LMFDB - Embedded newform 4544.2.a.bf.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4544,2,Mod(1,4544)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4544, 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("4544.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4544 = 2^{6} \cdot 71 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4544.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:	$36.2840226785$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.2373841.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 9x^{3} + 15x^{2} + 19x - 28 $ x^5 - 2x^4 - 9x^3 + 15x^2 + 19x - 28
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 568)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.20959$ of defining polynomial
Character	$\chi$	$=$	4544.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.20959 q^{3} -3.50442 q^{5} -0.937455 q^{7} +1.88227 q^{9} +O(q^{10})$	$q-2.20959 q^{3} -3.50442 q^{5} -0.937455 q^{7} +1.88227 q^{9} -1.06254 q^{13} +7.74333 q^{15} +0.235452 q^{17} -0.882274 q^{19} +2.07139 q^{21} -0.827093 q^{23} +7.28098 q^{25} +2.46971 q^{27} -2.13820 q^{29} -2.07139 q^{31} +3.28524 q^{35} -8.23890 q^{37} +2.34778 q^{39} +10.0586 q^{41} +6.86105 q^{43} -6.59628 q^{45} +3.83594 q^{47} -6.12118 q^{49} -0.520251 q^{51} +7.19257 q^{53} +1.94946 q^{57} +7.31766 q^{59} +7.30293 q^{61} -1.76455 q^{63} +3.72360 q^{65} +15.4867 q^{67} +1.82753 q^{69} +1.00000 q^{71} +3.61675 q^{73} -16.0879 q^{75} +10.2778 q^{79} -11.1039 q^{81} +4.36399 q^{83} -0.825123 q^{85} +4.72453 q^{87} -11.5272 q^{89} +0.996088 q^{91} +4.57691 q^{93} +3.09186 q^{95} -16.5472 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{3} - 7 q^{5} + q^{7} + 7 q^{9}+O(q^{10}) $ 5 * q + 2 * q^3 - 7 * q^5 + q^7 + 7 * q^9	$ 5 q + 2 q^{3} - 7 q^{5} + q^{7} + 7 q^{9} - 11 q^{13} - 6 q^{15} + 6 q^{17} - 2 q^{19} - 5 q^{21} - 5 q^{23} + 8 q^{25} + 5 q^{27} - 13 q^{29} + 5 q^{31} - 7 q^{37} + q^{39} + 8 q^{41} - 8 q^{43} - 7 q^{45} - q^{47} + 6 q^{49} - 14 q^{51} - 16 q^{53} - 9 q^{57} - 4 q^{59} - 22 q^{61} - 4 q^{63} + 14 q^{65} - 12 q^{67} - 13 q^{69} + 5 q^{71} - 8 q^{73} - 17 q^{75} + 15 q^{79} - 27 q^{81} - q^{83} - 14 q^{85} + 17 q^{87} - 4 q^{89} - 43 q^{91} + q^{93} - 8 q^{97}+O(q^{100}) $ 5 * q + 2 * q^3 - 7 * q^5 + q^7 + 7 * q^9 - 11 * q^13 - 6 * q^15 + 6 * q^17 - 2 * q^19 - 5 * q^21 - 5 * q^23 + 8 * q^25 + 5 * q^27 - 13 * q^29 + 5 * q^31 - 7 * q^37 + q^39 + 8 * q^41 - 8 * q^43 - 7 * q^45 - q^47 + 6 * q^49 - 14 * q^51 - 16 * q^53 - 9 * q^57 - 4 * q^59 - 22 * q^61 - 4 * q^63 + 14 * q^65 - 12 * q^67 - 13 * q^69 + 5 * q^71 - 8 * q^73 - 17 * q^75 + 15 * q^79 - 27 * q^81 - q^83 - 14 * q^85 + 17 * q^87 - 4 * q^89 - 43 * q^91 + q^93 - 8 * q^97

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
$3$	−2.20959	−1.27571	−0.637853	−	0.770158i	$-0.720177\pi$
$3$	−2.20959	−1.27571	−0.637853	+	0.770158i	$0.720177\pi$
$4$	0	0
$5$	−3.50442	−1.56723	−0.783613	−	0.621250i	$-0.786625\pi$
$5$	−3.50442	−1.56723	−0.783613	+	0.621250i	$0.786625\pi$
$6$	0	0
$7$	−0.937455	−0.354325	−0.177162	−	0.984182i	$-0.556692\pi$
$7$	−0.937455	−0.354325	−0.177162	+	0.984182i	$0.556692\pi$
$8$	0	0
$9$	1.88227	0.627425
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−1.06254	−0.294697	−0.147348	−	0.989085i	$-0.547074\pi$
$13$	−1.06254	−0.294697	−0.147348	+	0.989085i	$0.547074\pi$
$14$	0	0
$15$	7.74333	1.99932
$16$	0	0
$17$	0.235452	0.0571055	0.0285527	−	0.999592i	$-0.490910\pi$
$17$	0.235452	0.0571055	0.0285527	+	0.999592i	$0.490910\pi$
$18$	0	0
$19$	−0.882274	−0.202408	−0.101204	−	0.994866i	$-0.532269\pi$
$19$	−0.882274	−0.202408	−0.101204	+	0.994866i	$0.532269\pi$
$20$	0	0
$21$	2.07139	0.452014
$22$	0	0
$23$	−0.827093	−0.172461	−0.0862304	−	0.996275i	$-0.527482\pi$
$23$	−0.827093	−0.172461	−0.0862304	+	0.996275i	$0.527482\pi$
$24$	0	0
$25$	7.28098	1.45620
$26$	0	0
$27$	2.46971	0.475296
$28$	0	0
$29$	−2.13820	−0.397053	−0.198527	−	0.980095i	$-0.563616\pi$
$29$	−2.13820	−0.397053	−0.198527	+	0.980095i	$0.563616\pi$
$30$	0	0
$31$	−2.07139	−0.372032	−0.186016	−	0.982547i	$-0.559558\pi$
$31$	−2.07139	−0.372032	−0.186016	+	0.982547i	$0.559558\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.28524	0.555307
$36$	0	0
$37$	−8.23890	−1.35447	−0.677234	−	0.735768i	$-0.736821\pi$
$37$	−8.23890	−1.35447	−0.677234	+	0.735768i	$0.736821\pi$
$38$	0	0
$39$	2.34778	0.375946
$40$	0	0
$41$	10.0586	1.57089	0.785447	−	0.618929i	$-0.212433\pi$
$41$	10.0586	1.57089	0.785447	+	0.618929i	$0.212433\pi$
$42$	0	0
$43$	6.86105	1.04630	0.523150	−	0.852240i	$-0.324757\pi$
$43$	6.86105	1.04630	0.523150	+	0.852240i	$0.324757\pi$
$44$	0	0
$45$	−6.59628	−0.983316
$46$	0	0
$47$	3.83594	0.559529	0.279764	−	0.960069i	$-0.409744\pi$
$47$	3.83594	0.559529	0.279764	+	0.960069i	$0.409744\pi$
$48$	0	0
$49$	−6.12118	−0.874454
$50$	0	0
$51$	−0.520251	−0.0728498
$52$	0	0
$53$	7.19257	0.987975	0.493987	−	0.869469i	$-0.335539\pi$
$53$	7.19257	0.987975	0.493987	+	0.869469i	$0.335539\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.94946	0.258212
$58$	0	0
$59$	7.31766	0.952678	0.476339	−	0.879262i	$-0.341964\pi$
$59$	7.31766	0.952678	0.476339	+	0.879262i	$0.341964\pi$
$60$	0	0
$61$	7.30293	0.935044	0.467522	−	0.883981i	$-0.345147\pi$
$61$	7.30293	0.935044	0.467522	+	0.883981i	$0.345147\pi$
$62$	0	0
$63$	−1.76455	−0.222312
$64$	0	0
$65$	3.72360	0.461856
$66$	0	0
$67$	15.4867	1.89200	0.945998	−	0.324173i	$-0.105086\pi$
$67$	15.4867	1.89200	0.945998	+	0.324173i	$0.105086\pi$
$68$	0	0
$69$	1.82753	0.220009
$70$	0	0
$71$	1.00000	0.118678
$71$	1.00000	0.118678
$72$	0	0
$73$	3.61675	0.423309	0.211655	−	0.977345i	$-0.432115\pi$
$73$	3.61675	0.423309	0.211655	+	0.977345i	$0.432115\pi$
$74$	0	0
$75$	−16.0879	−1.85768
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.2778	1.15634	0.578172	−	0.815915i	$-0.303766\pi$
$79$	10.2778	1.15634	0.578172	+	0.815915i	$0.303766\pi$
$80$	0	0
$81$	−11.1039	−1.23376
$82$	0	0
$83$	4.36399	0.479010	0.239505	−	0.970895i	$-0.423015\pi$
$83$	4.36399	0.479010	0.239505	+	0.970895i	$0.423015\pi$
$84$	0	0
$85$	−0.825123	−0.0894971
$86$	0	0
$87$	4.72453	0.506523
$88$	0	0
$89$	−11.5272	−1.22189	−0.610943	−	0.791675i	$-0.709209\pi$
$89$	−11.5272	−1.22189	−0.610943	+	0.791675i	$0.709209\pi$
$90$	0	0
$91$	0.996088	0.104418
$92$	0	0
$93$	4.57691	0.474604
$94$	0	0
$95$	3.09186	0.317218
$96$	0	0
$97$	−16.5472	−1.68012	−0.840058	−	0.542496i	$-0.817479\pi$
$97$	−16.5472	−1.68012	−0.840058	+	0.542496i	$0.817479\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.2566	−1.41858	−0.709292	−	0.704915i	$-0.750985\pi$
$101$	−14.2566	−1.41858	−0.709292	+	0.704915i	$0.750985\pi$
$102$	0	0
$103$	11.6766	1.15053	0.575264	−	0.817968i	$-0.304899\pi$
$103$	11.6766	1.15053	0.575264	+	0.817968i	$0.304899\pi$
$104$	0	0
$105$	−7.25902	−0.708408
$106$	0	0
$107$	−8.30052	−0.802441	−0.401221	−	0.915981i	$-0.631414\pi$
$107$	−8.30052	−0.802441	−0.401221	+	0.915981i	$0.631414\pi$
$108$	0	0
$109$	4.21156	0.403394	0.201697	−	0.979448i	$-0.435354\pi$
$109$	4.21156	0.403394	0.201697	+	0.979448i	$0.435354\pi$
$110$	0	0
$111$	18.2046	1.72790
$112$	0	0
$113$	2.24386	0.211084	0.105542	−	0.994415i	$-0.466342\pi$
$113$	2.24386	0.211084	0.105542	+	0.994415i	$0.466342\pi$
$114$	0	0
$115$	2.89848	0.270285
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	−0.220726	−0.0202339
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	−22.2254	−2.00400
$124$	0	0
$125$	−7.99350	−0.714961
$126$	0	0
$127$	−8.56195	−0.759750	−0.379875	−	0.925038i	$-0.624033\pi$
$127$	−8.56195	−0.759750	−0.379875	+	0.925038i	$0.624033\pi$
$128$	0	0
$129$	−15.1601	−1.33477
$130$	0	0
$131$	−9.86990	−0.862337	−0.431168	−	0.902271i	$-0.641899\pi$
$131$	−9.86990	−0.862337	−0.431168	+	0.902271i	$0.641899\pi$
$132$	0	0
$133$	0.827093	0.0717180
$134$	0	0
$135$	−8.65492	−0.744897
$136$	0	0
$137$	12.3265	1.05312	0.526562	−	0.850137i	$-0.323481\pi$
$137$	12.3265	1.05312	0.526562	+	0.850137i	$0.323481\pi$
$138$	0	0
$139$	−7.83397	−0.664468	−0.332234	−	0.943197i	$-0.607802\pi$
$139$	−7.83397	−0.664468	−0.332234	+	0.943197i	$0.607802\pi$
$140$	0	0
$141$	−8.47584	−0.713794
$142$	0	0
$143$	0	0
$144$	0	0
$145$	7.49315	0.622272
$146$	0	0
$147$	13.5253	1.11555
$148$	0	0
$149$	8.79937	0.720873	0.360436	−	0.932784i	$-0.382628\pi$
$149$	8.79937	0.720873	0.360436	+	0.932784i	$0.382628\pi$
$150$	0	0
$151$	8.91159	0.725215	0.362607	−	0.931942i	$-0.381887\pi$
$151$	8.91159	0.725215	0.362607	+	0.931942i	$0.381887\pi$
$152$	0	0
$153$	0.443185	0.0358294
$154$	0	0
$155$	7.25902	0.583059
$156$	0	0
$157$	−18.9248	−1.51036	−0.755182	−	0.655515i	$-0.772452\pi$
$157$	−18.9248	−1.51036	−0.755182	+	0.655515i	$0.772452\pi$
$158$	0	0
$159$	−15.8926	−1.26037
$160$	0	0
$161$	0.775363	0.0611071
$162$	0	0
$163$	−9.35860	−0.733022	−0.366511	−	0.930414i	$-0.619448\pi$
$163$	−9.35860	−0.733022	−0.366511	+	0.930414i	$0.619448\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.7033	−1.44731	−0.723653	−	0.690164i	$-0.757538\pi$
$167$	−18.7033	−1.44731	−0.723653	+	0.690164i	$0.757538\pi$
$168$	0	0
$169$	−11.8710	−0.913154
$170$	0	0
$171$	−1.66068	−0.126995
$172$	0	0
$173$	6.71969	0.510889	0.255444	−	0.966824i	$-0.417778\pi$
$173$	6.71969	0.510889	0.255444	+	0.966824i	$0.417778\pi$
$174$	0	0
$175$	−6.82559	−0.515966
$176$	0	0
$177$	−16.1690	−1.21534
$178$	0	0
$179$	21.3990	1.59944	0.799718	−	0.600376i	$-0.204982\pi$
$179$	21.3990	1.59944	0.799718	+	0.600376i	$0.204982\pi$
$180$	0	0
$181$	−5.16659	−0.384029	−0.192015	−	0.981392i	$-0.561502\pi$
$181$	−5.16659	−0.384029	−0.192015	+	0.981392i	$0.561502\pi$
$182$	0	0
$183$	−16.1365	−1.19284
$184$	0	0
$185$	28.8726	2.12276
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−2.31525	−0.168409
$190$	0	0
$191$	5.27509	0.381692	0.190846	−	0.981620i	$-0.438877\pi$
$191$	5.27509	0.381692	0.190846	+	0.981620i	$0.438877\pi$
$192$	0	0
$193$	−10.7734	−0.775486	−0.387743	−	0.921768i	$-0.626745\pi$
$193$	−10.7734	−0.775486	−0.387743	+	0.921768i	$0.626745\pi$
$194$	0	0
$195$	−8.22763	−0.589193
$196$	0	0
$197$	1.05622	0.0752527	0.0376264	−	0.999292i	$-0.488020\pi$
$197$	1.05622	0.0752527	0.0376264	+	0.999292i	$0.488020\pi$
$198$	0	0
$199$	18.0211	1.27748	0.638742	−	0.769421i	$-0.279455\pi$
$199$	18.0211	1.27748	0.638742	+	0.769421i	$0.279455\pi$
$200$	0	0
$201$	−34.2191	−2.41363
$202$	0	0
$203$	2.00447	0.140686
$204$	0	0
$205$	−35.2497	−2.46195
$206$	0	0
$207$	−1.55681	−0.108206
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−13.6117	−0.937071	−0.468535	−	0.883445i	$-0.655218\pi$
$211$	−13.6117	−0.937071	−0.468535	+	0.883445i	$0.655218\pi$
$212$	0	0
$213$	−2.20959	−0.151398
$214$	0	0
$215$	−24.0440	−1.63979
$216$	0	0
$217$	1.94184	0.131820
$218$	0	0
$219$	−7.99153	−0.540018
$220$	0	0
$221$	−0.250178	−0.0168288
$222$	0	0
$223$	−4.94526	−0.331159	−0.165580	−	0.986196i	$-0.552949\pi$
$223$	−4.94526	−0.331159	−0.165580	+	0.986196i	$0.552949\pi$
$224$	0	0
$225$	13.7048	0.913653
$226$	0	0
$227$	19.6395	1.30352	0.651758	−	0.758427i	$-0.274032\pi$
$227$	19.6395	1.30352	0.651758	+	0.758427i	$0.274032\pi$
$228$	0	0
$229$	0.670895	0.0443340	0.0221670	−	0.999754i	$-0.492943\pi$
$229$	0.670895	0.0443340	0.0221670	+	0.999754i	$0.492943\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.64682	−0.304423	−0.152212	−	0.988348i	$-0.548640\pi$
$233$	−4.64682	−0.304423	−0.152212	+	0.988348i	$0.548640\pi$
$234$	0	0
$235$	−13.4427	−0.876908
$236$	0	0
$237$	−22.7097	−1.47516
$238$	0	0
$239$	−10.7020	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$239$	−10.7020	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$240$	0	0
$241$	−29.3763	−1.89229	−0.946147	−	0.323739i	$-0.895060\pi$
$241$	−29.3763	−1.89229	−0.946147	+	0.323739i	$0.895060\pi$
$242$	0	0
$243$	17.1258	1.09862
$244$	0	0
$245$	21.4512	1.37047
$246$	0	0
$247$	0.937455	0.0596489
$248$	0	0
$249$	−9.64262	−0.611076
$250$	0	0
$251$	−7.84571	−0.495217	−0.247608	−	0.968860i	$-0.579645\pi$
$251$	−7.84571	−0.495217	−0.247608	+	0.968860i	$0.579645\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	1.82318	0.114172
$256$	0	0
$257$	−9.95755	−0.621135	−0.310568	−	0.950551i	$-0.600519\pi$
$257$	−9.95755	−0.621135	−0.310568	+	0.950551i	$0.600519\pi$
$258$	0	0
$259$	7.72360	0.479921
$260$	0	0
$261$	−4.02467	−0.249121
$262$	0	0
$263$	16.9457	1.04491	0.522457	−	0.852665i	$-0.325015\pi$
$263$	16.9457	1.04491	0.522457	+	0.852665i	$0.325015\pi$
$264$	0	0
$265$	−25.2058	−1.54838
$266$	0	0
$267$	25.4704	1.55877
$268$	0	0
$269$	−3.45771	−0.210820	−0.105410	−	0.994429i	$-0.533616\pi$
$269$	−3.45771	−0.210820	−0.105410	+	0.994429i	$0.533616\pi$
$270$	0	0
$271$	11.4519	0.695654	0.347827	−	0.937559i	$-0.386920\pi$
$271$	11.4519	0.695654	0.347827	+	0.937559i	$0.386920\pi$
$272$	0	0
$273$	−2.20094	−0.133207
$274$	0	0
$275$	0	0
$276$	0	0
$277$	25.2230	1.51550	0.757751	−	0.652543i	$-0.226298\pi$
$277$	25.2230	1.51550	0.757751	+	0.652543i	$0.226298\pi$
$278$	0	0
$279$	−3.89892	−0.233422
$280$	0	0
$281$	8.05023	0.480236	0.240118	−	0.970744i	$-0.422814\pi$
$281$	8.05023	0.480236	0.240118	+	0.970744i	$0.422814\pi$
$282$	0	0
$283$	−26.8885	−1.59835	−0.799176	−	0.601097i	$-0.794731\pi$
$283$	−26.8885	−1.59835	−0.799176	+	0.601097i	$0.794731\pi$
$284$	0	0
$285$	−6.83174	−0.404677
$286$	0	0
$287$	−9.42952	−0.556607
$288$	0	0
$289$	−16.9446	−0.996739
$290$	0	0
$291$	36.5625	2.14333
$292$	0	0
$293$	−0.911474	−0.0532489	−0.0266245	−	0.999646i	$-0.508476\pi$
$293$	−0.911474	−0.0532489	−0.0266245	+	0.999646i	$0.508476\pi$
$294$	0	0
$295$	−25.6442	−1.49306
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.878823	0.0508236
$300$	0	0
$301$	−6.43193	−0.370730
$302$	0	0
$303$	31.5012	1.80970
$304$	0	0
$305$	−25.5925	−1.46543
$306$	0	0
$307$	−4.60736	−0.262956	−0.131478	−	0.991319i	$-0.541972\pi$
$307$	−4.60736	−0.262956	−0.131478	+	0.991319i	$0.541972\pi$
$308$	0	0
$309$	−25.8004	−1.46773
$310$	0	0
$311$	−33.5983	−1.90518	−0.952591	−	0.304253i	$-0.901593\pi$
$311$	−33.5983	−1.90518	−0.952591	+	0.304253i	$0.901593\pi$
$312$	0	0
$313$	12.5689	0.710435	0.355217	−	0.934784i	$-0.384407\pi$
$313$	12.5689	0.710435	0.355217	+	0.934784i	$0.384407\pi$
$314$	0	0
$315$	6.18372	0.348413
$316$	0	0
$317$	1.80549	0.101406	0.0507032	−	0.998714i	$-0.483854\pi$
$317$	1.80549	0.101406	0.0507032	+	0.998714i	$0.483854\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	18.3407	1.02368
$322$	0	0
$323$	−0.207733	−0.0115586
$324$	0	0
$325$	−7.73636	−0.429136
$326$	0	0
$327$	−9.30580	−0.514612
$328$	0	0
$329$	−3.59602	−0.198255
$330$	0	0
$331$	−25.2251	−1.38650	−0.693249	−	0.720699i	$-0.743821\pi$
$331$	−25.2251	−1.38650	−0.693249	+	0.720699i	$0.743821\pi$
$332$	0	0
$333$	−15.5079	−0.849826
$334$	0	0
$335$	−54.2718	−2.96518
$336$	0	0
$337$	32.6383	1.77792	0.888960	−	0.457985i	$-0.151429\pi$
$337$	32.6383	1.77792	0.888960	+	0.457985i	$0.151429\pi$
$338$	0	0
$339$	−4.95800	−0.269281
$340$	0	0
$341$	0	0
$342$	0	0
$343$	12.3005	0.664166
$344$	0	0
$345$	−6.40445	−0.344804
$346$	0	0
$347$	14.3846	0.772203	0.386102	−	0.922456i	$-0.373821\pi$
$347$	14.3846	0.772203	0.386102	+	0.922456i	$0.373821\pi$
$348$	0	0
$349$	−14.6286	−0.783053	−0.391527	−	0.920167i	$-0.628053\pi$
$349$	−14.6286	−0.783053	−0.391527	+	0.920167i	$0.628053\pi$
$350$	0	0
$351$	−2.62418	−0.140068
$352$	0	0
$353$	8.86903	0.472051	0.236025	−	0.971747i	$-0.424155\pi$
$353$	8.86903	0.472051	0.236025	+	0.971747i	$0.424155\pi$
$354$	0	0
$355$	−3.50442	−0.185995
$356$	0	0
$357$	0.487713	0.0258125
$358$	0	0
$359$	35.6224	1.88008	0.940040	−	0.341063i	$-0.110787\pi$
$359$	35.6224	1.88008	0.940040	+	0.341063i	$0.110787\pi$
$360$	0	0
$361$	−18.2216	−0.959031
$362$	0	0
$363$	24.3055	1.27571
$364$	0	0
$365$	−12.6746	−0.663421
$366$	0	0
$367$	−4.68562	−0.244587	−0.122294	−	0.992494i	$-0.539025\pi$
$367$	−4.68562	−0.244587	−0.122294	+	0.992494i	$0.539025\pi$
$368$	0	0
$369$	18.9331	0.985618
$370$	0	0
$371$	−6.74271	−0.350064
$372$	0	0
$373$	−3.29965	−0.170850	−0.0854248	−	0.996345i	$-0.527225\pi$
$373$	−3.29965	−0.170850	−0.0854248	+	0.996345i	$0.527225\pi$
$374$	0	0
$375$	17.6623	0.912079
$376$	0	0
$377$	2.27193	0.117010
$378$	0	0
$379$	1.10610	0.0568165	0.0284082	−	0.999596i	$-0.490956\pi$
$379$	1.10610	0.0568165	0.0284082	+	0.999596i	$0.490956\pi$
$380$	0	0
$381$	18.9184	0.969218
$382$	0	0
$383$	−26.1330	−1.33533	−0.667667	−	0.744460i	$-0.732707\pi$
$383$	−26.1330	−1.33533	−0.667667	+	0.744460i	$0.732707\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	12.9144	0.656475
$388$	0	0
$389$	30.4072	1.54170	0.770852	−	0.637014i	$-0.219831\pi$
$389$	30.4072	1.54170	0.770852	+	0.637014i	$0.219831\pi$
$390$	0	0
$391$	−0.194741	−0.00984845
$392$	0	0
$393$	21.8084	1.10009
$394$	0	0
$395$	−36.0178	−1.81225
$396$	0	0
$397$	10.7412	0.539086	0.269543	−	0.962988i	$-0.413127\pi$
$397$	10.7412	0.539086	0.269543	+	0.962988i	$0.413127\pi$
$398$	0	0
$399$	−1.82753	−0.0914911
$400$	0	0
$401$	−7.30687	−0.364888	−0.182444	−	0.983216i	$-0.558401\pi$
$401$	−7.30687	−0.364888	−0.182444	+	0.983216i	$0.558401\pi$
$402$	0	0
$403$	2.20094	0.109637
$404$	0	0
$405$	38.9126	1.93358
$406$	0	0
$407$	0	0
$408$	0	0
$409$	25.5321	1.26248	0.631240	−	0.775588i	$-0.282546\pi$
$409$	25.5321	1.26248	0.631240	+	0.775588i	$0.282546\pi$
$410$	0	0
$411$	−27.2365	−1.34348
$412$	0	0
$413$	−6.85998	−0.337557
$414$	0	0
$415$	−15.2933	−0.750717
$416$	0	0
$417$	17.3098	0.847666
$418$	0	0
$419$	1.65889	0.0810421	0.0405210	−	0.999179i	$-0.487098\pi$
$419$	1.65889	0.0810421	0.0405210	+	0.999179i	$0.487098\pi$
$420$	0	0
$421$	−29.3939	−1.43257	−0.716285	−	0.697808i	$-0.754159\pi$
$421$	−29.3939	−1.43257	−0.716285	+	0.697808i	$0.754159\pi$
$422$	0	0
$423$	7.22029	0.351062
$424$	0	0
$425$	1.71432	0.0831567
$426$	0	0
$427$	−6.84617	−0.331309
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.90388	−0.0917064	−0.0458532	−	0.998948i	$-0.514601\pi$
$431$	−1.90388	−0.0917064	−0.0458532	+	0.998948i	$0.514601\pi$
$432$	0	0
$433$	−13.5970	−0.653431	−0.326715	−	0.945123i	$-0.605942\pi$
$433$	−13.5970	−0.653431	−0.326715	+	0.945123i	$0.605942\pi$
$434$	0	0
$435$	−16.5568	−0.793836
$436$	0	0
$437$	0.729722	0.0349074
$438$	0	0
$439$	−24.9556	−1.19107	−0.595533	−	0.803331i	$-0.703059\pi$
$439$	−24.9556	−1.19107	−0.595533	+	0.803331i	$0.703059\pi$
$440$	0	0
$441$	−11.5217	−0.548654
$442$	0	0
$443$	24.0099	1.14074	0.570372	−	0.821387i	$-0.306799\pi$
$443$	24.0099	1.14074	0.570372	+	0.821387i	$0.306799\pi$
$444$	0	0
$445$	40.3963	1.91497
$446$	0	0
$447$	−19.4430	−0.919621
$448$	0	0
$449$	−5.70441	−0.269208	−0.134604	−	0.990899i	$-0.542976\pi$
$449$	−5.70441	−0.269208	−0.134604	+	0.990899i	$0.542976\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−19.6909	−0.925161
$454$	0	0
$455$	−3.49071	−0.163647
$456$	0	0
$457$	17.2659	0.807666	0.403833	−	0.914833i	$-0.367678\pi$
$457$	17.2659	0.807666	0.403833	+	0.914833i	$0.367678\pi$
$458$	0	0
$459$	0.581499	0.0271420
$460$	0	0
$461$	32.7467	1.52516	0.762582	−	0.646891i	$-0.223931\pi$
$461$	32.7467	1.52516	0.762582	+	0.646891i	$0.223931\pi$
$462$	0	0
$463$	22.6063	1.05061	0.525303	−	0.850915i	$-0.323952\pi$
$463$	22.6063	1.05061	0.525303	+	0.850915i	$0.323952\pi$
$464$	0	0
$465$	−16.0394	−0.743811
$466$	0	0
$467$	10.3883	0.480711	0.240356	−	0.970685i	$-0.422736\pi$
$467$	10.3883	0.480711	0.240356	+	0.970685i	$0.422736\pi$
$468$	0	0
$469$	−14.5180	−0.670381
$470$	0	0
$471$	41.8160	1.92678
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−6.42382	−0.294745
$476$	0	0
$477$	13.5384	0.619880
$478$	0	0
$479$	−33.4738	−1.52946	−0.764728	−	0.644353i	$-0.777127\pi$
$479$	−33.4738	−1.52946	−0.764728	+	0.644353i	$0.777127\pi$
$480$	0	0
$481$	8.75420	0.399157
$482$	0	0
$483$	−1.71323	−0.0779547
$484$	0	0
$485$	57.9885	2.63312
$486$	0	0
$487$	11.9933	0.543470	0.271735	−	0.962372i	$-0.412403\pi$
$487$	11.9933	0.543470	0.271735	+	0.962372i	$0.412403\pi$
$488$	0	0
$489$	20.6786	0.935120
$490$	0	0
$491$	−35.6659	−1.60958	−0.804790	−	0.593560i	$-0.797722\pi$
$491$	−35.6659	−1.60958	−0.804790	+	0.593560i	$0.797722\pi$
$492$	0	0
$493$	−0.503443	−0.0226739
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.937455	−0.0420506
$498$	0	0
$499$	−4.90034	−0.219369	−0.109685	−	0.993966i	$-0.534984\pi$
$499$	−4.90034	−0.219369	−0.109685	+	0.993966i	$0.534984\pi$
$500$	0	0
$501$	41.3266	1.84634
$502$	0	0
$503$	19.0629	0.849973	0.424986	−	0.905200i	$-0.360279\pi$
$503$	19.0629	0.849973	0.424986	+	0.905200i	$0.360279\pi$
$504$	0	0
$505$	49.9611	2.22324
$506$	0	0
$507$	26.2300	1.16492
$508$	0	0
$509$	−28.4631	−1.26160	−0.630802	−	0.775944i	$-0.717274\pi$
$509$	−28.4631	−1.26160	−0.630802	+	0.775944i	$0.717274\pi$
$510$	0	0
$511$	−3.39055	−0.149989
$512$	0	0
$513$	−2.17896	−0.0962036
$514$	0	0
$515$	−40.9197	−1.80314
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−14.8477	−0.651744
$520$	0	0
$521$	−1.01189	−0.0443317	−0.0221658	−	0.999754i	$-0.507056\pi$
$521$	−1.01189	−0.0443317	−0.0221658	+	0.999754i	$0.507056\pi$
$522$	0	0
$523$	−18.1383	−0.793133	−0.396567	−	0.918006i	$-0.629798\pi$
$523$	−18.1383	−0.793133	−0.396567	+	0.918006i	$0.629798\pi$
$524$	0	0
$525$	15.0817	0.658221
$526$	0	0
$527$	−0.487713	−0.0212451
$528$	0	0
$529$	−22.3159	−0.970257
$530$	0	0
$531$	13.7738	0.597733
$532$	0	0
$533$	−10.6877	−0.462938
$534$	0	0
$535$	29.0885	1.25761
$536$	0	0
$537$	−47.2829	−2.04041
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−42.8313	−1.84146	−0.920730	−	0.390200i	$-0.872406\pi$
$541$	−42.8313	−1.84146	−0.920730	+	0.390200i	$0.872406\pi$
$542$	0	0
$543$	11.4160	0.489908
$544$	0	0
$545$	−14.7591	−0.632209
$546$	0	0
$547$	−30.8190	−1.31773	−0.658863	−	0.752263i	$-0.728962\pi$
$547$	−30.8190	−1.31773	−0.658863	+	0.752263i	$0.728962\pi$
$548$	0	0
$549$	13.7461	0.586670
$550$	0	0
$551$	1.88648	0.0803666
$552$	0	0
$553$	−9.63499	−0.409722
$554$	0	0
$555$	−63.7965	−2.70801
$556$	0	0
$557$	−4.74406	−0.201012	−0.100506	−	0.994936i	$-0.532046\pi$
$557$	−4.74406	−0.201012	−0.100506	+	0.994936i	$0.532046\pi$
$558$	0	0
$559$	−7.29017	−0.308341
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0.0260938	0.00109972	0.000549862	−	1.00000i	$-0.499825\pi$
$563$	0.0260938	0.00109972	0.000549862		1.00000i	$0.499825\pi$
$564$	0	0
$565$	−7.86342	−0.330817
$566$	0	0
$567$	10.4094	0.437153
$568$	0	0
$569$	34.5702	1.44926	0.724628	−	0.689140i	$-0.242011\pi$
$569$	34.5702	1.44926	0.724628	+	0.689140i	$0.242011\pi$
$570$	0	0
$571$	−32.2437	−1.34936	−0.674679	−	0.738111i	$-0.735718\pi$
$571$	−32.2437	−1.34936	−0.674679	+	0.738111i	$0.735718\pi$
$572$	0	0
$573$	−11.6558	−0.486927
$574$	0	0
$575$	−6.02204	−0.251136
$576$	0	0
$577$	26.6266	1.10848	0.554240	−	0.832357i	$-0.313009\pi$
$577$	26.6266	1.10848	0.554240	+	0.832357i	$0.313009\pi$
$578$	0	0
$579$	23.8047	0.989292
$580$	0	0
$581$	−4.09105	−0.169725
$582$	0	0
$583$	0	0
$584$	0	0
$585$	7.00884	0.289780
$586$	0	0
$587$	19.2705	0.795379	0.397689	−	0.917520i	$-0.369812\pi$
$587$	19.2705	0.795379	0.397689	+	0.917520i	$0.369812\pi$
$588$	0	0
$589$	1.82753	0.0753022
$590$	0	0
$591$	−2.33382	−0.0960004
$592$	0	0
$593$	−6.50355	−0.267069	−0.133534	−	0.991044i	$-0.542633\pi$
$593$	−6.50355	−0.267069	−0.133534	+	0.991044i	$0.542633\pi$
$594$	0	0
$595$	0.773516	0.0317111
$596$	0	0
$597$	−39.8193	−1.62969
$598$	0	0
$599$	48.2298	1.97062	0.985309	−	0.170783i	$-0.0546296\pi$
$599$	48.2298	1.97062	0.985309	+	0.170783i	$0.0546296\pi$
$600$	0	0
$601$	27.3128	1.11411	0.557056	−	0.830475i	$-0.311931\pi$
$601$	27.3128	1.11411	0.557056	+	0.830475i	$0.311931\pi$
$602$	0	0
$603$	29.1501	1.18708
$604$	0	0
$605$	38.5486	1.56723
$606$	0	0
$607$	12.2571	0.497500	0.248750	−	0.968568i	$-0.419980\pi$
$607$	12.2571	0.497500	0.248750	+	0.968568i	$0.419980\pi$
$608$	0	0
$609$	−4.42904	−0.179474
$610$	0	0
$611$	−4.07585	−0.164891
$612$	0	0
$613$	−26.4277	−1.06740	−0.533702	−	0.845672i	$-0.679200\pi$
$613$	−26.4277	−1.06740	−0.533702	+	0.845672i	$0.679200\pi$
$614$	0	0
$615$	77.8873	3.14072
$616$	0	0
$617$	31.6521	1.27427	0.637133	−	0.770753i	$-0.280120\pi$
$617$	31.6521	1.27427	0.637133	+	0.770753i	$0.280120\pi$
$618$	0	0
$619$	39.1263	1.57262	0.786310	−	0.617832i	$-0.211989\pi$
$619$	39.1263	1.57262	0.786310	+	0.617832i	$0.211989\pi$
$620$	0	0
$621$	−2.04268	−0.0819700
$622$	0	0
$623$	10.8063	0.432944
$624$	0	0
$625$	−8.39227	−0.335691
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.93987	−0.0773475
$630$	0	0
$631$	1.47122	0.0585684	0.0292842	−	0.999571i	$-0.490677\pi$
$631$	1.47122	0.0585684	0.0292842	+	0.999571i	$0.490677\pi$
$632$	0	0
$633$	30.0763	1.19543
$634$	0	0
$635$	30.0047	1.19070
$636$	0	0
$637$	6.50402	0.257699
$638$	0	0
$639$	1.88227	0.0744616
$640$	0	0
$641$	22.4633	0.887248	0.443624	−	0.896213i	$-0.353693\pi$
$641$	22.4633	0.887248	0.443624	+	0.896213i	$0.353693\pi$
$642$	0	0
$643$	−18.6335	−0.734832	−0.367416	−	0.930057i	$-0.619757\pi$
$643$	−18.6335	−0.734832	−0.367416	+	0.930057i	$0.619757\pi$
$644$	0	0
$645$	53.1274	2.09189
$646$	0	0
$647$	9.32212	0.366490	0.183245	−	0.983067i	$-0.441340\pi$
$647$	9.32212	0.366490	0.183245	+	0.983067i	$0.441340\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−4.29065	−0.168164
$652$	0	0
$653$	−0.751764	−0.0294188	−0.0147094	−	0.999892i	$-0.504682\pi$
$653$	−0.751764	−0.0294188	−0.0147094	+	0.999892i	$0.504682\pi$
$654$	0	0
$655$	34.5883	1.35148
$656$	0	0
$657$	6.80772	0.265595
$658$	0	0
$659$	−35.2043	−1.37136	−0.685682	−	0.727901i	$-0.740496\pi$
$659$	−35.2043	−1.37136	−0.685682	+	0.727901i	$0.740496\pi$
$660$	0	0
$661$	−39.1653	−1.52335	−0.761677	−	0.647957i	$-0.775624\pi$
$661$	−39.1653	−1.52335	−0.761677	+	0.647957i	$0.775624\pi$
$662$	0	0
$663$	0.552790	0.0214686
$664$	0	0
$665$	−2.89848	−0.112398
$666$	0	0
$667$	1.76849	0.0684761
$668$	0	0
$669$	10.9270	0.422461
$670$	0	0
$671$	0	0
$672$	0	0
$673$	14.7929	0.570226	0.285113	−	0.958494i	$-0.407969\pi$
$673$	14.7929	0.570226	0.285113	+	0.958494i	$0.407969\pi$
$674$	0	0
$675$	17.9819	0.692124
$676$	0	0
$677$	−6.82042	−0.262130	−0.131065	−	0.991374i	$-0.541840\pi$
$677$	−6.82042	−0.262130	−0.131065	+	0.991374i	$0.541840\pi$
$678$	0	0
$679$	15.5123	0.595307
$680$	0	0
$681$	−43.3951	−1.66290
$682$	0	0
$683$	−30.5565	−1.16921	−0.584606	−	0.811317i	$-0.698751\pi$
$683$	−30.5565	−1.16921	−0.584606	+	0.811317i	$0.698751\pi$
$684$	0	0
$685$	−43.1973	−1.65048
$686$	0	0
$687$	−1.48240	−0.0565571
$688$	0	0
$689$	−7.64242	−0.291153
$690$	0	0
$691$	0.796647	0.0303059	0.0151529	−	0.999885i	$-0.495176\pi$
$691$	0.796647	0.0303059	0.0151529	+	0.999885i	$0.495176\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	27.4535	1.04137
$696$	0	0
$697$	2.36832	0.0897067
$698$	0	0
$699$	10.2676	0.388355
$700$	0	0
$701$	32.8740	1.24163	0.620816	−	0.783956i	$-0.286801\pi$
$701$	32.8740	1.24163	0.620816	+	0.783956i	$0.286801\pi$
$702$	0	0
$703$	7.26897	0.274154
$704$	0	0
$705$	29.7029	1.11868
$706$	0	0
$707$	13.3649	0.502640
$708$	0	0
$709$	2.32697	0.0873911	0.0436956	−	0.999045i	$-0.486087\pi$
$709$	2.32697	0.0873911	0.0436956	+	0.999045i	$0.486087\pi$
$710$	0	0
$711$	19.3457	0.725519
$712$	0	0
$713$	1.71323	0.0641610
$714$	0	0
$715$	0	0
$716$	0	0
$717$	23.6470	0.883114
$718$	0	0
$719$	−27.9156	−1.04108	−0.520538	−	0.853838i	$-0.674269\pi$
$719$	−27.9156	−1.04108	−0.520538	+	0.853838i	$0.674269\pi$
$720$	0	0
$721$	−10.9463	−0.407660
$722$	0	0
$723$	64.9095	2.41401
$724$	0	0
$725$	−15.5682	−0.578187
$726$	0	0
$727$	−46.6078	−1.72859	−0.864294	−	0.502986i	$-0.832235\pi$
$727$	−46.6078	−1.72859	−0.864294	+	0.502986i	$0.832235\pi$
$728$	0	0
$729$	−4.52939	−0.167755
$730$	0	0
$731$	1.61545	0.0597495
$732$	0	0
$733$	6.39157	0.236078	0.118039	−	0.993009i	$-0.462339\pi$
$733$	6.39157	0.236078	0.118039	+	0.993009i	$0.462339\pi$
$734$	0	0
$735$	−47.3983	−1.74831
$736$	0	0
$737$	0	0
$738$	0	0
$739$	30.3573	1.11671	0.558356	−	0.829602i	$-0.311432\pi$
$739$	30.3573	1.11671	0.558356	+	0.829602i	$0.311432\pi$
$740$	0	0
$741$	−2.07139	−0.0760944
$742$	0	0
$743$	11.0523	0.405469	0.202734	−	0.979234i	$-0.435017\pi$
$743$	11.0523	0.405469	0.202734	+	0.979234i	$0.435017\pi$
$744$	0	0
$745$	−30.8367	−1.12977
$746$	0	0
$747$	8.21423	0.300543
$748$	0	0
$749$	7.78137	0.284325
$750$	0	0
$751$	0.653745	0.0238555	0.0119277	−	0.999929i	$-0.496203\pi$
$751$	0.653745	0.0238555	0.0119277	+	0.999929i	$0.496203\pi$
$752$	0	0
$753$	17.3358	0.631751
$754$	0	0
$755$	−31.2300	−1.13658
$756$	0	0
$757$	8.96541	0.325853	0.162927	−	0.986638i	$-0.447907\pi$
$757$	8.96541	0.325853	0.162927	+	0.986638i	$0.447907\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	45.3755	1.64486	0.822431	−	0.568865i	$-0.192617\pi$
$761$	45.3755	1.64486	0.822431	+	0.568865i	$0.192617\pi$
$762$	0	0
$763$	−3.94815	−0.142933
$764$	0	0
$765$	−1.55311	−0.0561527
$766$	0	0
$767$	−7.77533	−0.280751
$768$	0	0
$769$	−14.4454	−0.520914	−0.260457	−	0.965485i	$-0.583873\pi$
$769$	−14.4454	−0.520914	−0.260457	+	0.965485i	$0.583873\pi$
$770$	0	0
$771$	22.0021	0.792386
$772$	0	0
$773$	−54.7942	−1.97081	−0.985405	−	0.170229i	$-0.945549\pi$
$773$	−54.7942	−1.97081	−0.985405	+	0.170229i	$0.945549\pi$
$774$	0	0
$775$	−15.0817	−0.541752
$776$	0	0
$777$	−17.0660	−0.612238
$778$	0	0
$779$	−8.87447	−0.317961
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−5.28073	−0.188718
$784$	0	0
$785$	66.3205	2.36708
$786$	0	0
$787$	13.4291	0.478697	0.239348	−	0.970934i	$-0.423066\pi$
$787$	13.4291	0.478697	0.239348	+	0.970934i	$0.423066\pi$
$788$	0	0
$789$	−37.4429	−1.33300
$790$	0	0
$791$	−2.10352	−0.0747924
$792$	0	0
$793$	−7.75969	−0.275555
$794$	0	0
$795$	55.6944	1.97528
$796$	0	0
$797$	36.2428	1.28379	0.641894	−	0.766794i	$-0.278149\pi$
$797$	36.2428	1.28379	0.641894	+	0.766794i	$0.278149\pi$
$798$	0	0
$799$	0.903179	0.0319522
$800$	0	0
$801$	−21.6974	−0.766641
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−2.71720	−0.0957686
$806$	0	0
$807$	7.64010	0.268944
$808$	0	0
$809$	−20.4757	−0.719888	−0.359944	−	0.932974i	$-0.617204\pi$
$809$	−20.4757	−0.719888	−0.359944	+	0.932974i	$0.617204\pi$
$810$	0	0
$811$	−28.9330	−1.01598	−0.507988	−	0.861364i	$-0.669611\pi$
$811$	−28.9330	−1.01598	−0.507988	+	0.861364i	$0.669611\pi$
$812$	0	0
$813$	−25.3040	−0.887450
$814$	0	0
$815$	32.7965	1.14881
$816$	0	0
$817$	−6.05333	−0.211779
$818$	0	0
$819$	1.87491	0.0655147
$820$	0	0
$821$	18.7569	0.654622	0.327311	−	0.944917i	$-0.393858\pi$
$821$	18.7569	0.654622	0.327311	+	0.944917i	$0.393858\pi$
$822$	0	0
$823$	−17.0378	−0.593900	−0.296950	−	0.954893i	$-0.595969\pi$
$823$	−17.0378	−0.593900	−0.296950	+	0.954893i	$0.595969\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	40.0030	1.39104	0.695520	−	0.718507i	$-0.255174\pi$
$827$	40.0030	1.39104	0.695520	+	0.718507i	$0.255174\pi$
$828$	0	0
$829$	−38.6990	−1.34407	−0.672036	−	0.740518i	$-0.734580\pi$
$829$	−38.6990	−1.34407	−0.672036	+	0.740518i	$0.734580\pi$
$830$	0	0
$831$	−55.7324	−1.93334
$832$	0	0
$833$	−1.44124	−0.0499361
$834$	0	0
$835$	65.5443	2.26825
$836$	0	0
$837$	−5.11574	−0.176826
$838$	0	0
$839$	−2.52052	−0.0870180	−0.0435090	−	0.999053i	$-0.513854\pi$
$839$	−2.52052	−0.0870180	−0.0435090	+	0.999053i	$0.513854\pi$
$840$	0	0
$841$	−24.4281	−0.842349
$842$	0	0
$843$	−17.7877	−0.612640
$844$	0	0
$845$	41.6010	1.43112
$846$	0	0
$847$	10.3120	0.354325
$848$	0	0
$849$	59.4124	2.03903
$850$	0	0
$851$	6.81434	0.233592
$852$	0	0
$853$	−40.8908	−1.40007	−0.700037	−	0.714107i	$-0.746833\pi$
$853$	−40.8908	−1.40007	−0.700037	+	0.714107i	$0.746833\pi$
$854$	0	0
$855$	5.81973	0.199031
$856$	0	0
$857$	−31.9563	−1.09161	−0.545803	−	0.837914i	$-0.683775\pi$
$857$	−31.9563	−1.09161	−0.545803	+	0.837914i	$0.683775\pi$
$858$	0	0
$859$	−21.0271	−0.717436	−0.358718	−	0.933446i	$-0.616786\pi$
$859$	−21.0271	−0.717436	−0.358718	+	0.933446i	$0.616786\pi$
$860$	0	0
$861$	20.8353	0.710067
$862$	0	0
$863$	−48.7452	−1.65930	−0.829652	−	0.558280i	$-0.811461\pi$
$863$	−48.7452	−1.65930	−0.829652	+	0.558280i	$0.811461\pi$
$864$	0	0
$865$	−23.5486	−0.800678
$866$	0	0
$867$	37.4405	1.27155
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−16.4553	−0.557565
$872$	0	0
$873$	−31.1464	−1.05415
$874$	0	0
$875$	7.49355	0.253328
$876$	0	0
$877$	6.05501	0.204463	0.102232	−	0.994761i	$-0.467402\pi$
$877$	6.05501	0.204463	0.102232	+	0.994761i	$0.467402\pi$
$878$	0	0
$879$	2.01398	0.0679299
$880$	0	0
$881$	−41.8984	−1.41159	−0.705797	−	0.708414i	$-0.749411\pi$
$881$	−41.8984	−1.41159	−0.705797	+	0.708414i	$0.749411\pi$
$882$	0	0
$883$	−36.4747	−1.22747	−0.613735	−	0.789512i	$-0.710334\pi$
$883$	−36.4747	−1.22747	−0.613735	+	0.789512i	$0.710334\pi$
$884$	0	0
$885$	56.6630	1.90471
$886$	0	0
$887$	18.1482	0.609356	0.304678	−	0.952455i	$-0.401451\pi$
$887$	18.1482	0.609356	0.304678	+	0.952455i	$0.401451\pi$
$888$	0	0
$889$	8.02645	0.269198
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3.38435	−0.113253
$894$	0	0
$895$	−74.9911	−2.50668
$896$	0	0
$897$	−1.94184	−0.0648360
$898$	0	0
$899$	4.42904	0.147717
$900$	0	0
$901$	1.69350	0.0564188
$902$	0	0
$903$	14.2119	0.472943
$904$	0	0
$905$	18.1059	0.601860
$906$	0	0
$907$	26.2809	0.872642	0.436321	−	0.899791i	$-0.356281\pi$
$907$	26.2809	0.872642	0.436321	+	0.899791i	$0.356281\pi$
$908$	0	0
$909$	−26.8348	−0.890055
$910$	0	0
$911$	−27.1374	−0.899101	−0.449551	−	0.893255i	$-0.648416\pi$
$911$	−27.1374	−0.899101	−0.449551	+	0.893255i	$0.648416\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	56.5490	1.86945
$916$	0	0
$917$	9.25259	0.305547
$918$	0	0
$919$	−0.625913	−0.0206470	−0.0103235	−	0.999947i	$-0.503286\pi$
$919$	−0.625913	−0.0206470	−0.0103235	+	0.999947i	$0.503286\pi$
$920$	0	0
$921$	10.1804	0.335454
$922$	0	0
$923$	−1.06254	−0.0349741
$924$	0	0
$925$	−59.9873	−1.97237
$926$	0	0
$927$	21.9785	0.721869
$928$	0	0
$929$	−24.0211	−0.788108	−0.394054	−	0.919087i	$-0.628928\pi$
$929$	−24.0211	−0.788108	−0.394054	+	0.919087i	$0.628928\pi$
$930$	0	0
$931$	5.40056	0.176996
$932$	0	0
$933$	74.2383	2.43045
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−39.4333	−1.28823	−0.644115	−	0.764929i	$-0.722774\pi$
$937$	−39.4333	−1.28823	−0.644115	+	0.764929i	$0.722774\pi$
$938$	0	0
$939$	−27.7720	−0.906306
$940$	0	0
$941$	−0.726616	−0.0236870	−0.0118435	−	0.999930i	$-0.503770\pi$
$941$	−0.726616	−0.0236870	−0.0118435	+	0.999930i	$0.503770\pi$
$942$	0	0
$943$	−8.31942	−0.270918
$944$	0	0
$945$	8.11360	0.263935
$946$	0	0
$947$	−14.8876	−0.483781	−0.241890	−	0.970304i	$-0.577767\pi$
$947$	−14.8876	−0.483781	−0.241890	+	0.970304i	$0.577767\pi$
$948$	0	0
$949$	−3.84296	−0.124748
$950$	0	0
$951$	−3.98939	−0.129365
$952$	0	0
$953$	18.6345	0.603630	0.301815	−	0.953367i	$-0.402408\pi$
$953$	18.6345	0.603630	0.301815	+	0.953367i	$0.402408\pi$
$954$	0	0
$955$	−18.4862	−0.598198
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−11.5555	−0.373148
$960$	0	0
$961$	−26.7093	−0.861592
$962$	0	0
$963$	−15.6239	−0.503472
$964$	0	0
$965$	37.7545	1.21536
$966$	0	0
$967$	50.5925	1.62694	0.813472	−	0.581604i	$-0.197575\pi$
$967$	50.5925	1.62694	0.813472	+	0.581604i	$0.197575\pi$
$968$	0	0
$969$	0.459004	0.0147453
$970$	0	0
$971$	38.2147	1.22637	0.613185	−	0.789939i	$-0.289888\pi$
$971$	38.2147	1.22637	0.613185	+	0.789939i	$0.289888\pi$
$972$	0	0
$973$	7.34400	0.235438
$974$	0	0
$975$	17.0942	0.547451
$976$	0	0
$977$	17.7455	0.567728	0.283864	−	0.958865i	$-0.408384\pi$
$977$	17.7455	0.567728	0.283864	+	0.958865i	$0.408384\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	7.92730	0.253099
$982$	0	0
$983$	−7.20048	−0.229660	−0.114830	−	0.993385i	$-0.536632\pi$
$983$	−7.20048	−0.229660	−0.114830	+	0.993385i	$0.536632\pi$
$984$	0	0
$985$	−3.70145	−0.117938
$986$	0	0
$987$	7.94572	0.252915
$988$	0	0
$989$	−5.67473	−0.180446
$990$	0	0
$991$	14.0912	0.447620	0.223810	−	0.974633i	$-0.428150\pi$
$991$	14.0912	0.447620	0.223810	+	0.974633i	$0.428150\pi$
$992$	0	0
$993$	55.7371	1.76876
$994$	0	0
$995$	−63.1537	−2.00211
$996$	0	0
$997$	52.1633	1.65203	0.826015	−	0.563648i	$-0.190603\pi$
$997$	52.1633	1.65203	0.826015	+	0.563648i	$0.190603\pi$
$998$	0	0
$999$	−20.3477	−0.643773

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4544.2.a.bf.1.1		5
4.3	odd	2		4544.2.a.be.1.5		5
8.3	odd	2		568.2.a.e.1.1	✓	5
8.5	even	2		1136.2.a.n.1.5		5
24.11	even	2		5112.2.a.n.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
568.2.a.e.1.1	✓	5	8.3	odd	2
1136.2.a.n.1.5		5	8.5	even	2
4544.2.a.be.1.5		5	4.3	odd	2
4544.2.a.bf.1.1		5	1.1	even	1	trivial
5112.2.a.n.1.2		5	24.11	even	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:	\( N \)	\(=\)	\( 4544 = 2^{6} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4544.a (trivial)

\(f(q)\)	\(=\)	\(q-2.20959 q^{3} -3.50442 q^{5} -0.937455 q^{7} +1.88227 q^{9} +O(q^{10})\)	\(q-2.20959 q^{3} -3.50442 q^{5} -0.937455 q^{7} +1.88227 q^{9} -1.06254 q^{13} +7.74333 q^{15} +0.235452 q^{17} -0.882274 q^{19} +2.07139 q^{21} -0.827093 q^{23} +7.28098 q^{25} +2.46971 q^{27} -2.13820 q^{29} -2.07139 q^{31} +3.28524 q^{35} -8.23890 q^{37} +2.34778 q^{39} +10.0586 q^{41} +6.86105 q^{43} -6.59628 q^{45} +3.83594 q^{47} -6.12118 q^{49} -0.520251 q^{51} +7.19257 q^{53} +1.94946 q^{57} +7.31766 q^{59} +7.30293 q^{61} -1.76455 q^{63} +3.72360 q^{65} +15.4867 q^{67} +1.82753 q^{69} +1.00000 q^{71} +3.61675 q^{73} -16.0879 q^{75} +10.2778 q^{79} -11.1039 q^{81} +4.36399 q^{83} -0.825123 q^{85} +4.72453 q^{87} -11.5272 q^{89} +0.996088 q^{91} +4.57691 q^{93} +3.09186 q^{95} -16.5472 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{3} - 7 q^{5} + q^{7} + 7 q^{9}+O(q^{10}) \) 5 * q + 2 * q^3 - 7 * q^5 + q^7 + 7 * q^9	\( 5 q + 2 q^{3} - 7 q^{5} + q^{7} + 7 q^{9} - 11 q^{13} - 6 q^{15} + 6 q^{17} - 2 q^{19} - 5 q^{21} - 5 q^{23} + 8 q^{25} + 5 q^{27} - 13 q^{29} + 5 q^{31} - 7 q^{37} + q^{39} + 8 q^{41} - 8 q^{43} - 7 q^{45} - q^{47} + 6 q^{49} - 14 q^{51} - 16 q^{53} - 9 q^{57} - 4 q^{59} - 22 q^{61} - 4 q^{63} + 14 q^{65} - 12 q^{67} - 13 q^{69} + 5 q^{71} - 8 q^{73} - 17 q^{75} + 15 q^{79} - 27 q^{81} - q^{83} - 14 q^{85} + 17 q^{87} - 4 q^{89} - 43 q^{91} + q^{93} - 8 q^{97}+O(q^{100}) \) 5 * q + 2 * q^3 - 7 * q^5 + q^7 + 7 * q^9 - 11 * q^13 - 6 * q^15 + 6 * q^17 - 2 * q^19 - 5 * q^21 - 5 * q^23 + 8 * q^25 + 5 * q^27 - 13 * q^29 + 5 * q^31 - 7 * q^37 + q^39 + 8 * q^41 - 8 * q^43 - 7 * q^45 - q^47 + 6 * q^49 - 14 * q^51 - 16 * q^53 - 9 * q^57 - 4 * q^59 - 22 * q^61 - 4 * q^63 + 14 * q^65 - 12 * q^67 - 13 * q^69 + 5 * q^71 - 8 * q^73 - 17 * q^75 + 15 * q^79 - 27 * q^81 - q^83 - 14 * q^85 + 17 * q^87 - 4 * q^89 - 43 * q^91 + q^93 - 8 * q^97

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