LMFDB - Embedded newform 480.4.a.o.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [480,4,Mod(1,480)] 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("480.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(480, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	480.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-6,0,10,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$28.3209168028$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{89}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 22 $ x^2 - x - 22
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.21699$ of defining polynomial
Character	$\chi$	$=$	480.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +5.00000 q^{5} -24.8680 q^{7} +9.00000 q^{9} +25.7359 q^{11} +60.6039 q^{13} -15.0000 q^{15} -28.6039 q^{17} -86.6039 q^{19} +74.6039 q^{21} +52.3398 q^{23} +25.0000 q^{25} -27.0000 q^{27} +6.00000 q^{29} -84.8680 q^{31} -77.2078 q^{33} -124.340 q^{35} -448.227 q^{37} -181.812 q^{39} +183.208 q^{41} -252.000 q^{43} +45.0000 q^{45} -41.9243 q^{47} +275.416 q^{49} +85.8117 q^{51} -228.792 q^{53} +128.680 q^{55} +259.812 q^{57} -179.849 q^{59} +480.039 q^{61} -223.812 q^{63} +303.019 q^{65} -855.472 q^{67} -157.019 q^{69} -675.775 q^{71} -621.623 q^{73} -75.0000 q^{75} -640.000 q^{77} +513.699 q^{79} +81.0000 q^{81} -1284.30 q^{83} -143.019 q^{85} -18.0000 q^{87} -1004.87 q^{89} -1507.10 q^{91} +254.604 q^{93} -433.019 q^{95} +300.416 q^{97} +231.623 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 10 q^{5} - 12 q^{7} + 18 q^{9} - 24 q^{11} + 8 q^{13} - 30 q^{15} + 56 q^{17} - 60 q^{19} + 36 q^{21} - 84 q^{23} + 50 q^{25} - 54 q^{27} + 12 q^{29} - 132 q^{31} + 72 q^{33} - 60 q^{35}+ \cdots - 216 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 10 * q^5 - 12 * q^7 + 18 * q^9 - 24 * q^11 + 8 * q^13 - 30 * q^15 + 56 * q^17 - 60 * q^19 + 36 * q^21 - 84 * q^23 + 50 * q^25 - 54 * q^27 + 12 * q^29 - 132 * q^31 + 72 * q^33 - 60 * q^35 - 104 * q^37 - 24 * q^39 + 140 * q^41 - 504 * q^43 + 90 * q^45 - 348 * q^47 + 98 * q^49 - 168 * q^51 - 684 * q^53 - 120 * q^55 + 180 * q^57 - 888 * q^59 - 172 * q^61 - 108 * q^63 + 40 * q^65 - 1560 * q^67 + 252 * q^69 - 144 * q^71 - 564 * q^73 - 150 * q^75 - 1280 * q^77 + 84 * q^79 + 162 * q^81 - 1512 * q^83 + 280 * q^85 - 36 * q^87 + 28 * q^89 - 2184 * q^91 + 396 * q^93 - 300 * q^95 + 148 * q^97 - 216 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−24.8680	−1.34274	−0.671372	−	0.741121i	$-0.734295\pi$
$7$	−24.8680	−1.34274	−0.671372	+	0.741121i	$0.734295\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	25.7359	0.705425	0.352712	−	0.935732i	$-0.385259\pi$
$11$	25.7359	0.705425	0.352712	+	0.935732i	$0.385259\pi$
$12$	0	0
$13$	60.6039	1.29296	0.646481	−	0.762930i	$-0.276240\pi$
$13$	60.6039	1.29296	0.646481	+	0.762930i	$0.276240\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	−28.6039	−0.408086	−0.204043	−	0.978962i	$-0.565408\pi$
$17$	−28.6039	−0.408086	−0.204043	+	0.978962i	$0.565408\pi$
$18$	0	0
$19$	−86.6039	−1.04570	−0.522850	−	0.852425i	$-0.675131\pi$
$19$	−86.6039	−1.04570	−0.522850	+	0.852425i	$0.675131\pi$
$20$	0	0
$21$	74.6039	0.775233
$22$	0	0
$23$	52.3398	0.474505	0.237252	−	0.971448i	$-0.423753\pi$
$23$	52.3398	0.474505	0.237252	+	0.971448i	$0.423753\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	6.00000	0.0384197	0.0192099	−	0.999815i	$-0.493885\pi$
$29$	6.00000	0.0384197	0.0192099	+	0.999815i	$0.493885\pi$
$30$	0	0
$31$	−84.8680	−0.491701	−0.245851	−	0.969308i	$-0.579067\pi$
$31$	−84.8680	−0.491701	−0.245851	+	0.969308i	$0.579067\pi$
$32$	0	0
$33$	−77.2078	−0.407277
$34$	0	0
$35$	−124.340	−0.600493
$36$	0	0
$37$	−448.227	−1.99157	−0.995785	−	0.0917169i	$-0.970765\pi$
$37$	−448.227	−1.99157	−0.995785	+	0.0917169i	$0.970765\pi$
$38$	0	0
$39$	−181.812	−0.746491
$40$	0	0
$41$	183.208	0.697860	0.348930	−	0.937149i	$-0.386545\pi$
$41$	183.208	0.697860	0.348930	+	0.937149i	$0.386545\pi$
$42$	0	0
$43$	−252.000	−0.893713	−0.446856	−	0.894606i	$-0.647456\pi$
$43$	−252.000	−0.893713	−0.446856	+	0.894606i	$0.647456\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	−41.9243	−0.130112	−0.0650562	−	0.997882i	$-0.520723\pi$
$47$	−41.9243	−0.130112	−0.0650562	+	0.997882i	$0.520723\pi$
$48$	0	0
$49$	275.416	0.802961
$50$	0	0
$51$	85.8117	0.235609
$52$	0	0
$53$	−228.792	−0.592963	−0.296481	−	0.955039i	$-0.595813\pi$
$53$	−228.792	−0.592963	−0.296481	+	0.955039i	$0.595813\pi$
$54$	0	0
$55$	128.680	0.315476
$56$	0	0
$57$	259.812	0.603735
$58$	0	0
$59$	−179.849	−0.396852	−0.198426	−	0.980116i	$-0.563583\pi$
$59$	−179.849	−0.396852	−0.198426	+	0.980116i	$0.563583\pi$
$60$	0	0
$61$	480.039	1.00758	0.503792	−	0.863825i	$-0.331938\pi$
$61$	480.039	1.00758	0.503792	+	0.863825i	$0.331938\pi$
$62$	0	0
$63$	−223.812	−0.447581
$64$	0	0
$65$	303.019	0.578230
$66$	0	0
$67$	−855.472	−1.55989	−0.779944	−	0.625849i	$-0.784753\pi$
$67$	−855.472	−1.55989	−0.779944	+	0.625849i	$0.784753\pi$
$68$	0	0
$69$	−157.019	−0.273955
$70$	0	0
$71$	−675.775	−1.12957	−0.564787	−	0.825237i	$-0.691042\pi$
$71$	−675.775	−1.12957	−0.564787	+	0.825237i	$0.691042\pi$
$72$	0	0
$73$	−621.623	−0.996651	−0.498325	−	0.866990i	$-0.666052\pi$
$73$	−621.623	−0.996651	−0.498325	+	0.866990i	$0.666052\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	−640.000	−0.947205
$78$	0	0
$79$	513.699	0.731591	0.365795	−	0.930695i	$-0.380797\pi$
$79$	513.699	0.731591	0.365795	+	0.930695i	$0.380797\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1284.30	−1.69844	−0.849220	−	0.528039i	$-0.822928\pi$
$83$	−1284.30	−1.69844	−0.849220	+	0.528039i	$0.822928\pi$
$84$	0	0
$85$	−143.019	−0.182502
$86$	0	0
$87$	−18.0000	−0.0221816
$88$	0	0
$89$	−1004.87	−1.19681	−0.598405	−	0.801194i	$-0.704198\pi$
$89$	−1004.87	−1.19681	−0.598405	+	0.801194i	$0.704198\pi$
$90$	0	0
$91$	−1507.10	−1.73612
$92$	0	0
$93$	254.604	0.283884
$94$	0	0
$95$	−433.019	−0.467651
$96$	0	0
$97$	300.416	0.314460	0.157230	−	0.987562i	$-0.449744\pi$
$97$	300.416	0.314460	0.157230	+	0.987562i	$0.449744\pi$
$98$	0	0
$99$	231.623	0.235142
$100$	0	0
$101$	−1649.25	−1.62481	−0.812407	−	0.583091i	$-0.801843\pi$
$101$	−1649.25	−1.62481	−0.812407	+	0.583091i	$0.801843\pi$
$102$	0	0
$103$	1326.38	1.26885	0.634427	−	0.772983i	$-0.281236\pi$
$103$	1326.38	1.26885	0.634427	+	0.772983i	$0.281236\pi$
$104$	0	0
$105$	373.019	0.346695
$106$	0	0
$107$	−1189.43	−1.07464	−0.537322	−	0.843377i	$-0.680564\pi$
$107$	−1189.43	−1.07464	−0.537322	+	0.843377i	$0.680564\pi$
$108$	0	0
$109$	1864.42	1.63834	0.819168	−	0.573554i	$-0.194436\pi$
$109$	1864.42	1.63834	0.819168	+	0.573554i	$0.194436\pi$
$110$	0	0
$111$	1344.68	1.14983
$112$	0	0
$113$	−1218.19	−1.01414	−0.507069	−	0.861906i	$-0.669271\pi$
$113$	−1218.19	−1.01414	−0.507069	+	0.861906i	$0.669271\pi$
$114$	0	0
$115$	261.699	0.212205
$116$	0	0
$117$	545.435	0.430987
$118$	0	0
$119$	711.320	0.547955
$120$	0	0
$121$	−668.662	−0.502376
$122$	0	0
$123$	−549.623	−0.402909
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−1740.19	−1.21588	−0.607941	−	0.793982i	$-0.708004\pi$
$127$	−1740.19	−1.21588	−0.607941	+	0.793982i	$0.708004\pi$
$128$	0	0
$129$	756.000	0.515985
$130$	0	0
$131$	456.303	0.304331	0.152166	−	0.988355i	$-0.451375\pi$
$131$	456.303	0.304331	0.152166	+	0.988355i	$0.451375\pi$
$132$	0	0
$133$	2153.66	1.40411
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	1329.51	0.829109	0.414555	−	0.910024i	$-0.363937\pi$
$137$	1329.51	0.829109	0.414555	+	0.910024i	$0.363937\pi$
$138$	0	0
$139$	−1097.25	−0.669550	−0.334775	−	0.942298i	$-0.608660\pi$
$139$	−1097.25	−0.669550	−0.334775	+	0.942298i	$0.608660\pi$
$140$	0	0
$141$	125.773	0.0751204
$142$	0	0
$143$	1559.70	0.912087
$144$	0	0
$145$	30.0000	0.0171818
$146$	0	0
$147$	−826.247	−0.463590
$148$	0	0
$149$	1590.16	0.874299	0.437150	−	0.899389i	$-0.355988\pi$
$149$	1590.16	0.874299	0.437150	+	0.899389i	$0.355988\pi$
$150$	0	0
$151$	1166.60	0.628721	0.314361	−	0.949304i	$-0.398210\pi$
$151$	1166.60	0.628721	0.314361	+	0.949304i	$0.398210\pi$
$152$	0	0
$153$	−257.435	−0.136029
$154$	0	0
$155$	−424.340	−0.219895
$156$	0	0
$157$	1770.57	0.900041	0.450021	−	0.893018i	$-0.351417\pi$
$157$	1770.57	0.900041	0.450021	+	0.893018i	$0.351417\pi$
$158$	0	0
$159$	686.377	0.342347
$160$	0	0
$161$	−1301.58	−0.637138
$162$	0	0
$163$	4018.35	1.93093	0.965464	−	0.260538i	$-0.0838999\pi$
$163$	4018.35	1.93093	0.965464	+	0.260538i	$0.0838999\pi$
$164$	0	0
$165$	−386.039	−0.182140
$166$	0	0
$167$	2899.74	1.34365	0.671823	−	0.740712i	$-0.265512\pi$
$167$	2899.74	1.34365	0.671823	+	0.740712i	$0.265512\pi$
$168$	0	0
$169$	1475.83	0.671748
$170$	0	0
$171$	−779.435	−0.348567
$172$	0	0
$173$	−3127.74	−1.37455	−0.687276	−	0.726396i	$-0.741194\pi$
$173$	−3127.74	−1.37455	−0.687276	+	0.726396i	$0.741194\pi$
$174$	0	0
$175$	−621.699	−0.268549
$176$	0	0
$177$	539.546	0.229123
$178$	0	0
$179$	2852.53	1.19111	0.595553	−	0.803316i	$-0.296933\pi$
$179$	2852.53	1.19111	0.595553	+	0.803316i	$0.296933\pi$
$180$	0	0
$181$	−1666.91	−0.684532	−0.342266	−	0.939603i	$-0.611194\pi$
$181$	−1666.91	−0.684532	−0.342266	+	0.939603i	$0.611194\pi$
$182$	0	0
$183$	−1440.12	−0.581729
$184$	0	0
$185$	−2241.14	−0.890657
$186$	0	0
$187$	−736.147	−0.287874
$188$	0	0
$189$	671.435	0.258411
$190$	0	0
$191$	3144.00	1.19106	0.595528	−	0.803334i	$-0.296943\pi$
$191$	3144.00	1.19106	0.595528	+	0.803334i	$0.296943\pi$
$192$	0	0
$193$	−1063.13	−0.396507	−0.198253	−	0.980151i	$-0.563527\pi$
$193$	−1063.13	−0.396507	−0.198253	+	0.980151i	$0.563527\pi$
$194$	0	0
$195$	−909.058	−0.333841
$196$	0	0
$197$	−4600.87	−1.66395	−0.831976	−	0.554812i	$-0.812790\pi$
$197$	−4600.87	−1.66395	−0.831976	+	0.554812i	$0.812790\pi$
$198$	0	0
$199$	1834.83	0.653606	0.326803	−	0.945092i	$-0.394029\pi$
$199$	1834.83	0.653606	0.326803	+	0.945092i	$0.394029\pi$
$200$	0	0
$201$	2566.42	0.900602
$202$	0	0
$203$	−149.208	−0.0515878
$204$	0	0
$205$	916.039	0.312092
$206$	0	0
$207$	471.058	0.158168
$208$	0	0
$209$	−2228.83	−0.737663
$210$	0	0
$211$	−1896.80	−0.618867	−0.309434	−	0.950921i	$-0.600139\pi$
$211$	−1896.80	−0.618867	−0.309434	+	0.950921i	$0.600139\pi$
$212$	0	0
$213$	2027.32	0.652160
$214$	0	0
$215$	−1260.00	−0.399680
$216$	0	0
$217$	2110.49	0.660229
$218$	0	0
$219$	1864.87	0.575417
$220$	0	0
$221$	−1733.51	−0.527639
$222$	0	0
$223$	5967.06	1.79186	0.895928	−	0.444200i	$-0.146512\pi$
$223$	5967.06	1.79186	0.895928	+	0.444200i	$0.146512\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−6375.03	−1.86399	−0.931995	−	0.362472i	$-0.881933\pi$
$227$	−6375.03	−1.86399	−0.931995	+	0.362472i	$0.881933\pi$
$228$	0	0
$229$	−1360.12	−0.392485	−0.196242	−	0.980555i	$-0.562874\pi$
$229$	−1360.12	−0.392485	−0.196242	+	0.980555i	$0.562874\pi$
$230$	0	0
$231$	1920.00	0.546869
$232$	0	0
$233$	96.7594	0.0272056	0.0136028	−	0.999907i	$-0.495670\pi$
$233$	96.7594	0.0272056	0.0136028	+	0.999907i	$0.495670\pi$
$234$	0	0
$235$	−209.621	−0.0581880
$236$	0	0
$237$	−1541.10	−0.422384
$238$	0	0
$239$	6068.00	1.64229	0.821143	−	0.570722i	$-0.193337\pi$
$239$	6068.00	1.64229	0.821143	+	0.570722i	$0.193337\pi$
$240$	0	0
$241$	−542.753	−0.145070	−0.0725349	−	0.997366i	$-0.523109\pi$
$241$	−542.753	−0.145070	−0.0725349	+	0.997366i	$0.523109\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	1377.08	0.359095
$246$	0	0
$247$	−5248.53	−1.35205
$248$	0	0
$249$	3852.91	0.980595
$250$	0	0
$251$	4658.42	1.17146	0.585731	−	0.810505i	$-0.300807\pi$
$251$	4658.42	1.17146	0.585731	+	0.810505i	$0.300807\pi$
$252$	0	0
$253$	1347.01	0.334727
$254$	0	0
$255$	429.058	0.105367
$256$	0	0
$257$	−6519.33	−1.58235	−0.791176	−	0.611588i	$-0.790531\pi$
$257$	−6519.33	−1.58235	−0.791176	+	0.611588i	$0.790531\pi$
$258$	0	0
$259$	11146.5	2.67417
$260$	0	0
$261$	54.0000	0.0128066
$262$	0	0
$263$	1205.17	0.282562	0.141281	−	0.989970i	$-0.454878\pi$
$263$	1205.17	0.282562	0.141281	+	0.989970i	$0.454878\pi$
$264$	0	0
$265$	−1143.96	−0.265181
$266$	0	0
$267$	3014.61	0.690978
$268$	0	0
$269$	−6083.66	−1.37891	−0.689456	−	0.724327i	$-0.742150\pi$
$269$	−6083.66	−1.37891	−0.689456	+	0.724327i	$0.742150\pi$
$270$	0	0
$271$	3151.36	0.706389	0.353194	−	0.935550i	$-0.385095\pi$
$271$	3151.36	0.706389	0.353194	+	0.935550i	$0.385095\pi$
$272$	0	0
$273$	4521.29	1.00235
$274$	0	0
$275$	643.398	0.141085
$276$	0	0
$277$	6780.54	1.47077	0.735385	−	0.677650i	$-0.237002\pi$
$277$	6780.54	1.47077	0.735385	+	0.677650i	$0.237002\pi$
$278$	0	0
$279$	−763.812	−0.163900
$280$	0	0
$281$	5163.29	1.09614	0.548071	−	0.836432i	$-0.315362\pi$
$281$	5163.29	1.09614	0.548071	+	0.836432i	$0.315362\pi$
$282$	0	0
$283$	−6543.64	−1.37448	−0.687242	−	0.726429i	$-0.741179\pi$
$283$	−6543.64	−1.37448	−0.687242	+	0.726429i	$0.741179\pi$
$284$	0	0
$285$	1299.06	0.269999
$286$	0	0
$287$	−4556.00	−0.937047
$288$	0	0
$289$	−4094.82	−0.833466
$290$	0	0
$291$	−901.247	−0.181553
$292$	0	0
$293$	267.208	0.0532780	0.0266390	−	0.999645i	$-0.491520\pi$
$293$	267.208	0.0532780	0.0266390	+	0.999645i	$0.491520\pi$
$294$	0	0
$295$	−899.243	−0.177478
$296$	0	0
$297$	−694.870	−0.135759
$298$	0	0
$299$	3172.00	0.613516
$300$	0	0
$301$	6266.73	1.20003
$302$	0	0
$303$	4947.74	0.938087
$304$	0	0
$305$	2400.19	0.450606
$306$	0	0
$307$	646.800	0.120244	0.0601219	−	0.998191i	$-0.480851\pi$
$307$	646.800	0.120244	0.0601219	+	0.998191i	$0.480851\pi$
$308$	0	0
$309$	−3979.14	−0.732573
$310$	0	0
$311$	−5855.48	−1.06763	−0.533816	−	0.845601i	$-0.679243\pi$
$311$	−5855.48	−1.06763	−0.533816	+	0.845601i	$0.679243\pi$
$312$	0	0
$313$	−3939.52	−0.711421	−0.355710	−	0.934596i	$-0.615761\pi$
$313$	−3939.52	−0.711421	−0.355710	+	0.934596i	$0.615761\pi$
$314$	0	0
$315$	−1119.06	−0.200164
$316$	0	0
$317$	16.1943	0.00286929	0.00143464	−	0.999999i	$-0.499543\pi$
$317$	16.1943	0.00286929	0.00143464	+	0.999999i	$0.499543\pi$
$318$	0	0
$319$	154.416	0.0271022
$320$	0	0
$321$	3568.30	0.620446
$322$	0	0
$323$	2477.21	0.426735
$324$	0	0
$325$	1515.10	0.258592
$326$	0	0
$327$	−5593.25	−0.945894
$328$	0	0
$329$	1042.57	0.174708
$330$	0	0
$331$	−1946.45	−0.323223	−0.161611	−	0.986854i	$-0.551669\pi$
$331$	−1946.45	−0.323223	−0.161611	+	0.986854i	$0.551669\pi$
$332$	0	0
$333$	−4034.04	−0.663857
$334$	0	0
$335$	−4277.36	−0.697603
$336$	0	0
$337$	−3713.10	−0.600195	−0.300097	−	0.953909i	$-0.597019\pi$
$337$	−3713.10	−0.600195	−0.300097	+	0.953909i	$0.597019\pi$
$338$	0	0
$339$	3654.57	0.585512
$340$	0	0
$341$	−2184.16	−0.346858
$342$	0	0
$343$	1680.69	0.264573
$344$	0	0
$345$	−785.097	−0.122517
$346$	0	0
$347$	−12200.6	−1.88750	−0.943751	−	0.330658i	$-0.892729\pi$
$347$	−12200.6	−1.88750	−0.943751	+	0.330658i	$0.892729\pi$
$348$	0	0
$349$	−1455.52	−0.223244	−0.111622	−	0.993751i	$-0.535605\pi$
$349$	−1455.52	−0.223244	−0.111622	+	0.993751i	$0.535605\pi$
$350$	0	0
$351$	−1636.30	−0.248830
$352$	0	0
$353$	364.526	0.0549625	0.0274813	−	0.999622i	$-0.491251\pi$
$353$	364.526	0.0549625	0.0274813	+	0.999622i	$0.491251\pi$
$354$	0	0
$355$	−3378.87	−0.505161
$356$	0	0
$357$	−2133.96	−0.316362
$358$	0	0
$359$	−11180.1	−1.64363	−0.821814	−	0.569756i	$-0.807038\pi$
$359$	−11180.1	−1.64363	−0.821814	+	0.569756i	$0.807038\pi$
$360$	0	0
$361$	641.233	0.0934879
$362$	0	0
$363$	2005.99	0.290047
$364$	0	0
$365$	−3108.12	−0.445716
$366$	0	0
$367$	−5558.37	−0.790585	−0.395292	−	0.918555i	$-0.629357\pi$
$367$	−5558.37	−0.790585	−0.395292	+	0.918555i	$0.629357\pi$
$368$	0	0
$369$	1648.87	0.232620
$370$	0	0
$371$	5689.60	0.796197
$372$	0	0
$373$	9598.80	1.33246	0.666229	−	0.745747i	$-0.267907\pi$
$373$	9598.80	1.33246	0.666229	+	0.745747i	$0.267907\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	363.623	0.0496752
$378$	0	0
$379$	5518.37	0.747915	0.373958	−	0.927446i	$-0.378001\pi$
$379$	5518.37	0.747915	0.373958	+	0.927446i	$0.378001\pi$
$380$	0	0
$381$	5220.58	0.701990
$382$	0	0
$383$	7208.20	0.961675	0.480838	−	0.876810i	$-0.340333\pi$
$383$	7208.20	0.961675	0.480838	+	0.876810i	$0.340333\pi$
$384$	0	0
$385$	−3200.00	−0.423603
$386$	0	0
$387$	−2268.00	−0.297904
$388$	0	0
$389$	−8580.13	−1.11833	−0.559164	−	0.829057i	$-0.688878\pi$
$389$	−8580.13	−1.11833	−0.559164	+	0.829057i	$0.688878\pi$
$390$	0	0
$391$	−1497.12	−0.193639
$392$	0	0
$393$	−1368.91	−0.175706
$394$	0	0
$395$	2568.50	0.327177
$396$	0	0
$397$	10323.7	1.30512	0.652558	−	0.757738i	$-0.273696\pi$
$397$	10323.7	1.30512	0.652558	+	0.757738i	$0.273696\pi$
$398$	0	0
$399$	−6460.99	−0.810661
$400$	0	0
$401$	6130.00	0.763386	0.381693	−	0.924289i	$-0.375341\pi$
$401$	6130.00	0.763386	0.381693	+	0.924289i	$0.375341\pi$
$402$	0	0
$403$	−5143.33	−0.635750
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	−11535.5	−1.40490
$408$	0	0
$409$	4015.45	0.485456	0.242728	−	0.970094i	$-0.421958\pi$
$409$	4015.45	0.485456	0.242728	+	0.970094i	$0.421958\pi$
$410$	0	0
$411$	−3988.54	−0.478686
$412$	0	0
$413$	4472.47	0.532871
$414$	0	0
$415$	−6421.51	−0.759566
$416$	0	0
$417$	3291.75	0.386565
$418$	0	0
$419$	11923.3	1.39020	0.695098	−	0.718915i	$-0.255361\pi$
$419$	11923.3	1.39020	0.695098	+	0.718915i	$0.255361\pi$
$420$	0	0
$421$	−12939.9	−1.49799	−0.748993	−	0.662577i	$-0.769463\pi$
$421$	−12939.9	−1.49799	−0.748993	+	0.662577i	$0.769463\pi$
$422$	0	0
$423$	−377.318	−0.0433708
$424$	0	0
$425$	−715.097	−0.0816172
$426$	0	0
$427$	−11937.6	−1.35293
$428$	0	0
$429$	−4679.09	−0.526594
$430$	0	0
$431$	−5958.20	−0.665885	−0.332942	−	0.942947i	$-0.608041\pi$
$431$	−5958.20	−0.665885	−0.332942	+	0.942947i	$0.608041\pi$
$432$	0	0
$433$	8063.83	0.894972	0.447486	−	0.894291i	$-0.352319\pi$
$433$	8063.83	0.894972	0.447486	+	0.894291i	$0.352319\pi$
$434$	0	0
$435$	−90.0000	−0.00991993
$436$	0	0
$437$	−4532.83	−0.496189
$438$	0	0
$439$	−6065.39	−0.659420	−0.329710	−	0.944082i	$-0.606951\pi$
$439$	−6065.39	−0.659420	−0.329710	+	0.944082i	$0.606951\pi$
$440$	0	0
$441$	2478.74	0.267654
$442$	0	0
$443$	5019.57	0.538345	0.269172	−	0.963092i	$-0.413250\pi$
$443$	5019.57	0.538345	0.269172	+	0.963092i	$0.413250\pi$
$444$	0	0
$445$	−5024.35	−0.535229
$446$	0	0
$447$	−4770.47	−0.504777
$448$	0	0
$449$	1830.39	0.192386	0.0961931	−	0.995363i	$-0.469333\pi$
$449$	1830.39	0.192386	0.0961931	+	0.995363i	$0.469333\pi$
$450$	0	0
$451$	4715.02	0.492288
$452$	0	0
$453$	−3499.81	−0.362992
$454$	0	0
$455$	−7535.48	−0.776414
$456$	0	0
$457$	−7111.06	−0.727881	−0.363940	−	0.931422i	$-0.618569\pi$
$457$	−7111.06	−0.727881	−0.363940	+	0.931422i	$0.618569\pi$
$458$	0	0
$459$	772.305	0.0785362
$460$	0	0
$461$	2347.58	0.237176	0.118588	−	0.992944i	$-0.462163\pi$
$461$	2347.58	0.237176	0.118588	+	0.992944i	$0.462163\pi$
$462$	0	0
$463$	−8669.58	−0.870215	−0.435107	−	0.900379i	$-0.643290\pi$
$463$	−8669.58	−0.870215	−0.435107	+	0.900379i	$0.643290\pi$
$464$	0	0
$465$	1273.02	0.126957
$466$	0	0
$467$	6448.02	0.638927	0.319463	−	0.947599i	$-0.396497\pi$
$467$	6448.02	0.638927	0.319463	+	0.947599i	$0.396497\pi$
$468$	0	0
$469$	21273.8	2.09453
$470$	0	0
$471$	−5311.70	−0.519639
$472$	0	0
$473$	−6485.45	−0.630447
$474$	0	0
$475$	−2165.10	−0.209140
$476$	0	0
$477$	−2059.13	−0.197654
$478$	0	0
$479$	7603.95	0.725330	0.362665	−	0.931920i	$-0.381867\pi$
$479$	7603.95	0.725330	0.362665	+	0.931920i	$0.381867\pi$
$480$	0	0
$481$	−27164.3	−2.57502
$482$	0	0
$483$	3904.75	0.367852
$484$	0	0
$485$	1502.08	0.140631
$486$	0	0
$487$	−14109.5	−1.31286	−0.656429	−	0.754388i	$-0.727934\pi$
$487$	−14109.5	−1.31286	−0.656429	+	0.754388i	$0.727934\pi$
$488$	0	0
$489$	−12055.0	−1.11482
$490$	0	0
$491$	−15808.3	−1.45299	−0.726493	−	0.687173i	$-0.758851\pi$
$491$	−15808.3	−1.45299	−0.726493	+	0.687173i	$0.758851\pi$
$492$	0	0
$493$	−171.623	−0.0156786
$494$	0	0
$495$	1158.12	0.105159
$496$	0	0
$497$	16805.1	1.51673
$498$	0	0
$499$	14653.7	1.31461	0.657304	−	0.753625i	$-0.271697\pi$
$499$	14653.7	1.31461	0.657304	+	0.753625i	$0.271697\pi$
$500$	0	0
$501$	−8699.23	−0.775754
$502$	0	0
$503$	−323.889	−0.0287108	−0.0143554	−	0.999897i	$-0.504570\pi$
$503$	−323.889	−0.0287108	−0.0143554	+	0.999897i	$0.504570\pi$
$504$	0	0
$505$	−8246.23	−0.726639
$506$	0	0
$507$	−4427.49	−0.387834
$508$	0	0
$509$	2178.99	0.189748	0.0948741	−	0.995489i	$-0.469755\pi$
$509$	2178.99	0.189748	0.0948741	+	0.995489i	$0.469755\pi$
$510$	0	0
$511$	15458.5	1.33825
$512$	0	0
$513$	2338.30	0.201245
$514$	0	0
$515$	6631.89	0.567449
$516$	0	0
$517$	−1078.96	−0.0917845
$518$	0	0
$519$	9383.22	0.793599
$520$	0	0
$521$	−12371.2	−1.04029	−0.520145	−	0.854078i	$-0.674122\pi$
$521$	−12371.2	−1.04029	−0.520145	+	0.854078i	$0.674122\pi$
$522$	0	0
$523$	−3963.16	−0.331351	−0.165676	−	0.986180i	$-0.552980\pi$
$523$	−3963.16	−0.331351	−0.165676	+	0.986180i	$0.552980\pi$
$524$	0	0
$525$	1865.10	0.155047
$526$	0	0
$527$	2427.55	0.200656
$528$	0	0
$529$	−9427.54	−0.774845
$530$	0	0
$531$	−1618.64	−0.132284
$532$	0	0
$533$	11103.1	0.902305
$534$	0	0
$535$	−5947.16	−0.480595
$536$	0	0
$537$	−8557.58	−0.687685
$538$	0	0
$539$	7088.07	0.566428
$540$	0	0
$541$	2426.23	0.192813	0.0964066	−	0.995342i	$-0.469265\pi$
$541$	2426.23	0.192813	0.0964066	+	0.995342i	$0.469265\pi$
$542$	0	0
$543$	5000.73	0.395215
$544$	0	0
$545$	9322.08	0.732686
$546$	0	0
$547$	1636.29	0.127902	0.0639512	−	0.997953i	$-0.479630\pi$
$547$	1636.29	0.127902	0.0639512	+	0.997953i	$0.479630\pi$
$548$	0	0
$549$	4320.35	0.335862
$550$	0	0
$551$	−519.623	−0.0401755
$552$	0	0
$553$	−12774.6	−0.982339
$554$	0	0
$555$	6723.41	0.514221
$556$	0	0
$557$	−4160.34	−0.316480	−0.158240	−	0.987401i	$-0.550582\pi$
$557$	−4160.34	−0.316480	−0.158240	+	0.987401i	$0.550582\pi$
$558$	0	0
$559$	−15272.2	−1.15554
$560$	0	0
$561$	2208.44	0.166204
$562$	0	0
$563$	−16665.2	−1.24752	−0.623762	−	0.781614i	$-0.714397\pi$
$563$	−16665.2	−1.24752	−0.623762	+	0.781614i	$0.714397\pi$
$564$	0	0
$565$	−6090.94	−0.453536
$566$	0	0
$567$	−2014.30	−0.149194
$568$	0	0
$569$	12324.4	0.908025	0.454013	−	0.890995i	$-0.349992\pi$
$569$	12324.4	0.908025	0.454013	+	0.890995i	$0.349992\pi$
$570$	0	0
$571$	14208.0	1.04131	0.520653	−	0.853768i	$-0.325688\pi$
$571$	14208.0	1.04131	0.520653	+	0.853768i	$0.325688\pi$
$572$	0	0
$573$	−9432.00	−0.687657
$574$	0	0
$575$	1308.50	0.0949009
$576$	0	0
$577$	13281.1	0.958232	0.479116	−	0.877752i	$-0.340957\pi$
$577$	13281.1	0.958232	0.479116	+	0.877752i	$0.340957\pi$
$578$	0	0
$579$	3189.39	0.228923
$580$	0	0
$581$	31938.0	2.28057
$582$	0	0
$583$	−5888.18	−0.418291
$584$	0	0
$585$	2727.17	0.192743
$586$	0	0
$587$	−24241.4	−1.70451	−0.852256	−	0.523124i	$-0.824766\pi$
$587$	−24241.4	−1.70451	−0.852256	+	0.523124i	$0.824766\pi$
$588$	0	0
$589$	7349.90	0.514172
$590$	0	0
$591$	13802.6	0.960683
$592$	0	0
$593$	18184.5	1.25927	0.629637	−	0.776889i	$-0.283204\pi$
$593$	18184.5	1.25927	0.629637	+	0.776889i	$0.283204\pi$
$594$	0	0
$595$	3556.60	0.245053
$596$	0	0
$597$	−5504.49	−0.377360
$598$	0	0
$599$	−10505.1	−0.716569	−0.358285	−	0.933612i	$-0.616638\pi$
$599$	−10505.1	−0.716569	−0.358285	+	0.933612i	$0.616638\pi$
$600$	0	0
$601$	−4822.88	−0.327337	−0.163669	−	0.986515i	$-0.552333\pi$
$601$	−4822.88	−0.327337	−0.163669	+	0.986515i	$0.552333\pi$
$602$	0	0
$603$	−7699.25	−0.519963
$604$	0	0
$605$	−3343.31	−0.224669
$606$	0	0
$607$	27083.2	1.81099	0.905496	−	0.424355i	$-0.139499\pi$
$607$	27083.2	1.81099	0.905496	+	0.424355i	$0.139499\pi$
$608$	0	0
$609$	447.623	0.0297843
$610$	0	0
$611$	−2540.77	−0.168230
$612$	0	0
$613$	−10078.3	−0.664042	−0.332021	−	0.943272i	$-0.607731\pi$
$613$	−10078.3	−0.664042	−0.332021	+	0.943272i	$0.607731\pi$
$614$	0	0
$615$	−2748.12	−0.180187
$616$	0	0
$617$	24141.2	1.57518	0.787591	−	0.616199i	$-0.211328\pi$
$617$	24141.2	1.57518	0.787591	+	0.616199i	$0.211328\pi$
$618$	0	0
$619$	22858.7	1.48428	0.742140	−	0.670245i	$-0.233811\pi$
$619$	22858.7	1.48428	0.742140	+	0.670245i	$0.233811\pi$
$620$	0	0
$621$	−1413.17	−0.0913184
$622$	0	0
$623$	24989.1	1.60701
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	6686.49	0.425890
$628$	0	0
$629$	12821.0	0.812732
$630$	0	0
$631$	−569.330	−0.0359187	−0.0179593	−	0.999839i	$-0.505717\pi$
$631$	−569.330	−0.0359187	−0.0179593	+	0.999839i	$0.505717\pi$
$632$	0	0
$633$	5690.39	0.357303
$634$	0	0
$635$	−8700.96	−0.543759
$636$	0	0
$637$	16691.3	1.03820
$638$	0	0
$639$	−6081.97	−0.376524
$640$	0	0
$641$	25927.2	1.59760	0.798802	−	0.601594i	$-0.205468\pi$
$641$	25927.2	1.59760	0.798802	+	0.601594i	$0.205468\pi$
$642$	0	0
$643$	19555.1	1.19935	0.599673	−	0.800245i	$-0.295297\pi$
$643$	19555.1	1.19935	0.599673	+	0.800245i	$0.295297\pi$
$644$	0	0
$645$	3780.00	0.230756
$646$	0	0
$647$	8653.85	0.525839	0.262919	−	0.964818i	$-0.415315\pi$
$647$	8653.85	0.525839	0.262919	+	0.964818i	$0.415315\pi$
$648$	0	0
$649$	−4628.57	−0.279949
$650$	0	0
$651$	−6331.48	−0.381183
$652$	0	0
$653$	−27732.5	−1.66195	−0.830976	−	0.556308i	$-0.812218\pi$
$653$	−27732.5	−1.66195	−0.830976	+	0.556308i	$0.812218\pi$
$654$	0	0
$655$	2281.51	0.136101
$656$	0	0
$657$	−5594.61	−0.332217
$658$	0	0
$659$	30254.7	1.78840	0.894201	−	0.447665i	$-0.147744\pi$
$659$	30254.7	1.78840	0.894201	+	0.447665i	$0.147744\pi$
$660$	0	0
$661$	−20714.2	−1.21890	−0.609448	−	0.792826i	$-0.708609\pi$
$661$	−20714.2	−1.21890	−0.609448	+	0.792826i	$0.708609\pi$
$662$	0	0
$663$	5200.52	0.304633
$664$	0	0
$665$	10768.3	0.627936
$666$	0	0
$667$	314.039	0.0182303
$668$	0	0
$669$	−17901.2	−1.03453
$670$	0	0
$671$	12354.2	0.710775
$672$	0	0
$673$	−5092.56	−0.291685	−0.145842	−	0.989308i	$-0.546589\pi$
$673$	−5092.56	−0.291685	−0.145842	+	0.989308i	$0.546589\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	−27189.5	−1.54354	−0.771771	−	0.635901i	$-0.780629\pi$
$677$	−27189.5	−1.54354	−0.771771	+	0.635901i	$0.780629\pi$
$678$	0	0
$679$	−7470.72	−0.422239
$680$	0	0
$681$	19125.1	1.07617
$682$	0	0
$683$	−7705.95	−0.431713	−0.215856	−	0.976425i	$-0.569254\pi$
$683$	−7705.95	−0.431713	−0.215856	+	0.976425i	$0.569254\pi$
$684$	0	0
$685$	6647.56	0.370789
$686$	0	0
$687$	4080.35	0.226601
$688$	0	0
$689$	−13865.7	−0.766678
$690$	0	0
$691$	−15821.6	−0.871033	−0.435516	−	0.900181i	$-0.643434\pi$
$691$	−15821.6	−0.871033	−0.435516	+	0.900181i	$0.643434\pi$
$692$	0	0
$693$	−5760.00	−0.315735
$694$	0	0
$695$	−5486.24	−0.299432
$696$	0	0
$697$	−5240.45	−0.284787
$698$	0	0
$699$	−290.278	−0.0157072
$700$	0	0
$701$	−2108.00	−0.113578	−0.0567888	−	0.998386i	$-0.518086\pi$
$701$	−2108.00	−0.113578	−0.0567888	+	0.998386i	$0.518086\pi$
$702$	0	0
$703$	38818.2	2.08258
$704$	0	0
$705$	628.864	0.0335949
$706$	0	0
$707$	41013.4	2.18171
$708$	0	0
$709$	301.312	0.0159605	0.00798027	−	0.999968i	$-0.497460\pi$
$709$	301.312	0.0159605	0.00798027	+	0.999968i	$0.497460\pi$
$710$	0	0
$711$	4623.29	0.243864
$712$	0	0
$713$	−4441.97	−0.233314
$714$	0	0
$715$	7798.49	0.407898
$716$	0	0
$717$	−18204.0	−0.948175
$718$	0	0
$719$	−12744.9	−0.661061	−0.330530	−	0.943795i	$-0.607228\pi$
$719$	−12744.9	−0.661061	−0.330530	+	0.943795i	$0.607228\pi$
$720$	0	0
$721$	−32984.3	−1.70375
$722$	0	0
$723$	1628.26	0.0837561
$724$	0	0
$725$	150.000	0.00768395
$726$	0	0
$727$	3416.70	0.174303	0.0871515	−	0.996195i	$-0.472224\pi$
$727$	3416.70	0.174303	0.0871515	+	0.996195i	$0.472224\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	7208.18	0.364712
$732$	0	0
$733$	−5965.41	−0.300597	−0.150298	−	0.988641i	$-0.548023\pi$
$733$	−5965.41	−0.300597	−0.150298	+	0.988641i	$0.548023\pi$
$734$	0	0
$735$	−4131.23	−0.207324
$736$	0	0
$737$	−22016.4	−1.10038
$738$	0	0
$739$	−8546.50	−0.425424	−0.212712	−	0.977115i	$-0.568230\pi$
$739$	−8546.50	−0.425424	−0.212712	+	0.977115i	$0.568230\pi$
$740$	0	0
$741$	15745.6	0.780606
$742$	0	0
$743$	33899.8	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$743$	33899.8	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$744$	0	0
$745$	7950.78	0.390999
$746$	0	0
$747$	−11558.7	−0.566147
$748$	0	0
$749$	29578.8	1.44297
$750$	0	0
$751$	18256.9	0.887087	0.443544	−	0.896253i	$-0.353721\pi$
$751$	18256.9	0.887087	0.443544	+	0.896253i	$0.353721\pi$
$752$	0	0
$753$	−13975.3	−0.676344
$754$	0	0
$755$	5833.02	0.281173
$756$	0	0
$757$	1117.87	0.0536717	0.0268359	−	0.999640i	$-0.491457\pi$
$757$	1117.87	0.0536717	0.0268359	+	0.999640i	$0.491457\pi$
$758$	0	0
$759$	−4041.04	−0.193255
$760$	0	0
$761$	28438.3	1.35465	0.677324	−	0.735685i	$-0.263140\pi$
$761$	28438.3	1.35465	0.677324	+	0.735685i	$0.263140\pi$
$762$	0	0
$763$	−46364.2	−2.19987
$764$	0	0
$765$	−1287.17	−0.0608339
$766$	0	0
$767$	−10899.5	−0.513115
$768$	0	0
$769$	4566.26	0.214127	0.107063	−	0.994252i	$-0.465855\pi$
$769$	4566.26	0.214127	0.107063	+	0.994252i	$0.465855\pi$
$770$	0	0
$771$	19558.0	0.913572
$772$	0	0
$773$	29259.6	1.36144	0.680720	−	0.732543i	$-0.261667\pi$
$773$	29259.6	1.36144	0.680720	+	0.732543i	$0.261667\pi$
$774$	0	0
$775$	−2121.70	−0.0983402
$776$	0	0
$777$	−33439.5	−1.54393
$778$	0	0
$779$	−15866.5	−0.729752
$780$	0	0
$781$	−17391.7	−0.796829
$782$	0	0
$783$	−162.000	−0.00739388
$784$	0	0
$785$	8852.83	0.402511
$786$	0	0
$787$	−7865.23	−0.356246	−0.178123	−	0.984008i	$-0.557002\pi$
$787$	−7865.23	−0.356246	−0.178123	+	0.984008i	$0.557002\pi$
$788$	0	0
$789$	−3615.50	−0.163137
$790$	0	0
$791$	30293.9	1.36173
$792$	0	0
$793$	29092.2	1.30277
$794$	0	0
$795$	3431.88	0.153102
$796$	0	0
$797$	11603.9	0.515724	0.257862	−	0.966182i	$-0.416982\pi$
$797$	11603.9	0.515724	0.257862	+	0.966182i	$0.416982\pi$
$798$	0	0
$799$	1199.20	0.0530970
$800$	0	0
$801$	−9043.83	−0.398936
$802$	0	0
$803$	−15998.1	−0.703062
$804$	0	0
$805$	−6507.92	−0.284937
$806$	0	0
$807$	18251.0	0.796115
$808$	0	0
$809$	−6069.60	−0.263777	−0.131889	−	0.991265i	$-0.542104\pi$
$809$	−6069.60	−0.263777	−0.131889	+	0.991265i	$0.542104\pi$
$810$	0	0
$811$	−26995.1	−1.16883	−0.584417	−	0.811453i	$-0.698677\pi$
$811$	−26995.1	−1.16883	−0.584417	+	0.811453i	$0.698677\pi$
$812$	0	0
$813$	−9454.07	−0.407834
$814$	0	0
$815$	20091.7	0.863537
$816$	0	0
$817$	21824.2	0.934555
$818$	0	0
$819$	−13563.9	−0.578705
$820$	0	0
$821$	23158.9	0.984472	0.492236	−	0.870462i	$-0.336180\pi$
$821$	23158.9	0.984472	0.492236	+	0.870462i	$0.336180\pi$
$822$	0	0
$823$	30150.7	1.27702	0.638511	−	0.769613i	$-0.279551\pi$
$823$	30150.7	1.27702	0.638511	+	0.769613i	$0.279551\pi$
$824$	0	0
$825$	−1930.19	−0.0814554
$826$	0	0
$827$	33988.7	1.42914	0.714572	−	0.699562i	$-0.246622\pi$
$827$	33988.7	1.42914	0.714572	+	0.699562i	$0.246622\pi$
$828$	0	0
$829$	−11198.3	−0.469158	−0.234579	−	0.972097i	$-0.575371\pi$
$829$	−11198.3	−0.469158	−0.234579	+	0.972097i	$0.575371\pi$
$830$	0	0
$831$	−20341.6	−0.849149
$832$	0	0
$833$	−7877.96	−0.327677
$834$	0	0
$835$	14498.7	0.600896
$836$	0	0
$837$	2291.43	0.0946279
$838$	0	0
$839$	−20126.3	−0.828174	−0.414087	−	0.910237i	$-0.635899\pi$
$839$	−20126.3	−0.828174	−0.414087	+	0.910237i	$0.635899\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	0	0
$843$	−15489.9	−0.632858
$844$	0	0
$845$	7379.16	0.300415
$846$	0	0
$847$	16628.3	0.674562
$848$	0	0
$849$	19630.9	0.793558
$850$	0	0
$851$	−23460.1	−0.945009
$852$	0	0
$853$	−24676.2	−0.990500	−0.495250	−	0.868751i	$-0.664923\pi$
$853$	−24676.2	−0.990500	−0.495250	+	0.868751i	$0.664923\pi$
$854$	0	0
$855$	−3897.17	−0.155884
$856$	0	0
$857$	−44115.5	−1.75841	−0.879204	−	0.476446i	$-0.841925\pi$
$857$	−44115.5	−1.75841	−0.879204	+	0.476446i	$0.841925\pi$
$858$	0	0
$859$	408.822	0.0162385	0.00811923	−	0.999967i	$-0.497416\pi$
$859$	408.822	0.0162385	0.00811923	+	0.999967i	$0.497416\pi$
$860$	0	0
$861$	13668.0	0.541004
$862$	0	0
$863$	−34369.3	−1.35567	−0.677836	−	0.735213i	$-0.737082\pi$
$863$	−34369.3	−1.35567	−0.677836	+	0.735213i	$0.737082\pi$
$864$	0	0
$865$	−15638.7	−0.614719
$866$	0	0
$867$	12284.5	0.481202
$868$	0	0
$869$	13220.5	0.516082
$870$	0	0
$871$	−51844.9	−2.01687
$872$	0	0
$873$	2703.74	0.104820
$874$	0	0
$875$	−3108.50	−0.120099
$876$	0	0
$877$	41868.4	1.61208	0.806040	−	0.591861i	$-0.201606\pi$
$877$	41868.4	1.61208	0.806040	+	0.591861i	$0.201606\pi$
$878$	0	0
$879$	−801.623	−0.0307600
$880$	0	0
$881$	−2174.40	−0.0831526	−0.0415763	−	0.999135i	$-0.513238\pi$
$881$	−2174.40	−0.0831526	−0.0415763	+	0.999135i	$0.513238\pi$
$882$	0	0
$883$	−19480.7	−0.742442	−0.371221	−	0.928545i	$-0.621061\pi$
$883$	−19480.7	−0.742442	−0.371221	+	0.928545i	$0.621061\pi$
$884$	0	0
$885$	2697.73	0.102467
$886$	0	0
$887$	14546.7	0.550653	0.275326	−	0.961351i	$-0.411214\pi$
$887$	14546.7	0.550653	0.275326	+	0.961351i	$0.411214\pi$
$888$	0	0
$889$	43275.0	1.63262
$890$	0	0
$891$	2084.61	0.0783805
$892$	0	0
$893$	3630.80	0.136058
$894$	0	0
$895$	14262.6	0.532679
$896$	0	0
$897$	−9515.99	−0.354214
$898$	0	0
$899$	−509.208	−0.0188910
$900$	0	0
$901$	6544.35	0.241980
$902$	0	0
$903$	−18800.2	−0.692836
$904$	0	0
$905$	−8334.54	−0.306132
$906$	0	0
$907$	−22357.6	−0.818492	−0.409246	−	0.912424i	$-0.634208\pi$
$907$	−22357.6	−0.818492	−0.409246	+	0.912424i	$0.634208\pi$
$908$	0	0
$909$	−14843.2	−0.541605
$910$	0	0
$911$	32284.2	1.17412	0.587060	−	0.809544i	$-0.300285\pi$
$911$	32284.2	1.17412	0.587060	+	0.809544i	$0.300285\pi$
$912$	0	0
$913$	−33052.7	−1.19812
$914$	0	0
$915$	−7200.58	−0.260157
$916$	0	0
$917$	−11347.3	−0.408639
$918$	0	0
$919$	−12767.9	−0.458295	−0.229147	−	0.973392i	$-0.573594\pi$
$919$	−12767.9	−0.458295	−0.229147	+	0.973392i	$0.573594\pi$
$920$	0	0
$921$	−1940.40	−0.0694228
$922$	0	0
$923$	−40954.6	−1.46049
$924$	0	0
$925$	−11205.7	−0.398314
$926$	0	0
$927$	11937.4	0.422951
$928$	0	0
$929$	−17645.4	−0.623171	−0.311586	−	0.950218i	$-0.600860\pi$
$929$	−17645.4	−0.623171	−0.311586	+	0.950218i	$0.600860\pi$
$930$	0	0
$931$	−23852.1	−0.839656
$932$	0	0
$933$	17566.4	0.616398
$934$	0	0
$935$	−3680.74	−0.128741
$936$	0	0
$937$	45735.0	1.59455	0.797277	−	0.603614i	$-0.206273\pi$
$937$	45735.0	1.59455	0.797277	+	0.603614i	$0.206273\pi$
$938$	0	0
$939$	11818.6	0.410739
$940$	0	0
$941$	−14982.6	−0.519042	−0.259521	−	0.965737i	$-0.583565\pi$
$941$	−14982.6	−0.519042	−0.259521	+	0.965737i	$0.583565\pi$
$942$	0	0
$943$	9589.06	0.331138
$944$	0	0
$945$	3357.17	0.115565
$946$	0	0
$947$	−31821.4	−1.09193	−0.545964	−	0.837809i	$-0.683836\pi$
$947$	−31821.4	−1.09193	−0.545964	+	0.837809i	$0.683836\pi$
$948$	0	0
$949$	−37672.8	−1.28863
$950$	0	0
$951$	−48.5830	−0.00165658
$952$	0	0
$953$	19627.5	0.667154	0.333577	−	0.942723i	$-0.391744\pi$
$953$	19627.5	0.667154	0.333577	+	0.942723i	$0.391744\pi$
$954$	0	0
$955$	15720.0	0.532657
$956$	0	0
$957$	−463.247	−0.0156475
$958$	0	0
$959$	−33062.3	−1.11328
$960$	0	0
$961$	−22588.4	−0.758230
$962$	0	0
$963$	−10704.9	−0.358214
$964$	0	0
$965$	−5315.65	−0.177323
$966$	0	0
$967$	34374.9	1.14315	0.571573	−	0.820551i	$-0.306333\pi$
$967$	34374.9	1.14315	0.571573	+	0.820551i	$0.306333\pi$
$968$	0	0
$969$	−7431.62	−0.246376
$970$	0	0
$971$	−33737.5	−1.11503	−0.557513	−	0.830169i	$-0.688244\pi$
$971$	−33737.5	−1.11503	−0.557513	+	0.830169i	$0.688244\pi$
$972$	0	0
$973$	27286.3	0.899034
$974$	0	0
$975$	−4545.29	−0.149298
$976$	0	0
$977$	38438.8	1.25872	0.629358	−	0.777116i	$-0.283318\pi$
$977$	38438.8	1.25872	0.629358	+	0.777116i	$0.283318\pi$
$978$	0	0
$979$	−25861.3	−0.844259
$980$	0	0
$981$	16779.7	0.546112
$982$	0	0
$983$	−18714.3	−0.607217	−0.303609	−	0.952797i	$-0.598192\pi$
$983$	−18714.3	−0.607217	−0.303609	+	0.952797i	$0.598192\pi$
$984$	0	0
$985$	−23004.3	−0.744142
$986$	0	0
$987$	−3127.71	−0.100867
$988$	0	0
$989$	−13189.6	−0.424071
$990$	0	0
$991$	23659.4	0.758392	0.379196	−	0.925316i	$-0.376201\pi$
$991$	23659.4	0.758392	0.379196	+	0.925316i	$0.376201\pi$
$992$	0	0
$993$	5839.36	0.186613
$994$	0	0
$995$	9174.15	0.292301
$996$	0	0
$997$	52118.0	1.65556	0.827780	−	0.561053i	$-0.189604\pi$
$997$	52118.0	1.65556	0.827780	+	0.561053i	$0.189604\pi$
$998$	0	0
$999$	12102.1	0.383278

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.4.a.o.1.1	✓	2
3.2	odd	2		1440.4.a.s.1.1		2
4.3	odd	2		480.4.a.r.1.2	yes	2
5.4	even	2		2400.4.a.bb.1.2		2
8.3	odd	2		960.4.a.bk.1.2		2
8.5	even	2		960.4.a.bn.1.1		2
12.11	even	2		1440.4.a.y.1.2		2
20.19	odd	2		2400.4.a.y.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.a.o.1.1	✓	2	1.1	even	1	trivial
480.4.a.r.1.2	yes	2	4.3	odd	2
960.4.a.bk.1.2		2	8.3	odd	2
960.4.a.bn.1.1		2	8.5	even	2
1440.4.a.s.1.1		2	3.2	odd	2
1440.4.a.y.1.2		2	12.11	even	2
2400.4.a.y.1.1		2	20.19	odd	2
2400.4.a.bb.1.2		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:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	480.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +5.00000 q^{5} -24.8680 q^{7} +9.00000 q^{9} +25.7359 q^{11} +60.6039 q^{13} -15.0000 q^{15} -28.6039 q^{17} -86.6039 q^{19} +74.6039 q^{21} +52.3398 q^{23} +25.0000 q^{25} -27.0000 q^{27} +6.00000 q^{29} -84.8680 q^{31} -77.2078 q^{33} -124.340 q^{35} -448.227 q^{37} -181.812 q^{39} +183.208 q^{41} -252.000 q^{43} +45.0000 q^{45} -41.9243 q^{47} +275.416 q^{49} +85.8117 q^{51} -228.792 q^{53} +128.680 q^{55} +259.812 q^{57} -179.849 q^{59} +480.039 q^{61} -223.812 q^{63} +303.019 q^{65} -855.472 q^{67} -157.019 q^{69} -675.775 q^{71} -621.623 q^{73} -75.0000 q^{75} -640.000 q^{77} +513.699 q^{79} +81.0000 q^{81} -1284.30 q^{83} -143.019 q^{85} -18.0000 q^{87} -1004.87 q^{89} -1507.10 q^{91} +254.604 q^{93} -433.019 q^{95} +300.416 q^{97} +231.623 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 10 q^{5} - 12 q^{7} + 18 q^{9} - 24 q^{11} + 8 q^{13} - 30 q^{15} + 56 q^{17} - 60 q^{19} + 36 q^{21} - 84 q^{23} + 50 q^{25} - 54 q^{27} + 12 q^{29} - 132 q^{31} + 72 q^{33} - 60 q^{35}+ \cdots - 216 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 10 * q^5 - 12 * q^7 + 18 * q^9 - 24 * q^11 + 8 * q^13 - 30 * q^15 + 56 * q^17 - 60 * q^19 + 36 * q^21 - 84 * q^23 + 50 * q^25 - 54 * q^27 + 12 * q^29 - 132 * q^31 + 72 * q^33 - 60 * q^35 - 104 * q^37 - 24 * q^39 + 140 * q^41 - 504 * q^43 + 90 * q^45 - 348 * q^47 + 98 * q^49 - 168 * q^51 - 684 * q^53 - 120 * q^55 + 180 * q^57 - 888 * q^59 - 172 * q^61 - 108 * q^63 + 40 * q^65 - 1560 * q^67 + 252 * q^69 - 144 * q^71 - 564 * q^73 - 150 * q^75 - 1280 * q^77 + 84 * q^79 + 162 * q^81 - 1512 * q^83 + 280 * q^85 - 36 * q^87 + 28 * q^89 - 2184 * q^91 + 396 * q^93 - 300 * q^95 + 148 * q^97 - 216 * q^99

Introduction

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