LMFDB - Embedded newform 460.4.a.b.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [460,4,Mod(1,460)] 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("460.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(460, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 460 = 2^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	460.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$27.1408786026$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 80x^{3} + 121x^{2} + 1212x + 1044 $ x^5 - 2x^4 - 80x^3 + 121x^2 + 1212x + 1044
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.04116$ of defining polynomial
Character	$\chi$	$=$	460.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.04116 q^{3} -5.00000 q^{5} +21.6113 q^{7} -22.8337 q^{9} -30.3495 q^{11} +69.6786 q^{13} +10.2058 q^{15} -43.4506 q^{17} +14.3892 q^{19} -44.1120 q^{21} -23.0000 q^{23} +25.0000 q^{25} +101.718 q^{27} -158.105 q^{29} -46.4782 q^{31} +61.9480 q^{33} -108.056 q^{35} +39.6849 q^{37} -142.225 q^{39} -201.771 q^{41} +17.8927 q^{43} +114.168 q^{45} -0.702668 q^{47} +124.048 q^{49} +88.6894 q^{51} -661.161 q^{53} +151.748 q^{55} -29.3705 q^{57} -732.679 q^{59} -875.997 q^{61} -493.465 q^{63} -348.393 q^{65} +746.803 q^{67} +46.9466 q^{69} -275.246 q^{71} -499.633 q^{73} -51.0289 q^{75} -655.892 q^{77} -797.970 q^{79} +408.887 q^{81} -1150.10 q^{83} +217.253 q^{85} +322.716 q^{87} +1166.89 q^{89} +1505.85 q^{91} +94.8693 q^{93} -71.9458 q^{95} +403.209 q^{97} +692.991 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{3} - 25 q^{5} + 8 q^{7} + 30 q^{9} + 7 q^{11} + 5 q^{13} + 15 q^{15} - 24 q^{17} - 13 q^{19} - 115 q^{23} + 125 q^{25} - 204 q^{27} - 253 q^{29} - 98 q^{31} - 473 q^{33} - 40 q^{35} - 435 q^{37}+ \cdots + 111 q^{99}+O(q^{100}) $ 5 * q - 3 * q^3 - 25 * q^5 + 8 * q^7 + 30 * q^9 + 7 * q^11 + 5 * q^13 + 15 * q^15 - 24 * q^17 - 13 * q^19 - 115 * q^23 + 125 * q^25 - 204 * q^27 - 253 * q^29 - 98 * q^31 - 473 * q^33 - 40 * q^35 - 435 * q^37 - 410 * q^39 - 774 * q^41 - 498 * q^43 - 150 * q^45 - 572 * q^47 - 683 * q^49 - 657 * q^51 - 665 * q^53 - 35 * q^55 - 932 * q^57 - 763 * q^59 + 337 * q^61 - 1527 * q^63 - 25 * q^65 - 305 * q^67 + 69 * q^69 - 1504 * q^71 - 1304 * q^73 - 75 * q^75 + 182 * q^77 + 626 * q^79 - 959 * q^81 - 1703 * q^83 + 120 * q^85 - 1354 * q^87 + 646 * q^89 - 767 * q^91 - 452 * q^93 + 65 * q^95 - 233 * q^97 + 111 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.04116	−0.392821	−0.196410	−	0.980522i	$-0.562928\pi$
$3$	−2.04116	−0.392821	−0.196410	+	0.980522i	$0.562928\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	21.6113	1.16690	0.583450	−	0.812149i	$-0.301702\pi$
$7$	21.6113	1.16690	0.583450	+	0.812149i	$0.301702\pi$
$8$	0	0
$9$	−22.8337	−0.845692
$10$	0	0
$11$	−30.3495	−0.831884	−0.415942	−	0.909391i	$-0.636548\pi$
$11$	−30.3495	−0.831884	−0.415942	+	0.909391i	$0.636548\pi$
$12$	0	0
$13$	69.6786	1.48657	0.743284	−	0.668976i	$-0.233267\pi$
$13$	69.6786	1.48657	0.743284	+	0.668976i	$0.233267\pi$
$14$	0	0
$15$	10.2058	0.175675
$16$	0	0
$17$	−43.4506	−0.619901	−0.309950	−	0.950753i	$-0.600312\pi$
$17$	−43.4506	−0.619901	−0.309950	+	0.950753i	$0.600312\pi$
$18$	0	0
$19$	14.3892	0.173742	0.0868710	−	0.996220i	$-0.472313\pi$
$19$	14.3892	0.173742	0.0868710	+	0.996220i	$0.472313\pi$
$20$	0	0
$21$	−44.1120	−0.458382
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	101.718	0.725026
$28$	0	0
$29$	−158.105	−1.01239	−0.506195	−	0.862419i	$-0.668948\pi$
$29$	−158.105	−1.01239	−0.506195	+	0.862419i	$0.668948\pi$
$30$	0	0
$31$	−46.4782	−0.269282	−0.134641	−	0.990894i	$-0.542988\pi$
$31$	−46.4782	−0.269282	−0.134641	+	0.990894i	$0.542988\pi$
$32$	0	0
$33$	61.9480	0.326781
$34$	0	0
$35$	−108.056	−0.521853
$36$	0	0
$37$	39.6849	0.176328	0.0881642	−	0.996106i	$-0.471900\pi$
$37$	39.6849	0.176328	0.0881642	+	0.996106i	$0.471900\pi$
$38$	0	0
$39$	−142.225	−0.583954
$40$	0	0
$41$	−201.771	−0.768571	−0.384285	−	0.923214i	$-0.625552\pi$
$41$	−201.771	−0.768571	−0.384285	+	0.923214i	$0.625552\pi$
$42$	0	0
$43$	17.8927	0.0634562	0.0317281	−	0.999497i	$-0.489899\pi$
$43$	17.8927	0.0634562	0.0317281	+	0.999497i	$0.489899\pi$
$44$	0	0
$45$	114.168	0.378205
$46$	0	0
$47$	−0.702668	−0.00218074	−0.00109037	−	0.999999i	$-0.500347\pi$
$47$	−0.702668	−0.00218074	−0.00109037	+	0.999999i	$0.500347\pi$
$48$	0	0
$49$	124.048	0.361655
$50$	0	0
$51$	88.6894	0.243510
$52$	0	0
$53$	−661.161	−1.71354	−0.856769	−	0.515701i	$-0.827532\pi$
$53$	−661.161	−1.71354	−0.856769	+	0.515701i	$0.827532\pi$
$54$	0	0
$55$	151.748	0.372030
$56$	0	0
$57$	−29.3705	−0.0682495
$58$	0	0
$59$	−732.679	−1.61672	−0.808361	−	0.588686i	$-0.799645\pi$
$59$	−732.679	−1.61672	−0.808361	+	0.588686i	$0.799645\pi$
$60$	0	0
$61$	−875.997	−1.83869	−0.919344	−	0.393454i	$-0.871280\pi$
$61$	−875.997	−1.83869	−0.919344	+	0.393454i	$0.871280\pi$
$62$	0	0
$63$	−493.465	−0.986838
$64$	0	0
$65$	−348.393	−0.664813
$66$	0	0
$67$	746.803	1.36174	0.680870	−	0.732404i	$-0.261602\pi$
$67$	746.803	1.36174	0.680870	+	0.732404i	$0.261602\pi$
$68$	0	0
$69$	46.9466	0.0819087
$70$	0	0
$71$	−275.246	−0.460079	−0.230040	−	0.973181i	$-0.573886\pi$
$71$	−275.246	−0.460079	−0.230040	+	0.973181i	$0.573886\pi$
$72$	0	0
$73$	−499.633	−0.801063	−0.400532	−	0.916283i	$-0.631175\pi$
$73$	−499.633	−0.801063	−0.400532	+	0.916283i	$0.631175\pi$
$74$	0	0
$75$	−51.0289	−0.0785641
$76$	0	0
$77$	−655.892	−0.970725
$78$	0	0
$79$	−797.970	−1.13644	−0.568219	−	0.822877i	$-0.692367\pi$
$79$	−797.970	−1.13644	−0.568219	+	0.822877i	$0.692367\pi$
$80$	0	0
$81$	408.887	0.560887
$82$	0	0
$83$	−1150.10	−1.52097	−0.760483	−	0.649358i	$-0.775038\pi$
$83$	−1150.10	−1.52097	−0.760483	+	0.649358i	$0.775038\pi$
$84$	0	0
$85$	217.253	0.277228
$86$	0	0
$87$	322.716	0.397687
$88$	0	0
$89$	1166.89	1.38978	0.694890	−	0.719116i	$-0.255453\pi$
$89$	1166.89	1.38978	0.694890	+	0.719116i	$0.255453\pi$
$90$	0	0
$91$	1505.85	1.73468
$92$	0	0
$93$	94.8693	0.105779
$94$	0	0
$95$	−71.9458	−0.0776998
$96$	0	0
$97$	403.209	0.422058	0.211029	−	0.977480i	$-0.432318\pi$
$97$	403.209	0.422058	0.211029	+	0.977480i	$0.432318\pi$
$98$	0	0
$99$	692.991	0.703517
$100$	0	0
$101$	484.319	0.477144	0.238572	−	0.971125i	$-0.423321\pi$
$101$	484.319	0.477144	0.238572	+	0.971125i	$0.423321\pi$
$102$	0	0
$103$	−2064.69	−1.97514	−0.987572	−	0.157167i	$-0.949764\pi$
$103$	−2064.69	−1.97514	−0.987572	+	0.157167i	$0.949764\pi$
$104$	0	0
$105$	220.560	0.204995
$106$	0	0
$107$	−1436.43	−1.29780	−0.648902	−	0.760872i	$-0.724772\pi$
$107$	−1436.43	−1.29780	−0.648902	+	0.760872i	$0.724772\pi$
$108$	0	0
$109$	1870.62	1.64379	0.821895	−	0.569639i	$-0.192917\pi$
$109$	1870.62	1.64379	0.821895	+	0.569639i	$0.192917\pi$
$110$	0	0
$111$	−81.0030	−0.0692654
$112$	0	0
$113$	1606.67	1.33755	0.668775	−	0.743465i	$-0.266819\pi$
$113$	1606.67	1.33755	0.668775	+	0.743465i	$0.266819\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	−1591.02	−1.25718
$118$	0	0
$119$	−939.023	−0.723362
$120$	0	0
$121$	−409.908	−0.307970
$122$	0	0
$123$	411.847	0.301910
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−2031.66	−1.41953	−0.709766	−	0.704438i	$-0.751199\pi$
$127$	−2031.66	−1.41953	−0.709766	+	0.704438i	$0.751199\pi$
$128$	0	0
$129$	−36.5219	−0.0249269
$130$	0	0
$131$	−434.691	−0.289917	−0.144958	−	0.989438i	$-0.546305\pi$
$131$	−434.691	−0.289917	−0.144958	+	0.989438i	$0.546305\pi$
$132$	0	0
$133$	310.968	0.202740
$134$	0	0
$135$	−508.591	−0.324241
$136$	0	0
$137$	939.729	0.586033	0.293016	−	0.956107i	$-0.405341\pi$
$137$	939.729	0.586033	0.293016	+	0.956107i	$0.405341\pi$
$138$	0	0
$139$	−124.609	−0.0760376	−0.0380188	−	0.999277i	$-0.512105\pi$
$139$	−124.609	−0.0760376	−0.0380188	+	0.999277i	$0.512105\pi$
$140$	0	0
$141$	1.43426	0.000856639 0
$142$	0	0
$143$	−2114.71	−1.23665
$144$	0	0
$145$	790.523	0.452754
$146$	0	0
$147$	−253.201	−0.142066
$148$	0	0
$149$	1375.14	0.756079	0.378039	−	0.925789i	$-0.376598\pi$
$149$	1375.14	0.756079	0.378039	+	0.925789i	$0.376598\pi$
$150$	0	0
$151$	−1196.51	−0.644838	−0.322419	−	0.946597i	$-0.604496\pi$
$151$	−1196.51	−0.644838	−0.322419	+	0.946597i	$0.604496\pi$
$152$	0	0
$153$	992.137	0.524245
$154$	0	0
$155$	232.391	0.120426
$156$	0	0
$157$	−212.340	−0.107940	−0.0539699	−	0.998543i	$-0.517188\pi$
$157$	−212.340	−0.107940	−0.0539699	+	0.998543i	$0.517188\pi$
$158$	0	0
$159$	1349.53	0.673113
$160$	0	0
$161$	−497.060	−0.243315
$162$	0	0
$163$	1390.16	0.668013	0.334006	−	0.942571i	$-0.391599\pi$
$163$	1390.16	0.668013	0.334006	+	0.942571i	$0.391599\pi$
$164$	0	0
$165$	−309.740	−0.146141
$166$	0	0
$167$	362.349	0.167901	0.0839503	−	0.996470i	$-0.473246\pi$
$167$	362.349	0.167901	0.0839503	+	0.996470i	$0.473246\pi$
$168$	0	0
$169$	2658.11	1.20988
$170$	0	0
$171$	−328.557	−0.146932
$172$	0	0
$173$	986.596	0.433581	0.216790	−	0.976218i	$-0.430441\pi$
$173$	986.596	0.433581	0.216790	+	0.976218i	$0.430441\pi$
$174$	0	0
$175$	540.282	0.233380
$176$	0	0
$177$	1495.51	0.635082
$178$	0	0
$179$	3977.44	1.66083	0.830413	−	0.557149i	$-0.188105\pi$
$179$	3977.44	1.66083	0.830413	+	0.557149i	$0.188105\pi$
$180$	0	0
$181$	2512.70	1.03186	0.515932	−	0.856630i	$-0.327446\pi$
$181$	2512.70	1.03186	0.515932	+	0.856630i	$0.327446\pi$
$182$	0	0
$183$	1788.05	0.722275
$184$	0	0
$185$	−198.424	−0.0788565
$186$	0	0
$187$	1318.70	0.515685
$188$	0	0
$189$	2198.26	0.846032
$190$	0	0
$191$	1977.49	0.749142	0.374571	−	0.927198i	$-0.377790\pi$
$191$	1977.49	0.749142	0.374571	+	0.927198i	$0.377790\pi$
$192$	0	0
$193$	−1241.30	−0.462957	−0.231478	−	0.972840i	$-0.574356\pi$
$193$	−1241.30	−0.462957	−0.231478	+	0.972840i	$0.574356\pi$
$194$	0	0
$195$	711.125	0.261152
$196$	0	0
$197$	1969.91	0.712436	0.356218	−	0.934403i	$-0.384066\pi$
$197$	1969.91	0.712436	0.356218	+	0.934403i	$0.384066\pi$
$198$	0	0
$199$	−4053.09	−1.44380	−0.721899	−	0.691998i	$-0.756731\pi$
$199$	−4053.09	−1.44380	−0.721899	+	0.691998i	$0.756731\pi$
$200$	0	0
$201$	−1524.34	−0.534919
$202$	0	0
$203$	−3416.85	−1.18136
$204$	0	0
$205$	1008.86	0.343715
$206$	0	0
$207$	525.175	0.176339
$208$	0	0
$209$	−436.704	−0.144533
$210$	0	0
$211$	374.110	0.122061	0.0610303	−	0.998136i	$-0.480561\pi$
$211$	374.110	0.122061	0.0610303	+	0.998136i	$0.480561\pi$
$212$	0	0
$213$	561.819	0.180729
$214$	0	0
$215$	−89.4637	−0.0283785
$216$	0	0
$217$	−1004.45	−0.314225
$218$	0	0
$219$	1019.83	0.314674
$220$	0	0
$221$	−3027.58	−0.921524
$222$	0	0
$223$	3226.90	0.969010	0.484505	−	0.874788i	$-0.339000\pi$
$223$	3226.90	0.969010	0.484505	+	0.874788i	$0.339000\pi$
$224$	0	0
$225$	−570.842	−0.169138
$226$	0	0
$227$	3020.61	0.883194	0.441597	−	0.897213i	$-0.354412\pi$
$227$	3020.61	0.883194	0.441597	+	0.897213i	$0.354412\pi$
$228$	0	0
$229$	934.761	0.269741	0.134871	−	0.990863i	$-0.456938\pi$
$229$	934.761	0.269741	0.134871	+	0.990863i	$0.456938\pi$
$230$	0	0
$231$	1338.78	0.381321
$232$	0	0
$233$	−3916.88	−1.10130	−0.550651	−	0.834736i	$-0.685620\pi$
$233$	−3916.88	−1.10130	−0.550651	+	0.834736i	$0.685620\pi$
$234$	0	0
$235$	3.51334	0.000975256 0
$236$	0	0
$237$	1628.78	0.446416
$238$	0	0
$239$	3200.06	0.866086	0.433043	−	0.901373i	$-0.357440\pi$
$239$	3200.06	0.866086	0.433043	+	0.901373i	$0.357440\pi$
$240$	0	0
$241$	−3774.24	−1.00880	−0.504399	−	0.863471i	$-0.668286\pi$
$241$	−3774.24	−1.00880	−0.504399	+	0.863471i	$0.668286\pi$
$242$	0	0
$243$	−3581.00	−0.945354
$244$	0	0
$245$	−620.238	−0.161737
$246$	0	0
$247$	1002.62	0.258279
$248$	0	0
$249$	2347.54	0.597467
$250$	0	0
$251$	3342.01	0.840420	0.420210	−	0.907427i	$-0.361956\pi$
$251$	3342.01	0.840420	0.420210	+	0.907427i	$0.361956\pi$
$252$	0	0
$253$	698.039	0.173460
$254$	0	0
$255$	−443.447	−0.108901
$256$	0	0
$257$	−2899.20	−0.703685	−0.351842	−	0.936059i	$-0.614445\pi$
$257$	−2899.20	−0.703685	−0.351842	+	0.936059i	$0.614445\pi$
$258$	0	0
$259$	857.641	0.205758
$260$	0	0
$261$	3610.11	0.856170
$262$	0	0
$263$	−6001.83	−1.40718	−0.703591	−	0.710605i	$-0.748421\pi$
$263$	−6001.83	−1.40718	−0.703591	+	0.710605i	$0.748421\pi$
$264$	0	0
$265$	3305.81	0.766317
$266$	0	0
$267$	−2381.81	−0.545934
$268$	0	0
$269$	8308.29	1.88314	0.941571	−	0.336813i	$-0.109349\pi$
$269$	8308.29	1.88314	0.941571	+	0.336813i	$0.109349\pi$
$270$	0	0
$271$	−2395.28	−0.536911	−0.268456	−	0.963292i	$-0.586513\pi$
$271$	−2395.28	−0.536911	−0.268456	+	0.963292i	$0.586513\pi$
$272$	0	0
$273$	−3073.66	−0.681416
$274$	0	0
$275$	−758.738	−0.166377
$276$	0	0
$277$	7466.49	1.61956	0.809780	−	0.586733i	$-0.199586\pi$
$277$	7466.49	1.61956	0.809780	+	0.586733i	$0.199586\pi$
$278$	0	0
$279$	1061.27	0.227729
$280$	0	0
$281$	−5570.87	−1.18267	−0.591335	−	0.806426i	$-0.701399\pi$
$281$	−5570.87	−1.18267	−0.591335	+	0.806426i	$0.701399\pi$
$282$	0	0
$283$	−5273.48	−1.10769	−0.553845	−	0.832620i	$-0.686840\pi$
$283$	−5273.48	−1.10769	−0.553845	+	0.832620i	$0.686840\pi$
$284$	0	0
$285$	146.853	0.0305221
$286$	0	0
$287$	−4360.54	−0.896845
$288$	0	0
$289$	−3025.05	−0.615723
$290$	0	0
$291$	−823.012	−0.165793
$292$	0	0
$293$	−511.514	−0.101990	−0.0509948	−	0.998699i	$-0.516239\pi$
$293$	−511.514	−0.101990	−0.0509948	+	0.998699i	$0.516239\pi$
$294$	0	0
$295$	3663.39	0.723020
$296$	0	0
$297$	−3087.10	−0.603137
$298$	0	0
$299$	−1602.61	−0.309971
$300$	0	0
$301$	386.685	0.0740471
$302$	0	0
$303$	−988.570	−0.187432
$304$	0	0
$305$	4379.99	0.822286
$306$	0	0
$307$	4157.46	0.772894	0.386447	−	0.922312i	$-0.373702\pi$
$307$	4157.46	0.772894	0.386447	+	0.922312i	$0.373702\pi$
$308$	0	0
$309$	4214.35	0.775877
$310$	0	0
$311$	−3408.45	−0.621465	−0.310733	−	0.950497i	$-0.600574\pi$
$311$	−3408.45	−0.621465	−0.310733	+	0.950497i	$0.600574\pi$
$312$	0	0
$313$	4516.35	0.815589	0.407794	−	0.913074i	$-0.366298\pi$
$313$	4516.35	0.815589	0.407794	+	0.913074i	$0.366298\pi$
$314$	0	0
$315$	2467.33	0.441327
$316$	0	0
$317$	2693.62	0.477251	0.238626	−	0.971112i	$-0.423303\pi$
$317$	2693.62	0.477251	0.238626	+	0.971112i	$0.423303\pi$
$318$	0	0
$319$	4798.40	0.842190
$320$	0	0
$321$	2931.98	0.509804
$322$	0	0
$323$	−625.217	−0.107703
$324$	0	0
$325$	1741.97	0.297314
$326$	0	0
$327$	−3818.23	−0.645715
$328$	0	0
$329$	−15.1856	−0.00254470
$330$	0	0
$331$	−11024.4	−1.83068	−0.915338	−	0.402687i	$-0.868076\pi$
$331$	−11024.4	−1.83068	−0.915338	+	0.402687i	$0.868076\pi$
$332$	0	0
$333$	−906.152	−0.149120
$334$	0	0
$335$	−3734.02	−0.608989
$336$	0	0
$337$	5218.80	0.843579	0.421790	−	0.906694i	$-0.361402\pi$
$337$	5218.80	0.843579	0.421790	+	0.906694i	$0.361402\pi$
$338$	0	0
$339$	−3279.47	−0.525417
$340$	0	0
$341$	1410.59	0.224011
$342$	0	0
$343$	−4731.84	−0.744885
$344$	0	0
$345$	−234.733	−0.0366307
$346$	0	0
$347$	93.8512	0.0145193	0.00725965	−	0.999974i	$-0.497689\pi$
$347$	93.8512	0.0145193	0.00725965	+	0.999974i	$0.497689\pi$
$348$	0	0
$349$	4540.27	0.696376	0.348188	−	0.937425i	$-0.386797\pi$
$349$	4540.27	0.696376	0.348188	+	0.937425i	$0.386797\pi$
$350$	0	0
$351$	7087.59	1.07780
$352$	0	0
$353$	4287.12	0.646403	0.323202	−	0.946330i	$-0.395241\pi$
$353$	4287.12	0.646403	0.323202	+	0.946330i	$0.395241\pi$
$354$	0	0
$355$	1376.23	0.205754
$356$	0	0
$357$	1916.69	0.284151
$358$	0	0
$359$	−176.418	−0.0259359	−0.0129680	−	0.999916i	$-0.504128\pi$
$359$	−176.418	−0.0259359	−0.0129680	+	0.999916i	$0.504128\pi$
$360$	0	0
$361$	−6651.95	−0.969814
$362$	0	0
$363$	836.686	0.120977
$364$	0	0
$365$	2498.17	0.358246
$366$	0	0
$367$	−7745.87	−1.10172	−0.550860	−	0.834598i	$-0.685700\pi$
$367$	−7745.87	−1.10172	−0.550860	+	0.834598i	$0.685700\pi$
$368$	0	0
$369$	4607.18	0.649974
$370$	0	0
$371$	−14288.5	−1.99953
$372$	0	0
$373$	−8268.02	−1.14773	−0.573863	−	0.818951i	$-0.694556\pi$
$373$	−8268.02	−1.14773	−0.573863	+	0.818951i	$0.694556\pi$
$374$	0	0
$375$	255.144	0.0351349
$376$	0	0
$377$	−11016.5	−1.50499
$378$	0	0
$379$	7229.18	0.979783	0.489892	−	0.871783i	$-0.337036\pi$
$379$	7229.18	0.979783	0.489892	+	0.871783i	$0.337036\pi$
$380$	0	0
$381$	4146.93	0.557621
$382$	0	0
$383$	−519.390	−0.0692940	−0.0346470	−	0.999400i	$-0.511031\pi$
$383$	−519.390	−0.0692940	−0.0346470	+	0.999400i	$0.511031\pi$
$384$	0	0
$385$	3279.46	0.434121
$386$	0	0
$387$	−408.557	−0.0536644
$388$	0	0
$389$	−10746.4	−1.40068	−0.700341	−	0.713809i	$-0.746969\pi$
$389$	−10746.4	−1.40068	−0.700341	+	0.713809i	$0.746969\pi$
$390$	0	0
$391$	999.363	0.129258
$392$	0	0
$393$	887.271	0.113885
$394$	0	0
$395$	3989.85	0.508231
$396$	0	0
$397$	−11738.8	−1.48401	−0.742004	−	0.670395i	$-0.766125\pi$
$397$	−11738.8	−1.48401	−0.742004	+	0.670395i	$0.766125\pi$
$398$	0	0
$399$	−634.734	−0.0796403
$400$	0	0
$401$	−94.4786	−0.0117657	−0.00588284	−	0.999983i	$-0.501873\pi$
$401$	−94.4786	−0.0117657	−0.00588284	+	0.999983i	$0.501873\pi$
$402$	0	0
$403$	−3238.54	−0.400306
$404$	0	0
$405$	−2044.43	−0.250836
$406$	0	0
$407$	−1204.42	−0.146685
$408$	0	0
$409$	−13712.5	−1.65780	−0.828901	−	0.559396i	$-0.811033\pi$
$409$	−13712.5	−1.65780	−0.828901	+	0.559396i	$0.811033\pi$
$410$	0	0
$411$	−1918.13	−0.230206
$412$	0	0
$413$	−15834.1	−1.88655
$414$	0	0
$415$	5750.51	0.680197
$416$	0	0
$417$	254.347	0.0298691
$418$	0	0
$419$	4392.60	0.512154	0.256077	−	0.966656i	$-0.417570\pi$
$419$	4392.60	0.512154	0.256077	+	0.966656i	$0.417570\pi$
$420$	0	0
$421$	112.287	0.0129989	0.00649943	−	0.999979i	$-0.497931\pi$
$421$	112.287	0.0129989	0.00649943	+	0.999979i	$0.497931\pi$
$422$	0	0
$423$	16.0445	0.00184423
$424$	0	0
$425$	−1086.26	−0.123980
$426$	0	0
$427$	−18931.4	−2.14556
$428$	0	0
$429$	4316.46	0.485782
$430$	0	0
$431$	−7035.30	−0.786261	−0.393131	−	0.919483i	$-0.628608\pi$
$431$	−7035.30	−0.786261	−0.393131	+	0.919483i	$0.628608\pi$
$432$	0	0
$433$	12046.0	1.33693	0.668467	−	0.743742i	$-0.266951\pi$
$433$	12046.0	1.33693	0.668467	+	0.743742i	$0.266951\pi$
$434$	0	0
$435$	−1613.58	−0.177851
$436$	0	0
$437$	−330.951	−0.0362277
$438$	0	0
$439$	−2089.02	−0.227115	−0.113557	−	0.993531i	$-0.536225\pi$
$439$	−2089.02	−0.227115	−0.113557	+	0.993531i	$0.536225\pi$
$440$	0	0
$441$	−2832.47	−0.305849
$442$	0	0
$443$	−7436.42	−0.797551	−0.398776	−	0.917049i	$-0.630565\pi$
$443$	−7436.42	−0.797551	−0.398776	+	0.917049i	$0.630565\pi$
$444$	0	0
$445$	−5834.46	−0.621528
$446$	0	0
$447$	−2806.87	−0.297003
$448$	0	0
$449$	1796.33	0.188806	0.0944032	−	0.995534i	$-0.469906\pi$
$449$	1796.33	0.188806	0.0944032	+	0.995534i	$0.469906\pi$
$450$	0	0
$451$	6123.66	0.639361
$452$	0	0
$453$	2442.26	0.253306
$454$	0	0
$455$	−7529.23	−0.775770
$456$	0	0
$457$	−3232.73	−0.330899	−0.165450	−	0.986218i	$-0.552907\pi$
$457$	−3232.73	−0.330899	−0.165450	+	0.986218i	$0.552907\pi$
$458$	0	0
$459$	−4419.72	−0.449444
$460$	0	0
$461$	−10750.2	−1.08609	−0.543044	−	0.839704i	$-0.682728\pi$
$461$	−10750.2	−1.08609	−0.543044	+	0.839704i	$0.682728\pi$
$462$	0	0
$463$	6598.81	0.662360	0.331180	−	0.943568i	$-0.392553\pi$
$463$	6598.81	0.662360	0.331180	+	0.943568i	$0.392553\pi$
$464$	0	0
$465$	−474.346	−0.0473060
$466$	0	0
$467$	8153.38	0.807909	0.403954	−	0.914779i	$-0.367635\pi$
$467$	8153.38	0.807909	0.403954	+	0.914779i	$0.367635\pi$
$468$	0	0
$469$	16139.4	1.58901
$470$	0	0
$471$	433.418	0.0424010
$472$	0	0
$473$	−543.036	−0.0527882
$474$	0	0
$475$	359.729	0.0347484
$476$	0	0
$477$	15096.8	1.44913
$478$	0	0
$479$	15056.7	1.43624	0.718120	−	0.695919i	$-0.245003\pi$
$479$	15056.7	1.43624	0.718120	+	0.695919i	$0.245003\pi$
$480$	0	0
$481$	2765.19	0.262124
$482$	0	0
$483$	1014.58	0.0955793
$484$	0	0
$485$	−2016.04	−0.188750
$486$	0	0
$487$	2098.90	0.195298	0.0976491	−	0.995221i	$-0.468868\pi$
$487$	2098.90	0.195298	0.0976491	+	0.995221i	$0.468868\pi$
$488$	0	0
$489$	−2837.54	−0.262409
$490$	0	0
$491$	−2245.71	−0.206410	−0.103205	−	0.994660i	$-0.532910\pi$
$491$	−2245.71	−0.206410	−0.103205	+	0.994660i	$0.532910\pi$
$492$	0	0
$493$	6869.74	0.627581
$494$	0	0
$495$	−3464.95	−0.314622
$496$	0	0
$497$	−5948.41	−0.536866
$498$	0	0
$499$	3216.32	0.288542	0.144271	−	0.989538i	$-0.453916\pi$
$499$	3216.32	0.288542	0.144271	+	0.989538i	$0.453916\pi$
$500$	0	0
$501$	−739.611	−0.0659548
$502$	0	0
$503$	−15459.5	−1.37039	−0.685193	−	0.728361i	$-0.740282\pi$
$503$	−15459.5	−1.37039	−0.685193	+	0.728361i	$0.740282\pi$
$504$	0	0
$505$	−2421.59	−0.213385
$506$	0	0
$507$	−5425.62	−0.475267
$508$	0	0
$509$	12581.8	1.09564	0.547818	−	0.836597i	$-0.315459\pi$
$509$	12581.8	1.09564	0.547818	+	0.836597i	$0.315459\pi$
$510$	0	0
$511$	−10797.7	−0.934761
$512$	0	0
$513$	1463.64	0.125967
$514$	0	0
$515$	10323.4	0.883311
$516$	0	0
$517$	21.3256	0.00181412
$518$	0	0
$519$	−2013.80	−0.170320
$520$	0	0
$521$	2681.76	0.225509	0.112754	−	0.993623i	$-0.464033\pi$
$521$	2681.76	0.225509	0.112754	+	0.993623i	$0.464033\pi$
$522$	0	0
$523$	−9009.09	−0.753232	−0.376616	−	0.926370i	$-0.622912\pi$
$523$	−9009.09	−0.753232	−0.376616	+	0.926370i	$0.622912\pi$
$524$	0	0
$525$	−1102.80	−0.0916764
$526$	0	0
$527$	2019.50	0.166928
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	16729.8	1.36725
$532$	0	0
$533$	−14059.2	−1.14253
$534$	0	0
$535$	7182.16	0.580396
$536$	0	0
$537$	−8118.57	−0.652406
$538$	0	0
$539$	−3764.78	−0.300855
$540$	0	0
$541$	22789.6	1.81109	0.905546	−	0.424249i	$-0.139462\pi$
$541$	22789.6	1.81109	0.905546	+	0.424249i	$0.139462\pi$
$542$	0	0
$543$	−5128.80	−0.405337
$544$	0	0
$545$	−9353.11	−0.735125
$546$	0	0
$547$	20709.0	1.61874	0.809371	−	0.587297i	$-0.199808\pi$
$547$	20709.0	1.61874	0.809371	+	0.587297i	$0.199808\pi$
$548$	0	0
$549$	20002.3	1.55496
$550$	0	0
$551$	−2274.99	−0.175895
$552$	0	0
$553$	−17245.2	−1.32611
$554$	0	0
$555$	405.015	0.0309764
$556$	0	0
$557$	586.443	0.0446111	0.0223055	−	0.999751i	$-0.492899\pi$
$557$	586.443	0.0446111	0.0223055	+	0.999751i	$0.492899\pi$
$558$	0	0
$559$	1246.74	0.0943320
$560$	0	0
$561$	−2691.68	−0.202572
$562$	0	0
$563$	−4988.10	−0.373399	−0.186699	−	0.982417i	$-0.559779\pi$
$563$	−4988.10	−0.373399	−0.186699	+	0.982417i	$0.559779\pi$
$564$	0	0
$565$	−8033.37	−0.598170
$566$	0	0
$567$	8836.57	0.654499
$568$	0	0
$569$	5548.19	0.408773	0.204387	−	0.978890i	$-0.434480\pi$
$569$	5548.19	0.408773	0.204387	+	0.978890i	$0.434480\pi$
$570$	0	0
$571$	−2684.93	−0.196779	−0.0983893	−	0.995148i	$-0.531369\pi$
$571$	−2684.93	−0.196779	−0.0983893	+	0.995148i	$0.531369\pi$
$572$	0	0
$573$	−4036.36	−0.294278
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	1462.86	0.105546	0.0527728	−	0.998607i	$-0.483194\pi$
$577$	1462.86	0.105546	0.0527728	+	0.998607i	$0.483194\pi$
$578$	0	0
$579$	2533.68	0.181859
$580$	0	0
$581$	−24855.2	−1.77481
$582$	0	0
$583$	20065.9	1.42546
$584$	0	0
$585$	7955.10	0.562227
$586$	0	0
$587$	−11020.7	−0.774912	−0.387456	−	0.921888i	$-0.626646\pi$
$587$	−11020.7	−0.774912	−0.387456	+	0.921888i	$0.626646\pi$
$588$	0	0
$589$	−668.782	−0.0467856
$590$	0	0
$591$	−4020.88	−0.279860
$592$	0	0
$593$	7341.16	0.508373	0.254186	−	0.967155i	$-0.418192\pi$
$593$	7341.16	0.508373	0.254186	+	0.967155i	$0.418192\pi$
$594$	0	0
$595$	4695.11	0.323497
$596$	0	0
$597$	8272.98	0.567154
$598$	0	0
$599$	−12642.5	−0.862366	−0.431183	−	0.902264i	$-0.641904\pi$
$599$	−12642.5	−0.862366	−0.431183	+	0.902264i	$0.641904\pi$
$600$	0	0
$601$	7426.60	0.504056	0.252028	−	0.967720i	$-0.418903\pi$
$601$	7426.60	0.504056	0.252028	+	0.967720i	$0.418903\pi$
$602$	0	0
$603$	−17052.3	−1.15161
$604$	0	0
$605$	2049.54	0.137728
$606$	0	0
$607$	−7506.30	−0.501929	−0.250965	−	0.967996i	$-0.580748\pi$
$607$	−7506.30	−0.501929	−0.250965	+	0.967996i	$0.580748\pi$
$608$	0	0
$609$	6974.31	0.464061
$610$	0	0
$611$	−48.9610	−0.00324182
$612$	0	0
$613$	23467.7	1.54625	0.773124	−	0.634254i	$-0.218693\pi$
$613$	23467.7	1.54625	0.773124	+	0.634254i	$0.218693\pi$
$614$	0	0
$615$	−2059.23	−0.135018
$616$	0	0
$617$	1436.56	0.0937337	0.0468669	−	0.998901i	$-0.485076\pi$
$617$	1436.56	0.0937337	0.0468669	+	0.998901i	$0.485076\pi$
$618$	0	0
$619$	12664.4	0.822332	0.411166	−	0.911561i	$-0.365122\pi$
$619$	12664.4	0.822332	0.411166	+	0.911561i	$0.365122\pi$
$620$	0	0
$621$	−2339.52	−0.151178
$622$	0	0
$623$	25218.1	1.62173
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	891.380	0.0567756
$628$	0	0
$629$	−1724.33	−0.109306
$630$	0	0
$631$	5383.68	0.339653	0.169826	−	0.985474i	$-0.445679\pi$
$631$	5383.68	0.339653	0.169826	+	0.985474i	$0.445679\pi$
$632$	0	0
$633$	−763.616	−0.0479479
$634$	0	0
$635$	10158.3	0.634834
$636$	0	0
$637$	8643.47	0.537625
$638$	0	0
$639$	6284.87	0.389085
$640$	0	0
$641$	31245.1	1.92529	0.962643	−	0.270775i	$-0.0872799\pi$
$641$	31245.1	1.92529	0.962643	+	0.270775i	$0.0872799\pi$
$642$	0	0
$643$	22381.8	1.37271	0.686354	−	0.727268i	$-0.259210\pi$
$643$	22381.8	1.37271	0.686354	+	0.727268i	$0.259210\pi$
$644$	0	0
$645$	182.609	0.0111477
$646$	0	0
$647$	13062.8	0.793741	0.396870	−	0.917875i	$-0.370096\pi$
$647$	13062.8	0.793741	0.396870	+	0.917875i	$0.370096\pi$
$648$	0	0
$649$	22236.4	1.34493
$650$	0	0
$651$	2050.25	0.123434
$652$	0	0
$653$	−17486.1	−1.04791	−0.523955	−	0.851746i	$-0.675544\pi$
$653$	−17486.1	−1.04791	−0.523955	+	0.851746i	$0.675544\pi$
$654$	0	0
$655$	2173.45	0.129655
$656$	0	0
$657$	11408.5	0.677453
$658$	0	0
$659$	−29707.1	−1.75603	−0.878016	−	0.478631i	$-0.841133\pi$
$659$	−29707.1	−1.75603	−0.878016	+	0.478631i	$0.841133\pi$
$660$	0	0
$661$	−9905.74	−0.582888	−0.291444	−	0.956588i	$-0.594136\pi$
$661$	−9905.74	−0.582888	−0.291444	+	0.956588i	$0.594136\pi$
$662$	0	0
$663$	6179.76	0.361994
$664$	0	0
$665$	−1554.84	−0.0906679
$666$	0	0
$667$	3636.41	0.211098
$668$	0	0
$669$	−6586.61	−0.380647
$670$	0	0
$671$	26586.1	1.52957
$672$	0	0
$673$	−7803.48	−0.446957	−0.223478	−	0.974709i	$-0.571741\pi$
$673$	−7803.48	−0.446957	−0.223478	+	0.974709i	$0.571741\pi$
$674$	0	0
$675$	2542.96	0.145005
$676$	0	0
$677$	29359.1	1.66671	0.833354	−	0.552740i	$-0.186418\pi$
$677$	29359.1	1.66671	0.833354	+	0.552740i	$0.186418\pi$
$678$	0	0
$679$	8713.86	0.492500
$680$	0	0
$681$	−6165.54	−0.346937
$682$	0	0
$683$	−26055.8	−1.45973	−0.729866	−	0.683591i	$-0.760417\pi$
$683$	−26055.8	−1.45973	−0.729866	+	0.683591i	$0.760417\pi$
$684$	0	0
$685$	−4698.65	−0.262082
$686$	0	0
$687$	−1907.99	−0.105960
$688$	0	0
$689$	−46068.8	−2.54729
$690$	0	0
$691$	15166.4	0.834959	0.417480	−	0.908686i	$-0.362914\pi$
$691$	15166.4	0.834959	0.417480	+	0.908686i	$0.362914\pi$
$692$	0	0
$693$	14976.4	0.820934
$694$	0	0
$695$	623.047	0.0340051
$696$	0	0
$697$	8767.08	0.476437
$698$	0	0
$699$	7994.95	0.432614
$700$	0	0
$701$	−26527.8	−1.42930	−0.714650	−	0.699482i	$-0.753414\pi$
$701$	−26527.8	−1.42930	−0.714650	+	0.699482i	$0.753414\pi$
$702$	0	0
$703$	571.032	0.0306357
$704$	0	0
$705$	−7.17128	−0.000383101 0
$706$	0	0
$707$	10466.7	0.556779
$708$	0	0
$709$	21268.1	1.12657	0.563287	−	0.826261i	$-0.309536\pi$
$709$	21268.1	1.12657	0.563287	+	0.826261i	$0.309536\pi$
$710$	0	0
$711$	18220.6	0.961077
$712$	0	0
$713$	1069.00	0.0561491
$714$	0	0
$715$	10573.6	0.553047
$716$	0	0
$717$	−6531.82	−0.340216
$718$	0	0
$719$	−27640.4	−1.43368	−0.716838	−	0.697240i	$-0.754411\pi$
$719$	−27640.4	−1.43368	−0.716838	+	0.697240i	$0.754411\pi$
$720$	0	0
$721$	−44620.6	−2.30480
$722$	0	0
$723$	7703.81	0.396276
$724$	0	0
$725$	−3952.62	−0.202478
$726$	0	0
$727$	−10860.2	−0.554036	−0.277018	−	0.960865i	$-0.589346\pi$
$727$	−10860.2	−0.554036	−0.277018	+	0.960865i	$0.589346\pi$
$728$	0	0
$729$	−3730.57	−0.189533
$730$	0	0
$731$	−777.450	−0.0393366
$732$	0	0
$733$	−2575.74	−0.129791	−0.0648956	−	0.997892i	$-0.520671\pi$
$733$	−2575.74	−0.129791	−0.0648956	+	0.997892i	$0.520671\pi$
$734$	0	0
$735$	1266.00	0.0635336
$736$	0	0
$737$	−22665.1	−1.13281
$738$	0	0
$739$	−25649.2	−1.27675	−0.638377	−	0.769724i	$-0.720394\pi$
$739$	−25649.2	−1.27675	−0.638377	+	0.769724i	$0.720394\pi$
$740$	0	0
$741$	−2046.50	−0.101457
$742$	0	0
$743$	3767.84	0.186041	0.0930207	−	0.995664i	$-0.470348\pi$
$743$	3767.84	0.186041	0.0930207	+	0.995664i	$0.470348\pi$
$744$	0	0
$745$	−6875.69	−0.338129
$746$	0	0
$747$	26261.1	1.28627
$748$	0	0
$749$	−31043.1	−1.51441
$750$	0	0
$751$	16030.6	0.778913	0.389456	−	0.921045i	$-0.372663\pi$
$751$	16030.6	0.778913	0.389456	+	0.921045i	$0.372663\pi$
$752$	0	0
$753$	−6821.55	−0.330134
$754$	0	0
$755$	5982.55	0.288380
$756$	0	0
$757$	−3408.80	−0.163666	−0.0818329	−	0.996646i	$-0.526077\pi$
$757$	−3408.80	−0.163666	−0.0818329	+	0.996646i	$0.526077\pi$
$758$	0	0
$759$	−1424.81	−0.0681385
$760$	0	0
$761$	28208.1	1.34368	0.671841	−	0.740696i	$-0.265504\pi$
$761$	28208.1	1.34368	0.671841	+	0.740696i	$0.265504\pi$
$762$	0	0
$763$	40426.6	1.91814
$764$	0	0
$765$	−4960.68	−0.234450
$766$	0	0
$767$	−51052.1	−2.40337
$768$	0	0
$769$	26349.3	1.23560	0.617802	−	0.786334i	$-0.288023\pi$
$769$	26349.3	1.23560	0.617802	+	0.786334i	$0.288023\pi$
$770$	0	0
$771$	5917.71	0.276422
$772$	0	0
$773$	29491.9	1.37225	0.686126	−	0.727482i	$-0.259310\pi$
$773$	29491.9	1.37225	0.686126	+	0.727482i	$0.259310\pi$
$774$	0	0
$775$	−1161.96	−0.0538563
$776$	0	0
$777$	−1750.58	−0.0808258
$778$	0	0
$779$	−2903.32	−0.133533
$780$	0	0
$781$	8353.56	0.382732
$782$	0	0
$783$	−16082.1	−0.734009
$784$	0	0
$785$	1061.70	0.0482722
$786$	0	0
$787$	18485.5	0.837276	0.418638	−	0.908153i	$-0.362508\pi$
$787$	18485.5	0.837276	0.418638	+	0.908153i	$0.362508\pi$
$788$	0	0
$789$	12250.7	0.552770
$790$	0	0
$791$	34722.3	1.56079
$792$	0	0
$793$	−61038.3	−2.73333
$794$	0	0
$795$	−6747.67	−0.301025
$796$	0	0
$797$	36684.3	1.63039	0.815197	−	0.579184i	$-0.196629\pi$
$797$	36684.3	1.63039	0.815197	+	0.579184i	$0.196629\pi$
$798$	0	0
$799$	30.5313	0.00135184
$800$	0	0
$801$	−26644.5	−1.17533
$802$	0	0
$803$	15163.6	0.666391
$804$	0	0
$805$	2485.30	0.108814
$806$	0	0
$807$	−16958.5	−0.739737
$808$	0	0
$809$	216.567	0.00941173	0.00470586	−	0.999989i	$-0.498502\pi$
$809$	216.567	0.00941173	0.00470586	+	0.999989i	$0.498502\pi$
$810$	0	0
$811$	25897.5	1.12131	0.560657	−	0.828048i	$-0.310549\pi$
$811$	25897.5	1.12131	0.560657	+	0.828048i	$0.310549\pi$
$812$	0	0
$813$	4889.14	0.210910
$814$	0	0
$815$	−6950.82	−0.298744
$816$	0	0
$817$	257.462	0.0110250
$818$	0	0
$819$	−34384.0	−1.46700
$820$	0	0
$821$	12735.6	0.541384	0.270692	−	0.962666i	$-0.412748\pi$
$821$	12735.6	0.541384	0.270692	+	0.962666i	$0.412748\pi$
$822$	0	0
$823$	25798.2	1.09267	0.546336	−	0.837566i	$-0.316022\pi$
$823$	25798.2	1.09267	0.546336	+	0.837566i	$0.316022\pi$
$824$	0	0
$825$	1548.70	0.0653562
$826$	0	0
$827$	21295.2	0.895413	0.447706	−	0.894181i	$-0.352241\pi$
$827$	21295.2	0.895413	0.447706	+	0.894181i	$0.352241\pi$
$828$	0	0
$829$	−43456.4	−1.82063	−0.910315	−	0.413915i	$-0.864161\pi$
$829$	−43456.4	−1.82063	−0.910315	+	0.413915i	$0.864161\pi$
$830$	0	0
$831$	−15240.3	−0.636197
$832$	0	0
$833$	−5389.94	−0.224190
$834$	0	0
$835$	−1811.75	−0.0750875
$836$	0	0
$837$	−4727.68	−0.195236
$838$	0	0
$839$	14245.5	0.586184	0.293092	−	0.956084i	$-0.405316\pi$
$839$	14245.5	0.586184	0.293092	+	0.956084i	$0.405316\pi$
$840$	0	0
$841$	608.090	0.0249330
$842$	0	0
$843$	11371.0	0.464577
$844$	0	0
$845$	−13290.6	−0.541076
$846$	0	0
$847$	−8858.64	−0.359370
$848$	0	0
$849$	10764.0	0.435123
$850$	0	0
$851$	−912.752	−0.0367670
$852$	0	0
$853$	1967.69	0.0789830	0.0394915	−	0.999220i	$-0.487426\pi$
$853$	1967.69	0.0789830	0.0394915	+	0.999220i	$0.487426\pi$
$854$	0	0
$855$	1642.79	0.0657101
$856$	0	0
$857$	−32176.3	−1.28252	−0.641262	−	0.767322i	$-0.721589\pi$
$857$	−32176.3	−1.28252	−0.641262	+	0.767322i	$0.721589\pi$
$858$	0	0
$859$	−1273.26	−0.0505741	−0.0252871	−	0.999680i	$-0.508050\pi$
$859$	−1273.26	−0.0505741	−0.0252871	+	0.999680i	$0.508050\pi$
$860$	0	0
$861$	8900.54	0.352299
$862$	0	0
$863$	69.6881	0.00274880	0.00137440	−	0.999999i	$-0.499563\pi$
$863$	69.6881	0.00274880	0.00137440	+	0.999999i	$0.499563\pi$
$864$	0	0
$865$	−4932.98	−0.193903
$866$	0	0
$867$	6174.59	0.241869
$868$	0	0
$869$	24218.0	0.945384
$870$	0	0
$871$	52036.3	2.02432
$872$	0	0
$873$	−9206.75	−0.356931
$874$	0	0
$875$	−2701.41	−0.104371
$876$	0	0
$877$	−20284.8	−0.781038	−0.390519	−	0.920595i	$-0.627704\pi$
$877$	−20284.8	−0.781038	−0.390519	+	0.920595i	$0.627704\pi$
$878$	0	0
$879$	1044.08	0.0400636
$880$	0	0
$881$	−29637.5	−1.13339	−0.566693	−	0.823929i	$-0.691778\pi$
$881$	−29637.5	−1.13339	−0.566693	+	0.823929i	$0.691778\pi$
$882$	0	0
$883$	−16749.3	−0.638344	−0.319172	−	0.947697i	$-0.603405\pi$
$883$	−16749.3	−0.638344	−0.319172	+	0.947697i	$0.603405\pi$
$884$	0	0
$885$	−7477.56	−0.284017
$886$	0	0
$887$	−22904.8	−0.867042	−0.433521	−	0.901143i	$-0.642729\pi$
$887$	−22904.8	−0.867042	−0.433521	+	0.901143i	$0.642729\pi$
$888$	0	0
$889$	−43906.7	−1.65645
$890$	0	0
$891$	−12409.5	−0.466593
$892$	0	0
$893$	−10.1108	−0.000378886 0
$894$	0	0
$895$	−19887.2	−0.742744
$896$	0	0
$897$	3271.17	0.121763
$898$	0	0
$899$	7348.42	0.272618
$900$	0	0
$901$	28727.8	1.06222
$902$	0	0
$903$	−789.285	−0.0290872
$904$	0	0
$905$	−12563.5	−0.461463
$906$	0	0
$907$	−20869.7	−0.764020	−0.382010	−	0.924158i	$-0.624768\pi$
$907$	−20869.7	−0.764020	−0.382010	+	0.924158i	$0.624768\pi$
$908$	0	0
$909$	−11058.8	−0.403517
$910$	0	0
$911$	−42361.0	−1.54060	−0.770298	−	0.637684i	$-0.779892\pi$
$911$	−42361.0	−1.54060	−0.770298	+	0.637684i	$0.779892\pi$
$912$	0	0
$913$	34905.0	1.26527
$914$	0	0
$915$	−8940.24	−0.323011
$916$	0	0
$917$	−9394.22	−0.338304
$918$	0	0
$919$	14136.3	0.507413	0.253707	−	0.967281i	$-0.418350\pi$
$919$	14136.3	0.507413	0.253707	+	0.967281i	$0.418350\pi$
$920$	0	0
$921$	−8486.01	−0.303609
$922$	0	0
$923$	−19178.7	−0.683939
$924$	0	0
$925$	992.122	0.0352657
$926$	0	0
$927$	47144.5	1.67036
$928$	0	0
$929$	−7897.49	−0.278911	−0.139455	−	0.990228i	$-0.544535\pi$
$929$	−7897.49	−0.278911	−0.139455	+	0.990228i	$0.544535\pi$
$930$	0	0
$931$	1784.94	0.0628347
$932$	0	0
$933$	6957.19	0.244124
$934$	0	0
$935$	−6593.52	−0.230621
$936$	0	0
$937$	21499.1	0.749567	0.374783	−	0.927112i	$-0.377717\pi$
$937$	21499.1	0.749567	0.374783	+	0.927112i	$0.377717\pi$
$938$	0	0
$939$	−9218.58	−0.320380
$940$	0	0
$941$	30258.7	1.04825	0.524125	−	0.851641i	$-0.324392\pi$
$941$	30258.7	1.04825	0.524125	+	0.851641i	$0.324392\pi$
$942$	0	0
$943$	4640.74	0.160258
$944$	0	0
$945$	−10991.3	−0.378357
$946$	0	0
$947$	−12769.1	−0.438163	−0.219082	−	0.975707i	$-0.570306\pi$
$947$	−12769.1	−0.438163	−0.219082	+	0.975707i	$0.570306\pi$
$948$	0	0
$949$	−34813.8	−1.19083
$950$	0	0
$951$	−5498.09	−0.187474
$952$	0	0
$953$	28174.8	0.957683	0.478841	−	0.877901i	$-0.341057\pi$
$953$	28174.8	0.957683	0.478841	+	0.877901i	$0.341057\pi$
$954$	0	0
$955$	−9887.45	−0.335027
$956$	0	0
$957$	−9794.28	−0.330830
$958$	0	0
$959$	20308.8	0.683841
$960$	0	0
$961$	−27630.8	−0.927487
$962$	0	0
$963$	32799.0	1.09754
$964$	0	0
$965$	6206.50	0.207041
$966$	0	0
$967$	−2696.72	−0.0896800	−0.0448400	−	0.998994i	$-0.514278\pi$
$967$	−2696.72	−0.0896800	−0.0448400	+	0.998994i	$0.514278\pi$
$968$	0	0
$969$	1276.17	0.0423079
$970$	0	0
$971$	−29192.7	−0.964817	−0.482408	−	0.875946i	$-0.660238\pi$
$971$	−29192.7	−0.964817	−0.482408	+	0.875946i	$0.660238\pi$
$972$	0	0
$973$	−2692.97	−0.0887283
$974$	0	0
$975$	−3555.62	−0.116791
$976$	0	0
$977$	−1150.88	−0.0376867	−0.0188433	−	0.999822i	$-0.505998\pi$
$977$	−1150.88	−0.0376867	−0.0188433	+	0.999822i	$0.505998\pi$
$978$	0	0
$979$	−35414.6	−1.15613
$980$	0	0
$981$	−42713.2	−1.39014
$982$	0	0
$983$	−9924.92	−0.322030	−0.161015	−	0.986952i	$-0.551477\pi$
$983$	−9924.92	−0.322030	−0.161015	+	0.986952i	$0.551477\pi$
$984$	0	0
$985$	−9849.53	−0.318611
$986$	0	0
$987$	30.9961	0.000999612 0
$988$	0	0
$989$	−411.533	−0.0132315
$990$	0	0
$991$	32446.8	1.04007	0.520034	−	0.854146i	$-0.325919\pi$
$991$	32446.8	1.04007	0.520034	+	0.854146i	$0.325919\pi$
$992$	0	0
$993$	22502.4	0.719127
$994$	0	0
$995$	20265.4	0.645686
$996$	0	0
$997$	49280.6	1.56543	0.782715	−	0.622381i	$-0.213834\pi$
$997$	49280.6	1.56543	0.782715	+	0.622381i	$0.213834\pi$
$998$	0	0
$999$	4036.68	0.127843

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.4.a.b.1.3	✓	5
4.3	odd	2		1840.4.a.o.1.3		5
5.2	odd	4		2300.4.c.d.1749.6		10
5.3	odd	4		2300.4.c.d.1749.5		10
5.4	even	2		2300.4.a.c.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.4.a.b.1.3	✓	5	1.1	even	1	trivial
1840.4.a.o.1.3		5	4.3	odd	2
2300.4.a.c.1.3		5	5.4	even	2
2300.4.c.d.1749.5		10	5.3	odd	4
2300.4.c.d.1749.6		10	5.2	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:	\( N \)	\(=\)	\( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	460.a (trivial)

\(f(q)\)	\(=\)	\(q-2.04116 q^{3} -5.00000 q^{5} +21.6113 q^{7} -22.8337 q^{9} -30.3495 q^{11} +69.6786 q^{13} +10.2058 q^{15} -43.4506 q^{17} +14.3892 q^{19} -44.1120 q^{21} -23.0000 q^{23} +25.0000 q^{25} +101.718 q^{27} -158.105 q^{29} -46.4782 q^{31} +61.9480 q^{33} -108.056 q^{35} +39.6849 q^{37} -142.225 q^{39} -201.771 q^{41} +17.8927 q^{43} +114.168 q^{45} -0.702668 q^{47} +124.048 q^{49} +88.6894 q^{51} -661.161 q^{53} +151.748 q^{55} -29.3705 q^{57} -732.679 q^{59} -875.997 q^{61} -493.465 q^{63} -348.393 q^{65} +746.803 q^{67} +46.9466 q^{69} -275.246 q^{71} -499.633 q^{73} -51.0289 q^{75} -655.892 q^{77} -797.970 q^{79} +408.887 q^{81} -1150.10 q^{83} +217.253 q^{85} +322.716 q^{87} +1166.89 q^{89} +1505.85 q^{91} +94.8693 q^{93} -71.9458 q^{95} +403.209 q^{97} +692.991 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{3} - 25 q^{5} + 8 q^{7} + 30 q^{9} + 7 q^{11} + 5 q^{13} + 15 q^{15} - 24 q^{17} - 13 q^{19} - 115 q^{23} + 125 q^{25} - 204 q^{27} - 253 q^{29} - 98 q^{31} - 473 q^{33} - 40 q^{35} - 435 q^{37}+ \cdots + 111 q^{99}+O(q^{100}) \) 5 * q - 3 * q^3 - 25 * q^5 + 8 * q^7 + 30 * q^9 + 7 * q^11 + 5 * q^13 + 15 * q^15 - 24 * q^17 - 13 * q^19 - 115 * q^23 + 125 * q^25 - 204 * q^27 - 253 * q^29 - 98 * q^31 - 473 * q^33 - 40 * q^35 - 435 * q^37 - 410 * q^39 - 774 * q^41 - 498 * q^43 - 150 * q^45 - 572 * q^47 - 683 * q^49 - 657 * q^51 - 665 * q^53 - 35 * q^55 - 932 * q^57 - 763 * q^59 + 337 * q^61 - 1527 * q^63 - 25 * q^65 - 305 * q^67 + 69 * q^69 - 1504 * q^71 - 1304 * q^73 - 75 * q^75 + 182 * q^77 + 626 * q^79 - 959 * q^81 - 1703 * q^83 + 120 * q^85 - 1354 * q^87 + 646 * q^89 - 767 * q^91 - 452 * q^93 + 65 * q^95 - 233 * q^97 + 111 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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