
$799$	−81.3363			−2.87747
$800$		0			0
$801$	0.139976			0.00494580
$802$		0			0
$803$		−	9.79722i		−	0.345736i
$804$		0			0
$805$	−15.0919	+	16.1299i	−0.531919	+	0.568505i
$806$		0			0
$807$		−	31.1274i		−	1.09574i
$808$		0			0
$809$	−49.2211			−1.73052			−0.865260	−	0.501323i	$-0.832847\pi$
$809$	−49.2211			−1.73052			−0.865260	+	0.501323i	$0.832847\pi$
$810$		0			0
$811$	8.76291			0.307707			0.153854	−	0.988094i	$-0.450832\pi$
$811$	8.76291			0.307707			0.153854	+	0.988094i	$0.450832\pi$
$812$		0			0
$813$			21.0043i			0.736653i
$814$		0			0
$815$	−11.9419	−	11.1733i	−0.418305	−	0.391385i
$816$		0			0
$817$			49.2843i			1.72424i
$818$		0			0
$819$	2.82843			0.0988332
$820$		0			0
$821$	24.3060			0.848284			0.424142	−	0.905596i	$-0.360576\pi$
$821$	24.3060			0.848284			0.424142	+	0.905596i	$0.360576\pi$
$822$		0			0
$823$		−	43.1586i		−	1.50442i	−0.658926	−	0.752208i	$-0.728989\pi$
$823$		−	43.1586i		−	1.50442i	0.658926	−	0.752208i	$-0.271011\pi$
$824$		0			0
$825$	5.26561	−	0.350520i	0.183325	−	0.0122036i
$826$		0			0
$827$		−	27.7189i		−	0.963882i	−0.876204	−	0.481941i	$-0.839932\pi$
$827$		−	27.7189i		−	0.963882i	0.876204	−	0.481941i	$-0.160068\pi$
$828$		0			0
$829$	−23.2094			−0.806096			−0.403048	−	0.915179i	$-0.632049\pi$
$829$	−23.2094			−0.806096			−0.403048	+	0.915179i	$0.632049\pi$
$830$		0			0
$831$	−13.2145			−0.458404
$832$		0			0
$833$		−	7.14949i		−	0.247715i
$834$		0			0
$835$	−25.0518	−	23.4396i	−0.866952	−	0.811160i
$836$		0			0
$837$			9.65685i			0.333790i
$838$		0			0
$839$	25.2796			0.872748			0.436374	−	0.899765i	$-0.356262\pi$
$839$	25.2796			0.872748			0.436374	+	0.899765i	$0.356262\pi$
$840$		0			0
$841$	−5.68629			−0.196079
$842$		0			0
$843$			1.83057i			0.0630481i
$844$		0			0
$845$	1.52773	−	1.63280i	0.0525554	−	0.0561702i
$846$		0			0
$847$			27.9619i			0.960782i
$848$		0			0
$849$	27.5355			0.945017
$850$		0			0
$851$	−28.3284			−0.971086
$852$		0			0
$853$		−	4.43934i		−	0.152000i	−0.997108	−	0.0760001i	$-0.975785\pi$
$853$		−	4.43934i		−	0.152000i	0.997108	−	0.0760001i	$-0.0242149\pi$
$854$		0			0
$855$	10.1109	−	10.8063i	0.345786	−	0.369569i
$856$		0			0
$857$			31.3609i			1.07127i	0.844451	+	0.535633i	$0.179927\pi$
$857$			31.3609i			1.07127i	−0.844451	+	0.535633i	$0.820073\pi$
$858$		0			0
$859$	28.7457			0.980791			0.490395	−	0.871500i	$-0.336852\pi$
$859$	28.7457			0.980791			0.490395	+	0.871500i	$0.336852\pi$
$860$		0			0
$861$	−9.23654			−0.314780
$862$		0			0
$863$		−	36.3471i		−	1.23727i	−0.785679	−	0.618634i	$-0.787686\pi$
$863$		−	36.3471i		−	1.23727i	0.785679	−	0.618634i	$-0.212314\pi$
$864$		0			0
$865$	30.9752	+	28.9818i	1.05319	+	0.985412i
$866$		0			0
$867$		−	34.1152i		−	1.15861i
$868$		0			0
$869$	−7.23226			−0.245338
$870$		0			0
$871$	1.46878			0.0497677
$872$		0			0
$873$			6.93933i			0.234861i
$874$		0			0
$875$	20.0239	−	24.4754i	0.676930	−	0.827420i
$876$		0			0
$877$		−	21.9896i		−	0.742535i	−0.928526	−	0.371268i	$-0.878923\pi$
$877$		−	21.9896i		−	0.742535i	0.928526	−	0.371268i	$-0.121077\pi$
$878$		0			0
$879$	3.95190			0.133294
$880$		0			0
$881$	−44.5348			−1.50041			−0.750207	−	0.661203i	$-0.770046\pi$
$881$	−44.5348			−1.50041			−0.750207	+	0.661203i	$0.770046\pi$
$882$		0			0
$883$		−	6.05277i		−	0.203692i	−0.994800	−	0.101846i	$-0.967525\pi$
$883$		−	6.05277i		−	0.203692i	0.994800	−	0.101846i	$-0.0324749\pi$
$884$		0			0
$885$	1.72335	+	1.61244i	0.0579297	+	0.0542017i
$886$		0			0
$887$			35.9700i			1.20775i	0.797077	+	0.603877i	$0.206378\pi$
$887$			35.9700i			1.20775i	−0.797077	+	0.603877i	$0.793622\pi$
$888$		0			0
$889$	37.6360			1.26227
$890$		0			0
$891$	1.05545			0.0353590
$892$		0			0
$893$			75.2929i			2.51958i
$894$		0			0
$895$	−24.3284	+	26.0018i	−0.813210	+	0.869143i
$896$		0			0
$897$			3.49264i			0.116616i
$898$		0			0
$899$	46.6274			1.55511
$900$		0			0
$901$	32.3959			1.07926
$902$		0			0
$903$			21.0624i			0.700914i
$904$		0			0
$905$	−24.5615	+	26.2509i	−0.816452	+	0.872609i
$906$		0			0
$907$			39.2695i			1.30392i	0.758252	+	0.651962i	$0.226054\pi$
$907$			39.2695i			1.30392i	−0.758252	+	0.651962i	$0.773946\pi$
$908$		0			0
$909$	16.1421			0.535401
$910$		0			0
$911$	−26.4583			−0.876604			−0.438302	−	0.898828i	$-0.644420\pi$
$911$	−26.4583			−0.876604			−0.438302	+	0.898828i	$0.644420\pi$
$912$		0			0
$913$		−	8.41294i		−	0.278428i
$914$		0			0
$915$	13.0264	+	12.1881i	0.430639	+	0.402925i
$916$		0			0
$917$		−	22.8112i		−	0.753291i
$918$		0			0
$919$	37.1513			1.22551			0.612754	−	0.790274i	$-0.290062\pi$
$919$	37.1513			1.22551			0.612754	+	0.790274i	$0.290062\pi$
$920$		0			0
$921$	14.9271			0.491866
$922$		0			0
$923$		−	0.845296i		−	0.0278232i
$924$		0			0
$925$	40.4650	−	2.69367i	1.33048	−	0.0885672i
$926$		0			0
$927$		−	16.7530i		−	0.550242i
$928$		0			0
$929$	−15.2412			−0.500048			−0.250024	−	0.968240i	$-0.580438\pi$
$929$	−15.2412			−0.500048			−0.250024	+	0.968240i	$0.580438\pi$
$930$		0			0
$931$	−6.61827			−0.216905
$932$		0			0
$933$			4.87437i			0.159580i
$934$		0			0
$935$	−12.3211	−	11.5281i	−0.402942	−	0.377011i
$936$		0			0
$937$			28.1850i			0.920764i	0.887721	+	0.460382i	$0.152288\pi$
$937$			28.1850i			0.920764i	−0.887721	+	0.460382i	$0.847712\pi$
$938$		0			0
$939$	27.2145			0.888110
$940$		0			0
$941$	42.8752			1.39769			0.698846	−	0.715272i	$-0.253697\pi$
$941$	42.8752			1.39769			0.698846	+	0.715272i	$0.253697\pi$
$942$		0			0
$943$		−	11.4056i		−	0.371417i
$944$		0			0
$945$	4.32106	−	4.61827i	0.140564	−	0.150232i
$946$		0			0
$947$			11.3428i			0.368591i	0.982871	+	0.184295i	$0.0590003\pi$
$947$			11.3428i			0.368591i	−0.982871	+	0.184295i	$0.941000\pi$
$948$		0			0
$949$	−9.28248			−0.301322
$950$		0			0
$951$	−3.68592			−0.119524
$952$		0			0
$953$			0.754313i			0.0244346i	0.999925	+	0.0122173i	$0.00388898\pi$
$953$			0.754313i			0.0244346i	−0.999925	+	0.0122173i	$0.996111\pi$
$954$		0			0
$955$	−9.62312	+	10.2850i	−0.311397	+	0.332815i
$956$		0			0
$957$		−	5.09618i		−	0.164736i
$958$		0			0
$959$	30.6751			0.990552
$960$		0			0
$961$	62.2548			2.00822
$962$		0			0
$963$		−	4.87437i		−	0.157074i
$964$		0			0
$965$	−38.8110	−	36.3133i	−1.24937	−	1.16897i
$966$		0			0
$967$			17.2530i			0.554820i	0.960752	+	0.277410i	$0.0894760\pi$
$967$			17.2530i			0.554820i	−0.960752	+	0.277410i	$0.910524\pi$
$968$		0			0
$969$	−47.3173			−1.52005
$970$		0			0
$971$	−35.2825			−1.13227			−0.566134	−	0.824313i	$-0.691562\pi$
$971$	−35.2825			−1.13227			−0.566134	+	0.824313i	$0.691562\pi$
$972$		0			0
$973$			26.1249i			0.837525i
$974$		0			0
$975$	−0.332104	−	4.98896i	−0.0106358	−	0.159775i
$976$		0			0
$977$		−	9.04808i		−	0.289474i	−0.989470	−	0.144737i	$-0.953766\pi$
$977$		−	9.04808i		−	0.289474i	0.989470	−	0.144737i	$-0.0462336\pi$
$978$		0			0
$979$	−0.147738			−0.00472172
$980$		0			0
$981$	−7.14949			−0.228266
$982$		0			0
$983$			30.5255i			0.973611i	0.873510	+	0.486805i	$0.161838\pi$
$983$			30.5255i			0.973611i	−0.873510	+	0.486805i	$0.838162\pi$
$984$		0			0
$985$	12.4547	+	11.6532i	0.396839	+	0.371301i
$986$		0			0
$987$			32.1776i			1.02423i
$988$		0			0
$989$	−26.0086			−0.827025
$990$		0			0
$991$	−6.10048			−0.193788			−0.0968940	−	0.995295i	$-0.530891\pi$
$991$	−6.10048			−0.193788			−0.0968940	+	0.995295i	$0.530891\pi$
$992$		0			0
$993$		−	24.6850i		−	0.783355i
$994$		0			0
$995$	10.4684	−	11.1884i	0.331871	−	0.354697i
$996$		0			0
$997$			31.4510i			0.996062i	0.867159	+	0.498031i	$0.165944\pi$
$997$			31.4510i			0.996062i	−0.867159	+	0.498031i	$0.834056\pi$
$998$		0			0
$999$	8.11091			0.256618

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1560.2.l.e.1249.5	yes	8
3.2	odd	2		4680.2.l.f.2809.7		8
4.3	odd	2		3120.2.l.o.1249.1		8
5.2	odd	4		7800.2.a.bw.1.4		4
5.3	odd	4		7800.2.a.bv.1.2		4
5.4	even	2	inner	1560.2.l.e.1249.1	✓	8
15.14	odd	2		4680.2.l.f.2809.8		8
20.19	odd	2		3120.2.l.o.1249.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.e.1249.1	✓	8	5.4	even	2	inner
1560.2.l.e.1249.5	yes	8	1.1	even	1	trivial
3120.2.l.o.1249.1		8	4.3	odd	2
3120.2.l.o.1249.5		8	20.19	odd	2
4680.2.l.f.2809.7		8	3.2	odd	2
4680.2.l.f.2809.8		8	15.14	odd	2
7800.2.a.bv.1.2		4	5.3	odd	4
7800.2.a.bw.1.4		4	5.2	odd	4

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:	\( N \)	\(=\)	\( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1560.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-1.52773 + 1.63280i) q^{5} -2.82843i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(-1.52773 + 1.63280i) q^{5} -2.82843i q^{7} -1.00000 q^{9} +1.05545 q^{11} -1.00000i q^{13} +(-1.63280 - 1.52773i) q^{15} +7.14949i q^{17} +6.61827 q^{19} +2.82843 q^{21} +3.49264i q^{23} +(-0.332104 - 4.98896i) q^{25} -1.00000i q^{27} -4.82843 q^{29} -9.65685 q^{31} +1.05545i q^{33} +(4.61827 + 4.32106i) q^{35} +8.11091i q^{37} +1.00000 q^{39} -3.26561 q^{41} +7.44670i q^{43} +(1.52773 - 1.63280i) q^{45} +11.3765i q^{47} -1.00000 q^{49} -7.14949 q^{51} -4.53122i q^{53} +(-1.61244 + 1.72335i) q^{55} +6.61827i q^{57} -1.05545 q^{59} -7.97792 q^{61} +2.82843i q^{63} +(1.63280 + 1.52773i) q^{65} +1.46878i q^{67} -3.49264 q^{69} +0.845296 q^{71} -9.28248i q^{73} +(4.98896 - 0.332104i) q^{75} -2.98527i q^{77} -6.85228 q^{79} +1.00000 q^{81} -7.97093i q^{83} +(-11.6737 - 10.9225i) q^{85} -4.82843i q^{87} -0.139976 q^{89} -2.82843 q^{91} -9.65685i q^{93} +(-10.1109 + 10.8063i) q^{95} -6.93933i q^{97} -1.05545 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} - 16 q^{11} - 4 q^{15} + 24 q^{19} - 4 q^{25} - 16 q^{29} - 32 q^{31} + 8 q^{35} + 8 q^{39} - 8 q^{41} - 8 q^{49} + 8 q^{51} - 36 q^{55} + 16 q^{59} + 24 q^{61} + 4 q^{65} - 8 q^{69} - 24 q^{71} - 4 q^{75} + 24 q^{79} + 8 q^{81} - 40 q^{85} + 8 q^{89} - 32 q^{95} + 16 q^{99}+O(q^{100}) \) 8 * q - 8 * q^9 - 16 * q^11 - 4 * q^15 + 24 * q^19 - 4 * q^25 - 16 * q^29 - 32 * q^31 + 8 * q^35 + 8 * q^39 - 8 * q^41 - 8 * q^49 + 8 * q^51 - 36 * q^55 + 16 * q^59 + 24 * q^61 + 4 * q^65 - 8 * q^69 - 24 * q^71 - 4 * q^75 + 24 * q^79 + 8 * q^81 - 40 * q^85 + 8 * q^89 - 32 * q^95 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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