LMFDB - Embedded newform 1440.2.m.c.719.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,2,Mod(719,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 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("1440.719");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.m (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 17x^{12} - 104x^{10} + 713x^{8} + 238x^{6} + 1004x^{4} - 152x^{2} + 64 $ x^16 - 17x^12 - 104x^10 + 713x^8 + 238x^6 + 1004x^4 - 152x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			719.5
Root			$-0.877859 + 2.23141i$ of defining polynomial
Character	$\chi$	$=$	1440.719
Dual form			1440.2.m.c.719.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.64901 - 1.51022i) q^{5} -0.936426 q^{7} +2.20837i q^{11} +3.33513 q^{13} +1.54417 q^{17} +3.12311 q^{19} -3.39228i q^{23} +(0.438447 + 4.98074i) q^{25} -8.44804 q^{29} -8.30571i q^{31} +(1.54417 + 1.41421i) q^{35} +7.60669 q^{37} -5.83095i q^{41} -7.77769i q^{43} -10.7575i q^{47} -6.12311 q^{49} -5.08842i q^{53} +(3.33513 - 3.64162i) q^{55} -10.6937i q^{59} +(-5.49966 - 5.03680i) q^{65} +12.1453i q^{67} -11.7460 q^{71} -5.59390i q^{73} -2.06798i q^{77} -1.02248i q^{79} +14.0877 q^{83} +(-2.54635 - 2.33205i) q^{85} +13.0761i q^{89} -3.12311 q^{91} +(-5.15002 - 4.71659i) q^{95} -2.18379i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{19} + 40 q^{25} - 32 q^{49} + 16 q^{91}+O(q^{100}) $ 16 * q - 16 * q^19 + 40 * q^25 - 32 * q^49 + 16 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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			0
$3$		0			0
$4$		0			0
$5$	−1.64901	−	1.51022i	−0.737458	−	0.675393i
$5$	−1.64901	−	1.51022i	−0.737458	−	0.675393i
$6$		0			0
$7$	−0.936426			−0.353936			−0.176968	−	0.984217i	$-0.556629\pi$
$7$	−0.936426			−0.353936			−0.176968	+	0.984217i	$0.556629\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			2.20837i			0.665848i	0.942954	+	0.332924i	$0.108035\pi$
$11$			2.20837i			0.665848i	−0.942954	+	0.332924i	$0.891965\pi$
$12$		0			0
$13$	3.33513			0.924999			0.462500	−	0.886619i	$-0.346953\pi$
$13$	3.33513			0.924999			0.462500	+	0.886619i	$0.346953\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.54417			0.374517			0.187259	−	0.982311i	$-0.440040\pi$
$17$	1.54417			0.374517			0.187259	+	0.982311i	$0.440040\pi$
$18$		0			0
$19$	3.12311			0.716490			0.358245	−	0.933628i	$-0.383375\pi$
$19$	3.12311			0.716490			0.358245	+	0.933628i	$0.383375\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	3.39228i		−	0.707340i	−0.935370	−	0.353670i	$-0.884934\pi$
$23$		−	3.39228i		−	0.707340i	0.935370	−	0.353670i	$-0.115066\pi$
$24$		0			0
$25$	0.438447	+	4.98074i	0.0876894	+	0.996148i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−8.44804			−1.56876			−0.784380	−	0.620280i	$-0.787019\pi$
$29$	−8.44804			−1.56876			−0.784380	+	0.620280i	$0.787019\pi$
$30$		0			0
$31$		−	8.30571i		−	1.49175i	−0.666086	−	0.745875i	$-0.732032\pi$
$31$		−	8.30571i		−	1.49175i	0.666086	−	0.745875i	$-0.267968\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.54417	+	1.41421i	0.261013	+	0.239046i
$36$		0			0
$37$	7.60669			1.25053			0.625266	−	0.780412i	$-0.284990\pi$
$37$	7.60669			1.25053			0.625266	+	0.780412i	$0.284990\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	5.83095i		−	0.910642i	−0.890327	−	0.455321i	$-0.849525\pi$
$41$		−	5.83095i		−	0.910642i	0.890327	−	0.455321i	$-0.150475\pi$
$42$		0			0
$43$		−	7.77769i		−	1.18609i	−0.805171	−	0.593043i	$-0.797926\pi$
$43$		−	7.77769i		−	1.18609i	0.805171	−	0.593043i	$-0.202074\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	10.7575i		−	1.56914i	−0.620040	−	0.784570i	$-0.712884\pi$
$47$		−	10.7575i		−	1.56914i	0.620040	−	0.784570i	$-0.287116\pi$
$48$		0			0
$49$	−6.12311			−0.874729
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	5.08842i		−	0.698949i	−0.936946	−	0.349474i	$-0.886360\pi$
$53$		−	5.08842i		−	0.698949i	0.936946	−	0.349474i	$-0.113640\pi$
$54$		0			0
$55$	3.33513	−	3.64162i	0.449709	−	0.491035i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	10.6937i		−	1.39219i	−0.717947	−	0.696097i	$-0.754918\pi$
$59$		−	10.6937i		−	1.39219i	0.717947	−	0.696097i	$-0.245082\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−5.49966	−	5.03680i	−0.682148	−	0.624738i
$66$		0			0
$67$			12.1453i			1.48378i	0.670521	+	0.741890i	$0.266071\pi$
$67$			12.1453i			1.48378i	−0.670521	+	0.741890i	$0.733929\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−11.7460			−1.39400			−0.697000	−	0.717071i	$-0.745482\pi$
$71$	−11.7460			−1.39400			−0.697000	+	0.717071i	$0.745482\pi$
$72$		0			0
$73$		−	5.59390i		−	0.654716i	−0.944900	−	0.327358i	$-0.893842\pi$
$73$		−	5.59390i		−	0.654716i	0.944900	−	0.327358i	$-0.106158\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	2.06798i		−	0.235668i
$78$		0			0
$79$		−	1.02248i		−	0.115038i	−0.998344	−	0.0575190i	$-0.981681\pi$
$79$		−	1.02248i		−	0.115038i	0.998344	−	0.0575190i	$-0.0183190\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	14.0877			1.54632			0.773161	−	0.634210i	$-0.218675\pi$
$83$	14.0877			1.54632			0.773161	+	0.634210i	$0.218675\pi$
$84$		0			0
$85$	−2.54635	−	2.33205i	−0.276191	−	0.252946i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			13.0761i			1.38607i	0.720906	+	0.693033i	$0.243726\pi$
$89$			13.0761i			1.38607i	−0.720906	+	0.693033i	$0.756274\pi$
$90$		0			0
$91$	−3.12311			−0.327390
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.15002	−	4.71659i	−0.528381	−	0.483912i
$96$		0			0
$97$		−	2.18379i		−	0.221730i	−0.993835	−	0.110865i	$-0.964638\pi$
$97$		−	2.18379i		−	0.221730i	0.993835	−	0.110865i	$-0.0353622\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.29801			−0.328165			−0.164082	−	0.986447i	$-0.552466\pi$
$101$	−3.29801			−0.328165			−0.164082	+	0.986447i	$0.552466\pi$
$102$		0			0
$103$	18.0227			1.77583			0.887913	−	0.460012i	$-0.152155\pi$
$103$	18.0227			1.77583			0.887913	+	0.460012i	$0.152155\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−10.9993			−1.06334			−0.531672	−	0.846950i	$-0.678436\pi$
$107$	−10.9993			−1.06334			−0.531672	+	0.846950i	$0.678436\pi$
$108$		0			0
$109$		−	11.9473i		−	1.14435i	−0.820133	−	0.572173i	$-0.806100\pi$
$109$		−	11.9473i		−	1.14435i	0.820133	−	0.572173i	$-0.193900\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	12.5435			1.17999			0.589996	−	0.807406i	$-0.299129\pi$
$113$	12.5435			1.17999			0.589996	+	0.807406i	$0.299129\pi$
$114$		0			0
$115$	−5.12311	+	5.59390i	−0.477732	+	0.521634i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.44600			−0.132555
$120$		0			0
$121$	6.12311			0.556646
$122$		0			0
$123$		0			0
$124$		0			0
$125$	6.79903	−	8.87543i	0.608124	−	0.793842i
$126$		0			0
$127$	−4.68213			−0.415472			−0.207736	−	0.978185i	$-0.566609\pi$
$127$	−4.68213			−0.415472			−0.207736	+	0.978185i	$0.566609\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			0.620058i			0.0541747i	0.999633	+	0.0270874i	$0.00862323\pi$
$131$			0.620058i			0.0541747i	−0.999633	+	0.0270874i	$0.991377\pi$
$132$		0			0
$133$	−2.92456			−0.253591
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.54417			0.131928			0.0659638	−	0.997822i	$-0.478988\pi$
$137$	1.54417			0.131928			0.0659638	+	0.997822i	$0.478988\pi$
$138$		0			0
$139$	12.0000			1.01783			0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			7.36520i			0.615909i
$144$		0			0
$145$	13.9309	+	12.7584i	1.15690	+	1.05953i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.85201			−0.151722			−0.0758612	−	0.997118i	$-0.524171\pi$
$149$	−1.85201			−0.151722			−0.0758612	+	0.997118i	$0.524171\pi$
$150$		0			0
$151$			17.6339i			1.43503i	0.696545	+	0.717513i	$0.254720\pi$
$151$			17.6339i			1.43503i	−0.696545	+	0.717513i	$0.745280\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−12.5435	+	13.6962i	−1.00752	+	1.10010i
$156$		0			0
$157$	−2.80928			−0.224205			−0.112102	−	0.993697i	$-0.535758\pi$
$157$	−2.80928			−0.224205			−0.112102	+	0.993697i	$0.535758\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			3.17662i			0.250353i
$162$		0			0
$163$		−	15.5554i		−	1.21839i	−0.793020	−	0.609195i	$-0.791492\pi$
$163$		−	15.5554i		−	1.21839i	0.793020	−	0.609195i	$-0.208508\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			9.43318i			0.729961i	0.931015	+	0.364981i	$0.118924\pi$
$167$			9.43318i			0.729961i	−0.931015	+	0.364981i	$0.881076\pi$
$168$		0			0
$169$	−1.87689			−0.144376
$170$		0			0
$171$		0			0
$172$		0			0
$173$			1.69614i			0.128955i	0.997919	+	0.0644776i	$0.0205381\pi$
$173$			1.69614i			0.128955i	−0.997919	+	0.0644776i	$0.979462\pi$
$174$		0			0
$175$	−0.410574	−	4.66410i	−0.0310364	−	0.352572i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	8.21342i		−	0.613900i	−0.951726	−	0.306950i	$-0.900692\pi$
$179$		−	8.21342i		−	0.613900i	0.951726	−	0.306950i	$-0.0993084\pi$
$180$		0			0
$181$		−	4.66410i		−	0.346680i	−0.984862	−	0.173340i	$-0.944544\pi$
$181$		−	4.66410i		−	0.346680i	0.984862	−	0.173340i	$-0.0554559\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−12.5435	−	11.4878i	−0.922215	−	0.844600i
$186$		0			0
$187$			3.41011i			0.249372i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	13.1921			0.954544			0.477272	−	0.878756i	$-0.341626\pi$
$191$	13.1921			0.954544			0.477272	+	0.878756i	$0.341626\pi$
$192$		0			0
$193$		−	23.3331i		−	1.67955i	−0.542934	−	0.839775i	$-0.682687\pi$
$193$		−	23.3331i		−	1.67955i	0.542934	−	0.839775i	$-0.317313\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	4.92539i		−	0.350920i	−0.984487	−	0.175460i	$-0.943859\pi$
$197$		−	4.92539i		−	0.350920i	0.984487	−	0.175460i	$-0.0561412\pi$
$198$		0			0
$199$			17.6339i			1.25003i	0.780611	+	0.625017i	$0.214908\pi$
$199$			17.6339i			1.25003i	−0.780611	+	0.625017i	$0.785092\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	7.91096			0.555241
$204$		0			0
$205$	−8.80604	+	9.61528i	−0.615041	+	0.671560i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			6.89697i			0.477073i
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−11.7460	+	12.8255i	−0.801074	+	0.874689i
$216$		0			0
$217$			7.77769i			0.527984i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.15002			0.346428
$222$		0			0
$223$	−21.9989			−1.47316			−0.736578	−	0.676352i	$-0.763560\pi$
$223$	−21.9989			−1.47316			−0.736578	+	0.676352i	$0.763560\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3.08835			−0.204981			−0.102490	−	0.994734i	$-0.532681\pi$
$227$	−3.08835			−0.204981			−0.102490	+	0.994734i	$0.532681\pi$
$228$		0			0
$229$		−	2.61914i		−	0.173077i	−0.996248	−	0.0865387i	$-0.972419\pi$
$229$		−	2.61914i		−	0.173077i	0.996248	−	0.0865387i	$-0.0275806\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	20.4544			1.34002			0.670008	−	0.742354i	$-0.266291\pi$
$233$	20.4544			1.34002			0.670008	+	0.742354i	$0.266291\pi$
$234$		0			0
$235$	−16.2462	+	17.7392i	−1.05979	+	1.15718i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	23.4921			1.51958			0.759789	−	0.650170i	$-0.225302\pi$
$239$	23.4921			1.51958			0.759789	+	0.650170i	$0.225302\pi$
$240$		0			0
$241$	−4.00000			−0.257663			−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000			−0.257663			−0.128831	+	0.991667i	$0.541123\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	10.0970	+	9.24726i	0.645076	+	0.590786i
$246$		0			0
$247$	10.4160			0.662752
$248$		0			0
$249$		0			0
$250$		0			0
$251$			3.44849i			0.217666i	0.994060	+	0.108833i	$0.0347114\pi$
$251$			3.44849i			0.217666i	−0.994060	+	0.108833i	$0.965289\pi$
$252$		0			0
$253$	7.49141			0.470981
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−15.6318			−0.975087			−0.487543	−	0.873099i	$-0.662107\pi$
$257$	−15.6318			−0.975087			−0.487543	+	0.873099i	$0.662107\pi$
$258$		0			0
$259$	−7.12311			−0.442608
$260$		0			0
$261$		0			0
$262$		0			0
$263$			14.7304i			0.908316i	0.890921	+	0.454158i	$0.150060\pi$
$263$			14.7304i			0.908316i	−0.890921	+	0.454158i	$0.849940\pi$
$264$		0			0
$265$	−7.68466	+	8.39084i	−0.472065	+	0.515445i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	15.0441			0.917253			0.458626	−	0.888629i	$-0.348342\pi$
$269$	15.0441			0.917253			0.458626	+	0.888629i	$0.348342\pi$
$270$		0			0
$271$		−	10.3507i		−	0.628758i	−0.949297	−	0.314379i	$-0.898204\pi$
$271$		−	10.3507i		−	0.628758i	0.949297	−	0.314379i	$-0.101796\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−10.9993	+	0.968253i	−0.663283	+	0.0583879i
$276$		0			0
$277$	−23.3459			−1.40272			−0.701360	−	0.712807i	$-0.747424\pi$
$277$	−23.3459			−1.40272			−0.701360	+	0.712807i	$0.747424\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	3.00252i		−	0.179116i	−0.995982	−	0.0895578i	$-0.971455\pi$
$281$		−	3.00252i		−	0.179116i	0.995982	−	0.0895578i	$-0.0285454\pi$
$282$		0			0
$283$		−	23.3331i		−	1.38701i	−0.720453	−	0.693503i	$-0.756066\pi$
$283$		−	23.3331i		−	1.38701i	0.720453	−	0.693503i	$-0.243934\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			5.46026i			0.322309i
$288$		0			0
$289$	−14.6155			−0.859737
$290$		0			0
$291$		0			0
$292$		0			0
$293$			29.8326i			1.74284i	0.490536	+	0.871421i	$0.336801\pi$
$293$			29.8326i			1.74284i	−0.490536	+	0.871421i	$0.663199\pi$
$294$		0			0
$295$	−16.1498	+	17.6339i	−0.940278	+	1.02669i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	11.3137i		−	0.654289i
$300$		0			0
$301$			7.28323i			0.419799i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	4.36758i		−	0.249271i	−0.992203	−	0.124636i	$-0.960224\pi$
$307$		−	4.36758i		−	0.249271i	0.992203	−	0.124636i	$-0.0397762\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	10.3000			0.584062			0.292031	−	0.956409i	$-0.405669\pi$
$311$	10.3000			0.584062			0.292031	+	0.956409i	$0.405669\pi$
$312$		0			0
$313$		−	32.0682i		−	1.81260i	−0.422631	−	0.906302i	$-0.638893\pi$
$313$		−	32.0682i		−	1.81260i	0.422631	−	0.906302i	$-0.361107\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			24.5354i			1.37805i	0.724739	+	0.689023i	$0.241960\pi$
$317$			24.5354i			1.37805i	−0.724739	+	0.689023i	$0.758040\pi$
$318$		0			0
$319$		−	18.6564i		−	1.04456i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.82262			0.268338
$324$		0			0
$325$	1.46228	+	16.6114i	0.0811127	+	0.921436i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			10.0736i			0.555375i
$330$		0			0
$331$	1.36932			0.0752645			0.0376322	−	0.999292i	$-0.488018\pi$
$331$	1.36932			0.0752645			0.0376322	+	0.999292i	$0.488018\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	18.3421	−	20.0276i	1.00213	−	1.09423i
$336$		0			0
$337$			29.8844i			1.62791i	0.580929	+	0.813954i	$0.302690\pi$
$337$			29.8844i			1.62791i	−0.580929	+	0.813954i	$0.697310\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	18.3421			0.993279
$342$		0			0
$343$	12.2888			0.663534
$344$		0			0
$345$		0			0
$346$		0			0
$347$	14.0877			0.756265			0.378133	−	0.925751i	$-0.376566\pi$
$347$	14.0877			0.756265			0.378133	+	0.925751i	$0.376566\pi$
$348$		0			0
$349$			33.2228i			1.77838i	0.457540	+	0.889189i	$0.348731\pi$
$349$			33.2228i			1.77838i	−0.457540	+	0.889189i	$0.651269\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.72087			−0.410940			−0.205470	−	0.978663i	$-0.565872\pi$
$353$	−7.72087			−0.410940			−0.205470	+	0.978663i	$0.565872\pi$
$354$		0			0
$355$	19.3693	+	17.7392i	1.02802	+	0.941497i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.44600			−0.0763172			−0.0381586	−	0.999272i	$-0.512149\pi$
$359$	−1.44600			−0.0763172			−0.0381586	+	0.999272i	$0.512149\pi$
$360$		0			0
$361$	−9.24621			−0.486643
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−8.44804	+	9.22437i	−0.442190	+	0.482826i
$366$		0			0
$367$	−32.4149			−1.69204			−0.846022	−	0.533148i	$-0.821009\pi$
$367$	−32.4149			−1.69204			−0.846022	+	0.533148i	$0.821009\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			4.76493i			0.247383i
$372$		0			0
$373$	9.47954			0.490832			0.245416	−	0.969418i	$-0.421075\pi$
$373$	9.47954			0.490832			0.245416	+	0.969418i	$0.421075\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−28.1753			−1.45110
$378$		0			0
$379$	16.4924			0.847159			0.423579	−	0.905859i	$-0.360773\pi$
$379$	16.4924			0.847159			0.423579	+	0.905859i	$0.360773\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	10.0138i		−	0.511682i	−0.966719	−	0.255841i	$-0.917648\pi$
$383$		−	10.0138i		−	0.511682i	0.966719	−	0.255841i	$-0.0823524\pi$
$384$		0			0
$385$	−3.12311	+	3.41011i	−0.159168	+	0.173795i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9.89404			−0.501648			−0.250824	−	0.968033i	$-0.580702\pi$
$389$	−9.89404			−0.501648			−0.250824	+	0.968033i	$0.580702\pi$
$390$		0			0
$391$		−	5.23827i		−	0.264911i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.54417	+	1.68608i	−0.0776958	+	0.0848357i
$396$		0			0
$397$	−10.3007			−0.516977			−0.258488	−	0.966014i	$-0.583224\pi$
$397$	−10.3007			−0.516977			−0.258488	+	0.966014i	$0.583224\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	0.522293i		−	0.0260821i	−0.999915	−	0.0130410i	$-0.995849\pi$
$401$		−	0.522293i		−	0.0260821i	0.999915	−	0.0130410i	$-0.00415121\pi$
$402$		0			0
$403$		−	27.7006i		−	1.37987i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			16.7984i			0.832665i
$408$		0			0
$409$	3.12311			0.154428			0.0772138	−	0.997015i	$-0.475398\pi$
$409$	3.12311			0.154428			0.0772138	+	0.997015i	$0.475398\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			10.0138i			0.492748i
$414$		0			0
$415$	−23.2306	−	21.2755i	−1.14035	−	1.04437i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			32.4291i			1.58427i	0.610348	+	0.792134i	$0.291030\pi$
$419$			32.4291i			1.58427i	−0.610348	+	0.792134i	$0.708970\pi$
$420$		0			0
$421$			30.6037i			1.49153i	0.666207	+	0.745767i	$0.267917\pi$
$421$			30.6037i			1.49153i	−0.666207	+	0.745767i	$0.732083\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.677039	+	7.69113i	0.0328412	+	0.373074i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.44600			0.0696516			0.0348258	−	0.999393i	$-0.488912\pi$
$431$	1.44600			0.0696516			0.0348258	+	0.999393i	$0.488912\pi$
$432$		0			0
$433$		−	15.5554i		−	0.747544i	−0.927521	−	0.373772i	$-0.878064\pi$
$433$		−	15.5554i		−	0.747544i	0.927521	−	0.373772i	$-0.121936\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	10.5945i		−	0.506802i
$438$		0			0
$439$			26.9621i			1.28683i	0.765517	+	0.643415i	$0.222483\pi$
$439$			26.9621i			1.28683i	−0.765517	+	0.643415i	$0.777517\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	25.0870			1.19192			0.595959	−	0.803015i	$-0.296772\pi$
$443$	25.0870			1.19192			0.595959	+	0.803015i	$0.296772\pi$
$444$		0			0
$445$	19.7479	−	21.5626i	0.936139	−	1.02217i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	9.89949i		−	0.467186i	−0.972334	−	0.233593i	$-0.924952\pi$
$449$		−	9.89949i		−	0.467186i	0.972334	−	0.233593i	$-0.0750483\pi$
$450$		0			0
$451$	12.8769			0.606349
$452$		0			0
$453$		0			0
$454$		0			0
$455$	5.15002	+	4.71659i	0.241437	+	0.221117i
$456$		0			0
$457$			13.3716i			0.625496i	0.949836	+	0.312748i	$0.101250\pi$
$457$			13.3716i			0.625496i	−0.949836	+	0.312748i	$0.898750\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.85201			−0.0862567			−0.0431283	−	0.999070i	$-0.513732\pi$
$461$	−1.85201			−0.0862567			−0.0431283	+	0.999070i	$0.513732\pi$
$462$		0			0
$463$	−13.2252			−0.614629			−0.307315	−	0.951608i	$-0.599430\pi$
$463$	−13.2252			−0.614629			−0.307315	+	0.951608i	$0.599430\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−18.9103			−0.875063			−0.437532	−	0.899203i	$-0.644147\pi$
$467$	−18.9103			−0.875063			−0.437532	+	0.899203i	$0.644147\pi$
$468$		0			0
$469$		−	11.3732i		−	0.525163i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	17.1760			0.789754
$474$		0			0
$475$	1.36932	+	15.5554i	0.0628286	+	0.713730i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	14.6381			0.668830			0.334415	−	0.942426i	$-0.391461\pi$
$479$	14.6381			0.668830			0.334415	+	0.942426i	$0.391461\pi$
$480$		0			0
$481$	25.3693			1.15674
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.29801	+	3.60109i	−0.149755	+	0.163517i
$486$		0			0
$487$	29.4903			1.33633			0.668167	−	0.744011i	$-0.267079\pi$
$487$	29.4903			1.33633			0.668167	+	0.744011i	$0.267079\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			16.3505i			0.737888i	0.929452	+	0.368944i	$0.120281\pi$
$491$			16.3505i			0.737888i	−0.929452	+	0.368944i	$0.879719\pi$
$492$		0			0
$493$	−13.0452			−0.587528
$494$		0			0
$495$		0			0
$496$		0			0
$497$	10.9993			0.493387
$498$		0			0
$499$	−16.8769			−0.755514			−0.377757	−	0.925905i	$-0.623304\pi$
$499$	−16.8769			−0.755514			−0.377757	+	0.925905i	$0.623304\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			16.0547i			0.715844i	0.933751	+	0.357922i	$0.116515\pi$
$503$			16.0547i			0.715844i	−0.933751	+	0.357922i	$0.883485\pi$
$504$		0			0
$505$	5.43845	+	4.98074i	0.242008	+	0.221640i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3.29801			−0.146182			−0.0730909	−	0.997325i	$-0.523286\pi$
$509$	−3.29801			−0.146182			−0.0730909	+	0.997325i	$0.523286\pi$
$510$		0			0
$511$			5.23827i			0.231728i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−29.7195	−	27.2183i	−1.30960	−	1.19938i
$516$		0			0
$517$	23.7565			1.04481
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	2.65433i		−	0.116288i	−0.998308	−	0.0581441i	$-0.981482\pi$
$521$		−	2.65433i		−	0.116288i	0.998308	−	0.0581441i	$-0.0185183\pi$
$522$		0			0
$523$			4.36758i			0.190981i	0.995430	+	0.0954905i	$0.0304420\pi$
$523$			4.36758i			0.190981i	−0.995430	+	0.0954905i	$0.969558\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	12.8255i		−	0.558686i
$528$		0			0
$529$	11.4924			0.499671
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	19.4470i		−	0.842343i
$534$		0			0
$535$	18.1379	+	16.6114i	0.784172	+	0.718174i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	13.5221i		−	0.582437i
$540$		0			0
$541$		−	9.90237i		−	0.425736i	−0.977081	−	0.212868i	$-0.931720\pi$
$541$		−	9.90237i		−	0.425736i	0.977081	−	0.212868i	$-0.0682804\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−18.0431	+	19.7012i	−0.772883	+	0.843908i
$546$		0			0
$547$		−	16.5129i		−	0.706039i	−0.935616	−	0.353019i	$-0.885155\pi$
$547$		−	16.5129i		−	0.706039i	0.935616	−	0.353019i	$-0.114845\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−26.3841			−1.12400
$552$		0			0
$553$			0.957477i			0.0407161i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	32.6443i		−	1.38318i	−0.722289	−	0.691591i	$-0.756910\pi$
$557$		−	32.6443i		−	1.38318i	0.722289	−	0.691591i	$-0.243090\pi$
$558$		0			0
$559$		−	25.9396i		−	1.09713i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−14.0877			−0.593724			−0.296862	−	0.954920i	$-0.595940\pi$
$563$	−14.0877			−0.593724			−0.296862	+	0.954920i	$0.595940\pi$
$564$		0			0
$565$	−20.6843	−	18.9435i	−0.870195	−	0.796958i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	30.0467i		−	1.25962i	−0.776748	−	0.629811i	$-0.783132\pi$
$569$		−	30.0467i		−	1.25962i	0.776748	−	0.629811i	$-0.216868\pi$
$570$		0			0
$571$	−33.8617			−1.41707			−0.708535	−	0.705676i	$-0.750644\pi$
$571$	−33.8617			−1.41707			−0.708535	+	0.705676i	$0.750644\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	16.8961	−	1.48734i	0.704615	−	0.0620262i
$576$		0			0
$577$		−	7.77769i		−	0.323789i	−0.986808	−	0.161895i	$-0.948240\pi$
$577$		−	7.77769i		−	0.323789i	0.986808	−	0.161895i	$-0.0517605\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−13.1921			−0.547299
$582$		0			0
$583$	11.2371			0.465394
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10.9993			0.453990			0.226995	−	0.973896i	$-0.427110\pi$
$587$	10.9993			0.453990			0.226995	+	0.973896i	$0.427110\pi$
$588$		0			0
$589$		−	25.9396i		−	1.06882i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	43.8071			1.79894			0.899472	−	0.436978i	$-0.143951\pi$
$593$	43.8071			1.79894			0.899472	+	0.436978i	$0.143951\pi$
$594$		0			0
$595$	2.38447	+	2.18379i	0.0977538	+	0.0895267i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−45.5382			−1.86064			−0.930320	−	0.366749i	$-0.880471\pi$
$599$	−45.5382			−1.86064			−0.930320	+	0.366749i	$0.880471\pi$
$600$		0			0
$601$	−23.3693			−0.953254			−0.476627	−	0.879106i	$-0.658141\pi$
$601$	−23.3693			−0.953254			−0.476627	+	0.879106i	$0.658141\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−10.0970	−	9.24726i	−0.410503	−	0.375955i
$606$		0			0
$607$	9.71010			0.394121			0.197060	−	0.980391i	$-0.436860\pi$
$607$	9.71010			0.394121			0.197060	+	0.980391i	$0.436860\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	35.8776i		−	1.45145i
$612$		0			0
$613$	23.8718			0.964172			0.482086	−	0.876124i	$-0.339879\pi$
$613$	23.8718			0.964172			0.482086	+	0.876124i	$0.339879\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	15.6318			0.629314			0.314657	−	0.949205i	$-0.398111\pi$
$617$	15.6318			0.629314			0.314657	+	0.949205i	$0.398111\pi$
$618$		0			0
$619$	−18.7386			−0.753169			−0.376585	−	0.926382i	$-0.622902\pi$
$619$	−18.7386			−0.753169			−0.376585	+	0.926382i	$0.622902\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	12.2448i		−	0.490578i
$624$		0			0
$625$	−24.6155	+	4.36758i	−0.984621	+	0.174703i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	11.7460			0.468346
$630$		0			0
$631$			17.6339i			0.701995i	0.936376	+	0.350997i	$0.114157\pi$
$631$			17.6339i			0.701995i	−0.936376	+	0.350997i	$0.885843\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	7.72087	+	7.07107i	0.306393	+	0.280607i
$636$		0			0
$637$	−20.4214			−0.809124
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	7.07107i		−	0.279290i	−0.990202	−	0.139645i	$-0.955404\pi$
$641$		−	7.07107i		−	0.279290i	0.990202	−	0.139645i	$-0.0445962\pi$
$642$		0			0
$643$			8.73516i			0.344481i	0.985055	+	0.172241i	$0.0551007\pi$
$643$			8.73516i			0.344481i	−0.985055	+	0.172241i	$0.944899\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			1.32431i			0.0520639i	0.999661	+	0.0260319i	$0.00828716\pi$
$647$			1.32431i			0.0520639i	−0.999661	+	0.0260319i	$0.991713\pi$
$648$		0			0
$649$	23.6155			0.926991
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	15.8459i		−	0.620098i	−0.950721	−	0.310049i	$-0.899654\pi$
$653$		−	15.8459i		−	0.620098i	0.950721	−	0.310049i	$-0.100346\pi$
$654$		0			0
$655$	0.936426	−	1.02248i	0.0365892	−	0.0399516i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			33.3211i			1.29800i	0.760786	+	0.649002i	$0.224813\pi$
$659$			33.3211i			1.29800i	−0.760786	+	0.649002i	$0.775187\pi$
$660$		0			0
$661$		−	44.5960i		−	1.73458i	−0.497800	−	0.867292i	$-0.665859\pi$
$661$		−	44.5960i		−	1.73458i	0.497800	−	0.867292i	$-0.334141\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.82262	+	4.41674i	0.187013	+	0.171274i
$666$		0			0
$667$			28.6581i			1.10965i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	5.25145i		−	0.201830i	−0.994895	−	0.100915i	$-0.967823\pi$
$677$		−	5.25145i		−	0.201830i	0.994895	−	0.100915i	$-0.0321770\pi$
$678$		0			0
$679$			2.04496i			0.0784783i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	20.2644			0.775394			0.387697	−	0.921787i	$-0.373271\pi$
$683$	20.2644			0.775394			0.387697	+	0.921787i	$0.373271\pi$
$684$		0			0
$685$	−2.54635	−	2.33205i	−0.0972911	−	0.0891030i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	16.9706i		−	0.646527i
$690$		0			0
$691$	23.1231			0.879644			0.439822	−	0.898085i	$-0.355041\pi$
$691$	23.1231			0.879644			0.439822	+	0.898085i	$0.355041\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−19.7881	−	18.1227i	−0.750605	−	0.687433i
$696$		0			0
$697$		−	9.00400i		−	0.341051i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−20.1941			−0.762720			−0.381360	−	0.924427i	$-0.624544\pi$
$701$	−20.1941			−0.762720			−0.381360	+	0.924427i	$0.624544\pi$
$702$		0			0
$703$	23.7565			0.895993
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.08835			0.116149
$708$		0			0
$709$			30.6037i			1.14935i	0.818383	+	0.574673i	$0.194871\pi$
$709$			30.6037i			1.14935i	−0.818383	+	0.574673i	$0.805129\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−28.1753			−1.05517
$714$		0			0
$715$	11.1231	−	12.1453i	0.415981	−	0.454207i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−32.3461			−1.20631			−0.603154	−	0.797625i	$-0.706089\pi$
$719$	−32.3461			−1.20631			−0.603154	+	0.797625i	$0.706089\pi$
$720$		0			0
$721$	−16.8769			−0.628528
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.70402	−	42.0775i	−0.137564	−	1.56272i
$726$		0			0
$727$	9.47954			0.351577			0.175788	−	0.984428i	$-0.443753\pi$
$727$	9.47954			0.351577			0.175788	+	0.984428i	$0.443753\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	12.0101i		−	0.444210i
$732$		0			0
$733$	19.6002			0.723951			0.361975	−	0.932188i	$-0.382102\pi$
$733$	19.6002			0.723951			0.361975	+	0.932188i	$0.382102\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−26.8212			−0.987973
$738$		0			0
$739$	−35.6155			−1.31014			−0.655069	−	0.755569i	$-0.727361\pi$
$739$	−35.6155			−1.31014			−0.655069	+	0.755569i	$0.727361\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	10.0138i		−	0.367371i	−0.982985	−	0.183686i	$-0.941197\pi$
$743$		−	10.0138i		−	0.367371i	0.982985	−	0.183686i	$-0.0588028\pi$
$744$		0			0
$745$	3.05398	+	2.79695i	0.111889	+	0.102472i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10.3000			0.376355
$750$		0			0
$751$		−	41.5286i		−	1.51540i	−0.652604	−	0.757699i	$-0.726323\pi$
$751$		−	41.5286i		−	1.51540i	0.652604	−	0.757699i	$-0.273677\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	26.6311	−	29.0784i	0.969207	−	1.05827i
$756$		0			0
$757$	3.33513			0.121217			0.0606087	−	0.998162i	$-0.480696\pi$
$757$	3.33513			0.121217			0.0606087	+	0.998162i	$0.480696\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			49.3019i			1.78719i	0.448870	+	0.893597i	$0.351827\pi$
$761$			49.3019i			1.78719i	−0.448870	+	0.893597i	$0.648173\pi$
$762$		0			0
$763$			11.1878i			0.405025i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	35.6647i		−	1.28778i
$768$		0			0
$769$	25.6155			0.923720			0.461860	−	0.886953i	$-0.347182\pi$
$769$	25.6155			0.923720			0.461860	+	0.886953i	$0.347182\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			2.27678i			0.0818901i	0.999161	+	0.0409450i	$0.0130369\pi$
$773$			2.27678i			0.0818901i	−0.999161	+	0.0409450i	$0.986963\pi$
$774$		0			0
$775$	41.3686	−	3.64162i	1.48600	−	0.130811i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	18.2107i		−	0.652465i
$780$		0			0
$781$		−	25.9396i		−	0.928192i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	4.63252	+	4.24264i	0.165342	+	0.151426i
$786$		0			0
$787$			19.9230i			0.710177i	0.934833	+	0.355088i	$0.115549\pi$
$787$			19.9230i			0.710177i	−0.934833	+	0.355088i	$0.884451\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−11.7460			−0.417641
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$			15.6829i			0.555516i	0.960651	+	0.277758i	$0.0895913\pi$
$797$			15.6829i			0.555516i	−0.960651	+	0.277758i	$0.910409\pi$
$798$		0			0
$799$		−	16.6114i		−	0.587670i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	12.3534			0.435942
$804$		0			0
$805$	4.79741	−	5.23827i	0.169087	−	0.184625i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	20.6695i		−	0.726700i	−0.931653	−	0.363350i	$-0.881633\pi$
$809$		−	20.6695i		−	0.726700i	0.931653	−	0.363350i	$-0.118367\pi$
$810$		0			0
$811$	49.4773			1.73738			0.868691	−	0.495354i	$-0.164962\pi$
$811$	49.4773			1.73738			0.868691	+	0.495354i	$0.164962\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−23.4921	+	25.6509i	−0.822892	+	0.898513i
$816$		0			0
$817$		−	24.2905i		−	0.849818i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−12.1521			−0.424110			−0.212055	−	0.977258i	$-0.568016\pi$
$821$	−12.1521			−0.424110			−0.212055	+	0.977258i	$0.568016\pi$
$822$		0			0
$823$	28.6692			0.999345			0.499673	−	0.866214i	$-0.333454\pi$
$823$	28.6692			0.999345			0.499673	+	0.866214i	$0.333454\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−39.1746			−1.36224			−0.681118	−	0.732174i	$-0.738506\pi$
$827$	−39.1746			−1.36224			−0.681118	+	0.732174i	$0.738506\pi$
$828$		0			0
$829$		−	4.66410i		−	0.161991i	−0.996714	−	0.0809954i	$-0.974190\pi$
$829$		−	4.66410i		−	0.161991i	0.996714	−	0.0809954i	$-0.0258099\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−9.45514			−0.327601
$834$		0			0
$835$	14.2462	−	15.5554i	0.493010	−	0.538316i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−24.9381			−0.860959			−0.430479	−	0.902600i	$-0.641656\pi$
$839$	−24.9381			−0.860959			−0.430479	+	0.902600i	$0.641656\pi$
$840$		0			0
$841$	42.3693			1.46101
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.09501	+	2.83453i	0.106472	+	0.0975108i
$846$		0			0
$847$	−5.73384			−0.197017
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	25.8040i		−	0.884551i
$852$		0			0
$853$	−45.5249			−1.55874			−0.779371	−	0.626563i	$-0.784461\pi$
$853$	−45.5249			−1.55874			−0.779371	+	0.626563i	$0.784461\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	40.7188			1.39093			0.695464	−	0.718561i	$-0.255199\pi$
$857$	40.7188			1.39093			0.695464	+	0.718561i	$0.255199\pi$
$858$		0			0
$859$	33.7538			1.15166			0.575832	−	0.817568i	$-0.304678\pi$
$859$	33.7538			1.15166			0.575832	+	0.817568i	$0.304678\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	45.6786i		−	1.55492i	−0.628935	−	0.777458i	$-0.716509\pi$
$863$		−	45.6786i		−	1.55492i	0.628935	−	0.777458i	$-0.283491\pi$
$864$		0			0
$865$	2.56155	−	2.79695i	0.0870954	−	0.0950991i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.25801			0.0765978
$870$		0			0
$871$			40.5061i			1.37250i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−6.36679	+	8.31118i	−0.215237	+	0.280969i
$876$		0			0
$877$	46.5766			1.57278			0.786389	−	0.617731i	$-0.211948\pi$
$877$	46.5766			1.57278			0.786389	+	0.617731i	$0.211948\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			5.83095i			0.196450i	0.995164	+	0.0982249i	$0.0313164\pi$
$881$			5.83095i			0.196450i	−0.995164	+	0.0982249i	$0.968684\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	16.2177i		−	0.544538i	−0.962221	−	0.272269i	$-0.912226\pi$
$887$		−	16.2177i		−	0.544538i	0.962221	−	0.272269i	$-0.0877741\pi$
$888$		0			0
$889$	4.38447			0.147050
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	33.5968i		−	1.12427i
$894$		0			0
$895$	−12.4041	+	13.5440i	−0.414623	+	0.452725i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			70.1670i			2.34020i
$900$		0			0
$901$		−	7.85741i		−	0.261768i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−7.04383	+	7.69113i	−0.234145	+	0.255662i
$906$		0			0
$907$			16.5129i			0.548300i	0.961687	+	0.274150i	$0.0883965\pi$
$907$			16.5129i			0.548300i	−0.961687	+	0.274150i	$0.911603\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	36.6842			1.21540			0.607700	−	0.794167i	$-0.292092\pi$
$911$	36.6842			1.21540			0.607700	+	0.794167i	$0.292092\pi$
$912$		0			0
$913$			31.1107i			1.02962i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	0.580639i		−	0.0191744i
$918$		0			0
$919$		−	30.1554i		−	0.994735i	−0.867540	−	0.497368i	$-0.834300\pi$
$919$		−	30.1554i		−	0.994735i	0.867540	−	0.497368i	$-0.165700\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−39.1746			−1.28945
$924$		0			0
$925$	3.33513	+	37.8869i	0.109658	+	1.24571i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			3.00252i			0.0985096i	0.998786	+	0.0492548i	$0.0156846\pi$
$929$			3.00252i			0.0985096i	−0.998786	+	0.0492548i	$0.984315\pi$
$930$		0			0
$931$	−19.1231			−0.626734
$932$		0			0
$933$		0			0
$934$		0			0
$935$	5.15002	−	5.62329i	0.168424	−	0.183901i
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3.29801			0.107512			0.0537561	−	0.998554i	$-0.482881\pi$
$941$	3.29801			0.107512			0.0537561	+	0.998554i	$0.482881\pi$
$942$		0			0
$943$	−19.7802			−0.644133
$944$		0			0
$945$		0			0
$946$		0			0
$947$	32.9979			1.07229			0.536144	−	0.844126i	$-0.319880\pi$
$947$	32.9979			1.07229			0.536144	+	0.844126i	$0.319880\pi$
$948$		0			0
$949$		−	18.6564i		−	0.605612i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−12.5435			−0.406323			−0.203162	−	0.979145i	$-0.565122\pi$
$953$	−12.5435			−0.406323			−0.203162	+	0.979145i	$0.565122\pi$
$954$		0			0
$955$	−21.7538	−	19.9230i	−0.703936	−	0.644692i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.44600			−0.0466939
$960$		0			0
$961$	−37.9848			−1.22532
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−35.2381	+	38.4764i	−1.13436	+	1.23860i
$966$		0			0
$967$	−15.0981			−0.485522			−0.242761	−	0.970086i	$-0.578053\pi$
$967$	−15.0981			−0.485522			−0.242761	+	0.970086i	$0.578053\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			19.8753i			0.637829i	0.947783	+	0.318915i	$0.103318\pi$
$971$			19.8753i			0.637829i	−0.947783	+	0.318915i	$0.896682\pi$
$972$		0			0
$973$	−11.2371			−0.360245
$974$		0			0
$975$		0			0
$976$		0			0
$977$	37.6305			1.20390			0.601952	−	0.798532i	$-0.294390\pi$
$977$	37.6305			1.20390			0.601952	+	0.798532i	$0.294390\pi$
$978$		0			0
$979$	−28.8769			−0.922910
$980$		0			0
$981$		0			0
$982$		0			0
$983$			53.0438i			1.69183i	0.533315	+	0.845917i	$0.320946\pi$
$983$			53.0438i			1.69183i	−0.533315	+	0.845917i	$0.679054\pi$
$984$		0			0
$985$	−7.43845	+	8.12201i	−0.237009	+	0.258789i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−26.3841			−0.838966
$990$		0			0
$991$		−	17.6339i		−	0.560159i	−0.959977	−	0.280080i	$-0.909639\pi$
$991$		−	17.6339i		−	0.560159i	0.959977	−	0.280080i	$-0.0903609\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	26.6311	−	29.0784i	0.844264	−	0.921848i
$996$		0			0
$997$	25.5141			0.808039			0.404019	−	0.914750i	$-0.367613\pi$
$997$	25.5141			0.808039			0.404019	+	0.914750i	$0.367613\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.m.c.719.5		16
3.2	odd	2	inner	1440.2.m.c.719.11		16
4.3	odd	2		360.2.m.c.179.13	yes	16
5.2	odd	4		7200.2.b.i.4751.8		16
5.3	odd	4		7200.2.b.i.4751.11		16
5.4	even	2	inner	1440.2.m.c.719.8		16
8.3	odd	2	inner	1440.2.m.c.719.12		16
8.5	even	2		360.2.m.c.179.16	yes	16
12.11	even	2		360.2.m.c.179.4	yes	16
15.2	even	4		7200.2.b.i.4751.5		16
15.8	even	4		7200.2.b.i.4751.10		16
15.14	odd	2	inner	1440.2.m.c.719.10		16
20.3	even	4		1800.2.b.g.251.5		16
20.7	even	4		1800.2.b.g.251.12		16
20.19	odd	2		360.2.m.c.179.3	yes	16
24.5	odd	2		360.2.m.c.179.1	✓	16
24.11	even	2	inner	1440.2.m.c.719.6		16
40.3	even	4		7200.2.b.i.4751.7		16
40.13	odd	4		1800.2.b.g.251.10		16
40.19	odd	2	inner	1440.2.m.c.719.9		16
40.27	even	4		7200.2.b.i.4751.12		16
40.29	even	2		360.2.m.c.179.2	yes	16
40.37	odd	4		1800.2.b.g.251.7		16
60.23	odd	4		1800.2.b.g.251.11		16
60.47	odd	4		1800.2.b.g.251.6		16
60.59	even	2		360.2.m.c.179.14	yes	16
120.29	odd	2		360.2.m.c.179.15	yes	16
120.53	even	4		1800.2.b.g.251.8		16
120.59	even	2	inner	1440.2.m.c.719.7		16
120.77	even	4		1800.2.b.g.251.9		16
120.83	odd	4		7200.2.b.i.4751.6		16
120.107	odd	4		7200.2.b.i.4751.9		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.m.c.179.1	✓	16	24.5	odd	2
360.2.m.c.179.2	yes	16	40.29	even	2
360.2.m.c.179.3	yes	16	20.19	odd	2
360.2.m.c.179.4	yes	16	12.11	even	2
360.2.m.c.179.13	yes	16	4.3	odd	2
360.2.m.c.179.14	yes	16	60.59	even	2
360.2.m.c.179.15	yes	16	120.29	odd	2
360.2.m.c.179.16	yes	16	8.5	even	2
1440.2.m.c.719.5		16	1.1	even	1	trivial
1440.2.m.c.719.6		16	24.11	even	2	inner
1440.2.m.c.719.7		16	120.59	even	2	inner
1440.2.m.c.719.8		16	5.4	even	2	inner
1440.2.m.c.719.9		16	40.19	odd	2	inner
1440.2.m.c.719.10		16	15.14	odd	2	inner
1440.2.m.c.719.11		16	3.2	odd	2	inner
1440.2.m.c.719.12		16	8.3	odd	2	inner
1800.2.b.g.251.5		16	20.3	even	4
1800.2.b.g.251.6		16	60.47	odd	4
1800.2.b.g.251.7		16	40.37	odd	4
1800.2.b.g.251.8		16	120.53	even	4
1800.2.b.g.251.9		16	120.77	even	4
1800.2.b.g.251.10		16	40.13	odd	4
1800.2.b.g.251.11		16	60.23	odd	4
1800.2.b.g.251.12		16	20.7	even	4
7200.2.b.i.4751.5		16	15.2	even	4
7200.2.b.i.4751.6		16	120.83	odd	4
7200.2.b.i.4751.7		16	40.3	even	4
7200.2.b.i.4751.8		16	5.2	odd	4
7200.2.b.i.4751.9		16	120.107	odd	4
7200.2.b.i.4751.10		16	15.8	even	4
7200.2.b.i.4751.11		16	5.3	odd	4
7200.2.b.i.4751.12		16	40.27	even	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:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.64901 - 1.51022i) q^{5} -0.936426 q^{7} +2.20837i q^{11} +3.33513 q^{13} +1.54417 q^{17} +3.12311 q^{19} -3.39228i q^{23} +(0.438447 + 4.98074i) q^{25} -8.44804 q^{29} -8.30571i q^{31} +(1.54417 + 1.41421i) q^{35} +7.60669 q^{37} -5.83095i q^{41} -7.77769i q^{43} -10.7575i q^{47} -6.12311 q^{49} -5.08842i q^{53} +(3.33513 - 3.64162i) q^{55} -10.6937i q^{59} +(-5.49966 - 5.03680i) q^{65} +12.1453i q^{67} -11.7460 q^{71} -5.59390i q^{73} -2.06798i q^{77} -1.02248i q^{79} +14.0877 q^{83} +(-2.54635 - 2.33205i) q^{85} +13.0761i q^{89} -3.12311 q^{91} +(-5.15002 - 4.71659i) q^{95} -2.18379i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{19} + 40 q^{25} - 32 q^{49} + 16 q^{91}+O(q^{100}) \) 16 * q - 16 * q^19 + 40 * q^25 - 32 * q^49 + 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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