LMFDB - Embedded newform 531.2.d.a.530.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [531,2,Mod(530,531)] 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("531.530"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(531, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$4.24005634733$
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} - 10 x^{18} + 139 x^{16} - 476 x^{14} + 4681 x^{12} - 666 x^{10} + 82273 x^{8} + 168944 x^{6} + \cdots + 3374569 $ x^20 - 10x^18 + 139x^16 - 476x^14 + 4681x^12 - 666x^10 + 82273x^8 + 168944x^6 + 845867x^4 + 1613394*x^2 + 3374569
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			530.1
Root			$2.59182 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	531.530
Dual form			531.2.d.a.530.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.59182 q^{2} +4.71754 q^{4} -1.33755i q^{5} -1.10356 q^{7} -7.04337 q^{8} +3.46670i q^{10} -1.27883 q^{11} +2.34848i q^{13} +2.86024 q^{14} +8.82007 q^{16} +6.38662i q^{17} -5.60912 q^{19} -6.30996i q^{20} +3.31451 q^{22} +2.40913 q^{23} +3.21095 q^{25} -6.08684i q^{26} -5.20610 q^{28} -0.508101i q^{29} -7.41366i q^{31} -8.77331 q^{32} -16.5530i q^{34} +1.47608i q^{35} +4.89565i q^{37} +14.5378 q^{38} +9.42089i q^{40} +4.17932i q^{41} +10.0589i q^{43} -6.03295 q^{44} -6.24402 q^{46} +4.73253 q^{47} -5.78215 q^{49} -8.32220 q^{50} +11.0790i q^{52} +10.2917i q^{53} +1.71051i q^{55} +7.77280 q^{56} +1.31691i q^{58} +(5.91308 + 4.90261i) q^{59} +7.67203i q^{61} +19.2149i q^{62} +5.09871 q^{64} +3.14122 q^{65} +3.56735i q^{67} +30.1291i q^{68} -3.82573i q^{70} -1.48942i q^{71} +5.22551i q^{73} -12.6886i q^{74} -26.4612 q^{76} +1.41127 q^{77} +13.7447 q^{79} -11.7973i q^{80} -10.8321i q^{82} -6.03295 q^{83} +8.54246 q^{85} -26.0708i q^{86} +9.00730 q^{88} -12.7980 q^{89} -2.59170i q^{91} +11.3651 q^{92} -12.2659 q^{94} +7.50251i q^{95} -16.8000i q^{97} +14.9863 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{4} - 8 q^{7} + 4 q^{16} - 8 q^{22} + 4 q^{25} + 8 q^{28} + 44 q^{49} + 36 q^{64} - 96 q^{76} - 24 q^{85} + 16 q^{88} - 112 q^{94}+O(q^{100}) $ 20 * q + 20 * q^4 - 8 * q^7 + 4 * q^16 - 8 * q^22 + 4 * q^25 + 8 * q^28 + 44 * q^49 + 36 * q^64 - 96 * q^76 - 24 * q^85 + 16 * q^88 - 112 * q^94

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 $

$2$	−2.59182			−1.83269			−0.916347	−	0.400385i	$-0.868876\pi$
$2$	−2.59182			−1.83269			−0.916347	+	0.400385i	$0.868876\pi$
$3$		0			0
$3$		0			0
$4$	4.71754			2.35877
$5$		−	1.33755i		−	0.598173i	−0.954226	−	0.299086i	$-0.903318\pi$
$5$		−	1.33755i		−	0.598173i	0.954226	−	0.299086i	$-0.0966819\pi$
$6$		0			0
$7$	−1.10356			−0.417108			−0.208554	−	0.978011i	$-0.566876\pi$
$7$	−1.10356			−0.417108			−0.208554	+	0.978011i	$0.566876\pi$
$8$	−7.04337			−2.49021
$9$		0			0
$10$			3.46670i			1.09627i
$11$	−1.27883			−0.385583			−0.192792	−	0.981240i	$-0.561754\pi$
$11$	−1.27883			−0.385583			−0.192792	+	0.981240i	$0.561754\pi$
$12$		0			0
$13$			2.34848i			0.651352i	0.945482	+	0.325676i	$0.105592\pi$
$13$			2.34848i			0.651352i	−0.945482	+	0.325676i	$0.894408\pi$
$14$	2.86024			0.764431
$15$		0			0
$16$	8.82007			2.20502
$17$			6.38662i			1.54898i	0.632584	+	0.774492i	$0.281994\pi$
$17$			6.38662i			1.54898i	−0.632584	+	0.774492i	$0.718006\pi$
$18$		0			0
$19$	−5.60912			−1.28682			−0.643411	−	0.765521i	$-0.722481\pi$
$19$	−5.60912			−1.28682			−0.643411	+	0.765521i	$0.722481\pi$
$20$		−	6.30996i		−	1.41095i
$21$		0			0
$22$	3.31451			0.706656
$23$	2.40913			0.502337			0.251169	−	0.967943i	$-0.419185\pi$
$23$	2.40913			0.502337			0.251169	+	0.967943i	$0.419185\pi$
$24$		0			0
$25$	3.21095			0.642189
$26$		−	6.08684i		−	1.19373i
$27$		0			0
$28$	−5.20610			−0.983860
$29$		−	0.508101i		−	0.0943520i	−0.998887	−	0.0471760i	$-0.984978\pi$
$29$		−	0.508101i		−	0.0943520i	0.998887	−	0.0471760i	$-0.0150222\pi$
$30$		0			0
$31$		−	7.41366i		−	1.33153i	−0.746161	−	0.665766i	$-0.768105\pi$
$31$		−	7.41366i		−	1.33153i	0.746161	−	0.665766i	$-0.231895\pi$
$32$	−8.77331			−1.55092
$33$		0			0
$34$		−	16.5530i		−	2.83881i
$35$			1.47608i			0.249502i
$36$		0			0
$37$			4.89565i			0.804840i	0.915455	+	0.402420i	$0.131831\pi$
$37$			4.89565i			0.804840i	−0.915455	+	0.402420i	$0.868169\pi$
$38$	14.5378			2.35835
$39$		0			0
$40$			9.42089i			1.48957i
$41$			4.17932i			0.652701i	0.945249	+	0.326350i	$0.105819\pi$
$41$			4.17932i			0.652701i	−0.945249	+	0.326350i	$0.894181\pi$
$42$		0			0
$43$			10.0589i			1.53396i	0.641670	+	0.766981i	$0.278242\pi$
$43$			10.0589i			1.53396i	−0.641670	+	0.766981i	$0.721758\pi$
$44$	−6.03295			−0.909501
$45$		0			0
$46$	−6.24402			−0.920631
$47$	4.73253			0.690310			0.345155	−	0.938546i	$-0.387826\pi$
$47$	4.73253			0.690310			0.345155	+	0.938546i	$0.387826\pi$
$48$		0			0
$49$	−5.78215			−0.826021
$50$	−8.32220			−1.17694
$51$		0			0
$52$			11.0790i			1.53639i
$53$			10.2917i			1.41367i	0.707379	+	0.706834i	$0.249877\pi$
$53$			10.2917i			1.41367i	−0.707379	+	0.706834i	$0.750123\pi$
$54$		0			0
$55$			1.71051i			0.230645i
$56$	7.77280			1.03868
$57$		0			0
$58$			1.31691i			0.172918i
$59$	5.91308	+	4.90261i	0.769817	+	0.638265i
$59$	5.91308	+	4.90261i	0.769817	+	0.638265i
$60$		0			0
$61$			7.67203i			0.982302i	0.871074	+	0.491151i	$0.163424\pi$
$61$			7.67203i			0.982302i	−0.871074	+	0.491151i	$0.836576\pi$
$62$			19.2149i			2.44029i
$63$		0			0
$64$	5.09871			0.637339
$65$	3.14122			0.389621
$66$		0			0
$67$			3.56735i			0.435822i	0.975969	+	0.217911i	$0.0699242\pi$
$67$			3.56735i			0.435822i	−0.975969	+	0.217911i	$0.930076\pi$
$68$			30.1291i			3.65369i
$69$		0			0
$70$		−	3.82573i		−	0.457262i
$71$		−	1.48942i		−	0.176761i	−0.996087	−	0.0883807i	$-0.971831\pi$
$71$		−	1.48942i		−	0.176761i	0.996087	−	0.0883807i	$-0.0281692\pi$
$72$		0			0
$73$			5.22551i			0.611600i	0.952096	+	0.305800i	$0.0989239\pi$
$73$			5.22551i			0.611600i	−0.952096	+	0.305800i	$0.901076\pi$
$74$		−	12.6886i		−	1.47502i
$75$		0			0
$76$	−26.4612			−3.03531
$77$	1.41127			0.160830
$78$		0			0
$79$	13.7447			1.54640			0.773202	−	0.634160i	$-0.218654\pi$
$79$	13.7447			1.54640			0.773202	+	0.634160i	$0.218654\pi$
$80$		−	11.7973i		−	1.31898i
$81$		0			0
$82$		−	10.8321i		−	1.19620i
$83$	−6.03295			−0.662202			−0.331101	−	0.943595i	$-0.607420\pi$
$83$	−6.03295			−0.662202			−0.331101	+	0.943595i	$0.607420\pi$
$84$		0			0
$85$	8.54246			0.926559
$86$		−	26.0708i		−	2.81128i
$87$		0			0
$88$	9.00730			0.960181
$89$	−12.7980			−1.35658			−0.678292	−	0.734792i	$-0.737280\pi$
$89$	−12.7980			−1.35658			−0.678292	+	0.734792i	$0.737280\pi$
$90$		0			0
$91$		−	2.59170i		−	0.271684i
$92$	11.3651			1.18490
$93$		0			0
$94$	−12.2659			−1.26513
$95$			7.50251i			0.769741i
$96$		0			0
$97$		−	16.8000i		−	1.70578i	−0.522093	−	0.852889i	$-0.674848\pi$
$97$		−	16.8000i		−	1.70578i	0.522093	−	0.852889i	$-0.325152\pi$
$98$	14.9863			1.51384
$99$		0			0
$100$	15.1478			1.51478
$101$	1.30042			0.129397			0.0646983	−	0.997905i	$-0.479392\pi$
$101$	1.30042			0.129397			0.0646983	+	0.997905i	$0.479392\pi$
$102$		0			0
$103$			11.5593i			1.13897i	0.822001	+	0.569486i	$0.192858\pi$
$103$			11.5593i			1.13897i	−0.822001	+	0.569486i	$0.807142\pi$
$104$		−	16.5412i		−	1.62200i
$105$		0			0
$106$		−	26.6741i		−	2.59082i
$107$		−	7.55469i		−	0.730339i	−0.930941	−	0.365170i	$-0.881011\pi$
$107$		−	7.55469i		−	0.730339i	0.930941	−	0.365170i	$-0.118989\pi$
$108$		0			0
$109$			18.6431i			1.78568i	0.450370	+	0.892842i	$0.351292\pi$
$109$			18.6431i			1.78568i	−0.450370	+	0.892842i	$0.648708\pi$
$110$		−	4.43334i		−	0.422702i
$111$		0			0
$112$	−9.73351			−0.919730
$113$	−13.7842			−1.29670			−0.648352	−	0.761340i	$-0.724542\pi$
$113$	−13.7842			−1.29670			−0.648352	+	0.761340i	$0.724542\pi$
$114$		0			0
$115$		−	3.22234i		−	0.300484i
$116$		−	2.39698i		−	0.222554i
$117$		0			0
$118$	−15.3256	−	12.7067i	−1.41084	−	1.16974i
$119$		−	7.04804i		−	0.646093i
$120$		0			0
$121$	−9.36458			−0.851326
$122$		−	19.8845i		−	1.80026i
$123$		0			0
$124$		−	34.9742i		−	3.14077i
$125$		−	10.9826i		−	0.982313i
$126$		0			0
$127$	−11.2241			−0.995980			−0.497990	−	0.867183i	$-0.665928\pi$
$127$	−11.2241			−0.995980			−0.497990	+	0.867183i	$0.665928\pi$
$128$	4.33167			0.382869
$129$		0			0
$130$	−8.14149			−0.714056
$131$	16.5704			1.44776			0.723880	−	0.689926i	$-0.242357\pi$
$131$	16.5704			1.44776			0.723880	+	0.689926i	$0.242357\pi$
$132$		0			0
$133$	6.19002			0.536743
$134$		−	9.24594i		−	0.798728i
$135$		0			0
$136$		−	44.9833i		−	3.85729i
$137$			18.5776i			1.58719i	0.608447	+	0.793594i	$0.291793\pi$
$137$			18.5776i			1.58719i	−0.608447	+	0.793594i	$0.708207\pi$
$138$		0			0
$139$	3.97525			0.337177			0.168588	−	0.985687i	$-0.446079\pi$
$139$	3.97525			0.337177			0.168588	+	0.985687i	$0.446079\pi$
$140$			6.96344i			0.588518i
$141$		0			0
$142$			3.86031i			0.323950i
$143$		−	3.00332i		−	0.251150i
$144$		0			0
$145$	−0.679613			−0.0564388
$146$		−	13.5436i		−	1.12088i
$147$		0			0
$148$			23.0954i			1.89843i
$149$	−1.33175			−0.109102			−0.0545508	−	0.998511i	$-0.517373\pi$
$149$	−1.33175			−0.109102			−0.0545508	+	0.998511i	$0.517373\pi$
$150$		0			0
$151$		−	10.2270i		−	0.832263i	−0.909304	−	0.416131i	$-0.863386\pi$
$151$		−	10.2270i		−	0.832263i	0.909304	−	0.416131i	$-0.136614\pi$
$152$	39.5071			3.20445
$153$		0			0
$154$	−3.65777			−0.294752
$155$	−9.91617			−0.796486
$156$		0			0
$157$		−	4.51008i		−	0.359944i	−0.983672	−	0.179972i	$-0.942399\pi$
$157$		−	4.51008i		−	0.359944i	0.983672	−	0.179972i	$-0.0576007\pi$
$158$	−35.6239			−2.83408
$159$		0			0
$160$			11.7348i			0.927716i
$161$	−2.65862			−0.209529
$162$		0			0
$163$	7.47402			0.585411			0.292705	−	0.956203i	$-0.405445\pi$
$163$	7.47402			0.585411			0.292705	+	0.956203i	$0.405445\pi$
$164$			19.7161i			1.53957i
$165$		0			0
$166$	15.6363			1.21361
$167$		−	16.3746i		−	1.26710i	−0.773701	−	0.633550i	$-0.781597\pi$
$167$		−	16.3746i		−	1.26710i	0.773701	−	0.633550i	$-0.218403\pi$
$168$		0			0
$169$	7.48464			0.575741
$170$	−22.1405			−1.69810
$171$		0			0
$172$			47.4530i			3.61826i
$173$	−18.2573			−1.38808			−0.694038	−	0.719938i	$-0.744170\pi$
$173$	−18.2573			−1.38808			−0.694038	+	0.719938i	$0.744170\pi$
$174$		0			0
$175$	−3.54348			−0.267862
$176$	−11.2794			−0.850218
$177$		0			0
$178$	33.1701			2.48620
$179$	16.9631			1.26788			0.633940	−	0.773382i	$-0.281437\pi$
$179$	16.9631			1.26788			0.633940	+	0.773382i	$0.281437\pi$
$180$		0			0
$181$	18.5575			1.37937			0.689684	−	0.724110i	$-0.257749\pi$
$181$	18.5575			1.37937			0.689684	+	0.724110i	$0.257749\pi$
$182$			6.71722i			0.497913i
$183$		0			0
$184$	−16.9683			−1.25092
$185$	6.54820			0.481433
$186$		0			0
$187$		−	8.16743i		−	0.597262i
$188$	22.3259			1.62828
$189$		0			0
$190$		−	19.4452i		−	1.41070i
$191$	−26.8946			−1.94603			−0.973013	−	0.230748i	$-0.925883\pi$
$191$	−26.8946			−1.94603			−0.973013	+	0.230748i	$0.925883\pi$
$192$		0			0
$193$	12.3495			0.888939			0.444469	−	0.895794i	$-0.353392\pi$
$193$	12.3495			0.888939			0.444469	+	0.895794i	$0.353392\pi$
$194$			43.5425i			3.12617i
$195$		0			0
$196$	−27.2775			−1.94839
$197$		−	9.09516i		−	0.648003i	−0.946056	−	0.324002i	$-0.894972\pi$
$197$		−	9.09516i		−	0.648003i	0.946056	−	0.324002i	$-0.105028\pi$
$198$		0			0
$199$	−11.2099			−0.794650			−0.397325	−	0.917678i	$-0.630061\pi$
$199$	−11.2099			−0.794650			−0.397325	+	0.917678i	$0.630061\pi$
$200$	−22.6159			−1.59918
$201$		0			0
$202$	−3.37045			−0.237144
$203$			0.560722i			0.0393549i
$204$		0			0
$205$	5.59008			0.390428
$206$		−	29.9596i		−	2.08739i
$207$		0			0
$208$			20.7138i			1.43624i
$209$	7.17314			0.496177
$210$		0			0
$211$		−	9.38631i		−	0.646180i	−0.946368	−	0.323090i	$-0.895278\pi$
$211$		−	9.38631i		−	0.646180i	0.946368	−	0.323090i	$-0.104722\pi$
$212$			48.5513i			3.33451i
$213$		0			0
$214$			19.5804i			1.33849i
$215$	13.4543			0.917574
$216$		0			0
$217$			8.18144i			0.555392i
$218$		−	48.3195i		−	3.27261i
$219$		0			0
$220$			8.06940i			0.544039i
$221$	−14.9989			−1.00893
$222$		0			0
$223$	−0.566357			−0.0379261			−0.0189630	−	0.999820i	$-0.506036\pi$
$223$	−0.566357			−0.0379261			−0.0189630	+	0.999820i	$0.506036\pi$
$224$	9.68191			0.646899
$225$		0			0
$226$	35.7261			2.37646
$227$	−5.44940			−0.361689			−0.180845	−	0.983512i	$-0.557883\pi$
$227$	−5.44940			−0.361689			−0.180845	+	0.983512i	$0.557883\pi$
$228$		0			0
$229$			20.2184i			1.33607i	0.744131	+	0.668034i	$0.232864\pi$
$229$			20.2184i			1.33607i	−0.744131	+	0.668034i	$0.767136\pi$
$230$			8.35172i			0.550696i
$231$		0			0
$232$			3.57874i			0.234956i
$233$	−16.4532			−1.07788			−0.538941	−	0.842344i	$-0.681175\pi$
$233$	−16.4532			−1.07788			−0.538941	+	0.842344i	$0.681175\pi$
$234$		0			0
$235$		−	6.33002i		−	0.412925i
$236$	27.8951	+	23.1282i	1.81582	+	1.50552i
$237$		0			0
$238$			18.2673i			1.18409i
$239$		−	0.297314i		−	0.0192317i	−0.999954	−	0.00961583i	$-0.996939\pi$
$239$		−	0.297314i		−	0.0192317i	0.999954	−	0.00961583i	$-0.00306086\pi$
$240$		0			0
$241$	3.52113			0.226816			0.113408	−	0.993549i	$-0.463823\pi$
$241$	3.52113			0.226816			0.113408	+	0.993549i	$0.463823\pi$
$242$	24.2713			1.56022
$243$		0			0
$244$			36.1931i			2.31702i
$245$			7.73394i			0.494103i
$246$		0			0
$247$		−	13.1729i		−	0.838173i
$248$			52.2171i			3.31579i
$249$		0			0
$250$			28.4649i			1.80028i
$251$			7.13256i			0.450203i	0.974335	+	0.225102i	$0.0722714\pi$
$251$			7.13256i			0.450203i	−0.974335	+	0.225102i	$0.927729\pi$
$252$		0			0
$253$	−3.08087			−0.193693
$254$	29.0909			1.82533
$255$		0			0
$256$	−21.4243			−1.33902
$257$		−	17.3343i		−	1.08128i	−0.841253	−	0.540642i	$-0.818181\pi$
$257$		−	17.3343i		−	1.08128i	0.841253	−	0.540642i	$-0.181819\pi$
$258$		0			0
$259$		−	5.40266i		−	0.335705i
$260$	14.8188			0.919025
$261$		0			0
$262$	−42.9474			−2.65330
$263$			20.3963i			1.25769i	0.777531	+	0.628844i	$0.216472\pi$
$263$			20.3963i			1.25769i	−0.777531	+	0.628844i	$0.783528\pi$
$264$		0			0
$265$	13.7657			0.845618
$266$	−16.0434			−0.983686
$267$		0			0
$268$			16.8291i			1.02800i
$269$	18.1302			1.10542			0.552708	−	0.833375i	$-0.313594\pi$
$269$	18.1302			1.10542			0.552708	+	0.833375i	$0.313594\pi$
$270$		0			0
$271$	−10.8287			−0.657799			−0.328900	−	0.944365i	$-0.606678\pi$
$271$	−10.8287			−0.657799			−0.328900	+	0.944365i	$0.606678\pi$
$272$			56.3305i			3.41554i
$273$		0			0
$274$		−	48.1497i		−	2.90883i
$275$	−4.10627			−0.247617
$276$		0			0
$277$	1.76618			0.106120			0.0530598	−	0.998591i	$-0.483103\pi$
$277$	1.76618			0.106120			0.0530598	+	0.998591i	$0.483103\pi$
$278$	−10.3031			−0.617942
$279$		0			0
$280$		−	10.3965i		−	0.621312i
$281$		−	27.0494i		−	1.61363i	−0.590802	−	0.806817i	$-0.701188\pi$
$281$		−	27.0494i		−	1.61363i	0.590802	−	0.806817i	$-0.298812\pi$
$282$		0			0
$283$		−	30.5394i		−	1.81538i	−0.419645	−	0.907688i	$-0.637845\pi$
$283$		−	30.5394i		−	1.81538i	0.419645	−	0.907688i	$-0.362155\pi$
$284$		−	7.02639i		−	0.416939i
$285$		0			0
$286$			7.78407i			0.460281i
$287$		−	4.61215i		−	0.272247i
$288$		0			0
$289$	−23.7889			−1.39935
$290$	1.76144			0.103435
$291$		0			0
$292$			24.6515i			1.44262i
$293$			2.32134i			0.135614i	0.997698	+	0.0678070i	$0.0216002\pi$
$293$			2.32134i			0.135614i	−0.997698	+	0.0678070i	$0.978400\pi$
$294$		0			0
$295$	6.55750	−	7.90906i	0.381793	−	0.460483i
$296$		−	34.4818i		−	2.00422i
$297$		0			0
$298$	3.45167			0.199950
$299$			5.65779i			0.327198i
$300$		0			0
$301$		−	11.1006i		−	0.639827i
$302$			26.5066i			1.52528i
$303$		0			0
$304$	−49.4729			−2.83746
$305$	10.2618			0.587586
$306$		0			0
$307$	−14.1536			−0.807791			−0.403895	−	0.914805i	$-0.632344\pi$
$307$	−14.1536			−0.807791			−0.403895	+	0.914805i	$0.632344\pi$
$308$	6.65774			0.379360
$309$		0			0
$310$	25.7009			1.45972
$311$		−	24.0962i		−	1.36637i	−0.730245	−	0.683185i	$-0.760594\pi$
$311$		−	24.0962i		−	1.36637i	0.730245	−	0.683185i	$-0.239406\pi$
$312$		0			0
$313$			2.81096i			0.158885i	0.996839	+	0.0794424i	$0.0253140\pi$
$313$			2.81096i			0.158885i	−0.996839	+	0.0794424i	$0.974686\pi$
$314$			11.6893i			0.659667i
$315$		0			0
$316$	64.8413			3.64761
$317$			7.10636i			0.399133i	0.979884	+	0.199566i	$0.0639534\pi$
$317$			7.10636i			0.399133i	−0.979884	+	0.199566i	$0.936047\pi$
$318$		0			0
$319$			0.649777i			0.0363805i
$320$		−	6.81981i		−	0.381239i
$321$		0			0
$322$	6.89067			0.384002
$323$		−	35.8233i		−	1.99326i
$324$		0			0
$325$			7.54085i			0.418291i
$326$	−19.3713			−1.07288
$327$		0			0
$328$		−	29.4365i		−	1.62536i
$329$	−5.22264			−0.287934
$330$		0			0
$331$	−22.2809			−1.22467			−0.612335	−	0.790598i	$-0.709770\pi$
$331$	−22.2809			−1.22467			−0.612335	+	0.790598i	$0.709770\pi$
$332$	−28.4606			−1.56198
$333$		0			0
$334$			42.4399i			2.32221i
$335$	4.77153			0.260697
$336$		0			0
$337$			10.6139i			0.578179i	0.957302	+	0.289089i	$0.0933524\pi$
$337$			10.6139i			0.578179i	−0.957302	+	0.289089i	$0.906648\pi$
$338$	−19.3988			−1.05516
$339$		0			0
$340$	40.2993			2.18554
$341$			9.48084i			0.513416i
$342$		0			0
$343$	14.1059			0.761647
$344$		−	70.8482i		−	3.81988i
$345$		0			0
$346$	47.3196			2.54392
$347$	−16.9371			−0.909229			−0.454615	−	0.890688i	$-0.650223\pi$
$347$	−16.9371			−0.909229			−0.454615	+	0.890688i	$0.650223\pi$
$348$		0			0
$349$		−	16.5987i		−	0.888506i	−0.895901	−	0.444253i	$-0.853469\pi$
$349$		−	16.5987i		−	0.888506i	0.895901	−	0.444253i	$-0.146531\pi$
$350$	9.18407			0.490909
$351$		0			0
$352$	11.2196			0.598008
$353$	1.72462			0.0917922			0.0458961	−	0.998946i	$-0.485386\pi$
$353$	1.72462			0.0917922			0.0458961	+	0.998946i	$0.485386\pi$
$354$		0			0
$355$	−1.99218			−0.105734
$356$	−60.3750			−3.19987
$357$		0			0
$358$	−43.9652			−2.32364
$359$			26.2732i			1.38665i	0.720626	+	0.693324i	$0.243855\pi$
$359$			26.2732i			1.38665i	−0.720626	+	0.693324i	$0.756145\pi$
$360$		0			0
$361$	12.4623			0.655909
$362$	−48.0977			−2.52796
$363$		0			0
$364$		−	12.2264i		−	0.640839i
$365$	6.98941			0.365842
$366$		0			0
$367$			14.6670i			0.765610i	0.923829	+	0.382805i	$0.125042\pi$
$367$			14.6670i			0.765610i	−0.923829	+	0.382805i	$0.874958\pi$
$368$	21.2487			1.10766
$369$		0			0
$370$	−16.9718			−0.882320
$371$		−	11.3575i		−	0.589652i
$372$		0			0
$373$	13.0388			0.675122			0.337561	−	0.941304i	$-0.390398\pi$
$373$	13.0388			0.675122			0.337561	+	0.941304i	$0.390398\pi$
$374$			21.1685i			1.09460i
$375$		0			0
$376$	−33.3329			−1.71901
$377$	1.19327			0.0614563
$378$		0			0
$379$	−30.8296			−1.58361			−0.791806	−	0.610772i	$-0.790859\pi$
$379$	−30.8296			−1.58361			−0.791806	+	0.610772i	$0.790859\pi$
$380$			35.3934i			1.81564i
$381$		0			0
$382$	69.7061			3.56647
$383$		−	12.8038i		−	0.654246i	−0.944982	−	0.327123i	$-0.893921\pi$
$383$		−	12.8038i		−	0.654246i	0.944982	−	0.327123i	$-0.106079\pi$
$384$		0			0
$385$		−	1.88766i		−	0.0962039i
$386$	−32.0078			−1.62915
$387$		0			0
$388$		−	79.2544i		−	4.02353i
$389$			5.21207i			0.264262i	0.991232	+	0.132131i	$0.0421820\pi$
$389$			5.21207i			0.264262i	−0.991232	+	0.132131i	$0.957818\pi$
$390$		0			0
$391$			15.3862i			0.778112i
$392$	40.7258			2.05696
$393$		0			0
$394$			23.5730i			1.18759i
$395$		−	18.3843i		−	0.925016i
$396$		0			0
$397$			28.3858i			1.42464i	0.701854	+	0.712321i	$0.252356\pi$
$397$			28.3858i			1.42464i	−0.701854	+	0.712321i	$0.747644\pi$
$398$	29.0541			1.45635
$399$		0			0
$400$	28.3208			1.41604
$401$	30.0635			1.50130			0.750649	−	0.660701i	$-0.229741\pi$
$401$	30.0635			1.50130			0.750649	+	0.660701i	$0.229741\pi$
$402$		0			0
$403$	17.4108			0.867295
$404$	6.13477			0.305216
$405$		0			0
$406$		−	1.45329i		−	0.0721256i
$407$		−	6.26073i		−	0.310333i
$408$		0			0
$409$		−	20.6179i		−	1.01949i	−0.860325	−	0.509746i	$-0.829740\pi$
$409$		−	20.6179i		−	1.01949i	0.860325	−	0.509746i	$-0.170260\pi$
$410$	−14.4885			−0.715535
$411$		0			0
$412$			54.5314i			2.68657i
$413$	−6.52545	−	5.41034i	−0.321096	−	0.266225i
$414$		0			0
$415$			8.06940i			0.396111i
$416$		−	20.6040i		−	1.01019i
$417$		0			0
$418$	−18.5915			−0.909340
$419$	20.2821			0.990847			0.495424	−	0.868651i	$-0.335013\pi$
$419$	20.2821			0.990847			0.495424	+	0.868651i	$0.335013\pi$
$420$		0			0
$421$		−	10.2119i		−	0.497696i	−0.968543	−	0.248848i	$-0.919948\pi$
$421$		−	10.2119i		−	0.497696i	0.968543	−	0.248848i	$-0.0800519\pi$
$422$			24.3276i			1.18425i
$423$		0			0
$424$		−	72.4879i		−	3.52032i
$425$			20.5071i			0.994741i
$426$		0			0
$427$		−	8.46657i		−	0.409726i
$428$		−	35.6395i		−	1.72270i
$429$		0			0
$430$	−34.8711			−1.68163
$431$	2.48805			0.119845			0.0599226	−	0.998203i	$-0.480915\pi$
$431$	2.48805			0.119845			0.0599226	+	0.998203i	$0.480915\pi$
$432$		0			0
$433$	2.62436			0.126118			0.0630592	−	0.998010i	$-0.479914\pi$
$433$	2.62436			0.126118			0.0630592	+	0.998010i	$0.479914\pi$
$434$		−	21.2048i		−	1.01786i
$435$		0			0
$436$			87.9494i			4.21201i
$437$	−13.5131			−0.646418
$438$		0			0
$439$	9.35004			0.446253			0.223126	−	0.974790i	$-0.428374\pi$
$439$	9.35004			0.446253			0.223126	+	0.974790i	$0.428374\pi$
$440$		−	12.0478i		−	0.574354i
$441$		0			0
$442$	38.8744			1.84906
$443$	21.9494			1.04285			0.521424	−	0.853298i	$-0.325401\pi$
$443$	21.9494			1.04285			0.521424	+	0.853298i	$0.325401\pi$
$444$		0			0
$445$			17.1180i			0.811472i
$446$	1.46790			0.0695069
$447$		0			0
$448$	−5.62675			−0.265839
$449$			33.5977i			1.58558i	0.609498	+	0.792788i	$0.291371\pi$
$449$			33.5977i			1.58558i	−0.609498	+	0.792788i	$0.708629\pi$
$450$		0			0
$451$		−	5.34467i		−	0.251670i
$452$	−65.0273			−3.05863
$453$		0			0
$454$	14.1239			0.662865
$455$	−3.46654			−0.162514
$456$		0			0
$457$		−	7.05058i		−	0.329812i	−0.986309	−	0.164906i	$-0.947268\pi$
$457$		−	7.05058i		−	0.329812i	0.986309	−	0.164906i	$-0.0527321\pi$
$458$		−	52.4024i		−	2.44860i
$459$		0			0
$460$		−	15.2015i		−	0.708773i
$461$			10.6981i			0.498258i	0.968470	+	0.249129i	$0.0801443\pi$
$461$			10.6981i			0.498258i	−0.968470	+	0.249129i	$0.919856\pi$
$462$		0			0
$463$		−	41.8771i		−	1.94619i	−0.230399	−	0.973096i	$-0.574003\pi$
$463$		−	41.8771i		−	1.94619i	0.230399	−	0.973096i	$-0.425997\pi$
$464$		−	4.48149i		−	0.208048i
$465$		0			0
$466$	42.6436			1.97543
$467$	12.6008			0.583095			0.291548	−	0.956556i	$-0.405830\pi$
$467$	12.6008			0.583095			0.291548	+	0.956556i	$0.405830\pi$
$468$		0			0
$469$		−	3.93680i		−	0.181785i
$470$			16.4063i			0.756765i
$471$		0			0
$472$	−41.6479	−	34.5308i	−1.91700	−	1.58941i
$473$		−	12.8636i		−	0.591470i
$474$		0			0
$475$	−18.0106			−0.826383
$476$		−	33.2494i		−	1.52398i
$477$		0			0
$478$			0.770586i			0.0352458i
$479$			25.0009i			1.14232i	0.820838	+	0.571161i	$0.193507\pi$
$479$			25.0009i			1.14232i	−0.820838	+	0.571161i	$0.806493\pi$
$480$		0			0
$481$	−11.4973			−0.524234
$482$	−9.12614			−0.415684
$483$		0			0
$484$	−44.1777			−2.00808
$485$	−22.4709			−1.02035
$486$		0			0
$487$	−20.8899			−0.946613			−0.473307	−	0.880898i	$-0.656940\pi$
$487$	−20.8899			−0.946613			−0.473307	+	0.880898i	$0.656940\pi$
$488$		−	54.0369i		−	2.44614i
$489$		0			0
$490$		−	20.0450i		−	0.905540i
$491$			6.70459i			0.302574i	0.988490	+	0.151287i	$0.0483418\pi$
$491$			6.70459i			0.302574i	−0.988490	+	0.151287i	$0.951658\pi$
$492$		0			0
$493$	3.24505			0.146150
$494$			34.1419i			1.53611i
$495$		0			0
$496$		−	65.3890i		−	2.93605i
$497$			1.64367i			0.0737286i
$498$		0			0
$499$	−6.58141			−0.294624			−0.147312	−	0.989090i	$-0.547062\pi$
$499$	−6.58141			−0.294624			−0.147312	+	0.989090i	$0.547062\pi$
$500$		−	51.8108i		−	2.31705i
$501$		0			0
$502$		−	18.4863i		−	0.825085i
$503$	−4.80083			−0.214059			−0.107029	−	0.994256i	$-0.534134\pi$
$503$	−4.80083			−0.214059			−0.107029	+	0.994256i	$0.534134\pi$
$504$		0			0
$505$		−	1.73938i		−	0.0774015i
$506$	7.98507			0.354980
$507$		0			0
$508$	−52.9502			−2.34929
$509$	20.9734			0.929630			0.464815	−	0.885408i	$-0.346121\pi$
$509$	20.9734			0.929630			0.464815	+	0.885408i	$0.346121\pi$
$510$		0			0
$511$		−	5.76668i		−	0.255103i
$512$	46.8647			2.07115
$513$		0			0
$514$			44.9274i			1.98166i
$515$	15.4612			0.681301
$516$		0			0
$517$	−6.05212			−0.266172
$518$			14.0027i			0.615244i
$519$		0			0
$520$	−22.1248			−0.970236
$521$		−	33.0424i		−	1.44762i	−0.690002	−	0.723808i	$-0.742390\pi$
$521$		−	33.0424i		−	1.44762i	0.690002	−	0.723808i	$-0.257610\pi$
$522$		0			0
$523$	−26.4689			−1.15740			−0.578702	−	0.815539i	$-0.696440\pi$
$523$	−26.4689			−1.15740			−0.578702	+	0.815539i	$0.696440\pi$
$524$	78.1713			3.41493
$525$		0			0
$526$		−	52.8635i		−	2.30496i
$527$	47.3482			2.06252
$528$		0			0
$529$	−17.1961			−0.747657
$530$	−35.6781			−1.54976
$531$		0			0
$532$	29.2017			1.26605
$533$	−9.81507			−0.425138
$534$		0			0
$535$	−10.1048			−0.436869
$536$		−	25.1262i		−	1.08529i
$537$		0			0
$538$	−46.9902			−2.02589
$539$	7.39441			0.318500
$540$		0			0
$541$		−	15.5849i		−	0.670045i	−0.942210	−	0.335023i	$-0.891256\pi$
$541$		−	15.5849i		−	0.670045i	0.942210	−	0.335023i	$-0.108744\pi$
$542$	28.0662			1.20554
$543$		0			0
$544$		−	56.0318i		−	2.40234i
$545$	24.9362			1.06815
$546$		0			0
$547$	37.2794			1.59395			0.796975	−	0.604012i	$-0.206432\pi$
$547$	37.2794			1.59395			0.796975	+	0.604012i	$0.206432\pi$
$548$			87.6403i			3.74381i
$549$		0			0
$550$	10.6427			0.453807
$551$			2.85000i			0.121414i
$552$		0			0
$553$	−15.1682			−0.645017
$554$	−4.57763			−0.194485
$555$		0			0
$556$	18.7534			0.795321
$557$			15.9475i			0.675717i	0.941197	+	0.337859i	$0.109703\pi$
$557$			15.9475i			0.675717i	−0.941197	+	0.337859i	$0.890297\pi$
$558$		0			0
$559$	−23.6230			−0.999148
$560$			13.0191i			0.550157i
$561$		0			0
$562$			70.1073i			2.95730i
$563$	−23.2133			−0.978324			−0.489162	−	0.872193i	$-0.662697\pi$
$563$	−23.2133			−0.978324			−0.489162	+	0.872193i	$0.662697\pi$
$564$		0			0
$565$			18.4371i			0.775653i
$566$			79.1526i			3.32703i
$567$		0			0
$568$			10.4905i			0.440172i
$569$	−6.74997			−0.282973			−0.141487	−	0.989940i	$-0.545188\pi$
$569$	−6.74997			−0.282973			−0.141487	+	0.989940i	$0.545188\pi$
$570$		0			0
$571$		−	4.88193i		−	0.204303i	−0.994769	−	0.102151i	$-0.967427\pi$
$571$		−	4.88193i		−	0.204303i	0.994769	−	0.102151i	$-0.0325726\pi$
$572$		−	14.1683i		−	0.592405i
$573$		0			0
$574$			11.9539i			0.498945i
$575$	7.73557			0.322596
$576$		0			0
$577$	−18.7442			−0.780332			−0.390166	−	0.920745i	$-0.627582\pi$
$577$	−18.7442			−0.780332			−0.390166	+	0.920745i	$0.627582\pi$
$578$	61.6566			2.56458
$579$		0			0
$580$	−3.20610			−0.133126
$581$	6.65774			0.276210
$582$		0			0
$583$		−	13.1613i		−	0.545087i
$584$		−	36.8052i		−	1.52301i
$585$		0			0
$586$		−	6.01649i		−	0.248539i
$587$	8.21903			0.339235			0.169618	−	0.985510i	$-0.445747\pi$
$587$	8.21903			0.339235			0.169618	+	0.985510i	$0.445747\pi$
$588$		0			0
$589$			41.5841i			1.71344i
$590$	−16.9959	+	20.4989i	−0.699709	+	0.843925i
$591$		0			0
$592$			43.1800i			1.77469i
$593$			19.2976i			0.792456i	0.918152	+	0.396228i	$0.129681\pi$
$593$			19.2976i			0.792456i	−0.918152	+	0.396228i	$0.870319\pi$
$594$		0			0
$595$	−9.42714			−0.386475
$596$	−6.28260			−0.257345
$597$		0			0
$598$		−	14.6640i		−	0.599654i
$599$			5.55790i			0.227090i	0.993533	+	0.113545i	$0.0362206\pi$
$599$			5.55790i			0.227090i	−0.993533	+	0.113545i	$0.963779\pi$
$600$		0			0
$601$			26.2502i			1.07077i	0.844608	+	0.535384i	$0.179833\pi$
$601$			26.2502i			1.07077i	−0.844608	+	0.535384i	$0.820167\pi$
$602$			28.7707i			1.17261i
$603$		0			0
$604$		−	48.2463i		−	1.96311i
$605$			12.5256i			0.509240i
$606$		0			0
$607$	2.50904			0.101839			0.0509195	−	0.998703i	$-0.483785\pi$
$607$	2.50904			0.101839			0.0509195	+	0.998703i	$0.483785\pi$
$608$	49.2106			1.99575
$609$		0			0
$610$	−26.5966			−1.07687
$611$			11.1143i			0.449635i
$612$		0			0
$613$			42.3201i			1.70929i	0.519210	+	0.854647i	$0.326226\pi$
$613$			42.3201i			1.70929i	−0.519210	+	0.854647i	$0.673774\pi$
$614$	36.6837			1.48043
$615$		0			0
$616$	−9.94012			−0.400499
$617$			21.3951i			0.861335i	0.902511	+	0.430667i	$0.141722\pi$
$617$			21.3951i			0.861335i	−0.902511	+	0.430667i	$0.858278\pi$
$618$		0			0
$619$	−3.27219			−0.131520			−0.0657602	−	0.997835i	$-0.520947\pi$
$619$	−3.27219			−0.131520			−0.0657602	+	0.997835i	$0.520947\pi$
$620$	−46.7799			−1.87873
$621$		0			0
$622$			62.4530i			2.50414i
$623$	14.1234			0.565842
$624$		0			0
$625$	1.36492			0.0545967
$626$		−	7.28550i		−	0.291187i
$627$		0			0
$628$		−	21.2765i		−	0.849024i
$629$	−31.2667			−1.24668
$630$		0			0
$631$	19.2561			0.766572			0.383286	−	0.923630i	$-0.374792\pi$
$631$	19.2561			0.766572			0.383286	+	0.923630i	$0.374792\pi$
$632$	−96.8092			−3.85086
$633$		0			0
$634$		−	18.4184i		−	0.731488i
$635$			15.0129i			0.595768i
$636$		0			0
$637$		−	13.5793i		−	0.538030i
$638$		−	1.68411i		−	0.0666744i
$639$		0			0
$640$		−	5.79385i		−	0.229022i
$641$		−	41.3576i		−	1.63353i	−0.576974	−	0.816763i	$-0.695767\pi$
$641$		−	41.3576i		−	1.63353i	0.576974	−	0.816763i	$-0.304233\pi$
$642$		0			0
$643$	−22.0837			−0.870897			−0.435449	−	0.900214i	$-0.643410\pi$
$643$	−22.0837			−0.870897			−0.435449	+	0.900214i	$0.643410\pi$
$644$	−12.5421			−0.494230
$645$		0			0
$646$			92.8477i			3.65304i
$647$			5.27457i			0.207365i	0.994610	+	0.103682i	$0.0330625\pi$
$647$			5.27457i			0.207365i	−0.994610	+	0.103682i	$0.966937\pi$
$648$		0			0
$649$	−7.56185	−	6.26962i	−0.296828	−	0.246104i
$650$		−	19.5445i		−	0.766600i
$651$		0			0
$652$	35.2590			1.38085
$653$			23.8696i			0.934089i	0.884234	+	0.467044i	$0.154681\pi$
$653$			23.8696i			0.934089i	−0.884234	+	0.467044i	$0.845319\pi$
$654$		0			0
$655$		−	22.1638i		−	0.866010i
$656$			36.8619i			1.43922i
$657$		0			0
$658$	13.5362			0.527694
$659$	39.0516			1.52123			0.760617	−	0.649201i	$-0.224897\pi$
$659$	39.0516			1.52123			0.760617	+	0.649201i	$0.224897\pi$
$660$		0			0
$661$	43.8662			1.70620			0.853099	−	0.521750i	$-0.174720\pi$
$661$	43.8662			1.70620			0.853099	+	0.521750i	$0.174720\pi$
$662$	57.7481			2.24445
$663$		0			0
$664$	42.4923			1.64902
$665$		−	8.27949i		−	0.321065i
$666$		0			0
$667$		−	1.22408i		−	0.0473965i
$668$		−	77.2475i		−	2.98880i
$669$		0			0
$670$	−12.3670			−0.477777
$671$		−	9.81125i		−	0.378759i
$672$		0			0
$673$			21.8356i			0.841699i	0.907131	+	0.420849i	$0.138268\pi$
$673$			21.8356i			0.841699i	−0.907131	+	0.420849i	$0.861732\pi$
$674$		−	27.5095i		−	1.05962i
$675$		0			0
$676$	35.3090			1.35804
$677$			18.1653i			0.698150i	0.937095	+	0.349075i	$0.113504\pi$
$677$			18.1653i			0.698150i	−0.937095	+	0.349075i	$0.886496\pi$
$678$		0			0
$679$			18.5398i			0.711493i
$680$	−60.1676			−2.30732
$681$		0			0
$682$		−	24.5726i		−	0.940935i
$683$	−0.927513			−0.0354903			−0.0177451	−	0.999843i	$-0.505649\pi$
$683$	−0.927513			−0.0354903			−0.0177451	+	0.999843i	$0.505649\pi$
$684$		0			0
$685$	24.8485			0.949413
$686$	−36.5600			−1.39587
$687$		0			0
$688$			88.7198i			3.38241i
$689$	−24.1698			−0.920795
$690$		0			0
$691$			16.1812i			0.615562i	0.951457	+	0.307781i	$0.0995864\pi$
$691$			16.1812i			0.615562i	−0.951457	+	0.307781i	$0.900414\pi$
$692$	−86.1294			−3.27415
$693$		0			0
$694$	43.8978			1.66634
$695$		−	5.31712i		−	0.201690i
$696$		0			0
$697$	−26.6918			−1.01102
$698$			43.0207i			1.62836i
$699$		0			0
$700$	−16.7165			−0.631825
$701$	−25.0850			−0.947448			−0.473724	−	0.880673i	$-0.657091\pi$
$701$	−25.0850			−0.947448			−0.473724	+	0.880673i	$0.657091\pi$
$702$		0			0
$703$		−	27.4603i		−	1.03568i
$704$	−6.52041			−0.245747
$705$		0			0
$706$	−4.46991			−0.168227
$707$	−1.43509			−0.0539723
$708$		0			0
$709$	−1.10014			−0.0413165			−0.0206583	−	0.999787i	$-0.506576\pi$
$709$	−1.10014			−0.0413165			−0.0206583	+	0.999787i	$0.506576\pi$
$710$	5.16337			0.193778
$711$		0			0
$712$	90.1409			3.37817
$713$		−	17.8604i		−	0.668878i
$714$		0			0
$715$	−4.01710			−0.150231
$716$	80.0239			2.99063
$717$		0			0
$718$		−	68.0955i		−	2.54130i
$719$	12.2771			0.457859			0.228929	−	0.973443i	$-0.426477\pi$
$719$	12.2771			0.457859			0.228929	+	0.973443i	$0.426477\pi$
$720$		0			0
$721$		−	12.7564i		−	0.475074i
$722$	−32.3000			−1.20208
$723$		0			0
$724$	87.5457			3.25361
$725$		−	1.63149i		−	0.0605919i
$726$		0			0
$727$	−48.3269			−1.79235			−0.896173	−	0.443704i	$-0.853664\pi$
$727$	−48.3269			−1.79235			−0.896173	+	0.443704i	$0.853664\pi$
$728$			18.2543i			0.676548i
$729$		0			0
$730$	−18.1153			−0.670477
$731$	−64.2421			−2.37608
$732$		0			0
$733$	41.3111			1.52586			0.762930	−	0.646481i	$-0.223760\pi$
$733$	41.3111			1.52586			0.762930	+	0.646481i	$0.223760\pi$
$734$		−	38.0142i		−	1.40313i
$735$		0			0
$736$	−21.1360			−0.779084
$737$		−	4.56206i		−	0.168046i
$738$		0			0
$739$			36.2934i			1.33508i	0.744576	+	0.667538i	$0.232652\pi$
$739$			36.2934i			1.33508i	−0.744576	+	0.667538i	$0.767348\pi$
$740$	30.8914			1.13559
$741$		0			0
$742$			29.4366i			1.08065i
$743$		−	34.1669i		−	1.25346i	−0.779236	−	0.626731i	$-0.784393\pi$
$743$		−	34.1669i		−	1.25346i	0.779236	−	0.626731i	$-0.215607\pi$
$744$		0			0
$745$			1.78129i			0.0652616i
$746$	−33.7941			−1.23729
$747$		0			0
$748$		−	38.5302i		−	1.40880i
$749$			8.33707i			0.304630i
$750$		0			0
$751$		−	15.7840i		−	0.575968i	−0.957635	−	0.287984i	$-0.907015\pi$
$751$		−	15.7840i		−	0.575968i	0.957635	−	0.287984i	$-0.0929850\pi$
$752$	41.7412			1.52215
$753$		0			0
$754$	−3.09273			−0.112631
$755$	−13.6792			−0.497837
$756$		0			0
$757$	−10.8100			−0.392897			−0.196449	−	0.980514i	$-0.562941\pi$
$757$	−10.8100			−0.392897			−0.196449	+	0.980514i	$0.562941\pi$
$758$	79.9049			2.90228
$759$		0			0
$760$		−	52.8429i		−	1.91681i
$761$			27.2877i			0.989178i	0.869127	+	0.494589i	$0.164681\pi$
$761$			27.2877i			0.989178i	−0.869127	+	0.494589i	$0.835319\pi$
$762$		0			0
$763$		−	20.5738i		−	0.744822i
$764$	−126.876			−4.59023
$765$		0			0
$766$			33.1853i			1.19903i
$767$	−11.5137	+	13.8867i	−0.415735	+	0.501421i
$768$		0			0
$769$		−	51.8241i		−	1.86882i	−0.356193	−	0.934412i	$-0.615925\pi$
$769$		−	51.8241i		−	1.86882i	0.356193	−	0.934412i	$-0.384075\pi$
$770$			4.89247i			0.176312i
$771$		0			0
$772$	58.2594			2.09680
$773$	14.0810			0.506458			0.253229	−	0.967406i	$-0.418507\pi$
$773$	14.0810			0.506458			0.253229	+	0.967406i	$0.418507\pi$
$774$		0			0
$775$		−	23.8049i		−	0.855096i
$776$			118.328i			4.24774i
$777$		0			0
$778$		−	13.5088i		−	0.484312i
$779$		−	23.4423i		−	0.839909i
$780$		0			0
$781$			1.90472i			0.0681563i
$782$		−	39.8782i		−	1.42604i
$783$		0			0
$784$	−50.9990			−1.82139
$785$	−6.03248			−0.215309
$786$		0			0
$787$	30.3109			1.08047			0.540233	−	0.841515i	$-0.318336\pi$
$787$	30.3109			1.08047			0.540233	+	0.841515i	$0.318336\pi$
$788$		−	42.9067i		−	1.52849i
$789$		0			0
$790$			47.6489i			1.69527i
$791$	15.2117			0.540866
$792$		0			0
$793$	−18.0176			−0.639824
$794$		−	73.5709i		−	2.61093i
$795$		0			0
$796$	−52.8832			−1.87440
$797$	30.4986			1.08032			0.540159	−	0.841563i	$-0.318364\pi$
$797$	30.4986			1.08032			0.540159	+	0.841563i	$0.318364\pi$
$798$		0			0
$799$			30.2249i			1.06928i
$800$	−28.1706			−0.995983
$801$		0			0
$802$	−77.9192			−2.75142
$803$		−	6.68257i		−	0.235823i
$804$		0			0
$805$			3.55605i			0.125334i
$806$	−45.1258			−1.58949
$807$		0			0
$808$	−9.15933			−0.322224
$809$	37.3336			1.31258			0.656289	−	0.754510i	$-0.272125\pi$
$809$	37.3336			1.31258			0.656289	+	0.754510i	$0.272125\pi$
$810$		0			0
$811$		−	23.8189i		−	0.836393i	−0.908356	−	0.418197i	$-0.862662\pi$
$811$		−	23.8189i		−	0.836393i	0.908356	−	0.418197i	$-0.137338\pi$
$812$			2.64522i			0.0928292i
$813$		0			0
$814$			16.2267i			0.568745i
$815$		−	9.99691i		−	0.350177i
$816$		0			0
$817$		−	56.4214i		−	1.97393i
$818$			53.4380i			1.86842i
$819$		0			0
$820$	26.3714			0.920929
$821$	32.3859			1.13027			0.565137	−	0.824997i	$-0.308823\pi$
$821$	32.3859			1.13027			0.565137	+	0.824997i	$0.308823\pi$
$822$		0			0
$823$			40.1246i			1.39866i	0.714800	+	0.699329i	$0.246518\pi$
$823$			40.1246i			1.39866i	−0.714800	+	0.699329i	$0.753482\pi$
$824$		−	81.4163i		−	2.83627i
$825$		0			0
$826$	16.9128	+	14.0226i	0.588472	+	0.487909i
$827$			5.45719i			0.189765i	0.995488	+	0.0948825i	$0.0302475\pi$
$827$			5.45719i			0.189765i	−0.995488	+	0.0948825i	$0.969752\pi$
$828$		0			0
$829$	18.6131			0.646458			0.323229	−	0.946321i	$-0.395232\pi$
$829$	18.6131			0.646458			0.323229	+	0.946321i	$0.395232\pi$
$830$		−	20.9144i		−	0.725951i
$831$		0			0
$832$			11.9742i			0.415132i
$833$		−	36.9284i		−	1.27949i
$834$		0			0
$835$	−21.9019			−0.757945
$836$	33.8396			1.17037
$837$		0			0
$838$	−52.5677			−1.81592
$839$	−50.4625			−1.74216			−0.871080	−	0.491141i	$-0.836580\pi$
$839$	−50.4625			−1.74216			−0.871080	+	0.491141i	$0.836580\pi$
$840$		0			0
$841$	28.7418			0.991098
$842$			26.4673i			0.912124i
$843$		0			0
$844$		−	44.2802i		−	1.52419i
$845$		−	10.0111i		−	0.344393i
$846$		0			0
$847$	10.3344			0.355094
$848$			90.7732i			3.11716i
$849$		0			0
$850$		−	53.1507i		−	1.82306i
$851$			11.7942i			0.404301i
$852$		0			0
$853$	7.16129			0.245198			0.122599	−	0.992456i	$-0.460877\pi$
$853$	7.16129			0.245198			0.122599	+	0.992456i	$0.460877\pi$
$854$			21.9438i			0.750902i
$855$		0			0
$856$			53.2104i			1.81869i
$857$	−36.4853			−1.24632			−0.623158	−	0.782096i	$-0.714150\pi$
$857$	−36.4853			−1.24632			−0.623158	+	0.782096i	$0.714150\pi$
$858$		0			0
$859$		−	37.4842i		−	1.27894i	−0.768815	−	0.639471i	$-0.779153\pi$
$859$		−	37.4842i		−	1.27894i	0.768815	−	0.639471i	$-0.220847\pi$
$860$	63.4710			2.16434
$861$		0			0
$862$	−6.44858			−0.219640
$863$	23.2826			0.792550			0.396275	−	0.918132i	$-0.370303\pi$
$863$	23.2826			0.792550			0.396275	+	0.918132i	$0.370303\pi$
$864$		0			0
$865$			24.4201i			0.830309i
$866$	−6.80186			−0.231137
$867$		0			0
$868$			38.5962i			1.31004i
$869$	−17.5772			−0.596267
$870$		0			0
$871$	−8.37787			−0.283873
$872$		−	131.310i		−	4.44672i
$873$		0			0
$874$	35.0235			1.18469
$875$			12.1200i			0.409730i
$876$		0			0
$877$	35.1282			1.18619			0.593097	−	0.805131i	$-0.297905\pi$
$877$	35.1282			1.18619			0.593097	+	0.805131i	$0.297905\pi$
$878$	−24.2336			−0.817845
$879$		0			0
$880$			15.0868i			0.508577i
$881$	30.5941			1.03074			0.515371	−	0.856967i	$-0.327654\pi$
$881$	30.5941			1.03074			0.515371	+	0.856967i	$0.327654\pi$
$882$		0			0
$883$	56.0462			1.88610			0.943052	−	0.332646i	$-0.107941\pi$
$883$	56.0462			1.88610			0.943052	+	0.332646i	$0.107941\pi$
$884$	−70.7577			−2.37984
$885$		0			0
$886$	−56.8890			−1.91122
$887$	33.5532			1.12661			0.563304	−	0.826250i	$-0.309530\pi$
$887$	33.5532			1.12661			0.563304	+	0.826250i	$0.309530\pi$
$888$		0			0
$889$	12.3865			0.415431
$890$		−	44.3668i		−	1.48718i
$891$		0			0
$892$	−2.67181			−0.0894588
$893$	−26.5453			−0.888306
$894$		0			0
$895$		−	22.6890i		−	0.758411i
$896$	−4.78027			−0.159698
$897$		0			0
$898$		−	87.0793i		−	2.90588i
$899$	−3.76689			−0.125633
$900$		0			0
$901$	−65.7289			−2.18975
$902$			13.8524i			0.461235i
$903$		0			0
$904$	97.0869			3.22906
$905$		−	24.8217i		−	0.825100i
$906$		0			0
$907$	55.3302			1.83721			0.918605	−	0.395177i	$-0.129317\pi$
$907$	55.3302			1.83721			0.918605	+	0.395177i	$0.129317\pi$
$908$	−25.7077			−0.853140
$909$		0			0
$910$	8.98464			0.297838
$911$			2.36437i			0.0783350i	0.999233	+	0.0391675i	$0.0124706\pi$
$911$			2.36437i			0.0783350i	−0.999233	+	0.0391675i	$0.987529\pi$
$912$		0			0
$913$	7.71514			0.255334
$914$			18.2738i			0.604445i
$915$		0			0
$916$			95.3809i			3.15147i
$917$	−18.2864			−0.603871
$918$		0			0
$919$			58.4248i			1.92726i	0.267244	+	0.963629i	$0.413887\pi$
$919$			58.4248i			1.92726i	−0.267244	+	0.963629i	$0.586113\pi$
$920$			22.6961i			0.748268i
$921$		0			0
$922$		−	27.7274i		−	0.913155i
$923$	3.49787			0.115134
$924$		0			0
$925$			15.7197i			0.516860i
$926$			108.538i			3.56678i
$927$		0			0
$928$			4.45773i			0.146332i
$929$	−0.743626			−0.0243976			−0.0121988	−	0.999926i	$-0.503883\pi$
$929$	−0.743626			−0.0243976			−0.0121988	+	0.999926i	$0.503883\pi$
$930$		0			0
$931$	32.4328			1.06294
$932$	−77.6183			−2.54247
$933$		0			0
$934$	−32.6590			−1.06864
$935$	−10.9244			−0.357266
$936$		0			0
$937$		−	1.30437i		−	0.0426119i	−0.999773	−	0.0213060i	$-0.993218\pi$
$937$		−	1.30437i		−	0.0426119i	0.999773	−	0.0213060i	$-0.00678241\pi$
$938$			10.2035i			0.333156i
$939$		0			0
$940$		−	29.8621i		−	0.973994i
$941$	−57.1702			−1.86370			−0.931848	−	0.362849i	$-0.881804\pi$
$941$	−57.1702			−1.86370			−0.931848	+	0.362849i	$0.881804\pi$
$942$		0			0
$943$			10.0685i			0.327876i
$944$	52.1537	+	43.2413i	1.69746	+	1.40739i
$945$		0			0
$946$			33.3402i			1.08398i
$947$		−	3.77389i		−	0.122635i	−0.998118	−	0.0613175i	$-0.980470\pi$
$947$		−	3.77389i		−	0.122635i	0.998118	−	0.0613175i	$-0.0195302\pi$
$948$		0			0
$949$	−12.2720			−0.398366
$950$	46.6803			1.51451
$951$		0			0
$952$			49.6419i			1.60890i
$953$		−	2.21032i		−	0.0715993i	−0.999359	−	0.0357996i	$-0.988602\pi$
$953$		−	2.21032i		−	0.0715993i	0.999359	−	0.0357996i	$-0.0113978\pi$
$954$		0			0
$955$			35.9730i			1.16406i
$956$		−	1.40259i		−	0.0453630i
$957$		0			0
$958$		−	64.7979i		−	2.09353i
$959$		−	20.5015i		−	0.662029i
$960$		0			0
$961$	−23.9623			−0.772977
$962$	29.7990			0.960760
$963$		0			0
$964$	16.6111			0.535006
$965$		−	16.5182i		−	0.531739i
$966$		0			0
$967$			1.62577i			0.0522813i	0.999658	+	0.0261407i	$0.00832178\pi$
$967$			1.62577i			0.0522813i	−0.999658	+	0.0261407i	$0.991678\pi$
$968$	65.9582			2.11998
$969$		0			0
$970$	58.2405			1.86999
$971$		−	47.1990i		−	1.51469i	−0.653016	−	0.757344i	$-0.726496\pi$
$971$		−	47.1990i		−	1.51469i	0.653016	−	0.757344i	$-0.273504\pi$
$972$		0			0
$973$	−4.38694			−0.140639
$974$	54.1430			1.73485
$975$		0			0
$976$			67.6678i			2.16599i
$977$	15.0769			0.482353			0.241177	−	0.970481i	$-0.422467\pi$
$977$	15.0769			0.482353			0.241177	+	0.970481i	$0.422467\pi$
$978$		0			0
$979$	16.3665			0.523076
$980$			36.4851i			1.16547i
$981$		0			0
$982$		−	17.3771i		−	0.554525i
$983$	−41.5648			−1.32571			−0.662855	−	0.748748i	$-0.730655\pi$
$983$	−41.5648			−1.32571			−0.662855	+	0.748748i	$0.730655\pi$
$984$		0			0
$985$	−12.1653			−0.387618
$986$	−8.41059			−0.267848
$987$		0			0
$988$		−	62.1437i		−	1.97706i
$989$			24.2330i			0.770566i
$990$		0			0
$991$		−	28.9530i		−	0.919723i	−0.887991	−	0.459862i	$-0.847899\pi$
$991$		−	28.9530i		−	0.919723i	0.887991	−	0.459862i	$-0.152101\pi$
$992$			65.0423i			2.06510i
$993$		0			0
$994$		−	4.26009i		−	0.135122i
$995$			14.9939i			0.475338i
$996$		0			0
$997$	30.1055			0.953451			0.476726	−	0.879052i	$-0.341824\pi$
$997$	30.1055			0.953451			0.476726	+	0.879052i	$0.341824\pi$
$998$	17.0578			0.539956
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.2.d.a.530.1	✓	20
3.2	odd	2	inner	531.2.d.a.530.20	yes	20
59.58	odd	2	inner	531.2.d.a.530.19	yes	20
177.176	even	2	inner	531.2.d.a.530.2	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.2.d.a.530.1	✓	20	1.1	even	1	trivial
531.2.d.a.530.2	yes	20	177.176	even	2	inner
531.2.d.a.530.19	yes	20	59.58	odd	2	inner
531.2.d.a.530.20	yes	20	3.2	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-2.59182 q^{2} +4.71754 q^{4} -1.33755i q^{5} -1.10356 q^{7} -7.04337 q^{8} +3.46670i q^{10} -1.27883 q^{11} +2.34848i q^{13} +2.86024 q^{14} +8.82007 q^{16} +6.38662i q^{17} -5.60912 q^{19} -6.30996i q^{20} +3.31451 q^{22} +2.40913 q^{23} +3.21095 q^{25} -6.08684i q^{26} -5.20610 q^{28} -0.508101i q^{29} -7.41366i q^{31} -8.77331 q^{32} -16.5530i q^{34} +1.47608i q^{35} +4.89565i q^{37} +14.5378 q^{38} +9.42089i q^{40} +4.17932i q^{41} +10.0589i q^{43} -6.03295 q^{44} -6.24402 q^{46} +4.73253 q^{47} -5.78215 q^{49} -8.32220 q^{50} +11.0790i q^{52} +10.2917i q^{53} +1.71051i q^{55} +7.77280 q^{56} +1.31691i q^{58} +(5.91308 + 4.90261i) q^{59} +7.67203i q^{61} +19.2149i q^{62} +5.09871 q^{64} +3.14122 q^{65} +3.56735i q^{67} +30.1291i q^{68} -3.82573i q^{70} -1.48942i q^{71} +5.22551i q^{73} -12.6886i q^{74} -26.4612 q^{76} +1.41127 q^{77} +13.7447 q^{79} -11.7973i q^{80} -10.8321i q^{82} -6.03295 q^{83} +8.54246 q^{85} -26.0708i q^{86} +9.00730 q^{88} -12.7980 q^{89} -2.59170i q^{91} +11.3651 q^{92} -12.2659 q^{94} +7.50251i q^{95} -16.8000i q^{97} +14.9863 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{4} - 8 q^{7} + 4 q^{16} - 8 q^{22} + 4 q^{25} + 8 q^{28} + 44 q^{49} + 36 q^{64} - 96 q^{76} - 24 q^{85} + 16 q^{88} - 112 q^{94}+O(q^{100}) \) 20 * q + 20 * q^4 - 8 * q^7 + 4 * q^16 - 8 * q^22 + 4 * q^25 + 8 * q^28 + 44 * q^49 + 36 * q^64 - 96 * q^76 - 24 * q^85 + 16 * q^88 - 112 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more