LMFDB - Embedded newform 48.4.c.b.47.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,4,Mod(47,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("48.47");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	48.c (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:	$2.83209168028$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			47.3
Root			$-1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	48.47
Dual form			48.4.c.b.47.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.89898 - 1.73205i) q^{3} -16.9706i q^{5} +17.3205i q^{7} +(21.0000 - 16.9706i) q^{9} +O(q^{10})$	\(q+(4.89898 - 1.73205i) q^{3} -16.9706i q^{5} +17.3205i q^{7} +(21.0000 - 16.9706i) q^{9} +29.3939 q^{11} -26.0000 q^{13} +(-29.3939 - 83.1384i) q^{15} +67.8823i q^{17} +107.387i q^{19} +(30.0000 + 84.8528i) q^{21} -176.363 q^{23} -163.000 q^{25} +(73.4847 - 119.512i) q^{27} +16.9706i q^{29} +31.1769i q^{31} +(144.000 - 50.9117i) q^{33} +293.939 q^{35} +206.000 q^{37} +(-127.373 + 45.0333i) q^{39} -305.470i q^{41} +93.5307i q^{43} +(-288.000 - 356.382i) q^{45} +117.576 q^{47} +43.0000 q^{49} +(117.576 + 332.554i) q^{51} +50.9117i q^{53} -498.831i q^{55} +(186.000 + 526.087i) q^{57} -558.484 q^{59} +278.000 q^{61} +(293.939 + 363.731i) q^{63} +441.235i q^{65} -890.274i q^{67} +(-864.000 + 305.470i) q^{69} -58.7878 q^{71} -422.000 q^{73} +(-798.534 + 282.324i) q^{75} +509.117i q^{77} +668.572i q^{79} +(153.000 - 712.764i) q^{81} -29.3939 q^{83} +1152.00 q^{85} +(29.3939 + 83.1384i) q^{87} -373.352i q^{89} -450.333i q^{91} +(54.0000 + 152.735i) q^{93} +1822.42 q^{95} -1070.00 q^{97} +(617.271 - 498.831i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 84 q^{9}+O(q^{10}) $ 4 * q + 84 * q^9	$ 4 q + 84 q^{9} - 104 q^{13} + 120 q^{21} - 652 q^{25} + 576 q^{33} + 824 q^{37} - 1152 q^{45} + 172 q^{49} + 744 q^{57} + 1112 q^{61} - 3456 q^{69} - 1688 q^{73} + 612 q^{81} + 4608 q^{85} + 216 q^{93} - 4280 q^{97}+O(q^{100}) $ 4 * q + 84 * q^9 - 104 * q^13 + 120 * q^21 - 652 * q^25 + 576 * q^33 + 824 * q^37 - 1152 * q^45 + 172 * q^49 + 744 * q^57 + 1112 * q^61 - 3456 * q^69 - 1688 * q^73 + 612 * q^81 + 4608 * q^85 + 216 * q^93 - 4280 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	4.89898	−	1.73205i	0.942809	−	0.333333i
$3$	4.89898	−	1.73205i	0.942809	−	0.333333i
$4$		0			0
$5$		−	16.9706i		−	1.51789i	−0.651153	−	0.758947i	$-0.725714\pi$
$5$		−	16.9706i		−	1.51789i	0.651153	−	0.758947i	$-0.274286\pi$
$6$		0			0
$7$			17.3205i			0.935220i	0.883935	+	0.467610i	$0.154885\pi$
$7$			17.3205i			0.935220i	−0.883935	+	0.467610i	$0.845115\pi$
$8$		0			0
$9$	21.0000	−	16.9706i	0.777778	−	0.628539i
$10$		0			0
$11$	29.3939			0.805690			0.402845	−	0.915268i	$-0.368021\pi$
$11$	29.3939			0.805690			0.402845	+	0.915268i	$0.368021\pi$
$12$		0			0
$13$	−26.0000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−26.0000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$	−29.3939	−	83.1384i	−0.505964	−	1.43108i
$16$		0			0
$17$			67.8823i			0.968463i	0.874940	+	0.484231i	$0.160901\pi$
$17$			67.8823i			0.968463i	−0.874940	+	0.484231i	$0.839099\pi$
$18$		0			0
$19$			107.387i			1.29665i	0.761365	+	0.648324i	$0.224530\pi$
$19$			107.387i			1.29665i	−0.761365	+	0.648324i	$0.775470\pi$
$20$		0			0
$21$	30.0000	+	84.8528i	0.311740	+	0.881733i
$22$		0			0
$23$	−176.363			−1.59888			−0.799441	−	0.600745i	$-0.794871\pi$
$23$	−176.363			−1.59888			−0.799441	+	0.600745i	$0.794871\pi$
$24$		0			0
$25$	−163.000			−1.30400
$26$		0			0
$27$	73.4847	−	119.512i	0.523783	−	0.851852i
$28$		0			0
$29$			16.9706i			0.108667i	0.998523	+	0.0543337i	$0.0173035\pi$
$29$			16.9706i			0.108667i	−0.998523	+	0.0543337i	$0.982697\pi$
$30$		0			0
$31$			31.1769i			0.180630i	0.995913	+	0.0903151i	$0.0287874\pi$
$31$			31.1769i			0.180630i	−0.995913	+	0.0903151i	$0.971213\pi$
$32$		0			0
$33$	144.000	−	50.9117i	0.759612	−	0.268563i
$34$		0			0
$35$	293.939			1.41956
$36$		0			0
$37$	206.000			0.915302			0.457651	−	0.889132i	$-0.348691\pi$
$37$	206.000			0.915302			0.457651	+	0.889132i	$0.348691\pi$
$38$		0			0
$39$	−127.373	+	45.0333i	−0.522976	+	0.184900i
$40$		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
$43$			93.5307i			0.331705i	0.986151	+	0.165852i	$0.0530375\pi$
$43$			93.5307i			0.331705i	−0.986151	+	0.165852i	$0.946962\pi$
$44$		0			0
$45$	−288.000	−	356.382i	−0.954056	−	1.18058i
$46$		0			0
$47$	117.576			0.364897			0.182448	−	0.983215i	$-0.441598\pi$
$47$	117.576			0.364897			0.182448	+	0.983215i	$0.441598\pi$
$48$		0			0
$49$	43.0000			0.125364
$50$		0			0
$51$	117.576	+	332.554i	0.322821	+	0.913075i
$52$		0			0
$53$			50.9117i			0.131948i	0.997821	+	0.0659741i	$0.0210155\pi$
$53$			50.9117i			0.131948i	−0.997821	+	0.0659741i	$0.978985\pi$
$54$		0			0
$55$		−	498.831i		−	1.22295i
$56$		0			0
$57$	186.000	+	526.087i	0.432216	+	1.22249i
$58$		0			0
$59$	−558.484			−1.23235			−0.616173	−	0.787611i	$-0.711318\pi$
$59$	−558.484			−1.23235			−0.616173	+	0.787611i	$0.711318\pi$
$60$		0			0
$61$	278.000			0.583512			0.291756	−	0.956493i	$-0.405760\pi$
$61$	278.000			0.583512			0.291756	+	0.956493i	$0.405760\pi$
$62$		0			0
$63$	293.939	+	363.731i	0.587822	+	0.727393i
$64$		0			0
$65$			441.235i			0.841976i
$66$		0			0
$67$		−	890.274i		−	1.62335i	−0.584111	−	0.811674i	$-0.698557\pi$
$67$		−	890.274i		−	1.62335i	0.584111	−	0.811674i	$-0.301443\pi$
$68$		0			0
$69$	−864.000	+	305.470i	−1.50744	+	0.532961i
$70$		0			0
$71$	−58.7878			−0.0982651			−0.0491326	−	0.998792i	$-0.515646\pi$
$71$	−58.7878			−0.0982651			−0.0491326	+	0.998792i	$0.515646\pi$
$72$		0			0
$73$	−422.000			−0.676594			−0.338297	−	0.941039i	$-0.609851\pi$
$73$	−422.000			−0.676594			−0.338297	+	0.941039i	$0.609851\pi$
$74$		0			0
$75$	−798.534	+	282.324i	−1.22942	+	0.434667i
$76$		0			0
$77$			509.117i			0.753497i
$78$		0			0
$79$			668.572i			0.952154i	0.879404	+	0.476077i	$0.157942\pi$
$79$			668.572i			0.952154i	−0.879404	+	0.476077i	$0.842058\pi$
$80$		0			0
$81$	153.000	−	712.764i	0.209877	−	0.977728i
$82$		0			0
$83$	−29.3939			−0.0388723			−0.0194361	−	0.999811i	$-0.506187\pi$
$83$	−29.3939			−0.0388723			−0.0194361	+	0.999811i	$0.506187\pi$
$84$		0			0
$85$	1152.00			1.47002
$86$		0			0
$87$	29.3939	+	83.1384i	0.0362225	+	0.102453i
$88$		0			0
$89$		−	373.352i		−	0.444666i	−0.974971	−	0.222333i	$-0.928633\pi$
$89$		−	373.352i		−	0.444666i	0.974971	−	0.222333i	$-0.0713672\pi$
$90$		0			0
$91$		−	450.333i		−	0.518766i
$92$		0			0
$93$	54.0000	+	152.735i	0.0602101	+	0.170300i
$94$		0			0
$95$	1822.42			1.96817
$96$		0			0
$97$	−1070.00			−1.12002			−0.560011	−	0.828486i	$-0.689203\pi$
$97$	−1070.00			−1.12002			−0.560011	+	0.828486i	$0.689203\pi$
$98$		0			0
$99$	617.271	−	498.831i	0.626648	−	0.506408i
$100$		0			0
$101$		−	1781.91i		−	1.75551i	−0.479109	−	0.877755i	$-0.659040\pi$
$101$		−	1781.91i		−	1.75551i	0.479109	−	0.877755i	$-0.340960\pi$
$102$		0			0
$103$			17.3205i			0.0165693i	0.999966	+	0.00828466i	$0.00263712\pi$
$103$			17.3205i			0.0165693i	−0.999966	+	0.00828466i	$0.997363\pi$
$104$		0			0
$105$	1440.00	−	509.117i	1.33838	−	0.473188i
$106$		0			0
$107$	−1381.51			−1.24819			−0.624093	−	0.781350i	$-0.714531\pi$
$107$	−1381.51			−1.24819			−0.624093	+	0.781350i	$0.714531\pi$
$108$		0			0
$109$	358.000			0.314589			0.157294	−	0.987552i	$-0.449723\pi$
$109$	358.000			0.314589			0.157294	+	0.987552i	$0.449723\pi$
$110$		0			0
$111$	1009.19	−	356.802i	0.862955	−	0.305101i
$112$		0			0
$113$			67.8823i			0.0565117i	0.999601	+	0.0282559i	$0.00899532\pi$
$113$			67.8823i			0.0565117i	−0.999601	+	0.0282559i	$0.991005\pi$
$114$		0			0
$115$			2992.98i			2.42693i
$116$		0			0
$117$	−546.000	+	441.235i	−0.431433	+	0.348651i
$118$		0			0
$119$	−1175.76			−0.905725
$120$		0			0
$121$	−467.000			−0.350864
$122$		0			0
$123$	−529.090	−	1496.49i	−0.387857	−	1.09703i
$124$		0			0
$125$			644.881i			0.461440i
$126$		0			0
$127$			31.1769i			0.0217835i	0.999941	+	0.0108917i	$0.00346702\pi$
$127$			31.1769i			0.0217835i	−0.999941	+	0.0108917i	$0.996533\pi$
$128$		0			0
$129$	162.000	+	458.205i	0.110568	+	0.312734i
$130$		0			0
$131$	1734.24			1.15665			0.578325	−	0.815806i	$-0.303707\pi$
$131$	1734.24			1.15665			0.578325	+	0.815806i	$0.303707\pi$
$132$		0			0
$133$	−1860.00			−1.21265
$134$		0			0
$135$	−2028.18	−	1247.08i	−1.29302	−	0.795046i
$136$		0			0
$137$			2274.06i			1.41814i	0.705136	+	0.709072i	$0.250886\pi$
$137$			2274.06i			1.41814i	−0.705136	+	0.709072i	$0.749114\pi$
$138$		0			0
$139$		−	1541.53i		−	0.940651i	−0.882493	−	0.470325i	$-0.844137\pi$
$139$		−	1541.53i		−	0.940651i	0.882493	−	0.470325i	$-0.155863\pi$
$140$		0			0
$141$	576.000	−	203.647i	0.344028	−	0.121632i
$142$		0			0
$143$	−764.241			−0.446916
$144$		0			0
$145$	288.000			0.164946
$146$		0			0
$147$	210.656	−	74.4782i	0.118195	−	0.0417881i
$148$		0			0
$149$			2358.91i			1.29698i	0.761225	+	0.648488i	$0.224598\pi$
$149$			2358.91i			1.29698i	−0.761225	+	0.648488i	$0.775402\pi$
$150$		0			0
$151$			2040.36i			1.09961i	0.835292	+	0.549807i	$0.185299\pi$
$151$			2040.36i			1.09961i	−0.835292	+	0.549807i	$0.814701\pi$
$152$		0			0
$153$	1152.00	+	1425.53i	0.608717	+	0.753249i
$154$		0			0
$155$	529.090			0.274178
$156$		0			0
$157$	−106.000			−0.0538836			−0.0269418	−	0.999637i	$-0.508577\pi$
$157$	−106.000			−0.0538836			−0.0269418	+	0.999637i	$0.508577\pi$
$158$		0			0
$159$	88.1816	+	249.415i	0.0439828	+	0.124402i
$160$		0			0
$161$		−	3054.70i		−	1.49531i
$162$		0			0
$163$			79.6743i			0.0382857i	0.999817	+	0.0191429i	$0.00609374\pi$
$163$			79.6743i			0.0382857i	−0.999817	+	0.0191429i	$0.993906\pi$
$164$		0			0
$165$	−864.000	−	2443.76i	−0.407650	−	1.15301i
$166$		0			0
$167$	−3115.75			−1.44374			−0.721868	−	0.692030i	$-0.756716\pi$
$167$	−3115.75			−1.44374			−0.721868	+	0.692030i	$0.756716\pi$
$168$		0			0
$169$	−1521.00			−0.692308
$170$		0			0
$171$	1822.42	+	2255.13i	0.814994	+	1.00850i
$172$		0			0
$173$		−	2223.14i		−	0.977009i	−0.872561	−	0.488504i	$-0.837543\pi$
$173$		−	2223.14i		−	0.977009i	0.872561	−	0.488504i	$-0.162457\pi$
$174$		0			0
$175$		−	2823.24i		−	1.21953i
$176$		0			0
$177$	−2736.00	+	967.322i	−1.16187	+	0.410782i
$178$		0			0
$179$	2792.42			1.16601			0.583003	−	0.812470i	$-0.301878\pi$
$179$	2792.42			1.16601			0.583003	+	0.812470i	$0.301878\pi$
$180$		0			0
$181$	4510.00			1.85208			0.926038	−	0.377431i	$-0.123192\pi$
$181$	4510.00			1.85208			0.926038	+	0.377431i	$0.123192\pi$
$182$		0			0
$183$	1361.92	−	481.510i	0.550141	−	0.194504i
$184$		0			0
$185$		−	3495.94i		−	1.38933i
$186$		0			0
$187$			1995.32i			0.780280i
$188$		0			0
$189$	2070.00	+	1272.79i	0.796668	+	0.489852i
$190$		0			0
$191$	1881.21			0.712667			0.356334	−	0.934359i	$-0.384027\pi$
$191$	1881.21			0.712667			0.356334	+	0.934359i	$0.384027\pi$
$192$		0			0
$193$	4994.00			1.86257			0.931285	−	0.364292i	$-0.118689\pi$
$193$	4994.00			1.86257			0.931285	+	0.364292i	$0.118689\pi$
$194$		0			0
$195$	764.241	+	2161.60i	0.280659	+	0.793822i
$196$		0			0
$197$			2155.26i			0.779472i	0.920927	+	0.389736i	$0.127434\pi$
$197$			2155.26i			0.779472i	−0.920927	+	0.389736i	$0.872566\pi$
$198$		0			0
$199$		−	3973.32i		−	1.41538i	−0.706521	−	0.707692i	$-0.749736\pi$
$199$		−	3973.32i		−	1.41538i	0.706521	−	0.707692i	$-0.250264\pi$
$200$		0			0
$201$	−1542.00	−	4361.43i	−0.541116	−	1.53051i
$202$		0			0
$203$	−293.939			−0.101628
$204$		0			0
$205$	−5184.00			−1.76618
$206$		0			0
$207$	−3703.63	+	2992.98i	−1.24357	+	1.00496i
$208$		0			0
$209$			3156.52i			1.04470i
$210$		0			0
$211$		−	4908.63i		−	1.60154i	−0.598974	−	0.800768i	$-0.704425\pi$
$211$		−	4908.63i		−	1.60154i	0.598974	−	0.800768i	$-0.295575\pi$
$212$		0			0
$213$	−288.000	+	101.823i	−0.0926452	+	0.0327550i
$214$		0			0
$215$	1587.27			0.503492
$216$		0			0
$217$	−540.000			−0.168929
$218$		0			0
$219$	−2067.37	+	730.925i	−0.637899	+	0.225531i
$220$		0			0
$221$		−	1764.94i		−	0.537206i
$222$		0			0
$223$			4021.82i			1.20772i	0.797091	+	0.603859i	$0.206371\pi$
$223$			4021.82i			1.20772i	−0.797091	+	0.603859i	$0.793629\pi$
$224$		0			0
$225$	−3423.00	+	2766.20i	−1.01422	+	0.819615i
$226$		0			0
$227$	−2968.78			−0.868039			−0.434020	−	0.900903i	$-0.642905\pi$
$227$	−2968.78			−0.868039			−0.434020	+	0.900903i	$0.642905\pi$
$228$		0			0
$229$	430.000			0.124084			0.0620419	−	0.998074i	$-0.480239\pi$
$229$	430.000			0.124084			0.0620419	+	0.998074i	$0.480239\pi$
$230$		0			0
$231$	881.816	+	2494.15i	0.251166	+	0.710404i
$232$		0			0
$233$			6346.99i			1.78457i	0.451471	+	0.892286i	$0.350899\pi$
$233$			6346.99i			1.78457i	−0.451471	+	0.892286i	$0.649101\pi$
$234$		0			0
$235$		−	1995.32i		−	0.553874i
$236$		0			0
$237$	1158.00	+	3275.32i	0.317385	+	0.897700i
$238$		0			0
$239$	−4115.14			−1.11375			−0.556875	−	0.830596i	$-0.688000\pi$
$239$	−4115.14			−1.11375			−0.556875	+	0.830596i	$0.688000\pi$
$240$		0			0
$241$	4690.00			1.25357			0.626783	−	0.779194i	$-0.284371\pi$
$241$	4690.00			1.25357			0.626783	+	0.779194i	$0.284371\pi$
$242$		0			0
$243$	−484.999	−	3756.82i	−0.128036	−	0.991770i
$244$		0			0
$245$		−	729.734i		−	0.190290i
$246$		0			0
$247$		−	2792.07i		−	0.719251i
$248$		0			0
$249$	−144.000	+	50.9117i	−0.0366491	+	0.0129574i
$250$		0			0
$251$	−1263.94			−0.317845			−0.158922	−	0.987291i	$-0.550802\pi$
$251$	−1263.94			−0.317845			−0.158922	+	0.987291i	$0.550802\pi$
$252$		0			0
$253$	−5184.00			−1.28820
$254$		0			0
$255$	5643.62	−	1995.32i	1.38595	−	0.490008i
$256$		0			0
$257$		−	271.529i		−	0.0659047i	−0.999457	−	0.0329524i	$-0.989509\pi$
$257$		−	271.529i		−	0.0659047i	0.999457	−	0.0329524i	$-0.0104910\pi$
$258$		0			0
$259$			3568.02i			0.856009i
$260$		0			0
$261$	288.000	+	356.382i	0.0683017	+	0.0845191i
$262$		0			0
$263$	5114.53			1.19915			0.599574	−	0.800320i	$-0.295337\pi$
$263$	5114.53			1.19915			0.599574	+	0.800320i	$0.295337\pi$
$264$		0			0
$265$	864.000			0.200283
$266$		0			0
$267$	−646.665	−	1829.05i	−0.148222	−	0.419235i
$268$		0			0
$269$		−	3173.50i		−	0.719299i	−0.933087	−	0.359649i	$-0.882896\pi$
$269$		−	3173.50i		−	0.719299i	0.933087	−	0.359649i	$-0.117104\pi$
$270$		0			0
$271$			668.572i			0.149863i	0.997189	+	0.0749314i	$0.0238738\pi$
$271$			668.572i			0.149863i	−0.997189	+	0.0749314i	$0.976126\pi$
$272$		0			0
$273$	−780.000	−	2206.17i	−0.172922	−	0.489098i
$274$		0			0
$275$	−4791.20			−1.05062
$276$		0			0
$277$	−2018.00			−0.437725			−0.218863	−	0.975756i	$-0.570235\pi$
$277$	−2018.00			−0.437725			−0.218863	+	0.975756i	$0.570235\pi$
$278$		0			0
$279$	529.090	+	654.715i	0.113533	+	0.140490i
$280$		0			0
$281$		−	4717.82i		−	1.00157i	−0.865572	−	0.500785i	$-0.833045\pi$
$281$		−	4717.82i		−	1.00157i	0.865572	−	0.500785i	$-0.166955\pi$
$282$		0			0
$283$			5081.84i			1.06743i	0.845663	+	0.533717i	$0.179205\pi$
$283$			5081.84i			1.06743i	−0.845663	+	0.533717i	$0.820795\pi$
$284$		0			0
$285$	8928.00	−	3156.52i	1.85561	−	0.656057i
$286$		0			0
$287$	5290.90			1.08819
$288$		0			0
$289$	305.000			0.0620802
$290$		0			0
$291$	−5241.91	+	1853.29i	−1.05597	+	0.373340i
$292$		0			0
$293$		−	424.264i		−	0.0845931i	−0.999105	−	0.0422965i	$-0.986533\pi$
$293$		−	424.264i		−	0.0845931i	0.999105	−	0.0422965i	$-0.0134674\pi$
$294$		0			0
$295$			9477.78i			1.87057i
$296$		0			0
$297$	2160.00	−	3512.91i	0.422006	−	0.686328i
$298$		0			0
$299$	4585.44			0.886900
$300$		0			0
$301$	−1620.00			−0.310217
$302$		0			0
$303$	−3086.36	−	8729.54i	−0.585170	−	1.65511i
$304$		0			0
$305$		−	4717.82i		−	0.885709i
$306$		0			0
$307$		−	3522.99i		−	0.654944i	−0.944861	−	0.327472i	$-0.893803\pi$
$307$		−	3522.99i		−	0.654944i	0.944861	−	0.327472i	$-0.106197\pi$
$308$		0			0
$309$	30.0000	+	84.8528i	0.00552311	+	0.0156217i
$310$		0			0
$311$	9464.83			1.72573			0.862864	−	0.505437i	$-0.168669\pi$
$311$	9464.83			1.72573			0.862864	+	0.505437i	$0.168669\pi$
$312$		0			0
$313$	−3718.00			−0.671418			−0.335709	−	0.941966i	$-0.608976\pi$
$313$	−3718.00			−0.671418			−0.335709	+	0.941966i	$0.608976\pi$
$314$		0			0
$315$	6172.71	−	4988.31i	1.10410	−	0.892251i
$316$		0			0
$317$			7212.49i			1.27790i	0.769249	+	0.638949i	$0.220631\pi$
$317$			7212.49i			1.27790i	−0.769249	+	0.638949i	$0.779369\pi$
$318$		0			0
$319$			498.831i			0.0875522i
$320$		0			0
$321$	−6768.00	+	2392.85i	−1.17680	+	0.416062i
$322$		0			0
$323$	−7289.68			−1.25575
$324$		0			0
$325$	4238.00			0.723329
$326$		0			0
$327$	1753.83	−	620.074i	0.296597	−	0.104863i
$328$		0			0
$329$			2036.47i			0.341259i
$330$		0			0
$331$		−	1541.53i		−	0.255982i	−0.991775	−	0.127991i	$-0.959147\pi$
$331$		−	1541.53i		−	0.255982i	0.991775	−	0.127991i	$-0.0408528\pi$
$332$		0			0
$333$	4326.00	−	3495.94i	0.711902	−	0.575304i
$334$		0			0
$335$	−15108.5			−2.46407
$336$		0			0
$337$	2530.00			0.408955			0.204478	−	0.978871i	$-0.434450\pi$
$337$	2530.00			0.408955			0.204478	+	0.978871i	$0.434450\pi$
$338$		0			0
$339$	117.576	+	332.554i	0.0188372	+	0.0532798i
$340$		0			0
$341$			916.410i			0.145532i
$342$		0			0
$343$			6685.72i			1.05246i
$344$		0			0
$345$	5184.00	+	14662.6i	0.808977	+	2.28813i
$346$		0			0
$347$	5555.44			0.859458			0.429729	−	0.902958i	$-0.358609\pi$
$347$	5555.44			0.859458			0.429729	+	0.902958i	$0.358609\pi$
$348$		0			0
$349$	−7786.00			−1.19420			−0.597099	−	0.802168i	$-0.703680\pi$
$349$	−7786.00			−1.19420			−0.597099	+	0.802168i	$0.703680\pi$
$350$		0			0
$351$	−1910.60	+	3107.30i	−0.290542	+	0.472522i
$352$		0			0
$353$		−	407.294i		−	0.0614109i	−0.999528	−	0.0307054i	$-0.990225\pi$
$353$		−	407.294i		−	0.0614109i	0.999528	−	0.0307054i	$-0.00977538\pi$
$354$		0			0
$355$			997.661i			0.149156i
$356$		0			0
$357$	−5760.00	+	2036.47i	−0.853926	+	0.301908i
$358$		0			0
$359$	−8759.38			−1.28775			−0.643875	−	0.765131i	$-0.722674\pi$
$359$	−8759.38			−1.28775			−0.643875	+	0.765131i	$0.722674\pi$
$360$		0			0
$361$	−4673.00			−0.681295
$362$		0			0
$363$	−2287.82	+	808.868i	−0.330798	+	0.116955i
$364$		0			0
$365$			7161.58i			1.02700i
$366$		0			0
$367$		−	12578.2i		−	1.78903i	−0.447037	−	0.894515i	$-0.647521\pi$
$367$		−	12578.2i		−	1.78903i	0.447037	−	0.894515i	$-0.352479\pi$
$368$		0			0
$369$	−5184.00	−	6414.87i	−0.731350	−	0.905000i
$370$		0			0
$371$	−881.816			−0.123401
$372$		0			0
$373$	−4258.00			−0.591075			−0.295537	−	0.955331i	$-0.595499\pi$
$373$	−4258.00			−0.591075			−0.295537	+	0.955331i	$0.595499\pi$
$374$		0			0
$375$	1116.97	+	3159.26i	0.153813	+	0.435049i
$376$		0			0
$377$		−	441.235i		−	0.0602778i
$378$		0			0
$379$			2726.25i			0.369493i	0.982786	+	0.184747i	$0.0591465\pi$
$379$			2726.25i			0.369493i	−0.982786	+	0.184747i	$0.940854\pi$
$380$		0			0
$381$	54.0000	+	152.735i	0.00726116	+	0.0205377i
$382$		0			0
$383$	4232.72			0.564704			0.282352	−	0.959311i	$-0.408885\pi$
$383$	4232.72			0.564704			0.282352	+	0.959311i	$0.408885\pi$
$384$		0			0
$385$	8640.00			1.14373
$386$		0			0
$387$	1587.27	+	1964.15i	0.208489	+	0.257993i
$388$		0			0
$389$		−	5447.55i		−	0.710030i	−0.934861	−	0.355015i	$-0.884476\pi$
$389$		−	5447.55i		−	0.710030i	0.934861	−	0.355015i	$-0.115524\pi$
$390$		0			0
$391$		−	11971.9i		−	1.54846i
$392$		0			0
$393$	8496.00	−	3003.79i	1.09050	−	0.385550i
$394$		0			0
$395$	11346.0			1.44527
$396$		0			0
$397$	13574.0			1.71602			0.858009	−	0.513634i	$-0.171701\pi$
$397$	13574.0			1.71602			0.858009	+	0.513634i	$0.171701\pi$
$398$		0			0
$399$	−9112.10	+	3221.61i	−1.14330	+	0.404217i
$400$		0			0
$401$			11743.6i			1.46247i	0.682128	+	0.731233i	$0.261055\pi$
$401$			11743.6i			1.46247i	−0.682128	+	0.731233i	$0.738945\pi$
$402$		0			0
$403$		−	810.600i		−	0.100196i
$404$		0			0
$405$	−12096.0	−	2596.50i	−1.48409	−	0.318570i
$406$		0			0
$407$	6055.14			0.737450
$408$		0			0
$409$	2890.00			0.349392			0.174696	−	0.984622i	$-0.444106\pi$
$409$	2890.00			0.349392			0.174696	+	0.984622i	$0.444106\pi$
$410$		0			0
$411$	3938.78	+	11140.6i	0.472715	+	1.33704i
$412$		0			0
$413$		−	9673.22i		−	1.15251i
$414$		0			0
$415$			498.831i			0.0590039i
$416$		0			0
$417$	−2670.00	−	7551.90i	−0.313550	−	0.886854i
$418$		0			0
$419$	8318.47			0.969890			0.484945	−	0.874545i	$-0.338840\pi$
$419$	8318.47			0.969890			0.484945	+	0.874545i	$0.338840\pi$
$420$		0			0
$421$	10414.0			1.20558			0.602788	−	0.797902i	$-0.294057\pi$
$421$	10414.0			1.20558			0.602788	+	0.797902i	$0.294057\pi$
$422$		0			0
$423$	2469.09	−	1995.32i	0.283809	−	0.229352i
$424$		0			0
$425$		−	11064.8i		−	1.26288i
$426$		0			0
$427$			4815.10i			0.545712i
$428$		0			0
$429$	−3744.00	+	1323.70i	−0.421357	+	0.148972i
$430$		0			0
$431$	−13050.9			−1.45856			−0.729279	−	0.684216i	$-0.760145\pi$
$431$	−13050.9			−1.45856			−0.729279	+	0.684216i	$0.760145\pi$
$432$		0			0
$433$	−9214.00			−1.02262			−0.511312	−	0.859395i	$-0.670841\pi$
$433$	−9214.00			−1.02262			−0.511312	+	0.859395i	$0.670841\pi$
$434$		0			0
$435$	1410.91	−	498.831i	0.155512	−	0.0549818i
$436$		0			0
$437$		−	18939.1i		−	2.07319i
$438$		0			0
$439$		−	7326.57i		−	0.796534i	−0.917270	−	0.398267i	$-0.869612\pi$
$439$		−	7326.57i		−	0.796534i	0.917270	−	0.398267i	$-0.130388\pi$
$440$		0			0
$441$	903.000	−	729.734i	0.0975057	−	0.0787965i
$442$		0			0
$443$	−2204.54			−0.236435			−0.118218	−	0.992988i	$-0.537718\pi$
$443$	−2204.54			−0.236435			−0.118218	+	0.992988i	$0.537718\pi$
$444$		0			0
$445$	−6336.00			−0.674956
$446$		0			0
$447$	4085.75	+	11556.2i	0.432325	+	1.22280i
$448$		0			0
$449$			2851.05i			0.299665i	0.988711	+	0.149832i	$0.0478734\pi$
$449$			2851.05i			0.299665i	−0.988711	+	0.149832i	$0.952127\pi$
$450$		0			0
$451$		−	8978.95i		−	0.937477i
$452$		0			0
$453$	3534.00	+	9995.66i	0.366538	+	1.03673i
$454$		0			0
$455$	−7642.41			−0.787432
$456$		0			0
$457$	−38.0000			−0.00388964			−0.00194482	−	0.999998i	$-0.500619\pi$
$457$	−38.0000			−0.00388964			−0.00194482	+	0.999998i	$0.500619\pi$
$458$		0			0
$459$	8112.71	+	4988.31i	0.824987	+	0.507264i
$460$		0			0
$461$			15833.5i			1.59966i	0.600230	+	0.799828i	$0.295076\pi$
$461$			15833.5i			1.59966i	−0.600230	+	0.799828i	$0.704924\pi$
$462$		0			0
$463$			11365.7i			1.14084i	0.821353	+	0.570421i	$0.193220\pi$
$463$			11365.7i			1.14084i	−0.821353	+	0.570421i	$0.806780\pi$
$464$		0			0
$465$	2592.00	−	916.410i	0.258497	−	0.0913925i
$466$		0			0
$467$	−499.696			−0.0495143			−0.0247571	−	0.999693i	$-0.507881\pi$
$467$	−499.696			−0.0495143			−0.0247571	+	0.999693i	$0.507881\pi$
$468$		0			0
$469$	15420.0			1.51819
$470$		0			0
$471$	−519.292	+	183.597i	−0.0508019	+	0.0179612i
$472$		0			0
$473$			2749.23i			0.267251i
$474$		0			0
$475$		−	17504.1i		−	1.69083i
$476$		0			0
$477$	864.000	+	1069.15i	0.0829347	+	0.102626i
$478$		0			0
$479$	8700.59			0.829937			0.414969	−	0.909836i	$-0.363793\pi$
$479$	8700.59			0.829937			0.414969	+	0.909836i	$0.363793\pi$
$480$		0			0
$481$	−5356.00			−0.507718
$482$		0			0
$483$	−5290.90	−	14964.9i	−0.498435	−	1.40979i
$484$		0			0
$485$			18158.5i			1.70007i
$486$		0			0
$487$			4007.97i			0.372933i	0.982461	+	0.186466i	$0.0597035\pi$
$487$			4007.97i			0.372933i	−0.982461	+	0.186466i	$0.940296\pi$
$488$		0			0
$489$	138.000	+	390.323i	0.0127619	+	0.0360961i
$490$		0			0
$491$	−16901.5			−1.55347			−0.776734	−	0.629828i	$-0.783125\pi$
$491$	−16901.5			−1.55347			−0.776734	+	0.629828i	$0.783125\pi$
$492$		0			0
$493$	−1152.00			−0.105240
$494$		0			0
$495$	−8465.44	−	10475.4i	−0.768673	−	0.951184i
$496$		0			0
$497$		−	1018.23i		−	0.0918994i
$498$		0			0
$499$			12439.6i			1.11598i	0.829849	+	0.557988i	$0.188427\pi$
$499$			12439.6i			1.11598i	−0.829849	+	0.557988i	$0.811573\pi$
$500$		0			0
$501$	−15264.0	+	5396.64i	−1.36117	+	0.481246i
$502$		0			0
$503$	12756.9			1.13082			0.565411	−	0.824809i	$-0.308717\pi$
$503$	12756.9			1.13082			0.565411	+	0.824809i	$0.308717\pi$
$504$		0			0
$505$	−30240.0			−2.66468
$506$		0			0
$507$	−7451.35	+	2634.45i	−0.652714	+	0.230769i
$508$		0			0
$509$		−	5549.37i		−	0.483245i	−0.970370	−	0.241622i	$-0.922320\pi$
$509$		−	5549.37i		−	0.483245i	0.970370	−	0.241622i	$-0.0776796\pi$
$510$		0			0
$511$		−	7309.25i		−	0.632764i
$512$		0			0
$513$	12834.0	+	7891.31i	1.10455	+	0.679162i
$514$		0			0
$515$	293.939			0.0251505
$516$		0			0
$517$	3456.00			0.293994
$518$		0			0
$519$	−3850.60	−	10891.1i	−0.325670	−	0.921133i
$520$		0			0
$521$			11506.0i			0.967541i	0.875195	+	0.483770i	$0.160733\pi$
$521$			11506.0i			0.967541i	−0.875195	+	0.483770i	$0.839267\pi$
$522$		0			0
$523$			17441.8i			1.45827i	0.684371	+	0.729134i	$0.260077\pi$
$523$			17441.8i			1.45827i	−0.684371	+	0.729134i	$0.739923\pi$
$524$		0			0
$525$	−4890.00	−	13831.0i	−0.406509	−	1.14978i
$526$		0			0
$527$	−2116.36			−0.174934
$528$		0			0
$529$	18937.0			1.55642
$530$		0			0
$531$	−11728.2	+	9477.78i	−0.958491	+	0.774578i
$532$		0			0
$533$			7942.22i			0.645433i
$534$		0			0
$535$			23445.0i			1.89461i
$536$		0			0
$537$	13680.0	−	4836.61i	1.09932	−	0.388669i
$538$		0			0
$539$	1263.94			0.101005
$540$		0			0
$541$	−6794.00			−0.539920			−0.269960	−	0.962871i	$-0.587011\pi$
$541$	−6794.00			−0.539920			−0.269960	+	0.962871i	$0.587011\pi$
$542$		0			0
$543$	22094.4	−	7811.55i	1.74615	−	0.617358i
$544$		0			0
$545$		−	6075.46i		−	0.477512i
$546$		0			0
$547$		−	3910.97i		−	0.305706i	−0.988249	−	0.152853i	$-0.951154\pi$
$547$		−	3910.97i		−	0.305706i	0.988249	−	0.152853i	$-0.0488461\pi$
$548$		0			0
$549$	5838.00	−	4717.82i	0.453843	−	0.366760i
$550$		0			0
$551$	−1822.42			−0.140903
$552$		0			0
$553$	−11580.0			−0.890473
$554$		0			0
$555$	−6055.14	−	17126.5i	−0.463110	−	1.30987i
$556$		0			0
$557$		−	22587.8i		−	1.71827i	−0.511749	−	0.859135i	$-0.671002\pi$
$557$		−	22587.8i		−	1.71827i	0.511749	−	0.859135i	$-0.328998\pi$
$558$		0			0
$559$		−	2431.80i		−	0.183997i
$560$		0			0
$561$	3456.00	+	9775.04i	0.260093	+	0.735655i
$562$		0			0
$563$	−8729.98			−0.653508			−0.326754	−	0.945109i	$-0.605955\pi$
$563$	−8729.98			−0.653508			−0.326754	+	0.945109i	$0.605955\pi$
$564$		0			0
$565$	1152.00			0.0857788
$566$		0			0
$567$	12345.4	+	2650.04i	0.914390	+	0.196281i
$568$		0			0
$569$		−	16122.0i		−	1.18782i	−0.804531	−	0.593911i	$-0.797583\pi$
$569$		−	16122.0i		−	1.18782i	0.804531	−	0.593911i	$-0.202417\pi$
$570$		0			0
$571$		−	9495.10i		−	0.695898i	−0.937513	−	0.347949i	$-0.886878\pi$
$571$		−	9495.10i		−	0.695898i	0.937513	−	0.347949i	$-0.113122\pi$
$572$		0			0
$573$	9216.00	−	3258.35i	0.671909	−	0.237556i
$574$		0			0
$575$	28747.2			2.08494
$576$		0			0
$577$	−8974.00			−0.647474			−0.323737	−	0.946147i	$-0.604939\pi$
$577$	−8974.00			−0.647474			−0.323737	+	0.946147i	$0.604939\pi$
$578$		0			0
$579$	24465.5	−	8649.86i	1.75605	−	0.620857i
$580$		0			0
$581$		−	509.117i		−	0.0363541i
$582$		0			0
$583$			1496.49i			0.106309i
$584$		0			0
$585$	7488.00	+	9265.93i	0.529215	+	0.654870i
$586$		0			0
$587$	−4908.78			−0.345157			−0.172578	−	0.984996i	$-0.555210\pi$
$587$	−4908.78			−0.345157			−0.172578	+	0.984996i	$0.555210\pi$
$588$		0			0
$589$	−3348.00			−0.234214
$590$		0			0
$591$	3733.02	+	10558.6i	0.259824	+	0.734893i
$592$		0			0
$593$		−	14730.4i		−	1.02008i	−0.860151	−	0.510040i	$-0.829631\pi$
$593$		−	14730.4i		−	1.02008i	0.860151	−	0.510040i	$-0.170369\pi$
$594$		0			0
$595$			19953.2i			1.37479i
$596$		0			0
$597$	−6882.00	−	19465.2i	−0.471795	−	1.33444i
$598$		0			0
$599$	−10052.7			−0.685714			−0.342857	−	0.939388i	$-0.611395\pi$
$599$	−10052.7			−0.685714			−0.342857	+	0.939388i	$0.611395\pi$
$600$		0			0
$601$	23690.0			1.60788			0.803939	−	0.594711i	$-0.202734\pi$
$601$	23690.0			1.60788			0.803939	+	0.594711i	$0.202734\pi$
$602$		0			0
$603$	−15108.5	−	18695.8i	−1.02034	−	1.26260i
$604$		0			0
$605$			7925.25i			0.532574i
$606$		0			0
$607$			22589.4i			1.51050i	0.655435	+	0.755252i	$0.272485\pi$
$607$			22589.4i			1.51050i	−0.655435	+	0.755252i	$0.727515\pi$
$608$		0			0
$609$	−1440.00	+	509.117i	−0.0958157	+	0.0338760i
$610$		0			0
$611$	−3056.96			−0.202408
$612$		0			0
$613$	9422.00			0.620801			0.310400	−	0.950606i	$-0.399537\pi$
$613$	9422.00			0.620801			0.310400	+	0.950606i	$0.399537\pi$
$614$		0			0
$615$	−25396.3	+	8978.95i	−1.66517	+	0.588726i
$616$		0			0
$617$		−	984.293i		−	0.0642239i	−0.999484	−	0.0321119i	$-0.989777\pi$
$617$		−	984.293i		−	0.0642239i	0.999484	−	0.0321119i	$-0.0102233\pi$
$618$		0			0
$619$		−	12238.7i		−	0.794691i	−0.917669	−	0.397345i	$-0.869931\pi$
$619$		−	12238.7i		−	0.794691i	0.917669	−	0.397345i	$-0.130069\pi$
$620$		0			0
$621$	−12960.0	+	21077.4i	−0.837467	+	1.36201i
$622$		0			0
$623$	6466.65			0.415860
$624$		0			0
$625$	−9431.00			−0.603584
$626$		0			0
$627$	5467.26	+	15463.7i	0.348232	+	0.984948i
$628$		0			0
$629$			13983.7i			0.886436i
$630$		0			0
$631$		−	13922.2i		−	0.878344i	−0.898403	−	0.439172i	$-0.855272\pi$
$631$		−	13922.2i		−	0.878344i	0.898403	−	0.439172i	$-0.144728\pi$
$632$		0			0
$633$	−8502.00	−	24047.3i	−0.533845	−	1.50994i
$634$		0			0
$635$	529.090			0.0330650
$636$		0			0
$637$	−1118.00			−0.0695397
$638$		0			0
$639$	−1234.54	+	997.661i	−0.0764284	+	0.0617635i
$640$		0			0
$641$			16427.5i			1.01224i	0.862462	+	0.506121i	$0.168921\pi$
$641$			16427.5i			1.01224i	−0.862462	+	0.506121i	$0.831079\pi$
$642$		0			0
$643$			1465.31i			0.0898700i	0.998990	+	0.0449350i	$0.0143081\pi$
$643$			1465.31i			0.0898700i	−0.998990	+	0.0449350i	$0.985692\pi$
$644$		0			0
$645$	7776.00	−	2749.23i	0.474697	−	0.167831i
$646$		0			0
$647$	−12286.6			−0.746581			−0.373290	−	0.927715i	$-0.621770\pi$
$647$	−12286.6			−0.746581			−0.373290	+	0.927715i	$0.621770\pi$
$648$		0			0
$649$	−16416.0			−0.992888
$650$		0			0
$651$	−2645.45	+	935.307i	−0.159268	+	0.0563097i
$652$		0			0
$653$			8502.25i			0.509523i	0.967004	+	0.254762i	$0.0819970\pi$
$653$			8502.25i			0.509523i	−0.967004	+	0.254762i	$0.918003\pi$
$654$		0			0
$655$		−	29431.0i		−	1.75567i
$656$		0			0
$657$	−8862.00	+	7161.58i	−0.526240	+	0.425266i
$658$		0			0
$659$	17136.6			1.01297			0.506486	−	0.862248i	$-0.330944\pi$
$659$	17136.6			1.01297			0.506486	+	0.862248i	$0.330944\pi$
$660$		0			0
$661$	−10018.0			−0.589493			−0.294747	−	0.955575i	$-0.595235\pi$
$661$	−10018.0			−0.589493			−0.294747	+	0.955575i	$0.595235\pi$
$662$		0			0
$663$	−3056.96	−	8646.40i	−0.179069	−	0.506483i
$664$		0			0
$665$			31565.2i			1.84067i
$666$		0			0
$667$		−	2992.98i		−	0.173746i
$668$		0			0
$669$	6966.00	+	19702.8i	0.402573	+	1.13865i
$670$		0			0
$671$	8171.50			0.470130
$672$		0			0
$673$	1682.00			0.0963393			0.0481696	−	0.998839i	$-0.484661\pi$
$673$	1682.00			0.0963393			0.0481696	+	0.998839i	$0.484661\pi$
$674$		0			0
$675$	−11978.0	+	19480.4i	−0.683013	+	1.11081i
$676$		0			0
$677$		−	9113.19i		−	0.517354i	−0.965964	−	0.258677i	$-0.916714\pi$
$677$		−	9113.19i		−	0.517354i	0.965964	−	0.258677i	$-0.0832865\pi$
$678$		0			0
$679$		−	18532.9i		−	1.04747i
$680$		0			0
$681$	−14544.0	+	5142.08i	−0.818395	+	0.289346i
$682$		0			0
$683$	−23250.6			−1.30257			−0.651287	−	0.758832i	$-0.725771\pi$
$683$	−23250.6			−1.30257			−0.651287	+	0.758832i	$0.725771\pi$
$684$		0			0
$685$	38592.0			2.15259
$686$		0			0
$687$	2106.56	−	744.782i	0.116987	−	0.0413613i
$688$		0			0
$689$		−	1323.70i		−	0.0731917i
$690$		0			0
$691$		−	25222.1i		−	1.38856i	−0.719705	−	0.694280i	$-0.755723\pi$
$691$		−	25222.1i		−	1.38856i	0.719705	−	0.694280i	$-0.244277\pi$
$692$		0			0
$693$	8640.00	+	10691.5i	0.473602	+	0.586053i
$694$		0			0
$695$	−26160.6			−1.42781
$696$		0			0
$697$	20736.0			1.12688
$698$		0			0
$699$	10993.3	+	31093.8i	0.594857	+	1.68251i
$700$		0			0
$701$		−	16410.5i		−	0.884190i	−0.896968	−	0.442095i	$-0.854235\pi$
$701$		−	16410.5i		−	0.884190i	0.896968	−	0.442095i	$-0.145765\pi$
$702$		0			0
$703$			22121.8i			1.18682i
$704$		0			0
$705$	−3456.00	−	9775.04i	−0.184625	−	0.522198i
$706$		0			0
$707$	30863.6			1.64179
$708$		0			0
$709$	−27602.0			−1.46208			−0.731040	−	0.682334i	$-0.760965\pi$
$709$	−27602.0			−1.46208			−0.731040	+	0.682334i	$0.760965\pi$
$710$		0			0
$711$	11346.0	+	14040.0i	0.598466	+	0.740564i
$712$		0			0
$713$		−	5498.46i		−	0.288806i
$714$		0			0
$715$			12969.6i			0.678371i
$716$		0			0
$717$	−20160.0	+	7127.64i	−1.05005	+	0.371250i
$718$		0			0
$719$	−31627.8			−1.64050			−0.820249	−	0.572006i	$-0.806165\pi$
$719$	−31627.8			−1.64050			−0.820249	+	0.572006i	$0.806165\pi$
$720$		0			0
$721$	−300.000			−0.0154960
$722$		0			0
$723$	22976.2	−	8123.32i	1.18187	−	0.417855i
$724$		0			0
$725$		−	2766.20i		−	0.141702i
$726$		0			0
$727$		−	27279.8i		−	1.39168i	−0.718197	−	0.695840i	$-0.755032\pi$
$727$		−	27279.8i		−	1.39168i	0.718197	−	0.695840i	$-0.244968\pi$
$728$		0			0
$729$	−8883.00	−	17564.5i	−0.451303	−	0.892371i
$730$		0			0
$731$	−6349.08			−0.321244
$732$		0			0
$733$	−15914.0			−0.801906			−0.400953	−	0.916099i	$-0.631321\pi$
$733$	−15914.0			−0.801906			−0.400953	+	0.916099i	$0.631321\pi$
$734$		0			0
$735$	−1263.94	−	3574.95i	−0.0634299	−	0.179407i
$736$		0			0
$737$		−	26168.6i		−	1.30791i
$738$		0			0
$739$		−	2275.91i		−	0.113289i	−0.998394	−	0.0566447i	$-0.981960\pi$
$739$		−	2275.91i		−	0.113289i	0.998394	−	0.0566447i	$-0.0180402\pi$
$740$		0			0
$741$	−4836.00	−	13678.3i	−0.239750	−	0.678116i
$742$		0			0
$743$	8641.80			0.426698			0.213349	−	0.976976i	$-0.431563\pi$
$743$	8641.80			0.426698			0.213349	+	0.976976i	$0.431563\pi$
$744$		0			0
$745$	40032.0			1.96867
$746$		0			0
$747$	−617.271	+	498.831i	−0.0302340	+	0.0244327i
$748$		0			0
$749$		−	23928.5i		−	1.16733i
$750$		0			0
$751$			16631.2i			0.808095i	0.914738	+	0.404048i	$0.132397\pi$
$751$			16631.2i			0.808095i	−0.914738	+	0.404048i	$0.867603\pi$
$752$		0			0
$753$	−6192.00	+	2189.20i	−0.299667	+	0.105948i
$754$		0			0
$755$	34626.0			1.66910
$756$		0			0
$757$	11422.0			0.548401			0.274201	−	0.961673i	$-0.411587\pi$
$757$	11422.0			0.548401			0.274201	+	0.961673i	$0.411587\pi$
$758$		0			0
$759$	−25396.3	+	8978.95i	−1.21453	+	0.429401i
$760$		0			0
$761$			12660.0i			0.603057i	0.953457	+	0.301528i	$0.0974968\pi$
$761$			12660.0i			0.603057i	−0.953457	+	0.301528i	$0.902503\pi$
$762$		0			0
$763$			6200.74i			0.294210i
$764$		0			0
$765$	24192.0	−	19550.1i	1.14335	−	0.923967i
$766$		0			0
$767$	14520.6			0.683582
$768$		0			0
$769$	18818.0			0.882437			0.441219	−	0.897400i	$-0.354546\pi$
$769$	18818.0			0.882437			0.441219	+	0.897400i	$0.354546\pi$
$770$		0			0
$771$	−470.302	−	1330.22i	−0.0219682	−	0.0621356i
$772$		0			0
$773$		−	1917.67i		−	0.0892289i	−0.999004	−	0.0446144i	$-0.985794\pi$
$773$		−	1917.67i		−	0.0892289i	0.999004	−	0.0446144i	$-0.0142059\pi$
$774$		0			0
$775$		−	5081.84i		−	0.235542i
$776$		0			0
$777$	6180.00	+	17479.7i	0.285336	+	0.807053i
$778$		0			0
$779$	32803.6			1.50874
$780$		0			0
$781$	−1728.00			−0.0791712
$782$		0			0
$783$	2028.18	+	1247.08i	0.0925685	+	0.0569181i
$784$		0			0
$785$			1798.88i			0.0817895i
$786$		0			0
$787$		−	19845.8i		−	0.898892i	−0.893308	−	0.449446i	$-0.851621\pi$
$787$		−	19845.8i		−	0.898892i	0.893308	−	0.449446i	$-0.148379\pi$
$788$		0			0
$789$	25056.0	−	8858.63i	1.13057	−	0.399716i
$790$		0			0
$791$	−1175.76			−0.0528509
$792$		0			0
$793$	−7228.00			−0.323674
$794$		0			0
$795$	4232.72	−	1496.49i	0.188829	−	0.0667611i
$796$		0			0
$797$			30971.3i			1.37649i	0.725480	+	0.688243i	$0.241618\pi$
$797$			30971.3i			1.37649i	−0.725480	+	0.688243i	$0.758382\pi$
$798$		0			0
$799$			7981.29i			0.353389i
$800$		0			0
$801$	−6336.00	−	7840.40i	−0.279490	−	0.345851i
$802$		0			0
$803$	−12404.2			−0.545125
$804$		0			0
$805$	−51840.0			−2.26971
$806$		0			0
$807$	−5496.65	−	15546.9i	−0.239766	−	0.678162i
$808$		0			0
$809$		−	9537.46i		−	0.414486i	−0.978290	−	0.207243i	$-0.933551\pi$
$809$		−	9537.46i		−	0.414486i	0.978290	−	0.207243i	$-0.0664491\pi$
$810$		0			0
$811$		−	3897.11i		−	0.168738i	−0.996435	−	0.0843688i	$-0.973113\pi$
$811$		−	3897.11i		−	0.168738i	0.996435	−	0.0843688i	$-0.0268874\pi$
$812$		0			0
$813$	1158.00	+	3275.32i	0.0499543	+	0.141292i
$814$		0			0
$815$	1352.12			0.0581137
$816$		0			0
$817$	−10044.0			−0.430104
$818$		0			0
$819$	−7642.41	−	9457.00i	−0.326065	−	0.403485i
$820$		0			0
$821$		−	13254.0i		−	0.563420i	−0.959500	−	0.281710i	$-0.909098\pi$
$821$		−	13254.0i		−	0.563420i	0.959500	−	0.281710i	$-0.0909017\pi$
$822$		0			0
$823$			10021.6i			0.424462i	0.977219	+	0.212231i	$0.0680730\pi$
$823$			10021.6i			0.424462i	−0.977219	+	0.212231i	$0.931927\pi$
$824$		0			0
$825$	−23472.0	+	8298.61i	−0.990534	+	0.350206i
$826$		0			0
$827$	−31128.1			−1.30886			−0.654432	−	0.756121i	$-0.727092\pi$
$827$	−31128.1			−1.30886			−0.654432	+	0.756121i	$0.727092\pi$
$828$		0			0
$829$	−12554.0			−0.525957			−0.262979	−	0.964802i	$-0.584705\pi$
$829$	−12554.0			−0.525957			−0.262979	+	0.964802i	$0.584705\pi$
$830$		0			0
$831$	−9886.14	+	3495.28i	−0.412691	+	0.145908i
$832$		0			0
$833$			2918.94i			0.121411i
$834$		0			0
$835$			52876.0i			2.19144i
$836$		0			0
$837$	3726.00	+	2291.03i	0.153870	+	0.0946110i
$838$		0			0
$839$	−21692.7			−0.892628			−0.446314	−	0.894877i	$-0.647263\pi$
$839$	−21692.7			−0.892628			−0.446314	+	0.894877i	$0.647263\pi$
$840$		0			0
$841$	24101.0			0.988191
$842$		0			0
$843$	−8171.50	−	23112.5i	−0.333857	−	0.944290i
$844$		0			0
$845$			25812.2i			1.05085i
$846$		0			0
$847$		−	8088.68i		−	0.328135i
$848$		0			0
$849$	8802.00	+	24895.8i	0.355811	+	1.00639i
$850$		0			0
$851$	−36330.8			−1.46346
$852$		0			0
$853$	−4546.00			−0.182476			−0.0912380	−	0.995829i	$-0.529082\pi$
$853$	−4546.00			−0.182476			−0.0912380	+	0.995829i	$0.529082\pi$
$854$		0			0
$855$	38270.8	−	30927.5i	1.53080	−	1.23707i
$856$		0			0
$857$			32617.4i			1.30010i	0.759889	+	0.650052i	$0.225253\pi$
$857$			32617.4i			1.30010i	−0.759889	+	0.650052i	$0.774747\pi$
$858$		0			0
$859$			21404.7i			0.850196i	0.905147	+	0.425098i	$0.139760\pi$
$859$			21404.7i			0.850196i	−0.905147	+	0.425098i	$0.860240\pi$
$860$		0			0
$861$	25920.0	−	9164.10i	1.02596	−	0.362732i
$862$		0			0
$863$	10111.5			0.398840			0.199420	−	0.979914i	$-0.436094\pi$
$863$	10111.5			0.398840			0.199420	+	0.979914i	$0.436094\pi$
$864$		0			0
$865$	−37728.0			−1.48299
$866$		0			0
$867$	1494.19	−	528.275i	0.0585298	−	0.0206934i
$868$		0			0
$869$			19651.9i			0.767141i
$870$		0			0
$871$			23147.1i			0.900471i
$872$		0			0
$873$	−22470.0	+	18158.5i	−0.871127	+	0.703977i
$874$		0			0
$875$	−11169.7			−0.431547
$876$		0			0
$877$	20870.0			0.803569			0.401784	−	0.915734i	$-0.368390\pi$
$877$	20870.0			0.803569			0.401784	+	0.915734i	$0.368390\pi$
$878$		0			0
$879$	−734.847	−	2078.46i	−0.0281977	−	0.0797551i
$880$		0			0
$881$		−	28714.2i		−	1.09808i	−0.835797	−	0.549038i	$-0.814994\pi$
$881$		−	28714.2i		−	1.09808i	0.835797	−	0.549038i	$-0.185006\pi$
$882$		0			0
$883$			2823.24i			0.107599i	0.998552	+	0.0537994i	$0.0171331\pi$
$883$			2823.24i			0.107599i	−0.998552	+	0.0537994i	$0.982867\pi$
$884$		0			0
$885$	16416.0	+	46431.5i	0.623523	+	1.76359i
$886$		0			0
$887$	2410.30			0.0912400			0.0456200	−	0.998959i	$-0.485474\pi$
$887$	2410.30			0.0912400			0.0456200	+	0.998959i	$0.485474\pi$
$888$		0			0
$889$	−540.000			−0.0203723
$890$		0			0
$891$	4497.26	−	20950.9i	0.169095	−	0.787745i
$892$		0			0
$893$			12626.1i			0.473143i
$894$		0			0
$895$		−	47388.9i		−	1.76987i
$896$		0			0
$897$	22464.0	−	7942.22i	0.836177	−	0.295633i
$898$		0			0
$899$	−529.090			−0.0196286
$900$		0			0
$901$	−3456.00			−0.127787
$902$		0			0
$903$	−7936.35	+	2805.92i	−0.292475	+	0.103406i
$904$		0			0
$905$		−	76537.2i		−	2.81125i
$906$		0			0
$907$		−	18474.1i		−	0.676319i	−0.941089	−	0.338159i	$-0.890196\pi$
$907$		−	18474.1i		−	0.676319i	0.941089	−	0.338159i	$-0.109804\pi$
$908$		0			0
$909$	−30240.0	−	37420.1i	−1.10341	−	1.36540i
$910$		0			0
$911$	44561.1			1.62061			0.810305	−	0.586008i	$-0.199301\pi$
$911$	44561.1			1.62061			0.810305	+	0.586008i	$0.199301\pi$
$912$		0			0
$913$	−864.000			−0.0313190
$914$		0			0
$915$	−8171.50	−	23112.5i	−0.295236	−	0.835055i
$916$		0			0
$917$			30037.9i			1.08172i
$918$		0			0
$919$			40561.2i			1.45592i	0.685620	+	0.727959i	$0.259531\pi$
$919$			40561.2i			1.45592i	−0.685620	+	0.727959i	$0.740469\pi$
$920$		0			0
$921$	−6102.00	−	17259.1i	−0.218315	−	0.617487i
$922$		0			0
$923$	1528.48			0.0545077
$924$		0			0
$925$	−33578.0			−1.19355
$926$		0			0
$927$	293.939	+	363.731i	0.0104145	+	0.0128873i
$928$		0			0
$929$			56478.0i			1.99460i	0.0734296	+	0.997300i	$0.476606\pi$
$929$			56478.0i			1.99460i	−0.0734296	+	0.997300i	$0.523394\pi$
$930$		0			0
$931$			4617.65i			0.162553i
$932$		0			0
$933$	46368.0	−	16393.6i	1.62703	−	0.575242i
$934$		0			0
$935$	33861.7			1.18438
$936$		0			0
$937$	−7030.00			−0.245101			−0.122551	−	0.992462i	$-0.539107\pi$
$937$	−7030.00			−0.245101			−0.122551	+	0.992462i	$0.539107\pi$
$938$		0			0
$939$	−18214.4	+	6439.76i	−0.633019	+	0.223806i
$940$		0			0
$941$		−	27475.3i		−	0.951828i	−0.879492	−	0.475914i	$-0.842117\pi$
$941$		−	27475.3i		−	0.951828i	0.879492	−	0.475914i	$-0.157883\pi$
$942$		0			0
$943$			53873.7i			1.86041i
$944$		0			0
$945$	21600.0	−	35129.1i	0.743543	−	1.20926i
$946$		0			0
$947$	3262.72			0.111958			0.0559790	−	0.998432i	$-0.482172\pi$
$947$	3262.72			0.111958			0.0559790	+	0.998432i	$0.482172\pi$
$948$		0			0
$949$	10972.0			0.375307
$950$		0			0
$951$	12492.4	+	35333.8i	0.425966	+	1.20481i
$952$		0			0
$953$		−	8519.22i		−	0.289575i	−0.989463	−	0.144787i	$-0.953750\pi$
$953$		−	8519.22i		−	0.289575i	0.989463	−	0.144787i	$-0.0462498\pi$
$954$		0			0
$955$		−	31925.2i		−	1.08175i
$956$		0			0
$957$	864.000	+	2443.76i	0.0291841	+	0.0825450i
$958$		0			0
$959$	−39387.8			−1.32628
$960$		0			0
$961$	28819.0			0.967373
$962$		0			0
$963$	−29011.8	+	23445.0i	−0.970811	+	0.784534i
$964$		0			0
$965$		−	84751.0i		−	2.82718i
$966$		0			0
$967$			29226.6i			0.971939i	0.873976	+	0.485969i	$0.161533\pi$
$967$			29226.6i			0.971939i	−0.873976	+	0.485969i	$0.838467\pi$
$968$		0			0
$969$	−35712.0	+	12626.1i	−1.18394	+	0.418585i
$970$		0			0
$971$	47059.6			1.55532			0.777659	−	0.628686i	$-0.216407\pi$
$971$	47059.6			1.55532			0.777659	+	0.628686i	$0.216407\pi$
$972$		0			0
$973$	26700.0			0.879715
$974$		0			0
$975$	20761.9	−	7340.43i	0.681961	−	0.241110i
$976$		0			0
$977$			27492.3i			0.900263i	0.892962	+	0.450132i	$0.148623\pi$
$977$			27492.3i			0.900263i	−0.892962	+	0.450132i	$0.851377\pi$
$978$		0			0
$979$		−	10974.3i		−	0.358263i
$980$		0			0
$981$	7518.00	−	6075.46i	0.244680	−	0.197731i
$982$		0			0
$983$	37447.8			1.21506			0.607528	−	0.794299i	$-0.292161\pi$
$983$	37447.8			1.21506			0.607528	+	0.794299i	$0.292161\pi$
$984$		0			0
$985$	36576.0			1.18316
$986$		0			0
$987$	3527.27	+	9976.61i	0.113753	+	0.321742i
$988$		0			0
$989$		−	16495.4i		−	0.530357i
$990$		0			0
$991$		−	37270.3i		−	1.19468i	−0.801988	−	0.597341i	$-0.796224\pi$
$991$		−	37270.3i		−	1.19468i	0.801988	−	0.597341i	$-0.203776\pi$
$992$		0			0
$993$	−2670.00	−	7551.90i	−0.0853272	−	0.241342i
$994$		0			0
$995$	−67429.6			−2.14840
$996$		0			0
$997$	−45106.0			−1.43282			−0.716410	−	0.697680i	$-0.754216\pi$
$997$	−45106.0			−1.43282			−0.716410	+	0.697680i	$0.754216\pi$
$998$		0			0
$999$	15137.8	−	24619.4i	0.479420	−	0.779702i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.4.c.b.47.3	yes	4
3.2	odd	2	inner	48.4.c.b.47.1	✓	4
4.3	odd	2	inner	48.4.c.b.47.2	yes	4
8.3	odd	2		192.4.c.c.191.3		4
8.5	even	2		192.4.c.c.191.2		4
12.11	even	2	inner	48.4.c.b.47.4	yes	4
16.3	odd	4		768.4.f.b.383.1		8
16.5	even	4		768.4.f.b.383.2		8
16.11	odd	4		768.4.f.b.383.8		8
16.13	even	4		768.4.f.b.383.7		8
24.5	odd	2		192.4.c.c.191.4		4
24.11	even	2		192.4.c.c.191.1		4
48.5	odd	4		768.4.f.b.383.3		8
48.11	even	4		768.4.f.b.383.5		8
48.29	odd	4		768.4.f.b.383.6		8
48.35	even	4		768.4.f.b.383.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.4.c.b.47.1	✓	4	3.2	odd	2	inner
48.4.c.b.47.2	yes	4	4.3	odd	2	inner
48.4.c.b.47.3	yes	4	1.1	even	1	trivial
48.4.c.b.47.4	yes	4	12.11	even	2	inner
192.4.c.c.191.1		4	24.11	even	2
192.4.c.c.191.2		4	8.5	even	2
192.4.c.c.191.3		4	8.3	odd	2
192.4.c.c.191.4		4	24.5	odd	2
768.4.f.b.383.1		8	16.3	odd	4
768.4.f.b.383.2		8	16.5	even	4
768.4.f.b.383.3		8	48.5	odd	4
768.4.f.b.383.4		8	48.35	even	4
768.4.f.b.383.5		8	48.11	even	4
768.4.f.b.383.6		8	48.29	odd	4
768.4.f.b.383.7		8	16.13	even	4
768.4.f.b.383.8		8	16.11	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	48.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(4.89898 - 1.73205i) q^{3} -16.9706i q^{5} +17.3205i q^{7} +(21.0000 - 16.9706i) q^{9} +O(q^{10})\)	\(q+(4.89898 - 1.73205i) q^{3} -16.9706i q^{5} +17.3205i q^{7} +(21.0000 - 16.9706i) q^{9} +29.3939 q^{11} -26.0000 q^{13} +(-29.3939 - 83.1384i) q^{15} +67.8823i q^{17} +107.387i q^{19} +(30.0000 + 84.8528i) q^{21} -176.363 q^{23} -163.000 q^{25} +(73.4847 - 119.512i) q^{27} +16.9706i q^{29} +31.1769i q^{31} +(144.000 - 50.9117i) q^{33} +293.939 q^{35} +206.000 q^{37} +(-127.373 + 45.0333i) q^{39} -305.470i q^{41} +93.5307i q^{43} +(-288.000 - 356.382i) q^{45} +117.576 q^{47} +43.0000 q^{49} +(117.576 + 332.554i) q^{51} +50.9117i q^{53} -498.831i q^{55} +(186.000 + 526.087i) q^{57} -558.484 q^{59} +278.000 q^{61} +(293.939 + 363.731i) q^{63} +441.235i q^{65} -890.274i q^{67} +(-864.000 + 305.470i) q^{69} -58.7878 q^{71} -422.000 q^{73} +(-798.534 + 282.324i) q^{75} +509.117i q^{77} +668.572i q^{79} +(153.000 - 712.764i) q^{81} -29.3939 q^{83} +1152.00 q^{85} +(29.3939 + 83.1384i) q^{87} -373.352i q^{89} -450.333i q^{91} +(54.0000 + 152.735i) q^{93} +1822.42 q^{95} -1070.00 q^{97} +(617.271 - 498.831i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 84 q^{9}+O(q^{10}) \) 4 * q + 84 * q^9	\( 4 q + 84 q^{9} - 104 q^{13} + 120 q^{21} - 652 q^{25} + 576 q^{33} + 824 q^{37} - 1152 q^{45} + 172 q^{49} + 744 q^{57} + 1112 q^{61} - 3456 q^{69} - 1688 q^{73} + 612 q^{81} + 4608 q^{85} + 216 q^{93} - 4280 q^{97}+O(q^{100}) \) 4 * q + 84 * q^9 - 104 * q^13 + 120 * q^21 - 652 * q^25 + 576 * q^33 + 824 * q^37 - 1152 * q^45 + 172 * q^49 + 744 * q^57 + 1112 * q^61 - 3456 * q^69 - 1688 * q^73 + 612 * q^81 + 4608 * q^85 + 216 * q^93 - 4280 * q^97

Introduction

L-functions

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