LMFDB - Embedded newform 9280.2.a.ct.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9280,2,Mod(1,9280)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9280.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9280 = 2^{6} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9280.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,-10,0,0,0,32,0,0,0,-18,0,0,0,18,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	yes
Analytic conductor:	$74.1011730757$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 31x^{8} + 353x^{6} - 1760x^{4} + 3412x^{2} - 1152 $ x^10 - 31x^8 + 353x^6 - 1760x^4 + 3412x^2 - 1152
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 4640)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.649617$ of defining polynomial
Character	$\chi$	$=$	9280.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.649617 q^{3} -1.00000 q^{5} -1.74317 q^{7} -2.57800 q^{9} +6.54379 q^{11} -6.26444 q^{13} -0.649617 q^{15} +2.27867 q^{17} +7.79921 q^{19} -1.13239 q^{21} +4.43505 q^{23} +1.00000 q^{25} -3.62356 q^{27} -1.00000 q^{29} -0.487747 q^{31} +4.25096 q^{33} +1.74317 q^{35} -4.85338 q^{37} -4.06949 q^{39} -10.7798 q^{41} +5.57241 q^{43} +2.57800 q^{45} +2.71455 q^{47} -3.96135 q^{49} +1.48026 q^{51} +7.64779 q^{53} -6.54379 q^{55} +5.06650 q^{57} -2.10874 q^{59} -4.89156 q^{61} +4.49389 q^{63} +6.26444 q^{65} -6.49998 q^{67} +2.88108 q^{69} +11.1010 q^{71} -3.34517 q^{73} +0.649617 q^{75} -11.4070 q^{77} +3.17962 q^{79} +5.38006 q^{81} -8.73090 q^{83} -2.27867 q^{85} -0.649617 q^{87} -6.52887 q^{89} +10.9200 q^{91} -0.316849 q^{93} -7.79921 q^{95} +10.5671 q^{97} -16.8699 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{5} + 32 q^{9} - 18 q^{13} + 18 q^{17} - 16 q^{21} + 10 q^{25} - 10 q^{29} + 8 q^{33} - 4 q^{37} + 16 q^{41} - 32 q^{45} + 48 q^{49} + 2 q^{53} + 52 q^{57} + 22 q^{61} + 18 q^{65} + 14 q^{69}+ \cdots - 10 q^{97}+O(q^{100}) $ 10 * q - 10 * q^5 + 32 * q^9 - 18 * q^13 + 18 * q^17 - 16 * q^21 + 10 * q^25 - 10 * q^29 + 8 * q^33 - 4 * q^37 + 16 * q^41 - 32 * q^45 + 48 * q^49 + 2 * q^53 + 52 * q^57 + 22 * q^61 + 18 * q^65 + 14 * q^69 - 30 * q^73 + 36 * q^77 + 42 * q^81 - 18 * q^85 + 24 * q^89 + 28 * q^93 - 10 * q^97

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
$3$	0.649617	0.375057	0.187528	−	0.982259i	$-0.439952\pi$
$3$	0.649617	0.375057	0.187528	+	0.982259i	$0.439952\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.74317	−0.658857	−0.329429	−	0.944180i	$-0.606856\pi$
$7$	−1.74317	−0.658857	−0.329429	+	0.944180i	$0.606856\pi$
$8$	0	0
$9$	−2.57800	−0.859333
$10$	0	0
$11$	6.54379	1.97303	0.986513	−	0.163682i	$-0.0523371\pi$
$11$	6.54379	1.97303	0.986513	+	0.163682i	$0.0523371\pi$
$12$	0	0
$13$	−6.26444	−1.73744	−0.868721	−	0.495302i	$-0.835058\pi$
$13$	−6.26444	−1.73744	−0.868721	+	0.495302i	$0.835058\pi$
$14$	0	0
$15$	−0.649617	−0.167730
$16$	0	0
$17$	2.27867	0.552658	0.276329	−	0.961063i	$-0.410882\pi$
$17$	2.27867	0.552658	0.276329	+	0.961063i	$0.410882\pi$
$18$	0	0
$19$	7.79921	1.78926	0.894631	−	0.446805i	$-0.147438\pi$
$19$	7.79921	1.78926	0.894631	+	0.446805i	$0.147438\pi$
$20$	0	0
$21$	−1.13239	−0.247109
$22$	0	0
$23$	4.43505	0.924772	0.462386	−	0.886679i	$-0.346993\pi$
$23$	4.43505	0.924772	0.462386	+	0.886679i	$0.346993\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−3.62356	−0.697355
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−0.487747	−0.0876019	−0.0438010	−	0.999040i	$-0.513947\pi$
$31$	−0.487747	−0.0876019	−0.0438010	+	0.999040i	$0.513947\pi$
$32$	0	0
$33$	4.25096	0.739997
$34$	0	0
$35$	1.74317	0.294650
$36$	0	0
$37$	−4.85338	−0.797890	−0.398945	−	0.916975i	$-0.630624\pi$
$37$	−4.85338	−0.797890	−0.398945	+	0.916975i	$0.630624\pi$
$38$	0	0
$39$	−4.06949	−0.651639
$40$	0	0
$41$	−10.7798	−1.68353	−0.841763	−	0.539847i	$-0.818482\pi$
$41$	−10.7798	−1.68353	−0.841763	+	0.539847i	$0.818482\pi$
$42$	0	0
$43$	5.57241	0.849785	0.424893	−	0.905244i	$-0.360312\pi$
$43$	5.57241	0.849785	0.424893	+	0.905244i	$0.360312\pi$
$44$	0	0
$45$	2.57800	0.384305
$46$	0	0
$47$	2.71455	0.395957	0.197979	−	0.980206i	$-0.436562\pi$
$47$	2.71455	0.395957	0.197979	+	0.980206i	$0.436562\pi$
$48$	0	0
$49$	−3.96135	−0.565907
$50$	0	0
$51$	1.48026	0.207278
$52$	0	0
$53$	7.64779	1.05050	0.525252	−	0.850947i	$-0.323971\pi$
$53$	7.64779	1.05050	0.525252	+	0.850947i	$0.323971\pi$
$54$	0	0
$55$	−6.54379	−0.882364
$56$	0	0
$57$	5.06650	0.671075
$58$	0	0
$59$	−2.10874	−0.274534	−0.137267	−	0.990534i	$-0.543832\pi$
$59$	−2.10874	−0.274534	−0.137267	+	0.990534i	$0.543832\pi$
$60$	0	0
$61$	−4.89156	−0.626300	−0.313150	−	0.949704i	$-0.601384\pi$
$61$	−4.89156	−0.626300	−0.313150	+	0.949704i	$0.601384\pi$
$62$	0	0
$63$	4.49389	0.566178
$64$	0	0
$65$	6.26444	0.777008
$66$	0	0
$67$	−6.49998	−0.794099	−0.397049	−	0.917797i	$-0.629966\pi$
$67$	−6.49998	−0.794099	−0.397049	+	0.917797i	$0.629966\pi$
$68$	0	0
$69$	2.88108	0.346842
$70$	0	0
$71$	11.1010	1.31745	0.658724	−	0.752384i	$-0.271096\pi$
$71$	11.1010	1.31745	0.658724	+	0.752384i	$0.271096\pi$
$72$	0	0
$73$	−3.34517	−0.391522	−0.195761	−	0.980652i	$-0.562718\pi$
$73$	−3.34517	−0.391522	−0.195761	+	0.980652i	$0.562718\pi$
$74$	0	0
$75$	0.649617	0.0750113
$76$	0	0
$77$	−11.4070	−1.29994
$78$	0	0
$79$	3.17962	0.357736	0.178868	−	0.983873i	$-0.442757\pi$
$79$	3.17962	0.357736	0.178868	+	0.983873i	$0.442757\pi$
$80$	0	0
$81$	5.38006	0.597785
$82$	0	0
$83$	−8.73090	−0.958341	−0.479170	−	0.877722i	$-0.659062\pi$
$83$	−8.73090	−0.958341	−0.479170	+	0.877722i	$0.659062\pi$
$84$	0	0
$85$	−2.27867	−0.247156
$86$	0	0
$87$	−0.649617	−0.0696463
$88$	0	0
$89$	−6.52887	−0.692059	−0.346030	−	0.938224i	$-0.612470\pi$
$89$	−6.52887	−0.692059	−0.346030	+	0.938224i	$0.612470\pi$
$90$	0	0
$91$	10.9200	1.14473
$92$	0	0
$93$	−0.316849	−0.0328557
$94$	0	0
$95$	−7.79921	−0.800182
$96$	0	0
$97$	10.5671	1.07292	0.536461	−	0.843925i	$-0.319761\pi$
$97$	10.5671	1.07292	0.536461	+	0.843925i	$0.319761\pi$
$98$	0	0
$99$	−16.8699	−1.69549
$100$	0	0
$101$	−19.0443	−1.89498	−0.947488	−	0.319792i	$-0.896387\pi$
$101$	−19.0443	−1.89498	−0.947488	+	0.319792i	$0.896387\pi$
$102$	0	0
$103$	6.58760	0.649095	0.324548	−	0.945869i	$-0.394788\pi$
$103$	6.58760	0.649095	0.324548	+	0.945869i	$0.394788\pi$
$104$	0	0
$105$	1.13239	0.110510
$106$	0	0
$107$	−16.3018	−1.57595	−0.787976	−	0.615707i	$-0.788871\pi$
$107$	−16.3018	−1.57595	−0.787976	+	0.615707i	$0.788871\pi$
$108$	0	0
$109$	11.5322	1.10458	0.552290	−	0.833652i	$-0.313754\pi$
$109$	11.5322	1.10458	0.552290	+	0.833652i	$0.313754\pi$
$110$	0	0
$111$	−3.15284	−0.299254
$112$	0	0
$113$	10.5671	0.994065	0.497032	−	0.867732i	$-0.334423\pi$
$113$	10.5671	0.994065	0.497032	+	0.867732i	$0.334423\pi$
$114$	0	0
$115$	−4.43505	−0.413571
$116$	0	0
$117$	16.1497	1.49304
$118$	0	0
$119$	−3.97211	−0.364123
$120$	0	0
$121$	31.8212	2.89283
$122$	0	0
$123$	−7.00276	−0.631418
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	14.7473	1.30861	0.654303	−	0.756232i	$-0.272962\pi$
$127$	14.7473	1.30861	0.654303	+	0.756232i	$0.272962\pi$
$128$	0	0
$129$	3.61994	0.318718
$130$	0	0
$131$	17.4004	1.52028	0.760142	−	0.649757i	$-0.225129\pi$
$131$	17.4004	1.52028	0.760142	+	0.649757i	$0.225129\pi$
$132$	0	0
$133$	−13.5954	−1.17887
$134$	0	0
$135$	3.62356	0.311867
$136$	0	0
$137$	−9.36877	−0.800428	−0.400214	−	0.916422i	$-0.631064\pi$
$137$	−9.36877	−0.800428	−0.400214	+	0.916422i	$0.631064\pi$
$138$	0	0
$139$	3.50138	0.296983	0.148492	−	0.988914i	$-0.452558\pi$
$139$	3.50138	0.296983	0.148492	+	0.988914i	$0.452558\pi$
$140$	0	0
$141$	1.76342	0.148506
$142$	0	0
$143$	−40.9931	−3.42802
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	−2.57336	−0.212247
$148$	0	0
$149$	17.1275	1.40314	0.701571	−	0.712599i	$-0.252482\pi$
$149$	17.1275	1.40314	0.701571	+	0.712599i	$0.252482\pi$
$150$	0	0
$151$	−2.82910	−0.230229	−0.115114	−	0.993352i	$-0.536723\pi$
$151$	−2.82910	−0.230229	−0.115114	+	0.993352i	$0.536723\pi$
$152$	0	0
$153$	−5.87439	−0.474917
$154$	0	0
$155$	0.487747	0.0391768
$156$	0	0
$157$	−11.9265	−0.951835	−0.475917	−	0.879490i	$-0.657884\pi$
$157$	−11.9265	−0.951835	−0.475917	+	0.879490i	$0.657884\pi$
$158$	0	0
$159$	4.96813	0.393999
$160$	0	0
$161$	−7.73106	−0.609293
$162$	0	0
$163$	14.7473	1.15509	0.577547	−	0.816357i	$-0.304010\pi$
$163$	14.7473	1.15509	0.577547	+	0.816357i	$0.304010\pi$
$164$	0	0
$165$	−4.25096	−0.330937
$166$	0	0
$167$	−15.7624	−1.21973	−0.609867	−	0.792504i	$-0.708777\pi$
$167$	−15.7624	−1.21973	−0.609867	+	0.792504i	$0.708777\pi$
$168$	0	0
$169$	26.2432	2.01870
$170$	0	0
$171$	−20.1064	−1.53757
$172$	0	0
$173$	−9.08656	−0.690838	−0.345419	−	0.938449i	$-0.612263\pi$
$173$	−9.08656	−0.690838	−0.345419	+	0.938449i	$0.612263\pi$
$174$	0	0
$175$	−1.74317	−0.131771
$176$	0	0
$177$	−1.36987	−0.102966
$178$	0	0
$179$	6.14945	0.459632	0.229816	−	0.973234i	$-0.426188\pi$
$179$	6.14945	0.459632	0.229816	+	0.973234i	$0.426188\pi$
$180$	0	0
$181$	−4.18117	−0.310784	−0.155392	−	0.987853i	$-0.549664\pi$
$181$	−4.18117	−0.310784	−0.155392	+	0.987853i	$0.549664\pi$
$182$	0	0
$183$	−3.17764	−0.234898
$184$	0	0
$185$	4.85338	0.356827
$186$	0	0
$187$	14.9111	1.09041
$188$	0	0
$189$	6.31650	0.459457
$190$	0	0
$191$	14.0996	1.02021	0.510106	−	0.860111i	$-0.329606\pi$
$191$	14.0996	1.02021	0.510106	+	0.860111i	$0.329606\pi$
$192$	0	0
$193$	19.9674	1.43729	0.718644	−	0.695379i	$-0.244763\pi$
$193$	19.9674	1.43729	0.718644	+	0.695379i	$0.244763\pi$
$194$	0	0
$195$	4.06949	0.291422
$196$	0	0
$197$	3.42529	0.244042	0.122021	−	0.992528i	$-0.461063\pi$
$197$	3.42529	0.244042	0.122021	+	0.992528i	$0.461063\pi$
$198$	0	0
$199$	23.3007	1.65174	0.825872	−	0.563857i	$-0.190683\pi$
$199$	23.3007	1.65174	0.825872	+	0.563857i	$0.190683\pi$
$200$	0	0
$201$	−4.22250	−0.297832
$202$	0	0
$203$	1.74317	0.122347
$204$	0	0
$205$	10.7798	0.752896
$206$	0	0
$207$	−11.4335	−0.794687
$208$	0	0
$209$	51.0364	3.53026
$210$	0	0
$211$	−6.54379	−0.450493	−0.225246	−	0.974302i	$-0.572319\pi$
$211$	−6.54379	−0.450493	−0.225246	+	0.974302i	$0.572319\pi$
$212$	0	0
$213$	7.21141	0.494118
$214$	0	0
$215$	−5.57241	−0.380035
$216$	0	0
$217$	0.850227	0.0577172
$218$	0	0
$219$	−2.17308	−0.146843
$220$	0	0
$221$	−14.2746	−0.960211
$222$	0	0
$223$	25.1811	1.68625	0.843126	−	0.537716i	$-0.180713\pi$
$223$	25.1811	1.68625	0.843126	+	0.537716i	$0.180713\pi$
$224$	0	0
$225$	−2.57800	−0.171867
$226$	0	0
$227$	−11.2856	−0.749049	−0.374524	−	0.927217i	$-0.622194\pi$
$227$	−11.2856	−0.749049	−0.374524	+	0.927217i	$0.622194\pi$
$228$	0	0
$229$	25.6055	1.69206	0.846029	−	0.533136i	$-0.178987\pi$
$229$	25.6055	1.69206	0.846029	+	0.533136i	$0.178987\pi$
$230$	0	0
$231$	−7.41015	−0.487552
$232$	0	0
$233$	19.8463	1.30018	0.650088	−	0.759859i	$-0.274732\pi$
$233$	19.8463	1.30018	0.650088	+	0.759859i	$0.274732\pi$
$234$	0	0
$235$	−2.71455	−0.177077
$236$	0	0
$237$	2.06554	0.134171
$238$	0	0
$239$	26.3757	1.70610	0.853051	−	0.521827i	$-0.174749\pi$
$239$	26.3757	1.70610	0.853051	+	0.521827i	$0.174749\pi$
$240$	0	0
$241$	13.1979	0.850154	0.425077	−	0.905157i	$-0.360247\pi$
$241$	13.1979	0.850154	0.425077	+	0.905157i	$0.360247\pi$
$242$	0	0
$243$	14.3657	0.921558
$244$	0	0
$245$	3.96135	0.253081
$246$	0	0
$247$	−48.8577	−3.10874
$248$	0	0
$249$	−5.67174	−0.359432
$250$	0	0
$251$	19.9551	1.25956	0.629778	−	0.776776i	$-0.283146\pi$
$251$	19.9551	1.25956	0.629778	+	0.776776i	$0.283146\pi$
$252$	0	0
$253$	29.0220	1.82460
$254$	0	0
$255$	−1.48026	−0.0926975
$256$	0	0
$257$	5.66517	0.353383	0.176692	−	0.984266i	$-0.443460\pi$
$257$	5.66517	0.353383	0.176692	+	0.984266i	$0.443460\pi$
$258$	0	0
$259$	8.46027	0.525696
$260$	0	0
$261$	2.57800	0.159574
$262$	0	0
$263$	9.77485	0.602743	0.301372	−	0.953507i	$-0.402556\pi$
$263$	9.77485	0.602743	0.301372	+	0.953507i	$0.402556\pi$
$264$	0	0
$265$	−7.64779	−0.469800
$266$	0	0
$267$	−4.24127	−0.259561
$268$	0	0
$269$	−1.43842	−0.0877018	−0.0438509	−	0.999038i	$-0.513963\pi$
$269$	−1.43842	−0.0877018	−0.0438509	+	0.999038i	$0.513963\pi$
$270$	0	0
$271$	20.1995	1.22703	0.613515	−	0.789683i	$-0.289755\pi$
$271$	20.1995	1.22703	0.613515	+	0.789683i	$0.289755\pi$
$272$	0	0
$273$	7.09382	0.429337
$274$	0	0
$275$	6.54379	0.394605
$276$	0	0
$277$	2.08292	0.125150	0.0625752	−	0.998040i	$-0.480069\pi$
$277$	2.08292	0.125150	0.0625752	+	0.998040i	$0.480069\pi$
$278$	0	0
$279$	1.25741	0.0752792
$280$	0	0
$281$	3.26601	0.194834	0.0974168	−	0.995244i	$-0.468942\pi$
$281$	3.26601	0.194834	0.0974168	+	0.995244i	$0.468942\pi$
$282$	0	0
$283$	−17.8809	−1.06291	−0.531456	−	0.847086i	$-0.678355\pi$
$283$	−17.8809	−1.06291	−0.531456	+	0.847086i	$0.678355\pi$
$284$	0	0
$285$	−5.06650	−0.300114
$286$	0	0
$287$	18.7911	1.10920
$288$	0	0
$289$	−11.8077	−0.694570
$290$	0	0
$291$	6.86454	0.402406
$292$	0	0
$293$	−3.36912	−0.196826	−0.0984131	−	0.995146i	$-0.531377\pi$
$293$	−3.36912	−0.196826	−0.0984131	+	0.995146i	$0.531377\pi$
$294$	0	0
$295$	2.10874	0.122775
$296$	0	0
$297$	−23.7118	−1.37590
$298$	0	0
$299$	−27.7831	−1.60674
$300$	0	0
$301$	−9.71368	−0.559887
$302$	0	0
$303$	−12.3715	−0.710723
$304$	0	0
$305$	4.89156	0.280090
$306$	0	0
$307$	−10.3730	−0.592020	−0.296010	−	0.955185i	$-0.595656\pi$
$307$	−10.3730	−0.592020	−0.296010	+	0.955185i	$0.595656\pi$
$308$	0	0
$309$	4.27942	0.243447
$310$	0	0
$311$	−2.16032	−0.122501	−0.0612503	−	0.998122i	$-0.519509\pi$
$311$	−2.16032	−0.122501	−0.0612503	+	0.998122i	$0.519509\pi$
$312$	0	0
$313$	−21.4702	−1.21357	−0.606783	−	0.794868i	$-0.707540\pi$
$313$	−21.4702	−1.21357	−0.606783	+	0.794868i	$0.707540\pi$
$314$	0	0
$315$	−4.49389	−0.253202
$316$	0	0
$317$	18.8315	1.05768	0.528841	−	0.848721i	$-0.322627\pi$
$317$	18.8315	1.05768	0.528841	+	0.848721i	$0.322627\pi$
$318$	0	0
$319$	−6.54379	−0.366382
$320$	0	0
$321$	−10.5899	−0.591071
$322$	0	0
$323$	17.7718	0.988850
$324$	0	0
$325$	−6.26444	−0.347488
$326$	0	0
$327$	7.49149	0.414280
$328$	0	0
$329$	−4.73192	−0.260879
$330$	0	0
$331$	14.7266	0.809445	0.404723	−	0.914439i	$-0.367368\pi$
$331$	14.7266	0.809445	0.404723	+	0.914439i	$0.367368\pi$
$332$	0	0
$333$	12.5120	0.685653
$334$	0	0
$335$	6.49998	0.355132
$336$	0	0
$337$	−10.5012	−0.572035	−0.286017	−	0.958224i	$-0.592332\pi$
$337$	−10.5012	−0.572035	−0.286017	+	0.958224i	$0.592332\pi$
$338$	0	0
$339$	6.86454	0.372831
$340$	0	0
$341$	−3.19171	−0.172841
$342$	0	0
$343$	19.1075	1.03171
$344$	0	0
$345$	−2.88108	−0.155112
$346$	0	0
$347$	36.9602	1.98413	0.992064	−	0.125734i	$-0.0401287\pi$
$347$	36.9602	1.98413	0.992064	+	0.125734i	$0.0401287\pi$
$348$	0	0
$349$	−21.0721	−1.12796	−0.563982	−	0.825787i	$-0.690731\pi$
$349$	−21.0721	−1.12796	−0.563982	+	0.825787i	$0.690731\pi$
$350$	0	0
$351$	22.6996	1.21161
$352$	0	0
$353$	−12.0973	−0.643874	−0.321937	−	0.946761i	$-0.604334\pi$
$353$	−12.0973	−0.643874	−0.321937	+	0.946761i	$0.604334\pi$
$354$	0	0
$355$	−11.1010	−0.589181
$356$	0	0
$357$	−2.58035	−0.136567
$358$	0	0
$359$	8.00901	0.422700	0.211350	−	0.977410i	$-0.432214\pi$
$359$	8.00901	0.422700	0.211350	+	0.977410i	$0.432214\pi$
$360$	0	0
$361$	41.8277	2.20146
$362$	0	0
$363$	20.6716	1.08498
$364$	0	0
$365$	3.34517	0.175094
$366$	0	0
$367$	−9.16141	−0.478222	−0.239111	−	0.970992i	$-0.576856\pi$
$367$	−9.16141	−0.478222	−0.239111	+	0.970992i	$0.576856\pi$
$368$	0	0
$369$	27.7904	1.44671
$370$	0	0
$371$	−13.3314	−0.692133
$372$	0	0
$373$	−23.5357	−1.21863	−0.609316	−	0.792927i	$-0.708556\pi$
$373$	−23.5357	−1.21863	−0.609316	+	0.792927i	$0.708556\pi$
$374$	0	0
$375$	−0.649617	−0.0335461
$376$	0	0
$377$	6.26444	0.322635
$378$	0	0
$379$	24.0056	1.23308	0.616542	−	0.787322i	$-0.288533\pi$
$379$	24.0056	1.23308	0.616542	+	0.787322i	$0.288533\pi$
$380$	0	0
$381$	9.58007	0.490802
$382$	0	0
$383$	−6.20501	−0.317061	−0.158531	−	0.987354i	$-0.550676\pi$
$383$	−6.20501	−0.317061	−0.158531	+	0.987354i	$0.550676\pi$
$384$	0	0
$385$	11.4070	0.581352
$386$	0	0
$387$	−14.3657	−0.730248
$388$	0	0
$389$	16.5157	0.837382	0.418691	−	0.908129i	$-0.362489\pi$
$389$	16.5157	0.837382	0.418691	+	0.908129i	$0.362489\pi$
$390$	0	0
$391$	10.1060	0.511082
$392$	0	0
$393$	11.3036	0.570193
$394$	0	0
$395$	−3.17962	−0.159984
$396$	0	0
$397$	−20.8454	−1.04620	−0.523100	−	0.852272i	$-0.675224\pi$
$397$	−20.8454	−1.04620	−0.523100	+	0.852272i	$0.675224\pi$
$398$	0	0
$399$	−8.83179	−0.442142
$400$	0	0
$401$	−19.9427	−0.995892	−0.497946	−	0.867208i	$-0.665912\pi$
$401$	−19.9427	−0.995892	−0.497946	+	0.867208i	$0.665912\pi$
$402$	0	0
$403$	3.05546	0.152203
$404$	0	0
$405$	−5.38006	−0.267338
$406$	0	0
$407$	−31.7595	−1.57426
$408$	0	0
$409$	11.9577	0.591271	0.295635	−	0.955301i	$-0.404469\pi$
$409$	11.9577	0.591271	0.295635	+	0.955301i	$0.404469\pi$
$410$	0	0
$411$	−6.08611	−0.300206
$412$	0	0
$413$	3.67589	0.180879
$414$	0	0
$415$	8.73090	0.428583
$416$	0	0
$417$	2.27456	0.111386
$418$	0	0
$419$	−9.03183	−0.441234	−0.220617	−	0.975361i	$-0.570807\pi$
$419$	−9.03183	−0.441234	−0.220617	+	0.975361i	$0.570807\pi$
$420$	0	0
$421$	1.63041	0.0794613	0.0397307	−	0.999210i	$-0.487350\pi$
$421$	1.63041	0.0794613	0.0397307	+	0.999210i	$0.487350\pi$
$422$	0	0
$423$	−6.99809	−0.340259
$424$	0	0
$425$	2.27867	0.110532
$426$	0	0
$427$	8.52683	0.412642
$428$	0	0
$429$	−26.6298	−1.28570
$430$	0	0
$431$	−19.2716	−0.928280	−0.464140	−	0.885762i	$-0.653637\pi$
$431$	−19.2716	−0.928280	−0.464140	+	0.885762i	$0.653637\pi$
$432$	0	0
$433$	22.4422	1.07850	0.539252	−	0.842145i	$-0.318707\pi$
$433$	22.4422	1.07850	0.539252	+	0.842145i	$0.318707\pi$
$434$	0	0
$435$	0.649617	0.0311468
$436$	0	0
$437$	34.5899	1.65466
$438$	0	0
$439$	−6.40306	−0.305601	−0.152801	−	0.988257i	$-0.548829\pi$
$439$	−6.40306	−0.305601	−0.152801	+	0.988257i	$0.548829\pi$
$440$	0	0
$441$	10.2123	0.486302
$442$	0	0
$443$	−19.0787	−0.906455	−0.453227	−	0.891395i	$-0.649727\pi$
$443$	−19.0787	−0.906455	−0.453227	+	0.891395i	$0.649727\pi$
$444$	0	0
$445$	6.52887	0.309498
$446$	0	0
$447$	11.1263	0.526258
$448$	0	0
$449$	7.06650	0.333489	0.166744	−	0.986000i	$-0.446675\pi$
$449$	7.06650	0.333489	0.166744	+	0.986000i	$0.446675\pi$
$450$	0	0
$451$	−70.5409	−3.32164
$452$	0	0
$453$	−1.83783	−0.0863487
$454$	0	0
$455$	−10.9200	−0.511937
$456$	0	0
$457$	−41.6492	−1.94827	−0.974133	−	0.225977i	$-0.927442\pi$
$457$	−41.6492	−1.94827	−0.974133	+	0.225977i	$0.927442\pi$
$458$	0	0
$459$	−8.25689	−0.385399
$460$	0	0
$461$	−6.55283	−0.305196	−0.152598	−	0.988288i	$-0.548764\pi$
$461$	−6.55283	−0.305196	−0.152598	+	0.988288i	$0.548764\pi$
$462$	0	0
$463$	18.3005	0.850498	0.425249	−	0.905076i	$-0.360187\pi$
$463$	18.3005	0.850498	0.425249	+	0.905076i	$0.360187\pi$
$464$	0	0
$465$	0.316849	0.0146935
$466$	0	0
$467$	5.94149	0.274939	0.137470	−	0.990506i	$-0.456103\pi$
$467$	5.94149	0.274939	0.137470	+	0.990506i	$0.456103\pi$
$468$	0	0
$469$	11.3306	0.523198
$470$	0	0
$471$	−7.74763	−0.356992
$472$	0	0
$473$	36.4647	1.67665
$474$	0	0
$475$	7.79921	0.357852
$476$	0	0
$477$	−19.7160	−0.902733
$478$	0	0
$479$	4.44084	0.202907	0.101454	−	0.994840i	$-0.467651\pi$
$479$	4.44084	0.202907	0.101454	+	0.994840i	$0.467651\pi$
$480$	0	0
$481$	30.4037	1.38629
$482$	0	0
$483$	−5.02223	−0.228519
$484$	0	0
$485$	−10.5671	−0.479825
$486$	0	0
$487$	12.7809	0.579156	0.289578	−	0.957154i	$-0.406485\pi$
$487$	12.7809	0.579156	0.289578	+	0.957154i	$0.406485\pi$
$488$	0	0
$489$	9.58007	0.433226
$490$	0	0
$491$	−16.6693	−0.752276	−0.376138	−	0.926564i	$-0.622748\pi$
$491$	−16.6693	−0.752276	−0.376138	+	0.926564i	$0.622748\pi$
$492$	0	0
$493$	−2.27867	−0.102626
$494$	0	0
$495$	16.8699	0.758244
$496$	0	0
$497$	−19.3510	−0.868011
$498$	0	0
$499$	36.3791	1.62855	0.814275	−	0.580479i	$-0.197135\pi$
$499$	36.3791	1.62855	0.814275	+	0.580479i	$0.197135\pi$
$500$	0	0
$501$	−10.2395	−0.457469
$502$	0	0
$503$	14.8349	0.661454	0.330727	−	0.943726i	$-0.392706\pi$
$503$	14.8349	0.661454	0.330727	+	0.943726i	$0.392706\pi$
$504$	0	0
$505$	19.0443	0.847459
$506$	0	0
$507$	17.0480	0.757129
$508$	0	0
$509$	−21.8442	−0.968225	−0.484112	−	0.875006i	$-0.660857\pi$
$509$	−21.8442	−0.968225	−0.484112	+	0.875006i	$0.660857\pi$
$510$	0	0
$511$	5.83121	0.257957
$512$	0	0
$513$	−28.2609	−1.24775
$514$	0	0
$515$	−6.58760	−0.290284
$516$	0	0
$517$	17.7634	0.781234
$518$	0	0
$519$	−5.90278	−0.259103
$520$	0	0
$521$	35.5277	1.55649	0.778247	−	0.627958i	$-0.216109\pi$
$521$	35.5277	1.55649	0.778247	+	0.627958i	$0.216109\pi$
$522$	0	0
$523$	−25.9468	−1.13457	−0.567287	−	0.823520i	$-0.692007\pi$
$523$	−25.9468	−1.13457	−0.567287	+	0.823520i	$0.692007\pi$
$524$	0	0
$525$	−1.13239	−0.0494218
$526$	0	0
$527$	−1.11141	−0.0484139
$528$	0	0
$529$	−3.33033	−0.144797
$530$	0	0
$531$	5.43632	0.235916
$532$	0	0
$533$	67.5296	2.92503
$534$	0	0
$535$	16.3018	0.704787
$536$	0	0
$537$	3.99479	0.172388
$538$	0	0
$539$	−25.9222	−1.11655
$540$	0	0
$541$	−36.2596	−1.55892	−0.779461	−	0.626451i	$-0.784507\pi$
$541$	−36.2596	−1.55892	−0.779461	+	0.626451i	$0.784507\pi$
$542$	0	0
$543$	−2.71616	−0.116561
$544$	0	0
$545$	−11.5322	−0.493983
$546$	0	0
$547$	−22.4660	−0.960575	−0.480287	−	0.877111i	$-0.659468\pi$
$547$	−22.4660	−0.960575	−0.480287	+	0.877111i	$0.659468\pi$
$548$	0	0
$549$	12.6104	0.538200
$550$	0	0
$551$	−7.79921	−0.332258
$552$	0	0
$553$	−5.54263	−0.235697
$554$	0	0
$555$	3.15284	0.133830
$556$	0	0
$557$	12.2506	0.519075	0.259537	−	0.965733i	$-0.416430\pi$
$557$	12.2506	0.519075	0.259537	+	0.965733i	$0.416430\pi$
$558$	0	0
$559$	−34.9080	−1.47645
$560$	0	0
$561$	9.68651	0.408965
$562$	0	0
$563$	11.7862	0.496729	0.248365	−	0.968667i	$-0.420107\pi$
$563$	11.7862	0.496729	0.248365	+	0.968667i	$0.420107\pi$
$564$	0	0
$565$	−10.5671	−0.444559
$566$	0	0
$567$	−9.37838	−0.393855
$568$	0	0
$569$	−5.49863	−0.230514	−0.115257	−	0.993336i	$-0.536769\pi$
$569$	−5.49863	−0.230514	−0.115257	+	0.993336i	$0.536769\pi$
$570$	0	0
$571$	38.9775	1.63116	0.815579	−	0.578645i	$-0.196418\pi$
$571$	38.9775	1.63116	0.815579	+	0.578645i	$0.196418\pi$
$572$	0	0
$573$	9.15935	0.382638
$574$	0	0
$575$	4.43505	0.184954
$576$	0	0
$577$	−3.58833	−0.149384	−0.0746920	−	0.997207i	$-0.523797\pi$
$577$	−3.58833	−0.149384	−0.0746920	+	0.997207i	$0.523797\pi$
$578$	0	0
$579$	12.9712	0.539064
$580$	0	0
$581$	15.2195	0.631410
$582$	0	0
$583$	50.0455	2.07267
$584$	0	0
$585$	−16.1497	−0.667708
$586$	0	0
$587$	−9.74196	−0.402094	−0.201047	−	0.979582i	$-0.564434\pi$
$587$	−9.74196	−0.402094	−0.201047	+	0.979582i	$0.564434\pi$
$588$	0	0
$589$	−3.80404	−0.156743
$590$	0	0
$591$	2.22513	0.0915295
$592$	0	0
$593$	38.4647	1.57956	0.789778	−	0.613393i	$-0.210196\pi$
$593$	38.4647	1.57956	0.789778	+	0.613393i	$0.210196\pi$
$594$	0	0
$595$	3.97211	0.162841
$596$	0	0
$597$	15.1365	0.619498
$598$	0	0
$599$	−35.8129	−1.46328	−0.731638	−	0.681693i	$-0.761244\pi$
$599$	−35.8129	−1.46328	−0.731638	+	0.681693i	$0.761244\pi$
$600$	0	0
$601$	29.4070	1.19953	0.599767	−	0.800174i	$-0.295260\pi$
$601$	29.4070	1.19953	0.599767	+	0.800174i	$0.295260\pi$
$602$	0	0
$603$	16.7569	0.682395
$604$	0	0
$605$	−31.8212	−1.29371
$606$	0	0
$607$	−28.4823	−1.15606	−0.578031	−	0.816015i	$-0.696179\pi$
$607$	−28.4823	−1.15606	−0.578031	+	0.816015i	$0.696179\pi$
$608$	0	0
$609$	1.13239	0.0458870
$610$	0	0
$611$	−17.0051	−0.687953
$612$	0	0
$613$	41.2904	1.66770	0.833851	−	0.551989i	$-0.186131\pi$
$613$	41.2904	1.66770	0.833851	+	0.551989i	$0.186131\pi$
$614$	0	0
$615$	7.00276	0.282379
$616$	0	0
$617$	−20.5077	−0.825610	−0.412805	−	0.910819i	$-0.635451\pi$
$617$	−20.5077	−0.825610	−0.412805	+	0.910819i	$0.635451\pi$
$618$	0	0
$619$	10.5860	0.425488	0.212744	−	0.977108i	$-0.431760\pi$
$619$	10.5860	0.425488	0.212744	+	0.977108i	$0.431760\pi$
$620$	0	0
$621$	−16.0707	−0.644894
$622$	0	0
$623$	11.3810	0.455968
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	33.1541	1.32405
$628$	0	0
$629$	−11.0592	−0.440960
$630$	0	0
$631$	25.7638	1.02564	0.512820	−	0.858496i	$-0.328601\pi$
$631$	25.7638	1.02564	0.512820	+	0.858496i	$0.328601\pi$
$632$	0	0
$633$	−4.25096	−0.168960
$634$	0	0
$635$	−14.7473	−0.585227
$636$	0	0
$637$	24.8156	0.983231
$638$	0	0
$639$	−28.6184	−1.13213
$640$	0	0
$641$	16.1737	0.638821	0.319410	−	0.947616i	$-0.396515\pi$
$641$	16.1737	0.638821	0.319410	+	0.947616i	$0.396515\pi$
$642$	0	0
$643$	12.1281	0.478286	0.239143	−	0.970984i	$-0.423134\pi$
$643$	12.1281	0.478286	0.239143	+	0.970984i	$0.423134\pi$
$644$	0	0
$645$	−3.61994	−0.142535
$646$	0	0
$647$	25.2595	0.993053	0.496526	−	0.868022i	$-0.334609\pi$
$647$	25.2595	0.993053	0.496526	+	0.868022i	$0.334609\pi$
$648$	0	0
$649$	−13.7991	−0.541663
$650$	0	0
$651$	0.552322	0.0216472
$652$	0	0
$653$	−48.7532	−1.90786	−0.953931	−	0.300027i	$-0.903004\pi$
$653$	−48.7532	−1.90786	−0.953931	+	0.300027i	$0.903004\pi$
$654$	0	0
$655$	−17.4004	−0.679892
$656$	0	0
$657$	8.62384	0.336448
$658$	0	0
$659$	−38.3526	−1.49400	−0.747002	−	0.664822i	$-0.768507\pi$
$659$	−38.3526	−1.49400	−0.747002	+	0.664822i	$0.768507\pi$
$660$	0	0
$661$	8.15271	0.317104	0.158552	−	0.987351i	$-0.449318\pi$
$661$	8.15271	0.317104	0.158552	+	0.987351i	$0.449318\pi$
$662$	0	0
$663$	−9.27300	−0.360133
$664$	0	0
$665$	13.5954	0.527206
$666$	0	0
$667$	−4.43505	−0.171726
$668$	0	0
$669$	16.3581	0.632440
$670$	0	0
$671$	−32.0093	−1.23571
$672$	0	0
$673$	−21.8237	−0.841243	−0.420622	−	0.907236i	$-0.638188\pi$
$673$	−21.8237	−0.841243	−0.420622	+	0.907236i	$0.638188\pi$
$674$	0	0
$675$	−3.62356	−0.139471
$676$	0	0
$677$	10.9873	0.422278	0.211139	−	0.977456i	$-0.432283\pi$
$677$	10.9873	0.422278	0.211139	+	0.977456i	$0.432283\pi$
$678$	0	0
$679$	−18.4202	−0.706902
$680$	0	0
$681$	−7.33129	−0.280936
$682$	0	0
$683$	10.1731	0.389265	0.194632	−	0.980876i	$-0.437649\pi$
$683$	10.1731	0.389265	0.194632	+	0.980876i	$0.437649\pi$
$684$	0	0
$685$	9.36877	0.357962
$686$	0	0
$687$	16.6338	0.634618
$688$	0	0
$689$	−47.9091	−1.82519
$690$	0	0
$691$	−28.3457	−1.07832	−0.539161	−	0.842203i	$-0.681258\pi$
$691$	−28.3457	−1.07832	−0.539161	+	0.842203i	$0.681258\pi$
$692$	0	0
$693$	29.4071	1.11708
$694$	0	0
$695$	−3.50138	−0.132815
$696$	0	0
$697$	−24.5636	−0.930414
$698$	0	0
$699$	12.8925	0.487640
$700$	0	0
$701$	43.6620	1.64909	0.824546	−	0.565795i	$-0.191431\pi$
$701$	43.6620	1.64909	0.824546	+	0.565795i	$0.191431\pi$
$702$	0	0
$703$	−37.8525	−1.42763
$704$	0	0
$705$	−1.76342	−0.0664141
$706$	0	0
$707$	33.1974	1.24852
$708$	0	0
$709$	−4.65462	−0.174808	−0.0874040	−	0.996173i	$-0.527857\pi$
$709$	−4.65462	−0.174808	−0.0874040	+	0.996173i	$0.527857\pi$
$710$	0	0
$711$	−8.19706	−0.307414
$712$	0	0
$713$	−2.16318	−0.0810118
$714$	0	0
$715$	40.9931	1.53306
$716$	0	0
$717$	17.1341	0.639885
$718$	0	0
$719$	−9.56963	−0.356887	−0.178443	−	0.983950i	$-0.557106\pi$
$719$	−9.56963	−0.356887	−0.178443	+	0.983950i	$0.557106\pi$
$720$	0	0
$721$	−11.4833	−0.427661
$722$	0	0
$723$	8.57360	0.318856
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−2.23406	−0.0828567	−0.0414283	−	0.999141i	$-0.513191\pi$
$727$	−2.23406	−0.0828567	−0.0414283	+	0.999141i	$0.513191\pi$
$728$	0	0
$729$	−6.80801	−0.252148
$730$	0	0
$731$	12.6977	0.469640
$732$	0	0
$733$	−14.1083	−0.521103	−0.260551	−	0.965460i	$-0.583904\pi$
$733$	−14.1083	−0.521103	−0.260551	+	0.965460i	$0.583904\pi$
$734$	0	0
$735$	2.57336	0.0949198
$736$	0	0
$737$	−42.5345	−1.56678
$738$	0	0
$739$	0.335788	0.0123522	0.00617609	−	0.999981i	$-0.498034\pi$
$739$	0.335788	0.0123522	0.00617609	+	0.999981i	$0.498034\pi$
$740$	0	0
$741$	−31.7388	−1.16595
$742$	0	0
$743$	27.1460	0.995889	0.497945	−	0.867209i	$-0.334088\pi$
$743$	27.1460	0.995889	0.497945	+	0.867209i	$0.334088\pi$
$744$	0	0
$745$	−17.1275	−0.627504
$746$	0	0
$747$	22.5082	0.823533
$748$	0	0
$749$	28.4168	1.03833
$750$	0	0
$751$	−49.2190	−1.79603	−0.898013	−	0.439969i	$-0.854990\pi$
$751$	−49.2190	−1.79603	−0.898013	+	0.439969i	$0.854990\pi$
$752$	0	0
$753$	12.9632	0.472404
$754$	0	0
$755$	2.82910	0.102961
$756$	0	0
$757$	−3.87855	−0.140968	−0.0704841	−	0.997513i	$-0.522454\pi$
$757$	−3.87855	−0.140968	−0.0704841	+	0.997513i	$0.522454\pi$
$758$	0	0
$759$	18.8532	0.684328
$760$	0	0
$761$	−21.3742	−0.774816	−0.387408	−	0.921908i	$-0.626629\pi$
$761$	−21.3742	−0.774816	−0.387408	+	0.921908i	$0.626629\pi$
$762$	0	0
$763$	−20.1025	−0.727761
$764$	0	0
$765$	5.87439	0.212389
$766$	0	0
$767$	13.2101	0.476987
$768$	0	0
$769$	1.18445	0.0427125	0.0213562	−	0.999772i	$-0.493202\pi$
$769$	1.18445	0.0427125	0.0213562	+	0.999772i	$0.493202\pi$
$770$	0	0
$771$	3.68019	0.132539
$772$	0	0
$773$	35.2570	1.26811	0.634054	−	0.773289i	$-0.281390\pi$
$773$	35.2570	1.26811	0.634054	+	0.773289i	$0.281390\pi$
$774$	0	0
$775$	−0.487747	−0.0175204
$776$	0	0
$777$	5.49594	0.197166
$778$	0	0
$779$	−84.0742	−3.01227
$780$	0	0
$781$	72.6427	2.59936
$782$	0	0
$783$	3.62356	0.129496
$784$	0	0
$785$	11.9265	0.425673
$786$	0	0
$787$	−12.7853	−0.455748	−0.227874	−	0.973691i	$-0.573178\pi$
$787$	−12.7853	−0.455748	−0.227874	+	0.973691i	$0.573178\pi$
$788$	0	0
$789$	6.34991	0.226063
$790$	0	0
$791$	−18.4202	−0.654947
$792$	0	0
$793$	30.6429	1.08816
$794$	0	0
$795$	−4.96813	−0.176202
$796$	0	0
$797$	13.1522	0.465876	0.232938	−	0.972492i	$-0.425166\pi$
$797$	13.1522	0.465876	0.232938	+	0.972492i	$0.425166\pi$
$798$	0	0
$799$	6.18554	0.218829
$800$	0	0
$801$	16.8314	0.594709
$802$	0	0
$803$	−21.8901	−0.772484
$804$	0	0
$805$	7.73106	0.272484
$806$	0	0
$807$	−0.934420	−0.0328931
$808$	0	0
$809$	−16.7667	−0.589486	−0.294743	−	0.955577i	$-0.595234\pi$
$809$	−16.7667	−0.589486	−0.294743	+	0.955577i	$0.595234\pi$
$810$	0	0
$811$	24.1298	0.847310	0.423655	−	0.905824i	$-0.360747\pi$
$811$	24.1298	0.847310	0.423655	+	0.905824i	$0.360747\pi$
$812$	0	0
$813$	13.1219	0.460206
$814$	0	0
$815$	−14.7473	−0.516574
$816$	0	0
$817$	43.4605	1.52049
$818$	0	0
$819$	−28.1517	−0.983701
$820$	0	0
$821$	32.0698	1.11924	0.559622	−	0.828748i	$-0.310946\pi$
$821$	32.0698	1.11924	0.559622	+	0.828748i	$0.310946\pi$
$822$	0	0
$823$	30.0712	1.04822	0.524109	−	0.851651i	$-0.324399\pi$
$823$	30.0712	1.04822	0.524109	+	0.851651i	$0.324399\pi$
$824$	0	0
$825$	4.25096	0.147999
$826$	0	0
$827$	−25.8235	−0.897970	−0.448985	−	0.893539i	$-0.648214\pi$
$827$	−25.8235	−0.897970	−0.448985	+	0.893539i	$0.648214\pi$
$828$	0	0
$829$	19.3689	0.672710	0.336355	−	0.941735i	$-0.390806\pi$
$829$	19.3689	0.672710	0.336355	+	0.941735i	$0.390806\pi$
$830$	0	0
$831$	1.35310	0.0469385
$832$	0	0
$833$	−9.02659	−0.312753
$834$	0	0
$835$	15.7624	0.545482
$836$	0	0
$837$	1.76738	0.0610896
$838$	0	0
$839$	−48.0628	−1.65931	−0.829655	−	0.558276i	$-0.811463\pi$
$839$	−48.0628	−1.65931	−0.829655	+	0.558276i	$0.811463\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	2.12165	0.0730737
$844$	0	0
$845$	−26.2432	−0.902792
$846$	0	0
$847$	−55.4698	−1.90596
$848$	0	0
$849$	−11.6158	−0.398652
$850$	0	0
$851$	−21.5250	−0.737866
$852$	0	0
$853$	31.9540	1.09408	0.547041	−	0.837106i	$-0.315754\pi$
$853$	31.9540	1.09408	0.547041	+	0.837106i	$0.315754\pi$
$854$	0	0
$855$	20.1064	0.687623
$856$	0	0
$857$	−22.6612	−0.774092	−0.387046	−	0.922060i	$-0.626504\pi$
$857$	−22.6612	−0.774092	−0.387046	+	0.922060i	$0.626504\pi$
$858$	0	0
$859$	44.9878	1.53496	0.767481	−	0.641071i	$-0.221510\pi$
$859$	44.9878	1.53496	0.767481	+	0.641071i	$0.221510\pi$
$860$	0	0
$861$	12.2070	0.416014
$862$	0	0
$863$	−55.8537	−1.90128	−0.950641	−	0.310294i	$-0.899572\pi$
$863$	−55.8537	−1.90128	−0.950641	+	0.310294i	$0.899572\pi$
$864$	0	0
$865$	9.08656	0.308952
$866$	0	0
$867$	−7.67047	−0.260503
$868$	0	0
$869$	20.8068	0.705822
$870$	0	0
$871$	40.7187	1.37970
$872$	0	0
$873$	−27.2418	−0.921997
$874$	0	0
$875$	1.74317	0.0589300
$876$	0	0
$877$	−40.7999	−1.37771	−0.688857	−	0.724897i	$-0.741887\pi$
$877$	−40.7999	−1.37771	−0.688857	+	0.724897i	$0.741887\pi$
$878$	0	0
$879$	−2.18864	−0.0738210
$880$	0	0
$881$	34.7986	1.17239	0.586197	−	0.810169i	$-0.300625\pi$
$881$	34.7986	1.17239	0.586197	+	0.810169i	$0.300625\pi$
$882$	0	0
$883$	−53.9991	−1.81721	−0.908607	−	0.417653i	$-0.862853\pi$
$883$	−53.9991	−1.81721	−0.908607	+	0.417653i	$0.862853\pi$
$884$	0	0
$885$	1.36987	0.0460477
$886$	0	0
$887$	31.5913	1.06073	0.530367	−	0.847768i	$-0.322054\pi$
$887$	31.5913	1.06073	0.530367	+	0.847768i	$0.322054\pi$
$888$	0	0
$889$	−25.7070	−0.862185
$890$	0	0
$891$	35.2060	1.17945
$892$	0	0
$893$	21.1713	0.708471
$894$	0	0
$895$	−6.14945	−0.205553
$896$	0	0
$897$	−18.0484	−0.602618
$898$	0	0
$899$	0.487747	0.0162673
$900$	0	0
$901$	17.4268	0.580569
$902$	0	0
$903$	−6.31017	−0.209989
$904$	0	0
$905$	4.18117	0.138987
$906$	0	0
$907$	−7.34817	−0.243992	−0.121996	−	0.992531i	$-0.538929\pi$
$907$	−7.34817	−0.243992	−0.121996	+	0.992531i	$0.538929\pi$
$908$	0	0
$909$	49.0961	1.62841
$910$	0	0
$911$	−2.29724	−0.0761111	−0.0380556	−	0.999276i	$-0.512116\pi$
$911$	−2.29724	−0.0761111	−0.0380556	+	0.999276i	$0.512116\pi$
$912$	0	0
$913$	−57.1332	−1.89083
$914$	0	0
$915$	3.17764	0.105050
$916$	0	0
$917$	−30.3320	−1.00165
$918$	0	0
$919$	12.0683	0.398096	0.199048	−	0.979990i	$-0.436215\pi$
$919$	12.0683	0.398096	0.199048	+	0.979990i	$0.436215\pi$
$920$	0	0
$921$	−6.73850	−0.222041
$922$	0	0
$923$	−69.5416	−2.28899
$924$	0	0
$925$	−4.85338	−0.159578
$926$	0	0
$927$	−16.9828	−0.557789
$928$	0	0
$929$	2.00301	0.0657165	0.0328582	−	0.999460i	$-0.489539\pi$
$929$	2.00301	0.0657165	0.0328582	+	0.999460i	$0.489539\pi$
$930$	0	0
$931$	−30.8954	−1.01256
$932$	0	0
$933$	−1.40338	−0.0459447
$934$	0	0
$935$	−14.9111	−0.487645
$936$	0	0
$937$	−49.5937	−1.62016	−0.810078	−	0.586322i	$-0.800575\pi$
$937$	−49.5937	−1.62016	−0.810078	+	0.586322i	$0.800575\pi$
$938$	0	0
$939$	−13.9474	−0.455156
$940$	0	0
$941$	2.07487	0.0676387	0.0338194	−	0.999428i	$-0.489233\pi$
$941$	2.07487	0.0676387	0.0338194	+	0.999428i	$0.489233\pi$
$942$	0	0
$943$	−47.8091	−1.55688
$944$	0	0
$945$	−6.31650	−0.205476
$946$	0	0
$947$	−3.92167	−0.127437	−0.0637186	−	0.997968i	$-0.520296\pi$
$947$	−3.92167	−0.127437	−0.0637186	+	0.997968i	$0.520296\pi$
$948$	0	0
$949$	20.9556	0.680247
$950$	0	0
$951$	12.2333	0.396691
$952$	0	0
$953$	7.42008	0.240360	0.120180	−	0.992752i	$-0.461653\pi$
$953$	7.42008	0.240360	0.120180	+	0.992752i	$0.461653\pi$
$954$	0	0
$955$	−14.0996	−0.456253
$956$	0	0
$957$	−4.25096	−0.137414
$958$	0	0
$959$	16.3314	0.527368
$960$	0	0
$961$	−30.7621	−0.992326
$962$	0	0
$963$	42.0259	1.35427
$964$	0	0
$965$	−19.9674	−0.642774
$966$	0	0
$967$	51.0617	1.64203	0.821016	−	0.570905i	$-0.193407\pi$
$967$	51.0617	1.64203	0.821016	+	0.570905i	$0.193407\pi$
$968$	0	0
$969$	11.5449	0.370875
$970$	0	0
$971$	37.1710	1.19287	0.596437	−	0.802660i	$-0.296582\pi$
$971$	37.1710	1.19287	0.596437	+	0.802660i	$0.296582\pi$
$972$	0	0
$973$	−6.10351	−0.195670
$974$	0	0
$975$	−4.06949	−0.130328
$976$	0	0
$977$	−11.7627	−0.376322	−0.188161	−	0.982138i	$-0.560253\pi$
$977$	−11.7627	−0.376322	−0.188161	+	0.982138i	$0.560253\pi$
$978$	0	0
$979$	−42.7236	−1.36545
$980$	0	0
$981$	−29.7299	−0.949202
$982$	0	0
$983$	21.1859	0.675725	0.337862	−	0.941196i	$-0.390296\pi$
$983$	21.1859	0.675725	0.337862	+	0.941196i	$0.390296\pi$
$984$	0	0
$985$	−3.42529	−0.109139
$986$	0	0
$987$	−3.07394	−0.0978445
$988$	0	0
$989$	24.7139	0.785858
$990$	0	0
$991$	28.5546	0.907066	0.453533	−	0.891239i	$-0.350163\pi$
$991$	28.5546	0.907066	0.453533	+	0.891239i	$0.350163\pi$
$992$	0	0
$993$	9.56663	0.303588
$994$	0	0
$995$	−23.3007	−0.738682
$996$	0	0
$997$	−57.5493	−1.82261	−0.911303	−	0.411737i	$-0.864922\pi$
$997$	−57.5493	−1.82261	−0.911303	+	0.411737i	$0.864922\pi$
$998$	0	0
$999$	17.5865	0.556413

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9280.2.a.ct.1.6		10
4.3	odd	2	inner	9280.2.a.ct.1.5		10
8.3	odd	2		4640.2.a.ba.1.6	yes	10
8.5	even	2		4640.2.a.ba.1.5	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4640.2.a.ba.1.5	✓	10	8.5	even	2
4640.2.a.ba.1.6	yes	10	8.3	odd	2
9280.2.a.ct.1.5		10	4.3	odd	2	inner
9280.2.a.ct.1.6		10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9280 = 2^{6} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9280.a (trivial)

\(f(q)\)	\(=\)	\(q+0.649617 q^{3} -1.00000 q^{5} -1.74317 q^{7} -2.57800 q^{9} +6.54379 q^{11} -6.26444 q^{13} -0.649617 q^{15} +2.27867 q^{17} +7.79921 q^{19} -1.13239 q^{21} +4.43505 q^{23} +1.00000 q^{25} -3.62356 q^{27} -1.00000 q^{29} -0.487747 q^{31} +4.25096 q^{33} +1.74317 q^{35} -4.85338 q^{37} -4.06949 q^{39} -10.7798 q^{41} +5.57241 q^{43} +2.57800 q^{45} +2.71455 q^{47} -3.96135 q^{49} +1.48026 q^{51} +7.64779 q^{53} -6.54379 q^{55} +5.06650 q^{57} -2.10874 q^{59} -4.89156 q^{61} +4.49389 q^{63} +6.26444 q^{65} -6.49998 q^{67} +2.88108 q^{69} +11.1010 q^{71} -3.34517 q^{73} +0.649617 q^{75} -11.4070 q^{77} +3.17962 q^{79} +5.38006 q^{81} -8.73090 q^{83} -2.27867 q^{85} -0.649617 q^{87} -6.52887 q^{89} +10.9200 q^{91} -0.316849 q^{93} -7.79921 q^{95} +10.5671 q^{97} -16.8699 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{5} + 32 q^{9} - 18 q^{13} + 18 q^{17} - 16 q^{21} + 10 q^{25} - 10 q^{29} + 8 q^{33} - 4 q^{37} + 16 q^{41} - 32 q^{45} + 48 q^{49} + 2 q^{53} + 52 q^{57} + 22 q^{61} + 18 q^{65} + 14 q^{69}+ \cdots - 10 q^{97}+O(q^{100}) \) 10 * q - 10 * q^5 + 32 * q^9 - 18 * q^13 + 18 * q^17 - 16 * q^21 + 10 * q^25 - 10 * q^29 + 8 * q^33 - 4 * q^37 + 16 * q^41 - 32 * q^45 + 48 * q^49 + 2 * q^53 + 52 * q^57 + 22 * q^61 + 18 * q^65 + 14 * q^69 - 30 * q^73 + 36 * q^77 + 42 * q^81 - 18 * q^85 + 24 * q^89 + 28 * q^93 - 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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