LMFDB - Embedded newform 9248.2.a.y.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9248,2,Mod(1,9248)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9248, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9248.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9248 = 2^{5} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9248.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-2,0,2,0,-6,0,6,0,0,0,2,0,8,0,0,0,-6,0,12,0,-6,0,2,0,-8, 0,2,0,0,0,-10,0,-14,0,0,0,-26,0,-16,0,8,0,-20,0,8,0,14,0,0,0,-6,0,-22, 0,6,0,-16,0,20,0,-24,0,4,0,0,0,-4,0,0,0,4,0,-8,0,-2,0,-24,0,32,0,26,0, 0,0,-10,0,0,0,-32,0,26,0,32,0,-22,0,50,0,-4,0,-32,0,-14,0,-32,0,28,0,32, 0,10,0,10,0,2,0,0,0,-8,0,-8,0,36,0,-8,0,-46,0,-16,0,4,0,38,0,24,0,-22, 0,2,0,-2,0,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(145)] == traces)

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

Self dual:	yes
Analytic conductor:	$73.8456517893$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.14272.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + 2x + 3 $ x^4 - 2x^3 - 5x^2 + 2*x + 3
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			1.4
Root			$-0.716360$ of defining polynomial
Character	$\chi$	$=$	9248.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.52559 q^{3} +1.05411 q^{5} -0.244878 q^{7} +3.37861 q^{9} +0.662250 q^{11} -4.99707 q^{13} +2.66225 q^{15} -5.85009 q^{19} -0.618463 q^{21} -5.18784 q^{23} -3.88885 q^{25} +0.956213 q^{27} -0.0541097 q^{29} +4.28071 q^{31} +1.67257 q^{33} -0.258129 q^{35} +8.48390 q^{37} -12.6206 q^{39} -7.54094 q^{41} -5.82165 q^{43} +3.56143 q^{45} +6.20319 q^{47} -6.94003 q^{49} -6.91955 q^{53} +0.698084 q^{55} -14.7749 q^{57} -7.54094 q^{59} +1.38154 q^{61} -0.827348 q^{63} -5.26746 q^{65} +1.40202 q^{67} -13.1024 q^{69} +3.54094 q^{71} +13.0248 q^{73} -9.82165 q^{75} -0.162171 q^{77} -4.67550 q^{79} -7.72083 q^{81} +10.6338 q^{83} -0.136659 q^{87} -13.4269 q^{89} +1.22367 q^{91} +10.8113 q^{93} -6.16664 q^{95} -9.81133 q^{97} +2.23748 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 6 q^{9} + 2 q^{13} + 8 q^{15} - 6 q^{19} + 12 q^{21} - 6 q^{23} + 2 q^{25} - 8 q^{27} + 2 q^{29} - 10 q^{33} - 14 q^{35} - 26 q^{39} - 16 q^{41} + 8 q^{43} - 20 q^{45}+ \cdots + 50 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 6 * q^9 + 2 * q^13 + 8 * q^15 - 6 * q^19 + 12 * q^21 - 6 * q^23 + 2 * q^25 - 8 * q^27 + 2 * q^29 - 10 * q^33 - 14 * q^35 - 26 * q^39 - 16 * q^41 + 8 * q^43 - 20 * q^45 + 8 * q^47 + 14 * q^49 - 6 * q^53 - 22 * q^55 + 6 * q^57 - 16 * q^59 + 20 * q^61 - 24 * q^63 + 4 * q^65 - 4 * q^69 + 4 * q^73 - 8 * q^75 - 2 * q^77 - 24 * q^79 + 32 * q^81 + 26 * q^83 - 10 * q^87 - 32 * q^91 + 26 * q^93 + 32 * q^95 - 22 * q^97 + 50 * q^99

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
$3$	2.52559	1.45815	0.729075	−	0.684433i	$-0.239950\pi$
$3$	2.52559	1.45815	0.729075	+	0.684433i	$0.239950\pi$
$4$	0	0
$5$	1.05411	0.471412	0.235706	−	0.971824i	$-0.424260\pi$
$5$	1.05411	0.471412	0.235706	+	0.971824i	$0.424260\pi$
$6$	0	0
$7$	−0.244878	−0.0925553	−0.0462777	−	0.998929i	$-0.514736\pi$
$7$	−0.244878	−0.0925553	−0.0462777	+	0.998929i	$0.514736\pi$
$8$	0	0
$9$	3.37861	1.12620
$10$	0	0
$11$	0.662250	0.199676	0.0998379	−	0.995004i	$-0.468168\pi$
$11$	0.662250	0.199676	0.0998379	+	0.995004i	$0.468168\pi$
$12$	0	0
$13$	−4.99707	−1.38594	−0.692969	−	0.720967i	$-0.743698\pi$
$13$	−4.99707	−1.38594	−0.692969	+	0.720967i	$0.743698\pi$
$14$	0	0
$15$	2.66225	0.687390
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−5.85009	−1.34210	−0.671051	−	0.741411i	$-0.734157\pi$
$19$	−5.85009	−1.34210	−0.671051	+	0.741411i	$0.734157\pi$
$20$	0	0
$21$	−0.618463	−0.134960
$22$	0	0
$23$	−5.18784	−1.08174	−0.540870	−	0.841106i	$-0.681905\pi$
$23$	−5.18784	−1.08174	−0.540870	+	0.841106i	$0.681905\pi$
$24$	0	0
$25$	−3.88885	−0.777771
$26$	0	0
$27$	0.956213	0.184023
$28$	0	0
$29$	−0.0541097	−0.0100479	−0.00502396	−	0.999987i	$-0.501599\pi$
$29$	−0.0541097	−0.0100479	−0.00502396	+	0.999987i	$0.501599\pi$
$30$	0	0
$31$	4.28071	0.768839	0.384419	−	0.923159i	$-0.374402\pi$
$31$	4.28071	0.768839	0.384419	+	0.923159i	$0.374402\pi$
$32$	0	0
$33$	1.67257	0.291157
$34$	0	0
$35$	−0.258129	−0.0436317
$36$	0	0
$37$	8.48390	1.39474	0.697372	−	0.716709i	$-0.254352\pi$
$37$	8.48390	1.39474	0.697372	+	0.716709i	$0.254352\pi$
$38$	0	0
$39$	−12.6206	−2.02091
$40$	0	0
$41$	−7.54094	−1.17770	−0.588848	−	0.808244i	$-0.700418\pi$
$41$	−7.54094	−1.17770	−0.588848	+	0.808244i	$0.700418\pi$
$42$	0	0
$43$	−5.82165	−0.887793	−0.443897	−	0.896078i	$-0.646404\pi$
$43$	−5.82165	−0.887793	−0.443897	+	0.896078i	$0.646404\pi$
$44$	0	0
$45$	3.56143	0.530906
$46$	0	0
$47$	6.20319	0.904828	0.452414	−	0.891808i	$-0.350563\pi$
$47$	6.20319	0.904828	0.452414	+	0.891808i	$0.350563\pi$
$48$	0	0
$49$	−6.94003	−0.991434
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.91955	−0.950473	−0.475237	−	0.879858i	$-0.657638\pi$
$53$	−6.91955	−0.950473	−0.475237	+	0.879858i	$0.657638\pi$
$54$	0	0
$55$	0.698084	0.0941296
$56$	0	0
$57$	−14.7749	−1.95699
$58$	0	0
$59$	−7.54094	−0.981747	−0.490873	−	0.871231i	$-0.663322\pi$
$59$	−7.54094	−0.981747	−0.490873	+	0.871231i	$0.663322\pi$
$60$	0	0
$61$	1.38154	0.176888	0.0884439	−	0.996081i	$-0.471811\pi$
$61$	1.38154	0.176888	0.0884439	+	0.996081i	$0.471811\pi$
$62$	0	0
$63$	−0.827348	−0.104236
$64$	0	0
$65$	−5.26746	−0.653348
$66$	0	0
$67$	1.40202	0.171284	0.0856422	−	0.996326i	$-0.472706\pi$
$67$	1.40202	0.171284	0.0856422	+	0.996326i	$0.472706\pi$
$68$	0	0
$69$	−13.1024	−1.57734
$70$	0	0
$71$	3.54094	0.420232	0.210116	−	0.977676i	$-0.432616\pi$
$71$	3.54094	0.420232	0.210116	+	0.977676i	$0.432616\pi$
$72$	0	0
$73$	13.0248	1.52444	0.762221	−	0.647317i	$-0.224109\pi$
$73$	13.0248	1.52444	0.762221	+	0.647317i	$0.224109\pi$
$74$	0	0
$75$	−9.82165	−1.13411
$76$	0	0
$77$	−0.162171	−0.0184811
$78$	0	0
$79$	−4.67550	−0.526035	−0.263017	−	0.964791i	$-0.584718\pi$
$79$	−4.67550	−0.526035	−0.263017	+	0.964791i	$0.584718\pi$
$80$	0	0
$81$	−7.72083	−0.857870
$82$	0	0
$83$	10.6338	1.16721	0.583606	−	0.812037i	$-0.301641\pi$
$83$	10.6338	1.16721	0.583606	+	0.812037i	$0.301641\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.136659	−0.0146514
$88$	0	0
$89$	−13.4269	−1.42324	−0.711622	−	0.702562i	$-0.752039\pi$
$89$	−13.4269	−1.42324	−0.711622	+	0.702562i	$0.752039\pi$
$90$	0	0
$91$	1.22367	0.128276
$92$	0	0
$93$	10.8113	1.12108
$94$	0	0
$95$	−6.16664	−0.632684
$96$	0	0
$97$	−9.81133	−0.996189	−0.498095	−	0.867123i	$-0.665967\pi$
$97$	−9.81133	−0.996189	−0.498095	+	0.867123i	$0.665967\pi$
$98$	0	0
$99$	2.23748	0.224876
$100$	0	0
$101$	17.3698	1.72836	0.864181	−	0.503181i	$-0.167837\pi$
$101$	17.3698	1.72836	0.864181	+	0.503181i	$0.167837\pi$
$102$	0	0
$103$	−18.7779	−1.85024	−0.925119	−	0.379677i	$-0.876035\pi$
$103$	−18.7779	−1.85024	−0.925119	+	0.379677i	$0.876035\pi$
$104$	0	0
$105$	−0.651927	−0.0636216
$106$	0	0
$107$	−3.79681	−0.367052	−0.183526	−	0.983015i	$-0.558751\pi$
$107$	−3.79681	−0.367052	−0.183526	+	0.983015i	$0.558751\pi$
$108$	0	0
$109$	8.32450	0.797342	0.398671	−	0.917094i	$-0.369471\pi$
$109$	8.32450	0.797342	0.398671	+	0.917094i	$0.369471\pi$
$110$	0	0
$111$	21.4269	2.03375
$112$	0	0
$113$	6.67257	0.627703	0.313851	−	0.949472i	$-0.398381\pi$
$113$	6.67257	0.627703	0.313851	+	0.949472i	$0.398381\pi$
$114$	0	0
$115$	−5.46855	−0.509945
$116$	0	0
$117$	−16.8832	−1.56085
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.5614	−0.960130
$122$	0	0
$123$	−19.0453	−1.71726
$124$	0	0
$125$	−9.36983	−0.838063
$126$	0	0
$127$	5.82165	0.516588	0.258294	−	0.966066i	$-0.416840\pi$
$127$	5.82165	0.516588	0.258294	+	0.966066i	$0.416840\pi$
$128$	0	0
$129$	−14.7031	−1.29454
$130$	0	0
$131$	−19.7208	−1.72302	−0.861508	−	0.507743i	$-0.830480\pi$
$131$	−19.7208	−1.72302	−0.861508	+	0.507743i	$0.830480\pi$
$132$	0	0
$133$	1.43256	0.124219
$134$	0	0
$135$	1.00795	0.0867508
$136$	0	0
$137$	−0.267462	−0.0228508	−0.0114254	−	0.999935i	$-0.503637\pi$
$137$	−0.267462	−0.0228508	−0.0114254	+	0.999935i	$0.503637\pi$
$138$	0	0
$139$	−19.0738	−1.61782	−0.808908	−	0.587935i	$-0.799941\pi$
$139$	−19.0738	−1.61782	−0.808908	+	0.587935i	$0.799941\pi$
$140$	0	0
$141$	15.6667	1.31938
$142$	0	0
$143$	−3.30931	−0.276738
$144$	0	0
$145$	−0.0570376	−0.00473671
$146$	0	0
$147$	−17.5277	−1.44566
$148$	0	0
$149$	−23.2618	−1.90568	−0.952839	−	0.303476i	$-0.901853\pi$
$149$	−23.2618	−1.90568	−0.952839	+	0.303476i	$0.901853\pi$
$150$	0	0
$151$	−6.30555	−0.513139	−0.256569	−	0.966526i	$-0.582592\pi$
$151$	−6.30555	−0.513139	−0.256569	+	0.966526i	$0.582592\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.51234	0.362440
$156$	0	0
$157$	−8.55850	−0.683042	−0.341521	−	0.939874i	$-0.610942\pi$
$157$	−8.55850	−0.683042	−0.341521	+	0.939874i	$0.610942\pi$
$158$	0	0
$159$	−17.4759	−1.38593
$160$	0	0
$161$	1.27039	0.100121
$162$	0	0
$163$	21.0321	1.64736	0.823680	−	0.567055i	$-0.191917\pi$
$163$	21.0321	1.64736	0.823680	+	0.567055i	$0.191917\pi$
$164$	0	0
$165$	1.76307	0.137255
$166$	0	0
$167$	3.66015	0.283231	0.141616	−	0.989922i	$-0.454770\pi$
$167$	3.66015	0.283231	0.141616	+	0.989922i	$0.454770\pi$
$168$	0	0
$169$	11.9707	0.920825
$170$	0	0
$171$	−19.7652	−1.51148
$172$	0	0
$173$	22.9108	1.74187	0.870937	−	0.491395i	$-0.163513\pi$
$173$	22.9108	1.74187	0.870937	+	0.491395i	$0.163513\pi$
$174$	0	0
$175$	0.952296	0.0719868
$176$	0	0
$177$	−19.0453	−1.43153
$178$	0	0
$179$	4.48750	0.335412	0.167706	−	0.985837i	$-0.446364\pi$
$179$	4.48750	0.335412	0.167706	+	0.985837i	$0.446364\pi$
$180$	0	0
$181$	−4.69599	−0.349050	−0.174525	−	0.984653i	$-0.555839\pi$
$181$	−4.69599	−0.349050	−0.174525	+	0.984653i	$0.555839\pi$
$182$	0	0
$183$	3.48920	0.257929
$184$	0	0
$185$	8.94296	0.657500
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−0.234156	−0.0170323
$190$	0	0
$191$	−7.09287	−0.513222	−0.256611	−	0.966515i	$-0.582606\pi$
$191$	−7.09287	−0.513222	−0.256611	+	0.966515i	$0.582606\pi$
$192$	0	0
$193$	−12.0819	−0.869673	−0.434836	−	0.900510i	$-0.643194\pi$
$193$	−12.0819	−0.869673	−0.434836	+	0.900510i	$0.643194\pi$
$194$	0	0
$195$	−13.3035	−0.952680
$196$	0	0
$197$	8.83767	0.629658	0.314829	−	0.949148i	$-0.398053\pi$
$197$	8.83767	0.629658	0.314829	+	0.949148i	$0.398053\pi$
$198$	0	0
$199$	17.8975	1.26872	0.634361	−	0.773037i	$-0.281263\pi$
$199$	17.8975	1.26872	0.634361	+	0.773037i	$0.281263\pi$
$200$	0	0
$201$	3.54094	0.249759
$202$	0	0
$203$	0.0132503	0.000929989 0
$204$	0	0
$205$	−7.94898	−0.555181
$206$	0	0
$207$	−17.5277	−1.21826
$208$	0	0
$209$	−3.87422	−0.267986
$210$	0	0
$211$	16.0533	1.10515	0.552577	−	0.833462i	$-0.313645\pi$
$211$	16.0533	1.10515	0.552577	+	0.833462i	$0.313645\pi$
$212$	0	0
$213$	8.94296	0.612762
$214$	0	0
$215$	−6.13666	−0.418517
$216$	0	0
$217$	−1.04825	−0.0711601
$218$	0	0
$219$	32.8954	2.22287
$220$	0	0
$221$	0	0
$222$	0	0
$223$	24.5313	1.64274	0.821369	−	0.570396i	$-0.193210\pi$
$223$	24.5313	1.64274	0.821369	+	0.570396i	$0.193210\pi$
$224$	0	0
$225$	−13.1389	−0.875928
$226$	0	0
$227$	15.4650	1.02645	0.513223	−	0.858256i	$-0.328452\pi$
$227$	15.4650	1.02645	0.513223	+	0.858256i	$0.328452\pi$
$228$	0	0
$229$	−21.9473	−1.45032	−0.725160	−	0.688581i	$-0.758234\pi$
$229$	−21.9473	−1.45032	−0.725160	+	0.688581i	$0.758234\pi$
$230$	0	0
$231$	−0.409577	−0.0269482
$232$	0	0
$233$	4.57148	0.299487	0.149744	−	0.988725i	$-0.452155\pi$
$233$	4.57148	0.299487	0.149744	+	0.988725i	$0.452155\pi$
$234$	0	0
$235$	6.53884	0.426547
$236$	0	0
$237$	−11.8084	−0.767038
$238$	0	0
$239$	−12.4458	−0.805053	−0.402526	−	0.915408i	$-0.631868\pi$
$239$	−12.4458	−0.805053	−0.402526	+	0.915408i	$0.631868\pi$
$240$	0	0
$241$	26.2587	1.69147	0.845735	−	0.533603i	$-0.179162\pi$
$241$	26.2587	1.69147	0.845735	+	0.533603i	$0.179162\pi$
$242$	0	0
$243$	−22.3683	−1.43493
$244$	0	0
$245$	−7.31556	−0.467374
$246$	0	0
$247$	29.2333	1.86007
$248$	0	0
$249$	26.8567	1.70197
$250$	0	0
$251$	−23.4217	−1.47837	−0.739183	−	0.673505i	$-0.764788\pi$
$251$	−23.4217	−1.47837	−0.739183	+	0.673505i	$0.764788\pi$
$252$	0	0
$253$	−3.43565	−0.215997
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.07895	0.316816	0.158408	−	0.987374i	$-0.449364\pi$
$257$	5.07895	0.316816	0.158408	+	0.987374i	$0.449364\pi$
$258$	0	0
$259$	−2.07752	−0.129091
$260$	0	0
$261$	−0.182816	−0.0113160
$262$	0	0
$263$	8.71553	0.537423	0.268711	−	0.963221i	$-0.413402\pi$
$263$	8.71553	0.537423	0.268711	+	0.963221i	$0.413402\pi$
$264$	0	0
$265$	−7.29396	−0.448065
$266$	0	0
$267$	−33.9108	−2.07531
$268$	0	0
$269$	−13.4810	−0.821949	−0.410975	−	0.911647i	$-0.634812\pi$
$269$	−13.4810	−0.821949	−0.410975	+	0.911647i	$0.634812\pi$
$270$	0	0
$271$	22.5330	1.36878	0.684391	−	0.729115i	$-0.260068\pi$
$271$	22.5330	1.36878	0.684391	+	0.729115i	$0.260068\pi$
$272$	0	0
$273$	3.09050	0.187046
$274$	0	0
$275$	−2.57539	−0.155302
$276$	0	0
$277$	9.66949	0.580983	0.290492	−	0.956878i	$-0.406181\pi$
$277$	9.66949	0.580983	0.290492	+	0.956878i	$0.406181\pi$
$278$	0	0
$279$	14.4629	0.865869
$280$	0	0
$281$	7.12578	0.425088	0.212544	−	0.977151i	$-0.431825\pi$
$281$	7.12578	0.425088	0.212544	+	0.977151i	$0.431825\pi$
$282$	0	0
$283$	−25.8859	−1.53876	−0.769379	−	0.638792i	$-0.779434\pi$
$283$	−25.8859	−1.53876	−0.769379	+	0.638792i	$0.779434\pi$
$284$	0	0
$285$	−15.5744	−0.922548
$286$	0	0
$287$	1.84661	0.109002
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−24.7794	−1.45259
$292$	0	0
$293$	18.5249	1.08223	0.541117	−	0.840947i	$-0.318001\pi$
$293$	18.5249	1.08223	0.541117	+	0.840947i	$0.318001\pi$
$294$	0	0
$295$	−7.94898	−0.462807
$296$	0	0
$297$	0.633252	0.0367450
$298$	0	0
$299$	25.9240	1.49922
$300$	0	0
$301$	1.42560	0.0821700
$302$	0	0
$303$	43.8691	2.52021
$304$	0	0
$305$	1.45629	0.0833870
$306$	0	0
$307$	−18.1001	−1.03303	−0.516514	−	0.856279i	$-0.672771\pi$
$307$	−18.1001	−1.03303	−0.516514	+	0.856279i	$0.672771\pi$
$308$	0	0
$309$	−47.4252	−2.69793
$310$	0	0
$311$	−29.0505	−1.64730	−0.823650	−	0.567098i	$-0.808066\pi$
$311$	−29.0505	−1.64730	−0.823650	+	0.567098i	$0.808066\pi$
$312$	0	0
$313$	3.13290	0.177082	0.0885410	−	0.996073i	$-0.471780\pi$
$313$	3.13290	0.177082	0.0885410	+	0.996073i	$0.471780\pi$
$314$	0	0
$315$	−0.872116	−0.0491382
$316$	0	0
$317$	−15.4839	−0.869663	−0.434831	−	0.900512i	$-0.643192\pi$
$317$	−15.4839	−0.869663	−0.434831	+	0.900512i	$0.643192\pi$
$318$	0	0
$319$	−0.0358341	−0.00200633
$320$	0	0
$321$	−9.58919	−0.535217
$322$	0	0
$323$	0	0
$324$	0	0
$325$	19.4329	1.07794
$326$	0	0
$327$	21.0243	1.16265
$328$	0	0
$329$	−1.51903	−0.0837467
$330$	0	0
$331$	8.46132	0.465076	0.232538	−	0.972587i	$-0.425297\pi$
$331$	8.46132	0.465076	0.232538	+	0.972587i	$0.425297\pi$
$332$	0	0
$333$	28.6638	1.57077
$334$	0	0
$335$	1.47789	0.0807456
$336$	0	0
$337$	16.5015	0.898892	0.449446	−	0.893308i	$-0.351621\pi$
$337$	16.5015	0.898892	0.449446	+	0.893308i	$0.351621\pi$
$338$	0	0
$339$	16.8522	0.915285
$340$	0	0
$341$	2.83490	0.153519
$342$	0	0
$343$	3.41361	0.184318
$344$	0	0
$345$	−13.8113	−0.743577
$346$	0	0
$347$	−17.3545	−0.931637	−0.465819	−	0.884880i	$-0.654240\pi$
$347$	−17.3545	−0.931637	−0.465819	+	0.884880i	$0.654240\pi$
$348$	0	0
$349$	17.3698	0.929785	0.464893	−	0.885367i	$-0.346093\pi$
$349$	17.3698	0.929785	0.464893	+	0.885367i	$0.346093\pi$
$350$	0	0
$351$	−4.77826	−0.255045
$352$	0	0
$353$	13.2574	0.705621	0.352810	−	0.935695i	$-0.385226\pi$
$353$	13.2574	0.705621	0.352810	+	0.935695i	$0.385226\pi$
$354$	0	0
$355$	3.73254	0.198103
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−33.6966	−1.77844	−0.889219	−	0.457482i	$-0.848751\pi$
$359$	−33.6966	−1.77844	−0.889219	+	0.457482i	$0.848751\pi$
$360$	0	0
$361$	15.2236	0.801240
$362$	0	0
$363$	−26.6738	−1.40001
$364$	0	0
$365$	13.7296	0.718641
$366$	0	0
$367$	−5.46341	−0.285188	−0.142594	−	0.989781i	$-0.545544\pi$
$367$	−5.46341	−0.285188	−0.142594	+	0.989781i	$0.545544\pi$
$368$	0	0
$369$	−25.4779	−1.32633
$370$	0	0
$371$	1.69445	0.0879713
$372$	0	0
$373$	5.30401	0.274631	0.137316	−	0.990527i	$-0.456153\pi$
$373$	5.30401	0.274631	0.137316	+	0.990527i	$0.456153\pi$
$374$	0	0
$375$	−23.6643	−1.22202
$376$	0	0
$377$	0.270390	0.0139258
$378$	0	0
$379$	2.01159	0.103328	0.0516642	−	0.998665i	$-0.483547\pi$
$379$	2.01159	0.103328	0.0516642	+	0.998665i	$0.483547\pi$
$380$	0	0
$381$	14.7031	0.753263
$382$	0	0
$383$	6.71719	0.343232	0.171616	−	0.985164i	$-0.445101\pi$
$383$	6.71719	0.343232	0.171616	+	0.985164i	$0.445101\pi$
$384$	0	0
$385$	−0.170946	−0.00871220
$386$	0	0
$387$	−19.6691	−0.999836
$388$	0	0
$389$	−11.7631	−0.596412	−0.298206	−	0.954502i	$-0.596388\pi$
$389$	−11.7631	−0.596412	−0.298206	+	0.954502i	$0.596388\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−49.8067	−2.51242
$394$	0	0
$395$	−4.92849	−0.247979
$396$	0	0
$397$	7.29396	0.366074	0.183037	−	0.983106i	$-0.441407\pi$
$397$	7.29396	0.366074	0.183037	+	0.983106i	$0.441407\pi$
$398$	0	0
$399$	3.61806	0.181130
$400$	0	0
$401$	−23.0748	−1.15230	−0.576149	−	0.817345i	$-0.695445\pi$
$401$	−23.0748	−1.15230	−0.576149	+	0.817345i	$0.695445\pi$
$402$	0	0
$403$	−21.3910	−1.06556
$404$	0	0
$405$	−8.13860	−0.404410
$406$	0	0
$407$	5.61846	0.278497
$408$	0	0
$409$	23.8554	1.17957	0.589787	−	0.807559i	$-0.299212\pi$
$409$	23.8554	1.17957	0.589787	+	0.807559i	$0.299212\pi$
$410$	0	0
$411$	−0.675500	−0.0333200
$412$	0	0
$413$	1.84661	0.0908659
$414$	0	0
$415$	11.2092	0.550238
$416$	0	0
$417$	−48.1725	−2.35902
$418$	0	0
$419$	9.50119	0.464163	0.232082	−	0.972696i	$-0.425446\pi$
$419$	9.50119	0.464163	0.232082	+	0.972696i	$0.425446\pi$
$420$	0	0
$421$	12.9749	0.632359	0.316180	−	0.948699i	$-0.397600\pi$
$421$	12.9749	0.632359	0.316180	+	0.948699i	$0.397600\pi$
$422$	0	0
$423$	20.9582	1.01902
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−0.338309	−0.0163719
$428$	0	0
$429$	−8.35796	−0.403526
$430$	0	0
$431$	−25.6293	−1.23452	−0.617261	−	0.786759i	$-0.711758\pi$
$431$	−25.6293	−1.23452	−0.617261	+	0.786759i	$0.711758\pi$
$432$	0	0
$433$	2.43256	0.116901	0.0584507	−	0.998290i	$-0.481384\pi$
$433$	2.43256	0.116901	0.0584507	+	0.998290i	$0.481384\pi$
$434$	0	0
$435$	−0.144054	−0.00690684
$436$	0	0
$437$	30.3493	1.45181
$438$	0	0
$439$	−0.179888	−0.00858557	−0.00429279	−	0.999991i	$-0.501366\pi$
$439$	−0.179888	−0.00858557	−0.00429279	+	0.999991i	$0.501366\pi$
$440$	0	0
$441$	−23.4477	−1.11656
$442$	0	0
$443$	29.1958	1.38714	0.693568	−	0.720391i	$-0.256038\pi$
$443$	29.1958	1.38714	0.693568	+	0.720391i	$0.256038\pi$
$444$	0	0
$445$	−14.1534	−0.670935
$446$	0	0
$447$	−58.7497	−2.77877
$448$	0	0
$449$	17.9649	0.847815	0.423907	−	0.905706i	$-0.360658\pi$
$449$	17.9649	0.847815	0.423907	+	0.905706i	$0.360658\pi$
$450$	0	0
$451$	−4.99399	−0.235158
$452$	0	0
$453$	−15.9252	−0.748233
$454$	0	0
$455$	1.28989	0.0604709
$456$	0	0
$457$	33.0162	1.54443	0.772217	−	0.635359i	$-0.219148\pi$
$457$	33.0162	1.54443	0.772217	+	0.635359i	$0.219148\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	29.4269	1.37055	0.685273	−	0.728286i	$-0.259683\pi$
$461$	29.4269	1.37055	0.685273	+	0.728286i	$0.259683\pi$
$462$	0	0
$463$	−4.51008	−0.209601	−0.104801	−	0.994493i	$-0.533420\pi$
$463$	−4.51008	−0.209601	−0.104801	+	0.994493i	$0.533420\pi$
$464$	0	0
$465$	11.3963	0.528492
$466$	0	0
$467$	−28.7229	−1.32914	−0.664569	−	0.747227i	$-0.731385\pi$
$467$	−28.7229	−1.32914	−0.664569	+	0.747227i	$0.731385\pi$
$468$	0	0
$469$	−0.343325	−0.0158533
$470$	0	0
$471$	−21.6153	−0.995979
$472$	0	0
$473$	−3.85539	−0.177271
$474$	0	0
$475$	22.7501	1.04385
$476$	0	0
$477$	−23.3785	−1.07043
$478$	0	0
$479$	−13.0951	−0.598332	−0.299166	−	0.954201i	$-0.596708\pi$
$479$	−13.0951	−0.598332	−0.299166	+	0.954201i	$0.596708\pi$
$480$	0	0
$481$	−42.3947	−1.93303
$482$	0	0
$483$	3.20849	0.145991
$484$	0	0
$485$	−10.3422	−0.469616
$486$	0	0
$487$	−23.8400	−1.08030	−0.540148	−	0.841570i	$-0.681632\pi$
$487$	−23.8400	−1.08030	−0.540148	+	0.841570i	$0.681632\pi$
$488$	0	0
$489$	53.1184	2.40210
$490$	0	0
$491$	37.0886	1.67378	0.836892	−	0.547369i	$-0.184370\pi$
$491$	37.0886	1.67378	0.836892	+	0.547369i	$0.184370\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	2.35855	0.106009
$496$	0	0
$497$	−0.867099	−0.0388947
$498$	0	0
$499$	4.75362	0.212801	0.106401	−	0.994323i	$-0.466067\pi$
$499$	4.75362	0.212801	0.106401	+	0.994323i	$0.466067\pi$
$500$	0	0
$501$	9.24405	0.412994
$502$	0	0
$503$	−15.0421	−0.670695	−0.335348	−	0.942094i	$-0.608854\pi$
$503$	−15.0421	−0.670695	−0.335348	+	0.942094i	$0.608854\pi$
$504$	0	0
$505$	18.3097	0.814771
$506$	0	0
$507$	30.2332	1.34270
$508$	0	0
$509$	−12.9983	−0.576141	−0.288071	−	0.957609i	$-0.593014\pi$
$509$	−12.9983	−0.576141	−0.288071	+	0.957609i	$0.593014\pi$
$510$	0	0
$511$	−3.18950	−0.141095
$512$	0	0
$513$	−5.59393	−0.246978
$514$	0	0
$515$	−19.7939	−0.872225
$516$	0	0
$517$	4.10806	0.180672
$518$	0	0
$519$	57.8632	2.53991
$520$	0	0
$521$	−25.6171	−1.12231	−0.561153	−	0.827712i	$-0.689642\pi$
$521$	−25.6171	−1.12231	−0.561153	+	0.827712i	$0.689642\pi$
$522$	0	0
$523$	15.5277	0.678978	0.339489	−	0.940610i	$-0.389746\pi$
$523$	15.5277	0.678978	0.339489	+	0.940610i	$0.389746\pi$
$524$	0	0
$525$	2.40511	0.104968
$526$	0	0
$527$	0	0
$528$	0	0
$529$	3.91369	0.170161
$530$	0	0
$531$	−25.4779	−1.10565
$532$	0	0
$533$	37.6826	1.63222
$534$	0	0
$535$	−4.00226	−0.173033
$536$	0	0
$537$	11.3336	0.489081
$538$	0	0
$539$	−4.59604	−0.197965
$540$	0	0
$541$	0.158134	0.00679873	0.00339936	−	0.999994i	$-0.498918\pi$
$541$	0.158134	0.00679873	0.00339936	+	0.999994i	$0.498918\pi$
$542$	0	0
$543$	−11.8601	−0.508967
$544$	0	0
$545$	8.77494	0.375877
$546$	0	0
$547$	−15.6317	−0.668364	−0.334182	−	0.942509i	$-0.608460\pi$
$547$	−15.6317	−0.668364	−0.334182	+	0.942509i	$0.608460\pi$
$548$	0	0
$549$	4.66768	0.199212
$550$	0	0
$551$	0.316547	0.0134853
$552$	0	0
$553$	1.14493	0.0486873
$554$	0	0
$555$	22.5863	0.958734
$556$	0	0
$557$	15.8332	0.670876	0.335438	−	0.942062i	$-0.391116\pi$
$557$	15.8332	0.670876	0.335438	+	0.942062i	$0.391116\pi$
$558$	0	0
$559$	29.0912	1.23043
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−25.8232	−1.08832	−0.544159	−	0.838982i	$-0.683151\pi$
$563$	−25.8232	−1.08832	−0.544159	+	0.838982i	$0.683151\pi$
$564$	0	0
$565$	7.03362	0.295907
$566$	0	0
$567$	1.89066	0.0794004
$568$	0	0
$569$	−34.9517	−1.46525	−0.732627	−	0.680631i	$-0.761706\pi$
$569$	−34.9517	−1.46525	−0.732627	+	0.680631i	$0.761706\pi$
$570$	0	0
$571$	−16.6073	−0.694994	−0.347497	−	0.937681i	$-0.612968\pi$
$571$	−16.6073	−0.694994	−0.347497	+	0.937681i	$0.612968\pi$
$572$	0	0
$573$	−17.9137	−0.748355
$574$	0	0
$575$	20.1747	0.841345
$576$	0	0
$577$	2.61846	0.109008	0.0545040	−	0.998514i	$-0.482642\pi$
$577$	2.61846	0.109008	0.0545040	+	0.998514i	$0.482642\pi$
$578$	0	0
$579$	−30.5139	−1.26811
$580$	0	0
$581$	−2.60399	−0.108032
$582$	0	0
$583$	−4.58247	−0.189787
$584$	0	0
$585$	−17.7967	−0.735803
$586$	0	0
$587$	−1.14841	−0.0473998	−0.0236999	−	0.999719i	$-0.507545\pi$
$587$	−1.14841	−0.0473998	−0.0236999	+	0.999719i	$0.507545\pi$
$588$	0	0
$589$	−25.0426	−1.03186
$590$	0	0
$591$	22.3203	0.918136
$592$	0	0
$593$	40.8742	1.67850	0.839251	−	0.543744i	$-0.182994\pi$
$593$	40.8742	1.67850	0.839251	+	0.543744i	$0.182994\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	45.2018	1.84999
$598$	0	0
$599$	−27.8875	−1.13945	−0.569725	−	0.821835i	$-0.692950\pi$
$599$	−27.8875	−1.13945	−0.569725	+	0.821835i	$0.692950\pi$
$600$	0	0
$601$	17.0468	0.695351	0.347676	−	0.937615i	$-0.386971\pi$
$601$	17.0468	0.695351	0.347676	+	0.937615i	$0.386971\pi$
$602$	0	0
$603$	4.73689	0.192901
$604$	0	0
$605$	−11.1329	−0.452617
$606$	0	0
$607$	−17.6562	−0.716645	−0.358322	−	0.933598i	$-0.616651\pi$
$607$	−17.6562	−0.716645	−0.358322	+	0.933598i	$0.616651\pi$
$608$	0	0
$609$	0.0334648	0.00135606
$610$	0	0
$611$	−30.9978	−1.25404
$612$	0	0
$613$	5.39031	0.217713	0.108856	−	0.994057i	$-0.465281\pi$
$613$	5.39031	0.217713	0.108856	+	0.994057i	$0.465281\pi$
$614$	0	0
$615$	−20.0759	−0.809537
$616$	0	0
$617$	−12.6960	−0.511121	−0.255561	−	0.966793i	$-0.582260\pi$
$617$	−12.6960	−0.511121	−0.255561	+	0.966793i	$0.582260\pi$
$618$	0	0
$619$	−0.162216	−0.00652000	−0.00326000	−	0.999995i	$-0.501038\pi$
$619$	−0.162216	−0.00652000	−0.00326000	+	0.999995i	$0.501038\pi$
$620$	0	0
$621$	−4.96068	−0.199065
$622$	0	0
$623$	3.28795	0.131729
$624$	0	0
$625$	9.56744	0.382698
$626$	0	0
$627$	−9.78470	−0.390763
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−9.14421	−0.364025	−0.182013	−	0.983296i	$-0.558261\pi$
$631$	−9.14421	−0.364025	−0.182013	+	0.983296i	$0.558261\pi$
$632$	0	0
$633$	40.5440	1.61148
$634$	0	0
$635$	6.13666	0.243526
$636$	0	0
$637$	34.6799	1.37407
$638$	0	0
$639$	11.9634	0.473267
$640$	0	0
$641$	−5.73531	−0.226531	−0.113266	−	0.993565i	$-0.536131\pi$
$641$	−5.73531	−0.226531	−0.113266	+	0.993565i	$0.536131\pi$
$642$	0	0
$643$	−42.7497	−1.68588	−0.842942	−	0.538005i	$-0.819178\pi$
$643$	−42.7497	−1.68588	−0.842942	+	0.538005i	$0.819178\pi$
$644$	0	0
$645$	−15.4987	−0.610260
$646$	0	0
$647$	−15.6566	−0.615523	−0.307761	−	0.951464i	$-0.599580\pi$
$647$	−15.6566	−0.615523	−0.307761	+	0.951464i	$0.599580\pi$
$648$	0	0
$649$	−4.99399	−0.196031
$650$	0	0
$651$	−2.64746	−0.103762
$652$	0	0
$653$	40.6477	1.59067	0.795334	−	0.606171i	$-0.207295\pi$
$653$	40.6477	1.59067	0.795334	+	0.606171i	$0.207295\pi$
$654$	0	0
$655$	−20.7879	−0.812251
$656$	0	0
$657$	44.0058	1.71683
$658$	0	0
$659$	19.8975	0.775097	0.387549	−	0.921849i	$-0.373322\pi$
$659$	19.8975	0.775097	0.387549	+	0.921849i	$0.373322\pi$
$660$	0	0
$661$	38.3449	1.49144	0.745722	−	0.666257i	$-0.232105\pi$
$661$	38.3449	1.49144	0.745722	+	0.666257i	$0.232105\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.51008	0.0585582
$666$	0	0
$667$	0.280713	0.0108692
$668$	0	0
$669$	61.9561	2.39536
$670$	0	0
$671$	0.914923	0.0353202
$672$	0	0
$673$	−24.5879	−0.947795	−0.473897	−	0.880580i	$-0.657153\pi$
$673$	−24.5879	−0.947795	−0.473897	+	0.880580i	$0.657153\pi$
$674$	0	0
$675$	−3.71857	−0.143128
$676$	0	0
$677$	38.3448	1.47371	0.736854	−	0.676052i	$-0.236310\pi$
$677$	38.3448	1.47371	0.736854	+	0.676052i	$0.236310\pi$
$678$	0	0
$679$	2.40258	0.0922026
$680$	0	0
$681$	39.0581	1.49671
$682$	0	0
$683$	−18.1016	−0.692640	−0.346320	−	0.938116i	$-0.612569\pi$
$683$	−18.1016	−0.692640	−0.346320	+	0.938116i	$0.612569\pi$
$684$	0	0
$685$	−0.281935	−0.0107722
$686$	0	0
$687$	−55.4299	−2.11478
$688$	0	0
$689$	34.5775	1.31730
$690$	0	0
$691$	−38.9678	−1.48241	−0.741203	−	0.671281i	$-0.765744\pi$
$691$	−38.9678	−1.48241	−0.741203	+	0.671281i	$0.765744\pi$
$692$	0	0
$693$	−0.547911	−0.0208134
$694$	0	0
$695$	−20.1058	−0.762658
$696$	0	0
$697$	0	0
$698$	0	0
$699$	11.5457	0.436698
$700$	0	0
$701$	19.5352	0.737836	0.368918	−	0.929462i	$-0.379728\pi$
$701$	19.5352	0.737836	0.368918	+	0.929462i	$0.379728\pi$
$702$	0	0
$703$	−49.6316	−1.87189
$704$	0	0
$705$	16.5144	0.621970
$706$	0	0
$707$	−4.25349	−0.159969
$708$	0	0
$709$	18.6362	0.699896	0.349948	−	0.936769i	$-0.386199\pi$
$709$	18.6362	0.699896	0.349948	+	0.936769i	$0.386199\pi$
$710$	0	0
$711$	−15.7967	−0.592422
$712$	0	0
$713$	−22.2077	−0.831683
$714$	0	0
$715$	−3.48838	−0.130458
$716$	0	0
$717$	−31.4330	−1.17389
$718$	0	0
$719$	47.1841	1.75967	0.879835	−	0.475279i	$-0.157653\pi$
$719$	47.1841	1.75967	0.879835	+	0.475279i	$0.157653\pi$
$720$	0	0
$721$	4.59829	0.171249
$722$	0	0
$723$	66.3187	2.46642
$724$	0	0
$725$	0.210425	0.00781498
$726$	0	0
$727$	−14.9413	−0.554142	−0.277071	−	0.960849i	$-0.589364\pi$
$727$	−14.9413	−0.554142	−0.277071	+	0.960849i	$0.589364\pi$
$728$	0	0
$729$	−33.3307	−1.23447
$730$	0	0
$731$	0	0
$732$	0	0
$733$	41.7691	1.54278	0.771388	−	0.636365i	$-0.219563\pi$
$733$	41.7691	1.54278	0.771388	+	0.636365i	$0.219563\pi$
$734$	0	0
$735$	−18.4761	−0.681501
$736$	0	0
$737$	0.928490	0.0342014
$738$	0	0
$739$	−4.54817	−0.167307	−0.0836537	−	0.996495i	$-0.526659\pi$
$739$	−4.54817	−0.167307	−0.0836537	+	0.996495i	$0.526659\pi$
$740$	0	0
$741$	73.8314	2.71227
$742$	0	0
$743$	−41.6549	−1.52817	−0.764085	−	0.645116i	$-0.776809\pi$
$743$	−41.6549	−1.52817	−0.764085	+	0.645116i	$0.776809\pi$
$744$	0	0
$745$	−24.5205	−0.898360
$746$	0	0
$747$	35.9275	1.31452
$748$	0	0
$749$	0.929757	0.0339726
$750$	0	0
$751$	8.76461	0.319825	0.159913	−	0.987131i	$-0.448879\pi$
$751$	8.76461	0.319825	0.159913	+	0.987131i	$0.448879\pi$
$752$	0	0
$753$	−59.1537	−2.15568
$754$	0	0
$755$	−6.64674	−0.241900
$756$	0	0
$757$	−6.85974	−0.249322	−0.124661	−	0.992199i	$-0.539784\pi$
$757$	−6.85974	−0.249322	−0.124661	+	0.992199i	$0.539784\pi$
$758$	0	0
$759$	−8.67704	−0.314957
$760$	0	0
$761$	19.7369	0.715462	0.357731	−	0.933825i	$-0.383551\pi$
$761$	19.7369	0.715462	0.357731	+	0.933825i	$0.383551\pi$
$762$	0	0
$763$	−2.03849	−0.0737983
$764$	0	0
$765$	0	0
$766$	0	0
$767$	37.6826	1.36064
$768$	0	0
$769$	8.40361	0.303042	0.151521	−	0.988454i	$-0.451583\pi$
$769$	8.40361	0.303042	0.151521	+	0.988454i	$0.451583\pi$
$770$	0	0
$771$	12.8273	0.461966
$772$	0	0
$773$	−25.0759	−0.901916	−0.450958	−	0.892545i	$-0.648918\pi$
$773$	−25.0759	−0.901916	−0.450958	+	0.892545i	$0.648918\pi$
$774$	0	0
$775$	−16.6471	−0.597980
$776$	0	0
$777$	−5.24698	−0.188234
$778$	0	0
$779$	44.1152	1.58059
$780$	0	0
$781$	2.34499	0.0839102
$782$	0	0
$783$	−0.0517404	−0.00184905
$784$	0	0
$785$	−9.02159	−0.321995
$786$	0	0
$787$	−25.1477	−0.896418	−0.448209	−	0.893929i	$-0.647938\pi$
$787$	−25.1477	−0.896418	−0.448209	+	0.893929i	$0.647938\pi$
$788$	0	0
$789$	22.0119	0.783643
$790$	0	0
$791$	−1.63397	−0.0580972
$792$	0	0
$793$	−6.90364	−0.245156
$794$	0	0
$795$	−18.4216	−0.653346
$796$	0	0
$797$	33.2544	1.17793	0.588966	−	0.808158i	$-0.299535\pi$
$797$	33.2544	1.17793	0.588966	+	0.808158i	$0.299535\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−45.3641	−1.60286
$802$	0	0
$803$	8.62570	0.304394
$804$	0	0
$805$	1.33913	0.0471981
$806$	0	0
$807$	−34.0474	−1.19853
$808$	0	0
$809$	−23.2618	−0.817840	−0.408920	−	0.912570i	$-0.634095\pi$
$809$	−23.2618	−0.817840	−0.408920	+	0.912570i	$0.634095\pi$
$810$	0	0
$811$	19.9987	0.702248	0.351124	−	0.936329i	$-0.385800\pi$
$811$	19.9987	0.702248	0.351124	+	0.936329i	$0.385800\pi$
$812$	0	0
$813$	56.9091	1.99589
$814$	0	0
$815$	22.1701	0.776585
$816$	0	0
$817$	34.0572	1.19151
$818$	0	0
$819$	4.13432	0.144465
$820$	0	0
$821$	0.109722	0.00382932	0.00191466	−	0.999998i	$-0.499391\pi$
$821$	0.109722	0.00382932	0.00191466	+	0.999998i	$0.499391\pi$
$822$	0	0
$823$	34.3566	1.19759	0.598797	−	0.800900i	$-0.295645\pi$
$823$	34.3566	1.19759	0.598797	+	0.800900i	$0.295645\pi$
$824$	0	0
$825$	−6.50439	−0.226454
$826$	0	0
$827$	−12.8156	−0.445643	−0.222822	−	0.974859i	$-0.571527\pi$
$827$	−12.8156	−0.445643	−0.222822	+	0.974859i	$0.571527\pi$
$828$	0	0
$829$	−4.29365	−0.149124	−0.0745622	−	0.997216i	$-0.523756\pi$
$829$	−4.29365	−0.149124	−0.0745622	+	0.997216i	$0.523756\pi$
$830$	0	0
$831$	24.4212	0.847161
$832$	0	0
$833$	0	0
$834$	0	0
$835$	3.85820	0.133519
$836$	0	0
$837$	4.09327	0.141484
$838$	0	0
$839$	49.5276	1.70988	0.854941	−	0.518726i	$-0.173593\pi$
$839$	49.5276	1.70988	0.854941	+	0.518726i	$0.173593\pi$
$840$	0	0
$841$	−28.9971	−0.999899
$842$	0	0
$843$	17.9968	0.619843
$844$	0	0
$845$	12.6185	0.434088
$846$	0	0
$847$	2.58626	0.0888651
$848$	0	0
$849$	−65.3773	−2.24374
$850$	0	0
$851$	−44.0131	−1.50875
$852$	0	0
$853$	−5.70168	−0.195222	−0.0976110	−	0.995225i	$-0.531120\pi$
$853$	−5.70168	−0.195222	−0.0976110	+	0.995225i	$0.531120\pi$
$854$	0	0
$855$	−20.8347	−0.712530
$856$	0	0
$857$	−55.0131	−1.87921	−0.939606	−	0.342257i	$-0.888809\pi$
$857$	−55.0131	−1.87921	−0.939606	+	0.342257i	$0.888809\pi$
$858$	0	0
$859$	−20.6389	−0.704192	−0.352096	−	0.935964i	$-0.614531\pi$
$859$	−20.6389	−0.704192	−0.352096	+	0.935964i	$0.614531\pi$
$860$	0	0
$861$	4.66379	0.158941
$862$	0	0
$863$	−50.7829	−1.72867	−0.864335	−	0.502917i	$-0.832260\pi$
$863$	−50.7829	−1.72867	−0.864335	+	0.502917i	$0.832260\pi$
$864$	0	0
$865$	24.1505	0.821140
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.09635	−0.105036
$870$	0	0
$871$	−7.00601	−0.237390
$872$	0	0
$873$	−33.1486	−1.12191
$874$	0	0
$875$	2.29447	0.0775672
$876$	0	0
$877$	10.0395	0.339009	0.169505	−	0.985529i	$-0.445783\pi$
$877$	10.0395	0.339009	0.169505	+	0.985529i	$0.445783\pi$
$878$	0	0
$879$	46.7863	1.57806
$880$	0	0
$881$	−4.07895	−0.137423	−0.0687116	−	0.997637i	$-0.521889\pi$
$881$	−4.07895	−0.137423	−0.0687116	+	0.997637i	$0.521889\pi$
$882$	0	0
$883$	−29.5124	−0.993171	−0.496585	−	0.867988i	$-0.665413\pi$
$883$	−29.5124	−0.993171	−0.496585	+	0.867988i	$0.665413\pi$
$884$	0	0
$885$	−20.0759	−0.674843
$886$	0	0
$887$	20.9759	0.704302	0.352151	−	0.935943i	$-0.385450\pi$
$887$	20.9759	0.704302	0.352151	+	0.935943i	$0.385450\pi$
$888$	0	0
$889$	−1.42560	−0.0478130
$890$	0	0
$891$	−5.11312	−0.171296
$892$	0	0
$893$	−36.2892	−1.21437
$894$	0	0
$895$	4.73032	0.158117
$896$	0	0
$897$	65.4735	2.18610
$898$	0	0
$899$	−0.231628	−0.00772523
$900$	0	0
$901$	0	0
$902$	0	0
$903$	3.60047	0.119816
$904$	0	0
$905$	−4.95009	−0.164546
$906$	0	0
$907$	−48.2978	−1.60370	−0.801850	−	0.597525i	$-0.796151\pi$
$907$	−48.2978	−1.60370	−0.801850	+	0.597525i	$0.796151\pi$
$908$	0	0
$909$	58.6859	1.94649
$910$	0	0
$911$	58.2360	1.92945	0.964723	−	0.263268i	$-0.0848004\pi$
$911$	58.2360	1.92945	0.964723	+	0.263268i	$0.0848004\pi$
$912$	0	0
$913$	7.04224	0.233064
$914$	0	0
$915$	3.67800	0.121591
$916$	0	0
$917$	4.82920	0.159474
$918$	0	0
$919$	−15.5299	−0.512286	−0.256143	−	0.966639i	$-0.582452\pi$
$919$	−15.5299	−0.512286	−0.256143	+	0.966639i	$0.582452\pi$
$920$	0	0
$921$	−45.7135	−1.50631
$922$	0	0
$923$	−17.6943	−0.582416
$924$	0	0
$925$	−32.9926	−1.08479
$926$	0	0
$927$	−63.4431	−2.08374
$928$	0	0
$929$	−29.8243	−0.978503	−0.489252	−	0.872143i	$-0.662730\pi$
$929$	−29.8243	−0.978503	−0.489252	+	0.872143i	$0.662730\pi$
$930$	0	0
$931$	40.5998	1.33061
$932$	0	0
$933$	−73.3696	−2.40201
$934$	0	0
$935$	0	0
$936$	0	0
$937$	15.8038	0.516288	0.258144	−	0.966106i	$-0.416889\pi$
$937$	15.8038	0.516288	0.258144	+	0.966106i	$0.416889\pi$
$938$	0	0
$939$	7.91243	0.258212
$940$	0	0
$941$	17.1888	0.560340	0.280170	−	0.959950i	$-0.409609\pi$
$941$	17.1888	0.560340	0.280170	+	0.959950i	$0.409609\pi$
$942$	0	0
$943$	39.1212	1.27396
$944$	0	0
$945$	−0.246826	−0.00802925
$946$	0	0
$947$	23.8875	0.776238	0.388119	−	0.921609i	$-0.373125\pi$
$947$	23.8875	0.776238	0.388119	+	0.921609i	$0.373125\pi$
$948$	0	0
$949$	−65.0861	−2.11278
$950$	0	0
$951$	−39.1060	−1.26810
$952$	0	0
$953$	5.53350	0.179248	0.0896238	−	0.995976i	$-0.471434\pi$
$953$	5.53350	0.179248	0.0896238	+	0.995976i	$0.471434\pi$
$954$	0	0
$955$	−7.47666	−0.241939
$956$	0	0
$957$	−0.0905024	−0.00292553
$958$	0	0
$959$	0.0654957	0.00211497
$960$	0	0
$961$	−12.6755	−0.408887
$962$	0	0
$963$	−12.8279	−0.413375
$964$	0	0
$965$	−12.7356	−0.409974
$966$	0	0
$967$	−43.6249	−1.40288	−0.701441	−	0.712727i	$-0.747460\pi$
$967$	−43.6249	−1.40288	−0.701441	+	0.712727i	$0.747460\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	55.2417	1.77279	0.886395	−	0.462929i	$-0.153201\pi$
$971$	55.2417	1.77279	0.886395	+	0.462929i	$0.153201\pi$
$972$	0	0
$973$	4.67075	0.149737
$974$	0	0
$975$	49.0795	1.57180
$976$	0	0
$977$	−37.1550	−1.18869	−0.594346	−	0.804209i	$-0.702589\pi$
$977$	−37.1550	−1.18869	−0.594346	+	0.804209i	$0.702589\pi$
$978$	0	0
$979$	−8.89194	−0.284188
$980$	0	0
$981$	28.1252	0.897970
$982$	0	0
$983$	−46.8158	−1.49319	−0.746597	−	0.665277i	$-0.768313\pi$
$983$	−46.8158	−1.49319	−0.746597	+	0.665277i	$0.768313\pi$
$984$	0	0
$985$	9.31587	0.296828
$986$	0	0
$987$	−3.83644	−0.122115
$988$	0	0
$989$	30.2018	0.960361
$990$	0	0
$991$	10.9847	0.348939	0.174469	−	0.984663i	$-0.444179\pi$
$991$	10.9847	0.348939	0.174469	+	0.984663i	$0.444179\pi$
$992$	0	0
$993$	21.3698	0.678151
$994$	0	0
$995$	18.8659	0.598091
$996$	0	0
$997$	−40.3932	−1.27927	−0.639633	−	0.768680i	$-0.720914\pi$
$997$	−40.3932	−1.27927	−0.639633	+	0.768680i	$0.720914\pi$
$998$	0	0
$999$	8.11241	0.256665

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9248.2.a.y.1.4	yes	4
4.3	odd	2		9248.2.a.bk.1.1	yes	4
17.16	even	2		9248.2.a.bj.1.1	yes	4
68.67	odd	2		9248.2.a.x.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9248.2.a.x.1.4	✓	4	68.67	odd	2
9248.2.a.y.1.4	yes	4	1.1	even	1	trivial
9248.2.a.bj.1.1	yes	4	17.16	even	2
9248.2.a.bk.1.1	yes	4	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9248 = 2^{5} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9248.a (trivial)

\(f(q)\)	\(=\)	\(q+2.52559 q^{3} +1.05411 q^{5} -0.244878 q^{7} +3.37861 q^{9} +0.662250 q^{11} -4.99707 q^{13} +2.66225 q^{15} -5.85009 q^{19} -0.618463 q^{21} -5.18784 q^{23} -3.88885 q^{25} +0.956213 q^{27} -0.0541097 q^{29} +4.28071 q^{31} +1.67257 q^{33} -0.258129 q^{35} +8.48390 q^{37} -12.6206 q^{39} -7.54094 q^{41} -5.82165 q^{43} +3.56143 q^{45} +6.20319 q^{47} -6.94003 q^{49} -6.91955 q^{53} +0.698084 q^{55} -14.7749 q^{57} -7.54094 q^{59} +1.38154 q^{61} -0.827348 q^{63} -5.26746 q^{65} +1.40202 q^{67} -13.1024 q^{69} +3.54094 q^{71} +13.0248 q^{73} -9.82165 q^{75} -0.162171 q^{77} -4.67550 q^{79} -7.72083 q^{81} +10.6338 q^{83} -0.136659 q^{87} -13.4269 q^{89} +1.22367 q^{91} +10.8113 q^{93} -6.16664 q^{95} -9.81133 q^{97} +2.23748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 6 q^{9} + 2 q^{13} + 8 q^{15} - 6 q^{19} + 12 q^{21} - 6 q^{23} + 2 q^{25} - 8 q^{27} + 2 q^{29} - 10 q^{33} - 14 q^{35} - 26 q^{39} - 16 q^{41} + 8 q^{43} - 20 q^{45}+ \cdots + 50 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 6 * q^9 + 2 * q^13 + 8 * q^15 - 6 * q^19 + 12 * q^21 - 6 * q^23 + 2 * q^25 - 8 * q^27 + 2 * q^29 - 10 * q^33 - 14 * q^35 - 26 * q^39 - 16 * q^41 + 8 * q^43 - 20 * q^45 + 8 * q^47 + 14 * q^49 - 6 * q^53 - 22 * q^55 + 6 * q^57 - 16 * q^59 + 20 * q^61 - 24 * q^63 + 4 * q^65 - 4 * q^69 + 4 * q^73 - 8 * q^75 - 2 * q^77 - 24 * q^79 + 32 * q^81 + 26 * q^83 - 10 * q^87 - 32 * q^91 + 26 * q^93 + 32 * q^95 - 22 * q^97 + 50 * q^99

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