LMFDB - Embedded newform 720.2.x.g.703.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [720,2,Mod(127,720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("720.127"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	720.x (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$5.74922894553$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.3317760000.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 25x^{4} + 625 $ x^8 - 25*x^4 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			703.4
Root			$2.15988 - 0.578737i$ of defining polynomial
Character	$\chi$	$=$	720.703
Dual form			720.2.x.g.127.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.58114 + 1.58114i) q^{5} +(1.73205 + 1.73205i) q^{7} +5.47723i q^{11} +(-2.00000 - 2.00000i) q^{13} +(3.16228 - 3.16228i) q^{17} -3.46410 q^{19} +(-5.47723 + 5.47723i) q^{23} +5.00000i q^{25} +3.16228i q^{29} -6.92820i q^{31} +5.47723i q^{35} +(2.00000 - 2.00000i) q^{37} +6.32456 q^{41} +(3.46410 - 3.46410i) q^{43} +(5.47723 + 5.47723i) q^{47} -1.00000i q^{49} +(6.32456 + 6.32456i) q^{53} +(-8.66025 + 8.66025i) q^{55} -5.47723 q^{59} -6.00000 q^{61} -6.32456i q^{65} +(10.3923 + 10.3923i) q^{67} +(-1.00000 - 1.00000i) q^{73} +(-9.48683 + 9.48683i) q^{77} +(5.47723 - 5.47723i) q^{83} +10.0000 q^{85} -12.6491i q^{89} -6.92820i q^{91} +(-5.47723 - 5.47723i) q^{95} +(11.0000 - 11.0000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{13} + 16 q^{37} - 48 q^{61} - 8 q^{73} + 80 q^{85} + 88 q^{97}+O(q^{100}) $ 8 * q - 16 * q^13 + 16 * q^37 - 48 * q^61 - 8 * q^73 + 80 * q^85 + 88 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$	$1$

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
$5$	1.58114	+	1.58114i	0.707107	+	0.707107i
$5$	1.58114	+	1.58114i	0.707107	+	0.707107i
$6$		0			0
$7$	1.73205	+	1.73205i	0.654654	+	0.654654i	0.954110	−	0.299456i	$-0.0968053\pi$
$7$	1.73205	+	1.73205i	0.654654	+	0.654654i	−0.299456	+	0.954110i	$0.596805\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			5.47723i			1.65145i	0.564076	+	0.825723i	$0.309232\pi$
$11$			5.47723i			1.65145i	−0.564076	+	0.825723i	$0.690768\pi$
$12$		0			0
$13$	−2.00000	−	2.00000i	−0.554700	−	0.554700i	0.373094	−	0.927794i	$-0.378297\pi$
$13$	−2.00000	−	2.00000i	−0.554700	−	0.554700i	−0.927794	+	0.373094i	$0.878297\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.16228	−	3.16228i	0.766965	−	0.766965i	−0.210606	−	0.977571i	$-0.567544\pi$
$17$	3.16228	−	3.16228i	0.766965	−	0.766965i	0.977571	+	0.210606i	$0.0675437\pi$
$18$		0			0
$19$	−3.46410			−0.794719			−0.397360	−	0.917663i	$-0.630073\pi$
$19$	−3.46410			−0.794719			−0.397360	+	0.917663i	$0.630073\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−5.47723	+	5.47723i	−1.14208	+	1.14208i	−0.154011	+	0.988069i	$0.549219\pi$
$23$	−5.47723	+	5.47723i	−1.14208	+	1.14208i	−0.988069	+	0.154011i	$0.950781\pi$
$24$		0			0
$25$			5.00000i			1.00000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			3.16228i			0.587220i	0.955925	+	0.293610i	$0.0948567\pi$
$29$			3.16228i			0.587220i	−0.955925	+	0.293610i	$0.905143\pi$
$30$		0			0
$31$		−	6.92820i		−	1.24434i	−0.782881	−	0.622171i	$-0.786251\pi$
$31$		−	6.92820i		−	1.24434i	0.782881	−	0.622171i	$-0.213749\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			5.47723i			0.925820i
$36$		0			0
$37$	2.00000	−	2.00000i	0.328798	−	0.328798i	−0.523331	−	0.852129i	$-0.675311\pi$
$37$	2.00000	−	2.00000i	0.328798	−	0.328798i	0.852129	+	0.523331i	$0.175311\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.32456			0.987730			0.493865	−	0.869539i	$-0.335584\pi$
$41$	6.32456			0.987730			0.493865	+	0.869539i	$0.335584\pi$
$42$		0			0
$43$	3.46410	−	3.46410i	0.528271	−	0.528271i	−0.391786	−	0.920056i	$-0.628143\pi$
$43$	3.46410	−	3.46410i	0.528271	−	0.528271i	0.920056	+	0.391786i	$0.128143\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.47723	+	5.47723i	0.798935	+	0.798935i	0.982928	−	0.183992i	$-0.0589021\pi$
$47$	5.47723	+	5.47723i	0.798935	+	0.798935i	−0.183992	+	0.982928i	$0.558902\pi$
$48$		0			0
$49$		−	1.00000i		−	0.142857i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.32456	+	6.32456i	0.868744	+	0.868744i	0.992333	−	0.123589i	$-0.0394404\pi$
$53$	6.32456	+	6.32456i	0.868744	+	0.868744i	−0.123589	+	0.992333i	$0.539440\pi$
$54$		0			0
$55$	−8.66025	+	8.66025i	−1.16775	+	1.16775i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.47723			−0.713074			−0.356537	−	0.934281i	$-0.616043\pi$
$59$	−5.47723			−0.713074			−0.356537	+	0.934281i	$0.616043\pi$
$60$		0			0
$61$	−6.00000			−0.768221			−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000			−0.768221			−0.384111	+	0.923287i	$0.625492\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	6.32456i		−	0.784465i
$66$		0			0
$67$	10.3923	+	10.3923i	1.26962	+	1.26962i	0.946283	+	0.323339i	$0.104805\pi$
$67$	10.3923	+	10.3923i	1.26962	+	1.26962i	0.323339	+	0.946283i	$0.395195\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−1.00000	−	1.00000i	−0.117041	−	0.117041i	0.646160	−	0.763202i	$-0.276374\pi$
$73$	−1.00000	−	1.00000i	−0.117041	−	0.117041i	−0.763202	+	0.646160i	$0.776374\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−9.48683	+	9.48683i	−1.08112	+	1.08112i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.47723	−	5.47723i	0.601204	−	0.601204i	−0.339428	−	0.940632i	$-0.610234\pi$
$83$	5.47723	−	5.47723i	0.601204	−	0.601204i	0.940632	+	0.339428i	$0.110234\pi$
$84$		0			0
$85$	10.0000			1.08465
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	12.6491i		−	1.34080i	−0.741999	−	0.670402i	$-0.766122\pi$
$89$		−	12.6491i		−	1.34080i	0.741999	−	0.670402i	$-0.233878\pi$
$90$		0			0
$91$		−	6.92820i		−	0.726273i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.47723	−	5.47723i	−0.561951	−	0.561951i
$96$		0			0
$97$	11.0000	−	11.0000i	1.11688	−	1.11688i	0.124684	−	0.992196i	$-0.460208\pi$
$97$	11.0000	−	11.0000i	1.11688	−	1.11688i	0.992196	−	0.124684i	$-0.0397918\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.16228			−0.314658			−0.157329	−	0.987546i	$-0.550288\pi$
$101$	−3.16228			−0.314658			−0.157329	+	0.987546i	$0.550288\pi$
$102$		0			0
$103$	8.66025	−	8.66025i	0.853320	−	0.853320i	−0.137220	−	0.990541i	$-0.543817\pi$
$103$	8.66025	−	8.66025i	0.853320	−	0.853320i	0.990541	+	0.137220i	$0.0438168\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.47723	−	5.47723i	−0.529503	−	0.529503i	0.390921	−	0.920424i	$-0.372157\pi$
$107$	−5.47723	−	5.47723i	−0.529503	−	0.529503i	−0.920424	+	0.390921i	$0.872157\pi$
$108$		0			0
$109$			18.0000i			1.72409i	0.506834	+	0.862044i	$0.330816\pi$
$109$			18.0000i			1.72409i	−0.506834	+	0.862044i	$0.669184\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−9.48683	−	9.48683i	−0.892446	−	0.892446i	0.102307	−	0.994753i	$-0.467378\pi$
$113$	−9.48683	−	9.48683i	−0.892446	−	0.892446i	−0.994753	+	0.102307i	$0.967378\pi$
$114$		0			0
$115$	−17.3205			−1.61515
$116$		0			0
$117$		0			0
$118$		0			0
$119$	10.9545			1.00419
$120$		0			0
$121$	−19.0000			−1.72727
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.90569	+	7.90569i	−0.707107	+	0.707107i
$126$		0			0
$127$	8.66025	+	8.66025i	0.768473	+	0.768473i	0.977838	−	0.209364i	$-0.0671395\pi$
$127$	8.66025	+	8.66025i	0.768473	+	0.768473i	−0.209364	+	0.977838i	$0.567139\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	16.4317i		−	1.43564i	−0.696228	−	0.717821i	$-0.745140\pi$
$131$		−	16.4317i		−	1.43564i	0.696228	−	0.717821i	$-0.254860\pi$
$132$		0			0
$133$	−6.00000	−	6.00000i	−0.520266	−	0.520266i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	9.48683	−	9.48683i	0.810515	−	0.810515i	−0.174196	−	0.984711i	$-0.555733\pi$
$137$	9.48683	−	9.48683i	0.810515	−	0.810515i	0.984711	+	0.174196i	$0.0557327\pi$
$138$		0			0
$139$	−17.3205			−1.46911			−0.734553	−	0.678551i	$-0.762608\pi$
$139$	−17.3205			−1.46911			−0.734553	+	0.678551i	$0.762608\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	10.9545	−	10.9545i	0.916057	−	0.916057i
$144$		0			0
$145$	−5.00000	+	5.00000i	−0.415227	+	0.415227i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			22.1359i			1.81345i	0.421725	+	0.906724i	$0.361425\pi$
$149$			22.1359i			1.81345i	−0.421725	+	0.906724i	$0.638575\pi$
$150$		0			0
$151$		−	10.3923i		−	0.845714i	−0.906196	−	0.422857i	$-0.861027\pi$
$151$		−	10.3923i		−	0.845714i	0.906196	−	0.422857i	$-0.138973\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	10.9545	−	10.9545i	0.879883	−	0.879883i
$156$		0			0
$157$	4.00000	−	4.00000i	0.319235	−	0.319235i	−0.529238	−	0.848473i	$-0.677522\pi$
$157$	4.00000	−	4.00000i	0.319235	−	0.319235i	0.848473	+	0.529238i	$0.177522\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−18.9737			−1.49533
$162$		0			0
$163$	13.8564	−	13.8564i	1.08532	−	1.08532i	0.0893140	−	0.996004i	$-0.471533\pi$
$163$	13.8564	−	13.8564i	1.08532	−	1.08532i	0.996004	−	0.0893140i	$-0.0284675\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−16.4317	−	16.4317i	−1.27152	−	1.27152i	−0.945290	−	0.326231i	$-0.894221\pi$
$167$	−16.4317	−	16.4317i	−1.27152	−	1.27152i	−0.326231	−	0.945290i	$-0.605779\pi$
$168$		0			0
$169$		−	5.00000i		−	0.384615i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−9.48683	−	9.48683i	−0.721271	−	0.721271i	0.247593	−	0.968864i	$-0.420360\pi$
$173$	−9.48683	−	9.48683i	−0.721271	−	0.721271i	−0.968864	+	0.247593i	$0.920360\pi$
$174$		0			0
$175$	−8.66025	+	8.66025i	−0.654654	+	0.654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	5.47723			0.409387			0.204694	−	0.978826i	$-0.434380\pi$
$179$	5.47723			0.409387			0.204694	+	0.978826i	$0.434380\pi$
$180$		0			0
$181$	6.00000			0.445976			0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000			0.445976			0.222988	+	0.974821i	$0.428419\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	6.32456			0.464991
$186$		0			0
$187$	17.3205	+	17.3205i	1.26660	+	1.26660i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	10.9545i		−	0.792636i	−0.918113	−	0.396318i	$-0.870288\pi$
$191$		−	10.9545i		−	0.792636i	0.918113	−	0.396318i	$-0.129712\pi$
$192$		0			0
$193$	7.00000	+	7.00000i	0.503871	+	0.503871i	0.912639	−	0.408768i	$-0.134041\pi$
$193$	7.00000	+	7.00000i	0.503871	+	0.503871i	−0.408768	+	0.912639i	$0.634041\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$	−3.46410			−0.245564			−0.122782	−	0.992434i	$-0.539182\pi$
$199$	−3.46410			−0.245564			−0.122782	+	0.992434i	$0.539182\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.47723	+	5.47723i	−0.384426	+	0.384426i
$204$		0			0
$205$	10.0000	+	10.0000i	0.698430	+	0.698430i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	18.9737i		−	1.31244i
$210$		0			0
$211$		−	10.3923i		−	0.715436i	−0.933830	−	0.357718i	$-0.883555\pi$
$211$		−	10.3923i		−	0.715436i	0.933830	−	0.357718i	$-0.116445\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	10.9545			0.747087
$216$		0			0
$217$	12.0000	−	12.0000i	0.814613	−	0.814613i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12.6491			−0.850871
$222$		0			0
$223$	−5.19615	+	5.19615i	−0.347960	+	0.347960i	−0.859349	−	0.511389i	$-0.829131\pi$
$223$	−5.19615	+	5.19615i	−0.347960	+	0.347960i	0.511389	+	0.859349i	$0.329131\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.9545	+	10.9545i	0.727072	+	0.727072i	0.970036	−	0.242963i	$-0.0781194\pi$
$227$	10.9545	+	10.9545i	0.727072	+	0.727072i	−0.242963	+	0.970036i	$0.578119\pi$
$228$		0			0
$229$		−	2.00000i		−	0.132164i	−0.997814	−	0.0660819i	$-0.978950\pi$
$229$		−	2.00000i		−	0.132164i	0.997814	−	0.0660819i	$-0.0210498\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.16228	+	3.16228i	0.207168	+	0.207168i	0.803063	−	0.595895i	$-0.203202\pi$
$233$	3.16228	+	3.16228i	0.207168	+	0.207168i	−0.595895	+	0.803063i	$0.703202\pi$
$234$		0			0
$235$			17.3205i			1.12987i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−21.9089			−1.41717			−0.708585	−	0.705626i	$-0.750666\pi$
$239$	−21.9089			−1.41717			−0.708585	+	0.705626i	$0.750666\pi$
$240$		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
$245$	1.58114	−	1.58114i	0.101015	−	0.101015i
$246$		0			0
$247$	6.92820	+	6.92820i	0.440831	+	0.440831i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			5.47723i			0.345719i	0.984946	+	0.172860i	$0.0553007\pi$
$251$			5.47723i			0.345719i	−0.984946	+	0.172860i	$0.944699\pi$
$252$		0			0
$253$	−30.0000	−	30.0000i	−1.88608	−	1.88608i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−9.48683	+	9.48683i	−0.591772	+	0.591772i	−0.938110	−	0.346338i	$-0.887425\pi$
$257$	−9.48683	+	9.48683i	−0.591772	+	0.591772i	0.346338	+	0.938110i	$0.387425\pi$
$258$		0			0
$259$	6.92820			0.430498
$260$		0			0
$261$		0			0
$262$		0			0
$263$	5.47723	−	5.47723i	0.337740	−	0.337740i	−0.517776	−	0.855516i	$-0.673240\pi$
$263$	5.47723	−	5.47723i	0.337740	−	0.337740i	0.855516	+	0.517776i	$0.173240\pi$
$264$		0			0
$265$			20.0000i			1.22859i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	3.16228i		−	0.192807i	−0.995342	−	0.0964037i	$-0.969266\pi$
$269$		−	3.16228i		−	0.192807i	0.995342	−	0.0964037i	$-0.0307340\pi$
$270$		0			0
$271$		−	17.3205i		−	1.05215i	−0.850439	−	0.526073i	$-0.823664\pi$
$271$		−	17.3205i		−	1.05215i	0.850439	−	0.526073i	$-0.176336\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−27.3861			−1.65145
$276$		0			0
$277$	−14.0000	+	14.0000i	−0.841178	+	0.841178i	−0.989012	−	0.147834i	$-0.952770\pi$
$277$	−14.0000	+	14.0000i	−0.841178	+	0.841178i	0.147834	+	0.989012i	$0.452770\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	31.6228			1.88646			0.943228	−	0.332145i	$-0.107772\pi$
$281$	31.6228			1.88646			0.943228	+	0.332145i	$0.107772\pi$
$282$		0			0
$283$	−3.46410	+	3.46410i	−0.205919	+	0.205919i	−0.802531	−	0.596611i	$-0.796514\pi$
$283$	−3.46410	+	3.46410i	−0.205919	+	0.205919i	0.596611	+	0.802531i	$0.296514\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	10.9545	+	10.9545i	0.646621	+	0.646621i
$288$		0			0
$289$		−	3.00000i		−	0.176471i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	18.9737	+	18.9737i	1.10845	+	1.10845i	0.993354	+	0.115099i	$0.0367187\pi$
$293$	18.9737	+	18.9737i	1.10845	+	1.10845i	0.115099	+	0.993354i	$0.463281\pi$
$294$		0			0
$295$	−8.66025	−	8.66025i	−0.504219	−	0.504219i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	21.9089			1.26702
$300$		0			0
$301$	12.0000			0.691669
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−9.48683	−	9.48683i	−0.543214	−	0.543214i
$306$		0			0
$307$	−10.3923	−	10.3923i	−0.593120	−	0.593120i	0.345353	−	0.938473i	$-0.387759\pi$
$307$	−10.3923	−	10.3923i	−0.593120	−	0.593120i	−0.938473	+	0.345353i	$0.887759\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			21.9089i			1.24234i	0.783676	+	0.621170i	$0.213342\pi$
$311$			21.9089i			1.24234i	−0.783676	+	0.621170i	$0.786658\pi$
$312$		0			0
$313$	17.0000	+	17.0000i	0.960897	+	0.960897i	0.999264	−	0.0383669i	$-0.0122156\pi$
$313$	17.0000	+	17.0000i	0.960897	+	0.960897i	−0.0383669	+	0.999264i	$0.512216\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.16228	+	3.16228i	−0.177611	+	0.177611i	−0.790314	−	0.612702i	$-0.790082\pi$
$317$	−3.16228	+	3.16228i	−0.177611	+	0.177611i	0.612702	+	0.790314i	$0.290082\pi$
$318$		0			0
$319$	−17.3205			−0.969762
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−10.9545	+	10.9545i	−0.609522	+	0.609522i
$324$		0			0
$325$	10.0000	−	10.0000i	0.554700	−	0.554700i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			18.9737i			1.04605i
$330$		0			0
$331$		−	17.3205i		−	0.952021i	−0.879440	−	0.476011i	$-0.842082\pi$
$331$		−	17.3205i		−	0.952021i	0.879440	−	0.476011i	$-0.157918\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			32.8634i			1.79552i
$336$		0			0
$337$	1.00000	−	1.00000i	0.0544735	−	0.0544735i	−0.679345	−	0.733819i	$-0.737736\pi$
$337$	1.00000	−	1.00000i	0.0544735	−	0.0544735i	0.733819	+	0.679345i	$0.237736\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	37.9473			2.05496
$342$		0			0
$343$	13.8564	−	13.8564i	0.748176	−	0.748176i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$			6.00000i			0.321173i	0.987022	+	0.160586i	$0.0513385\pi$
$349$			6.00000i			0.321173i	−0.987022	+	0.160586i	$0.948662\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3.16228	+	3.16228i	0.168311	+	0.168311i	0.786237	−	0.617926i	$-0.212027\pi$
$353$	3.16228	+	3.16228i	0.168311	+	0.168311i	−0.617926	+	0.786237i	$0.712027\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10.9545			−0.578154			−0.289077	−	0.957306i	$-0.593348\pi$
$359$	−10.9545			−0.578154			−0.289077	+	0.957306i	$0.593348\pi$
$360$		0			0
$361$	−7.00000			−0.368421
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	3.16228i		−	0.165521i
$366$		0			0
$367$	15.5885	+	15.5885i	0.813711	+	0.813711i	0.985188	−	0.171477i	$-0.0548540\pi$
$367$	15.5885	+	15.5885i	0.813711	+	0.813711i	−0.171477	+	0.985188i	$0.554854\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			21.9089i			1.13745i
$372$		0			0
$373$	−8.00000	−	8.00000i	−0.414224	−	0.414224i	0.468983	−	0.883207i	$-0.344621\pi$
$373$	−8.00000	−	8.00000i	−0.414224	−	0.414224i	−0.883207	+	0.468983i	$0.844621\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6.32456	−	6.32456i	0.325731	−	0.325731i
$378$		0			0
$379$	3.46410			0.177939			0.0889695	−	0.996034i	$-0.471643\pi$
$379$	3.46410			0.177939			0.0889695	+	0.996034i	$0.471643\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−16.4317	+	16.4317i	−0.839619	+	0.839619i	−0.988809	−	0.149189i	$-0.952334\pi$
$383$	−16.4317	+	16.4317i	−0.839619	+	0.839619i	0.149189	+	0.988809i	$0.452334\pi$
$384$		0			0
$385$	−30.0000			−1.52894
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	22.1359i		−	1.12234i	−0.827702	−	0.561168i	$-0.810352\pi$
$389$		−	22.1359i		−	1.12234i	0.827702	−	0.561168i	$-0.189648\pi$
$390$		0			0
$391$			34.6410i			1.75187i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	4.00000	−	4.00000i	0.200754	−	0.200754i	−0.599569	−	0.800323i	$-0.704661\pi$
$397$	4.00000	−	4.00000i	0.200754	−	0.200754i	0.800323	+	0.599569i	$0.204661\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−12.6491			−0.631666			−0.315833	−	0.948815i	$-0.602284\pi$
$401$	−12.6491			−0.631666			−0.315833	+	0.948815i	$0.602284\pi$
$402$		0			0
$403$	−13.8564	+	13.8564i	−0.690237	+	0.690237i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	10.9545	+	10.9545i	0.542992	+	0.542992i
$408$		0			0
$409$		−	12.0000i		−	0.593362i	−0.954977	−	0.296681i	$-0.904120\pi$
$409$		−	12.0000i		−	0.593362i	0.954977	−	0.296681i	$-0.0958798\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−9.48683	−	9.48683i	−0.466817	−	0.466817i
$414$		0			0
$415$	17.3205			0.850230
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−27.3861			−1.33790			−0.668950	−	0.743307i	$-0.733256\pi$
$419$	−27.3861			−1.33790			−0.668950	+	0.743307i	$0.733256\pi$
$420$		0			0
$421$	−2.00000			−0.0974740			−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000			−0.0974740			−0.0487370	+	0.998812i	$0.515520\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	15.8114	+	15.8114i	0.766965	+	0.766965i
$426$		0			0
$427$	−10.3923	−	10.3923i	−0.502919	−	0.502919i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			10.9545i			0.527657i	0.964570	+	0.263829i	$0.0849854\pi$
$431$			10.9545i			0.527657i	−0.964570	+	0.263829i	$0.915015\pi$
$432$		0			0
$433$	17.0000	+	17.0000i	0.816968	+	0.816968i	0.985668	−	0.168700i	$-0.0539568\pi$
$433$	17.0000	+	17.0000i	0.816968	+	0.816968i	−0.168700	+	0.985668i	$0.553957\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	18.9737	−	18.9737i	0.907634	−	0.907634i
$438$		0			0
$439$	3.46410			0.165333			0.0826663	−	0.996577i	$-0.473656\pi$
$439$	3.46410			0.165333			0.0826663	+	0.996577i	$0.473656\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−10.9545	+	10.9545i	−0.520462	+	0.520462i	−0.917711	−	0.397249i	$-0.869965\pi$
$443$	−10.9545	+	10.9545i	−0.520462	+	0.520462i	0.397249	+	0.917711i	$0.369965\pi$
$444$		0			0
$445$	20.0000	−	20.0000i	0.948091	−	0.948091i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			31.6228i			1.49237i	0.665738	+	0.746186i	$0.268117\pi$
$449$			31.6228i			1.49237i	−0.665738	+	0.746186i	$0.731883\pi$
$450$		0			0
$451$			34.6410i			1.63118i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	10.9545	−	10.9545i	0.513553	−	0.513553i
$456$		0			0
$457$	23.0000	−	23.0000i	1.07589	−	1.07589i	0.0790217	−	0.996873i	$-0.474820\pi$
$457$	23.0000	−	23.0000i	1.07589	−	1.07589i	0.996873	−	0.0790217i	$-0.0251796\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−34.7851			−1.62010			−0.810051	−	0.586360i	$-0.800560\pi$
$461$	−34.7851			−1.62010			−0.810051	+	0.586360i	$0.800560\pi$
$462$		0			0
$463$	8.66025	−	8.66025i	0.402476	−	0.402476i	−0.476629	−	0.879105i	$-0.658141\pi$
$463$	8.66025	−	8.66025i	0.402476	−	0.402476i	0.879105	+	0.476629i	$0.158141\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5.47723	+	5.47723i	0.253456	+	0.253456i	0.822386	−	0.568930i	$-0.192643\pi$
$467$	5.47723	+	5.47723i	0.253456	+	0.253456i	−0.568930	+	0.822386i	$0.692643\pi$
$468$		0			0
$469$			36.0000i			1.66233i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	18.9737	+	18.9737i	0.872410	+	0.872410i
$474$		0			0
$475$		−	17.3205i		−	0.794719i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	21.9089			1.00104			0.500522	−	0.865724i	$-0.333142\pi$
$479$	21.9089			1.00104			0.500522	+	0.865724i	$0.333142\pi$
$480$		0			0
$481$	−8.00000			−0.364769
$482$		0			0
$483$		0			0
$484$		0			0
$485$	34.7851			1.57951
$486$		0			0
$487$	−19.0526	−	19.0526i	−0.863354	−	0.863354i	0.128372	−	0.991726i	$-0.459025\pi$
$487$	−19.0526	−	19.0526i	−0.863354	−	0.863354i	−0.991726	+	0.128372i	$0.959025\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	5.47723i		−	0.247184i	−0.992333	−	0.123592i	$-0.960559\pi$
$491$		−	5.47723i		−	0.247184i	0.992333	−	0.123592i	$-0.0394414\pi$
$492$		0			0
$493$	10.0000	+	10.0000i	0.450377	+	0.450377i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	31.1769			1.39567			0.697835	−	0.716258i	$-0.254147\pi$
$499$	31.1769			1.39567			0.697835	+	0.716258i	$0.254147\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−5.47723	+	5.47723i	−0.244217	+	0.244217i	−0.818592	−	0.574375i	$-0.805245\pi$
$503$	−5.47723	+	5.47723i	−0.244217	+	0.244217i	0.574375	+	0.818592i	$0.305245\pi$
$504$		0			0
$505$	−5.00000	−	5.00000i	−0.222497	−	0.222497i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	22.1359i		−	0.981158i	−0.871397	−	0.490579i	$-0.836785\pi$
$509$		−	22.1359i		−	0.981158i	0.871397	−	0.490579i	$-0.163215\pi$
$510$		0			0
$511$		−	3.46410i		−	0.153243i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	27.3861			1.20678
$516$		0			0
$517$	−30.0000	+	30.0000i	−1.31940	+	1.31940i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−6.32456			−0.277084			−0.138542	−	0.990357i	$-0.544242\pi$
$521$	−6.32456			−0.277084			−0.138542	+	0.990357i	$0.544242\pi$
$522$		0			0
$523$	−3.46410	+	3.46410i	−0.151475	+	0.151475i	−0.778776	−	0.627302i	$-0.784159\pi$
$523$	−3.46410	+	3.46410i	−0.151475	+	0.151475i	0.627302	+	0.778776i	$0.284159\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−21.9089	−	21.9089i	−0.954367	−	0.954367i
$528$		0			0
$529$		−	37.0000i		−	1.60870i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−12.6491	−	12.6491i	−0.547894	−	0.547894i
$534$		0			0
$535$		−	17.3205i		−	0.748831i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5.47723			0.235921
$540$		0			0
$541$	6.00000			0.257960			0.128980	−	0.991647i	$-0.458830\pi$
$541$	6.00000			0.257960			0.128980	+	0.991647i	$0.458830\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−28.4605	+	28.4605i	−1.21911	+	1.21911i
$546$		0			0
$547$	−17.3205	−	17.3205i	−0.740571	−	0.740571i	0.232117	−	0.972688i	$-0.425435\pi$
$547$	−17.3205	−	17.3205i	−0.740571	−	0.740571i	−0.972688	+	0.232117i	$0.925435\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	10.9545i		−	0.466675i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	6.32456	−	6.32456i	0.267980	−	0.267980i	−0.560306	−	0.828286i	$-0.689316\pi$
$557$	6.32456	−	6.32456i	0.267980	−	0.267980i	0.828286	+	0.560306i	$0.189316\pi$
$558$		0			0
$559$	−13.8564			−0.586064
$560$		0			0
$561$		0			0
$562$		0			0
$563$	21.9089	−	21.9089i	0.923350	−	0.923350i	−0.0739144	−	0.997265i	$-0.523549\pi$
$563$	21.9089	−	21.9089i	0.923350	−	0.923350i	0.997265	+	0.0739144i	$0.0235492\pi$
$564$		0			0
$565$		−	30.0000i		−	1.26211i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			25.2982i			1.06056i	0.847824	+	0.530278i	$0.177913\pi$
$569$			25.2982i			1.06056i	−0.847824	+	0.530278i	$0.822087\pi$
$570$		0			0
$571$		−	10.3923i		−	0.434904i	−0.976071	−	0.217452i	$-0.930225\pi$
$571$		−	10.3923i		−	0.434904i	0.976071	−	0.217452i	$-0.0697746\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−27.3861	−	27.3861i	−1.14208	−	1.14208i
$576$		0			0
$577$	−31.0000	+	31.0000i	−1.29055	+	1.29055i	−0.356098	+	0.934448i	$0.615893\pi$
$577$	−31.0000	+	31.0000i	−1.29055	+	1.29055i	−0.934448	+	0.356098i	$0.884107\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	18.9737			0.787160
$582$		0			0
$583$	−34.6410	+	34.6410i	−1.43468	+	1.43468i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−16.4317	−	16.4317i	−0.678208	−	0.678208i	0.281387	−	0.959594i	$-0.409206\pi$
$587$	−16.4317	−	16.4317i	−0.678208	−	0.678208i	−0.959594	+	0.281387i	$0.909206\pi$
$588$		0			0
$589$			24.0000i			0.988903i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−28.4605	−	28.4605i	−1.16873	−	1.16873i	−0.982507	−	0.186226i	$-0.940374\pi$
$593$	−28.4605	−	28.4605i	−1.16873	−	1.16873i	−0.186226	−	0.982507i	$-0.559626\pi$
$594$		0			0
$595$	17.3205	+	17.3205i	0.710072	+	0.710072i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	10.9545			0.447587			0.223793	−	0.974637i	$-0.428156\pi$
$599$	10.9545			0.447587			0.223793	+	0.974637i	$0.428156\pi$
$600$		0			0
$601$	−6.00000			−0.244745			−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000			−0.244745			−0.122373	+	0.992484i	$0.539050\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−30.0416	−	30.0416i	−1.22137	−	1.22137i
$606$		0			0
$607$	−25.9808	−	25.9808i	−1.05453	−	1.05453i	−0.998425	−	0.0561015i	$-0.982133\pi$
$607$	−25.9808	−	25.9808i	−1.05453	−	1.05453i	−0.0561015	−	0.998425i	$-0.517867\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	21.9089i		−	0.886339i
$612$		0			0
$613$	−4.00000	−	4.00000i	−0.161558	−	0.161558i	0.621698	−	0.783257i	$-0.286443\pi$
$613$	−4.00000	−	4.00000i	−0.161558	−	0.161558i	−0.783257	+	0.621698i	$0.786443\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−22.1359	+	22.1359i	−0.891160	+	0.891160i	−0.994632	−	0.103473i	$-0.967005\pi$
$617$	−22.1359	+	22.1359i	−0.891160	+	0.891160i	0.103473	+	0.994632i	$0.467005\pi$
$618$		0			0
$619$	31.1769			1.25311			0.626553	−	0.779379i	$-0.284465\pi$
$619$	31.1769			1.25311			0.626553	+	0.779379i	$0.284465\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	21.9089	−	21.9089i	0.877762	−	0.877762i
$624$		0			0
$625$	−25.0000			−1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	12.6491i		−	0.504353i
$630$		0			0
$631$			27.7128i			1.10323i	0.834099	+	0.551615i	$0.185988\pi$
$631$			27.7128i			1.10323i	−0.834099	+	0.551615i	$0.814012\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			27.3861i			1.08679i
$636$		0			0
$637$	−2.00000	+	2.00000i	−0.0792429	+	0.0792429i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12.6491			−0.499610			−0.249805	−	0.968296i	$-0.580366\pi$
$641$	−12.6491			−0.499610			−0.249805	+	0.968296i	$0.580366\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	5.47723	+	5.47723i	0.215332	+	0.215332i	0.806528	−	0.591196i	$-0.201344\pi$
$647$	5.47723	+	5.47723i	0.215332	+	0.215332i	−0.591196	+	0.806528i	$0.701344\pi$
$648$		0			0
$649$		−	30.0000i		−	1.17760i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15.8114	−	15.8114i	−0.618747	−	0.618747i	0.326463	−	0.945210i	$-0.394143\pi$
$653$	−15.8114	−	15.8114i	−0.618747	−	0.618747i	−0.945210	+	0.326463i	$0.894143\pi$
$654$		0			0
$655$	25.9808	−	25.9808i	1.01515	−	1.01515i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−5.47723			−0.213362			−0.106681	−	0.994293i	$-0.534022\pi$
$659$	−5.47723			−0.213362			−0.106681	+	0.994293i	$0.534022\pi$
$660$		0			0
$661$	42.0000			1.63361			0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000			1.63361			0.816805	+	0.576913i	$0.195743\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	18.9737i		−	0.735767i
$666$		0			0
$667$	−17.3205	−	17.3205i	−0.670653	−	0.670653i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	32.8634i		−	1.26868i
$672$		0			0
$673$	31.0000	+	31.0000i	1.19496	+	1.19496i	0.975656	+	0.219306i	$0.0703793\pi$
$673$	31.0000	+	31.0000i	1.19496	+	1.19496i	0.219306	+	0.975656i	$0.429621\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.16228	−	3.16228i	0.121536	−	0.121536i	−0.643723	−	0.765259i	$-0.722611\pi$
$677$	3.16228	−	3.16228i	0.121536	−	0.121536i	0.765259	+	0.643723i	$0.222611\pi$
$678$		0			0
$679$	38.1051			1.46234
$680$		0			0
$681$		0			0
$682$		0			0
$683$	27.3861	−	27.3861i	1.04790	−	1.04790i	0.0491076	−	0.998793i	$-0.484362\pi$
$683$	27.3861	−	27.3861i	1.04790	−	1.04790i	0.998793	−	0.0491076i	$-0.0156377\pi$
$684$		0			0
$685$	30.0000			1.14624
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	25.2982i		−	0.963785i
$690$		0			0
$691$			17.3205i			0.658903i	0.944172	+	0.329452i	$0.106864\pi$
$691$			17.3205i			0.658903i	−0.944172	+	0.329452i	$0.893136\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−27.3861	−	27.3861i	−1.03882	−	1.03882i
$696$		0			0
$697$	20.0000	−	20.0000i	0.757554	−	0.757554i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−3.16228			−0.119438			−0.0597188	−	0.998215i	$-0.519020\pi$
$701$	−3.16228			−0.119438			−0.0597188	+	0.998215i	$0.519020\pi$
$702$		0			0
$703$	−6.92820	+	6.92820i	−0.261302	+	0.261302i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5.47723	−	5.47723i	−0.205992	−	0.205992i
$708$		0			0
$709$		−	22.0000i		−	0.826227i	−0.910679	−	0.413114i	$-0.864441\pi$
$709$		−	22.0000i		−	0.826227i	0.910679	−	0.413114i	$-0.135559\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	37.9473	+	37.9473i	1.42114	+	1.42114i
$714$		0			0
$715$	34.6410			1.29550
$716$		0			0
$717$		0			0
$718$		0			0
$719$	21.9089			0.817064			0.408532	−	0.912744i	$-0.366041\pi$
$719$	21.9089			0.817064			0.408532	+	0.912744i	$0.366041\pi$
$720$		0			0
$721$	30.0000			1.11726
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−15.8114			−0.587220
$726$		0			0
$727$	1.73205	+	1.73205i	0.0642382	+	0.0642382i	0.738496	−	0.674258i	$-0.235536\pi$
$727$	1.73205	+	1.73205i	0.0642382	+	0.0642382i	−0.674258	+	0.738496i	$0.735536\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	21.9089i		−	0.810330i
$732$		0			0
$733$	−26.0000	−	26.0000i	−0.960332	−	0.960332i	0.0389108	−	0.999243i	$-0.487611\pi$
$733$	−26.0000	−	26.0000i	−0.960332	−	0.960332i	−0.999243	+	0.0389108i	$0.987611\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−56.9210	+	56.9210i	−2.09671	+	2.09671i
$738$		0			0
$739$	−51.9615			−1.91144			−0.955718	−	0.294285i	$-0.904919\pi$
$739$	−51.9615			−1.91144			−0.955718	+	0.294285i	$0.904919\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	5.47723	−	5.47723i	0.200940	−	0.200940i	−0.599463	−	0.800403i	$-0.704619\pi$
$743$	5.47723	−	5.47723i	0.200940	−	0.200940i	0.800403	+	0.599463i	$0.204619\pi$
$744$		0			0
$745$	−35.0000	+	35.0000i	−1.28230	+	1.28230i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	18.9737i		−	0.693283i
$750$		0			0
$751$			34.6410i			1.26407i	0.774940	+	0.632034i	$0.217780\pi$
$751$			34.6410i			1.26407i	−0.774940	+	0.632034i	$0.782220\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	16.4317	−	16.4317i	0.598010	−	0.598010i
$756$		0			0
$757$	−34.0000	+	34.0000i	−1.23575	+	1.23575i	−0.274030	+	0.961721i	$0.588357\pi$
$757$	−34.0000	+	34.0000i	−1.23575	+	1.23575i	−0.961721	+	0.274030i	$0.911643\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6.32456			−0.229265			−0.114632	−	0.993408i	$-0.536569\pi$
$761$	−6.32456			−0.229265			−0.114632	+	0.993408i	$0.536569\pi$
$762$		0			0
$763$	−31.1769	+	31.1769i	−1.12868	+	1.12868i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	10.9545	+	10.9545i	0.395542	+	0.395542i
$768$		0			0
$769$		−	18.0000i		−	0.649097i	−0.945869	−	0.324548i	$-0.894788\pi$
$769$		−	18.0000i		−	0.649097i	0.945869	−	0.324548i	$-0.105212\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	3.16228	+	3.16228i	0.113739	+	0.113739i	0.761686	−	0.647947i	$-0.224372\pi$
$773$	3.16228	+	3.16228i	0.113739	+	0.113739i	−0.647947	+	0.761686i	$0.724372\pi$
$774$		0			0
$775$	34.6410			1.24434
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−21.9089			−0.784968
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	12.6491			0.451466
$786$		0			0
$787$	−10.3923	−	10.3923i	−0.370446	−	0.370446i	0.497194	−	0.867639i	$-0.334364\pi$
$787$	−10.3923	−	10.3923i	−0.370446	−	0.370446i	−0.867639	+	0.497194i	$0.834364\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	32.8634i		−	1.16849i
$792$		0			0
$793$	12.0000	+	12.0000i	0.426132	+	0.426132i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	9.48683	−	9.48683i	0.336041	−	0.336041i	−0.518834	−	0.854875i	$-0.673634\pi$
$797$	9.48683	−	9.48683i	0.336041	−	0.336041i	0.854875	+	0.518834i	$0.173634\pi$
$798$		0			0
$799$	34.6410			1.22551
$800$		0			0
$801$		0			0
$802$		0			0
$803$	5.47723	−	5.47723i	0.193287	−	0.193287i
$804$		0			0
$805$	−30.0000	−	30.0000i	−1.05736	−	1.05736i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			12.6491i			0.444719i	0.974965	+	0.222360i	$0.0713759\pi$
$809$			12.6491i			0.444719i	−0.974965	+	0.222360i	$0.928624\pi$
$810$		0			0
$811$			24.2487i			0.851487i	0.904844	+	0.425744i	$0.139987\pi$
$811$			24.2487i			0.851487i	−0.904844	+	0.425744i	$0.860013\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	43.8178			1.53487
$816$		0			0
$817$	−12.0000	+	12.0000i	−0.419827	+	0.419827i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3.16228			0.110364			0.0551821	−	0.998476i	$-0.482426\pi$
$821$	3.16228			0.110364			0.0551821	+	0.998476i	$0.482426\pi$
$822$		0			0
$823$	−5.19615	+	5.19615i	−0.181126	+	0.181126i	−0.791847	−	0.610720i	$-0.790880\pi$
$823$	−5.19615	+	5.19615i	−0.181126	+	0.181126i	0.610720	+	0.791847i	$0.290880\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$		−	6.00000i		−	0.208389i	−0.994557	−	0.104194i	$-0.966774\pi$
$829$		−	6.00000i		−	0.208389i	0.994557	−	0.104194i	$-0.0332264\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3.16228	−	3.16228i	−0.109566	−	0.109566i
$834$		0			0
$835$		−	51.9615i		−	1.79820i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−32.8634			−1.13457			−0.567284	−	0.823522i	$-0.692006\pi$
$839$	−32.8634			−1.13457			−0.567284	+	0.823522i	$0.692006\pi$
$840$		0			0
$841$	19.0000			0.655172
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7.90569	−	7.90569i	0.271964	−	0.271964i
$846$		0			0
$847$	−32.9090	−	32.9090i	−1.13077	−	1.13077i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			21.9089i			0.751027i
$852$		0			0
$853$	−8.00000	−	8.00000i	−0.273915	−	0.273915i	0.556759	−	0.830674i	$-0.312045\pi$
$853$	−8.00000	−	8.00000i	−0.273915	−	0.273915i	−0.830674	+	0.556759i	$0.812045\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	22.1359	−	22.1359i	0.756149	−	0.756149i	−0.219470	−	0.975619i	$-0.570433\pi$
$857$	22.1359	−	22.1359i	0.756149	−	0.756149i	0.975619	+	0.219470i	$0.0704328\pi$
$858$		0			0
$859$	−17.3205			−0.590968			−0.295484	−	0.955348i	$-0.595481\pi$
$859$	−17.3205			−0.590968			−0.295484	+	0.955348i	$0.595481\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−38.3406	+	38.3406i	−1.30513	+	1.30513i	−0.380241	+	0.924887i	$0.624159\pi$
$863$	−38.3406	+	38.3406i	−1.30513	+	1.30513i	−0.924887	+	0.380241i	$0.875841\pi$
$864$		0			0
$865$		−	30.0000i		−	1.02003i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		−	41.5692i		−	1.40852i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−27.3861			−0.925820
$876$		0			0
$877$	−16.0000	+	16.0000i	−0.540282	+	0.540282i	−0.923611	−	0.383330i	$-0.874777\pi$
$877$	−16.0000	+	16.0000i	−0.540282	+	0.540282i	0.383330	+	0.923611i	$0.374777\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−25.2982			−0.852319			−0.426159	−	0.904648i	$-0.640134\pi$
$881$	−25.2982			−0.852319			−0.426159	+	0.904648i	$0.640134\pi$
$882$		0			0
$883$	20.7846	−	20.7846i	0.699458	−	0.699458i	−0.264836	−	0.964294i	$-0.585318\pi$
$883$	20.7846	−	20.7846i	0.699458	−	0.699458i	0.964294	+	0.264836i	$0.0853177\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	16.4317	+	16.4317i	0.551722	+	0.551722i	0.926937	−	0.375216i	$-0.122431\pi$
$887$	16.4317	+	16.4317i	0.551722	+	0.551722i	−0.375216	+	0.926937i	$0.622431\pi$
$888$		0			0
$889$			30.0000i			1.00617i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−18.9737	−	18.9737i	−0.634930	−	0.634930i
$894$		0			0
$895$	8.66025	+	8.66025i	0.289480	+	0.289480i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	21.9089			0.730703
$900$		0			0
$901$	40.0000			1.33259
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9.48683	+	9.48683i	0.315353	+	0.315353i
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			32.8634i			1.08881i	0.838822	+	0.544406i	$0.183245\pi$
$911$			32.8634i			1.08881i	−0.838822	+	0.544406i	$0.816755\pi$
$912$		0			0
$913$	30.0000	+	30.0000i	0.992855	+	0.992855i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	28.4605	−	28.4605i	0.939848	−	0.939848i
$918$		0			0
$919$	−20.7846			−0.685621			−0.342811	−	0.939405i	$-0.611379\pi$
$919$	−20.7846			−0.685621			−0.342811	+	0.939405i	$0.611379\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	10.0000	+	10.0000i	0.328798	+	0.328798i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			31.6228i			1.03751i	0.854923	+	0.518755i	$0.173604\pi$
$929$			31.6228i			1.03751i	−0.854923	+	0.518755i	$0.826396\pi$
$930$		0			0
$931$			3.46410i			0.113531i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			54.7723i			1.79124i
$936$		0			0
$937$	−11.0000	+	11.0000i	−0.359354	+	0.359354i	−0.863575	−	0.504221i	$-0.831780\pi$
$937$	−11.0000	+	11.0000i	−0.359354	+	0.359354i	0.504221	+	0.863575i	$0.331780\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3.16228			−0.103087			−0.0515437	−	0.998671i	$-0.516414\pi$
$941$	−3.16228			−0.103087			−0.0515437	+	0.998671i	$0.516414\pi$
$942$		0			0
$943$	−34.6410	+	34.6410i	−1.12807	+	1.12807i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−32.8634	−	32.8634i	−1.06792	−	1.06792i	−0.997519	−	0.0703964i	$-0.977574\pi$
$947$	−32.8634	−	32.8634i	−1.06792	−	1.06792i	−0.0703964	−	0.997519i	$-0.522426\pi$
$948$		0			0
$949$			4.00000i			0.129845i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−9.48683	−	9.48683i	−0.307309	−	0.307309i	0.536556	−	0.843865i	$-0.319725\pi$
$953$	−9.48683	−	9.48683i	−0.307309	−	0.307309i	−0.843865	+	0.536556i	$0.819725\pi$
$954$		0			0
$955$	17.3205	−	17.3205i	0.560478	−	0.560478i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	32.8634			1.06121
$960$		0			0
$961$	−17.0000			−0.548387
$962$		0			0
$963$		0			0
$964$		0			0
$965$			22.1359i			0.712581i
$966$		0			0
$967$	−25.9808	−	25.9808i	−0.835485	−	0.835485i	0.152776	−	0.988261i	$-0.451179\pi$
$967$	−25.9808	−	25.9808i	−0.835485	−	0.835485i	−0.988261	+	0.152776i	$0.951179\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			38.3406i			1.23041i	0.788368	+	0.615204i	$0.210926\pi$
$971$			38.3406i			1.23041i	−0.788368	+	0.615204i	$0.789074\pi$
$972$		0			0
$973$	−30.0000	−	30.0000i	−0.961756	−	0.961756i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	41.1096	−	41.1096i	1.31521	−	1.31521i	0.397695	−	0.917517i	$-0.369810\pi$
$977$	41.1096	−	41.1096i	1.31521	−	1.31521i	0.917517	−	0.397695i	$-0.130190\pi$
$978$		0			0
$979$	69.2820			2.21426
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−16.4317	+	16.4317i	−0.524089	+	0.524089i	−0.918804	−	0.394715i	$-0.870843\pi$
$983$	−16.4317	+	16.4317i	−0.524089	+	0.524089i	0.394715	+	0.918804i	$0.370843\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			37.9473i			1.20665i
$990$		0			0
$991$			45.0333i			1.43053i	0.698853	+	0.715265i	$0.253694\pi$
$991$			45.0333i			1.43053i	−0.698853	+	0.715265i	$0.746306\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−5.47723	−	5.47723i	−0.173640	−	0.173640i
$996$		0			0
$997$	2.00000	−	2.00000i	0.0633406	−	0.0633406i	−0.674727	−	0.738068i	$-0.735739\pi$
$997$	2.00000	−	2.00000i	0.0633406	−	0.0633406i	0.738068	+	0.674727i	$0.235739\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.2.x.g.703.4	yes	8
3.2	odd	2	inner	720.2.x.g.703.2	yes	8
4.3	odd	2	inner	720.2.x.g.703.3	yes	8
5.2	odd	4	inner	720.2.x.g.127.3	yes	8
5.3	odd	4		3600.2.x.o.3007.4		8
5.4	even	2		3600.2.x.o.2143.2		8
12.11	even	2	inner	720.2.x.g.703.1	yes	8
15.2	even	4	inner	720.2.x.g.127.1	✓	8
15.8	even	4		3600.2.x.o.3007.3		8
15.14	odd	2		3600.2.x.o.2143.1		8
20.3	even	4		3600.2.x.o.3007.1		8
20.7	even	4	inner	720.2.x.g.127.4	yes	8
20.19	odd	2		3600.2.x.o.2143.3		8
60.23	odd	4		3600.2.x.o.3007.2		8
60.47	odd	4	inner	720.2.x.g.127.2	yes	8
60.59	even	2		3600.2.x.o.2143.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.x.g.127.1	✓	8	15.2	even	4	inner
720.2.x.g.127.2	yes	8	60.47	odd	4	inner
720.2.x.g.127.3	yes	8	5.2	odd	4	inner
720.2.x.g.127.4	yes	8	20.7	even	4	inner
720.2.x.g.703.1	yes	8	12.11	even	2	inner
720.2.x.g.703.2	yes	8	3.2	odd	2	inner
720.2.x.g.703.3	yes	8	4.3	odd	2	inner
720.2.x.g.703.4	yes	8	1.1	even	1	trivial
3600.2.x.o.2143.1		8	15.14	odd	2
3600.2.x.o.2143.2		8	5.4	even	2
3600.2.x.o.2143.3		8	20.19	odd	2
3600.2.x.o.2143.4		8	60.59	even	2
3600.2.x.o.3007.1		8	20.3	even	4
3600.2.x.o.3007.2		8	60.23	odd	4
3600.2.x.o.3007.3		8	15.8	even	4
3600.2.x.o.3007.4		8	5.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	720.x (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.58114 + 1.58114i) q^{5} +(1.73205 + 1.73205i) q^{7} +5.47723i q^{11} +(-2.00000 - 2.00000i) q^{13} +(3.16228 - 3.16228i) q^{17} -3.46410 q^{19} +(-5.47723 + 5.47723i) q^{23} +5.00000i q^{25} +3.16228i q^{29} -6.92820i q^{31} +5.47723i q^{35} +(2.00000 - 2.00000i) q^{37} +6.32456 q^{41} +(3.46410 - 3.46410i) q^{43} +(5.47723 + 5.47723i) q^{47} -1.00000i q^{49} +(6.32456 + 6.32456i) q^{53} +(-8.66025 + 8.66025i) q^{55} -5.47723 q^{59} -6.00000 q^{61} -6.32456i q^{65} +(10.3923 + 10.3923i) q^{67} +(-1.00000 - 1.00000i) q^{73} +(-9.48683 + 9.48683i) q^{77} +(5.47723 - 5.47723i) q^{83} +10.0000 q^{85} -12.6491i q^{89} -6.92820i q^{91} +(-5.47723 - 5.47723i) q^{95} +(11.0000 - 11.0000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{13} + 16 q^{37} - 48 q^{61} - 8 q^{73} + 80 q^{85} + 88 q^{97}+O(q^{100}) \) 8 * q - 16 * q^13 + 16 * q^37 - 48 * q^61 - 8 * q^73 + 80 * q^85 + 88 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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