LMFDB - Embedded newform 1120.2.e.a.1119.32

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1120,2,Mod(1119,1120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1120, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1120.1119");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1120.e (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:	$8.94324502638$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1119.32
Character	$\chi$	$=$	1120.1119
Dual form			1120.2.e.a.1119.31

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.568337i q^{3} +(2.03890 + 0.918082i) q^{5} +(-1.76634 + 1.96978i) q^{7} +2.67699 q^{9} +O(q^{10})$	\(q+0.568337i q^{3} +(2.03890 + 0.918082i) q^{5} +(-1.76634 + 1.96978i) q^{7} +2.67699 q^{9} -0.417558i q^{11} +2.13777 q^{13} +(-0.521780 + 1.15878i) q^{15} +0.314177 q^{17} +2.19560 q^{19} +(-1.11950 - 1.00388i) q^{21} -0.992696 q^{23} +(3.31425 + 3.74376i) q^{25} +3.22645i q^{27} -4.35539 q^{29} +4.22727 q^{31} +0.237314 q^{33} +(-5.40982 + 2.39455i) q^{35} +6.83413i q^{37} +1.21498i q^{39} -3.82702i q^{41} -6.04572 q^{43} +(5.45813 + 2.45770i) q^{45} +4.98866i q^{47} +(-0.760076 - 6.95861i) q^{49} +0.178558i q^{51} +7.14952i q^{53} +(0.383352 - 0.851360i) q^{55} +1.24784i q^{57} +9.50211 q^{59} -0.783607i q^{61} +(-4.72848 + 5.27309i) q^{63} +(4.35871 + 1.96265i) q^{65} -11.1044 q^{67} -0.564186i q^{69} +6.55657i q^{71} +5.11659 q^{73} +(-2.12772 + 1.88361i) q^{75} +(0.822498 + 0.737550i) q^{77} -11.4459i q^{79} +6.19727 q^{81} +5.49750i q^{83} +(0.640576 + 0.288440i) q^{85} -2.47533i q^{87} -12.5863i q^{89} +(-3.77604 + 4.21094i) q^{91} +2.40252i q^{93} +(4.47662 + 2.01574i) q^{95} +0.314177 q^{97} -1.11780i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{9}+O(q^{10}) $ 48 * q - 48 * q^9	$ 48 q - 48 q^{9} + 8 q^{21} - 16 q^{25} + 16 q^{29} + 24 q^{49} - 16 q^{65} - 32 q^{81} + 32 q^{85}+O(q^{100}) $ 48 * q - 48 * q^9 + 8 * q^21 - 16 * q^25 + 16 * q^29 + 24 * q^49 - 16 * q^65 - 32 * q^81 + 32 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$			0.568337i			0.328130i	0.986450	+	0.164065i	$0.0524606\pi$
$3$			0.568337i			0.328130i	−0.986450	+	0.164065i	$0.947539\pi$
$4$		0			0
$5$	2.03890	+	0.918082i	0.911825	+	0.410579i
$5$	2.03890	+	0.918082i	0.911825	+	0.410579i
$6$		0			0
$7$	−1.76634	+	1.96978i	−0.667614	+	0.744507i
$7$	−1.76634	+	1.96978i	−0.667614	+	0.744507i
$8$		0			0
$9$	2.67699			0.892331
$10$		0			0
$11$		−	0.417558i		−	0.125898i	−0.998017	−	0.0629492i	$-0.979949\pi$
$11$		−	0.417558i		−	0.125898i	0.998017	−	0.0629492i	$-0.0200506\pi$
$12$		0			0
$13$	2.13777			0.592911			0.296456	−	0.955047i	$-0.404195\pi$
$13$	2.13777			0.592911			0.296456	+	0.955047i	$0.404195\pi$
$14$		0			0
$15$	−0.521780	+	1.15878i	−0.134723	+	0.299197i
$16$		0			0
$17$	0.314177			0.0761990			0.0380995	−	0.999274i	$-0.487870\pi$
$17$	0.314177			0.0761990			0.0380995	+	0.999274i	$0.487870\pi$
$18$		0			0
$19$	2.19560			0.503705			0.251853	−	0.967766i	$-0.418960\pi$
$19$	2.19560			0.503705			0.251853	+	0.967766i	$0.418960\pi$
$20$		0			0
$21$	−1.11950	−	1.00388i	−0.244295	−	0.219064i
$22$		0			0
$23$	−0.992696			−0.206991			−0.103496	−	0.994630i	$-0.533003\pi$
$23$	−0.992696			−0.206991			−0.103496	+	0.994630i	$0.533003\pi$
$24$		0			0
$25$	3.31425	+	3.74376i	0.662850	+	0.748752i
$26$		0			0
$27$			3.22645i			0.620930i
$28$		0			0
$29$	−4.35539			−0.808776			−0.404388	−	0.914588i	$-0.632515\pi$
$29$	−4.35539			−0.808776			−0.404388	+	0.914588i	$0.632515\pi$
$30$		0			0
$31$	4.22727			0.759240			0.379620	−	0.925142i	$-0.376055\pi$
$31$	4.22727			0.759240			0.379620	+	0.925142i	$0.376055\pi$
$32$		0			0
$33$	0.237314			0.0413110
$34$		0			0
$35$	−5.40982	+	2.39455i	−0.914426	+	0.404752i
$36$		0			0
$37$			6.83413i			1.12352i	0.827299	+	0.561762i	$0.189876\pi$
$37$			6.83413i			1.12352i	−0.827299	+	0.561762i	$0.810124\pi$
$38$		0			0
$39$			1.21498i			0.194552i
$40$		0			0
$41$		−	3.82702i		−	0.597680i	−0.954303	−	0.298840i	$-0.903400\pi$
$41$		−	3.82702i		−	0.597680i	0.954303	−	0.298840i	$-0.0965996\pi$
$42$		0			0
$43$	−6.04572			−0.921963			−0.460981	−	0.887410i	$-0.652503\pi$
$43$	−6.04572			−0.921963			−0.460981	+	0.887410i	$0.652503\pi$
$44$		0			0
$45$	5.45813	+	2.45770i	0.813650	+	0.366372i
$46$		0			0
$47$			4.98866i			0.727671i	0.931463	+	0.363835i	$0.118533\pi$
$47$			4.98866i			0.727671i	−0.931463	+	0.363835i	$0.881467\pi$
$48$		0			0
$49$	−0.760076	−	6.95861i	−0.108582	−	0.994087i
$50$		0			0
$51$			0.178558i			0.0250032i
$52$		0			0
$53$			7.14952i			0.982063i	0.871142	+	0.491031i	$0.163380\pi$
$53$			7.14952i			0.982063i	−0.871142	+	0.491031i	$0.836620\pi$
$54$		0			0
$55$	0.383352	−	0.851360i	0.0516912	−	0.114797i
$56$		0			0
$57$			1.24784i			0.165281i
$58$		0			0
$59$	9.50211			1.23707			0.618534	−	0.785758i	$-0.287727\pi$
$59$	9.50211			1.23707			0.618534	+	0.785758i	$0.287727\pi$
$60$		0			0
$61$		−	0.783607i		−	0.100331i	−0.998741	−	0.0501653i	$-0.984025\pi$
$61$		−	0.783607i		−	0.100331i	0.998741	−	0.0501653i	$-0.0159748\pi$
$62$		0			0
$63$	−4.72848	+	5.27309i	−0.595733	+	0.664347i
$64$		0			0
$65$	4.35871	+	1.96265i	0.540631	+	0.243437i
$66$		0			0
$67$	−11.1044			−1.35661			−0.678307	−	0.734779i	$-0.737286\pi$
$67$	−11.1044			−1.35661			−0.678307	+	0.734779i	$0.737286\pi$
$68$		0			0
$69$		−	0.564186i		−	0.0679200i
$70$		0			0
$71$			6.55657i			0.778121i	0.921212	+	0.389061i	$0.127200\pi$
$71$			6.55657i			0.778121i	−0.921212	+	0.389061i	$0.872800\pi$
$72$		0			0
$73$	5.11659			0.598852			0.299426	−	0.954120i	$-0.403205\pi$
$73$	5.11659			0.598852			0.299426	+	0.954120i	$0.403205\pi$
$74$		0			0
$75$	−2.12772	+	1.88361i	−0.245688	+	0.217501i
$76$		0			0
$77$	0.822498	+	0.737550i	0.0937323	+	0.0840516i
$78$		0			0
$79$		−	11.4459i		−	1.28777i	−0.765124	−	0.643883i	$-0.777322\pi$
$79$		−	11.4459i		−	1.28777i	0.765124	−	0.643883i	$-0.222678\pi$
$80$		0			0
$81$	6.19727			0.688586
$82$		0			0
$83$			5.49750i			0.603429i	0.953398	+	0.301715i	$0.0975590\pi$
$83$			5.49750i			0.603429i	−0.953398	+	0.301715i	$0.902441\pi$
$84$		0			0
$85$	0.640576	+	0.288440i	0.0694802	+	0.0312857i
$86$		0			0
$87$		−	2.47533i		−	0.265383i
$88$		0			0
$89$		−	12.5863i		−	1.33415i	−0.744991	−	0.667074i	$-0.767546\pi$
$89$		−	12.5863i		−	1.33415i	0.744991	−	0.667074i	$-0.232454\pi$
$90$		0			0
$91$	−3.77604	+	4.21094i	−0.395836	+	0.441427i
$92$		0			0
$93$			2.40252i			0.249129i
$94$		0			0
$95$	4.47662	+	2.01574i	0.459291	+	0.206811i
$96$		0			0
$97$	0.314177			0.0318998			0.0159499	−	0.999873i	$-0.494923\pi$
$97$	0.314177			0.0318998			0.0159499	+	0.999873i	$0.494923\pi$
$98$		0			0
$99$		−	1.11780i		−	0.112343i
$100$		0			0
$101$			17.1970i			1.71116i	0.517670	+	0.855581i	$0.326800\pi$
$101$			17.1970i			1.71116i	−0.517670	+	0.855581i	$0.673200\pi$
$102$		0			0
$103$			1.34147i			0.132179i	0.997814	+	0.0660895i	$0.0210523\pi$
$103$			1.34147i			0.132179i	−0.997814	+	0.0660895i	$0.978948\pi$
$104$		0			0
$105$	−1.36091	−	3.07460i	−0.132811	−	0.300050i
$106$		0			0
$107$	11.1044			1.07350			0.536750	−	0.843742i	$-0.319652\pi$
$107$	11.1044			1.07350			0.536750	+	0.843742i	$0.319652\pi$
$108$		0			0
$109$	−7.87554			−0.754340			−0.377170	−	0.926144i	$-0.623103\pi$
$109$	−7.87554			−0.754340			−0.377170	+	0.926144i	$0.623103\pi$
$110$		0			0
$111$	−3.88409			−0.368661
$112$		0			0
$113$		−	13.4598i		−	1.26619i	−0.774075	−	0.633094i	$-0.781785\pi$
$113$		−	13.4598i		−	1.26619i	0.774075	−	0.633094i	$-0.218215\pi$
$114$		0			0
$115$	−2.02401	−	0.911376i	−0.188740	−	0.0849863i
$116$		0			0
$117$	5.72280			0.529073
$118$		0			0
$119$	−0.554943	+	0.618859i	−0.0508716	+	0.0567307i
$120$		0			0
$121$	10.8256			0.984150
$122$		0			0
$123$	2.17504			0.196116
$124$		0			0
$125$	3.32036	+	10.6759i	0.296982	+	0.954883i
$126$		0			0
$127$	2.48912			0.220874			0.110437	−	0.993883i	$-0.464775\pi$
$127$	2.48912			0.220874			0.110437	+	0.993883i	$0.464775\pi$
$128$		0			0
$129$		−	3.43600i		−	0.302523i
$130$		0			0
$131$	−16.9444			−1.48044			−0.740219	−	0.672366i	$-0.765278\pi$
$131$	−16.9444			−1.48044			−0.740219	+	0.672366i	$0.765278\pi$
$132$		0			0
$133$	−3.87818	+	4.32485i	−0.336281	+	0.375012i
$134$		0			0
$135$	−2.96214	+	6.57841i	−0.254941	+	0.566179i
$136$		0			0
$137$		−	8.90368i		−	0.760693i	−0.924844	−	0.380346i	$-0.875805\pi$
$137$		−	8.90368i		−	0.760693i	0.924844	−	0.380346i	$-0.124195\pi$
$138$		0			0
$139$	16.8468			1.42892			0.714462	−	0.699674i	$-0.246672\pi$
$139$	16.8468			1.42892			0.714462	+	0.699674i	$0.246672\pi$
$140$		0			0
$141$	−2.83524			−0.238770
$142$		0			0
$143$		−	0.892644i		−	0.0746466i
$144$		0			0
$145$	−8.88022	−	3.99861i	−0.737462	−	0.332066i
$146$		0			0
$147$	3.95484	−	0.431979i	0.326190	−	0.0356290i
$148$		0			0
$149$	−13.1069			−1.07376			−0.536880	−	0.843658i	$-0.680397\pi$
$149$	−13.1069			−1.07376			−0.536880	+	0.843658i	$0.680397\pi$
$150$		0			0
$151$			9.12834i			0.742854i	0.928462	+	0.371427i	$0.121131\pi$
$151$			9.12834i			0.742854i	−0.928462	+	0.371427i	$0.878869\pi$
$152$		0			0
$153$	0.841049			0.0679948
$154$		0			0
$155$	8.61900	+	3.88098i	0.692294	+	0.311728i
$156$		0			0
$157$	6.99791			0.558494			0.279247	−	0.960219i	$-0.409915\pi$
$157$	6.99791			0.558494			0.279247	+	0.960219i	$0.409915\pi$
$158$		0			0
$159$	−4.06334			−0.322244
$160$		0			0
$161$	1.75344	−	1.95539i	0.138190	−	0.154107i
$162$		0			0
$163$	1.88529			0.147667			0.0738336	−	0.997271i	$-0.476477\pi$
$163$	1.88529			0.147667			0.0738336	+	0.997271i	$0.476477\pi$
$164$		0			0
$165$	0.483860	+	0.217873i	0.0376684	+	0.0169614i
$166$		0			0
$167$		−	19.5972i		−	1.51647i	−0.651979	−	0.758237i	$-0.726061\pi$
$167$		−	19.5972i		−	1.51647i	0.651979	−	0.758237i	$-0.273939\pi$
$168$		0			0
$169$	−8.42993			−0.648456
$170$		0			0
$171$	5.87761			0.449472
$172$		0			0
$173$	−18.5244			−1.40838			−0.704191	−	0.710011i	$-0.748690\pi$
$173$	−18.5244			−1.40838			−0.704191	+	0.710011i	$0.748690\pi$
$174$		0			0
$175$	−13.2285	−	0.0844107i	−0.999980	−	0.00638085i
$176$		0			0
$177$			5.40040i			0.405919i
$178$		0			0
$179$		−	21.2960i		−	1.59174i	−0.605468	−	0.795870i	$-0.707014\pi$
$179$		−	21.2960i		−	1.59174i	0.605468	−	0.795870i	$-0.292986\pi$
$180$		0			0
$181$			8.09936i			0.602020i	0.953621	+	0.301010i	$0.0973238\pi$
$181$			8.09936i			0.602020i	−0.953621	+	0.301010i	$0.902676\pi$
$182$		0			0
$183$	0.445353			0.0329214
$184$		0			0
$185$	−6.27429	+	13.9341i	−0.461295	+	1.02446i
$186$		0			0
$187$		−	0.131187i		−	0.00959334i
$188$		0			0
$189$	−6.35539	−	5.69901i	−0.462287	−	0.414542i
$190$		0			0
$191$		−	4.97598i		−	0.360049i	−0.983662	−	0.180024i	$-0.942382\pi$
$191$		−	4.97598i		−	0.360049i	0.983662	−	0.180024i	$-0.0576177\pi$
$192$		0			0
$193$		−	0.670962i		−	0.0482969i	−0.999708	−	0.0241484i	$-0.992313\pi$
$193$		−	0.670962i		−	0.0482969i	0.999708	−	0.0241484i	$-0.00768744\pi$
$194$		0			0
$195$	−1.11545	+	2.47722i	−0.0798788	+	0.177397i
$196$		0			0
$197$		−	10.2280i		−	0.728717i	−0.931259	−	0.364359i	$-0.881288\pi$
$197$		−	10.2280i		−	0.728717i	0.931259	−	0.364359i	$-0.118712\pi$
$198$		0			0
$199$	26.3489			1.86782			0.933911	−	0.357506i	$-0.116373\pi$
$199$	26.3489			1.86782			0.933911	+	0.357506i	$0.116373\pi$
$200$		0			0
$201$		−	6.31102i		−	0.445145i
$202$		0			0
$203$	7.69311	−	8.57917i	0.539950	−	0.602140i
$204$		0			0
$205$	3.51352	−	7.80292i	0.245395	−	0.544979i
$206$		0			0
$207$	−2.65744			−0.184705
$208$		0			0
$209$		−	0.916791i		−	0.0634157i
$210$		0			0
$211$		−	10.4904i		−	0.722188i	−0.932529	−	0.361094i	$-0.882403\pi$
$211$		−	10.4904i		−	0.722188i	0.932529	−	0.361094i	$-0.117597\pi$
$212$		0			0
$213$	−3.72634			−0.255325
$214$		0			0
$215$	−12.3266	−	5.55046i	−0.840669	−	0.378538i
$216$		0			0
$217$	−7.46680	+	8.32680i	−0.506880	+	0.565260i
$218$		0			0
$219$			2.90795i			0.196501i
$220$		0			0
$221$	0.671638			0.0451793
$222$		0			0
$223$		−	16.5160i		−	1.10599i	−0.833183	−	0.552997i	$-0.813484\pi$
$223$		−	16.5160i		−	1.10599i	0.833183	−	0.552997i	$-0.186516\pi$
$224$		0			0
$225$	8.87223	+	10.0220i	0.591482	+	0.668135i
$226$		0			0
$227$		−	25.5638i		−	1.69673i	−0.529415	−	0.848363i	$-0.677589\pi$
$227$		−	25.5638i		−	1.69673i	0.529415	−	0.848363i	$-0.322411\pi$
$228$		0			0
$229$		−	23.9007i		−	1.57940i	−0.613492	−	0.789701i	$-0.710236\pi$
$229$		−	23.9007i		−	1.57940i	0.613492	−	0.789701i	$-0.289764\pi$
$230$		0			0
$231$	−0.419177	+	0.467456i	−0.0275798	+	0.0307563i
$232$		0			0
$233$		−	27.7990i		−	1.82117i	−0.413319	−	0.910586i	$-0.635631\pi$
$233$		−	27.7990i		−	1.82117i	0.413319	−	0.910586i	$-0.364369\pi$
$234$		0			0
$235$	−4.58000	+	10.1714i	−0.298766	+	0.663509i
$236$		0			0
$237$	6.50514			0.422554
$238$		0			0
$239$		−	27.6363i		−	1.78765i	−0.448420	−	0.893823i	$-0.648013\pi$
$239$		−	27.6363i		−	1.78765i	0.448420	−	0.893823i	$-0.351987\pi$
$240$		0			0
$241$			22.8312i			1.47069i	0.677694	+	0.735344i	$0.262980\pi$
$241$			22.8312i			1.47069i	−0.677694	+	0.735344i	$0.737020\pi$
$242$		0			0
$243$			13.2015i			0.846875i
$244$		0			0
$245$	4.83886	−	14.8857i	0.309143	−	0.951015i
$246$		0			0
$247$	4.69370			0.298653
$248$		0			0
$249$	−3.12444			−0.198003
$250$		0			0
$251$	4.13678			0.261111			0.130556	−	0.991441i	$-0.458324\pi$
$251$	4.13678			0.261111			0.130556	+	0.991441i	$0.458324\pi$
$252$		0			0
$253$			0.414508i			0.0260599i
$254$		0			0
$255$	−0.163931	+	0.364063i	−0.0102658	+	0.0227985i
$256$		0			0
$257$	−23.0124			−1.43547			−0.717736	−	0.696316i	$-0.754821\pi$
$257$	−23.0124			−1.43547			−0.717736	+	0.696316i	$0.754821\pi$
$258$		0			0
$259$	−13.4617	−	12.0714i	−0.836472	−	0.750081i
$260$		0			0
$261$	−11.6594			−0.721696
$262$		0			0
$263$	22.7561			1.40320			0.701600	−	0.712571i	$-0.252470\pi$
$263$	22.7561			1.40320			0.701600	+	0.712571i	$0.252470\pi$
$264$		0			0
$265$	−6.56385	+	14.5772i	−0.403214	+	0.895469i
$266$		0			0
$267$	7.15328			0.437774
$268$		0			0
$269$			14.4798i			0.882851i	0.897298	+	0.441425i	$0.145527\pi$
$269$			14.4798i			0.882851i	−0.897298	+	0.441425i	$0.854473\pi$
$270$		0			0
$271$	−29.3183			−1.78096			−0.890480	−	0.455022i	$-0.849631\pi$
$271$	−29.3183			−1.78096			−0.890480	+	0.455022i	$0.849631\pi$
$272$		0			0
$273$	−2.39324	−	2.14606i	−0.144845	−	0.129886i
$274$		0			0
$275$	1.56324	−	1.38389i	0.0942667	−	0.0834518i
$276$		0			0
$277$		−	18.7307i		−	1.12542i	−0.826656	−	0.562708i	$-0.809759\pi$
$277$		−	18.7307i		−	1.12542i	0.826656	−	0.562708i	$-0.190241\pi$
$278$		0			0
$279$	11.3164			0.677494
$280$		0			0
$281$	−1.59471			−0.0951324			−0.0475662	−	0.998868i	$-0.515147\pi$
$281$	−1.59471			−0.0951324			−0.0475662	+	0.998868i	$0.515147\pi$
$282$		0			0
$283$		−	27.1464i		−	1.61369i	−0.590764	−	0.806844i	$-0.701174\pi$
$283$		−	27.1464i		−	1.61369i	0.590764	−	0.806844i	$-0.298826\pi$
$284$		0			0
$285$	−1.14562	+	2.54423i	−0.0678607	+	0.150707i
$286$		0			0
$287$	7.53839	+	6.75982i	0.444977	+	0.399019i
$288$		0			0
$289$	−16.9013			−0.994194
$290$		0			0
$291$			0.178558i			0.0104673i
$292$		0			0
$293$	−22.1716			−1.29528			−0.647638	−	0.761948i	$-0.724243\pi$
$293$	−22.1716			−1.29528			−0.647638	+	0.761948i	$0.724243\pi$
$294$		0			0
$295$	19.3739	+	8.72371i	1.12799	+	0.507914i
$296$		0			0
$297$	1.34723			0.0781741
$298$		0			0
$299$	−2.12216			−0.122728
$300$		0			0
$301$	10.6788	−	11.9087i	0.615516	−	0.686408i
$302$		0			0
$303$	−9.77367			−0.561483
$304$		0			0
$305$	0.719416	−	1.59770i	0.0411936	−	0.0914840i
$306$		0			0
$307$			20.8032i			1.18730i	0.804722	+	0.593652i	$0.202314\pi$
$307$			20.8032i			1.18730i	−0.804722	+	0.593652i	$0.797686\pi$
$308$		0			0
$309$	−0.762407			−0.0433718
$310$		0			0
$311$	21.7937			1.23581			0.617905	−	0.786253i	$-0.287982\pi$
$311$	21.7937			1.23581			0.617905	+	0.786253i	$0.287982\pi$
$312$		0			0
$313$	20.2049			1.14205			0.571026	−	0.820932i	$-0.306546\pi$
$313$	20.2049			1.14205			0.571026	+	0.820932i	$0.306546\pi$
$314$		0			0
$315$	−14.4820	+	6.41018i	−0.815971	+	0.361173i
$316$		0			0
$317$		−	18.5623i		−	1.04257i	−0.853384	−	0.521283i	$-0.825454\pi$
$317$		−	18.5623i		−	1.04257i	0.853384	−	0.521283i	$-0.174546\pi$
$318$		0			0
$319$			1.81863i			0.101824i
$320$		0			0
$321$			6.31102i			0.352247i
$322$		0			0
$323$	0.689807			0.0383819
$324$		0			0
$325$	7.08511	+	8.00331i	0.393011	+	0.443944i
$326$		0			0
$327$		−	4.47596i		−	0.247521i
$328$		0			0
$329$	−9.82657	−	8.81168i	−0.541756	−	0.485804i
$330$		0			0
$331$			21.3958i			1.17602i	0.808853	+	0.588010i	$0.200089\pi$
$331$			21.3958i			1.17602i	−0.808853	+	0.588010i	$0.799911\pi$
$332$		0			0
$333$			18.2949i			1.00256i
$334$		0			0
$335$	−22.6407	−	10.1947i	−1.23699	−	0.556997i
$336$		0			0
$337$			6.81153i			0.371048i	0.982640	+	0.185524i	$0.0593982\pi$
$337$			6.81153i			0.371048i	−0.982640	+	0.185524i	$0.940602\pi$
$338$		0			0
$339$	7.64969			0.415474
$340$		0			0
$341$		−	1.76513i		−	0.0955872i
$342$		0			0
$343$	15.0495	+	10.7941i	0.812596	+	0.582827i
$344$		0			0
$345$	0.517969	−	1.15032i	0.0278865	−	0.0619312i
$346$		0			0
$347$	3.10677			0.166780			0.0833901	−	0.996517i	$-0.473425\pi$
$347$	3.10677			0.166780			0.0833901	+	0.996517i	$0.473425\pi$
$348$		0			0
$349$		−	12.7904i		−	0.684656i	−0.939580	−	0.342328i	$-0.888785\pi$
$349$		−	12.7904i		−	0.684656i	0.939580	−	0.342328i	$-0.111215\pi$
$350$		0			0
$351$			6.89741i			0.368156i
$352$		0			0
$353$	22.3980			1.19212			0.596062	−	0.802938i	$-0.296731\pi$
$353$	22.3980			1.19212			0.596062	+	0.802938i	$0.296731\pi$
$354$		0			0
$355$	−6.01947	+	13.3682i	−0.319480	+	0.709511i
$356$		0			0
$357$	−0.351721	−	0.315395i	−0.0186150	−	0.0166925i
$358$		0			0
$359$			5.07412i			0.267802i	0.990995	+	0.133901i	$0.0427503\pi$
$359$			5.07412i			0.267802i	−0.990995	+	0.133901i	$0.957250\pi$
$360$		0			0
$361$	−14.1793			−0.746281
$362$		0			0
$363$			6.15262i			0.322929i
$364$		0			0
$365$	10.4322	+	4.69745i	0.546048	+	0.245876i
$366$		0			0
$367$			25.1559i			1.31313i	0.754270	+	0.656564i	$0.227991\pi$
$367$			25.1559i			1.31313i	−0.754270	+	0.656564i	$0.772009\pi$
$368$		0			0
$369$		−	10.2449i		−	0.533328i
$370$		0			0
$371$	−14.0830	−	12.6285i	−0.731153	−	0.655639i
$372$		0			0
$373$			10.0597i			0.520872i	0.965491	+	0.260436i	$0.0838664\pi$
$373$			10.0597i			0.520872i	−0.965491	+	0.260436i	$0.916134\pi$
$374$		0			0
$375$	−6.06752	+	1.88708i	−0.313325	+	0.0974485i
$376$		0			0
$377$	−9.31084			−0.479533
$378$		0			0
$379$			11.0716i			0.568707i	0.958719	+	0.284354i	$0.0917790\pi$
$379$			11.0716i			0.568707i	−0.958719	+	0.284354i	$0.908221\pi$
$380$		0			0
$381$			1.41466i			0.0724753i
$382$		0			0
$383$		−	9.17507i		−	0.468824i	−0.972137	−	0.234412i	$-0.924684\pi$
$383$		−	9.17507i		−	0.468824i	0.972137	−	0.234412i	$-0.0753165\pi$
$384$		0			0
$385$	0.999862	+	2.25891i	0.0509577	+	0.115125i
$386$		0			0
$387$	−16.1843			−0.822696
$388$		0			0
$389$	−2.66261			−0.135000			−0.0674999	−	0.997719i	$-0.521502\pi$
$389$	−2.66261			−0.135000			−0.0674999	+	0.997719i	$0.521502\pi$
$390$		0			0
$391$	−0.311882			−0.0157725
$392$		0			0
$393$		−	9.63012i		−	0.485775i
$394$		0			0
$395$	10.5083	−	23.3371i	0.528729	−	1.17422i
$396$		0			0
$397$	−11.4822			−0.576273			−0.288136	−	0.957589i	$-0.593036\pi$
$397$	−11.4822			−0.576273			−0.288136	+	0.957589i	$0.593036\pi$
$398$		0			0
$399$	−2.45798	−	2.20411i	−0.123053	−	0.110344i
$400$		0			0
$401$	−15.5028			−0.774171			−0.387086	−	0.922044i	$-0.626518\pi$
$401$	−15.5028			−0.774171			−0.387086	+	0.922044i	$0.626518\pi$
$402$		0			0
$403$	9.03694			0.450162
$404$		0			0
$405$	12.6356	+	5.68960i	0.627870	+	0.282719i
$406$		0			0
$407$	2.85364			0.141450
$408$		0			0
$409$			16.4612i			0.813953i	0.913439	+	0.406977i	$0.133417\pi$
$409$			16.4612i			0.813953i	−0.913439	+	0.406977i	$0.866583\pi$
$410$		0			0
$411$	5.06029			0.249606
$412$		0			0
$413$	−16.7840	+	18.7171i	−0.825885	+	0.921007i
$414$		0			0
$415$	−5.04716	+	11.2089i	−0.247755	+	0.550222i
$416$		0			0
$417$			9.57464i			0.468872i
$418$		0			0
$419$	2.09799			0.102493			0.0512467	−	0.998686i	$-0.483681\pi$
$419$	2.09799			0.102493			0.0512467	+	0.998686i	$0.483681\pi$
$420$		0			0
$421$	31.4382			1.53220			0.766101	−	0.642720i	$-0.222194\pi$
$421$	31.4382			1.53220			0.766101	+	0.642720i	$0.222194\pi$
$422$		0			0
$423$			13.3546i			0.649323i
$424$		0			0
$425$	1.04126	+	1.17620i	0.0505085	+	0.0570542i
$426$		0			0
$427$	1.54353	+	1.38412i	0.0746969	+	0.0669822i
$428$		0			0
$429$	0.507323			0.0244938
$430$		0			0
$431$		−	18.3661i		−	0.884662i	−0.896852	−	0.442331i	$-0.854152\pi$
$431$		−	18.3661i		−	0.884662i	0.896852	−	0.442331i	$-0.145848\pi$
$432$		0			0
$433$	38.7519			1.86230			0.931149	−	0.364639i	$-0.118808\pi$
$433$	38.7519			1.86230			0.931149	+	0.364639i	$0.118808\pi$
$434$		0			0
$435$	2.27256	−	5.04696i	0.108961	−	0.241983i
$436$		0			0
$437$	−2.17956			−0.104263
$438$		0			0
$439$	14.5414			0.694021			0.347011	−	0.937861i	$-0.387197\pi$
$439$	14.5414			0.694021			0.347011	+	0.937861i	$0.387197\pi$
$440$		0			0
$441$	−2.03472	−	18.6282i	−0.0968913	−	0.887055i
$442$		0			0
$443$	−11.1257			−0.528597			−0.264299	−	0.964441i	$-0.585140\pi$
$443$	−11.1257			−0.528597			−0.264299	+	0.964441i	$0.585140\pi$
$444$		0			0
$445$	11.5553	−	25.6623i	0.547773	−	1.21651i
$446$		0			0
$447$		−	7.44915i		−	0.352333i
$448$		0			0
$449$	−6.80764			−0.321272			−0.160636	−	0.987014i	$-0.551355\pi$
$449$	−6.80764			−0.321272			−0.160636	+	0.987014i	$0.551355\pi$
$450$		0			0
$451$	−1.59800			−0.0752469
$452$		0			0
$453$	−5.18798			−0.243752
$454$		0			0
$455$	−11.5650	+	5.11899i	−0.542174	+	0.239982i
$456$		0			0
$457$			12.1530i			0.568493i	0.958751	+	0.284246i	$0.0917434\pi$
$457$			12.1530i			0.568493i	−0.958751	+	0.284246i	$0.908257\pi$
$458$		0			0
$459$			1.01367i			0.0473143i
$460$		0			0
$461$		−	14.5294i		−	0.676700i	−0.941020	−	0.338350i	$-0.890131\pi$
$461$		−	14.5294i		−	0.676700i	0.941020	−	0.338350i	$-0.109869\pi$
$462$		0			0
$463$	−34.8827			−1.62114			−0.810568	−	0.585644i	$-0.800842\pi$
$463$	−34.8827			−1.62114			−0.810568	+	0.585644i	$0.800842\pi$
$464$		0			0
$465$	−2.20571	+	4.89850i	−0.102287	+	0.227162i
$466$		0			0
$467$			8.87768i			0.410810i	0.978677	+	0.205405i	$0.0658511\pi$
$467$			8.87768i			0.410810i	−0.978677	+	0.205405i	$0.934149\pi$
$468$		0			0
$469$	19.6141	−	21.8732i	0.905695	−	1.01001i
$470$		0			0
$471$			3.97717i			0.183259i
$472$		0			0
$473$			2.52444i			0.116074i
$474$		0			0
$475$	7.27677	+	8.21981i	0.333881	+	0.377151i
$476$		0			0
$477$			19.1392i			0.876325i
$478$		0			0
$479$	26.1849			1.19642			0.598210	−	0.801339i	$-0.295879\pi$
$479$	26.1849			1.19642			0.598210	+	0.801339i	$0.295879\pi$
$480$		0			0
$481$			14.6098i			0.666150i
$482$		0			0
$483$	1.11132	+	0.996545i	0.0505669	+	0.0453444i
$484$		0			0
$485$	0.640576	+	0.288440i	0.0290870	+	0.0130974i
$486$		0			0
$487$	−25.7184			−1.16541			−0.582707	−	0.812683i	$-0.698006\pi$
$487$	−25.7184			−1.16541			−0.582707	+	0.812683i	$0.698006\pi$
$488$		0			0
$489$			1.07148i			0.0484540i
$490$		0			0
$491$		−	8.43083i		−	0.380478i	−0.981738	−	0.190239i	$-0.939074\pi$
$491$		−	8.43083i		−	0.380478i	0.981738	−	0.190239i	$-0.0609263\pi$
$492$		0			0
$493$	−1.36836			−0.0616280
$494$		0			0
$495$	1.02623	−	2.27908i	0.0461257	−	0.102437i
$496$		0			0
$497$	−12.9150	−	11.5811i	−0.579317	−	0.519485i
$498$		0			0
$499$		−	27.5094i		−	1.23149i	−0.787946	−	0.615744i	$-0.788856\pi$
$499$		−	27.5094i		−	1.23149i	0.787946	−	0.615744i	$-0.211144\pi$
$500$		0			0
$501$	11.1378			0.497600
$502$		0			0
$503$		−	20.0825i		−	0.895434i	−0.894175	−	0.447717i	$-0.852237\pi$
$503$		−	20.0825i		−	0.895434i	0.894175	−	0.447717i	$-0.147763\pi$
$504$		0			0
$505$	−15.7882	+	35.0629i	−0.702566	+	1.56028i
$506$		0			0
$507$		−	4.79104i		−	0.212778i
$508$		0			0
$509$		−	16.0628i		−	0.711970i	−0.934492	−	0.355985i	$-0.884145\pi$
$509$		−	16.0628i		−	0.711970i	0.934492	−	0.355985i	$-0.115855\pi$
$510$		0			0
$511$	−9.03765	+	10.0786i	−0.399802	+	0.445850i
$512$		0			0
$513$			7.08399i			0.312766i
$514$		0			0
$515$	−1.23158	+	2.73513i	−0.0542699	+	0.120524i
$516$		0			0
$517$	2.08305			0.0916126
$518$		0			0
$519$		−	10.5281i		−	0.462132i
$520$		0			0
$521$			24.3375i			1.06625i	0.846038	+	0.533123i	$0.178982\pi$
$521$			24.3375i			1.06625i	−0.846038	+	0.533123i	$0.821018\pi$
$522$		0			0
$523$			9.55887i			0.417980i	0.977918	+	0.208990i	$0.0670176\pi$
$523$			9.55887i			0.417980i	−0.977918	+	0.208990i	$0.932982\pi$
$524$		0			0
$525$	0.0479738	−	7.51824i	0.00209375	−	0.328123i
$526$		0			0
$527$	1.32811			0.0578534
$528$		0			0
$529$	−22.0146			−0.957155
$530$		0			0
$531$	25.4371			1.10387
$532$		0			0
$533$		−	8.18129i		−	0.354371i
$534$		0			0
$535$	22.6407	+	10.1947i	0.978844	+	0.440756i
$536$		0			0
$537$	12.1033			0.522297
$538$		0			0
$539$	−2.90562	+	0.317376i	−0.125154	+	0.0136703i
$540$		0			0
$541$	32.4093			1.39338			0.696692	−	0.717370i	$-0.254654\pi$
$541$	32.4093			1.39338			0.696692	+	0.717370i	$0.254654\pi$
$542$		0			0
$543$	−4.60316			−0.197541
$544$		0			0
$545$	−16.0575	−	7.23040i	−0.687826	−	0.309716i
$546$		0			0
$547$	−11.9222			−0.509756			−0.254878	−	0.966973i	$-0.582035\pi$
$547$	−11.9222			−0.509756			−0.254878	+	0.966973i	$0.582035\pi$
$548$		0			0
$549$		−	2.09771i		−	0.0895281i
$550$		0			0
$551$	−9.56271			−0.407385
$552$		0			0
$553$	22.5459	+	20.2174i	0.958751	+	0.859731i
$554$		0			0
$555$	−7.91928	−	3.56591i	−0.336155	−	0.151365i
$556$		0			0
$557$		−	8.18888i		−	0.346974i	−0.984836	−	0.173487i	$-0.944497\pi$
$557$		−	8.18888i		−	0.346974i	0.984836	−	0.173487i	$-0.0555034\pi$
$558$		0			0
$559$	−12.9244			−0.546642
$560$		0			0
$561$	0.0745584			0.00314786
$562$		0			0
$563$		−	15.1994i		−	0.640577i	−0.947320	−	0.320289i	$-0.896220\pi$
$563$		−	15.1994i		−	0.640577i	0.947320	−	0.320289i	$-0.103780\pi$
$564$		0			0
$565$	12.3572	−	27.4432i	0.519870	−	1.15454i
$566$		0			0
$567$	−10.9465	+	12.2073i	−0.459710	+	0.512657i
$568$		0			0
$569$	5.51352			0.231139			0.115569	−	0.993299i	$-0.463131\pi$
$569$	5.51352			0.231139			0.115569	+	0.993299i	$0.463131\pi$
$570$		0			0
$571$			34.4092i			1.43998i	0.693985	+	0.719989i	$0.255853\pi$
$571$			34.4092i			1.43998i	−0.693985	+	0.719989i	$0.744147\pi$
$572$		0			0
$573$	2.82803			0.118143
$574$		0			0
$575$	−3.29004	−	3.71642i	−0.137204	−	0.154985i
$576$		0			0
$577$	38.2066			1.59056			0.795281	−	0.606240i	$-0.207323\pi$
$577$	38.2066			1.59056			0.795281	+	0.606240i	$0.207323\pi$
$578$		0			0
$579$	0.381333			0.0158476
$580$		0			0
$581$	−10.8289	−	9.71047i	−0.449258	−	0.402858i
$582$		0			0
$583$	2.98534			0.123640
$584$		0			0
$585$	11.6682	+	5.25400i	0.482422	+	0.217226i
$586$		0			0
$587$			29.5651i			1.22028i	0.792293	+	0.610141i	$0.208887\pi$
$587$			29.5651i			1.22028i	−0.792293	+	0.610141i	$0.791113\pi$
$588$		0			0
$589$	9.28140			0.382434
$590$		0			0
$591$	5.81297			0.239114
$592$		0			0
$593$	−3.15891			−0.129721			−0.0648604	−	0.997894i	$-0.520660\pi$
$593$	−3.15891			−0.129721			−0.0648604	+	0.997894i	$0.520660\pi$
$594$		0			0
$595$	−1.69964	+	0.752311i	−0.0696784	+	0.0308417i
$596$		0			0
$597$			14.9750i			0.612888i
$598$		0			0
$599$			13.0969i			0.535123i	0.963541	+	0.267562i	$0.0862179\pi$
$599$			13.0969i			0.535123i	−0.963541	+	0.267562i	$0.913782\pi$
$600$		0			0
$601$			21.3097i			0.869241i	0.900614	+	0.434621i	$0.143118\pi$
$601$			21.3097i			0.869241i	−0.900614	+	0.434621i	$0.856882\pi$
$602$		0			0
$603$	−29.7263			−1.21055
$604$		0			0
$605$	22.0724	+	9.93883i	0.897372	+	0.404071i
$606$		0			0
$607$			16.8102i			0.682303i	0.940008	+	0.341152i	$0.110817\pi$
$607$			16.8102i			0.682303i	−0.940008	+	0.341152i	$0.889183\pi$
$608$		0			0
$609$	4.87586	+	4.37228i	0.197580	+	0.177174i
$610$		0			0
$611$			10.6646i			0.431444i
$612$		0			0
$613$			32.3717i			1.30748i	0.756718	+	0.653741i	$0.226801\pi$
$613$			32.3717i			1.30748i	−0.756718	+	0.653741i	$0.773199\pi$
$614$		0			0
$615$	4.43469	+	1.99686i	0.178824	+	0.0805212i
$616$		0			0
$617$			30.4363i			1.22532i	0.790347	+	0.612659i	$0.209900\pi$
$617$			30.4363i			1.22532i	−0.790347	+	0.612659i	$0.790100\pi$
$618$		0			0
$619$	10.0232			0.402868			0.201434	−	0.979502i	$-0.435440\pi$
$619$	10.0232			0.402868			0.201434	+	0.979502i	$0.435440\pi$
$620$		0			0
$621$		−	3.20288i		−	0.128527i
$622$		0			0
$623$	24.7923	+	22.2318i	0.993283	+	0.890697i
$624$		0			0
$625$	−3.03148	+	24.8155i	−0.121259	+	0.992621i
$626$		0			0
$627$	0.521046			0.0208086
$628$		0			0
$629$			2.14712i			0.0856114i
$630$		0			0
$631$			38.8327i			1.54591i	0.634464	+	0.772953i	$0.281221\pi$
$631$			38.8327i			1.54591i	−0.634464	+	0.772953i	$0.718779\pi$
$632$		0			0
$633$	5.96208			0.236971
$634$		0			0
$635$	5.07508	+	2.28522i	0.201398	+	0.0906861i
$636$		0			0
$637$	−1.62487	−	14.8759i	−0.0643796	−	0.589406i
$638$		0			0
$639$			17.5519i			0.694342i
$640$		0			0
$641$	9.67246			0.382039			0.191020	−	0.981586i	$-0.438821\pi$
$641$	9.67246			0.382039			0.191020	+	0.981586i	$0.438821\pi$
$642$		0			0
$643$			41.6742i			1.64347i	0.569869	+	0.821736i	$0.306994\pi$
$643$			41.6742i			1.64347i	−0.569869	+	0.821736i	$0.693006\pi$
$644$		0			0
$645$	3.15453	−	7.00568i	0.124210	−	0.275848i
$646$		0			0
$647$		−	0.700919i		−	0.0275560i	−0.999905	−	0.0137780i	$-0.995614\pi$
$647$		−	0.700919i		−	0.0275560i	0.999905	−	0.0137780i	$-0.00438581\pi$
$648$		0			0
$649$		−	3.96768i		−	0.155745i
$650$		0			0
$651$	−4.73243	−	4.24366i	−0.185479	−	0.166322i
$652$		0			0
$653$			0.607909i			0.0237893i	0.999929	+	0.0118947i	$0.00378628\pi$
$653$			0.607909i			0.0237893i	−0.999929	+	0.0118947i	$0.996214\pi$
$654$		0			0
$655$	−34.5479	−	15.5563i	−1.34990	−	0.607836i
$656$		0			0
$657$	13.6971			0.534374
$658$		0			0
$659$			30.7316i			1.19713i	0.801073	+	0.598566i	$0.204263\pi$
$659$			30.7316i			1.19713i	−0.801073	+	0.598566i	$0.795737\pi$
$660$		0			0
$661$		−	2.19337i		−	0.0853123i	−0.999090	−	0.0426561i	$-0.986418\pi$
$661$		−	2.19337i		−	0.0853123i	0.999090	−	0.0426561i	$-0.0135820\pi$
$662$		0			0
$663$			0.381717i			0.0148247i
$664$		0			0
$665$	−11.8778	+	5.25747i	−0.460602	+	0.203876i
$666$		0			0
$667$	4.32358			0.167410
$668$		0			0
$669$	9.38667			0.362910
$670$		0			0
$671$	−0.327201			−0.0126315
$672$		0			0
$673$		−	40.8919i		−	1.57627i	−0.615503	−	0.788134i	$-0.711047\pi$
$673$		−	40.8919i		−	1.57627i	0.615503	−	0.788134i	$-0.288953\pi$
$674$		0			0
$675$	−12.0790	+	10.6933i	−0.464922	+	0.411583i
$676$		0			0
$677$	24.3019			0.933998			0.466999	−	0.884258i	$-0.345335\pi$
$677$	24.3019			0.933998			0.466999	+	0.884258i	$0.345335\pi$
$678$		0			0
$679$	−0.554943	+	0.618859i	−0.0212968	+	0.0237496i
$680$		0			0
$681$	14.5288			0.556746
$682$		0			0
$683$	29.6731			1.13541			0.567704	−	0.823232i	$-0.307832\pi$
$683$	29.6731			1.13541			0.567704	+	0.823232i	$0.307832\pi$
$684$		0			0
$685$	8.17431	−	18.1537i	0.312324	−	0.693619i
$686$		0			0
$687$	13.5836			0.518249
$688$		0			0
$689$			15.2841i			0.582276i
$690$		0			0
$691$	−19.9801			−0.760080			−0.380040	−	0.924970i	$-0.624090\pi$
$691$	−19.9801			−0.760080			−0.380040	+	0.924970i	$0.624090\pi$
$692$		0			0
$693$	2.20182	+	1.97442i	0.0836402	+	0.0750018i
$694$		0			0
$695$	34.3489	+	15.4667i	1.30293	+	0.586686i
$696$		0			0
$697$		−	1.20236i		−	0.0455426i
$698$		0			0
$699$	15.7992			0.597581
$700$		0			0
$701$	−29.7677			−1.12431			−0.562155	−	0.827032i	$-0.690028\pi$
$701$	−29.7677			−1.12431			−0.562155	+	0.827032i	$0.690028\pi$
$702$		0			0
$703$			15.0050i			0.565925i
$704$		0			0
$705$	−5.78078	−	2.60298i	−0.217717	−	0.0980341i
$706$		0			0
$707$	−33.8742	−	30.3757i	−1.27397	−	1.14240i
$708$		0			0
$709$	24.2654			0.911307			0.455653	−	0.890157i	$-0.349406\pi$
$709$	24.2654			0.911307			0.455653	+	0.890157i	$0.349406\pi$
$710$		0			0
$711$		−	30.6406i		−	1.14911i
$712$		0			0
$713$	−4.19639			−0.157156
$714$		0			0
$715$	0.819520	−	1.82001i	0.0306483	−	0.0680647i
$716$		0			0
$717$	15.7068			0.586580
$718$		0			0
$719$	−35.3909			−1.31986			−0.659928	−	0.751329i	$-0.729413\pi$
$719$	−35.3909			−1.31986			−0.659928	+	0.751329i	$0.729413\pi$
$720$		0			0
$721$	−2.64240	−	2.36949i	−0.0984082	−	0.0882445i
$722$		0			0
$723$	−12.9758			−0.482577
$724$		0			0
$725$	−14.4349	−	16.3055i	−0.536097	−	0.605573i
$726$		0			0
$727$		−	48.3779i		−	1.79424i	−0.441791	−	0.897118i	$-0.645657\pi$
$727$		−	48.3779i		−	1.79424i	0.441791	−	0.897118i	$-0.354343\pi$
$728$		0			0
$729$	11.0889			0.410701
$730$		0			0
$731$	−1.89942			−0.0702527
$732$		0			0
$733$	−19.0097			−0.702140			−0.351070	−	0.936349i	$-0.614182\pi$
$733$	−19.0097			−0.702140			−0.351070	+	0.936349i	$0.614182\pi$
$734$		0			0
$735$	8.46012	+	2.75010i	0.312056	+	0.101439i
$736$		0			0
$737$			4.63671i			0.170796i
$738$		0			0
$739$			6.76661i			0.248914i	0.992225	+	0.124457i	$0.0397189\pi$
$739$			6.76661i			0.248914i	−0.992225	+	0.124457i	$0.960281\pi$
$740$		0			0
$741$			2.66760i			0.0979968i
$742$		0			0
$743$	−38.1728			−1.40042			−0.700212	−	0.713935i	$-0.746911\pi$
$743$	−38.1728			−1.40042			−0.700212	+	0.713935i	$0.746911\pi$
$744$		0			0
$745$	−26.7237	−	12.0332i	−0.979082	−	0.440863i
$746$		0			0
$747$			14.7168i			0.538459i
$748$		0			0
$749$	−19.6141	+	21.8732i	−0.716683	+	0.799228i
$750$		0			0
$751$		−	46.2870i		−	1.68904i	−0.535528	−	0.844518i	$-0.679887\pi$
$751$		−	46.2870i		−	1.68904i	0.535528	−	0.844518i	$-0.320113\pi$
$752$		0			0
$753$			2.35109i			0.0856784i
$754$		0			0
$755$	−8.38057	+	18.6118i	−0.305000	+	0.677353i
$756$		0			0
$757$			11.1203i			0.404174i	0.979368	+	0.202087i	$0.0647724\pi$
$757$			11.1203i			0.404174i	−0.979368	+	0.202087i	$0.935228\pi$
$758$		0			0
$759$	−0.235580			−0.00855102
$760$		0			0
$761$		−	17.1255i		−	0.620798i	−0.950606	−	0.310399i	$-0.899537\pi$
$761$		−	17.1255i		−	0.620798i	0.950606	−	0.310399i	$-0.100463\pi$
$762$		0			0
$763$	13.9109	−	15.5131i	0.503608	−	0.561612i
$764$		0			0
$765$	1.71482	+	0.772152i	0.0619993	+	0.0279172i
$766$		0			0
$767$	20.3133			0.733472
$768$		0			0
$769$			12.6701i			0.456896i	0.973556	+	0.228448i	$0.0733650\pi$
$769$			12.6701i			0.456896i	−0.973556	+	0.228448i	$0.926635\pi$
$770$		0			0
$771$		−	13.0788i		−	0.471021i
$772$		0			0
$773$	−16.5444			−0.595059			−0.297530	−	0.954713i	$-0.596163\pi$
$773$	−16.5444			−0.595059			−0.297530	+	0.954713i	$0.596163\pi$
$774$		0			0
$775$	14.0102	+	15.8259i	0.503263	+	0.568483i
$776$		0			0
$777$	6.86063	−	7.65081i	0.246124	−	0.274471i
$778$		0			0
$779$		−	8.40260i		−	0.301054i
$780$		0			0
$781$	2.73775			0.0979643
$782$		0			0
$783$		−	14.0524i		−	0.502193i
$784$		0			0
$785$	14.2681	+	6.42466i	0.509249	+	0.229306i
$786$		0			0
$787$		−	18.5644i		−	0.661749i	−0.943675	−	0.330874i	$-0.892656\pi$
$787$		−	18.5644i		−	0.661749i	0.943675	−	0.330874i	$-0.107344\pi$
$788$		0			0
$789$			12.9331i			0.460431i
$790$		0			0
$791$	26.5128	+	23.7746i	0.942687	+	0.845326i
$792$		0			0
$793$		−	1.67517i		−	0.0594872i
$794$		0			0
$795$	−8.28476	−	3.73048i	−0.293830	−	0.132306i
$796$		0			0
$797$	0.331862			0.0117552			0.00587758	−	0.999983i	$-0.498129\pi$
$797$	0.331862			0.0117552			0.00587758	+	0.999983i	$0.498129\pi$
$798$		0			0
$799$			1.56732i			0.0554478i
$800$		0			0
$801$		−	33.6935i		−	1.19050i
$802$		0			0
$803$		−	2.13647i		−	0.0753945i
$804$		0			0
$805$	5.37031	−	2.37706i	0.189278	−	0.0837802i
$806$		0			0
$807$	−8.22942			−0.289689
$808$		0			0
$809$	27.7319			0.975000			0.487500	−	0.873123i	$-0.337909\pi$
$809$	27.7319			0.975000			0.487500	+	0.873123i	$0.337909\pi$
$810$		0			0
$811$	35.4515			1.24487			0.622435	−	0.782672i	$-0.286144\pi$
$811$	35.4515			1.24487			0.622435	+	0.782672i	$0.286144\pi$
$812$		0			0
$813$		−	16.6627i		−	0.584386i
$814$		0			0
$815$	3.84392	+	1.73085i	0.134647	+	0.0606290i
$816$		0			0
$817$	−13.2740			−0.464398
$818$		0			0
$819$	−10.1084	+	11.2727i	−0.353217	+	0.393899i
$820$		0			0
$821$	−20.9000			−0.729415			−0.364708	−	0.931122i	$-0.618831\pi$
$821$	−20.9000			−0.729415			−0.364708	+	0.931122i	$0.618831\pi$
$822$		0			0
$823$	4.56084			0.158981			0.0794904	−	0.996836i	$-0.474671\pi$
$823$	4.56084			0.158981			0.0794904	+	0.996836i	$0.474671\pi$
$824$		0			0
$825$	0.786517	+	0.888445i	0.0273830	+	0.0309317i
$826$		0			0
$827$	38.4325			1.33643			0.668214	−	0.743969i	$-0.267059\pi$
$827$	38.4325			1.33643			0.668214	+	0.743969i	$0.267059\pi$
$828$		0			0
$829$		−	20.0377i		−	0.695939i	−0.937506	−	0.347970i	$-0.886871\pi$
$829$		−	20.0377i		−	0.695939i	0.937506	−	0.347970i	$-0.113129\pi$
$830$		0			0
$831$	10.6453			0.369282
$832$		0			0
$833$	−0.238798	−	2.18623i	−0.00827386	−	0.0757485i
$834$		0			0
$835$	17.9918	−	39.9567i	0.622632	−	1.38276i
$836$		0			0
$837$			13.6391i			0.471435i
$838$		0			0
$839$	−49.2372			−1.69986			−0.849928	−	0.526899i	$-0.823355\pi$
$839$	−49.2372			−1.69986			−0.849928	+	0.526899i	$0.823355\pi$
$840$		0			0
$841$	−10.0306			−0.345881
$842$		0			0
$843$		−	0.906333i		−	0.0312158i
$844$		0			0
$845$	−17.1878	−	7.73937i	−0.591279	−	0.266242i
$846$		0			0
$847$	−19.1218	+	21.3242i	−0.657032	+	0.732707i
$848$		0			0
$849$	15.4283			0.529499
$850$		0			0
$851$		−	6.78421i		−	0.232560i
$852$		0			0
$853$	−16.8478			−0.576856			−0.288428	−	0.957502i	$-0.593133\pi$
$853$	−16.8478			−0.576856			−0.288428	+	0.957502i	$0.593133\pi$
$854$		0			0
$855$	11.9839	+	5.39613i	0.409840	+	0.184544i
$856$		0			0
$857$	−32.6207			−1.11430			−0.557150	−	0.830412i	$-0.688105\pi$
$857$	−32.6207			−1.11430			−0.557150	+	0.830412i	$0.688105\pi$
$858$		0			0
$859$	−1.67446			−0.0571319			−0.0285660	−	0.999592i	$-0.509094\pi$
$859$	−1.67446			−0.0571319			−0.0285660	+	0.999592i	$0.509094\pi$
$860$		0			0
$861$	−3.84186	+	4.28434i	−0.130930	+	0.146010i
$862$		0			0
$863$	5.08818			0.173204			0.0866019	−	0.996243i	$-0.472399\pi$
$863$	5.08818			0.173204			0.0866019	+	0.996243i	$0.472399\pi$
$864$		0			0
$865$	−37.7694	−	17.0069i	−1.28420	−	0.578252i
$866$		0			0
$867$		−	9.60563i		−	0.326224i
$868$		0			0
$869$	−4.77933			−0.162128
$870$		0			0
$871$	−23.7386			−0.804352
$872$		0			0
$873$	0.841049			0.0284652
$874$		0			0
$875$	−26.8941	−	12.3169i	−0.909187	−	0.416389i
$876$		0			0
$877$		−	2.64486i		−	0.0893106i	−0.999002	−	0.0446553i	$-0.985781\pi$
$877$		−	2.64486i		−	0.0893106i	0.999002	−	0.0446553i	$-0.0142190\pi$
$878$		0			0
$879$		−	12.6009i		−	0.425019i
$880$		0			0
$881$		−	30.3075i		−	1.02109i	−0.859852	−	0.510543i	$-0.829445\pi$
$881$		−	30.3075i		−	1.02109i	0.859852	−	0.510543i	$-0.170555\pi$
$882$		0			0
$883$	−41.7294			−1.40431			−0.702153	−	0.712027i	$-0.747778\pi$
$883$	−41.7294			−1.40431			−0.702153	+	0.712027i	$0.747778\pi$
$884$		0			0
$885$	−4.95801	+	11.0109i	−0.166662	+	0.370127i
$886$		0			0
$887$		−	36.1150i		−	1.21262i	−0.795228	−	0.606311i	$-0.792649\pi$
$887$		−	36.1150i		−	1.21262i	0.795228	−	0.606311i	$-0.207351\pi$
$888$		0			0
$889$	−4.39664	+	4.90303i	−0.147459	+	0.164442i
$890$		0			0
$891$		−	2.58772i		−	0.0866918i
$892$		0			0
$893$			10.9531i			0.366532i
$894$		0			0
$895$	19.5515	−	43.4205i	0.653534	−	1.45139i
$896$		0			0
$897$		−	1.20610i		−	0.0402705i
$898$		0			0
$899$	−18.4114			−0.614055
$900$		0			0
$901$			2.24621i			0.0748322i
$902$		0			0
$903$	6.76818	+	6.06916i	0.225231	+	0.201969i
$904$		0			0
$905$	−7.43587	+	16.5138i	−0.247177	+	0.548937i
$906$		0			0
$907$	25.9915			0.863033			0.431517	−	0.902105i	$-0.357979\pi$
$907$	25.9915			0.863033			0.431517	+	0.902105i	$0.357979\pi$
$908$		0			0
$909$			46.0361i			1.52692i
$910$		0			0
$911$		−	16.9582i		−	0.561850i	−0.959730	−	0.280925i	$-0.909359\pi$
$911$		−	16.9582i		−	0.561850i	0.959730	−	0.280925i	$-0.0906412\pi$
$912$		0			0
$913$	2.29553			0.0759708
$914$		0			0
$915$	0.908032	+	0.408871i	0.0300186	+	0.0135168i
$916$		0			0
$917$	29.9296	−	33.3767i	0.988361	−	1.10220i
$918$		0			0
$919$			29.9513i			0.988001i	0.869462	+	0.494000i	$0.164466\pi$
$919$			29.9513i			0.988001i	−0.869462	+	0.494000i	$0.835534\pi$
$920$		0			0
$921$	−11.8233			−0.389589
$922$		0			0
$923$			14.0164i			0.461357i
$924$		0			0
$925$	−25.5853	+	22.6500i	−0.841241	+	0.744728i
$926$		0			0
$927$			3.59110i			0.117947i
$928$		0			0
$929$			15.1272i			0.496308i	0.968721	+	0.248154i	$0.0798240\pi$
$929$			15.1272i			0.496308i	−0.968721	+	0.248154i	$0.920176\pi$
$930$		0			0
$931$	−1.66882	−	15.2783i	−0.0546935	−	0.500727i
$932$		0			0
$933$			12.3862i			0.405506i
$934$		0			0
$935$	0.120440	−	0.267477i	0.00393882	−	0.00874745i
$936$		0			0
$937$	3.96137			0.129412			0.0647061	−	0.997904i	$-0.479389\pi$
$937$	3.96137			0.129412			0.0647061	+	0.997904i	$0.479389\pi$
$938$		0			0
$939$			11.4832i			0.374741i
$940$		0			0
$941$		−	2.09097i		−	0.0681638i	−0.999419	−	0.0340819i	$-0.989149\pi$
$941$		−	2.09097i		−	0.0681638i	0.999419	−	0.0340819i	$-0.0108507\pi$
$942$		0			0
$943$			3.79906i			0.123715i
$944$		0			0
$945$	−7.72587	−	17.4545i	−0.251323	−	0.567795i
$946$		0			0
$947$	5.00636			0.162685			0.0813425	−	0.996686i	$-0.474079\pi$
$947$	5.00636			0.162685			0.0813425	+	0.996686i	$0.474079\pi$
$948$		0			0
$949$	10.9381			0.355066
$950$		0			0
$951$	10.5497			0.342097
$952$		0			0
$953$		−	55.9362i		−	1.81195i	−0.423330	−	0.905975i	$-0.639139\pi$
$953$		−	55.9362i		−	1.81195i	0.423330	−	0.905975i	$-0.360861\pi$
$954$		0			0
$955$	4.56835	−	10.1455i	0.147828	−	0.328302i
$956$		0			0
$957$	−1.03359			−0.0334114
$958$		0			0
$959$	17.5383	+	15.7269i	0.566341	+	0.507849i
$960$		0			0
$961$	−13.1302			−0.423554
$962$		0			0
$963$	29.7263			0.957917
$964$		0			0
$965$	0.615998	−	1.36803i	0.0198297	−	0.0440383i
$966$		0			0
$967$	−0.335414			−0.0107862			−0.00539310	−	0.999985i	$-0.501717\pi$
$967$	−0.335414			−0.0107862			−0.00539310	+	0.999985i	$0.501717\pi$
$968$		0			0
$969$			0.392043i			0.0125942i
$970$		0			0
$971$	7.07794			0.227142			0.113571	−	0.993530i	$-0.463771\pi$
$971$	7.07794			0.227142			0.113571	+	0.993530i	$0.463771\pi$
$972$		0			0
$973$	−29.7571	+	33.1844i	−0.953970	+	1.06384i
$974$		0			0
$975$	−4.54858	+	4.02673i	−0.145671	+	0.128959i
$976$		0			0
$977$			6.33879i			0.202796i	0.994846	+	0.101398i	$0.0323315\pi$
$977$			6.33879i			0.202796i	−0.994846	+	0.101398i	$0.967668\pi$
$978$		0			0
$979$	−5.25552			−0.167967
$980$		0			0
$981$	−21.0828			−0.673121
$982$		0			0
$983$		−	1.53083i		−	0.0488259i	−0.999702	−	0.0244130i	$-0.992228\pi$
$983$		−	1.53083i		−	0.0488259i	0.999702	−	0.0244130i	$-0.00777166\pi$
$984$		0			0
$985$	9.39017	−	20.8540i	0.299196	−	0.664463i
$986$		0			0
$987$	5.00800	−	5.58480i	0.159407	−	0.177766i
$988$		0			0
$989$	6.00156			0.190838
$990$		0			0
$991$			18.9430i			0.601746i	0.953664	+	0.300873i	$0.0972780\pi$
$991$			18.9430i			0.601746i	−0.953664	+	0.300873i	$0.902722\pi$
$992$		0			0
$993$	−12.1600			−0.385887
$994$		0			0
$995$	53.7228	+	24.1904i	1.70313	+	0.766888i
$996$		0			0
$997$	−52.7696			−1.67123			−0.835616	−	0.549315i	$-0.814889\pi$
$997$	−52.7696			−1.67123			−0.835616	+	0.549315i	$0.814889\pi$
$998$		0			0
$999$	−22.0499			−0.697630

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.2.e.a.1119.32	yes	48
4.3	odd	2	inner	1120.2.e.a.1119.35	yes	48
5.4	even	2	inner	1120.2.e.a.1119.37	yes	48
7.6	odd	2	inner	1120.2.e.a.1119.29	✓	48
8.3	odd	2		2240.2.e.g.2239.30		48
8.5	even	2		2240.2.e.g.2239.37		48
20.19	odd	2	inner	1120.2.e.a.1119.30	yes	48
28.27	even	2	inner	1120.2.e.a.1119.38	yes	48
35.34	odd	2	inner	1120.2.e.a.1119.36	yes	48
40.19	odd	2		2240.2.e.g.2239.35		48
40.29	even	2		2240.2.e.g.2239.32		48
56.13	odd	2		2240.2.e.g.2239.36		48
56.27	even	2		2240.2.e.g.2239.31		48
140.139	even	2	inner	1120.2.e.a.1119.31	yes	48
280.69	odd	2		2240.2.e.g.2239.29		48
280.139	even	2		2240.2.e.g.2239.38		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.e.a.1119.29	✓	48	7.6	odd	2	inner
1120.2.e.a.1119.30	yes	48	20.19	odd	2	inner
1120.2.e.a.1119.31	yes	48	140.139	even	2	inner
1120.2.e.a.1119.32	yes	48	1.1	even	1	trivial
1120.2.e.a.1119.35	yes	48	4.3	odd	2	inner
1120.2.e.a.1119.36	yes	48	35.34	odd	2	inner
1120.2.e.a.1119.37	yes	48	5.4	even	2	inner
1120.2.e.a.1119.38	yes	48	28.27	even	2	inner
2240.2.e.g.2239.29		48	280.69	odd	2
2240.2.e.g.2239.30		48	8.3	odd	2
2240.2.e.g.2239.31		48	56.27	even	2
2240.2.e.g.2239.32		48	40.29	even	2
2240.2.e.g.2239.35		48	40.19	odd	2
2240.2.e.g.2239.36		48	56.13	odd	2
2240.2.e.g.2239.37		48	8.5	even	2
2240.2.e.g.2239.38		48	280.139	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1120.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.568337i q^{3} +(2.03890 + 0.918082i) q^{5} +(-1.76634 + 1.96978i) q^{7} +2.67699 q^{9} +O(q^{10})\)	\(q+0.568337i q^{3} +(2.03890 + 0.918082i) q^{5} +(-1.76634 + 1.96978i) q^{7} +2.67699 q^{9} -0.417558i q^{11} +2.13777 q^{13} +(-0.521780 + 1.15878i) q^{15} +0.314177 q^{17} +2.19560 q^{19} +(-1.11950 - 1.00388i) q^{21} -0.992696 q^{23} +(3.31425 + 3.74376i) q^{25} +3.22645i q^{27} -4.35539 q^{29} +4.22727 q^{31} +0.237314 q^{33} +(-5.40982 + 2.39455i) q^{35} +6.83413i q^{37} +1.21498i q^{39} -3.82702i q^{41} -6.04572 q^{43} +(5.45813 + 2.45770i) q^{45} +4.98866i q^{47} +(-0.760076 - 6.95861i) q^{49} +0.178558i q^{51} +7.14952i q^{53} +(0.383352 - 0.851360i) q^{55} +1.24784i q^{57} +9.50211 q^{59} -0.783607i q^{61} +(-4.72848 + 5.27309i) q^{63} +(4.35871 + 1.96265i) q^{65} -11.1044 q^{67} -0.564186i q^{69} +6.55657i q^{71} +5.11659 q^{73} +(-2.12772 + 1.88361i) q^{75} +(0.822498 + 0.737550i) q^{77} -11.4459i q^{79} +6.19727 q^{81} +5.49750i q^{83} +(0.640576 + 0.288440i) q^{85} -2.47533i q^{87} -12.5863i q^{89} +(-3.77604 + 4.21094i) q^{91} +2.40252i q^{93} +(4.47662 + 2.01574i) q^{95} +0.314177 q^{97} -1.11780i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{9}+O(q^{10}) \) 48 * q - 48 * q^9	\( 48 q - 48 q^{9} + 8 q^{21} - 16 q^{25} + 16 q^{29} + 24 q^{49} - 16 q^{65} - 32 q^{81} + 32 q^{85}+O(q^{100}) \) 48 * q - 48 * q^9 + 8 * q^21 - 16 * q^25 + 16 * q^29 + 24 * q^49 - 16 * q^65 - 32 * q^81 + 32 * q^85

Introduction

L-functions

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