LMFDB - Embedded newform 1280.4.a.bd.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,4,Mod(1,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1280.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1280.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$75.5224448073$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 21x^{4} + 38x^{3} + 93x^{2} - 108x - 60 $ x^6 - 2x^5 - 21x^4 + 38x^3 + 93x^2 - 108*x - 60
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.46129$ of defining polynomial
Character	$\chi$	$=$	1280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.25785 q^{3} +5.00000 q^{5} -34.6280 q^{7} +12.1606 q^{9} +O(q^{10})$	$q+6.25785 q^{3} +5.00000 q^{5} -34.6280 q^{7} +12.1606 q^{9} -7.91595 q^{11} -11.0346 q^{13} +31.2892 q^{15} +57.6152 q^{17} +141.133 q^{19} -216.696 q^{21} +129.328 q^{23} +25.0000 q^{25} -92.8625 q^{27} -89.1664 q^{29} +78.3307 q^{31} -49.5368 q^{33} -173.140 q^{35} -249.332 q^{37} -69.0530 q^{39} -91.8705 q^{41} +194.184 q^{43} +60.8031 q^{45} +72.6149 q^{47} +856.096 q^{49} +360.547 q^{51} +456.782 q^{53} -39.5797 q^{55} +883.187 q^{57} +341.098 q^{59} -217.067 q^{61} -421.098 q^{63} -55.1731 q^{65} -529.237 q^{67} +809.314 q^{69} +381.540 q^{71} +876.902 q^{73} +156.446 q^{75} +274.113 q^{77} +203.950 q^{79} -909.456 q^{81} +996.654 q^{83} +288.076 q^{85} -557.989 q^{87} -172.456 q^{89} +382.107 q^{91} +490.182 q^{93} +705.664 q^{95} +1058.49 q^{97} -96.2629 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 30 q^{5} + 14 q^{7} + 54 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 + 30 * q^5 + 14 * q^7 + 54 * q^9	$ 6 q + 6 q^{3} + 30 q^{5} + 14 q^{7} + 54 q^{9} + 44 q^{11} + 30 q^{15} + 152 q^{19} + 4 q^{21} + 302 q^{23} + 150 q^{25} + 216 q^{27} + 132 q^{31} - 116 q^{33} + 70 q^{35} - 68 q^{37} + 300 q^{39} - 20 q^{41} + 602 q^{43} + 270 q^{45} + 470 q^{47} + 654 q^{49} + 612 q^{51} + 528 q^{53} + 220 q^{55} + 340 q^{57} + 472 q^{59} - 476 q^{61} + 650 q^{63} + 1206 q^{67} - 980 q^{69} - 796 q^{71} - 216 q^{73} + 150 q^{75} + 412 q^{77} - 1008 q^{79} + 1254 q^{81} + 1778 q^{83} - 984 q^{87} + 212 q^{89} + 3652 q^{91} - 1392 q^{93} + 760 q^{95} - 792 q^{97} + 5516 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 + 30 * q^5 + 14 * q^7 + 54 * q^9 + 44 * q^11 + 30 * q^15 + 152 * q^19 + 4 * q^21 + 302 * q^23 + 150 * q^25 + 216 * q^27 + 132 * q^31 - 116 * q^33 + 70 * q^35 - 68 * q^37 + 300 * q^39 - 20 * q^41 + 602 * q^43 + 270 * q^45 + 470 * q^47 + 654 * q^49 + 612 * q^51 + 528 * q^53 + 220 * q^55 + 340 * q^57 + 472 * q^59 - 476 * q^61 + 650 * q^63 + 1206 * q^67 - 980 * q^69 - 796 * q^71 - 216 * q^73 + 150 * q^75 + 412 * q^77 - 1008 * q^79 + 1254 * q^81 + 1778 * q^83 - 984 * q^87 + 212 * q^89 + 3652 * q^91 - 1392 * q^93 + 760 * q^95 - 792 * q^97 + 5516 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	6.25785	1.20432	0.602161	−	0.798374i	$-0.294306\pi$
$3$	6.25785	1.20432	0.602161	+	0.798374i	$0.294306\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−34.6280	−1.86973	−0.934867	−	0.354998i	$-0.884482\pi$
$7$	−34.6280	−1.86973	−0.934867	+	0.354998i	$0.884482\pi$
$8$	0	0
$9$	12.1606	0.450394
$10$	0	0
$11$	−7.91595	−0.216977	−0.108489	−	0.994098i	$-0.534601\pi$
$11$	−7.91595	−0.216977	−0.108489	+	0.994098i	$0.534601\pi$
$12$	0	0
$13$	−11.0346	−0.235420	−0.117710	−	0.993048i	$-0.537555\pi$
$13$	−11.0346	−0.235420	−0.117710	+	0.993048i	$0.537555\pi$
$14$	0	0
$15$	31.2892	0.538590
$16$	0	0
$17$	57.6152	0.821985	0.410992	−	0.911639i	$-0.365182\pi$
$17$	57.6152	0.821985	0.410992	+	0.911639i	$0.365182\pi$
$18$	0	0
$19$	141.133	1.70411	0.852055	−	0.523453i	$-0.175356\pi$
$19$	141.133	1.70411	0.852055	+	0.523453i	$0.175356\pi$
$20$	0	0
$21$	−216.696	−2.25176
$22$	0	0
$23$	129.328	1.17247	0.586233	−	0.810143i	$-0.300610\pi$
$23$	129.328	1.17247	0.586233	+	0.810143i	$0.300610\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−92.8625	−0.661903
$28$	0	0
$29$	−89.1664	−0.570958	−0.285479	−	0.958385i	$-0.592153\pi$
$29$	−89.1664	−0.570958	−0.285479	+	0.958385i	$0.592153\pi$
$30$	0	0
$31$	78.3307	0.453826	0.226913	−	0.973915i	$-0.427137\pi$
$31$	78.3307	0.453826	0.226913	+	0.973915i	$0.427137\pi$
$32$	0	0
$33$	−49.5368	−0.261310
$34$	0	0
$35$	−173.140	−0.836171
$36$	0	0
$37$	−249.332	−1.10783	−0.553917	−	0.832572i	$-0.686868\pi$
$37$	−249.332	−1.10783	−0.553917	+	0.832572i	$0.686868\pi$
$38$	0	0
$39$	−69.0530	−0.283521
$40$	0	0
$41$	−91.8705	−0.349946	−0.174973	−	0.984573i	$-0.555984\pi$
$41$	−91.8705	−0.349946	−0.174973	+	0.984573i	$0.555984\pi$
$42$	0	0
$43$	194.184	0.688668	0.344334	−	0.938847i	$-0.388105\pi$
$43$	194.184	0.688668	0.344334	+	0.938847i	$0.388105\pi$
$44$	0	0
$45$	60.8031	0.201422
$46$	0	0
$47$	72.6149	0.225361	0.112680	−	0.993631i	$-0.464056\pi$
$47$	72.6149	0.225361	0.112680	+	0.993631i	$0.464056\pi$
$48$	0	0
$49$	856.096	2.49591
$50$	0	0
$51$	360.547	0.989935
$52$	0	0
$53$	456.782	1.18385	0.591923	−	0.805995i	$-0.298369\pi$
$53$	456.782	1.18385	0.591923	+	0.805995i	$0.298369\pi$
$54$	0	0
$55$	−39.5797	−0.0970351
$56$	0	0
$57$	883.187	2.05230
$58$	0	0
$59$	341.098	0.752664	0.376332	−	0.926485i	$-0.377185\pi$
$59$	341.098	0.752664	0.376332	+	0.926485i	$0.377185\pi$
$60$	0	0
$61$	−217.067	−0.455616	−0.227808	−	0.973706i	$-0.573156\pi$
$61$	−217.067	−0.455616	−0.227808	+	0.973706i	$0.573156\pi$
$62$	0	0
$63$	−421.098	−0.842117
$64$	0	0
$65$	−55.1731	−0.105283
$66$	0	0
$67$	−529.237	−0.965024	−0.482512	−	0.875889i	$-0.660276\pi$
$67$	−529.237	−0.965024	−0.482512	+	0.875889i	$0.660276\pi$
$68$	0	0
$69$	809.314	1.41203
$70$	0	0
$71$	381.540	0.637754	0.318877	−	0.947796i	$-0.396694\pi$
$71$	381.540	0.637754	0.318877	+	0.947796i	$0.396694\pi$
$72$	0	0
$73$	876.902	1.40594	0.702970	−	0.711220i	$-0.251857\pi$
$73$	876.902	1.40594	0.702970	+	0.711220i	$0.251857\pi$
$74$	0	0
$75$	156.446	0.240865
$76$	0	0
$77$	274.113	0.405690
$78$	0	0
$79$	203.950	0.290458	0.145229	−	0.989398i	$-0.453608\pi$
$79$	203.950	0.290458	0.145229	+	0.989398i	$0.453608\pi$
$80$	0	0
$81$	−909.456	−1.24754
$82$	0	0
$83$	996.654	1.31804	0.659018	−	0.752127i	$-0.270972\pi$
$83$	996.654	1.31804	0.659018	+	0.752127i	$0.270972\pi$
$84$	0	0
$85$	288.076	0.367603
$86$	0	0
$87$	−557.989	−0.687618
$88$	0	0
$89$	−172.456	−0.205396	−0.102698	−	0.994713i	$-0.532748\pi$
$89$	−172.456	−0.205396	−0.102698	+	0.994713i	$0.532748\pi$
$90$	0	0
$91$	382.107	0.440172
$92$	0	0
$93$	490.182	0.546553
$94$	0	0
$95$	705.664	0.762101
$96$	0	0
$97$	1058.49	1.10797	0.553984	−	0.832527i	$-0.313107\pi$
$97$	1058.49	1.10797	0.553984	+	0.832527i	$0.313107\pi$
$98$	0	0
$99$	−96.2629	−0.0977251
$100$	0	0
$101$	689.992	0.679770	0.339885	−	0.940467i	$-0.389612\pi$
$101$	689.992	0.679770	0.339885	+	0.940467i	$0.389612\pi$
$102$	0	0
$103$	1289.96	1.23402	0.617009	−	0.786956i	$-0.288344\pi$
$103$	1289.96	1.23402	0.617009	+	0.786956i	$0.288344\pi$
$104$	0	0
$105$	−1083.48	−1.00702
$106$	0	0
$107$	1843.74	1.66580	0.832902	−	0.553420i	$-0.186678\pi$
$107$	1843.74	1.66580	0.832902	+	0.553420i	$0.186678\pi$
$108$	0	0
$109$	340.598	0.299297	0.149648	−	0.988739i	$-0.452186\pi$
$109$	340.598	0.299297	0.149648	+	0.988739i	$0.452186\pi$
$110$	0	0
$111$	−1560.28	−1.33419
$112$	0	0
$113$	157.831	0.131393	0.0656967	−	0.997840i	$-0.479073\pi$
$113$	157.831	0.131393	0.0656967	+	0.997840i	$0.479073\pi$
$114$	0	0
$115$	646.639	0.524343
$116$	0	0
$117$	−134.188	−0.106031
$118$	0	0
$119$	−1995.10	−1.53689
$120$	0	0
$121$	−1268.34	−0.952921
$122$	0	0
$123$	−574.912	−0.421447
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−494.704	−0.345652	−0.172826	−	0.984952i	$-0.555290\pi$
$127$	−494.704	−0.345652	−0.172826	+	0.984952i	$0.555290\pi$
$128$	0	0
$129$	1215.17	0.829379
$130$	0	0
$131$	488.783	0.325993	0.162997	−	0.986627i	$-0.447884\pi$
$131$	488.783	0.325993	0.162997	+	0.986627i	$0.447884\pi$
$132$	0	0
$133$	−4887.14	−3.18623
$134$	0	0
$135$	−464.312	−0.296012
$136$	0	0
$137$	1532.13	0.955464	0.477732	−	0.878506i	$-0.341459\pi$
$137$	1532.13	0.955464	0.477732	+	0.878506i	$0.341459\pi$
$138$	0	0
$139$	755.095	0.460765	0.230382	−	0.973100i	$-0.426002\pi$
$139$	755.095	0.460765	0.230382	+	0.973100i	$0.426002\pi$
$140$	0	0
$141$	454.413	0.271407
$142$	0	0
$143$	87.3495	0.0510807
$144$	0	0
$145$	−445.832	−0.255340
$146$	0	0
$147$	5357.32	3.00588
$148$	0	0
$149$	1446.67	0.795410	0.397705	−	0.917513i	$-0.369807\pi$
$149$	1446.67	0.795410	0.397705	+	0.917513i	$0.369807\pi$
$150$	0	0
$151$	−1230.20	−0.662998	−0.331499	−	0.943456i	$-0.607554\pi$
$151$	−1230.20	−0.662998	−0.331499	+	0.943456i	$0.607554\pi$
$152$	0	0
$153$	700.637	0.370217
$154$	0	0
$155$	391.654	0.202957
$156$	0	0
$157$	−1773.74	−0.901657	−0.450828	−	0.892611i	$-0.648871\pi$
$157$	−1773.74	−0.901657	−0.450828	+	0.892611i	$0.648871\pi$
$158$	0	0
$159$	2858.47	1.42573
$160$	0	0
$161$	−4478.36	−2.19220
$162$	0	0
$163$	1711.28	0.822319	0.411159	−	0.911563i	$-0.365124\pi$
$163$	1711.28	0.822319	0.411159	+	0.911563i	$0.365124\pi$
$164$	0	0
$165$	−247.684	−0.116862
$166$	0	0
$167$	3913.26	1.81328	0.906638	−	0.421909i	$-0.138640\pi$
$167$	3913.26	1.81328	0.906638	+	0.421909i	$0.138640\pi$
$168$	0	0
$169$	−2075.24	−0.944578
$170$	0	0
$171$	1716.26	0.767520
$172$	0	0
$173$	−3985.34	−1.75144	−0.875722	−	0.482815i	$-0.839614\pi$
$173$	−3985.34	−1.75144	−0.875722	+	0.482815i	$0.839614\pi$
$174$	0	0
$175$	−865.699	−0.373947
$176$	0	0
$177$	2134.54	0.906450
$178$	0	0
$179$	−884.734	−0.369431	−0.184715	−	0.982792i	$-0.559136\pi$
$179$	−884.734	−0.369431	−0.184715	+	0.982792i	$0.559136\pi$
$180$	0	0
$181$	−2810.77	−1.15427	−0.577134	−	0.816649i	$-0.695829\pi$
$181$	−2810.77	−1.15427	−0.577134	+	0.816649i	$0.695829\pi$
$182$	0	0
$183$	−1358.37	−0.548709
$184$	0	0
$185$	−1246.66	−0.495439
$186$	0	0
$187$	−456.079	−0.178352
$188$	0	0
$189$	3215.64	1.23758
$190$	0	0
$191$	749.321	0.283869	0.141935	−	0.989876i	$-0.454668\pi$
$191$	749.321	0.283869	0.141935	+	0.989876i	$0.454668\pi$
$192$	0	0
$193$	−3969.83	−1.48060	−0.740298	−	0.672279i	$-0.765315\pi$
$193$	−3969.83	−1.48060	−0.740298	+	0.672279i	$0.765315\pi$
$194$	0	0
$195$	−345.265	−0.126795
$196$	0	0
$197$	2243.20	0.811275	0.405638	−	0.914034i	$-0.367049\pi$
$197$	2243.20	0.811275	0.405638	+	0.914034i	$0.367049\pi$
$198$	0	0
$199$	−672.030	−0.239392	−0.119696	−	0.992811i	$-0.538192\pi$
$199$	−672.030	−0.239392	−0.119696	+	0.992811i	$0.538192\pi$
$200$	0	0
$201$	−3311.89	−1.16220
$202$	0	0
$203$	3087.65	1.06754
$204$	0	0
$205$	−459.353	−0.156500
$206$	0	0
$207$	1572.71	0.528071
$208$	0	0
$209$	−1117.20	−0.369753
$210$	0	0
$211$	1935.07	0.631356	0.315678	−	0.948866i	$-0.397768\pi$
$211$	1935.07	0.631356	0.315678	+	0.948866i	$0.397768\pi$
$212$	0	0
$213$	2387.62	0.768061
$214$	0	0
$215$	970.918	0.307982
$216$	0	0
$217$	−2712.43	−0.848535
$218$	0	0
$219$	5487.52	1.69321
$220$	0	0
$221$	−635.762	−0.193511
$222$	0	0
$223$	−2492.56	−0.748494	−0.374247	−	0.927329i	$-0.622099\pi$
$223$	−2492.56	−0.748494	−0.374247	+	0.927329i	$0.622099\pi$
$224$	0	0
$225$	304.016	0.0900787
$226$	0	0
$227$	−3155.27	−0.922567	−0.461283	−	0.887253i	$-0.652611\pi$
$227$	−3155.27	−0.922567	−0.461283	+	0.887253i	$0.652611\pi$
$228$	0	0
$229$	2299.43	0.663539	0.331770	−	0.943360i	$-0.392354\pi$
$229$	2299.43	0.663539	0.331770	+	0.943360i	$0.392354\pi$
$230$	0	0
$231$	1715.36	0.488581
$232$	0	0
$233$	−741.991	−0.208624	−0.104312	−	0.994545i	$-0.533264\pi$
$233$	−741.991	−0.208624	−0.104312	+	0.994545i	$0.533264\pi$
$234$	0	0
$235$	363.074	0.100785
$236$	0	0
$237$	1276.29	0.349806
$238$	0	0
$239$	−5786.01	−1.56597	−0.782984	−	0.622042i	$-0.786303\pi$
$239$	−5786.01	−1.56597	−0.782984	+	0.622042i	$0.786303\pi$
$240$	0	0
$241$	265.054	0.0708449	0.0354224	−	0.999372i	$-0.488722\pi$
$241$	265.054	0.0708449	0.0354224	+	0.999372i	$0.488722\pi$
$242$	0	0
$243$	−3183.95	−0.840537
$244$	0	0
$245$	4280.48	1.11620
$246$	0	0
$247$	−1557.35	−0.401181
$248$	0	0
$249$	6236.91	1.58734
$250$	0	0
$251$	1762.02	0.443098	0.221549	−	0.975149i	$-0.428889\pi$
$251$	1762.02	0.443098	0.221549	+	0.975149i	$0.428889\pi$
$252$	0	0
$253$	−1023.75	−0.254398
$254$	0	0
$255$	1802.74	0.442713
$256$	0	0
$257$	−1507.84	−0.365980	−0.182990	−	0.983115i	$-0.558578\pi$
$257$	−1507.84	−0.365980	−0.182990	+	0.983115i	$0.558578\pi$
$258$	0	0
$259$	8633.85	2.07136
$260$	0	0
$261$	−1084.32	−0.257156
$262$	0	0
$263$	2772.54	0.650046	0.325023	−	0.945706i	$-0.394628\pi$
$263$	2772.54	0.650046	0.325023	+	0.945706i	$0.394628\pi$
$264$	0	0
$265$	2283.91	0.529432
$266$	0	0
$267$	−1079.20	−0.247364
$268$	0	0
$269$	5166.61	1.17106	0.585528	−	0.810652i	$-0.300887\pi$
$269$	5166.61	1.17106	0.585528	+	0.810652i	$0.300887\pi$
$270$	0	0
$271$	1458.79	0.326994	0.163497	−	0.986544i	$-0.447723\pi$
$271$	1458.79	0.326994	0.163497	+	0.986544i	$0.447723\pi$
$272$	0	0
$273$	2391.16	0.530109
$274$	0	0
$275$	−197.899	−0.0433954
$276$	0	0
$277$	−1994.60	−0.432650	−0.216325	−	0.976321i	$-0.569407\pi$
$277$	−1994.60	−0.432650	−0.216325	+	0.976321i	$0.569407\pi$
$278$	0	0
$279$	952.551	0.204400
$280$	0	0
$281$	311.583	0.0661477	0.0330739	−	0.999453i	$-0.489470\pi$
$281$	311.583	0.0661477	0.0330739	+	0.999453i	$0.489470\pi$
$282$	0	0
$283$	6072.33	1.27549	0.637743	−	0.770249i	$-0.279868\pi$
$283$	6072.33	1.27549	0.637743	+	0.770249i	$0.279868\pi$
$284$	0	0
$285$	4415.93	0.917815
$286$	0	0
$287$	3181.29	0.654305
$288$	0	0
$289$	−1593.49	−0.324341
$290$	0	0
$291$	6623.84	1.33435
$292$	0	0
$293$	−2321.06	−0.462791	−0.231395	−	0.972860i	$-0.574329\pi$
$293$	−2321.06	−0.462791	−0.231395	+	0.972860i	$0.574329\pi$
$294$	0	0
$295$	1705.49	0.336602
$296$	0	0
$297$	735.095	0.143618
$298$	0	0
$299$	−1427.08	−0.276021
$300$	0	0
$301$	−6724.19	−1.28763
$302$	0	0
$303$	4317.86	0.818662
$304$	0	0
$305$	−1085.34	−0.203758
$306$	0	0
$307$	−2499.43	−0.464658	−0.232329	−	0.972637i	$-0.574635\pi$
$307$	−2499.43	−0.464658	−0.232329	+	0.972637i	$0.574635\pi$
$308$	0	0
$309$	8072.39	1.48616
$310$	0	0
$311$	−3052.83	−0.556624	−0.278312	−	0.960491i	$-0.589775\pi$
$311$	−3052.83	−0.556624	−0.278312	+	0.960491i	$0.589775\pi$
$312$	0	0
$313$	−6179.23	−1.11588	−0.557941	−	0.829881i	$-0.688408\pi$
$313$	−6179.23	−1.11588	−0.557941	+	0.829881i	$0.688408\pi$
$314$	0	0
$315$	−2105.49	−0.376606
$316$	0	0
$317$	3116.20	0.552124	0.276062	−	0.961140i	$-0.410971\pi$
$317$	3116.20	0.552124	0.276062	+	0.961140i	$0.410971\pi$
$318$	0	0
$319$	705.836	0.123885
$320$	0	0
$321$	11537.8	2.00617
$322$	0	0
$323$	8131.39	1.40075
$324$	0	0
$325$	−275.866	−0.0470839
$326$	0	0
$327$	2131.41	0.360450
$328$	0	0
$329$	−2514.51	−0.421365
$330$	0	0
$331$	−4157.26	−0.690343	−0.345171	−	0.938540i	$-0.612179\pi$
$331$	−4157.26	−0.690343	−0.345171	+	0.938540i	$0.612179\pi$
$332$	0	0
$333$	−3032.03	−0.498962
$334$	0	0
$335$	−2646.19	−0.431572
$336$	0	0
$337$	−8123.42	−1.31309	−0.656544	−	0.754287i	$-0.727983\pi$
$337$	−8123.42	−1.31309	−0.656544	+	0.754287i	$0.727983\pi$
$338$	0	0
$339$	987.679	0.158240
$340$	0	0
$341$	−620.062	−0.0984699
$342$	0	0
$343$	−17767.5	−2.79695
$344$	0	0
$345$	4046.57	0.631478
$346$	0	0
$347$	4620.55	0.714824	0.357412	−	0.933947i	$-0.383659\pi$
$347$	4620.55	0.714824	0.357412	+	0.933947i	$0.383659\pi$
$348$	0	0
$349$	5560.98	0.852929	0.426465	−	0.904504i	$-0.359759\pi$
$349$	5560.98	0.852929	0.426465	+	0.904504i	$0.359759\pi$
$350$	0	0
$351$	1024.70	0.155825
$352$	0	0
$353$	12031.8	1.81413	0.907066	−	0.420989i	$-0.138317\pi$
$353$	12031.8	1.81413	0.907066	+	0.420989i	$0.138317\pi$
$354$	0	0
$355$	1907.70	0.285212
$356$	0	0
$357$	−12485.0	−1.85092
$358$	0	0
$359$	1533.24	0.225408	0.112704	−	0.993629i	$-0.464049\pi$
$359$	1533.24	0.225408	0.112704	+	0.993629i	$0.464049\pi$
$360$	0	0
$361$	13059.5	1.90399
$362$	0	0
$363$	−7937.06	−1.14762
$364$	0	0
$365$	4384.51	0.628755
$366$	0	0
$367$	5480.04	0.779444	0.389722	−	0.920933i	$-0.372571\pi$
$367$	5480.04	0.779444	0.389722	+	0.920933i	$0.372571\pi$
$368$	0	0
$369$	−1117.20	−0.157613
$370$	0	0
$371$	−15817.4	−2.21348
$372$	0	0
$373$	−6225.70	−0.864221	−0.432111	−	0.901821i	$-0.642231\pi$
$373$	−6225.70	−0.864221	−0.432111	+	0.901821i	$0.642231\pi$
$374$	0	0
$375$	782.231	0.107718
$376$	0	0
$377$	983.917	0.134415
$378$	0	0
$379$	−11172.0	−1.51416	−0.757078	−	0.653325i	$-0.773374\pi$
$379$	−11172.0	−1.51416	−0.757078	+	0.653325i	$0.773374\pi$
$380$	0	0
$381$	−3095.78	−0.416277
$382$	0	0
$383$	−7621.03	−1.01675	−0.508376	−	0.861135i	$-0.669754\pi$
$383$	−7621.03	−1.01675	−0.508376	+	0.861135i	$0.669754\pi$
$384$	0	0
$385$	1370.57	0.181430
$386$	0	0
$387$	2361.40	0.310172
$388$	0	0
$389$	−5546.31	−0.722903	−0.361451	−	0.932391i	$-0.617719\pi$
$389$	−5546.31	−0.722903	−0.361451	+	0.932391i	$0.617719\pi$
$390$	0	0
$391$	7451.25	0.963749
$392$	0	0
$393$	3058.73	0.392601
$394$	0	0
$395$	1019.75	0.129897
$396$	0	0
$397$	−11025.2	−1.39380	−0.696900	−	0.717169i	$-0.745438\pi$
$397$	−11025.2	−1.39380	−0.696900	+	0.717169i	$0.745438\pi$
$398$	0	0
$399$	−30583.0	−3.83725
$400$	0	0
$401$	10522.3	1.31037	0.655186	−	0.755467i	$-0.272590\pi$
$401$	10522.3	1.31037	0.655186	+	0.755467i	$0.272590\pi$
$402$	0	0
$403$	−864.350	−0.106840
$404$	0	0
$405$	−4547.28	−0.557916
$406$	0	0
$407$	1973.70	0.240375
$408$	0	0
$409$	−2320.17	−0.280502	−0.140251	−	0.990116i	$-0.544791\pi$
$409$	−2320.17	−0.280502	−0.140251	+	0.990116i	$0.544791\pi$
$410$	0	0
$411$	9587.82	1.15069
$412$	0	0
$413$	−11811.5	−1.40728
$414$	0	0
$415$	4983.27	0.589444
$416$	0	0
$417$	4725.27	0.554910
$418$	0	0
$419$	−9212.17	−1.07409	−0.537045	−	0.843554i	$-0.680459\pi$
$419$	−9212.17	−1.07409	−0.537045	+	0.843554i	$0.680459\pi$
$420$	0	0
$421$	−6967.70	−0.806615	−0.403308	−	0.915064i	$-0.632140\pi$
$421$	−6967.70	−0.806615	−0.403308	+	0.915064i	$0.632140\pi$
$422$	0	0
$423$	883.042	0.101501
$424$	0	0
$425$	1440.38	0.164397
$426$	0	0
$427$	7516.59	0.851882
$428$	0	0
$429$	546.620	0.0615176
$430$	0	0
$431$	11247.1	1.25697	0.628486	−	0.777821i	$-0.283675\pi$
$431$	11247.1	1.25697	0.628486	+	0.777821i	$0.283675\pi$
$432$	0	0
$433$	2589.27	0.287372	0.143686	−	0.989623i	$-0.454104\pi$
$433$	2589.27	0.287372	0.143686	+	0.989623i	$0.454104\pi$
$434$	0	0
$435$	−2789.95	−0.307512
$436$	0	0
$437$	18252.4	1.99801
$438$	0	0
$439$	4220.01	0.458793	0.229396	−	0.973333i	$-0.426325\pi$
$439$	4220.01	0.458793	0.229396	+	0.973333i	$0.426325\pi$
$440$	0	0
$441$	10410.7	1.12414
$442$	0	0
$443$	−9764.04	−1.04719	−0.523593	−	0.851969i	$-0.675409\pi$
$443$	−9764.04	−1.04719	−0.523593	+	0.851969i	$0.675409\pi$
$444$	0	0
$445$	−862.279	−0.0918560
$446$	0	0
$447$	9053.06	0.957931
$448$	0	0
$449$	−17159.3	−1.80355	−0.901777	−	0.432202i	$-0.857737\pi$
$449$	−17159.3	−1.80355	−0.901777	+	0.432202i	$0.857737\pi$
$450$	0	0
$451$	727.242	0.0759302
$452$	0	0
$453$	−7698.43	−0.798463
$454$	0	0
$455$	1910.53	0.196851
$456$	0	0
$457$	−13027.3	−1.33346	−0.666730	−	0.745299i	$-0.732307\pi$
$457$	−13027.3	−1.33346	−0.666730	+	0.745299i	$0.732307\pi$
$458$	0	0
$459$	−5350.29	−0.544075
$460$	0	0
$461$	−313.396	−0.0316623	−0.0158312	−	0.999875i	$-0.505039\pi$
$461$	−313.396	−0.0316623	−0.0158312	+	0.999875i	$0.505039\pi$
$462$	0	0
$463$	12166.5	1.22123	0.610613	−	0.791930i	$-0.290923\pi$
$463$	12166.5	1.22123	0.610613	+	0.791930i	$0.290923\pi$
$464$	0	0
$465$	2450.91	0.244426
$466$	0	0
$467$	1844.42	0.182761	0.0913806	−	0.995816i	$-0.470872\pi$
$467$	1844.42	0.182761	0.0913806	+	0.995816i	$0.470872\pi$
$468$	0	0
$469$	18326.4	1.80434
$470$	0	0
$471$	−11099.8	−1.08589
$472$	0	0
$473$	−1537.15	−0.149425
$474$	0	0
$475$	3528.32	0.340822
$476$	0	0
$477$	5554.75	0.533196
$478$	0	0
$479$	16355.9	1.56017	0.780084	−	0.625675i	$-0.215176\pi$
$479$	16355.9	1.56017	0.780084	+	0.625675i	$0.215176\pi$
$480$	0	0
$481$	2751.28	0.260806
$482$	0	0
$483$	−28024.9	−2.64012
$484$	0	0
$485$	5292.43	0.495499
$486$	0	0
$487$	11824.6	1.10025	0.550126	−	0.835082i	$-0.314580\pi$
$487$	11824.6	1.10025	0.550126	+	0.835082i	$0.314580\pi$
$488$	0	0
$489$	10708.9	0.990337
$490$	0	0
$491$	−7931.40	−0.729000	−0.364500	−	0.931203i	$-0.618760\pi$
$491$	−7931.40	−0.729000	−0.364500	+	0.931203i	$0.618760\pi$
$492$	0	0
$493$	−5137.34	−0.469319
$494$	0	0
$495$	−481.315	−0.0437040
$496$	0	0
$497$	−13212.0	−1.19243
$498$	0	0
$499$	−2734.69	−0.245333	−0.122667	−	0.992448i	$-0.539145\pi$
$499$	−2734.69	−0.245333	−0.122667	+	0.992448i	$0.539145\pi$
$500$	0	0
$501$	24488.6	2.18377
$502$	0	0
$503$	7297.64	0.646890	0.323445	−	0.946247i	$-0.395159\pi$
$503$	7297.64	0.646890	0.323445	+	0.946247i	$0.395159\pi$
$504$	0	0
$505$	3449.96	0.304002
$506$	0	0
$507$	−12986.5	−1.13758
$508$	0	0
$509$	10972.4	0.955486	0.477743	−	0.878500i	$-0.341455\pi$
$509$	10972.4	0.955486	0.477743	+	0.878500i	$0.341455\pi$
$510$	0	0
$511$	−30365.3	−2.62873
$512$	0	0
$513$	−13105.9	−1.12796
$514$	0	0
$515$	6449.81	0.551869
$516$	0	0
$517$	−574.815	−0.0488982
$518$	0	0
$519$	−24939.7	−2.10931
$520$	0	0
$521$	−7693.43	−0.646939	−0.323470	−	0.946239i	$-0.604849\pi$
$521$	−7693.43	−0.646939	−0.323470	+	0.946239i	$0.604849\pi$
$522$	0	0
$523$	14535.3	1.21527	0.607634	−	0.794217i	$-0.292119\pi$
$523$	14535.3	1.21527	0.607634	+	0.794217i	$0.292119\pi$
$524$	0	0
$525$	−5417.41	−0.450353
$526$	0	0
$527$	4513.04	0.373038
$528$	0	0
$529$	4558.69	0.374676
$530$	0	0
$531$	4147.97	0.338995
$532$	0	0
$533$	1013.76	0.0823840
$534$	0	0
$535$	9218.70	0.744970
$536$	0	0
$537$	−5536.53	−0.444914
$538$	0	0
$539$	−6776.81	−0.541555
$540$	0	0
$541$	19131.8	1.52040	0.760202	−	0.649686i	$-0.225100\pi$
$541$	19131.8	1.52040	0.760202	+	0.649686i	$0.225100\pi$
$542$	0	0
$543$	−17589.3	−1.39011
$544$	0	0
$545$	1702.99	0.133850
$546$	0	0
$547$	−7142.78	−0.558324	−0.279162	−	0.960244i	$-0.590057\pi$
$547$	−7142.78	−0.558324	−0.279162	+	0.960244i	$0.590057\pi$
$548$	0	0
$549$	−2639.67	−0.205207
$550$	0	0
$551$	−12584.3	−0.972974
$552$	0	0
$553$	−7062.39	−0.543080
$554$	0	0
$555$	−7801.40	−0.596668
$556$	0	0
$557$	11347.8	0.863236	0.431618	−	0.902057i	$-0.357943\pi$
$557$	11347.8	0.863236	0.431618	+	0.902057i	$0.357943\pi$
$558$	0	0
$559$	−2142.74	−0.162126
$560$	0	0
$561$	−2854.07	−0.214793
$562$	0	0
$563$	8802.19	0.658913	0.329456	−	0.944171i	$-0.393135\pi$
$563$	8802.19	0.658913	0.329456	+	0.944171i	$0.393135\pi$
$564$	0	0
$565$	789.153	0.0587609
$566$	0	0
$567$	31492.6	2.33257
$568$	0	0
$569$	−16714.7	−1.23149	−0.615744	−	0.787946i	$-0.711145\pi$
$569$	−16714.7	−1.23149	−0.615744	+	0.787946i	$0.711145\pi$
$570$	0	0
$571$	−17386.8	−1.27428	−0.637142	−	0.770746i	$-0.719884\pi$
$571$	−17386.8	−1.27428	−0.637142	+	0.770746i	$0.719884\pi$
$572$	0	0
$573$	4689.14	0.341870
$574$	0	0
$575$	3233.20	0.234493
$576$	0	0
$577$	2260.52	0.163097	0.0815483	−	0.996669i	$-0.474014\pi$
$577$	2260.52	0.163097	0.0815483	+	0.996669i	$0.474014\pi$
$578$	0	0
$579$	−24842.6	−1.78311
$580$	0	0
$581$	−34512.1	−2.46438
$582$	0	0
$583$	−3615.86	−0.256867
$584$	0	0
$585$	−670.940	−0.0474187
$586$	0	0
$587$	−25591.0	−1.79941	−0.899705	−	0.436498i	$-0.856219\pi$
$587$	−25591.0	−1.79941	−0.899705	+	0.436498i	$0.856219\pi$
$588$	0	0
$589$	11055.0	0.773369
$590$	0	0
$591$	14037.6	0.977037
$592$	0	0
$593$	−10053.7	−0.696218	−0.348109	−	0.937454i	$-0.613176\pi$
$593$	−10053.7	−0.696218	−0.348109	+	0.937454i	$0.613176\pi$
$594$	0	0
$595$	−9975.49	−0.687320
$596$	0	0
$597$	−4205.46	−0.288305
$598$	0	0
$599$	−7086.68	−0.483395	−0.241698	−	0.970352i	$-0.577704\pi$
$599$	−7086.68	−0.483395	−0.241698	+	0.970352i	$0.577704\pi$
$600$	0	0
$601$	−6673.11	−0.452915	−0.226458	−	0.974021i	$-0.572714\pi$
$601$	−6673.11	−0.452915	−0.226458	+	0.974021i	$0.572714\pi$
$602$	0	0
$603$	−6435.86	−0.434641
$604$	0	0
$605$	−6341.69	−0.426159
$606$	0	0
$607$	15205.6	1.01677	0.508383	−	0.861131i	$-0.330243\pi$
$607$	15205.6	1.01677	0.508383	+	0.861131i	$0.330243\pi$
$608$	0	0
$609$	19322.0	1.28566
$610$	0	0
$611$	−801.278	−0.0530544
$612$	0	0
$613$	8147.64	0.536836	0.268418	−	0.963303i	$-0.413499\pi$
$613$	8147.64	0.536836	0.268418	+	0.963303i	$0.413499\pi$
$614$	0	0
$615$	−2874.56	−0.188477
$616$	0	0
$617$	−449.110	−0.0293039	−0.0146519	−	0.999893i	$-0.504664\pi$
$617$	−449.110	−0.0293039	−0.0146519	+	0.999893i	$0.504664\pi$
$618$	0	0
$619$	27800.4	1.80516	0.902579	−	0.430524i	$-0.141671\pi$
$619$	27800.4	1.80516	0.902579	+	0.430524i	$0.141671\pi$
$620$	0	0
$621$	−12009.7	−0.776059
$622$	0	0
$623$	5971.79	0.384037
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−6991.26	−0.445302
$628$	0	0
$629$	−14365.3	−0.910623
$630$	0	0
$631$	14793.5	0.933315	0.466657	−	0.884438i	$-0.345458\pi$
$631$	14793.5	0.933315	0.466657	+	0.884438i	$0.345458\pi$
$632$	0	0
$633$	12109.4	0.760356
$634$	0	0
$635$	−2473.52	−0.154580
$636$	0	0
$637$	−9446.70	−0.587585
$638$	0	0
$639$	4639.77	0.287240
$640$	0	0
$641$	−13853.8	−0.853655	−0.426828	−	0.904333i	$-0.640369\pi$
$641$	−13853.8	−0.853655	−0.426828	+	0.904333i	$0.640369\pi$
$642$	0	0
$643$	4978.60	0.305345	0.152672	−	0.988277i	$-0.451212\pi$
$643$	4978.60	0.305345	0.152672	+	0.988277i	$0.451212\pi$
$644$	0	0
$645$	6075.86	0.370910
$646$	0	0
$647$	−1903.39	−0.115657	−0.0578284	−	0.998327i	$-0.518418\pi$
$647$	−1903.39	−0.115657	−0.0578284	+	0.998327i	$0.518418\pi$
$648$	0	0
$649$	−2700.11	−0.163311
$650$	0	0
$651$	−16974.0	−1.02191
$652$	0	0
$653$	10877.0	0.651839	0.325920	−	0.945397i	$-0.394326\pi$
$653$	10877.0	0.651839	0.325920	+	0.945397i	$0.394326\pi$
$654$	0	0
$655$	2443.91	0.145789
$656$	0	0
$657$	10663.7	0.633226
$658$	0	0
$659$	30611.3	1.80948	0.904739	−	0.425965i	$-0.140065\pi$
$659$	30611.3	1.80948	0.904739	+	0.425965i	$0.140065\pi$
$660$	0	0
$661$	16098.8	0.947310	0.473655	−	0.880710i	$-0.342934\pi$
$661$	16098.8	0.947310	0.473655	+	0.880710i	$0.342934\pi$
$662$	0	0
$663$	−3978.50	−0.233050
$664$	0	0
$665$	−24435.7	−1.42493
$666$	0	0
$667$	−11531.7	−0.669429
$668$	0	0
$669$	−15598.0	−0.901428
$670$	0	0
$671$	1718.29	0.0988583
$672$	0	0
$673$	−23837.8	−1.36535	−0.682673	−	0.730724i	$-0.739183\pi$
$673$	−23837.8	−1.36535	−0.682673	+	0.730724i	$0.739183\pi$
$674$	0	0
$675$	−2321.56	−0.132381
$676$	0	0
$677$	−25235.3	−1.43260	−0.716302	−	0.697791i	$-0.754167\pi$
$677$	−25235.3	−1.43260	−0.716302	+	0.697791i	$0.754167\pi$
$678$	0	0
$679$	−36653.2	−2.07161
$680$	0	0
$681$	−19745.2	−1.11107
$682$	0	0
$683$	−19975.3	−1.11909	−0.559543	−	0.828802i	$-0.689023\pi$
$683$	−19975.3	−1.11909	−0.559543	+	0.828802i	$0.689023\pi$
$684$	0	0
$685$	7660.64	0.427297
$686$	0	0
$687$	14389.5	0.799115
$688$	0	0
$689$	−5040.42	−0.278700
$690$	0	0
$691$	−13772.3	−0.758209	−0.379104	−	0.925354i	$-0.623768\pi$
$691$	−13772.3	−0.758209	−0.379104	+	0.925354i	$0.623768\pi$
$692$	0	0
$693$	3333.39	0.182720
$694$	0	0
$695$	3775.48	0.206060
$696$	0	0
$697$	−5293.14	−0.287650
$698$	0	0
$699$	−4643.26	−0.251251
$700$	0	0
$701$	−6347.90	−0.342021	−0.171011	−	0.985269i	$-0.554703\pi$
$701$	−6347.90	−0.342021	−0.171011	+	0.985269i	$0.554703\pi$
$702$	0	0
$703$	−35188.9	−1.88787
$704$	0	0
$705$	2272.06	0.121377
$706$	0	0
$707$	−23893.0	−1.27099
$708$	0	0
$709$	−17910.5	−0.948719	−0.474360	−	0.880331i	$-0.657320\pi$
$709$	−17910.5	−0.948719	−0.474360	+	0.880331i	$0.657320\pi$
$710$	0	0
$711$	2480.16	0.130821
$712$	0	0
$713$	10130.3	0.532096
$714$	0	0
$715$	436.748	0.0228440
$716$	0	0
$717$	−36208.0	−1.88593
$718$	0	0
$719$	−36601.6	−1.89849	−0.949243	−	0.314545i	$-0.898148\pi$
$719$	−36601.6	−1.89849	−0.949243	+	0.314545i	$0.898148\pi$
$720$	0	0
$721$	−44668.8	−2.30728
$722$	0	0
$723$	1658.67	0.0853201
$724$	0	0
$725$	−2229.16	−0.114192
$726$	0	0
$727$	2644.18	0.134893	0.0674466	−	0.997723i	$-0.478515\pi$
$727$	2644.18	0.134893	0.0674466	+	0.997723i	$0.478515\pi$
$728$	0	0
$729$	4630.66	0.235262
$730$	0	0
$731$	11187.9	0.566075
$732$	0	0
$733$	33452.1	1.68565	0.842825	−	0.538188i	$-0.180891\pi$
$733$	33452.1	1.68565	0.842825	+	0.538188i	$0.180891\pi$
$734$	0	0
$735$	26786.6	1.34427
$736$	0	0
$737$	4189.42	0.209388
$738$	0	0
$739$	−25834.9	−1.28600	−0.642999	−	0.765867i	$-0.722310\pi$
$739$	−25834.9	−1.28600	−0.642999	+	0.765867i	$0.722310\pi$
$740$	0	0
$741$	−9745.64	−0.483151
$742$	0	0
$743$	7625.09	0.376497	0.188249	−	0.982121i	$-0.439719\pi$
$743$	7625.09	0.376497	0.188249	+	0.982121i	$0.439719\pi$
$744$	0	0
$745$	7233.37	0.355718
$746$	0	0
$747$	12119.9	0.593635
$748$	0	0
$749$	−63845.0	−3.11461
$750$	0	0
$751$	−25362.7	−1.23235	−0.616177	−	0.787607i	$-0.711320\pi$
$751$	−25362.7	−1.23235	−0.616177	+	0.787607i	$0.711320\pi$
$752$	0	0
$753$	11026.4	0.533633
$754$	0	0
$755$	−6151.02	−0.296502
$756$	0	0
$757$	38948.0	1.87000	0.935000	−	0.354648i	$-0.115399\pi$
$757$	38948.0	1.87000	0.935000	+	0.354648i	$0.115399\pi$
$758$	0	0
$759$	−6406.48	−0.306378
$760$	0	0
$761$	13722.2	0.653654	0.326827	−	0.945084i	$-0.394021\pi$
$761$	13722.2	0.653654	0.326827	+	0.945084i	$0.394021\pi$
$762$	0	0
$763$	−11794.2	−0.559606
$764$	0	0
$765$	3503.19	0.165566
$766$	0	0
$767$	−3763.89	−0.177192
$768$	0	0
$769$	18689.5	0.876414	0.438207	−	0.898874i	$-0.355614\pi$
$769$	18689.5	0.876414	0.438207	+	0.898874i	$0.355614\pi$
$770$	0	0
$771$	−9435.86	−0.440758
$772$	0	0
$773$	−6386.49	−0.297162	−0.148581	−	0.988900i	$-0.547471\pi$
$773$	−6386.49	−0.297162	−0.148581	+	0.988900i	$0.547471\pi$
$774$	0	0
$775$	1958.27	0.0907653
$776$	0	0
$777$	54029.3	2.49458
$778$	0	0
$779$	−12965.9	−0.596345
$780$	0	0
$781$	−3020.25	−0.138378
$782$	0	0
$783$	8280.21	0.377919
$784$	0	0
$785$	−8868.71	−0.403233
$786$	0	0
$787$	19410.3	0.879167	0.439583	−	0.898202i	$-0.355126\pi$
$787$	19410.3	0.879167	0.439583	+	0.898202i	$0.355126\pi$
$788$	0	0
$789$	17350.1	0.782865
$790$	0	0
$791$	−5465.35	−0.245671
$792$	0	0
$793$	2395.25	0.107261
$794$	0	0
$795$	14292.3	0.637607
$796$	0	0
$797$	30740.8	1.36624	0.683121	−	0.730305i	$-0.260622\pi$
$797$	30740.8	1.36624	0.683121	+	0.730305i	$0.260622\pi$
$798$	0	0
$799$	4183.72	0.185243
$800$	0	0
$801$	−2097.17	−0.0925092
$802$	0	0
$803$	−6941.51	−0.305057
$804$	0	0
$805$	−22391.8	−0.980382
$806$	0	0
$807$	32331.9	1.41033
$808$	0	0
$809$	−31805.5	−1.38223	−0.691114	−	0.722746i	$-0.742880\pi$
$809$	−31805.5	−1.38223	−0.691114	+	0.722746i	$0.742880\pi$
$810$	0	0
$811$	31014.7	1.34288	0.671439	−	0.741060i	$-0.265677\pi$
$811$	31014.7	1.34288	0.671439	+	0.741060i	$0.265677\pi$
$812$	0	0
$813$	9128.90	0.393806
$814$	0	0
$815$	8556.41	0.367752
$816$	0	0
$817$	27405.7	1.17357
$818$	0	0
$819$	4646.66	0.198251
$820$	0	0
$821$	12021.4	0.511021	0.255511	−	0.966806i	$-0.417756\pi$
$821$	12021.4	0.511021	0.255511	+	0.966806i	$0.417756\pi$
$822$	0	0
$823$	15489.0	0.656032	0.328016	−	0.944672i	$-0.393620\pi$
$823$	15489.0	0.656032	0.328016	+	0.944672i	$0.393620\pi$
$824$	0	0
$825$	−1238.42	−0.0522621
$826$	0	0
$827$	17097.2	0.718898	0.359449	−	0.933165i	$-0.382965\pi$
$827$	17097.2	0.718898	0.359449	+	0.933165i	$0.382965\pi$
$828$	0	0
$829$	−9885.97	−0.414178	−0.207089	−	0.978322i	$-0.566399\pi$
$829$	−9885.97	−0.414178	−0.207089	+	0.978322i	$0.566399\pi$
$830$	0	0
$831$	−12481.9	−0.521050
$832$	0	0
$833$	49324.2	2.05160
$834$	0	0
$835$	19566.3	0.810922
$836$	0	0
$837$	−7273.99	−0.300389
$838$	0	0
$839$	−39204.3	−1.61321	−0.806606	−	0.591090i	$-0.798698\pi$
$839$	−39204.3	−1.61321	−0.806606	+	0.591090i	$0.798698\pi$
$840$	0	0
$841$	−16438.4	−0.674007
$842$	0	0
$843$	1949.84	0.0796632
$844$	0	0
$845$	−10376.2	−0.422428
$846$	0	0
$847$	43920.0	1.78171
$848$	0	0
$849$	37999.7	1.53610
$850$	0	0
$851$	−32245.5	−1.29890
$852$	0	0
$853$	45054.1	1.80847	0.904234	−	0.427038i	$-0.140443\pi$
$853$	45054.1	1.80847	0.904234	+	0.427038i	$0.140443\pi$
$854$	0	0
$855$	8581.31	0.343245
$856$	0	0
$857$	48464.2	1.93174	0.965872	−	0.259021i	$-0.0834000\pi$
$857$	48464.2	1.93174	0.965872	+	0.259021i	$0.0834000\pi$
$858$	0	0
$859$	1000.76	0.0397503	0.0198751	−	0.999802i	$-0.493673\pi$
$859$	1000.76	0.0397503	0.0198751	+	0.999802i	$0.493673\pi$
$860$	0	0
$861$	19908.0	0.787995
$862$	0	0
$863$	−22485.5	−0.886922	−0.443461	−	0.896294i	$-0.646250\pi$
$863$	−22485.5	−0.886922	−0.443461	+	0.896294i	$0.646250\pi$
$864$	0	0
$865$	−19926.7	−0.783270
$866$	0	0
$867$	−9971.79	−0.390611
$868$	0	0
$869$	−1614.46	−0.0630228
$870$	0	0
$871$	5839.94	0.227186
$872$	0	0
$873$	12871.9	0.499022
$874$	0	0
$875$	−4328.50	−0.167234
$876$	0	0
$877$	43726.2	1.68361	0.841806	−	0.539780i	$-0.181493\pi$
$877$	43726.2	1.68361	0.841806	+	0.539780i	$0.181493\pi$
$878$	0	0
$879$	−14524.8	−0.557349
$880$	0	0
$881$	20290.0	0.775923	0.387962	−	0.921675i	$-0.373179\pi$
$881$	20290.0	0.775923	0.387962	+	0.921675i	$0.373179\pi$
$882$	0	0
$883$	14887.3	0.567382	0.283691	−	0.958916i	$-0.408441\pi$
$883$	14887.3	0.567382	0.283691	+	0.958916i	$0.408441\pi$
$884$	0	0
$885$	10672.7	0.405377
$886$	0	0
$887$	34359.3	1.30065	0.650323	−	0.759658i	$-0.274633\pi$
$887$	34359.3	1.30065	0.650323	+	0.759658i	$0.274633\pi$
$888$	0	0
$889$	17130.6	0.646278
$890$	0	0
$891$	7199.21	0.270687
$892$	0	0
$893$	10248.3	0.384040
$894$	0	0
$895$	−4423.67	−0.165214
$896$	0	0
$897$	−8930.47	−0.332419
$898$	0	0
$899$	−6984.47	−0.259116
$900$	0	0
$901$	26317.6	0.973103
$902$	0	0
$903$	−42078.9	−1.55072
$904$	0	0
$905$	−14053.8	−0.516205
$906$	0	0
$907$	53363.9	1.95360	0.976802	−	0.214144i	$-0.0686963\pi$
$907$	53363.9	1.95360	0.976802	+	0.214144i	$0.0686963\pi$
$908$	0	0
$909$	8390.73	0.306164
$910$	0	0
$911$	25213.9	0.916984	0.458492	−	0.888699i	$-0.348390\pi$
$911$	25213.9	0.916984	0.458492	+	0.888699i	$0.348390\pi$
$912$	0	0
$913$	−7889.46	−0.285984
$914$	0	0
$915$	−6791.86	−0.245390
$916$	0	0
$917$	−16925.5	−0.609521
$918$	0	0
$919$	3747.02	0.134497	0.0672485	−	0.997736i	$-0.478578\pi$
$919$	3747.02	0.134497	0.0672485	+	0.997736i	$0.478578\pi$
$920$	0	0
$921$	−15641.1	−0.559599
$922$	0	0
$923$	−4210.15	−0.150140
$924$	0	0
$925$	−6233.29	−0.221567
$926$	0	0
$927$	15686.8	0.555794
$928$	0	0
$929$	−27726.7	−0.979209	−0.489604	−	0.871945i	$-0.662859\pi$
$929$	−27726.7	−0.979209	−0.489604	+	0.871945i	$0.662859\pi$
$930$	0	0
$931$	120823.	4.25330
$932$	0	0
$933$	−19104.1	−0.670355
$934$	0	0
$935$	−2280.40	−0.0797614
$936$	0	0
$937$	−29333.2	−1.02270	−0.511352	−	0.859371i	$-0.670855\pi$
$937$	−29333.2	−1.02270	−0.511352	+	0.859371i	$0.670855\pi$
$938$	0	0
$939$	−38668.7	−1.34388
$940$	0	0
$941$	24218.4	0.838998	0.419499	−	0.907756i	$-0.362206\pi$
$941$	24218.4	0.838998	0.419499	+	0.907756i	$0.362206\pi$
$942$	0	0
$943$	−11881.4	−0.410299
$944$	0	0
$945$	16078.2	0.553464
$946$	0	0
$947$	50824.6	1.74401	0.872005	−	0.489497i	$-0.162820\pi$
$947$	50824.6	1.74401	0.872005	+	0.489497i	$0.162820\pi$
$948$	0	0
$949$	−9676.28	−0.330986
$950$	0	0
$951$	19500.7	0.664935
$952$	0	0
$953$	10155.1	0.345180	0.172590	−	0.984994i	$-0.444786\pi$
$953$	10155.1	0.345180	0.172590	+	0.984994i	$0.444786\pi$
$954$	0	0
$955$	3746.61	0.126950
$956$	0	0
$957$	4417.01	0.149197
$958$	0	0
$959$	−53054.5	−1.78646
$960$	0	0
$961$	−23655.3	−0.794042
$962$	0	0
$963$	22421.0	0.750268
$964$	0	0
$965$	−19849.2	−0.662142
$966$	0	0
$967$	−28225.7	−0.938652	−0.469326	−	0.883025i	$-0.655503\pi$
$967$	−28225.7	−0.938652	−0.469326	+	0.883025i	$0.655503\pi$
$968$	0	0
$969$	50885.0	1.68696
$970$	0	0
$971$	−47631.4	−1.57422	−0.787109	−	0.616814i	$-0.788423\pi$
$971$	−47631.4	−1.57422	−0.787109	+	0.616814i	$0.788423\pi$
$972$	0	0
$973$	−26147.4	−0.861508
$974$	0	0
$975$	−1726.32	−0.0567042
$976$	0	0
$977$	13542.4	0.443460	0.221730	−	0.975108i	$-0.428830\pi$
$977$	13542.4	0.443460	0.221730	+	0.975108i	$0.428830\pi$
$978$	0	0
$979$	1365.15	0.0445663
$980$	0	0
$981$	4141.88	0.134801
$982$	0	0
$983$	−35108.5	−1.13915	−0.569576	−	0.821938i	$-0.692893\pi$
$983$	−35108.5	−1.13915	−0.569576	+	0.821938i	$0.692893\pi$
$984$	0	0
$985$	11216.0	0.362813
$986$	0	0
$987$	−15735.4	−0.507460
$988$	0	0
$989$	25113.4	0.807440
$990$	0	0
$991$	42816.1	1.37245	0.686225	−	0.727390i	$-0.259267\pi$
$991$	42816.1	1.37245	0.686225	+	0.727390i	$0.259267\pi$
$992$	0	0
$993$	−26015.5	−0.831396
$994$	0	0
$995$	−3360.15	−0.107059
$996$	0	0
$997$	13029.0	0.413873	0.206936	−	0.978354i	$-0.433651\pi$
$997$	13029.0	0.413873	0.206936	+	0.978354i	$0.433651\pi$
$998$	0	0
$999$	23153.6	0.733280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.4.a.bd.1.5		6
4.3	odd	2		1280.4.a.bb.1.2		6
8.3	odd	2		1280.4.a.bc.1.5		6
8.5	even	2		1280.4.a.ba.1.2		6
16.3	odd	4		40.4.d.a.21.1	✓	12
16.5	even	4		160.4.d.a.81.3		12
16.11	odd	4		40.4.d.a.21.2	yes	12
16.13	even	4		160.4.d.a.81.10		12
48.5	odd	4		1440.4.k.c.721.6		12
48.11	even	4		360.4.k.c.181.11		12
48.29	odd	4		1440.4.k.c.721.12		12
48.35	even	4		360.4.k.c.181.12		12
80.3	even	4		200.4.f.c.149.4		12
80.13	odd	4		800.4.f.b.49.3		12
80.19	odd	4		200.4.d.b.101.12		12
80.27	even	4		200.4.f.c.149.3		12
80.29	even	4		800.4.d.d.401.3		12
80.37	odd	4		800.4.f.b.49.4		12
80.43	even	4		200.4.f.b.149.10		12
80.53	odd	4		800.4.f.c.49.9		12
80.59	odd	4		200.4.d.b.101.11		12
80.67	even	4		200.4.f.b.149.9		12
80.69	even	4		800.4.d.d.401.10		12
80.77	odd	4		800.4.f.c.49.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.d.a.21.1	✓	12	16.3	odd	4
40.4.d.a.21.2	yes	12	16.11	odd	4
160.4.d.a.81.3		12	16.5	even	4
160.4.d.a.81.10		12	16.13	even	4
200.4.d.b.101.11		12	80.59	odd	4
200.4.d.b.101.12		12	80.19	odd	4
200.4.f.b.149.9		12	80.67	even	4
200.4.f.b.149.10		12	80.43	even	4
200.4.f.c.149.3		12	80.27	even	4
200.4.f.c.149.4		12	80.3	even	4
360.4.k.c.181.11		12	48.11	even	4
360.4.k.c.181.12		12	48.35	even	4
800.4.d.d.401.3		12	80.29	even	4
800.4.d.d.401.10		12	80.69	even	4
800.4.f.b.49.3		12	80.13	odd	4
800.4.f.b.49.4		12	80.37	odd	4
800.4.f.c.49.9		12	80.53	odd	4
800.4.f.c.49.10		12	80.77	odd	4
1280.4.a.ba.1.2		6	8.5	even	2
1280.4.a.bb.1.2		6	4.3	odd	2
1280.4.a.bc.1.5		6	8.3	odd	2
1280.4.a.bd.1.5		6	1.1	even	1	trivial
1440.4.k.c.721.6		12	48.5	odd	4
1440.4.k.c.721.12		12	48.29	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1280.a (trivial)

\(f(q)\)	\(=\)	\(q+6.25785 q^{3} +5.00000 q^{5} -34.6280 q^{7} +12.1606 q^{9} +O(q^{10})\)	\(q+6.25785 q^{3} +5.00000 q^{5} -34.6280 q^{7} +12.1606 q^{9} -7.91595 q^{11} -11.0346 q^{13} +31.2892 q^{15} +57.6152 q^{17} +141.133 q^{19} -216.696 q^{21} +129.328 q^{23} +25.0000 q^{25} -92.8625 q^{27} -89.1664 q^{29} +78.3307 q^{31} -49.5368 q^{33} -173.140 q^{35} -249.332 q^{37} -69.0530 q^{39} -91.8705 q^{41} +194.184 q^{43} +60.8031 q^{45} +72.6149 q^{47} +856.096 q^{49} +360.547 q^{51} +456.782 q^{53} -39.5797 q^{55} +883.187 q^{57} +341.098 q^{59} -217.067 q^{61} -421.098 q^{63} -55.1731 q^{65} -529.237 q^{67} +809.314 q^{69} +381.540 q^{71} +876.902 q^{73} +156.446 q^{75} +274.113 q^{77} +203.950 q^{79} -909.456 q^{81} +996.654 q^{83} +288.076 q^{85} -557.989 q^{87} -172.456 q^{89} +382.107 q^{91} +490.182 q^{93} +705.664 q^{95} +1058.49 q^{97} -96.2629 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 30 q^{5} + 14 q^{7} + 54 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 + 30 * q^5 + 14 * q^7 + 54 * q^9	\( 6 q + 6 q^{3} + 30 q^{5} + 14 q^{7} + 54 q^{9} + 44 q^{11} + 30 q^{15} + 152 q^{19} + 4 q^{21} + 302 q^{23} + 150 q^{25} + 216 q^{27} + 132 q^{31} - 116 q^{33} + 70 q^{35} - 68 q^{37} + 300 q^{39} - 20 q^{41} + 602 q^{43} + 270 q^{45} + 470 q^{47} + 654 q^{49} + 612 q^{51} + 528 q^{53} + 220 q^{55} + 340 q^{57} + 472 q^{59} - 476 q^{61} + 650 q^{63} + 1206 q^{67} - 980 q^{69} - 796 q^{71} - 216 q^{73} + 150 q^{75} + 412 q^{77} - 1008 q^{79} + 1254 q^{81} + 1778 q^{83} - 984 q^{87} + 212 q^{89} + 3652 q^{91} - 1392 q^{93} + 760 q^{95} - 792 q^{97} + 5516 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 + 30 * q^5 + 14 * q^7 + 54 * q^9 + 44 * q^11 + 30 * q^15 + 152 * q^19 + 4 * q^21 + 302 * q^23 + 150 * q^25 + 216 * q^27 + 132 * q^31 - 116 * q^33 + 70 * q^35 - 68 * q^37 + 300 * q^39 - 20 * q^41 + 602 * q^43 + 270 * q^45 + 470 * q^47 + 654 * q^49 + 612 * q^51 + 528 * q^53 + 220 * q^55 + 340 * q^57 + 472 * q^59 - 476 * q^61 + 650 * q^63 + 1206 * q^67 - 980 * q^69 - 796 * q^71 - 216 * q^73 + 150 * q^75 + 412 * q^77 - 1008 * q^79 + 1254 * q^81 + 1778 * q^83 - 984 * q^87 + 212 * q^89 + 3652 * q^91 - 1392 * q^93 + 760 * q^95 - 792 * q^97 + 5516 * q^99

Introduction

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