LMFDB - Embedded newform 480.4.a.j.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,-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:	$0$
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)

$f(q)$

$=$

$q+3.00000 q^{3} +5.00000 q^{5} -12.0000 q^{7} +9.00000 q^{9} -24.0000 q^{11} +38.0000 q^{13} +15.0000 q^{15} -6.00000 q^{17} +104.000 q^{19} -36.0000 q^{21} +100.000 q^{23} +25.0000 q^{25} +27.0000 q^{27} +230.000 q^{29} -56.0000 q^{31} -72.0000 q^{33} -60.0000 q^{35} +190.000 q^{37} +114.000 q^{39} +202.000 q^{41} -148.000 q^{43} +45.0000 q^{45} +124.000 q^{47} -199.000 q^{49} -18.0000 q^{51} +206.000 q^{53} -120.000 q^{55} +312.000 q^{57} -128.000 q^{59} +190.000 q^{61} -108.000 q^{63} +190.000 q^{65} -204.000 q^{67} +300.000 q^{69} -440.000 q^{71} +1210.00 q^{73} +75.0000 q^{75} +288.000 q^{77} +816.000 q^{79} +81.0000 q^{81} -1412.00 q^{83} -30.0000 q^{85} +690.000 q^{87} -214.000 q^{89} -456.000 q^{91} -168.000 q^{93} +520.000 q^{95} +1202.00 q^{97} -216.000 q^{99} +O(q^{100})$

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
$7$	−12.0000	−0.647939	−0.323970	−	0.946068i	$-0.605018\pi$
$7$	−12.0000	−0.647939	−0.323970	+	0.946068i	$0.605018\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−24.0000	−0.657843	−0.328921	−	0.944357i	$-0.606685\pi$
$11$	−24.0000	−0.657843	−0.328921	+	0.944357i	$0.606685\pi$
$12$	0	0
$13$	38.0000	0.810716	0.405358	−	0.914158i	$-0.367147\pi$
$13$	38.0000	0.810716	0.405358	+	0.914158i	$0.367147\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−6.00000	−0.0856008	−0.0428004	−	0.999084i	$-0.513628\pi$
$17$	−6.00000	−0.0856008	−0.0428004	+	0.999084i	$0.513628\pi$
$18$	0	0
$19$	104.000	1.25575	0.627875	−	0.778314i	$-0.283925\pi$
$19$	104.000	1.25575	0.627875	+	0.778314i	$0.283925\pi$
$20$	0	0
$21$	−36.0000	−0.374088
$22$	0	0
$23$	100.000	0.906584	0.453292	−	0.891362i	$-0.350249\pi$
$23$	100.000	0.906584	0.453292	+	0.891362i	$0.350249\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	230.000	1.47276	0.736378	−	0.676570i	$-0.236535\pi$
$29$	230.000	1.47276	0.736378	+	0.676570i	$0.236535\pi$
$30$	0	0
$31$	−56.0000	−0.324448	−0.162224	−	0.986754i	$-0.551867\pi$
$31$	−56.0000	−0.324448	−0.162224	+	0.986754i	$0.551867\pi$
$32$	0	0
$33$	−72.0000	−0.379806
$34$	0	0
$35$	−60.0000	−0.289767
$36$	0	0
$37$	190.000	0.844211	0.422106	−	0.906547i	$-0.361291\pi$
$37$	190.000	0.844211	0.422106	+	0.906547i	$0.361291\pi$
$38$	0	0
$39$	114.000	0.468067
$40$	0	0
$41$	202.000	0.769441	0.384721	−	0.923033i	$-0.374298\pi$
$41$	202.000	0.769441	0.384721	+	0.923033i	$0.374298\pi$
$42$	0	0
$43$	−148.000	−0.524879	−0.262439	−	0.964948i	$-0.584527\pi$
$43$	−148.000	−0.524879	−0.262439	+	0.964948i	$0.584527\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	124.000	0.384835	0.192418	−	0.981313i	$-0.438367\pi$
$47$	124.000	0.384835	0.192418	+	0.981313i	$0.438367\pi$
$48$	0	0
$49$	−199.000	−0.580175
$50$	0	0
$51$	−18.0000	−0.0494217
$52$	0	0
$53$	206.000	0.533892	0.266946	−	0.963711i	$-0.413985\pi$
$53$	206.000	0.533892	0.266946	+	0.963711i	$0.413985\pi$
$54$	0	0
$55$	−120.000	−0.294196
$56$	0	0
$57$	312.000	0.725007
$58$	0	0
$59$	−128.000	−0.282444	−0.141222	−	0.989978i	$-0.545103\pi$
$59$	−128.000	−0.282444	−0.141222	+	0.989978i	$0.545103\pi$
$60$	0	0
$61$	190.000	0.398803	0.199402	−	0.979918i	$-0.436100\pi$
$61$	190.000	0.398803	0.199402	+	0.979918i	$0.436100\pi$
$62$	0	0
$63$	−108.000	−0.215980
$64$	0	0
$65$	190.000	0.362563
$66$	0	0
$67$	−204.000	−0.371979	−0.185989	−	0.982552i	$-0.559549\pi$
$67$	−204.000	−0.371979	−0.185989	+	0.982552i	$0.559549\pi$
$68$	0	0
$69$	300.000	0.523417
$70$	0	0
$71$	−440.000	−0.735470	−0.367735	−	0.929931i	$-0.619867\pi$
$71$	−440.000	−0.735470	−0.367735	+	0.929931i	$0.619867\pi$
$72$	0	0
$73$	1210.00	1.94000	0.969999	−	0.243111i	$-0.0781678\pi$
$73$	1210.00	1.94000	0.969999	+	0.243111i	$0.0781678\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	288.000	0.426242
$78$	0	0
$79$	816.000	1.16212	0.581058	−	0.813862i	$-0.302639\pi$
$79$	816.000	1.16212	0.581058	+	0.813862i	$0.302639\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1412.00	−1.86731	−0.933657	−	0.358167i	$-0.883402\pi$
$83$	−1412.00	−1.86731	−0.933657	+	0.358167i	$0.883402\pi$
$84$	0	0
$85$	−30.0000	−0.0382818
$86$	0	0
$87$	690.000	0.850296
$88$	0	0
$89$	−214.000	−0.254876	−0.127438	−	0.991847i	$-0.540675\pi$
$89$	−214.000	−0.254876	−0.127438	+	0.991847i	$0.540675\pi$
$90$	0	0
$91$	−456.000	−0.525294
$92$	0	0
$93$	−168.000	−0.187320
$94$	0	0
$95$	520.000	0.561588
$96$	0	0
$97$	1202.00	1.25819	0.629096	−	0.777328i	$-0.283425\pi$
$97$	1202.00	1.25819	0.629096	+	0.777328i	$0.283425\pi$
$98$	0	0
$99$	−216.000	−0.219281
$100$	0	0
$101$	1342.00	1.32212	0.661059	−	0.750334i	$-0.270107\pi$
$101$	1342.00	1.32212	0.661059	+	0.750334i	$0.270107\pi$
$102$	0	0
$103$	908.000	0.868620	0.434310	−	0.900763i	$-0.356992\pi$
$103$	908.000	0.868620	0.434310	+	0.900763i	$0.356992\pi$
$104$	0	0
$105$	−180.000	−0.167297
$106$	0	0
$107$	−876.000	−0.791459	−0.395730	−	0.918367i	$-0.629508\pi$
$107$	−876.000	−0.791459	−0.395730	+	0.918367i	$0.629508\pi$
$108$	0	0
$109$	302.000	0.265379	0.132690	−	0.991158i	$-0.457639\pi$
$109$	302.000	0.265379	0.132690	+	0.991158i	$0.457639\pi$
$110$	0	0
$111$	570.000	0.487405
$112$	0	0
$113$	−998.000	−0.830831	−0.415416	−	0.909632i	$-0.636364\pi$
$113$	−998.000	−0.830831	−0.415416	+	0.909632i	$0.636364\pi$
$114$	0	0
$115$	500.000	0.405437
$116$	0	0
$117$	342.000	0.270239
$118$	0	0
$119$	72.0000	0.0554641
$120$	0	0
$121$	−755.000	−0.567243
$122$	0	0
$123$	606.000	0.444237
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	836.000	0.584118	0.292059	−	0.956400i	$-0.405660\pi$
$127$	836.000	0.584118	0.292059	+	0.956400i	$0.405660\pi$
$128$	0	0
$129$	−444.000	−0.303039
$130$	0	0
$131$	−1480.00	−0.987085	−0.493543	−	0.869722i	$-0.664298\pi$
$131$	−1480.00	−0.987085	−0.493543	+	0.869722i	$0.664298\pi$
$132$	0	0
$133$	−1248.00	−0.813649
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	1346.00	0.839391	0.419695	−	0.907665i	$-0.362137\pi$
$137$	1346.00	0.839391	0.419695	+	0.907665i	$0.362137\pi$
$138$	0	0
$139$	824.000	0.502811	0.251406	−	0.967882i	$-0.419107\pi$
$139$	824.000	0.502811	0.251406	+	0.967882i	$0.419107\pi$
$140$	0	0
$141$	372.000	0.222185
$142$	0	0
$143$	−912.000	−0.533324
$144$	0	0
$145$	1150.00	0.658637
$146$	0	0
$147$	−597.000	−0.334964
$148$	0	0
$149$	−2450.00	−1.34706	−0.673530	−	0.739160i	$-0.735223\pi$
$149$	−2450.00	−1.34706	−0.673530	+	0.739160i	$0.735223\pi$
$150$	0	0
$151$	2696.00	1.45296	0.726481	−	0.687186i	$-0.241154\pi$
$151$	2696.00	1.45296	0.726481	+	0.687186i	$0.241154\pi$
$152$	0	0
$153$	−54.0000	−0.0285336
$154$	0	0
$155$	−280.000	−0.145098
$156$	0	0
$157$	−554.000	−0.281618	−0.140809	−	0.990037i	$-0.544970\pi$
$157$	−554.000	−0.281618	−0.140809	+	0.990037i	$0.544970\pi$
$158$	0	0
$159$	618.000	0.308243
$160$	0	0
$161$	−1200.00	−0.587411
$162$	0	0
$163$	−1364.00	−0.655440	−0.327720	−	0.944775i	$-0.606280\pi$
$163$	−1364.00	−0.655440	−0.327720	+	0.944775i	$0.606280\pi$
$164$	0	0
$165$	−360.000	−0.169854
$166$	0	0
$167$	−2004.00	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−2004.00	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−753.000	−0.342740
$170$	0	0
$171$	936.000	0.418583
$172$	0	0
$173$	−1546.00	−0.679423	−0.339712	−	0.940530i	$-0.610329\pi$
$173$	−1546.00	−0.679423	−0.339712	+	0.940530i	$0.610329\pi$
$174$	0	0
$175$	−300.000	−0.129588
$176$	0	0
$177$	−384.000	−0.163069
$178$	0	0
$179$	−1072.00	−0.447626	−0.223813	−	0.974632i	$-0.571850\pi$
$179$	−1072.00	−0.447626	−0.223813	+	0.974632i	$0.571850\pi$
$180$	0	0
$181$	−3754.00	−1.54162	−0.770808	−	0.637067i	$-0.780147\pi$
$181$	−3754.00	−1.54162	−0.770808	+	0.637067i	$0.780147\pi$
$182$	0	0
$183$	570.000	0.230249
$184$	0	0
$185$	950.000	0.377543
$186$	0	0
$187$	144.000	0.0563119
$188$	0	0
$189$	−324.000	−0.124696
$190$	0	0
$191$	1224.00	0.463694	0.231847	−	0.972752i	$-0.425523\pi$
$191$	1224.00	0.463694	0.231847	+	0.972752i	$0.425523\pi$
$192$	0	0
$193$	−1694.00	−0.631797	−0.315898	−	0.948793i	$-0.602306\pi$
$193$	−1694.00	−0.631797	−0.315898	+	0.948793i	$0.602306\pi$
$194$	0	0
$195$	570.000	0.209326
$196$	0	0
$197$	3134.00	1.13344	0.566721	−	0.823909i	$-0.308212\pi$
$197$	3134.00	1.13344	0.566721	+	0.823909i	$0.308212\pi$
$198$	0	0
$199$	2560.00	0.911928	0.455964	−	0.889998i	$-0.349295\pi$
$199$	2560.00	0.911928	0.455964	+	0.889998i	$0.349295\pi$
$200$	0	0
$201$	−612.000	−0.214762
$202$	0	0
$203$	−2760.00	−0.954256
$204$	0	0
$205$	1010.00	0.344105
$206$	0	0
$207$	900.000	0.302195
$208$	0	0
$209$	−2496.00	−0.826086
$210$	0	0
$211$	−1856.00	−0.605556	−0.302778	−	0.953061i	$-0.597914\pi$
$211$	−1856.00	−0.605556	−0.302778	+	0.953061i	$0.597914\pi$
$212$	0	0
$213$	−1320.00	−0.424624
$214$	0	0
$215$	−740.000	−0.234733
$216$	0	0
$217$	672.000	0.210223
$218$	0	0
$219$	3630.00	1.12006
$220$	0	0
$221$	−228.000	−0.0693979
$222$	0	0
$223$	1596.00	0.479265	0.239632	−	0.970864i	$-0.422973\pi$
$223$	1596.00	0.479265	0.239632	+	0.970864i	$0.422973\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−3996.00	−1.16839	−0.584193	−	0.811614i	$-0.698589\pi$
$227$	−3996.00	−1.16839	−0.584193	+	0.811614i	$0.698589\pi$
$228$	0	0
$229$	−6090.00	−1.75737	−0.878687	−	0.477399i	$-0.841580\pi$
$229$	−6090.00	−1.75737	−0.878687	+	0.477399i	$0.841580\pi$
$230$	0	0
$231$	864.000	0.246091
$232$	0	0
$233$	−894.000	−0.251364	−0.125682	−	0.992071i	$-0.540112\pi$
$233$	−894.000	−0.251364	−0.125682	+	0.992071i	$0.540112\pi$
$234$	0	0
$235$	620.000	0.172104
$236$	0	0
$237$	2448.00	0.670948
$238$	0	0
$239$	−6336.00	−1.71482	−0.857410	−	0.514635i	$-0.827928\pi$
$239$	−6336.00	−1.71482	−0.857410	+	0.514635i	$0.827928\pi$
$240$	0	0
$241$	338.000	0.0903423	0.0451711	−	0.998979i	$-0.485617\pi$
$241$	338.000	0.0903423	0.0451711	+	0.998979i	$0.485617\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−995.000	−0.259462
$246$	0	0
$247$	3952.00	1.01806
$248$	0	0
$249$	−4236.00	−1.07809
$250$	0	0
$251$	−4872.00	−1.22517	−0.612585	−	0.790404i	$-0.709870\pi$
$251$	−4872.00	−1.22517	−0.612585	+	0.790404i	$0.709870\pi$
$252$	0	0
$253$	−2400.00	−0.596390
$254$	0	0
$255$	−90.0000	−0.0221020
$256$	0	0
$257$	−1878.00	−0.455823	−0.227911	−	0.973682i	$-0.573190\pi$
$257$	−1878.00	−0.455823	−0.227911	+	0.973682i	$0.573190\pi$
$258$	0	0
$259$	−2280.00	−0.546997
$260$	0	0
$261$	2070.00	0.490919
$262$	0	0
$263$	2244.00	0.526125	0.263063	−	0.964779i	$-0.415267\pi$
$263$	2244.00	0.526125	0.263063	+	0.964779i	$0.415267\pi$
$264$	0	0
$265$	1030.00	0.238764
$266$	0	0
$267$	−642.000	−0.147153
$268$	0	0
$269$	−4314.00	−0.977804	−0.488902	−	0.872339i	$-0.662602\pi$
$269$	−4314.00	−0.977804	−0.488902	+	0.872339i	$0.662602\pi$
$270$	0	0
$271$	−6392.00	−1.43279	−0.716395	−	0.697694i	$-0.754209\pi$
$271$	−6392.00	−1.43279	−0.716395	+	0.697694i	$0.754209\pi$
$272$	0	0
$273$	−1368.00	−0.303279
$274$	0	0
$275$	−600.000	−0.131569
$276$	0	0
$277$	4398.00	0.953972	0.476986	−	0.878911i	$-0.341729\pi$
$277$	4398.00	0.953972	0.476986	+	0.878911i	$0.341729\pi$
$278$	0	0
$279$	−504.000	−0.108149
$280$	0	0
$281$	−7622.00	−1.61812	−0.809058	−	0.587729i	$-0.800022\pi$
$281$	−7622.00	−1.61812	−0.809058	+	0.587729i	$0.800022\pi$
$282$	0	0
$283$	1020.00	0.214250	0.107125	−	0.994246i	$-0.465836\pi$
$283$	1020.00	0.214250	0.107125	+	0.994246i	$0.465836\pi$
$284$	0	0
$285$	1560.00	0.324233
$286$	0	0
$287$	−2424.00	−0.498551
$288$	0	0
$289$	−4877.00	−0.992673
$290$	0	0
$291$	3606.00	0.726417
$292$	0	0
$293$	−3746.00	−0.746907	−0.373453	−	0.927649i	$-0.621826\pi$
$293$	−3746.00	−0.746907	−0.373453	+	0.927649i	$0.621826\pi$
$294$	0	0
$295$	−640.000	−0.126313
$296$	0	0
$297$	−648.000	−0.126602
$298$	0	0
$299$	3800.00	0.734982
$300$	0	0
$301$	1776.00	0.340089
$302$	0	0
$303$	4026.00	0.763326
$304$	0	0
$305$	950.000	0.178350
$306$	0	0
$307$	9700.00	1.80328	0.901642	−	0.432483i	$-0.142362\pi$
$307$	9700.00	1.80328	0.901642	+	0.432483i	$0.142362\pi$
$308$	0	0
$309$	2724.00	0.501498
$310$	0	0
$311$	4152.00	0.757036	0.378518	−	0.925594i	$-0.376434\pi$
$311$	4152.00	0.757036	0.378518	+	0.925594i	$0.376434\pi$
$312$	0	0
$313$	6362.00	1.14889	0.574443	−	0.818544i	$-0.305219\pi$
$313$	6362.00	1.14889	0.574443	+	0.818544i	$0.305219\pi$
$314$	0	0
$315$	−540.000	−0.0965891
$316$	0	0
$317$	10886.0	1.92877	0.964383	−	0.264511i	$-0.0852104\pi$
$317$	10886.0	1.92877	0.964383	+	0.264511i	$0.0852104\pi$
$318$	0	0
$319$	−5520.00	−0.968842
$320$	0	0
$321$	−2628.00	−0.456949
$322$	0	0
$323$	−624.000	−0.107493
$324$	0	0
$325$	950.000	0.162143
$326$	0	0
$327$	906.000	0.153217
$328$	0	0
$329$	−1488.00	−0.249350
$330$	0	0
$331$	4128.00	0.685485	0.342742	−	0.939429i	$-0.388644\pi$
$331$	4128.00	0.685485	0.342742	+	0.939429i	$0.388644\pi$
$332$	0	0
$333$	1710.00	0.281404
$334$	0	0
$335$	−1020.00	−0.166354
$336$	0	0
$337$	12002.0	1.94003	0.970016	−	0.243042i	$-0.0781453\pi$
$337$	12002.0	1.94003	0.970016	+	0.243042i	$0.0781453\pi$
$338$	0	0
$339$	−2994.00	−0.479681
$340$	0	0
$341$	1344.00	0.213436
$342$	0	0
$343$	6504.00	1.02386
$344$	0	0
$345$	1500.00	0.234079
$346$	0	0
$347$	6276.00	0.970932	0.485466	−	0.874256i	$-0.338650\pi$
$347$	6276.00	0.970932	0.485466	+	0.874256i	$0.338650\pi$
$348$	0	0
$349$	−9362.00	−1.43592	−0.717960	−	0.696084i	$-0.754924\pi$
$349$	−9362.00	−1.43592	−0.717960	+	0.696084i	$0.754924\pi$
$350$	0	0
$351$	1026.00	0.156022
$352$	0	0
$353$	−838.000	−0.126352	−0.0631760	−	0.998002i	$-0.520123\pi$
$353$	−838.000	−0.126352	−0.0631760	+	0.998002i	$0.520123\pi$
$354$	0	0
$355$	−2200.00	−0.328912
$356$	0	0
$357$	216.000	0.0320222
$358$	0	0
$359$	−6896.00	−1.01381	−0.506904	−	0.862003i	$-0.669210\pi$
$359$	−6896.00	−1.01381	−0.506904	+	0.862003i	$0.669210\pi$
$360$	0	0
$361$	3957.00	0.576906
$362$	0	0
$363$	−2265.00	−0.327498
$364$	0	0
$365$	6050.00	0.867593
$366$	0	0
$367$	−1132.00	−0.161008	−0.0805040	−	0.996754i	$-0.525653\pi$
$367$	−1132.00	−0.161008	−0.0805040	+	0.996754i	$0.525653\pi$
$368$	0	0
$369$	1818.00	0.256480
$370$	0	0
$371$	−2472.00	−0.345930
$372$	0	0
$373$	−12578.0	−1.74602	−0.873008	−	0.487705i	$-0.837834\pi$
$373$	−12578.0	−1.74602	−0.873008	+	0.487705i	$0.837834\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	8740.00	1.19399
$378$	0	0
$379$	11752.0	1.59277	0.796385	−	0.604790i	$-0.206743\pi$
$379$	11752.0	1.59277	0.796385	+	0.604790i	$0.206743\pi$
$380$	0	0
$381$	2508.00	0.337241
$382$	0	0
$383$	−7372.00	−0.983529	−0.491764	−	0.870728i	$-0.663648\pi$
$383$	−7372.00	−0.983529	−0.491764	+	0.870728i	$0.663648\pi$
$384$	0	0
$385$	1440.00	0.190621
$386$	0	0
$387$	−1332.00	−0.174960
$388$	0	0
$389$	6654.00	0.867278	0.433639	−	0.901087i	$-0.357229\pi$
$389$	6654.00	0.867278	0.433639	+	0.901087i	$0.357229\pi$
$390$	0	0
$391$	−600.000	−0.0776044
$392$	0	0
$393$	−4440.00	−0.569894
$394$	0	0
$395$	4080.00	0.519714
$396$	0	0
$397$	4278.00	0.540823	0.270411	−	0.962745i	$-0.412840\pi$
$397$	4278.00	0.540823	0.270411	+	0.962745i	$0.412840\pi$
$398$	0	0
$399$	−3744.00	−0.469761
$400$	0	0
$401$	9074.00	1.13001	0.565005	−	0.825088i	$-0.308874\pi$
$401$	9074.00	1.13001	0.565005	+	0.825088i	$0.308874\pi$
$402$	0	0
$403$	−2128.00	−0.263035
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	−4560.00	−0.555358
$408$	0	0
$409$	6682.00	0.807833	0.403916	−	0.914796i	$-0.367649\pi$
$409$	6682.00	0.807833	0.403916	+	0.914796i	$0.367649\pi$
$410$	0	0
$411$	4038.00	0.484623
$412$	0	0
$413$	1536.00	0.183006
$414$	0	0
$415$	−7060.00	−0.835089
$416$	0	0
$417$	2472.00	0.290298
$418$	0	0
$419$	4832.00	0.563386	0.281693	−	0.959505i	$-0.409104\pi$
$419$	4832.00	0.563386	0.281693	+	0.959505i	$0.409104\pi$
$420$	0	0
$421$	3974.00	0.460050	0.230025	−	0.973185i	$-0.426119\pi$
$421$	3974.00	0.460050	0.230025	+	0.973185i	$0.426119\pi$
$422$	0	0
$423$	1116.00	0.128278
$424$	0	0
$425$	−150.000	−0.0171202
$426$	0	0
$427$	−2280.00	−0.258400
$428$	0	0
$429$	−2736.00	−0.307915
$430$	0	0
$431$	1112.00	0.124276	0.0621382	−	0.998068i	$-0.480208\pi$
$431$	1112.00	0.124276	0.0621382	+	0.998068i	$0.480208\pi$
$432$	0	0
$433$	1106.00	0.122751	0.0613753	−	0.998115i	$-0.480451\pi$
$433$	1106.00	0.122751	0.0613753	+	0.998115i	$0.480451\pi$
$434$	0	0
$435$	3450.00	0.380264
$436$	0	0
$437$	10400.0	1.13844
$438$	0	0
$439$	−9280.00	−1.00891	−0.504454	−	0.863439i	$-0.668306\pi$
$439$	−9280.00	−1.00891	−0.504454	+	0.863439i	$0.668306\pi$
$440$	0	0
$441$	−1791.00	−0.193392
$442$	0	0
$443$	7004.00	0.751174	0.375587	−	0.926787i	$-0.377441\pi$
$443$	7004.00	0.751174	0.375587	+	0.926787i	$0.377441\pi$
$444$	0	0
$445$	−1070.00	−0.113984
$446$	0	0
$447$	−7350.00	−0.777725
$448$	0	0
$449$	−11502.0	−1.20894	−0.604469	−	0.796629i	$-0.706615\pi$
$449$	−11502.0	−1.20894	−0.604469	+	0.796629i	$0.706615\pi$
$450$	0	0
$451$	−4848.00	−0.506172
$452$	0	0
$453$	8088.00	0.838868
$454$	0	0
$455$	−2280.00	−0.234919
$456$	0	0
$457$	11578.0	1.18511	0.592556	−	0.805529i	$-0.298119\pi$
$457$	11578.0	1.18511	0.592556	+	0.805529i	$0.298119\pi$
$458$	0	0
$459$	−162.000	−0.0164739
$460$	0	0
$461$	−6362.00	−0.642750	−0.321375	−	0.946952i	$-0.604145\pi$
$461$	−6362.00	−0.642750	−0.321375	+	0.946952i	$0.604145\pi$
$462$	0	0
$463$	2892.00	0.290286	0.145143	−	0.989411i	$-0.453636\pi$
$463$	2892.00	0.290286	0.145143	+	0.989411i	$0.453636\pi$
$464$	0	0
$465$	−840.000	−0.0837722
$466$	0	0
$467$	−11036.0	−1.09354	−0.546772	−	0.837281i	$-0.684144\pi$
$467$	−11036.0	−1.09354	−0.546772	+	0.837281i	$0.684144\pi$
$468$	0	0
$469$	2448.00	0.241019
$470$	0	0
$471$	−1662.00	−0.162592
$472$	0	0
$473$	3552.00	0.345288
$474$	0	0
$475$	2600.00	0.251150
$476$	0	0
$477$	1854.00	0.177964
$478$	0	0
$479$	−13664.0	−1.30339	−0.651695	−	0.758481i	$-0.725942\pi$
$479$	−13664.0	−1.30339	−0.651695	+	0.758481i	$0.725942\pi$
$480$	0	0
$481$	7220.00	0.684415
$482$	0	0
$483$	−3600.00	−0.339142
$484$	0	0
$485$	6010.00	0.562680
$486$	0	0
$487$	4820.00	0.448491	0.224245	−	0.974533i	$-0.428008\pi$
$487$	4820.00	0.448491	0.224245	+	0.974533i	$0.428008\pi$
$488$	0	0
$489$	−4092.00	−0.378418
$490$	0	0
$491$	17464.0	1.60517	0.802586	−	0.596537i	$-0.203457\pi$
$491$	17464.0	1.60517	0.802586	+	0.596537i	$0.203457\pi$
$492$	0	0
$493$	−1380.00	−0.126069
$494$	0	0
$495$	−1080.00	−0.0980654
$496$	0	0
$497$	5280.00	0.476540
$498$	0	0
$499$	13960.0	1.25238	0.626188	−	0.779672i	$-0.284614\pi$
$499$	13960.0	1.25238	0.626188	+	0.779672i	$0.284614\pi$
$500$	0	0
$501$	−6012.00	−0.536120
$502$	0	0
$503$	20388.0	1.80727	0.903634	−	0.428305i	$-0.140889\pi$
$503$	20388.0	1.80727	0.903634	+	0.428305i	$0.140889\pi$
$504$	0	0
$505$	6710.00	0.591269
$506$	0	0
$507$	−2259.00	−0.197881
$508$	0	0
$509$	−12954.0	−1.12805	−0.564024	−	0.825759i	$-0.690747\pi$
$509$	−12954.0	−1.12805	−0.564024	+	0.825759i	$0.690747\pi$
$510$	0	0
$511$	−14520.0	−1.25700
$512$	0	0
$513$	2808.00	0.241669
$514$	0	0
$515$	4540.00	0.388459
$516$	0	0
$517$	−2976.00	−0.253161
$518$	0	0
$519$	−4638.00	−0.392265
$520$	0	0
$521$	−3542.00	−0.297846	−0.148923	−	0.988849i	$-0.547581\pi$
$521$	−3542.00	−0.297846	−0.148923	+	0.988849i	$0.547581\pi$
$522$	0	0
$523$	1532.00	0.128087	0.0640437	−	0.997947i	$-0.479600\pi$
$523$	1532.00	0.128087	0.0640437	+	0.997947i	$0.479600\pi$
$524$	0	0
$525$	−900.000	−0.0748176
$526$	0	0
$527$	336.000	0.0277730
$528$	0	0
$529$	−2167.00	−0.178105
$530$	0	0
$531$	−1152.00	−0.0941479
$532$	0	0
$533$	7676.00	0.623798
$534$	0	0
$535$	−4380.00	−0.353951
$536$	0	0
$537$	−3216.00	−0.258437
$538$	0	0
$539$	4776.00	0.381664
$540$	0	0
$541$	−11826.0	−0.939814	−0.469907	−	0.882716i	$-0.655713\pi$
$541$	−11826.0	−0.939814	−0.469907	+	0.882716i	$0.655713\pi$
$542$	0	0
$543$	−11262.0	−0.890053
$544$	0	0
$545$	1510.00	0.118681
$546$	0	0
$547$	−11260.0	−0.880151	−0.440076	−	0.897961i	$-0.645048\pi$
$547$	−11260.0	−0.880151	−0.440076	+	0.897961i	$0.645048\pi$
$548$	0	0
$549$	1710.00	0.132934
$550$	0	0
$551$	23920.0	1.84941
$552$	0	0
$553$	−9792.00	−0.752980
$554$	0	0
$555$	2850.00	0.217974
$556$	0	0
$557$	4374.00	0.332733	0.166367	−	0.986064i	$-0.446797\pi$
$557$	4374.00	0.332733	0.166367	+	0.986064i	$0.446797\pi$
$558$	0	0
$559$	−5624.00	−0.425527
$560$	0	0
$561$	432.000	0.0325117
$562$	0	0
$563$	−11604.0	−0.868651	−0.434325	−	0.900756i	$-0.643013\pi$
$563$	−11604.0	−0.868651	−0.434325	+	0.900756i	$0.643013\pi$
$564$	0	0
$565$	−4990.00	−0.371559
$566$	0	0
$567$	−972.000	−0.0719932
$568$	0	0
$569$	−7990.00	−0.588679	−0.294339	−	0.955701i	$-0.595100\pi$
$569$	−7990.00	−0.588679	−0.294339	+	0.955701i	$0.595100\pi$
$570$	0	0
$571$	−26080.0	−1.91141	−0.955704	−	0.294329i	$-0.904904\pi$
$571$	−26080.0	−1.91141	−0.955704	+	0.294329i	$0.904904\pi$
$572$	0	0
$573$	3672.00	0.267714
$574$	0	0
$575$	2500.00	0.181317
$576$	0	0
$577$	13922.0	1.00447	0.502236	−	0.864731i	$-0.332511\pi$
$577$	13922.0	1.00447	0.502236	+	0.864731i	$0.332511\pi$
$578$	0	0
$579$	−5082.00	−0.364768
$580$	0	0
$581$	16944.0	1.20991
$582$	0	0
$583$	−4944.00	−0.351217
$584$	0	0
$585$	1710.00	0.120854
$586$	0	0
$587$	26340.0	1.85208	0.926038	−	0.377431i	$-0.123193\pi$
$587$	26340.0	1.85208	0.926038	+	0.377431i	$0.123193\pi$
$588$	0	0
$589$	−5824.00	−0.407426
$590$	0	0
$591$	9402.00	0.654394
$592$	0	0
$593$	−9478.00	−0.656349	−0.328174	−	0.944617i	$-0.606433\pi$
$593$	−9478.00	−0.656349	−0.328174	+	0.944617i	$0.606433\pi$
$594$	0	0
$595$	360.000	0.0248043
$596$	0	0
$597$	7680.00	0.526502
$598$	0	0
$599$	6528.00	0.445287	0.222643	−	0.974900i	$-0.428531\pi$
$599$	6528.00	0.445287	0.222643	+	0.974900i	$0.428531\pi$
$600$	0	0
$601$	2090.00	0.141852	0.0709259	−	0.997482i	$-0.477405\pi$
$601$	2090.00	0.141852	0.0709259	+	0.997482i	$0.477405\pi$
$602$	0	0
$603$	−1836.00	−0.123993
$604$	0	0
$605$	−3775.00	−0.253679
$606$	0	0
$607$	8788.00	0.587634	0.293817	−	0.955862i	$-0.405074\pi$
$607$	8788.00	0.587634	0.293817	+	0.955862i	$0.405074\pi$
$608$	0	0
$609$	−8280.00	−0.550940
$610$	0	0
$611$	4712.00	0.311992
$612$	0	0
$613$	−2626.00	−0.173023	−0.0865115	−	0.996251i	$-0.527572\pi$
$613$	−2626.00	−0.173023	−0.0865115	+	0.996251i	$0.527572\pi$
$614$	0	0
$615$	3030.00	0.198669
$616$	0	0
$617$	−29214.0	−1.90618	−0.953089	−	0.302691i	$-0.902115\pi$
$617$	−29214.0	−1.90618	−0.953089	+	0.302691i	$0.902115\pi$
$618$	0	0
$619$	22984.0	1.49242	0.746208	−	0.665713i	$-0.231873\pi$
$619$	22984.0	1.49242	0.746208	+	0.665713i	$0.231873\pi$
$620$	0	0
$621$	2700.00	0.174472
$622$	0	0
$623$	2568.00	0.165144
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−7488.00	−0.476941
$628$	0	0
$629$	−1140.00	−0.0722651
$630$	0	0
$631$	4472.00	0.282136	0.141068	−	0.990000i	$-0.454946\pi$
$631$	4472.00	0.282136	0.141068	+	0.990000i	$0.454946\pi$
$632$	0	0
$633$	−5568.00	−0.349618
$634$	0	0
$635$	4180.00	0.261226
$636$	0	0
$637$	−7562.00	−0.470357
$638$	0	0
$639$	−3960.00	−0.245157
$640$	0	0
$641$	−8798.00	−0.542122	−0.271061	−	0.962562i	$-0.587374\pi$
$641$	−8798.00	−0.542122	−0.271061	+	0.962562i	$0.587374\pi$
$642$	0	0
$643$	−29428.0	−1.80486	−0.902432	−	0.430833i	$-0.858220\pi$
$643$	−29428.0	−1.80486	−0.902432	+	0.430833i	$0.858220\pi$
$644$	0	0
$645$	−2220.00	−0.135523
$646$	0	0
$647$	4860.00	0.295311	0.147656	−	0.989039i	$-0.452827\pi$
$647$	4860.00	0.295311	0.147656	+	0.989039i	$0.452827\pi$
$648$	0	0
$649$	3072.00	0.185804
$650$	0	0
$651$	2016.00	0.121372
$652$	0	0
$653$	−6570.00	−0.393727	−0.196864	−	0.980431i	$-0.563076\pi$
$653$	−6570.00	−0.393727	−0.196864	+	0.980431i	$0.563076\pi$
$654$	0	0
$655$	−7400.00	−0.441438
$656$	0	0
$657$	10890.0	0.646666
$658$	0	0
$659$	26496.0	1.56622	0.783109	−	0.621885i	$-0.213633\pi$
$659$	26496.0	1.56622	0.783109	+	0.621885i	$0.213633\pi$
$660$	0	0
$661$	−19642.0	−1.15580	−0.577901	−	0.816107i	$-0.696128\pi$
$661$	−19642.0	−1.15580	−0.577901	+	0.816107i	$0.696128\pi$
$662$	0	0
$663$	−684.000	−0.0400669
$664$	0	0
$665$	−6240.00	−0.363875
$666$	0	0
$667$	23000.0	1.33518
$668$	0	0
$669$	4788.00	0.276704
$670$	0	0
$671$	−4560.00	−0.262350
$672$	0	0
$673$	−19582.0	−1.12159	−0.560795	−	0.827954i	$-0.689505\pi$
$673$	−19582.0	−1.12159	−0.560795	+	0.827954i	$0.689505\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	−22914.0	−1.30082	−0.650411	−	0.759582i	$-0.725403\pi$
$677$	−22914.0	−1.30082	−0.650411	+	0.759582i	$0.725403\pi$
$678$	0	0
$679$	−14424.0	−0.815232
$680$	0	0
$681$	−11988.0	−0.674569
$682$	0	0
$683$	−13764.0	−0.771105	−0.385553	−	0.922686i	$-0.625989\pi$
$683$	−13764.0	−0.771105	−0.385553	+	0.922686i	$0.625989\pi$
$684$	0	0
$685$	6730.00	0.375387
$686$	0	0
$687$	−18270.0	−1.01462
$688$	0	0
$689$	7828.00	0.432835
$690$	0	0
$691$	−34688.0	−1.90969	−0.954843	−	0.297109i	$-0.903977\pi$
$691$	−34688.0	−1.90969	−0.954843	+	0.297109i	$0.903977\pi$
$692$	0	0
$693$	2592.00	0.142081
$694$	0	0
$695$	4120.00	0.224864
$696$	0	0
$697$	−1212.00	−0.0658648
$698$	0	0
$699$	−2682.00	−0.145125
$700$	0	0
$701$	−17226.0	−0.928127	−0.464064	−	0.885802i	$-0.653609\pi$
$701$	−17226.0	−0.928127	−0.464064	+	0.885802i	$0.653609\pi$
$702$	0	0
$703$	19760.0	1.06012
$704$	0	0
$705$	1860.00	0.0993640
$706$	0	0
$707$	−16104.0	−0.856652
$708$	0	0
$709$	−12970.0	−0.687022	−0.343511	−	0.939149i	$-0.611616\pi$
$709$	−12970.0	−0.687022	−0.343511	+	0.939149i	$0.611616\pi$
$710$	0	0
$711$	7344.00	0.387372
$712$	0	0
$713$	−5600.00	−0.294140
$714$	0	0
$715$	−4560.00	−0.238510
$716$	0	0
$717$	−19008.0	−0.990051
$718$	0	0
$719$	10832.0	0.561843	0.280922	−	0.959731i	$-0.409360\pi$
$719$	10832.0	0.561843	0.280922	+	0.959731i	$0.409360\pi$
$720$	0	0
$721$	−10896.0	−0.562813
$722$	0	0
$723$	1014.00	0.0521592
$724$	0	0
$725$	5750.00	0.294551
$726$	0	0
$727$	35588.0	1.81552	0.907762	−	0.419486i	$-0.137790\pi$
$727$	35588.0	1.81552	0.907762	+	0.419486i	$0.137790\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	888.000	0.0449300
$732$	0	0
$733$	19238.0	0.969402	0.484701	−	0.874680i	$-0.338928\pi$
$733$	19238.0	0.969402	0.484701	+	0.874680i	$0.338928\pi$
$734$	0	0
$735$	−2985.00	−0.149801
$736$	0	0
$737$	4896.00	0.244703
$738$	0	0
$739$	−4072.00	−0.202694	−0.101347	−	0.994851i	$-0.532315\pi$
$739$	−4072.00	−0.202694	−0.101347	+	0.994851i	$0.532315\pi$
$740$	0	0
$741$	11856.0	0.587775
$742$	0	0
$743$	31268.0	1.54389	0.771946	−	0.635688i	$-0.219284\pi$
$743$	31268.0	1.54389	0.771946	+	0.635688i	$0.219284\pi$
$744$	0	0
$745$	−12250.0	−0.602423
$746$	0	0
$747$	−12708.0	−0.622438
$748$	0	0
$749$	10512.0	0.512817
$750$	0	0
$751$	−28216.0	−1.37099	−0.685497	−	0.728075i	$-0.740415\pi$
$751$	−28216.0	−1.37099	−0.685497	+	0.728075i	$0.740415\pi$
$752$	0	0
$753$	−14616.0	−0.707353
$754$	0	0
$755$	13480.0	0.649785
$756$	0	0
$757$	−33874.0	−1.62638	−0.813191	−	0.581997i	$-0.802272\pi$
$757$	−33874.0	−1.62638	−0.813191	+	0.581997i	$0.802272\pi$
$758$	0	0
$759$	−7200.00	−0.344326
$760$	0	0
$761$	5466.00	0.260371	0.130186	−	0.991490i	$-0.458443\pi$
$761$	5466.00	0.260371	0.130186	+	0.991490i	$0.458443\pi$
$762$	0	0
$763$	−3624.00	−0.171950
$764$	0	0
$765$	−270.000	−0.0127606
$766$	0	0
$767$	−4864.00	−0.228982
$768$	0	0
$769$	−8878.00	−0.416318	−0.208159	−	0.978095i	$-0.566747\pi$
$769$	−8878.00	−0.416318	−0.208159	+	0.978095i	$0.566747\pi$
$770$	0	0
$771$	−5634.00	−0.263169
$772$	0	0
$773$	−9538.00	−0.443801	−0.221900	−	0.975069i	$-0.571226\pi$
$773$	−9538.00	−0.443801	−0.221900	+	0.975069i	$0.571226\pi$
$774$	0	0
$775$	−1400.00	−0.0648897
$776$	0	0
$777$	−6840.00	−0.315809
$778$	0	0
$779$	21008.0	0.966226
$780$	0	0
$781$	10560.0	0.483824
$782$	0	0
$783$	6210.00	0.283432
$784$	0	0
$785$	−2770.00	−0.125943
$786$	0	0
$787$	−5404.00	−0.244767	−0.122384	−	0.992483i	$-0.539054\pi$
$787$	−5404.00	−0.244767	−0.122384	+	0.992483i	$0.539054\pi$
$788$	0	0
$789$	6732.00	0.303759
$790$	0	0
$791$	11976.0	0.538328
$792$	0	0
$793$	7220.00	0.323316
$794$	0	0
$795$	3090.00	0.137850
$796$	0	0
$797$	16326.0	0.725592	0.362796	−	0.931869i	$-0.381822\pi$
$797$	16326.0	0.725592	0.362796	+	0.931869i	$0.381822\pi$
$798$	0	0
$799$	−744.000	−0.0329422
$800$	0	0
$801$	−1926.00	−0.0849586
$802$	0	0
$803$	−29040.0	−1.27621
$804$	0	0
$805$	−6000.00	−0.262698
$806$	0	0
$807$	−12942.0	−0.564535
$808$	0	0
$809$	26202.0	1.13871	0.569353	−	0.822093i	$-0.307194\pi$
$809$	26202.0	1.13871	0.569353	+	0.822093i	$0.307194\pi$
$810$	0	0
$811$	−26208.0	−1.13476	−0.567378	−	0.823457i	$-0.692042\pi$
$811$	−26208.0	−1.13476	−0.567378	+	0.823457i	$0.692042\pi$
$812$	0	0
$813$	−19176.0	−0.827222
$814$	0	0
$815$	−6820.00	−0.293122
$816$	0	0
$817$	−15392.0	−0.659116
$818$	0	0
$819$	−4104.00	−0.175098
$820$	0	0
$821$	−1986.00	−0.0844237	−0.0422119	−	0.999109i	$-0.513440\pi$
$821$	−1986.00	−0.0844237	−0.0422119	+	0.999109i	$0.513440\pi$
$822$	0	0
$823$	−5236.00	−0.221769	−0.110884	−	0.993833i	$-0.535368\pi$
$823$	−5236.00	−0.221769	−0.110884	+	0.993833i	$0.535368\pi$
$824$	0	0
$825$	−1800.00	−0.0759612
$826$	0	0
$827$	−26044.0	−1.09509	−0.547545	−	0.836777i	$-0.684437\pi$
$827$	−26044.0	−1.09509	−0.547545	+	0.836777i	$0.684437\pi$
$828$	0	0
$829$	21246.0	0.890113	0.445057	−	0.895502i	$-0.353184\pi$
$829$	21246.0	0.890113	0.445057	+	0.895502i	$0.353184\pi$
$830$	0	0
$831$	13194.0	0.550776
$832$	0	0
$833$	1194.00	0.0496634
$834$	0	0
$835$	−10020.0	−0.415277
$836$	0	0
$837$	−1512.00	−0.0624401
$838$	0	0
$839$	−17440.0	−0.717635	−0.358817	−	0.933408i	$-0.616820\pi$
$839$	−17440.0	−0.717635	−0.358817	+	0.933408i	$0.616820\pi$
$840$	0	0
$841$	28511.0	1.16901
$842$	0	0
$843$	−22866.0	−0.934219
$844$	0	0
$845$	−3765.00	−0.153278
$846$	0	0
$847$	9060.00	0.367539
$848$	0	0
$849$	3060.00	0.123697
$850$	0	0
$851$	19000.0	0.765349
$852$	0	0
$853$	33582.0	1.34798	0.673989	−	0.738741i	$-0.264579\pi$
$853$	33582.0	1.34798	0.673989	+	0.738741i	$0.264579\pi$
$854$	0	0
$855$	4680.00	0.187196
$856$	0	0
$857$	−18942.0	−0.755013	−0.377507	−	0.926007i	$-0.623218\pi$
$857$	−18942.0	−0.755013	−0.377507	+	0.926007i	$0.623218\pi$
$858$	0	0
$859$	25720.0	1.02160	0.510800	−	0.859699i	$-0.329349\pi$
$859$	25720.0	1.02160	0.510800	+	0.859699i	$0.329349\pi$
$860$	0	0
$861$	−7272.00	−0.287839
$862$	0	0
$863$	32436.0	1.27941	0.639707	−	0.768619i	$-0.279056\pi$
$863$	32436.0	1.27941	0.639707	+	0.768619i	$0.279056\pi$
$864$	0	0
$865$	−7730.00	−0.303847
$866$	0	0
$867$	−14631.0	−0.573120
$868$	0	0
$869$	−19584.0	−0.764490
$870$	0	0
$871$	−7752.00	−0.301569
$872$	0	0
$873$	10818.0	0.419397
$874$	0	0
$875$	−1500.00	−0.0579534
$876$	0	0
$877$	8646.00	0.332902	0.166451	−	0.986050i	$-0.446769\pi$
$877$	8646.00	0.332902	0.166451	+	0.986050i	$0.446769\pi$
$878$	0	0
$879$	−11238.0	−0.431227
$880$	0	0
$881$	27442.0	1.04943	0.524713	−	0.851279i	$-0.324173\pi$
$881$	27442.0	1.04943	0.524713	+	0.851279i	$0.324173\pi$
$882$	0	0
$883$	−14116.0	−0.537986	−0.268993	−	0.963142i	$-0.586691\pi$
$883$	−14116.0	−0.537986	−0.268993	+	0.963142i	$0.586691\pi$
$884$	0	0
$885$	−1920.00	−0.0729267
$886$	0	0
$887$	4124.00	0.156111	0.0780554	−	0.996949i	$-0.475129\pi$
$887$	4124.00	0.156111	0.0780554	+	0.996949i	$0.475129\pi$
$888$	0	0
$889$	−10032.0	−0.378473
$890$	0	0
$891$	−1944.00	−0.0730937
$892$	0	0
$893$	12896.0	0.483257
$894$	0	0
$895$	−5360.00	−0.200184
$896$	0	0
$897$	11400.0	0.424342
$898$	0	0
$899$	−12880.0	−0.477833
$900$	0	0
$901$	−1236.00	−0.0457016
$902$	0	0
$903$	5328.00	0.196351
$904$	0	0
$905$	−18770.0	−0.689432
$906$	0	0
$907$	42100.0	1.54124	0.770622	−	0.637293i	$-0.219946\pi$
$907$	42100.0	1.54124	0.770622	+	0.637293i	$0.219946\pi$
$908$	0	0
$909$	12078.0	0.440706
$910$	0	0
$911$	34152.0	1.24205	0.621024	−	0.783791i	$-0.286717\pi$
$911$	34152.0	1.24205	0.621024	+	0.783791i	$0.286717\pi$
$912$	0	0
$913$	33888.0	1.22840
$914$	0	0
$915$	2850.00	0.102971
$916$	0	0
$917$	17760.0	0.639571
$918$	0	0
$919$	41984.0	1.50699	0.753495	−	0.657453i	$-0.228366\pi$
$919$	41984.0	1.50699	0.753495	+	0.657453i	$0.228366\pi$
$920$	0	0
$921$	29100.0	1.04113
$922$	0	0
$923$	−16720.0	−0.596257
$924$	0	0
$925$	4750.00	0.168842
$926$	0	0
$927$	8172.00	0.289540
$928$	0	0
$929$	48434.0	1.71051	0.855257	−	0.518204i	$-0.173399\pi$
$929$	48434.0	1.71051	0.855257	+	0.518204i	$0.173399\pi$
$930$	0	0
$931$	−20696.0	−0.728554
$932$	0	0
$933$	12456.0	0.437075
$934$	0	0
$935$	720.000	0.0251834
$936$	0	0
$937$	−25590.0	−0.892197	−0.446099	−	0.894984i	$-0.647187\pi$
$937$	−25590.0	−0.892197	−0.446099	+	0.894984i	$0.647187\pi$
$938$	0	0
$939$	19086.0	0.663310
$940$	0	0
$941$	−18906.0	−0.654961	−0.327480	−	0.944858i	$-0.606200\pi$
$941$	−18906.0	−0.654961	−0.327480	+	0.944858i	$0.606200\pi$
$942$	0	0
$943$	20200.0	0.697564
$944$	0	0
$945$	−1620.00	−0.0557657
$946$	0	0
$947$	−36316.0	−1.24616	−0.623079	−	0.782159i	$-0.714118\pi$
$947$	−36316.0	−1.24616	−0.623079	+	0.782159i	$0.714118\pi$
$948$	0	0
$949$	45980.0	1.57279
$950$	0	0
$951$	32658.0	1.11357
$952$	0	0
$953$	15890.0	0.540113	0.270056	−	0.962844i	$-0.412958\pi$
$953$	15890.0	0.540113	0.270056	+	0.962844i	$0.412958\pi$
$954$	0	0
$955$	6120.00	0.207370
$956$	0	0
$957$	−16560.0	−0.559361
$958$	0	0
$959$	−16152.0	−0.543874
$960$	0	0
$961$	−26655.0	−0.894733
$962$	0	0
$963$	−7884.00	−0.263820
$964$	0	0
$965$	−8470.00	−0.282548
$966$	0	0
$967$	116.000	0.00385761	0.00192880	−	0.999998i	$-0.499386\pi$
$967$	116.000	0.00385761	0.00192880	+	0.999998i	$0.499386\pi$
$968$	0	0
$969$	−1872.00	−0.0620612
$970$	0	0
$971$	4744.00	0.156789	0.0783945	−	0.996922i	$-0.475021\pi$
$971$	4744.00	0.156789	0.0783945	+	0.996922i	$0.475021\pi$
$972$	0	0
$973$	−9888.00	−0.325791
$974$	0	0
$975$	2850.00	0.0936134
$976$	0	0
$977$	−18374.0	−0.601675	−0.300837	−	0.953675i	$-0.597266\pi$
$977$	−18374.0	−0.601675	−0.300837	+	0.953675i	$0.597266\pi$
$978$	0	0
$979$	5136.00	0.167668
$980$	0	0
$981$	2718.00	0.0884598
$982$	0	0
$983$	−13036.0	−0.422974	−0.211487	−	0.977381i	$-0.567831\pi$
$983$	−13036.0	−0.422974	−0.211487	+	0.977381i	$0.567831\pi$
$984$	0	0
$985$	15670.0	0.506891
$986$	0	0
$987$	−4464.00	−0.143962
$988$	0	0
$989$	−14800.0	−0.475847
$990$	0	0
$991$	15224.0	0.487998	0.243999	−	0.969775i	$-0.421541\pi$
$991$	15224.0	0.487998	0.243999	+	0.969775i	$0.421541\pi$
$992$	0	0
$993$	12384.0	0.395765
$994$	0	0
$995$	12800.0	0.407826
$996$	0	0
$997$	−43794.0	−1.39114	−0.695572	−	0.718457i	$-0.744849\pi$
$997$	−43794.0	−1.39114	−0.695572	+	0.718457i	$0.744849\pi$
$998$	0	0
$999$	5130.00	0.162468

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.4.a.j.1.1	yes	1
3.2	odd	2		1440.4.a.d.1.1		1
4.3	odd	2		480.4.a.e.1.1	✓	1
5.4	even	2		2400.4.a.g.1.1		1
8.3	odd	2		960.4.a.y.1.1		1
8.5	even	2		960.4.a.d.1.1		1
12.11	even	2		1440.4.a.g.1.1		1
20.19	odd	2		2400.4.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.a.e.1.1	✓	1	4.3	odd	2
480.4.a.j.1.1	yes	1	1.1	even	1	trivial
960.4.a.d.1.1		1	8.5	even	2
960.4.a.y.1.1		1	8.3	odd	2
1440.4.a.d.1.1		1	3.2	odd	2
1440.4.a.g.1.1		1	12.11	even	2
2400.4.a.g.1.1		1	5.4	even	2
2400.4.a.p.1.1		1	20.19	odd	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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