LMFDB - Embedded newform 480.4.a.f.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 = [1,0,-3,0,5,0,16] 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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q-3.00000 q^{3} +5.00000 q^{5} +16.0000 q^{7} +9.00000 q^{9} +24.0000 q^{11} -14.0000 q^{13} -15.0000 q^{15} -18.0000 q^{17} +36.0000 q^{19} -48.0000 q^{21} +104.000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -250.000 q^{29} -28.0000 q^{31} -72.0000 q^{33} +80.0000 q^{35} -54.0000 q^{37} +42.0000 q^{39} +354.000 q^{41} +228.000 q^{43} +45.0000 q^{45} +408.000 q^{47} -87.0000 q^{49} +54.0000 q^{51} +262.000 q^{53} +120.000 q^{55} -108.000 q^{57} -64.0000 q^{59} +374.000 q^{61} +144.000 q^{63} -70.0000 q^{65} +300.000 q^{67} -312.000 q^{69} +1016.00 q^{71} +274.000 q^{73} -75.0000 q^{75} +384.000 q^{77} +788.000 q^{79} +81.0000 q^{81} -396.000 q^{83} -90.0000 q^{85} +750.000 q^{87} +786.000 q^{89} -224.000 q^{91} +84.0000 q^{93} +180.000 q^{95} -1086.00 q^{97} +216.000 q^{99} +O(q^{100})$

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
$7$	16.0000	0.863919	0.431959	−	0.901893i	$-0.357822\pi$
$7$	16.0000	0.863919	0.431959	+	0.901893i	$0.357822\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	24.0000	0.657843	0.328921	−	0.944357i	$-0.393315\pi$
$11$	24.0000	0.657843	0.328921	+	0.944357i	$0.393315\pi$
$12$	0	0
$13$	−14.0000	−0.298685	−0.149342	−	0.988786i	$-0.547716\pi$
$13$	−14.0000	−0.298685	−0.149342	+	0.988786i	$0.547716\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	−18.0000	−0.256802	−0.128401	−	0.991722i	$-0.540985\pi$
$17$	−18.0000	−0.256802	−0.128401	+	0.991722i	$0.540985\pi$
$18$	0	0
$19$	36.0000	0.434682	0.217341	−	0.976096i	$-0.430262\pi$
$19$	36.0000	0.434682	0.217341	+	0.976096i	$0.430262\pi$
$20$	0	0
$21$	−48.0000	−0.498784
$22$	0	0
$23$	104.000	0.942848	0.471424	−	0.881907i	$-0.343740\pi$
$23$	104.000	0.942848	0.471424	+	0.881907i	$0.343740\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−250.000	−1.60082	−0.800411	−	0.599452i	$-0.795385\pi$
$29$	−250.000	−1.60082	−0.800411	+	0.599452i	$0.795385\pi$
$30$	0	0
$31$	−28.0000	−0.162224	−0.0811121	−	0.996705i	$-0.525847\pi$
$31$	−28.0000	−0.162224	−0.0811121	+	0.996705i	$0.525847\pi$
$32$	0	0
$33$	−72.0000	−0.379806
$34$	0	0
$35$	80.0000	0.386356
$36$	0	0
$37$	−54.0000	−0.239934	−0.119967	−	0.992778i	$-0.538279\pi$
$37$	−54.0000	−0.239934	−0.119967	+	0.992778i	$0.538279\pi$
$38$	0	0
$39$	42.0000	0.172446
$40$	0	0
$41$	354.000	1.34843	0.674214	−	0.738536i	$-0.264483\pi$
$41$	354.000	1.34843	0.674214	+	0.738536i	$0.264483\pi$
$42$	0	0
$43$	228.000	0.808597	0.404299	−	0.914627i	$-0.367516\pi$
$43$	228.000	0.808597	0.404299	+	0.914627i	$0.367516\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	408.000	1.26623	0.633116	−	0.774057i	$-0.281776\pi$
$47$	408.000	1.26623	0.633116	+	0.774057i	$0.281776\pi$
$48$	0	0
$49$	−87.0000	−0.253644
$50$	0	0
$51$	54.0000	0.148265
$52$	0	0
$53$	262.000	0.679028	0.339514	−	0.940601i	$-0.389737\pi$
$53$	262.000	0.679028	0.339514	+	0.940601i	$0.389737\pi$
$54$	0	0
$55$	120.000	0.294196
$56$	0	0
$57$	−108.000	−0.250964
$58$	0	0
$59$	−64.0000	−0.141222	−0.0706109	−	0.997504i	$-0.522495\pi$
$59$	−64.0000	−0.141222	−0.0706109	+	0.997504i	$0.522495\pi$
$60$	0	0
$61$	374.000	0.785013	0.392507	−	0.919749i	$-0.371608\pi$
$61$	374.000	0.785013	0.392507	+	0.919749i	$0.371608\pi$
$62$	0	0
$63$	144.000	0.287973
$64$	0	0
$65$	−70.0000	−0.133576
$66$	0	0
$67$	300.000	0.547027	0.273514	−	0.961868i	$-0.411814\pi$
$67$	300.000	0.547027	0.273514	+	0.961868i	$0.411814\pi$
$68$	0	0
$69$	−312.000	−0.544353
$70$	0	0
$71$	1016.00	1.69827	0.849134	−	0.528178i	$-0.177124\pi$
$71$	1016.00	1.69827	0.849134	+	0.528178i	$0.177124\pi$
$72$	0	0
$73$	274.000	0.439305	0.219653	−	0.975578i	$-0.429508\pi$
$73$	274.000	0.439305	0.219653	+	0.975578i	$0.429508\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	384.000	0.568323
$78$	0	0
$79$	788.000	1.12224	0.561120	−	0.827735i	$-0.310371\pi$
$79$	788.000	1.12224	0.561120	+	0.827735i	$0.310371\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−396.000	−0.523695	−0.261847	−	0.965109i	$-0.584332\pi$
$83$	−396.000	−0.523695	−0.261847	+	0.965109i	$0.584332\pi$
$84$	0	0
$85$	−90.0000	−0.114846
$86$	0	0
$87$	750.000	0.924235
$88$	0	0
$89$	786.000	0.936133	0.468066	−	0.883693i	$-0.344951\pi$
$89$	786.000	0.936133	0.468066	+	0.883693i	$0.344951\pi$
$90$	0	0
$91$	−224.000	−0.258039
$92$	0	0
$93$	84.0000	0.0936602
$94$	0	0
$95$	180.000	0.194396
$96$	0	0
$97$	−1086.00	−1.13677	−0.568385	−	0.822763i	$-0.692431\pi$
$97$	−1086.00	−1.13677	−0.568385	+	0.822763i	$0.692431\pi$
$98$	0	0
$99$	216.000	0.219281
$100$	0	0
$101$	78.0000	0.0768445	0.0384222	−	0.999262i	$-0.487767\pi$
$101$	78.0000	0.0768445	0.0384222	+	0.999262i	$0.487767\pi$
$102$	0	0
$103$	1208.00	1.15561	0.577805	−	0.816175i	$-0.303910\pi$
$103$	1208.00	1.15561	0.577805	+	0.816175i	$0.303910\pi$
$104$	0	0
$105$	−240.000	−0.223063
$106$	0	0
$107$	−44.0000	−0.0397537	−0.0198768	−	0.999802i	$-0.506327\pi$
$107$	−44.0000	−0.0397537	−0.0198768	+	0.999802i	$0.506327\pi$
$108$	0	0
$109$	−1122.00	−0.985946	−0.492973	−	0.870045i	$-0.664090\pi$
$109$	−1122.00	−0.985946	−0.492973	+	0.870045i	$0.664090\pi$
$110$	0	0
$111$	162.000	0.138526
$112$	0	0
$113$	606.000	0.504493	0.252246	−	0.967663i	$-0.418831\pi$
$113$	606.000	0.504493	0.252246	+	0.967663i	$0.418831\pi$
$114$	0	0
$115$	520.000	0.421654
$116$	0	0
$117$	−126.000	−0.0995616
$118$	0	0
$119$	−288.000	−0.221856
$120$	0	0
$121$	−755.000	−0.567243
$122$	0	0
$123$	−1062.00	−0.778515
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1744.00	1.21854	0.609272	−	0.792962i	$-0.291462\pi$
$127$	1744.00	1.21854	0.609272	+	0.792962i	$0.291462\pi$
$128$	0	0
$129$	−684.000	−0.466844
$130$	0	0
$131$	−480.000	−0.320136	−0.160068	−	0.987106i	$-0.551171\pi$
$131$	−480.000	−0.320136	−0.160068	+	0.987106i	$0.551171\pi$
$132$	0	0
$133$	576.000	0.375530
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	1598.00	0.996543	0.498271	−	0.867021i	$-0.333968\pi$
$137$	1598.00	0.996543	0.498271	+	0.867021i	$0.333968\pi$
$138$	0	0
$139$	−2964.00	−1.80866	−0.904328	−	0.426838i	$-0.859627\pi$
$139$	−2964.00	−1.80866	−0.904328	+	0.426838i	$0.859627\pi$
$140$	0	0
$141$	−1224.00	−0.731060
$142$	0	0
$143$	−336.000	−0.196488
$144$	0	0
$145$	−1250.00	−0.715909
$146$	0	0
$147$	261.000	0.146442
$148$	0	0
$149$	334.000	0.183640	0.0918200	−	0.995776i	$-0.470732\pi$
$149$	334.000	0.183640	0.0918200	+	0.995776i	$0.470732\pi$
$150$	0	0
$151$	−1148.00	−0.618695	−0.309347	−	0.950949i	$-0.600111\pi$
$151$	−1148.00	−0.618695	−0.309347	+	0.950949i	$0.600111\pi$
$152$	0	0
$153$	−162.000	−0.0856008
$154$	0	0
$155$	−140.000	−0.0725488
$156$	0	0
$157$	906.000	0.460552	0.230276	−	0.973125i	$-0.426037\pi$
$157$	906.000	0.460552	0.230276	+	0.973125i	$0.426037\pi$
$158$	0	0
$159$	−786.000	−0.392037
$160$	0	0
$161$	1664.00	0.814544
$162$	0	0
$163$	−1916.00	−0.920691	−0.460346	−	0.887740i	$-0.652275\pi$
$163$	−1916.00	−0.920691	−0.460346	+	0.887740i	$0.652275\pi$
$164$	0	0
$165$	−360.000	−0.169854
$166$	0	0
$167$	1152.00	0.533799	0.266900	−	0.963724i	$-0.414001\pi$
$167$	1152.00	0.533799	0.266900	+	0.963724i	$0.414001\pi$
$168$	0	0
$169$	−2001.00	−0.910787
$170$	0	0
$171$	324.000	0.144894
$172$	0	0
$173$	3142.00	1.38082	0.690410	−	0.723418i	$-0.257430\pi$
$173$	3142.00	1.38082	0.690410	+	0.723418i	$0.257430\pi$
$174$	0	0
$175$	400.000	0.172784
$176$	0	0
$177$	192.000	0.0815345
$178$	0	0
$179$	−1032.00	−0.430923	−0.215462	−	0.976512i	$-0.569126\pi$
$179$	−1032.00	−0.430923	−0.215462	+	0.976512i	$0.569126\pi$
$180$	0	0
$181$	−1562.00	−0.641451	−0.320725	−	0.947172i	$-0.603927\pi$
$181$	−1562.00	−0.641451	−0.320725	+	0.947172i	$0.603927\pi$
$182$	0	0
$183$	−1122.00	−0.453227
$184$	0	0
$185$	−270.000	−0.107302
$186$	0	0
$187$	−432.000	−0.168936
$188$	0	0
$189$	−432.000	−0.166261
$190$	0	0
$191$	1960.00	0.742516	0.371258	−	0.928530i	$-0.378926\pi$
$191$	1960.00	0.742516	0.371258	+	0.928530i	$0.378926\pi$
$192$	0	0
$193$	−4006.00	−1.49408	−0.747042	−	0.664777i	$-0.768527\pi$
$193$	−4006.00	−1.49408	−0.747042	+	0.664777i	$0.768527\pi$
$194$	0	0
$195$	210.000	0.0771201
$196$	0	0
$197$	2118.00	0.765996	0.382998	−	0.923749i	$-0.374892\pi$
$197$	2118.00	0.765996	0.382998	+	0.923749i	$0.374892\pi$
$198$	0	0
$199$	−3748.00	−1.33512	−0.667559	−	0.744556i	$-0.732661\pi$
$199$	−3748.00	−1.33512	−0.667559	+	0.744556i	$0.732661\pi$
$200$	0	0
$201$	−900.000	−0.315826
$202$	0	0
$203$	−4000.00	−1.38298
$204$	0	0
$205$	1770.00	0.603035
$206$	0	0
$207$	936.000	0.314283
$208$	0	0
$209$	864.000	0.285953
$210$	0	0
$211$	−4796.00	−1.56479	−0.782394	−	0.622784i	$-0.786002\pi$
$211$	−4796.00	−1.56479	−0.782394	+	0.622784i	$0.786002\pi$
$212$	0	0
$213$	−3048.00	−0.980495
$214$	0	0
$215$	1140.00	0.361616
$216$	0	0
$217$	−448.000	−0.140148
$218$	0	0
$219$	−822.000	−0.253633
$220$	0	0
$221$	252.000	0.0767030
$222$	0	0
$223$	2560.00	0.768746	0.384373	−	0.923178i	$-0.374418\pi$
$223$	2560.00	0.768746	0.384373	+	0.923178i	$0.374418\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	3500.00	1.02336	0.511681	−	0.859176i	$-0.329023\pi$
$227$	3500.00	1.02336	0.511681	+	0.859176i	$0.329023\pi$
$228$	0	0
$229$	1966.00	0.567323	0.283661	−	0.958924i	$-0.408451\pi$
$229$	1966.00	0.567323	0.283661	+	0.958924i	$0.408451\pi$
$230$	0	0
$231$	−1152.00	−0.328121
$232$	0	0
$233$	3246.00	0.912672	0.456336	−	0.889808i	$-0.349162\pi$
$233$	3246.00	0.912672	0.456336	+	0.889808i	$0.349162\pi$
$234$	0	0
$235$	2040.00	0.566276
$236$	0	0
$237$	−2364.00	−0.647925
$238$	0	0
$239$	−7320.00	−1.98114	−0.990568	−	0.137023i	$-0.956247\pi$
$239$	−7320.00	−1.98114	−0.990568	+	0.137023i	$0.956247\pi$
$240$	0	0
$241$	3490.00	0.932824	0.466412	−	0.884568i	$-0.345546\pi$
$241$	3490.00	0.932824	0.466412	+	0.884568i	$0.345546\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−435.000	−0.113433
$246$	0	0
$247$	−504.000	−0.129833
$248$	0	0
$249$	1188.00	0.302355
$250$	0	0
$251$	−7456.00	−1.87497	−0.937487	−	0.348020i	$-0.886854\pi$
$251$	−7456.00	−1.87497	−0.937487	+	0.348020i	$0.886854\pi$
$252$	0	0
$253$	2496.00	0.620246
$254$	0	0
$255$	270.000	0.0663061
$256$	0	0
$257$	4558.00	1.10630	0.553152	−	0.833080i	$-0.313425\pi$
$257$	4558.00	1.10630	0.553152	+	0.833080i	$0.313425\pi$
$258$	0	0
$259$	−864.000	−0.207283
$260$	0	0
$261$	−2250.00	−0.533607
$262$	0	0
$263$	−2848.00	−0.667738	−0.333869	−	0.942619i	$-0.608354\pi$
$263$	−2848.00	−0.667738	−0.333869	+	0.942619i	$0.608354\pi$
$264$	0	0
$265$	1310.00	0.303670
$266$	0	0
$267$	−2358.00	−0.540477
$268$	0	0
$269$	3110.00	0.704907	0.352454	−	0.935829i	$-0.385347\pi$
$269$	3110.00	0.704907	0.352454	+	0.935829i	$0.385347\pi$
$270$	0	0
$271$	1700.00	0.381061	0.190531	−	0.981681i	$-0.438979\pi$
$271$	1700.00	0.381061	0.190531	+	0.981681i	$0.438979\pi$
$272$	0	0
$273$	672.000	0.148979
$274$	0	0
$275$	600.000	0.131569
$276$	0	0
$277$	−6494.00	−1.40862	−0.704308	−	0.709895i	$-0.748743\pi$
$277$	−6494.00	−1.40862	−0.704308	+	0.709895i	$0.748743\pi$
$278$	0	0
$279$	−252.000	−0.0540747
$280$	0	0
$281$	2498.00	0.530314	0.265157	−	0.964205i	$-0.414576\pi$
$281$	2498.00	0.530314	0.265157	+	0.964205i	$0.414576\pi$
$282$	0	0
$283$	−5324.00	−1.11830	−0.559150	−	0.829066i	$-0.688872\pi$
$283$	−5324.00	−1.11830	−0.559150	+	0.829066i	$0.688872\pi$
$284$	0	0
$285$	−540.000	−0.112235
$286$	0	0
$287$	5664.00	1.16493
$288$	0	0
$289$	−4589.00	−0.934053
$290$	0	0
$291$	3258.00	0.656314
$292$	0	0
$293$	−522.000	−0.104080	−0.0520402	−	0.998645i	$-0.516572\pi$
$293$	−522.000	−0.104080	−0.0520402	+	0.998645i	$0.516572\pi$
$294$	0	0
$295$	−320.000	−0.0631563
$296$	0	0
$297$	−648.000	−0.126602
$298$	0	0
$299$	−1456.00	−0.281614
$300$	0	0
$301$	3648.00	0.698562
$302$	0	0
$303$	−234.000	−0.0443662
$304$	0	0
$305$	1870.00	0.351068
$306$	0	0
$307$	7844.00	1.45824	0.729122	−	0.684384i	$-0.239929\pi$
$307$	7844.00	1.45824	0.729122	+	0.684384i	$0.239929\pi$
$308$	0	0
$309$	−3624.00	−0.667191
$310$	0	0
$311$	−3248.00	−0.592210	−0.296105	−	0.955155i	$-0.595688\pi$
$311$	−3248.00	−0.592210	−0.296105	+	0.955155i	$0.595688\pi$
$312$	0	0
$313$	−5374.00	−0.970468	−0.485234	−	0.874384i	$-0.661266\pi$
$313$	−5374.00	−0.970468	−0.485234	+	0.874384i	$0.661266\pi$
$314$	0	0
$315$	720.000	0.128785
$316$	0	0
$317$	−6786.00	−1.20233	−0.601167	−	0.799124i	$-0.705297\pi$
$317$	−6786.00	−1.20233	−0.601167	+	0.799124i	$0.705297\pi$
$318$	0	0
$319$	−6000.00	−1.05309
$320$	0	0
$321$	132.000	0.0229518
$322$	0	0
$323$	−648.000	−0.111628
$324$	0	0
$325$	−350.000	−0.0597369
$326$	0	0
$327$	3366.00	0.569236
$328$	0	0
$329$	6528.00	1.09392
$330$	0	0
$331$	6596.00	1.09531	0.547657	−	0.836703i	$-0.315520\pi$
$331$	6596.00	1.09531	0.547657	+	0.836703i	$0.315520\pi$
$332$	0	0
$333$	−486.000	−0.0799779
$334$	0	0
$335$	1500.00	0.244638
$336$	0	0
$337$	−5830.00	−0.942375	−0.471187	−	0.882033i	$-0.656174\pi$
$337$	−5830.00	−0.942375	−0.471187	+	0.882033i	$0.656174\pi$
$338$	0	0
$339$	−1818.00	−0.291269
$340$	0	0
$341$	−672.000	−0.106718
$342$	0	0
$343$	−6880.00	−1.08305
$344$	0	0
$345$	−1560.00	−0.243442
$346$	0	0
$347$	11732.0	1.81501	0.907503	−	0.420047i	$-0.137986\pi$
$347$	11732.0	1.81501	0.907503	+	0.420047i	$0.137986\pi$
$348$	0	0
$349$	1014.00	0.155525	0.0777624	−	0.996972i	$-0.475222\pi$
$349$	1014.00	0.155525	0.0777624	+	0.996972i	$0.475222\pi$
$350$	0	0
$351$	378.000	0.0574819
$352$	0	0
$353$	−8202.00	−1.23668	−0.618341	−	0.785910i	$-0.712195\pi$
$353$	−8202.00	−1.23668	−0.618341	+	0.785910i	$0.712195\pi$
$354$	0	0
$355$	5080.00	0.759488
$356$	0	0
$357$	864.000	0.128089
$358$	0	0
$359$	−8160.00	−1.19963	−0.599817	−	0.800138i	$-0.704760\pi$
$359$	−8160.00	−1.19963	−0.599817	+	0.800138i	$0.704760\pi$
$360$	0	0
$361$	−5563.00	−0.811051
$362$	0	0
$363$	2265.00	0.327498
$364$	0	0
$365$	1370.00	0.196463
$366$	0	0
$367$	12360.0	1.75800	0.879001	−	0.476820i	$-0.158211\pi$
$367$	12360.0	1.75800	0.879001	+	0.476820i	$0.158211\pi$
$368$	0	0
$369$	3186.00	0.449476
$370$	0	0
$371$	4192.00	0.586625
$372$	0	0
$373$	930.000	0.129098	0.0645490	−	0.997915i	$-0.479439\pi$
$373$	930.000	0.129098	0.0645490	+	0.997915i	$0.479439\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	3500.00	0.478141
$378$	0	0
$379$	4228.00	0.573028	0.286514	−	0.958076i	$-0.407503\pi$
$379$	4228.00	0.573028	0.286514	+	0.958076i	$0.407503\pi$
$380$	0	0
$381$	−5232.00	−0.703526
$382$	0	0
$383$	8384.00	1.11854	0.559272	−	0.828984i	$-0.311081\pi$
$383$	8384.00	1.11854	0.559272	+	0.828984i	$0.311081\pi$
$384$	0	0
$385$	1920.00	0.254162
$386$	0	0
$387$	2052.00	0.269532
$388$	0	0
$389$	5534.00	0.721298	0.360649	−	0.932702i	$-0.382555\pi$
$389$	5534.00	0.721298	0.360649	+	0.932702i	$0.382555\pi$
$390$	0	0
$391$	−1872.00	−0.242126
$392$	0	0
$393$	1440.00	0.184831
$394$	0	0
$395$	3940.00	0.501881
$396$	0	0
$397$	5426.00	0.685952	0.342976	−	0.939344i	$-0.388565\pi$
$397$	5426.00	0.685952	0.342976	+	0.939344i	$0.388565\pi$
$398$	0	0
$399$	−1728.00	−0.216813
$400$	0	0
$401$	−78.0000	−0.00971355	−0.00485678	−	0.999988i	$-0.501546\pi$
$401$	−78.0000	−0.00971355	−0.00485678	+	0.999988i	$0.501546\pi$
$402$	0	0
$403$	392.000	0.0484539
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	−1296.00	−0.157839
$408$	0	0
$409$	−454.000	−0.0548872	−0.0274436	−	0.999623i	$-0.508737\pi$
$409$	−454.000	−0.0548872	−0.0274436	+	0.999623i	$0.508737\pi$
$410$	0	0
$411$	−4794.00	−0.575354
$412$	0	0
$413$	−1024.00	−0.122004
$414$	0	0
$415$	−1980.00	−0.234203
$416$	0	0
$417$	8892.00	1.04423
$418$	0	0
$419$	12296.0	1.43365	0.716824	−	0.697254i	$-0.245595\pi$
$419$	12296.0	1.43365	0.716824	+	0.697254i	$0.245595\pi$
$420$	0	0
$421$	12798.0	1.48156	0.740780	−	0.671748i	$-0.234456\pi$
$421$	12798.0	1.48156	0.740780	+	0.671748i	$0.234456\pi$
$422$	0	0
$423$	3672.00	0.422077
$424$	0	0
$425$	−450.000	−0.0513605
$426$	0	0
$427$	5984.00	0.678187
$428$	0	0
$429$	1008.00	0.113442
$430$	0	0
$431$	9912.00	1.10776	0.553880	−	0.832597i	$-0.313147\pi$
$431$	9912.00	1.10776	0.553880	+	0.832597i	$0.313147\pi$
$432$	0	0
$433$	−6774.00	−0.751819	−0.375910	−	0.926656i	$-0.622670\pi$
$433$	−6774.00	−0.751819	−0.375910	+	0.926656i	$0.622670\pi$
$434$	0	0
$435$	3750.00	0.413330
$436$	0	0
$437$	3744.00	0.409839
$438$	0	0
$439$	−16628.0	−1.80777	−0.903885	−	0.427775i	$-0.859297\pi$
$439$	−16628.0	−1.80777	−0.903885	+	0.427775i	$0.859297\pi$
$440$	0	0
$441$	−783.000	−0.0845481
$442$	0	0
$443$	−940.000	−0.100814	−0.0504072	−	0.998729i	$-0.516052\pi$
$443$	−940.000	−0.100814	−0.0504072	+	0.998729i	$0.516052\pi$
$444$	0	0
$445$	3930.00	0.418651
$446$	0	0
$447$	−1002.00	−0.106025
$448$	0	0
$449$	−1662.00	−0.174687	−0.0873437	−	0.996178i	$-0.527838\pi$
$449$	−1662.00	−0.174687	−0.0873437	+	0.996178i	$0.527838\pi$
$450$	0	0
$451$	8496.00	0.887053
$452$	0	0
$453$	3444.00	0.357204
$454$	0	0
$455$	−1120.00	−0.115399
$456$	0	0
$457$	−13942.0	−1.42709	−0.713544	−	0.700610i	$-0.752911\pi$
$457$	−13942.0	−1.42709	−0.713544	+	0.700610i	$0.752911\pi$
$458$	0	0
$459$	486.000	0.0494217
$460$	0	0
$461$	−16170.0	−1.63365	−0.816824	−	0.576887i	$-0.804267\pi$
$461$	−16170.0	−1.63365	−0.816824	+	0.576887i	$0.804267\pi$
$462$	0	0
$463$	1048.00	0.105194	0.0525969	−	0.998616i	$-0.483250\pi$
$463$	1048.00	0.105194	0.0525969	+	0.998616i	$0.483250\pi$
$464$	0	0
$465$	420.000	0.0418861
$466$	0	0
$467$	13716.0	1.35910	0.679551	−	0.733628i	$-0.262175\pi$
$467$	13716.0	1.35910	0.679551	+	0.733628i	$0.262175\pi$
$468$	0	0
$469$	4800.00	0.472587
$470$	0	0
$471$	−2718.00	−0.265900
$472$	0	0
$473$	5472.00	0.531930
$474$	0	0
$475$	900.000	0.0869365
$476$	0	0
$477$	2358.00	0.226343
$478$	0	0
$479$	−8832.00	−0.842473	−0.421236	−	0.906951i	$-0.638404\pi$
$479$	−8832.00	−0.842473	−0.421236	+	0.906951i	$0.638404\pi$
$480$	0	0
$481$	756.000	0.0716645
$482$	0	0
$483$	−4992.00	−0.470277
$484$	0	0
$485$	−5430.00	−0.508379
$486$	0	0
$487$	10120.0	0.941645	0.470822	−	0.882228i	$-0.343957\pi$
$487$	10120.0	0.941645	0.470822	+	0.882228i	$0.343957\pi$
$488$	0	0
$489$	5748.00	0.531561
$490$	0	0
$491$	−4376.00	−0.402212	−0.201106	−	0.979569i	$-0.564454\pi$
$491$	−4376.00	−0.402212	−0.201106	+	0.979569i	$0.564454\pi$
$492$	0	0
$493$	4500.00	0.411095
$494$	0	0
$495$	1080.00	0.0980654
$496$	0	0
$497$	16256.0	1.46717
$498$	0	0
$499$	12364.0	1.10920	0.554598	−	0.832119i	$-0.312872\pi$
$499$	12364.0	1.10920	0.554598	+	0.832119i	$0.312872\pi$
$500$	0	0
$501$	−3456.00	−0.308189
$502$	0	0
$503$	−1248.00	−0.110627	−0.0553137	−	0.998469i	$-0.517616\pi$
$503$	−1248.00	−0.110627	−0.0553137	+	0.998469i	$0.517616\pi$
$504$	0	0
$505$	390.000	0.0343659
$506$	0	0
$507$	6003.00	0.525843
$508$	0	0
$509$	−12730.0	−1.10854	−0.554270	−	0.832337i	$-0.687003\pi$
$509$	−12730.0	−1.10854	−0.554270	+	0.832337i	$0.687003\pi$
$510$	0	0
$511$	4384.00	0.379524
$512$	0	0
$513$	−972.000	−0.0836547
$514$	0	0
$515$	6040.00	0.516804
$516$	0	0
$517$	9792.00	0.832982
$518$	0	0
$519$	−9426.00	−0.797217
$520$	0	0
$521$	−13286.0	−1.11722	−0.558609	−	0.829431i	$-0.688665\pi$
$521$	−13286.0	−1.11722	−0.558609	+	0.829431i	$0.688665\pi$
$522$	0	0
$523$	−15892.0	−1.32870	−0.664349	−	0.747423i	$-0.731291\pi$
$523$	−15892.0	−1.32870	−0.664349	+	0.747423i	$0.731291\pi$
$524$	0	0
$525$	−1200.00	−0.0997567
$526$	0	0
$527$	504.000	0.0416596
$528$	0	0
$529$	−1351.00	−0.111038
$530$	0	0
$531$	−576.000	−0.0470740
$532$	0	0
$533$	−4956.00	−0.402755
$534$	0	0
$535$	−220.000	−0.0177784
$536$	0	0
$537$	3096.00	0.248794
$538$	0	0
$539$	−2088.00	−0.166858
$540$	0	0
$541$	9662.00	0.767841	0.383920	−	0.923366i	$-0.374574\pi$
$541$	9662.00	0.767841	0.383920	+	0.923366i	$0.374574\pi$
$542$	0	0
$543$	4686.00	0.370342
$544$	0	0
$545$	−5610.00	−0.440928
$546$	0	0
$547$	9596.00	0.750083	0.375041	−	0.927008i	$-0.377628\pi$
$547$	9596.00	0.750083	0.375041	+	0.927008i	$0.377628\pi$
$548$	0	0
$549$	3366.00	0.261671
$550$	0	0
$551$	−9000.00	−0.695849
$552$	0	0
$553$	12608.0	0.969524
$554$	0	0
$555$	810.000	0.0619506
$556$	0	0
$557$	−4458.00	−0.339123	−0.169562	−	0.985520i	$-0.554235\pi$
$557$	−4458.00	−0.339123	−0.169562	+	0.985520i	$0.554235\pi$
$558$	0	0
$559$	−3192.00	−0.241516
$560$	0	0
$561$	1296.00	0.0975350
$562$	0	0
$563$	−4708.00	−0.352431	−0.176215	−	0.984352i	$-0.556386\pi$
$563$	−4708.00	−0.352431	−0.176215	+	0.984352i	$0.556386\pi$
$564$	0	0
$565$	3030.00	0.225616
$566$	0	0
$567$	1296.00	0.0959910
$568$	0	0
$569$	−12358.0	−0.910500	−0.455250	−	0.890364i	$-0.650450\pi$
$569$	−12358.0	−0.910500	−0.455250	+	0.890364i	$0.650450\pi$
$570$	0	0
$571$	−7532.00	−0.552022	−0.276011	−	0.961155i	$-0.589013\pi$
$571$	−7532.00	−0.552022	−0.276011	+	0.961155i	$0.589013\pi$
$572$	0	0
$573$	−5880.00	−0.428692
$574$	0	0
$575$	2600.00	0.188570
$576$	0	0
$577$	−18878.0	−1.36205	−0.681024	−	0.732261i	$-0.738465\pi$
$577$	−18878.0	−1.36205	−0.681024	+	0.732261i	$0.738465\pi$
$578$	0	0
$579$	12018.0	0.862610
$580$	0	0
$581$	−6336.00	−0.452430
$582$	0	0
$583$	6288.00	0.446694
$584$	0	0
$585$	−630.000	−0.0445253
$586$	0	0
$587$	−22380.0	−1.57363	−0.786816	−	0.617188i	$-0.788272\pi$
$587$	−22380.0	−1.57363	−0.786816	+	0.617188i	$0.788272\pi$
$588$	0	0
$589$	−1008.00	−0.0705160
$590$	0	0
$591$	−6354.00	−0.442248
$592$	0	0
$593$	7726.00	0.535023	0.267512	−	0.963555i	$-0.413799\pi$
$593$	7726.00	0.535023	0.267512	+	0.963555i	$0.413799\pi$
$594$	0	0
$595$	−1440.00	−0.0992172
$596$	0	0
$597$	11244.0	0.770831
$598$	0	0
$599$	−21232.0	−1.44827	−0.724137	−	0.689656i	$-0.757762\pi$
$599$	−21232.0	−1.44827	−0.724137	+	0.689656i	$0.757762\pi$
$600$	0	0
$601$	18954.0	1.28644	0.643219	−	0.765682i	$-0.277598\pi$
$601$	18954.0	1.28644	0.643219	+	0.765682i	$0.277598\pi$
$602$	0	0
$603$	2700.00	0.182342
$604$	0	0
$605$	−3775.00	−0.253679
$606$	0	0
$607$	1896.00	0.126781	0.0633907	−	0.997989i	$-0.479809\pi$
$607$	1896.00	0.126781	0.0633907	+	0.997989i	$0.479809\pi$
$608$	0	0
$609$	12000.0	0.798464
$610$	0	0
$611$	−5712.00	−0.378204
$612$	0	0
$613$	−9862.00	−0.649792	−0.324896	−	0.945750i	$-0.605329\pi$
$613$	−9862.00	−0.649792	−0.324896	+	0.945750i	$0.605329\pi$
$614$	0	0
$615$	−5310.00	−0.348162
$616$	0	0
$617$	−20434.0	−1.33329	−0.666647	−	0.745374i	$-0.732271\pi$
$617$	−20434.0	−1.33329	−0.666647	+	0.745374i	$0.732271\pi$
$618$	0	0
$619$	−12644.0	−0.821010	−0.410505	−	0.911858i	$-0.634648\pi$
$619$	−12644.0	−0.821010	−0.410505	+	0.911858i	$0.634648\pi$
$620$	0	0
$621$	−2808.00	−0.181451
$622$	0	0
$623$	12576.0	0.808743
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−2592.00	−0.165095
$628$	0	0
$629$	972.000	0.0616155
$630$	0	0
$631$	−4660.00	−0.293996	−0.146998	−	0.989137i	$-0.546961\pi$
$631$	−4660.00	−0.293996	−0.146998	+	0.989137i	$0.546961\pi$
$632$	0	0
$633$	14388.0	0.903431
$634$	0	0
$635$	8720.00	0.544949
$636$	0	0
$637$	1218.00	0.0757597
$638$	0	0
$639$	9144.00	0.566089
$640$	0	0
$641$	−8598.00	−0.529798	−0.264899	−	0.964276i	$-0.585339\pi$
$641$	−8598.00	−0.529798	−0.264899	+	0.964276i	$0.585339\pi$
$642$	0	0
$643$	−1836.00	−0.112605	−0.0563023	−	0.998414i	$-0.517931\pi$
$643$	−1836.00	−0.112605	−0.0563023	+	0.998414i	$0.517931\pi$
$644$	0	0
$645$	−3420.00	−0.208779
$646$	0	0
$647$	1696.00	0.103055	0.0515275	−	0.998672i	$-0.483591\pi$
$647$	1696.00	0.103055	0.0515275	+	0.998672i	$0.483591\pi$
$648$	0	0
$649$	−1536.00	−0.0929018
$650$	0	0
$651$	1344.00	0.0809148
$652$	0	0
$653$	−24730.0	−1.48202	−0.741010	−	0.671493i	$-0.765653\pi$
$653$	−24730.0	−1.48202	−0.741010	+	0.671493i	$0.765653\pi$
$654$	0	0
$655$	−2400.00	−0.143169
$656$	0	0
$657$	2466.00	0.146435
$658$	0	0
$659$	4800.00	0.283735	0.141868	−	0.989886i	$-0.454689\pi$
$659$	4800.00	0.283735	0.141868	+	0.989886i	$0.454689\pi$
$660$	0	0
$661$	32174.0	1.89323	0.946614	−	0.322370i	$-0.104479\pi$
$661$	32174.0	1.89323	0.946614	+	0.322370i	$0.104479\pi$
$662$	0	0
$663$	−756.000	−0.0442845
$664$	0	0
$665$	2880.00	0.167942
$666$	0	0
$667$	−26000.0	−1.50933
$668$	0	0
$669$	−7680.00	−0.443836
$670$	0	0
$671$	8976.00	0.516415
$672$	0	0
$673$	7114.00	0.407466	0.203733	−	0.979026i	$-0.434693\pi$
$673$	7114.00	0.407466	0.203733	+	0.979026i	$0.434693\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	−20466.0	−1.16185	−0.580925	−	0.813957i	$-0.697309\pi$
$677$	−20466.0	−1.16185	−0.580925	+	0.813957i	$0.697309\pi$
$678$	0	0
$679$	−17376.0	−0.982076
$680$	0	0
$681$	−10500.0	−0.590838
$682$	0	0
$683$	−34068.0	−1.90860	−0.954301	−	0.298846i	$-0.903398\pi$
$683$	−34068.0	−1.90860	−0.954301	+	0.298846i	$0.903398\pi$
$684$	0	0
$685$	7990.00	0.445667
$686$	0	0
$687$	−5898.00	−0.327544
$688$	0	0
$689$	−3668.00	−0.202815
$690$	0	0
$691$	−21340.0	−1.17484	−0.587418	−	0.809284i	$-0.699856\pi$
$691$	−21340.0	−1.17484	−0.587418	+	0.809284i	$0.699856\pi$
$692$	0	0
$693$	3456.00	0.189441
$694$	0	0
$695$	−14820.0	−0.808856
$696$	0	0
$697$	−6372.00	−0.346279
$698$	0	0
$699$	−9738.00	−0.526931
$700$	0	0
$701$	−5370.00	−0.289333	−0.144666	−	0.989481i	$-0.546211\pi$
$701$	−5370.00	−0.289333	−0.144666	+	0.989481i	$0.546211\pi$
$702$	0	0
$703$	−1944.00	−0.104295
$704$	0	0
$705$	−6120.00	−0.326940
$706$	0	0
$707$	1248.00	0.0663874
$708$	0	0
$709$	−18690.0	−0.990011	−0.495005	−	0.868890i	$-0.664834\pi$
$709$	−18690.0	−0.990011	−0.495005	+	0.868890i	$0.664834\pi$
$710$	0	0
$711$	7092.00	0.374080
$712$	0	0
$713$	−2912.00	−0.152953
$714$	0	0
$715$	−1680.00	−0.0878719
$716$	0	0
$717$	21960.0	1.14381
$718$	0	0
$719$	−14328.0	−0.743177	−0.371588	−	0.928398i	$-0.621187\pi$
$719$	−14328.0	−0.743177	−0.371588	+	0.928398i	$0.621187\pi$
$720$	0	0
$721$	19328.0	0.998353
$722$	0	0
$723$	−10470.0	−0.538566
$724$	0	0
$725$	−6250.00	−0.320164
$726$	0	0
$727$	−14488.0	−0.739106	−0.369553	−	0.929210i	$-0.620489\pi$
$727$	−14488.0	−0.739106	−0.369553	+	0.929210i	$0.620489\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−4104.00	−0.207650
$732$	0	0
$733$	25354.0	1.27759	0.638794	−	0.769378i	$-0.279434\pi$
$733$	25354.0	1.27759	0.638794	+	0.769378i	$0.279434\pi$
$734$	0	0
$735$	1305.00	0.0654907
$736$	0	0
$737$	7200.00	0.359858
$738$	0	0
$739$	33100.0	1.64764	0.823818	−	0.566854i	$-0.191840\pi$
$739$	33100.0	1.64764	0.823818	+	0.566854i	$0.191840\pi$
$740$	0	0
$741$	1512.00	0.0749591
$742$	0	0
$743$	−4456.00	−0.220020	−0.110010	−	0.993930i	$-0.535088\pi$
$743$	−4456.00	−0.220020	−0.110010	+	0.993930i	$0.535088\pi$
$744$	0	0
$745$	1670.00	0.0821263
$746$	0	0
$747$	−3564.00	−0.174565
$748$	0	0
$749$	−704.000	−0.0343439
$750$	0	0
$751$	23268.0	1.13057	0.565287	−	0.824894i	$-0.308765\pi$
$751$	23268.0	1.13057	0.565287	+	0.824894i	$0.308765\pi$
$752$	0	0
$753$	22368.0	1.08252
$754$	0	0
$755$	−5740.00	−0.276689
$756$	0	0
$757$	−35726.0	−1.71530	−0.857651	−	0.514232i	$-0.828077\pi$
$757$	−35726.0	−1.71530	−0.857651	+	0.514232i	$0.828077\pi$
$758$	0	0
$759$	−7488.00	−0.358099
$760$	0	0
$761$	−12278.0	−0.584858	−0.292429	−	0.956287i	$-0.594464\pi$
$761$	−12278.0	−0.584858	−0.292429	+	0.956287i	$0.594464\pi$
$762$	0	0
$763$	−17952.0	−0.851777
$764$	0	0
$765$	−810.000	−0.0382818
$766$	0	0
$767$	896.000	0.0421808
$768$	0	0
$769$	−26542.0	−1.24464	−0.622321	−	0.782763i	$-0.713810\pi$
$769$	−26542.0	−1.24464	−0.622321	+	0.782763i	$0.713810\pi$
$770$	0	0
$771$	−13674.0	−0.638725
$772$	0	0
$773$	9942.00	0.462599	0.231299	−	0.972883i	$-0.425702\pi$
$773$	9942.00	0.462599	0.231299	+	0.972883i	$0.425702\pi$
$774$	0	0
$775$	−700.000	−0.0324448
$776$	0	0
$777$	2592.00	0.119675
$778$	0	0
$779$	12744.0	0.586138
$780$	0	0
$781$	24384.0	1.11719
$782$	0	0
$783$	6750.00	0.308078
$784$	0	0
$785$	4530.00	0.205965
$786$	0	0
$787$	−11132.0	−0.504210	−0.252105	−	0.967700i	$-0.581123\pi$
$787$	−11132.0	−0.504210	−0.252105	+	0.967700i	$0.581123\pi$
$788$	0	0
$789$	8544.00	0.385519
$790$	0	0
$791$	9696.00	0.435841
$792$	0	0
$793$	−5236.00	−0.234471
$794$	0	0
$795$	−3930.00	−0.175324
$796$	0	0
$797$	23910.0	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	23910.0	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	0	0
$799$	−7344.00	−0.325172
$800$	0	0
$801$	7074.00	0.312044
$802$	0	0
$803$	6576.00	0.288994
$804$	0	0
$805$	8320.00	0.364275
$806$	0	0
$807$	−9330.00	−0.406978
$808$	0	0
$809$	−15934.0	−0.692472	−0.346236	−	0.938148i	$-0.612540\pi$
$809$	−15934.0	−0.692472	−0.346236	+	0.938148i	$0.612540\pi$
$810$	0	0
$811$	−23756.0	−1.02859	−0.514295	−	0.857614i	$-0.671946\pi$
$811$	−23756.0	−1.02859	−0.514295	+	0.857614i	$0.671946\pi$
$812$	0	0
$813$	−5100.00	−0.220006
$814$	0	0
$815$	−9580.00	−0.411746
$816$	0	0
$817$	8208.00	0.351483
$818$	0	0
$819$	−2016.00	−0.0860131
$820$	0	0
$821$	−114.000	−0.00484607	−0.00242304	−	0.999997i	$-0.500771\pi$
$821$	−114.000	−0.00484607	−0.00242304	+	0.999997i	$0.500771\pi$
$822$	0	0
$823$	43784.0	1.85445	0.927226	−	0.374502i	$-0.122186\pi$
$823$	43784.0	1.85445	0.927226	+	0.374502i	$0.122186\pi$
$824$	0	0
$825$	−1800.00	−0.0759612
$826$	0	0
$827$	−17044.0	−0.716660	−0.358330	−	0.933595i	$-0.616654\pi$
$827$	−17044.0	−0.716660	−0.358330	+	0.933595i	$0.616654\pi$
$828$	0	0
$829$	−21682.0	−0.908380	−0.454190	−	0.890905i	$-0.650071\pi$
$829$	−21682.0	−0.908380	−0.454190	+	0.890905i	$0.650071\pi$
$830$	0	0
$831$	19482.0	0.813265
$832$	0	0
$833$	1566.00	0.0651365
$834$	0	0
$835$	5760.00	0.238722
$836$	0	0
$837$	756.000	0.0312201
$838$	0	0
$839$	39488.0	1.62488	0.812442	−	0.583042i	$-0.198138\pi$
$839$	39488.0	1.62488	0.812442	+	0.583042i	$0.198138\pi$
$840$	0	0
$841$	38111.0	1.56263
$842$	0	0
$843$	−7494.00	−0.306177
$844$	0	0
$845$	−10005.0	−0.407317
$846$	0	0
$847$	−12080.0	−0.490052
$848$	0	0
$849$	15972.0	0.645651
$850$	0	0
$851$	−5616.00	−0.226221
$852$	0	0
$853$	−14182.0	−0.569264	−0.284632	−	0.958637i	$-0.591871\pi$
$853$	−14182.0	−0.569264	−0.284632	+	0.958637i	$0.591871\pi$
$854$	0	0
$855$	1620.00	0.0647986
$856$	0	0
$857$	27094.0	1.07995	0.539973	−	0.841682i	$-0.318435\pi$
$857$	27094.0	1.07995	0.539973	+	0.841682i	$0.318435\pi$
$858$	0	0
$859$	26692.0	1.06021	0.530104	−	0.847932i	$-0.322153\pi$
$859$	26692.0	1.06021	0.530104	+	0.847932i	$0.322153\pi$
$860$	0	0
$861$	−16992.0	−0.672574
$862$	0	0
$863$	38872.0	1.53328	0.766639	−	0.642079i	$-0.221928\pi$
$863$	38872.0	1.53328	0.766639	+	0.642079i	$0.221928\pi$
$864$	0	0
$865$	15710.0	0.617521
$866$	0	0
$867$	13767.0	0.539275
$868$	0	0
$869$	18912.0	0.738257
$870$	0	0
$871$	−4200.00	−0.163389
$872$	0	0
$873$	−9774.00	−0.378923
$874$	0	0
$875$	2000.00	0.0772712
$876$	0	0
$877$	6490.00	0.249888	0.124944	−	0.992164i	$-0.460125\pi$
$877$	6490.00	0.249888	0.124944	+	0.992164i	$0.460125\pi$
$878$	0	0
$879$	1566.00	0.0600909
$880$	0	0
$881$	−35766.0	−1.36775	−0.683875	−	0.729600i	$-0.739706\pi$
$881$	−35766.0	−1.36775	−0.683875	+	0.729600i	$0.739706\pi$
$882$	0	0
$883$	1316.00	0.0501551	0.0250775	−	0.999686i	$-0.492017\pi$
$883$	1316.00	0.0501551	0.0250775	+	0.999686i	$0.492017\pi$
$884$	0	0
$885$	960.000	0.0364633
$886$	0	0
$887$	6656.00	0.251958	0.125979	−	0.992033i	$-0.459793\pi$
$887$	6656.00	0.251958	0.125979	+	0.992033i	$0.459793\pi$
$888$	0	0
$889$	27904.0	1.05272
$890$	0	0
$891$	1944.00	0.0730937
$892$	0	0
$893$	14688.0	0.550409
$894$	0	0
$895$	−5160.00	−0.192715
$896$	0	0
$897$	4368.00	0.162590
$898$	0	0
$899$	7000.00	0.259692
$900$	0	0
$901$	−4716.00	−0.174376
$902$	0	0
$903$	−10944.0	−0.403315
$904$	0	0
$905$	−7810.00	−0.286865
$906$	0	0
$907$	−15772.0	−0.577399	−0.288699	−	0.957420i	$-0.593223\pi$
$907$	−15772.0	−0.577399	−0.288699	+	0.957420i	$0.593223\pi$
$908$	0	0
$909$	702.000	0.0256148
$910$	0	0
$911$	15168.0	0.551634	0.275817	−	0.961210i	$-0.411052\pi$
$911$	15168.0	0.551634	0.275817	+	0.961210i	$0.411052\pi$
$912$	0	0
$913$	−9504.00	−0.344509
$914$	0	0
$915$	−5610.00	−0.202689
$916$	0	0
$917$	−7680.00	−0.276571
$918$	0	0
$919$	7148.00	0.256573	0.128287	−	0.991737i	$-0.459052\pi$
$919$	7148.00	0.256573	0.128287	+	0.991737i	$0.459052\pi$
$920$	0	0
$921$	−23532.0	−0.841917
$922$	0	0
$923$	−14224.0	−0.507247
$924$	0	0
$925$	−1350.00	−0.0479867
$926$	0	0
$927$	10872.0	0.385203
$928$	0	0
$929$	−8206.00	−0.289806	−0.144903	−	0.989446i	$-0.546287\pi$
$929$	−8206.00	−0.289806	−0.144903	+	0.989446i	$0.546287\pi$
$930$	0	0
$931$	−3132.00	−0.110255
$932$	0	0
$933$	9744.00	0.341912
$934$	0	0
$935$	−2160.00	−0.0755503
$936$	0	0
$937$	−55574.0	−1.93759	−0.968796	−	0.247860i	$-0.920273\pi$
$937$	−55574.0	−1.93759	−0.968796	+	0.247860i	$0.920273\pi$
$938$	0	0
$939$	16122.0	0.560300
$940$	0	0
$941$	−3690.00	−0.127833	−0.0639163	−	0.997955i	$-0.520359\pi$
$941$	−3690.00	−0.127833	−0.0639163	+	0.997955i	$0.520359\pi$
$942$	0	0
$943$	36816.0	1.27136
$944$	0	0
$945$	−2160.00	−0.0743543
$946$	0	0
$947$	−46700.0	−1.60248	−0.801239	−	0.598345i	$-0.795825\pi$
$947$	−46700.0	−1.60248	−0.801239	+	0.598345i	$0.795825\pi$
$948$	0	0
$949$	−3836.00	−0.131214
$950$	0	0
$951$	20358.0	0.694168
$952$	0	0
$953$	−40018.0	−1.36024	−0.680121	−	0.733100i	$-0.738073\pi$
$953$	−40018.0	−1.36024	−0.680121	+	0.733100i	$0.738073\pi$
$954$	0	0
$955$	9800.00	0.332063
$956$	0	0
$957$	18000.0	0.608001
$958$	0	0
$959$	25568.0	0.860932
$960$	0	0
$961$	−29007.0	−0.973683
$962$	0	0
$963$	−396.000	−0.0132512
$964$	0	0
$965$	−20030.0	−0.668175
$966$	0	0
$967$	−1064.00	−0.0353836	−0.0176918	−	0.999843i	$-0.505632\pi$
$967$	−1064.00	−0.0353836	−0.0176918	+	0.999843i	$0.505632\pi$
$968$	0	0
$969$	1944.00	0.0644482
$970$	0	0
$971$	5664.00	0.187195	0.0935975	−	0.995610i	$-0.470163\pi$
$971$	5664.00	0.187195	0.0935975	+	0.995610i	$0.470163\pi$
$972$	0	0
$973$	−47424.0	−1.56253
$974$	0	0
$975$	1050.00	0.0344891
$976$	0	0
$977$	33870.0	1.10911	0.554553	−	0.832148i	$-0.312889\pi$
$977$	33870.0	1.10911	0.554553	+	0.832148i	$0.312889\pi$
$978$	0	0
$979$	18864.0	0.615828
$980$	0	0
$981$	−10098.0	−0.328649
$982$	0	0
$983$	19976.0	0.648154	0.324077	−	0.946031i	$-0.394946\pi$
$983$	19976.0	0.648154	0.324077	+	0.946031i	$0.394946\pi$
$984$	0	0
$985$	10590.0	0.342564
$986$	0	0
$987$	−19584.0	−0.631576
$988$	0	0
$989$	23712.0	0.762384
$990$	0	0
$991$	28748.0	0.921504	0.460752	−	0.887529i	$-0.347580\pi$
$991$	28748.0	0.921504	0.460752	+	0.887529i	$0.347580\pi$
$992$	0	0
$993$	−19788.0	−0.632380
$994$	0	0
$995$	−18740.0	−0.597083
$996$	0	0
$997$	−16830.0	−0.534615	−0.267308	−	0.963611i	$-0.586134\pi$
$997$	−16830.0	−0.534615	−0.267308	+	0.963611i	$0.586134\pi$
$998$	0	0
$999$	1458.00	0.0461753

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.4.a.f.1.1	✓	1
3.2	odd	2		1440.4.a.h.1.1		1
4.3	odd	2		480.4.a.i.1.1	yes	1
5.4	even	2		2400.4.a.o.1.1		1
8.3	odd	2		960.4.a.c.1.1		1
8.5	even	2		960.4.a.z.1.1		1
12.11	even	2		1440.4.a.c.1.1		1
20.19	odd	2		2400.4.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.a.f.1.1	✓	1	1.1	even	1	trivial
480.4.a.i.1.1	yes	1	4.3	odd	2
960.4.a.c.1.1		1	8.3	odd	2
960.4.a.z.1.1		1	8.5	even	2
1440.4.a.c.1.1		1	12.11	even	2
1440.4.a.h.1.1		1	3.2	odd	2
2400.4.a.h.1.1		1	20.19	odd	2
2400.4.a.o.1.1		1	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)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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