LMFDB - Embedded newform 4864.2.a.bp.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4864,2,Mod(1,4864)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4864, 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("4864.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4864 = 2^{8} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4864.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,8,0,-4,0,12,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$38.8392355432$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 13x^{6} + 24x^{5} + 48x^{4} - 68x^{3} - 62x^{2} + 32x + 16 $ x^8 - 2x^7 - 13x^6 + 24x^5 + 48x^4 - 68x^3 - 62x^2 + 32*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.81542$ of defining polynomial
Character	$\chi$	$=$	4864.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.70663 q^{3} -1.66222 q^{5} -1.99556 q^{7} -0.0874066 q^{9} -2.08741 q^{11} +4.77322 q^{13} +2.83680 q^{15} +2.10657 q^{17} -1.00000 q^{19} +3.40569 q^{21} -4.84437 q^{23} -2.23703 q^{25} +5.26907 q^{27} -0.695946 q^{29} -9.77587 q^{31} +3.56244 q^{33} +3.31706 q^{35} +0.0772780 q^{37} -8.14614 q^{39} -10.7743 q^{41} +1.43840 q^{43} +0.145289 q^{45} +2.88133 q^{47} -3.01774 q^{49} -3.59513 q^{51} +9.00264 q^{53} +3.46973 q^{55} +1.70663 q^{57} -11.5253 q^{59} -8.82910 q^{61} +0.174425 q^{63} -7.93414 q^{65} -1.27859 q^{67} +8.26756 q^{69} -4.66655 q^{71} -4.44578 q^{73} +3.81779 q^{75} +4.16555 q^{77} -1.10044 q^{79} -8.73014 q^{81} -2.47419 q^{83} -3.50157 q^{85} +1.18772 q^{87} +15.8259 q^{89} -9.52526 q^{91} +16.6838 q^{93} +1.66222 q^{95} -13.9046 q^{97} +0.182453 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5} - 4 q^{7} + 12 q^{9} - 4 q^{11} + 8 q^{13} - 4 q^{17} - 8 q^{19} + 16 q^{21} + 12 q^{25} + 28 q^{29} - 8 q^{31} + 12 q^{35} + 4 q^{37} + 4 q^{39} - 8 q^{41} + 4 q^{43} + 24 q^{45} - 12 q^{47}+ \cdots + 76 q^{99}+O(q^{100}) $ 8 * q + 8 * q^5 - 4 * q^7 + 12 * q^9 - 4 * q^11 + 8 * q^13 - 4 * q^17 - 8 * q^19 + 16 * q^21 + 12 * q^25 + 28 * q^29 - 8 * q^31 + 12 * q^35 + 4 * q^37 + 4 * q^39 - 8 * q^41 + 4 * q^43 + 24 * q^45 - 12 * q^47 + 12 * q^49 - 12 * q^51 + 32 * q^53 + 8 * q^55 - 12 * q^59 + 8 * q^61 + 16 * q^63 + 8 * q^65 + 4 * q^67 + 28 * q^69 + 24 * q^71 + 24 * q^77 + 24 * q^79 - 8 * q^81 - 40 * q^83 + 24 * q^85 - 24 * q^87 + 8 * q^89 + 4 * q^91 + 32 * q^93 - 8 * q^95 + 16 * q^97 + 76 * 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
$3$	−1.70663	−0.985325	−0.492662	−	0.870221i	$-0.663976\pi$
$3$	−1.70663	−0.985325	−0.492662	+	0.870221i	$0.663976\pi$
$4$	0	0
$5$	−1.66222	−0.743367	−0.371683	−	0.928360i	$-0.621219\pi$
$5$	−1.66222	−0.743367	−0.371683	+	0.928360i	$0.621219\pi$
$6$	0	0
$7$	−1.99556	−0.754251	−0.377125	−	0.926162i	$-0.623087\pi$
$7$	−1.99556	−0.754251	−0.377125	+	0.926162i	$0.623087\pi$
$8$	0	0
$9$	−0.0874066	−0.0291355
$10$	0	0
$11$	−2.08741	−0.629377	−0.314688	−	0.949195i	$-0.601900\pi$
$11$	−2.08741	−0.629377	−0.314688	+	0.949195i	$0.601900\pi$
$12$	0	0
$13$	4.77322	1.32385	0.661927	−	0.749568i	$-0.269739\pi$
$13$	4.77322	1.32385	0.661927	+	0.749568i	$0.269739\pi$
$14$	0	0
$15$	2.83680	0.732457
$16$	0	0
$17$	2.10657	0.510917	0.255459	−	0.966820i	$-0.417774\pi$
$17$	2.10657	0.510917	0.255459	+	0.966820i	$0.417774\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	3.40569	0.743182
$22$	0	0
$23$	−4.84437	−1.01012	−0.505061	−	0.863084i	$-0.668530\pi$
$23$	−4.84437	−1.01012	−0.505061	+	0.863084i	$0.668530\pi$
$24$	0	0
$25$	−2.23703	−0.447406
$26$	0	0
$27$	5.26907	1.01403
$28$	0	0
$29$	−0.695946	−0.129234	−0.0646170	−	0.997910i	$-0.520583\pi$
$29$	−0.695946	−0.129234	−0.0646170	+	0.997910i	$0.520583\pi$
$30$	0	0
$31$	−9.77587	−1.75580	−0.877899	−	0.478846i	$-0.841055\pi$
$31$	−9.77587	−1.75580	−0.877899	+	0.478846i	$0.841055\pi$
$32$	0	0
$33$	3.56244	0.620140
$34$	0	0
$35$	3.31706	0.560685
$36$	0	0
$37$	0.0772780	0.0127044	0.00635221	−	0.999980i	$-0.497978\pi$
$37$	0.0772780	0.0127044	0.00635221	+	0.999980i	$0.497978\pi$
$38$	0	0
$39$	−8.14614	−1.30443
$40$	0	0
$41$	−10.7743	−1.68267	−0.841334	−	0.540516i	$-0.818229\pi$
$41$	−10.7743	−1.68267	−0.841334	+	0.540516i	$0.818229\pi$
$42$	0	0
$43$	1.43840	0.219354	0.109677	−	0.993967i	$-0.465018\pi$
$43$	1.43840	0.219354	0.109677	+	0.993967i	$0.465018\pi$
$44$	0	0
$45$	0.145289	0.0216584
$46$	0	0
$47$	2.88133	0.420285	0.210143	−	0.977671i	$-0.432607\pi$
$47$	2.88133	0.420285	0.210143	+	0.977671i	$0.432607\pi$
$48$	0	0
$49$	−3.01774	−0.431106
$50$	0	0
$51$	−3.59513	−0.503419
$52$	0	0
$53$	9.00264	1.23661	0.618304	−	0.785939i	$-0.287820\pi$
$53$	9.00264	1.23661	0.618304	+	0.785939i	$0.287820\pi$
$54$	0	0
$55$	3.46973	0.467858
$56$	0	0
$57$	1.70663	0.226049
$58$	0	0
$59$	−11.5253	−1.50046	−0.750230	−	0.661177i	$-0.770057\pi$
$59$	−11.5253	−1.50046	−0.750230	+	0.661177i	$0.770057\pi$
$60$	0	0
$61$	−8.82910	−1.13045	−0.565225	−	0.824937i	$-0.691211\pi$
$61$	−8.82910	−1.13045	−0.565225	+	0.824937i	$0.691211\pi$
$62$	0	0
$63$	0.174425	0.0219755
$64$	0	0
$65$	−7.93414	−0.984109
$66$	0	0
$67$	−1.27859	−0.156205	−0.0781023	−	0.996945i	$-0.524886\pi$
$67$	−1.27859	−0.156205	−0.0781023	+	0.996945i	$0.524886\pi$
$68$	0	0
$69$	8.26756	0.995297
$70$	0	0
$71$	−4.66655	−0.553818	−0.276909	−	0.960896i	$-0.589310\pi$
$71$	−4.66655	−0.553818	−0.276909	+	0.960896i	$0.589310\pi$
$72$	0	0
$73$	−4.44578	−0.520339	−0.260170	−	0.965563i	$-0.583779\pi$
$73$	−4.44578	−0.520339	−0.260170	+	0.965563i	$0.583779\pi$
$74$	0	0
$75$	3.81779	0.440840
$76$	0	0
$77$	4.16555	0.474708
$78$	0	0
$79$	−1.10044	−0.123809	−0.0619044	−	0.998082i	$-0.519717\pi$
$79$	−1.10044	−0.123809	−0.0619044	+	0.998082i	$0.519717\pi$
$80$	0	0
$81$	−8.73014	−0.970016
$82$	0	0
$83$	−2.47419	−0.271578	−0.135789	−	0.990738i	$-0.543357\pi$
$83$	−2.47419	−0.271578	−0.135789	+	0.990738i	$0.543357\pi$
$84$	0	0
$85$	−3.50157	−0.379799
$86$	0	0
$87$	1.18772	0.127337
$88$	0	0
$89$	15.8259	1.67755	0.838773	−	0.544481i	$-0.183273\pi$
$89$	15.8259	1.67755	0.838773	+	0.544481i	$0.183273\pi$
$90$	0	0
$91$	−9.52526	−0.998518
$92$	0	0
$93$	16.6838	1.73003
$94$	0	0
$95$	1.66222	0.170540
$96$	0	0
$97$	−13.9046	−1.41180	−0.705898	−	0.708313i	$-0.749456\pi$
$97$	−13.9046	−1.41180	−0.705898	+	0.708313i	$0.749456\pi$
$98$	0	0
$99$	0.182453	0.0183372
$100$	0	0
$101$	1.00493	0.0999945	0.0499973	−	0.998749i	$-0.484079\pi$
$101$	1.00493	0.0999945	0.0499973	+	0.998749i	$0.484079\pi$
$102$	0	0
$103$	12.4581	1.22753	0.613767	−	0.789487i	$-0.289653\pi$
$103$	12.4581	1.22753	0.613767	+	0.789487i	$0.289653\pi$
$104$	0	0
$105$	−5.66100	−0.552457
$106$	0	0
$107$	−15.3004	−1.47915	−0.739573	−	0.673076i	$-0.764973\pi$
$107$	−15.3004	−1.47915	−0.739573	+	0.673076i	$0.764973\pi$
$108$	0	0
$109$	7.65850	0.733551	0.366775	−	0.930309i	$-0.380462\pi$
$109$	7.65850	0.733551	0.366775	+	0.930309i	$0.380462\pi$
$110$	0	0
$111$	−0.131885	−0.0125180
$112$	0	0
$113$	17.3938	1.63627	0.818134	−	0.575028i	$-0.195009\pi$
$113$	17.3938	1.63627	0.818134	+	0.575028i	$0.195009\pi$
$114$	0	0
$115$	8.05240	0.750891
$116$	0	0
$117$	−0.417211	−0.0385712
$118$	0	0
$119$	−4.20378	−0.385360
$120$	0	0
$121$	−6.64273	−0.603885
$122$	0	0
$123$	18.3878	1.65797
$124$	0	0
$125$	12.0295	1.07595
$126$	0	0
$127$	−18.1488	−1.61045	−0.805225	−	0.592970i	$-0.797955\pi$
$127$	−18.1488	−1.61045	−0.805225	+	0.592970i	$0.797955\pi$
$128$	0	0
$129$	−2.45482	−0.216135
$130$	0	0
$131$	9.70500	0.847930	0.423965	−	0.905679i	$-0.360638\pi$
$131$	9.70500	0.847930	0.423965	+	0.905679i	$0.360638\pi$
$132$	0	0
$133$	1.99556	0.173037
$134$	0	0
$135$	−8.75834	−0.753798
$136$	0	0
$137$	20.2253	1.72797	0.863983	−	0.503521i	$-0.167962\pi$
$137$	20.2253	1.72797	0.863983	+	0.503521i	$0.167962\pi$
$138$	0	0
$139$	16.7746	1.42280	0.711401	−	0.702786i	$-0.248061\pi$
$139$	16.7746	1.42280	0.711401	+	0.702786i	$0.248061\pi$
$140$	0	0
$141$	−4.91737	−0.414117
$142$	0	0
$143$	−9.96366	−0.833203
$144$	0	0
$145$	1.15681	0.0960682
$146$	0	0
$147$	5.15017	0.424779
$148$	0	0
$149$	13.6533	1.11853	0.559263	−	0.828991i	$-0.311084\pi$
$149$	13.6533	1.11853	0.559263	+	0.828991i	$0.311084\pi$
$150$	0	0
$151$	6.77602	0.551425	0.275712	−	0.961240i	$-0.411086\pi$
$151$	6.77602	0.551425	0.275712	+	0.961240i	$0.411086\pi$
$152$	0	0
$153$	−0.184128	−0.0148858
$154$	0	0
$155$	16.2496	1.30520
$156$	0	0
$157$	23.4178	1.86895	0.934473	−	0.356035i	$-0.115871\pi$
$157$	23.4178	1.86895	0.934473	+	0.356035i	$0.115871\pi$
$158$	0	0
$159$	−15.3642	−1.21846
$160$	0	0
$161$	9.66724	0.761885
$162$	0	0
$163$	1.06973	0.0837875	0.0418938	−	0.999122i	$-0.486661\pi$
$163$	1.06973	0.0837875	0.0418938	+	0.999122i	$0.486661\pi$
$164$	0	0
$165$	−5.92155	−0.460992
$166$	0	0
$167$	13.6217	1.05408	0.527039	−	0.849841i	$-0.323302\pi$
$167$	13.6217	1.05408	0.527039	+	0.849841i	$0.323302\pi$
$168$	0	0
$169$	9.78367	0.752590
$170$	0	0
$171$	0.0874066	0.00668415
$172$	0	0
$173$	−11.5684	−0.879527	−0.439763	−	0.898114i	$-0.644938\pi$
$173$	−11.5684	−0.879527	−0.439763	+	0.898114i	$0.644938\pi$
$174$	0	0
$175$	4.46413	0.337456
$176$	0	0
$177$	19.6694	1.47844
$178$	0	0
$179$	−20.4983	−1.53211	−0.766057	−	0.642772i	$-0.777784\pi$
$179$	−20.4983	−1.53211	−0.766057	+	0.642772i	$0.777784\pi$
$180$	0	0
$181$	18.7745	1.39550	0.697750	−	0.716341i	$-0.254185\pi$
$181$	18.7745	1.39550	0.697750	+	0.716341i	$0.254185\pi$
$182$	0	0
$183$	15.0680	1.11386
$184$	0	0
$185$	−0.128453	−0.00944404
$186$	0	0
$187$	−4.39726	−0.321560
$188$	0	0
$189$	−10.5147	−0.764835
$190$	0	0
$191$	14.0392	1.01584	0.507919	−	0.861405i	$-0.330415\pi$
$191$	14.0392	1.01584	0.507919	+	0.861405i	$0.330415\pi$
$192$	0	0
$193$	−23.8700	−1.71820	−0.859099	−	0.511809i	$-0.828976\pi$
$193$	−23.8700	−1.71820	−0.859099	+	0.511809i	$0.828976\pi$
$194$	0	0
$195$	13.5407	0.969667
$196$	0	0
$197$	1.84890	0.131729	0.0658645	−	0.997829i	$-0.479020\pi$
$197$	1.84890	0.131729	0.0658645	+	0.997829i	$0.479020\pi$
$198$	0	0
$199$	−18.3608	−1.30156	−0.650781	−	0.759266i	$-0.725558\pi$
$199$	−18.3608	−1.30156	−0.650781	+	0.759266i	$0.725558\pi$
$200$	0	0
$201$	2.18208	0.153912
$202$	0	0
$203$	1.38880	0.0974748
$204$	0	0
$205$	17.9093	1.25084
$206$	0	0
$207$	0.423430	0.0294304
$208$	0	0
$209$	2.08741	0.144389
$210$	0	0
$211$	−15.5147	−1.06808	−0.534039	−	0.845460i	$-0.679326\pi$
$211$	−15.5147	−1.06808	−0.534039	+	0.845460i	$0.679326\pi$
$212$	0	0
$213$	7.96409	0.545690
$214$	0	0
$215$	−2.39094	−0.163061
$216$	0	0
$217$	19.5083	1.32431
$218$	0	0
$219$	7.58731	0.512703
$220$	0	0
$221$	10.0551	0.676380
$222$	0	0
$223$	13.9260	0.932553	0.466276	−	0.884639i	$-0.345595\pi$
$223$	13.9260	0.932553	0.466276	+	0.884639i	$0.345595\pi$
$224$	0	0
$225$	0.195531	0.0130354
$226$	0	0
$227$	1.49624	0.0993088	0.0496544	−	0.998766i	$-0.484188\pi$
$227$	1.49624	0.0993088	0.0496544	+	0.998766i	$0.484188\pi$
$228$	0	0
$229$	1.03706	0.0685310	0.0342655	−	0.999413i	$-0.489091\pi$
$229$	1.03706	0.0685310	0.0342655	+	0.999413i	$0.489091\pi$
$230$	0	0
$231$	−7.10905	−0.467741
$232$	0	0
$233$	−4.93372	−0.323219	−0.161609	−	0.986855i	$-0.551668\pi$
$233$	−4.93372	−0.323219	−0.161609	+	0.986855i	$0.551668\pi$
$234$	0	0
$235$	−4.78940	−0.312426
$236$	0	0
$237$	1.87804	0.121992
$238$	0	0
$239$	3.84533	0.248733	0.124367	−	0.992236i	$-0.460310\pi$
$239$	3.84533	0.248733	0.124367	+	0.992236i	$0.460310\pi$
$240$	0	0
$241$	−1.41134	−0.0909123	−0.0454562	−	0.998966i	$-0.514474\pi$
$241$	−1.41134	−0.0909123	−0.0454562	+	0.998966i	$0.514474\pi$
$242$	0	0
$243$	−0.908064	−0.0582523
$244$	0	0
$245$	5.01614	0.320469
$246$	0	0
$247$	−4.77322	−0.303713
$248$	0	0
$249$	4.22253	0.267592
$250$	0	0
$251$	8.24640	0.520508	0.260254	−	0.965540i	$-0.416194\pi$
$251$	8.24640	0.520508	0.260254	+	0.965540i	$0.416194\pi$
$252$	0	0
$253$	10.1122	0.635747
$254$	0	0
$255$	5.97590	0.374225
$256$	0	0
$257$	15.0726	0.940200	0.470100	−	0.882613i	$-0.344218\pi$
$257$	15.0726	0.940200	0.470100	+	0.882613i	$0.344218\pi$
$258$	0	0
$259$	−0.154213	−0.00958232
$260$	0	0
$261$	0.0608303	0.00376530
$262$	0	0
$263$	−9.49141	−0.585265	−0.292633	−	0.956225i	$-0.594531\pi$
$263$	−9.49141	−0.585265	−0.292633	+	0.956225i	$0.594531\pi$
$264$	0	0
$265$	−14.9644	−0.919253
$266$	0	0
$267$	−27.0091	−1.65293
$268$	0	0
$269$	3.98549	0.243000	0.121500	−	0.992591i	$-0.461230\pi$
$269$	3.98549	0.243000	0.121500	+	0.992591i	$0.461230\pi$
$270$	0	0
$271$	6.38427	0.387817	0.193908	−	0.981020i	$-0.437884\pi$
$271$	6.38427	0.387817	0.193908	+	0.981020i	$0.437884\pi$
$272$	0	0
$273$	16.2561	0.983865
$274$	0	0
$275$	4.66959	0.281587
$276$	0	0
$277$	15.1758	0.911823	0.455912	−	0.890025i	$-0.349313\pi$
$277$	15.1758	0.911823	0.455912	+	0.890025i	$0.349313\pi$
$278$	0	0
$279$	0.854475	0.0511561
$280$	0	0
$281$	−14.9058	−0.889207	−0.444603	−	0.895728i	$-0.646655\pi$
$281$	−14.9058	−0.889207	−0.444603	+	0.895728i	$0.646655\pi$
$282$	0	0
$283$	−14.9931	−0.891250	−0.445625	−	0.895220i	$-0.647019\pi$
$283$	−14.9931	−0.891250	−0.445625	+	0.895220i	$0.647019\pi$
$284$	0	0
$285$	−2.83680	−0.168037
$286$	0	0
$287$	21.5008	1.26915
$288$	0	0
$289$	−12.5624	−0.738963
$290$	0	0
$291$	23.7300	1.39108
$292$	0	0
$293$	21.6983	1.26763	0.633814	−	0.773485i	$-0.281488\pi$
$293$	21.6983	1.26763	0.633814	+	0.773485i	$0.281488\pi$
$294$	0	0
$295$	19.1575	1.11539
$296$	0	0
$297$	−10.9987	−0.638208
$298$	0	0
$299$	−23.1233	−1.33725
$300$	0	0
$301$	−2.87042	−0.165448
$302$	0	0
$303$	−1.71505	−0.0985270
$304$	0	0
$305$	14.6759	0.840339
$306$	0	0
$307$	7.81449	0.445996	0.222998	−	0.974819i	$-0.428416\pi$
$307$	7.81449	0.445996	0.222998	+	0.974819i	$0.428416\pi$
$308$	0	0
$309$	−21.2614	−1.20952
$310$	0	0
$311$	12.8288	0.727455	0.363727	−	0.931505i	$-0.381504\pi$
$311$	12.8288	0.727455	0.363727	+	0.931505i	$0.381504\pi$
$312$	0	0
$313$	4.07578	0.230377	0.115188	−	0.993344i	$-0.463253\pi$
$313$	4.07578	0.230377	0.115188	+	0.993344i	$0.463253\pi$
$314$	0	0
$315$	−0.289933	−0.0163358
$316$	0	0
$317$	−2.26448	−0.127186	−0.0635931	−	0.997976i	$-0.520256\pi$
$317$	−2.26448	−0.127186	−0.0635931	+	0.997976i	$0.520256\pi$
$318$	0	0
$319$	1.45272	0.0813368
$320$	0	0
$321$	26.1122	1.45744
$322$	0	0
$323$	−2.10657	−0.117212
$324$	0	0
$325$	−10.6778	−0.592300
$326$	0	0
$327$	−13.0702	−0.722786
$328$	0	0
$329$	−5.74987	−0.317000
$330$	0	0
$331$	−7.31171	−0.401888	−0.200944	−	0.979603i	$-0.564401\pi$
$331$	−7.31171	−0.401888	−0.200944	+	0.979603i	$0.564401\pi$
$332$	0	0
$333$	−0.00675460	−0.000370150 0
$334$	0	0
$335$	2.12529	0.116117
$336$	0	0
$337$	−6.00356	−0.327035	−0.163517	−	0.986540i	$-0.552284\pi$
$337$	−6.00356	−0.327035	−0.163517	+	0.986540i	$0.552284\pi$
$338$	0	0
$339$	−29.6848	−1.61225
$340$	0	0
$341$	20.4062	1.10506
$342$	0	0
$343$	19.9910	1.07941
$344$	0	0
$345$	−13.7425	−0.739871
$346$	0	0
$347$	−6.66747	−0.357929	−0.178964	−	0.983856i	$-0.557275\pi$
$347$	−6.66747	−0.357929	−0.178964	+	0.983856i	$0.557275\pi$
$348$	0	0
$349$	−35.5072	−1.90066	−0.950329	−	0.311248i	$-0.899253\pi$
$349$	−35.5072	−1.90066	−0.950329	+	0.311248i	$0.899253\pi$
$350$	0	0
$351$	25.1504	1.34243
$352$	0	0
$353$	−27.8315	−1.48132	−0.740659	−	0.671881i	$-0.765487\pi$
$353$	−27.8315	−1.48132	−0.740659	+	0.671881i	$0.765487\pi$
$354$	0	0
$355$	7.75683	0.411690
$356$	0	0
$357$	7.17431	0.379705
$358$	0	0
$359$	10.6778	0.563551	0.281775	−	0.959480i	$-0.409077\pi$
$359$	10.6778	0.563551	0.281775	+	0.959480i	$0.409077\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	11.3367	0.595023
$364$	0	0
$365$	7.38986	0.386803
$366$	0	0
$367$	−16.8950	−0.881911	−0.440956	−	0.897529i	$-0.645360\pi$
$367$	−16.8950	−0.881911	−0.440956	+	0.897529i	$0.645360\pi$
$368$	0	0
$369$	0.941747	0.0490254
$370$	0	0
$371$	−17.9653	−0.932713
$372$	0	0
$373$	−33.4640	−1.73270	−0.866350	−	0.499438i	$-0.833540\pi$
$373$	−33.4640	−1.73270	−0.866350	+	0.499438i	$0.833540\pi$
$374$	0	0
$375$	−20.5300	−1.06016
$376$	0	0
$377$	−3.32191	−0.171087
$378$	0	0
$379$	37.0251	1.90185	0.950925	−	0.309423i	$-0.100136\pi$
$379$	37.0251	1.90185	0.950925	+	0.309423i	$0.100136\pi$
$380$	0	0
$381$	30.9734	1.58682
$382$	0	0
$383$	16.2264	0.829131	0.414566	−	0.910019i	$-0.363934\pi$
$383$	16.2264	0.829131	0.414566	+	0.910019i	$0.363934\pi$
$384$	0	0
$385$	−6.92405	−0.352882
$386$	0	0
$387$	−0.125726	−0.00639100
$388$	0	0
$389$	11.8131	0.598947	0.299473	−	0.954105i	$-0.403189\pi$
$389$	11.8131	0.598947	0.299473	+	0.954105i	$0.403189\pi$
$390$	0	0
$391$	−10.2050	−0.516089
$392$	0	0
$393$	−16.5629	−0.835486
$394$	0	0
$395$	1.82917	0.0920353
$396$	0	0
$397$	10.9654	0.550340	0.275170	−	0.961396i	$-0.411266\pi$
$397$	10.9654	0.550340	0.275170	+	0.961396i	$0.411266\pi$
$398$	0	0
$399$	−3.40569	−0.170498
$400$	0	0
$401$	11.9479	0.596650	0.298325	−	0.954464i	$-0.403572\pi$
$401$	11.9479	0.596650	0.298325	+	0.954464i	$0.403572\pi$
$402$	0	0
$403$	−46.6624	−2.32442
$404$	0	0
$405$	14.5114	0.721077
$406$	0	0
$407$	−0.161311	−0.00799587
$408$	0	0
$409$	1.28997	0.0637848	0.0318924	−	0.999491i	$-0.489847\pi$
$409$	1.28997	0.0637848	0.0318924	+	0.999491i	$0.489847\pi$
$410$	0	0
$411$	−34.5172	−1.70261
$412$	0	0
$413$	22.9993	1.13172
$414$	0	0
$415$	4.11265	0.201882
$416$	0	0
$417$	−28.6281	−1.40192
$418$	0	0
$419$	15.6802	0.766030	0.383015	−	0.923742i	$-0.374886\pi$
$419$	15.6802	0.766030	0.383015	+	0.923742i	$0.374886\pi$
$420$	0	0
$421$	26.2544	1.27956	0.639782	−	0.768557i	$-0.279025\pi$
$421$	26.2544	1.27956	0.639782	+	0.768557i	$0.279025\pi$
$422$	0	0
$423$	−0.251847	−0.0122452
$424$	0	0
$425$	−4.71245	−0.228587
$426$	0	0
$427$	17.6190	0.852643
$428$	0	0
$429$	17.0043	0.820975
$430$	0	0
$431$	33.4648	1.61194	0.805972	−	0.591954i	$-0.201643\pi$
$431$	33.4648	1.61194	0.805972	+	0.591954i	$0.201643\pi$
$432$	0	0
$433$	5.89119	0.283112	0.141556	−	0.989930i	$-0.454789\pi$
$433$	5.89119	0.283112	0.141556	+	0.989930i	$0.454789\pi$
$434$	0	0
$435$	−1.97426	−0.0946584
$436$	0	0
$437$	4.84437	0.231738
$438$	0	0
$439$	25.9770	1.23982	0.619908	−	0.784674i	$-0.287170\pi$
$439$	25.9770	1.23982	0.619908	+	0.784674i	$0.287170\pi$
$440$	0	0
$441$	0.263770	0.0125605
$442$	0	0
$443$	11.0652	0.525722	0.262861	−	0.964834i	$-0.415334\pi$
$443$	11.0652	0.525722	0.262861	+	0.964834i	$0.415334\pi$
$444$	0	0
$445$	−26.3062	−1.24703
$446$	0	0
$447$	−23.3012	−1.10211
$448$	0	0
$449$	3.56950	0.168455	0.0842274	−	0.996447i	$-0.473158\pi$
$449$	3.56950	0.168455	0.0842274	+	0.996447i	$0.473158\pi$
$450$	0	0
$451$	22.4904	1.05903
$452$	0	0
$453$	−11.5642	−0.543332
$454$	0	0
$455$	15.8331	0.742265
$456$	0	0
$457$	23.3258	1.09113	0.545567	−	0.838067i	$-0.316314\pi$
$457$	23.3258	1.09113	0.545567	+	0.838067i	$0.316314\pi$
$458$	0	0
$459$	11.0996	0.518087
$460$	0	0
$461$	8.16887	0.380462	0.190231	−	0.981739i	$-0.439076\pi$
$461$	8.16887	0.380462	0.190231	+	0.981739i	$0.439076\pi$
$462$	0	0
$463$	12.8015	0.594936	0.297468	−	0.954732i	$-0.403858\pi$
$463$	12.8015	0.594936	0.297468	+	0.954732i	$0.403858\pi$
$464$	0	0
$465$	−27.7321	−1.28605
$466$	0	0
$467$	−0.897425	−0.0415279	−0.0207639	−	0.999784i	$-0.506610\pi$
$467$	−0.897425	−0.0415279	−0.0207639	+	0.999784i	$0.506610\pi$
$468$	0	0
$469$	2.55150	0.117817
$470$	0	0
$471$	−39.9656	−1.84152
$472$	0	0
$473$	−3.00253	−0.138056
$474$	0	0
$475$	2.23703	0.102642
$476$	0	0
$477$	−0.786890	−0.0360292
$478$	0	0
$479$	−12.9960	−0.593803	−0.296901	−	0.954908i	$-0.595953\pi$
$479$	−12.9960	−0.593803	−0.296901	+	0.954908i	$0.595953\pi$
$480$	0	0
$481$	0.368865	0.0168188
$482$	0	0
$483$	−16.4984	−0.750704
$484$	0	0
$485$	23.1125	1.04948
$486$	0	0
$487$	12.0621	0.546588	0.273294	−	0.961931i	$-0.411887\pi$
$487$	12.0621	0.546588	0.273294	+	0.961931i	$0.411887\pi$
$488$	0	0
$489$	−1.82563	−0.0825579
$490$	0	0
$491$	30.5337	1.37797	0.688983	−	0.724778i	$-0.258058\pi$
$491$	30.5337	1.37797	0.688983	+	0.724778i	$0.258058\pi$
$492$	0	0
$493$	−1.46606	−0.0660279
$494$	0	0
$495$	−0.303277	−0.0136313
$496$	0	0
$497$	9.31239	0.417718
$498$	0	0
$499$	40.2124	1.80015	0.900077	−	0.435731i	$-0.143510\pi$
$499$	40.2124	1.80015	0.900077	+	0.435731i	$0.143510\pi$
$500$	0	0
$501$	−23.2472	−1.03861
$502$	0	0
$503$	35.7181	1.59259	0.796296	−	0.604907i	$-0.206790\pi$
$503$	35.7181	1.59259	0.796296	+	0.604907i	$0.206790\pi$
$504$	0	0
$505$	−1.67042	−0.0743326
$506$	0	0
$507$	−16.6971	−0.741545
$508$	0	0
$509$	35.3428	1.56654	0.783270	−	0.621681i	$-0.213550\pi$
$509$	35.3428	1.56654	0.783270	+	0.621681i	$0.213550\pi$
$510$	0	0
$511$	8.87183	0.392466
$512$	0	0
$513$	−5.26907	−0.232635
$514$	0	0
$515$	−20.7081	−0.912508
$516$	0	0
$517$	−6.01450	−0.264518
$518$	0	0
$519$	19.7429	0.866619
$520$	0	0
$521$	−2.50662	−0.109817	−0.0549086	−	0.998491i	$-0.517487\pi$
$521$	−2.50662	−0.109817	−0.0549086	+	0.998491i	$0.517487\pi$
$522$	0	0
$523$	−9.95330	−0.435227	−0.217614	−	0.976035i	$-0.569827\pi$
$523$	−9.95330	−0.435227	−0.217614	+	0.976035i	$0.569827\pi$
$524$	0	0
$525$	−7.61863	−0.332504
$526$	0	0
$527$	−20.5935	−0.897068
$528$	0	0
$529$	0.467940	0.0203452
$530$	0	0
$531$	1.00738	0.0437167
$532$	0	0
$533$	−51.4283	−2.22761
$534$	0	0
$535$	25.4326	1.09955
$536$	0	0
$537$	34.9831	1.50963
$538$	0	0
$539$	6.29925	0.271328
$540$	0	0
$541$	−34.7061	−1.49213	−0.746065	−	0.665873i	$-0.768059\pi$
$541$	−34.7061	−1.49213	−0.746065	+	0.665873i	$0.768059\pi$
$542$	0	0
$543$	−32.0412	−1.37502
$544$	0	0
$545$	−12.7301	−0.545297
$546$	0	0
$547$	7.21429	0.308461	0.154230	−	0.988035i	$-0.450710\pi$
$547$	7.21429	0.308461	0.154230	+	0.988035i	$0.450710\pi$
$548$	0	0
$549$	0.771721	0.0329363
$550$	0	0
$551$	0.695946	0.0296483
$552$	0	0
$553$	2.19599	0.0933829
$554$	0	0
$555$	0.219222	0.00930545
$556$	0	0
$557$	−4.27230	−0.181023	−0.0905116	−	0.995895i	$-0.528850\pi$
$557$	−4.27230	−0.181023	−0.0905116	+	0.995895i	$0.528850\pi$
$558$	0	0
$559$	6.86582	0.290393
$560$	0	0
$561$	7.50451	0.316840
$562$	0	0
$563$	−23.9662	−1.01005	−0.505027	−	0.863103i	$-0.668518\pi$
$563$	−23.9662	−1.01005	−0.505027	+	0.863103i	$0.668518\pi$
$564$	0	0
$565$	−28.9122	−1.21635
$566$	0	0
$567$	17.4215	0.731635
$568$	0	0
$569$	−6.53074	−0.273783	−0.136891	−	0.990586i	$-0.543711\pi$
$569$	−6.53074	−0.273783	−0.136891	+	0.990586i	$0.543711\pi$
$570$	0	0
$571$	13.8125	0.578035	0.289017	−	0.957324i	$-0.406671\pi$
$571$	13.8125	0.578035	0.289017	+	0.957324i	$0.406671\pi$
$572$	0	0
$573$	−23.9597	−1.00093
$574$	0	0
$575$	10.8370	0.451934
$576$	0	0
$577$	−1.75601	−0.0731035	−0.0365517	−	0.999332i	$-0.511637\pi$
$577$	−1.75601	−0.0731035	−0.0365517	+	0.999332i	$0.511637\pi$
$578$	0	0
$579$	40.7373	1.69298
$580$	0	0
$581$	4.93740	0.204838
$582$	0	0
$583$	−18.7922	−0.778293
$584$	0	0
$585$	0.693496	0.0286725
$586$	0	0
$587$	−22.1818	−0.915540	−0.457770	−	0.889071i	$-0.651352\pi$
$587$	−22.1818	−0.915540	−0.457770	+	0.889071i	$0.651352\pi$
$588$	0	0
$589$	9.77587	0.402808
$590$	0	0
$591$	−3.15540	−0.129796
$592$	0	0
$593$	−14.8347	−0.609186	−0.304593	−	0.952483i	$-0.598520\pi$
$593$	−14.8347	−0.609186	−0.304593	+	0.952483i	$0.598520\pi$
$594$	0	0
$595$	6.98760	0.286464
$596$	0	0
$597$	31.3351	1.28246
$598$	0	0
$599$	−19.9337	−0.814468	−0.407234	−	0.913324i	$-0.633507\pi$
$599$	−19.9337	−0.814468	−0.407234	+	0.913324i	$0.633507\pi$
$600$	0	0
$601$	−31.4765	−1.28395	−0.641976	−	0.766724i	$-0.721885\pi$
$601$	−31.4765	−1.28395	−0.641976	+	0.766724i	$0.721885\pi$
$602$	0	0
$603$	0.111757	0.00455110
$604$	0	0
$605$	11.0417	0.448908
$606$	0	0
$607$	−25.8245	−1.04818	−0.524092	−	0.851662i	$-0.675595\pi$
$607$	−25.8245	−1.04818	−0.524092	+	0.851662i	$0.675595\pi$
$608$	0	0
$609$	−2.37018	−0.0960443
$610$	0	0
$611$	13.7532	0.556396
$612$	0	0
$613$	0.844965	0.0341278	0.0170639	−	0.999854i	$-0.494568\pi$
$613$	0.844965	0.0341278	0.0170639	+	0.999854i	$0.494568\pi$
$614$	0	0
$615$	−30.5646	−1.23248
$616$	0	0
$617$	10.0344	0.403970	0.201985	−	0.979389i	$-0.435261\pi$
$617$	10.0344	0.403970	0.201985	+	0.979389i	$0.435261\pi$
$618$	0	0
$619$	−29.2896	−1.17725	−0.588625	−	0.808406i	$-0.700331\pi$
$619$	−29.2896	−1.17725	−0.588625	+	0.808406i	$0.700331\pi$
$620$	0	0
$621$	−25.5253	−1.02430
$622$	0	0
$623$	−31.5816	−1.26529
$624$	0	0
$625$	−8.81055	−0.352422
$626$	0	0
$627$	−3.56244	−0.142270
$628$	0	0
$629$	0.162791	0.00649091
$630$	0	0
$631$	−1.02437	−0.0407796	−0.0203898	−	0.999792i	$-0.506491\pi$
$631$	−1.02437	−0.0407796	−0.0203898	+	0.999792i	$0.506491\pi$
$632$	0	0
$633$	26.4779	1.05240
$634$	0	0
$635$	30.1673	1.19715
$636$	0	0
$637$	−14.4043	−0.570721
$638$	0	0
$639$	0.407887	0.0161358
$640$	0	0
$641$	−31.8367	−1.25747	−0.628737	−	0.777618i	$-0.716428\pi$
$641$	−31.8367	−1.25747	−0.628737	+	0.777618i	$0.716428\pi$
$642$	0	0
$643$	34.2996	1.35264	0.676322	−	0.736606i	$-0.263573\pi$
$643$	34.2996	1.35264	0.676322	+	0.736606i	$0.263573\pi$
$644$	0	0
$645$	4.08045	0.160668
$646$	0	0
$647$	9.45559	0.371738	0.185869	−	0.982575i	$-0.440490\pi$
$647$	9.45559	0.371738	0.185869	+	0.982575i	$0.440490\pi$
$648$	0	0
$649$	24.0579	0.944355
$650$	0	0
$651$	−33.2936	−1.30488
$652$	0	0
$653$	−6.33451	−0.247888	−0.123944	−	0.992289i	$-0.539554\pi$
$653$	−6.33451	−0.247888	−0.123944	+	0.992289i	$0.539554\pi$
$654$	0	0
$655$	−16.1318	−0.630323
$656$	0	0
$657$	0.388590	0.0151604
$658$	0	0
$659$	−7.22355	−0.281390	−0.140695	−	0.990053i	$-0.544934\pi$
$659$	−7.22355	−0.281390	−0.140695	+	0.990053i	$0.544934\pi$
$660$	0	0
$661$	−0.529621	−0.0205999	−0.0102999	−	0.999947i	$-0.503279\pi$
$661$	−0.529621	−0.0205999	−0.0102999	+	0.999947i	$0.503279\pi$
$662$	0	0
$663$	−17.1604	−0.666454
$664$	0	0
$665$	−3.31706	−0.128630
$666$	0	0
$667$	3.37142	0.130542
$668$	0	0
$669$	−23.7665	−0.918867
$670$	0	0
$671$	18.4299	0.711479
$672$	0	0
$673$	41.7434	1.60909	0.804544	−	0.593893i	$-0.202410\pi$
$673$	41.7434	1.60909	0.804544	+	0.593893i	$0.202410\pi$
$674$	0	0
$675$	−11.7871	−0.453684
$676$	0	0
$677$	−33.4165	−1.28430	−0.642151	−	0.766578i	$-0.721958\pi$
$677$	−33.4165	−1.28430	−0.642151	+	0.766578i	$0.721958\pi$
$678$	0	0
$679$	27.7474	1.06485
$680$	0	0
$681$	−2.55353	−0.0978514
$682$	0	0
$683$	−30.1765	−1.15467	−0.577336	−	0.816507i	$-0.695908\pi$
$683$	−30.1765	−1.15467	−0.577336	+	0.816507i	$0.695908\pi$
$684$	0	0
$685$	−33.6189	−1.28451
$686$	0	0
$687$	−1.76988	−0.0675252
$688$	0	0
$689$	42.9716	1.63709
$690$	0	0
$691$	−17.6002	−0.669544	−0.334772	−	0.942299i	$-0.608659\pi$
$691$	−17.6002	−0.669544	−0.334772	+	0.942299i	$0.608659\pi$
$692$	0	0
$693$	−0.364096	−0.0138309
$694$	0	0
$695$	−27.8830	−1.05766
$696$	0	0
$697$	−22.6968	−0.859704
$698$	0	0
$699$	8.42005	0.318475
$700$	0	0
$701$	30.6993	1.15950	0.579748	−	0.814796i	$-0.303151\pi$
$701$	30.6993	1.15950	0.579748	+	0.814796i	$0.303151\pi$
$702$	0	0
$703$	−0.0772780	−0.00291459
$704$	0	0
$705$	8.17374	0.307841
$706$	0	0
$707$	−2.00540	−0.0754210
$708$	0	0
$709$	−8.79437	−0.330279	−0.165140	−	0.986270i	$-0.552807\pi$
$709$	−8.79437	−0.330279	−0.165140	+	0.986270i	$0.552807\pi$
$710$	0	0
$711$	0.0961853	0.00360723
$712$	0	0
$713$	47.3579	1.77357
$714$	0	0
$715$	16.5618	0.619375
$716$	0	0
$717$	−6.56256	−0.245083
$718$	0	0
$719$	−12.5991	−0.469867	−0.234933	−	0.972011i	$-0.575487\pi$
$719$	−12.5991	−0.469867	−0.234933	+	0.972011i	$0.575487\pi$
$720$	0	0
$721$	−24.8609	−0.925869
$722$	0	0
$723$	2.40864	0.0895781
$724$	0	0
$725$	1.55685	0.0578200
$726$	0	0
$727$	−39.7756	−1.47519	−0.737597	−	0.675241i	$-0.764040\pi$
$727$	−39.7756	−1.47519	−0.737597	+	0.675241i	$0.764040\pi$
$728$	0	0
$729$	27.7402	1.02741
$730$	0	0
$731$	3.03009	0.112072
$732$	0	0
$733$	5.50154	0.203204	0.101602	−	0.994825i	$-0.467603\pi$
$733$	5.50154	0.203204	0.101602	+	0.994825i	$0.467603\pi$
$734$	0	0
$735$	−8.56071	−0.315766
$736$	0	0
$737$	2.66894	0.0983115
$738$	0	0
$739$	52.6937	1.93837	0.969185	−	0.246335i	$-0.0792263\pi$
$739$	52.6937	1.93837	0.969185	+	0.246335i	$0.0792263\pi$
$740$	0	0
$741$	8.14614	0.299256
$742$	0	0
$743$	−6.61647	−0.242735	−0.121367	−	0.992608i	$-0.538728\pi$
$743$	−6.61647	−0.242735	−0.121367	+	0.992608i	$0.538728\pi$
$744$	0	0
$745$	−22.6948	−0.831474
$746$	0	0
$747$	0.216261	0.00791256
$748$	0	0
$749$	30.5329	1.11565
$750$	0	0
$751$	39.9062	1.45620	0.728099	−	0.685472i	$-0.240404\pi$
$751$	39.9062	1.45620	0.728099	+	0.685472i	$0.240404\pi$
$752$	0	0
$753$	−14.0736	−0.512869
$754$	0	0
$755$	−11.2632	−0.409911
$756$	0	0
$757$	34.2645	1.24537	0.622683	−	0.782475i	$-0.286043\pi$
$757$	34.2645	1.24537	0.622683	+	0.782475i	$0.286043\pi$
$758$	0	0
$759$	−17.2578	−0.626417
$760$	0	0
$761$	12.4757	0.452243	0.226122	−	0.974099i	$-0.427395\pi$
$761$	12.4757	0.452243	0.226122	+	0.974099i	$0.427395\pi$
$762$	0	0
$763$	−15.2830	−0.553281
$764$	0	0
$765$	0.306060	0.0110656
$766$	0	0
$767$	−55.0126	−1.98639
$768$	0	0
$769$	3.32384	0.119861	0.0599304	−	0.998203i	$-0.480912\pi$
$769$	3.32384	0.119861	0.0599304	+	0.998203i	$0.480912\pi$
$770$	0	0
$771$	−25.7233	−0.926402
$772$	0	0
$773$	−14.4248	−0.518824	−0.259412	−	0.965767i	$-0.583529\pi$
$773$	−14.4248	−0.518824	−0.259412	+	0.965767i	$0.583529\pi$
$774$	0	0
$775$	21.8689	0.785555
$776$	0	0
$777$	0.263185	0.00944170
$778$	0	0
$779$	10.7743	0.386030
$780$	0	0
$781$	9.74099	0.348560
$782$	0	0
$783$	−3.66699	−0.131047
$784$	0	0
$785$	−38.9255	−1.38931
$786$	0	0
$787$	−38.6403	−1.37738	−0.688689	−	0.725057i	$-0.741813\pi$
$787$	−38.6403	−1.37738	−0.688689	+	0.725057i	$0.741813\pi$
$788$	0	0
$789$	16.1983	0.576676
$790$	0	0
$791$	−34.7103	−1.23416
$792$	0	0
$793$	−42.1433	−1.49655
$794$	0	0
$795$	25.5387	0.905763
$796$	0	0
$797$	−30.9316	−1.09565	−0.547826	−	0.836592i	$-0.684544\pi$
$797$	−30.9316	−1.09565	−0.547826	+	0.836592i	$0.684544\pi$
$798$	0	0
$799$	6.06971	0.214731
$800$	0	0
$801$	−1.38329	−0.0488762
$802$	0	0
$803$	9.28015	0.327490
$804$	0	0
$805$	−16.0691	−0.566360
$806$	0	0
$807$	−6.80177	−0.239434
$808$	0	0
$809$	−56.7269	−1.99441	−0.997206	−	0.0747056i	$-0.976198\pi$
$809$	−56.7269	−1.99441	−0.997206	+	0.0747056i	$0.976198\pi$
$810$	0	0
$811$	−4.91800	−0.172694	−0.0863472	−	0.996265i	$-0.527519\pi$
$811$	−4.91800	−0.172694	−0.0863472	+	0.996265i	$0.527519\pi$
$812$	0	0
$813$	−10.8956	−0.382125
$814$	0	0
$815$	−1.77812	−0.0622849
$816$	0	0
$817$	−1.43840	−0.0503233
$818$	0	0
$819$	0.832570	0.0290924
$820$	0	0
$821$	19.0797	0.665886	0.332943	−	0.942947i	$-0.391958\pi$
$821$	19.0797	0.665886	0.332943	+	0.942947i	$0.391958\pi$
$822$	0	0
$823$	−19.6263	−0.684131	−0.342066	−	0.939676i	$-0.611127\pi$
$823$	−19.6263	−0.684131	−0.342066	+	0.939676i	$0.611127\pi$
$824$	0	0
$825$	−7.96928	−0.277455
$826$	0	0
$827$	28.0107	0.974029	0.487014	−	0.873394i	$-0.338086\pi$
$827$	28.0107	0.974029	0.487014	+	0.873394i	$0.338086\pi$
$828$	0	0
$829$	−23.1932	−0.805533	−0.402766	−	0.915303i	$-0.631951\pi$
$829$	−23.1932	−0.805533	−0.402766	+	0.915303i	$0.631951\pi$
$830$	0	0
$831$	−25.8995	−0.898442
$832$	0	0
$833$	−6.35707	−0.220259
$834$	0	0
$835$	−22.6422	−0.783567
$836$	0	0
$837$	−51.5097	−1.78044
$838$	0	0
$839$	27.6363	0.954110	0.477055	−	0.878873i	$-0.341704\pi$
$839$	27.6363	0.954110	0.477055	+	0.878873i	$0.341704\pi$
$840$	0	0
$841$	−28.5157	−0.983299
$842$	0	0
$843$	25.4388	0.876157
$844$	0	0
$845$	−16.2626	−0.559450
$846$	0	0
$847$	13.2560	0.455481
$848$	0	0
$849$	25.5878	0.878171
$850$	0	0
$851$	−0.374363	−0.0128330
$852$	0	0
$853$	−53.0502	−1.81640	−0.908202	−	0.418532i	$-0.862545\pi$
$853$	−53.0502	−1.81640	−0.908202	+	0.418532i	$0.862545\pi$
$854$	0	0
$855$	−0.145289	−0.00496877
$856$	0	0
$857$	8.06832	0.275609	0.137804	−	0.990459i	$-0.455996\pi$
$857$	8.06832	0.275609	0.137804	+	0.990459i	$0.455996\pi$
$858$	0	0
$859$	10.1806	0.347357	0.173679	−	0.984802i	$-0.444435\pi$
$859$	10.1806	0.347357	0.173679	+	0.984802i	$0.444435\pi$
$860$	0	0
$861$	−36.6940	−1.25053
$862$	0	0
$863$	45.7074	1.55590	0.777949	−	0.628327i	$-0.216260\pi$
$863$	45.7074	1.55590	0.777949	+	0.628327i	$0.216260\pi$
$864$	0	0
$865$	19.2292	0.653811
$866$	0	0
$867$	21.4394	0.728119
$868$	0	0
$869$	2.29706	0.0779223
$870$	0	0
$871$	−6.10299	−0.206792
$872$	0	0
$873$	1.21535	0.0411334
$874$	0	0
$875$	−24.0056	−0.811539
$876$	0	0
$877$	−8.41970	−0.284313	−0.142157	−	0.989844i	$-0.545404\pi$
$877$	−8.41970	−0.284313	−0.142157	+	0.989844i	$0.545404\pi$
$878$	0	0
$879$	−37.0310	−1.24903
$880$	0	0
$881$	8.33625	0.280855	0.140428	−	0.990091i	$-0.455152\pi$
$881$	8.33625	0.280855	0.140428	+	0.990091i	$0.455152\pi$
$882$	0	0
$883$	−26.9797	−0.907941	−0.453970	−	0.891017i	$-0.649993\pi$
$883$	−26.9797	−0.907941	−0.453970	+	0.891017i	$0.649993\pi$
$884$	0	0
$885$	−32.6948	−1.09902
$886$	0	0
$887$	−25.8570	−0.868193	−0.434097	−	0.900866i	$-0.642932\pi$
$887$	−25.8570	−0.868193	−0.434097	+	0.900866i	$0.642932\pi$
$888$	0	0
$889$	36.2171	1.21468
$890$	0	0
$891$	18.2234	0.610505
$892$	0	0
$893$	−2.88133	−0.0964200
$894$	0	0
$895$	34.0726	1.13892
$896$	0	0
$897$	39.4629	1.31763
$898$	0	0
$899$	6.80348	0.226909
$900$	0	0
$901$	18.9647	0.631805
$902$	0	0
$903$	4.89875	0.163020
$904$	0	0
$905$	−31.2074	−1.03737
$906$	0	0
$907$	21.7649	0.722690	0.361345	−	0.932432i	$-0.382318\pi$
$907$	21.7649	0.722690	0.361345	+	0.932432i	$0.382318\pi$
$908$	0	0
$909$	−0.0878377	−0.00291339
$910$	0	0
$911$	13.3098	0.440974	0.220487	−	0.975390i	$-0.429235\pi$
$911$	13.3098	0.440974	0.220487	+	0.975390i	$0.429235\pi$
$912$	0	0
$913$	5.16464	0.170925
$914$	0	0
$915$	−25.0464	−0.828007
$916$	0	0
$917$	−19.3669	−0.639552
$918$	0	0
$919$	−29.7421	−0.981103	−0.490551	−	0.871412i	$-0.663205\pi$
$919$	−29.7421	−0.981103	−0.490551	+	0.871412i	$0.663205\pi$
$920$	0	0
$921$	−13.3365	−0.439451
$922$	0	0
$923$	−22.2745	−0.733174
$924$	0	0
$925$	−0.172873	−0.00568403
$926$	0	0
$927$	−1.08892	−0.0357648
$928$	0	0
$929$	−30.3181	−0.994705	−0.497352	−	0.867549i	$-0.665694\pi$
$929$	−30.3181	−0.994705	−0.497352	+	0.867549i	$0.665694\pi$
$930$	0	0
$931$	3.01774	0.0989024
$932$	0	0
$933$	−21.8941	−0.716779
$934$	0	0
$935$	7.30921	0.239037
$936$	0	0
$937$	8.81465	0.287962	0.143981	−	0.989580i	$-0.454010\pi$
$937$	8.81465	0.287962	0.143981	+	0.989580i	$0.454010\pi$
$938$	0	0
$939$	−6.95586	−0.226996
$940$	0	0
$941$	42.2936	1.37873	0.689366	−	0.724413i	$-0.257889\pi$
$941$	42.2936	1.37873	0.689366	+	0.724413i	$0.257889\pi$
$942$	0	0
$943$	52.1949	1.69970
$944$	0	0
$945$	17.4778	0.568553
$946$	0	0
$947$	30.2387	0.982626	0.491313	−	0.870983i	$-0.336517\pi$
$947$	30.2387	0.982626	0.491313	+	0.870983i	$0.336517\pi$
$948$	0	0
$949$	−21.2207	−0.688853
$950$	0	0
$951$	3.86464	0.125320
$952$	0	0
$953$	1.52937	0.0495411	0.0247706	−	0.999693i	$-0.492114\pi$
$953$	1.52937	0.0495411	0.0247706	+	0.999693i	$0.492114\pi$
$954$	0	0
$955$	−23.3362	−0.755141
$956$	0	0
$957$	−2.47926	−0.0801432
$958$	0	0
$959$	−40.3609	−1.30332
$960$	0	0
$961$	64.5676	2.08283
$962$	0	0
$963$	1.33736	0.0430957
$964$	0	0
$965$	39.6771	1.27725
$966$	0	0
$967$	−3.67723	−0.118252	−0.0591259	−	0.998251i	$-0.518831\pi$
$967$	−3.67723	−0.118252	−0.0591259	+	0.998251i	$0.518831\pi$
$968$	0	0
$969$	3.59513	0.115492
$970$	0	0
$971$	5.89852	0.189293	0.0946463	−	0.995511i	$-0.469828\pi$
$971$	5.89852	0.189293	0.0946463	+	0.995511i	$0.469828\pi$
$972$	0	0
$973$	−33.4747	−1.07315
$974$	0	0
$975$	18.2232	0.583608
$976$	0	0
$977$	−19.5003	−0.623871	−0.311935	−	0.950103i	$-0.600977\pi$
$977$	−19.5003	−0.623871	−0.311935	+	0.950103i	$0.600977\pi$
$978$	0	0
$979$	−33.0352	−1.05581
$980$	0	0
$981$	−0.669403	−0.0213724
$982$	0	0
$983$	−2.43384	−0.0776274	−0.0388137	−	0.999246i	$-0.512358\pi$
$983$	−2.43384	−0.0776274	−0.0388137	+	0.999246i	$0.512358\pi$
$984$	0	0
$985$	−3.07328	−0.0979229
$986$	0	0
$987$	9.81291	0.312348
$988$	0	0
$989$	−6.96815	−0.221574
$990$	0	0
$991$	15.1966	0.482736	0.241368	−	0.970434i	$-0.422404\pi$
$991$	15.1966	0.482736	0.241368	+	0.970434i	$0.422404\pi$
$992$	0	0
$993$	12.4784	0.395990
$994$	0	0
$995$	30.5196	0.967537
$996$	0	0
$997$	−42.5056	−1.34617	−0.673083	−	0.739567i	$-0.735030\pi$
$997$	−42.5056	−1.34617	−0.673083	+	0.739567i	$0.735030\pi$
$998$	0	0
$999$	0.407183	0.0128827

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.bp.1.3		8
4.3	odd	2		4864.2.a.bq.1.6		8
8.3	odd	2		4864.2.a.bo.1.3		8
8.5	even	2		4864.2.a.bn.1.6		8
16.3	odd	4		152.2.c.b.77.8	yes	16
16.5	even	4		608.2.c.b.305.11		16
16.11	odd	4		152.2.c.b.77.7	✓	16
16.13	even	4		608.2.c.b.305.6		16
48.5	odd	4		5472.2.g.b.2737.11		16
48.11	even	4		1368.2.g.b.685.10		16
48.29	odd	4		5472.2.g.b.2737.6		16
48.35	even	4		1368.2.g.b.685.9		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.c.b.77.7	✓	16	16.11	odd	4
152.2.c.b.77.8	yes	16	16.3	odd	4
608.2.c.b.305.6		16	16.13	even	4
608.2.c.b.305.11		16	16.5	even	4
1368.2.g.b.685.9		16	48.35	even	4
1368.2.g.b.685.10		16	48.11	even	4
4864.2.a.bn.1.6		8	8.5	even	2
4864.2.a.bo.1.3		8	8.3	odd	2
4864.2.a.bp.1.3		8	1.1	even	1	trivial
4864.2.a.bq.1.6		8	4.3	odd	2
5472.2.g.b.2737.6		16	48.29	odd	4
5472.2.g.b.2737.11		16	48.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4864 = 2^{8} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4864.a (trivial)

\(f(q)\)	\(=\)	\(q-1.70663 q^{3} -1.66222 q^{5} -1.99556 q^{7} -0.0874066 q^{9} -2.08741 q^{11} +4.77322 q^{13} +2.83680 q^{15} +2.10657 q^{17} -1.00000 q^{19} +3.40569 q^{21} -4.84437 q^{23} -2.23703 q^{25} +5.26907 q^{27} -0.695946 q^{29} -9.77587 q^{31} +3.56244 q^{33} +3.31706 q^{35} +0.0772780 q^{37} -8.14614 q^{39} -10.7743 q^{41} +1.43840 q^{43} +0.145289 q^{45} +2.88133 q^{47} -3.01774 q^{49} -3.59513 q^{51} +9.00264 q^{53} +3.46973 q^{55} +1.70663 q^{57} -11.5253 q^{59} -8.82910 q^{61} +0.174425 q^{63} -7.93414 q^{65} -1.27859 q^{67} +8.26756 q^{69} -4.66655 q^{71} -4.44578 q^{73} +3.81779 q^{75} +4.16555 q^{77} -1.10044 q^{79} -8.73014 q^{81} -2.47419 q^{83} -3.50157 q^{85} +1.18772 q^{87} +15.8259 q^{89} -9.52526 q^{91} +16.6838 q^{93} +1.66222 q^{95} -13.9046 q^{97} +0.182453 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5} - 4 q^{7} + 12 q^{9} - 4 q^{11} + 8 q^{13} - 4 q^{17} - 8 q^{19} + 16 q^{21} + 12 q^{25} + 28 q^{29} - 8 q^{31} + 12 q^{35} + 4 q^{37} + 4 q^{39} - 8 q^{41} + 4 q^{43} + 24 q^{45} - 12 q^{47}+ \cdots + 76 q^{99}+O(q^{100}) \) 8 * q + 8 * q^5 - 4 * q^7 + 12 * q^9 - 4 * q^11 + 8 * q^13 - 4 * q^17 - 8 * q^19 + 16 * q^21 + 12 * q^25 + 28 * q^29 - 8 * q^31 + 12 * q^35 + 4 * q^37 + 4 * q^39 - 8 * q^41 + 4 * q^43 + 24 * q^45 - 12 * q^47 + 12 * q^49 - 12 * q^51 + 32 * q^53 + 8 * q^55 - 12 * q^59 + 8 * q^61 + 16 * q^63 + 8 * q^65 + 4 * q^67 + 28 * q^69 + 24 * q^71 + 24 * q^77 + 24 * q^79 - 8 * q^81 - 40 * q^83 + 24 * q^85 - 24 * q^87 + 8 * q^89 + 4 * q^91 + 32 * q^93 - 8 * q^95 + 16 * q^97 + 76 * q^99

Introduction

L-functions

Modular forms

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