LMFDB - Embedded newform 572.2.f.b.441.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [572,2,Mod(441,572)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(572, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("572.441");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 572 = 2^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	572.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$4.56744299562$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 11x^{2} + 25 $ x^4 + 11*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			441.2
Root			$1.79129i$ of defining polynomial
Character	$\chi$	$=$	572.441
Dual form			572.2.f.b.441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.79129 q^{3} +0.791288i q^{7} +0.208712 q^{9} +O(q^{10})$	$q-1.79129 q^{3} +0.791288i q^{7} +0.208712 q^{9} -1.00000i q^{11} +(2.00000 - 3.00000i) q^{13} +1.58258 q^{17} +2.20871i q^{19} -1.41742i q^{21} -0.791288 q^{23} +5.00000 q^{25} +5.00000 q^{27} +7.58258 q^{29} -7.58258i q^{31} +1.79129i q^{33} -9.16515i q^{37} +(-3.58258 + 5.37386i) q^{39} -8.37386i q^{41} +2.00000 q^{43} +6.00000i q^{47} +6.37386 q^{49} -2.83485 q^{51} -2.20871 q^{53} -3.95644i q^{57} -6.00000i q^{59} +10.0000 q^{61} +0.165151i q^{63} -3.16515i q^{67} +1.41742 q^{69} +16.7477i q^{71} +12.9564i q^{73} -8.95644 q^{75} +0.791288 q^{77} -8.00000 q^{79} -9.58258 q^{81} -2.37386i q^{83} -13.5826 q^{87} -6.00000i q^{89} +(2.37386 + 1.58258i) q^{91} +13.5826i q^{93} +1.58258i q^{97} -0.208712i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 10 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 + 10 * q^9	$ 4 q + 2 q^{3} + 10 q^{9} + 8 q^{13} - 12 q^{17} + 6 q^{23} + 20 q^{25} + 20 q^{27} + 12 q^{29} + 4 q^{39} + 8 q^{43} - 2 q^{49} - 48 q^{51} - 18 q^{53} + 40 q^{61} + 24 q^{69} + 10 q^{75} - 6 q^{77} - 32 q^{79} - 20 q^{81} - 36 q^{87} - 18 q^{91}+O(q^{100}) $ 4 * q + 2 * q^3 + 10 * q^9 + 8 * q^13 - 12 * q^17 + 6 * q^23 + 20 * q^25 + 20 * q^27 + 12 * q^29 + 4 * q^39 + 8 * q^43 - 2 * q^49 - 48 * q^51 - 18 * q^53 + 40 * q^61 + 24 * q^69 + 10 * q^75 - 6 * q^77 - 32 * q^79 - 20 * q^81 - 36 * q^87 - 18 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/572\mathbb{Z}\right)^\times$.

$n$	$287$	$353$	$365$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.79129			−1.03420			−0.517100	−	0.855925i	$-0.672989\pi$
$3$	−1.79129			−1.03420			−0.517100	+	0.855925i	$0.672989\pi$
$4$		0			0
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$			0.791288i			0.299079i	0.988756	+	0.149539i	$0.0477791\pi$
$7$			0.791288i			0.299079i	−0.988756	+	0.149539i	$0.952221\pi$
$8$		0			0
$9$	0.208712			0.0695707
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	2.00000	−	3.00000i	0.554700	−	0.832050i
$13$	2.00000	−	3.00000i	0.554700	−	0.832050i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.58258			0.383831			0.191915	−	0.981411i	$-0.438530\pi$
$17$	1.58258			0.383831			0.191915	+	0.981411i	$0.438530\pi$
$18$		0			0
$19$			2.20871i			0.506713i	0.967373	+	0.253357i	$0.0815346\pi$
$19$			2.20871i			0.506713i	−0.967373	+	0.253357i	$0.918465\pi$
$20$		0			0
$21$		−	1.41742i		−	0.309307i
$22$		0			0
$23$	−0.791288			−0.164995			−0.0824975	−	0.996591i	$-0.526290\pi$
$23$	−0.791288			−0.164995			−0.0824975	+	0.996591i	$0.526290\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$	5.00000			0.962250
$28$		0			0
$29$	7.58258			1.40805			0.704024	−	0.710176i	$-0.251385\pi$
$29$	7.58258			1.40805			0.704024	+	0.710176i	$0.251385\pi$
$30$		0			0
$31$		−	7.58258i		−	1.36187i	−0.732343	−	0.680935i	$-0.761573\pi$
$31$		−	7.58258i		−	1.36187i	0.732343	−	0.680935i	$-0.238427\pi$
$32$		0			0
$33$			1.79129i			0.311823i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	9.16515i		−	1.50674i	−0.657596	−	0.753371i	$-0.728427\pi$
$37$		−	9.16515i		−	1.50674i	0.657596	−	0.753371i	$-0.271573\pi$
$38$		0			0
$39$	−3.58258	+	5.37386i	−0.573671	+	0.860507i
$40$		0			0
$41$		−	8.37386i		−	1.30778i	−0.756591	−	0.653889i	$-0.773136\pi$
$41$		−	8.37386i		−	1.30778i	0.756591	−	0.653889i	$-0.226864\pi$
$42$		0			0
$43$	2.00000			0.304997			0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000			0.304997			0.152499	+	0.988304i	$0.451268\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			6.00000i			0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$			6.00000i			0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$		0			0
$49$	6.37386			0.910552
$50$		0			0
$51$	−2.83485			−0.396958
$52$		0			0
$53$	−2.20871			−0.303390			−0.151695	−	0.988427i	$-0.548473\pi$
$53$	−2.20871			−0.303390			−0.151695	+	0.988427i	$0.548473\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	3.95644i		−	0.524043i
$58$		0			0
$59$		−	6.00000i		−	0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$		−	6.00000i		−	0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$			0.165151i			0.0208071i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	3.16515i		−	0.386685i	−0.981131	−	0.193342i	$-0.938067\pi$
$67$		−	3.16515i		−	0.386685i	0.981131	−	0.193342i	$-0.0619328\pi$
$68$		0			0
$69$	1.41742			0.170638
$70$		0			0
$71$			16.7477i			1.98759i	0.111229	+	0.993795i	$0.464521\pi$
$71$			16.7477i			1.98759i	−0.111229	+	0.993795i	$0.535479\pi$
$72$		0			0
$73$			12.9564i			1.51644i	0.652001	+	0.758218i	$0.273930\pi$
$73$			12.9564i			1.51644i	−0.652001	+	0.758218i	$0.726070\pi$
$74$		0			0
$75$	−8.95644			−1.03420
$76$		0			0
$77$	0.791288			0.0901756
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	−9.58258			−1.06473
$82$		0			0
$83$		−	2.37386i		−	0.260565i	−0.991477	−	0.130283i	$-0.958412\pi$
$83$		−	2.37386i		−	0.260565i	0.991477	−	0.130283i	$-0.0415885\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−13.5826			−1.45620
$88$		0			0
$89$		−	6.00000i		−	0.635999i	−0.948091	−	0.317999i	$-0.896989\pi$
$89$		−	6.00000i		−	0.635999i	0.948091	−	0.317999i	$-0.103011\pi$
$90$		0			0
$91$	2.37386	+	1.58258i	0.248849	+	0.165899i
$92$		0			0
$93$			13.5826i			1.40845i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			1.58258i			0.160686i	0.996767	+	0.0803431i	$0.0256016\pi$
$97$			1.58258i			0.160686i	−0.996767	+	0.0803431i	$0.974398\pi$
$98$		0			0
$99$		−	0.208712i		−	0.0209764i
$100$		0			0
$101$	−3.16515			−0.314944			−0.157472	−	0.987523i	$-0.550334\pi$
$101$	−3.16515			−0.314944			−0.157472	+	0.987523i	$0.550334\pi$
$102$		0			0
$103$	−6.37386			−0.628035			−0.314018	−	0.949417i	$-0.601675\pi$
$103$	−6.37386			−0.628035			−0.314018	+	0.949417i	$0.601675\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.41742			−0.427049			−0.213524	−	0.976938i	$-0.568494\pi$
$107$	−4.41742			−0.427049			−0.213524	+	0.976938i	$0.568494\pi$
$108$		0			0
$109$		−	14.2087i		−	1.36095i	−0.732772	−	0.680474i	$-0.761774\pi$
$109$		−	14.2087i		−	1.36095i	0.732772	−	0.680474i	$-0.238226\pi$
$110$		0			0
$111$			16.4174i			1.55827i
$112$		0			0
$113$	−17.3739			−1.63440			−0.817198	−	0.576357i	$-0.804474\pi$
$113$	−17.3739			−1.63440			−0.817198	+	0.576357i	$0.804474\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.417424	−	0.626136i	0.0385909	−	0.0578863i
$118$		0			0
$119$			1.25227i			0.114796i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			15.0000i			1.35250i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−2.00000			−0.177471			−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000			−0.177471			−0.0887357	+	0.996055i	$0.528283\pi$
$128$		0			0
$129$	−3.58258			−0.315428
$130$		0			0
$131$	−4.41742			−0.385952			−0.192976	−	0.981203i	$-0.561814\pi$
$131$	−4.41742			−0.385952			−0.192976	+	0.981203i	$0.561814\pi$
$132$		0			0
$133$	−1.74773			−0.151547
$134$		0			0
$135$		0			0
$136$		0			0
$137$			12.0000i			1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$			12.0000i			1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$		0			0
$139$	12.7477			1.08125			0.540624	−	0.841264i	$-0.318188\pi$
$139$	12.7477			1.08125			0.540624	+	0.841264i	$0.318188\pi$
$140$		0			0
$141$		−	10.7477i		−	0.905122i
$142$		0			0
$143$	−3.00000	−	2.00000i	−0.250873	−	0.167248i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−11.4174			−0.941693
$148$		0			0
$149$			20.3739i			1.66909i	0.550938	+	0.834546i	$0.314270\pi$
$149$			20.3739i			1.66909i	−0.550938	+	0.834546i	$0.685730\pi$
$150$		0			0
$151$		−	12.0000i		−	0.976546i	−0.872691	−	0.488273i	$-0.837627\pi$
$151$		−	12.0000i		−	0.976546i	0.872691	−	0.488273i	$-0.162373\pi$
$152$		0			0
$153$	0.330303			0.0267034
$154$		0			0
$155$		0			0
$156$		0			0
$157$	16.3739			1.30678			0.653388	−	0.757023i	$-0.273347\pi$
$157$	16.3739			1.30678			0.653388	+	0.757023i	$0.273347\pi$
$158$		0			0
$159$	3.95644			0.313766
$160$		0			0
$161$		−	0.626136i		−	0.0493465i
$162$		0			0
$163$		−	3.16515i		−	0.247914i	−0.992288	−	0.123957i	$-0.960442\pi$
$163$		−	3.16515i		−	0.247914i	0.992288	−	0.123957i	$-0.0395585\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	17.3739i		−	1.34443i	−0.740356	−	0.672215i	$-0.765343\pi$
$167$		−	17.3739i		−	1.34443i	0.740356	−	0.672215i	$-0.234657\pi$
$168$		0			0
$169$	−5.00000	−	12.0000i	−0.384615	−	0.923077i
$170$		0			0
$171$			0.460985i			0.0352524i
$172$		0			0
$173$	−13.5826			−1.03266			−0.516332	−	0.856388i	$-0.672703\pi$
$173$	−13.5826			−1.03266			−0.516332	+	0.856388i	$0.672703\pi$
$174$		0			0
$175$			3.95644i			0.299079i
$176$		0			0
$177$			10.7477i			0.807849i
$178$		0			0
$179$	18.3303			1.37007			0.685036	−	0.728510i	$-0.259787\pi$
$179$	18.3303			1.37007			0.685036	+	0.728510i	$0.259787\pi$
$180$		0			0
$181$	13.3739			0.994071			0.497036	−	0.867730i	$-0.334422\pi$
$181$	13.3739			0.994071			0.497036	+	0.867730i	$0.334422\pi$
$182$		0			0
$183$	−17.9129			−1.32416
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	1.58258i		−	0.115729i
$188$		0			0
$189$			3.95644i			0.287789i
$190$		0			0
$191$	5.37386			0.388839			0.194420	−	0.980918i	$-0.437718\pi$
$191$	5.37386			0.388839			0.194420	+	0.980918i	$0.437718\pi$
$192$		0			0
$193$			6.79129i			0.488848i	0.969669	+	0.244424i	$0.0785988\pi$
$193$			6.79129i			0.488848i	−0.969669	+	0.244424i	$0.921401\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	5.37386i		−	0.382872i	−0.981505	−	0.191436i	$-0.938686\pi$
$197$		−	5.37386i		−	0.382872i	0.981505	−	0.191436i	$-0.0613144\pi$
$198$		0			0
$199$	9.37386			0.664496			0.332248	−	0.943192i	$-0.392193\pi$
$199$	9.37386			0.664496			0.332248	+	0.943192i	$0.392193\pi$
$200$		0			0
$201$			5.66970i			0.399910i
$202$		0			0
$203$			6.00000i			0.421117i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.165151			−0.0114788
$208$		0			0
$209$	2.20871			0.152780
$210$		0			0
$211$	16.0000			1.10149			0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000			1.10149			0.550743	+	0.834675i	$0.314345\pi$
$212$		0			0
$213$		−	30.0000i		−	2.05557i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	6.00000			0.407307
$218$		0			0
$219$		−	23.2087i		−	1.56830i
$220$		0			0
$221$	3.16515	−	4.74773i	0.212911	−	0.319367i
$222$		0			0
$223$			15.1652i			1.01553i	0.861495	+	0.507767i	$0.169529\pi$
$223$			15.1652i			1.01553i	−0.861495	+	0.507767i	$0.830471\pi$
$224$		0			0
$225$	1.04356			0.0695707
$226$		0			0
$227$			5.37386i			0.356676i	0.983969	+	0.178338i	$0.0570720\pi$
$227$			5.37386i			0.356676i	−0.983969	+	0.178338i	$0.942928\pi$
$228$		0			0
$229$			4.74773i			0.313739i	0.987619	+	0.156869i	$0.0501401\pi$
$229$			4.74773i			0.313739i	−0.987619	+	0.156869i	$0.949860\pi$
$230$		0			0
$231$	−1.41742			−0.0932597
$232$		0			0
$233$	−16.7477			−1.09718			−0.548590	−	0.836091i	$-0.684835\pi$
$233$	−16.7477			−1.09718			−0.548590	+	0.836091i	$0.684835\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	14.3303			0.930853
$238$		0			0
$239$		−	5.37386i		−	0.347606i	−0.984780	−	0.173803i	$-0.944394\pi$
$239$		−	5.37386i		−	0.347606i	0.984780	−	0.173803i	$-0.0556057\pi$
$240$		0			0
$241$		−	8.37386i		−	0.539408i	−0.962943	−	0.269704i	$-0.913074\pi$
$241$		−	8.37386i		−	0.539408i	0.962943	−	0.269704i	$-0.0869258\pi$
$242$		0			0
$243$	2.16515			0.138895
$244$		0			0
$245$		0			0
$246$		0			0
$247$	6.62614	+	4.41742i	0.421611	+	0.281074i
$248$		0			0
$249$			4.25227i			0.269477i
$250$		0			0
$251$	−0.791288			−0.0499456			−0.0249728	−	0.999688i	$-0.507950\pi$
$251$	−0.791288			−0.0499456			−0.0249728	+	0.999688i	$0.507950\pi$
$252$		0			0
$253$			0.791288i			0.0497478i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−13.1216			−0.818502			−0.409251	−	0.912422i	$-0.634210\pi$
$257$	−13.1216			−0.818502			−0.409251	+	0.912422i	$0.634210\pi$
$258$		0			0
$259$	7.25227			0.450634
$260$		0			0
$261$	1.58258			0.0979590
$262$		0			0
$263$	−19.5826			−1.20751			−0.603757	−	0.797169i	$-0.706330\pi$
$263$	−19.5826			−1.20751			−0.603757	+	0.797169i	$0.706330\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			10.7477i			0.657750i
$268$		0			0
$269$	15.6261			0.952742			0.476371	−	0.879244i	$-0.341952\pi$
$269$	15.6261			0.952742			0.476371	+	0.879244i	$0.341952\pi$
$270$		0			0
$271$		−	17.3739i		−	1.05539i	−0.849435	−	0.527694i	$-0.823057\pi$
$271$		−	17.3739i		−	1.05539i	0.849435	−	0.527694i	$-0.176943\pi$
$272$		0			0
$273$	−4.25227	−	2.83485i	−0.257359	−	0.171573i
$274$		0			0
$275$		−	5.00000i		−	0.301511i
$276$		0			0
$277$	18.7477			1.12644			0.563221	−	0.826306i	$-0.309562\pi$
$277$	18.7477			1.12644			0.563221	+	0.826306i	$0.309562\pi$
$278$		0			0
$279$		−	1.58258i		−	0.0947463i
$280$		0			0
$281$			16.1216i			0.961733i	0.876794	+	0.480867i	$0.159678\pi$
$281$			16.1216i			0.961733i	−0.876794	+	0.480867i	$0.840322\pi$
$282$		0			0
$283$	−6.74773			−0.401111			−0.200555	−	0.979682i	$-0.564275\pi$
$283$	−6.74773			−0.401111			−0.200555	+	0.979682i	$0.564275\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6.62614			0.391128
$288$		0			0
$289$	−14.4955			−0.852674
$290$		0			0
$291$		−	2.83485i		−	0.166182i
$292$		0			0
$293$			6.00000i			0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$			6.00000i			0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	5.00000i		−	0.290129i
$298$		0			0
$299$	−1.58258	+	2.37386i	−0.0915227	+	0.137284i
$300$		0			0
$301$			1.58258i			0.0912181i
$302$		0			0
$303$	5.66970			0.325716
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	30.3303i		−	1.73104i	−0.500873	−	0.865521i	$-0.666988\pi$
$307$		−	30.3303i		−	1.73104i	0.500873	−	0.865521i	$-0.333012\pi$
$308$		0			0
$309$	11.4174			0.649515
$310$		0			0
$311$	8.53901			0.484203			0.242102	−	0.970251i	$-0.422163\pi$
$311$	8.53901			0.484203			0.242102	+	0.970251i	$0.422163\pi$
$312$		0			0
$313$	−4.37386			−0.247225			−0.123613	−	0.992331i	$-0.539448\pi$
$313$	−4.37386			−0.247225			−0.123613	+	0.992331i	$0.539448\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	1.25227i		−	0.0703347i	−0.999381	−	0.0351673i	$-0.988804\pi$
$317$		−	1.25227i		−	0.0703347i	0.999381	−	0.0351673i	$-0.0111964\pi$
$318$		0			0
$319$		−	7.58258i		−	0.424543i
$320$		0			0
$321$	7.91288			0.441654
$322$		0			0
$323$			3.49545i			0.194492i
$324$		0			0
$325$	10.0000	−	15.0000i	0.554700	−	0.832050i
$326$		0			0
$327$			25.4519i			1.40749i
$328$		0			0
$329$	−4.74773			−0.261751
$330$		0			0
$331$			21.1652i			1.16334i	0.813424	+	0.581671i	$0.197601\pi$
$331$			21.1652i			1.16334i	−0.813424	+	0.581671i	$0.802399\pi$
$332$		0			0
$333$		−	1.91288i		−	0.104825i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−2.00000			−0.108947			−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000			−0.108947			−0.0544735	+	0.998515i	$0.517348\pi$
$338$		0			0
$339$	31.1216			1.69029
$340$		0			0
$341$	−7.58258			−0.410619
$342$		0			0
$343$			10.5826i			0.571405i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.0000			−0.644194			−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000			−0.644194			−0.322097	+	0.946707i	$0.604388\pi$
$348$		0			0
$349$		−	33.9564i		−	1.81765i	−0.417181	−	0.908823i	$-0.636982\pi$
$349$		−	33.9564i		−	1.81765i	0.417181	−	0.908823i	$-0.363018\pi$
$350$		0			0
$351$	10.0000	−	15.0000i	0.533761	−	0.800641i
$352$		0			0
$353$			34.7477i			1.84943i	0.380654	+	0.924717i	$0.375699\pi$
$353$			34.7477i			1.84943i	−0.380654	+	0.924717i	$0.624301\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	2.24318i		−	0.118722i
$358$		0			0
$359$		−	31.1216i		−	1.64253i	−0.570544	−	0.821267i	$-0.693267\pi$
$359$		−	31.1216i		−	1.64253i	0.570544	−	0.821267i	$-0.306733\pi$
$360$		0			0
$361$	14.1216			0.743242
$362$		0			0
$363$	1.79129			0.0940182
$364$		0			0
$365$		0			0
$366$		0			0
$367$	24.1216			1.25914			0.629568	−	0.776945i	$-0.283232\pi$
$367$	24.1216			1.25914			0.629568	+	0.776945i	$0.283232\pi$
$368$		0			0
$369$		−	1.74773i		−	0.0909830i
$370$		0			0
$371$		−	1.74773i		−	0.0907375i
$372$		0			0
$373$	−29.4955			−1.52722			−0.763608	−	0.645680i	$-0.776574\pi$
$373$	−29.4955			−1.52722			−0.763608	+	0.645680i	$0.776574\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	15.1652	−	22.7477i	0.781045	−	1.17157i
$378$		0			0
$379$			6.00000i			0.308199i	0.988055	+	0.154100i	$0.0492477\pi$
$379$			6.00000i			0.308199i	−0.988055	+	0.154100i	$0.950752\pi$
$380$		0			0
$381$	3.58258			0.183541
$382$		0			0
$383$			28.7477i			1.46894i	0.678641	+	0.734470i	$0.262569\pi$
$383$			28.7477i			1.46894i	−0.678641	+	0.734470i	$0.737431\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.417424			0.0212189
$388$		0			0
$389$	−11.0436			−0.559931			−0.279965	−	0.960010i	$-0.590323\pi$
$389$	−11.0436			−0.559931			−0.279965	+	0.960010i	$0.590323\pi$
$390$		0			0
$391$	−1.25227			−0.0633302
$392$		0			0
$393$	7.91288			0.399152
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	10.4174i		−	0.522836i	−0.965226	−	0.261418i	$-0.915810\pi$
$397$		−	10.4174i		−	0.522836i	0.965226	−	0.261418i	$-0.0841900\pi$
$398$		0			0
$399$	3.13068			0.156730
$400$		0			0
$401$			6.00000i			0.299626i	0.988714	+	0.149813i	$0.0478671\pi$
$401$			6.00000i			0.299626i	−0.988714	+	0.149813i	$0.952133\pi$
$402$		0			0
$403$	−22.7477	−	15.1652i	−1.13314	−	0.755430i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−9.16515			−0.454300
$408$		0			0
$409$		−	9.16515i		−	0.453188i	−0.973989	−	0.226594i	$-0.927241\pi$
$409$		−	9.16515i		−	0.453188i	0.973989	−	0.226594i	$-0.0727590\pi$
$410$		0			0
$411$		−	21.4955i		−	1.06029i
$412$		0			0
$413$	4.74773			0.233620
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−22.8348			−1.11823
$418$		0			0
$419$	21.6261			1.05651			0.528253	−	0.849087i	$-0.322847\pi$
$419$	21.6261			1.05651			0.528253	+	0.849087i	$0.322847\pi$
$420$		0			0
$421$			10.7477i			0.523812i	0.965093	+	0.261906i	$0.0843511\pi$
$421$			10.7477i			0.523812i	−0.965093	+	0.261906i	$0.915649\pi$
$422$		0			0
$423$			1.25227i			0.0608876i
$424$		0			0
$425$	7.91288			0.383831
$426$		0			0
$427$			7.91288i			0.382931i
$428$		0			0
$429$	5.37386	+	3.58258i	0.259453	+	0.172968i
$430$		0			0
$431$			14.3739i			0.692365i	0.938167	+	0.346182i	$0.112522\pi$
$431$			14.3739i			0.692365i	−0.938167	+	0.346182i	$0.887478\pi$
$432$		0			0
$433$	9.37386			0.450479			0.225240	−	0.974303i	$-0.427684\pi$
$433$	9.37386			0.450479			0.225240	+	0.974303i	$0.427684\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	1.74773i		−	0.0836051i
$438$		0			0
$439$	12.7477			0.608416			0.304208	−	0.952606i	$-0.401608\pi$
$439$	12.7477			0.608416			0.304208	+	0.952606i	$0.401608\pi$
$440$		0			0
$441$	1.33030			0.0633478
$442$		0			0
$443$	0.791288			0.0375952			0.0187976	−	0.999823i	$-0.494016\pi$
$443$	0.791288			0.0375952			0.0187976	+	0.999823i	$0.494016\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	36.4955i		−	1.72618i
$448$		0			0
$449$			21.4955i			1.01443i	0.861819	+	0.507217i	$0.169326\pi$
$449$			21.4955i			1.01443i	−0.861819	+	0.507217i	$0.830674\pi$
$450$		0			0
$451$	−8.37386			−0.394310
$452$		0			0
$453$			21.4955i			1.00994i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			29.3739i			1.37405i	0.726633	+	0.687026i	$0.241084\pi$
$457$			29.3739i			1.37405i	−0.726633	+	0.687026i	$0.758916\pi$
$458$		0			0
$459$	7.91288			0.369342
$460$		0			0
$461$		−	16.1216i		−	0.750857i	−0.926851	−	0.375429i	$-0.877496\pi$
$461$		−	16.1216i		−	0.750857i	0.926851	−	0.375429i	$-0.122504\pi$
$462$		0			0
$463$		−	16.7477i		−	0.778333i	−0.921167	−	0.389166i	$-0.872763\pi$
$463$		−	16.7477i		−	0.778333i	0.921167	−	0.389166i	$-0.127237\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6.33030			−0.292931			−0.146466	−	0.989216i	$-0.546790\pi$
$467$	−6.33030			−0.292931			−0.146466	+	0.989216i	$0.546790\pi$
$468$		0			0
$469$	2.50455			0.115649
$470$		0			0
$471$	−29.3303			−1.35147
$472$		0			0
$473$		−	2.00000i		−	0.0919601i
$474$		0			0
$475$			11.0436i			0.506713i
$476$		0			0
$477$	−0.460985			−0.0211071
$478$		0			0
$479$		−	33.4955i		−	1.53045i	−0.643765	−	0.765223i	$-0.722629\pi$
$479$		−	33.4955i		−	1.53045i	0.643765	−	0.765223i	$-0.277371\pi$
$480$		0			0
$481$	−27.4955	−	18.3303i	−1.25368	−	0.835790i
$482$		0			0
$483$			1.12159i			0.0510341i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			34.7477i			1.57457i	0.616589	+	0.787285i	$0.288514\pi$
$487$			34.7477i			1.57457i	−0.616589	+	0.787285i	$0.711486\pi$
$488$		0			0
$489$			5.66970i			0.256393i
$490$		0			0
$491$	−35.0780			−1.58305			−0.791525	−	0.611137i	$-0.790712\pi$
$491$	−35.0780			−1.58305			−0.791525	+	0.611137i	$0.790712\pi$
$492$		0			0
$493$	12.0000			0.540453
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−13.2523			−0.594446
$498$		0			0
$499$			35.0780i			1.57031i	0.619300	+	0.785154i	$0.287416\pi$
$499$			35.0780i			1.57031i	−0.619300	+	0.785154i	$0.712584\pi$
$500$		0			0
$501$			31.1216i			1.39041i
$502$		0			0
$503$	31.9129			1.42292			0.711462	−	0.702724i	$-0.248033\pi$
$503$	31.9129			1.42292			0.711462	+	0.702724i	$0.248033\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	8.95644	+	21.4955i	0.397769	+	0.954647i
$508$		0			0
$509$			15.4955i			0.686824i	0.939185	+	0.343412i	$0.111583\pi$
$509$			15.4955i			0.686824i	−0.939185	+	0.343412i	$0.888417\pi$
$510$		0			0
$511$	−10.2523			−0.453534
$512$		0			0
$513$			11.0436i			0.487585i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	6.00000			0.263880
$518$		0			0
$519$	24.3303			1.06798
$520$		0			0
$521$	−30.7913			−1.34899			−0.674495	−	0.738279i	$-0.735639\pi$
$521$	−30.7913			−1.34899			−0.674495	+	0.738279i	$0.735639\pi$
$522$		0			0
$523$	−20.7477			−0.907235			−0.453617	−	0.891197i	$-0.649867\pi$
$523$	−20.7477			−0.907235			−0.453617	+	0.891197i	$0.649867\pi$
$524$		0			0
$525$		−	7.08712i		−	0.309307i
$526$		0			0
$527$		−	12.0000i		−	0.522728i
$528$		0			0
$529$	−22.3739			−0.972777
$530$		0			0
$531$		−	1.25227i		−	0.0543440i
$532$		0			0
$533$	−25.1216	−	16.7477i	−1.08814	−	0.725425i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−32.8348			−1.41693
$538$		0			0
$539$		−	6.37386i		−	0.274542i
$540$		0			0
$541$			8.37386i			0.360021i	0.983665	+	0.180010i	$0.0576131\pi$
$541$			8.37386i			0.360021i	−0.983665	+	0.180010i	$0.942387\pi$
$542$		0			0
$543$	−23.9564			−1.02807
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−2.74773			−0.117484			−0.0587422	−	0.998273i	$-0.518709\pi$
$547$	−2.74773			−0.117484			−0.0587422	+	0.998273i	$0.518709\pi$
$548$		0			0
$549$	2.08712			0.0890762
$550$		0			0
$551$			16.7477i			0.713477i
$552$		0			0
$553$		−	6.33030i		−	0.269192i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			15.6261i			0.662101i	0.943613	+	0.331050i	$0.107403\pi$
$557$			15.6261i			0.662101i	−0.943613	+	0.331050i	$0.892597\pi$
$558$		0			0
$559$	4.00000	−	6.00000i	0.169182	−	0.253773i
$560$		0			0
$561$			2.83485i			0.119687i
$562$		0			0
$563$	1.58258			0.0666976			0.0333488	−	0.999444i	$-0.489383\pi$
$563$	1.58258			0.0666976			0.0333488	+	0.999444i	$0.489383\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	7.58258i		−	0.318438i
$568$		0			0
$569$	−40.7477			−1.70823			−0.854117	−	0.520081i	$-0.825902\pi$
$569$	−40.7477			−1.70823			−0.854117	+	0.520081i	$0.825902\pi$
$570$		0			0
$571$	−36.7477			−1.53784			−0.768922	−	0.639342i	$-0.779207\pi$
$571$	−36.7477			−1.53784			−0.768922	+	0.639342i	$0.779207\pi$
$572$		0			0
$573$	−9.62614			−0.402138
$574$		0			0
$575$	−3.95644			−0.164995
$576$		0			0
$577$			36.3303i			1.51245i	0.654311	+	0.756225i	$0.272959\pi$
$577$			36.3303i			1.51245i	−0.654311	+	0.756225i	$0.727041\pi$
$578$		0			0
$579$		−	12.1652i		−	0.505566i
$580$		0			0
$581$	1.87841			0.0779296
$582$		0			0
$583$			2.20871i			0.0914755i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			4.74773i			0.195960i	0.995188	+	0.0979798i	$0.0312381\pi$
$587$			4.74773i			0.195960i	−0.995188	+	0.0979798i	$0.968762\pi$
$588$		0			0
$589$	16.7477			0.690078
$590$		0			0
$591$			9.62614i			0.395966i
$592$		0			0
$593$			37.1216i			1.52440i	0.647341	+	0.762201i	$0.275881\pi$
$593$			37.1216i			1.52440i	−0.647341	+	0.762201i	$0.724119\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−16.7913			−0.687222
$598$		0			0
$599$	17.3739			0.709877			0.354938	−	0.934890i	$-0.384502\pi$
$599$	17.3739			0.709877			0.354938	+	0.934890i	$0.384502\pi$
$600$		0			0
$601$	14.7477			0.601572			0.300786	−	0.953692i	$-0.402751\pi$
$601$	14.7477			0.601572			0.300786	+	0.953692i	$0.402751\pi$
$602$		0			0
$603$		−	0.660606i		−	0.0269019i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	8.00000			0.324710			0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000			0.324710			0.162355	+	0.986732i	$0.448091\pi$
$608$		0			0
$609$		−	10.7477i		−	0.435520i
$610$		0			0
$611$	18.0000	+	12.0000i	0.728202	+	0.485468i
$612$		0			0
$613$			26.2087i			1.05856i	0.848447	+	0.529280i	$0.177538\pi$
$613$			26.2087i			1.05856i	−0.848447	+	0.529280i	$0.822462\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			13.2523i			0.533516i	0.963763	+	0.266758i	$0.0859525\pi$
$617$			13.2523i			0.533516i	−0.963763	+	0.266758i	$0.914047\pi$
$618$		0			0
$619$		−	28.7477i		−	1.15547i	−0.816225	−	0.577734i	$-0.803937\pi$
$619$		−	28.7477i		−	1.15547i	0.816225	−	0.577734i	$-0.196063\pi$
$620$		0			0
$621$	−3.95644			−0.158766
$622$		0			0
$623$	4.74773			0.190214
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$	−3.95644			−0.158005
$628$		0			0
$629$		−	14.5045i		−	0.578334i
$630$		0			0
$631$			10.7477i			0.427860i	0.976849	+	0.213930i	$0.0686265\pi$
$631$			10.7477i			0.427860i	−0.976849	+	0.213930i	$0.931374\pi$
$632$		0			0
$633$	−28.6606			−1.13916
$634$		0			0
$635$		0			0
$636$		0			0
$637$	12.7477	−	19.1216i	0.505083	−	0.757625i
$638$		0			0
$639$			3.49545i			0.138278i
$640$		0			0
$641$	−35.5390			−1.40371			−0.701853	−	0.712321i	$-0.747644\pi$
$641$	−35.5390			−1.40371			−0.701853	+	0.712321i	$0.747644\pi$
$642$		0			0
$643$		−	33.1652i		−	1.30791i	−0.756535	−	0.653953i	$-0.773109\pi$
$643$		−	33.1652i		−	1.30791i	0.756535	−	0.653953i	$-0.226891\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	35.7042			1.40368			0.701838	−	0.712337i	$-0.252363\pi$
$647$	35.7042			1.40368			0.701838	+	0.712337i	$0.252363\pi$
$648$		0			0
$649$	−6.00000			−0.235521
$650$		0			0
$651$	−10.7477			−0.421237
$652$		0			0
$653$	12.3303			0.482522			0.241261	−	0.970460i	$-0.422439\pi$
$653$	12.3303			0.482522			0.241261	+	0.970460i	$0.422439\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			2.70417i			0.105500i
$658$		0			0
$659$	−24.6606			−0.960641			−0.480320	−	0.877093i	$-0.659480\pi$
$659$	−24.6606			−0.960641			−0.480320	+	0.877093i	$0.659480\pi$
$660$		0			0
$661$			3.16515i			0.123110i	0.998104	+	0.0615551i	$0.0196060\pi$
$661$			3.16515i			0.123110i	−0.998104	+	0.0615551i	$0.980394\pi$
$662$		0			0
$663$	−5.66970	+	8.50455i	−0.220193	+	0.330289i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−6.00000			−0.232321
$668$		0			0
$669$		−	27.1652i		−	1.05027i
$670$		0			0
$671$		−	10.0000i		−	0.386046i
$672$		0			0
$673$	−26.7477			−1.03105			−0.515525	−	0.856875i	$-0.672403\pi$
$673$	−26.7477			−1.03105			−0.515525	+	0.856875i	$0.672403\pi$
$674$		0			0
$675$	25.0000			0.962250
$676$		0			0
$677$	−18.0000			−0.691796			−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000			−0.691796			−0.345898	+	0.938272i	$0.612426\pi$
$678$		0			0
$679$	−1.25227			−0.0480578
$680$		0			0
$681$		−	9.62614i		−	0.368874i
$682$		0			0
$683$		−	9.49545i		−	0.363333i	−0.983360	−	0.181667i	$-0.941851\pi$
$683$		−	9.49545i		−	0.363333i	0.983360	−	0.181667i	$-0.0581492\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	8.50455i		−	0.324469i
$688$		0			0
$689$	−4.41742	+	6.62614i	−0.168290	+	0.252436i
$690$		0			0
$691$			25.5826i			0.973207i	0.873623	+	0.486604i	$0.161764\pi$
$691$			25.5826i			0.973207i	−0.873623	+	0.486604i	$0.838236\pi$
$692$		0			0
$693$	0.165151			0.00627358
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	13.2523i		−	0.501966i
$698$		0			0
$699$	30.0000			1.13470
$700$		0			0
$701$	19.2523			0.727148			0.363574	−	0.931565i	$-0.381556\pi$
$701$	19.2523			0.727148			0.363574	+	0.931565i	$0.381556\pi$
$702$		0			0
$703$	20.2432			0.763486
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	2.50455i		−	0.0941931i
$708$		0			0
$709$		−	7.25227i		−	0.272365i	−0.990684	−	0.136182i	$-0.956517\pi$
$709$		−	7.25227i		−	0.272365i	0.990684	−	0.136182i	$-0.0434833\pi$
$710$		0			0
$711$	−1.66970			−0.0626185
$712$		0			0
$713$			6.00000i			0.224702i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			9.62614i			0.359495i
$718$		0			0
$719$	51.1652			1.90814			0.954069	−	0.299587i	$-0.0968488\pi$
$719$	51.1652			1.90814			0.954069	+	0.299587i	$0.0968488\pi$
$720$		0			0
$721$		−	5.04356i		−	0.187832i
$722$		0			0
$723$			15.0000i			0.557856i
$724$		0			0
$725$	37.9129			1.40805
$726$		0			0
$727$	−0.878409			−0.0325784			−0.0162892	−	0.999867i	$-0.505185\pi$
$727$	−0.878409			−0.0325784			−0.0162892	+	0.999867i	$0.505185\pi$
$728$		0			0
$729$	24.8693			0.921086
$730$		0			0
$731$	3.16515			0.117067
$732$		0			0
$733$			15.6261i			0.577165i	0.957455	+	0.288582i	$0.0931839\pi$
$733$			15.6261i			0.577165i	−0.957455	+	0.288582i	$0.906816\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.16515			−0.116590
$738$		0			0
$739$			8.04356i			0.295887i	0.988996	+	0.147944i	$0.0472654\pi$
$739$			8.04356i			0.295887i	−0.988996	+	0.147944i	$0.952735\pi$
$740$		0			0
$741$	−11.8693	−	7.91288i	−0.436030	−	0.290687i
$742$		0			0
$743$		−	33.4955i		−	1.22883i	−0.788983	−	0.614415i	$-0.789392\pi$
$743$		−	33.4955i		−	1.22883i	0.788983	−	0.614415i	$-0.210608\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	0.495454i		−	0.0181277i
$748$		0			0
$749$		−	3.49545i		−	0.127721i
$750$		0			0
$751$	45.6170			1.66459			0.832295	−	0.554333i	$-0.187027\pi$
$751$	45.6170			1.66459			0.832295	+	0.554333i	$0.187027\pi$
$752$		0			0
$753$	1.41742			0.0516538
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−40.3739			−1.46741			−0.733707	−	0.679467i	$-0.762211\pi$
$757$	−40.3739			−1.46741			−0.733707	+	0.679467i	$0.762211\pi$
$758$		0			0
$759$		−	1.41742i		−	0.0514492i
$760$		0			0
$761$		−	41.8693i		−	1.51776i	−0.651230	−	0.758881i	$-0.725747\pi$
$761$		−	41.8693i		−	1.51776i	0.651230	−	0.758881i	$-0.274253\pi$
$762$		0			0
$763$	11.2432			0.407030
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−18.0000	−	12.0000i	−0.649942	−	0.433295i
$768$		0			0
$769$			17.2087i			0.620562i	0.950645	+	0.310281i	$0.100423\pi$
$769$			17.2087i			0.620562i	−0.950645	+	0.310281i	$0.899577\pi$
$770$		0			0
$771$	23.5045			0.846496
$772$		0			0
$773$		−	25.2523i		−	0.908261i	−0.890935	−	0.454131i	$-0.849950\pi$
$773$		−	25.2523i		−	0.908261i	0.890935	−	0.454131i	$-0.150050\pi$
$774$		0			0
$775$		−	37.9129i		−	1.36187i
$776$		0			0
$777$	−12.9909			−0.466046
$778$		0			0
$779$	18.4955			0.662668
$780$		0			0
$781$	16.7477			0.599281
$782$		0			0
$783$	37.9129			1.35490
$784$		0			0
$785$		0			0
$786$		0			0
$787$			4.87841i			0.173897i	0.996213	+	0.0869483i	$0.0277115\pi$
$787$			4.87841i			0.173897i	−0.996213	+	0.0869483i	$0.972289\pi$
$788$		0			0
$789$	35.0780			1.24881
$790$		0			0
$791$		−	13.7477i		−	0.488813i
$792$		0			0
$793$	20.0000	−	30.0000i	0.710221	−	1.06533i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	18.0000			0.637593			0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000			0.637593			0.318796	+	0.947823i	$0.396721\pi$
$798$		0			0
$799$			9.49545i			0.335925i
$800$		0			0
$801$		−	1.25227i		−	0.0442469i
$802$		0			0
$803$	12.9564			0.457223
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−27.9909			−0.985327
$808$		0			0
$809$	21.1652			0.744127			0.372064	−	0.928207i	$-0.378650\pi$
$809$	21.1652			0.744127			0.372064	+	0.928207i	$0.378650\pi$
$810$		0			0
$811$			22.2867i			0.782593i	0.920265	+	0.391297i	$0.127973\pi$
$811$			22.2867i			0.782593i	−0.920265	+	0.391297i	$0.872027\pi$
$812$		0			0
$813$			31.1216i			1.09148i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			4.41742i			0.154546i
$818$		0			0
$819$	0.495454	+	0.330303i	0.0173126	+	0.0115417i
$820$		0			0
$821$			15.4955i			0.540795i	0.962749	+	0.270398i	$0.0871551\pi$
$821$			15.4955i			0.540795i	−0.962749	+	0.270398i	$0.912845\pi$
$822$		0			0
$823$	−47.1216			−1.64256			−0.821278	−	0.570529i	$-0.806738\pi$
$823$	−47.1216			−1.64256			−0.821278	+	0.570529i	$0.806738\pi$
$824$		0			0
$825$			8.95644i			0.311823i
$826$		0			0
$827$		−	18.6261i		−	0.647694i	−0.946109	−	0.323847i	$-0.895024\pi$
$827$		−	18.6261i		−	0.647694i	0.946109	−	0.323847i	$-0.104976\pi$
$828$		0			0
$829$	−4.37386			−0.151911			−0.0759553	−	0.997111i	$-0.524201\pi$
$829$	−4.37386			−0.151911			−0.0759553	+	0.997111i	$0.524201\pi$
$830$		0			0
$831$	−33.5826			−1.16497
$832$		0			0
$833$	10.0871			0.349498
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	37.9129i		−	1.31046i
$838$		0			0
$839$			2.50455i			0.0864665i	0.999065	+	0.0432333i	$0.0137659\pi$
$839$			2.50455i			0.0864665i	−0.999065	+	0.0432333i	$0.986234\pi$
$840$		0			0
$841$	28.4955			0.982602
$842$		0			0
$843$		−	28.8784i		−	0.994625i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	0.791288i		−	0.0271890i
$848$		0			0
$849$	12.0871			0.414829
$850$		0			0
$851$			7.25227i			0.248605i
$852$		0			0
$853$			12.9564i			0.443620i	0.975090	+	0.221810i	$0.0711965\pi$
$853$			12.9564i			0.443620i	−0.975090	+	0.221810i	$0.928804\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	30.3303			1.03606			0.518032	−	0.855361i	$-0.326665\pi$
$857$	30.3303			1.03606			0.518032	+	0.855361i	$0.326665\pi$
$858$		0			0
$859$	−48.1216			−1.64189			−0.820944	−	0.571009i	$-0.806552\pi$
$859$	−48.1216			−1.64189			−0.820944	+	0.571009i	$0.806552\pi$
$860$		0			0
$861$	−11.8693			−0.404505
$862$		0			0
$863$		−	16.7477i		−	0.570099i	−0.958513	−	0.285050i	$-0.907990\pi$
$863$		−	16.7477i		−	0.570099i	0.958513	−	0.285050i	$-0.0920101\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	25.9655			0.881836
$868$		0			0
$869$			8.00000i			0.271381i
$870$		0			0
$871$	−9.49545	−	6.33030i	−0.321741	−	0.214494i
$872$		0			0
$873$			0.330303i			0.0111791i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	20.5390i		−	0.693553i	−0.937948	−	0.346777i	$-0.887276\pi$
$877$		−	20.5390i		−	0.693553i	0.937948	−	0.346777i	$-0.112724\pi$
$878$		0			0
$879$		−	10.7477i		−	0.362512i
$880$		0			0
$881$	5.37386			0.181050			0.0905250	−	0.995894i	$-0.471145\pi$
$881$	5.37386			0.181050			0.0905250	+	0.995894i	$0.471145\pi$
$882$		0			0
$883$	−1.37386			−0.0462342			−0.0231171	−	0.999733i	$-0.507359\pi$
$883$	−1.37386			−0.0462342			−0.0231171	+	0.999733i	$0.507359\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	28.7477			0.965254			0.482627	−	0.875826i	$-0.339683\pi$
$887$	28.7477			0.965254			0.482627	+	0.875826i	$0.339683\pi$
$888$		0			0
$889$		−	1.58258i		−	0.0530779i
$890$		0			0
$891$			9.58258i			0.321028i
$892$		0			0
$893$	−13.2523			−0.443470
$894$		0			0
$895$		0			0
$896$		0			0
$897$	2.83485	−	4.25227i	0.0946528	−	0.141979i
$898$		0			0
$899$		−	57.4955i		−	1.91758i
$900$		0			0
$901$	−3.49545			−0.116450
$902$		0			0
$903$		−	2.83485i		−	0.0943379i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−51.1216			−1.69746			−0.848732	−	0.528823i	$-0.822634\pi$
$907$	−51.1216			−1.69746			−0.848732	+	0.528823i	$0.822634\pi$
$908$		0			0
$909$	−0.660606			−0.0219109
$910$		0			0
$911$	39.9564			1.32382			0.661908	−	0.749585i	$-0.269747\pi$
$911$	39.9564			1.32382			0.661908	+	0.749585i	$0.269747\pi$
$912$		0			0
$913$	−2.37386			−0.0785634
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	3.49545i		−	0.115430i
$918$		0			0
$919$	−15.2523			−0.503126			−0.251563	−	0.967841i	$-0.580945\pi$
$919$	−15.2523			−0.503126			−0.251563	+	0.967841i	$0.580945\pi$
$920$		0			0
$921$			54.3303i			1.79024i
$922$		0			0
$923$	50.2432	+	33.4955i	1.65377	+	1.10252i
$924$		0			0
$925$		−	45.8258i		−	1.50674i
$926$		0			0
$927$	−1.33030			−0.0436929
$928$		0			0
$929$			38.2432i			1.25472i	0.778730	+	0.627359i	$0.215864\pi$
$929$			38.2432i			1.25472i	−0.778730	+	0.627359i	$0.784136\pi$
$930$		0			0
$931$			14.0780i			0.461389i
$932$		0			0
$933$	−15.2958			−0.500763
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−29.4955			−0.963574			−0.481787	−	0.876288i	$-0.660012\pi$
$937$	−29.4955			−0.963574			−0.481787	+	0.876288i	$0.660012\pi$
$938$		0			0
$939$	7.83485			0.255681
$940$		0			0
$941$		−	18.6261i		−	0.607195i	−0.952800	−	0.303597i	$-0.901812\pi$
$941$		−	18.6261i		−	0.607195i	0.952800	−	0.303597i	$-0.0981878\pi$
$942$		0			0
$943$			6.62614i			0.215777i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			18.0000i			0.584921i	0.956278	+	0.292461i	$0.0944741\pi$
$947$			18.0000i			0.584921i	−0.956278	+	0.292461i	$0.905526\pi$
$948$		0			0
$949$	38.8693	+	25.9129i	1.26175	+	0.841168i
$950$		0			0
$951$			2.24318i			0.0727401i
$952$		0			0
$953$	47.4083			1.53571			0.767853	−	0.640626i	$-0.221325\pi$
$953$	47.4083			1.53571			0.767853	+	0.640626i	$0.221325\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			13.5826i			0.439062i
$958$		0			0
$959$	−9.49545			−0.306624
$960$		0			0
$961$	−26.4955			−0.854692
$962$		0			0
$963$	−0.921970			−0.0297101
$964$		0			0
$965$		0			0
$966$		0			0
$967$			38.2087i			1.22871i	0.789030	+	0.614355i	$0.210584\pi$
$967$			38.2087i			1.22871i	−0.789030	+	0.614355i	$0.789416\pi$
$968$		0			0
$969$		−	6.26136i		−	0.201144i
$970$		0			0
$971$	−54.0345			−1.73405			−0.867024	−	0.498266i	$-0.833970\pi$
$971$	−54.0345			−1.73405			−0.867024	+	0.498266i	$0.833970\pi$
$972$		0			0
$973$			10.0871i			0.323378i
$974$		0			0
$975$	−17.9129	+	26.8693i	−0.573671	+	0.860507i
$976$		0			0
$977$		−	51.4955i		−	1.64749i	−0.566964	−	0.823743i	$-0.691882\pi$
$977$		−	51.4955i		−	1.64749i	0.566964	−	0.823743i	$-0.308118\pi$
$978$		0			0
$979$	−6.00000			−0.191761
$980$		0			0
$981$		−	2.96553i		−	0.0946821i
$982$		0			0
$983$			34.7477i			1.10828i	0.832423	+	0.554140i	$0.186953\pi$
$983$			34.7477i			1.10828i	−0.832423	+	0.554140i	$0.813047\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	8.50455			0.270703
$988$		0			0
$989$	−1.58258			−0.0503230
$990$		0			0
$991$	−20.1216			−0.639183			−0.319592	−	0.947555i	$-0.603546\pi$
$991$	−20.1216			−0.639183			−0.319592	+	0.947555i	$0.603546\pi$
$992$		0			0
$993$		−	37.9129i		−	1.20313i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	50.9909			1.61490			0.807449	−	0.589937i	$-0.200848\pi$
$997$	50.9909			1.61490			0.807449	+	0.589937i	$0.200848\pi$
$998$		0			0
$999$		−	45.8258i		−	1.44986i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.f.b.441.2	yes	4
3.2	odd	2		5148.2.e.b.1585.3		4
4.3	odd	2		2288.2.j.f.1585.3		4
13.5	odd	4		7436.2.a.j.1.1		2
13.8	odd	4		7436.2.a.i.1.1		2
13.12	even	2	inner	572.2.f.b.441.1	✓	4
39.38	odd	2		5148.2.e.b.1585.2		4
52.51	odd	2		2288.2.j.f.1585.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.f.b.441.1	✓	4	13.12	even	2	inner
572.2.f.b.441.2	yes	4	1.1	even	1	trivial
2288.2.j.f.1585.3		4	4.3	odd	2
2288.2.j.f.1585.4		4	52.51	odd	2
5148.2.e.b.1585.2		4	39.38	odd	2
5148.2.e.b.1585.3		4	3.2	odd	2
7436.2.a.i.1.1		2	13.8	odd	4
7436.2.a.j.1.1		2	13.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.79129 q^{3} +0.791288i q^{7} +0.208712 q^{9} +O(q^{10})\)	\(q-1.79129 q^{3} +0.791288i q^{7} +0.208712 q^{9} -1.00000i q^{11} +(2.00000 - 3.00000i) q^{13} +1.58258 q^{17} +2.20871i q^{19} -1.41742i q^{21} -0.791288 q^{23} +5.00000 q^{25} +5.00000 q^{27} +7.58258 q^{29} -7.58258i q^{31} +1.79129i q^{33} -9.16515i q^{37} +(-3.58258 + 5.37386i) q^{39} -8.37386i q^{41} +2.00000 q^{43} +6.00000i q^{47} +6.37386 q^{49} -2.83485 q^{51} -2.20871 q^{53} -3.95644i q^{57} -6.00000i q^{59} +10.0000 q^{61} +0.165151i q^{63} -3.16515i q^{67} +1.41742 q^{69} +16.7477i q^{71} +12.9564i q^{73} -8.95644 q^{75} +0.791288 q^{77} -8.00000 q^{79} -9.58258 q^{81} -2.37386i q^{83} -13.5826 q^{87} -6.00000i q^{89} +(2.37386 + 1.58258i) q^{91} +13.5826i q^{93} +1.58258i q^{97} -0.208712i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 10 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 + 10 * q^9	\( 4 q + 2 q^{3} + 10 q^{9} + 8 q^{13} - 12 q^{17} + 6 q^{23} + 20 q^{25} + 20 q^{27} + 12 q^{29} + 4 q^{39} + 8 q^{43} - 2 q^{49} - 48 q^{51} - 18 q^{53} + 40 q^{61} + 24 q^{69} + 10 q^{75} - 6 q^{77} - 32 q^{79} - 20 q^{81} - 36 q^{87} - 18 q^{91}+O(q^{100}) \) 4 * q + 2 * q^3 + 10 * q^9 + 8 * q^13 - 12 * q^17 + 6 * q^23 + 20 * q^25 + 20 * q^27 + 12 * q^29 + 4 * q^39 + 8 * q^43 - 2 * q^49 - 48 * q^51 - 18 * q^53 + 40 * q^61 + 24 * q^69 + 10 * q^75 - 6 * q^77 - 32 * q^79 - 20 * q^81 - 36 * q^87 - 18 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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