LMFDB - Embedded newform 531.3.c.c.235.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,3,Mod(235,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("531.235");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	531.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:	$14.4687020375$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 60 x^{18} + 1522 x^{16} + 21286 x^{14} + 179593 x^{12} + 941588 x^{10} + 3057025 x^{8} + \cdots + 570861 $ x^20 + 60x^18 + 1522x^16 + 21286x^14 + 179593x^12 + 941588x^10 + 3057025x^8 + 5962230x^6 + 6517782x^4 + 3428244*x^2 + 570861
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11}\cdot 3 $
Twist minimal:	no (minimal twist has level 177)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			235.20
Root			$3.65759i$ of defining polynomial
Character	$\chi$	$=$	531.235
Dual form			531.3.c.c.235.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.65759i q^{2} -9.37798 q^{4} -0.395869 q^{5} -6.20847 q^{7} -19.6705i q^{8} +O(q^{10})$	\(q+3.65759i q^{2} -9.37798 q^{4} -0.395869 q^{5} -6.20847 q^{7} -19.6705i q^{8} -1.44793i q^{10} -8.54756i q^{11} +0.0887419i q^{13} -22.7081i q^{14} +34.4346 q^{16} +8.10940 q^{17} +10.5067 q^{19} +3.71245 q^{20} +31.2635 q^{22} -14.0307i q^{23} -24.8433 q^{25} -0.324582 q^{26} +58.2229 q^{28} +56.9344 q^{29} -0.471045i q^{31} +47.2659i q^{32} +29.6609i q^{34} +2.45774 q^{35} -43.8322i q^{37} +38.4292i q^{38} +7.78692i q^{40} -13.8735 q^{41} +53.7675i q^{43} +80.1588i q^{44} +51.3187 q^{46} -34.1786i q^{47} -10.4549 q^{49} -90.8666i q^{50} -0.832220i q^{52} +79.3757 q^{53} +3.38371i q^{55} +122.123i q^{56} +208.243i q^{58} +(41.5316 - 41.9061i) q^{59} +45.7782i q^{61} +1.72289 q^{62} -35.1409 q^{64} -0.0351301i q^{65} -104.453i q^{67} -76.0498 q^{68} +8.98940i q^{70} -74.0325 q^{71} -109.458i q^{73} +160.320 q^{74} -98.5316 q^{76} +53.0673i q^{77} -74.2528 q^{79} -13.6316 q^{80} -50.7436i q^{82} +88.8936i q^{83} -3.21026 q^{85} -196.660 q^{86} -168.134 q^{88} -142.879i q^{89} -0.550951i q^{91} +131.580i q^{92} +125.012 q^{94} -4.15927 q^{95} -81.4737i q^{97} -38.2398i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 40 q^{4} - 8 q^{7}+O(q^{10}) $ 20 * q - 40 * q^4 - 8 * q^7	$ 20 q - 40 q^{4} - 8 q^{7} - 8 q^{16} - 16 q^{17} - 60 q^{19} + 164 q^{20} + 40 q^{22} + 100 q^{25} + 156 q^{26} + 200 q^{28} + 60 q^{29} + 32 q^{35} - 28 q^{41} + 180 q^{46} + 284 q^{49} + 8 q^{53} + 152 q^{59} + 8 q^{62} + 204 q^{64} - 384 q^{68} - 92 q^{71} - 104 q^{74} + 120 q^{76} - 420 q^{79} - 376 q^{80} - 348 q^{85} - 232 q^{86} - 212 q^{88} + 152 q^{94} - 788 q^{95}+O(q^{100}) $ 20 * q - 40 * q^4 - 8 * q^7 - 8 * q^16 - 16 * q^17 - 60 * q^19 + 164 * q^20 + 40 * q^22 + 100 * q^25 + 156 * q^26 + 200 * q^28 + 60 * q^29 + 32 * q^35 - 28 * q^41 + 180 * q^46 + 284 * q^49 + 8 * q^53 + 152 * q^59 + 8 * q^62 + 204 * q^64 - 384 * q^68 - 92 * q^71 - 104 * q^74 + 120 * q^76 - 420 * q^79 - 376 * q^80 - 348 * q^85 - 232 * q^86 - 212 * q^88 + 152 * q^94 - 788 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/531\mathbb{Z}\right)^\times$.

$n$	$119$	$415$
$\chi(n)$	$1$	$-1$

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 $

$2$			3.65759i			1.82880i	0.404816	+	0.914398i	$0.367335\pi$
$2$			3.65759i			1.82880i	−0.404816	+	0.914398i	$0.632665\pi$
$3$		0			0
$3$		0			0
$4$	−9.37798			−2.34450
$5$	−0.395869			−0.0791737			−0.0395869	−	0.999216i	$-0.512604\pi$
$5$	−0.395869			−0.0791737			−0.0395869	+	0.999216i	$0.512604\pi$
$6$		0			0
$7$	−6.20847			−0.886924			−0.443462	−	0.896293i	$-0.646250\pi$
$7$	−6.20847			−0.886924			−0.443462	+	0.896293i	$0.646250\pi$
$8$		−	19.6705i		−	2.45881i
$9$		0			0
$10$		−	1.44793i		−	0.144793i
$11$		−	8.54756i		−	0.777051i	−0.921438	−	0.388525i	$-0.872985\pi$
$11$		−	8.54756i		−	0.777051i	0.921438	−	0.388525i	$-0.127015\pi$
$12$		0			0
$13$			0.0887419i			0.00682630i	0.999994	+	0.00341315i	$0.00108644\pi$
$13$			0.0887419i			0.00682630i	−0.999994	+	0.00341315i	$0.998914\pi$
$14$		−	22.7081i		−	1.62200i
$15$		0			0
$16$	34.4346			2.15216
$17$	8.10940			0.477023			0.238512	−	0.971140i	$-0.423340\pi$
$17$	8.10940			0.477023			0.238512	+	0.971140i	$0.423340\pi$
$18$		0			0
$19$	10.5067			0.552984			0.276492	−	0.961016i	$-0.410828\pi$
$19$	10.5067			0.552984			0.276492	+	0.961016i	$0.410828\pi$
$20$	3.71245			0.185622
$21$		0			0
$22$	31.2635			1.42107
$23$		−	14.0307i		−	0.610032i	−0.952347	−	0.305016i	$-0.901338\pi$
$23$		−	14.0307i		−	0.610032i	0.952347	−	0.305016i	$-0.0986618\pi$
$24$		0			0
$25$	−24.8433			−0.993732
$26$	−0.324582			−0.0124839
$27$		0			0
$28$	58.2229			2.07939
$29$	56.9344			1.96325			0.981627	−	0.190808i	$-0.0611108\pi$
$29$	56.9344			1.96325			0.981627	+	0.190808i	$0.0611108\pi$
$30$		0			0
$31$		−	0.471045i		−	0.0151950i	−0.999971	−	0.00759751i	$-0.997582\pi$
$31$		−	0.471045i		−	0.0151950i	0.999971	−	0.00759751i	$-0.00241838\pi$
$32$			47.2659i			1.47706i
$33$		0			0
$34$			29.6609i			0.872378i
$35$	2.45774			0.0702211
$36$		0			0
$37$		−	43.8322i		−	1.18465i	−0.805698	−	0.592327i	$-0.798209\pi$
$37$		−	43.8322i		−	1.18465i	0.805698	−	0.592327i	$-0.201791\pi$
$38$			38.4292i			1.01129i
$39$		0			0
$40$			7.78692i			0.194673i
$41$	−13.8735			−0.338378			−0.169189	−	0.985584i	$-0.554115\pi$
$41$	−13.8735			−0.338378			−0.169189	+	0.985584i	$0.554115\pi$
$42$		0			0
$43$			53.7675i			1.25041i	0.780462	+	0.625204i	$0.214984\pi$
$43$			53.7675i			1.25041i	−0.780462	+	0.625204i	$0.785016\pi$
$44$			80.1588i			1.82179i
$45$		0			0
$46$	51.3187			1.11562
$47$		−	34.1786i		−	0.727205i	−0.931554	−	0.363602i	$-0.881547\pi$
$47$		−	34.1786i		−	0.727205i	0.931554	−	0.363602i	$-0.118453\pi$
$48$		0			0
$49$	−10.4549			−0.213365
$50$		−	90.8666i		−	1.81733i
$51$		0			0
$52$		−	0.832220i		−	0.0160042i
$53$	79.3757			1.49765			0.748827	−	0.662765i	$-0.230617\pi$
$53$	79.3757			1.49765			0.748827	+	0.662765i	$0.230617\pi$
$54$		0			0
$55$			3.38371i			0.0615220i
$56$			122.123i			2.18078i
$57$		0			0
$58$			208.243i			3.59039i
$59$	41.5316	−	41.9061i	0.703926	−	0.710273i
$59$	41.5316	−	41.9061i	0.703926	−	0.710273i
$60$		0			0
$61$			45.7782i			0.750462i	0.926931	+	0.375231i	$0.122437\pi$
$61$			45.7782i			0.750462i	−0.926931	+	0.375231i	$0.877563\pi$
$62$	1.72289			0.0277886
$63$		0			0
$64$	−35.1409			−0.549077
$65$		−	0.0351301i		−	0.000540463i
$66$		0			0
$67$		−	104.453i		−	1.55900i	−0.626403	−	0.779499i	$-0.715474\pi$
$67$		−	104.453i		−	1.55900i	0.626403	−	0.779499i	$-0.284526\pi$
$68$	−76.0498			−1.11838
$69$		0			0
$70$			8.98940i			0.128420i
$71$	−74.0325			−1.04271			−0.521356	−	0.853340i	$-0.674573\pi$
$71$	−74.0325			−1.04271			−0.521356	+	0.853340i	$0.674573\pi$
$72$		0			0
$73$		−	109.458i		−	1.49943i	−0.661761	−	0.749715i	$-0.730190\pi$
$73$		−	109.458i		−	1.49943i	0.661761	−	0.749715i	$-0.269810\pi$
$74$	160.320			2.16649
$75$		0			0
$76$	−98.5316			−1.29647
$77$			53.0673i			0.689185i
$78$		0			0
$79$	−74.2528			−0.939909			−0.469954	−	0.882691i	$-0.655730\pi$
$79$	−74.2528			−0.939909			−0.469954	+	0.882691i	$0.655730\pi$
$80$	−13.6316			−0.170395
$81$		0			0
$82$		−	50.7436i		−	0.618825i
$83$			88.8936i			1.07101i	0.844533	+	0.535503i	$0.179878\pi$
$83$			88.8936i			1.07101i	−0.844533	+	0.535503i	$0.820122\pi$
$84$		0			0
$85$	−3.21026			−0.0377677
$86$	−196.660			−2.28674
$87$		0			0
$88$	−168.134			−1.91062
$89$		−	142.879i		−	1.60538i	−0.596394	−	0.802692i	$-0.703400\pi$
$89$		−	142.879i		−	1.60538i	0.596394	−	0.802692i	$-0.296600\pi$
$90$		0			0
$91$		−	0.550951i		−	0.00605441i
$92$			131.580i			1.43022i
$93$		0			0
$94$	125.012			1.32991
$95$	−4.15927			−0.0437818
$96$		0			0
$97$		−	81.4737i		−	0.839935i	−0.907539	−	0.419968i	$-0.862041\pi$
$97$		−	81.4737i		−	0.839935i	0.907539	−	0.419968i	$-0.137959\pi$
$98$		−	38.2398i		−	0.390202i
$99$		0			0
$100$	232.980			2.32980
$101$			92.0885i			0.911768i	0.890039	+	0.455884i	$0.150677\pi$
$101$			92.0885i			0.911768i	−0.890039	+	0.455884i	$0.849323\pi$
$102$		0			0
$103$		−	147.520i		−	1.43223i	−0.697982	−	0.716115i	$-0.745918\pi$
$103$		−	147.520i		−	1.43223i	0.697982	−	0.716115i	$-0.254082\pi$
$104$	1.74559			0.0167846
$105$		0			0
$106$			290.324i			2.73891i
$107$	−68.9648			−0.644530			−0.322265	−	0.946649i	$-0.604444\pi$
$107$	−68.9648			−0.644530			−0.322265	+	0.946649i	$0.604444\pi$
$108$		0			0
$109$		−	57.3299i		−	0.525963i	−0.964801	−	0.262981i	$-0.915294\pi$
$109$		−	57.3299i		−	0.525963i	0.964801	−	0.262981i	$-0.0847058\pi$
$110$	−12.3762			−0.112511
$111$		0			0
$112$	−213.786			−1.90881
$113$			19.6848i			0.174201i	0.996200	+	0.0871007i	$0.0277602\pi$
$113$			19.6848i			0.174201i	−0.996200	+	0.0871007i	$0.972240\pi$
$114$		0			0
$115$			5.55433i			0.0482985i
$116$	−533.930			−4.60284
$117$		0			0
$118$	153.276	+	151.906i	1.29894	+	1.28734i
$119$	−50.3469			−0.423084
$120$		0			0
$121$	47.9392			0.396192
$122$	−167.438			−1.37244
$123$		0			0
$124$			4.41745i			0.0356246i
$125$	19.7314			0.157851
$126$		0			0
$127$	5.95404			0.0468822			0.0234411	−	0.999725i	$-0.492538\pi$
$127$	5.95404			0.0468822			0.0234411	+	0.999725i	$0.492538\pi$
$128$			60.5324i			0.472909i
$129$		0			0
$130$	0.128492			0.000988397
$131$		−	207.532i		−	1.58421i	−0.610384	−	0.792106i	$-0.708985\pi$
$131$		−	207.532i		−	1.58421i	0.610384	−	0.792106i	$-0.291015\pi$
$132$		0			0
$133$	−65.2305			−0.490455
$134$	382.046			2.85109
$135$		0			0
$136$		−	159.516i		−	1.17291i
$137$	62.5295			0.456419			0.228210	−	0.973612i	$-0.426713\pi$
$137$	62.5295			0.456419			0.228210	+	0.973612i	$0.426713\pi$
$138$		0			0
$139$	−101.500			−0.730217			−0.365108	−	0.930965i	$-0.618968\pi$
$139$	−101.500			−0.730217			−0.365108	+	0.930965i	$0.618968\pi$
$140$	−23.0486			−0.164633
$141$		0			0
$142$		−	270.781i		−	1.90691i
$143$	0.758526			0.00530438
$144$		0			0
$145$	−22.5385			−0.155438
$146$	400.354			2.74215
$147$		0			0
$148$			411.058i			2.77742i
$149$			15.5533i			0.104385i	0.998637	+	0.0521924i	$0.0166209\pi$
$149$			15.5533i			0.104385i	−0.998637	+	0.0521924i	$0.983379\pi$
$150$		0			0
$151$		−	48.2391i		−	0.319464i	−0.987160	−	0.159732i	$-0.948937\pi$
$151$		−	48.2391i		−	0.319464i	0.987160	−	0.159732i	$-0.0510631\pi$
$152$		−	206.671i		−	1.35968i
$153$		0			0
$154$	−194.098			−1.26038
$155$			0.186472i			0.00120305i
$156$		0			0
$157$			218.253i			1.39014i	0.718940	+	0.695072i	$0.244628\pi$
$157$			218.253i			1.39014i	−0.718940	+	0.695072i	$0.755372\pi$
$158$		−	271.586i		−	1.71890i
$159$		0			0
$160$		−	18.7111i		−	0.116944i
$161$			87.1094i			0.541052i
$162$		0			0
$163$	300.810			1.84546			0.922729	−	0.385449i	$-0.125953\pi$
$163$	300.810			1.84546			0.922729	+	0.385449i	$0.125953\pi$
$164$	130.106			0.793326
$165$		0			0
$166$	−325.136			−1.95865
$167$	285.372			1.70881			0.854407	−	0.519605i	$-0.173921\pi$
$167$	285.372			1.70881			0.854407	+	0.519605i	$0.173921\pi$
$168$		0			0
$169$	168.992			0.999953
$170$		−	11.7418i		−	0.0690694i
$171$		0			0
$172$		−	504.231i		−	2.93157i
$173$		−	115.450i		−	0.667344i	−0.942689	−	0.333672i	$-0.891712\pi$
$173$		−	115.450i		−	0.667344i	0.942689	−	0.333672i	$-0.108288\pi$
$174$		0			0
$175$	154.239			0.881365
$176$		−	294.332i		−	1.67234i
$177$		0			0
$178$	522.594			2.93592
$179$		−	145.888i		−	0.815016i	−0.913202	−	0.407508i	$-0.866398\pi$
$179$		−	145.888i		−	0.815016i	0.913202	−	0.407508i	$-0.133602\pi$
$180$		0			0
$181$	−40.9343			−0.226157			−0.113078	−	0.993586i	$-0.536071\pi$
$181$	−40.9343			−0.226157			−0.113078	+	0.993586i	$0.536071\pi$
$182$	2.01516			0.0110723
$183$		0			0
$184$	−275.991			−1.49995
$185$			17.3518i			0.0937935i
$186$		0			0
$187$		−	69.3155i		−	0.370671i
$188$			320.527i			1.70493i
$189$		0			0
$190$		−	15.2129i		−	0.0800680i
$191$		−	281.121i		−	1.47184i	−0.677070	−	0.735919i	$-0.736751\pi$
$191$		−	281.121i		−	1.47184i	0.677070	−	0.735919i	$-0.263249\pi$
$192$		0			0
$193$	46.9362			0.243193			0.121596	−	0.992580i	$-0.461199\pi$
$193$	46.9362			0.243193			0.121596	+	0.992580i	$0.461199\pi$
$194$	297.998			1.53607
$195$		0			0
$196$	98.0459			0.500234
$197$	−20.8170			−0.105670			−0.0528350	−	0.998603i	$-0.516826\pi$
$197$	−20.8170			−0.105670			−0.0528350	+	0.998603i	$0.516826\pi$
$198$		0			0
$199$	−206.378			−1.03707			−0.518537	−	0.855055i	$-0.673523\pi$
$199$	−206.378			−1.03707			−0.518537	+	0.855055i	$0.673523\pi$
$200$			488.679i			2.44339i
$201$		0			0
$202$	−336.822			−1.66744
$203$	−353.475			−1.74126
$204$		0			0
$205$	5.49209			0.0267907
$206$	539.567			2.61926
$207$		0			0
$208$			3.05579i			0.0146913i
$209$		−	89.8066i		−	0.429697i
$210$		0			0
$211$			200.660i			0.950993i	0.879718	+	0.475497i	$0.157732\pi$
$211$			200.660i			0.950993i	−0.879718	+	0.475497i	$0.842268\pi$
$212$	−744.384			−3.51124
$213$		0			0
$214$		−	252.245i		−	1.17871i
$215$		−	21.2849i		−	0.0989994i
$216$		0			0
$217$			2.92447i			0.0134768i
$218$	209.690			0.961879
$219$		0			0
$220$		−	31.7324i		−	0.144238i
$221$			0.719643i			0.00325630i
$222$		0			0
$223$	−271.603			−1.21795			−0.608976	−	0.793189i	$-0.708419\pi$
$223$	−271.603			−1.21795			−0.608976	+	0.793189i	$0.708419\pi$
$224$		−	293.449i		−	1.31004i
$225$		0			0
$226$	−71.9989			−0.318579
$227$		−	124.072i		−	0.546571i	−0.961933	−	0.273286i	$-0.911890\pi$
$227$		−	124.072i		−	0.546571i	0.961933	−	0.273286i	$-0.0881104\pi$
$228$		0			0
$229$		−	269.653i		−	1.17752i	−0.808307	−	0.588762i	$-0.799616\pi$
$229$		−	269.653i		−	1.17752i	0.808307	−	0.588762i	$-0.200384\pi$
$230$	−20.3155			−0.0883281
$231$		0			0
$232$		−	1119.93i		−	4.82727i
$233$			432.383i			1.85572i	0.372928	+	0.927860i	$0.378354\pi$
$233$			432.383i			1.85572i	−0.372928	+	0.927860i	$0.621646\pi$
$234$		0			0
$235$			13.5302i			0.0575755i
$236$	−389.483	+	392.995i	−1.65035	+	1.66523i
$237$		0			0
$238$		−	184.149i		−	0.773733i
$239$	−345.014			−1.44357			−0.721787	−	0.692115i	$-0.756679\pi$
$239$	−345.014			−1.44357			−0.721787	+	0.692115i	$0.756679\pi$
$240$		0			0
$241$	−395.883			−1.64267			−0.821334	−	0.570447i	$-0.806770\pi$
$241$	−395.883			−1.64267			−0.821334	+	0.570447i	$0.806770\pi$
$242$			175.342i			0.724554i
$243$		0			0
$244$		−	429.307i		−	1.75945i
$245$	4.13877			0.0168929
$246$		0			0
$247$			0.932384i			0.00377483i
$248$	−9.26568			−0.0373616
$249$		0			0
$250$			72.1694i			0.288678i
$251$	−98.2846			−0.391572			−0.195786	−	0.980647i	$-0.562726\pi$
$251$	−98.2846			−0.391572			−0.195786	+	0.980647i	$0.562726\pi$
$252$		0			0
$253$	−119.929			−0.474026
$254$			21.7774i			0.0857379i
$255$		0			0
$256$	−361.966			−1.41393
$257$	248.162			0.965609			0.482805	−	0.875728i	$-0.339618\pi$
$257$	248.162			0.965609			0.482805	+	0.875728i	$0.339618\pi$
$258$		0			0
$259$			272.131i			1.05070i
$260$			0.329450i			0.00126711i
$261$		0			0
$262$	759.066			2.89720
$263$	−194.073			−0.737919			−0.368959	−	0.929446i	$-0.620286\pi$
$263$	−194.073			−0.737919			−0.368959	+	0.929446i	$0.620286\pi$
$264$		0			0
$265$	−31.4224			−0.118575
$266$		−	238.587i		−	0.896942i
$267$		0			0
$268$			979.557i			3.65506i
$269$		−	213.661i		−	0.794278i	−0.917758	−	0.397139i	$-0.870003\pi$
$269$		−	213.661i		−	0.794278i	0.917758	−	0.397139i	$-0.129997\pi$
$270$		0			0
$271$	−56.3572			−0.207960			−0.103980	−	0.994579i	$-0.533158\pi$
$271$	−56.3572			−0.207960			−0.103980	+	0.994579i	$0.533158\pi$
$272$	279.244			1.02663
$273$		0			0
$274$			228.707i			0.834698i
$275$			212.349i			0.772180i
$276$		0			0
$277$	170.342			0.614953			0.307476	−	0.951556i	$-0.400515\pi$
$277$	170.342			0.614953			0.307476	+	0.951556i	$0.400515\pi$
$278$		−	371.246i		−	1.33542i
$279$		0			0
$280$		−	48.3448i		−	0.172660i
$281$	39.9961			0.142335			0.0711674	−	0.997464i	$-0.477328\pi$
$281$	39.9961			0.142335			0.0711674	+	0.997464i	$0.477328\pi$
$282$		0			0
$283$			357.475i			1.26316i	0.775309	+	0.631581i	$0.217594\pi$
$283$			357.475i			1.26316i	−0.775309	+	0.631581i	$0.782406\pi$
$284$	694.275			2.44463
$285$		0			0
$286$			2.77438i			0.00970063i
$287$	86.1333			0.300116
$288$		0			0
$289$	−223.238			−0.772449
$290$		−	82.4368i		−	0.284265i
$291$		0			0
$292$			1026.50i			3.51541i
$293$	−67.1472			−0.229171			−0.114586	−	0.993413i	$-0.536554\pi$
$293$	−67.1472			−0.229171			−0.114586	+	0.993413i	$0.536554\pi$
$294$		0			0
$295$	−16.4411	+	16.5893i	−0.0557324	+	0.0562350i
$296$	−862.200			−2.91284
$297$		0			0
$298$	−56.8878			−0.190899
$299$	1.24511			0.00416426
$300$		0			0
$301$		−	333.814i		−	1.10902i
$302$	176.439			0.584235
$303$		0			0
$304$	361.794			1.19011
$305$		−	18.1221i		−	0.0594169i
$306$		0			0
$307$	123.299			0.401626			0.200813	−	0.979630i	$-0.435642\pi$
$307$	123.299			0.401626			0.200813	+	0.979630i	$0.435642\pi$
$308$		−	497.664i		−	1.61579i
$309$		0			0
$310$	−0.682039			−0.00220013
$311$	246.230			0.791737			0.395868	−	0.918307i	$-0.370444\pi$
$311$	246.230			0.791737			0.395868	+	0.918307i	$0.370444\pi$
$312$		0			0
$313$			489.577i			1.56414i	0.623189	+	0.782071i	$0.285837\pi$
$313$			489.577i			1.56414i	−0.623189	+	0.782071i	$0.714163\pi$
$314$	−798.280			−2.54229
$315$		0			0
$316$	696.341			2.20361
$317$	−319.318			−1.00731			−0.503656	−	0.863904i	$-0.668012\pi$
$317$	−319.318			−1.00731			−0.503656	+	0.863904i	$0.668012\pi$
$318$		0			0
$319$		−	486.650i		−	1.52555i
$320$	13.9112			0.0434725
$321$		0			0
$322$	−318.611			−0.989474
$323$	85.2029			0.263786
$324$		0			0
$325$		−	2.20464i		−	0.00678351i
$326$			1100.24i			3.37497i
$327$		0			0
$328$			272.898i			0.832007i
$329$			212.197i			0.644976i
$330$		0			0
$331$	−222.382			−0.671850			−0.335925	−	0.941889i	$-0.609049\pi$
$331$	−222.382			−0.671850			−0.335925	+	0.941889i	$0.609049\pi$
$332$		−	833.642i		−	2.51097i
$333$		0			0
$334$			1043.77i			3.12507i
$335$			41.3496i			0.123432i
$336$		0			0
$337$			285.028i			0.845781i	0.906181	+	0.422890i	$0.138984\pi$
$337$			285.028i			0.845781i	−0.906181	+	0.422890i	$0.861016\pi$
$338$			618.104i			1.82871i
$339$		0			0
$340$	30.1057			0.0885462
$341$	−4.02629			−0.0118073
$342$		0			0
$343$	369.124			1.07616
$344$	1057.63			3.07451
$345$		0			0
$346$	422.271			1.22044
$347$			62.2130i			0.179288i	0.995974	+	0.0896441i	$0.0285730\pi$
$347$			62.2130i			0.179288i	−0.995974	+	0.0896441i	$0.971427\pi$
$348$		0			0
$349$			291.454i			0.835112i	0.908651	+	0.417556i	$0.137113\pi$
$349$			291.454i			0.835112i	−0.908651	+	0.417556i	$0.862887\pi$
$350$			564.143i			1.61184i
$351$		0			0
$352$	404.008			1.14775
$353$			306.106i			0.867156i	0.901116	+	0.433578i	$0.142749\pi$
$353$			306.106i			0.867156i	−0.901116	+	0.433578i	$0.857251\pi$
$354$		0			0
$355$	29.3071			0.0825553
$356$			1339.92i			3.76382i
$357$		0			0
$358$	533.598			1.49050
$359$	−461.119			−1.28445			−0.642226	−	0.766515i	$-0.721989\pi$
$359$	−461.119			−1.28445			−0.642226	+	0.766515i	$0.721989\pi$
$360$		0			0
$361$	−250.609			−0.694209
$362$		−	149.721i		−	0.413594i
$363$		0			0
$364$			5.16681i			0.0141945i
$365$			43.3312i			0.118715i
$366$		0			0
$367$		−	232.591i		−	0.633763i	−0.948465	−	0.316881i	$-0.897364\pi$
$367$		−	232.591i		−	0.633763i	0.948465	−	0.316881i	$-0.102636\pi$
$368$		−	483.143i		−	1.31289i
$369$		0			0
$370$	−63.4658			−0.171529
$371$	−492.802			−1.32831
$372$		0			0
$373$	−36.2600			−0.0972117			−0.0486058	−	0.998818i	$-0.515478\pi$
$373$	−36.2600			−0.0972117			−0.0486058	+	0.998818i	$0.515478\pi$
$374$	253.528			0.677882
$375$		0			0
$376$	−672.309			−1.78806
$377$			5.05246i			0.0134018i
$378$		0			0
$379$	−37.4494			−0.0988110			−0.0494055	−	0.998779i	$-0.515733\pi$
$379$	−37.4494			−0.0988110			−0.0494055	+	0.998779i	$0.515733\pi$
$380$	39.0056			0.102646
$381$		0			0
$382$	1028.23			2.69169
$383$	487.151			1.27193			0.635967	−	0.771716i	$-0.280601\pi$
$383$	487.151			1.27193			0.635967	+	0.771716i	$0.280601\pi$
$384$		0			0
$385$		−	21.0077i		−	0.0545654i
$386$			171.673i			0.444750i
$387$		0			0
$388$			764.059i			1.96922i
$389$	−377.266			−0.969834			−0.484917	−	0.874560i	$-0.661150\pi$
$389$	−377.266			−0.969834			−0.484917	+	0.874560i	$0.661150\pi$
$390$		0			0
$391$		−	113.781i		−	0.290999i
$392$			205.653i			0.524624i
$393$		0			0
$394$		−	76.1400i		−	0.193249i
$395$	29.3944			0.0744161
$396$		0			0
$397$		−	457.106i		−	1.15140i	−0.817661	−	0.575700i	$-0.804730\pi$
$397$		−	457.106i		−	1.15140i	0.817661	−	0.575700i	$-0.195270\pi$
$398$		−	754.845i		−	1.89660i
$399$		0			0
$400$	−855.469			−2.13867
$401$			242.627i			0.605056i	0.953141	+	0.302528i	$0.0978305\pi$
$401$			242.627i			0.605056i	−0.953141	+	0.302528i	$0.902169\pi$
$402$		0			0
$403$	0.0418014			0.000103726
$404$		−	863.605i		−	2.13764i
$405$		0			0
$406$		−	1292.87i		−	3.18441i
$407$	−374.658			−0.920537
$408$		0			0
$409$		−	362.561i		−	0.886458i	−0.896408	−	0.443229i	$-0.853833\pi$
$409$		−	362.561i		−	0.886458i	0.896408	−	0.443229i	$-0.146167\pi$
$410$			20.0878i			0.0489947i
$411$		0			0
$412$			1383.44i			3.35786i
$413$	−257.848	+	260.173i	−0.624329	+	0.629959i
$414$		0			0
$415$		−	35.1902i		−	0.0847956i
$416$	−4.19446			−0.0100828
$417$		0			0
$418$	328.476			0.785827
$419$			818.928i			1.95448i	0.212133	+	0.977241i	$0.431959\pi$
$419$			818.928i			1.95448i	−0.212133	+	0.977241i	$0.568041\pi$
$420$		0			0
$421$		−	329.980i		−	0.783801i	−0.920008	−	0.391901i	$-0.871818\pi$
$421$		−	329.980i		−	0.783801i	0.920008	−	0.391901i	$-0.128182\pi$
$422$	−733.931			−1.73917
$423$		0			0
$424$		−	1561.36i		−	3.68244i
$425$	−201.464			−0.474033
$426$		0			0
$427$		−	284.213i		−	0.665603i
$428$	646.750			1.51110
$429$		0			0
$430$	77.8514			0.181050
$431$		−	713.566i		−	1.65561i	−0.561019	−	0.827803i	$-0.689590\pi$
$431$		−	713.566i		−	1.65561i	0.561019	−	0.827803i	$-0.310410\pi$
$432$		0			0
$433$	625.618			1.44485			0.722423	−	0.691452i	$-0.243029\pi$
$433$	625.618			1.44485			0.722423	+	0.691452i	$0.243029\pi$
$434$	−10.6965			−0.0246464
$435$		0			0
$436$			537.639i			1.23312i
$437$		−	147.417i		−	0.337338i
$438$		0			0
$439$	−461.737			−1.05179			−0.525897	−	0.850548i	$-0.676270\pi$
$439$	−461.737			−1.05179			−0.525897	+	0.850548i	$0.676270\pi$
$440$	66.5591			0.151271
$441$		0			0
$442$	−2.63216			−0.00595511
$443$		−	255.246i		−	0.576175i	−0.957604	−	0.288088i	$-0.906981\pi$
$443$		−	255.246i		−	0.576175i	0.957604	−	0.288088i	$-0.0930195\pi$
$444$		0			0
$445$			56.5614i			0.127104i
$446$		−	993.413i		−	2.22738i
$447$		0			0
$448$	218.171			0.486989
$449$	224.485			0.499966			0.249983	−	0.968250i	$-0.419575\pi$
$449$	224.485			0.499966			0.249983	+	0.968250i	$0.419575\pi$
$450$		0			0
$451$			118.585i			0.262937i
$452$		−	184.603i		−	0.408415i
$453$		0			0
$454$	453.804			0.999568
$455$			0.218104i			0.000479350i
$456$		0			0
$457$		−	64.2869i		−	0.140671i	−0.997523	−	0.0703357i	$-0.977593\pi$
$457$		−	64.2869i		−	0.140671i	0.997523	−	0.0703357i	$-0.0224071\pi$
$458$	986.281			2.15345
$459$		0			0
$460$		−	52.0884i		−	0.113236i
$461$	212.758			0.461513			0.230757	−	0.973012i	$-0.425880\pi$
$461$	212.758			0.461513			0.230757	+	0.973012i	$0.425880\pi$
$462$		0			0
$463$		−	291.329i		−	0.629221i	−0.949221	−	0.314610i	$-0.898126\pi$
$463$		−	291.329i		−	0.629221i	0.949221	−	0.314610i	$-0.101874\pi$
$464$	1960.51			4.22524
$465$		0			0
$466$	−1581.48			−3.39373
$467$		−	31.5294i		−	0.0675148i	−0.999430	−	0.0337574i	$-0.989253\pi$
$467$		−	31.5294i		−	0.0675148i	0.999430	−	0.0337574i	$-0.0107474\pi$
$468$		0			0
$469$			648.493i			1.38271i
$470$	−49.4881			−0.105294
$471$		0			0
$472$	−824.313	−	816.946i	−1.74643	−	1.73082i
$473$	459.581			0.971630
$474$		0			0
$475$	−261.021			−0.549518
$476$	472.153			0.991917
$477$		0			0
$478$		−	1261.92i		−	2.64000i
$479$	172.294			0.359695			0.179848	−	0.983694i	$-0.442440\pi$
$479$	172.294			0.359695			0.179848	+	0.983694i	$0.442440\pi$
$480$		0			0
$481$	3.88975			0.00808680
$482$		−	1447.98i		−	3.00411i
$483$		0			0
$484$	−449.573			−0.928870
$485$			32.2529i			0.0665008i
$486$		0			0
$487$	−403.087			−0.827694			−0.413847	−	0.910347i	$-0.635815\pi$
$487$	−403.087			−0.827694			−0.413847	+	0.910347i	$0.635815\pi$
$488$	900.478			1.84524
$489$		0			0
$490$			15.1379i			0.0308937i
$491$	−294.608			−0.600017			−0.300008	−	0.953937i	$-0.596989\pi$
$491$	−294.608			−0.600017			−0.300008	+	0.953937i	$0.596989\pi$
$492$		0			0
$493$	461.703			0.936518
$494$	−3.41028			−0.00690340
$495$		0			0
$496$		−	16.2203i		−	0.0327021i
$497$	459.628			0.924806
$498$		0			0
$499$	515.505			1.03308			0.516538	−	0.856264i	$-0.327220\pi$
$499$	515.505			1.03308			0.516538	+	0.856264i	$0.327220\pi$
$500$	−185.041			−0.370081
$501$		0			0
$502$		−	359.485i		−	0.716105i
$503$			618.687i			1.22999i	0.788529	+	0.614997i	$0.210843\pi$
$503$			618.687i			1.22999i	−0.788529	+	0.614997i	$0.789157\pi$
$504$		0			0
$505$		−	36.4550i		−	0.0721881i
$506$		−	438.650i		−	0.866896i
$507$		0			0
$508$	−55.8368			−0.109915
$509$			597.596i			1.17406i	0.809566	+	0.587029i	$0.199703\pi$
$509$			597.596i			1.17406i	−0.809566	+	0.587029i	$0.800297\pi$
$510$		0			0
$511$			679.569i			1.32988i
$512$		−	1081.80i		−	2.11288i
$513$		0			0
$514$			907.674i			1.76590i
$515$			58.3984i			0.113395i
$516$		0			0
$517$	−292.144			−0.565075
$518$	−995.344			−1.92151
$519$		0			0
$520$	−0.691026			−0.00132890
$521$	190.298			0.365256			0.182628	−	0.983182i	$-0.441540\pi$
$521$	190.298			0.365256			0.182628	+	0.983182i	$0.441540\pi$
$522$		0			0
$523$	−579.438			−1.10791			−0.553956	−	0.832546i	$-0.686882\pi$
$523$	−579.438			−1.10791			−0.553956	+	0.832546i	$0.686882\pi$
$524$			1946.23i			3.71418i
$525$		0			0
$526$		−	709.838i		−	1.34950i
$527$		−	3.81989i		−	0.00724837i
$528$		0			0
$529$	332.139			0.627861
$530$		−	114.930i		−	0.216849i
$531$		0			0
$532$	611.730			1.14987
$533$		−	1.23116i		−	0.00230987i
$534$		0			0
$535$	27.3010			0.0510299
$536$	−2054.64			−3.83328
$537$		0			0
$538$	781.484			1.45257
$539$			89.3639i			0.165796i
$540$		0			0
$541$			808.858i			1.49512i	0.664196	+	0.747558i	$0.268774\pi$
$541$			808.858i			1.49512i	−0.664196	+	0.747558i	$0.731226\pi$
$542$		−	206.132i		−	0.380317i
$543$		0			0
$544$			383.298i			0.704591i
$545$			22.6951i			0.0416424i
$546$		0			0
$547$	270.832			0.495122			0.247561	−	0.968872i	$-0.420371\pi$
$547$	270.832			0.495122			0.247561	+	0.968872i	$0.420371\pi$
$548$	−586.400			−1.07007
$549$		0			0
$550$	−776.688			−1.41216
$551$	598.192			1.08565
$552$		0			0
$553$	460.996			0.833628
$554$			623.041i			1.12462i
$555$		0			0
$556$	951.866			1.71199
$557$	519.832			0.933270			0.466635	−	0.884450i	$-0.345466\pi$
$557$	519.832			0.933270			0.466635	+	0.884450i	$0.345466\pi$
$558$		0			0
$559$	−4.77143			−0.00853565
$560$	84.6312			0.151127
$561$		0			0
$562$			146.289i			0.260301i
$563$		−	417.978i		−	0.742411i	−0.928551	−	0.371206i	$-0.878944\pi$
$563$		−	417.978i		−	0.742411i	0.928551	−	0.371206i	$-0.121056\pi$
$564$		0			0
$565$		−	7.79258i		−	0.0137922i
$566$	−1307.50			−2.31007
$567$		0			0
$568$			1456.25i			2.56383i
$569$			285.720i			0.502144i	0.967968	+	0.251072i	$0.0807831\pi$
$569$			285.720i			0.502144i	−0.967968	+	0.251072i	$0.919217\pi$
$570$		0			0
$571$		−	69.7739i		−	0.122196i	−0.998132	−	0.0610980i	$-0.980540\pi$
$571$		−	69.7739i		−	0.122196i	0.998132	−	0.0610980i	$-0.0194602\pi$
$572$	−7.11345			−0.0124361
$573$		0			0
$574$			315.040i			0.548851i
$575$			348.569i			0.606208i
$576$		0			0
$577$	114.805			0.198968			0.0994840	−	0.995039i	$-0.468281\pi$
$577$	114.805			0.198968			0.0994840	+	0.995039i	$0.468281\pi$
$578$		−	816.512i		−	1.41265i
$579$		0			0
$580$	211.366			0.364424
$581$		−	551.893i		−	0.949902i
$582$		0			0
$583$		−	678.469i		−	1.16375i
$584$	−2153.10			−3.68681
$585$		0			0
$586$		−	245.597i		−	0.419108i
$587$		−	861.663i		−	1.46791i	−0.679199	−	0.733954i	$-0.737673\pi$
$587$		−	861.663i		−	1.46791i	0.679199	−	0.733954i	$-0.262327\pi$
$588$		0			0
$589$		−	4.94913i		−	0.00840260i
$590$	−60.6770	−	60.1347i	−0.102842	−	0.101923i
$591$		0			0
$592$		−	1509.34i		−	2.54957i
$593$	−903.733			−1.52400			−0.762001	−	0.647576i	$-0.775783\pi$
$593$	−903.733			−1.52400			−0.762001	+	0.647576i	$0.775783\pi$
$594$		0			0
$595$	19.9308			0.0334971
$596$		−	145.859i		−	0.244730i
$597$		0			0
$598$			4.55412i			0.00761558i
$599$	−236.884			−0.395466			−0.197733	−	0.980256i	$-0.563358\pi$
$599$	−236.884			−0.395466			−0.197733	+	0.980256i	$0.563358\pi$
$600$		0			0
$601$			681.352i			1.13370i	0.823822	+	0.566848i	$0.191837\pi$
$601$			681.352i			1.13370i	−0.823822	+	0.566848i	$0.808163\pi$
$602$	1220.96			2.02817
$603$		0			0
$604$			452.386i			0.748983i
$605$	−18.9776			−0.0313680
$606$		0			0
$607$	−855.559			−1.40949			−0.704744	−	0.709462i	$-0.748938\pi$
$607$	−855.559			−1.40949			−0.704744	+	0.709462i	$0.748938\pi$
$608$			496.608i			0.816790i
$609$		0			0
$610$	66.2834			0.108661
$611$	3.03308			0.00496412
$612$		0			0
$613$			629.123i			1.02630i	0.858299	+	0.513151i	$0.171522\pi$
$613$			629.123i			1.02630i	−0.858299	+	0.513151i	$0.828478\pi$
$614$			450.978i			0.734492i
$615$		0			0
$616$	1043.86			1.69457
$617$	909.047			1.47333			0.736667	−	0.676256i	$-0.236398\pi$
$617$	909.047			1.47333			0.736667	+	0.676256i	$0.236398\pi$
$618$		0			0
$619$	437.303			0.706466			0.353233	−	0.935535i	$-0.385082\pi$
$619$	437.303			0.706466			0.353233	+	0.935535i	$0.385082\pi$
$620$		−	1.74873i		−	0.00282053i
$621$		0			0
$622$			900.609i			1.44792i
$623$			887.061i			1.42385i
$624$		0			0
$625$	613.271			0.981234
$626$	−1790.67			−2.86050
$627$		0			0
$628$		−	2046.77i		−	3.25919i
$629$		−	355.453i		−	0.565108i
$630$		0			0
$631$	−548.459			−0.869190			−0.434595	−	0.900626i	$-0.643108\pi$
$631$	−548.459			−0.869190			−0.434595	+	0.900626i	$0.643108\pi$
$632$			1460.59i			2.31106i
$633$		0			0
$634$		−	1167.93i		−	1.84217i
$635$	−2.35702			−0.00371184
$636$		0			0
$637$		−	0.927788i		−	0.00145650i
$638$	1779.97			2.78992
$639$		0			0
$640$		−	23.9629i		−	0.0374420i
$641$	1064.88			1.66128			0.830640	−	0.556810i	$-0.187975\pi$
$641$	1064.88			1.66128			0.830640	+	0.556810i	$0.187975\pi$
$642$		0			0
$643$	1042.17			1.62079			0.810395	−	0.585884i	$-0.199252\pi$
$643$	1042.17			1.62079			0.810395	+	0.585884i	$0.199252\pi$
$644$		−	816.910i		−	1.26849i
$645$		0			0
$646$			311.638i			0.482411i
$647$	144.371			0.223138			0.111569	−	0.993757i	$-0.464412\pi$
$647$	144.371			0.223138			0.111569	+	0.993757i	$0.464412\pi$
$648$		0			0
$649$	−358.195	−	354.994i	−0.551918	−	0.546986i
$650$	8.06367			0.0124057
$651$		0			0
$652$	−2820.99			−4.32667
$653$	816.374			1.25019			0.625095	−	0.780549i	$-0.285060\pi$
$653$	816.374			1.25019			0.625095	+	0.780549i	$0.285060\pi$
$654$		0			0
$655$			82.1553i			0.125428i
$656$	−477.729			−0.728245
$657$		0			0
$658$	−776.130			−1.17953
$659$		−	317.201i		−	0.481337i	−0.970607	−	0.240669i	$-0.922633\pi$
$659$		−	317.201i		−	0.481337i	0.970607	−	0.240669i	$-0.0773667\pi$
$660$		0			0
$661$	307.730			0.465552			0.232776	−	0.972530i	$-0.425219\pi$
$661$	307.730			0.465552			0.232776	+	0.972530i	$0.425219\pi$
$662$		−	813.384i		−	1.22868i
$663$		0			0
$664$	1748.58			2.63340
$665$	25.8227			0.0388311
$666$		0			0
$667$		−	798.831i		−	1.19765i
$668$	−2676.21			−4.00631
$669$		0			0
$670$	−151.240			−0.225731
$671$	391.292			0.583147
$672$		0			0
$673$		−	1083.95i		−	1.61062i	−0.592853	−	0.805311i	$-0.701998\pi$
$673$		−	1083.95i		−	1.61062i	0.592853	−	0.805311i	$-0.298002\pi$
$674$	−1042.52			−1.54676
$675$		0			0
$676$	−1584.80			−2.34439
$677$	815.853			1.20510			0.602550	−	0.798081i	$-0.294151\pi$
$677$	815.853			1.20510			0.602550	+	0.798081i	$0.294151\pi$
$678$		0			0
$679$			505.827i			0.744959i
$680$			63.1472i			0.0928635i
$681$		0			0
$682$		−	14.7265i		−	0.0215931i
$683$		−	1151.50i		−	1.68594i	−0.537957	−	0.842972i	$-0.680804\pi$
$683$		−	1151.50i		−	1.68594i	0.537957	−	0.842972i	$-0.319196\pi$
$684$		0			0
$685$	−24.7534			−0.0361364
$686$			1350.10i			1.96808i
$687$		0			0
$688$			1851.46i			2.69108i
$689$			7.04395i			0.0102234i
$690$		0			0
$691$			939.727i			1.35995i	0.733234	+	0.679976i	$0.238010\pi$
$691$			939.727i			1.35995i	−0.733234	+	0.679976i	$0.761990\pi$
$692$			1082.69i			1.56458i
$693$		0			0
$694$	−227.550			−0.327882
$695$	40.1807			0.0578140
$696$		0			0
$697$	−112.506			−0.161414
$698$	−1066.02			−1.52725
$699$		0			0
$700$	−1446.45			−2.06636
$701$		−	782.026i		−	1.11559i	−0.829980	−	0.557793i	$-0.811648\pi$
$701$		−	782.026i		−	1.11559i	0.829980	−	0.557793i	$-0.188352\pi$
$702$		0			0
$703$		−	460.532i		−	0.655095i
$704$			300.369i			0.426661i
$705$		0			0
$706$	−1119.61			−1.58585
$707$		−	571.729i		−	0.808669i
$708$		0			0
$709$	−461.863			−0.651429			−0.325715	−	0.945468i	$-0.605605\pi$
$709$	−461.863			−0.651429			−0.325715	+	0.945468i	$0.605605\pi$
$710$			107.194i			0.150977i
$711$		0			0
$712$	−2810.50			−3.94733
$713$	−6.60911			−0.00926944
$714$		0			0
$715$	−0.300277			−0.000419968
$716$			1368.13i			1.91080i
$717$		0			0
$718$		−	1686.58i		−	2.34900i
$719$			398.240i			0.553881i	0.960887	+	0.276940i	$0.0893205\pi$
$719$			398.240i			0.553881i	−0.960887	+	0.276940i	$0.910680\pi$
$720$		0			0
$721$			915.872i			1.27028i
$722$		−	916.627i		−	1.26957i
$723$		0			0
$724$	383.881			0.530223
$725$	−1414.44			−1.95095
$726$		0			0
$727$	−402.797			−0.554053			−0.277027	−	0.960862i	$-0.589349\pi$
$727$	−402.797			−0.554053			−0.277027	+	0.960862i	$0.589349\pi$
$728$	−10.8375			−0.0148866
$729$		0			0
$730$	−158.488			−0.217106
$731$			436.022i			0.596473i
$732$		0			0
$733$	701.565			0.957114			0.478557	−	0.878056i	$-0.341160\pi$
$733$	701.565			0.957114			0.478557	+	0.878056i	$0.341160\pi$
$734$	850.723			1.15902
$735$		0			0
$736$	663.175			0.901053
$737$	−892.817			−1.21142
$738$		0			0
$739$			952.579i			1.28901i	0.764599	+	0.644506i	$0.222937\pi$
$739$			952.579i			1.28901i	−0.764599	+	0.644506i	$0.777063\pi$
$740$		−	162.725i		−	0.219898i
$741$		0			0
$742$		−	1802.47i		−	2.42920i
$743$	681.496			0.917222			0.458611	−	0.888637i	$-0.348347\pi$
$743$	681.496			0.917222			0.458611	+	0.888637i	$0.348347\pi$
$744$		0			0
$745$		−	6.15708i		−	0.00826454i
$746$		−	132.624i		−	0.177780i
$747$		0			0
$748$			650.040i			0.869037i
$749$	428.166			0.571650
$750$		0			0
$751$		−	1302.06i		−	1.73376i	−0.498514	−	0.866882i	$-0.666121\pi$
$751$		−	1302.06i		−	1.73376i	0.498514	−	0.866882i	$-0.333879\pi$
$752$		−	1176.93i		−	1.56506i
$753$		0			0
$754$	−18.4799			−0.0245091
$755$			19.0964i			0.0252932i
$756$		0			0
$757$	653.976			0.863904			0.431952	−	0.901896i	$-0.357825\pi$
$757$	653.976			0.863904			0.431952	+	0.901896i	$0.357825\pi$
$758$		−	136.975i		−	0.180705i
$759$		0			0
$760$			81.8148i			0.107651i
$761$	−638.164			−0.838586			−0.419293	−	0.907851i	$-0.637722\pi$
$761$	−638.164			−0.838586			−0.419293	+	0.907851i	$0.637722\pi$
$762$		0			0
$763$			355.931i			0.466489i
$764$			2636.35i			3.45072i
$765$		0			0
$766$			1781.80i			2.32611i
$767$	3.71883	+	3.68560i	0.00484854	+	0.00480521i
$768$		0			0
$769$			1305.23i			1.69730i	0.528951	+	0.848652i	$0.322586\pi$
$769$			1305.23i			1.69730i	−0.528951	+	0.848652i	$0.677414\pi$
$770$	76.8375			0.0997889
$771$		0			0
$772$	−440.167			−0.570164
$773$		−	155.891i		−	0.201671i	−0.994903	−	0.100835i	$-0.967848\pi$
$773$		−	155.891i		−	0.201671i	0.994903	−	0.100835i	$-0.0321515\pi$
$774$		0			0
$775$			11.7023i			0.0150998i
$776$	−1602.63			−2.06524
$777$		0			0
$778$		−	1379.88i		−	1.77363i
$779$	−145.765			−0.187118
$780$		0			0
$781$			632.797i			0.810240i
$782$	416.164			0.532178
$783$		0			0
$784$	−360.010			−0.459197
$785$		−	86.3994i		−	0.110063i
$786$		0			0
$787$	389.653			0.495111			0.247556	−	0.968874i	$-0.420373\pi$
$787$	389.653			0.495111			0.247556	+	0.968874i	$0.420373\pi$
$788$	195.221			0.247743
$789$		0			0
$790$			107.513i			0.136092i
$791$		−	122.212i		−	0.154504i
$792$		0			0
$793$	−4.06244			−0.00512288
$794$	1671.91			2.10568
$795$		0			0
$796$	1935.41			2.43141
$797$		−	48.4897i		−	0.0608403i	−0.999537	−	0.0304201i	$-0.990315\pi$
$797$		−	48.4897i		−	0.0608403i	0.999537	−	0.0304201i	$-0.00968453\pi$
$798$		0			0
$799$		−	277.168i		−	0.346894i
$800$		−	1174.24i		−	1.46780i
$801$		0			0
$802$	−887.432			−1.10652
$803$	−935.602			−1.16513
$804$		0			0
$805$		−	34.4839i		−	0.0428371i
$806$			0.152893i			0.000189693i
$807$		0			0
$808$	1811.42			2.24186
$809$		−	1140.26i		−	1.40947i	−0.709472	−	0.704734i	$-0.751066\pi$
$809$		−	1140.26i		−	1.40947i	0.709472	−	0.704734i	$-0.248934\pi$
$810$		0			0
$811$		−	30.8005i		−	0.0379784i	−0.999820	−	0.0189892i	$-0.993955\pi$
$811$		−	30.8005i		−	0.0379784i	0.999820	−	0.0189892i	$-0.00604481\pi$
$812$	3314.89			4.08237
$813$		0			0
$814$		−	1370.35i		−	1.68347i
$815$	−119.081			−0.146112
$816$		0			0
$817$			564.919i			0.691455i
$818$	1326.10			1.62115
$819$		0			0
$820$	−51.5047			−0.0628106
$821$			892.226i			1.08675i	0.839489	+	0.543377i	$0.182855\pi$
$821$			892.226i			1.08675i	−0.839489	+	0.543377i	$0.817145\pi$
$822$		0			0
$823$		−	1300.11i		−	1.57972i	−0.613290	−	0.789858i	$-0.710154\pi$
$823$		−	1300.11i		−	1.57972i	0.613290	−	0.789858i	$-0.289846\pi$
$824$	−2901.78			−3.52158
$825$		0			0
$826$	−951.606	−	943.103i	−1.15207	−	1.14177i
$827$	1617.28			1.95559			0.977797	−	0.209556i	$-0.0672019\pi$
$827$	1617.28			1.95559			0.977797	+	0.209556i	$0.0672019\pi$
$828$		0			0
$829$	353.866			0.426859			0.213430	−	0.976958i	$-0.431537\pi$
$829$	353.866			0.426859			0.213430	+	0.976958i	$0.431537\pi$
$830$	128.711			0.155074
$831$		0			0
$832$		−	3.11847i		−	0.00374816i
$833$	−84.7829			−0.101780
$834$		0			0
$835$	−112.970			−0.135293
$836$			842.204i			1.00742i
$837$		0			0
$838$	−2995.30			−3.57435
$839$		−	968.406i		−	1.15424i	−0.816660	−	0.577119i	$-0.804177\pi$
$839$		−	968.406i		−	1.15424i	0.816660	−	0.577119i	$-0.195823\pi$
$840$		0			0
$841$	2400.52			2.85437
$842$	1206.93			1.43341
$843$		0			0
$844$		−	1881.78i		−	2.22960i
$845$	−66.8987			−0.0791700
$846$		0			0
$847$	−297.629			−0.351392
$848$	2733.27			3.22320
$849$		0			0
$850$		−	736.873i		−	0.866910i
$851$	−614.998			−0.722677
$852$		0			0
$853$	−303.366			−0.355646			−0.177823	−	0.984063i	$-0.556905\pi$
$853$	−303.366			−0.355646			−0.177823	+	0.984063i	$0.556905\pi$
$854$	1039.53			1.21725
$855$		0			0
$856$			1356.57i			1.58478i
$857$			662.303i			0.772816i	0.922328	+	0.386408i	$0.126284\pi$
$857$			662.303i			0.772816i	−0.922328	+	0.386408i	$0.873716\pi$
$858$		0			0
$859$		−	1317.31i		−	1.53354i	−0.641924	−	0.766769i	$-0.721863\pi$
$859$		−	1317.31i		−	1.53354i	0.641924	−	0.766769i	$-0.278137\pi$
$860$			199.609i			0.232104i
$861$		0			0
$862$	2609.93			3.02777
$863$			248.154i			0.287548i	0.989611	+	0.143774i	$0.0459238\pi$
$863$			248.154i			0.287548i	−0.989611	+	0.143774i	$0.954076\pi$
$864$		0			0
$865$			45.7032i			0.0528361i
$866$			2288.26i			2.64233i
$867$		0			0
$868$		−	27.4256i		−	0.0315963i
$869$			634.680i			0.730357i
$870$		0			0
$871$	9.26935			0.0106422
$872$	−1127.71			−1.29324
$873$		0			0
$874$	539.190			0.616922
$875$	−122.502			−0.140002
$876$		0			0
$877$	1210.94			1.38078			0.690390	−	0.723437i	$-0.257439\pi$
$877$	1210.94			1.38078			0.690390	+	0.723437i	$0.257439\pi$
$878$		−	1688.85i		−	1.92352i
$879$		0			0
$880$			116.517i			0.132405i
$881$		−	821.372i		−	0.932318i	−0.884701	−	0.466159i	$-0.845637\pi$
$881$		−	821.372i		−	0.932318i	0.884701	−	0.466159i	$-0.154363\pi$
$882$		0			0
$883$	−121.125			−0.137175			−0.0685874	−	0.997645i	$-0.521849\pi$
$883$	−121.125			−0.137175			−0.0685874	+	0.997645i	$0.521849\pi$
$884$		−	6.74880i		−	0.00763439i
$885$		0			0
$886$	933.585			1.05371
$887$		−	385.806i		−	0.434956i	−0.976065	−	0.217478i	$-0.930217\pi$
$887$		−	385.806i		−	0.434956i	0.976065	−	0.217478i	$-0.0697830\pi$
$888$		0			0
$889$	−36.9655			−0.0415809
$890$	−206.878			−0.232448
$891$		0			0
$892$	2547.09			2.85548
$893$		−	359.104i		−	0.402133i
$894$		0			0
$895$			57.7524i			0.0645278i
$896$		−	375.813i		−	0.419435i
$897$		0			0
$898$			821.073i			0.914336i
$899$		−	26.8187i		−	0.0298317i
$900$		0			0
$901$	643.689			0.714416
$902$	−433.734			−0.480858
$903$		0			0
$904$	387.208			0.428328
$905$	16.2046			0.0179057
$906$		0			0
$907$	−836.228			−0.921971			−0.460986	−	0.887408i	$-0.652504\pi$
$907$	−836.228			−0.921971			−0.460986	+	0.887408i	$0.652504\pi$
$908$			1163.54i			1.28143i
$909$		0			0
$910$	−0.797737			−0.000876634
$911$	−26.1105			−0.0286614			−0.0143307	−	0.999897i	$-0.504562\pi$
$911$	−26.1105			−0.0286614			−0.0143307	+	0.999897i	$0.504562\pi$
$912$		0			0
$913$	759.823			0.832227
$914$	235.135			0.257259
$915$		0			0
$916$			2528.80i			2.76070i
$917$			1288.45i			1.40508i
$918$		0			0
$919$		−	1116.36i		−	1.21476i	−0.794412	−	0.607380i	$-0.792221\pi$
$919$		−	1116.36i		−	1.21476i	0.794412	−	0.607380i	$-0.207779\pi$
$920$	109.256			0.118757
$921$		0			0
$922$			778.180i			0.844013i
$923$		−	6.56978i		−	0.00711786i
$924$		0			0
$925$			1088.94i			1.17723i
$926$	1065.56			1.15072
$927$		0			0
$928$			2691.05i			2.89984i
$929$			1393.02i			1.49949i	0.661730	+	0.749743i	$0.269823\pi$
$929$			1393.02i			1.49949i	−0.661730	+	0.749743i	$0.730177\pi$
$930$		0			0
$931$	−109.846			−0.117988
$932$		−	4054.88i		−	4.35073i
$933$		0			0
$934$	115.322			0.123471
$935$			27.4398i			0.0293474i
$936$		0			0
$937$			1420.84i			1.51637i	0.652041	+	0.758184i	$0.273913\pi$
$937$			1420.84i			1.51637i	−0.652041	+	0.758184i	$0.726087\pi$
$938$	−2371.92			−2.52870
$939$		0			0
$940$		−	126.886i		−	0.134986i
$941$		−	968.807i		−	1.02955i	−0.857325	−	0.514775i	$-0.827875\pi$
$941$		−	968.807i		−	1.02955i	0.857325	−	0.514775i	$-0.172125\pi$
$942$		0			0
$943$			194.655i			0.206422i
$944$	1430.13	−	1443.02i	1.51496	−	1.52862i
$945$		0			0
$946$			1680.96i			1.77691i
$947$	573.028			0.605098			0.302549	−	0.953134i	$-0.402162\pi$
$947$	573.028			0.605098			0.302549	+	0.953134i	$0.402162\pi$
$948$		0			0
$949$	9.71355			0.0102356
$950$		−	954.708i		−	1.00496i
$951$		0			0
$952$			990.347i			1.04028i
$953$	−834.685			−0.875850			−0.437925	−	0.899011i	$-0.644287\pi$
$953$	−834.685			−0.875850			−0.437925	+	0.899011i	$0.644287\pi$
$954$		0			0
$955$			111.287i			0.116531i
$956$	3235.54			3.38445
$957$		0			0
$958$			630.182i			0.657810i
$959$	−388.212			−0.404809
$960$		0			0
$961$	960.778			0.999769
$962$			14.2271i			0.0147891i
$963$		0			0
$964$	3712.58			3.85123
$965$	−18.5806			−0.0192545
$966$		0			0
$967$			128.620i			0.133009i	0.997786	+	0.0665046i	$0.0211847\pi$
$967$			128.620i			0.133009i	−0.997786	+	0.0665046i	$0.978815\pi$
$968$		−	942.987i		−	0.974160i
$969$		0			0
$970$	−117.968			−0.121616
$971$	−1485.57			−1.52994			−0.764969	−	0.644067i	$-0.777246\pi$
$971$	−1485.57			−1.52994			−0.764969	+	0.644067i	$0.777246\pi$
$972$		0			0
$973$	630.160			0.647647
$974$		−	1474.33i		−	1.51368i
$975$		0			0
$976$			1576.35i			1.61512i
$977$		−	1.03419i		−	0.00105854i	−1.00000		0.000529271i	$-0.999832\pi$
$977$		−	1.03419i		−	0.00105854i	1.00000		0.000529271i	$-0.000168472\pi$
$978$		0			0
$979$	−1221.27			−1.24747
$980$	−38.8133			−0.0396054
$981$		0			0
$982$		−	1077.56i		−	1.09731i
$983$			941.272i			0.957550i	0.877938	+	0.478775i	$0.158919\pi$
$983$			941.272i			0.957550i	−0.877938	+	0.478775i	$0.841081\pi$
$984$		0			0
$985$	8.24079			0.00836628
$986$			1688.72i			1.71270i
$987$		0			0
$988$		−	8.74388i		−	0.00885008i
$989$	754.398			0.762788
$990$		0			0
$991$		−	1077.04i		−	1.08682i	−0.839467	−	0.543410i	$-0.817133\pi$
$991$		−	1077.04i		−	1.08682i	0.839467	−	0.543410i	$-0.182867\pi$
$992$	22.2644			0.0224439
$993$		0			0
$994$			1681.13i			1.69128i
$995$	81.6984			0.0821090
$996$		0			0
$997$	−1366.02			−1.37013			−0.685063	−	0.728484i	$-0.740225\pi$
$997$	−1366.02			−1.37013			−0.685063	+	0.728484i	$0.740225\pi$
$998$			1885.51i			1.88929i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.3.c.c.235.20		20
3.2	odd	2		177.3.c.a.58.1	✓	20
59.58	odd	2	inner	531.3.c.c.235.1		20
177.176	even	2		177.3.c.a.58.20	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.c.a.58.1	✓	20	3.2	odd	2
177.3.c.a.58.20	yes	20	177.176	even	2
531.3.c.c.235.1		20	59.58	odd	2	inner
531.3.c.c.235.20		20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	531.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.65759i q^{2} -9.37798 q^{4} -0.395869 q^{5} -6.20847 q^{7} -19.6705i q^{8} +O(q^{10})\)	\(q+3.65759i q^{2} -9.37798 q^{4} -0.395869 q^{5} -6.20847 q^{7} -19.6705i q^{8} -1.44793i q^{10} -8.54756i q^{11} +0.0887419i q^{13} -22.7081i q^{14} +34.4346 q^{16} +8.10940 q^{17} +10.5067 q^{19} +3.71245 q^{20} +31.2635 q^{22} -14.0307i q^{23} -24.8433 q^{25} -0.324582 q^{26} +58.2229 q^{28} +56.9344 q^{29} -0.471045i q^{31} +47.2659i q^{32} +29.6609i q^{34} +2.45774 q^{35} -43.8322i q^{37} +38.4292i q^{38} +7.78692i q^{40} -13.8735 q^{41} +53.7675i q^{43} +80.1588i q^{44} +51.3187 q^{46} -34.1786i q^{47} -10.4549 q^{49} -90.8666i q^{50} -0.832220i q^{52} +79.3757 q^{53} +3.38371i q^{55} +122.123i q^{56} +208.243i q^{58} +(41.5316 - 41.9061i) q^{59} +45.7782i q^{61} +1.72289 q^{62} -35.1409 q^{64} -0.0351301i q^{65} -104.453i q^{67} -76.0498 q^{68} +8.98940i q^{70} -74.0325 q^{71} -109.458i q^{73} +160.320 q^{74} -98.5316 q^{76} +53.0673i q^{77} -74.2528 q^{79} -13.6316 q^{80} -50.7436i q^{82} +88.8936i q^{83} -3.21026 q^{85} -196.660 q^{86} -168.134 q^{88} -142.879i q^{89} -0.550951i q^{91} +131.580i q^{92} +125.012 q^{94} -4.15927 q^{95} -81.4737i q^{97} -38.2398i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 40 q^{4} - 8 q^{7}+O(q^{10}) \) 20 * q - 40 * q^4 - 8 * q^7	\( 20 q - 40 q^{4} - 8 q^{7} - 8 q^{16} - 16 q^{17} - 60 q^{19} + 164 q^{20} + 40 q^{22} + 100 q^{25} + 156 q^{26} + 200 q^{28} + 60 q^{29} + 32 q^{35} - 28 q^{41} + 180 q^{46} + 284 q^{49} + 8 q^{53} + 152 q^{59} + 8 q^{62} + 204 q^{64} - 384 q^{68} - 92 q^{71} - 104 q^{74} + 120 q^{76} - 420 q^{79} - 376 q^{80} - 348 q^{85} - 232 q^{86} - 212 q^{88} + 152 q^{94} - 788 q^{95}+O(q^{100}) \) 20 * q - 40 * q^4 - 8 * q^7 - 8 * q^16 - 16 * q^17 - 60 * q^19 + 164 * q^20 + 40 * q^22 + 100 * q^25 + 156 * q^26 + 200 * q^28 + 60 * q^29 + 32 * q^35 - 28 * q^41 + 180 * q^46 + 284 * q^49 + 8 * q^53 + 152 * q^59 + 8 * q^62 + 204 * q^64 - 384 * q^68 - 92 * q^71 - 104 * q^74 + 120 * q^76 - 420 * q^79 - 376 * q^80 - 348 * q^85 - 232 * q^86 - 212 * q^88 + 152 * q^94 - 788 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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