LMFDB - Embedded newform 900.3.l.b.793.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [900,3,Mod(757,900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("900.757"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	900.l (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-20,0,0,0,16,0,0,0,0,0,-40,0,0,0,0,0,-40,0,0,0,0, 0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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

Self dual:	no
Analytic conductor:	$24.5232237924$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			793.1
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	900.793
Dual form			900.3.l.b.757.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-7.44949 + 7.44949i) q^{7} +16.2474 q^{11} +(-12.2474 - 12.2474i) q^{13} +(-7.55051 + 7.55051i) q^{17} +14.4949i q^{19} +(-2.65153 - 2.65153i) q^{23} -34.2474i q^{29} -20.4949 q^{31} +(-7.34847 + 7.34847i) q^{37} -25.5051 q^{41} +(-25.1010 - 25.1010i) q^{43} +(22.0454 - 22.0454i) q^{47} -61.9898i q^{49} +(-35.3031 - 35.3031i) q^{53} -88.7423i q^{59} -102.495 q^{61} +(24.6969 - 24.6969i) q^{67} -77.9796 q^{71} +(44.1918 + 44.1918i) q^{73} +(-121.035 + 121.035i) q^{77} +48.4949i q^{79} +(-101.641 - 101.641i) q^{83} +156.969i q^{89} +182.474 q^{91} +(-55.4041 + 55.4041i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{7} + 16 q^{11} - 40 q^{17} - 40 q^{23} + 16 q^{31} - 200 q^{41} - 120 q^{43} - 200 q^{53} - 312 q^{61} + 40 q^{67} + 80 q^{71} + 20 q^{73} - 200 q^{77} - 240 q^{83} + 240 q^{91} - 300 q^{97}+O(q^{100}) $ 4 * q - 20 * q^7 + 16 * q^11 - 40 * q^17 - 40 * q^23 + 16 * q^31 - 200 * q^41 - 120 * q^43 - 200 * q^53 - 312 * q^61 + 40 * q^67 + 80 * q^71 + 20 * q^73 - 200 * q^77 - 240 * q^83 + 240 * q^91 - 300 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−7.44949	+	7.44949i	−1.06421	+	1.06421i	−0.0664211	+	0.997792i	$0.521158\pi$
$7$	−7.44949	+	7.44949i	−1.06421	+	1.06421i	−0.997792	+	0.0664211i	$0.978842\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	16.2474			1.47704			0.738520	−	0.674231i	$-0.235525\pi$
$11$	16.2474			1.47704			0.738520	+	0.674231i	$0.235525\pi$
$12$		0			0
$13$	−12.2474	−	12.2474i	−0.942111	−	0.942111i	0.0563023	−	0.998414i	$-0.482069\pi$
$13$	−12.2474	−	12.2474i	−0.942111	−	0.942111i	−0.998414	+	0.0563023i	$0.982069\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−7.55051	+	7.55051i	−0.444148	+	0.444148i	−0.893403	−	0.449256i	$-0.851689\pi$
$17$	−7.55051	+	7.55051i	−0.444148	+	0.444148i	0.449256	+	0.893403i	$0.351689\pi$
$18$		0			0
$19$			14.4949i			0.762889i	0.924392	+	0.381445i	$0.124573\pi$
$19$			14.4949i			0.762889i	−0.924392	+	0.381445i	$0.875427\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.65153	−	2.65153i	−0.115284	−	0.115284i	0.647111	−	0.762395i	$-0.275977\pi$
$23$	−2.65153	−	2.65153i	−0.115284	−	0.115284i	−0.762395	+	0.647111i	$0.775977\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	34.2474i		−	1.18095i	−0.807057	−	0.590473i	$-0.798941\pi$
$29$		−	34.2474i		−	1.18095i	0.807057	−	0.590473i	$-0.201059\pi$
$30$		0			0
$31$	−20.4949			−0.661126			−0.330563	−	0.943784i	$-0.607239\pi$
$31$	−20.4949			−0.661126			−0.330563	+	0.943784i	$0.607239\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−7.34847	+	7.34847i	−0.198607	+	0.198607i	−0.799403	−	0.600795i	$-0.794851\pi$
$37$	−7.34847	+	7.34847i	−0.198607	+	0.198607i	0.600795	+	0.799403i	$0.294851\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−25.5051			−0.622076			−0.311038	−	0.950398i	$-0.600677\pi$
$41$	−25.5051			−0.622076			−0.311038	+	0.950398i	$0.600677\pi$
$42$		0			0
$43$	−25.1010	−	25.1010i	−0.583745	−	0.583745i	0.352186	−	0.935930i	$-0.385439\pi$
$43$	−25.1010	−	25.1010i	−0.583745	−	0.583745i	−0.935930	+	0.352186i	$0.885439\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	22.0454	−	22.0454i	0.469051	−	0.469051i	−0.432556	−	0.901607i	$-0.642388\pi$
$47$	22.0454	−	22.0454i	0.469051	−	0.469051i	0.901607	+	0.432556i	$0.142388\pi$
$48$		0			0
$49$		−	61.9898i		−	1.26510i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−35.3031	−	35.3031i	−0.666096	−	0.666096i	0.290714	−	0.956810i	$-0.406107\pi$
$53$	−35.3031	−	35.3031i	−0.666096	−	0.666096i	−0.956810	+	0.290714i	$0.906107\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	88.7423i		−	1.50411i	−0.659102	−	0.752054i	$-0.729063\pi$
$59$		−	88.7423i		−	1.50411i	0.659102	−	0.752054i	$-0.270937\pi$
$60$		0			0
$61$	−102.495			−1.68024			−0.840122	−	0.542397i	$-0.817517\pi$
$61$	−102.495			−1.68024			−0.840122	+	0.542397i	$0.817517\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	24.6969	−	24.6969i	0.368611	−	0.368611i	−0.498359	−	0.866970i	$-0.666064\pi$
$67$	24.6969	−	24.6969i	0.368611	−	0.368611i	0.866970	+	0.498359i	$0.166064\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−77.9796			−1.09830			−0.549152	−	0.835722i	$-0.685049\pi$
$71$	−77.9796			−1.09830			−0.549152	+	0.835722i	$0.685049\pi$
$72$		0			0
$73$	44.1918	+	44.1918i	0.605368	+	0.605368i	0.941732	−	0.336364i	$-0.109197\pi$
$73$	44.1918	+	44.1918i	0.605368	+	0.605368i	−0.336364	+	0.941732i	$0.609197\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−121.035	+	121.035i	−1.57189	+	1.57189i
$78$		0			0
$79$			48.4949i			0.613859i	0.951732	+	0.306930i	$0.0993017\pi$
$79$			48.4949i			0.613859i	−0.951732	+	0.306930i	$0.900698\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−101.641	−	101.641i	−1.22459	−	1.22459i	−0.965981	−	0.258613i	$-0.916734\pi$
$83$	−101.641	−	101.641i	−1.22459	−	1.22459i	−0.258613	−	0.965981i	$-0.583266\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			156.969i			1.76370i	0.471529	+	0.881850i	$0.343702\pi$
$89$			156.969i			1.76370i	−0.471529	+	0.881850i	$0.656298\pi$
$90$		0			0
$91$	182.474			2.00521
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−55.4041	+	55.4041i	−0.571176	+	0.571176i	−0.932457	−	0.361281i	$-0.882340\pi$
$97$	−55.4041	+	55.4041i	−0.571176	+	0.571176i	0.361281	+	0.932457i	$0.382340\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−45.7526			−0.452996			−0.226498	−	0.974012i	$-0.572728\pi$
$101$	−45.7526			−0.452996			−0.226498	+	0.974012i	$0.572728\pi$
$102$		0			0
$103$	−23.1566	−	23.1566i	−0.224822	−	0.224822i	0.585704	−	0.810525i	$-0.300818\pi$
$103$	−23.1566	−	23.1566i	−0.224822	−	0.224822i	−0.810525	+	0.585704i	$0.800818\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	16.3383	−	16.3383i	0.152694	−	0.152694i	−0.626626	−	0.779320i	$-0.715565\pi$
$107$	16.3383	−	16.3383i	0.152694	−	0.152694i	0.779320	+	0.626626i	$0.215565\pi$
$108$		0			0
$109$		−	159.485i		−	1.46316i	−0.681754	−	0.731581i	$-0.738783\pi$
$109$		−	159.485i		−	1.46316i	0.681754	−	0.731581i	$-0.261217\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	13.0556	+	13.0556i	0.115536	+	0.115536i	0.762511	−	0.646975i	$-0.223966\pi$
$113$	13.0556	+	13.0556i	0.115536	+	0.115536i	−0.646975	+	0.762511i	$0.723966\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	112.495i		−	0.945335i
$120$		0			0
$121$	142.980			1.18165
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−7.44949	+	7.44949i	−0.0586574	+	0.0586574i	−0.735827	−	0.677170i	$-0.763206\pi$
$127$	−7.44949	+	7.44949i	−0.0586574	+	0.0586574i	0.677170	+	0.735827i	$0.263206\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−19.7526			−0.150783			−0.0753914	−	0.997154i	$-0.524021\pi$
$131$	−19.7526			−0.150783			−0.0753914	+	0.997154i	$0.524021\pi$
$132$		0			0
$133$	−107.980	−	107.980i	−0.811877	−	0.811877i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−101.641	+	101.641i	−0.741907	+	0.741907i	−0.972945	−	0.231037i	$-0.925788\pi$
$137$	−101.641	+	101.641i	−0.741907	+	0.741907i	0.231037	+	0.972945i	$0.425788\pi$
$138$		0			0
$139$		−	24.0204i		−	0.172809i	−0.996260	−	0.0864044i	$-0.972462\pi$
$139$		−	24.0204i		−	0.172809i	0.996260	−	0.0864044i	$-0.0275377\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−198.990	−	198.990i	−1.39154	−	1.39154i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	199.732i		−	1.34048i	−0.742143	−	0.670242i	$-0.766190\pi$
$149$		−	199.732i		−	1.34048i	0.742143	−	0.670242i	$-0.233810\pi$
$150$		0			0
$151$	−158.990			−1.05291			−0.526456	−	0.850202i	$-0.676480\pi$
$151$	−158.990			−1.05291			−0.526456	+	0.850202i	$0.676480\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	37.3485	−	37.3485i	0.237888	−	0.237888i	−0.578087	−	0.815975i	$-0.696201\pi$
$157$	37.3485	−	37.3485i	0.237888	−	0.237888i	0.815975	+	0.578087i	$0.196201\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	39.5051			0.245373
$162$		0			0
$163$	−8.98979	−	8.98979i	−0.0551521	−	0.0551521i	0.678993	−	0.734145i	$-0.262417\pi$
$163$	−8.98979	−	8.98979i	−0.0551521	−	0.0551521i	−0.734145	+	0.678993i	$0.762417\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	88.3587	−	88.3587i	0.529094	−	0.529094i	−0.391208	−	0.920302i	$-0.627943\pi$
$167$	88.3587	−	88.3587i	0.529094	−	0.529094i	0.920302	+	0.391208i	$0.127943\pi$
$168$		0			0
$169$			131.000i			0.775148i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	170.631	+	170.631i	0.986307	+	0.986307i	0.999908	−	0.0136005i	$-0.00432929\pi$
$173$	170.631	+	170.631i	0.986307	+	0.986307i	−0.0136005	+	0.999908i	$0.504329\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$			75.7526i			0.423199i	0.977357	+	0.211599i	$0.0678672\pi$
$179$			75.7526i			0.423199i	−0.977357	+	0.211599i	$0.932133\pi$
$180$		0			0
$181$	−279.444			−1.54389			−0.771944	−	0.635690i	$-0.780716\pi$
$181$	−279.444			−1.54389			−0.771944	+	0.635690i	$0.780716\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−122.677	+	122.677i	−0.656024	+	0.656024i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	69.4847			0.363794			0.181897	−	0.983318i	$-0.441776\pi$
$191$	69.4847			0.363794			0.181897	+	0.983318i	$0.441776\pi$
$192$		0			0
$193$	53.5857	+	53.5857i	0.277646	+	0.277646i	0.832169	−	0.554523i	$-0.187099\pi$
$193$	53.5857	+	53.5857i	0.277646	+	0.277646i	−0.554523	+	0.832169i	$0.687099\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	260.252	−	260.252i	1.32108	−	1.32108i	0.408171	−	0.912906i	$-0.366167\pi$
$197$	260.252	−	260.252i	1.32108	−	1.32108i	0.912906	−	0.408171i	$-0.133833\pi$
$198$		0			0
$199$			65.0102i			0.326684i	0.986569	+	0.163342i	$0.0522275\pi$
$199$			65.0102i			0.326684i	−0.986569	+	0.163342i	$0.947773\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	255.126	+	255.126i	1.25678	+	1.25678i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$			235.505i			1.12682i
$210$		0			0
$211$	12.0204			0.0569688			0.0284844	−	0.999594i	$-0.490932\pi$
$211$	12.0204			0.0569688			0.0284844	+	0.999594i	$0.490932\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	152.677	−	152.677i	0.703578	−	0.703578i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	184.949			0.836873
$222$		0			0
$223$	239.722	+	239.722i	1.07499	+	1.07499i	0.996951	+	0.0780357i	$0.0248648\pi$
$223$	239.722	+	239.722i	1.07499	+	1.07499i	0.0780357	+	0.996951i	$0.475135\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.02041	−	2.02041i	0.00890049	−	0.00890049i	−0.702643	−	0.711543i	$-0.747997\pi$
$227$	2.02041	−	2.02041i	0.00890049	−	0.00890049i	0.711543	+	0.702643i	$0.247997\pi$
$228$		0			0
$229$			284.969i			1.24441i	0.782855	+	0.622204i	$0.213763\pi$
$229$			284.969i			1.24441i	−0.782855	+	0.622204i	$0.786237\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	151.237	+	151.237i	0.649087	+	0.649087i	0.952772	−	0.303685i	$-0.0982172\pi$
$233$	151.237	+	151.237i	0.649087	+	0.649087i	−0.303685	+	0.952772i	$0.598217\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	50.9694i		−	0.213261i	−0.994299	−	0.106631i	$-0.965994\pi$
$239$		−	50.9694i		−	0.213261i	0.994299	−	0.106631i	$-0.0340062\pi$
$240$		0			0
$241$	−32.0000			−0.132780			−0.0663900	−	0.997794i	$-0.521148\pi$
$241$	−32.0000			−0.132780			−0.0663900	+	0.997794i	$0.521148\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	177.526	−	177.526i	0.718727	−	0.718727i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	373.237			1.48700			0.743500	−	0.668735i	$-0.233164\pi$
$251$	373.237			1.48700			0.743500	+	0.668735i	$0.233164\pi$
$252$		0			0
$253$	−43.0806	−	43.0806i	−0.170279	−	0.170279i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−206.136	+	206.136i	−0.802086	+	0.802086i	−0.983421	−	0.181335i	$-0.941958\pi$
$257$	−206.136	+	206.136i	−0.802086	+	0.802086i	0.181335	+	0.983421i	$0.441958\pi$
$258$		0			0
$259$		−	109.485i		−	0.422721i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	210.833	+	210.833i	0.801647	+	0.801647i	0.983353	−	0.181706i	$-0.0581619\pi$
$263$	210.833	+	210.833i	0.801647	+	0.801647i	−0.181706	+	0.983353i	$0.558162\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			205.258i			0.763040i	0.924361	+	0.381520i	$0.124599\pi$
$269$			205.258i			0.763040i	−0.924361	+	0.381520i	$0.875401\pi$
$270$		0			0
$271$	−66.0000			−0.243542			−0.121771	−	0.992558i	$-0.538857\pi$
$271$	−66.0000			−0.243542			−0.121771	+	0.992558i	$0.538857\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−148.157	+	148.157i	−0.534861	+	0.534861i	−0.922015	−	0.387154i	$-0.873458\pi$
$277$	−148.157	+	148.157i	−0.534861	+	0.534861i	0.387154	+	0.922015i	$0.373458\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−324.434			−1.15457			−0.577284	−	0.816543i	$-0.695887\pi$
$281$	−324.434			−1.15457			−0.577284	+	0.816543i	$0.695887\pi$
$282$		0			0
$283$	−304.747	−	304.747i	−1.07684	−	1.07684i	−0.996791	−	0.0800537i	$-0.974491\pi$
$283$	−304.747	−	304.747i	−1.07684	−	1.07684i	−0.0800537	−	0.996791i	$-0.525509\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	190.000	−	190.000i	0.662021	−	0.662021i
$288$		0			0
$289$			174.980i			0.605466i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−145.151	−	145.151i	−0.495396	−	0.495396i	0.414605	−	0.910001i	$-0.363920\pi$
$293$	−145.151	−	145.151i	−0.495396	−	0.495396i	−0.910001	+	0.414605i	$0.863920\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$			64.9490i			0.217221i
$300$		0			0
$301$	373.980			1.24246
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−120.606	+	120.606i	−0.392854	+	0.392854i	−0.875703	−	0.482850i	$-0.839602\pi$
$307$	−120.606	+	120.606i	−0.392854	+	0.392854i	0.482850	+	0.875703i	$0.339602\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	18.0204			0.0579434			0.0289717	−	0.999580i	$-0.490777\pi$
$311$	18.0204			0.0579434			0.0289717	+	0.999580i	$0.490777\pi$
$312$		0			0
$313$	323.586	+	323.586i	1.03382	+	1.03382i	0.999408	+	0.0344125i	$0.0109560\pi$
$313$	323.586	+	323.586i	1.03382	+	1.03382i	0.0344125	+	0.999408i	$0.489044\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	62.4495	−	62.4495i	0.197002	−	0.197002i	−0.601712	−	0.798713i	$-0.705514\pi$
$317$	62.4495	−	62.4495i	0.197002	−	0.197002i	0.798713	+	0.601712i	$0.205514\pi$
$318$		0			0
$319$		−	556.434i		−	1.74431i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−109.444	−	109.444i	−0.338836	−	0.338836i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$			328.454i			0.998341i
$330$		0			0
$331$	618.413			1.86832			0.934159	−	0.356857i	$-0.116152\pi$
$331$	618.413			1.86832			0.934159	+	0.356857i	$0.116152\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	55.4041	−	55.4041i	0.164404	−	0.164404i	−0.620111	−	0.784514i	$-0.712912\pi$
$337$	55.4041	−	55.4041i	0.164404	−	0.164404i	0.784514	+	0.620111i	$0.212912\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−332.990			−0.976510
$342$		0			0
$343$	96.7673	+	96.7673i	0.282121	+	0.282121i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	413.081	−	413.081i	1.19043	−	1.19043i	0.213488	−	0.976946i	$-0.431517\pi$
$347$	413.081	−	413.081i	1.19043	−	1.19043i	0.976946	−	0.213488i	$-0.0684826\pi$
$348$		0			0
$349$			258.454i			0.740556i	0.928921	+	0.370278i	$0.120738\pi$
$349$			258.454i			0.740556i	−0.928921	+	0.370278i	$0.879262\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−83.9138	−	83.9138i	−0.237716	−	0.237716i	0.578188	−	0.815904i	$-0.303760\pi$
$353$	−83.9138	−	83.9138i	−0.237716	−	0.237716i	−0.815904	+	0.578188i	$0.803760\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	138.515i		−	0.385837i	−0.981215	−	0.192918i	$-0.938205\pi$
$359$		−	138.515i		−	0.385837i	0.981215	−	0.192918i	$-0.0617952\pi$
$360$		0			0
$361$	150.898			0.418000
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	408.964	−	408.964i	1.11434	−	1.11434i	0.121786	−	0.992556i	$-0.461138\pi$
$367$	408.964	−	408.964i	1.11434	−	1.11434i	0.992556	−	0.121786i	$-0.0388623\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	525.980			1.41773
$372$		0			0
$373$	143.106	+	143.106i	0.383661	+	0.383661i	0.872419	−	0.488758i	$-0.162550\pi$
$373$	143.106	+	143.106i	0.383661	+	0.383661i	−0.488758	+	0.872419i	$0.662550\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−419.444	+	419.444i	−1.11258	+	1.11258i
$378$		0			0
$379$			194.000i			0.511873i	0.966694	+	0.255937i	$0.0823839\pi$
$379$			194.000i			0.511873i	−0.966694	+	0.255937i	$0.917616\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	51.4893	+	51.4893i	0.134437	+	0.134437i	0.771123	−	0.636686i	$-0.219695\pi$
$383$	51.4893	+	51.4893i	0.134437	+	0.134437i	−0.636686	+	0.771123i	$0.719695\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	35.1964i		−	0.0904792i	−0.998976	−	0.0452396i	$-0.985595\pi$
$389$		−	35.1964i		−	0.0904792i	0.998976	−	0.0452396i	$-0.0144051\pi$
$390$		0			0
$391$	40.0408			0.102406
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−409.267	+	409.267i	−1.03090	+	1.03090i	−0.0313917	+	0.999507i	$0.509994\pi$
$397$	−409.267	+	409.267i	−1.03090	+	1.03090i	−0.999507	+	0.0313917i	$0.990006\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−375.898			−0.937401			−0.468701	−	0.883357i	$-0.655278\pi$
$401$	−375.898			−0.937401			−0.468701	+	0.883357i	$0.655278\pi$
$402$		0			0
$403$	251.010	+	251.010i	0.622854	+	0.622854i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−119.394	+	119.394i	−0.293351	+	0.293351i
$408$		0			0
$409$		−	207.959i		−	0.508458i	−0.967144	−	0.254229i	$-0.918178\pi$
$409$		−	207.959i		−	0.508458i	0.967144	−	0.254229i	$-0.0818216\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	661.085	+	661.085i	1.60069	+	1.60069i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			810.186i			1.93362i	0.255499	+	0.966809i	$0.417760\pi$
$419$			810.186i			1.93362i	−0.255499	+	0.966809i	$0.582240\pi$
$420$		0			0
$421$	−505.959			−1.20180			−0.600902	−	0.799323i	$-0.705192\pi$
$421$	−505.959			−1.20180			−0.600902	+	0.799323i	$0.705192\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	763.535	−	763.535i	1.78814	−	1.78814i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	92.4541			0.214511			0.107255	−	0.994232i	$-0.465794\pi$
$431$	92.4541			0.214511			0.107255	+	0.994232i	$0.465794\pi$
$432$		0			0
$433$	−69.8990	−	69.8990i	−0.161430	−	0.161430i	0.621770	−	0.783200i	$-0.286414\pi$
$433$	−69.8990	−	69.8990i	−0.161430	−	0.161430i	−0.783200	+	0.621770i	$0.786414\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	38.4337	−	38.4337i	0.0879489	−	0.0879489i
$438$		0			0
$439$		−	128.929i		−	0.293687i	−0.989160	−	0.146843i	$-0.953089\pi$
$439$		−	128.929i		−	0.293687i	0.989160	−	0.146843i	$-0.0469114\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−335.555	−	335.555i	−0.757461	−	0.757461i	0.218399	−	0.975860i	$-0.429917\pi$
$443$	−335.555	−	335.555i	−0.757461	−	0.757461i	−0.975860	+	0.218399i	$0.929917\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	521.423i		−	1.16130i	−0.814153	−	0.580650i	$-0.802799\pi$
$449$		−	521.423i		−	1.16130i	0.814153	−	0.580650i	$-0.197201\pi$
$450$		0			0
$451$	−414.393			−0.918831
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−548.939	+	548.939i	−1.20118	+	1.20118i	−0.227371	+	0.973808i	$0.573013\pi$
$457$	−548.939	+	548.939i	−1.20118	+	1.20118i	−0.973808	+	0.227371i	$0.926987\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−142.288			−0.308651			−0.154326	−	0.988020i	$-0.549320\pi$
$461$	−142.288			−0.308651			−0.154326	+	0.988020i	$0.549320\pi$
$462$		0			0
$463$	−537.045	−	537.045i	−1.15993	−	1.15993i	−0.984491	−	0.175434i	$-0.943867\pi$
$463$	−537.045	−	537.045i	−1.15993	−	1.15993i	−0.175434	−	0.984491i	$-0.556133\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−367.044	+	367.044i	−0.785962	+	0.785962i	−0.980830	−	0.194867i	$-0.937572\pi$
$467$	−367.044	+	367.044i	−0.785962	+	0.785962i	0.194867	+	0.980830i	$0.437572\pi$
$468$		0			0
$469$			367.959i			0.784561i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−407.828	−	407.828i	−0.862215	−	0.862215i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$			94.9694i			0.198266i	0.995074	+	0.0991330i	$0.0316069\pi$
$479$			94.9694i			0.198266i	−0.995074	+	0.0991330i	$0.968393\pi$
$480$		0			0
$481$	180.000			0.374220
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−244.621	+	244.621i	−0.502302	+	0.502302i	−0.912153	−	0.409851i	$-0.865581\pi$
$487$	−244.621	+	244.621i	−0.502302	+	0.502302i	0.409851	+	0.912153i	$0.365581\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−163.217			−0.332417			−0.166209	−	0.986091i	$-0.553153\pi$
$491$	−163.217			−0.332417			−0.166209	+	0.986091i	$0.553153\pi$
$492$		0			0
$493$	258.586	+	258.586i	0.524515	+	0.524515i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	580.908	−	580.908i	1.16883	−	1.16883i
$498$		0			0
$499$		−	8.55613i		−	0.0171465i	−0.999963	−	0.00857327i	$-0.997271\pi$
$499$		−	8.55613i		−	0.0171465i	0.999963	−	0.00857327i	$-0.00272899\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	46.5903	+	46.5903i	0.0926249	+	0.0926249i	0.751901	−	0.659276i	$-0.229137\pi$
$503$	46.5903	+	46.5903i	0.0926249	+	0.0926249i	−0.659276	+	0.751901i	$0.729137\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	336.227i		−	0.660564i	−0.943882	−	0.330282i	$-0.892856\pi$
$509$		−	336.227i		−	0.660564i	0.943882	−	0.330282i	$-0.107144\pi$
$510$		0			0
$511$	−658.413			−1.28848
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	358.182	−	358.182i	0.692808	−	0.692808i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−252.556			−0.484753			−0.242376	−	0.970182i	$-0.577927\pi$
$521$	−252.556			−0.484753			−0.242376	+	0.970182i	$0.577927\pi$
$522$		0			0
$523$	33.6867	+	33.6867i	0.0644106	+	0.0644106i	0.738578	−	0.674168i	$-0.235498\pi$
$523$	33.6867	+	33.6867i	0.0644106	+	0.0644106i	−0.674168	+	0.738578i	$0.735498\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	154.747	−	154.747i	0.293637	−	0.293637i
$528$		0			0
$529$		−	514.939i		−	0.973419i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	312.372	+	312.372i	0.586065	+	0.586065i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	1007.18i		−	1.86860i
$540$		0			0
$541$	−604.474			−1.11733			−0.558664	−	0.829394i	$-0.688686\pi$
$541$	−604.474			−1.11733			−0.558664	+	0.829394i	$0.688686\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−615.353	+	615.353i	−1.12496	+	1.12496i	−0.133975	+	0.990985i	$0.542774\pi$
$547$	−615.353	+	615.353i	−1.12496	+	1.12496i	−0.990985	+	0.133975i	$0.957226\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	496.413			0.900931
$552$		0			0
$553$	−361.262	−	361.262i	−0.653277	−	0.653277i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−439.040	+	439.040i	−0.788222	+	0.788222i	−0.981203	−	0.192980i	$-0.938185\pi$
$557$	−439.040	+	439.040i	−0.788222	+	0.788222i	0.192980	+	0.981203i	$0.438185\pi$
$558$		0			0
$559$			614.847i			1.09991i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−52.5765	−	52.5765i	−0.0933864	−	0.0933864i	0.658870	−	0.752257i	$-0.271035\pi$
$563$	−52.5765	−	52.5765i	−0.0933864	−	0.0933864i	−0.752257	+	0.658870i	$0.771035\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	680.929i		−	1.19671i	−0.801231	−	0.598356i	$-0.795821\pi$
$569$		−	680.929i		−	1.19671i	0.801231	−	0.598356i	$-0.204179\pi$
$570$		0			0
$571$	254.495			0.445700			0.222850	−	0.974853i	$-0.428464\pi$
$571$	254.495			0.445700			0.222850	+	0.974853i	$0.428464\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	279.999	−	279.999i	0.485267	−	0.485267i	−0.421542	−	0.906809i	$-0.638511\pi$
$577$	279.999	−	279.999i	0.485267	−	0.485267i	0.906809	+	0.421542i	$0.138511\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1514.35			2.60646
$582$		0			0
$583$	−573.585	−	573.585i	−0.983850	−	0.983850i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	312.347	−	312.347i	0.532108	−	0.532108i	−0.389091	−	0.921199i	$-0.627211\pi$
$587$	312.347	−	312.347i	0.532108	−	0.532108i	0.921199	+	0.389091i	$0.127211\pi$
$588$		0			0
$589$		−	297.071i		−	0.504366i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−424.570	−	424.570i	−0.715969	−	0.715969i	0.251808	−	0.967777i	$-0.418975\pi$
$593$	−424.570	−	424.570i	−0.715969	−	0.715969i	−0.967777	+	0.251808i	$0.918975\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	396.536i		−	0.661996i	−0.943631	−	0.330998i	$-0.892615\pi$
$599$		−	396.536i		−	0.661996i	0.943631	−	0.330998i	$-0.107385\pi$
$600$		0			0
$601$	238.908			0.397518			0.198759	−	0.980048i	$-0.436309\pi$
$601$	238.908			0.397518			0.198759	+	0.980048i	$0.436309\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−92.8025	+	92.8025i	−0.152887	+	0.152887i	−0.779406	−	0.626519i	$-0.784479\pi$
$607$	−92.8025	+	92.8025i	−0.152887	+	0.152887i	0.626519	+	0.779406i	$0.284479\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−540.000			−0.883797
$612$		0			0
$613$	−301.943	−	301.943i	−0.492567	−	0.492567i	0.416547	−	0.909114i	$-0.363240\pi$
$613$	−301.943	−	301.943i	−0.492567	−	0.492567i	−0.909114	+	0.416547i	$0.863240\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−372.904	+	372.904i	−0.604382	+	0.604382i	−0.941472	−	0.337090i	$-0.890557\pi$
$617$	−372.904	+	372.904i	−0.604382	+	0.604382i	0.337090	+	0.941472i	$0.390557\pi$
$618$		0			0
$619$			815.939i			1.31816i	0.752074	+	0.659078i	$0.229053\pi$
$619$			815.939i			1.31816i	−0.752074	+	0.659078i	$0.770947\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1169.34	−	1169.34i	−1.87695	−	1.87695i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	110.969i		−	0.176422i
$630$		0			0
$631$	550.434			0.872320			0.436160	−	0.899869i	$-0.356338\pi$
$631$	550.434			0.872320			0.436160	+	0.899869i	$0.356338\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−759.217	+	759.217i	−1.19186	+	1.19186i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−927.939			−1.44764			−0.723821	−	0.689988i	$-0.757616\pi$
$641$	−927.939			−1.44764			−0.723821	+	0.689988i	$0.757616\pi$
$642$		0			0
$643$	−190.960	−	190.960i	−0.296983	−	0.296983i	0.542848	−	0.839831i	$-0.317346\pi$
$643$	−190.960	−	190.960i	−0.296983	−	0.296983i	−0.839831	+	0.542848i	$0.817346\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−637.146	+	637.146i	−0.984770	+	0.984770i	−0.999886	−	0.0151154i	$-0.995188\pi$
$647$	−637.146	+	637.146i	−0.984770	+	0.984770i	0.0151154	+	0.999886i	$0.495188\pi$
$648$		0			0
$649$		−	1441.84i		−	2.22163i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−404.116	−	404.116i	−0.618860	−	0.618860i	0.326379	−	0.945239i	$-0.394172\pi$
$653$	−404.116	−	404.116i	−0.618860	−	0.618860i	−0.945239	+	0.326379i	$0.894172\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			453.196i			0.687703i	0.939024	+	0.343852i	$0.111732\pi$
$659$			453.196i			0.687703i	−0.939024	+	0.343852i	$0.888268\pi$
$660$		0			0
$661$	717.546			1.08555			0.542773	−	0.839879i	$-0.317374\pi$
$661$	717.546			1.08555			0.542773	+	0.839879i	$0.317374\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−90.8082	+	90.8082i	−0.136144	+	0.136144i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1665.28			−2.48179
$672$		0			0
$673$	121.413	+	121.413i	0.180406	+	0.180406i	0.791533	−	0.611127i	$-0.209283\pi$
$673$	121.413	+	121.413i	0.180406	+	0.180406i	−0.611127	+	0.791533i	$0.709283\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−671.943	+	671.943i	−0.992531	+	0.992531i	−0.999972	−	0.00744150i	$-0.997631\pi$
$677$	−671.943	+	671.943i	−0.992531	+	0.992531i	0.00744150	+	0.999972i	$0.497631\pi$
$678$		0			0
$679$		−	825.464i		−	1.21571i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	802.095	+	802.095i	1.17437	+	1.17437i	0.981157	+	0.193214i	$0.0618913\pi$
$683$	802.095	+	802.095i	1.17437	+	1.17437i	0.193214	+	0.981157i	$0.438109\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$			864.745i			1.25507i
$690$		0			0
$691$	−1146.49			−1.65918			−0.829591	−	0.558371i	$-0.811426\pi$
$691$	−1146.49			−1.65918			−0.829591	+	0.558371i	$0.811426\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	192.577	−	192.577i	0.276293	−	0.276293i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	673.217			0.960366			0.480183	−	0.877168i	$-0.340570\pi$
$701$	673.217			0.960366			0.480183	+	0.877168i	$0.340570\pi$
$702$		0			0
$703$	−106.515	−	106.515i	−0.151515	−	0.151515i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	340.833	−	340.833i	0.482084	−	0.482084i
$708$		0			0
$709$			955.939i			1.34829i	0.738598	+	0.674146i	$0.235488\pi$
$709$			955.939i			1.34829i	−0.738598	+	0.674146i	$0.764512\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	54.3429	+	54.3429i	0.0762172	+	0.0762172i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$			89.0306i			0.123826i	0.998082	+	0.0619128i	$0.0197201\pi$
$719$			89.0306i			0.123826i	−0.998082	+	0.0619128i	$0.980280\pi$
$720$		0			0
$721$	345.010			0.478516
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−332.703	+	332.703i	−0.457638	+	0.457638i	−0.897879	−	0.440242i	$-0.854893\pi$
$727$	−332.703	+	332.703i	−0.457638	+	0.457638i	0.440242	+	0.897879i	$0.354893\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	379.051			0.518538
$732$		0			0
$733$	−229.823	−	229.823i	−0.313537	−	0.313537i	0.532741	−	0.846278i	$-0.321162\pi$
$733$	−229.823	−	229.823i	−0.313537	−	0.313537i	−0.846278	+	0.532741i	$0.821162\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	401.262	−	401.262i	0.544454	−	0.544454i
$738$		0			0
$739$		−	923.281i		−	1.24936i	−0.780879	−	0.624682i	$-0.785228\pi$
$739$		−	923.281i		−	1.24936i	0.780879	−	0.624682i	$-0.214772\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−585.782	−	585.782i	−0.788401	−	0.788401i	0.192831	−	0.981232i	$-0.438233\pi$
$743$	−585.782	−	585.782i	−0.788401	−	0.788401i	−0.981232	+	0.192831i	$0.938233\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$			243.423i			0.324998i
$750$		0			0
$751$	1354.27			1.80329			0.901645	−	0.432477i	$-0.142361\pi$
$751$	1354.27			1.80329			0.901645	+	0.432477i	$0.142361\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	985.832	−	985.832i	1.30229	−	1.30229i	0.375442	−	0.926846i	$-0.377491\pi$
$757$	985.832	−	985.832i	1.30229	−	1.30229i	0.926846	−	0.375442i	$-0.122509\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−206.495			−0.271347			−0.135673	−	0.990754i	$-0.543320\pi$
$761$	−206.495			−0.271347			−0.135673	+	0.990754i	$0.543320\pi$
$762$		0			0
$763$	1188.08	+	1188.08i	1.55712	+	1.55712i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1086.87	+	1086.87i	−1.41704	+	1.41704i
$768$		0			0
$769$			852.969i			1.10919i	0.832119	+	0.554596i	$0.187127\pi$
$769$			852.969i			1.10919i	−0.832119	+	0.554596i	$0.812873\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−491.035	−	491.035i	−0.635233	−	0.635233i	0.314143	−	0.949376i	$-0.398283\pi$
$773$	−491.035	−	491.035i	−0.635233	−	0.635233i	−0.949376	+	0.314143i	$0.898283\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	369.694i		−	0.474575i
$780$		0			0
$781$	−1266.97			−1.62224
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	958.586	−	958.586i	1.21803	−	1.21803i	0.249702	−	0.968323i	$-0.419667\pi$
$787$	958.586	−	958.586i	1.21803	−	1.21803i	0.968323	−	0.249702i	$-0.0803328\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−194.515			−0.245911
$792$		0			0
$793$	1255.30	+	1255.30i	1.58298	+	1.58298i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	169.065	−	169.065i	0.212126	−	0.212126i	−0.593044	−	0.805170i	$-0.702074\pi$
$797$	169.065	−	169.065i	0.212126	−	0.212126i	0.805170	+	0.593044i	$0.202074\pi$
$798$		0			0
$799$			332.908i			0.416656i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	718.005	+	718.005i	0.894153	+	0.894153i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	126.082i		−	0.155849i	−0.996959	−	0.0779244i	$-0.975171\pi$
$809$		−	126.082i		−	0.155849i	0.996959	−	0.0779244i	$-0.0248293\pi$
$810$		0			0
$811$	32.9286			0.0406024			0.0203012	−	0.999794i	$-0.493537\pi$
$811$	32.9286			0.0406024			0.0203012	+	0.999794i	$0.493537\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	363.837	−	363.837i	0.445333	−	0.445333i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−980.064			−1.19374			−0.596872	−	0.802337i	$-0.703590\pi$
$821$	−980.064			−1.19374			−0.596872	+	0.802337i	$0.703590\pi$
$822$		0			0
$823$	503.863	+	503.863i	0.612227	+	0.612227i	0.943526	−	0.331299i	$-0.107487\pi$
$823$	503.863	+	503.863i	0.612227	+	0.612227i	−0.331299	+	0.943526i	$0.607487\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1078.84	−	1078.84i	1.30452	−	1.30452i	0.379208	−	0.925311i	$-0.376196\pi$
$827$	1078.84	−	1078.84i	1.30452	−	1.30452i	0.925311	−	0.379208i	$-0.123804\pi$
$828$		0			0
$829$		−	124.352i		−	0.150002i	−0.997183	−	0.0750012i	$-0.976104\pi$
$829$		−	124.352i		−	0.150002i	0.997183	−	0.0750012i	$-0.0238961\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	468.055	+	468.055i	0.561890	+	0.561890i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1125.48i		−	1.34146i	−0.741702	−	0.670730i	$-0.765981\pi$
$839$		−	1125.48i		−	1.34146i	0.741702	−	0.670730i	$-0.234019\pi$
$840$		0			0
$841$	−331.888			−0.394635
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−1065.12	+	1065.12i	−1.25753	+	1.25753i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	38.9694			0.0457925
$852$		0			0
$853$	−1039.87	−	1039.87i	−1.21908	−	1.21908i	−0.967955	−	0.251122i	$-0.919201\pi$
$853$	−1039.87	−	1039.87i	−1.21908	−	1.21908i	−0.251122	−	0.967955i	$-0.580799\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	620.177	−	620.177i	0.723660	−	0.723660i	−0.245688	−	0.969349i	$-0.579014\pi$
$857$	620.177	−	620.177i	0.723660	−	0.723660i	0.969349	+	0.245688i	$0.0790140\pi$
$858$		0			0
$859$		−	948.888i		−	1.10464i	−0.833631	−	0.552321i	$-0.813742\pi$
$859$		−	948.888i		−	1.10464i	0.833631	−	0.552321i	$-0.186258\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1064.42	−	1064.42i	−1.23339	−	1.23339i	−0.962652	−	0.270740i	$-0.912732\pi$
$863$	−1064.42	−	1064.42i	−1.23339	−	1.23339i	−0.270740	−	0.962652i	$-0.587268\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$			787.918i			0.906695i
$870$		0			0
$871$	−604.949			−0.694545
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	329.975	−	329.975i	0.376254	−	0.376254i	−0.493495	−	0.869749i	$-0.664281\pi$
$877$	329.975	−	329.975i	0.376254	−	0.376254i	0.869749	+	0.493495i	$0.164281\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−135.857			−0.154208			−0.0771039	−	0.997023i	$-0.524567\pi$
$881$	−135.857			−0.154208			−0.0771039	+	0.997023i	$0.524567\pi$
$882$		0			0
$883$	46.8694	+	46.8694i	0.0530797	+	0.0530797i	0.733148	−	0.680069i	$-0.238050\pi$
$883$	46.8694	+	46.8694i	0.0530797	+	0.0530797i	−0.680069	+	0.733148i	$0.738050\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−43.5097	+	43.5097i	−0.0490526	+	0.0490526i	−0.731208	−	0.682155i	$-0.761043\pi$
$887$	−43.5097	+	43.5097i	−0.0490526	+	0.0490526i	0.682155	+	0.731208i	$0.261043\pi$
$888$		0			0
$889$		−	110.990i		−	0.124848i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	319.546	+	319.546i	0.357834	+	0.357834i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$			701.898i			0.780754i
$900$		0			0
$901$	533.112			0.591690
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−638.938	+	638.938i	−0.704452	+	0.704452i	−0.965363	−	0.260911i	$-0.915977\pi$
$907$	−638.938	+	638.938i	−0.704452	+	0.704452i	0.260911	+	0.965363i	$0.415977\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−347.546			−0.381499			−0.190750	−	0.981639i	$-0.561092\pi$
$911$	−347.546			−0.381499			−0.190750	+	0.981639i	$0.561092\pi$
$912$		0			0
$913$	−1651.41	−	1651.41i	−1.80878	−	1.80878i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	147.146	−	147.146i	0.160465	−	0.160465i
$918$		0			0
$919$		−	27.6684i		−	0.0301071i	−0.999887	−	0.0150535i	$-0.995208\pi$
$919$		−	27.6684i		−	0.0301071i	0.999887	−	0.0150535i	$-0.00479187\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	955.051	+	955.051i	1.03472	+	1.03472i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1070.49i			1.15231i	0.817341	+	0.576154i	$0.195447\pi$
$929$			1070.49i			1.15231i	−0.817341	+	0.576154i	$0.804553\pi$
$930$		0			0
$931$	898.536			0.965130
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−60.5551	+	60.5551i	−0.0646266	+	0.0646266i	−0.738681	−	0.674055i	$-0.764551\pi$
$937$	−60.5551	+	60.5551i	−0.0646266	+	0.0646266i	0.674055	+	0.738681i	$0.264551\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	264.064			0.280620			0.140310	−	0.990108i	$-0.455190\pi$
$941$	264.064			0.280620			0.140310	+	0.990108i	$0.455190\pi$
$942$		0			0
$943$	67.6276	+	67.6276i	0.0717153	+	0.0717153i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−115.455	+	115.455i	−0.121917	+	0.121917i	−0.765433	−	0.643516i	$-0.777475\pi$
$947$	−115.455	+	115.455i	−0.121917	+	0.121917i	0.643516	+	0.765433i	$0.277475\pi$
$948$		0			0
$949$		−	1082.47i		−	1.14065i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−378.359	−	378.359i	−0.397019	−	0.397019i	0.480162	−	0.877180i	$-0.340578\pi$
$953$	−378.359	−	378.359i	−0.397019	−	0.397019i	−0.877180	+	0.480162i	$0.840578\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	1514.35i		−	1.57909i
$960$		0			0
$961$	−540.959			−0.562913
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	52.1964	−	52.1964i	0.0539777	−	0.0539777i	−0.679603	−	0.733580i	$-0.737848\pi$
$967$	52.1964	−	52.1964i	0.0539777	−	0.0539777i	0.733580	+	0.679603i	$0.237848\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1545.61			1.59177			0.795886	−	0.605447i	$-0.207006\pi$
$971$	1545.61			1.59177			0.795886	+	0.605447i	$0.207006\pi$
$972$		0			0
$973$	178.940	+	178.940i	0.183905	+	0.183905i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−843.206	+	843.206i	−0.863056	+	0.863056i	−0.991692	−	0.128636i	$-0.958940\pi$
$977$	−843.206	+	843.206i	−0.863056	+	0.863056i	0.128636	+	0.991692i	$0.458940\pi$
$978$		0			0
$979$			2550.35i			2.60506i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1241.39	+	1241.39i	1.26286	+	1.26286i	0.949703	+	0.313153i	$0.101385\pi$
$983$	1241.39	+	1241.39i	1.26286	+	1.26286i	0.313153	+	0.949703i	$0.398615\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			133.112i			0.134593i
$990$		0			0
$991$	219.816			0.221813			0.110906	−	0.993831i	$-0.464625\pi$
$991$	219.816			0.221813			0.110906	+	0.993831i	$0.464625\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−599.571	+	599.571i	−0.601375	+	0.601375i	−0.940677	−	0.339302i	$-0.889809\pi$
$997$	−599.571	+	599.571i	−0.601375	+	0.601375i	0.339302	+	0.940677i	$0.389809\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.3.l.b.793.1		4
3.2	odd	2		300.3.k.a.193.2		4
5.2	odd	4	inner	900.3.l.b.757.1		4
5.3	odd	4		180.3.l.b.37.1		4
5.4	even	2		180.3.l.b.73.1		4
12.11	even	2		1200.3.bg.o.193.1		4
15.2	even	4		300.3.k.a.157.2		4
15.8	even	4		60.3.k.a.37.1	yes	4
15.14	odd	2		60.3.k.a.13.1	✓	4
20.3	even	4		720.3.bh.f.577.1		4
20.19	odd	2		720.3.bh.f.433.1		4
60.23	odd	4		240.3.bg.d.97.2		4
60.47	odd	4		1200.3.bg.o.1057.1		4
60.59	even	2		240.3.bg.d.193.2		4
120.29	odd	2		960.3.bg.b.193.2		4
120.53	even	4		960.3.bg.b.577.2		4
120.59	even	2		960.3.bg.a.193.1		4
120.83	odd	4		960.3.bg.a.577.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.3.k.a.13.1	✓	4	15.14	odd	2
60.3.k.a.37.1	yes	4	15.8	even	4
180.3.l.b.37.1		4	5.3	odd	4
180.3.l.b.73.1		4	5.4	even	2
240.3.bg.d.97.2		4	60.23	odd	4
240.3.bg.d.193.2		4	60.59	even	2
300.3.k.a.157.2		4	15.2	even	4
300.3.k.a.193.2		4	3.2	odd	2
720.3.bh.f.433.1		4	20.19	odd	2
720.3.bh.f.577.1		4	20.3	even	4
900.3.l.b.757.1		4	5.2	odd	4	inner
900.3.l.b.793.1		4	1.1	even	1	trivial
960.3.bg.a.193.1		4	120.59	even	2
960.3.bg.a.577.1		4	120.83	odd	4
960.3.bg.b.193.2		4	120.29	odd	2
960.3.bg.b.577.2		4	120.53	even	4
1200.3.bg.o.193.1		4	12.11	even	2
1200.3.bg.o.1057.1		4	60.47	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	900.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-7.44949 + 7.44949i) q^{7} +16.2474 q^{11} +(-12.2474 - 12.2474i) q^{13} +(-7.55051 + 7.55051i) q^{17} +14.4949i q^{19} +(-2.65153 - 2.65153i) q^{23} -34.2474i q^{29} -20.4949 q^{31} +(-7.34847 + 7.34847i) q^{37} -25.5051 q^{41} +(-25.1010 - 25.1010i) q^{43} +(22.0454 - 22.0454i) q^{47} -61.9898i q^{49} +(-35.3031 - 35.3031i) q^{53} -88.7423i q^{59} -102.495 q^{61} +(24.6969 - 24.6969i) q^{67} -77.9796 q^{71} +(44.1918 + 44.1918i) q^{73} +(-121.035 + 121.035i) q^{77} +48.4949i q^{79} +(-101.641 - 101.641i) q^{83} +156.969i q^{89} +182.474 q^{91} +(-55.4041 + 55.4041i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{7} + 16 q^{11} - 40 q^{17} - 40 q^{23} + 16 q^{31} - 200 q^{41} - 120 q^{43} - 200 q^{53} - 312 q^{61} + 40 q^{67} + 80 q^{71} + 20 q^{73} - 200 q^{77} - 240 q^{83} + 240 q^{91} - 300 q^{97}+O(q^{100}) \) 4 * q - 20 * q^7 + 16 * q^11 - 40 * q^17 - 40 * q^23 + 16 * q^31 - 200 * q^41 - 120 * q^43 - 200 * q^53 - 312 * q^61 + 40 * q^67 + 80 * q^71 + 20 * q^73 - 200 * q^77 - 240 * q^83 + 240 * q^91 - 300 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more