LMFDB - Embedded newform 240.4.a.f.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [240,4,Mod(1,240)] 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("240.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	240.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,3,0,-5,0,-20,0,9,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$14.1604584014$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	240.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} -5.00000 q^{5} -20.0000 q^{7} +9.00000 q^{9} +24.0000 q^{11} +74.0000 q^{13} -15.0000 q^{15} +54.0000 q^{17} +124.000 q^{19} -60.0000 q^{21} +120.000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -78.0000 q^{29} -200.000 q^{31} +72.0000 q^{33} +100.000 q^{35} -70.0000 q^{37} +222.000 q^{39} +330.000 q^{41} -92.0000 q^{43} -45.0000 q^{45} +24.0000 q^{47} +57.0000 q^{49} +162.000 q^{51} +450.000 q^{53} -120.000 q^{55} +372.000 q^{57} -24.0000 q^{59} -322.000 q^{61} -180.000 q^{63} -370.000 q^{65} +196.000 q^{67} +360.000 q^{69} +288.000 q^{71} -430.000 q^{73} +75.0000 q^{75} -480.000 q^{77} +520.000 q^{79} +81.0000 q^{81} -156.000 q^{83} -270.000 q^{85} -234.000 q^{87} +1026.00 q^{89} -1480.00 q^{91} -600.000 q^{93} -620.000 q^{95} -286.000 q^{97} +216.000 q^{99} +O(q^{100})$

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
$7$	−20.0000	−1.07990	−0.539949	−	0.841698i	$-0.681557\pi$
$7$	−20.0000	−1.07990	−0.539949	+	0.841698i	$0.681557\pi$
$8$	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
$13$	74.0000	1.57876	0.789381	−	0.613904i	$-0.210402\pi$
$13$	74.0000	1.57876	0.789381	+	0.613904i	$0.210402\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	54.0000	0.770407	0.385204	−	0.922832i	$-0.374131\pi$
$17$	54.0000	0.770407	0.385204	+	0.922832i	$0.374131\pi$
$18$	0	0
$19$	124.000	1.49724	0.748620	−	0.663000i	$-0.230717\pi$
$19$	124.000	1.49724	0.748620	+	0.663000i	$0.230717\pi$
$20$	0	0
$21$	−60.0000	−0.623480
$22$	0	0
$23$	120.000	1.08790	0.543951	−	0.839117i	$-0.316928\pi$
$23$	120.000	1.08790	0.543951	+	0.839117i	$0.316928\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−78.0000	−0.499456	−0.249728	−	0.968316i	$-0.580341\pi$
$29$	−78.0000	−0.499456	−0.249728	+	0.968316i	$0.580341\pi$
$30$	0	0
$31$	−200.000	−1.15874	−0.579372	−	0.815063i	$-0.696702\pi$
$31$	−200.000	−1.15874	−0.579372	+	0.815063i	$0.696702\pi$
$32$	0	0
$33$	72.0000	0.379806
$34$	0	0
$35$	100.000	0.482945
$36$	0	0
$37$	−70.0000	−0.311025	−0.155513	−	0.987834i	$-0.549703\pi$
$37$	−70.0000	−0.311025	−0.155513	+	0.987834i	$0.549703\pi$
$38$	0	0
$39$	222.000	0.911499
$40$	0	0
$41$	330.000	1.25701	0.628504	−	0.777806i	$-0.283668\pi$
$41$	330.000	1.25701	0.628504	+	0.777806i	$0.283668\pi$
$42$	0	0
$43$	−92.0000	−0.326276	−0.163138	−	0.986603i	$-0.552162\pi$
$43$	−92.0000	−0.326276	−0.163138	+	0.986603i	$0.552162\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	24.0000	0.0744843	0.0372421	−	0.999306i	$-0.488143\pi$
$47$	24.0000	0.0744843	0.0372421	+	0.999306i	$0.488143\pi$
$48$	0	0
$49$	57.0000	0.166181
$50$	0	0
$51$	162.000	0.444795
$52$	0	0
$53$	450.000	1.16627	0.583134	−	0.812376i	$-0.301826\pi$
$53$	450.000	1.16627	0.583134	+	0.812376i	$0.301826\pi$
$54$	0	0
$55$	−120.000	−0.294196
$56$	0	0
$57$	372.000	0.864432
$58$	0	0
$59$	−24.0000	−0.0529582	−0.0264791	−	0.999649i	$-0.508430\pi$
$59$	−24.0000	−0.0529582	−0.0264791	+	0.999649i	$0.508430\pi$
$60$	0	0
$61$	−322.000	−0.675867	−0.337933	−	0.941170i	$-0.609728\pi$
$61$	−322.000	−0.675867	−0.337933	+	0.941170i	$0.609728\pi$
$62$	0	0
$63$	−180.000	−0.359966
$64$	0	0
$65$	−370.000	−0.706044
$66$	0	0
$67$	196.000	0.357391	0.178696	−	0.983904i	$-0.442812\pi$
$67$	196.000	0.357391	0.178696	+	0.983904i	$0.442812\pi$
$68$	0	0
$69$	360.000	0.628100
$70$	0	0
$71$	288.000	0.481399	0.240699	−	0.970600i	$-0.422623\pi$
$71$	288.000	0.481399	0.240699	+	0.970600i	$0.422623\pi$
$72$	0	0
$73$	−430.000	−0.689420	−0.344710	−	0.938709i	$-0.612023\pi$
$73$	−430.000	−0.689420	−0.344710	+	0.938709i	$0.612023\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	−480.000	−0.710404
$78$	0	0
$79$	520.000	0.740564	0.370282	−	0.928919i	$-0.379261\pi$
$79$	520.000	0.740564	0.370282	+	0.928919i	$0.379261\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−156.000	−0.206304	−0.103152	−	0.994666i	$-0.532893\pi$
$83$	−156.000	−0.206304	−0.103152	+	0.994666i	$0.532893\pi$
$84$	0	0
$85$	−270.000	−0.344537
$86$	0	0
$87$	−234.000	−0.288361
$88$	0	0
$89$	1026.00	1.22198	0.610988	−	0.791640i	$-0.290773\pi$
$89$	1026.00	1.22198	0.610988	+	0.791640i	$0.290773\pi$
$90$	0	0
$91$	−1480.00	−1.70490
$92$	0	0
$93$	−600.000	−0.669001
$94$	0	0
$95$	−620.000	−0.669586
$96$	0	0
$97$	−286.000	−0.299370	−0.149685	−	0.988734i	$-0.547826\pi$
$97$	−286.000	−0.299370	−0.149685	+	0.988734i	$0.547826\pi$
$98$	0	0
$99$	216.000	0.219281
$100$	0	0
$101$	−1734.00	−1.70831	−0.854156	−	0.520017i	$-0.825925\pi$
$101$	−1734.00	−1.70831	−0.854156	+	0.520017i	$0.825925\pi$
$102$	0	0
$103$	−452.000	−0.432397	−0.216198	−	0.976349i	$-0.569366\pi$
$103$	−452.000	−0.432397	−0.216198	+	0.976349i	$0.569366\pi$
$104$	0	0
$105$	300.000	0.278829
$106$	0	0
$107$	1404.00	1.26850	0.634251	−	0.773127i	$-0.281308\pi$
$107$	1404.00	1.26850	0.634251	+	0.773127i	$0.281308\pi$
$108$	0	0
$109$	−1474.00	−1.29526	−0.647631	−	0.761954i	$-0.724240\pi$
$109$	−1474.00	−1.29526	−0.647631	+	0.761954i	$0.724240\pi$
$110$	0	0
$111$	−210.000	−0.179570
$112$	0	0
$113$	1086.00	0.904091	0.452046	−	0.891995i	$-0.350694\pi$
$113$	1086.00	0.904091	0.452046	+	0.891995i	$0.350694\pi$
$114$	0	0
$115$	−600.000	−0.486524
$116$	0	0
$117$	666.000	0.526254
$118$	0	0
$119$	−1080.00	−0.831962
$120$	0	0
$121$	−755.000	−0.567243
$122$	0	0
$123$	990.000	0.725734
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1244.00	−0.869190	−0.434595	−	0.900626i	$-0.643109\pi$
$127$	−1244.00	−0.869190	−0.434595	+	0.900626i	$0.643109\pi$
$128$	0	0
$129$	−276.000	−0.188376
$130$	0	0
$131$	−2328.00	−1.55266	−0.776329	−	0.630327i	$-0.782921\pi$
$131$	−2328.00	−1.55266	−0.776329	+	0.630327i	$0.782921\pi$
$132$	0	0
$133$	−2480.00	−1.61687
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	2118.00	1.32082	0.660412	−	0.750903i	$-0.270382\pi$
$137$	2118.00	1.32082	0.660412	+	0.750903i	$0.270382\pi$
$138$	0	0
$139$	−2324.00	−1.41812	−0.709062	−	0.705147i	$-0.750881\pi$
$139$	−2324.00	−1.41812	−0.709062	+	0.705147i	$0.750881\pi$
$140$	0	0
$141$	72.0000	0.0430035
$142$	0	0
$143$	1776.00	1.03858
$144$	0	0
$145$	390.000	0.223364
$146$	0	0
$147$	171.000	0.0959445
$148$	0	0
$149$	258.000	0.141854	0.0709268	−	0.997482i	$-0.477404\pi$
$149$	258.000	0.141854	0.0709268	+	0.997482i	$0.477404\pi$
$150$	0	0
$151$	808.000	0.435458	0.217729	−	0.976009i	$-0.430135\pi$
$151$	808.000	0.435458	0.217729	+	0.976009i	$0.430135\pi$
$152$	0	0
$153$	486.000	0.256802
$154$	0	0
$155$	1000.00	0.518206
$156$	0	0
$157$	2378.00	1.20882	0.604411	−	0.796673i	$-0.293408\pi$
$157$	2378.00	1.20882	0.604411	+	0.796673i	$0.293408\pi$
$158$	0	0
$159$	1350.00	0.673346
$160$	0	0
$161$	−2400.00	−1.17482
$162$	0	0
$163$	52.0000	0.0249874	0.0124937	−	0.999922i	$-0.496023\pi$
$163$	52.0000	0.0249874	0.0124937	+	0.999922i	$0.496023\pi$
$164$	0	0
$165$	−360.000	−0.169854
$166$	0	0
$167$	3720.00	1.72373	0.861863	−	0.507141i	$-0.169298\pi$
$167$	3720.00	1.72373	0.861863	+	0.507141i	$0.169298\pi$
$168$	0	0
$169$	3279.00	1.49249
$170$	0	0
$171$	1116.00	0.499080
$172$	0	0
$173$	426.000	0.187215	0.0936075	−	0.995609i	$-0.470160\pi$
$173$	426.000	0.187215	0.0936075	+	0.995609i	$0.470160\pi$
$174$	0	0
$175$	−500.000	−0.215980
$176$	0	0
$177$	−72.0000	−0.0305754
$178$	0	0
$179$	1440.00	0.601289	0.300644	−	0.953736i	$-0.402798\pi$
$179$	1440.00	0.601289	0.300644	+	0.953736i	$0.402798\pi$
$180$	0	0
$181$	−3130.00	−1.28537	−0.642683	−	0.766133i	$-0.722179\pi$
$181$	−3130.00	−1.28537	−0.642683	+	0.766133i	$0.722179\pi$
$182$	0	0
$183$	−966.000	−0.390212
$184$	0	0
$185$	350.000	0.139095
$186$	0	0
$187$	1296.00	0.506807
$188$	0	0
$189$	−540.000	−0.207827
$190$	0	0
$191$	−3576.00	−1.35471	−0.677357	−	0.735655i	$-0.736875\pi$
$191$	−3576.00	−1.35471	−0.677357	+	0.735655i	$0.736875\pi$
$192$	0	0
$193$	2666.00	0.994315	0.497158	−	0.867660i	$-0.334377\pi$
$193$	2666.00	0.994315	0.497158	+	0.867660i	$0.334377\pi$
$194$	0	0
$195$	−1110.00	−0.407635
$196$	0	0
$197$	−2718.00	−0.982992	−0.491496	−	0.870880i	$-0.663550\pi$
$197$	−2718.00	−0.982992	−0.491496	+	0.870880i	$0.663550\pi$
$198$	0	0
$199$	3832.00	1.36504	0.682521	−	0.730866i	$-0.260884\pi$
$199$	3832.00	1.36504	0.682521	+	0.730866i	$0.260884\pi$
$200$	0	0
$201$	588.000	0.206340
$202$	0	0
$203$	1560.00	0.539362
$204$	0	0
$205$	−1650.00	−0.562151
$206$	0	0
$207$	1080.00	0.362634
$208$	0	0
$209$	2976.00	0.984948
$210$	0	0
$211$	−1100.00	−0.358896	−0.179448	−	0.983767i	$-0.557431\pi$
$211$	−1100.00	−0.358896	−0.179448	+	0.983767i	$0.557431\pi$
$212$	0	0
$213$	864.000	0.277936
$214$	0	0
$215$	460.000	0.145915
$216$	0	0
$217$	4000.00	1.25133
$218$	0	0
$219$	−1290.00	−0.398037
$220$	0	0
$221$	3996.00	1.21629
$222$	0	0
$223$	−1964.00	−0.589772	−0.294886	−	0.955532i	$-0.595282\pi$
$223$	−1964.00	−0.589772	−0.294886	+	0.955532i	$0.595282\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−660.000	−0.192977	−0.0964884	−	0.995334i	$-0.530761\pi$
$227$	−660.000	−0.192977	−0.0964884	+	0.995334i	$0.530761\pi$
$228$	0	0
$229$	−1906.00	−0.550009	−0.275004	−	0.961443i	$-0.588679\pi$
$229$	−1906.00	−0.550009	−0.275004	+	0.961443i	$0.588679\pi$
$230$	0	0
$231$	−1440.00	−0.410152
$232$	0	0
$233$	−1458.00	−0.409943	−0.204972	−	0.978768i	$-0.565710\pi$
$233$	−1458.00	−0.409943	−0.204972	+	0.978768i	$0.565710\pi$
$234$	0	0
$235$	−120.000	−0.0333104
$236$	0	0
$237$	1560.00	0.427565
$238$	0	0
$239$	−1176.00	−0.318281	−0.159140	−	0.987256i	$-0.550872\pi$
$239$	−1176.00	−0.318281	−0.159140	+	0.987256i	$0.550872\pi$
$240$	0	0
$241$	866.000	0.231469	0.115734	−	0.993280i	$-0.463078\pi$
$241$	866.000	0.231469	0.115734	+	0.993280i	$0.463078\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−285.000	−0.0743183
$246$	0	0
$247$	9176.00	2.36379
$248$	0	0
$249$	−468.000	−0.119110
$250$	0	0
$251$	−432.000	−0.108636	−0.0543179	−	0.998524i	$-0.517298\pi$
$251$	−432.000	−0.108636	−0.0543179	+	0.998524i	$0.517298\pi$
$252$	0	0
$253$	2880.00	0.715668
$254$	0	0
$255$	−810.000	−0.198918
$256$	0	0
$257$	2526.00	0.613103	0.306552	−	0.951854i	$-0.400825\pi$
$257$	2526.00	0.613103	0.306552	+	0.951854i	$0.400825\pi$
$258$	0	0
$259$	1400.00	0.335876
$260$	0	0
$261$	−702.000	−0.166485
$262$	0	0
$263$	−5448.00	−1.27733	−0.638666	−	0.769484i	$-0.720513\pi$
$263$	−5448.00	−1.27733	−0.638666	+	0.769484i	$0.720513\pi$
$264$	0	0
$265$	−2250.00	−0.521571
$266$	0	0
$267$	3078.00	0.705508
$268$	0	0
$269$	−2574.00	−0.583418	−0.291709	−	0.956507i	$-0.594224\pi$
$269$	−2574.00	−0.583418	−0.291709	+	0.956507i	$0.594224\pi$
$270$	0	0
$271$	3184.00	0.713706	0.356853	−	0.934161i	$-0.383850\pi$
$271$	3184.00	0.713706	0.356853	+	0.934161i	$0.383850\pi$
$272$	0	0
$273$	−4440.00	−0.984326
$274$	0	0
$275$	600.000	0.131569
$276$	0	0
$277$	3962.00	0.859399	0.429699	−	0.902972i	$-0.358620\pi$
$277$	3962.00	0.859399	0.429699	+	0.902972i	$0.358620\pi$
$278$	0	0
$279$	−1800.00	−0.386248
$280$	0	0
$281$	−8286.00	−1.75908	−0.879540	−	0.475825i	$-0.842149\pi$
$281$	−8286.00	−1.75908	−0.879540	+	0.475825i	$0.842149\pi$
$282$	0	0
$283$	2716.00	0.570493	0.285246	−	0.958454i	$-0.407925\pi$
$283$	2716.00	0.570493	0.285246	+	0.958454i	$0.407925\pi$
$284$	0	0
$285$	−1860.00	−0.386586
$286$	0	0
$287$	−6600.00	−1.35744
$288$	0	0
$289$	−1997.00	−0.406473
$290$	0	0
$291$	−858.000	−0.172841
$292$	0	0
$293$	6018.00	1.19992	0.599958	−	0.800032i	$-0.295184\pi$
$293$	6018.00	1.19992	0.599958	+	0.800032i	$0.295184\pi$
$294$	0	0
$295$	120.000	0.0236836
$296$	0	0
$297$	648.000	0.126602
$298$	0	0
$299$	8880.00	1.71754
$300$	0	0
$301$	1840.00	0.352345
$302$	0	0
$303$	−5202.00	−0.986294
$304$	0	0
$305$	1610.00	0.302257
$306$	0	0
$307$	−9236.00	−1.71702	−0.858512	−	0.512793i	$-0.828611\pi$
$307$	−9236.00	−1.71702	−0.858512	+	0.512793i	$0.828611\pi$
$308$	0	0
$309$	−1356.00	−0.249644
$310$	0	0
$311$	−1536.00	−0.280060	−0.140030	−	0.990147i	$-0.544720\pi$
$311$	−1536.00	−0.280060	−0.140030	+	0.990147i	$0.544720\pi$
$312$	0	0
$313$	−7342.00	−1.32586	−0.662930	−	0.748681i	$-0.730687\pi$
$313$	−7342.00	−1.32586	−0.662930	+	0.748681i	$0.730687\pi$
$314$	0	0
$315$	900.000	0.160982
$316$	0	0
$317$	−3894.00	−0.689933	−0.344967	−	0.938615i	$-0.612110\pi$
$317$	−3894.00	−0.689933	−0.344967	+	0.938615i	$0.612110\pi$
$318$	0	0
$319$	−1872.00	−0.328564
$320$	0	0
$321$	4212.00	0.732370
$322$	0	0
$323$	6696.00	1.15348
$324$	0	0
$325$	1850.00	0.315752
$326$	0	0
$327$	−4422.00	−0.747820
$328$	0	0
$329$	−480.000	−0.0804354
$330$	0	0
$331$	−3692.00	−0.613084	−0.306542	−	0.951857i	$-0.599172\pi$
$331$	−3692.00	−0.613084	−0.306542	+	0.951857i	$0.599172\pi$
$332$	0	0
$333$	−630.000	−0.103675
$334$	0	0
$335$	−980.000	−0.159830
$336$	0	0
$337$	−8998.00	−1.45446	−0.727229	−	0.686395i	$-0.759192\pi$
$337$	−8998.00	−1.45446	−0.727229	+	0.686395i	$0.759192\pi$
$338$	0	0
$339$	3258.00	0.521977
$340$	0	0
$341$	−4800.00	−0.762271
$342$	0	0
$343$	5720.00	0.900440
$344$	0	0
$345$	−1800.00	−0.280895
$346$	0	0
$347$	−5244.00	−0.811276	−0.405638	−	0.914034i	$-0.632951\pi$
$347$	−5244.00	−0.811276	−0.405638	+	0.914034i	$0.632951\pi$
$348$	0	0
$349$	6302.00	0.966585	0.483293	−	0.875459i	$-0.339441\pi$
$349$	6302.00	0.966585	0.483293	+	0.875459i	$0.339441\pi$
$350$	0	0
$351$	1998.00	0.303833
$352$	0	0
$353$	3414.00	0.514756	0.257378	−	0.966311i	$-0.417141\pi$
$353$	3414.00	0.514756	0.257378	+	0.966311i	$0.417141\pi$
$354$	0	0
$355$	−1440.00	−0.215288
$356$	0	0
$357$	−3240.00	−0.480333
$358$	0	0
$359$	−4824.00	−0.709195	−0.354597	−	0.935019i	$-0.615382\pi$
$359$	−4824.00	−0.709195	−0.354597	+	0.935019i	$0.615382\pi$
$360$	0	0
$361$	8517.00	1.24173
$362$	0	0
$363$	−2265.00	−0.327498
$364$	0	0
$365$	2150.00	0.308318
$366$	0	0
$367$	3508.00	0.498954	0.249477	−	0.968381i	$-0.419741\pi$
$367$	3508.00	0.498954	0.249477	+	0.968381i	$0.419741\pi$
$368$	0	0
$369$	2970.00	0.419003
$370$	0	0
$371$	−9000.00	−1.25945
$372$	0	0
$373$	10802.0	1.49948	0.749740	−	0.661732i	$-0.230178\pi$
$373$	10802.0	1.49948	0.749740	+	0.661732i	$0.230178\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	−5772.00	−0.788523
$378$	0	0
$379$	−1460.00	−0.197876	−0.0989382	−	0.995094i	$-0.531545\pi$
$379$	−1460.00	−0.197876	−0.0989382	+	0.995094i	$0.531545\pi$
$380$	0	0
$381$	−3732.00	−0.501827
$382$	0	0
$383$	4872.00	0.649994	0.324997	−	0.945715i	$-0.394637\pi$
$383$	4872.00	0.649994	0.324997	+	0.945715i	$0.394637\pi$
$384$	0	0
$385$	2400.00	0.317702
$386$	0	0
$387$	−828.000	−0.108759
$388$	0	0
$389$	−14046.0	−1.83075	−0.915373	−	0.402606i	$-0.868104\pi$
$389$	−14046.0	−1.83075	−0.915373	+	0.402606i	$0.868104\pi$
$390$	0	0
$391$	6480.00	0.838127
$392$	0	0
$393$	−6984.00	−0.896428
$394$	0	0
$395$	−2600.00	−0.331190
$396$	0	0
$397$	−2734.00	−0.345631	−0.172816	−	0.984954i	$-0.555286\pi$
$397$	−2734.00	−0.345631	−0.172816	+	0.984954i	$0.555286\pi$
$398$	0	0
$399$	−7440.00	−0.933498
$400$	0	0
$401$	−15942.0	−1.98530	−0.992650	−	0.121019i	$-0.961384\pi$
$401$	−15942.0	−1.98530	−0.992650	+	0.121019i	$0.961384\pi$
$402$	0	0
$403$	−14800.0	−1.82938
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	−1680.00	−0.204606
$408$	0	0
$409$	8714.00	1.05350	0.526748	−	0.850022i	$-0.323411\pi$
$409$	8714.00	1.05350	0.526748	+	0.850022i	$0.323411\pi$
$410$	0	0
$411$	6354.00	0.762578
$412$	0	0
$413$	480.000	0.0571895
$414$	0	0
$415$	780.000	0.0922619
$416$	0	0
$417$	−6972.00	−0.818754
$418$	0	0
$419$	−11976.0	−1.39634	−0.698169	−	0.715933i	$-0.746002\pi$
$419$	−11976.0	−1.39634	−0.698169	+	0.715933i	$0.746002\pi$
$420$	0	0
$421$	11054.0	1.27967	0.639833	−	0.768514i	$-0.279004\pi$
$421$	11054.0	1.27967	0.639833	+	0.768514i	$0.279004\pi$
$422$	0	0
$423$	216.000	0.0248281
$424$	0	0
$425$	1350.00	0.154081
$426$	0	0
$427$	6440.00	0.729868
$428$	0	0
$429$	5328.00	0.599623
$430$	0	0
$431$	−720.000	−0.0804668	−0.0402334	−	0.999190i	$-0.512810\pi$
$431$	−720.000	−0.0804668	−0.0402334	+	0.999190i	$0.512810\pi$
$432$	0	0
$433$	−15622.0	−1.73382	−0.866912	−	0.498462i	$-0.833898\pi$
$433$	−15622.0	−1.73382	−0.866912	+	0.498462i	$0.833898\pi$
$434$	0	0
$435$	1170.00	0.128959
$436$	0	0
$437$	14880.0	1.62885
$438$	0	0
$439$	9880.00	1.07414	0.537069	−	0.843538i	$-0.319531\pi$
$439$	9880.00	1.07414	0.537069	+	0.843538i	$0.319531\pi$
$440$	0	0
$441$	513.000	0.0553936
$442$	0	0
$443$	16116.0	1.72843	0.864215	−	0.503123i	$-0.167816\pi$
$443$	16116.0	1.72843	0.864215	+	0.503123i	$0.167816\pi$
$444$	0	0
$445$	−5130.00	−0.546484
$446$	0	0
$447$	774.000	0.0818992
$448$	0	0
$449$	9018.00	0.947852	0.473926	−	0.880565i	$-0.342836\pi$
$449$	9018.00	0.947852	0.473926	+	0.880565i	$0.342836\pi$
$450$	0	0
$451$	7920.00	0.826914
$452$	0	0
$453$	2424.00	0.251412
$454$	0	0
$455$	7400.00	0.762456
$456$	0	0
$457$	−3670.00	−0.375657	−0.187829	−	0.982202i	$-0.560145\pi$
$457$	−3670.00	−0.375657	−0.187829	+	0.982202i	$0.560145\pi$
$458$	0	0
$459$	1458.00	0.148265
$460$	0	0
$461$	17562.0	1.77428	0.887141	−	0.461499i	$-0.152688\pi$
$461$	17562.0	1.77428	0.887141	+	0.461499i	$0.152688\pi$
$462$	0	0
$463$	−1172.00	−0.117640	−0.0588202	−	0.998269i	$-0.518734\pi$
$463$	−1172.00	−0.117640	−0.0588202	+	0.998269i	$0.518734\pi$
$464$	0	0
$465$	3000.00	0.299186
$466$	0	0
$467$	−6876.00	−0.681335	−0.340667	−	0.940184i	$-0.610653\pi$
$467$	−6876.00	−0.681335	−0.340667	+	0.940184i	$0.610653\pi$
$468$	0	0
$469$	−3920.00	−0.385946
$470$	0	0
$471$	7134.00	0.697914
$472$	0	0
$473$	−2208.00	−0.214638
$474$	0	0
$475$	3100.00	0.299448
$476$	0	0
$477$	4050.00	0.388756
$478$	0	0
$479$	−2280.00	−0.217486	−0.108743	−	0.994070i	$-0.534683\pi$
$479$	−2280.00	−0.217486	−0.108743	+	0.994070i	$0.534683\pi$
$480$	0	0
$481$	−5180.00	−0.491035
$482$	0	0
$483$	−7200.00	−0.678284
$484$	0	0
$485$	1430.00	0.133882
$486$	0	0
$487$	3076.00	0.286215	0.143108	−	0.989707i	$-0.454290\pi$
$487$	3076.00	0.286215	0.143108	+	0.989707i	$0.454290\pi$
$488$	0	0
$489$	156.000	0.0144265
$490$	0	0
$491$	18912.0	1.73826	0.869131	−	0.494582i	$-0.164679\pi$
$491$	18912.0	1.73826	0.869131	+	0.494582i	$0.164679\pi$
$492$	0	0
$493$	−4212.00	−0.384785
$494$	0	0
$495$	−1080.00	−0.0980654
$496$	0	0
$497$	−5760.00	−0.519862
$498$	0	0
$499$	−9956.00	−0.893170	−0.446585	−	0.894741i	$-0.647360\pi$
$499$	−9956.00	−0.893170	−0.446585	+	0.894741i	$0.647360\pi$
$500$	0	0
$501$	11160.0	0.995194
$502$	0	0
$503$	10656.0	0.944588	0.472294	−	0.881441i	$-0.343426\pi$
$503$	10656.0	0.944588	0.472294	+	0.881441i	$0.343426\pi$
$504$	0	0
$505$	8670.00	0.763980
$506$	0	0
$507$	9837.00	0.861689
$508$	0	0
$509$	−2766.00	−0.240866	−0.120433	−	0.992721i	$-0.538428\pi$
$509$	−2766.00	−0.240866	−0.120433	+	0.992721i	$0.538428\pi$
$510$	0	0
$511$	8600.00	0.744504
$512$	0	0
$513$	3348.00	0.288144
$514$	0	0
$515$	2260.00	0.193374
$516$	0	0
$517$	576.000	0.0489989
$518$	0	0
$519$	1278.00	0.108089
$520$	0	0
$521$	10530.0	0.885466	0.442733	−	0.896654i	$-0.354009\pi$
$521$	10530.0	0.885466	0.442733	+	0.896654i	$0.354009\pi$
$522$	0	0
$523$	−12692.0	−1.06115	−0.530576	−	0.847637i	$-0.678024\pi$
$523$	−12692.0	−1.06115	−0.530576	+	0.847637i	$0.678024\pi$
$524$	0	0
$525$	−1500.00	−0.124696
$526$	0	0
$527$	−10800.0	−0.892705
$528$	0	0
$529$	2233.00	0.183529
$530$	0	0
$531$	−216.000	−0.0176527
$532$	0	0
$533$	24420.0	1.98452
$534$	0	0
$535$	−7020.00	−0.567292
$536$	0	0
$537$	4320.00	0.347154
$538$	0	0
$539$	1368.00	0.109321
$540$	0	0
$541$	18110.0	1.43920	0.719602	−	0.694386i	$-0.244324\pi$
$541$	18110.0	1.43920	0.719602	+	0.694386i	$0.244324\pi$
$542$	0	0
$543$	−9390.00	−0.742106
$544$	0	0
$545$	7370.00	0.579259
$546$	0	0
$547$	−3620.00	−0.282962	−0.141481	−	0.989941i	$-0.545186\pi$
$547$	−3620.00	−0.282962	−0.141481	+	0.989941i	$0.545186\pi$
$548$	0	0
$549$	−2898.00	−0.225289
$550$	0	0
$551$	−9672.00	−0.747806
$552$	0	0
$553$	−10400.0	−0.799734
$554$	0	0
$555$	1050.00	0.0803063
$556$	0	0
$557$	−14166.0	−1.07762	−0.538809	−	0.842428i	$-0.681125\pi$
$557$	−14166.0	−1.07762	−0.538809	+	0.842428i	$0.681125\pi$
$558$	0	0
$559$	−6808.00	−0.515112
$560$	0	0
$561$	3888.00	0.292605
$562$	0	0
$563$	13404.0	1.00339	0.501697	−	0.865043i	$-0.332709\pi$
$563$	13404.0	1.00339	0.501697	+	0.865043i	$0.332709\pi$
$564$	0	0
$565$	−5430.00	−0.404322
$566$	0	0
$567$	−1620.00	−0.119989
$568$	0	0
$569$	−18654.0	−1.37437	−0.687185	−	0.726483i	$-0.741154\pi$
$569$	−18654.0	−1.37437	−0.687185	+	0.726483i	$0.741154\pi$
$570$	0	0
$571$	7684.00	0.563162	0.281581	−	0.959537i	$-0.409141\pi$
$571$	7684.00	0.563162	0.281581	+	0.959537i	$0.409141\pi$
$572$	0	0
$573$	−10728.0	−0.782144
$574$	0	0
$575$	3000.00	0.217580
$576$	0	0
$577$	−1726.00	−0.124531	−0.0622654	−	0.998060i	$-0.519833\pi$
$577$	−1726.00	−0.124531	−0.0622654	+	0.998060i	$0.519833\pi$
$578$	0	0
$579$	7998.00	0.574068
$580$	0	0
$581$	3120.00	0.222787
$582$	0	0
$583$	10800.0	0.767222
$584$	0	0
$585$	−3330.00	−0.235348
$586$	0	0
$587$	−10596.0	−0.745049	−0.372524	−	0.928022i	$-0.621508\pi$
$587$	−10596.0	−0.745049	−0.372524	+	0.928022i	$0.621508\pi$
$588$	0	0
$589$	−24800.0	−1.73492
$590$	0	0
$591$	−8154.00	−0.567531
$592$	0	0
$593$	2862.00	0.198193	0.0990963	−	0.995078i	$-0.468405\pi$
$593$	2862.00	0.198193	0.0990963	+	0.995078i	$0.468405\pi$
$594$	0	0
$595$	5400.00	0.372065
$596$	0	0
$597$	11496.0	0.788107
$598$	0	0
$599$	23592.0	1.60925	0.804627	−	0.593781i	$-0.202365\pi$
$599$	23592.0	1.60925	0.804627	+	0.593781i	$0.202365\pi$
$600$	0	0
$601$	−9574.00	−0.649803	−0.324902	−	0.945748i	$-0.605331\pi$
$601$	−9574.00	−0.649803	−0.324902	+	0.945748i	$0.605331\pi$
$602$	0	0
$603$	1764.00	0.119130
$604$	0	0
$605$	3775.00	0.253679
$606$	0	0
$607$	−17444.0	−1.16644	−0.583221	−	0.812314i	$-0.698208\pi$
$607$	−17444.0	−1.16644	−0.583221	+	0.812314i	$0.698208\pi$
$608$	0	0
$609$	4680.00	0.311401
$610$	0	0
$611$	1776.00	0.117593
$612$	0	0
$613$	−2374.00	−0.156419	−0.0782096	−	0.996937i	$-0.524920\pi$
$613$	−2374.00	−0.156419	−0.0782096	+	0.996937i	$0.524920\pi$
$614$	0	0
$615$	−4950.00	−0.324558
$616$	0	0
$617$	−12162.0	−0.793555	−0.396778	−	0.917915i	$-0.629872\pi$
$617$	−12162.0	−0.793555	−0.396778	+	0.917915i	$0.629872\pi$
$618$	0	0
$619$	−8804.00	−0.571668	−0.285834	−	0.958279i	$-0.592271\pi$
$619$	−8804.00	−0.571668	−0.285834	+	0.958279i	$0.592271\pi$
$620$	0	0
$621$	3240.00	0.209367
$622$	0	0
$623$	−20520.0	−1.31961
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	8928.00	0.568660
$628$	0	0
$629$	−3780.00	−0.239616
$630$	0	0
$631$	12688.0	0.800478	0.400239	−	0.916411i	$-0.368927\pi$
$631$	12688.0	0.800478	0.400239	+	0.916411i	$0.368927\pi$
$632$	0	0
$633$	−3300.00	−0.207209
$634$	0	0
$635$	6220.00	0.388714
$636$	0	0
$637$	4218.00	0.262360
$638$	0	0
$639$	2592.00	0.160466
$640$	0	0
$641$	−9150.00	−0.563812	−0.281906	−	0.959442i	$-0.590967\pi$
$641$	−9150.00	−0.563812	−0.281906	+	0.959442i	$0.590967\pi$
$642$	0	0
$643$	−25292.0	−1.55120	−0.775598	−	0.631227i	$-0.782552\pi$
$643$	−25292.0	−1.55120	−0.775598	+	0.631227i	$0.782552\pi$
$644$	0	0
$645$	1380.00	0.0842441
$646$	0	0
$647$	2736.00	0.166249	0.0831246	−	0.996539i	$-0.473510\pi$
$647$	2736.00	0.166249	0.0831246	+	0.996539i	$0.473510\pi$
$648$	0	0
$649$	−576.000	−0.0348382
$650$	0	0
$651$	12000.0	0.722453
$652$	0	0
$653$	22218.0	1.33148	0.665741	−	0.746183i	$-0.268116\pi$
$653$	22218.0	1.33148	0.665741	+	0.746183i	$0.268116\pi$
$654$	0	0
$655$	11640.0	0.694370
$656$	0	0
$657$	−3870.00	−0.229807
$658$	0	0
$659$	−14520.0	−0.858299	−0.429149	−	0.903234i	$-0.641187\pi$
$659$	−14520.0	−0.858299	−0.429149	+	0.903234i	$0.641187\pi$
$660$	0	0
$661$	−10618.0	−0.624799	−0.312400	−	0.949951i	$-0.601133\pi$
$661$	−10618.0	−0.624799	−0.312400	+	0.949951i	$0.601133\pi$
$662$	0	0
$663$	11988.0	0.702225
$664$	0	0
$665$	12400.0	0.723085
$666$	0	0
$667$	−9360.00	−0.543359
$668$	0	0
$669$	−5892.00	−0.340505
$670$	0	0
$671$	−7728.00	−0.444614
$672$	0	0
$673$	1370.00	0.0784690	0.0392345	−	0.999230i	$-0.487508\pi$
$673$	1370.00	0.0784690	0.0392345	+	0.999230i	$0.487508\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	−13758.0	−0.781038	−0.390519	−	0.920595i	$-0.627704\pi$
$677$	−13758.0	−0.781038	−0.390519	+	0.920595i	$0.627704\pi$
$678$	0	0
$679$	5720.00	0.323289
$680$	0	0
$681$	−1980.00	−0.111415
$682$	0	0
$683$	−11988.0	−0.671608	−0.335804	−	0.941932i	$-0.609008\pi$
$683$	−11988.0	−0.671608	−0.335804	+	0.941932i	$0.609008\pi$
$684$	0	0
$685$	−10590.0	−0.590691
$686$	0	0
$687$	−5718.00	−0.317548
$688$	0	0
$689$	33300.0	1.84126
$690$	0	0
$691$	−32996.0	−1.81654	−0.908268	−	0.418388i	$-0.862595\pi$
$691$	−32996.0	−1.81654	−0.908268	+	0.418388i	$0.862595\pi$
$692$	0	0
$693$	−4320.00	−0.236801
$694$	0	0
$695$	11620.0	0.634204
$696$	0	0
$697$	17820.0	0.968408
$698$	0	0
$699$	−4374.00	−0.236681
$700$	0	0
$701$	−25902.0	−1.39558	−0.697792	−	0.716300i	$-0.745834\pi$
$701$	−25902.0	−1.39558	−0.697792	+	0.716300i	$0.745834\pi$
$702$	0	0
$703$	−8680.00	−0.465679
$704$	0	0
$705$	−360.000	−0.0192318
$706$	0	0
$707$	34680.0	1.84480
$708$	0	0
$709$	−27394.0	−1.45106	−0.725531	−	0.688189i	$-0.758406\pi$
$709$	−27394.0	−1.45106	−0.725531	+	0.688189i	$0.758406\pi$
$710$	0	0
$711$	4680.00	0.246855
$712$	0	0
$713$	−24000.0	−1.26060
$714$	0	0
$715$	−8880.00	−0.464466
$716$	0	0
$717$	−3528.00	−0.183760
$718$	0	0
$719$	−34848.0	−1.80753	−0.903763	−	0.428033i	$-0.859207\pi$
$719$	−34848.0	−1.80753	−0.903763	+	0.428033i	$0.859207\pi$
$720$	0	0
$721$	9040.00	0.466945
$722$	0	0
$723$	2598.00	0.133639
$724$	0	0
$725$	−1950.00	−0.0998913
$726$	0	0
$727$	−28028.0	−1.42985	−0.714925	−	0.699201i	$-0.753539\pi$
$727$	−28028.0	−1.42985	−0.714925	+	0.699201i	$0.753539\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−4968.00	−0.251365
$732$	0	0
$733$	18002.0	0.907120	0.453560	−	0.891226i	$-0.350154\pi$
$733$	18002.0	0.907120	0.453560	+	0.891226i	$0.350154\pi$
$734$	0	0
$735$	−855.000	−0.0429077
$736$	0	0
$737$	4704.00	0.235107
$738$	0	0
$739$	−15284.0	−0.760800	−0.380400	−	0.924822i	$-0.624214\pi$
$739$	−15284.0	−0.760800	−0.380400	+	0.924822i	$0.624214\pi$
$740$	0	0
$741$	27528.0	1.36473
$742$	0	0
$743$	18768.0	0.926691	0.463345	−	0.886178i	$-0.346649\pi$
$743$	18768.0	0.926691	0.463345	+	0.886178i	$0.346649\pi$
$744$	0	0
$745$	−1290.00	−0.0634388
$746$	0	0
$747$	−1404.00	−0.0687680
$748$	0	0
$749$	−28080.0	−1.36985
$750$	0	0
$751$	−8696.00	−0.422532	−0.211266	−	0.977429i	$-0.567759\pi$
$751$	−8696.00	−0.422532	−0.211266	+	0.977429i	$0.567759\pi$
$752$	0	0
$753$	−1296.00	−0.0627209
$754$	0	0
$755$	−4040.00	−0.194743
$756$	0	0
$757$	−38662.0	−1.85627	−0.928134	−	0.372247i	$-0.878587\pi$
$757$	−38662.0	−1.85627	−0.928134	+	0.372247i	$0.878587\pi$
$758$	0	0
$759$	8640.00	0.413191
$760$	0	0
$761$	23874.0	1.13723	0.568615	−	0.822604i	$-0.307479\pi$
$761$	23874.0	1.13723	0.568615	+	0.822604i	$0.307479\pi$
$762$	0	0
$763$	29480.0	1.39875
$764$	0	0
$765$	−2430.00	−0.114846
$766$	0	0
$767$	−1776.00	−0.0836084
$768$	0	0
$769$	23618.0	1.10753	0.553763	−	0.832675i	$-0.313192\pi$
$769$	23618.0	1.10753	0.553763	+	0.832675i	$0.313192\pi$
$770$	0	0
$771$	7578.00	0.353975
$772$	0	0
$773$	11538.0	0.536860	0.268430	−	0.963299i	$-0.413495\pi$
$773$	11538.0	0.536860	0.268430	+	0.963299i	$0.413495\pi$
$774$	0	0
$775$	−5000.00	−0.231749
$776$	0	0
$777$	4200.00	0.193918
$778$	0	0
$779$	40920.0	1.88204
$780$	0	0
$781$	6912.00	0.316685
$782$	0	0
$783$	−2106.00	−0.0961204
$784$	0	0
$785$	−11890.0	−0.540602
$786$	0	0
$787$	14884.0	0.674152	0.337076	−	0.941478i	$-0.390562\pi$
$787$	14884.0	0.674152	0.337076	+	0.941478i	$0.390562\pi$
$788$	0	0
$789$	−16344.0	−0.737467
$790$	0	0
$791$	−21720.0	−0.976327
$792$	0	0
$793$	−23828.0	−1.06703
$794$	0	0
$795$	−6750.00	−0.301129
$796$	0	0
$797$	−11334.0	−0.503728	−0.251864	−	0.967763i	$-0.581043\pi$
$797$	−11334.0	−0.503728	−0.251864	+	0.967763i	$0.581043\pi$
$798$	0	0
$799$	1296.00	0.0573832
$800$	0	0
$801$	9234.00	0.407325
$802$	0	0
$803$	−10320.0	−0.453530
$804$	0	0
$805$	12000.0	0.525397
$806$	0	0
$807$	−7722.00	−0.336837
$808$	0	0
$809$	44730.0	1.94391	0.971955	−	0.235167i	$-0.0755638\pi$
$809$	44730.0	1.94391	0.971955	+	0.235167i	$0.0755638\pi$
$810$	0	0
$811$	42748.0	1.85091	0.925453	−	0.378862i	$-0.123684\pi$
$811$	42748.0	1.85091	0.925453	+	0.378862i	$0.123684\pi$
$812$	0	0
$813$	9552.00	0.412058
$814$	0	0
$815$	−260.000	−0.0111747
$816$	0	0
$817$	−11408.0	−0.488513
$818$	0	0
$819$	−13320.0	−0.568301
$820$	0	0
$821$	−31686.0	−1.34695	−0.673477	−	0.739208i	$-0.735200\pi$
$821$	−31686.0	−1.34695	−0.673477	+	0.739208i	$0.735200\pi$
$822$	0	0
$823$	−11036.0	−0.467425	−0.233713	−	0.972306i	$-0.575087\pi$
$823$	−11036.0	−0.467425	−0.233713	+	0.972306i	$0.575087\pi$
$824$	0	0
$825$	1800.00	0.0759612
$826$	0	0
$827$	−25884.0	−1.08836	−0.544181	−	0.838968i	$-0.683159\pi$
$827$	−25884.0	−1.08836	−0.544181	+	0.838968i	$0.683159\pi$
$828$	0	0
$829$	15950.0	0.668234	0.334117	−	0.942532i	$-0.391562\pi$
$829$	15950.0	0.668234	0.334117	+	0.942532i	$0.391562\pi$
$830$	0	0
$831$	11886.0	0.496174
$832$	0	0
$833$	3078.00	0.128027
$834$	0	0
$835$	−18600.0	−0.770874
$836$	0	0
$837$	−5400.00	−0.223000
$838$	0	0
$839$	−13800.0	−0.567853	−0.283927	−	0.958846i	$-0.591637\pi$
$839$	−13800.0	−0.567853	−0.283927	+	0.958846i	$0.591637\pi$
$840$	0	0
$841$	−18305.0	−0.750543
$842$	0	0
$843$	−24858.0	−1.01560
$844$	0	0
$845$	−16395.0	−0.667462
$846$	0	0
$847$	15100.0	0.612565
$848$	0	0
$849$	8148.00	0.329374
$850$	0	0
$851$	−8400.00	−0.338365
$852$	0	0
$853$	−27862.0	−1.11838	−0.559189	−	0.829040i	$-0.688887\pi$
$853$	−27862.0	−1.11838	−0.559189	+	0.829040i	$0.688887\pi$
$854$	0	0
$855$	−5580.00	−0.223195
$856$	0	0
$857$	−7314.00	−0.291530	−0.145765	−	0.989319i	$-0.546564\pi$
$857$	−7314.00	−0.291530	−0.145765	+	0.989319i	$0.546564\pi$
$858$	0	0
$859$	28780.0	1.14314	0.571572	−	0.820552i	$-0.306334\pi$
$859$	28780.0	1.14314	0.571572	+	0.820552i	$0.306334\pi$
$860$	0	0
$861$	−19800.0	−0.783719
$862$	0	0
$863$	32688.0	1.28935	0.644677	−	0.764455i	$-0.276992\pi$
$863$	32688.0	1.28935	0.644677	+	0.764455i	$0.276992\pi$
$864$	0	0
$865$	−2130.00	−0.0837251
$866$	0	0
$867$	−5991.00	−0.234677
$868$	0	0
$869$	12480.0	0.487175
$870$	0	0
$871$	14504.0	0.564236
$872$	0	0
$873$	−2574.00	−0.0997900
$874$	0	0
$875$	2500.00	0.0965891
$876$	0	0
$877$	36650.0	1.41115	0.705577	−	0.708633i	$-0.250688\pi$
$877$	36650.0	1.41115	0.705577	+	0.708633i	$0.250688\pi$
$878$	0	0
$879$	18054.0	0.692772
$880$	0	0
$881$	−2646.00	−0.101187	−0.0505936	−	0.998719i	$-0.516111\pi$
$881$	−2646.00	−0.101187	−0.0505936	+	0.998719i	$0.516111\pi$
$882$	0	0
$883$	−10892.0	−0.415113	−0.207557	−	0.978223i	$-0.566551\pi$
$883$	−10892.0	−0.415113	−0.207557	+	0.978223i	$0.566551\pi$
$884$	0	0
$885$	360.000	0.0136737
$886$	0	0
$887$	43464.0	1.64530	0.822648	−	0.568550i	$-0.192496\pi$
$887$	43464.0	1.64530	0.822648	+	0.568550i	$0.192496\pi$
$888$	0	0
$889$	24880.0	0.938637
$890$	0	0
$891$	1944.00	0.0730937
$892$	0	0
$893$	2976.00	0.111521
$894$	0	0
$895$	−7200.00	−0.268904
$896$	0	0
$897$	26640.0	0.991621
$898$	0	0
$899$	15600.0	0.578742
$900$	0	0
$901$	24300.0	0.898502
$902$	0	0
$903$	5520.00	0.203426
$904$	0	0
$905$	15650.0	0.574833
$906$	0	0
$907$	14884.0	0.544890	0.272445	−	0.962171i	$-0.412168\pi$
$907$	14884.0	0.544890	0.272445	+	0.962171i	$0.412168\pi$
$908$	0	0
$909$	−15606.0	−0.569437
$910$	0	0
$911$	1248.00	0.0453876	0.0226938	−	0.999742i	$-0.492776\pi$
$911$	1248.00	0.0453876	0.0226938	+	0.999742i	$0.492776\pi$
$912$	0	0
$913$	−3744.00	−0.135716
$914$	0	0
$915$	4830.00	0.174508
$916$	0	0
$917$	46560.0	1.67671
$918$	0	0
$919$	6640.00	0.238339	0.119169	−	0.992874i	$-0.461977\pi$
$919$	6640.00	0.238339	0.119169	+	0.992874i	$0.461977\pi$
$920$	0	0
$921$	−27708.0	−0.991324
$922$	0	0
$923$	21312.0	0.760014
$924$	0	0
$925$	−1750.00	−0.0622050
$926$	0	0
$927$	−4068.00	−0.144132
$928$	0	0
$929$	29946.0	1.05758	0.528792	−	0.848751i	$-0.322645\pi$
$929$	29946.0	1.05758	0.528792	+	0.848751i	$0.322645\pi$
$930$	0	0
$931$	7068.00	0.248812
$932$	0	0
$933$	−4608.00	−0.161693
$934$	0	0
$935$	−6480.00	−0.226651
$936$	0	0
$937$	45002.0	1.56900	0.784499	−	0.620130i	$-0.212920\pi$
$937$	45002.0	1.56900	0.784499	+	0.620130i	$0.212920\pi$
$938$	0	0
$939$	−22026.0	−0.765486
$940$	0	0
$941$	6090.00	0.210976	0.105488	−	0.994421i	$-0.466360\pi$
$941$	6090.00	0.210976	0.105488	+	0.994421i	$0.466360\pi$
$942$	0	0
$943$	39600.0	1.36750
$944$	0	0
$945$	2700.00	0.0929429
$946$	0	0
$947$	−56388.0	−1.93491	−0.967457	−	0.253035i	$-0.918571\pi$
$947$	−56388.0	−1.93491	−0.967457	+	0.253035i	$0.918571\pi$
$948$	0	0
$949$	−31820.0	−1.08843
$950$	0	0
$951$	−11682.0	−0.398333
$952$	0	0
$953$	10854.0	0.368936	0.184468	−	0.982839i	$-0.440944\pi$
$953$	10854.0	0.368936	0.184468	+	0.982839i	$0.440944\pi$
$954$	0	0
$955$	17880.0	0.605846
$956$	0	0
$957$	−5616.00	−0.189696
$958$	0	0
$959$	−42360.0	−1.42636
$960$	0	0
$961$	10209.0	0.342687
$962$	0	0
$963$	12636.0	0.422834
$964$	0	0
$965$	−13330.0	−0.444671
$966$	0	0
$967$	42316.0	1.40723	0.703615	−	0.710582i	$-0.251568\pi$
$967$	42316.0	1.40723	0.703615	+	0.710582i	$0.251568\pi$
$968$	0	0
$969$	20088.0	0.665964
$970$	0	0
$971$	−24480.0	−0.809063	−0.404532	−	0.914524i	$-0.632565\pi$
$971$	−24480.0	−0.809063	−0.404532	+	0.914524i	$0.632565\pi$
$972$	0	0
$973$	46480.0	1.53143
$974$	0	0
$975$	5550.00	0.182300
$976$	0	0
$977$	−6906.00	−0.226144	−0.113072	−	0.993587i	$-0.536069\pi$
$977$	−6906.00	−0.226144	−0.113072	+	0.993587i	$0.536069\pi$
$978$	0	0
$979$	24624.0	0.803868
$980$	0	0
$981$	−13266.0	−0.431754
$982$	0	0
$983$	−6960.00	−0.225829	−0.112914	−	0.993605i	$-0.536019\pi$
$983$	−6960.00	−0.225829	−0.112914	+	0.993605i	$0.536019\pi$
$984$	0	0
$985$	13590.0	0.439608
$986$	0	0
$987$	−1440.00	−0.0464394
$988$	0	0
$989$	−11040.0	−0.354956
$990$	0	0
$991$	−47792.0	−1.53195	−0.765975	−	0.642870i	$-0.777744\pi$
$991$	−47792.0	−1.53195	−0.765975	+	0.642870i	$0.777744\pi$
$992$	0	0
$993$	−11076.0	−0.353964
$994$	0	0
$995$	−19160.0	−0.610465
$996$	0	0
$997$	9938.00	0.315687	0.157843	−	0.987464i	$-0.449546\pi$
$997$	9938.00	0.315687	0.157843	+	0.987464i	$0.449546\pi$
$998$	0	0
$999$	−1890.00	−0.0598568

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.4.a.f.1.1		1
3.2	odd	2		720.4.a.r.1.1		1
4.3	odd	2		15.4.a.b.1.1	✓	1
5.2	odd	4		1200.4.f.m.49.1		2
5.3	odd	4		1200.4.f.m.49.2		2
5.4	even	2		1200.4.a.o.1.1		1
8.3	odd	2		960.4.a.bi.1.1		1
8.5	even	2		960.4.a.l.1.1		1
12.11	even	2		45.4.a.b.1.1		1
20.3	even	4		75.4.b.a.49.1		2
20.7	even	4		75.4.b.a.49.2		2
20.19	odd	2		75.4.a.a.1.1		1
28.27	even	2		735.4.a.i.1.1		1
36.7	odd	6		405.4.e.d.271.1		2
36.11	even	6		405.4.e.k.271.1		2
36.23	even	6		405.4.e.k.136.1		2
36.31	odd	6		405.4.e.d.136.1		2
44.43	even	2		1815.4.a.a.1.1		1
60.23	odd	4		225.4.b.d.199.2		2
60.47	odd	4		225.4.b.d.199.1		2
60.59	even	2		225.4.a.g.1.1		1
84.83	odd	2		2205.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.a.b.1.1	✓	1	4.3	odd	2
45.4.a.b.1.1		1	12.11	even	2
75.4.a.a.1.1		1	20.19	odd	2
75.4.b.a.49.1		2	20.3	even	4
75.4.b.a.49.2		2	20.7	even	4
225.4.a.g.1.1		1	60.59	even	2
225.4.b.d.199.1		2	60.47	odd	4
225.4.b.d.199.2		2	60.23	odd	4
240.4.a.f.1.1		1	1.1	even	1	trivial
405.4.e.d.136.1		2	36.31	odd	6
405.4.e.d.271.1		2	36.7	odd	6
405.4.e.k.136.1		2	36.23	even	6
405.4.e.k.271.1		2	36.11	even	6
720.4.a.r.1.1		1	3.2	odd	2
735.4.a.i.1.1		1	28.27	even	2
960.4.a.l.1.1		1	8.5	even	2
960.4.a.bi.1.1		1	8.3	odd	2
1200.4.a.o.1.1		1	5.4	even	2
1200.4.f.m.49.1		2	5.2	odd	4
1200.4.f.m.49.2		2	5.3	odd	4
1815.4.a.a.1.1		1	44.43	even	2
2205.4.a.c.1.1		1	84.83	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:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	240.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more