LMFDB - Embedded newform 624.4.d.b.287.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(287,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("624.287");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.d (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:	$36.8171918436$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 120x^{10} + 5196x^{8} + 96803x^{6} + 702900x^{4} + 976752x^{2} + 254016 $ x^12 + 120x^10 + 5196x^8 + 96803x^6 + 702900x^4 + 976752*x^2 + 254016
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			287.1
Root			$4.65383i$ of defining polynomial
Character	$\chi$	$=$	624.287
Dual form			624.4.d.b.287.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-4.84460 - 1.87879i) q^{3} +18.9647i q^{5} -27.0499i q^{7} +(19.9403 + 18.2040i) q^{9} +9.39288 q^{11} -13.0000 q^{13} +(35.6306 - 91.8764i) q^{15} -1.52149i q^{17} -31.1981i q^{19} +(-50.8209 + 131.046i) q^{21} -145.367 q^{23} -234.660 q^{25} +(-62.4015 - 125.654i) q^{27} +32.9938i q^{29} +73.9450i q^{31} +(-45.5048 - 17.6472i) q^{33} +512.992 q^{35} +392.429 q^{37} +(62.9798 + 24.4242i) q^{39} -323.179i q^{41} +305.902i q^{43} +(-345.232 + 378.162i) q^{45} +29.9279 q^{47} -388.695 q^{49} +(-2.85856 + 7.37101i) q^{51} -415.331i q^{53} +178.133i q^{55} +(-58.6146 + 151.142i) q^{57} +765.208 q^{59} -89.5213 q^{61} +(492.414 - 539.383i) q^{63} -246.541i q^{65} -34.1410i q^{67} +(704.244 + 273.113i) q^{69} +534.416 q^{71} +730.823 q^{73} +(1136.83 + 440.876i) q^{75} -254.076i q^{77} -784.929i q^{79} +(66.2324 + 725.985i) q^{81} -782.388 q^{83} +28.8546 q^{85} +(61.9883 - 159.842i) q^{87} +1148.57i q^{89} +351.648i q^{91} +(138.927 - 358.234i) q^{93} +591.663 q^{95} +1665.46 q^{97} +(187.297 + 170.988i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 54 q^{9} - 156 q^{13} - 54 q^{21} - 408 q^{25} - 360 q^{33} + 636 q^{37} - 810 q^{45} + 336 q^{49} - 1260 q^{57} + 960 q^{61} - 252 q^{69} + 3216 q^{73} - 2538 q^{81} + 2196 q^{85} - 1116 q^{93}+ \cdots + 4800 q^{97}+O(q^{100}) $ 12 * q + 54 * q^9 - 156 * q^13 - 54 * q^21 - 408 * q^25 - 360 * q^33 + 636 * q^37 - 810 * q^45 + 336 * q^49 - 1260 * q^57 + 960 * q^61 - 252 * q^69 + 3216 * q^73 - 2538 * q^81 + 2196 * q^85 - 1116 * q^93 + 4800 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	−4.84460	−	1.87879i	−0.932344	−	0.361573i
$3$	−4.84460	−	1.87879i	−0.932344	−	0.361573i
$4$		0			0
$5$			18.9647i			1.69625i	0.529793	+	0.848127i	$0.322270\pi$
$5$			18.9647i			1.69625i	−0.529793	+	0.848127i	$0.677730\pi$
$6$		0			0
$7$		−	27.0499i		−	1.46056i	−0.683150	−	0.730278i	$-0.739391\pi$
$7$		−	27.0499i		−	1.46056i	0.683150	−	0.730278i	$-0.260609\pi$
$8$		0			0
$9$	19.9403	+	18.2040i	0.738530	+	0.674220i
$10$		0			0
$11$	9.39288			0.257460			0.128730	−	0.991680i	$-0.458910\pi$
$11$	9.39288			0.257460			0.128730	+	0.991680i	$0.458910\pi$
$12$		0			0
$13$	−13.0000			−0.277350
$13$	−13.0000			−0.277350
$14$		0			0
$15$	35.6306	−	91.8764i	0.613319	−	1.58149i
$16$		0			0
$17$		−	1.52149i		−	0.0217068i	−0.999941	−	0.0108534i	$-0.996545\pi$
$17$		−	1.52149i		−	0.0217068i	0.999941	−	0.0108534i	$-0.00345481\pi$
$18$		0			0
$19$		−	31.1981i		−	0.376702i	−0.982102	−	0.188351i	$-0.939686\pi$
$19$		−	31.1981i		−	0.376702i	0.982102	−	0.188351i	$-0.0603143\pi$
$20$		0			0
$21$	−50.8209	+	131.046i	−0.528097	+	1.36174i
$22$		0			0
$23$	−145.367			−1.31787			−0.658936	−	0.752199i	$-0.728993\pi$
$23$	−145.367			−1.31787			−0.658936	+	0.752199i	$0.728993\pi$
$24$		0			0
$25$	−234.660			−1.87728
$26$		0			0
$27$	−62.4015	−	125.654i	−0.444784	−	0.895638i
$28$		0			0
$29$			32.9938i			0.211269i	0.994405	+	0.105634i	$0.0336873\pi$
$29$			32.9938i			0.211269i	−0.994405	+	0.105634i	$0.966313\pi$
$30$		0			0
$31$			73.9450i			0.428417i	0.976788	+	0.214208i	$0.0687171\pi$
$31$			73.9450i			0.428417i	−0.976788	+	0.214208i	$0.931283\pi$
$32$		0			0
$33$	−45.5048	−	17.6472i	−0.240041	−	0.0930905i
$34$		0			0
$35$	512.992			2.47747
$36$		0			0
$37$	392.429			1.74365			0.871824	−	0.489820i	$-0.162937\pi$
$37$	392.429			1.74365			0.871824	+	0.489820i	$0.162937\pi$
$38$		0			0
$39$	62.9798	+	24.4242i	0.258586	+	0.100282i
$40$		0			0
$41$		−	323.179i		−	1.23103i	−0.788127	−	0.615513i	$-0.788949\pi$
$41$		−	323.179i		−	1.23103i	0.788127	−	0.615513i	$-0.211051\pi$
$42$		0			0
$43$			305.902i			1.08487i	0.840096	+	0.542437i	$0.182498\pi$
$43$			305.902i			1.08487i	−0.840096	+	0.542437i	$0.817502\pi$
$44$		0			0
$45$	−345.232	+	378.162i	−1.14365	+	1.25273i
$46$		0			0
$47$	29.9279			0.0928815			0.0464407	−	0.998921i	$-0.485212\pi$
$47$	29.9279			0.0928815			0.0464407	+	0.998921i	$0.485212\pi$
$48$		0			0
$49$	−388.695			−1.13322
$50$		0			0
$51$	−2.85856	+	7.37101i	−0.00784859	+	0.0202382i
$52$		0			0
$53$		−	415.331i		−	1.07642i	−0.842812	−	0.538208i	$-0.819101\pi$
$53$		−	415.331i		−	1.07642i	0.842812	−	0.538208i	$-0.180899\pi$
$54$		0			0
$55$			178.133i			0.436717i
$56$		0			0
$57$	−58.6146	+	151.142i	−0.136205	+	0.351216i
$58$		0			0
$59$	765.208			1.68850			0.844251	−	0.535948i	$-0.180046\pi$
$59$	765.208			1.68850			0.844251	+	0.535948i	$0.180046\pi$
$60$		0			0
$61$	−89.5213			−0.187902			−0.0939510	−	0.995577i	$-0.529950\pi$
$61$	−89.5213			−0.187902			−0.0939510	+	0.995577i	$0.529950\pi$
$62$		0			0
$63$	492.414	−	539.383i	0.984736	−	1.07866i
$64$		0			0
$65$		−	246.541i		−	0.470456i
$66$		0			0
$67$		−	34.1410i		−	0.0622535i	−0.999515	−	0.0311267i	$-0.990090\pi$
$67$		−	34.1410i		−	0.0622535i	0.999515	−	0.0311267i	$-0.00990954\pi$
$68$		0			0
$69$	704.244	+	273.113i	1.22871	+	0.476507i
$70$		0			0
$71$	534.416			0.893290			0.446645	−	0.894711i	$-0.352619\pi$
$71$	534.416			0.893290			0.446645	+	0.894711i	$0.352619\pi$
$72$		0			0
$73$	730.823			1.17173			0.585865	−	0.810408i	$-0.300755\pi$
$73$	730.823			1.17173			0.585865	+	0.810408i	$0.300755\pi$
$74$		0			0
$75$	1136.83	+	440.876i	1.75027	+	0.678772i
$76$		0			0
$77$		−	254.076i		−	0.376035i
$78$		0			0
$79$		−	784.929i		−	1.11787i	−0.829213	−	0.558933i	$-0.811211\pi$
$79$		−	784.929i		−	1.11787i	0.829213	−	0.558933i	$-0.188789\pi$
$80$		0			0
$81$	66.2324	+	725.985i	0.0908537	+	0.995864i
$82$		0			0
$83$	−782.388			−1.03468			−0.517339	−	0.855781i	$-0.673077\pi$
$83$	−782.388			−1.03468			−0.517339	+	0.855781i	$0.673077\pi$
$84$		0			0
$85$	28.8546			0.0368202
$86$		0			0
$87$	61.9883	−	159.842i	0.0763890	−	0.196975i
$88$		0			0
$89$			1148.57i			1.36796i	0.729502	+	0.683979i	$0.239752\pi$
$89$			1148.57i			1.36796i	−0.729502	+	0.683979i	$0.760248\pi$
$90$		0			0
$91$			351.648i			0.405085i
$92$		0			0
$93$	138.927	−	358.234i	0.154904	−	0.399432i
$94$		0			0
$95$	591.663			0.638982
$96$		0			0
$97$	1665.46			1.74332			0.871660	−	0.490111i	$-0.163044\pi$
$97$	1665.46			1.74332			0.871660	+	0.490111i	$0.163044\pi$
$98$		0			0
$99$	187.297	+	170.988i	0.190142	+	0.173585i
$100$		0			0
$101$		−	1115.43i		−	1.09891i	−0.835525	−	0.549453i	$-0.814836\pi$
$101$		−	1115.43i		−	1.09891i	0.835525	−	0.549453i	$-0.185164\pi$
$102$		0			0
$103$		−	1318.22i		−	1.26105i	−0.776168	−	0.630527i	$-0.782839\pi$
$103$		−	1318.22i		−	1.26105i	0.776168	−	0.630527i	$-0.217161\pi$
$104$		0			0
$105$	−2485.24	−	963.804i	−2.30986	−	0.895787i
$106$		0			0
$107$	1822.87			1.64695			0.823474	−	0.567354i	$-0.192033\pi$
$107$	1822.87			1.64695			0.823474	+	0.567354i	$0.192033\pi$
$108$		0			0
$109$	2001.88			1.75913			0.879564	−	0.475780i	$-0.157834\pi$
$109$	2001.88			1.75913			0.879564	+	0.475780i	$0.157834\pi$
$110$		0			0
$111$	−1901.16	−	737.291i	−1.62568	−	0.630456i
$112$		0			0
$113$		−	646.361i		−	0.538093i	−0.963127	−	0.269047i	$-0.913291\pi$
$113$		−	646.361i		−	0.538093i	0.963127	−	0.269047i	$-0.0867085\pi$
$114$		0			0
$115$		−	2756.84i		−	2.23545i
$116$		0			0
$117$	−259.224	−	236.651i	−0.204831	−	0.186995i
$118$		0			0
$119$	−41.1561			−0.0317040
$120$		0			0
$121$	−1242.77			−0.933714
$122$		0			0
$123$	−607.185	+	1565.67i	−0.445106	+	1.14774i
$124$		0			0
$125$		−	2079.66i		−	1.48809i
$126$		0			0
$127$		−	1825.33i		−	1.27537i	−0.770298	−	0.637684i	$-0.779893\pi$
$127$		−	1825.33i		−	1.27537i	0.770298	−	0.637684i	$-0.220107\pi$
$128$		0			0
$129$	574.724	−	1481.97i	0.392261	−	1.01148i
$130$		0			0
$131$	2253.25			1.50280			0.751402	−	0.659844i	$-0.229378\pi$
$131$	2253.25			1.50280			0.751402	+	0.659844i	$0.229378\pi$
$132$		0			0
$133$	−843.905			−0.550194
$134$		0			0
$135$	2383.00	−	1183.43i	1.51923	−	0.754467i
$136$		0			0
$137$			1546.24i			0.964265i	0.876098	+	0.482132i	$0.160138\pi$
$137$			1546.24i			0.964265i	−0.876098	+	0.482132i	$0.839862\pi$
$138$		0			0
$139$			747.775i			0.456298i	0.973626	+	0.228149i	$0.0732674\pi$
$139$			747.775i			0.456298i	−0.973626	+	0.228149i	$0.926733\pi$
$140$		0			0
$141$	−144.989	−	56.2281i	−0.0865975	−	0.0335834i
$142$		0			0
$143$	−122.107			−0.0714065
$144$		0			0
$145$	−625.717			−0.358365
$146$		0			0
$147$	1883.07	+	730.276i	1.05655	+	0.409742i
$148$		0			0
$149$			2308.42i			1.26922i	0.772833	+	0.634609i	$0.218839\pi$
$149$			2308.42i			1.26922i	−0.772833	+	0.634609i	$0.781161\pi$
$150$		0			0
$151$			1897.60i			1.02268i	0.859378	+	0.511340i	$0.170851\pi$
$151$			1897.60i			1.02268i	−0.859378	+	0.511340i	$0.829149\pi$
$152$		0			0
$153$	27.6971	−	30.3390i	0.0146352	−	0.0160311i
$154$		0			0
$155$	−1402.34			−0.726703
$156$		0			0
$157$	1538.82			0.782237			0.391118	−	0.920340i	$-0.372088\pi$
$157$	1538.82			0.782237			0.391118	+	0.920340i	$0.372088\pi$
$158$		0			0
$159$	−780.318	+	2012.11i	−0.389203	+	1.00359i
$160$		0			0
$161$			3932.15i			1.92483i
$162$		0			0
$163$		−	3933.13i		−	1.88998i	−0.327099	−	0.944990i	$-0.606071\pi$
$163$		−	3933.13i		−	1.88998i	0.327099	−	0.944990i	$-0.393929\pi$
$164$		0			0
$165$	334.674	−	862.984i	0.157905	−	0.407171i
$166$		0			0
$167$	−274.828			−0.127346			−0.0636732	−	0.997971i	$-0.520282\pi$
$167$	−274.828			−0.127346			−0.0636732	+	0.997971i	$0.520282\pi$
$168$		0			0
$169$	169.000			0.0769231
$170$		0			0
$171$	567.929	−	622.100i	0.253980	−	0.278206i
$172$		0			0
$173$		−	1086.77i		−	0.477607i	−0.971068	−	0.238803i	$-0.923245\pi$
$173$		−	1086.77i		−	0.477607i	0.971068	−	0.238803i	$-0.0767551\pi$
$174$		0			0
$175$			6347.51i			2.74187i
$176$		0			0
$177$	−3707.13	−	1437.66i	−1.57426	−	0.610516i
$178$		0			0
$179$	−730.004			−0.304821			−0.152411	−	0.988317i	$-0.548704\pi$
$179$	−730.004			−0.304821			−0.152411	+	0.988317i	$0.548704\pi$
$180$		0			0
$181$	−1845.20			−0.757751			−0.378875	−	0.925448i	$-0.623689\pi$
$181$	−1845.20			−0.757751			−0.378875	+	0.925448i	$0.623689\pi$
$182$		0			0
$183$	433.695	+	168.191i	0.175189	+	0.0679403i
$184$		0			0
$185$			7442.30i			2.95767i
$186$		0			0
$187$		−	14.2912i		−	0.00558863i
$188$		0			0
$189$	−3398.94	+	1687.95i	−1.30813	+	0.649632i
$190$		0			0
$191$	1806.56			0.684389			0.342195	−	0.939629i	$-0.388830\pi$
$191$	1806.56			0.684389			0.342195	+	0.939629i	$0.388830\pi$
$192$		0			0
$193$	2597.91			0.968920			0.484460	−	0.874813i	$-0.339016\pi$
$193$	2597.91			0.968920			0.484460	+	0.874813i	$0.339016\pi$
$194$		0			0
$195$	−463.198	+	1194.39i	−0.170104	+	0.438627i
$196$		0			0
$197$		−	573.841i		−	0.207535i	−0.994602	−	0.103768i	$-0.966910\pi$
$197$		−	573.841i		−	0.207535i	0.994602	−	0.103768i	$-0.0330899\pi$
$198$		0			0
$199$			5308.02i			1.89083i	0.325864	+	0.945417i	$0.394345\pi$
$199$			5308.02i			1.89083i	−0.325864	+	0.945417i	$0.605655\pi$
$200$		0			0
$201$	−64.1436	+	165.399i	−0.0225092	+	0.0580416i
$202$		0			0
$203$	892.477			0.308570
$204$		0			0
$205$	6128.99			2.08813
$206$		0			0
$207$	−2898.66	−	2646.25i	−0.973288	−	0.888536i
$208$		0			0
$209$		−	293.040i		−	0.0969857i
$210$		0			0
$211$			1494.94i			0.487755i	0.969806	+	0.243877i	$0.0784194\pi$
$211$			1494.94i			0.487755i	−0.969806	+	0.243877i	$0.921581\pi$
$212$		0			0
$213$	−2589.03	−	1004.05i	−0.832853	−	0.322989i
$214$		0			0
$215$	−5801.33			−1.84022
$216$		0			0
$217$	2000.20			0.625726
$218$		0			0
$219$	−3540.55	−	1373.06i	−1.09246	−	0.423666i
$220$		0			0
$221$			19.7794i			0.00602038i
$222$		0			0
$223$			2016.71i			0.605599i	0.953054	+	0.302800i	$0.0979213\pi$
$223$			2016.71i			0.605599i	−0.953054	+	0.302800i	$0.902079\pi$
$224$		0			0
$225$	−4679.19	−	4271.73i	−1.38643	−	1.26570i
$226$		0			0
$227$	3386.54			0.990189			0.495094	−	0.868839i	$-0.335134\pi$
$227$	3386.54			0.990189			0.495094	+	0.868839i	$0.335134\pi$
$228$		0			0
$229$	−1306.25			−0.376942			−0.188471	−	0.982079i	$-0.560353\pi$
$229$	−1306.25			−0.376942			−0.188471	+	0.982079i	$0.560353\pi$
$230$		0			0
$231$	−477.355	+	1230.90i	−0.135964	+	0.350594i
$232$		0			0
$233$			965.637i			0.271506i	0.990743	+	0.135753i	$0.0433454\pi$
$233$			965.637i			0.271506i	−0.990743	+	0.135753i	$0.956655\pi$
$234$		0			0
$235$			567.573i			0.157551i
$236$		0			0
$237$	−1474.71	+	3802.67i	−0.404190	+	1.04224i
$238$		0			0
$239$	−4920.68			−1.33177			−0.665884	−	0.746055i	$-0.731945\pi$
$239$	−4920.68			−1.33177			−0.665884	+	0.746055i	$0.731945\pi$
$240$		0			0
$241$	−621.817			−0.166202			−0.0831011	−	0.996541i	$-0.526482\pi$
$241$	−621.817			−0.166202			−0.0831011	+	0.996541i	$0.526482\pi$
$242$		0			0
$243$	1043.10	−	3641.54i	0.275371	−	0.961338i
$244$		0			0
$245$		−	7371.49i		−	1.92223i
$246$		0			0
$247$			405.575i			0.104478i
$248$		0			0
$249$	3790.36	+	1469.94i	0.964675	+	0.374111i
$250$		0			0
$251$	−4513.14			−1.13493			−0.567464	−	0.823398i	$-0.692075\pi$
$251$	−4513.14			−1.13493			−0.567464	+	0.823398i	$0.692075\pi$
$252$		0			0
$253$	−1365.41			−0.339299
$254$		0			0
$255$	−139.789	−	54.2116i	−0.0343291	−	0.0133132i
$256$		0			0
$257$		−	3835.60i		−	0.930965i	−0.885057	−	0.465483i	$-0.845881\pi$
$257$		−	3835.60i		−	0.930965i	0.885057	−	0.465483i	$-0.154119\pi$
$258$		0			0
$259$		−	10615.2i		−	2.54669i
$260$		0			0
$261$	−600.617	+	657.906i	−0.142442	+	0.156028i
$262$		0			0
$263$	−2295.90			−0.538293			−0.269147	−	0.963099i	$-0.586742\pi$
$263$	−2295.90			−0.538293			−0.269147	+	0.963099i	$0.586742\pi$
$264$		0			0
$265$	7876.62			1.82588
$266$		0			0
$267$	2157.92	−	5564.37i	0.494616	−	1.27541i
$268$		0			0
$269$			4332.30i			0.981952i	0.871173	+	0.490976i	$0.163360\pi$
$269$			4332.30i			0.981952i	−0.871173	+	0.490976i	$0.836640\pi$
$270$		0			0
$271$		−	7712.96i		−	1.72889i	−0.502727	−	0.864445i	$-0.667670\pi$
$271$		−	7712.96i		−	1.72889i	0.502727	−	0.864445i	$-0.332330\pi$
$272$		0			0
$273$	660.672	−	1703.60i	0.146468	−	0.377679i
$274$		0			0
$275$	−2204.13			−0.483324
$276$		0			0
$277$	−707.242			−0.153408			−0.0767040	−	0.997054i	$-0.524440\pi$
$277$	−707.242			−0.153408			−0.0767040	+	0.997054i	$0.524440\pi$
$278$		0			0
$279$	−1346.09	+	1474.49i	−0.288847	+	0.316399i
$280$		0			0
$281$		−	8445.74i		−	1.79299i	−0.443052	−	0.896496i	$-0.646104\pi$
$281$		−	8445.74i		−	1.79299i	0.443052	−	0.896496i	$-0.353896\pi$
$282$		0			0
$283$			5277.23i			1.10848i	0.832358	+	0.554238i	$0.186990\pi$
$283$			5277.23i			1.10848i	−0.832358	+	0.554238i	$0.813010\pi$
$284$		0			0
$285$	−2866.37	−	1111.61i	−0.595751	−	0.231039i
$286$		0			0
$287$	−8741.95			−1.79798
$288$		0			0
$289$	4910.69			0.999529
$290$		0			0
$291$	−8068.50	−	3129.05i	−1.62537	−	0.630337i
$292$		0			0
$293$		−	2522.55i		−	0.502966i	−0.967862	−	0.251483i	$-0.919082\pi$
$293$		−	2522.55i		−	0.502966i	0.967862	−	0.251483i	$-0.0809182\pi$
$294$		0			0
$295$			14511.9i			2.86413i
$296$		0			0
$297$	−586.130	−	1180.26i	−0.114514	−	0.230591i
$298$		0			0
$299$	1889.77			0.365512
$300$		0			0
$301$	8274.60			1.58452
$302$		0			0
$303$	−2095.66	+	5403.82i	−0.397334	+	1.02456i
$304$		0			0
$305$		−	1697.74i		−	0.318730i
$306$		0			0
$307$		−	3496.71i		−	0.650057i	−0.945704	−	0.325029i	$-0.894626\pi$
$307$		−	3496.71i		−	0.650057i	0.945704	−	0.325029i	$-0.105374\pi$
$308$		0			0
$309$	−2476.66	+	6386.27i	−0.455963	+	1.17574i
$310$		0			0
$311$	−4140.52			−0.754943			−0.377471	−	0.926021i	$-0.623206\pi$
$311$	−4140.52			−0.754943			−0.377471	+	0.926021i	$0.623206\pi$
$312$		0			0
$313$	−8963.98			−1.61877			−0.809383	−	0.587280i	$-0.800199\pi$
$313$	−8963.98			−1.61877			−0.809383	+	0.587280i	$0.800199\pi$
$314$		0			0
$315$	10229.2	+	9338.49i	1.82969	+	1.67036i
$316$		0			0
$317$		−	4352.50i		−	0.771170i	−0.922672	−	0.385585i	$-0.874000\pi$
$317$		−	4352.50i		−	0.771170i	0.922672	−	0.385585i	$-0.126000\pi$
$318$		0			0
$319$			309.907i			0.0543932i
$320$		0			0
$321$	−8831.08	−	3424.79i	−1.53552	−	0.595492i
$322$		0			0
$323$	−47.4676			−0.00817699
$324$		0			0
$325$	3050.58			0.520663
$326$		0			0
$327$	−9698.29	−	3761.10i	−1.64011	−	0.636053i
$328$		0			0
$329$		−	809.545i		−	0.135659i
$330$		0			0
$331$			329.402i			0.0546996i	0.999626	+	0.0273498i	$0.00870680\pi$
$331$			329.402i			0.0546996i	−0.999626	+	0.0273498i	$0.991293\pi$
$332$		0			0
$333$	7825.16	+	7143.76i	1.28774	+	1.17560i
$334$		0			0
$335$	647.473			0.105598
$336$		0			0
$337$	7316.96			1.18273			0.591365	−	0.806404i	$-0.298589\pi$
$337$	7316.96			1.18273			0.591365	+	0.806404i	$0.298589\pi$
$338$		0			0
$339$	−1214.37	+	3131.36i	−0.194560	+	0.501688i
$340$		0			0
$341$			694.556i			0.110300i
$342$		0			0
$343$			1236.05i			0.194578i
$344$		0			0
$345$	−5179.51	+	13355.8i	−0.808276	+	2.08420i
$346$		0			0
$347$	4803.87			0.743185			0.371592	−	0.928396i	$-0.378812\pi$
$347$	4803.87			0.743185			0.371592	+	0.928396i	$0.378812\pi$
$348$		0			0
$349$	7003.62			1.07420			0.537099	−	0.843519i	$-0.319520\pi$
$349$	7003.62			1.07420			0.537099	+	0.843519i	$0.319520\pi$
$350$		0			0
$351$	811.220	+	1633.51i	0.123361	+	0.248405i
$352$		0			0
$353$		−	8180.33i		−	1.23341i	−0.787193	−	0.616707i	$-0.788466\pi$
$353$		−	8180.33i		−	1.23341i	0.787193	−	0.616707i	$-0.211534\pi$
$354$		0			0
$355$			10135.0i			1.51525i
$356$		0			0
$357$	199.385	+	77.3235i	0.0295590	+	0.0114633i
$358$		0			0
$359$	5651.15			0.830798			0.415399	−	0.909639i	$-0.363642\pi$
$359$	5651.15			0.830798			0.415399	+	0.909639i	$0.363642\pi$
$360$		0			0
$361$	5885.68			0.858096
$362$		0			0
$363$	6020.74	+	2334.91i	0.870543	+	0.337606i
$364$		0			0
$365$			13859.8i			1.98755i
$366$		0			0
$367$		−	7110.90i		−	1.01141i	−0.862708	−	0.505703i	$-0.831233\pi$
$367$		−	7110.90i		−	1.01141i	0.862708	−	0.505703i	$-0.168767\pi$
$368$		0			0
$369$	5883.13	−	6444.29i	0.829983	−	0.909150i
$370$		0			0
$371$	−11234.6			−1.57217
$372$		0			0
$373$	2779.65			0.385857			0.192928	−	0.981213i	$-0.438201\pi$
$373$	2779.65			0.385857			0.192928	+	0.981213i	$0.438201\pi$
$374$		0			0
$375$	−3907.24	+	10075.1i	−0.538051	+	1.38741i
$376$		0			0
$377$		−	428.919i		−	0.0585954i
$378$		0			0
$379$			3707.98i			0.502549i	0.967916	+	0.251274i	$0.0808496\pi$
$379$			3707.98i			0.502549i	−0.967916	+	0.251274i	$0.919150\pi$
$380$		0			0
$381$	−3429.40	+	8842.98i	−0.461138	+	1.18908i
$382$		0			0
$383$	−8269.75			−1.10330			−0.551651	−	0.834075i	$-0.686002\pi$
$383$	−8269.75			−1.10330			−0.551651	+	0.834075i	$0.686002\pi$
$384$		0			0
$385$	4818.48			0.637850
$386$		0			0
$387$	−5568.62	+	6099.78i	−0.731444	+	0.801212i
$388$		0			0
$389$			5001.39i			0.651878i	0.945391	+	0.325939i	$0.105680\pi$
$389$			5001.39i			0.651878i	−0.945391	+	0.325939i	$0.894320\pi$
$390$		0			0
$391$			221.174i			0.0286068i
$392$		0			0
$393$	−10916.1	−	4233.38i	−1.40113	−	0.543373i
$394$		0			0
$395$	14885.9			1.89618
$396$		0			0
$397$	−9115.07			−1.15232			−0.576161	−	0.817336i	$-0.695450\pi$
$397$	−9115.07			−1.15232			−0.576161	+	0.817336i	$0.695450\pi$
$398$		0			0
$399$	4088.38	+	1585.52i	0.512970	+	0.198935i
$400$		0			0
$401$			2574.10i			0.320559i	0.987072	+	0.160280i	$0.0512396\pi$
$401$			2574.10i			0.320559i	−0.987072	+	0.160280i	$0.948760\pi$
$402$		0			0
$403$		−	961.285i		−	0.118821i
$404$		0			0
$405$	−13768.1	+	1256.08i	−1.68924	+	0.154111i
$406$		0			0
$407$	3686.04			0.448919
$408$		0			0
$409$	−509.413			−0.0615864			−0.0307932	−	0.999526i	$-0.509803\pi$
$409$	−509.413			−0.0615864			−0.0307932	+	0.999526i	$0.509803\pi$
$410$		0			0
$411$	2905.06	−	7490.92i	0.348652	−	0.899026i
$412$		0			0
$413$		−	20698.8i		−	2.46615i
$414$		0			0
$415$		−	14837.8i		−	1.75508i
$416$		0			0
$417$	1404.91	−	3622.67i	0.164985	−	0.425427i
$418$		0			0
$419$	3747.50			0.436938			0.218469	−	0.975844i	$-0.429894\pi$
$419$	3747.50			0.436938			0.218469	+	0.975844i	$0.429894\pi$
$420$		0			0
$421$	−13871.8			−1.60587			−0.802933	−	0.596069i	$-0.796729\pi$
$421$	−13871.8			−1.60587			−0.802933	+	0.596069i	$0.796729\pi$
$422$		0			0
$423$	596.771	+	544.805i	0.0685958	+	0.0626226i
$424$		0			0
$425$			357.032i			0.0407497i
$426$		0			0
$427$			2421.54i			0.274441i
$428$		0			0
$429$	591.562	+	229.414i	0.0665755	+	0.0258187i
$430$		0			0
$431$	1861.83			0.208078			0.104039	−	0.994573i	$-0.466823\pi$
$431$	1861.83			0.208078			0.104039	+	0.994573i	$0.466823\pi$
$432$		0			0
$433$	9690.92			1.07556			0.537778	−	0.843086i	$-0.319264\pi$
$433$	9690.92			1.07556			0.537778	+	0.843086i	$0.319264\pi$
$434$		0			0
$435$	3031.35	+	1175.59i	0.334120	+	0.129575i
$436$		0			0
$437$			4535.17i			0.496445i
$438$		0			0
$439$		−	13719.8i		−	1.49159i	−0.666173	−	0.745797i	$-0.732069\pi$
$439$		−	13719.8i		−	1.49159i	0.666173	−	0.745797i	$-0.267931\pi$
$440$		0			0
$441$	−7750.71	−	7075.79i	−0.836919	−	0.764042i
$442$		0			0
$443$	5826.49			0.624886			0.312443	−	0.949936i	$-0.398853\pi$
$443$	5826.49			0.624886			0.312443	+	0.949936i	$0.398853\pi$
$444$		0			0
$445$	−21782.3			−2.32040
$446$		0			0
$447$	4337.04	−	11183.4i	0.458915	−	1.18335i
$448$		0			0
$449$			10377.1i			1.09071i	0.838206	+	0.545353i	$0.183605\pi$
$449$			10377.1i			1.09071i	−0.838206	+	0.545353i	$0.816395\pi$
$450$		0			0
$451$		−	3035.58i		−	0.316940i
$452$		0			0
$453$	3565.19	−	9193.13i	0.369773	−	0.953490i
$454$		0			0
$455$	−6668.90			−0.687127
$456$		0			0
$457$	5187.07			0.530942			0.265471	−	0.964119i	$-0.414472\pi$
$457$	5187.07			0.530942			0.265471	+	0.964119i	$0.414472\pi$
$458$		0			0
$459$	−191.182	+	94.9433i	−0.0194414	+	0.00965484i
$460$		0			0
$461$		−	6627.57i		−	0.669581i	−0.942293	−	0.334791i	$-0.891334\pi$
$461$		−	6627.57i		−	0.669581i	0.942293	−	0.334791i	$-0.108666\pi$
$462$		0			0
$463$		−	6406.83i		−	0.643089i	−0.946894	−	0.321545i	$-0.895798\pi$
$463$		−	6406.83i		−	0.643089i	0.946894	−	0.321545i	$-0.104202\pi$
$464$		0			0
$465$	6793.80	+	2634.71i	0.677537	+	0.262756i
$466$		0			0
$467$	9510.16			0.942350			0.471175	−	0.882040i	$-0.343830\pi$
$467$	9510.16			0.942350			0.471175	+	0.882040i	$0.343830\pi$
$468$		0			0
$469$	−923.508			−0.0909246
$470$		0			0
$471$	−7454.96	−	2891.11i	−0.729313	−	0.282835i
$472$		0			0
$473$			2873.30i			0.279312i
$474$		0			0
$475$			7320.94i			0.707174i
$476$		0			0
$477$	7560.66	−	8281.82i	0.725742	−	0.794966i
$478$		0			0
$479$	−20500.8			−1.95555			−0.977773	−	0.209668i	$-0.932762\pi$
$479$	−20500.8			−1.95555			−0.977773	+	0.209668i	$0.932762\pi$
$480$		0			0
$481$	−5101.58			−0.483601
$482$		0			0
$483$	7387.67	−	19049.7i	0.695964	−	1.79460i
$484$		0			0
$485$			31585.0i			2.95711i
$486$		0			0
$487$		−	3216.16i		−	0.299257i	−0.988742	−	0.149629i	$-0.952192\pi$
$487$		−	3216.16i		−	0.299257i	0.988742	−	0.149629i	$-0.0478078\pi$
$488$		0			0
$489$	−7389.52	+	19054.5i	−0.683365	+	1.76211i
$490$		0			0
$491$	14594.4			1.34141			0.670707	−	0.741723i	$-0.265991\pi$
$491$	14594.4			1.34141			0.670707	+	0.741723i	$0.265991\pi$
$492$		0			0
$493$	50.1997			0.00458596
$494$		0			0
$495$	−3242.73	+	3552.03i	−0.294444	+	0.322529i
$496$		0			0
$497$		−	14455.9i		−	1.30470i
$498$		0			0
$499$		−	369.555i		−	0.0331534i	−0.999863	−	0.0165767i	$-0.994723\pi$
$499$		−	369.555i		−	0.0331534i	0.999863	−	0.0165767i	$-0.00527678\pi$
$500$		0			0
$501$	1331.43	+	516.344i	0.118731	+	0.0460450i
$502$		0			0
$503$	−3667.35			−0.325088			−0.162544	−	0.986701i	$-0.551970\pi$
$503$	−3667.35			−0.325088			−0.162544	+	0.986701i	$0.551970\pi$
$504$		0			0
$505$	21153.8			1.86402
$506$		0			0
$507$	−818.738	−	317.515i	−0.0717188	−	0.0278133i
$508$		0			0
$509$		−	5884.32i		−	0.512412i	−0.966622	−	0.256206i	$-0.917527\pi$
$509$		−	5884.32i		−	0.512412i	0.966622	−	0.256206i	$-0.0824726\pi$
$510$		0			0
$511$		−	19768.7i		−	1.71138i
$512$		0			0
$513$	−3920.18	+	1946.81i	−0.337388	+	0.167551i
$514$		0			0
$515$	24999.7			2.13907
$516$		0			0
$517$	281.109			0.0239133
$518$		0			0
$519$	−2041.82	+	5264.99i	−0.172690	+	0.445294i
$520$		0			0
$521$			18495.2i			1.55526i	0.628722	+	0.777630i	$0.283578\pi$
$521$			18495.2i			1.55526i	−0.628722	+	0.777630i	$0.716422\pi$
$522$		0			0
$523$		−	10139.5i		−	0.847746i	−0.905722	−	0.423873i	$-0.860670\pi$
$523$		−	10139.5i		−	0.847746i	0.905722	−	0.423873i	$-0.139330\pi$
$524$		0			0
$525$	11925.6	−	30751.2i	0.991385	−	2.55636i
$526$		0			0
$527$	112.507			0.00929955
$528$		0			0
$529$	8964.49			0.736787
$530$		0			0
$531$	15258.5	+	13929.8i	1.24701	+	1.13842i
$532$		0			0
$533$			4201.33i			0.341425i
$534$		0			0
$535$			34570.2i			2.79364i
$536$		0			0
$537$	3536.58	+	1371.52i	0.284198	+	0.110215i
$538$		0			0
$539$	−3650.97			−0.291759
$540$		0			0
$541$	754.389			0.0599514			0.0299757	−	0.999551i	$-0.490457\pi$
$541$	754.389			0.0599514			0.0299757	+	0.999551i	$0.490457\pi$
$542$		0			0
$543$	8939.27	+	3466.75i	0.706484	+	0.273982i
$544$		0			0
$545$			37965.0i			2.98393i
$546$		0			0
$547$			7114.45i			0.556110i	0.960565	+	0.278055i	$0.0896897\pi$
$547$			7114.45i			0.556110i	−0.960565	+	0.278055i	$0.910310\pi$
$548$		0			0
$549$	−1785.08	−	1629.64i	−0.138771	−	0.126687i
$550$		0			0
$551$	1029.34			0.0795853
$552$		0			0
$553$	−21232.2			−1.63271
$554$		0			0
$555$	13982.5	−	36055.0i	1.06941	−	2.75756i
$556$		0			0
$557$		−	6758.58i		−	0.514130i	−0.966394	−	0.257065i	$-0.917245\pi$
$557$		−	6758.58i		−	0.514130i	0.966394	−	0.257065i	$-0.0827554\pi$
$558$		0			0
$559$		−	3976.72i		−	0.300890i
$560$		0			0
$561$	−26.8501	+	69.2350i	−0.00202070	+	0.00521052i
$562$		0			0
$563$	−11775.0			−0.881452			−0.440726	−	0.897642i	$-0.645279\pi$
$563$	−11775.0			−0.881452			−0.440726	+	0.897642i	$0.645279\pi$
$564$		0			0
$565$	12258.0			0.912742
$566$		0			0
$567$	19637.8	−	1791.58i	1.45451	−	0.132697i
$568$		0			0
$569$		−	24159.5i		−	1.78000i	−0.455962	−	0.889999i	$-0.650705\pi$
$569$		−	24159.5i		−	1.78000i	0.455962	−	0.889999i	$-0.349295\pi$
$570$		0			0
$571$			14791.5i			1.08408i	0.840354	+	0.542038i	$0.182347\pi$
$571$			14791.5i			1.08408i	−0.840354	+	0.542038i	$0.817653\pi$
$572$		0			0
$573$	−8752.08	−	3394.15i	−0.638086	−	0.247457i
$574$		0			0
$575$	34111.7			2.47401
$576$		0			0
$577$	−678.760			−0.0489725			−0.0244863	−	0.999700i	$-0.507795\pi$
$577$	−678.760			−0.0489725			−0.0244863	+	0.999700i	$0.507795\pi$
$578$		0			0
$579$	−12585.8	−	4880.92i	−0.903367	−	0.350335i
$580$		0			0
$581$			21163.5i			1.51120i
$582$		0			0
$583$		−	3901.15i		−	0.277134i
$584$		0			0
$585$	4488.02	−	4916.11i	0.317191	−	0.347446i
$586$		0			0
$587$	−17118.7			−1.20368			−0.601842	−	0.798615i	$-0.705566\pi$
$587$	−17118.7			−1.20368			−0.601842	+	0.798615i	$0.705566\pi$
$588$		0			0
$589$	2306.94			0.161385
$590$		0			0
$591$	−1078.13	+	2780.03i	−0.0750392	+	0.193494i
$592$		0			0
$593$		−	1677.81i		−	0.116188i	−0.998311	−	0.0580939i	$-0.981498\pi$
$593$		−	1677.81i		−	0.116188i	0.998311	−	0.0580939i	$-0.0185023\pi$
$594$		0			0
$595$		−	780.513i		−	0.0537780i
$596$		0			0
$597$	9972.65	−	25715.3i	0.683674	−	1.76291i
$598$		0			0
$599$	26951.4			1.83841			0.919203	−	0.393783i	$-0.128834\pi$
$599$	26951.4			1.83841			0.919203	+	0.393783i	$0.128834\pi$
$600$		0			0
$601$	−10467.5			−0.710448			−0.355224	−	0.934781i	$-0.615595\pi$
$601$	−10467.5			−0.710448			−0.355224	+	0.934781i	$0.615595\pi$
$602$		0			0
$603$	621.500	−	680.781i	0.0419725	−	0.0459761i
$604$		0			0
$605$		−	23568.8i		−	1.58382i
$606$		0			0
$607$			10455.4i			0.699131i	0.936912	+	0.349566i	$0.113671\pi$
$607$			10455.4i			0.699131i	−0.936912	+	0.349566i	$0.886329\pi$
$608$		0			0
$609$	−4323.70	−	1676.77i	−0.287693	−	0.111570i
$610$		0			0
$611$	−389.062			−0.0257607
$612$		0			0
$613$	3808.84			0.250959			0.125479	−	0.992096i	$-0.459953\pi$
$613$	3808.84			0.250959			0.125479	+	0.992096i	$0.459953\pi$
$614$		0			0
$615$	−29692.5	−	11515.1i	−1.94686	−	0.755012i
$616$		0			0
$617$		−	23475.0i		−	1.53171i	−0.643012	−	0.765856i	$-0.722316\pi$
$617$		−	23475.0i		−	1.53171i	0.643012	−	0.765856i	$-0.277684\pi$
$618$		0			0
$619$		−	3890.94i		−	0.252650i	−0.991989	−	0.126325i	$-0.959682\pi$
$619$		−	3890.94i		−	0.252650i	0.991989	−	0.126325i	$-0.0403181\pi$
$620$		0			0
$621$	9071.10	+	18266.0i	0.586169	+	1.18034i
$622$		0			0
$623$	31068.7			1.99798
$624$		0			0
$625$	10107.7			0.646893
$626$		0			0
$627$	−550.560	+	1419.66i	−0.0350674	+	0.0904240i
$628$		0			0
$629$		−	597.077i		−	0.0378490i
$630$		0			0
$631$			28914.1i			1.82417i	0.410003	+	0.912084i	$0.365528\pi$
$631$			28914.1i			1.82417i	−0.410003	+	0.912084i	$0.634472\pi$
$632$		0			0
$633$	2808.68	−	7242.41i	0.176359	−	0.454755i
$634$		0			0
$635$	34616.8			2.16335
$636$		0			0
$637$	5053.04			0.314299
$638$		0			0
$639$	10656.4	+	9728.49i	0.659721	+	0.602274i
$640$		0			0
$641$		−	18809.5i		−	1.15902i	−0.814966	−	0.579509i	$-0.803244\pi$
$641$		−	18809.5i		−	1.15902i	0.814966	−	0.579509i	$-0.196756\pi$
$642$		0			0
$643$		−	23057.2i		−	1.41413i	−0.707148	−	0.707066i	$-0.750018\pi$
$643$		−	23057.2i		−	1.41413i	0.707148	−	0.707066i	$-0.249982\pi$
$644$		0			0
$645$	28105.1	+	10899.5i	1.71572	+	0.665374i
$646$		0			0
$647$	−27308.1			−1.65934			−0.829670	−	0.558254i	$-0.811471\pi$
$647$	−27308.1			−1.65934			−0.829670	+	0.558254i	$0.811471\pi$
$648$		0			0
$649$	7187.51			0.434722
$650$		0			0
$651$	−9690.18	−	3757.95i	−0.583392	−	0.226246i
$652$		0			0
$653$			13032.8i			0.781028i	0.920597	+	0.390514i	$0.127703\pi$
$653$			13032.8i			0.781028i	−0.920597	+	0.390514i	$0.872297\pi$
$654$		0			0
$655$			42732.2i			2.54914i
$656$		0			0
$657$	14572.8	+	13303.9i	0.865359	+	0.790005i
$658$		0			0
$659$	14776.2			0.873441			0.436720	−	0.899597i	$-0.356140\pi$
$659$	14776.2			0.873441			0.436720	+	0.899597i	$0.356140\pi$
$660$		0			0
$661$	−32520.4			−1.91361			−0.956806	−	0.290726i	$-0.906103\pi$
$661$	−32520.4			−1.91361			−0.956806	+	0.290726i	$0.906103\pi$
$662$		0			0
$663$	37.1612	−	95.8231i	0.00217681	−	0.00561306i
$664$		0			0
$665$		−	16004.4i		−	0.933269i
$666$		0			0
$667$		−	4796.20i		−	0.278425i
$668$		0			0
$669$	3788.96	−	9770.14i	0.218968	−	0.564627i
$670$		0			0
$671$	−840.863			−0.0483773
$672$		0			0
$673$	−5999.36			−0.343623			−0.171811	−	0.985130i	$-0.554962\pi$
$673$	−5999.36			−0.343623			−0.171811	+	0.985130i	$0.554962\pi$
$674$		0			0
$675$	14643.1	+	29486.0i	0.834984	+	1.68136i
$676$		0			0
$677$			17024.7i			0.966485i	0.875486	+	0.483243i	$0.160541\pi$
$677$			17024.7i			0.966485i	−0.875486	+	0.483243i	$0.839459\pi$
$678$		0			0
$679$		−	45050.5i		−	2.54621i
$680$		0			0
$681$	−16406.5	−	6362.60i	−0.923196	−	0.358025i
$682$		0			0
$683$	5330.85			0.298652			0.149326	−	0.988788i	$-0.452290\pi$
$683$	5330.85			0.298652			0.149326	+	0.988788i	$0.452290\pi$
$684$		0			0
$685$	−29324.0			−1.63564
$686$		0			0
$687$	6328.27	+	2454.17i	0.351439	+	0.136292i
$688$		0			0
$689$			5399.30i			0.298544i
$690$		0			0
$691$		−	17083.2i		−	0.940486i	−0.882537	−	0.470243i	$-0.844166\pi$
$691$		−	17083.2i		−	0.940486i	0.882537	−	0.470243i	$-0.155834\pi$
$692$		0			0
$693$	4625.19	−	5066.36i	0.253530	−	0.277713i
$694$		0			0
$695$	−14181.3			−0.773998
$696$		0			0
$697$	−491.714			−0.0267216
$698$		0			0
$699$	1814.23	−	4678.13i	0.0981693	−	0.253137i
$700$		0			0
$701$			6292.35i			0.339028i	0.985528	+	0.169514i	$0.0542198\pi$
$701$			6292.35i			0.339028i	−0.985528	+	0.169514i	$0.945780\pi$
$702$		0			0
$703$		−	12243.0i		−	0.656836i
$704$		0			0
$705$	1066.35	−	2749.66i	0.0569660	−	0.146891i
$706$		0			0
$707$	−30172.2			−1.60501
$708$		0			0
$709$	5829.62			0.308795			0.154398	−	0.988009i	$-0.450656\pi$
$709$	5829.62			0.308795			0.154398	+	0.988009i	$0.450656\pi$
$710$		0			0
$711$	14288.8	−	15651.7i	0.753688	−	0.825578i
$712$		0			0
$713$		−	10749.1i		−	0.564598i
$714$		0			0
$715$		−	2315.73i		−	0.121124i
$716$		0			0
$717$	23838.8	+	9244.92i	1.24167	+	0.481531i
$718$		0			0
$719$	−27748.6			−1.43929			−0.719644	−	0.694343i	$-0.755695\pi$
$719$	−27748.6			−1.43929			−0.719644	+	0.694343i	$0.755695\pi$
$720$		0			0
$721$	−35657.8			−1.84184
$722$		0			0
$723$	3012.45	+	1168.26i	0.154958	+	0.0600942i
$724$		0			0
$725$		−	7742.31i		−	0.396610i
$726$		0			0
$727$			26119.3i			1.33248i	0.745739	+	0.666238i	$0.232097\pi$
$727$			26119.3i			1.33248i	−0.745739	+	0.666238i	$0.767903\pi$
$728$		0			0
$729$	−11895.1	+	15682.1i	−0.604334	+	0.796731i
$730$		0			0
$731$	465.426			0.0235491
$732$		0			0
$733$	29497.8			1.48639			0.743196	−	0.669073i	$-0.233309\pi$
$733$	29497.8			1.48639			0.743196	+	0.669073i	$0.233309\pi$
$734$		0			0
$735$	−13849.5	+	35711.9i	−0.695027	+	1.79218i
$736$		0			0
$737$		−	320.682i		−	0.0160278i
$738$		0			0
$739$			3129.82i			0.155795i	0.996961	+	0.0778974i	$0.0248207\pi$
$739$			3129.82i			0.155795i	−0.996961	+	0.0778974i	$0.975179\pi$
$740$		0			0
$741$	761.990	−	1964.85i	0.0377765	−	0.0974097i
$742$		0			0
$743$	−22557.0			−1.11378			−0.556889	−	0.830587i	$-0.688005\pi$
$743$	−22557.0			−1.11378			−0.556889	+	0.830587i	$0.688005\pi$
$744$		0			0
$745$	−43778.6			−2.15292
$746$		0			0
$747$	−15601.1	−	14242.6i	−0.764141	−	0.697601i
$748$		0			0
$749$		−	49308.4i		−	2.40546i
$750$		0			0
$751$		−	26473.3i		−	1.28632i	−0.765733	−	0.643158i	$-0.777624\pi$
$751$		−	26473.3i		−	1.28632i	0.765733	−	0.643158i	$-0.222376\pi$
$752$		0			0
$753$	21864.4	+	8479.23i	1.05814	+	0.410359i
$754$		0			0
$755$	−35987.5			−1.73473
$756$		0			0
$757$	19518.6			0.937142			0.468571	−	0.883426i	$-0.344769\pi$
$757$	19518.6			0.937142			0.468571	+	0.883426i	$0.344769\pi$
$758$		0			0
$759$	6614.88	+	2565.32i	0.316344	+	0.122681i
$760$		0			0
$761$		−	127.754i		−	0.00608553i	−0.999995	−	0.00304276i	$-0.999031\pi$
$761$		−	127.754i		−	0.00608553i	0.999995	−	0.00304276i	$-0.000968544\pi$
$762$		0			0
$763$		−	54150.5i		−	2.56930i
$764$		0			0
$765$	575.370	+	525.267i	0.0271929	+	0.0248249i
$766$		0			0
$767$	−9947.70			−0.468306
$768$		0			0
$769$	7997.51			0.375030			0.187515	−	0.982262i	$-0.439957\pi$
$769$	7997.51			0.375030			0.187515	+	0.982262i	$0.439957\pi$
$770$		0			0
$771$	−7206.28	+	18581.9i	−0.336612	+	0.867980i
$772$		0			0
$773$			36122.3i			1.68076i	0.541995	+	0.840382i	$0.317669\pi$
$773$			36122.3i			1.68076i	−0.541995	+	0.840382i	$0.682331\pi$
$774$		0			0
$775$		−	17351.9i		−	0.804257i
$776$		0			0
$777$	−19943.6	+	51426.2i	−0.920815	+	2.37439i
$778$		0			0
$779$	−10082.6			−0.463730
$780$		0			0
$781$	5019.71			0.229986
$782$		0			0
$783$	4145.82	−	2058.86i	0.189220	−	0.0939690i
$784$		0			0
$785$			29183.2i			1.32687i
$786$		0			0
$787$			36951.4i			1.67367i	0.547458	+	0.836833i	$0.315595\pi$
$787$			36951.4i			1.67367i	−0.547458	+	0.836833i	$0.684405\pi$
$788$		0			0
$789$	11122.7	+	4313.50i	0.501874	+	0.194632i
$790$		0			0
$791$	−17484.0			−0.785915
$792$		0			0
$793$	1163.78			0.0521147
$794$		0			0
$795$	−38159.1	−	14798.5i	−1.70234	−	0.660187i
$796$		0			0
$797$		−	7249.80i		−	0.322210i	−0.986937	−	0.161105i	$-0.948494\pi$
$797$		−	7249.80i		−	0.322210i	0.986937	−	0.161105i	$-0.0515057\pi$
$798$		0			0
$799$		−	45.5349i		−	0.00201616i
$800$		0			0
$801$	−20908.5	+	22902.9i	−0.922305	+	1.01028i
$802$		0			0
$803$	6864.53			0.301674
$804$		0			0
$805$	−74572.0			−3.26499
$806$		0			0
$807$	8139.47	−	20988.3i	0.355047	−	0.915517i
$808$		0			0
$809$		−	8810.75i		−	0.382904i	−0.981502	−	0.191452i	$-0.938680\pi$
$809$		−	8810.75i		−	0.382904i	0.981502	−	0.191452i	$-0.0613196\pi$
$810$		0			0
$811$		−	1190.22i		−	0.0515344i	−0.999668	−	0.0257672i	$-0.991797\pi$
$811$		−	1190.22i		−	0.0515344i	0.999668	−	0.0257672i	$-0.00820286\pi$
$812$		0			0
$813$	−14491.0	+	37366.2i	−0.625120	+	1.61192i
$814$		0			0
$815$	74590.7			3.20589
$816$		0			0
$817$	9543.56			0.408674
$818$		0			0
$819$	−6401.39	+	7011.98i	−0.273117	+	0.299168i
$820$		0			0
$821$			22524.3i			0.957497i	0.877952	+	0.478749i	$0.158909\pi$
$821$			22524.3i			0.957497i	−0.877952	+	0.478749i	$0.841091\pi$
$822$		0			0
$823$		−	11724.0i		−	0.496564i	−0.968688	−	0.248282i	$-0.920134\pi$
$823$		−	11724.0i		−	0.496564i	0.968688	−	0.248282i	$-0.0798659\pi$
$824$		0			0
$825$	10678.1	+	4141.09i	0.450624	+	0.174757i
$826$		0			0
$827$	41525.1			1.74603			0.873017	−	0.487689i	$-0.162160\pi$
$827$	41525.1			1.74603			0.873017	+	0.487689i	$0.162160\pi$
$828$		0			0
$829$	10398.6			0.435657			0.217828	−	0.975987i	$-0.430103\pi$
$829$	10398.6			0.435657			0.217828	+	0.975987i	$0.430103\pi$
$830$		0			0
$831$	3426.30	+	1328.76i	0.143029	+	0.0554682i
$832$		0			0
$833$			591.396i			0.0245986i
$834$		0			0
$835$		−	5212.03i		−	0.216012i
$836$		0			0
$837$	9291.52	−	4614.28i	0.383706	−	0.190553i
$838$		0			0
$839$	−12481.7			−0.513606			−0.256803	−	0.966464i	$-0.582669\pi$
$839$	−12481.7			−0.513606			−0.256803	+	0.966464i	$0.582669\pi$
$840$		0			0
$841$	23300.4			0.955366
$842$		0			0
$843$	−15867.8	+	40916.3i	−0.648297	+	1.67169i
$844$		0			0
$845$			3205.03i			0.130481i
$846$		0			0
$847$			33616.9i			1.36374i
$848$		0			0
$849$	9914.79	−	25566.1i	0.400795	−	1.03348i
$850$		0			0
$851$	−57046.1			−2.29790
$852$		0			0
$853$	−41076.9			−1.64882			−0.824411	−	0.565991i	$-0.808494\pi$
$853$	−41076.9			−1.64882			−0.824411	+	0.565991i	$0.808494\pi$
$854$		0			0
$855$	11797.9	+	10770.6i	0.471908	+	0.430815i
$856$		0			0
$857$		−	4204.50i		−	0.167588i	−0.996483	−	0.0837941i	$-0.973296\pi$
$857$		−	4204.50i		−	0.167588i	0.996483	−	0.0837941i	$-0.0267038\pi$
$858$		0			0
$859$		−	27745.2i		−	1.10204i	−0.834491	−	0.551021i	$-0.814238\pi$
$859$		−	27745.2i		−	1.10204i	0.834491	−	0.551021i	$-0.185762\pi$
$860$		0			0
$861$	42351.3	+	16424.3i	1.67634	+	0.650101i
$862$		0			0
$863$	25148.1			0.991947			0.495973	−	0.868338i	$-0.334811\pi$
$863$	25148.1			0.991947			0.495973	+	0.868338i	$0.334811\pi$
$864$		0			0
$865$	20610.4			0.810142
$866$		0			0
$867$	−23790.3	−	9226.13i	−0.931905	−	0.361402i
$868$		0			0
$869$		−	7372.74i		−	0.287806i
$870$		0			0
$871$			443.832i			0.0172660i
$872$		0			0
$873$	33209.8	+	30318.0i	1.28749	+	1.17538i
$874$		0			0
$875$	−56254.6			−2.17343
$876$		0			0
$877$	−20440.5			−0.787033			−0.393516	−	0.919318i	$-0.628742\pi$
$877$	−20440.5			−0.787033			−0.393516	+	0.919318i	$0.628742\pi$
$878$		0			0
$879$	−4739.33	+	12220.7i	−0.181859	+	0.468937i
$880$		0			0
$881$		−	16071.5i		−	0.614601i	−0.951613	−	0.307300i	$-0.900574\pi$
$881$		−	16071.5i		−	0.614601i	0.951613	−	0.307300i	$-0.0994256\pi$
$882$		0			0
$883$			49484.4i			1.88594i	0.332880	+	0.942969i	$0.391980\pi$
$883$			49484.4i			1.88594i	−0.332880	+	0.942969i	$0.608020\pi$
$884$		0			0
$885$	27264.8	−	70304.5i	1.03559	−	2.67035i
$886$		0			0
$887$	−27463.0			−1.03959			−0.519796	−	0.854290i	$-0.673992\pi$
$887$	−27463.0			−1.03959			−0.519796	+	0.854290i	$0.673992\pi$
$888$		0			0
$889$	−49374.9			−1.86274
$890$		0			0
$891$	622.113	+	6819.09i	0.0233912	+	0.256395i
$892$		0			0
$893$		−	933.693i		−	0.0349886i
$894$		0			0
$895$		−	13844.3i		−	0.517055i
$896$		0			0
$897$	−9155.17	−	3550.47i	−0.340783	−	0.132159i
$898$		0			0
$899$	−2439.72			−0.0905110
$900$		0			0
$901$	−631.921			−0.0233655
$902$		0			0
$903$	−40087.1	−	15546.2i	−1.47732	−	0.572919i
$904$		0			0
$905$		−	34993.7i		−	1.28534i
$906$		0			0
$907$			32344.3i			1.18410i	0.805903	+	0.592048i	$0.201681\pi$
$907$			32344.3i			1.18410i	−0.805903	+	0.592048i	$0.798319\pi$
$908$		0			0
$909$	20305.2	−	22242.0i	0.740905	−	0.811575i
$910$		0			0
$911$	−27199.1			−0.989182			−0.494591	−	0.869126i	$-0.664682\pi$
$911$	−27199.1			−0.989182			−0.494591	+	0.869126i	$0.664682\pi$
$912$		0			0
$913$	−7348.88			−0.266388
$914$		0			0
$915$	−3189.70	+	8224.89i	−0.115244	+	0.297166i
$916$		0			0
$917$		−	60950.1i		−	2.19493i
$918$		0			0
$919$			34349.3i			1.23295i	0.787376	+	0.616473i	$0.211439\pi$
$919$			34349.3i			1.23295i	−0.787376	+	0.616473i	$0.788561\pi$
$920$		0			0
$921$	−6569.57	+	16940.1i	−0.235043	+	0.606077i
$922$		0			0
$923$	−6947.41			−0.247754
$924$		0			0
$925$	−92087.3			−3.27331
$926$		0			0
$927$	23996.9	−	26285.8i	0.850228	−	0.931326i
$928$		0			0
$929$		−	13372.2i		−	0.472256i	−0.971722	−	0.236128i	$-0.924121\pi$
$929$		−	13372.2i		−	0.472256i	0.971722	−	0.236128i	$-0.0758785\pi$
$930$		0			0
$931$			12126.6i			0.426887i
$932$		0			0
$933$	20059.2	+	7779.15i	0.703866	+	0.272967i
$934$		0			0
$935$	271.028			0.00947973
$936$		0			0
$937$	14249.6			0.496812			0.248406	−	0.968656i	$-0.420093\pi$
$937$	14249.6			0.496812			0.248406	+	0.968656i	$0.420093\pi$
$938$		0			0
$939$	43426.9	+	16841.4i	1.50925	+	0.585302i
$940$		0			0
$941$			41885.8i			1.45105i	0.688196	+	0.725525i	$0.258403\pi$
$941$			41885.8i			1.45105i	−0.688196	+	0.725525i	$0.741597\pi$
$942$		0			0
$943$			46979.5i			1.62234i
$944$		0			0
$945$	−32011.5	−	64459.8i	−1.10194	−	2.21892i
$946$		0			0
$947$	−8510.92			−0.292046			−0.146023	−	0.989281i	$-0.546647\pi$
$947$	−8510.92			−0.292046			−0.146023	+	0.989281i	$0.546647\pi$
$948$		0			0
$949$	−9500.70			−0.324980
$950$		0			0
$951$	−8177.43	+	21086.1i	−0.278834	+	0.718996i
$952$		0			0
$953$			2613.59i			0.0888379i	0.999013	+	0.0444190i	$0.0141436\pi$
$953$			2613.59i			0.0888379i	−0.999013	+	0.0444190i	$0.985856\pi$
$954$		0			0
$955$			34260.9i			1.16090i
$956$		0			0
$957$	582.248	−	1501.37i	0.0196671	−	0.0507132i
$958$		0			0
$959$	41825.6			1.40836
$960$		0			0
$961$	24323.1			0.816459
$962$		0			0
$963$	36348.6	+	33183.4i	1.21632	+	1.11041i
$964$		0			0
$965$			49268.5i			1.64353i
$966$		0			0
$967$			10938.7i			0.363768i	0.983320	+	0.181884i	$0.0582195\pi$
$967$			10938.7i			0.363768i	−0.983320	+	0.181884i	$0.941780\pi$
$968$		0			0
$969$	229.962	+	89.1815i	0.00762377	+	0.00295658i
$970$		0			0
$971$	−13458.2			−0.444794			−0.222397	−	0.974956i	$-0.571388\pi$
$971$	−13458.2			−0.444794			−0.222397	+	0.974956i	$0.571388\pi$
$972$		0			0
$973$	20227.2			0.666449
$974$		0			0
$975$	−14778.8	−	5731.38i	−0.485437	−	0.188258i
$976$		0			0
$977$		−	16556.3i		−	0.542153i	−0.962558	−	0.271076i	$-0.912620\pi$
$977$		−	16556.3i		−	0.542153i	0.962558	−	0.271076i	$-0.0873796\pi$
$978$		0			0
$979$			10788.4i			0.352194i
$980$		0			0
$981$	39918.1	+	36442.1i	1.29917	+	1.18604i
$982$		0			0
$983$	42411.4			1.37611			0.688054	−	0.725659i	$-0.258465\pi$
$983$	42411.4			1.37611			0.688054	+	0.725659i	$0.258465\pi$
$984$		0			0
$985$	10882.7			0.352033
$986$		0			0
$987$	−1520.96	+	3921.92i	−0.0490504	+	0.126480i
$988$		0			0
$989$		−	44467.9i		−	1.42973i
$990$		0			0
$991$		−	19264.9i		−	0.617526i	−0.951139	−	0.308763i	$-0.900085\pi$
$991$		−	19264.9i		−	0.617526i	0.951139	−	0.308763i	$-0.0999150\pi$
$992$		0			0
$993$	618.876	−	1595.82i	0.0197779	−	0.0509988i
$994$		0			0
$995$	−100665.			−3.20733
$996$		0			0
$997$	20315.1			0.645323			0.322661	−	0.946514i	$-0.395423\pi$
$997$	20315.1			0.645323			0.322661	+	0.946514i	$0.395423\pi$
$998$		0			0
$999$	−24488.2	−	49310.5i	−0.775547	−	1.56168i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.d.b.287.1	✓	12
3.2	odd	2	inner	624.4.d.b.287.11	yes	12
4.3	odd	2	inner	624.4.d.b.287.12	yes	12
12.11	even	2	inner	624.4.d.b.287.2	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
624.4.d.b.287.1	✓	12	1.1	even	1	trivial
624.4.d.b.287.2	yes	12	12.11	even	2	inner
624.4.d.b.287.11	yes	12	3.2	odd	2	inner
624.4.d.b.287.12	yes	12	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.84460 - 1.87879i) q^{3} +18.9647i q^{5} -27.0499i q^{7} +(19.9403 + 18.2040i) q^{9} +9.39288 q^{11} -13.0000 q^{13} +(35.6306 - 91.8764i) q^{15} -1.52149i q^{17} -31.1981i q^{19} +(-50.8209 + 131.046i) q^{21} -145.367 q^{23} -234.660 q^{25} +(-62.4015 - 125.654i) q^{27} +32.9938i q^{29} +73.9450i q^{31} +(-45.5048 - 17.6472i) q^{33} +512.992 q^{35} +392.429 q^{37} +(62.9798 + 24.4242i) q^{39} -323.179i q^{41} +305.902i q^{43} +(-345.232 + 378.162i) q^{45} +29.9279 q^{47} -388.695 q^{49} +(-2.85856 + 7.37101i) q^{51} -415.331i q^{53} +178.133i q^{55} +(-58.6146 + 151.142i) q^{57} +765.208 q^{59} -89.5213 q^{61} +(492.414 - 539.383i) q^{63} -246.541i q^{65} -34.1410i q^{67} +(704.244 + 273.113i) q^{69} +534.416 q^{71} +730.823 q^{73} +(1136.83 + 440.876i) q^{75} -254.076i q^{77} -784.929i q^{79} +(66.2324 + 725.985i) q^{81} -782.388 q^{83} +28.8546 q^{85} +(61.9883 - 159.842i) q^{87} +1148.57i q^{89} +351.648i q^{91} +(138.927 - 358.234i) q^{93} +591.663 q^{95} +1665.46 q^{97} +(187.297 + 170.988i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 54 q^{9} - 156 q^{13} - 54 q^{21} - 408 q^{25} - 360 q^{33} + 636 q^{37} - 810 q^{45} + 336 q^{49} - 1260 q^{57} + 960 q^{61} - 252 q^{69} + 3216 q^{73} - 2538 q^{81} + 2196 q^{85} - 1116 q^{93}+ \cdots + 4800 q^{97}+O(q^{100}) \) 12 * q + 54 * q^9 - 156 * q^13 - 54 * q^21 - 408 * q^25 - 360 * q^33 + 636 * q^37 - 810 * q^45 + 336 * q^49 - 1260 * q^57 + 960 * q^61 - 252 * q^69 + 3216 * q^73 - 2538 * q^81 + 2196 * q^85 - 1116 * q^93 + 4800 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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