LMFDB - Embedded newform 572.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [572,2,Mod(1,572)]

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, 0]))

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

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

chi := DirichletCharacter("572.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 572 = 2^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	572.a (trivial)

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:	yes
Analytic conductor:	$4.56744299562$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.79129$ of defining polynomial
Character	$\chi$	$=$	572.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.79129 q^{3} -2.00000 q^{5} +3.79129 q^{7} +4.79129 q^{9} +O(q^{10})$	$q+2.79129 q^{3} -2.00000 q^{5} +3.79129 q^{7} +4.79129 q^{9} +1.00000 q^{11} +1.00000 q^{13} -5.58258 q^{15} -8.37386 q^{19} +10.5826 q^{21} +0.208712 q^{23} -1.00000 q^{25} +5.00000 q^{27} +7.58258 q^{29} +4.00000 q^{31} +2.79129 q^{33} -7.58258 q^{35} +8.00000 q^{37} +2.79129 q^{39} -5.79129 q^{41} +8.00000 q^{43} -9.58258 q^{45} -11.5826 q^{47} +7.37386 q^{49} -9.95644 q^{53} -2.00000 q^{55} -23.3739 q^{57} -9.58258 q^{59} +5.58258 q^{61} +18.1652 q^{63} -2.00000 q^{65} +8.00000 q^{67} +0.582576 q^{69} -8.00000 q^{71} -12.3739 q^{73} -2.79129 q^{75} +3.79129 q^{77} +2.41742 q^{79} -0.417424 q^{81} +3.37386 q^{83} +21.1652 q^{87} -13.5826 q^{89} +3.79129 q^{91} +11.1652 q^{93} +16.7477 q^{95} -11.5826 q^{97} +4.79129 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 4 q^{5} + 3 q^{7} + 5 q^{9}+O(q^{10}) $ 2 * q + q^3 - 4 * q^5 + 3 * q^7 + 5 * q^9	$ 2 q + q^{3} - 4 q^{5} + 3 q^{7} + 5 q^{9} + 2 q^{11} + 2 q^{13} - 2 q^{15} - 3 q^{19} + 12 q^{21} + 5 q^{23} - 2 q^{25} + 10 q^{27} + 6 q^{29} + 8 q^{31} + q^{33} - 6 q^{35} + 16 q^{37} + q^{39} - 7 q^{41} + 16 q^{43} - 10 q^{45} - 14 q^{47} + q^{49} + 3 q^{53} - 4 q^{55} - 33 q^{57} - 10 q^{59} + 2 q^{61} + 18 q^{63} - 4 q^{65} + 16 q^{67} - 8 q^{69} - 16 q^{71} - 11 q^{73} - q^{75} + 3 q^{77} + 14 q^{79} - 10 q^{81} - 7 q^{83} + 24 q^{87} - 18 q^{89} + 3 q^{91} + 4 q^{93} + 6 q^{95} - 14 q^{97} + 5 q^{99}+O(q^{100}) $ 2 * q + q^3 - 4 * q^5 + 3 * q^7 + 5 * q^9 + 2 * q^11 + 2 * q^13 - 2 * q^15 - 3 * q^19 + 12 * q^21 + 5 * q^23 - 2 * q^25 + 10 * q^27 + 6 * q^29 + 8 * q^31 + q^33 - 6 * q^35 + 16 * q^37 + q^39 - 7 * q^41 + 16 * q^43 - 10 * q^45 - 14 * q^47 + q^49 + 3 * q^53 - 4 * q^55 - 33 * q^57 - 10 * q^59 + 2 * q^61 + 18 * q^63 - 4 * q^65 + 16 * q^67 - 8 * q^69 - 16 * q^71 - 11 * q^73 - q^75 + 3 * q^77 + 14 * q^79 - 10 * q^81 - 7 * q^83 + 24 * q^87 - 18 * q^89 + 3 * q^91 + 4 * q^93 + 6 * q^95 - 14 * q^97 + 5 * q^99

Display 100 coefficients 10 coefficients

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
$3$	2.79129	1.61155	0.805775	−	0.592221i	$-0.201749\pi$
$3$	2.79129	1.61155	0.805775	+	0.592221i	$0.201749\pi$
$4$	0	0
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	3.79129	1.43297	0.716486	−	0.697601i	$-0.245749\pi$
$7$	3.79129	1.43297	0.716486	+	0.697601i	$0.245749\pi$
$8$	0	0
$9$	4.79129	1.59710
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−5.58258	−1.44141
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−8.37386	−1.92110	−0.960548	−	0.278114i	$-0.910291\pi$
$19$	−8.37386	−1.92110	−0.960548	+	0.278114i	$0.910291\pi$
$20$	0	0
$21$	10.5826	2.30931
$22$	0	0
$23$	0.208712	0.0435195	0.0217597	−	0.999763i	$-0.493073\pi$
$23$	0.208712	0.0435195	0.0217597	+	0.999763i	$0.493073\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	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
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	2.79129	0.485901
$34$	0	0
$35$	−7.58258	−1.28169
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	2.79129	0.446964
$40$	0	0
$41$	−5.79129	−0.904447	−0.452224	−	0.891905i	$-0.649369\pi$
$41$	−5.79129	−0.904447	−0.452224	+	0.891905i	$0.649369\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	−9.58258	−1.42849
$46$	0	0
$47$	−11.5826	−1.68949	−0.844746	−	0.535167i	$-0.820249\pi$
$47$	−11.5826	−1.68949	−0.844746	+	0.535167i	$0.820249\pi$
$48$	0	0
$49$	7.37386	1.05341
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.95644	−1.36762	−0.683811	−	0.729659i	$-0.739679\pi$
$53$	−9.95644	−1.36762	−0.683811	+	0.729659i	$0.739679\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	−23.3739	−3.09594
$58$	0	0
$59$	−9.58258	−1.24755	−0.623773	−	0.781606i	$-0.714401\pi$
$59$	−9.58258	−1.24755	−0.623773	+	0.781606i	$0.714401\pi$
$60$	0	0
$61$	5.58258	0.714776	0.357388	−	0.933956i	$-0.383667\pi$
$61$	5.58258	0.714776	0.357388	+	0.933956i	$0.383667\pi$
$62$	0	0
$63$	18.1652	2.28859
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	0.582576	0.0701339
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−12.3739	−1.44825	−0.724126	−	0.689668i	$-0.757756\pi$
$73$	−12.3739	−1.44825	−0.724126	+	0.689668i	$0.757756\pi$
$74$	0	0
$75$	−2.79129	−0.322310
$76$	0	0
$77$	3.79129	0.432057
$78$	0	0
$79$	2.41742	0.271981	0.135991	−	0.990710i	$-0.456578\pi$
$79$	2.41742	0.271981	0.135991	+	0.990710i	$0.456578\pi$
$80$	0	0
$81$	−0.417424	−0.0463805
$82$	0	0
$83$	3.37386	0.370330	0.185165	−	0.982707i	$-0.440718\pi$
$83$	3.37386	0.370330	0.185165	+	0.982707i	$0.440718\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	21.1652	2.26914
$88$	0	0
$89$	−13.5826	−1.43975	−0.719875	−	0.694104i	$-0.755801\pi$
$89$	−13.5826	−1.43975	−0.719875	+	0.694104i	$0.755801\pi$
$90$	0	0
$91$	3.79129	0.397435
$92$	0	0
$93$	11.1652	1.15777
$94$	0	0
$95$	16.7477	1.71828
$96$	0	0
$97$	−11.5826	−1.17603	−0.588016	−	0.808849i	$-0.700091\pi$
$97$	−11.5826	−1.17603	−0.588016	+	0.808849i	$0.700091\pi$
$98$	0	0
$99$	4.79129	0.481543
$100$	0	0
$101$	11.1652	1.11097	0.555487	−	0.831525i	$-0.312532\pi$
$101$	11.1652	1.11097	0.555487	+	0.831525i	$0.312532\pi$
$102$	0	0
$103$	−14.5390	−1.43257	−0.716286	−	0.697807i	$-0.754159\pi$
$103$	−14.5390	−1.43257	−0.716286	+	0.697807i	$0.754159\pi$
$104$	0	0
$105$	−21.1652	−2.06551
$106$	0	0
$107$	−5.58258	−0.539688	−0.269844	−	0.962904i	$-0.586972\pi$
$107$	−5.58258	−0.539688	−0.269844	+	0.962904i	$0.586972\pi$
$108$	0	0
$109$	8.37386	0.802071	0.401035	−	0.916063i	$-0.368650\pi$
$109$	8.37386	0.802071	0.401035	+	0.916063i	$0.368650\pi$
$110$	0	0
$111$	22.3303	2.11950
$112$	0	0
$113$	−6.79129	−0.638871	−0.319435	−	0.947608i	$-0.603493\pi$
$113$	−6.79129	−0.638871	−0.319435	+	0.947608i	$0.603493\pi$
$114$	0	0
$115$	−0.417424	−0.0389250
$116$	0	0
$117$	4.79129	0.442955
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−16.1652	−1.45756
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	1.58258	0.140431	0.0702154	−	0.997532i	$-0.477631\pi$
$127$	1.58258	0.140431	0.0702154	+	0.997532i	$0.477631\pi$
$128$	0	0
$129$	22.3303	1.96607
$130$	0	0
$131$	−1.16515	−0.101800	−0.0508999	−	0.998704i	$-0.516209\pi$
$131$	−1.16515	−0.101800	−0.0508999	+	0.998704i	$0.516209\pi$
$132$	0	0
$133$	−31.7477	−2.75288
$134$	0	0
$135$	−10.0000	−0.860663
$136$	0	0
$137$	3.16515	0.270417	0.135209	−	0.990817i	$-0.456830\pi$
$137$	3.16515	0.270417	0.135209	+	0.990817i	$0.456830\pi$
$138$	0	0
$139$	14.7477	1.25089	0.625443	−	0.780270i	$-0.284918\pi$
$139$	14.7477	1.25089	0.625443	+	0.780270i	$0.284918\pi$
$140$	0	0
$141$	−32.3303	−2.72270
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−15.1652	−1.25940
$146$	0	0
$147$	20.5826	1.69762
$148$	0	0
$149$	−13.3739	−1.09563	−0.547815	−	0.836600i	$-0.684540\pi$
$149$	−13.3739	−1.09563	−0.547815	+	0.836600i	$0.684540\pi$
$150$	0	0
$151$	0.834849	0.0679390	0.0339695	−	0.999423i	$-0.489185\pi$
$151$	0.834849	0.0679390	0.0339695	+	0.999423i	$0.489185\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	9.79129	0.781430	0.390715	−	0.920512i	$-0.372228\pi$
$157$	9.79129	0.781430	0.390715	+	0.920512i	$0.372228\pi$
$158$	0	0
$159$	−27.7913	−2.20399
$160$	0	0
$161$	0.791288	0.0623622
$162$	0	0
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	0	0
$165$	−5.58258	−0.434603
$166$	0	0
$167$	5.95644	0.460923	0.230462	−	0.973081i	$-0.425976\pi$
$167$	5.95644	0.460923	0.230462	+	0.973081i	$0.425976\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−40.1216	−3.06817
$172$	0	0
$173$	−18.7477	−1.42536	−0.712682	−	0.701488i	$-0.752520\pi$
$173$	−18.7477	−1.42536	−0.712682	+	0.701488i	$0.752520\pi$
$174$	0	0
$175$	−3.79129	−0.286594
$176$	0	0
$177$	−26.7477	−2.01048
$178$	0	0
$179$	−11.1652	−0.834523	−0.417261	−	0.908787i	$-0.637010\pi$
$179$	−11.1652	−0.834523	−0.417261	+	0.908787i	$0.637010\pi$
$180$	0	0
$181$	−17.9564	−1.33469	−0.667346	−	0.744748i	$-0.732570\pi$
$181$	−17.9564	−1.33469	−0.667346	+	0.744748i	$0.732570\pi$
$182$	0	0
$183$	15.5826	1.15190
$184$	0	0
$185$	−16.0000	−1.17634
$186$	0	0
$187$	0	0
$188$	0	0
$189$	18.9564	1.37888
$190$	0	0
$191$	21.9564	1.58871	0.794356	−	0.607452i	$-0.207808\pi$
$191$	21.9564	1.58871	0.794356	+	0.607452i	$0.207808\pi$
$192$	0	0
$193$	−1.37386	−0.0988929	−0.0494464	−	0.998777i	$-0.515746\pi$
$193$	−1.37386	−0.0988929	−0.0494464	+	0.998777i	$0.515746\pi$
$194$	0	0
$195$	−5.58258	−0.399777
$196$	0	0
$197$	9.20871	0.656094	0.328047	−	0.944661i	$-0.393610\pi$
$197$	9.20871	0.656094	0.328047	+	0.944661i	$0.393610\pi$
$198$	0	0
$199$	9.20871	0.652788	0.326394	−	0.945234i	$-0.394166\pi$
$199$	9.20871	0.652788	0.326394	+	0.945234i	$0.394166\pi$
$200$	0	0
$201$	22.3303	1.57506
$202$	0	0
$203$	28.7477	2.01769
$204$	0	0
$205$	11.5826	0.808962
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−8.37386	−0.579232
$210$	0	0
$211$	17.5826	1.21043	0.605217	−	0.796060i	$-0.293086\pi$
$211$	17.5826	1.21043	0.605217	+	0.796060i	$0.293086\pi$
$212$	0	0
$213$	−22.3303	−1.53005
$214$	0	0
$215$	−16.0000	−1.09119
$216$	0	0
$217$	15.1652	1.02948
$218$	0	0
$219$	−34.5390	−2.33393
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−9.58258	−0.641697	−0.320848	−	0.947131i	$-0.603968\pi$
$223$	−9.58258	−0.641697	−0.320848	+	0.947131i	$0.603968\pi$
$224$	0	0
$225$	−4.79129	−0.319419
$226$	0	0
$227$	−5.20871	−0.345714	−0.172857	−	0.984947i	$-0.555300\pi$
$227$	−5.20871	−0.345714	−0.172857	+	0.984947i	$0.555300\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	10.5826	0.696282
$232$	0	0
$233$	16.4174	1.07554	0.537771	−	0.843091i	$-0.319267\pi$
$233$	16.4174	1.07554	0.537771	+	0.843091i	$0.319267\pi$
$234$	0	0
$235$	23.1652	1.51113
$236$	0	0
$237$	6.74773	0.438312
$238$	0	0
$239$	12.3739	0.800399	0.400199	−	0.916428i	$-0.368941\pi$
$239$	12.3739	0.800399	0.400199	+	0.916428i	$0.368941\pi$
$240$	0	0
$241$	17.7913	1.14604	0.573019	−	0.819542i	$-0.305772\pi$
$241$	17.7913	1.14604	0.573019	+	0.819542i	$0.305772\pi$
$242$	0	0
$243$	−16.1652	−1.03699
$244$	0	0
$245$	−14.7477	−0.942198
$246$	0	0
$247$	−8.37386	−0.532816
$248$	0	0
$249$	9.41742	0.596805
$250$	0	0
$251$	14.5390	0.917694	0.458847	−	0.888515i	$-0.348263\pi$
$251$	14.5390	0.917694	0.458847	+	0.888515i	$0.348263\pi$
$252$	0	0
$253$	0.208712	0.0131216
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.5390	1.28119	0.640594	−	0.767880i	$-0.278688\pi$
$257$	20.5390	1.28119	0.640594	+	0.767880i	$0.278688\pi$
$258$	0	0
$259$	30.3303	1.88463
$260$	0	0
$261$	36.3303	2.24879
$262$	0	0
$263$	14.0000	0.863277	0.431638	−	0.902047i	$-0.357936\pi$
$263$	14.0000	0.863277	0.431638	+	0.902047i	$0.357936\pi$
$264$	0	0
$265$	19.9129	1.22324
$266$	0	0
$267$	−37.9129	−2.32023
$268$	0	0
$269$	9.37386	0.571535	0.285767	−	0.958299i	$-0.407752\pi$
$269$	9.37386	0.571535	0.285767	+	0.958299i	$0.407752\pi$
$270$	0	0
$271$	−6.04356	−0.367120	−0.183560	−	0.983008i	$-0.558762\pi$
$271$	−6.04356	−0.367120	−0.183560	+	0.983008i	$0.558762\pi$
$272$	0	0
$273$	10.5826	0.640487
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−11.1652	−0.670849	−0.335424	−	0.942067i	$-0.608880\pi$
$277$	−11.1652	−0.670849	−0.335424	+	0.942067i	$0.608880\pi$
$278$	0	0
$279$	19.1652	1.14739
$280$	0	0
$281$	26.7913	1.59823	0.799117	−	0.601175i	$-0.205301\pi$
$281$	26.7913	1.59823	0.799117	+	0.601175i	$0.205301\pi$
$282$	0	0
$283$	−8.33030	−0.495185	−0.247593	−	0.968864i	$-0.579639\pi$
$283$	−8.33030	−0.495185	−0.247593	+	0.968864i	$0.579639\pi$
$284$	0	0
$285$	46.7477	2.76910
$286$	0	0
$287$	−21.9564	−1.29605
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−32.3303	−1.89524
$292$	0	0
$293$	6.83485	0.399296	0.199648	−	0.979868i	$-0.436020\pi$
$293$	6.83485	0.399296	0.199648	+	0.979868i	$0.436020\pi$
$294$	0	0
$295$	19.1652	1.11584
$296$	0	0
$297$	5.00000	0.290129
$298$	0	0
$299$	0.208712	0.0120701
$300$	0	0
$301$	30.3303	1.74821
$302$	0	0
$303$	31.1652	1.79039
$304$	0	0
$305$	−11.1652	−0.639315
$306$	0	0
$307$	26.3303	1.50275	0.751375	−	0.659876i	$-0.229391\pi$
$307$	26.3303	1.50275	0.751375	+	0.659876i	$0.229391\pi$
$308$	0	0
$309$	−40.5826	−2.30866
$310$	0	0
$311$	27.5390	1.56159	0.780797	−	0.624785i	$-0.214813\pi$
$311$	27.5390	1.56159	0.780797	+	0.624785i	$0.214813\pi$
$312$	0	0
$313$	−21.7913	−1.23172	−0.615858	−	0.787857i	$-0.711191\pi$
$313$	−21.7913	−1.23172	−0.615858	+	0.787857i	$0.711191\pi$
$314$	0	0
$315$	−36.3303	−2.04698
$316$	0	0
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	0	0
$319$	7.58258	0.424543
$320$	0	0
$321$	−15.5826	−0.869735
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	23.3739	1.29258
$328$	0	0
$329$	−43.9129	−2.42100
$330$	0	0
$331$	19.1652	1.05341	0.526706	−	0.850048i	$-0.323427\pi$
$331$	19.1652	1.05341	0.526706	+	0.850048i	$0.323427\pi$
$332$	0	0
$333$	38.3303	2.10049
$334$	0	0
$335$	−16.0000	−0.874173
$336$	0	0
$337$	−23.4955	−1.27988	−0.639939	−	0.768425i	$-0.721041\pi$
$337$	−23.4955	−1.27988	−0.639939	+	0.768425i	$0.721041\pi$
$338$	0	0
$339$	−18.9564	−1.02957
$340$	0	0
$341$	4.00000	0.216612
$342$	0	0
$343$	1.41742	0.0765337
$344$	0	0
$345$	−1.16515	−0.0627296
$346$	0	0
$347$	28.3303	1.52085	0.760425	−	0.649426i	$-0.224991\pi$
$347$	28.3303	1.52085	0.760425	+	0.649426i	$0.224991\pi$
$348$	0	0
$349$	12.5390	0.671198	0.335599	−	0.942005i	$-0.391061\pi$
$349$	12.5390	0.671198	0.335599	+	0.942005i	$0.391061\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	11.9129	0.634059	0.317029	−	0.948416i	$-0.397315\pi$
$353$	11.9129	0.634059	0.317029	+	0.948416i	$0.397315\pi$
$354$	0	0
$355$	16.0000	0.849192
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−37.2867	−1.96792	−0.983959	−	0.178392i	$-0.942910\pi$
$359$	−37.2867	−1.96792	−0.983959	+	0.178392i	$0.942910\pi$
$360$	0	0
$361$	51.1216	2.69061
$362$	0	0
$363$	2.79129	0.146505
$364$	0	0
$365$	24.7477	1.29536
$366$	0	0
$367$	−21.9564	−1.14612	−0.573058	−	0.819515i	$-0.694243\pi$
$367$	−21.9564	−1.14612	−0.573058	+	0.819515i	$0.694243\pi$
$368$	0	0
$369$	−27.7477	−1.44449
$370$	0	0
$371$	−37.7477	−1.95976
$372$	0	0
$373$	−29.0780	−1.50560	−0.752802	−	0.658247i	$-0.771298\pi$
$373$	−29.0780	−1.50560	−0.752802	+	0.658247i	$0.771298\pi$
$374$	0	0
$375$	33.4955	1.72970
$376$	0	0
$377$	7.58258	0.390523
$378$	0	0
$379$	−34.7477	−1.78487	−0.892435	−	0.451175i	$-0.851005\pi$
$379$	−34.7477	−1.78487	−0.892435	+	0.451175i	$0.851005\pi$
$380$	0	0
$381$	4.41742	0.226312
$382$	0	0
$383$	10.7477	0.549183	0.274592	−	0.961561i	$-0.411457\pi$
$383$	10.7477	0.549183	0.274592	+	0.961561i	$0.411457\pi$
$384$	0	0
$385$	−7.58258	−0.386444
$386$	0	0
$387$	38.3303	1.94844
$388$	0	0
$389$	−4.37386	−0.221764	−0.110882	−	0.993834i	$-0.535368\pi$
$389$	−4.37386	−0.221764	−0.110882	+	0.993834i	$0.535368\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−3.25227	−0.164055
$394$	0	0
$395$	−4.83485	−0.243268
$396$	0	0
$397$	6.83485	0.343031	0.171516	−	0.985181i	$-0.445134\pi$
$397$	6.83485	0.343031	0.171516	+	0.985181i	$0.445134\pi$
$398$	0	0
$399$	−88.6170	−4.43640
$400$	0	0
$401$	−18.3303	−0.915372	−0.457686	−	0.889114i	$-0.651322\pi$
$401$	−18.3303	−0.915372	−0.457686	+	0.889114i	$0.651322\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	0.834849	0.0414840
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	−6.83485	−0.337962	−0.168981	−	0.985619i	$-0.554048\pi$
$409$	−6.83485	−0.337962	−0.168981	+	0.985619i	$0.554048\pi$
$410$	0	0
$411$	8.83485	0.435791
$412$	0	0
$413$	−36.3303	−1.78770
$414$	0	0
$415$	−6.74773	−0.331233
$416$	0	0
$417$	41.1652	2.01587
$418$	0	0
$419$	−30.9564	−1.51232	−0.756161	−	0.654386i	$-0.772927\pi$
$419$	−30.9564	−1.51232	−0.756161	+	0.654386i	$0.772927\pi$
$420$	0	0
$421$	10.3303	0.503468	0.251734	−	0.967796i	$-0.418999\pi$
$421$	10.3303	0.503468	0.251734	+	0.967796i	$0.418999\pi$
$422$	0	0
$423$	−55.4955	−2.69828
$424$	0	0
$425$	0	0
$426$	0	0
$427$	21.1652	1.02425
$428$	0	0
$429$	2.79129	0.134765
$430$	0	0
$431$	26.9564	1.29845	0.649223	−	0.760598i	$-0.275094\pi$
$431$	26.9564	1.29845	0.649223	+	0.760598i	$0.275094\pi$
$432$	0	0
$433$	24.3739	1.17133	0.585667	−	0.810552i	$-0.300833\pi$
$433$	24.3739	1.17133	0.585667	+	0.810552i	$0.300833\pi$
$434$	0	0
$435$	−42.3303	−2.02958
$436$	0	0
$437$	−1.74773	−0.0836051
$438$	0	0
$439$	−8.33030	−0.397584	−0.198792	−	0.980042i	$-0.563702\pi$
$439$	−8.33030	−0.397584	−0.198792	+	0.980042i	$0.563702\pi$
$440$	0	0
$441$	35.3303	1.68240
$442$	0	0
$443$	8.20871	0.390008	0.195004	−	0.980802i	$-0.437528\pi$
$443$	8.20871	0.390008	0.195004	+	0.980802i	$0.437528\pi$
$444$	0	0
$445$	27.1652	1.28775
$446$	0	0
$447$	−37.3303	−1.76566
$448$	0	0
$449$	−3.66970	−0.173184	−0.0865919	−	0.996244i	$-0.527598\pi$
$449$	−3.66970	−0.173184	−0.0865919	+	0.996244i	$0.527598\pi$
$450$	0	0
$451$	−5.79129	−0.272701
$452$	0	0
$453$	2.33030	0.109487
$454$	0	0
$455$	−7.58258	−0.355477
$456$	0	0
$457$	−25.2087	−1.17921	−0.589607	−	0.807690i	$-0.700717\pi$
$457$	−25.2087	−1.17921	−0.589607	+	0.807690i	$0.700717\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	17.1216	0.797432	0.398716	−	0.917074i	$-0.369456\pi$
$461$	17.1216	0.797432	0.398716	+	0.917074i	$0.369456\pi$
$462$	0	0
$463$	27.1652	1.26247	0.631236	−	0.775591i	$-0.282548\pi$
$463$	27.1652	1.26247	0.631236	+	0.775591i	$0.282548\pi$
$464$	0	0
$465$	−22.3303	−1.03554
$466$	0	0
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	0	0
$469$	30.3303	1.40052
$470$	0	0
$471$	27.3303	1.25931
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	8.37386	0.384219
$476$	0	0
$477$	−47.7042	−2.18422
$478$	0	0
$479$	−8.83485	−0.403675	−0.201837	−	0.979419i	$-0.564691\pi$
$479$	−8.83485	−0.403675	−0.201837	+	0.979419i	$0.564691\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	2.20871	0.100500
$484$	0	0
$485$	23.1652	1.05188
$486$	0	0
$487$	31.0780	1.40828	0.704140	−	0.710061i	$-0.251333\pi$
$487$	31.0780	1.40828	0.704140	+	0.710061i	$0.251333\pi$
$488$	0	0
$489$	16.7477	0.757358
$490$	0	0
$491$	−13.1652	−0.594135	−0.297067	−	0.954856i	$-0.596009\pi$
$491$	−13.1652	−0.594135	−0.297067	+	0.954856i	$0.596009\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−9.58258	−0.430705
$496$	0	0
$497$	−30.3303	−1.36050
$498$	0	0
$499$	13.1652	0.589353	0.294677	−	0.955597i	$-0.404788\pi$
$499$	13.1652	0.589353	0.294677	+	0.955597i	$0.404788\pi$
$500$	0	0
$501$	16.6261	0.742801
$502$	0	0
$503$	−3.16515	−0.141127	−0.0705636	−	0.997507i	$-0.522480\pi$
$503$	−3.16515	−0.141127	−0.0705636	+	0.997507i	$0.522480\pi$
$504$	0	0
$505$	−22.3303	−0.993685
$506$	0	0
$507$	2.79129	0.123965
$508$	0	0
$509$	41.0780	1.82075	0.910376	−	0.413782i	$-0.135793\pi$
$509$	41.0780	1.82075	0.910376	+	0.413782i	$0.135793\pi$
$510$	0	0
$511$	−46.9129	−2.07530
$512$	0	0
$513$	−41.8693	−1.84858
$514$	0	0
$515$	29.0780	1.28133
$516$	0	0
$517$	−11.5826	−0.509401
$518$	0	0
$519$	−52.3303	−2.29705
$520$	0	0
$521$	23.2867	1.02021	0.510105	−	0.860112i	$-0.329606\pi$
$521$	23.2867	1.02021	0.510105	+	0.860112i	$0.329606\pi$
$522$	0	0
$523$	9.16515	0.400764	0.200382	−	0.979718i	$-0.435782\pi$
$523$	9.16515	0.400764	0.200382	+	0.979718i	$0.435782\pi$
$524$	0	0
$525$	−10.5826	−0.461861
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−22.9564	−0.998106
$530$	0	0
$531$	−45.9129	−1.99245
$532$	0	0
$533$	−5.79129	−0.250849
$534$	0	0
$535$	11.1652	0.482712
$536$	0	0
$537$	−31.1652	−1.34488
$538$	0	0
$539$	7.37386	0.317615
$540$	0	0
$541$	−33.7913	−1.45280	−0.726400	−	0.687272i	$-0.758808\pi$
$541$	−33.7913	−1.45280	−0.726400	+	0.687272i	$0.758808\pi$
$542$	0	0
$543$	−50.1216	−2.15092
$544$	0	0
$545$	−16.7477	−0.717394
$546$	0	0
$547$	36.6606	1.56749	0.783747	−	0.621080i	$-0.213306\pi$
$547$	36.6606	1.56749	0.783747	+	0.621080i	$0.213306\pi$
$548$	0	0
$549$	26.7477	1.14157
$550$	0	0
$551$	−63.4955	−2.70500
$552$	0	0
$553$	9.16515	0.389742
$554$	0	0
$555$	−44.6606	−1.89574
$556$	0	0
$557$	40.1216	1.70001	0.850003	−	0.526778i	$-0.176600\pi$
$557$	40.1216	1.70001	0.850003	+	0.526778i	$0.176600\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2.74773	0.115803	0.0579014	−	0.998322i	$-0.481559\pi$
$563$	2.74773	0.115803	0.0579014	+	0.998322i	$0.481559\pi$
$564$	0	0
$565$	13.5826	0.571423
$566$	0	0
$567$	−1.58258	−0.0664619
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	2.74773	0.114989	0.0574944	−	0.998346i	$-0.481689\pi$
$571$	2.74773	0.114989	0.0574944	+	0.998346i	$0.481689\pi$
$572$	0	0
$573$	61.2867	2.56029
$574$	0	0
$575$	−0.208712	−0.00870390
$576$	0	0
$577$	13.5826	0.565450	0.282725	−	0.959201i	$-0.408762\pi$
$577$	13.5826	0.565450	0.282725	+	0.959201i	$0.408762\pi$
$578$	0	0
$579$	−3.83485	−0.159371
$580$	0	0
$581$	12.7913	0.530672
$582$	0	0
$583$	−9.95644	−0.412354
$584$	0	0
$585$	−9.58258	−0.396191
$586$	0	0
$587$	−13.2523	−0.546980	−0.273490	−	0.961875i	$-0.588178\pi$
$587$	−13.2523	−0.546980	−0.273490	+	0.961875i	$0.588178\pi$
$588$	0	0
$589$	−33.4955	−1.38016
$590$	0	0
$591$	25.7042	1.05733
$592$	0	0
$593$	28.9564	1.18910	0.594549	−	0.804059i	$-0.297331\pi$
$593$	28.9564	1.18910	0.594549	+	0.804059i	$0.297331\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	25.7042	1.05200
$598$	0	0
$599$	37.1216	1.51675	0.758374	−	0.651820i	$-0.225994\pi$
$599$	37.1216	1.51675	0.758374	+	0.651820i	$0.225994\pi$
$600$	0	0
$601$	−27.5826	−1.12512	−0.562558	−	0.826758i	$-0.690183\pi$
$601$	−27.5826	−1.12512	−0.562558	+	0.826758i	$0.690183\pi$
$602$	0	0
$603$	38.3303	1.56093
$604$	0	0
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	41.9129	1.70119	0.850596	−	0.525820i	$-0.176242\pi$
$607$	41.9129	1.70119	0.850596	+	0.525820i	$0.176242\pi$
$608$	0	0
$609$	80.2432	3.25162
$610$	0	0
$611$	−11.5826	−0.468581
$612$	0	0
$613$	−4.37386	−0.176659	−0.0883293	−	0.996091i	$-0.528153\pi$
$613$	−4.37386	−0.176659	−0.0883293	+	0.996091i	$0.528153\pi$
$614$	0	0
$615$	32.3303	1.30368
$616$	0	0
$617$	−12.4174	−0.499907	−0.249953	−	0.968258i	$-0.580415\pi$
$617$	−12.4174	−0.499907	−0.249953	+	0.968258i	$0.580415\pi$
$618$	0	0
$619$	32.6606	1.31274	0.656370	−	0.754439i	$-0.272091\pi$
$619$	32.6606	1.31274	0.656370	+	0.754439i	$0.272091\pi$
$620$	0	0
$621$	1.04356	0.0418767
$622$	0	0
$623$	−51.4955	−2.06312
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	−23.3739	−0.933462
$628$	0	0
$629$	0	0
$630$	0	0
$631$	27.1652	1.08143	0.540714	−	0.841207i	$-0.318154\pi$
$631$	27.1652	1.08143	0.540714	+	0.841207i	$0.318154\pi$
$632$	0	0
$633$	49.0780	1.95068
$634$	0	0
$635$	−3.16515	−0.125605
$636$	0	0
$637$	7.37386	0.292163
$638$	0	0
$639$	−38.3303	−1.51632
$640$	0	0
$641$	−39.7042	−1.56822	−0.784110	−	0.620622i	$-0.786880\pi$
$641$	−39.7042	−1.56822	−0.784110	+	0.620622i	$0.786880\pi$
$642$	0	0
$643$	18.7477	0.739338	0.369669	−	0.929163i	$-0.379471\pi$
$643$	18.7477	0.739338	0.369669	+	0.929163i	$0.379471\pi$
$644$	0	0
$645$	−44.6606	−1.75851
$646$	0	0
$647$	−16.2867	−0.640298	−0.320149	−	0.947367i	$-0.603733\pi$
$647$	−16.2867	−0.640298	−0.320149	+	0.947367i	$0.603733\pi$
$648$	0	0
$649$	−9.58258	−0.376149
$650$	0	0
$651$	42.3303	1.65906
$652$	0	0
$653$	20.3303	0.795586	0.397793	−	0.917475i	$-0.369776\pi$
$653$	20.3303	0.795586	0.397793	+	0.917475i	$0.369776\pi$
$654$	0	0
$655$	2.33030	0.0910525
$656$	0	0
$657$	−59.2867	−2.31300
$658$	0	0
$659$	−25.1652	−0.980295	−0.490148	−	0.871639i	$-0.663057\pi$
$659$	−25.1652	−0.980295	−0.490148	+	0.871639i	$0.663057\pi$
$660$	0	0
$661$	−34.2432	−1.33191	−0.665953	−	0.745994i	$-0.731975\pi$
$661$	−34.2432	−1.33191	−0.665953	+	0.745994i	$0.731975\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	63.4955	2.46225
$666$	0	0
$667$	1.58258	0.0612776
$668$	0	0
$669$	−26.7477	−1.03413
$670$	0	0
$671$	5.58258	0.215513
$672$	0	0
$673$	−41.4955	−1.59953	−0.799766	−	0.600312i	$-0.795043\pi$
$673$	−41.4955	−1.59953	−0.799766	+	0.600312i	$0.795043\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	−2.33030	−0.0895608	−0.0447804	−	0.998997i	$-0.514259\pi$
$677$	−2.33030	−0.0895608	−0.0447804	+	0.998997i	$0.514259\pi$
$678$	0	0
$679$	−43.9129	−1.68522
$680$	0	0
$681$	−14.5390	−0.557136
$682$	0	0
$683$	18.0000	0.688751	0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000	0.688751	0.344375	+	0.938832i	$0.388091\pi$
$684$	0	0
$685$	−6.33030	−0.241868
$686$	0	0
$687$	27.9129	1.06494
$688$	0	0
$689$	−9.95644	−0.379310
$690$	0	0
$691$	35.5826	1.35362	0.676812	−	0.736155i	$-0.263361\pi$
$691$	35.5826	1.35362	0.676812	+	0.736155i	$0.263361\pi$
$692$	0	0
$693$	18.1652	0.690037
$694$	0	0
$695$	−29.4955	−1.11883
$696$	0	0
$697$	0	0
$698$	0	0
$699$	45.8258	1.73329
$700$	0	0
$701$	−1.91288	−0.0722484	−0.0361242	−	0.999347i	$-0.511501\pi$
$701$	−1.91288	−0.0722484	−0.0361242	+	0.999347i	$0.511501\pi$
$702$	0	0
$703$	−66.9909	−2.52661
$704$	0	0
$705$	64.6606	2.43526
$706$	0	0
$707$	42.3303	1.59199
$708$	0	0
$709$	−22.3303	−0.838632	−0.419316	−	0.907840i	$-0.637730\pi$
$709$	−22.3303	−0.838632	−0.419316	+	0.907840i	$0.637730\pi$
$710$	0	0
$711$	11.5826	0.434381
$712$	0	0
$713$	0.834849	0.0312653
$714$	0	0
$715$	−2.00000	−0.0747958
$716$	0	0
$717$	34.5390	1.28988
$718$	0	0
$719$	13.6697	0.509794	0.254897	−	0.966968i	$-0.417958\pi$
$719$	13.6697	0.509794	0.254897	+	0.966968i	$0.417958\pi$
$720$	0	0
$721$	−55.1216	−2.05284
$722$	0	0
$723$	49.6606	1.84690
$724$	0	0
$725$	−7.58258	−0.281610
$726$	0	0
$727$	−30.5390	−1.13263	−0.566315	−	0.824189i	$-0.691631\pi$
$727$	−30.5390	−1.13263	−0.566315	+	0.824189i	$0.691631\pi$
$728$	0	0
$729$	−43.8693	−1.62479
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−4.95644	−0.183070	−0.0915351	−	0.995802i	$-0.529177\pi$
$733$	−4.95644	−0.183070	−0.0915351	+	0.995802i	$0.529177\pi$
$734$	0	0
$735$	−41.1652	−1.51840
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	−46.9564	−1.72732	−0.863660	−	0.504074i	$-0.831834\pi$
$739$	−46.9564	−1.72732	−0.863660	+	0.504074i	$0.831834\pi$
$740$	0	0
$741$	−23.3739	−0.858660
$742$	0	0
$743$	−15.1652	−0.556355	−0.278178	−	0.960530i	$-0.589730\pi$
$743$	−15.1652	−0.556355	−0.278178	+	0.960530i	$0.589730\pi$
$744$	0	0
$745$	26.7477	0.979961
$746$	0	0
$747$	16.1652	0.591452
$748$	0	0
$749$	−21.1652	−0.773358
$750$	0	0
$751$	−49.8693	−1.81976	−0.909879	−	0.414875i	$-0.863825\pi$
$751$	−49.8693	−1.81976	−0.909879	+	0.414875i	$0.863825\pi$
$752$	0	0
$753$	40.5826	1.47891
$754$	0	0
$755$	−1.66970	−0.0607665
$756$	0	0
$757$	−32.9564	−1.19782	−0.598911	−	0.800816i	$-0.704400\pi$
$757$	−32.9564	−1.19782	−0.598911	+	0.800816i	$0.704400\pi$
$758$	0	0
$759$	0.582576	0.0211462
$760$	0	0
$761$	−42.0345	−1.52375	−0.761874	−	0.647725i	$-0.775721\pi$
$761$	−42.0345	−1.52375	−0.761874	+	0.647725i	$0.775721\pi$
$762$	0	0
$763$	31.7477	1.14934
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.58258	−0.346007
$768$	0	0
$769$	−3.87841	−0.139859	−0.0699295	−	0.997552i	$-0.522277\pi$
$769$	−3.87841	−0.139859	−0.0699295	+	0.997552i	$0.522277\pi$
$770$	0	0
$771$	57.3303	2.06470
$772$	0	0
$773$	25.4955	0.917008	0.458504	−	0.888692i	$-0.348386\pi$
$773$	25.4955	0.917008	0.458504	+	0.888692i	$0.348386\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	84.6606	3.03718
$778$	0	0
$779$	48.4955	1.73753
$780$	0	0
$781$	−8.00000	−0.286263
$782$	0	0
$783$	37.9129	1.35490
$784$	0	0
$785$	−19.5826	−0.698932
$786$	0	0
$787$	31.3739	1.11836	0.559179	−	0.829047i	$-0.311117\pi$
$787$	31.3739	1.11836	0.559179	+	0.829047i	$0.311117\pi$
$788$	0	0
$789$	39.0780	1.39121
$790$	0	0
$791$	−25.7477	−0.915484
$792$	0	0
$793$	5.58258	0.198243
$794$	0	0
$795$	55.5826	1.97131
$796$	0	0
$797$	−43.4955	−1.54069	−0.770344	−	0.637628i	$-0.779916\pi$
$797$	−43.4955	−1.54069	−0.770344	+	0.637628i	$0.779916\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−65.0780	−2.29942
$802$	0	0
$803$	−12.3739	−0.436664
$804$	0	0
$805$	−1.58258	−0.0557785
$806$	0	0
$807$	26.1652	0.921057
$808$	0	0
$809$	−24.6606	−0.867021	−0.433510	−	0.901149i	$-0.642725\pi$
$809$	−24.6606	−0.867021	−0.433510	+	0.901149i	$0.642725\pi$
$810$	0	0
$811$	−41.2867	−1.44977	−0.724887	−	0.688868i	$-0.758108\pi$
$811$	−41.2867	−1.44977	−0.724887	+	0.688868i	$0.758108\pi$
$812$	0	0
$813$	−16.8693	−0.591633
$814$	0	0
$815$	−12.0000	−0.420342
$816$	0	0
$817$	−66.9909	−2.34372
$818$	0	0
$819$	18.1652	0.634742
$820$	0	0
$821$	−1.16515	−0.0406641	−0.0203320	−	0.999793i	$-0.506472\pi$
$821$	−1.16515	−0.0406641	−0.0203320	+	0.999793i	$0.506472\pi$
$822$	0	0
$823$	−10.1216	−0.352816	−0.176408	−	0.984317i	$-0.556448\pi$
$823$	−10.1216	−0.352816	−0.176408	+	0.984317i	$0.556448\pi$
$824$	0	0
$825$	−2.79129	−0.0971802
$826$	0	0
$827$	2.87841	0.100092	0.0500461	−	0.998747i	$-0.484063\pi$
$827$	2.87841	0.100092	0.0500461	+	0.998747i	$0.484063\pi$
$828$	0	0
$829$	−42.2087	−1.46597	−0.732985	−	0.680245i	$-0.761873\pi$
$829$	−42.2087	−1.46597	−0.732985	+	0.680245i	$0.761873\pi$
$830$	0	0
$831$	−31.1652	−1.08111
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−11.9129	−0.412262
$836$	0	0
$837$	20.0000	0.691301
$838$	0	0
$839$	27.4955	0.949248	0.474624	−	0.880189i	$-0.342584\pi$
$839$	27.4955	0.949248	0.474624	+	0.880189i	$0.342584\pi$
$840$	0	0
$841$	28.4955	0.982602
$842$	0	0
$843$	74.7822	2.57564
$844$	0	0
$845$	−2.00000	−0.0688021
$846$	0	0
$847$	3.79129	0.130270
$848$	0	0
$849$	−23.2523	−0.798016
$850$	0	0
$851$	1.66970	0.0572365
$852$	0	0
$853$	9.95644	0.340902	0.170451	−	0.985366i	$-0.445478\pi$
$853$	9.95644	0.340902	0.170451	+	0.985366i	$0.445478\pi$
$854$	0	0
$855$	80.2432	2.74426
$856$	0	0
$857$	28.6606	0.979028	0.489514	−	0.871996i	$-0.337174\pi$
$857$	28.6606	0.979028	0.489514	+	0.871996i	$0.337174\pi$
$858$	0	0
$859$	−18.7042	−0.638178	−0.319089	−	0.947725i	$-0.603377\pi$
$859$	−18.7042	−0.638178	−0.319089	+	0.947725i	$0.603377\pi$
$860$	0	0
$861$	−61.2867	−2.08865
$862$	0	0
$863$	12.3303	0.419728	0.209864	−	0.977731i	$-0.432698\pi$
$863$	12.3303	0.419728	0.209864	+	0.977731i	$0.432698\pi$
$864$	0	0
$865$	37.4955	1.27488
$866$	0	0
$867$	−47.4519	−1.61155
$868$	0	0
$869$	2.41742	0.0820055
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	−55.4955	−1.87824
$874$	0	0
$875$	45.4955	1.53803
$876$	0	0
$877$	−18.7913	−0.634537	−0.317268	−	0.948336i	$-0.602766\pi$
$877$	−18.7913	−0.634537	−0.317268	+	0.948336i	$0.602766\pi$
$878$	0	0
$879$	19.0780	0.643486
$880$	0	0
$881$	20.3739	0.686413	0.343206	−	0.939260i	$-0.388487\pi$
$881$	20.3739	0.686413	0.343206	+	0.939260i	$0.388487\pi$
$882$	0	0
$883$	12.2867	0.413482	0.206741	−	0.978396i	$-0.433714\pi$
$883$	12.2867	0.413482	0.206741	+	0.978396i	$0.433714\pi$
$884$	0	0
$885$	53.4955	1.79823
$886$	0	0
$887$	56.7477	1.90540	0.952701	−	0.303909i	$-0.0982918\pi$
$887$	56.7477	1.90540	0.952701	+	0.303909i	$0.0982918\pi$
$888$	0	0
$889$	6.00000	0.201234
$890$	0	0
$891$	−0.417424	−0.0139842
$892$	0	0
$893$	96.9909	3.24568
$894$	0	0
$895$	22.3303	0.746420
$896$	0	0
$897$	0.582576	0.0194516
$898$	0	0
$899$	30.3303	1.01157
$900$	0	0
$901$	0	0
$902$	0	0
$903$	84.6606	2.81733
$904$	0	0
$905$	35.9129	1.19378
$906$	0	0
$907$	32.2087	1.06947	0.534736	−	0.845019i	$-0.320411\pi$
$907$	32.2087	1.06947	0.534736	+	0.845019i	$0.320411\pi$
$908$	0	0
$909$	53.4955	1.77433
$910$	0	0
$911$	−6.95644	−0.230477	−0.115239	−	0.993338i	$-0.536763\pi$
$911$	−6.95644	−0.230477	−0.115239	+	0.993338i	$0.536763\pi$
$912$	0	0
$913$	3.37386	0.111659
$914$	0	0
$915$	−31.1652	−1.03029
$916$	0	0
$917$	−4.41742	−0.145876
$918$	0	0
$919$	15.6697	0.516896	0.258448	−	0.966025i	$-0.416789\pi$
$919$	15.6697	0.516896	0.258448	+	0.966025i	$0.416789\pi$
$920$	0	0
$921$	73.4955	2.42176
$922$	0	0
$923$	−8.00000	−0.263323
$924$	0	0
$925$	−8.00000	−0.263038
$926$	0	0
$927$	−69.6606	−2.28795
$928$	0	0
$929$	−11.1652	−0.366317	−0.183158	−	0.983083i	$-0.558632\pi$
$929$	−11.1652	−0.366317	−0.183158	+	0.983083i	$0.558632\pi$
$930$	0	0
$931$	−61.7477	−2.02370
$932$	0	0
$933$	76.8693	2.51659
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5.16515	0.168738	0.0843691	−	0.996435i	$-0.473113\pi$
$937$	5.16515	0.168738	0.0843691	+	0.996435i	$0.473113\pi$
$938$	0	0
$939$	−60.8258	−1.98497
$940$	0	0
$941$	26.7913	0.873371	0.436686	−	0.899614i	$-0.356152\pi$
$941$	26.7913	0.873371	0.436686	+	0.899614i	$0.356152\pi$
$942$	0	0
$943$	−1.20871	−0.0393611
$944$	0	0
$945$	−37.9129	−1.23331
$946$	0	0
$947$	−30.7477	−0.999167	−0.499583	−	0.866266i	$-0.666514\pi$
$947$	−30.7477	−0.999167	−0.499583	+	0.866266i	$0.666514\pi$
$948$	0	0
$949$	−12.3739	−0.401673
$950$	0	0
$951$	−33.4955	−1.08616
$952$	0	0
$953$	23.0780	0.747571	0.373785	−	0.927515i	$-0.378060\pi$
$953$	23.0780	0.747571	0.373785	+	0.927515i	$0.378060\pi$
$954$	0	0
$955$	−43.9129	−1.42099
$956$	0	0
$957$	21.1652	0.684172
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−26.7477	−0.861933
$964$	0	0
$965$	2.74773	0.0884525
$966$	0	0
$967$	38.6170	1.24184	0.620920	−	0.783874i	$-0.286759\pi$
$967$	38.6170	1.24184	0.620920	+	0.783874i	$0.286759\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	57.2087	1.83591	0.917957	−	0.396679i	$-0.129837\pi$
$971$	57.2087	1.83591	0.917957	+	0.396679i	$0.129837\pi$
$972$	0	0
$973$	55.9129	1.79248
$974$	0	0
$975$	−2.79129	−0.0893928
$976$	0	0
$977$	−52.6606	−1.68476	−0.842381	−	0.538882i	$-0.818847\pi$
$977$	−52.6606	−1.68476	−0.842381	+	0.538882i	$0.818847\pi$
$978$	0	0
$979$	−13.5826	−0.434101
$980$	0	0
$981$	40.1216	1.28098
$982$	0	0
$983$	−28.6606	−0.914131	−0.457066	−	0.889433i	$-0.651100\pi$
$983$	−28.6606	−0.914131	−0.457066	+	0.889433i	$0.651100\pi$
$984$	0	0
$985$	−18.4174	−0.586828
$986$	0	0
$987$	−122.573	−3.90156
$988$	0	0
$989$	1.66970	0.0530933
$990$	0	0
$991$	49.9564	1.58692	0.793459	−	0.608623i	$-0.208278\pi$
$991$	49.9564	1.58692	0.793459	+	0.608623i	$0.208278\pi$
$992$	0	0
$993$	53.4955	1.69763
$994$	0	0
$995$	−18.4174	−0.583872
$996$	0	0
$997$	15.9129	0.503966	0.251983	−	0.967732i	$-0.418917\pi$
$997$	15.9129	0.503966	0.251983	+	0.967732i	$0.418917\pi$
$998$	0	0
$999$	40.0000	1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.a.c.1.2	✓	2
3.2	odd	2		5148.2.a.k.1.2		2
4.3	odd	2		2288.2.a.m.1.1		2
8.3	odd	2		9152.2.a.bo.1.2		2
8.5	even	2		9152.2.a.bm.1.1		2
11.10	odd	2		6292.2.a.n.1.2		2
13.12	even	2		7436.2.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.a.c.1.2	✓	2	1.1	even	1	trivial
2288.2.a.m.1.1		2	4.3	odd	2
5148.2.a.k.1.2		2	3.2	odd	2
6292.2.a.n.1.2		2	11.10	odd	2
7436.2.a.k.1.2		2	13.12	even	2
9152.2.a.bm.1.1		2	8.5	even	2
9152.2.a.bo.1.2		2	8.3	odd	2

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.a (trivial)

\(f(q)\)	\(=\)	\(q+2.79129 q^{3} -2.00000 q^{5} +3.79129 q^{7} +4.79129 q^{9} +O(q^{10})\)	\(q+2.79129 q^{3} -2.00000 q^{5} +3.79129 q^{7} +4.79129 q^{9} +1.00000 q^{11} +1.00000 q^{13} -5.58258 q^{15} -8.37386 q^{19} +10.5826 q^{21} +0.208712 q^{23} -1.00000 q^{25} +5.00000 q^{27} +7.58258 q^{29} +4.00000 q^{31} +2.79129 q^{33} -7.58258 q^{35} +8.00000 q^{37} +2.79129 q^{39} -5.79129 q^{41} +8.00000 q^{43} -9.58258 q^{45} -11.5826 q^{47} +7.37386 q^{49} -9.95644 q^{53} -2.00000 q^{55} -23.3739 q^{57} -9.58258 q^{59} +5.58258 q^{61} +18.1652 q^{63} -2.00000 q^{65} +8.00000 q^{67} +0.582576 q^{69} -8.00000 q^{71} -12.3739 q^{73} -2.79129 q^{75} +3.79129 q^{77} +2.41742 q^{79} -0.417424 q^{81} +3.37386 q^{83} +21.1652 q^{87} -13.5826 q^{89} +3.79129 q^{91} +11.1652 q^{93} +16.7477 q^{95} -11.5826 q^{97} +4.79129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 4 q^{5} + 3 q^{7} + 5 q^{9}+O(q^{10}) \) 2 * q + q^3 - 4 * q^5 + 3 * q^7 + 5 * q^9	\( 2 q + q^{3} - 4 q^{5} + 3 q^{7} + 5 q^{9} + 2 q^{11} + 2 q^{13} - 2 q^{15} - 3 q^{19} + 12 q^{21} + 5 q^{23} - 2 q^{25} + 10 q^{27} + 6 q^{29} + 8 q^{31} + q^{33} - 6 q^{35} + 16 q^{37} + q^{39} - 7 q^{41} + 16 q^{43} - 10 q^{45} - 14 q^{47} + q^{49} + 3 q^{53} - 4 q^{55} - 33 q^{57} - 10 q^{59} + 2 q^{61} + 18 q^{63} - 4 q^{65} + 16 q^{67} - 8 q^{69} - 16 q^{71} - 11 q^{73} - q^{75} + 3 q^{77} + 14 q^{79} - 10 q^{81} - 7 q^{83} + 24 q^{87} - 18 q^{89} + 3 q^{91} + 4 q^{93} + 6 q^{95} - 14 q^{97} + 5 q^{99}+O(q^{100}) \) 2 * q + q^3 - 4 * q^5 + 3 * q^7 + 5 * q^9 + 2 * q^11 + 2 * q^13 - 2 * q^15 - 3 * q^19 + 12 * q^21 + 5 * q^23 - 2 * q^25 + 10 * q^27 + 6 * q^29 + 8 * q^31 + q^33 - 6 * q^35 + 16 * q^37 + q^39 - 7 * q^41 + 16 * q^43 - 10 * q^45 - 14 * q^47 + q^49 + 3 * q^53 - 4 * q^55 - 33 * q^57 - 10 * q^59 + 2 * q^61 + 18 * q^63 - 4 * q^65 + 16 * q^67 - 8 * q^69 - 16 * q^71 - 11 * q^73 - q^75 + 3 * q^77 + 14 * q^79 - 10 * q^81 - 7 * q^83 + 24 * q^87 - 18 * q^89 + 3 * q^91 + 4 * q^93 + 6 * q^95 - 14 * q^97 + 5 * q^99

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