LMFDB - Embedded newform 720.2.bm.e.109.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [720,2,Mod(109,720)] 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("720.109"); 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([0, 3, 0, 2])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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:	8.0.3317760000.5
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 7x^{4} + 16 $ x^8 - 7*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			109.3
Root			$0.178197 - 1.40294i$ of defining polynomial
Character	$\chi$	$=$	720.109
Dual form			720.2.bm.e.469.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.178197 - 1.40294i) q^{2} +(-1.93649 - 0.500000i) q^{4} +(1.58114 + 1.58114i) q^{5} +(-1.04655 + 2.62769i) q^{8} +(2.50000 - 1.93649i) q^{10} +(3.50000 + 1.93649i) q^{16} -8.06126i q^{17} +(5.87298 + 5.87298i) q^{19} +(-2.27129 - 3.85243i) q^{20} +8.77405 q^{23} +5.00000i q^{25} +7.74597 q^{31} +(3.34047 - 4.56522i) q^{32} +(-11.3095 - 1.43649i) q^{34} +(9.28600 - 7.19291i) q^{38} +(-5.80948 + 2.50000i) q^{40} +(1.56351 - 12.3095i) q^{46} +1.02391i q^{47} -7.00000 q^{49} +(7.01471 + 0.890985i) q^{50} +(-9.79796 - 9.79796i) q^{53} +(8.74597 + 8.74597i) q^{61} +(1.38031 - 10.8671i) q^{62} +(-5.80948 - 5.50000i) q^{64} +(-4.03063 + 15.6106i) q^{68} +(-8.43649 - 14.3095i) q^{76} -16.0000 q^{79} +(2.47212 + 8.59585i) q^{80} +(-2.44949 + 2.44949i) q^{83} +(12.7460 - 12.7460i) q^{85} +(-16.9909 - 4.38702i) q^{92} +(1.43649 + 0.182458i) q^{94} +18.5720i q^{95} +(-1.24738 + 9.82059i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 20 q^{10} + 28 q^{16} + 16 q^{19} - 44 q^{34} + 28 q^{46} - 56 q^{49} + 8 q^{61} - 52 q^{76} - 128 q^{79} + 40 q^{85} - 4 q^{94}+O(q^{100}) $ 8 * q + 20 * q^10 + 28 * q^16 + 16 * q^19 - 44 * q^34 + 28 * q^46 - 56 * q^49 + 8 * q^61 - 52 * q^76 - 128 * q^79 + 40 * q^85 - 4 * q^94

Character values

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

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

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 $

$2$	0.178197	−	1.40294i	0.126004	−	0.992030i
$2$	0.178197	−	1.40294i	0.126004	−	0.992030i
$3$		0			0
$3$		0			0
$4$	−1.93649	−	0.500000i	−0.968246	−	0.250000i
$5$	1.58114	+	1.58114i	0.707107	+	0.707107i
$5$	1.58114	+	1.58114i	0.707107	+	0.707107i
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	−1.04655	+	2.62769i	−0.370011	+	0.929028i
$9$		0			0
$10$	2.50000	−	1.93649i	0.790569	−	0.612372i
$11$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$11$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$12$		0			0
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		0			0
$16$	3.50000	+	1.93649i	0.875000	+	0.484123i
$17$		−	8.06126i		−	1.95514i	−0.210606	−	0.977571i	$-0.567544\pi$
$17$		−	8.06126i		−	1.95514i	0.210606	−	0.977571i	$-0.432456\pi$
$18$		0			0
$19$	5.87298	+	5.87298i	1.34735	+	1.34735i	0.888523	+	0.458831i	$0.151732\pi$
$19$	5.87298	+	5.87298i	1.34735	+	1.34735i	0.458831	+	0.888523i	$0.348268\pi$
$20$	−2.27129	−	3.85243i	−0.507877	−	0.861430i
$21$		0			0
$22$		0			0
$23$	8.77405			1.82951			0.914757	−	0.404004i	$-0.132382\pi$
$23$	8.77405			1.82951			0.914757	+	0.404004i	$0.132382\pi$
$24$		0			0
$25$			5.00000i			1.00000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$29$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$30$		0			0
$31$	7.74597			1.39122			0.695608	−	0.718421i	$-0.255135\pi$
$31$	7.74597			1.39122			0.695608	+	0.718421i	$0.255135\pi$
$32$	3.34047	−	4.56522i	0.590518	−	0.807024i
$33$		0			0
$34$	−11.3095	−	1.43649i	−1.93956	−	0.246356i
$35$		0			0
$36$		0			0
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$	9.28600	−	7.19291i	1.50639	−	1.16684i
$39$		0			0
$40$	−5.80948	+	2.50000i	−0.918559	+	0.395285i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$		0			0
$46$	1.56351	−	12.3095i	0.230527	−	1.81493i
$47$			1.02391i			0.149353i	0.997208	+	0.0746766i	$0.0237924\pi$
$47$			1.02391i			0.149353i	−0.997208	+	0.0746766i	$0.976208\pi$
$48$		0			0
$49$	−7.00000			−1.00000
$50$	7.01471	+	0.890985i	0.992030	+	0.126004i
$51$		0			0
$52$		0			0
$53$	−9.79796	−	9.79796i	−1.34585	−	1.34585i	−0.890113	−	0.455740i	$-0.849375\pi$
$53$	−9.79796	−	9.79796i	−1.34585	−	1.34585i	−0.455740	−	0.890113i	$-0.650625\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$59$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$60$		0			0
$61$	8.74597	+	8.74597i	1.11981	+	1.11981i	0.991769	+	0.128037i	$0.0408676\pi$
$61$	8.74597	+	8.74597i	1.11981	+	1.11981i	0.128037	+	0.991769i	$0.459132\pi$
$62$	1.38031	−	10.8671i	0.175299	−	1.38013i
$63$		0			0
$64$	−5.80948	−	5.50000i	−0.726184	−	0.687500i
$65$		0			0
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$	−4.03063	+	15.6106i	−0.488786	+	1.89306i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$		0			0
$76$	−8.43649	−	14.3095i	−0.967732	−	1.64141i
$77$		0			0
$78$		0			0
$79$	−16.0000			−1.80014			−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000			−1.80014			−0.900070	+	0.435745i	$0.856485\pi$
$80$	2.47212	+	8.59585i	0.276392	+	0.961045i
$81$		0			0
$82$		0			0
$83$	−2.44949	+	2.44949i	−0.268866	+	0.268866i	−0.828643	−	0.559777i	$-0.810887\pi$
$83$	−2.44949	+	2.44949i	−0.268866	+	0.268866i	0.559777	+	0.828643i	$0.310887\pi$
$84$		0			0
$85$	12.7460	−	12.7460i	1.38249	−	1.38249i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	−16.9909	−	4.38702i	−1.77142	−	0.457379i
$93$		0			0
$94$	1.43649	+	0.182458i	0.148163	+	0.0188191i
$95$			18.5720i			1.90545i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−1.24738	+	9.82059i	−0.126004	+	0.992030i
$99$		0			0
$100$	2.50000	−	9.68246i	0.250000	−	0.968246i
$101$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$101$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$	−15.4919	+	12.0000i	−1.50471	+	1.16554i
$107$	7.34847	+	7.34847i	0.710403	+	0.710403i	0.966620	−	0.256216i	$-0.0824759\pi$
$107$	7.34847	+	7.34847i	0.710403	+	0.710403i	−0.256216	+	0.966620i	$0.582476\pi$
$108$		0			0
$109$	0.745967	+	0.745967i	0.0714507	+	0.0714507i	0.741929	−	0.670478i	$-0.233911\pi$
$109$	0.745967	+	0.745967i	0.0714507	+	0.0714507i	−0.670478	+	0.741929i	$0.733911\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	11.5347i		−	1.08509i	−0.840027	−	0.542545i	$-0.817461\pi$
$113$		−	11.5347i		−	1.08509i	0.840027	−	0.542545i	$-0.182539\pi$
$114$		0			0
$115$	13.8730	+	13.8730i	1.29366	+	1.29366i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$			11.0000i			1.00000i
$122$	13.8286	−	10.7116i	1.25198	−	0.969781i
$123$		0			0
$124$	−15.0000	−	3.87298i	−1.34704	−	0.347804i
$125$	−7.90569	+	7.90569i	−0.707107	+	0.707107i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−8.75141	+	7.17027i	−0.773523	+	0.633769i
$129$		0			0
$130$		0			0
$131$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$131$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	21.1825	+	8.43649i	1.81638	+	0.723423i
$137$	−20.7104			−1.76941			−0.884703	−	0.466155i	$-0.845639\pi$
$137$	−20.7104			−1.76941			−0.884703	+	0.466155i	$0.845639\pi$
$138$		0			0
$139$	−9.61895	+	9.61895i	−0.815869	+	0.815869i	−0.985506	−	0.169638i	$-0.945740\pi$
$139$	−9.61895	+	9.61895i	−0.815869	+	0.815869i	0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$149$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$150$		0			0
$151$		−	23.2379i		−	1.89107i	−0.325515	−	0.945537i	$-0.605538\pi$
$151$		−	23.2379i		−	1.89107i	0.325515	−	0.945537i	$-0.394462\pi$
$152$	−21.5787	+	9.28600i	−1.75027	+	0.753194i
$153$		0			0
$154$		0			0
$155$	12.2474	+	12.2474i	0.983739	+	0.983739i
$156$		0			0
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$	−2.85115	+	22.4471i	−0.226825	+	1.78579i
$159$		0			0
$160$	12.5000	−	1.93649i	0.988212	−	0.153093i
$161$		0			0
$162$		0			0
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$	3.00000	+	3.87298i	0.232845	+	0.300602i
$167$	10.8219			0.837422			0.418711	−	0.908120i	$-0.362482\pi$
$167$	10.8219			0.837422			0.418711	+	0.908120i	$0.362482\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$	−15.6106	−	20.1531i	−1.19728	−	1.54568i
$171$		0			0
$172$		0			0
$173$	9.79796	−	9.79796i	0.744925	−	0.744925i	−0.228596	−	0.973521i	$-0.573414\pi$
$173$	9.79796	−	9.79796i	0.744925	−	0.744925i	0.973521	+	0.228596i	$0.0734136\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$179$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$180$		0			0
$181$	3.25403	−	3.25403i	0.241870	−	0.241870i	−0.575753	−	0.817624i	$-0.695291\pi$
$181$	3.25403	−	3.25403i	0.241870	−	0.241870i	0.817624	+	0.575753i	$0.195291\pi$
$182$		0			0
$183$		0			0
$184$	−9.18246	+	23.0554i	−0.676940	+	1.69967i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	0.511957	−	1.98280i	0.0373383	−	0.144611i
$189$		0			0
$190$	26.0554	+	3.30948i	1.89026	+	0.240095i
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	13.5554	+	3.50000i	0.968246	+	0.250000i
$197$	−3.16228	−	3.16228i	−0.225303	−	0.225303i	0.585424	−	0.810727i	$-0.300928\pi$
$197$	−3.16228	−	3.16228i	−0.225303	−	0.225303i	−0.810727	+	0.585424i	$0.800928\pi$
$198$		0			0
$199$		−	16.0000i		−	1.13421i	−0.823646	−	0.567105i	$-0.808063\pi$
$199$		−	16.0000i		−	1.13421i	0.823646	−	0.567105i	$-0.191937\pi$
$200$	−13.1384	−	5.23274i	−0.929028	−	0.370011i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	10.1270	+	10.1270i	0.697173	+	0.697173i	0.963800	−	0.266627i	$-0.0859092\pi$
$211$	10.1270	+	10.1270i	0.697173	+	0.697173i	−0.266627	+	0.963800i	$0.585909\pi$
$212$	14.0747	+	23.8726i	0.966653	+	1.63958i
$213$		0			0
$214$	11.6190	−	9.00000i	0.794255	−	0.615227i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	1.17948	−	0.913619i	0.0798843	−	0.0618781i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$		0			0
$226$	−16.1825	−	2.05544i	−1.07644	−	0.136726i
$227$	−17.1464	+	17.1464i	−1.13805	+	1.13805i	−0.149249	+	0.988800i	$0.547686\pi$
$227$	−17.1464	+	17.1464i	−1.13805	+	1.13805i	−0.988800	+	0.149249i	$0.952314\pi$
$228$		0			0
$229$	20.7460	−	20.7460i	1.37093	−	1.37093i	0.511868	−	0.859064i	$-0.328954\pi$
$229$	20.7460	−	20.7460i	1.37093	−	1.37093i	0.859064	−	0.511868i	$-0.171046\pi$
$230$	21.9351	−	16.9909i	1.44636	−	1.12034i
$231$		0			0
$232$		0			0
$233$	1.11445			0.0730100			0.0365050	−	0.999333i	$-0.488378\pi$
$233$	1.11445			0.0730100			0.0365050	+	0.999333i	$0.488378\pi$
$234$		0			0
$235$	−1.61895	+	1.61895i	−0.105609	+	0.105609i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−30.9839			−1.99585			−0.997923	−	0.0644157i	$-0.979482\pi$
$241$	−30.9839			−1.99585			−0.997923	+	0.0644157i	$0.979482\pi$
$242$	15.4324	+	1.96017i	0.992030	+	0.126004i
$243$		0			0
$244$	−12.5635	−	21.3095i	−0.804296	−	1.36420i
$245$	−11.0680	−	11.0680i	−0.707107	−	0.707107i
$246$		0			0
$247$		0			0
$248$	−8.10653	+	20.3540i	−0.514765	+	1.29248i
$249$		0			0
$250$	9.68246	+	12.5000i	0.612372	+	0.790569i
$251$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$251$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	8.50000	+	13.5554i	0.531250	+	0.847215i
$257$			17.2370i			1.07521i	0.843196	+	0.537606i	$0.180671\pi$
$257$			17.2370i			1.07521i	−0.843196	+	0.537606i	$0.819329\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−28.3700			−1.74937			−0.874683	−	0.484695i	$-0.838931\pi$
$263$	−28.3700			−1.74937			−0.874683	+	0.484695i	$0.838931\pi$
$264$		0			0
$265$		−	30.9839i		−	1.90332i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$269$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$270$		0			0
$271$	−32.0000			−1.94386			−0.971931	−	0.235267i	$-0.924404\pi$
$271$	−32.0000			−1.94386			−0.971931	+	0.235267i	$0.924404\pi$
$272$	15.6106	−	28.2144i	0.946529	−	1.71075i
$273$		0			0
$274$	−3.69052	+	29.0554i	−0.222953	+	1.75530i
$275$		0			0
$276$		0			0
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$	11.7808	+	15.2089i	0.706563	+	0.912169i
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−47.9839			−2.82258
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9.79796	−	9.79796i	−0.572403	−	0.572403i	0.360396	−	0.932799i	$-0.382641\pi$
$293$	−9.79796	−	9.79796i	−0.572403	−	0.572403i	−0.932799	+	0.360396i	$0.882641\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−32.6014	−	4.14092i	−1.87600	−	0.238283i
$303$		0			0
$304$	9.18246	+	31.9284i	0.526650	+	1.83122i
$305$			27.6572i			1.58365i
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$	19.3649	−	15.0000i	1.09985	−	0.851943i
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$		0			0
$316$	30.9839	+	8.00000i	1.74298	+	0.450035i
$317$	15.8114	−	15.8114i	0.888056	−	0.888056i	−0.106280	−	0.994336i	$-0.533894\pi$
$317$	15.8114	−	15.8114i	0.888056	−	0.888056i	0.994336	+	0.106280i	$0.0338940\pi$
$318$		0			0
$319$		0			0
$320$	−0.489323	−	17.8819i	−0.0273540	−	0.999626i
$321$		0			0
$322$		0			0
$323$	47.3436	−	47.3436i	2.63427	−	2.63427i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	25.6190	−	25.6190i	1.40814	−	1.40814i	0.638635	−	0.769510i	$-0.279499\pi$
$331$	25.6190	−	25.6190i	1.40814	−	1.40814i	0.769510	−	0.638635i	$-0.220501\pi$
$332$	5.96816	−	3.51867i	0.327545	−	0.193112i
$333$		0			0
$334$	1.92843	−	15.1825i	0.105519	−	0.830747i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	18.2382	+	2.31656i	0.992030	+	0.126004i
$339$		0			0
$340$	−31.0554	+	18.3095i	−1.68422	+	0.992971i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−12.0000	−	15.4919i	−0.645124	−	0.832851i
$347$	25.2982	+	25.2982i	1.35808	+	1.35808i	0.876283	+	0.481797i	$0.160016\pi$
$347$	25.2982	+	25.2982i	1.35808	+	1.35808i	0.481797	+	0.876283i	$0.339984\pi$
$348$		0			0
$349$	−24.7460	−	24.7460i	−1.32462	−	1.32462i	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−24.7460	−	24.7460i	−1.32462	−	1.32462i	−0.414632	−	0.909989i	$-0.636090\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	36.8329i		−	1.96042i	−0.197969	−	0.980208i	$-0.563435\pi$
$353$		−	36.8329i		−	1.96042i	0.197969	−	0.980208i	$-0.436565\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$			49.9839i			2.63073i
$362$	−3.98536	−	5.14508i	−0.209466	−	0.270419i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$	30.7092	+	16.9909i	1.60083	+	0.885710i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$		0			0
$376$	−2.69052	−	1.07157i	−0.138753	−	0.0552623i
$377$		0			0
$378$		0			0
$379$	−21.3649	+	21.3649i	−1.09744	+	1.09744i	−0.102733	+	0.994709i	$0.532759\pi$
$379$	−21.3649	+	21.3649i	−1.09744	+	1.09744i	−0.994709	+	0.102733i	$0.967241\pi$
$380$	9.28600	−	35.9645i	0.476362	−	1.84494i
$381$		0			0
$382$		0			0
$383$		−	20.6198i		−	1.05362i	−0.849982	−	0.526812i	$-0.823387\pi$
$383$		−	20.6198i		−	1.05362i	0.849982	−	0.526812i	$-0.176613\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$389$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$390$		0			0
$391$		−	70.7298i		−	3.57696i
$392$	7.32584	−	18.3938i	0.370011	−	0.929028i
$393$		0			0
$394$	−5.00000	+	3.87298i	−0.251896	+	0.195118i
$395$	−25.2982	−	25.2982i	−1.27289	−	1.27289i
$396$		0			0
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$	−22.4471	−	2.85115i	−1.12517	−	0.142915i
$399$		0			0
$400$	−9.68246	+	17.5000i	−0.484123	+	0.875000i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	30.9839i		−	1.53205i	−0.642809	−	0.766027i	$-0.722231\pi$
$409$		−	30.9839i		−	1.53205i	0.642809	−	0.766027i	$-0.277769\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−7.74597			−0.380235
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$419$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$420$		0			0
$421$	11.2540	−	11.2540i	0.548488	−	0.548488i	−0.377515	−	0.926003i	$-0.623221\pi$
$421$	11.2540	−	11.2540i	0.548488	−	0.548488i	0.926003	+	0.377515i	$0.123221\pi$
$422$	16.0122	−	12.4030i	0.779463	−	0.603769i
$423$		0			0
$424$	36.0000	−	15.4919i	1.74831	−	0.752355i
$425$	40.3063			1.95514
$426$		0			0
$427$		0			0
$428$	−10.5560	−	17.9045i	−0.510244	−	0.865446i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	−1.07157	−	1.81754i	−0.0513191	−	0.0870445i
$437$	51.5298	+	51.5298i	2.46501	+	2.46501i
$438$		0			0
$439$			16.0000i			0.763638i	0.924237	+	0.381819i	$0.124702\pi$
$439$			16.0000i			0.763638i	−0.924237	+	0.381819i	$0.875298\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−26.9444	−	26.9444i	−1.28017	−	1.28017i	−0.940572	−	0.339595i	$-0.889710\pi$
$443$	−26.9444	−	26.9444i	−1.28017	−	1.28017i	−0.339595	−	0.940572i	$-0.610290\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$	−5.76733	+	22.3368i	−0.271272	+	1.05063i
$453$		0			0
$454$	21.0000	+	27.1109i	0.985579	+	1.27238i
$455$		0			0
$456$		0			0
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$	−25.4085	−	32.8023i	−1.18726	−	1.53275i
$459$		0			0
$460$	−19.9284	−	33.8014i	−0.929168	−	1.57600i
$461$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$461$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	0.198592	−	1.56351i	0.00919958	−	0.0724281i
$467$	−25.2982	+	25.2982i	−1.17066	+	1.17066i	−0.188610	+	0.982052i	$0.560398\pi$
$467$	−25.2982	+	25.2982i	−1.17066	+	1.17066i	−0.982052	+	0.188610i	$0.939602\pi$
$468$		0			0
$469$		0			0
$470$	1.98280	+	2.55978i	0.0914598	+	0.118074i
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−29.3649	+	29.3649i	−1.34735	+	1.34735i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$	−5.52123	+	43.4686i	−0.251485	+	1.97994i
$483$		0			0
$484$	5.50000	−	21.3014i	0.250000	−	0.968246i
$485$		0			0
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$	−32.1347	+	13.8286i	−1.45467	+	0.625991i
$489$		0			0
$490$	−17.5000	+	13.5554i	−0.790569	+	0.612372i
$491$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$491$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	27.1109	+	15.0000i	1.21731	+	0.673520i
$497$		0			0
$498$		0			0
$499$	18.1270	+	18.1270i	0.811477	+	0.811477i	0.984855	−	0.173379i	$-0.0554684\pi$
$499$	18.1270	+	18.1270i	0.811477	+	0.811477i	−0.173379	+	0.984855i	$0.555468\pi$
$500$	19.2622	−	11.3565i	0.861430	−	0.507877i
$501$		0			0
$502$		0			0
$503$	34.0723			1.51921			0.759604	−	0.650386i	$-0.225393\pi$
$503$	34.0723			1.51921			0.759604	+	0.650386i	$0.225393\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$509$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$510$		0			0
$511$		0			0
$512$	20.5322	−	9.50947i	0.907402	−	0.420263i
$513$		0			0
$514$	24.1825	+	3.07157i	1.06664	+	0.135481i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$	−5.05544	+	39.8014i	−0.220428	+	1.73542i
$527$		−	62.4422i		−	2.72003i
$528$		0			0
$529$	53.9839			2.34712
$530$	−43.4686	−	5.52123i	−1.88815	−	0.239827i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$			23.2379i			1.00466i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	22.2379	+	22.2379i	0.956082	+	0.956082i	0.999075	−	0.0429934i	$-0.0136894\pi$
$541$	22.2379	+	22.2379i	0.956082	+	0.956082i	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	−5.70230	+	44.8941i	−0.244935	+	1.92837i
$543$		0			0
$544$	−36.8014	−	26.9284i	−1.57785	−	1.15455i
$545$			2.35895i			0.101046i
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	40.1055	+	10.3552i	1.71322	+	0.442352i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	23.4365	−	13.8175i	0.993929	−	0.585994i
$557$	−29.3939	+	29.3939i	−1.24546	+	1.24546i	−0.287754	+	0.957704i	$0.592909\pi$
$557$	−29.3939	+	29.3939i	−1.24546	+	1.24546i	−0.957704	+	0.287754i	$0.907091\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−25.2982	+	25.2982i	−1.06619	+	1.06619i	−0.0685449	+	0.997648i	$0.521836\pi$
$563$	−25.2982	+	25.2982i	−1.06619	+	1.06619i	−0.997648	+	0.0685449i	$0.978164\pi$
$564$		0			0
$565$	18.2379	−	18.2379i	0.767274	−	0.767274i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	5.36492	−	5.36492i	0.224515	−	0.224515i	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	5.36492	−	5.36492i	0.224515	−	0.224515i	0.810397	+	0.585882i	$0.199252\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			43.8702i			1.82951i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−8.55058	+	67.3186i	−0.355657	+	2.80008i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−15.4919	+	12.0000i	−0.639966	+	0.495715i
$587$	−31.8434	−	31.8434i	−1.31432	−	1.31432i	−0.918201	−	0.396116i	$-0.870358\pi$
$587$	−31.8434	−	31.8434i	−1.31432	−	1.31432i	−0.396116	−	0.918201i	$-0.629642\pi$
$588$		0			0
$589$	45.4919	+	45.4919i	1.87446	+	1.87446i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			31.1306i			1.27838i	0.769049	+	0.639190i	$0.220730\pi$
$593$			31.1306i			1.27838i	−0.769049	+	0.639190i	$0.779270\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$			30.9839i			1.26386i	0.775026	+	0.631929i	$0.217737\pi$
$601$			30.9839i			1.26386i	−0.775026	+	0.631929i	$0.782263\pi$
$602$		0			0
$603$		0			0
$604$	−11.6190	+	45.0000i	−0.472768	+	1.83102i
$605$	−17.3925	+	17.3925i	−0.707107	+	0.707107i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$	46.4300	−	7.19291i	1.88299	−	0.291711i
$609$		0			0
$610$	38.8014	+	4.92843i	1.57102	+	0.199546i
$611$		0			0
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−29.8861			−1.20317			−0.601584	−	0.798810i	$-0.705464\pi$
$617$	−29.8861			−1.20317			−0.601584	+	0.798810i	$0.705464\pi$
$618$		0			0
$619$	33.6190	−	33.6190i	1.35126	−	1.35126i	0.467005	−	0.884255i	$-0.345333\pi$
$619$	33.6190	−	33.6190i	1.35126	−	1.35126i	0.884255	−	0.467005i	$-0.154667\pi$
$620$	−17.5934	−	29.8408i	−0.706566	−	1.19844i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−25.0000			−1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$			32.0000i			1.27390i	0.770905	+	0.636950i	$0.219804\pi$
$631$			32.0000i			1.27390i	−0.770905	+	0.636950i	$0.780196\pi$
$632$	16.7448	−	42.0430i	0.666071	−	1.67238i
$633$		0			0
$634$	−19.3649	−	25.0000i	−0.769079	−	0.992877i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−25.1744	−	2.50000i	−0.995105	−	0.0988212i
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		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
$646$	−57.9839	−	74.8569i	−2.28134	−	2.94520i
$647$	−14.4763			−0.569124			−0.284562	−	0.958658i	$-0.591848\pi$
$647$	−14.4763			−0.569124			−0.284562	+	0.958658i	$0.591848\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.79796	+	9.79796i	−0.383424	+	0.383424i	−0.872334	−	0.488910i	$-0.837395\pi$
$653$	−9.79796	+	9.79796i	−0.383424	+	0.383424i	0.488910	+	0.872334i	$0.337395\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$659$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$660$		0			0
$661$	−34.2379	+	34.2379i	−1.33170	+	1.33170i	−0.427850	+	0.903850i	$0.640729\pi$
$661$	−34.2379	+	34.2379i	−1.33170	+	1.33170i	−0.903850	+	0.427850i	$0.859271\pi$
$662$	−31.3767	−	40.5071i	−1.21949	−	1.57435i
$663$		0			0
$664$	−3.87298	−	9.00000i	−0.150301	−	0.349268i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−20.9565	−	5.41094i	−0.810830	−	0.209355i
$669$		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
$676$	6.50000	−	25.1744i	0.250000	−	0.968246i
$677$	29.3939	+	29.3939i	1.12970	+	1.12970i	0.990226	+	0.139473i	$0.0445407\pi$
$677$	29.3939	+	29.3939i	1.12970	+	1.12970i	0.139473	+	0.990226i	$0.455459\pi$
$678$		0			0
$679$		0			0
$680$	20.1531	+	46.8317i	0.772838	+	1.79591i
$681$		0			0
$682$		0			0
$683$	−25.2982	−	25.2982i	−0.968010	−	0.968010i	0.0314944	−	0.999504i	$-0.489973\pi$
$683$	−25.2982	−	25.2982i	−0.968010	−	0.968010i	−0.999504	+	0.0314944i	$0.989973\pi$
$684$		0			0
$685$	−32.7460	−	32.7460i	−1.25116	−	1.25116i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−29.8730	−	29.8730i	−1.13642	−	1.13642i	−0.989087	−	0.147335i	$-0.952930\pi$
$691$	−29.8730	−	29.8730i	−1.13642	−	1.13642i	−0.147335	−	0.989087i	$-0.547070\pi$
$692$	−23.8726	+	14.0747i	−0.907502	+	0.535039i
$693$		0			0
$694$	40.0000	−	30.9839i	1.51838	−	1.17613i
$695$	−30.4178			−1.15381
$696$		0			0
$697$		0			0
$698$	−39.1268	+	30.3075i	−1.48097	+	1.14716i
$699$		0			0
$700$		0			0
$701$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$701$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−51.6744	−	6.56351i	−1.94479	−	0.247021i
$707$		0			0
$708$		0			0
$709$	10.2379	−	10.2379i	0.384492	−	0.384492i	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	10.2379	−	10.2379i	0.384492	−	0.384492i	0.872718	+	0.488225i	$0.162356\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	67.9635			2.54525
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	70.1245	+	8.90697i	2.60976	+	0.331483i
$723$		0			0
$724$	−7.92843	+	4.67439i	−0.294658	+	0.173722i
$725$		0			0
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$		0			0
$735$		0			0
$736$	29.3095	−	40.0554i	1.08036	−	1.47646i
$737$		0			0
$738$		0			0
$739$	25.1109	+	25.1109i	0.923719	+	0.923719i	0.997290	−	0.0735712i	$-0.0234396\pi$
$739$	25.1109	+	25.1109i	0.923719	+	0.923719i	−0.0735712	+	0.997290i	$0.523440\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−53.6682			−1.96889			−0.984447	−	0.175680i	$-0.943788\pi$
$743$	−53.6682			−1.96889			−0.984447	+	0.175680i	$0.943788\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	54.2218			1.97858			0.989290	−	0.145962i	$-0.0466277\pi$
$751$	54.2218			1.97858			0.989290	+	0.145962i	$0.0466277\pi$
$752$	−1.98280	+	3.58370i	−0.0723053	+	0.130684i
$753$		0			0
$754$		0			0
$755$	36.7423	−	36.7423i	1.33719	−	1.33719i
$756$		0			0
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$	26.1666	+	33.7809i	0.950413	+	1.22698i
$759$		0			0
$760$	−48.8014	−	19.4365i	−1.77021	−	0.705036i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−28.9284	−	3.67439i	−1.04523	−	0.132761i
$767$		0			0
$768$		0			0
$769$	30.9839			1.11731			0.558653	−	0.829401i	$-0.311318\pi$
$769$	30.9839			1.11731			0.558653	+	0.829401i	$0.311318\pi$
$770$		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
$775$			38.7298i			1.39122i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−99.2298	−	12.6038i	−3.54845	−	0.450712i
$783$		0			0
$784$	−24.5000	−	13.5554i	−0.875000	−	0.484123i
$785$		0			0
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$	4.54259	+	7.70486i	0.161823	+	0.274474i
$789$		0			0
$790$	−40.0000	+	30.9839i	−1.42314	+	1.10236i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−8.00000	+	30.9839i	−0.283552	+	1.09819i
$797$	34.7851	−	34.7851i	1.23215	−	1.23215i	0.269013	−	0.963136i	$-0.413302\pi$
$797$	34.7851	−	34.7851i	1.23215	−	1.23215i	0.963136	−	0.269013i	$-0.0866976\pi$
$798$		0			0
$799$	8.25403			0.292007
$800$	22.8261	+	16.7024i	0.807024	+	0.590518i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	−14.3810	+	14.3810i	−0.504987	+	0.504987i	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−14.3810	+	14.3810i	−0.504987	+	0.504987i	0.407997	+	0.912983i	$0.366228\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−43.4686	−	5.52123i	−1.51984	−	0.193045i
$819$		0			0
$820$		0			0
$821$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$821$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	25.2982	+	25.2982i	0.879705	+	0.879705i	0.993504	−	0.113799i	$-0.0363018\pi$
$827$	25.2982	+	25.2982i	0.879705	+	0.879705i	−0.113799	+	0.993504i	$0.536302\pi$
$828$		0			0
$829$	−6.23790	−	6.23790i	−0.216651	−	0.216651i	0.590434	−	0.807086i	$-0.298956\pi$
$829$	−6.23790	−	6.23790i	−0.216651	−	0.216651i	−0.807086	+	0.590434i	$0.798956\pi$
$830$	−1.38031	+	10.8671i	−0.0479112	+	0.377204i
$831$		0			0
$832$		0			0
$833$			56.4288i			1.95514i
$834$		0			0
$835$	17.1109	+	17.1109i	0.592147	+	0.592147i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$		−	29.0000i		−	1.00000i
$842$	−13.7833	−	17.7942i	−0.475005	−	0.613228i
$843$		0			0
$844$	−14.5474	−	24.6744i	−0.500741	−	0.849328i
$845$	−20.5548	+	20.5548i	−0.707107	+	0.707107i
$846$		0			0
$847$		0			0
$848$	−15.3192	−	53.2665i	−0.526063	−	1.82918i
$849$		0			0
$850$	7.18246	−	56.5474i	0.246356	−	1.93956i
$851$		0			0
$852$		0			0
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$		0			0
$855$		0			0
$856$	−27.0000	+	11.6190i	−0.922841	+	0.397128i
$857$	−46.0086			−1.57162			−0.785812	−	0.618466i	$-0.787755\pi$
$857$	−46.0086			−1.57162			−0.785812	+	0.618466i	$0.787755\pi$
$858$		0			0
$859$	−2.63508	+	2.63508i	−0.0899079	+	0.0899079i	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−2.63508	+	2.63508i	−0.0899079	+	0.0899079i	0.660722	+	0.750630i	$0.270250\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			4.67839i			0.159254i	0.996825	+	0.0796271i	$0.0253730\pi$
$863$			4.67839i			0.159254i	−0.996825	+	0.0796271i	$0.974627\pi$
$864$		0			0
$865$	30.9839			1.05348
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−2.74086	+	1.17948i	−0.0928171	+	0.0399421i
$873$		0			0
$874$	81.4758	−	63.1109i	2.75596	−	2.13476i
$875$		0			0
$876$		0			0
$877$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$877$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$878$	22.4471	+	2.85115i	0.757552	+	0.0962217i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$		0			0
$886$	−42.6028	+	33.0000i	−1.43127	+	1.10866i
$887$	47.9659			1.61054			0.805268	−	0.592911i	$-0.202021\pi$
$887$	47.9659			1.61054			0.805268	+	0.592911i	$0.202021\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.01343	+	6.01343i	−0.201232	+	0.201232i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−78.9839	+	78.9839i	−2.63133	+	2.63133i
$902$		0			0
$903$		0			0
$904$	30.3095	+	12.0716i	1.00808	+	0.401495i
$905$	10.2902			0.342056
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$	41.7771	−	24.6307i	1.38642	−	0.817399i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−50.5474	+	29.8014i	−1.67013	+	0.984666i
$917$		0			0
$918$		0			0
$919$			23.2379i			0.766548i	0.923635	+	0.383274i	$0.125203\pi$
$919$			23.2379i			0.766548i	−0.923635	+	0.383274i	$0.874797\pi$
$920$	−50.9726	+	21.9351i	−1.68052	+	0.723179i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−41.1109	−	41.1109i	−1.34735	−	1.34735i
$932$	−2.15812	−	0.557225i	−0.0706917	−	0.0182525i
$933$		0			0
$934$	30.9839	+	40.0000i	1.01382	+	1.30884i
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$	3.94456	−	2.32561i	0.128657	−	0.0758530i
$941$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$941$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−41.6413	+	41.6413i	−1.35316	+	1.35316i	−0.471060	+	0.882101i	$0.656129\pi$
$947$	−41.6413	+	41.6413i	−1.35316	+	1.35316i	−0.882101	+	0.471060i	$0.843871\pi$
$948$		0			0
$949$		0			0
$950$	35.9645	+	46.4300i	1.16684	+	1.50639i
$951$		0			0
$952$		0			0
$953$	26.4127			0.855590			0.427795	−	0.903876i	$-0.359290\pi$
$953$	26.4127			0.855590			0.427795	+	0.903876i	$0.359290\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	29.0000			0.935484
$962$		0			0
$963$		0			0
$964$	60.0000	+	15.4919i	1.93247	+	0.498962i
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	−28.9046	−	11.5120i	−0.929028	−	0.370011i
$969$		0			0
$970$		0			0
$971$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$971$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	13.6744	+	47.5474i	0.437707	+	1.52195i
$977$		−	47.2531i		−	1.51176i	−0.654710	−	0.755880i	$-0.727209\pi$
$977$		−	47.2531i		−	1.51176i	0.654710	−	0.755880i	$-0.272791\pi$
$978$		0			0
$979$		0			0
$980$	15.8990	+	26.9670i	0.507877	+	0.861430i
$981$		0			0
$982$		0			0
$983$	41.8224			1.33393			0.666964	−	0.745090i	$-0.267594\pi$
$983$	41.8224			1.33393			0.666964	+	0.745090i	$0.267594\pi$
$984$		0			0
$985$		−	10.0000i		−	0.318626i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	32.0000			1.01651			0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000			1.01651			0.508257	+	0.861206i	$0.330290\pi$
$992$	25.8752	−	35.3620i	0.821539	−	1.12275i
$993$		0			0
$994$		0			0
$995$	25.2982	−	25.2982i	0.802008	−	0.802008i
$996$		0			0
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$	28.6613	−	22.2010i	0.907259	−	0.702759i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.2.bm.e.109.3	yes	8
3.2	odd	2	inner	720.2.bm.e.109.2	✓	8
5.4	even	2	inner	720.2.bm.e.109.2	✓	8
15.14	odd	2	CM	720.2.bm.e.109.3	yes	8
16.5	even	4	inner	720.2.bm.e.469.2	yes	8
48.5	odd	4	inner	720.2.bm.e.469.3	yes	8
80.69	even	4	inner	720.2.bm.e.469.3	yes	8
240.149	odd	4	inner	720.2.bm.e.469.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.bm.e.109.2	✓	8	3.2	odd	2	inner
720.2.bm.e.109.2	✓	8	5.4	even	2	inner
720.2.bm.e.109.3	yes	8	1.1	even	1	trivial
720.2.bm.e.109.3	yes	8	15.14	odd	2	CM
720.2.bm.e.469.2	yes	8	16.5	even	4	inner
720.2.bm.e.469.2	yes	8	240.149	odd	4	inner
720.2.bm.e.469.3	yes	8	48.5	odd	4	inner
720.2.bm.e.469.3	yes	8	80.69	even	4	inner

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.bm (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.178197 - 1.40294i) q^{2} +(-1.93649 - 0.500000i) q^{4} +(1.58114 + 1.58114i) q^{5} +(-1.04655 + 2.62769i) q^{8} +(2.50000 - 1.93649i) q^{10} +(3.50000 + 1.93649i) q^{16} -8.06126i q^{17} +(5.87298 + 5.87298i) q^{19} +(-2.27129 - 3.85243i) q^{20} +8.77405 q^{23} +5.00000i q^{25} +7.74597 q^{31} +(3.34047 - 4.56522i) q^{32} +(-11.3095 - 1.43649i) q^{34} +(9.28600 - 7.19291i) q^{38} +(-5.80948 + 2.50000i) q^{40} +(1.56351 - 12.3095i) q^{46} +1.02391i q^{47} -7.00000 q^{49} +(7.01471 + 0.890985i) q^{50} +(-9.79796 - 9.79796i) q^{53} +(8.74597 + 8.74597i) q^{61} +(1.38031 - 10.8671i) q^{62} +(-5.80948 - 5.50000i) q^{64} +(-4.03063 + 15.6106i) q^{68} +(-8.43649 - 14.3095i) q^{76} -16.0000 q^{79} +(2.47212 + 8.59585i) q^{80} +(-2.44949 + 2.44949i) q^{83} +(12.7460 - 12.7460i) q^{85} +(-16.9909 - 4.38702i) q^{92} +(1.43649 + 0.182458i) q^{94} +18.5720i q^{95} +(-1.24738 + 9.82059i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 20 q^{10} + 28 q^{16} + 16 q^{19} - 44 q^{34} + 28 q^{46} - 56 q^{49} + 8 q^{61} - 52 q^{76} - 128 q^{79} + 40 q^{85} - 4 q^{94}+O(q^{100}) \) 8 * q + 20 * q^10 + 28 * q^16 + 16 * q^19 - 44 * q^34 + 28 * q^46 - 56 * q^49 + 8 * q^61 - 52 * q^76 - 128 * q^79 + 40 * q^85 - 4 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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