LMFDB - Embedded newform 531.2.d.a.530.5

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			530.5
Root			$1.45273 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	531.530
Dual form			531.2.d.a.530.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.45273 q^{2} +0.110435 q^{4} -2.26259i q^{5} -4.85531 q^{7} +2.74503 q^{8} +3.28694i q^{10} -2.57169 q^{11} +6.45604i q^{13} +7.05347 q^{14} -4.20867 q^{16} -3.42694i q^{17} +4.08935 q^{19} -0.249870i q^{20} +3.73598 q^{22} +7.50414 q^{23} -0.119321 q^{25} -9.37890i q^{26} -0.536196 q^{28} +5.85827i q^{29} -2.76745i q^{31} +0.624014 q^{32} +4.97842i q^{34} +10.9856i q^{35} +7.57065i q^{37} -5.94074 q^{38} -6.21089i q^{40} +0.574794i q^{41} -5.57355i q^{43} -0.284005 q^{44} -10.9015 q^{46} +3.35614 q^{47} +16.5740 q^{49} +0.173342 q^{50} +0.712973i q^{52} +11.6451i q^{53} +5.81869i q^{55} -13.3280 q^{56} -8.51051i q^{58} +(-7.67748 - 0.237302i) q^{59} +11.3254i q^{61} +4.02037i q^{62} +7.51082 q^{64} +14.6074 q^{65} +7.40114i q^{67} -0.378454i q^{68} -15.9591i q^{70} -12.1262i q^{71} +14.3250i q^{73} -10.9981i q^{74} +0.451608 q^{76} +12.4864 q^{77} +3.61237 q^{79} +9.52251i q^{80} -0.835023i q^{82} -0.284005 q^{83} -7.75376 q^{85} +8.09688i q^{86} -7.05938 q^{88} -7.67125 q^{89} -31.3460i q^{91} +0.828720 q^{92} -4.87558 q^{94} -9.25254i q^{95} +5.10048i q^{97} -24.0776 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.45273			−1.02724			−0.513619	−	0.858018i	$-0.671695\pi$
$2$	−1.45273			−1.02724			−0.513619	+	0.858018i	$0.671695\pi$
$3$		0			0
$3$		0			0
$4$	0.110435			0.0552175
$5$		−	2.26259i		−	1.01186i	−0.862574	−	0.505931i	$-0.831149\pi$
$5$		−	2.26259i		−	1.01186i	0.862574	−	0.505931i	$-0.168851\pi$
$6$		0			0
$7$	−4.85531			−1.83513			−0.917566	−	0.397582i	$-0.869849\pi$
$7$	−4.85531			−1.83513			−0.917566	+	0.397582i	$0.869849\pi$
$8$	2.74503			0.970516
$9$		0			0
$10$			3.28694i			1.03942i
$11$	−2.57169			−0.775394			−0.387697	−	0.921787i	$-0.626729\pi$
$11$	−2.57169			−0.775394			−0.387697	+	0.921787i	$0.626729\pi$
$12$		0			0
$13$			6.45604i			1.79058i	0.445482	+	0.895291i	$0.353032\pi$
$13$			6.45604i			1.79058i	−0.445482	+	0.895291i	$0.646968\pi$
$14$	7.05347			1.88512
$15$		0			0
$16$	−4.20867			−1.05217
$17$		−	3.42694i		−	0.831154i	−0.909558	−	0.415577i	$-0.863580\pi$
$17$		−	3.42694i		−	0.831154i	0.909558	−	0.415577i	$-0.136420\pi$
$18$		0			0
$19$	4.08935			0.938162			0.469081	−	0.883155i	$-0.344585\pi$
$19$	4.08935			0.938162			0.469081	+	0.883155i	$0.344585\pi$
$20$		−	0.249870i		−	0.0558725i
$21$		0			0
$22$	3.73598			0.796514
$23$	7.50414			1.56472			0.782360	−	0.622826i	$-0.214016\pi$
$23$	7.50414			1.56472			0.782360	+	0.622826i	$0.214016\pi$
$24$		0			0
$25$	−0.119321			−0.0238643
$26$		−	9.37890i		−	1.83935i
$27$		0			0
$28$	−0.536196			−0.101332
$29$			5.85827i			1.08785i	0.839133	+	0.543927i	$0.183063\pi$
$29$			5.85827i			1.08785i	−0.839133	+	0.543927i	$0.816937\pi$
$30$		0			0
$31$		−	2.76745i		−	0.497048i	−0.968626	−	0.248524i	$-0.920054\pi$
$31$		−	2.76745i		−	0.497048i	0.968626	−	0.248524i	$-0.0799456\pi$
$32$	0.624014			0.110311
$33$		0			0
$34$			4.97842i			0.853793i
$35$			10.9856i			1.85690i
$36$		0			0
$37$			7.57065i			1.24461i	0.782776	+	0.622304i	$0.213803\pi$
$37$			7.57065i			1.24461i	−0.782776	+	0.622304i	$0.786197\pi$
$38$	−5.94074			−0.963715
$39$		0			0
$40$		−	6.21089i		−	0.982028i
$41$			0.574794i			0.0897678i	0.998992	+	0.0448839i	$0.0142918\pi$
$41$			0.574794i			0.0897678i	−0.998992	+	0.0448839i	$0.985708\pi$
$42$		0			0
$43$		−	5.57355i		−	0.849958i	−0.905203	−	0.424979i	$-0.860281\pi$
$43$		−	5.57355i		−	0.849958i	0.905203	−	0.424979i	$-0.139719\pi$
$44$	−0.284005			−0.0428154
$45$		0			0
$46$	−10.9015			−1.60734
$47$	3.35614			0.489543			0.244772	−	0.969581i	$-0.421287\pi$
$47$	3.35614			0.489543			0.244772	+	0.969581i	$0.421287\pi$
$48$		0			0
$49$	16.5740			2.36771
$50$	0.173342			0.0245143
$51$		0			0
$52$			0.712973i			0.0988715i
$53$			11.6451i			1.59958i	0.600282	+	0.799789i	$0.295055\pi$
$53$			11.6451i			1.59958i	−0.600282	+	0.799789i	$0.704945\pi$
$54$		0			0
$55$			5.81869i			0.784592i
$56$	−13.3280			−1.78103
$57$		0			0
$58$		−	8.51051i		−	1.11748i
$59$	−7.67748	−	0.237302i	−0.999523	−	0.0308941i
$59$	−7.67748	−	0.237302i	−0.999523	−	0.0308941i
$60$		0			0
$61$			11.3254i			1.45007i	0.688711	+	0.725036i	$0.258177\pi$
$61$			11.3254i			1.45007i	−0.688711	+	0.725036i	$0.741823\pi$
$62$			4.02037i			0.510587i
$63$		0			0
$64$	7.51082			0.938853
$65$	14.6074			1.81182
$66$		0			0
$67$			7.40114i			0.904193i	0.891969	+	0.452096i	$0.149324\pi$
$67$			7.40114i			0.904193i	−0.891969	+	0.452096i	$0.850676\pi$
$68$		−	0.378454i		−	0.0458943i
$69$		0			0
$70$		−	15.9591i		−	1.90748i
$71$		−	12.1262i		−	1.43912i	−0.694433	−	0.719558i	$-0.744345\pi$
$71$		−	12.1262i		−	1.43912i	0.694433	−	0.719558i	$-0.255655\pi$
$72$		0			0
$73$			14.3250i			1.67661i	0.545198	+	0.838307i	$0.316454\pi$
$73$			14.3250i			1.67661i	−0.545198	+	0.838307i	$0.683546\pi$
$74$		−	10.9981i		−	1.27851i
$75$		0			0
$76$	0.451608			0.0518030
$77$	12.4864			1.42295
$78$		0			0
$79$	3.61237			0.406424			0.203212	−	0.979135i	$-0.434862\pi$
$79$	3.61237			0.406424			0.203212	+	0.979135i	$0.434862\pi$
$80$			9.52251i			1.06465i
$81$		0			0
$82$		−	0.835023i		−	0.0922129i
$83$	−0.284005			−0.0311736			−0.0155868	−	0.999879i	$-0.504962\pi$
$83$	−0.284005			−0.0311736			−0.0155868	+	0.999879i	$0.504962\pi$
$84$		0			0
$85$	−7.75376			−0.841013
$86$			8.09688i			0.873109i
$87$		0			0
$88$	−7.05938			−0.752533
$89$	−7.67125			−0.813150			−0.406575	−	0.913617i	$-0.633277\pi$
$89$	−7.67125			−0.813150			−0.406575	+	0.913617i	$0.633277\pi$
$90$		0			0
$91$		−	31.3460i		−	3.28596i
$92$	0.828720			0.0864000
$93$		0			0
$94$	−4.87558			−0.502877
$95$		−	9.25254i		−	0.949290i
$96$		0			0
$97$			5.10048i			0.517875i	0.965894	+	0.258937i	$0.0833723\pi$
$97$			5.10048i			0.517875i	−0.965894	+	0.258937i	$0.916628\pi$
$98$	−24.0776			−2.43220
$99$		0			0
$100$	−0.0131773			−0.00131773
$101$	−3.07213			−0.305689			−0.152844	−	0.988250i	$-0.548843\pi$
$101$	−3.07213			−0.305689			−0.152844	+	0.988250i	$0.548843\pi$
$102$		0			0
$103$		−	3.09347i		−	0.304808i	−0.988318	−	0.152404i	$-0.951298\pi$
$103$		−	3.09347i		−	0.304808i	0.988318	−	0.152404i	$-0.0487015\pi$
$104$			17.7220i			1.73779i
$105$		0			0
$106$		−	16.9172i		−	1.64315i
$107$			5.27987i			0.510424i	0.966885	+	0.255212i	$0.0821453\pi$
$107$			5.27987i			0.510424i	−0.966885	+	0.255212i	$0.917855\pi$
$108$		0			0
$109$		−	7.08037i		−	0.678176i	−0.940755	−	0.339088i	$-0.889882\pi$
$109$		−	7.08037i		−	0.678176i	0.940755	−	0.339088i	$-0.110118\pi$
$110$		−	8.45301i		−	0.805963i
$111$		0			0
$112$	20.4344			1.93087
$113$	7.40015			0.696148			0.348074	−	0.937467i	$-0.386836\pi$
$113$	7.40015			0.696148			0.348074	+	0.937467i	$0.386836\pi$
$114$		0			0
$115$		−	16.9788i		−	1.58328i
$116$			0.646959i			0.0600686i
$117$		0			0
$118$	11.1533	+	0.344737i	1.02675	+	0.0317356i
$119$			16.6388i			1.52528i
$120$		0			0
$121$	−4.38640			−0.398764
$122$		−	16.4528i		−	1.48957i
$123$		0			0
$124$		−	0.305623i		−	0.0274458i
$125$		−	11.0430i		−	0.987714i
$126$		0			0
$127$	−5.34019			−0.473865			−0.236933	−	0.971526i	$-0.576142\pi$
$127$	−5.34019			−0.473865			−0.236933	+	0.971526i	$0.576142\pi$
$128$	−12.1593			−1.07474
$129$		0			0
$130$	−21.2206			−1.86117
$131$	−1.90964			−0.166846			−0.0834230	−	0.996514i	$-0.526585\pi$
$131$	−1.90964			−0.166846			−0.0834230	+	0.996514i	$0.526585\pi$
$132$		0			0
$133$	−19.8551			−1.72165
$134$		−	10.7519i		−	0.928821i
$135$		0			0
$136$		−	9.40706i		−	0.806648i
$137$		−	8.62207i		−	0.736634i	−0.929700	−	0.368317i	$-0.879934\pi$
$137$		−	8.62207i		−	0.736634i	0.929700	−	0.368317i	$-0.120066\pi$
$138$		0			0
$139$	−7.90580			−0.670561			−0.335281	−	0.942118i	$-0.608831\pi$
$139$	−7.90580			−0.670561			−0.335281	+	0.942118i	$0.608831\pi$
$140$			1.21319i			0.102534i
$141$		0			0
$142$			17.6161i			1.47831i
$143$		−	16.6029i		−	1.38841i
$144$		0			0
$145$	13.2549			1.10076
$146$		−	20.8104i		−	1.72228i
$147$		0			0
$148$			0.836066i			0.0687242i
$149$	−4.29181			−0.351599			−0.175799	−	0.984426i	$-0.556251\pi$
$149$	−4.29181			−0.351599			−0.175799	+	0.984426i	$0.556251\pi$
$150$		0			0
$151$			18.8965i			1.53778i	0.639382	+	0.768889i	$0.279190\pi$
$151$			18.8965i			1.53778i	−0.639382	+	0.768889i	$0.720810\pi$
$152$	11.2254			0.910501
$153$		0			0
$154$	−18.1393			−1.46171
$155$	−6.26161			−0.502944
$156$		0			0
$157$		−	2.55047i		−	0.203549i	−0.994807	−	0.101775i	$-0.967548\pi$
$157$		−	2.55047i		−	0.203549i	0.994807	−	0.101775i	$-0.0324521\pi$
$158$	−5.24781			−0.417494
$159$		0			0
$160$		−	1.41189i		−	0.111620i
$161$	−36.4349			−2.87147
$162$		0			0
$163$	19.7606			1.54777			0.773885	−	0.633326i	$-0.218311\pi$
$163$	19.7606			1.54777			0.773885	+	0.633326i	$0.218311\pi$
$164$			0.0634775i			0.00495676i
$165$		0			0
$166$	0.412584			0.0320227
$167$			11.5091i			0.890598i	0.895382	+	0.445299i	$0.146903\pi$
$167$			11.5091i			0.890598i	−0.895382	+	0.445299i	$0.853097\pi$
$168$		0			0
$169$	−28.6804			−2.20618
$170$	11.2641			0.863920
$171$		0			0
$172$		−	0.615515i		−	0.0469326i
$173$	17.2417			1.31087			0.655433	−	0.755254i	$-0.272486\pi$
$173$	17.2417			1.31087			0.655433	+	0.755254i	$0.272486\pi$
$174$		0			0
$175$	0.579342			0.0437941
$176$	10.8234			0.815846
$177$		0			0
$178$	11.1443			0.835299
$179$	−10.7120			−0.800654			−0.400327	−	0.916372i	$-0.631103\pi$
$179$	−10.7120			−0.800654			−0.400327	+	0.916372i	$0.631103\pi$
$180$		0			0
$181$	11.4631			0.852044			0.426022	−	0.904713i	$-0.359915\pi$
$181$	11.4631			0.852044			0.426022	+	0.904713i	$0.359915\pi$
$182$			45.5374i			3.37546i
$183$		0			0
$184$	20.5991			1.51859
$185$	17.1293			1.25937
$186$		0			0
$187$			8.81302i			0.644472i
$188$	0.370635			0.0270314
$189$		0			0
$190$			13.4415i			0.975147i
$191$	13.5518			0.980576			0.490288	−	0.871560i	$-0.336892\pi$
$191$	13.5518			0.980576			0.490288	+	0.871560i	$0.336892\pi$
$192$		0			0
$193$	−1.83044			−0.131758			−0.0658790	−	0.997828i	$-0.520985\pi$
$193$	−1.83044			−0.131758			−0.0658790	+	0.997828i	$0.520985\pi$
$194$		−	7.40963i		−	0.531981i
$195$		0			0
$196$	1.83035			0.130739
$197$			11.3856i			0.811187i	0.914054	+	0.405594i	$0.132935\pi$
$197$			11.3856i			0.811187i	−0.914054	+	0.405594i	$0.867065\pi$
$198$		0			0
$199$	4.29374			0.304375			0.152187	−	0.988352i	$-0.451368\pi$
$199$	4.29374			0.304375			0.152187	+	0.988352i	$0.451368\pi$
$200$	−0.327541			−0.0231607
$201$		0			0
$202$	4.46299			0.314015
$203$		−	28.4437i		−	1.99636i
$204$		0			0
$205$	1.30052			0.0908326
$206$			4.49398i			0.313110i
$207$		0			0
$208$		−	27.1714i		−	1.88399i
$209$	−10.5166			−0.727445
$210$		0			0
$211$			7.86792i			0.541650i	0.962629	+	0.270825i	$0.0872965\pi$
$211$			7.86792i			0.541650i	−0.962629	+	0.270825i	$0.912703\pi$
$212$			1.28603i			0.0883247i
$213$		0			0
$214$		−	7.67025i		−	0.524327i
$215$	−12.6107			−0.860040
$216$		0			0
$217$			13.4368i			0.912150i
$218$			10.2859i			0.696648i
$219$		0			0
$220$			0.642588i			0.0433232i
$221$	22.1244			1.48825
$222$		0			0
$223$	3.43086			0.229748			0.114874	−	0.993380i	$-0.463354\pi$
$223$	3.43086			0.229748			0.114874	+	0.993380i	$0.463354\pi$
$224$	−3.02978			−0.202435
$225$		0			0
$226$	−10.7504			−0.715109
$227$	9.17998			0.609297			0.304648	−	0.952465i	$-0.401461\pi$
$227$	9.17998			0.609297			0.304648	+	0.952465i	$0.401461\pi$
$228$		0			0
$229$		−	7.03290i		−	0.464747i	−0.972627	−	0.232374i	$-0.925351\pi$
$229$		−	7.03290i		−	0.464747i	0.972627	−	0.232374i	$-0.0746492\pi$
$230$			24.6657i			1.62641i
$231$		0			0
$232$			16.0812i			1.05578i
$233$	−7.81506			−0.511982			−0.255991	−	0.966679i	$-0.582402\pi$
$233$	−7.81506			−0.511982			−0.255991	+	0.966679i	$0.582402\pi$
$234$		0			0
$235$		−	7.59357i		−	0.495350i
$236$	−0.847863	−	0.0262065i	−0.0551912	−	0.00170590i
$237$		0			0
$238$		−	24.1718i		−	1.56682i
$239$			4.92949i			0.318862i	0.987209	+	0.159431i	$0.0509660\pi$
$239$			4.92949i			0.318862i	−0.987209	+	0.159431i	$0.949034\pi$
$240$		0			0
$241$	−10.1051			−0.650926			−0.325463	−	0.945555i	$-0.605520\pi$
$241$	−10.1051			−0.650926			−0.325463	+	0.945555i	$0.605520\pi$
$242$	6.37227			0.409625
$243$		0			0
$244$			1.25072i			0.0800694i
$245$		−	37.5002i		−	2.39580i
$246$		0			0
$247$			26.4010i			1.67986i
$248$		−	7.59674i		−	0.482394i
$249$		0			0
$250$			16.0425i			1.01462i
$251$			10.2721i			0.648370i	0.945994	+	0.324185i	$0.105090\pi$
$251$			10.2721i			0.648370i	−0.945994	+	0.324185i	$0.894910\pi$
$252$		0			0
$253$	−19.2983			−1.21328
$254$	7.75788			0.486772
$255$		0			0
$256$	2.64251			0.165157
$257$			20.3096i			1.26688i	0.773794	+	0.633438i	$0.218357\pi$
$257$			20.3096i			1.26688i	−0.773794	+	0.633438i	$0.781643\pi$
$258$		0			0
$259$		−	36.7578i		−	2.28402i
$260$	1.61317			0.100044
$261$		0			0
$262$	2.77420			0.171390
$263$			3.30806i			0.203984i	0.994785	+	0.101992i	$0.0325216\pi$
$263$			3.30806i			0.203984i	−0.994785	+	0.101992i	$0.967478\pi$
$264$		0			0
$265$	26.3481			1.61855
$266$	28.8441			1.76855
$267$		0			0
$268$			0.817345i			0.0499273i
$269$	8.21596			0.500936			0.250468	−	0.968125i	$-0.419416\pi$
$269$	8.21596			0.500936			0.250468	+	0.968125i	$0.419416\pi$
$270$		0			0
$271$	11.6941			0.710367			0.355184	−	0.934797i	$-0.384418\pi$
$271$	11.6941			0.710367			0.355184	+	0.934797i	$0.384418\pi$
$272$			14.4229i			0.874514i
$273$		0			0
$274$			12.5256i			0.756698i
$275$	0.306858			0.0185042
$276$		0			0
$277$	21.3391			1.28214			0.641072	−	0.767481i	$-0.278490\pi$
$277$	21.3391			1.28214			0.641072	+	0.767481i	$0.278490\pi$
$278$	11.4850			0.688826
$279$		0			0
$280$			30.1558i			1.80215i
$281$		−	26.5150i		−	1.58175i	−0.611978	−	0.790875i	$-0.709626\pi$
$281$		−	26.5150i		−	1.58175i	0.611978	−	0.790875i	$-0.290374\pi$
$282$		0			0
$283$			29.0605i			1.72747i	0.503948	+	0.863734i	$0.331880\pi$
$283$			29.0605i			1.72747i	−0.503948	+	0.863734i	$0.668120\pi$
$284$		−	1.33916i		−	0.0794644i
$285$		0			0
$286$			24.1196i			1.42622i
$287$		−	2.79080i		−	0.164736i
$288$		0			0
$289$	5.25611			0.309183
$290$	−19.2558			−1.13074
$291$		0			0
$292$			1.58198i			0.0925785i
$293$			12.3707i			0.722705i	0.932429	+	0.361353i	$0.117685\pi$
$293$			12.3707i			0.722705i	−0.932429	+	0.361353i	$0.882315\pi$
$294$		0			0
$295$	−0.536918	+	17.3710i	−0.0312606	+	1.01138i
$296$			20.7817i			1.20791i
$297$		0			0
$298$	6.23486			0.361176
$299$			48.4470i			2.80176i
$300$		0			0
$301$			27.0613i			1.55979i
$302$		−	27.4516i		−	1.57966i
$303$		0			0
$304$	−17.2108			−0.987104
$305$	25.6248			1.46727
$306$		0			0
$307$	−12.5057			−0.713739			−0.356870	−	0.934154i	$-0.616156\pi$
$307$	−12.5057			−0.713739			−0.356870	+	0.934154i	$0.616156\pi$
$308$	1.37893			0.0785719
$309$		0			0
$310$	9.09645			0.516643
$311$		−	16.8176i		−	0.953641i	−0.879001	−	0.476820i	$-0.841789\pi$
$311$		−	16.8176i		−	0.953641i	0.879001	−	0.476820i	$-0.158211\pi$
$312$		0			0
$313$			5.41180i			0.305893i	0.988234	+	0.152947i	$0.0488763\pi$
$313$			5.41180i			0.305893i	−0.988234	+	0.152947i	$0.951124\pi$
$314$			3.70515i			0.209094i
$315$		0			0
$316$	0.398933			0.0224417
$317$			15.9111i			0.893658i	0.894619	+	0.446829i	$0.147447\pi$
$317$			15.9111i			0.893658i	−0.894619	+	0.446829i	$0.852553\pi$
$318$		0			0
$319$		−	15.0657i		−	0.843516i
$320$		−	16.9939i		−	0.949989i
$321$		0			0
$322$	52.9302			2.94968
$323$		−	14.0139i		−	0.779757i
$324$		0			0
$325$		−	0.770343i		−	0.0427310i
$326$	−28.7069			−1.58993
$327$		0			0
$328$			1.57783i			0.0871211i
$329$	−16.2951			−0.898377
$330$		0			0
$331$	−24.0924			−1.32424			−0.662119	−	0.749398i	$-0.730343\pi$
$331$	−24.0924			−1.32424			−0.662119	+	0.749398i	$0.730343\pi$
$332$	−0.0313641			−0.00172133
$333$		0			0
$334$		−	16.7196i		−	0.914856i
$335$	16.7458			0.914918
$336$		0			0
$337$		−	11.8126i		−	0.643473i	−0.946829	−	0.321737i	$-0.895733\pi$
$337$		−	11.8126i		−	0.643473i	0.946829	−	0.321737i	$-0.104267\pi$
$338$	41.6650			2.26628
$339$		0			0
$340$	−0.856287			−0.0464387
$341$			7.11703i			0.385409i
$342$		0			0
$343$	−46.4846			−2.50994
$344$		−	15.2996i		−	0.824898i
$345$		0			0
$346$	−25.0477			−1.34657
$347$	−17.2964			−0.928520			−0.464260	−	0.885699i	$-0.653680\pi$
$347$	−17.2964			−0.928520			−0.464260	+	0.885699i	$0.653680\pi$
$348$		0			0
$349$			13.3289i			0.713478i	0.934204	+	0.356739i	$0.116111\pi$
$349$			13.3289i			0.713478i	−0.934204	+	0.356739i	$0.883889\pi$
$350$	−0.841629			−0.0449870
$351$		0			0
$352$	−1.60477			−0.0855346
$353$	−33.2513			−1.76979			−0.884895	−	0.465790i	$-0.845770\pi$
$353$	−33.2513			−1.76979			−0.884895	+	0.465790i	$0.845770\pi$
$354$		0			0
$355$	−27.4366			−1.45619
$356$	−0.847175			−0.0449002
$357$		0			0
$358$	15.5617			0.822462
$359$			10.0929i			0.532685i	0.963878	+	0.266343i	$0.0858153\pi$
$359$			10.0929i			0.532685i	−0.963878	+	0.266343i	$0.914185\pi$
$360$		0			0
$361$	−2.27719			−0.119852
$362$	−16.6528			−0.875252
$363$		0			0
$364$		−	3.46170i		−	0.181442i
$365$	32.4116			1.69650
$366$		0			0
$367$		−	12.4787i		−	0.651382i	−0.945476	−	0.325691i	$-0.894403\pi$
$367$		−	12.4787i		−	0.651382i	0.945476	−	0.325691i	$-0.105597\pi$
$368$	−31.5825			−1.64635
$369$		0			0
$370$	−24.8843			−1.29367
$371$		−	56.5405i		−	2.93544i
$372$		0			0
$373$	21.9482			1.13644			0.568218	−	0.822878i	$-0.307633\pi$
$373$	21.9482			1.13644			0.568218	+	0.822878i	$0.307633\pi$
$374$		−	12.8030i		−	0.662026i
$375$		0			0
$376$	9.21272			0.475110
$377$	−37.8212			−1.94789
$378$		0			0
$379$	11.8951			0.611012			0.305506	−	0.952190i	$-0.401174\pi$
$379$	11.8951			0.611012			0.305506	+	0.952190i	$0.401174\pi$
$380$		−	1.02180i		−	0.0524175i
$381$		0			0
$382$	−19.6872			−1.00728
$383$			29.6662i			1.51587i	0.652329	+	0.757936i	$0.273792\pi$
$383$			29.6662i			1.51587i	−0.652329	+	0.757936i	$0.726208\pi$
$384$		0			0
$385$		−	28.2515i		−	1.43983i
$386$	2.65914			0.135347
$387$		0			0
$388$			0.563271i			0.0285958i
$389$		−	4.61602i		−	0.234041i	−0.993129	−	0.117021i	$-0.962666\pi$
$389$		−	4.61602i		−	0.234041i	0.993129	−	0.117021i	$-0.0373344\pi$
$390$		0			0
$391$		−	25.7162i		−	1.30052i
$392$	45.4962			2.29790
$393$		0			0
$394$		−	16.5402i		−	0.833282i
$395$		−	8.17332i		−	0.411244i
$396$		0			0
$397$		−	15.8459i		−	0.795284i	−0.917541	−	0.397642i	$-0.869829\pi$
$397$		−	15.8459i		−	0.795284i	0.917541	−	0.397642i	$-0.130171\pi$
$398$	−6.23766			−0.312665
$399$		0			0
$400$	0.502185			0.0251092
$401$	27.6101			1.37878			0.689391	−	0.724389i	$-0.257878\pi$
$401$	27.6101			1.37878			0.689391	+	0.724389i	$0.257878\pi$
$402$		0			0
$403$	17.8667			0.890006
$404$	−0.339271			−0.0168794
$405$		0			0
$406$			41.3211i			2.05073i
$407$		−	19.4694i		−	0.965062i
$408$		0			0
$409$		−	20.2371i		−	1.00066i	−0.865835	−	0.500331i	$-0.833212\pi$
$409$		−	20.2371i		−	1.00066i	0.865835	−	0.500331i	$-0.166788\pi$
$410$	−1.88932			−0.0933067
$411$		0			0
$412$		−	0.341627i		−	0.0168308i
$413$	37.2765	+	1.15218i	1.83426	+	0.0566948i
$414$		0			0
$415$			0.642588i			0.0315434i
$416$			4.02865i			0.197521i
$417$		0			0
$418$	15.2778			0.747260
$419$	−19.2363			−0.939755			−0.469878	−	0.882732i	$-0.655702\pi$
$419$	−19.2363			−0.939755			−0.469878	+	0.882732i	$0.655702\pi$
$420$		0			0
$421$			2.56453i			0.124988i	0.998045	+	0.0624938i	$0.0199054\pi$
$421$			2.56453i			0.124988i	−0.998045	+	0.0624938i	$0.980095\pi$
$422$		−	11.4300i		−	0.556404i
$423$		0			0
$424$			31.9662i			1.55242i
$425$			0.408907i			0.0198349i
$426$		0			0
$427$		−	54.9884i		−	2.66108i
$428$			0.583083i			0.0281844i
$429$		0			0
$430$	18.3199			0.883466
$431$	−18.7842			−0.904801			−0.452401	−	0.891815i	$-0.649432\pi$
$431$	−18.7842			−0.904801			−0.452401	+	0.891815i	$0.649432\pi$
$432$		0			0
$433$	14.7190			0.707350			0.353675	−	0.935368i	$-0.384932\pi$
$433$	14.7190			0.707350			0.353675	+	0.935368i	$0.384932\pi$
$434$		−	19.5201i		−	0.936995i
$435$		0			0
$436$		−	0.781921i		−	0.0374472i
$437$	30.6871			1.46796
$438$		0			0
$439$	0.352951			0.0168454			0.00842272	−	0.999965i	$-0.497319\pi$
$439$	0.352951			0.0168454			0.00842272	+	0.999965i	$0.497319\pi$
$440$			15.9725i			0.761459i
$441$		0			0
$442$	−32.1409			−1.52879
$443$	−31.8432			−1.51292			−0.756458	−	0.654043i	$-0.773072\pi$
$443$	−31.8432			−1.51292			−0.756458	+	0.654043i	$0.773072\pi$
$444$		0			0
$445$			17.3569i			0.822796i
$446$	−4.98413			−0.236005
$447$		0			0
$448$	−36.4673			−1.72292
$449$		−	24.9989i		−	1.17977i	−0.807487	−	0.589885i	$-0.799173\pi$
$449$		−	24.9989i		−	1.17977i	0.807487	−	0.589885i	$-0.200827\pi$
$450$		0			0
$451$		−	1.47819i		−	0.0696054i
$452$	0.817236			0.0384396
$453$		0			0
$454$	−13.3361			−0.625893
$455$	−70.9233			−3.32493
$456$		0			0
$457$			3.58018i			0.167474i	0.996488	+	0.0837368i	$0.0266855\pi$
$457$			3.58018i			0.167474i	−0.996488	+	0.0837368i	$0.973314\pi$
$458$			10.2169i			0.477406i
$459$		0			0
$460$		−	1.87506i		−	0.0874249i
$461$		−	18.7859i		−	0.874945i	−0.899232	−	0.437473i	$-0.855874\pi$
$461$		−	18.7859i		−	0.874945i	0.899232	−	0.437473i	$-0.144126\pi$
$462$		0			0
$463$			1.10207i			0.0512173i	0.999672	+	0.0256087i	$0.00815238\pi$
$463$			1.10207i			0.0512173i	−0.999672	+	0.0256087i	$0.991848\pi$
$464$		−	24.6556i		−	1.14461i
$465$		0			0
$466$	11.3532			0.525927
$467$	14.6881			0.679683			0.339841	−	0.940483i	$-0.389627\pi$
$467$	14.6881			0.679683			0.339841	+	0.940483i	$0.389627\pi$
$468$		0			0
$469$		−	35.9348i		−	1.65931i
$470$			11.0314i			0.508842i
$471$		0			0
$472$	−21.0749	−	0.651403i	−0.970053	−	0.0299833i
$473$			14.3334i			0.659053i
$474$		0			0
$475$	−0.487947			−0.0223886
$476$			1.83751i			0.0842221i
$477$		0			0
$478$		−	7.16124i		−	0.327547i
$479$			16.8642i			0.770546i	0.922803	+	0.385273i	$0.125893\pi$
$479$			16.8642i			0.770546i	−0.922803	+	0.385273i	$0.874107\pi$
$480$		0			0
$481$	−48.8764			−2.22857
$482$	14.6800			0.668655
$483$		0			0
$484$	−0.484412			−0.0220187
$485$	11.5403			0.524018
$486$		0			0
$487$	33.9330			1.53765			0.768826	−	0.639458i	$-0.220841\pi$
$487$	33.9330			1.53765			0.768826	+	0.639458i	$0.220841\pi$
$488$			31.0887i			1.40732i
$489$		0			0
$490$			54.4778i			2.46105i
$491$			13.9181i			0.628116i	0.949404	+	0.314058i	$0.101689\pi$
$491$			13.9181i			0.628116i	−0.949404	+	0.314058i	$0.898311\pi$
$492$		0			0
$493$	20.0759			0.904174
$494$		−	38.3536i		−	1.72561i
$495$		0			0
$496$			11.6473i			0.522979i
$497$			58.8764i			2.64097i
$498$		0			0
$499$	−11.7860			−0.527613			−0.263806	−	0.964576i	$-0.584978\pi$
$499$	−11.7860			−0.527613			−0.263806	+	0.964576i	$0.584978\pi$
$500$		−	1.21953i		−	0.0545392i
$501$		0			0
$502$		−	14.9227i		−	0.666031i
$503$	4.02516			0.179473			0.0897366	−	0.995966i	$-0.471397\pi$
$503$	4.02516			0.179473			0.0897366	+	0.995966i	$0.471397\pi$
$504$		0			0
$505$			6.95098i			0.309315i
$506$	28.0353			1.24632
$507$		0			0
$508$	−0.589745			−0.0261657
$509$	−30.1921			−1.33824			−0.669121	−	0.743153i	$-0.733329\pi$
$509$	−30.1921			−1.33824			−0.669121	+	0.743153i	$0.733329\pi$
$510$		0			0
$511$		−	69.5523i		−	3.07681i
$512$	20.4796			0.905081
$513$		0			0
$514$		−	29.5044i		−	1.30138i
$515$	−6.99925			−0.308424
$516$		0			0
$517$	−8.63096			−0.379589
$518$			53.3993i			2.34623i
$519$		0			0
$520$	40.0977			1.75840
$521$		−	18.7607i		−	0.821919i	−0.911654	−	0.410960i	$-0.865194\pi$
$521$		−	18.7607i		−	0.821919i	0.911654	−	0.410960i	$-0.134806\pi$
$522$		0			0
$523$	22.1115			0.966867			0.483434	−	0.875381i	$-0.339389\pi$
$523$	22.1115			0.966867			0.483434	+	0.875381i	$0.339389\pi$
$524$	−0.210891			−0.00921282
$525$		0			0
$526$		−	4.80573i		−	0.209540i
$527$	−9.48387			−0.413124
$528$		0			0
$529$	33.3121			1.44835
$530$	−38.2768			−1.66264
$531$		0			0
$532$	−2.19269			−0.0950654
$533$	−3.71089			−0.160737
$534$		0			0
$535$	11.9462			0.516479
$536$			20.3164i			0.877534i
$537$		0			0
$538$	−11.9356			−0.514580
$539$	−42.6232			−1.83591
$540$		0			0
$541$			31.0576i			1.33527i	0.744488	+	0.667636i	$0.232694\pi$
$541$			31.0576i			1.33527i	−0.744488	+	0.667636i	$0.767306\pi$
$542$	−16.9884			−0.729716
$543$		0			0
$544$		−	2.13845i		−	0.0916855i
$545$	−16.0200			−0.686220
$546$		0			0
$547$	19.9270			0.852016			0.426008	−	0.904719i	$-0.359920\pi$
$547$	19.9270			0.852016			0.426008	+	0.904719i	$0.359920\pi$
$548$		−	0.952180i		−	0.0406751i
$549$		0			0
$550$	−0.445783			−0.0190082
$551$			23.9565i			1.02058i
$552$		0			0
$553$	−17.5392			−0.745841
$554$	−31.0001			−1.31707
$555$		0			0
$556$	−0.873078			−0.0370268
$557$		−	3.35263i		−	0.142056i	−0.997474	−	0.0710278i	$-0.977372\pi$
$557$		−	3.35263i		−	0.142056i	0.997474	−	0.0710278i	$-0.0226279\pi$
$558$		0			0
$559$	35.9830			1.52192
$560$		−	46.2347i		−	1.95377i
$561$		0			0
$562$			38.5192i			1.62483i
$563$	−43.0536			−1.81449			−0.907245	−	0.420602i	$-0.861819\pi$
$563$	−43.0536			−1.81449			−0.907245	+	0.420602i	$0.861819\pi$
$564$		0			0
$565$		−	16.7435i		−	0.704405i
$566$		−	42.2172i		−	1.77452i
$567$		0			0
$568$		−	33.2868i		−	1.39669i
$569$	40.9929			1.71851			0.859256	−	0.511546i	$-0.170927\pi$
$569$	40.9929			1.71851			0.859256	+	0.511546i	$0.170927\pi$
$570$		0			0
$571$		−	4.00831i		−	0.167743i	−0.996477	−	0.0838714i	$-0.973272\pi$
$571$		−	4.00831i		−	0.167743i	0.996477	−	0.0838714i	$-0.0267285\pi$
$572$		−	1.83355i		−	0.0766644i
$573$		0			0
$574$			4.05429i			0.169223i
$575$	−0.895404			−0.0373409
$576$		0			0
$577$	−17.9896			−0.748918			−0.374459	−	0.927243i	$-0.622172\pi$
$577$	−17.9896			−0.748918			−0.374459	+	0.927243i	$0.622172\pi$
$578$	−7.63573			−0.317604
$579$		0			0
$580$	1.46380			0.0607812
$581$	1.37893			0.0572077
$582$		0			0
$583$		−	29.9476i		−	1.24030i
$584$			39.3226i			1.62718i
$585$		0			0
$586$		−	17.9714i		−	0.742390i
$587$	14.0726			0.580838			0.290419	−	0.956900i	$-0.406205\pi$
$587$	14.0726			0.580838			0.290419	+	0.956900i	$0.406205\pi$
$588$		0			0
$589$		−	11.3171i		−	0.466312i
$590$	0.779999	−	25.2354i	0.0321121	−	1.03893i
$591$		0			0
$592$		−	31.8624i		−	1.30954i
$593$			29.9437i			1.22964i	0.788668	+	0.614820i	$0.210771\pi$
$593$			29.9437i			1.22964i	−0.788668	+	0.614820i	$0.789229\pi$
$594$		0			0
$595$	37.6469			1.54337
$596$	−0.473967			−0.0194144
$597$		0			0
$598$		−	70.3806i		−	2.87807i
$599$			1.71365i			0.0700180i	0.999387	+	0.0350090i	$0.0111460\pi$
$599$			1.71365i			0.0700180i	−0.999387	+	0.0350090i	$0.988854\pi$
$600$		0			0
$601$			35.2521i			1.43796i	0.695030	+	0.718981i	$0.255391\pi$
$601$			35.2521i			1.43796i	−0.695030	+	0.718981i	$0.744609\pi$
$602$		−	39.3128i		−	1.60227i
$603$		0			0
$604$			2.08684i			0.0849124i
$605$			9.92463i			0.403494i
$606$		0			0
$607$	0.194178			0.00788142			0.00394071	−	0.999992i	$-0.498746\pi$
$607$	0.194178			0.00788142			0.00394071	+	0.999992i	$0.498746\pi$
$608$	2.55181			0.103490
$609$		0			0
$610$	−37.2260			−1.50724
$611$			21.6674i			0.876567i
$612$		0			0
$613$		−	20.1353i		−	0.813259i	−0.913593	−	0.406629i	$-0.866704\pi$
$613$		−	20.1353i		−	0.813259i	0.913593	−	0.406629i	$-0.133296\pi$
$614$	18.1675			0.733180
$615$		0			0
$616$	34.2755			1.38100
$617$			2.68078i			0.107924i	0.998543	+	0.0539620i	$0.0171850\pi$
$617$			2.68078i			0.107924i	−0.998543	+	0.0539620i	$0.982815\pi$
$618$		0			0
$619$	−34.3618			−1.38112			−0.690559	−	0.723277i	$-0.742635\pi$
$619$	−34.3618			−1.38112			−0.690559	+	0.723277i	$0.742635\pi$
$620$	−0.691501			−0.0277713
$621$		0			0
$622$			24.4316i			0.979616i
$623$	37.2462			1.49224
$624$		0			0
$625$	−25.5824			−1.02329
$626$		−	7.86191i		−	0.314225i
$627$		0			0
$628$		−	0.281661i		−	0.0112395i
$629$	25.9441			1.03446
$630$		0			0
$631$	−37.7495			−1.50278			−0.751391	−	0.659857i	$-0.770617\pi$
$631$	−37.7495			−1.50278			−0.751391	+	0.659857i	$0.770617\pi$
$632$	9.91608			0.394441
$633$		0			0
$634$		−	23.1146i		−	0.918000i
$635$			12.0827i			0.479486i
$636$		0			0
$637$			107.002i			4.23958i
$638$			21.8864i			0.866492i
$639$		0			0
$640$			27.5114i			1.08748i
$641$			28.7163i			1.13423i	0.823640	+	0.567113i	$0.191940\pi$
$641$			28.7163i			1.13423i	−0.823640	+	0.567113i	$0.808060\pi$
$642$		0			0
$643$	4.48994			0.177066			0.0885330	−	0.996073i	$-0.471782\pi$
$643$	4.48994			0.177066			0.0885330	+	0.996073i	$0.471782\pi$
$644$	−4.02369			−0.158556
$645$		0			0
$646$			20.3585i			0.800996i
$647$			22.0680i			0.867585i	0.901013	+	0.433792i	$0.142825\pi$
$647$			22.0680i			0.867585i	−0.901013	+	0.433792i	$0.857175\pi$
$648$		0			0
$649$	19.7441	+	0.610269i	0.775024	+	0.0239551i
$650$			1.11910i			0.0438949i
$651$		0			0
$652$	2.18226			0.0854640
$653$		−	19.2580i		−	0.753623i	−0.926290	−	0.376812i	$-0.877020\pi$
$653$		−	19.2580i		−	0.753623i	0.926290	−	0.376812i	$-0.122980\pi$
$654$		0			0
$655$			4.32073i			0.168825i
$656$		−	2.41912i		−	0.0944508i
$657$		0			0
$658$	23.6724			0.922847
$659$	34.0407			1.32604			0.663018	−	0.748603i	$-0.269275\pi$
$659$	34.0407			1.32604			0.663018	+	0.748603i	$0.269275\pi$
$660$		0			0
$661$	3.89624			0.151546			0.0757731	−	0.997125i	$-0.475858\pi$
$661$	3.89624			0.151546			0.0757731	+	0.997125i	$0.475858\pi$
$662$	34.9999			1.36031
$663$		0			0
$664$	−0.779604			−0.0302545
$665$			44.9239i			1.74207i
$666$		0			0
$667$			43.9613i			1.70219i
$668$			1.27100i			0.0491766i
$669$		0			0
$670$	−24.3271			−0.939838
$671$		−	29.1255i		−	1.12438i
$672$		0			0
$673$			9.67617i			0.372989i	0.982456	+	0.186494i	$0.0597126\pi$
$673$			9.67617i			0.372989i	−0.982456	+	0.186494i	$0.940287\pi$
$674$			17.1606i			0.661000i
$675$		0			0
$676$	−3.16732			−0.121820
$677$		−	16.5422i		−	0.635770i	−0.948129	−	0.317885i	$-0.897027\pi$
$677$		−	16.5422i		−	0.635770i	0.948129	−	0.317885i	$-0.102973\pi$
$678$		0			0
$679$		−	24.7644i		−	0.950369i
$680$	−21.2843			−0.816217
$681$		0			0
$682$		−	10.3391i		−	0.395906i
$683$	25.7358			0.984752			0.492376	−	0.870383i	$-0.336129\pi$
$683$	25.7358			0.984752			0.492376	+	0.870383i	$0.336129\pi$
$684$		0			0
$685$	−19.5082			−0.745371
$686$	67.5298			2.57830
$687$		0			0
$688$			23.4572i			0.894299i
$689$	−75.1812			−2.86417
$690$		0			0
$691$		−	28.3944i		−	1.08017i	−0.841610	−	0.540086i	$-0.818392\pi$
$691$		−	28.3944i		−	1.08017i	0.841610	−	0.540086i	$-0.181608\pi$
$692$	1.90409			0.0723828
$693$		0			0
$694$	25.1271			0.953810
$695$			17.8876i			0.678516i
$696$		0			0
$697$	1.96978			0.0746109
$698$		−	19.3633i		−	0.732911i
$699$		0			0
$700$	0.0639797			0.00241820
$701$	−4.87339			−0.184065			−0.0920326	−	0.995756i	$-0.529336\pi$
$701$	−4.87339			−0.184065			−0.0920326	+	0.995756i	$0.529336\pi$
$702$		0			0
$703$			30.9591i			1.16764i
$704$	−19.3155			−0.727981
$705$		0			0
$706$	48.3053			1.81800
$707$	14.9161			0.560979
$708$		0			0
$709$	26.0675			0.978984			0.489492	−	0.872008i	$-0.337182\pi$
$709$	26.0675			0.978984			0.489492	+	0.872008i	$0.337182\pi$
$710$	39.8581			1.49585
$711$		0			0
$712$	−21.0578			−0.789176
$713$		−	20.7673i		−	0.777742i
$714$		0			0
$715$	−37.5657			−1.40488
$716$	−1.18298			−0.0442101
$717$		0			0
$718$		−	14.6624i		−	0.547194i
$719$	12.9351			0.482400			0.241200	−	0.970475i	$-0.422459\pi$
$719$	12.9351			0.482400			0.241200	+	0.970475i	$0.422459\pi$
$720$		0			0
$721$			15.0197i			0.559364i
$722$	3.30816			0.123117
$723$		0			0
$724$	1.26593			0.0470478
$725$		−	0.699017i		−	0.0259609i
$726$		0			0
$727$	40.1070			1.48749			0.743744	−	0.668465i	$-0.233048\pi$
$727$	40.1070			1.48749			0.743744	+	0.668465i	$0.233048\pi$
$728$		−	86.0459i		−	3.18907i
$729$		0			0
$730$	−47.0855			−1.74271
$731$	−19.1002			−0.706446
$732$		0			0
$733$	10.8132			0.399395			0.199698	−	0.979858i	$-0.436004\pi$
$733$	10.8132			0.399395			0.199698	+	0.979858i	$0.436004\pi$
$734$			18.1282i			0.669125i
$735$		0			0
$736$	4.68268			0.172606
$737$		−	19.0334i		−	0.701106i
$738$		0			0
$739$		−	13.3226i		−	0.490079i	−0.969513	−	0.245039i	$-0.921199\pi$
$739$		−	13.3226i		−	0.490079i	0.969513	−	0.245039i	$-0.0788009\pi$
$740$	1.89168			0.0695394
$741$		0			0
$742$			82.1383i			3.01539i
$743$		−	13.0441i		−	0.478543i	−0.970953	−	0.239271i	$-0.923091\pi$
$743$		−	13.0441i		−	0.478543i	0.970953	−	0.239271i	$-0.0769085\pi$
$744$		0			0
$745$			9.71062i			0.355770i
$746$	−31.8849			−1.16739
$747$		0			0
$748$			0.973267i			0.0355862i
$749$		−	25.6354i		−	0.936697i
$750$		0			0
$751$		−	22.9341i		−	0.836876i	−0.908245	−	0.418438i	$-0.862578\pi$
$751$		−	22.9341i		−	0.836876i	0.908245	−	0.418438i	$-0.137422\pi$
$752$	−14.1249			−0.515082
$753$		0			0
$754$	54.9442			2.00095
$755$	42.7552			1.55602
$756$		0			0
$757$	−0.702841			−0.0255452			−0.0127726	−	0.999918i	$-0.504066\pi$
$757$	−0.702841			−0.0255452			−0.0127726	+	0.999918i	$0.504066\pi$
$758$	−17.2805			−0.627655
$759$		0			0
$760$		−	25.3985i		−	0.921302i
$761$		−	24.2259i		−	0.878189i	−0.898441	−	0.439095i	$-0.855299\pi$
$761$		−	24.2259i		−	0.878189i	0.898441	−	0.439095i	$-0.144701\pi$
$762$		0			0
$763$			34.3773i			1.24454i
$764$	1.49660			0.0541450
$765$		0			0
$766$		−	43.0971i		−	1.55716i
$767$	1.53203	−	49.5661i	0.0553185	−	1.78973i
$768$		0			0
$769$		−	21.1758i		−	0.763618i	−0.924241	−	0.381809i	$-0.875301\pi$
$769$		−	21.1758i		−	0.763618i	0.924241	−	0.381809i	$-0.124699\pi$
$770$			41.0419i			1.47905i
$771$		0			0
$772$	−0.202145			−0.00727535
$773$	−14.4343			−0.519167			−0.259583	−	0.965721i	$-0.583585\pi$
$773$	−14.4343			−0.519167			−0.259583	+	0.965721i	$0.583585\pi$
$774$		0			0
$775$			0.330216i			0.0118617i
$776$			14.0010i			0.502606i
$777$		0			0
$778$			6.70584i			0.240416i
$779$			2.35054i			0.0842167i
$780$		0			0
$781$			31.1849i			1.11588i
$782$			37.3588i			1.33595i
$783$		0			0
$784$	−69.7545			−2.49123
$785$	−5.77067			−0.205964
$786$		0			0
$787$	−12.7783			−0.455497			−0.227748	−	0.973720i	$-0.573136\pi$
$787$	−12.7783			−0.455497			−0.227748	+	0.973720i	$0.573136\pi$
$788$			1.25736i			0.0447918i
$789$		0			0
$790$			11.8737i			0.422446i
$791$	−35.9300			−1.27752
$792$		0			0
$793$	−73.1173			−2.59647
$794$			23.0199i			0.816946i
$795$		0			0
$796$	0.474179			0.0168068
$797$	10.6941			0.378803			0.189401	−	0.981900i	$-0.439345\pi$
$797$	10.6941			0.378803			0.189401	+	0.981900i	$0.439345\pi$
$798$		0			0
$799$		−	11.5013i		−	0.406886i
$800$	−0.0744582			−0.00263249
$801$		0			0
$802$	−40.1101			−1.41634
$803$		−	36.8395i		−	1.30004i
$804$		0			0
$805$			82.4373i			2.90553i
$806$	−25.9556			−0.914248
$807$		0			0
$808$	−8.43311			−0.296676
$809$	44.3215			1.55826			0.779131	−	0.626862i	$-0.215661\pi$
$809$	44.3215			1.55826			0.779131	+	0.626862i	$0.215661\pi$
$810$		0			0
$811$		−	21.9196i		−	0.769702i	−0.922979	−	0.384851i	$-0.874253\pi$
$811$		−	21.9196i		−	0.769702i	0.922979	−	0.384851i	$-0.125747\pi$
$812$		−	3.14118i		−	0.110234i
$813$		0			0
$814$			28.2838i			0.991348i
$815$		−	44.7102i		−	1.56613i
$816$		0			0
$817$		−	22.7922i		−	0.797398i
$818$			29.3991i			1.02792i
$819$		0			0
$820$	0.143624			0.00501555
$821$	3.38196			0.118031			0.0590157	−	0.998257i	$-0.481204\pi$
$821$	3.38196			0.118031			0.0590157	+	0.998257i	$0.481204\pi$
$822$		0			0
$823$		−	23.5824i		−	0.822029i	−0.911629	−	0.411015i	$-0.865175\pi$
$823$		−	23.5824i		−	0.822029i	0.911629	−	0.411015i	$-0.134825\pi$
$824$		−	8.49167i		−	0.295821i
$825$		0			0
$826$	−54.1528	−	1.67380i	−1.88422	−	0.0582391i
$827$		−	57.2583i		−	1.99107i	−0.0944133	−	0.995533i	$-0.530098\pi$
$827$		−	57.2583i		−	1.99107i	0.0944133	−	0.995533i	$-0.469902\pi$
$828$		0			0
$829$	45.0168			1.56350			0.781749	−	0.623593i	$-0.214328\pi$
$829$	45.0168			1.56350			0.781749	+	0.623593i	$0.214328\pi$
$830$		−	0.933508i		−	0.0324026i
$831$		0			0
$832$			48.4901i			1.68109i
$833$		−	56.7980i		−	1.96793i
$834$		0			0
$835$	26.0403			0.901162
$836$	−1.16140			−0.0401678
$837$		0			0
$838$	27.9452			0.965352
$839$	24.6050			0.849460			0.424730	−	0.905320i	$-0.360369\pi$
$839$	24.6050			0.849460			0.424730	+	0.905320i	$0.360369\pi$
$840$		0			0
$841$	−5.31937			−0.183426
$842$		−	3.72558i		−	0.128392i
$843$		0			0
$844$			0.868895i			0.0299086i
$845$			64.8920i			2.23235i
$846$		0			0
$847$	21.2973			0.731784
$848$		−	49.0104i		−	1.68303i
$849$		0			0
$850$		−	0.594033i		−	0.0203752i
$851$			56.8112i			1.94746i
$852$		0			0
$853$	−47.0831			−1.61209			−0.806047	−	0.591851i	$-0.798397\pi$
$853$	−47.0831			−1.61209			−0.806047	+	0.591851i	$0.798397\pi$
$854$			79.8835i			2.73356i
$855$		0			0
$856$			14.4934i			0.495375i
$857$	34.8653			1.19098			0.595488	−	0.803364i	$-0.296959\pi$
$857$	34.8653			1.19098			0.595488	+	0.803364i	$0.296959\pi$
$858$		0			0
$859$			13.8066i			0.471075i	0.971865	+	0.235537i	$0.0756850\pi$
$859$			13.8066i			0.471075i	−0.971865	+	0.235537i	$0.924315\pi$
$860$	−1.39266			−0.0474893
$861$		0			0
$862$	27.2884			0.929446
$863$	−24.9833			−0.850441			−0.425221	−	0.905090i	$-0.639804\pi$
$863$	−24.9833			−0.850441			−0.425221	+	0.905090i	$0.639804\pi$
$864$		0			0
$865$		−	39.0110i		−	1.32641i
$866$	−21.3828			−0.726616
$867$		0			0
$868$			1.48389i			0.0503667i
$869$	−9.28991			−0.315139
$870$		0			0
$871$	−47.7820			−1.61903
$872$		−	19.4358i		−	0.658181i
$873$		0			0
$874$	−44.5801			−1.50795
$875$			53.6171i			1.81259i
$876$		0			0
$877$	34.7540			1.17356			0.586779	−	0.809747i	$-0.300396\pi$
$877$	34.7540			1.17356			0.586779	+	0.809747i	$0.300396\pi$
$878$	−0.512744			−0.0173043
$879$		0			0
$880$		−	24.4890i		−	0.825523i
$881$	35.3540			1.19111			0.595553	−	0.803316i	$-0.296933\pi$
$881$	35.3540			1.19111			0.595553	+	0.803316i	$0.296933\pi$
$882$		0			0
$883$	−52.3914			−1.76311			−0.881556	−	0.472079i	$-0.843504\pi$
$883$	−52.3914			−1.76311			−0.881556	+	0.472079i	$0.843504\pi$
$884$	2.44331			0.0821775
$885$		0			0
$886$	46.2597			1.55412
$887$	−44.0877			−1.48032			−0.740160	−	0.672431i	$-0.765250\pi$
$887$	−44.0877			−1.48032			−0.740160	+	0.672431i	$0.765250\pi$
$888$		0			0
$889$	25.9283			0.869606
$890$		−	25.2149i		−	0.845207i
$891$		0			0
$892$	0.378888			0.0126861
$893$	13.7244			0.459271
$894$		0			0
$895$			24.2369i			0.810151i
$896$	59.0369			1.97228
$897$		0			0
$898$			36.3167i			1.21190i
$899$	16.2125			0.540716
$900$		0			0
$901$	39.9070			1.32950
$902$			2.14742i			0.0715013i
$903$		0			0
$904$	20.3137			0.675623
$905$		−	25.9363i		−	0.862151i
$906$		0			0
$907$	−34.5597			−1.14754			−0.573768	−	0.819018i	$-0.694519\pi$
$907$	−34.5597			−1.14754			−0.573768	+	0.819018i	$0.694519\pi$
$908$	1.01379			0.0336439
$909$		0			0
$910$	103.033			3.41550
$911$		−	19.6647i		−	0.651519i	−0.945453	−	0.325760i	$-0.894380\pi$
$911$		−	19.6647i		−	0.651519i	0.945453	−	0.325760i	$-0.105620\pi$
$912$		0			0
$913$	0.730374			0.0241718
$914$		−	5.20104i		−	0.172035i
$915$		0			0
$916$		−	0.776679i		−	0.0256622i
$917$	9.27188			0.306184
$918$		0			0
$919$			38.8259i			1.28075i	0.768063	+	0.640374i	$0.221221\pi$
$919$			38.8259i			1.28075i	−0.768063	+	0.640374i	$0.778779\pi$
$920$		−	46.6074i		−	1.53660i
$921$		0			0
$922$			27.2909i			0.898777i
$923$	78.2872			2.57685
$924$		0			0
$925$		−	0.903341i		−	0.0297017i
$926$		−	1.60101i		−	0.0526124i
$927$		0			0
$928$			3.65564i			0.120002i
$929$	−45.9332			−1.50702			−0.753510	−	0.657437i	$-0.771641\pi$
$929$	−45.9332			−1.50702			−0.753510	+	0.657437i	$0.771641\pi$
$930$		0			0
$931$	67.7769			2.22130
$932$	−0.863057			−0.0282704
$933$		0			0
$934$	−21.3378			−0.698196
$935$	19.9403			0.652117
$936$		0			0
$937$		−	21.3913i		−	0.698823i	−0.936969	−	0.349412i	$-0.886381\pi$
$937$		−	21.3913i		−	0.698823i	0.936969	−	0.349412i	$-0.113619\pi$
$938$			52.2037i			1.70451i
$939$		0			0
$940$		−	0.838597i		−	0.0273520i
$941$	−3.42668			−0.111707			−0.0558533	−	0.998439i	$-0.517788\pi$
$941$	−3.42668			−0.111707			−0.0558533	+	0.998439i	$0.517788\pi$
$942$		0			0
$943$			4.31333i			0.140462i
$944$	32.3120	+	0.998728i	1.05167	+	0.0325058i
$945$		0			0
$946$		−	20.8227i		−	0.677004i
$947$			43.7251i			1.42088i	0.703760	+	0.710438i	$0.251503\pi$
$947$			43.7251i			1.42088i	−0.703760	+	0.710438i	$0.748497\pi$
$948$		0			0
$949$	−92.4827			−3.00212
$950$	0.708858			0.0229984
$951$		0			0
$952$			45.6741i			1.48031i
$953$		−	47.1692i		−	1.52796i	−0.645241	−	0.763979i	$-0.723243\pi$
$953$		−	47.1692i		−	1.52796i	0.645241	−	0.763979i	$-0.276757\pi$
$954$		0			0
$955$		−	30.6623i		−	0.992207i
$956$			0.544389i			0.0176068i
$957$		0			0
$958$		−	24.4992i		−	0.791534i
$959$			41.8628i			1.35182i
$960$		0			0
$961$	23.3412			0.752943
$962$	71.0044			2.28927
$963$		0			0
$964$	−1.11596			−0.0359425
$965$			4.14154i			0.133321i
$966$		0			0
$967$			43.2197i			1.38985i	0.719082	+	0.694926i	$0.244563\pi$
$967$			43.2197i			1.38985i	−0.719082	+	0.694926i	$0.755437\pi$
$968$	−12.0408			−0.387006
$969$		0			0
$970$	−16.7650			−0.538291
$971$		−	27.9637i		−	0.897398i	−0.893683	−	0.448699i	$-0.851888\pi$
$971$		−	27.9637i		−	0.897398i	0.893683	−	0.448699i	$-0.148112\pi$
$972$		0			0
$973$	38.3851			1.23057
$974$	−49.2956			−1.57953
$975$		0			0
$976$		−	47.6650i		−	1.52572i
$977$	−16.4979			−0.527816			−0.263908	−	0.964548i	$-0.585012\pi$
$977$	−16.4979			−0.527816			−0.263908	+	0.964548i	$0.585012\pi$
$978$		0			0
$979$	19.7281			0.630512
$980$		−	4.14133i		−	0.132290i
$981$		0			0
$982$		−	20.2193i		−	0.645224i
$983$	−19.1629			−0.611201			−0.305600	−	0.952160i	$-0.598857\pi$
$983$	−19.1629			−0.611201			−0.305600	+	0.952160i	$0.598857\pi$
$984$		0			0
$985$	25.7609			0.820809
$986$	−29.1650			−0.928802
$987$		0			0
$988$			2.91560i			0.0927575i
$989$		−	41.8247i		−	1.32995i
$990$		0			0
$991$		−	44.4423i		−	1.41175i	−0.708334	−	0.705877i	$-0.750553\pi$
$991$		−	44.4423i		−	1.41175i	0.708334	−	0.705877i	$-0.249447\pi$
$992$		−	1.72693i		−	0.0548299i
$993$		0			0
$994$		−	85.5318i		−	2.71290i
$995$		−	9.71497i		−	0.307985i
$996$		0			0
$997$	11.2680			0.356863			0.178431	−	0.983952i	$-0.442898\pi$
$997$	11.2680			0.356863			0.178431	+	0.983952i	$0.442898\pi$
$998$	17.1219			0.541984
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.2.d.a.530.5	✓	20
3.2	odd	2	inner	531.2.d.a.530.16	yes	20
59.58	odd	2	inner	531.2.d.a.530.15	yes	20
177.176	even	2	inner	531.2.d.a.530.6	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.2.d.a.530.5	✓	20	1.1	even	1	trivial
531.2.d.a.530.6	yes	20	177.176	even	2	inner
531.2.d.a.530.15	yes	20	59.58	odd	2	inner
531.2.d.a.530.16	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)

\(f(q)\)	\(=\)	\(q-1.45273 q^{2} +0.110435 q^{4} -2.26259i q^{5} -4.85531 q^{7} +2.74503 q^{8} +3.28694i q^{10} -2.57169 q^{11} +6.45604i q^{13} +7.05347 q^{14} -4.20867 q^{16} -3.42694i q^{17} +4.08935 q^{19} -0.249870i q^{20} +3.73598 q^{22} +7.50414 q^{23} -0.119321 q^{25} -9.37890i q^{26} -0.536196 q^{28} +5.85827i q^{29} -2.76745i q^{31} +0.624014 q^{32} +4.97842i q^{34} +10.9856i q^{35} +7.57065i q^{37} -5.94074 q^{38} -6.21089i q^{40} +0.574794i q^{41} -5.57355i q^{43} -0.284005 q^{44} -10.9015 q^{46} +3.35614 q^{47} +16.5740 q^{49} +0.173342 q^{50} +0.712973i q^{52} +11.6451i q^{53} +5.81869i q^{55} -13.3280 q^{56} -8.51051i q^{58} +(-7.67748 - 0.237302i) q^{59} +11.3254i q^{61} +4.02037i q^{62} +7.51082 q^{64} +14.6074 q^{65} +7.40114i q^{67} -0.378454i q^{68} -15.9591i q^{70} -12.1262i q^{71} +14.3250i q^{73} -10.9981i q^{74} +0.451608 q^{76} +12.4864 q^{77} +3.61237 q^{79} +9.52251i q^{80} -0.835023i q^{82} -0.284005 q^{83} -7.75376 q^{85} +8.09688i q^{86} -7.05938 q^{88} -7.67125 q^{89} -31.3460i q^{91} +0.828720 q^{92} -4.87558 q^{94} -9.25254i q^{95} +5.10048i q^{97} -24.0776 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{4} - 8 q^{7} + 4 q^{16} - 8 q^{22} + 4 q^{25} + 8 q^{28} + 44 q^{49} + 36 q^{64} - 96 q^{76} - 24 q^{85} + 16 q^{88} - 112 q^{94}+O(q^{100}) \) 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

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