LMFDB - Embedded newform 576.4.k.a.145.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,4,Mod(145,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("576.145");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	576.k (of order $4$, degree $2$, 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:	$33.9851001633$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - x^{8} + 6x^{7} + 14x^{6} - 80x^{5} + 56x^{4} + 96x^{3} - 64x^{2} - 512x + 1024 $ x^10 - 2x^9 - x^8 + 6x^7 + 14x^6 - 80x^5 + 56x^4 + 96x^3 - 64x^2 - 512x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			145.2
Root			$-1.62580 + 1.16481i$ of defining polynomial
Character	$\chi$	$=$	576.145
Dual form			576.4.k.a.433.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.22587 + 8.22587i) q^{5} +2.67171i q^{7} +O(q^{10})$	$q+(-8.22587 + 8.22587i) q^{5} +2.67171i q^{7} +(-45.2213 + 45.2213i) q^{11} +(35.3968 + 35.3968i) q^{13} +72.4991 q^{17} +(-19.4427 - 19.4427i) q^{19} -139.462i q^{23} -10.3299i q^{25} +(-66.0434 - 66.0434i) q^{29} -188.682 q^{31} +(-21.9771 - 21.9771i) q^{35} +(-84.0653 + 84.0653i) q^{37} -104.629i q^{41} +(31.4857 - 31.4857i) q^{43} -488.151 q^{47} +335.862 q^{49} +(-149.560 + 149.560i) q^{53} -743.968i q^{55} +(284.698 - 284.698i) q^{59} +(-228.069 - 228.069i) q^{61} -582.338 q^{65} +(-139.151 - 139.151i) q^{67} -453.655i q^{71} -259.747i q^{73} +(-120.818 - 120.818i) q^{77} -323.190 q^{79} +(-563.897 - 563.897i) q^{83} +(-596.368 + 596.368i) q^{85} -866.853i q^{89} +(-94.5697 + 94.5697i) q^{91} +319.866 q^{95} -936.077 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{5}+O(q^{10}) $ 10 * q + 2 * q^5	$ 10 q + 2 q^{5} + 18 q^{11} - 2 q^{13} + 4 q^{17} + 26 q^{19} + 202 q^{29} - 368 q^{31} + 476 q^{35} - 10 q^{37} + 838 q^{43} - 944 q^{47} + 94 q^{49} + 378 q^{53} + 1706 q^{59} + 910 q^{61} + 492 q^{65} - 1942 q^{67} + 268 q^{77} + 4416 q^{79} - 2562 q^{83} - 12 q^{85} - 3332 q^{91} + 6900 q^{95} - 4 q^{97}+O(q^{100}) $ 10 * q + 2 * q^5 + 18 * q^11 - 2 * q^13 + 4 * q^17 + 26 * q^19 + 202 * q^29 - 368 * q^31 + 476 * q^35 - 10 * q^37 + 838 * q^43 - 944 * q^47 + 94 * q^49 + 378 * q^53 + 1706 * q^59 + 910 * q^61 + 492 * q^65 - 1942 * q^67 + 268 * q^77 + 4416 * q^79 - 2562 * q^83 - 12 * q^85 - 3332 * q^91 + 6900 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

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
$5$	−8.22587	+	8.22587i	−0.735744	+	0.735744i	−0.971751	−	0.236007i	$-0.924161\pi$
$5$	−8.22587	+	8.22587i	−0.735744	+	0.735744i	0.236007	+	0.971751i	$0.424161\pi$
$6$		0			0
$7$			2.67171i			0.144259i	0.997395	+	0.0721293i	$0.0229794\pi$
$7$			2.67171i			0.144259i	−0.997395	+	0.0721293i	$0.977021\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−45.2213	+	45.2213i	−1.23952	+	1.23952i	−0.279323	+	0.960197i	$0.590110\pi$
$11$	−45.2213	+	45.2213i	−1.23952	+	1.23952i	−0.960197	+	0.279323i	$0.909890\pi$
$12$		0			0
$13$	35.3968	+	35.3968i	0.755176	+	0.755176i	0.975440	−	0.220264i	$-0.0706918\pi$
$13$	35.3968	+	35.3968i	0.755176	+	0.755176i	−0.220264	+	0.975440i	$0.570692\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	72.4991			1.03433			0.517165	−	0.855886i	$-0.326987\pi$
$17$	72.4991			1.03433			0.517165	+	0.855886i	$0.326987\pi$
$18$		0			0
$19$	−19.4427	−	19.4427i	−0.234761	−	0.234761i	0.579916	−	0.814676i	$-0.303085\pi$
$19$	−19.4427	−	19.4427i	−0.234761	−	0.234761i	−0.814676	+	0.579916i	$0.803085\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	139.462i		−	1.26434i	−0.774830	−	0.632170i	$-0.782165\pi$
$23$		−	139.462i		−	1.26434i	0.774830	−	0.632170i	$-0.217835\pi$
$24$		0			0
$25$		−	10.3299i		−	0.0826390i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−66.0434	−	66.0434i	−0.422895	−	0.422895i	0.463304	−	0.886199i	$-0.346664\pi$
$29$	−66.0434	−	66.0434i	−0.422895	−	0.422895i	−0.886199	+	0.463304i	$0.846664\pi$
$30$		0			0
$31$	−188.682			−1.09317			−0.546584	−	0.837404i	$-0.684072\pi$
$31$	−188.682			−1.09317			−0.546584	+	0.837404i	$0.684072\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−21.9771	−	21.9771i	−0.106137	−	0.106137i
$36$		0			0
$37$	−84.0653	+	84.0653i	−0.373520	+	0.373520i	−0.868758	−	0.495237i	$-0.835081\pi$
$37$	−84.0653	+	84.0653i	−0.373520	+	0.373520i	0.495237	+	0.868758i	$0.335081\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	104.629i		−	0.398545i	−0.979944	−	0.199272i	$-0.936142\pi$
$41$		−	104.629i		−	0.398545i	0.979944	−	0.199272i	$-0.0638578\pi$
$42$		0			0
$43$	31.4857	−	31.4857i	0.111663	−	0.111663i	−0.649067	−	0.760731i	$-0.724841\pi$
$43$	31.4857	−	31.4857i	0.111663	−	0.111663i	0.760731	+	0.649067i	$0.224841\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−488.151			−1.51498			−0.757491	−	0.652846i	$-0.773575\pi$
$47$	−488.151			−1.51498			−0.757491	+	0.652846i	$0.773575\pi$
$48$		0			0
$49$	335.862			0.979189
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−149.560	+	149.560i	−0.387617	+	0.387617i	−0.873837	−	0.486220i	$-0.838376\pi$
$53$	−149.560	+	149.560i	−0.387617	+	0.387617i	0.486220	+	0.873837i	$0.338376\pi$
$54$		0			0
$55$		−	743.968i		−	1.82394i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	284.698	−	284.698i	0.628212	−	0.628212i	−0.319406	−	0.947618i	$-0.603483\pi$
$59$	284.698	−	284.698i	0.628212	−	0.628212i	0.947618	+	0.319406i	$0.103483\pi$
$60$		0			0
$61$	−228.069	−	228.069i	−0.478709	−	0.478709i	0.426010	−	0.904719i	$-0.359919\pi$
$61$	−228.069	−	228.069i	−0.478709	−	0.478709i	−0.904719	+	0.426010i	$0.859919\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−582.338			−1.11123
$66$		0			0
$67$	−139.151	−	139.151i	−0.253730	−	0.253730i	0.568768	−	0.822498i	$-0.307420\pi$
$67$	−139.151	−	139.151i	−0.253730	−	0.253730i	−0.822498	+	0.568768i	$0.807420\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	453.655i		−	0.758294i	−0.925336	−	0.379147i	$-0.876217\pi$
$71$		−	453.655i		−	0.758294i	0.925336	−	0.379147i	$-0.123783\pi$
$72$		0			0
$73$		−	259.747i		−	0.416454i	−0.978081	−	0.208227i	$-0.933231\pi$
$73$		−	259.747i		−	0.416454i	0.978081	−	0.208227i	$-0.0667692\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−120.818	−	120.818i	−0.178811	−	0.178811i
$78$		0			0
$79$	−323.190			−0.460275			−0.230138	−	0.973158i	$-0.573918\pi$
$79$	−323.190			−0.460275			−0.230138	+	0.973158i	$0.573918\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−563.897	−	563.897i	−0.745732	−	0.745732i	0.227943	−	0.973674i	$-0.426800\pi$
$83$	−563.897	−	563.897i	−0.745732	−	0.745732i	−0.973674	+	0.227943i	$0.926800\pi$
$84$		0			0
$85$	−596.368	+	596.368i	−0.761002	+	0.761002i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	866.853i		−	1.03243i	−0.856459	−	0.516215i	$-0.827341\pi$
$89$		−	866.853i		−	1.03243i	0.856459	−	0.516215i	$-0.172659\pi$
$90$		0			0
$91$	−94.5697	+	94.5697i	−0.108941	+	0.108941i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	319.866			0.345448
$96$		0			0
$97$	−936.077			−0.979837			−0.489919	−	0.871768i	$-0.662974\pi$
$97$	−936.077			−0.979837			−0.489919	+	0.871768i	$0.662974\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.58844	−	1.58844i	0.00156491	−	0.00156491i	−0.706324	−	0.707889i	$-0.749648\pi$
$101$	1.58844	−	1.58844i	0.00156491	−	0.00156491i	0.707889	+	0.706324i	$0.249648\pi$
$102$		0			0
$103$			1388.28i			1.32807i	0.747700	+	0.664036i	$0.231158\pi$
$103$			1388.28i			1.32807i	−0.747700	+	0.664036i	$0.768842\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−821.526	+	821.526i	−0.742243	+	0.742243i	−0.973009	−	0.230767i	$-0.925877\pi$
$107$	−821.526	+	821.526i	−0.742243	+	0.742243i	0.230767	+	0.973009i	$0.425877\pi$
$108$		0			0
$109$	532.797	+	532.797i	0.468190	+	0.468190i	0.901328	−	0.433138i	$-0.142594\pi$
$109$	532.797	+	532.797i	0.468190	+	0.468190i	−0.433138	+	0.901328i	$0.642594\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	67.2680			0.0560003			0.0280002	−	0.999608i	$-0.491086\pi$
$113$	67.2680			0.0560003			0.0280002	+	0.999608i	$0.491086\pi$
$114$		0			0
$115$	1147.19	+	1147.19i	0.930230	+	0.930230i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			193.696i			0.149211i
$120$		0			0
$121$		−	2758.92i		−	2.07282i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−943.262	−	943.262i	−0.674943	−	0.674943i
$126$		0			0
$127$	−1903.59			−1.33005			−0.665026	−	0.746820i	$-0.731579\pi$
$127$	−1903.59			−1.33005			−0.665026	+	0.746820i	$0.731579\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	918.430	+	918.430i	0.612546	+	0.612546i	0.943609	−	0.331062i	$-0.107407\pi$
$131$	918.430	+	918.430i	0.612546	+	0.612546i	−0.331062	+	0.943609i	$0.607407\pi$
$132$		0			0
$133$	51.9451	−	51.9451i	0.0338662	−	0.0338662i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	477.234i		−	0.297612i	−0.988866	−	0.148806i	$-0.952457\pi$
$137$		−	477.234i		−	0.297612i	0.988866	−	0.148806i	$-0.0475430\pi$
$138$		0			0
$139$	1513.89	−	1513.89i	0.923788	−	0.923788i	−0.0735064	−	0.997295i	$-0.523419\pi$
$139$	1513.89	−	1513.89i	0.923788	−	0.923788i	0.997295	+	0.0735064i	$0.0234189\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3201.37			−1.87211
$144$		0			0
$145$	1086.53			0.622285
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−375.353	+	375.353i	−0.206377	+	0.206377i	−0.802725	−	0.596349i	$-0.796618\pi$
$149$	−375.353	+	375.353i	−0.206377	+	0.206377i	0.596349	+	0.802725i	$0.296618\pi$
$150$		0			0
$151$		−	2997.52i		−	1.61546i	−0.589553	−	0.807730i	$-0.700696\pi$
$151$		−	2997.52i		−	1.61546i	0.589553	−	0.807730i	$-0.299304\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1552.07	−	1552.07i	0.804293	−	0.804293i
$156$		0			0
$157$	−1509.01	−	1509.01i	−0.767082	−	0.767082i	0.210510	−	0.977592i	$-0.432488\pi$
$157$	−1509.01	−	1509.01i	−0.767082	−	0.767082i	−0.977592	+	0.210510i	$0.932488\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	372.601			0.182392
$162$		0			0
$163$	1425.19	+	1425.19i	0.684844	+	0.684844i	0.961088	−	0.276244i	$-0.0890898\pi$
$163$	1425.19	+	1425.19i	0.684844	+	0.684844i	−0.276244	+	0.961088i	$0.589090\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	792.415i		−	0.367179i	−0.983003	−	0.183590i	$-0.941228\pi$
$167$		−	792.415i		−	0.367179i	0.983003	−	0.183590i	$-0.0587717\pi$
$168$		0			0
$169$			308.861i			0.140583i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	773.594	+	773.594i	0.339972	+	0.339972i	0.856357	−	0.516384i	$-0.172722\pi$
$173$	773.594	+	773.594i	0.339972	+	0.339972i	−0.516384	+	0.856357i	$0.672722\pi$
$174$		0			0
$175$	27.5984			0.0119214
$176$		0			0
$177$		0			0
$178$		0			0
$179$	426.050	+	426.050i	0.177902	+	0.177902i	0.790441	−	0.612539i	$-0.209852\pi$
$179$	426.050	+	426.050i	0.177902	+	0.177902i	−0.612539	+	0.790441i	$0.709852\pi$
$180$		0			0
$181$	−2618.06	+	2618.06i	−1.07513	+	1.07513i	−0.0781951	+	0.996938i	$0.524916\pi$
$181$	−2618.06	+	2618.06i	−1.07513	+	1.07513i	−0.996938	+	0.0781951i	$0.975084\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	1383.02i		−	0.549631i
$186$		0			0
$187$	−3278.50	+	3278.50i	−1.28207	+	1.28207i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3216.39			1.21848			0.609240	−	0.792986i	$-0.291475\pi$
$191$	3216.39			1.21848			0.609240	+	0.792986i	$0.291475\pi$
$192$		0			0
$193$	2852.57			1.06390			0.531950	−	0.846776i	$-0.321459\pi$
$193$	2852.57			1.06390			0.531950	+	0.846776i	$0.321459\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1609.02	+	1609.02i	−0.581918	+	0.581918i	−0.935430	−	0.353512i	$-0.884987\pi$
$197$	−1609.02	+	1609.02i	−0.581918	+	0.581918i	0.353512	+	0.935430i	$0.384987\pi$
$198$		0			0
$199$		−	747.136i		−	0.266146i	−0.991106	−	0.133073i	$-0.957516\pi$
$199$		−	747.136i		−	0.266146i	0.991106	−	0.133073i	$-0.0424845\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	176.449	−	176.449i	0.0610062	−	0.0610062i
$204$		0			0
$205$	860.666	+	860.666i	0.293227	+	0.293227i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1758.44			0.581981
$210$		0			0
$211$	2227.13	+	2227.13i	0.726645	+	0.726645i	0.969950	−	0.243305i	$-0.0782315\pi$
$211$	2227.13	+	2227.13i	0.726645	+	0.726645i	−0.243305	+	0.969950i	$0.578231\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			517.995i			0.164311i
$216$		0			0
$217$		−	504.102i		−	0.157699i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2566.23	+	2566.23i	0.781102	+	0.781102i
$222$		0			0
$223$	358.053			0.107520			0.0537601	−	0.998554i	$-0.482879\pi$
$223$	358.053			0.107520			0.0537601	+	0.998554i	$0.482879\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3455.40	+	3455.40i	1.01032	+	1.01032i	0.999946	+	0.0103741i	$0.00330223\pi$
$227$	3455.40	+	3455.40i	1.01032	+	1.01032i	0.0103741	+	0.999946i	$0.496698\pi$
$228$		0			0
$229$	−1430.03	+	1430.03i	−0.412659	+	0.412659i	−0.882664	−	0.470005i	$-0.844252\pi$
$229$	−1430.03	+	1430.03i	−0.412659	+	0.412659i	0.470005	+	0.882664i	$0.344252\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			926.479i			0.260496i	0.991481	+	0.130248i	$0.0415774\pi$
$233$			926.479i			0.260496i	−0.991481	+	0.130248i	$0.958423\pi$
$234$		0			0
$235$	4015.47	−	4015.47i	1.11464	−	1.11464i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−792.472			−0.214480			−0.107240	−	0.994233i	$-0.534201\pi$
$239$	−792.472			−0.214480			−0.107240	+	0.994233i	$0.534201\pi$
$240$		0			0
$241$	1449.01			0.387299			0.193650	−	0.981071i	$-0.437967\pi$
$241$	1449.01			0.387299			0.193650	+	0.981071i	$0.437967\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2762.76	+	2762.76i	−0.720433	+	0.720433i
$246$		0			0
$247$		−	1376.42i		−	0.354572i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3580.04	+	3580.04i	−0.900280	+	0.900280i	−0.995460	−	0.0951802i	$-0.969657\pi$
$251$	−3580.04	+	3580.04i	−0.900280	+	0.900280i	0.0951802	+	0.995460i	$0.469657\pi$
$252$		0			0
$253$	6306.64	+	6306.64i	1.56717	+	1.56717i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4708.87			1.14292			0.571461	−	0.820629i	$-0.306377\pi$
$257$	4708.87			1.14292			0.571461	+	0.820629i	$0.306377\pi$
$258$		0			0
$259$	−224.598	−	224.598i	−0.0538835	−	0.0538835i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	2967.82i		−	0.695830i	−0.937526	−	0.347915i	$-0.886890\pi$
$263$		−	2967.82i		−	0.695830i	0.937526	−	0.347915i	$-0.113110\pi$
$264$		0			0
$265$		−	2460.53i		−	0.570374i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	663.633	+	663.633i	0.150418	+	0.150418i	0.778305	−	0.627887i	$-0.216080\pi$
$269$	663.633	+	663.633i	0.150418	+	0.150418i	−0.627887	+	0.778305i	$0.716080\pi$
$270$		0			0
$271$	−8058.74			−1.80640			−0.903199	−	0.429223i	$-0.858788\pi$
$271$	−8058.74			−1.80640			−0.903199	+	0.429223i	$0.858788\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	467.130	+	467.130i	0.102433	+	0.102433i
$276$		0			0
$277$	482.477	−	482.477i	0.104654	−	0.104654i	−0.652841	−	0.757495i	$-0.726423\pi$
$277$	482.477	−	482.477i	0.104654	−	0.104654i	0.757495	+	0.652841i	$0.226423\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			5899.10i			1.25235i	0.779682	+	0.626175i	$0.215381\pi$
$281$			5899.10i			1.25235i	−0.779682	+	0.626175i	$0.784619\pi$
$282$		0			0
$283$	679.897	−	679.897i	0.142812	−	0.142812i	−0.632086	−	0.774898i	$-0.717801\pi$
$283$	679.897	−	679.897i	0.142812	−	0.142812i	0.774898	+	0.632086i	$0.217801\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	279.538			0.0574935
$288$		0			0
$289$	343.118			0.0698388
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3552.87	−	3552.87i	0.708398	−	0.708398i	−0.257800	−	0.966198i	$-0.582998\pi$
$293$	3552.87	−	3552.87i	0.708398	−	0.708398i	0.966198	+	0.257800i	$0.0829976\pi$
$294$		0			0
$295$			4683.78i			0.924407i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4936.50	−	4936.50i	0.954799	−	0.954799i
$300$		0			0
$301$	84.1205	+	84.1205i	0.0161084	+	0.0161084i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3752.13			0.704415
$306$		0			0
$307$	−2735.56	−	2735.56i	−0.508556	−	0.508556i	0.405527	−	0.914083i	$-0.367088\pi$
$307$	−2735.56	−	2735.56i	−0.508556	−	0.508556i	−0.914083	+	0.405527i	$0.867088\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			5796.70i			1.05692i	0.848960	+	0.528458i	$0.177229\pi$
$311$			5796.70i			1.05692i	−0.848960	+	0.528458i	$0.822771\pi$
$312$		0			0
$313$			8362.62i			1.51017i	0.655627	+	0.755085i	$0.272404\pi$
$313$			8362.62i			1.51017i	−0.655627	+	0.755085i	$0.727596\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	344.406	+	344.406i	0.0610214	+	0.0610214i	0.736959	−	0.675938i	$-0.236261\pi$
$317$	344.406	+	344.406i	0.0610214	+	0.0610214i	−0.675938	+	0.736959i	$0.736261\pi$
$318$		0			0
$319$	5973.13			1.04837
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1409.58	−	1409.58i	−0.242820	−	0.242820i
$324$		0			0
$325$	365.644	−	365.644i	0.0624071	−	0.0624071i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	1304.20i		−	0.218549i
$330$		0			0
$331$	−2687.86	+	2687.86i	−0.446339	+	0.446339i	−0.894135	−	0.447797i	$-0.852209\pi$
$331$	−2687.86	+	2687.86i	−0.446339	+	0.446339i	0.447797	+	0.894135i	$0.352209\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2289.27			0.373361
$336$		0			0
$337$	−1795.31			−0.290199			−0.145099	−	0.989417i	$-0.546350\pi$
$337$	−1795.31			−0.290199			−0.145099	+	0.989417i	$0.546350\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	8532.42	−	8532.42i	1.35500	−	1.35500i
$342$		0			0
$343$			1813.72i			0.285515i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1967.33	+	1967.33i	−0.304357	+	0.304357i	−0.842716	−	0.538359i	$-0.819045\pi$
$347$	−1967.33	+	1967.33i	−0.304357	+	0.304357i	0.538359	+	0.842716i	$0.319045\pi$
$348$		0			0
$349$	−7363.37	−	7363.37i	−1.12938	−	1.12938i	−0.990279	−	0.139097i	$-0.955580\pi$
$349$	−7363.37	−	7363.37i	−1.12938	−	1.12938i	−0.139097	−	0.990279i	$-0.544420\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−10644.3			−1.60493			−0.802466	−	0.596698i	$-0.796479\pi$
$353$	−10644.3			−1.60493			−0.802466	+	0.596698i	$0.796479\pi$
$354$		0			0
$355$	3731.70	+	3731.70i	0.557911	+	0.557911i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			7459.42i			1.09664i	0.836269	+	0.548319i	$0.184732\pi$
$359$			7459.42i			1.09664i	−0.836269	+	0.548319i	$0.815268\pi$
$360$		0			0
$361$		−	6102.96i		−	0.889775i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2136.65	+	2136.65i	0.306403	+	0.306403i
$366$		0			0
$367$	6251.35			0.889149			0.444574	−	0.895742i	$-0.353355\pi$
$367$	6251.35			0.889149			0.444574	+	0.895742i	$0.353355\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−399.582	−	399.582i	−0.0559171	−	0.0559171i
$372$		0			0
$373$	8911.86	−	8911.86i	1.23710	−	1.23710i	0.275921	−	0.961180i	$-0.411017\pi$
$373$	8911.86	−	8911.86i	1.23710	−	1.23710i	0.961180	−	0.275921i	$-0.0889827\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	4675.45i		−	0.638721i
$378$		0			0
$379$	−1184.03	+	1184.03i	−0.160473	+	0.160473i	−0.782776	−	0.622303i	$-0.786197\pi$
$379$	−1184.03	+	1184.03i	−0.160473	+	0.160473i	0.622303	+	0.782776i	$0.286197\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2880.38			−0.384283			−0.192142	−	0.981367i	$-0.561543\pi$
$383$	−2880.38			−0.384283			−0.192142	+	0.981367i	$0.561543\pi$
$384$		0			0
$385$	1987.66			0.263119
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9244.24	+	9244.24i	−1.20489	+	1.20489i	−0.232226	+	0.972662i	$0.574601\pi$
$389$	−9244.24	+	9244.24i	−1.20489	+	1.20489i	−0.972662	+	0.232226i	$0.925399\pi$
$390$		0			0
$391$		−	10110.9i		−	1.30774i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2658.52	−	2658.52i	0.338645	−	0.338645i
$396$		0			0
$397$	−4257.80	−	4257.80i	−0.538270	−	0.538270i	0.384751	−	0.923020i	$-0.374287\pi$
$397$	−4257.80	−	4257.80i	−0.538270	−	0.538270i	−0.923020	+	0.384751i	$0.874287\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−12722.6			−1.58437			−0.792187	−	0.610278i	$-0.791058\pi$
$401$	−12722.6			−1.58437			−0.792187	+	0.610278i	$0.791058\pi$
$402$		0			0
$403$	−6678.72	−	6678.72i	−0.825535	−	0.825535i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	7603.08i		−	0.925972i
$408$		0			0
$409$			232.991i			0.0281678i	0.999901	+	0.0140839i	$0.00448320\pi$
$409$			232.991i			0.0281678i	−0.999901	+	0.0140839i	$0.995517\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	760.629	+	760.629i	0.0906250	+	0.0906250i
$414$		0			0
$415$	9277.08			1.09734
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6125.69	−	6125.69i	−0.714223	−	0.714223i	0.253193	−	0.967416i	$-0.418519\pi$
$419$	−6125.69	−	6125.69i	−0.714223	−	0.714223i	−0.967416	+	0.253193i	$0.918519\pi$
$420$		0			0
$421$	−8308.44	+	8308.44i	−0.961825	+	0.961825i	−0.999298	−	0.0374725i	$-0.988069\pi$
$421$	−8308.44	+	8308.44i	−0.961825	+	0.961825i	0.0374725	+	0.999298i	$0.488069\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	748.907i		−	0.0854760i
$426$		0			0
$427$	609.333	−	609.333i	0.0690579	−	0.0690579i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8737.57			0.976506			0.488253	−	0.872702i	$-0.337634\pi$
$431$	8737.57			0.976506			0.488253	+	0.872702i	$0.337634\pi$
$432$		0			0
$433$	−11627.5			−1.29049			−0.645247	−	0.763974i	$-0.723245\pi$
$433$	−11627.5			−1.29049			−0.645247	+	0.763974i	$0.723245\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2711.51	+	2711.51i	−0.296817	+	0.296817i
$438$		0			0
$439$			17631.8i			1.91690i	0.285261	+	0.958450i	$0.407920\pi$
$439$			17631.8i			1.91690i	−0.285261	+	0.958450i	$0.592080\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4549.81	+	4549.81i	−0.487964	+	0.487964i	−0.907663	−	0.419699i	$-0.862136\pi$
$443$	−4549.81	+	4549.81i	−0.487964	+	0.487964i	0.419699	+	0.907663i	$0.362136\pi$
$444$		0			0
$445$	7130.62	+	7130.62i	0.759604	+	0.759604i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	12926.5			1.35867			0.679334	−	0.733830i	$-0.262269\pi$
$449$	12926.5			1.35867			0.679334	+	0.733830i	$0.262269\pi$
$450$		0			0
$451$	4731.46	+	4731.46i	0.494004	+	0.494004i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	1555.84i		−	0.160305i
$456$		0			0
$457$			9320.32i			0.954018i	0.878898	+	0.477009i	$0.158279\pi$
$457$			9320.32i			0.954018i	−0.878898	+	0.477009i	$0.841721\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−12885.0	−	12885.0i	−1.30177	−	1.30177i	−0.927200	−	0.374566i	$-0.877792\pi$
$461$	−12885.0	−	12885.0i	−1.30177	−	1.30177i	−0.374566	−	0.927200i	$-0.622208\pi$
$462$		0			0
$463$	−7038.37			−0.706482			−0.353241	−	0.935532i	$-0.614920\pi$
$463$	−7038.37			−0.706482			−0.353241	+	0.935532i	$0.614920\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6001.76	+	6001.76i	0.594707	+	0.594707i	0.938899	−	0.344192i	$-0.111847\pi$
$467$	6001.76	+	6001.76i	0.594707	+	0.594707i	−0.344192	+	0.938899i	$0.611847\pi$
$468$		0			0
$469$	371.769	−	371.769i	0.0366028	−	0.0366028i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			2847.65i			0.276818i
$474$		0			0
$475$	−200.840	+	200.840i	−0.0194004	+	0.0194004i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−587.317			−0.0560234			−0.0280117	−	0.999608i	$-0.508918\pi$
$479$	−587.317			−0.0560234			−0.0280117	+	0.999608i	$0.508918\pi$
$480$		0			0
$481$	−5951.28			−0.564148
$482$		0			0
$483$		0			0
$484$		0			0
$485$	7700.05	−	7700.05i	0.720910	−	0.720910i
$486$		0			0
$487$			8366.45i			0.778481i	0.921136	+	0.389240i	$0.127262\pi$
$487$			8366.45i			0.778481i	−0.921136	+	0.389240i	$0.872738\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1529.30	−	1529.30i	0.140563	−	0.140563i	−0.633324	−	0.773887i	$-0.718310\pi$
$491$	1529.30	−	1529.30i	0.140563	−	0.140563i	0.773887	+	0.633324i	$0.218310\pi$
$492$		0			0
$493$	−4788.09	−	4788.09i	−0.437413	−	0.437413i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1212.03			0.109390
$498$		0			0
$499$	−11364.5	−	11364.5i	−1.01952	−	1.01952i	−0.999806	−	0.0197191i	$-0.993723\pi$
$499$	−11364.5	−	11364.5i	−1.01952	−	1.01952i	−0.0197191	−	0.999806i	$-0.506277\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	12570.2i		−	1.11427i	−0.830421	−	0.557137i	$-0.811900\pi$
$503$		−	12570.2i		−	1.11427i	0.830421	−	0.557137i	$-0.188100\pi$
$504$		0			0
$505$			26.1326i			0.00230274i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−11880.4	−	11880.4i	−1.03456	−	1.03456i	−0.999381	−	0.0351750i	$-0.988801\pi$
$509$	−11880.4	−	11880.4i	−1.03456	−	1.03456i	−0.0351750	−	0.999381i	$-0.511199\pi$
$510$		0			0
$511$	693.968			0.0600770
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−11419.8	−	11419.8i	−0.977122	−	0.977122i
$516$		0			0
$517$	22074.8	−	22074.8i	1.87785	−	1.87785i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			6612.98i			0.556085i	0.960569	+	0.278042i	$0.0896856\pi$
$521$			6612.98i			0.556085i	−0.960569	+	0.278042i	$0.910314\pi$
$522$		0			0
$523$	−5129.30	+	5129.30i	−0.428850	+	0.428850i	−0.888236	−	0.459387i	$-0.848069\pi$
$523$	−5129.30	+	5129.30i	−0.428850	+	0.428850i	0.459387	+	0.888236i	$0.348069\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−13679.3			−1.13070
$528$		0			0
$529$	−7282.60			−0.598553
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3703.53	−	3703.53i	0.300972	−	0.300972i
$534$		0			0
$535$		−	13515.5i		−	1.09220i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−15188.1	+	15188.1i	−1.21372	+	1.21372i
$540$		0			0
$541$	−10968.5	−	10968.5i	−0.871672	−	0.871672i	0.120983	−	0.992655i	$-0.461395\pi$
$541$	−10968.5	−	10968.5i	−0.871672	−	0.871672i	−0.992655	+	0.120983i	$0.961395\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8765.44			−0.688936
$546$		0			0
$547$	−13088.8	−	13088.8i	−1.02311	−	1.02311i	−0.999727	−	0.0233784i	$-0.992558\pi$
$547$	−13088.8	−	13088.8i	−1.02311	−	1.02311i	−0.0233784	−	0.999727i	$-0.507442\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			2568.12i			0.198558i
$552$		0			0
$553$		−	863.469i		−	0.0663986i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5049.87	−	5049.87i	−0.384147	−	0.384147i	0.488447	−	0.872594i	$-0.337564\pi$
$557$	−5049.87	−	5049.87i	−0.384147	−	0.384147i	−0.872594	+	0.488447i	$0.837564\pi$
$558$		0			0
$559$	2228.98			0.168651
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3249.06	−	3249.06i	−0.243217	−	0.243217i	0.574962	−	0.818180i	$-0.305017\pi$
$563$	−3249.06	−	3249.06i	−0.243217	−	0.243217i	−0.818180	+	0.574962i	$0.805017\pi$
$564$		0			0
$565$	−553.338	+	553.338i	−0.0412019	+	0.0412019i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	2806.05i		−	0.206741i	−0.994643	−	0.103371i	$-0.967037\pi$
$569$		−	2806.05i		−	0.206741i	0.994643	−	0.103371i	$-0.0329628\pi$
$570$		0			0
$571$	12038.8	−	12038.8i	0.882324	−	0.882324i	−0.111446	−	0.993770i	$-0.535548\pi$
$571$	12038.8	−	12038.8i	0.882324	−	0.882324i	0.993770	+	0.111446i	$0.0355483\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1440.62			−0.104484
$576$		0			0
$577$	7206.84			0.519973			0.259987	−	0.965612i	$-0.416282\pi$
$577$	7206.84			0.519973			0.259987	+	0.965612i	$0.416282\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1506.57	−	1506.57i	0.107578	−	0.107578i
$582$		0			0
$583$		−	13526.6i		−	0.960918i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10377.0	−	10377.0i	0.729647	−	0.729647i	−0.240903	−	0.970549i	$-0.577443\pi$
$587$	10377.0	−	10377.0i	0.729647	−	0.729647i	0.970549	+	0.240903i	$0.0774435\pi$
$588$		0			0
$589$	3668.48	+	3668.48i	0.256633	+	0.256633i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	4758.60			0.329531			0.164766	−	0.986333i	$-0.447313\pi$
$593$	4758.60			0.329531			0.164766	+	0.986333i	$0.447313\pi$
$594$		0			0
$595$	−1593.32	−	1593.32i	−0.109781	−	0.109781i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			14256.4i			0.972455i	0.873832	+	0.486227i	$0.161627\pi$
$599$			14256.4i			0.972455i	−0.873832	+	0.486227i	$0.838373\pi$
$600$		0			0
$601$			10385.2i			0.704862i	0.935838	+	0.352431i	$0.114645\pi$
$601$			10385.2i			0.704862i	−0.935838	+	0.352431i	$0.885355\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	22694.5	+	22694.5i	1.52506	+	1.52506i
$606$		0			0
$607$	16243.6			1.08618			0.543088	−	0.839676i	$-0.317255\pi$
$607$	16243.6			1.08618			0.543088	+	0.839676i	$0.317255\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−17279.0	−	17279.0i	−1.14408	−	1.14408i
$612$		0			0
$613$	−500.502	+	500.502i	−0.0329773	+	0.0329773i	−0.723403	−	0.690426i	$-0.757423\pi$
$613$	−500.502	+	500.502i	−0.0329773	+	0.0329773i	0.690426	+	0.723403i	$0.257423\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	11575.9i		−	0.755316i	−0.925945	−	0.377658i	$-0.876729\pi$
$617$		−	11575.9i		−	0.755316i	0.925945	−	0.377658i	$-0.123271\pi$
$618$		0			0
$619$	−18356.1	+	18356.1i	−1.19191	+	1.19191i	−0.215380	+	0.976530i	$0.569099\pi$
$619$	−18356.1	+	18356.1i	−1.19191	+	1.19191i	−0.976530	+	0.215380i	$0.930901\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2315.98			0.148937
$624$		0			0
$625$	16809.5			1.07581
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−6094.66	+	6094.66i	−0.386343	+	0.386343i
$630$		0			0
$631$			10224.8i			0.645079i	0.946556	+	0.322539i	$0.104537\pi$
$631$			10224.8i			0.645079i	−0.946556	+	0.322539i	$0.895463\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	15658.7	−	15658.7i	0.978578	−	0.978578i
$636$		0			0
$637$	11888.4	+	11888.4i	0.739461	+	0.739461i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−19804.4			−1.22032			−0.610162	−	0.792277i	$-0.708896\pi$
$641$	−19804.4			−1.22032			−0.610162	+	0.792277i	$0.708896\pi$
$642$		0			0
$643$	−15680.7	−	15680.7i	−0.961723	−	0.961723i	0.0375712	−	0.999294i	$-0.488038\pi$
$643$	−15680.7	−	15680.7i	−0.961723	−	0.961723i	−0.999294	+	0.0375712i	$0.988038\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	9232.26i		−	0.560985i	−0.959856	−	0.280493i	$-0.909502\pi$
$647$		−	9232.26i		−	0.560985i	0.959856	−	0.280493i	$-0.0904978\pi$
$648$		0			0
$649$			25748.8i			1.55736i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	19697.9	+	19697.9i	1.18046	+	1.18046i	0.979626	+	0.200833i	$0.0643648\pi$
$653$	19697.9	+	19697.9i	1.18046	+	1.18046i	0.200833	+	0.979626i	$0.435635\pi$
$654$		0			0
$655$	−15109.8			−0.901355
$656$		0			0
$657$		0			0
$658$		0			0
$659$	3888.06	+	3888.06i	0.229829	+	0.229829i	0.812621	−	0.582792i	$-0.198040\pi$
$659$	3888.06	+	3888.06i	0.229829	+	0.229829i	−0.582792	+	0.812621i	$0.698040\pi$
$660$		0			0
$661$	−8110.20	+	8110.20i	−0.477232	+	0.477232i	−0.904245	−	0.427013i	$-0.859566\pi$
$661$	−8110.20	+	8110.20i	−0.477232	+	0.477232i	0.427013	+	0.904245i	$0.359566\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			854.587i			0.0498338i
$666$		0			0
$667$	−9210.54	+	9210.54i	−0.534683	+	0.534683i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	20627.1			1.18674
$672$		0			0
$673$	−28428.2			−1.62827			−0.814135	−	0.580676i	$-0.802788\pi$
$673$	−28428.2			−1.62827			−0.814135	+	0.580676i	$0.802788\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	16967.4	−	16967.4i	0.963235	−	0.963235i	−0.0361128	−	0.999348i	$-0.511498\pi$
$677$	16967.4	−	16967.4i	0.963235	−	0.963235i	0.999348	+	0.0361128i	$0.0114975\pi$
$678$		0			0
$679$		−	2500.92i		−	0.141350i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−9550.16	+	9550.16i	−0.535032	+	0.535032i	−0.922066	−	0.387034i	$-0.873500\pi$
$683$	−9550.16	+	9550.16i	−0.535032	+	0.535032i	0.387034	+	0.922066i	$0.373500\pi$
$684$		0			0
$685$	3925.66	+	3925.66i	0.218966	+	0.218966i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−10587.9			−0.585439
$690$		0			0
$691$	20859.7	+	20859.7i	1.14839	+	1.14839i	0.986868	+	0.161527i	$0.0516418\pi$
$691$	20859.7	+	20859.7i	1.14839	+	1.14839i	0.161527	+	0.986868i	$0.448358\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			24906.1i			1.35934i
$696$		0			0
$697$		−	7585.52i		−	0.412227i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−23495.4	−	23495.4i	−1.26592	−	1.26592i	−0.948178	−	0.317740i	$-0.897076\pi$
$701$	−23495.4	−	23495.4i	−1.26592	−	1.26592i	−0.317740	−	0.948178i	$-0.602924\pi$
$702$		0			0
$703$	3268.91			0.175376
$704$		0			0
$705$		0			0
$706$		0			0
$707$	4.24384	+	4.24384i	0.000225751	+	0.000225751i
$708$		0			0
$709$	4559.45	−	4559.45i	0.241515	−	0.241515i	−0.575962	−	0.817477i	$-0.695372\pi$
$709$	4559.45	−	4559.45i	0.241515	−	0.241515i	0.817477	+	0.575962i	$0.195372\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			26313.9i			1.38214i
$714$		0			0
$715$	26334.1	−	26334.1i	1.37740	−	1.37740i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6494.67			−0.336871			−0.168436	−	0.985713i	$-0.553872\pi$
$719$	−6494.67			−0.336871			−0.168436	+	0.985713i	$0.553872\pi$
$720$		0			0
$721$	−3709.08			−0.191586
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−682.221	+	682.221i	−0.0349476	+	0.0349476i
$726$		0			0
$727$			24866.4i			1.26856i	0.773103	+	0.634280i	$0.218704\pi$
$727$			24866.4i			1.26856i	−0.773103	+	0.634280i	$0.781296\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2282.68	−	2282.68i	0.115497	−	0.115497i
$732$		0			0
$733$	−14914.3	−	14914.3i	−0.751533	−	0.751533i	0.223232	−	0.974765i	$-0.428339\pi$
$733$	−14914.3	−	14914.3i	−0.751533	−	0.751533i	−0.974765	+	0.223232i	$0.928339\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	12585.1			0.629008
$738$		0			0
$739$	−8451.86	−	8451.86i	−0.420713	−	0.420713i	0.464737	−	0.885449i	$-0.346149\pi$
$739$	−8451.86	−	8451.86i	−0.420713	−	0.420713i	−0.885449	+	0.464737i	$0.846149\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	5622.43i		−	0.277614i	−0.990319	−	0.138807i	$-0.955673\pi$
$743$		−	5622.43i		−	0.277614i	0.990319	−	0.138807i	$-0.0443267\pi$
$744$		0			0
$745$		−	6175.21i		−	0.303681i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2194.88	−	2194.88i	−0.107075	−	0.107075i
$750$		0			0
$751$	32314.9			1.57016			0.785079	−	0.619396i	$-0.212622\pi$
$751$	32314.9			1.57016			0.785079	+	0.619396i	$0.212622\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	24657.2	+	24657.2i	1.18857	+	1.18857i
$756$		0			0
$757$	12692.8	−	12692.8i	0.609418	−	0.609418i	−0.333376	−	0.942794i	$-0.608188\pi$
$757$	12692.8	−	12692.8i	0.609418	−	0.609418i	0.942794	+	0.333376i	$0.108188\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			13108.2i			0.624404i	0.950016	+	0.312202i	$0.101067\pi$
$761$			13108.2i			0.624404i	−0.950016	+	0.312202i	$0.898933\pi$
$762$		0			0
$763$	−1423.48	+	1423.48i	−0.0675404	+	0.0675404i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	20154.8			0.948822
$768$		0			0
$769$	23661.2			1.10955			0.554776	−	0.832000i	$-0.312804\pi$
$769$	23661.2			1.10955			0.554776	+	0.832000i	$0.312804\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−21370.5	+	21370.5i	−0.994362	+	0.994362i	−0.999984	−	0.00562228i	$-0.998210\pi$
$773$	−21370.5	+	21370.5i	−0.994362	+	0.994362i	0.00562228	+	0.999984i	$0.498210\pi$
$774$		0			0
$775$			1949.06i			0.0903384i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2034.27	+	2034.27i	−0.0935627	+	0.0935627i
$780$		0			0
$781$	20514.8	+	20514.8i	0.939921	+	0.939921i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	24825.8			1.12875
$786$		0			0
$787$	20890.9	+	20890.9i	0.946226	+	0.946226i	0.998626	−	0.0524002i	$-0.0166872\pi$
$787$	20890.9	+	20890.9i	0.946226	+	0.946226i	−0.0524002	+	0.998626i	$0.516687\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			179.720i			0.00807853i
$792$		0			0
$793$		−	16145.8i		−	0.723020i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	6834.83	+	6834.83i	0.303767	+	0.303767i	0.842486	−	0.538719i	$-0.181092\pi$
$797$	6834.83	+	6834.83i	0.303767	+	0.303767i	−0.538719	+	0.842486i	$0.681092\pi$
$798$		0			0
$799$	−35390.5			−1.56699
$800$		0			0
$801$		0			0
$802$		0			0
$803$	11746.1	+	11746.1i	0.516203	+	0.516203i
$804$		0			0
$805$	−3064.97	+	3064.97i	−0.134194	+	0.134194i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			15807.0i			0.686952i	0.939162	+	0.343476i	$0.111604\pi$
$809$			15807.0i			0.686952i	−0.939162	+	0.343476i	$0.888396\pi$
$810$		0			0
$811$	11522.7	−	11522.7i	0.498910	−	0.498910i	−0.412189	−	0.911099i	$-0.635236\pi$
$811$	11522.7	−	11522.7i	0.498910	−	0.498910i	0.911099	+	0.412189i	$0.135236\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−23446.9			−1.00774
$816$		0			0
$817$	−1224.33			−0.0524284
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−308.824	+	308.824i	−0.0131279	+	0.0131279i	−0.713640	−	0.700512i	$-0.752955\pi$
$821$	−308.824	+	308.824i	−0.0131279	+	0.0131279i	0.700512	+	0.713640i	$0.252955\pi$
$822$		0			0
$823$			5633.49i			0.238604i	0.992858	+	0.119302i	$0.0380656\pi$
$823$			5633.49i			0.238604i	−0.992858	+	0.119302i	$0.961934\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−15835.5	+	15835.5i	−0.665844	+	0.665844i	−0.956751	−	0.290908i	$-0.906043\pi$
$827$	−15835.5	+	15835.5i	−0.665844	+	0.665844i	0.290908	+	0.956751i	$0.406043\pi$
$828$		0			0
$829$	−6200.19	−	6200.19i	−0.259761	−	0.259761i	0.565196	−	0.824957i	$-0.308801\pi$
$829$	−6200.19	−	6200.19i	−0.259761	−	0.259761i	−0.824957	+	0.565196i	$0.808801\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	24349.7			1.01281
$834$		0			0
$835$	6518.30	+	6518.30i	0.270150	+	0.270150i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	30644.3i		−	1.26098i	−0.776199	−	0.630488i	$-0.782855\pi$
$839$		−	30644.3i		−	1.26098i	0.776199	−	0.630488i	$-0.217145\pi$
$840$		0			0
$841$		−	15665.5i		−	0.642319i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2540.65	−	2540.65i	−0.103433	−	0.103433i
$846$		0			0
$847$	7371.03			0.299022
$848$		0			0
$849$		0			0
$850$		0			0
$851$	11723.9	+	11723.9i	0.472256	+	0.472256i
$852$		0			0
$853$	−30801.0	+	30801.0i	−1.23635	+	1.23635i	−0.274869	+	0.961482i	$0.588635\pi$
$853$	−30801.0	+	30801.0i	−1.23635	+	1.23635i	−0.961482	+	0.274869i	$0.911365\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	41788.5i		−	1.66566i	−0.553533	−	0.832828i	$-0.686721\pi$
$857$		−	41788.5i		−	1.66566i	0.553533	−	0.832828i	$-0.313279\pi$
$858$		0			0
$859$	−11914.5	+	11914.5i	−0.473243	+	0.473243i	−0.902963	−	0.429719i	$-0.858612\pi$
$859$	−11914.5	+	11914.5i	−0.473243	+	0.473243i	0.429719	+	0.902963i	$0.358612\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	27636.7			1.09011			0.545054	−	0.838401i	$-0.316509\pi$
$863$	27636.7			1.09011			0.545054	+	0.838401i	$0.316509\pi$
$864$		0			0
$865$	−12727.0			−0.500266
$866$		0			0
$867$		0			0
$868$		0			0
$869$	14615.1	−	14615.1i	0.570520	−	0.570520i
$870$		0			0
$871$		−	9850.95i		−	0.383223i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2520.12	−	2520.12i	0.0973663	−	0.0973663i
$876$		0			0
$877$	−11918.3	−	11918.3i	−0.458896	−	0.458896i	0.439397	−	0.898293i	$-0.355192\pi$
$877$	−11918.3	−	11918.3i	−0.458896	−	0.458896i	−0.898293	+	0.439397i	$0.855192\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	13330.0			0.509759			0.254880	−	0.966973i	$-0.417964\pi$
$881$	13330.0			0.509759			0.254880	+	0.966973i	$0.417964\pi$
$882$		0			0
$883$	25172.1	+	25172.1i	0.959353	+	0.959353i	0.999206	−	0.0398530i	$-0.0126890\pi$
$883$	25172.1	+	25172.1i	0.959353	+	0.959353i	−0.0398530	+	0.999206i	$0.512689\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			48821.3i			1.84810i	0.382278	+	0.924048i	$0.375140\pi$
$887$			48821.3i			1.84810i	−0.382278	+	0.924048i	$0.624860\pi$
$888$		0			0
$889$		−	5085.84i		−	0.191871i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	9490.96	+	9490.96i	0.355658	+	0.355658i
$894$		0			0
$895$	−7009.27			−0.261781
$896$		0			0
$897$		0			0
$898$		0			0
$899$	12461.2	+	12461.2i	0.462296	+	0.462296i
$900$		0			0
$901$	−10843.0	+	10843.0i	−0.400924	+	0.400924i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	43071.7i		−	1.58205i
$906$		0			0
$907$	−5320.20	+	5320.20i	−0.194768	+	0.194768i	−0.797753	−	0.602985i	$-0.793978\pi$
$907$	−5320.20	+	5320.20i	−0.194768	+	0.194768i	0.602985	+	0.797753i	$0.293978\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−26016.0			−0.946155			−0.473077	−	0.881021i	$-0.656857\pi$
$911$	−26016.0			−0.946155			−0.473077	+	0.881021i	$0.656857\pi$
$912$		0			0
$913$	51000.2			1.84870
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2453.77	+	2453.77i	−0.0883650	+	0.0883650i
$918$		0			0
$919$		−	24082.1i		−	0.864413i	−0.901775	−	0.432206i	$-0.857735\pi$
$919$		−	24082.1i		−	0.864413i	0.901775	−	0.432206i	$-0.142265\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	16057.9	−	16057.9i	0.572646	−	0.572646i
$924$		0			0
$925$	868.385	+	868.385i	0.0308674	+	0.0308674i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−9324.93			−0.329323			−0.164661	−	0.986350i	$-0.552653\pi$
$929$	−9324.93			−0.329323			−0.164661	+	0.986350i	$0.552653\pi$
$930$		0			0
$931$	−6530.06	−	6530.06i	−0.229875	−	0.229875i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	53937.0i		−	1.88656i
$936$		0			0
$937$		−	15535.2i		−	0.541636i	−0.962631	−	0.270818i	$-0.912706\pi$
$937$		−	15535.2i		−	0.541636i	0.962631	−	0.270818i	$-0.0872941\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	34024.8	+	34024.8i	1.17872	+	1.17872i	0.980070	+	0.198651i	$0.0636559\pi$
$941$	34024.8	+	34024.8i	1.17872	+	1.17872i	0.198651	+	0.980070i	$0.436344\pi$
$942$		0			0
$943$	−14591.8			−0.503896
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−3816.02	−	3816.02i	−0.130944	−	0.130944i	0.638597	−	0.769541i	$-0.279515\pi$
$947$	−3816.02	−	3816.02i	−0.130944	−	0.130944i	−0.769541	+	0.638597i	$0.779515\pi$
$948$		0			0
$949$	9194.21	−	9194.21i	0.314496	−	0.314496i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			21759.7i			0.739629i	0.929106	+	0.369815i	$0.120579\pi$
$953$			21759.7i			0.739629i	−0.929106	+	0.369815i	$0.879421\pi$
$954$		0			0
$955$	−26457.6	+	26457.6i	−0.896489	+	0.896489i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1275.03			0.0429331
$960$		0			0
$961$	5809.78			0.195018
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−23464.9	+	23464.9i	−0.782759	+	0.782759i
$966$		0			0
$967$		−	19338.5i		−	0.643108i	−0.946891	−	0.321554i	$-0.895795\pi$
$967$		−	19338.5i		−	0.643108i	0.946891	−	0.321554i	$-0.104205\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	4184.92	−	4184.92i	0.138311	−	0.138311i	−0.634561	−	0.772873i	$-0.718819\pi$
$971$	4184.92	−	4184.92i	0.138311	−	0.138311i	0.772873	+	0.634561i	$0.218819\pi$
$972$		0			0
$973$	4044.67	+	4044.67i	0.133264	+	0.133264i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−17841.9			−0.584249			−0.292125	−	0.956380i	$-0.594362\pi$
$977$	−17841.9			−0.584249			−0.292125	+	0.956380i	$0.594362\pi$
$978$		0			0
$979$	39200.2	+	39200.2i	1.27972	+	1.27972i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			61512.0i			1.99586i	0.0643304	+	0.997929i	$0.479509\pi$
$983$			61512.0i			1.99586i	−0.0643304	+	0.997929i	$0.520491\pi$
$984$		0			0
$985$		−	26471.2i		−	0.856286i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4391.05	−	4391.05i	−0.141180	−	0.141180i
$990$		0			0
$991$	−1827.73			−0.0585870			−0.0292935	−	0.999571i	$-0.509326\pi$
$991$	−1827.73			−0.0585870			−0.0292935	+	0.999571i	$0.509326\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	6145.84	+	6145.84i	0.195815	+	0.195815i
$996$		0			0
$997$	−25017.0	+	25017.0i	−0.794681	+	0.794681i	−0.982251	−	0.187570i	$-0.939939\pi$
$997$	−25017.0	+	25017.0i	−0.794681	+	0.794681i	0.187570	+	0.982251i	$0.439939\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.4.k.a.145.2		10
3.2	odd	2		64.4.e.a.17.3		10
4.3	odd	2		144.4.k.a.109.3		10
12.11	even	2		16.4.e.a.13.3	yes	10
16.5	even	4	inner	576.4.k.a.433.2		10
16.11	odd	4		144.4.k.a.37.3		10
24.5	odd	2		128.4.e.a.33.3		10
24.11	even	2		128.4.e.b.33.3		10
48.5	odd	4		64.4.e.a.49.3		10
48.11	even	4		16.4.e.a.5.3	✓	10
48.29	odd	4		128.4.e.a.97.3		10
48.35	even	4		128.4.e.b.97.3		10
96.5	odd	8		1024.4.a.m.1.6		10
96.11	even	8		1024.4.a.n.1.6		10
96.29	odd	8		1024.4.b.k.513.5		10
96.35	even	8		1024.4.b.j.513.6		10
96.53	odd	8		1024.4.a.m.1.5		10
96.59	even	8		1024.4.a.n.1.5		10
96.77	odd	8		1024.4.b.k.513.6		10
96.83	even	8		1024.4.b.j.513.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.4.e.a.5.3	✓	10	48.11	even	4
16.4.e.a.13.3	yes	10	12.11	even	2
64.4.e.a.17.3		10	3.2	odd	2
64.4.e.a.49.3		10	48.5	odd	4
128.4.e.a.33.3		10	24.5	odd	2
128.4.e.a.97.3		10	48.29	odd	4
128.4.e.b.33.3		10	24.11	even	2
128.4.e.b.97.3		10	48.35	even	4
144.4.k.a.37.3		10	16.11	odd	4
144.4.k.a.109.3		10	4.3	odd	2
576.4.k.a.145.2		10	1.1	even	1	trivial
576.4.k.a.433.2		10	16.5	even	4	inner
1024.4.a.m.1.5		10	96.53	odd	8
1024.4.a.m.1.6		10	96.5	odd	8
1024.4.a.n.1.5		10	96.59	even	8
1024.4.a.n.1.6		10	96.11	even	8
1024.4.b.j.513.5		10	96.83	even	8
1024.4.b.j.513.6		10	96.35	even	8
1024.4.b.k.513.5		10	96.29	odd	8
1024.4.b.k.513.6		10	96.77	odd	8

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:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	576.k (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-8.22587 + 8.22587i) q^{5} +2.67171i q^{7} +O(q^{10})\)	\(q+(-8.22587 + 8.22587i) q^{5} +2.67171i q^{7} +(-45.2213 + 45.2213i) q^{11} +(35.3968 + 35.3968i) q^{13} +72.4991 q^{17} +(-19.4427 - 19.4427i) q^{19} -139.462i q^{23} -10.3299i q^{25} +(-66.0434 - 66.0434i) q^{29} -188.682 q^{31} +(-21.9771 - 21.9771i) q^{35} +(-84.0653 + 84.0653i) q^{37} -104.629i q^{41} +(31.4857 - 31.4857i) q^{43} -488.151 q^{47} +335.862 q^{49} +(-149.560 + 149.560i) q^{53} -743.968i q^{55} +(284.698 - 284.698i) q^{59} +(-228.069 - 228.069i) q^{61} -582.338 q^{65} +(-139.151 - 139.151i) q^{67} -453.655i q^{71} -259.747i q^{73} +(-120.818 - 120.818i) q^{77} -323.190 q^{79} +(-563.897 - 563.897i) q^{83} +(-596.368 + 596.368i) q^{85} -866.853i q^{89} +(-94.5697 + 94.5697i) q^{91} +319.866 q^{95} -936.077 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{5}+O(q^{10}) \) 10 * q + 2 * q^5	\( 10 q + 2 q^{5} + 18 q^{11} - 2 q^{13} + 4 q^{17} + 26 q^{19} + 202 q^{29} - 368 q^{31} + 476 q^{35} - 10 q^{37} + 838 q^{43} - 944 q^{47} + 94 q^{49} + 378 q^{53} + 1706 q^{59} + 910 q^{61} + 492 q^{65} - 1942 q^{67} + 268 q^{77} + 4416 q^{79} - 2562 q^{83} - 12 q^{85} - 3332 q^{91} + 6900 q^{95} - 4 q^{97}+O(q^{100}) \) 10 * q + 2 * q^5 + 18 * q^11 - 2 * q^13 + 4 * q^17 + 26 * q^19 + 202 * q^29 - 368 * q^31 + 476 * q^35 - 10 * q^37 + 838 * q^43 - 944 * q^47 + 94 * q^49 + 378 * q^53 + 1706 * q^59 + 910 * q^61 + 492 * q^65 - 1942 * q^67 + 268 * q^77 + 4416 * q^79 - 2562 * q^83 - 12 * q^85 - 3332 * q^91 + 6900 * q^95 - 4 * q^97

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