LMFDB - Embedded newform 320.4.a.o.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [320,4,Mod(1,320)] 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("320.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	320.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-8,0,-10,0,8,0,26] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.8806112018$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 160)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	320.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.898979 q^{3} -5.00000 q^{5} +28.4949 q^{7} -26.1918 q^{9} -2.60612 q^{11} +33.1918 q^{13} -4.49490 q^{15} +119.576 q^{17} -143.192 q^{19} +25.6163 q^{21} +113.889 q^{23} +25.0000 q^{25} -47.8184 q^{27} +67.6163 q^{29} +117.394 q^{31} -2.34285 q^{33} -142.474 q^{35} +256.384 q^{37} +29.8388 q^{39} +245.959 q^{41} +369.262 q^{43} +130.959 q^{45} -76.3133 q^{47} +468.959 q^{49} +107.496 q^{51} -40.4245 q^{53} +13.0306 q^{55} -128.727 q^{57} -457.980 q^{59} +477.233 q^{61} -746.334 q^{63} -165.959 q^{65} +602.252 q^{67} +102.384 q^{69} -1091.98 q^{71} +117.878 q^{73} +22.4745 q^{75} -74.2612 q^{77} +858.624 q^{79} +664.192 q^{81} +565.748 q^{83} -597.878 q^{85} +60.7857 q^{87} -625.151 q^{89} +945.798 q^{91} +105.535 q^{93} +715.959 q^{95} +805.959 q^{97} +68.2591 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{3} - 10 q^{5} + 8 q^{7} + 26 q^{9} - 64 q^{11} - 12 q^{13} + 40 q^{15} + 4 q^{17} - 208 q^{19} + 208 q^{21} + 120 q^{23} + 50 q^{25} - 272 q^{27} + 292 q^{29} + 176 q^{31} + 544 q^{33} - 40 q^{35}+ \cdots - 3136 q^{99}+O(q^{100}) $ 2 * q - 8 * q^3 - 10 * q^5 + 8 * q^7 + 26 * q^9 - 64 * q^11 - 12 * q^13 + 40 * q^15 + 4 * q^17 - 208 * q^19 + 208 * q^21 + 120 * q^23 + 50 * q^25 - 272 * q^27 + 292 * q^29 + 176 * q^31 + 544 * q^33 - 40 * q^35 + 356 * q^37 + 432 * q^39 + 100 * q^41 + 376 * q^43 - 130 * q^45 - 280 * q^47 + 546 * q^49 + 1136 * q^51 - 316 * q^53 + 320 * q^55 + 448 * q^57 - 720 * q^59 + 1268 * q^61 - 1816 * q^63 + 60 * q^65 + 744 * q^67 + 48 * q^69 - 48 * q^71 - 940 * q^73 - 200 * q^75 + 1184 * q^77 + 32 * q^79 + 1250 * q^81 + 1592 * q^83 - 20 * q^85 - 1936 * q^87 - 780 * q^89 + 1872 * q^91 - 416 * q^93 + 1040 * q^95 + 1220 * q^97 - 3136 * 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
$3$	0.898979	0.173009	0.0865043	−	0.996251i	$-0.472430\pi$
$3$	0.898979	0.173009	0.0865043	+	0.996251i	$0.472430\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	28.4949	1.53858	0.769290	−	0.638900i	$-0.220610\pi$
$7$	28.4949	1.53858	0.769290	+	0.638900i	$0.220610\pi$
$8$	0	0
$9$	−26.1918	−0.970068
$10$	0	0
$11$	−2.60612	−0.0714342	−0.0357171	−	0.999362i	$-0.511372\pi$
$11$	−2.60612	−0.0714342	−0.0357171	+	0.999362i	$0.511372\pi$
$12$	0	0
$13$	33.1918	0.708135	0.354068	−	0.935220i	$-0.384798\pi$
$13$	33.1918	0.708135	0.354068	+	0.935220i	$0.384798\pi$
$14$	0	0
$15$	−4.49490	−0.0773718
$16$	0	0
$17$	119.576	1.70596	0.852980	−	0.521944i	$-0.174793\pi$
$17$	119.576	1.70596	0.852980	+	0.521944i	$0.174793\pi$
$18$	0	0
$19$	−143.192	−1.72897	−0.864486	−	0.502657i	$-0.832356\pi$
$19$	−143.192	−1.72897	−0.864486	+	0.502657i	$0.832356\pi$
$20$	0	0
$21$	25.6163	0.266188
$22$	0	0
$23$	113.889	1.03250	0.516249	−	0.856439i	$-0.327328\pi$
$23$	113.889	1.03250	0.516249	+	0.856439i	$0.327328\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−47.8184	−0.340839
$28$	0	0
$29$	67.6163	0.432967	0.216483	−	0.976286i	$-0.430541\pi$
$29$	67.6163	0.432967	0.216483	+	0.976286i	$0.430541\pi$
$30$	0	0
$31$	117.394	0.680147	0.340074	−	0.940399i	$-0.389548\pi$
$31$	117.394	0.680147	0.340074	+	0.940399i	$0.389548\pi$
$32$	0	0
$33$	−2.34285	−0.0123587
$34$	0	0
$35$	−142.474	−0.688074
$36$	0	0
$37$	256.384	1.13917	0.569584	−	0.821933i	$-0.307104\pi$
$37$	256.384	1.13917	0.569584	+	0.821933i	$0.307104\pi$
$38$	0	0
$39$	29.8388	0.122514
$40$	0	0
$41$	245.959	0.936887	0.468444	−	0.883493i	$-0.344815\pi$
$41$	245.959	0.936887	0.468444	+	0.883493i	$0.344815\pi$
$42$	0	0
$43$	369.262	1.30958	0.654790	−	0.755811i	$-0.272757\pi$
$43$	369.262	1.30958	0.654790	+	0.755811i	$0.272757\pi$
$44$	0	0
$45$	130.959	0.433828
$46$	0	0
$47$	−76.3133	−0.236839	−0.118420	−	0.992964i	$-0.537783\pi$
$47$	−76.3133	−0.236839	−0.118420	+	0.992964i	$0.537783\pi$
$48$	0	0
$49$	468.959	1.36723
$50$	0	0
$51$	107.496	0.295146
$52$	0	0
$53$	−40.4245	−0.104769	−0.0523843	−	0.998627i	$-0.516682\pi$
$53$	−40.4245	−0.104769	−0.0523843	+	0.998627i	$0.516682\pi$
$54$	0	0
$55$	13.0306	0.0319463
$56$	0	0
$57$	−128.727	−0.299127
$58$	0	0
$59$	−457.980	−1.01057	−0.505287	−	0.862951i	$-0.668613\pi$
$59$	−457.980	−1.01057	−0.505287	+	0.862951i	$0.668613\pi$
$60$	0	0
$61$	477.233	1.00169	0.500847	−	0.865536i	$-0.333022\pi$
$61$	477.233	1.00169	0.500847	+	0.865536i	$0.333022\pi$
$62$	0	0
$63$	−746.334	−1.49253
$64$	0	0
$65$	−165.959	−0.316688
$66$	0	0
$67$	602.252	1.09816	0.549081	−	0.835769i	$-0.314978\pi$
$67$	602.252	1.09816	0.549081	+	0.835769i	$0.314978\pi$
$68$	0	0
$69$	102.384	0.178631
$70$	0	0
$71$	−1091.98	−1.82527	−0.912633	−	0.408780i	$-0.865954\pi$
$71$	−1091.98	−1.82527	−0.912633	+	0.408780i	$0.865954\pi$
$72$	0	0
$73$	117.878	0.188993	0.0944967	−	0.995525i	$-0.469876\pi$
$73$	117.878	0.188993	0.0944967	+	0.995525i	$0.469876\pi$
$74$	0	0
$75$	22.4745	0.0346017
$76$	0	0
$77$	−74.2612	−0.109907
$78$	0	0
$79$	858.624	1.22282	0.611410	−	0.791314i	$-0.290603\pi$
$79$	858.624	1.22282	0.611410	+	0.791314i	$0.290603\pi$
$80$	0	0
$81$	664.192	0.911100
$82$	0	0
$83$	565.748	0.748180	0.374090	−	0.927392i	$-0.377955\pi$
$83$	565.748	0.748180	0.374090	+	0.927392i	$0.377955\pi$
$84$	0	0
$85$	−597.878	−0.762929
$86$	0	0
$87$	60.7857	0.0749070
$88$	0	0
$89$	−625.151	−0.744560	−0.372280	−	0.928120i	$-0.621424\pi$
$89$	−625.151	−0.744560	−0.372280	+	0.928120i	$0.621424\pi$
$90$	0	0
$91$	945.798	1.08952
$92$	0	0
$93$	105.535	0.117671
$94$	0	0
$95$	715.959	0.773220
$96$	0	0
$97$	805.959	0.843637	0.421818	−	0.906680i	$-0.361392\pi$
$97$	805.959	0.843637	0.421818	+	0.906680i	$0.361392\pi$
$98$	0	0
$99$	68.2591	0.0692960
$100$	0	0
$101$	260.988	0.257121	0.128561	−	0.991702i	$-0.458964\pi$
$101$	260.988	0.257121	0.128561	+	0.991702i	$0.458964\pi$
$102$	0	0
$103$	−1640.01	−1.56888	−0.784441	−	0.620203i	$-0.787050\pi$
$103$	−1640.01	−1.56888	−0.784441	+	0.620203i	$0.787050\pi$
$104$	0	0
$105$	−128.082	−0.119043
$106$	0	0
$107$	−679.001	−0.613472	−0.306736	−	0.951795i	$-0.599237\pi$
$107$	−679.001	−0.613472	−0.306736	+	0.951795i	$0.599237\pi$
$108$	0	0
$109$	840.220	0.738335	0.369168	−	0.929363i	$-0.379643\pi$
$109$	840.220	0.738335	0.369168	+	0.929363i	$0.379643\pi$
$110$	0	0
$111$	230.484	0.197086
$112$	0	0
$113$	−1609.15	−1.33961	−0.669806	−	0.742536i	$-0.733623\pi$
$113$	−1609.15	−1.33961	−0.669806	+	0.742536i	$0.733623\pi$
$114$	0	0
$115$	−569.444	−0.461747
$116$	0	0
$117$	−869.355	−0.686939
$118$	0	0
$119$	3407.29	2.62476
$120$	0	0
$121$	−1324.21	−0.994897
$122$	0	0
$123$	221.112	0.162090
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1149.60	−0.803234	−0.401617	−	0.915808i	$-0.631552\pi$
$127$	−1149.60	−0.803234	−0.401617	+	0.915808i	$0.631552\pi$
$128$	0	0
$129$	331.959	0.226569
$130$	0	0
$131$	−2436.99	−1.62535	−0.812673	−	0.582719i	$-0.801989\pi$
$131$	−2436.99	−1.62535	−0.812673	+	0.582719i	$0.801989\pi$
$132$	0	0
$133$	−4080.24	−2.66016
$134$	0	0
$135$	239.092	0.152428
$136$	0	0
$137$	−764.465	−0.476735	−0.238367	−	0.971175i	$-0.576612\pi$
$137$	−764.465	−0.476735	−0.238367	+	0.971175i	$0.576612\pi$
$138$	0	0
$139$	−2547.35	−1.55441	−0.777207	−	0.629245i	$-0.783364\pi$
$139$	−2547.35	−1.55441	−0.777207	+	0.629245i	$0.783364\pi$
$140$	0	0
$141$	−68.6041	−0.0409752
$142$	0	0
$143$	−86.5020	−0.0505850
$144$	0	0
$145$	−338.082	−0.193629
$146$	0	0
$147$	421.585	0.236542
$148$	0	0
$149$	−2565.76	−1.41070	−0.705352	−	0.708857i	$-0.749211\pi$
$149$	−2565.76	−1.41070	−0.705352	+	0.708857i	$0.749211\pi$
$150$	0	0
$151$	−524.949	−0.282912	−0.141456	−	0.989945i	$-0.545178\pi$
$151$	−524.949	−0.282912	−0.141456	+	0.989945i	$0.545178\pi$
$152$	0	0
$153$	−3131.90	−1.65490
$154$	0	0
$155$	−586.969	−0.304171
$156$	0	0
$157$	3569.59	1.81455	0.907275	−	0.420537i	$-0.138158\pi$
$157$	3569.59	1.81455	0.907275	+	0.420537i	$0.138158\pi$
$158$	0	0
$159$	−36.3408	−0.0181259
$160$	0	0
$161$	3245.25	1.58858
$162$	0	0
$163$	1450.96	0.697227	0.348613	−	0.937267i	$-0.386653\pi$
$163$	1450.96	0.697227	0.348613	+	0.937267i	$0.386653\pi$
$164$	0	0
$165$	11.7143	0.00552699
$166$	0	0
$167$	3281.38	1.52048	0.760242	−	0.649640i	$-0.225080\pi$
$167$	3281.38	1.52048	0.760242	+	0.649640i	$0.225080\pi$
$168$	0	0
$169$	−1095.30	−0.498544
$170$	0	0
$171$	3750.46	1.67722
$172$	0	0
$173$	−1387.01	−0.609553	−0.304776	−	0.952424i	$-0.598582\pi$
$173$	−1387.01	−0.609553	−0.304776	+	0.952424i	$0.598582\pi$
$174$	0	0
$175$	712.372	0.307716
$176$	0	0
$177$	−411.714	−0.174838
$178$	0	0
$179$	−3009.90	−1.25682	−0.628409	−	0.777883i	$-0.716293\pi$
$179$	−3009.90	−1.25682	−0.628409	+	0.777883i	$0.716293\pi$
$180$	0	0
$181$	2304.14	0.946217	0.473109	−	0.881004i	$-0.343132\pi$
$181$	2304.14	0.946217	0.473109	+	0.881004i	$0.343132\pi$
$182$	0	0
$183$	429.022	0.173302
$184$	0	0
$185$	−1281.92	−0.509451
$186$	0	0
$187$	−311.628	−0.121864
$188$	0	0
$189$	−1362.58	−0.524408
$190$	0	0
$191$	166.647	0.0631317	0.0315658	−	0.999502i	$-0.489951\pi$
$191$	166.647	0.0631317	0.0315658	+	0.999502i	$0.489951\pi$
$192$	0	0
$193$	484.669	0.180763	0.0903815	−	0.995907i	$-0.471191\pi$
$193$	484.669	0.180763	0.0903815	+	0.995907i	$0.471191\pi$
$194$	0	0
$195$	−149.194	−0.0547897
$196$	0	0
$197$	−2088.10	−0.755182	−0.377591	−	0.925973i	$-0.623247\pi$
$197$	−2088.10	−0.755182	−0.377591	+	0.925973i	$0.623247\pi$
$198$	0	0
$199$	−3401.93	−1.21184	−0.605921	−	0.795524i	$-0.707195\pi$
$199$	−3401.93	−1.21184	−0.605921	+	0.795524i	$0.707195\pi$
$200$	0	0
$201$	541.412	0.189991
$202$	0	0
$203$	1926.72	0.666154
$204$	0	0
$205$	−1229.80	−0.418989
$206$	0	0
$207$	−2982.96	−1.00159
$208$	0	0
$209$	373.176	0.123508
$210$	0	0
$211$	2246.24	0.732879	0.366439	−	0.930442i	$-0.380577\pi$
$211$	2246.24	0.732879	0.366439	+	0.930442i	$0.380577\pi$
$212$	0	0
$213$	−981.665	−0.315787
$214$	0	0
$215$	−1846.31	−0.585662
$216$	0	0
$217$	3345.13	1.04646
$218$	0	0
$219$	105.969	0.0326975
$220$	0	0
$221$	3968.93	1.20805
$222$	0	0
$223$	−1129.63	−0.339217	−0.169608	−	0.985512i	$-0.554250\pi$
$223$	−1129.63	−0.339217	−0.169608	+	0.985512i	$0.554250\pi$
$224$	0	0
$225$	−654.796	−0.194014
$226$	0	0
$227$	1693.88	0.495273	0.247637	−	0.968853i	$-0.420346\pi$
$227$	1693.88	0.495273	0.247637	+	0.968853i	$0.420346\pi$
$228$	0	0
$229$	5250.72	1.51518	0.757592	−	0.652728i	$-0.226376\pi$
$229$	5250.72	1.51518	0.757592	+	0.652728i	$0.226376\pi$
$230$	0	0
$231$	−66.7593	−0.0190149
$232$	0	0
$233$	−6042.60	−1.69899	−0.849493	−	0.527600i	$-0.823092\pi$
$233$	−6042.60	−1.69899	−0.849493	+	0.527600i	$0.823092\pi$
$234$	0	0
$235$	381.566	0.105918
$236$	0	0
$237$	771.886	0.211559
$238$	0	0
$239$	−385.416	−0.104312	−0.0521559	−	0.998639i	$-0.516609\pi$
$239$	−385.416	−0.104312	−0.0521559	+	0.998639i	$0.516609\pi$
$240$	0	0
$241$	4858.73	1.29866	0.649332	−	0.760505i	$-0.275049\pi$
$241$	4858.73	1.29866	0.649332	+	0.760505i	$0.275049\pi$
$242$	0	0
$243$	1888.19	0.498467
$244$	0	0
$245$	−2344.80	−0.611443
$246$	0	0
$247$	−4752.80	−1.22435
$248$	0	0
$249$	508.596	0.129442
$250$	0	0
$251$	−3784.74	−0.951757	−0.475878	−	0.879511i	$-0.657870\pi$
$251$	−3784.74	−0.951757	−0.475878	+	0.879511i	$0.657870\pi$
$252$	0	0
$253$	−296.808	−0.0737556
$254$	0	0
$255$	−537.480	−0.131993
$256$	0	0
$257$	3702.36	0.898626	0.449313	−	0.893374i	$-0.351669\pi$
$257$	3702.36	0.898626	0.449313	+	0.893374i	$0.351669\pi$
$258$	0	0
$259$	7305.63	1.75270
$260$	0	0
$261$	−1771.00	−0.420007
$262$	0	0
$263$	3517.77	0.824771	0.412385	−	0.911009i	$-0.364696\pi$
$263$	3517.77	0.824771	0.412385	+	0.911009i	$0.364696\pi$
$264$	0	0
$265$	202.122	0.0468539
$266$	0	0
$267$	−561.998	−0.128815
$268$	0	0
$269$	5564.09	1.26115	0.630573	−	0.776130i	$-0.282820\pi$
$269$	5564.09	1.26115	0.630573	+	0.776130i	$0.282820\pi$
$270$	0	0
$271$	2585.39	0.579525	0.289762	−	0.957099i	$-0.406424\pi$
$271$	2585.39	0.579525	0.289762	+	0.957099i	$0.406424\pi$
$272$	0	0
$273$	850.253	0.188497
$274$	0	0
$275$	−65.1531	−0.0142868
$276$	0	0
$277$	884.433	0.191843	0.0959213	−	0.995389i	$-0.469420\pi$
$277$	884.433	0.191843	0.0959213	+	0.995389i	$0.469420\pi$
$278$	0	0
$279$	−3074.76	−0.659789
$280$	0	0
$281$	3165.55	0.672032	0.336016	−	0.941856i	$-0.390920\pi$
$281$	3165.55	0.672032	0.336016	+	0.941856i	$0.390920\pi$
$282$	0	0
$283$	−4425.40	−0.929550	−0.464775	−	0.885429i	$-0.653865\pi$
$283$	−4425.40	−0.929550	−0.464775	+	0.885429i	$0.653865\pi$
$284$	0	0
$285$	643.633	0.133774
$286$	0	0
$287$	7008.58	1.44148
$288$	0	0
$289$	9385.30	1.91030
$290$	0	0
$291$	724.541	0.145956
$292$	0	0
$293$	−2871.70	−0.572582	−0.286291	−	0.958143i	$-0.592422\pi$
$293$	−2871.70	−0.572582	−0.286291	+	0.958143i	$0.592422\pi$
$294$	0	0
$295$	2289.90	0.451942
$296$	0	0
$297$	124.621	0.0243475
$298$	0	0
$299$	3780.18	0.731148
$300$	0	0
$301$	10522.1	2.01489
$302$	0	0
$303$	234.623	0.0444842
$304$	0	0
$305$	−2386.16	−0.447972
$306$	0	0
$307$	−3145.45	−0.584757	−0.292379	−	0.956303i	$-0.594447\pi$
$307$	−3145.45	−0.584757	−0.292379	+	0.956303i	$0.594447\pi$
$308$	0	0
$309$	−1474.33	−0.271430
$310$	0	0
$311$	306.614	0.0559052	0.0279526	−	0.999609i	$-0.491101\pi$
$311$	306.614	0.0559052	0.0279526	+	0.999609i	$0.491101\pi$
$312$	0	0
$313$	−3120.66	−0.563547	−0.281773	−	0.959481i	$-0.590923\pi$
$313$	−3120.66	−0.563547	−0.281773	+	0.959481i	$0.590923\pi$
$314$	0	0
$315$	3731.67	0.667478
$316$	0	0
$317$	8131.12	1.44066	0.720330	−	0.693632i	$-0.243990\pi$
$317$	8131.12	1.44066	0.720330	+	0.693632i	$0.243990\pi$
$318$	0	0
$319$	−176.216	−0.0309286
$320$	0	0
$321$	−610.408	−0.106136
$322$	0	0
$323$	−17122.2	−2.94956
$324$	0	0
$325$	829.796	0.141627
$326$	0	0
$327$	755.341	0.127738
$328$	0	0
$329$	−2174.54	−0.364396
$330$	0	0
$331$	−7877.80	−1.30817	−0.654083	−	0.756423i	$-0.726945\pi$
$331$	−7877.80	−1.30817	−0.654083	+	0.756423i	$0.726945\pi$
$332$	0	0
$333$	−6715.16	−1.10507
$334$	0	0
$335$	−3011.26	−0.491113
$336$	0	0
$337$	−2169.15	−0.350627	−0.175313	−	0.984513i	$-0.556094\pi$
$337$	−2169.15	−0.350627	−0.175313	+	0.984513i	$0.556094\pi$
$338$	0	0
$339$	−1446.59	−0.231765
$340$	0	0
$341$	−305.943	−0.0485857
$342$	0	0
$343$	3589.19	0.565009
$344$	0	0
$345$	−511.918	−0.0798863
$346$	0	0
$347$	−325.475	−0.0503527	−0.0251764	−	0.999683i	$-0.508015\pi$
$347$	−325.475	−0.0503527	−0.0251764	+	0.999683i	$0.508015\pi$
$348$	0	0
$349$	−9311.11	−1.42812	−0.714058	−	0.700087i	$-0.753145\pi$
$349$	−9311.11	−1.42812	−0.714058	+	0.700087i	$0.753145\pi$
$350$	0	0
$351$	−1587.18	−0.241360
$352$	0	0
$353$	3144.94	0.474188	0.237094	−	0.971487i	$-0.423805\pi$
$353$	3144.94	0.474188	0.237094	+	0.971487i	$0.423805\pi$
$354$	0	0
$355$	5459.89	0.816284
$356$	0	0
$357$	3063.09	0.454106
$358$	0	0
$359$	570.424	0.0838603	0.0419302	−	0.999121i	$-0.486649\pi$
$359$	570.424	0.0838603	0.0419302	+	0.999121i	$0.486649\pi$
$360$	0	0
$361$	13644.9	1.98934
$362$	0	0
$363$	−1190.44	−0.172126
$364$	0	0
$365$	−589.388	−0.0845204
$366$	0	0
$367$	1629.17	0.231722	0.115861	−	0.993265i	$-0.463037\pi$
$367$	1629.17	0.231722	0.115861	+	0.993265i	$0.463037\pi$
$368$	0	0
$369$	−6442.12	−0.908844
$370$	0	0
$371$	−1151.89	−0.161195
$372$	0	0
$373$	6985.76	0.969728	0.484864	−	0.874589i	$-0.338869\pi$
$373$	6985.76	0.969728	0.484864	+	0.874589i	$0.338869\pi$
$374$	0	0
$375$	−112.372	−0.0154744
$376$	0	0
$377$	2244.31	0.306599
$378$	0	0
$379$	11101.3	1.50458	0.752291	−	0.658831i	$-0.228949\pi$
$379$	11101.3	1.50458	0.752291	+	0.658831i	$0.228949\pi$
$380$	0	0
$381$	−1033.47	−0.138967
$382$	0	0
$383$	6608.71	0.881695	0.440848	−	0.897582i	$-0.354678\pi$
$383$	6608.71	0.881695	0.440848	+	0.897582i	$0.354678\pi$
$384$	0	0
$385$	371.306	0.0491520
$386$	0	0
$387$	−9671.66	−1.27038
$388$	0	0
$389$	−503.208	−0.0655878	−0.0327939	−	0.999462i	$-0.510440\pi$
$389$	−503.208	−0.0655878	−0.0327939	+	0.999462i	$0.510440\pi$
$390$	0	0
$391$	13618.3	1.76140
$392$	0	0
$393$	−2190.80	−0.281199
$394$	0	0
$395$	−4293.12	−0.546862
$396$	0	0
$397$	−1885.18	−0.238323	−0.119162	−	0.992875i	$-0.538021\pi$
$397$	−1885.18	−0.238323	−0.119162	+	0.992875i	$0.538021\pi$
$398$	0	0
$399$	−3668.05	−0.460231
$400$	0	0
$401$	−13212.7	−1.64541	−0.822706	−	0.568467i	$-0.807537\pi$
$401$	−13212.7	−1.64541	−0.822706	+	0.568467i	$0.807537\pi$
$402$	0	0
$403$	3896.52	0.481636
$404$	0	0
$405$	−3320.96	−0.407456
$406$	0	0
$407$	−668.167	−0.0813755
$408$	0	0
$409$	−8441.37	−1.02054	−0.510268	−	0.860016i	$-0.670454\pi$
$409$	−8441.37	−1.02054	−0.510268	+	0.860016i	$0.670454\pi$
$410$	0	0
$411$	−687.239	−0.0824793
$412$	0	0
$413$	−13050.1	−1.55485
$414$	0	0
$415$	−2828.74	−0.334596
$416$	0	0
$417$	−2290.02	−0.268927
$418$	0	0
$419$	12823.2	1.49512	0.747559	−	0.664195i	$-0.231226\pi$
$419$	12823.2	1.49512	0.747559	+	0.664195i	$0.231226\pi$
$420$	0	0
$421$	2474.24	0.286431	0.143215	−	0.989692i	$-0.454256\pi$
$421$	2474.24	0.286431	0.143215	+	0.989692i	$0.454256\pi$
$422$	0	0
$423$	1998.78	0.229750
$424$	0	0
$425$	2989.39	0.341192
$426$	0	0
$427$	13598.7	1.54119
$428$	0	0
$429$	−77.7635	−0.00875165
$430$	0	0
$431$	−12461.3	−1.39267	−0.696334	−	0.717718i	$-0.745187\pi$
$431$	−12461.3	−1.39267	−0.696334	+	0.717718i	$0.745187\pi$
$432$	0	0
$433$	−12063.0	−1.33883	−0.669414	−	0.742890i	$-0.733455\pi$
$433$	−12063.0	−1.33883	−0.669414	+	0.742890i	$0.733455\pi$
$434$	0	0
$435$	−303.928	−0.0334994
$436$	0	0
$437$	−16307.9	−1.78516
$438$	0	0
$439$	−16480.7	−1.79176	−0.895879	−	0.444299i	$-0.853453\pi$
$439$	−16480.7	−1.79176	−0.895879	+	0.444299i	$0.853453\pi$
$440$	0	0
$441$	−12282.9	−1.32630
$442$	0	0
$443$	2159.19	0.231572	0.115786	−	0.993274i	$-0.463061\pi$
$443$	2159.19	0.231572	0.115786	+	0.993274i	$0.463061\pi$
$444$	0	0
$445$	3125.76	0.332978
$446$	0	0
$447$	−2306.56	−0.244064
$448$	0	0
$449$	5552.38	0.583592	0.291796	−	0.956481i	$-0.405747\pi$
$449$	5552.38	0.583592	0.291796	+	0.956481i	$0.405747\pi$
$450$	0	0
$451$	−641.000	−0.0669257
$452$	0	0
$453$	−471.918	−0.0489463
$454$	0	0
$455$	−4728.99	−0.487249
$456$	0	0
$457$	16667.9	1.70611	0.853055	−	0.521820i	$-0.174747\pi$
$457$	16667.9	1.70611	0.853055	+	0.521820i	$0.174747\pi$
$458$	0	0
$459$	−5717.91	−0.581457
$460$	0	0
$461$	−15255.5	−1.54126	−0.770629	−	0.637284i	$-0.780058\pi$
$461$	−15255.5	−1.54126	−0.770629	+	0.637284i	$0.780058\pi$
$462$	0	0
$463$	6806.85	0.683242	0.341621	−	0.939838i	$-0.389024\pi$
$463$	6806.85	0.683242	0.341621	+	0.939838i	$0.389024\pi$
$464$	0	0
$465$	−527.673	−0.0526242
$466$	0	0
$467$	6455.78	0.639696	0.319848	−	0.947469i	$-0.396368\pi$
$467$	6455.78	0.639696	0.319848	+	0.947469i	$0.396368\pi$
$468$	0	0
$469$	17161.1	1.68961
$470$	0	0
$471$	3208.99	0.313933
$472$	0	0
$473$	−962.343	−0.0935488
$474$	0	0
$475$	−3579.80	−0.345794
$476$	0	0
$477$	1058.79	0.101633
$478$	0	0
$479$	−8481.34	−0.809024	−0.404512	−	0.914533i	$-0.632559\pi$
$479$	−8481.34	−0.809024	−0.404512	+	0.914533i	$0.632559\pi$
$480$	0	0
$481$	8509.84	0.806685
$482$	0	0
$483$	2917.41	0.274838
$484$	0	0
$485$	−4029.80	−0.377286
$486$	0	0
$487$	−10966.2	−1.02038	−0.510191	−	0.860061i	$-0.670425\pi$
$487$	−10966.2	−1.02038	−0.510191	+	0.860061i	$0.670425\pi$
$488$	0	0
$489$	1304.38	0.120626
$490$	0	0
$491$	6784.15	0.623553	0.311777	−	0.950155i	$-0.399076\pi$
$491$	6784.15	0.623553	0.311777	+	0.950155i	$0.399076\pi$
$492$	0	0
$493$	8085.26	0.738624
$494$	0	0
$495$	−341.296	−0.0309901
$496$	0	0
$497$	−31115.8	−2.80832
$498$	0	0
$499$	−17785.8	−1.59560	−0.797799	−	0.602924i	$-0.794002\pi$
$499$	−17785.8	−1.59560	−0.797799	+	0.602924i	$0.794002\pi$
$500$	0	0
$501$	2949.89	0.263057
$502$	0	0
$503$	−8440.25	−0.748175	−0.374087	−	0.927393i	$-0.622044\pi$
$503$	−8440.25	−0.748175	−0.374087	+	0.927393i	$0.622044\pi$
$504$	0	0
$505$	−1304.94	−0.114988
$506$	0	0
$507$	−984.654	−0.0862525
$508$	0	0
$509$	−10923.4	−0.951222	−0.475611	−	0.879656i	$-0.657773\pi$
$509$	−10923.4	−0.951222	−0.475611	+	0.879656i	$0.657773\pi$
$510$	0	0
$511$	3358.91	0.290782
$512$	0	0
$513$	6847.20	0.589301
$514$	0	0
$515$	8200.05	0.701626
$516$	0	0
$517$	198.882	0.0169184
$518$	0	0
$519$	−1246.90	−0.105458
$520$	0	0
$521$	−9879.76	−0.830787	−0.415394	−	0.909642i	$-0.636356\pi$
$521$	−9879.76	−0.830787	−0.415394	+	0.909642i	$0.636356\pi$
$522$	0	0
$523$	105.197	0.00879531	0.00439766	−	0.999990i	$-0.498600\pi$
$523$	105.197	0.00879531	0.00439766	+	0.999990i	$0.498600\pi$
$524$	0	0
$525$	640.408	0.0532375
$526$	0	0
$527$	14037.4	1.16030
$528$	0	0
$529$	803.653	0.0660519
$530$	0	0
$531$	11995.3	0.980325
$532$	0	0
$533$	8163.84	0.663443
$534$	0	0
$535$	3395.01	0.274353
$536$	0	0
$537$	−2705.84	−0.217440
$538$	0	0
$539$	−1222.17	−0.0976668
$540$	0	0
$541$	−19055.6	−1.51435	−0.757175	−	0.653212i	$-0.773421\pi$
$541$	−19055.6	−1.51435	−0.757175	+	0.653212i	$0.773421\pi$
$542$	0	0
$543$	2071.37	0.163704
$544$	0	0
$545$	−4201.10	−0.330193
$546$	0	0
$547$	−3910.22	−0.305647	−0.152824	−	0.988253i	$-0.548837\pi$
$547$	−3910.22	−0.305647	−0.152824	+	0.988253i	$0.548837\pi$
$548$	0	0
$549$	−12499.6	−0.971712
$550$	0	0
$551$	−9682.11	−0.748587
$552$	0	0
$553$	24466.4	1.88141
$554$	0	0
$555$	−1152.42	−0.0881395
$556$	0	0
$557$	−1727.52	−0.131413	−0.0657067	−	0.997839i	$-0.520930\pi$
$557$	−1727.52	−0.131413	−0.0657067	+	0.997839i	$0.520930\pi$
$558$	0	0
$559$	12256.5	0.927360
$560$	0	0
$561$	−280.148	−0.0210835
$562$	0	0
$563$	20905.0	1.56491	0.782453	−	0.622709i	$-0.213968\pi$
$563$	20905.0	1.56491	0.782453	+	0.622709i	$0.213968\pi$
$564$	0	0
$565$	8045.76	0.599093
$566$	0	0
$567$	18926.1	1.40180
$568$	0	0
$569$	−256.523	−0.0188998	−0.00944990	−	0.999955i	$-0.503008\pi$
$569$	−256.523	−0.0188998	−0.00944990	+	0.999955i	$0.503008\pi$
$570$	0	0
$571$	−15793.4	−1.15750	−0.578749	−	0.815506i	$-0.696459\pi$
$571$	−15793.4	−1.15750	−0.578749	+	0.815506i	$0.696459\pi$
$572$	0	0
$573$	149.812	0.0109223
$574$	0	0
$575$	2847.22	0.206500
$576$	0	0
$577$	−5923.65	−0.427391	−0.213696	−	0.976900i	$-0.568550\pi$
$577$	−5923.65	−0.427391	−0.213696	+	0.976900i	$0.568550\pi$
$578$	0	0
$579$	435.708	0.0312736
$580$	0	0
$581$	16120.9	1.15113
$582$	0	0
$583$	105.351	0.00748405
$584$	0	0
$585$	4346.78	0.307209
$586$	0	0
$587$	−11730.7	−0.824832	−0.412416	−	0.910996i	$-0.635315\pi$
$587$	−11730.7	−0.824832	−0.412416	+	0.910996i	$0.635315\pi$
$588$	0	0
$589$	−16809.8	−1.17596
$590$	0	0
$591$	−1877.16	−0.130653
$592$	0	0
$593$	2781.00	0.192584	0.0962919	−	0.995353i	$-0.469302\pi$
$593$	2781.00	0.192584	0.0962919	+	0.995353i	$0.469302\pi$
$594$	0	0
$595$	−17036.5	−1.17383
$596$	0	0
$597$	−3058.27	−0.209659
$598$	0	0
$599$	22999.7	1.56885	0.784425	−	0.620224i	$-0.212958\pi$
$599$	22999.7	1.56885	0.784425	+	0.620224i	$0.212958\pi$
$600$	0	0
$601$	−2680.42	−0.181925	−0.0909624	−	0.995854i	$-0.528994\pi$
$601$	−2680.42	−0.181925	−0.0909624	+	0.995854i	$0.528994\pi$
$602$	0	0
$603$	−15774.1	−1.06529
$604$	0	0
$605$	6621.04	0.444932
$606$	0	0
$607$	−16977.3	−1.13523	−0.567617	−	0.823293i	$-0.692135\pi$
$607$	−16977.3	−1.13523	−0.567617	+	0.823293i	$0.692135\pi$
$608$	0	0
$609$	1732.08	0.115250
$610$	0	0
$611$	−2532.98	−0.167714
$612$	0	0
$613$	24124.3	1.58951	0.794755	−	0.606930i	$-0.207599\pi$
$613$	24124.3	1.58951	0.794755	+	0.606930i	$0.207599\pi$
$614$	0	0
$615$	−1105.56	−0.0724887
$616$	0	0
$617$	12286.0	0.801644	0.400822	−	0.916156i	$-0.368725\pi$
$617$	12286.0	0.801644	0.400822	+	0.916156i	$0.368725\pi$
$618$	0	0
$619$	−6240.17	−0.405192	−0.202596	−	0.979262i	$-0.564938\pi$
$619$	−6240.17	−0.405192	−0.202596	+	0.979262i	$0.564938\pi$
$620$	0	0
$621$	−5445.98	−0.351915
$622$	0	0
$623$	−17813.6	−1.14557
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	335.477	0.0213679
$628$	0	0
$629$	30657.2	1.94338
$630$	0	0
$631$	8142.22	0.513687	0.256844	−	0.966453i	$-0.417317\pi$
$631$	8142.22	0.513687	0.256844	+	0.966453i	$0.417317\pi$
$632$	0	0
$633$	2019.32	0.126794
$634$	0	0
$635$	5748.02	0.359217
$636$	0	0
$637$	15565.6	0.968182
$638$	0	0
$639$	28600.9	1.77063
$640$	0	0
$641$	6380.36	0.393150	0.196575	−	0.980489i	$-0.437018\pi$
$641$	6380.36	0.393150	0.196575	+	0.980489i	$0.437018\pi$
$642$	0	0
$643$	−18308.0	−1.12286	−0.561430	−	0.827524i	$-0.689749\pi$
$643$	−18308.0	−1.12286	−0.561430	+	0.827524i	$0.689749\pi$
$644$	0	0
$645$	−1659.80	−0.101325
$646$	0	0
$647$	−21497.9	−1.30629	−0.653145	−	0.757233i	$-0.726551\pi$
$647$	−21497.9	−1.30629	−0.653145	+	0.757233i	$0.726551\pi$
$648$	0	0
$649$	1193.55	0.0721895
$650$	0	0
$651$	3007.20	0.181047
$652$	0	0
$653$	16035.4	0.960972	0.480486	−	0.877002i	$-0.340460\pi$
$653$	16035.4	0.960972	0.480486	+	0.877002i	$0.340460\pi$
$654$	0	0
$655$	12184.9	0.726877
$656$	0	0
$657$	−3087.43	−0.183337
$658$	0	0
$659$	−12259.7	−0.724688	−0.362344	−	0.932044i	$-0.618023\pi$
$659$	−12259.7	−0.724688	−0.362344	+	0.932044i	$0.618023\pi$
$660$	0	0
$661$	−7515.22	−0.442221	−0.221111	−	0.975249i	$-0.570968\pi$
$661$	−7515.22	−0.442221	−0.221111	+	0.975249i	$0.570968\pi$
$662$	0	0
$663$	3567.99	0.209003
$664$	0	0
$665$	20401.2	1.18966
$666$	0	0
$667$	7700.74	0.447037
$668$	0	0
$669$	−1015.51	−0.0586874
$670$	0	0
$671$	−1243.73	−0.0715552
$672$	0	0
$673$	17438.7	0.998829	0.499415	−	0.866363i	$-0.333548\pi$
$673$	17438.7	0.998829	0.499415	+	0.866363i	$0.333548\pi$
$674$	0	0
$675$	−1195.46	−0.0681678
$676$	0	0
$677$	−14679.1	−0.833331	−0.416665	−	0.909060i	$-0.636801\pi$
$677$	−14679.1	−0.833331	−0.416665	+	0.909060i	$0.636801\pi$
$678$	0	0
$679$	22965.7	1.29800
$680$	0	0
$681$	1522.77	0.0856866
$682$	0	0
$683$	30494.6	1.70841	0.854204	−	0.519937i	$-0.174045\pi$
$683$	30494.6	1.70841	0.854204	+	0.519937i	$0.174045\pi$
$684$	0	0
$685$	3822.33	0.213202
$686$	0	0
$687$	4720.29	0.262140
$688$	0	0
$689$	−1341.76	−0.0741903
$690$	0	0
$691$	−13887.9	−0.764576	−0.382288	−	0.924043i	$-0.624864\pi$
$691$	−13887.9	−0.764576	−0.382288	+	0.924043i	$0.624864\pi$
$692$	0	0
$693$	1945.04	0.106617
$694$	0	0
$695$	12736.8	0.695155
$696$	0	0
$697$	29410.7	1.59829
$698$	0	0
$699$	−5432.17	−0.293939
$700$	0	0
$701$	−11296.0	−0.608622	−0.304311	−	0.952573i	$-0.598426\pi$
$701$	−11296.0	−0.608622	−0.304311	+	0.952573i	$0.598426\pi$
$702$	0	0
$703$	−36712.0	−1.96959
$704$	0	0
$705$	343.020	0.0183247
$706$	0	0
$707$	7436.82	0.395602
$708$	0	0
$709$	24405.6	1.29277	0.646384	−	0.763013i	$-0.276281\pi$
$709$	24405.6	1.29277	0.646384	+	0.763013i	$0.276281\pi$
$710$	0	0
$711$	−22489.0	−1.18622
$712$	0	0
$713$	13369.8	0.702251
$714$	0	0
$715$	432.510	0.0226223
$716$	0	0
$717$	−346.481	−0.0180468
$718$	0	0
$719$	−11883.4	−0.616379	−0.308189	−	0.951325i	$-0.599723\pi$
$719$	−11883.4	−0.616379	−0.308189	+	0.951325i	$0.599723\pi$
$720$	0	0
$721$	−46731.9	−2.41385
$722$	0	0
$723$	4367.90	0.224680
$724$	0	0
$725$	1690.41	0.0865934
$726$	0	0
$727$	21154.5	1.07920	0.539599	−	0.841922i	$-0.318576\pi$
$727$	21154.5	1.07920	0.539599	+	0.841922i	$0.318576\pi$
$728$	0	0
$729$	−16235.7	−0.824861
$730$	0	0
$731$	44154.7	2.23409
$732$	0	0
$733$	5806.54	0.292591	0.146296	−	0.989241i	$-0.453265\pi$
$733$	5806.54	0.292591	0.146296	+	0.989241i	$0.453265\pi$
$734$	0	0
$735$	−2107.92	−0.105785
$736$	0	0
$737$	−1569.54	−0.0784462
$738$	0	0
$739$	14389.0	0.716248	0.358124	−	0.933674i	$-0.383416\pi$
$739$	14389.0	0.716248	0.358124	+	0.933674i	$0.383416\pi$
$740$	0	0
$741$	−4272.67	−0.211822
$742$	0	0
$743$	−18366.0	−0.906839	−0.453420	−	0.891297i	$-0.649796\pi$
$743$	−18366.0	−0.906839	−0.453420	+	0.891297i	$0.649796\pi$
$744$	0	0
$745$	12828.8	0.630886
$746$	0	0
$747$	−14818.0	−0.725785
$748$	0	0
$749$	−19348.1	−0.943876
$750$	0	0
$751$	19355.0	0.940444	0.470222	−	0.882548i	$-0.344174\pi$
$751$	19355.0	0.940444	0.470222	+	0.882548i	$0.344174\pi$
$752$	0	0
$753$	−3402.41	−0.164662
$754$	0	0
$755$	2624.74	0.126522
$756$	0	0
$757$	494.408	0.0237379	0.0118689	−	0.999930i	$-0.496222\pi$
$757$	494.408	0.0237379	0.0118689	+	0.999930i	$0.496222\pi$
$758$	0	0
$759$	−266.824	−0.0127604
$760$	0	0
$761$	−2310.20	−0.110045	−0.0550227	−	0.998485i	$-0.517523\pi$
$761$	−2310.20	−0.110045	−0.0550227	+	0.998485i	$0.517523\pi$
$762$	0	0
$763$	23942.0	1.13599
$764$	0	0
$765$	15659.5	0.740093
$766$	0	0
$767$	−15201.2	−0.715623
$768$	0	0
$769$	24590.9	1.15315	0.576573	−	0.817045i	$-0.304389\pi$
$769$	24590.9	1.15315	0.576573	+	0.817045i	$0.304389\pi$
$770$	0	0
$771$	3328.34	0.155470
$772$	0	0
$773$	22102.6	1.02843	0.514214	−	0.857662i	$-0.328084\pi$
$773$	22102.6	1.02843	0.514214	+	0.857662i	$0.328084\pi$
$774$	0	0
$775$	2934.85	0.136029
$776$	0	0
$777$	6567.61	0.303232
$778$	0	0
$779$	−35219.3	−1.61985
$780$	0	0
$781$	2845.83	0.130386
$782$	0	0
$783$	−3233.30	−0.147572
$784$	0	0
$785$	−17848.0	−0.811492
$786$	0	0
$787$	29715.0	1.34590	0.672952	−	0.739686i	$-0.265026\pi$
$787$	29715.0	1.34590	0.672952	+	0.739686i	$0.265026\pi$
$788$	0	0
$789$	3162.40	0.142693
$790$	0	0
$791$	−45852.6	−2.06110
$792$	0	0
$793$	15840.2	0.709335
$794$	0	0
$795$	181.704	0.00810613
$796$	0	0
$797$	1237.06	0.0549799	0.0274899	−	0.999622i	$-0.491249\pi$
$797$	1237.06	0.0549799	0.0274899	+	0.999622i	$0.491249\pi$
$798$	0	0
$799$	−9125.20	−0.404038
$800$	0	0
$801$	16373.9	0.722274
$802$	0	0
$803$	−307.203	−0.0135006
$804$	0	0
$805$	−16226.2	−0.710435
$806$	0	0
$807$	5002.00	0.218189
$808$	0	0
$809$	−4675.39	−0.203186	−0.101593	−	0.994826i	$-0.532394\pi$
$809$	−4675.39	−0.203186	−0.101593	+	0.994826i	$0.532394\pi$
$810$	0	0
$811$	−3243.36	−0.140431	−0.0702156	−	0.997532i	$-0.522369\pi$
$811$	−3243.36	−0.140431	−0.0702156	+	0.997532i	$0.522369\pi$
$812$	0	0
$813$	2324.21	0.100263
$814$	0	0
$815$	−7254.80	−0.311809
$816$	0	0
$817$	−52875.3	−2.26423
$818$	0	0
$819$	−24772.2	−1.05691
$820$	0	0
$821$	−33551.4	−1.42625	−0.713126	−	0.701036i	$-0.752721\pi$
$821$	−33551.4	−1.42625	−0.713126	+	0.701036i	$0.752721\pi$
$822$	0	0
$823$	−6365.66	−0.269615	−0.134807	−	0.990872i	$-0.543042\pi$
$823$	−6365.66	−0.269615	−0.134807	+	0.990872i	$0.543042\pi$
$824$	0	0
$825$	−58.5713	−0.00247175
$826$	0	0
$827$	−18980.7	−0.798094	−0.399047	−	0.916930i	$-0.630659\pi$
$827$	−18980.7	−0.798094	−0.399047	+	0.916930i	$0.630659\pi$
$828$	0	0
$829$	−33674.2	−1.41080	−0.705400	−	0.708809i	$-0.749233\pi$
$829$	−33674.2	−1.41080	−0.705400	+	0.708809i	$0.749233\pi$
$830$	0	0
$831$	795.087	0.0331904
$832$	0	0
$833$	56076.0	2.33244
$834$	0	0
$835$	−16406.9	−0.679981
$836$	0	0
$837$	−5613.58	−0.231821
$838$	0	0
$839$	4639.47	0.190909	0.0954544	−	0.995434i	$-0.469570\pi$
$839$	4639.47	0.190909	0.0954544	+	0.995434i	$0.469570\pi$
$840$	0	0
$841$	−19817.0	−0.812540
$842$	0	0
$843$	2845.77	0.116267
$844$	0	0
$845$	5476.51	0.222956
$846$	0	0
$847$	−37733.2	−1.53073
$848$	0	0
$849$	−3978.34	−0.160820
$850$	0	0
$851$	29199.2	1.17619
$852$	0	0
$853$	1646.37	0.0660850	0.0330425	−	0.999454i	$-0.489480\pi$
$853$	1646.37	0.0660850	0.0330425	+	0.999454i	$0.489480\pi$
$854$	0	0
$855$	−18752.3	−0.750076
$856$	0	0
$857$	−46767.8	−1.86413	−0.932064	−	0.362295i	$-0.881993\pi$
$857$	−46767.8	−1.86413	−0.932064	+	0.362295i	$0.881993\pi$
$858$	0	0
$859$	11466.7	0.455460	0.227730	−	0.973724i	$-0.426870\pi$
$859$	11466.7	0.455460	0.227730	+	0.973724i	$0.426870\pi$
$860$	0	0
$861$	6300.57	0.249388
$862$	0	0
$863$	42704.1	1.68443	0.842216	−	0.539140i	$-0.181250\pi$
$863$	42704.1	1.68443	0.842216	+	0.539140i	$0.181250\pi$
$864$	0	0
$865$	6935.06	0.272600
$866$	0	0
$867$	8437.19	0.330498
$868$	0	0
$869$	−2237.68	−0.0873511
$870$	0	0
$871$	19989.9	0.777647
$872$	0	0
$873$	−21109.6	−0.818385
$874$	0	0
$875$	−3561.86	−0.137615
$876$	0	0
$877$	18376.9	0.707574	0.353787	−	0.935326i	$-0.384894\pi$
$877$	18376.9	0.707574	0.353787	+	0.935326i	$0.384894\pi$
$878$	0	0
$879$	−2581.60	−0.0990616
$880$	0	0
$881$	−7024.26	−0.268619	−0.134310	−	0.990939i	$-0.542882\pi$
$881$	−7024.26	−0.268619	−0.134310	+	0.990939i	$0.542882\pi$
$882$	0	0
$883$	41747.1	1.59106	0.795528	−	0.605917i	$-0.207194\pi$
$883$	41747.1	1.59106	0.795528	+	0.605917i	$0.207194\pi$
$884$	0	0
$885$	2058.57	0.0781900
$886$	0	0
$887$	−5863.18	−0.221946	−0.110973	−	0.993823i	$-0.535397\pi$
$887$	−5863.18	−0.221946	−0.110973	+	0.993823i	$0.535397\pi$
$888$	0	0
$889$	−32757.8	−1.23584
$890$	0	0
$891$	−1730.97	−0.0650836
$892$	0	0
$893$	10927.4	0.409488
$894$	0	0
$895$	15049.5	0.562066
$896$	0	0
$897$	3398.30	0.126495
$898$	0	0
$899$	7937.74	0.294481
$900$	0	0
$901$	−4833.78	−0.178731
$902$	0	0
$903$	9459.14	0.348594
$904$	0	0
$905$	−11520.7	−0.423161
$906$	0	0
$907$	13945.6	0.510536	0.255268	−	0.966870i	$-0.417836\pi$
$907$	13945.6	0.510536	0.255268	+	0.966870i	$0.417836\pi$
$908$	0	0
$909$	−6835.75	−0.249425
$910$	0	0
$911$	38202.8	1.38937	0.694685	−	0.719314i	$-0.255544\pi$
$911$	38202.8	1.38937	0.694685	+	0.719314i	$0.255544\pi$
$912$	0	0
$913$	−1474.41	−0.0534456
$914$	0	0
$915$	−2145.11	−0.0775030
$916$	0	0
$917$	−69441.7	−2.50073
$918$	0	0
$919$	−8586.98	−0.308225	−0.154112	−	0.988053i	$-0.549252\pi$
$919$	−8586.98	−0.308225	−0.154112	+	0.988053i	$0.549252\pi$
$920$	0	0
$921$	−2827.70	−0.101168
$922$	0	0
$923$	−36244.7	−1.29254
$924$	0	0
$925$	6409.59	0.227834
$926$	0	0
$927$	42954.9	1.52192
$928$	0	0
$929$	50645.2	1.78861	0.894303	−	0.447461i	$-0.147672\pi$
$929$	50645.2	1.78861	0.894303	+	0.447461i	$0.147672\pi$
$930$	0	0
$931$	−67151.1	−2.36390
$932$	0	0
$933$	275.640	0.00967208
$934$	0	0
$935$	1558.14	0.0544992
$936$	0	0
$937$	26068.7	0.908887	0.454444	−	0.890775i	$-0.349838\pi$
$937$	26068.7	0.908887	0.454444	+	0.890775i	$0.349838\pi$
$938$	0	0
$939$	−2805.41	−0.0974985
$940$	0	0
$941$	4036.78	0.139846	0.0699230	−	0.997552i	$-0.477725\pi$
$941$	4036.78	0.139846	0.0699230	+	0.997552i	$0.477725\pi$
$942$	0	0
$943$	28012.0	0.967334
$944$	0	0
$945$	6812.90	0.234522
$946$	0	0
$947$	−47790.1	−1.63988	−0.819942	−	0.572446i	$-0.805995\pi$
$947$	−47790.1	−1.63988	−0.819942	+	0.572446i	$0.805995\pi$
$948$	0	0
$949$	3912.57	0.133833
$950$	0	0
$951$	7309.71	0.249247
$952$	0	0
$953$	−49284.8	−1.67523	−0.837613	−	0.546264i	$-0.816050\pi$
$953$	−49284.8	−1.67523	−0.837613	+	0.546264i	$0.816050\pi$
$954$	0	0
$955$	−833.235	−0.0282333
$956$	0	0
$957$	−158.415	−0.00535092
$958$	0	0
$959$	−21783.4	−0.733495
$960$	0	0
$961$	−16009.7	−0.537400
$962$	0	0
$963$	17784.3	0.595110
$964$	0	0
$965$	−2423.35	−0.0808397
$966$	0	0
$967$	19830.2	0.659459	0.329729	−	0.944076i	$-0.393043\pi$
$967$	19830.2	0.659459	0.329729	+	0.944076i	$0.393043\pi$
$968$	0	0
$969$	−15392.5	−0.510299
$970$	0	0
$971$	24538.6	0.811000	0.405500	−	0.914095i	$-0.367097\pi$
$971$	24538.6	0.811000	0.405500	+	0.914095i	$0.367097\pi$
$972$	0	0
$973$	−72586.5	−2.39159
$974$	0	0
$975$	745.969	0.0245027
$976$	0	0
$977$	−15113.6	−0.494909	−0.247454	−	0.968900i	$-0.579594\pi$
$977$	−15113.6	−0.494909	−0.247454	+	0.968900i	$0.579594\pi$
$978$	0	0
$979$	1629.22	0.0531870
$980$	0	0
$981$	−22006.9	−0.716235
$982$	0	0
$983$	−13262.7	−0.430330	−0.215165	−	0.976578i	$-0.569029\pi$
$983$	−13262.7	−0.430330	−0.215165	+	0.976578i	$0.569029\pi$
$984$	0	0
$985$	10440.5	0.337728
$986$	0	0
$987$	−1954.87	−0.0630436
$988$	0	0
$989$	42054.8	1.35214
$990$	0	0
$991$	7053.73	0.226104	0.113052	−	0.993589i	$-0.463937\pi$
$991$	7053.73	0.226104	0.113052	+	0.993589i	$0.463937\pi$
$992$	0	0
$993$	−7081.98	−0.226324
$994$	0	0
$995$	17009.7	0.541953
$996$	0	0
$997$	−38686.1	−1.22889	−0.614443	−	0.788961i	$-0.710619\pi$
$997$	−38686.1	−1.22889	−0.614443	+	0.788961i	$0.710619\pi$
$998$	0	0
$999$	−12259.8	−0.388273

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.4.a.o.1.2		2
4.3	odd	2		320.4.a.s.1.1		2
5.4	even	2		1600.4.a.cn.1.1		2
8.3	odd	2		160.4.a.c.1.2	✓	2
8.5	even	2		160.4.a.g.1.1	yes	2
16.3	odd	4		1280.4.d.x.641.2		4
16.5	even	4		1280.4.d.q.641.2		4
16.11	odd	4		1280.4.d.x.641.3		4
16.13	even	4		1280.4.d.q.641.3		4
20.19	odd	2		1600.4.a.cd.1.2		2
24.5	odd	2		1440.4.a.x.1.2		2
24.11	even	2		1440.4.a.t.1.1		2
40.3	even	4		800.4.c.i.449.3		4
40.13	odd	4		800.4.c.k.449.2		4
40.19	odd	2		800.4.a.s.1.1		2
40.27	even	4		800.4.c.i.449.2		4
40.29	even	2		800.4.a.m.1.2		2
40.37	odd	4		800.4.c.k.449.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.a.c.1.2	✓	2	8.3	odd	2
160.4.a.g.1.1	yes	2	8.5	even	2
320.4.a.o.1.2		2	1.1	even	1	trivial
320.4.a.s.1.1		2	4.3	odd	2
800.4.a.m.1.2		2	40.29	even	2
800.4.a.s.1.1		2	40.19	odd	2
800.4.c.i.449.2		4	40.27	even	4
800.4.c.i.449.3		4	40.3	even	4
800.4.c.k.449.2		4	40.13	odd	4
800.4.c.k.449.3		4	40.37	odd	4
1280.4.d.q.641.2		4	16.5	even	4
1280.4.d.q.641.3		4	16.13	even	4
1280.4.d.x.641.2		4	16.3	odd	4
1280.4.d.x.641.3		4	16.11	odd	4
1440.4.a.t.1.1		2	24.11	even	2
1440.4.a.x.1.2		2	24.5	odd	2
1600.4.a.cd.1.2		2	20.19	odd	2
1600.4.a.cn.1.1		2	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	320.a (trivial)

\(f(q)\)	\(=\)	\(q+0.898979 q^{3} -5.00000 q^{5} +28.4949 q^{7} -26.1918 q^{9} -2.60612 q^{11} +33.1918 q^{13} -4.49490 q^{15} +119.576 q^{17} -143.192 q^{19} +25.6163 q^{21} +113.889 q^{23} +25.0000 q^{25} -47.8184 q^{27} +67.6163 q^{29} +117.394 q^{31} -2.34285 q^{33} -142.474 q^{35} +256.384 q^{37} +29.8388 q^{39} +245.959 q^{41} +369.262 q^{43} +130.959 q^{45} -76.3133 q^{47} +468.959 q^{49} +107.496 q^{51} -40.4245 q^{53} +13.0306 q^{55} -128.727 q^{57} -457.980 q^{59} +477.233 q^{61} -746.334 q^{63} -165.959 q^{65} +602.252 q^{67} +102.384 q^{69} -1091.98 q^{71} +117.878 q^{73} +22.4745 q^{75} -74.2612 q^{77} +858.624 q^{79} +664.192 q^{81} +565.748 q^{83} -597.878 q^{85} +60.7857 q^{87} -625.151 q^{89} +945.798 q^{91} +105.535 q^{93} +715.959 q^{95} +805.959 q^{97} +68.2591 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{3} - 10 q^{5} + 8 q^{7} + 26 q^{9} - 64 q^{11} - 12 q^{13} + 40 q^{15} + 4 q^{17} - 208 q^{19} + 208 q^{21} + 120 q^{23} + 50 q^{25} - 272 q^{27} + 292 q^{29} + 176 q^{31} + 544 q^{33} - 40 q^{35}+ \cdots - 3136 q^{99}+O(q^{100}) \) 2 * q - 8 * q^3 - 10 * q^5 + 8 * q^7 + 26 * q^9 - 64 * q^11 - 12 * q^13 + 40 * q^15 + 4 * q^17 - 208 * q^19 + 208 * q^21 + 120 * q^23 + 50 * q^25 - 272 * q^27 + 292 * q^29 + 176 * q^31 + 544 * q^33 - 40 * q^35 + 356 * q^37 + 432 * q^39 + 100 * q^41 + 376 * q^43 - 130 * q^45 - 280 * q^47 + 546 * q^49 + 1136 * q^51 - 316 * q^53 + 320 * q^55 + 448 * q^57 - 720 * q^59 + 1268 * q^61 - 1816 * q^63 + 60 * q^65 + 744 * q^67 + 48 * q^69 - 48 * q^71 - 940 * q^73 - 200 * q^75 + 1184 * q^77 + 32 * q^79 + 1250 * q^81 + 1592 * q^83 - 20 * q^85 - 1936 * q^87 - 780 * q^89 + 1872 * q^91 - 416 * q^93 + 1040 * q^95 + 1220 * q^97 - 3136 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more