LMFDB - Embedded newform 384.4.f.c.191.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,Mod(191,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("384.191");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	384.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$22.6567334422$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	384.191
Dual form			384.4.f.c.191.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.24264 + 3.00000i) q^{3} +5.65685 q^{5} -16.9706i q^{7} +(9.00000 - 25.4558i) q^{9} +O(q^{10})$	\(q+(-4.24264 + 3.00000i) q^{3} +5.65685 q^{5} -16.9706i q^{7} +(9.00000 - 25.4558i) q^{9} -30.0000i q^{11} +72.0000i q^{13} +(-24.0000 + 16.9706i) q^{15} +50.9117i q^{17} -25.4558 q^{19} +(50.9117 + 72.0000i) q^{21} -144.000 q^{23} -93.0000 q^{25} +(38.1838 + 135.000i) q^{27} +5.65685 q^{29} -220.617i q^{31} +(90.0000 + 127.279i) q^{33} -96.0000i q^{35} -72.0000i q^{37} +(-216.000 - 305.470i) q^{39} -305.470i q^{41} +229.103 q^{43} +(50.9117 - 144.000i) q^{45} -576.000 q^{47} +55.0000 q^{49} +(-152.735 - 216.000i) q^{51} -514.774 q^{53} -169.706i q^{55} +(108.000 - 76.3675i) q^{57} +414.000i q^{59} -504.000i q^{61} +(-432.000 - 152.735i) q^{63} +407.294i q^{65} -789.131 q^{67} +(610.940 - 432.000i) q^{69} -720.000 q^{71} -178.000 q^{73} +(394.566 - 279.000i) q^{75} -509.117 q^{77} -967.322i q^{79} +(-567.000 - 458.205i) q^{81} -438.000i q^{83} +288.000i q^{85} +(-24.0000 + 16.9706i) q^{87} +865.499i q^{89} +1221.88 q^{91} +(661.852 + 936.000i) q^{93} -144.000 q^{95} +650.000 q^{97} +(-763.675 - 270.000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 36 q^{9}+O(q^{10}) $ 4 * q + 36 * q^9	$ 4 q + 36 q^{9} - 96 q^{15} - 576 q^{23} - 372 q^{25} + 360 q^{33} - 864 q^{39} - 2304 q^{47} + 220 q^{49} + 432 q^{57} - 1728 q^{63} - 2880 q^{71} - 712 q^{73} - 2268 q^{81} - 96 q^{87} - 576 q^{95} + 2600 q^{97}+O(q^{100}) $ 4 * q + 36 * q^9 - 96 * q^15 - 576 * q^23 - 372 * q^25 + 360 * q^33 - 864 * q^39 - 2304 * q^47 + 220 * q^49 + 432 * q^57 - 1728 * q^63 - 2880 * q^71 - 712 * q^73 - 2268 * q^81 - 96 * q^87 - 576 * q^95 + 2600 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−4.24264	+	3.00000i	−0.816497	+	0.577350i
$3$	−4.24264	+	3.00000i	−0.816497	+	0.577350i
$4$		0			0
$5$	5.65685			0.505964			0.252982	−	0.967471i	$-0.418589\pi$
$5$	5.65685			0.505964			0.252982	+	0.967471i	$0.418589\pi$
$6$		0			0
$7$		−	16.9706i		−	0.916324i	−0.888869	−	0.458162i	$-0.848508\pi$
$7$		−	16.9706i		−	0.916324i	0.888869	−	0.458162i	$-0.151492\pi$
$8$		0			0
$9$	9.00000	−	25.4558i	0.333333	−	0.942809i
$10$		0			0
$11$		−	30.0000i		−	0.822304i	−0.911567	−	0.411152i	$-0.865127\pi$
$11$		−	30.0000i		−	0.822304i	0.911567	−	0.411152i	$-0.134873\pi$
$12$		0			0
$13$			72.0000i			1.53609i	0.640394	+	0.768046i	$0.278771\pi$
$13$			72.0000i			1.53609i	−0.640394	+	0.768046i	$0.721229\pi$
$14$		0			0
$15$	−24.0000	+	16.9706i	−0.413118	+	0.292119i
$16$		0			0
$17$			50.9117i			0.726347i	0.931722	+	0.363173i	$0.118307\pi$
$17$			50.9117i			0.726347i	−0.931722	+	0.363173i	$0.881693\pi$
$18$		0			0
$19$	−25.4558			−0.307367			−0.153683	−	0.988120i	$-0.549114\pi$
$19$	−25.4558			−0.307367			−0.153683	+	0.988120i	$0.549114\pi$
$20$		0			0
$21$	50.9117	+	72.0000i	0.529040	+	0.748176i
$22$		0			0
$23$	−144.000			−1.30548			−0.652741	−	0.757581i	$-0.726381\pi$
$23$	−144.000			−1.30548			−0.652741	+	0.757581i	$0.726381\pi$
$24$		0			0
$25$	−93.0000			−0.744000
$26$		0			0
$27$	38.1838	+	135.000i	0.272166	+	0.962250i
$28$		0			0
$29$	5.65685			0.0362225			0.0181112	−	0.999836i	$-0.494235\pi$
$29$	5.65685			0.0362225			0.0181112	+	0.999836i	$0.494235\pi$
$30$		0			0
$31$		−	220.617i		−	1.27819i	−0.769126	−	0.639097i	$-0.779308\pi$
$31$		−	220.617i		−	1.27819i	0.769126	−	0.639097i	$-0.220692\pi$
$32$		0			0
$33$	90.0000	+	127.279i	0.474757	+	0.671408i
$34$		0			0
$35$		−	96.0000i		−	0.463627i
$36$		0			0
$37$		−	72.0000i		−	0.319912i	−0.987124	−	0.159956i	$-0.948865\pi$
$37$		−	72.0000i		−	0.319912i	0.987124	−	0.159956i	$-0.0511352\pi$
$38$		0			0
$39$	−216.000	−	305.470i	−0.886864	−	1.25421i
$40$		0			0
$41$		−	305.470i		−	1.16357i	−0.813342	−	0.581786i	$-0.802354\pi$
$41$		−	305.470i		−	1.16357i	0.813342	−	0.581786i	$-0.197646\pi$
$42$		0			0
$43$	229.103			0.812507			0.406254	−	0.913760i	$-0.366835\pi$
$43$	229.103			0.812507			0.406254	+	0.913760i	$0.366835\pi$
$44$		0			0
$45$	50.9117	−	144.000i	0.168655	−	0.477028i
$46$		0			0
$47$	−576.000			−1.78762			−0.893811	−	0.448444i	$-0.851978\pi$
$47$	−576.000			−1.78762			−0.893811	+	0.448444i	$0.851978\pi$
$48$		0			0
$49$	55.0000			0.160350
$50$		0			0
$51$	−152.735	−	216.000i	−0.419357	−	0.593060i
$52$		0			0
$53$	−514.774			−1.33414			−0.667072	−	0.744993i	$-0.732453\pi$
$53$	−514.774			−1.33414			−0.667072	+	0.744993i	$0.732453\pi$
$54$		0			0
$55$		−	169.706i		−	0.416056i
$56$		0			0
$57$	108.000	−	76.3675i	0.250964	−	0.177458i
$58$		0			0
$59$			414.000i			0.913529i	0.889588	+	0.456764i	$0.150992\pi$
$59$			414.000i			0.913529i	−0.889588	+	0.456764i	$0.849008\pi$
$60$		0			0
$61$		−	504.000i		−	1.05788i	−0.848660	−	0.528939i	$-0.822590\pi$
$61$		−	504.000i		−	1.05788i	0.848660	−	0.528939i	$-0.177410\pi$
$62$		0			0
$63$	−432.000	−	152.735i	−0.863919	−	0.305441i
$64$		0			0
$65$			407.294i			0.777208i
$66$		0			0
$67$	−789.131			−1.43892			−0.719461	−	0.694533i	$-0.755611\pi$
$67$	−789.131			−1.43892			−0.719461	+	0.694533i	$0.755611\pi$
$68$		0			0
$69$	610.940	−	432.000i	1.06592	−	0.753720i
$70$		0			0
$71$	−720.000			−1.20350			−0.601748	−	0.798686i	$-0.705529\pi$
$71$	−720.000			−1.20350			−0.601748	+	0.798686i	$0.705529\pi$
$72$		0			0
$73$	−178.000			−0.285388			−0.142694	−	0.989767i	$-0.545576\pi$
$73$	−178.000			−0.285388			−0.142694	+	0.989767i	$0.545576\pi$
$74$		0			0
$75$	394.566	−	279.000i	0.607473	−	0.429549i
$76$		0			0
$77$	−509.117			−0.753497
$78$		0			0
$79$		−	967.322i		−	1.37762i	−0.724940	−	0.688812i	$-0.758133\pi$
$79$		−	967.322i		−	1.37762i	0.724940	−	0.688812i	$-0.241867\pi$
$80$		0			0
$81$	−567.000	−	458.205i	−0.777778	−	0.628539i
$82$		0			0
$83$		−	438.000i		−	0.579238i	−0.957142	−	0.289619i	$-0.906471\pi$
$83$		−	438.000i		−	0.579238i	0.957142	−	0.289619i	$-0.0935286\pi$
$84$		0			0
$85$			288.000i			0.367506i
$86$		0			0
$87$	−24.0000	+	16.9706i	−0.0295755	+	0.0209130i
$88$		0			0
$89$			865.499i			1.03082i	0.856945	+	0.515408i	$0.172360\pi$
$89$			865.499i			1.03082i	−0.856945	+	0.515408i	$0.827640\pi$
$90$		0			0
$91$	1221.88			1.40756
$92$		0			0
$93$	661.852	+	936.000i	0.737966	+	1.04364i
$94$		0			0
$95$	−144.000			−0.155517
$96$		0			0
$97$	650.000			0.680387			0.340193	−	0.940356i	$-0.389507\pi$
$97$	650.000			0.680387			0.340193	+	0.940356i	$0.389507\pi$
$98$		0			0
$99$	−763.675	−	270.000i	−0.775275	−	0.274101i
$100$		0			0
$101$	−718.420			−0.707777			−0.353889	−	0.935288i	$-0.615141\pi$
$101$	−718.420			−0.707777			−0.353889	+	0.935288i	$0.615141\pi$
$102$		0			0
$103$		−	356.382i		−	0.340926i	−0.985364	−	0.170463i	$-0.945474\pi$
$103$		−	356.382i		−	0.340926i	0.985364	−	0.170463i	$-0.0545263\pi$
$104$		0			0
$105$	288.000	+	407.294i	0.267675	+	0.378550i
$106$		0			0
$107$			1662.00i			1.50160i	0.660527	+	0.750802i	$0.270333\pi$
$107$			1662.00i			1.50160i	−0.660527	+	0.750802i	$0.729667\pi$
$108$		0			0
$109$			1224.00i			1.07558i	0.843080	+	0.537789i	$0.180740\pi$
$109$			1224.00i			1.07558i	−0.843080	+	0.537789i	$0.819260\pi$
$110$		0			0
$111$	216.000	+	305.470i	0.184701	+	0.261207i
$112$		0			0
$113$		−	2341.94i		−	1.94965i	−0.222961	−	0.974827i	$-0.571572\pi$
$113$		−	2341.94i		−	1.94965i	0.222961	−	0.974827i	$-0.428428\pi$
$114$		0			0
$115$	−814.587			−0.660527
$116$		0			0
$117$	1832.82	+	648.000i	1.44824	+	0.512031i
$118$		0			0
$119$	864.000			0.665569
$120$		0			0
$121$	431.000			0.323817
$122$		0			0
$123$	916.410	+	1296.00i	0.671788	+	0.950052i
$124$		0			0
$125$	−1233.19			−0.882402
$126$		0			0
$127$			2630.44i			1.83790i	0.394372	+	0.918951i	$0.370962\pi$
$127$			2630.44i			1.83790i	−0.394372	+	0.918951i	$0.629038\pi$
$128$		0			0
$129$	−972.000	+	687.308i	−0.663410	+	0.469101i
$130$		0			0
$131$		−	2634.00i		−	1.75675i	−0.477976	−	0.878373i	$-0.658630\pi$
$131$		−	2634.00i		−	1.75675i	0.477976	−	0.878373i	$-0.341370\pi$
$132$		0			0
$133$			432.000i			0.281648i
$134$		0			0
$135$	216.000	+	763.675i	0.137706	+	0.486864i
$136$		0			0
$137$			203.647i			0.126998i	0.997982	+	0.0634990i	$0.0202260\pi$
$137$			203.647i			0.126998i	−0.997982	+	0.0634990i	$0.979774\pi$
$138$		0			0
$139$	992.778			0.605801			0.302900	−	0.953022i	$-0.402045\pi$
$139$	992.778			0.605801			0.302900	+	0.953022i	$0.402045\pi$
$140$		0			0
$141$	2443.76	−	1728.00i	1.45959	−	1.03208i
$142$		0			0
$143$	2160.00			1.26313
$144$		0			0
$145$	32.0000			0.0183273
$146$		0			0
$147$	−233.345	+	165.000i	−0.130925	+	0.0925780i
$148$		0			0
$149$	−2438.10			−1.34052			−0.670259	−	0.742127i	$-0.733817\pi$
$149$	−2438.10			−1.34052			−0.670259	+	0.742127i	$0.733817\pi$
$150$		0			0
$151$			729.734i			0.393278i	0.980476	+	0.196639i	$0.0630026\pi$
$151$			729.734i			0.393278i	−0.980476	+	0.196639i	$0.936997\pi$
$152$		0			0
$153$	1296.00	+	458.205i	0.684806	+	0.242116i
$154$		0			0
$155$		−	1248.00i		−	0.646721i
$156$		0			0
$157$		−	792.000i		−	0.402602i	−0.979529	−	0.201301i	$-0.935483\pi$
$157$		−	792.000i		−	0.402602i	0.979529	−	0.201301i	$-0.0645169\pi$
$158$		0			0
$159$	2184.00	−	1544.32i	1.08932	−	0.770268i
$160$		0			0
$161$			2443.76i			1.19624i
$162$		0			0
$163$	−2774.69			−1.33331			−0.666657	−	0.745364i	$-0.732276\pi$
$163$	−2774.69			−1.33331			−0.666657	+	0.745364i	$0.732276\pi$
$164$		0			0
$165$	509.117	+	720.000i	0.240210	+	0.339709i
$166$		0			0
$167$	2448.00			1.13432			0.567161	−	0.823607i	$-0.308042\pi$
$167$	2448.00			1.13432			0.567161	+	0.823607i	$0.308042\pi$
$168$		0			0
$169$	−2987.00			−1.35958
$170$		0			0
$171$	−229.103	+	648.000i	−0.102456	+	0.289788i
$172$		0			0
$173$	−3162.18			−1.38969			−0.694845	−	0.719160i	$-0.744527\pi$
$173$	−3162.18			−1.38969			−0.694845	+	0.719160i	$0.744527\pi$
$174$		0			0
$175$			1578.26i			0.681745i
$176$		0			0
$177$	−1242.00	−	1756.45i	−0.527426	−	0.745893i
$178$		0			0
$179$			2454.00i			1.02470i	0.858778	+	0.512348i	$0.171224\pi$
$179$			2454.00i			1.02470i	−0.858778	+	0.512348i	$0.828776\pi$
$180$		0			0
$181$		−	3240.00i		−	1.33054i	−0.746604	−	0.665269i	$-0.768317\pi$
$181$		−	3240.00i		−	1.33054i	0.746604	−	0.665269i	$-0.231683\pi$
$182$		0			0
$183$	1512.00	+	2138.29i	0.610766	+	0.863754i
$184$		0			0
$185$		−	407.294i		−	0.161864i
$186$		0			0
$187$	1527.35			0.597278
$188$		0			0
$189$	2291.03	−	648.000i	0.881733	−	0.249392i
$190$		0			0
$191$	2016.00			0.763731			0.381866	−	0.924218i	$-0.375282\pi$
$191$	2016.00			0.763731			0.381866	+	0.924218i	$0.375282\pi$
$192$		0			0
$193$	3382.00			1.26136			0.630678	−	0.776045i	$-0.282777\pi$
$193$	3382.00			1.26136			0.630678	+	0.776045i	$0.282777\pi$
$194$		0			0
$195$	−1221.88	−	1728.00i	−0.448721	−	0.634588i
$196$		0			0
$197$	−1736.65			−0.628079			−0.314039	−	0.949410i	$-0.601682\pi$
$197$	−1736.65			−0.628079			−0.314039	+	0.949410i	$0.601682\pi$
$198$		0			0
$199$			2358.91i			0.840294i	0.907456	+	0.420147i	$0.138022\pi$
$199$			2358.91i			0.840294i	−0.907456	+	0.420147i	$0.861978\pi$
$200$		0			0
$201$	3348.00	−	2367.39i	1.17487	−	0.830762i
$202$		0			0
$203$		−	96.0000i		−	0.0331915i
$204$		0			0
$205$		−	1728.00i		−	0.588726i
$206$		0			0
$207$	−1296.00	+	3665.64i	−0.435161	+	1.23082i
$208$		0			0
$209$			763.675i			0.252749i
$210$		0			0
$211$	3589.27			1.17107			0.585535	−	0.810647i	$-0.300884\pi$
$211$	3589.27			1.17107			0.585535	+	0.810647i	$0.300884\pi$
$212$		0			0
$213$	3054.70	−	2160.00i	0.982651	−	0.694839i
$214$		0			0
$215$	1296.00			0.411100
$216$		0			0
$217$	−3744.00			−1.17124
$218$		0			0
$219$	755.190	−	534.000i	0.233018	−	0.164769i
$220$		0			0
$221$	−3665.64			−1.11574
$222$		0			0
$223$		−	967.322i		−	0.290478i	−0.989397	−	0.145239i	$-0.953605\pi$
$223$		−	967.322i		−	0.290478i	0.989397	−	0.145239i	$-0.0463952\pi$
$224$		0			0
$225$	−837.000	+	2367.39i	−0.248000	+	0.701450i
$226$		0			0
$227$		−	198.000i		−	0.0578930i	−0.999581	−	0.0289465i	$-0.990785\pi$
$227$		−	198.000i		−	0.0578930i	0.999581	−	0.0289465i	$-0.00921525\pi$
$228$		0			0
$229$			2232.00i			0.644082i	0.946726	+	0.322041i	$0.104369\pi$
$229$			2232.00i			0.644082i	−0.946726	+	0.322041i	$0.895631\pi$
$230$		0			0
$231$	2160.00	−	1527.35i	0.615228	−	0.435032i
$232$		0			0
$233$		−	4022.02i		−	1.13086i	−0.824795	−	0.565432i	$-0.808709\pi$
$233$		−	4022.02i		−	1.13086i	0.824795	−	0.565432i	$-0.191291\pi$
$234$		0			0
$235$	−3258.35			−0.904473
$236$		0			0
$237$	2901.97	+	4104.00i	0.795371	+	1.12482i
$238$		0			0
$239$	−576.000			−0.155893			−0.0779463	−	0.996958i	$-0.524836\pi$
$239$	−576.000			−0.155893			−0.0779463	+	0.996958i	$0.524836\pi$
$240$		0			0
$241$	2342.00			0.625981			0.312991	−	0.949756i	$-0.398669\pi$
$241$	2342.00			0.625981			0.312991	+	0.949756i	$0.398669\pi$
$242$		0			0
$243$	3780.19	+	243.000i	0.997940	+	0.0641500i
$244$		0			0
$245$	311.127			0.0811313
$246$		0			0
$247$		−	1832.82i		−	0.472144i
$248$		0			0
$249$	1314.00	+	1858.28i	0.334423	+	0.472946i
$250$		0			0
$251$			4482.00i			1.12710i	0.826083	+	0.563548i	$0.190564\pi$
$251$			4482.00i			1.12710i	−0.826083	+	0.563548i	$0.809436\pi$
$252$		0			0
$253$			4320.00i			1.07350i
$254$		0			0
$255$	−864.000	−	1221.88i	−0.212180	−	0.300067i
$256$		0			0
$257$			1731.00i			0.420143i	0.977686	+	0.210071i	$0.0673696\pi$
$257$			1731.00i			0.420143i	−0.977686	+	0.210071i	$0.932630\pi$
$258$		0			0
$259$	−1221.88			−0.293143
$260$		0			0
$261$	50.9117	−	144.000i	0.0120742	−	0.0341509i
$262$		0			0
$263$	−1296.00			−0.303858			−0.151929	−	0.988391i	$-0.548549\pi$
$263$	−1296.00			−0.303858			−0.151929	+	0.988391i	$0.548549\pi$
$264$		0			0
$265$	−2912.00			−0.675029
$266$		0			0
$267$	−2596.50	−	3672.00i	−0.595142	−	0.841658i
$268$		0			0
$269$	−197.990			−0.0448760			−0.0224380	−	0.999748i	$-0.507143\pi$
$269$	−197.990			−0.0448760			−0.0224380	+	0.999748i	$0.507143\pi$
$270$		0			0
$271$		−	4361.43i		−	0.977632i	−0.872387	−	0.488816i	$-0.837429\pi$
$271$		−	4361.43i		−	0.977632i	0.872387	−	0.488816i	$-0.162571\pi$
$272$		0			0
$273$	−5184.00	+	3665.64i	−1.14927	+	0.812655i
$274$		0			0
$275$			2790.00i			0.611794i
$276$		0			0
$277$			7128.00i			1.54614i	0.634322	+	0.773069i	$0.281279\pi$
$277$			7128.00i			1.54614i	−0.634322	+	0.773069i	$0.718721\pi$
$278$		0			0
$279$	−5616.00	−	1985.56i	−1.20509	−	0.426065i
$280$		0			0
$281$		−	254.558i		−	0.0540416i	−0.999635	−	0.0270208i	$-0.991398\pi$
$281$		−	254.558i		−	0.0540416i	0.999635	−	0.0270208i	$-0.00860203\pi$
$282$		0			0
$283$	−2571.04			−0.540044			−0.270022	−	0.962854i	$-0.587031\pi$
$283$	−2571.04			−0.540044			−0.270022	+	0.962854i	$0.587031\pi$
$284$		0			0
$285$	610.940	−	432.000i	0.126979	−	0.0897876i
$286$		0			0
$287$	−5184.00			−1.06621
$288$		0			0
$289$	2321.00			0.472420
$290$		0			0
$291$	−2757.72	+	1950.00i	−0.555533	+	0.392821i
$292$		0			0
$293$	707.107			0.140988			0.0704942	−	0.997512i	$-0.477542\pi$
$293$	707.107			0.140988			0.0704942	+	0.997512i	$0.477542\pi$
$294$		0			0
$295$			2341.94i			0.462213i
$296$		0			0
$297$	4050.00	−	1145.51i	0.791262	−	0.223803i
$298$		0			0
$299$		−	10368.0i		−	2.00534i
$300$		0			0
$301$		−	3888.00i		−	0.744520i
$302$		0			0
$303$	3048.00	−	2155.26i	0.577898	−	0.408635i
$304$		0			0
$305$		−	2851.05i		−	0.535249i
$306$		0			0
$307$	10207.8			1.89769			0.948843	−	0.315749i	$-0.102256\pi$
$307$	10207.8			1.89769			0.948843	+	0.315749i	$0.102256\pi$
$308$		0			0
$309$	1069.15	+	1512.00i	0.196834	+	0.278365i
$310$		0			0
$311$	−7920.00			−1.44406			−0.722029	−	0.691863i	$-0.756790\pi$
$311$	−7920.00			−1.44406			−0.722029	+	0.691863i	$0.756790\pi$
$312$		0			0
$313$	−538.000			−0.0971551			−0.0485776	−	0.998819i	$-0.515469\pi$
$313$	−538.000			−0.0971551			−0.0485776	+	0.998819i	$0.515469\pi$
$314$		0			0
$315$	−2443.76	−	864.000i	−0.437112	−	0.154542i
$316$		0			0
$317$	4689.53			0.830884			0.415442	−	0.909620i	$-0.363627\pi$
$317$	4689.53			0.830884			0.415442	+	0.909620i	$0.363627\pi$
$318$		0			0
$319$		−	169.706i		−	0.0297859i
$320$		0			0
$321$	−4986.00	−	7051.27i	−0.866951	−	1.22605i
$322$		0			0
$323$		−	1296.00i		−	0.223255i
$324$		0			0
$325$		−	6696.00i		−	1.14285i
$326$		0			0
$327$	−3672.00	−	5192.99i	−0.620985	−	0.878205i
$328$		0			0
$329$			9775.04i			1.63804i
$330$		0			0
$331$	−7051.27			−1.17091			−0.585457	−	0.810703i	$-0.699085\pi$
$331$	−7051.27			−1.17091			−0.585457	+	0.810703i	$0.699085\pi$
$332$		0			0
$333$	−1832.82	−	648.000i	−0.301615	−	0.106637i
$334$		0			0
$335$	−4464.00			−0.728043
$336$		0			0
$337$	11086.0			1.79197			0.895984	−	0.444087i	$-0.146472\pi$
$337$	11086.0			1.79197			0.895984	+	0.444087i	$0.146472\pi$
$338$		0			0
$339$	7025.81	+	9936.00i	1.12563	+	1.59189i
$340$		0			0
$341$	−6618.52			−1.05106
$342$		0			0
$343$		−	6754.28i		−	1.06326i
$344$		0			0
$345$	3456.00	−	2443.76i	0.539318	−	0.381356i
$346$		0			0
$347$		−	1086.00i		−	0.168010i	−0.996465	−	0.0840051i	$-0.973229\pi$
$347$		−	1086.00i		−	0.168010i	0.996465	−	0.0840051i	$-0.0267712\pi$
$348$		0			0
$349$		−	1080.00i		−	0.165648i	−0.996564	−	0.0828239i	$-0.973606\pi$
$349$		−	1080.00i		−	0.165648i	0.996564	−	0.0828239i	$-0.0263939\pi$
$350$		0			0
$351$	−9720.00	+	2749.23i	−1.47811	+	0.418072i
$352$		0			0
$353$		−	6720.34i		−	1.01328i	−0.862158	−	0.506640i	$-0.830887\pi$
$353$		−	6720.34i		−	1.01328i	0.862158	−	0.506640i	$-0.169113\pi$
$354$		0			0
$355$	−4072.94			−0.608927
$356$		0			0
$357$	−3665.64	+	2592.00i	−0.543435	+	0.384267i
$358$		0			0
$359$	−4176.00			−0.613930			−0.306965	−	0.951721i	$-0.599313\pi$
$359$	−4176.00			−0.613930			−0.306965	+	0.951721i	$0.599313\pi$
$360$		0			0
$361$	−6211.00			−0.905526
$362$		0			0
$363$	−1828.58	+	1293.00i	−0.264395	+	0.186956i
$364$		0			0
$365$	−1006.92			−0.144396
$366$		0			0
$367$		−	6737.31i		−	0.958269i	−0.877741	−	0.479135i	$-0.840951\pi$
$367$		−	6737.31i		−	0.958269i	0.877741	−	0.479135i	$-0.159049\pi$
$368$		0			0
$369$	−7776.00	−	2749.23i	−1.09703	−	0.387857i
$370$		0			0
$371$			8736.00i			1.22251i
$372$		0			0
$373$		−	5832.00i		−	0.809570i	−0.914412	−	0.404785i	$-0.867346\pi$
$373$		−	5832.00i		−	0.809570i	0.914412	−	0.404785i	$-0.132654\pi$
$374$		0			0
$375$	5232.00	−	3699.58i	0.720478	−	0.509455i
$376$		0			0
$377$			407.294i			0.0556411i
$378$		0			0
$379$	636.396			0.0862519			0.0431260	−	0.999070i	$-0.486268\pi$
$379$	636.396			0.0862519			0.0431260	+	0.999070i	$0.486268\pi$
$380$		0			0
$381$	−7891.31	−	11160.0i	−1.06111	−	1.50064i
$382$		0			0
$383$	288.000			0.0384233			0.0192116	−	0.999815i	$-0.493884\pi$
$383$	288.000			0.0384233			0.0192116	+	0.999815i	$0.493884\pi$
$384$		0			0
$385$	−2880.00			−0.381243
$386$		0			0
$387$	2061.92	−	5832.00i	0.270836	−	0.766039i
$388$		0			0
$389$	10798.9			1.40753			0.703763	−	0.710435i	$-0.251502\pi$
$389$	10798.9			1.40753			0.703763	+	0.710435i	$0.251502\pi$
$390$		0			0
$391$		−	7331.28i		−	0.948233i
$392$		0			0
$393$	7902.00	+	11175.1i	1.01426	+	1.43438i
$394$		0			0
$395$		−	5472.00i		−	0.697028i
$396$		0			0
$397$			2952.00i			0.373191i	0.982437	+	0.186595i	$0.0597453\pi$
$397$			2952.00i			0.373191i	−0.982437	+	0.186595i	$0.940255\pi$
$398$		0			0
$399$	−1296.00	−	1832.82i	−0.162609	−	0.229964i
$400$		0			0
$401$			8502.25i			1.05881i	0.848370	+	0.529404i	$0.177585\pi$
$401$			8502.25i			1.05881i	−0.848370	+	0.529404i	$0.822415\pi$
$402$		0			0
$403$	15884.4			1.96343
$404$		0			0
$405$	−3207.44	−	2592.00i	−0.393528	−	0.318019i
$406$		0			0
$407$	−2160.00			−0.263064
$408$		0			0
$409$	−614.000			−0.0742307			−0.0371153	−	0.999311i	$-0.511817\pi$
$409$	−614.000			−0.0742307			−0.0371153	+	0.999311i	$0.511817\pi$
$410$		0			0
$411$	−610.940	−	864.000i	−0.0733223	−	0.103693i
$412$		0			0
$413$	7025.81			0.837089
$414$		0			0
$415$		−	2477.70i		−	0.293074i
$416$		0			0
$417$	−4212.00	+	2978.33i	−0.494634	+	0.349759i
$418$		0			0
$419$		−	4950.00i		−	0.577144i	−0.957458	−	0.288572i	$-0.906820\pi$
$419$		−	4950.00i		−	0.577144i	0.957458	−	0.288572i	$-0.0931805\pi$
$420$		0			0
$421$		−	5256.00i		−	0.608460i	−0.952599	−	0.304230i	$-0.901601\pi$
$421$		−	5256.00i		−	0.608460i	0.952599	−	0.304230i	$-0.0983992\pi$
$422$		0			0
$423$	−5184.00	+	14662.6i	−0.595874	+	1.68539i
$424$		0			0
$425$		−	4734.79i		−	0.540402i
$426$		0			0
$427$	−8553.16			−0.969360
$428$		0			0
$429$	−9164.10	+	6480.00i	−1.03135	+	0.729271i
$430$		0			0
$431$	−2880.00			−0.321867			−0.160934	−	0.986965i	$-0.551450\pi$
$431$	−2880.00			−0.321867			−0.160934	+	0.986965i	$0.551450\pi$
$432$		0			0
$433$	2554.00			0.283458			0.141729	−	0.989905i	$-0.454734\pi$
$433$	2554.00			0.283458			0.141729	+	0.989905i	$0.454734\pi$
$434$		0			0
$435$	−135.765	+	96.0000i	−0.0149642	+	0.0105813i
$436$		0			0
$437$	3665.64			0.401262
$438$		0			0
$439$			10504.8i			1.14206i	0.820928	+	0.571032i	$0.193457\pi$
$439$			10504.8i			1.14206i	−0.820928	+	0.571032i	$0.806543\pi$
$440$		0			0
$441$	495.000	−	1400.07i	0.0534500	−	0.151179i
$442$		0			0
$443$			3186.00i			0.341696i	0.985297	+	0.170848i	$0.0546507\pi$
$443$			3186.00i			0.341696i	−0.985297	+	0.170848i	$0.945349\pi$
$444$		0			0
$445$			4896.00i			0.521557i
$446$		0			0
$447$	10344.0	−	7314.31i	1.09453	−	0.773949i
$448$		0			0
$449$		−	12982.5i		−	1.36455i	−0.731098	−	0.682273i	$-0.760992\pi$
$449$		−	12982.5i		−	1.36455i	0.731098	−	0.682273i	$-0.239008\pi$
$450$		0			0
$451$	−9164.10			−0.956809
$452$		0			0
$453$	−2189.20	−	3096.00i	−0.227059	−	0.321110i
$454$		0			0
$455$	6912.00			0.712175
$456$		0			0
$457$	−8534.00			−0.873531			−0.436766	−	0.899575i	$-0.643876\pi$
$457$	−8534.00			−0.873531			−0.436766	+	0.899575i	$0.643876\pi$
$458$		0			0
$459$	−6873.08	+	1944.00i	−0.698928	+	0.197687i
$460$		0			0
$461$	12722.3			1.28533			0.642663	−	0.766149i	$-0.277830\pi$
$461$	12722.3			1.28533			0.642663	+	0.766149i	$0.277830\pi$
$462$		0			0
$463$			16614.2i			1.66766i	0.552021	+	0.833830i	$0.313857\pi$
$463$			16614.2i			1.66766i	−0.552021	+	0.833830i	$0.686143\pi$
$464$		0			0
$465$	3744.00	+	5294.82i	0.373385	+	0.528046i
$466$		0			0
$467$			12378.0i			1.22652i	0.789881	+	0.613261i	$0.210143\pi$
$467$			12378.0i			1.22652i	−0.789881	+	0.613261i	$0.789857\pi$
$468$		0			0
$469$			13392.0i			1.31852i
$470$		0			0
$471$	2376.00	+	3360.17i	0.232442	+	0.328723i
$472$		0			0
$473$		−	6873.08i		−	0.668128i
$474$		0			0
$475$	2367.39			0.228681
$476$		0			0
$477$	−4632.96	+	13104.0i	−0.444715	+	1.25784i
$478$		0			0
$479$	5184.00			0.494495			0.247247	−	0.968952i	$-0.420474\pi$
$479$	5184.00			0.494495			0.247247	+	0.968952i	$0.420474\pi$
$480$		0			0
$481$	5184.00			0.491414
$482$		0			0
$483$	−7331.28	−	10368.0i	−0.690652	−	0.976729i
$484$		0			0
$485$	3676.96			0.344251
$486$		0			0
$487$		−	6262.14i		−	0.582679i	−0.956620	−	0.291339i	$-0.905899\pi$
$487$		−	6262.14i		−	0.582679i	0.956620	−	0.291339i	$-0.0941009\pi$
$488$		0			0
$489$	11772.0	−	8324.06i	1.08865	−	0.769789i
$490$		0			0
$491$		−	114.000i		−	0.0104781i	−0.999986	−	0.00523905i	$-0.998332\pi$
$491$		−	114.000i		−	0.0104781i	0.999986	−	0.00523905i	$-0.00166765\pi$
$492$		0			0
$493$			288.000i			0.0263101i
$494$		0			0
$495$	−4320.00	−	1527.35i	−0.392262	−	0.138685i
$496$		0			0
$497$			12218.8i			1.10279i
$498$		0			0
$499$	4607.51			0.413347			0.206674	−	0.978410i	$-0.433736\pi$
$499$	4607.51			0.413347			0.206674	+	0.978410i	$0.433736\pi$
$500$		0			0
$501$	−10386.0	+	7344.00i	−0.926171	+	0.654902i
$502$		0			0
$503$	19728.0			1.74876			0.874382	−	0.485239i	$-0.161267\pi$
$503$	19728.0			1.74876			0.874382	+	0.485239i	$0.161267\pi$
$504$		0			0
$505$	−4064.00			−0.358110
$506$		0			0
$507$	12672.8	−	8961.00i	1.11009	−	0.784955i
$508$		0			0
$509$	6115.06			0.532505			0.266253	−	0.963903i	$-0.414214\pi$
$509$	6115.06			0.532505			0.266253	+	0.963903i	$0.414214\pi$
$510$		0			0
$511$			3020.76i			0.261508i
$512$		0			0
$513$	−972.000	−	3436.54i	−0.0836547	−	0.295764i
$514$		0			0
$515$		−	2016.00i		−	0.172496i
$516$		0			0
$517$			17280.0i			1.46997i
$518$		0			0
$519$	13416.0	−	9486.54i	1.13468	−	0.802337i
$520$		0			0
$521$			7331.28i			0.616486i	0.951308	+	0.308243i	$0.0997410\pi$
$521$			7331.28i			0.616486i	−0.951308	+	0.308243i	$0.900259\pi$
$522$		0			0
$523$	941.866			0.0787475			0.0393737	−	0.999225i	$-0.487464\pi$
$523$	941.866			0.0787475			0.0393737	+	0.999225i	$0.487464\pi$
$524$		0			0
$525$	−4734.79	−	6696.00i	−0.393606	−	0.556643i
$526$		0			0
$527$	11232.0			0.928413
$528$		0			0
$529$	8569.00			0.704282
$530$		0			0
$531$	10538.7	+	3726.00i	0.861283	+	0.304510i
$532$		0			0
$533$	21993.8			1.78735
$534$		0			0
$535$			9401.69i			0.759758i
$536$		0			0
$537$	−7362.00	−	10411.4i	−0.591608	−	0.836661i
$538$		0			0
$539$		−	1650.00i		−	0.131856i
$540$		0			0
$541$		−	17208.0i		−	1.36752i	−0.729706	−	0.683761i	$-0.760343\pi$
$541$		−	17208.0i		−	1.36752i	0.729706	−	0.683761i	$-0.239657\pi$
$542$		0			0
$543$	9720.00	+	13746.2i	0.768186	+	1.08638i
$544$		0			0
$545$			6923.99i			0.544204i
$546$		0			0
$547$	−18353.7			−1.43464			−0.717318	−	0.696746i	$-0.754631\pi$
$547$	−18353.7			−1.43464			−0.717318	+	0.696746i	$0.754631\pi$
$548$		0			0
$549$	−12829.7	−	4536.00i	−0.997377	−	0.352626i
$550$		0			0
$551$	−144.000			−0.0111336
$552$		0			0
$553$	−16416.0			−1.26235
$554$		0			0
$555$	1221.88	+	1728.00i	0.0934521	+	0.132161i
$556$		0			0
$557$	11704.0			0.890333			0.445167	−	0.895448i	$-0.353144\pi$
$557$	11704.0			0.890333			0.445167	+	0.895448i	$0.353144\pi$
$558$		0			0
$559$			16495.4i			1.24809i
$560$		0			0
$561$	−6480.00	+	4582.05i	−0.487675	+	0.344838i
$562$		0			0
$563$		−	102.000i		−	0.00763550i	−0.999993	−	0.00381775i	$-0.998785\pi$
$563$		−	102.000i		−	0.00763550i	0.999993	−	0.00381775i	$-0.00121523\pi$
$564$		0			0
$565$		−	13248.0i		−	0.986456i
$566$		0			0
$567$	−7776.00	+	9622.31i	−0.575946	+	0.712697i
$568$		0			0
$569$			1018.23i			0.0750204i	0.999296	+	0.0375102i	$0.0119427\pi$
$569$			1018.23i			0.0750204i	−0.999296	+	0.0375102i	$0.988057\pi$
$570$		0			0
$571$	22477.5			1.64738			0.823690	−	0.567040i	$-0.191911\pi$
$571$	22477.5			1.64738			0.823690	+	0.567040i	$0.191911\pi$
$572$		0			0
$573$	−8553.16	+	6048.00i	−0.623584	+	0.440940i
$574$		0			0
$575$	13392.0			0.971278
$576$		0			0
$577$	−3778.00			−0.272583			−0.136291	−	0.990669i	$-0.543518\pi$
$577$	−3778.00			−0.272583			−0.136291	+	0.990669i	$0.543518\pi$
$578$		0			0
$579$	−14348.6	+	10146.0i	−1.02989	+	0.728244i
$580$		0			0
$581$	−7433.11			−0.530770
$582$		0			0
$583$			15443.2i			1.09707i
$584$		0			0
$585$	10368.0	+	3665.64i	0.732759	+	0.259069i
$586$		0			0
$587$			18366.0i			1.29139i	0.763595	+	0.645695i	$0.223432\pi$
$587$			18366.0i			1.29139i	−0.763595	+	0.645695i	$0.776568\pi$
$588$		0			0
$589$			5616.00i			0.392875i
$590$		0			0
$591$	7368.00	−	5209.96i	0.512824	−	0.362621i
$592$		0			0
$593$			14255.3i			0.987174i	0.869697	+	0.493587i	$0.164314\pi$
$593$			14255.3i			0.987174i	−0.869697	+	0.493587i	$0.835686\pi$
$594$		0			0
$595$	4887.52			0.336754
$596$		0			0
$597$	−7076.72	−	10008.0i	−0.485144	−	0.686097i
$598$		0			0
$599$	1584.00			0.108048			0.0540238	−	0.998540i	$-0.482795\pi$
$599$	1584.00			0.108048			0.0540238	+	0.998540i	$0.482795\pi$
$600$		0			0
$601$	11230.0			0.762198			0.381099	−	0.924534i	$-0.375546\pi$
$601$	11230.0			0.762198			0.381099	+	0.924534i	$0.375546\pi$
$602$		0			0
$603$	−7102.18	+	20088.0i	−0.479640	+	1.35663i
$604$		0			0
$605$	2438.10			0.163840
$606$		0			0
$607$		−	8094.96i		−	0.541292i	−0.962679	−	0.270646i	$-0.912763\pi$
$607$		−	8094.96i		−	0.541292i	0.962679	−	0.270646i	$-0.0872373\pi$
$608$		0			0
$609$	288.000	+	407.294i	0.0191631	+	0.0271008i
$610$		0			0
$611$		−	41472.0i		−	2.74595i
$612$		0			0
$613$		−	1512.00i		−	0.0996233i	−0.998759	−	0.0498117i	$-0.984138\pi$
$613$		−	1512.00i		−	0.0996233i	0.998759	−	0.0498117i	$-0.0158621\pi$
$614$		0			0
$615$	5184.00	+	7331.28i	0.339901	+	0.480692i
$616$		0			0
$617$		−	25710.4i		−	1.67757i	−0.544461	−	0.838786i	$-0.683266\pi$
$617$		−	25710.4i		−	1.67757i	0.544461	−	0.838786i	$-0.316734\pi$
$618$		0			0
$619$	4963.89			0.322319			0.161160	−	0.986928i	$-0.448477\pi$
$619$	4963.89			0.322319			0.161160	+	0.986928i	$0.448477\pi$
$620$		0			0
$621$	−5498.46	−	19440.0i	−0.355307	−	1.25620i
$622$		0			0
$623$	14688.0			0.944562
$624$		0			0
$625$	4649.00			0.297536
$626$		0			0
$627$	−2291.03	−	3240.00i	−0.145925	−	0.206369i
$628$		0			0
$629$	3665.64			0.232367
$630$		0			0
$631$		−	5311.79i		−	0.335117i	−0.985862	−	0.167559i	$-0.946412\pi$
$631$		−	5311.79i		−	0.335117i	0.985862	−	0.167559i	$-0.0535883\pi$
$632$		0			0
$633$	−15228.0	+	10767.8i	−0.956175	+	0.676118i
$634$		0			0
$635$			14880.0i			0.929913i
$636$		0			0
$637$			3960.00i			0.246312i
$638$		0			0
$639$	−6480.00	+	18328.2i	−0.401166	+	1.13467i
$640$		0			0
$641$		−	11658.8i		−	0.718399i	−0.933261	−	0.359200i	$-0.883050\pi$
$641$		−	11658.8i		−	0.718399i	0.933261	−	0.359200i	$-0.116950\pi$
$642$		0			0
$643$	−15909.9			−0.975778			−0.487889	−	0.872906i	$-0.662233\pi$
$643$	−15909.9			−0.975778			−0.487889	+	0.872906i	$0.662233\pi$
$644$		0			0
$645$	−5498.46	+	3888.00i	−0.335662	+	0.237349i
$646$		0			0
$647$	−10224.0			−0.621247			−0.310624	−	0.950533i	$-0.600538\pi$
$647$	−10224.0			−0.621247			−0.310624	+	0.950533i	$0.600538\pi$
$648$		0			0
$649$	12420.0			0.751198
$650$		0			0
$651$	15884.4	−	11232.0i	0.956314	−	0.676216i
$652$		0			0
$653$	−18639.3			−1.11702			−0.558510	−	0.829498i	$-0.688627\pi$
$653$	−18639.3			−1.11702			−0.558510	+	0.829498i	$0.688627\pi$
$654$		0			0
$655$		−	14900.2i		−	0.888851i
$656$		0			0
$657$	−1602.00	+	4531.14i	−0.0951293	+	0.269066i
$658$		0			0
$659$		−	8490.00i		−	0.501857i	−0.968006	−	0.250928i	$-0.919264\pi$
$659$		−	8490.00i		−	0.501857i	0.968006	−	0.250928i	$-0.0807358\pi$
$660$		0			0
$661$		−	32904.0i		−	1.93618i	−0.250596	−	0.968092i	$-0.580627\pi$
$661$		−	32904.0i		−	1.93618i	0.250596	−	0.968092i	$-0.419373\pi$
$662$		0			0
$663$	15552.0	−	10996.9i	0.910995	−	0.644171i
$664$		0			0
$665$			2443.76i			0.142504i
$666$		0			0
$667$	−814.587			−0.0472878
$668$		0			0
$669$	2901.97	+	4104.00i	0.167708	+	0.237175i
$670$		0			0
$671$	−15120.0			−0.869897
$672$		0			0
$673$	−4214.00			−0.241364			−0.120682	−	0.992691i	$-0.538508\pi$
$673$	−4214.00			−0.241364			−0.120682	+	0.992691i	$0.538508\pi$
$674$		0			0
$675$	−3551.09	−	12555.0i	−0.202491	−	0.715914i
$676$		0			0
$677$	25665.1			1.45700			0.728502	−	0.685044i	$-0.240217\pi$
$677$	25665.1			1.45700			0.728502	+	0.685044i	$0.240217\pi$
$678$		0			0
$679$		−	11030.9i		−	0.623455i
$680$		0			0
$681$	594.000	+	840.043i	0.0334246	+	0.0472695i
$682$		0			0
$683$		−	16110.0i		−	0.902536i	−0.892389	−	0.451268i	$-0.850972\pi$
$683$		−	16110.0i		−	0.902536i	0.892389	−	0.451268i	$-0.149028\pi$
$684$		0			0
$685$			1152.00i			0.0642564i
$686$		0			0
$687$	−6696.00	−	9469.57i	−0.371861	−	0.525891i
$688$		0			0
$689$		−	37063.7i		−	2.04937i
$690$		0			0
$691$	−17844.5			−0.982400			−0.491200	−	0.871047i	$-0.663442\pi$
$691$	−17844.5			−0.982400			−0.491200	+	0.871047i	$0.663442\pi$
$692$		0			0
$693$	−4582.05	+	12960.0i	−0.251166	+	0.710404i
$694$		0			0
$695$	5616.00			0.306514
$696$		0			0
$697$	15552.0			0.845156
$698$		0			0
$699$	12066.1	+	17064.0i	0.652905	+	0.923347i
$700$		0			0
$701$	−3456.34			−0.186226			−0.0931128	−	0.995656i	$-0.529682\pi$
$701$	−3456.34			−0.186226			−0.0931128	+	0.995656i	$0.529682\pi$
$702$		0			0
$703$			1832.82i			0.0983302i
$704$		0			0
$705$	13824.0	−	9775.04i	0.738499	−	0.522198i
$706$		0			0
$707$			12192.0i			0.648554i
$708$		0			0
$709$		−	1224.00i		−	0.0648354i	−0.999474	−	0.0324177i	$-0.989679\pi$
$709$		−	1224.00i		−	0.0648354i	0.999474	−	0.0324177i	$-0.0103207\pi$
$710$		0			0
$711$	−24624.0	−	8705.90i	−1.29884	−	0.459208i
$712$		0			0
$713$			31768.9i			1.66866i
$714$		0			0
$715$	12218.8			0.639101
$716$		0			0
$717$	2443.76	−	1728.00i	0.127286	−	0.0900047i
$718$		0			0
$719$	−19872.0			−1.03074			−0.515369	−	0.856968i	$-0.672345\pi$
$719$	−19872.0			−1.03074			−0.515369	+	0.856968i	$0.672345\pi$
$720$		0			0
$721$	−6048.00			−0.312398
$722$		0			0
$723$	−9936.26	+	7026.00i	−0.511112	+	0.361410i
$724$		0			0
$725$	−526.087			−0.0269495
$726$		0			0
$727$			2223.14i			0.113414i	0.998391	+	0.0567069i	$0.0180601\pi$
$727$			2223.14i			0.113414i	−0.998391	+	0.0567069i	$0.981940\pi$
$728$		0			0
$729$	−16767.0	+	10309.6i	−0.851852	+	0.523783i
$730$		0			0
$731$			11664.0i			0.590162i
$732$		0			0
$733$			9288.00i			0.468022i	0.972234	+	0.234011i	$0.0751852\pi$
$733$			9288.00i			0.468022i	−0.972234	+	0.234011i	$0.924815\pi$
$734$		0			0
$735$	−1320.00	+	933.381i	−0.0662434	+	0.0468412i
$736$		0			0
$737$			23673.9i			1.18323i
$738$		0			0
$739$	−15349.9			−0.764079			−0.382039	−	0.924146i	$-0.624778\pi$
$739$	−15349.9			−0.764079			−0.382039	+	0.924146i	$0.624778\pi$
$740$		0			0
$741$	5498.46	+	7776.00i	0.272593	+	0.385504i
$742$		0			0
$743$	7632.00			0.376838			0.188419	−	0.982089i	$-0.439664\pi$
$743$	7632.00			0.376838			0.188419	+	0.982089i	$0.439664\pi$
$744$		0			0
$745$	−13792.0			−0.678255
$746$		0			0
$747$	−11149.7	−	3942.00i	−0.546111	−	0.193079i
$748$		0			0
$749$	28205.1			1.37596
$750$		0			0
$751$		−	10199.3i		−	0.495577i	−0.968814	−	0.247788i	$-0.920296\pi$
$751$		−	10199.3i		−	0.495577i	0.968814	−	0.247788i	$-0.0797037\pi$
$752$		0			0
$753$	−13446.0	−	19015.5i	−0.650730	−	0.920271i
$754$		0			0
$755$			4128.00i			0.198985i
$756$		0			0
$757$			38232.0i			1.83562i	0.397018	+	0.917811i	$0.370045\pi$
$757$			38232.0i			1.83562i	−0.397018	+	0.917811i	$0.629955\pi$
$758$		0			0
$759$	−12960.0	−	18328.2i	−0.619787	−	0.876511i
$760$		0			0
$761$			15171.7i			0.722698i	0.932431	+	0.361349i	$0.117684\pi$
$761$			15171.7i			0.722698i	−0.932431	+	0.361349i	$0.882316\pi$
$762$		0			0
$763$	20772.0			0.985578
$764$		0			0
$765$	7331.28	+	2592.00i	0.346488	+	0.122502i
$766$		0			0
$767$	−29808.0			−1.40327
$768$		0			0
$769$	−20810.0			−0.975849			−0.487924	−	0.872886i	$-0.662246\pi$
$769$	−20810.0			−0.975849			−0.487924	+	0.872886i	$0.662246\pi$
$770$		0			0
$771$	−5192.99	−	7344.00i	−0.242569	−	0.343045i
$772$		0			0
$773$	−34931.1			−1.62533			−0.812667	−	0.582728i	$-0.801985\pi$
$773$	−34931.1			−1.62533			−0.812667	+	0.582728i	$0.801985\pi$
$774$		0			0
$775$			20517.4i			0.950977i
$776$		0			0
$777$	5184.00	−	3665.64i	0.239350	−	0.169246i
$778$		0			0
$779$			7776.00i			0.357643i
$780$		0			0
$781$			21600.0i			0.989640i
$782$		0			0
$783$	216.000	+	763.675i	0.00985851	+	0.0348551i
$784$		0			0
$785$		−	4480.23i		−	0.203702i
$786$		0			0
$787$	13924.3			0.630685			0.315343	−	0.948978i	$-0.397881\pi$
$787$	13924.3			0.630685			0.315343	+	0.948978i	$0.397881\pi$
$788$		0			0
$789$	5498.46	−	3888.00i	0.248099	−	0.175433i
$790$		0			0
$791$	−39744.0			−1.78652
$792$		0			0
$793$	36288.0			1.62500
$794$		0			0
$795$	12354.6	−	8736.00i	0.551159	−	0.389728i
$796$		0			0
$797$	−22915.9			−1.01847			−0.509237	−	0.860626i	$-0.670072\pi$
$797$	−22915.9			−1.01847			−0.509237	+	0.860626i	$0.670072\pi$
$798$		0			0
$799$		−	29325.1i		−	1.29843i
$800$		0			0
$801$	22032.0	+	7789.49i	0.971863	+	0.343606i
$802$		0			0
$803$			5340.00i			0.234676i
$804$		0			0
$805$			13824.0i			0.605257i
$806$		0			0
$807$	840.000	−	593.970i	0.0366411	−	0.0259092i
$808$		0			0
$809$			7229.46i			0.314183i	0.987584	+	0.157092i	$0.0502118\pi$
$809$			7229.46i			0.314183i	−0.987584	+	0.157092i	$0.949788\pi$
$810$		0			0
$811$	10106.0			0.437569			0.218785	−	0.975773i	$-0.429791\pi$
$811$	10106.0			0.437569			0.218785	+	0.975773i	$0.429791\pi$
$812$		0			0
$813$	13084.3	+	18504.0i	0.564436	+	0.798233i
$814$		0			0
$815$	−15696.0			−0.674610
$816$		0			0
$817$	−5832.00			−0.249738
$818$		0			0
$819$	10996.9	−	31104.0i	0.469186	−	1.32706i
$820$		0			0
$821$	16795.2			0.713954			0.356977	−	0.934113i	$-0.383807\pi$
$821$	16795.2			0.713954			0.356977	+	0.934113i	$0.383807\pi$
$822$		0			0
$823$		−	13321.9i		−	0.564243i	−0.959379	−	0.282121i	$-0.908962\pi$
$823$		−	13321.9i		−	0.564243i	0.959379	−	0.282121i	$-0.0910381\pi$
$824$		0			0
$825$	−8370.00	−	11837.0i	−0.353219	−	0.499528i
$826$		0			0
$827$			2574.00i			0.108231i	0.998535	+	0.0541153i	$0.0172339\pi$
$827$			2574.00i			0.108231i	−0.998535	+	0.0541153i	$0.982766\pi$
$828$		0			0
$829$		−	24408.0i		−	1.02259i	−0.859406	−	0.511294i	$-0.829166\pi$
$829$		−	24408.0i		−	1.02259i	0.859406	−	0.511294i	$-0.170834\pi$
$830$		0			0
$831$	−21384.0	−	30241.5i	−0.892663	−	1.26242i
$832$		0			0
$833$			2800.14i			0.116470i
$834$		0			0
$835$	13848.0			0.573927
$836$		0			0
$837$	29783.3	−	8424.00i	1.22994	−	0.347881i
$838$		0			0
$839$	−8784.00			−0.361451			−0.180725	−	0.983534i	$-0.557845\pi$
$839$	−8784.00			−0.361451			−0.180725	+	0.983534i	$0.557845\pi$
$840$		0			0
$841$	−24357.0			−0.998688
$842$		0			0
$843$	763.675	+	1080.00i	0.0312009	+	0.0441248i
$844$		0			0
$845$	−16897.0			−0.687900
$846$		0			0
$847$		−	7314.31i		−	0.296721i
$848$		0			0
$849$	10908.0	−	7713.12i	0.440944	−	0.311795i
$850$		0			0
$851$			10368.0i			0.417639i
$852$		0			0
$853$			11448.0i			0.459522i	0.973247	+	0.229761i	$0.0737944\pi$
$853$			11448.0i			0.459522i	−0.973247	+	0.229761i	$0.926206\pi$
$854$		0			0
$855$	−1296.00	+	3665.64i	−0.0518389	+	0.146623i
$856$		0			0
$857$		−	21790.2i		−	0.868540i	−0.900783	−	0.434270i	$-0.857006\pi$
$857$		−	21790.2i		−	0.868540i	0.900783	−	0.434270i	$-0.142994\pi$
$858$		0			0
$859$	−31132.5			−1.23659			−0.618293	−	0.785948i	$-0.712175\pi$
$859$	−31132.5			−1.23659			−0.618293	+	0.785948i	$0.712175\pi$
$860$		0			0
$861$	21993.8	−	15552.0i	0.870556	−	0.615576i
$862$		0			0
$863$	31392.0			1.23823			0.619117	−	0.785299i	$-0.287491\pi$
$863$	31392.0			1.23823			0.619117	+	0.785299i	$0.287491\pi$
$864$		0			0
$865$	−17888.0			−0.703133
$866$		0			0
$867$	−9847.17	+	6963.00i	−0.385729	+	0.272752i
$868$		0			0
$869$	−29019.7			−1.13282
$870$		0			0
$871$		−	56817.4i		−	2.21032i
$872$		0			0
$873$	5850.00	−	16546.3i	0.226796	−	0.641475i
$874$		0			0
$875$			20928.0i			0.808566i
$876$		0			0
$877$		−	32472.0i		−	1.25029i	−0.780510	−	0.625143i	$-0.785040\pi$
$877$		−	32472.0i		−	1.25029i	0.780510	−	0.625143i	$-0.214960\pi$
$878$		0			0
$879$	−3000.00	+	2121.32i	−0.115117	+	0.0813997i
$880$		0			0
$881$		−	33907.2i		−	1.29667i	−0.761357	−	0.648333i	$-0.775467\pi$
$881$		−	33907.2i		−	1.29667i	0.761357	−	0.648333i	$-0.224533\pi$
$882$		0			0
$883$	−18557.3			−0.707252			−0.353626	−	0.935387i	$-0.615051\pi$
$883$	−18557.3			−0.707252			−0.353626	+	0.935387i	$0.615051\pi$
$884$		0			0
$885$	−7025.81	−	9936.00i	−0.266859	−	0.377395i
$886$		0			0
$887$	−31536.0			−1.19377			−0.596886	−	0.802326i	$-0.703595\pi$
$887$	−31536.0			−1.19377			−0.596886	+	0.802326i	$0.703595\pi$
$888$		0			0
$889$	44640.0			1.68411
$890$		0			0
$891$	−13746.2	+	17010.0i	−0.516850	+	0.639570i
$892$		0			0
$893$	14662.6			0.549456
$894$		0			0
$895$			13881.9i			0.518460i
$896$		0			0
$897$	31104.0	+	43987.7i	1.15778	+	1.63735i
$898$		0			0
$899$		−	1248.00i		−	0.0462994i
$900$		0			0
$901$		−	26208.0i		−	0.969051i
$902$		0			0
$903$	11664.0	+	16495.4i	0.429849	+	0.607898i
$904$		0			0
$905$		−	18328.2i		−	0.673205i
$906$		0			0
$907$	−15451.7			−0.565673			−0.282836	−	0.959168i	$-0.591275\pi$
$907$	−15451.7			−0.565673			−0.282836	+	0.959168i	$0.591275\pi$
$908$		0			0
$909$	−6465.78	+	18288.0i	−0.235926	+	0.667299i
$910$		0			0
$911$	−18432.0			−0.670340			−0.335170	−	0.942158i	$-0.608794\pi$
$911$	−18432.0			−0.670340			−0.335170	+	0.942158i	$0.608794\pi$
$912$		0			0
$913$	−13140.0			−0.476309
$914$		0			0
$915$	8553.16	+	12096.0i	0.309026	+	0.437029i
$916$		0			0
$917$	−44700.5			−1.60975
$918$		0			0
$919$		−	7619.78i		−	0.273508i	−0.990605	−	0.136754i	$-0.956333\pi$
$919$		−	7619.78i		−	0.273508i	0.990605	−	0.136754i	$-0.0436669\pi$
$920$		0			0
$921$	−43308.0	+	30623.4i	−1.54945	+	1.09563i
$922$		0			0
$923$		−	51840.0i		−	1.84868i
$924$		0			0
$925$			6696.00i			0.238014i
$926$		0			0
$927$	−9072.00	−	3207.44i	−0.321428	−	0.113642i
$928$		0			0
$929$			29070.6i			1.02667i	0.858189	+	0.513334i	$0.171590\pi$
$929$			29070.6i			1.02667i	−0.858189	+	0.513334i	$0.828410\pi$
$930$		0			0
$931$	−1400.07			−0.0492862
$932$		0			0
$933$	33601.7	−	23760.0i	1.17907	−	0.833727i
$934$		0			0
$935$	8640.00			0.302201
$936$		0			0
$937$	−8462.00			−0.295028			−0.147514	−	0.989060i	$-0.547127\pi$
$937$	−8462.00			−0.295028			−0.147514	+	0.989060i	$0.547127\pi$
$938$		0			0
$939$	2282.54	−	1614.00i	0.0793268	−	0.0560925i
$940$		0			0
$941$	−50408.2			−1.74629			−0.873146	−	0.487458i	$-0.837924\pi$
$941$	−50408.2			−1.74629			−0.873146	+	0.487458i	$0.837924\pi$
$942$		0			0
$943$			43987.7i			1.51902i
$944$		0			0
$945$	12960.0	−	3665.64i	0.446126	−	0.126183i
$946$		0			0
$947$		−	28794.0i		−	0.988046i	−0.869449	−	0.494023i	$-0.835526\pi$
$947$		−	28794.0i		−	0.988046i	0.869449	−	0.494023i	$-0.164474\pi$
$948$		0			0
$949$		−	12816.0i		−	0.438382i
$950$		0			0
$951$	−19896.0	+	14068.6i	−0.678414	+	0.479711i
$952$		0			0
$953$		−	6516.70i		−	0.221507i	−0.993848	−	0.110754i	$-0.964674\pi$
$953$		−	6516.70i		−	0.221507i	0.993848	−	0.110754i	$-0.0353265\pi$
$954$		0			0
$955$	11404.2			0.386421
$956$		0			0
$957$	509.117	+	720.000i	0.0171969	+	0.0243201i
$958$		0			0
$959$	3456.00			0.116371
$960$		0			0
$961$	−18881.0			−0.633782
$962$		0			0
$963$	42307.6	+	14958.0i	1.41573	+	0.500535i
$964$		0			0
$965$	19131.5			0.638201
$966$		0			0
$967$			10165.4i			0.338052i	0.985612	+	0.169026i	$0.0540622\pi$
$967$			10165.4i			0.338052i	−0.985612	+	0.169026i	$0.945938\pi$
$968$		0			0
$969$	3888.00	+	5498.46i	0.128896	+	0.182287i
$970$		0			0
$971$		−	3342.00i		−	0.110453i	−0.998474	−	0.0552265i	$-0.982412\pi$
$971$		−	3342.00i		−	0.110453i	0.998474	−	0.0552265i	$-0.0175881\pi$
$972$		0			0
$973$		−	16848.0i		−	0.555110i
$974$		0			0
$975$	20088.0	+	28408.7i	0.659827	+	0.933136i
$976$		0			0
$977$			30394.3i			0.995291i	0.867381	+	0.497645i	$0.165802\pi$
$977$			30394.3i			0.995291i	−0.867381	+	0.497645i	$0.834198\pi$
$978$		0			0
$979$	25965.0			0.847644
$980$		0			0
$981$	31158.0	+	11016.0i	1.01406	+	0.358526i
$982$		0			0
$983$	39312.0			1.27554			0.637771	−	0.770226i	$-0.279857\pi$
$983$	39312.0			1.27554			0.637771	+	0.770226i	$0.279857\pi$
$984$		0			0
$985$	−9824.00			−0.317785
$986$		0			0
$987$	−29325.1	−	41472.0i	−0.945724	−	1.33746i
$988$		0			0
$989$	−32990.8			−1.06071
$990$		0			0
$991$			1747.97i			0.0560303i	0.999607	+	0.0280152i	$0.00891867\pi$
$991$			1747.97i			0.0560303i	−0.999607	+	0.0280152i	$0.991081\pi$
$992$		0			0
$993$	29916.0	−	21153.8i	0.956048	−	0.676028i
$994$		0			0
$995$			13344.0i			0.425159i
$996$		0			0
$997$		−	37512.0i		−	1.19159i	−0.803136	−	0.595796i	$-0.796837\pi$
$997$		−	37512.0i		−	1.19159i	0.803136	−	0.595796i	$-0.203163\pi$
$998$		0			0
$999$	9720.00	−	2749.23i	0.307835	−	0.0870689i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.4.f.c.191.2	yes	4
3.2	odd	2		384.4.f.d.191.1	yes	4
4.3	odd	2		384.4.f.d.191.3	yes	4
8.3	odd	2		384.4.f.d.191.2	yes	4
8.5	even	2	inner	384.4.f.c.191.3	yes	4
12.11	even	2	inner	384.4.f.c.191.4	yes	4
16.3	odd	4		768.4.c.h.767.2		2
16.5	even	4		768.4.c.i.767.2		2
16.11	odd	4		768.4.c.c.767.1		2
16.13	even	4		768.4.c.b.767.1		2
24.5	odd	2		384.4.f.d.191.4	yes	4
24.11	even	2	inner	384.4.f.c.191.1	✓	4
48.5	odd	4		768.4.c.c.767.2		2
48.11	even	4		768.4.c.i.767.1		2
48.29	odd	4		768.4.c.h.767.1		2
48.35	even	4		768.4.c.b.767.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.f.c.191.1	✓	4	24.11	even	2	inner
384.4.f.c.191.2	yes	4	1.1	even	1	trivial
384.4.f.c.191.3	yes	4	8.5	even	2	inner
384.4.f.c.191.4	yes	4	12.11	even	2	inner
384.4.f.d.191.1	yes	4	3.2	odd	2
384.4.f.d.191.2	yes	4	8.3	odd	2
384.4.f.d.191.3	yes	4	4.3	odd	2
384.4.f.d.191.4	yes	4	24.5	odd	2
768.4.c.b.767.1		2	16.13	even	4
768.4.c.b.767.2		2	48.35	even	4
768.4.c.c.767.1		2	16.11	odd	4
768.4.c.c.767.2		2	48.5	odd	4
768.4.c.h.767.1		2	48.29	odd	4
768.4.c.h.767.2		2	16.3	odd	4
768.4.c.i.767.1		2	48.11	even	4
768.4.c.i.767.2		2	16.5	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.24264 + 3.00000i) q^{3} +5.65685 q^{5} -16.9706i q^{7} +(9.00000 - 25.4558i) q^{9} +O(q^{10})\)	\(q+(-4.24264 + 3.00000i) q^{3} +5.65685 q^{5} -16.9706i q^{7} +(9.00000 - 25.4558i) q^{9} -30.0000i q^{11} +72.0000i q^{13} +(-24.0000 + 16.9706i) q^{15} +50.9117i q^{17} -25.4558 q^{19} +(50.9117 + 72.0000i) q^{21} -144.000 q^{23} -93.0000 q^{25} +(38.1838 + 135.000i) q^{27} +5.65685 q^{29} -220.617i q^{31} +(90.0000 + 127.279i) q^{33} -96.0000i q^{35} -72.0000i q^{37} +(-216.000 - 305.470i) q^{39} -305.470i q^{41} +229.103 q^{43} +(50.9117 - 144.000i) q^{45} -576.000 q^{47} +55.0000 q^{49} +(-152.735 - 216.000i) q^{51} -514.774 q^{53} -169.706i q^{55} +(108.000 - 76.3675i) q^{57} +414.000i q^{59} -504.000i q^{61} +(-432.000 - 152.735i) q^{63} +407.294i q^{65} -789.131 q^{67} +(610.940 - 432.000i) q^{69} -720.000 q^{71} -178.000 q^{73} +(394.566 - 279.000i) q^{75} -509.117 q^{77} -967.322i q^{79} +(-567.000 - 458.205i) q^{81} -438.000i q^{83} +288.000i q^{85} +(-24.0000 + 16.9706i) q^{87} +865.499i q^{89} +1221.88 q^{91} +(661.852 + 936.000i) q^{93} -144.000 q^{95} +650.000 q^{97} +(-763.675 - 270.000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 36 q^{9}+O(q^{10}) \) 4 * q + 36 * q^9	\( 4 q + 36 q^{9} - 96 q^{15} - 576 q^{23} - 372 q^{25} + 360 q^{33} - 864 q^{39} - 2304 q^{47} + 220 q^{49} + 432 q^{57} - 1728 q^{63} - 2880 q^{71} - 712 q^{73} - 2268 q^{81} - 96 q^{87} - 576 q^{95} + 2600 q^{97}+O(q^{100}) \) 4 * q + 36 * q^9 - 96 * q^15 - 576 * q^23 - 372 * q^25 + 360 * q^33 - 864 * q^39 - 2304 * q^47 + 220 * q^49 + 432 * q^57 - 1728 * q^63 - 2880 * q^71 - 712 * q^73 - 2268 * q^81 - 96 * q^87 - 576 * q^95 + 2600 * q^97

Introduction

L-functions

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