LMFDB - Embedded newform 294.4.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [294,4,Mod(1,294)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(294, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("294.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$294 = 2 \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	294.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2,3,4,-6,-6,0,-8,9,12,12] 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:	$17.3465615417$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 6)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	294.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

f(q)

=

q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{5} -6.00000 q^{6} -8.00000 q^{8} +9.00000 q^{9} +12.0000 q^{10} +12.0000 q^{11} +12.0000 q^{12} -38.0000 q^{13} -18.0000 q^{15} +16.0000 q^{16} +126.000 q^{17} -18.0000 q^{18} -20.0000 q^{19} -24.0000 q^{20} -24.0000 q^{22} +168.000 q^{23} -24.0000 q^{24} -89.0000 q^{25} +76.0000 q^{26} +27.0000 q^{27} +30.0000 q^{29} +36.0000 q^{30} +88.0000 q^{31} -32.0000 q^{32} +36.0000 q^{33} -252.000 q^{34} +36.0000 q^{36} +254.000 q^{37} +40.0000 q^{38} -114.000 q^{39} +48.0000 q^{40} -42.0000 q^{41} -52.0000 q^{43} +48.0000 q^{44} -54.0000 q^{45} -336.000 q^{46} +96.0000 q^{47} +48.0000 q^{48} +178.000 q^{50} +378.000 q^{51} -152.000 q^{52} +198.000 q^{53} -54.0000 q^{54} -72.0000 q^{55} -60.0000 q^{57} -60.0000 q^{58} +660.000 q^{59} -72.0000 q^{60} +538.000 q^{61} -176.000 q^{62} +64.0000 q^{64} +228.000 q^{65} -72.0000 q^{66} +884.000 q^{67} +504.000 q^{68} +504.000 q^{69} +792.000 q^{71} -72.0000 q^{72} -218.000 q^{73} -508.000 q^{74} -267.000 q^{75} -80.0000 q^{76} +228.000 q^{78} -520.000 q^{79} -96.0000 q^{80} +81.0000 q^{81} +84.0000 q^{82} +492.000 q^{83} -756.000 q^{85} +104.000 q^{86} +90.0000 q^{87} -96.0000 q^{88} -810.000 q^{89} +108.000 q^{90} +672.000 q^{92} +264.000 q^{93} -192.000 q^{94} +120.000 q^{95} -96.0000 q^{96} -1154.00 q^{97} +108.000 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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	−6.00000	−0.536656	−0.268328	−	0.963328i	$-0.586471\pi$
$5$	−6.00000	−0.536656	−0.268328	+	0.963328i	$0.586471\pi$
$6$	−6.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	12.0000	0.379473
$11$	12.0000	0.328921	0.164461	−	0.986384i	$-0.447412\pi$
$11$	12.0000	0.328921	0.164461	+	0.986384i	$0.447412\pi$
$12$	12.0000	0.288675
$13$	−38.0000	−0.810716	−0.405358	−	0.914158i	$-0.632853\pi$
$13$	−38.0000	−0.810716	−0.405358	+	0.914158i	$0.632853\pi$
$14$	0	0
$15$	−18.0000	−0.309839
$16$	16.0000	0.250000
$17$	126.000	1.79762	0.898808	−	0.438342i	$-0.144434\pi$
$17$	126.000	1.79762	0.898808	+	0.438342i	$0.144434\pi$
$18$	−18.0000	−0.235702
$19$	−20.0000	−0.241490	−0.120745	−	0.992684i	$-0.538528\pi$
$19$	−20.0000	−0.241490	−0.120745	+	0.992684i	$0.538528\pi$
$20$	−24.0000	−0.268328
$21$	0	0
$22$	−24.0000	−0.232583
$23$	168.000	1.52306	0.761531	−	0.648129i	$-0.224448\pi$
$23$	168.000	1.52306	0.761531	+	0.648129i	$0.224448\pi$
$24$	−24.0000	−0.204124
$25$	−89.0000	−0.712000
$26$	76.0000	0.573263
$27$	27.0000	0.192450
$28$	0	0
$29$	30.0000	0.192099	0.0960493	−	0.995377i	$-0.469379\pi$
$29$	30.0000	0.192099	0.0960493	+	0.995377i	$0.469379\pi$
$30$	36.0000	0.219089
$31$	88.0000	0.509847	0.254924	−	0.966961i	$-0.417950\pi$
$31$	88.0000	0.509847	0.254924	+	0.966961i	$0.417950\pi$
$32$	−32.0000	−0.176777
$33$	36.0000	0.189903
$34$	−252.000	−1.27111
$35$	0	0
$36$	36.0000	0.166667
$37$	254.000	1.12858	0.564288	−	0.825578i	$-0.309151\pi$
$37$	254.000	1.12858	0.564288	+	0.825578i	$0.309151\pi$
$38$	40.0000	0.170759
$39$	−114.000	−0.468067
$40$	48.0000	0.189737
$41$	−42.0000	−0.159983	−0.0799914	−	0.996796i	$-0.525489\pi$
$41$	−42.0000	−0.159983	−0.0799914	+	0.996796i	$0.525489\pi$
$42$	0	0
$43$	−52.0000	−0.184417	−0.0922084	−	0.995740i	$-0.529393\pi$
$43$	−52.0000	−0.184417	−0.0922084	+	0.995740i	$0.529393\pi$
$44$	48.0000	0.164461
$45$	−54.0000	−0.178885
$46$	−336.000	−1.07697
$47$	96.0000	0.297937	0.148969	−	0.988842i	$-0.452405\pi$
$47$	96.0000	0.297937	0.148969	+	0.988842i	$0.452405\pi$
$48$	48.0000	0.144338
$49$	0	0
$50$	178.000	0.503460
$51$	378.000	1.03785
$52$	−152.000	−0.405358
$53$	198.000	0.513158	0.256579	−	0.966523i	$-0.417405\pi$
$53$	198.000	0.513158	0.256579	+	0.966523i	$0.417405\pi$
$54$	−54.0000	−0.136083
$55$	−72.0000	−0.176518
$56$	0	0
$57$	−60.0000	−0.139424
$58$	−60.0000	−0.135834
$59$	660.000	1.45635	0.728175	−	0.685391i	$-0.240369\pi$
$59$	660.000	1.45635	0.728175	+	0.685391i	$0.240369\pi$
$60$	−72.0000	−0.154919
$61$	538.000	1.12924	0.564622	−	0.825350i	$-0.309022\pi$
$61$	538.000	1.12924	0.564622	+	0.825350i	$0.309022\pi$
$62$	−176.000	−0.360516
$63$	0	0
$64$	64.0000	0.125000
$65$	228.000	0.435076
$66$	−72.0000	−0.134282
$67$	884.000	1.61191	0.805954	−	0.591979i	$-0.201653\pi$
$67$	884.000	1.61191	0.805954	+	0.591979i	$0.201653\pi$
$68$	504.000	0.898808
$69$	504.000	0.879340
$70$	0	0
$71$	792.000	1.32385	0.661923	−	0.749572i	$-0.269740\pi$
$71$	792.000	1.32385	0.661923	+	0.749572i	$0.269740\pi$
$72$	−72.0000	−0.117851
$73$	−218.000	−0.349520	−0.174760	−	0.984611i	$-0.555915\pi$
$73$	−218.000	−0.349520	−0.174760	+	0.984611i	$0.555915\pi$
$74$	−508.000	−0.798024
$75$	−267.000	−0.411073
$76$	−80.0000	−0.120745
$77$	0	0
$78$	228.000	0.330973
$79$	−520.000	−0.740564	−0.370282	−	0.928919i	$-0.620739\pi$
$79$	−520.000	−0.740564	−0.370282	+	0.928919i	$0.620739\pi$
$80$	−96.0000	−0.134164
$81$	81.0000	0.111111
$82$	84.0000	0.113125
$83$	492.000	0.650651	0.325325	−	0.945602i	$-0.394526\pi$
$83$	492.000	0.650651	0.325325	+	0.945602i	$0.394526\pi$
$84$	0	0
$85$	−756.000	−0.964703
$86$	104.000	0.130402
$87$	90.0000	0.110908
$88$	−96.0000	−0.116291
$89$	−810.000	−0.964717	−0.482359	−	0.875974i	$-0.660220\pi$
$89$	−810.000	−0.964717	−0.482359	+	0.875974i	$0.660220\pi$
$90$	108.000	0.126491
$91$	0	0
$92$	672.000	0.761531
$93$	264.000	0.294360
$94$	−192.000	−0.210673
$95$	120.000	0.129597
$96$	−96.0000	−0.102062
$97$	−1154.00	−1.20795	−0.603974	−	0.797004i	$-0.706417\pi$
$97$	−1154.00	−1.20795	−0.603974	+	0.797004i	$0.706417\pi$
$98$	0	0
$99$	108.000	0.109640
$100$	−356.000	−0.356000
$101$	618.000	0.608845	0.304422	−	0.952537i	$-0.401537\pi$
$101$	618.000	0.608845	0.304422	+	0.952537i	$0.401537\pi$
$102$	−756.000	−0.733874
$103$	−128.000	−0.122449	−0.0612243	−	0.998124i	$-0.519501\pi$
$103$	−128.000	−0.122449	−0.0612243	+	0.998124i	$0.519501\pi$
$104$	304.000	0.286631
$105$	0	0
$106$	−396.000	−0.362858
$107$	−1476.00	−1.33355	−0.666777	−	0.745257i	$-0.732327\pi$
$107$	−1476.00	−1.33355	−0.666777	+	0.745257i	$0.732327\pi$
$108$	108.000	0.0962250
$109$	1190.00	1.04570	0.522850	−	0.852425i	$-0.324869\pi$
$109$	1190.00	1.04570	0.522850	+	0.852425i	$0.324869\pi$
$110$	144.000	0.124817
$111$	762.000	0.651584
$112$	0	0
$113$	−462.000	−0.384613	−0.192307	−	0.981335i	$-0.561597\pi$
$113$	−462.000	−0.384613	−0.192307	+	0.981335i	$0.561597\pi$
$114$	120.000	0.0985880
$115$	−1008.00	−0.817361
$116$	120.000	0.0960493
$117$	−342.000	−0.270239
$118$	−1320.00	−1.02980
$119$	0	0
$120$	144.000	0.109545
$121$	−1187.00	−0.891811
$122$	−1076.00	−0.798496
$123$	−126.000	−0.0923662
$124$	352.000	0.254924
$125$	1284.00	0.918756
$126$	0	0
$127$	−2536.00	−1.77192	−0.885959	−	0.463763i	$-0.846499\pi$
$127$	−2536.00	−1.77192	−0.885959	+	0.463763i	$0.846499\pi$
$128$	−128.000	−0.0883883
$129$	−156.000	−0.106473
$130$	−456.000	−0.307645
$131$	−2292.00	−1.52865	−0.764324	−	0.644832i	$-0.776927\pi$
$131$	−2292.00	−1.52865	−0.764324	+	0.644832i	$0.776927\pi$
$132$	144.000	0.0949514
$133$	0	0
$134$	−1768.00	−1.13979
$135$	−162.000	−0.103280
$136$	−1008.00	−0.635554
$137$	−726.000	−0.452747	−0.226374	−	0.974041i	$-0.572687\pi$
$137$	−726.000	−0.452747	−0.226374	+	0.974041i	$0.572687\pi$
$138$	−1008.00	−0.621787
$139$	−380.000	−0.231879	−0.115939	−	0.993256i	$-0.536988\pi$
$139$	−380.000	−0.231879	−0.115939	+	0.993256i	$0.536988\pi$
$140$	0	0
$141$	288.000	0.172014
$142$	−1584.00	−0.936101
$143$	−456.000	−0.266662
$144$	144.000	0.0833333
$145$	−180.000	−0.103091
$146$	436.000	0.247148
$147$	0	0
$148$	1016.00	0.564288
$149$	1590.00	0.874214	0.437107	−	0.899410i	$-0.356003\pi$
$149$	1590.00	0.874214	0.437107	+	0.899410i	$0.356003\pi$
$150$	534.000	0.290673
$151$	2432.00	1.31068	0.655342	−	0.755332i	$-0.272524\pi$
$151$	2432.00	1.31068	0.655342	+	0.755332i	$0.272524\pi$
$152$	160.000	0.0853797
$153$	1134.00	0.599206
$154$	0	0
$155$	−528.000	−0.273613
$156$	−456.000	−0.234033
$157$	−614.000	−0.312118	−0.156059	−	0.987748i	$-0.549879\pi$
$157$	−614.000	−0.312118	−0.156059	+	0.987748i	$0.549879\pi$
$158$	1040.00	0.523658
$159$	594.000	0.296272
$160$	192.000	0.0948683
$161$	0	0
$162$	−162.000	−0.0785674
$163$	−1852.00	−0.889938	−0.444969	−	0.895546i	$-0.646785\pi$
$163$	−1852.00	−0.889938	−0.444969	+	0.895546i	$0.646785\pi$
$164$	−168.000	−0.0799914
$165$	−216.000	−0.101913
$166$	−984.000	−0.460080
$167$	2136.00	0.989752	0.494876	−	0.868964i	$-0.335213\pi$
$167$	2136.00	0.989752	0.494876	+	0.868964i	$0.335213\pi$
$168$	0	0
$169$	−753.000	−0.342740
$170$	1512.00	0.682148
$171$	−180.000	−0.0804967
$172$	−208.000	−0.0922084
$173$	−1758.00	−0.772591	−0.386296	−	0.922375i	$-0.626246\pi$
$173$	−1758.00	−0.772591	−0.386296	+	0.922375i	$0.626246\pi$
$174$	−180.000	−0.0784239
$175$	0	0
$176$	192.000	0.0822304
$177$	1980.00	0.840824
$178$	1620.00	0.682158
$179$	−540.000	−0.225483	−0.112742	−	0.993624i	$-0.535963\pi$
$179$	−540.000	−0.225483	−0.112742	+	0.993624i	$0.535963\pi$
$180$	−216.000	−0.0894427
$181$	−1982.00	−0.813928	−0.406964	−	0.913444i	$-0.633412\pi$
$181$	−1982.00	−0.813928	−0.406964	+	0.913444i	$0.633412\pi$
$182$	0	0
$183$	1614.00	0.651969
$184$	−1344.00	−0.538484
$185$	−1524.00	−0.605658
$186$	−528.000	−0.208144
$187$	1512.00	0.591275
$188$	384.000	0.148969
$189$	0	0
$190$	−240.000	−0.0916391
$191$	−2688.00	−1.01831	−0.509154	−	0.860675i	$-0.670042\pi$
$191$	−2688.00	−1.01831	−0.509154	+	0.860675i	$0.670042\pi$
$192$	192.000	0.0721688
$193$	−2302.00	−0.858557	−0.429279	−	0.903172i	$-0.641232\pi$
$193$	−2302.00	−0.858557	−0.429279	+	0.903172i	$0.641232\pi$
$194$	2308.00	0.854148
$195$	684.000	0.251191
$196$	0	0
$197$	4374.00	1.58190	0.790951	−	0.611880i	$-0.209586\pi$
$197$	4374.00	1.58190	0.790951	+	0.611880i	$0.209586\pi$
$198$	−216.000	−0.0775275
$199$	1600.00	0.569955	0.284977	−	0.958534i	$-0.408014\pi$
$199$	1600.00	0.569955	0.284977	+	0.958534i	$0.408014\pi$
$200$	712.000	0.251730
$201$	2652.00	0.930635
$202$	−1236.00	−0.430518
$203$	0	0
$204$	1512.00	0.518927
$205$	252.000	0.0858558
$206$	256.000	0.0865843
$207$	1512.00	0.507687
$208$	−608.000	−0.202679
$209$	−240.000	−0.0794313
$210$	0	0
$211$	3332.00	1.08713	0.543565	−	0.839367i	$-0.317074\pi$
$211$	3332.00	1.08713	0.543565	+	0.839367i	$0.317074\pi$
$212$	792.000	0.256579
$213$	2376.00	0.764323
$214$	2952.00	0.942965
$215$	312.000	0.0989685
$216$	−216.000	−0.0680414
$217$	0	0
$218$	−2380.00	−0.739422
$219$	−654.000	−0.201796
$220$	−288.000	−0.0882589
$221$	−4788.00	−1.45736
$222$	−1524.00	−0.460740
$223$	−2648.00	−0.795171	−0.397586	−	0.917565i	$-0.630152\pi$
$223$	−2648.00	−0.795171	−0.397586	+	0.917565i	$0.630152\pi$
$224$	0	0
$225$	−801.000	−0.237333
$226$	924.000	0.271963
$227$	−2244.00	−0.656121	−0.328061	−	0.944657i	$-0.606395\pi$
$227$	−2244.00	−0.656121	−0.328061	+	0.944657i	$0.606395\pi$
$228$	−240.000	−0.0697122
$229$	5650.00	1.63040	0.815202	−	0.579177i	$-0.196626\pi$
$229$	5650.00	1.63040	0.815202	+	0.579177i	$0.196626\pi$
$230$	2016.00	0.577961
$231$	0	0
$232$	−240.000	−0.0679171
$233$	4698.00	1.32093	0.660464	−	0.750858i	$-0.270360\pi$
$233$	4698.00	1.32093	0.660464	+	0.750858i	$0.270360\pi$
$234$	684.000	0.191088
$235$	−576.000	−0.159890
$236$	2640.00	0.728175
$237$	−1560.00	−0.427565
$238$	0	0
$239$	−1200.00	−0.324776	−0.162388	−	0.986727i	$-0.551920\pi$
$239$	−1200.00	−0.324776	−0.162388	+	0.986727i	$0.551920\pi$
$240$	−288.000	−0.0774597
$241$	718.000	0.191911	0.0959553	−	0.995386i	$-0.469409\pi$
$241$	718.000	0.191911	0.0959553	+	0.995386i	$0.469409\pi$
$242$	2374.00	0.630605
$243$	243.000	0.0641500
$244$	2152.00	0.564622
$245$	0	0
$246$	252.000	0.0653127
$247$	760.000	0.195780
$248$	−704.000	−0.180258
$249$	1476.00	0.375653
$250$	−2568.00	−0.649658
$251$	−6012.00	−1.51185	−0.755924	−	0.654659i	$-0.772812\pi$
$251$	−6012.00	−1.51185	−0.755924	+	0.654659i	$0.772812\pi$
$252$	0	0
$253$	2016.00	0.500968
$254$	5072.00	1.25294
$255$	−2268.00	−0.556971
$256$	256.000	0.0625000
$257$	2046.00	0.496599	0.248300	−	0.968683i	$-0.420128\pi$
$257$	2046.00	0.496599	0.248300	+	0.968683i	$0.420128\pi$
$258$	312.000	0.0752879
$259$	0	0
$260$	912.000	0.217538
$261$	270.000	0.0640329
$262$	4584.00	1.08092
$263$	−6072.00	−1.42363	−0.711817	−	0.702365i	$-0.752127\pi$
$263$	−6072.00	−1.42363	−0.711817	+	0.702365i	$0.752127\pi$
$264$	−288.000	−0.0671408
$265$	−1188.00	−0.275390
$266$	0	0
$267$	−2430.00	−0.556980
$268$	3536.00	0.805954
$269$	6930.00	1.57074	0.785371	−	0.619025i	$-0.212472\pi$
$269$	6930.00	1.57074	0.785371	+	0.619025i	$0.212472\pi$
$270$	324.000	0.0730297
$271$	−1352.00	−0.303056	−0.151528	−	0.988453i	$-0.548419\pi$
$271$	−1352.00	−0.303056	−0.151528	+	0.988453i	$0.548419\pi$
$272$	2016.00	0.449404
$273$	0	0
$274$	1452.00	0.320141
$275$	−1068.00	−0.234192
$276$	2016.00	0.439670
$277$	−1186.00	−0.257256	−0.128628	−	0.991693i	$-0.541057\pi$
$277$	−1186.00	−0.257256	−0.128628	+	0.991693i	$0.541057\pi$
$278$	760.000	0.163963
$279$	792.000	0.169949
$280$	0	0
$281$	2442.00	0.518425	0.259213	−	0.965820i	$-0.416537\pi$
$281$	2442.00	0.518425	0.259213	+	0.965820i	$0.416537\pi$
$282$	−576.000	−0.121632
$283$	−2828.00	−0.594018	−0.297009	−	0.954875i	$-0.595989\pi$
$283$	−2828.00	−0.594018	−0.297009	+	0.954875i	$0.595989\pi$
$284$	3168.00	0.661923
$285$	360.000	0.0748230
$286$	912.000	0.188558
$287$	0	0
$288$	−288.000	−0.0589256
$289$	10963.0	2.23143
$290$	360.000	0.0728963
$291$	−3462.00	−0.697409
$292$	−872.000	−0.174760
$293$	−4758.00	−0.948687	−0.474344	−	0.880340i	$-0.657315\pi$
$293$	−4758.00	−0.948687	−0.474344	+	0.880340i	$0.657315\pi$
$294$	0	0
$295$	−3960.00	−0.781560
$296$	−2032.00	−0.399012
$297$	324.000	0.0633010
$298$	−3180.00	−0.618163
$299$	−6384.00	−1.23477
$300$	−1068.00	−0.205537
$301$	0	0
$302$	−4864.00	−0.926794
$303$	1854.00	0.351517
$304$	−320.000	−0.0603726
$305$	−3228.00	−0.606016
$306$	−2268.00	−0.423702
$307$	8476.00	1.57574	0.787868	−	0.615844i	$-0.211185\pi$
$307$	8476.00	1.57574	0.787868	+	0.615844i	$0.211185\pi$
$308$	0	0
$309$	−384.000	−0.0706958
$310$	1056.00	0.193473
$311$	−4632.00	−0.844555	−0.422278	−	0.906467i	$-0.638769\pi$
$311$	−4632.00	−0.844555	−0.422278	+	0.906467i	$0.638769\pi$
$312$	912.000	0.165487
$313$	4822.00	0.870785	0.435392	−	0.900241i	$-0.356610\pi$
$313$	4822.00	0.870785	0.435392	+	0.900241i	$0.356610\pi$
$314$	1228.00	0.220701
$315$	0	0
$316$	−2080.00	−0.370282
$317$	−3426.00	−0.607014	−0.303507	−	0.952829i	$-0.598158\pi$
$317$	−3426.00	−0.607014	−0.303507	+	0.952829i	$0.598158\pi$
$318$	−1188.00	−0.209496
$319$	360.000	0.0631854
$320$	−384.000	−0.0670820
$321$	−4428.00	−0.769928
$322$	0	0
$323$	−2520.00	−0.434107
$324$	324.000	0.0555556
$325$	3382.00	0.577230
$326$	3704.00	0.629281
$327$	3570.00	0.603735
$328$	336.000	0.0565625
$329$	0	0
$330$	432.000	0.0720631
$331$	−2788.00	−0.462968	−0.231484	−	0.972839i	$-0.574358\pi$
$331$	−2788.00	−0.462968	−0.231484	+	0.972839i	$0.574358\pi$
$332$	1968.00	0.325325
$333$	2286.00	0.376192
$334$	−4272.00	−0.699861
$335$	−5304.00	−0.865040
$336$	0	0
$337$	434.000	0.0701528	0.0350764	−	0.999385i	$-0.488833\pi$
$337$	434.000	0.0701528	0.0350764	+	0.999385i	$0.488833\pi$
$338$	1506.00	0.242354
$339$	−1386.00	−0.222057
$340$	−3024.00	−0.482351
$341$	1056.00	0.167700
$342$	360.000	0.0569198
$343$	0	0
$344$	416.000	0.0652012
$345$	−3024.00	−0.471903
$346$	3516.00	0.546304
$347$	6684.00	1.03405	0.517026	−	0.855970i	$-0.327039\pi$
$347$	6684.00	1.03405	0.517026	+	0.855970i	$0.327039\pi$
$348$	360.000	0.0554541
$349$	−2630.00	−0.403383	−0.201692	−	0.979449i	$-0.564644\pi$
$349$	−2630.00	−0.403383	−0.201692	+	0.979449i	$0.564644\pi$
$350$	0	0
$351$	−1026.00	−0.156022
$352$	−384.000	−0.0581456
$353$	7422.00	1.11907	0.559537	−	0.828805i	$-0.310979\pi$
$353$	7422.00	1.11907	0.559537	+	0.828805i	$0.310979\pi$
$354$	−3960.00	−0.594553
$355$	−4752.00	−0.710451
$356$	−3240.00	−0.482359
$357$	0	0
$358$	1080.00	0.159441
$359$	−10440.0	−1.53482	−0.767412	−	0.641154i	$-0.778456\pi$
$359$	−10440.0	−1.53482	−0.767412	+	0.641154i	$0.778456\pi$
$360$	432.000	0.0632456
$361$	−6459.00	−0.941682
$362$	3964.00	0.575534
$363$	−3561.00	−0.514887
$364$	0	0
$365$	1308.00	0.187572
$366$	−3228.00	−0.461012
$367$	−10424.0	−1.48264	−0.741319	−	0.671153i	$-0.765800\pi$
$367$	−10424.0	−1.48264	−0.741319	+	0.671153i	$0.765800\pi$
$368$	2688.00	0.380765
$369$	−378.000	−0.0533276
$370$	3048.00	0.428265
$371$	0	0
$372$	1056.00	0.147180
$373$	3278.00	0.455036	0.227518	−	0.973774i	$-0.426939\pi$
$373$	3278.00	0.455036	0.227518	+	0.973774i	$0.426939\pi$
$374$	−3024.00	−0.418094
$375$	3852.00	0.530444
$376$	−768.000	−0.105337
$377$	−1140.00	−0.155737
$378$	0	0
$379$	6140.00	0.832165	0.416083	−	0.909327i	$-0.363403\pi$
$379$	6140.00	0.832165	0.416083	+	0.909327i	$0.363403\pi$
$380$	480.000	0.0647986
$381$	−7608.00	−1.02302
$382$	5376.00	0.720053
$383$	3072.00	0.409848	0.204924	−	0.978778i	$-0.434305\pi$
$383$	3072.00	0.409848	0.204924	+	0.978778i	$0.434305\pi$
$384$	−384.000	−0.0510310
$385$	0	0
$386$	4604.00	0.607092
$387$	−468.000	−0.0614723
$388$	−4616.00	−0.603974
$389$	6150.00	0.801587	0.400794	−	0.916168i	$-0.368734\pi$
$389$	6150.00	0.801587	0.400794	+	0.916168i	$0.368734\pi$
$390$	−1368.00	−0.177619
$391$	21168.0	2.73788
$392$	0	0
$393$	−6876.00	−0.882566
$394$	−8748.00	−1.11857
$395$	3120.00	0.397428
$396$	432.000	0.0548202
$397$	106.000	0.0134005	0.00670024	−	0.999978i	$-0.497867\pi$
$397$	106.000	0.0134005	0.00670024	+	0.999978i	$0.497867\pi$
$398$	−3200.00	−0.403019
$399$	0	0
$400$	−1424.00	−0.178000
$401$	−1758.00	−0.218929	−0.109464	−	0.993991i	$-0.534914\pi$
$401$	−1758.00	−0.218929	−0.109464	+	0.993991i	$0.534914\pi$
$402$	−5304.00	−0.658058
$403$	−3344.00	−0.413341
$404$	2472.00	0.304422
$405$	−486.000	−0.0596285
$406$	0	0
$407$	3048.00	0.371213
$408$	−3024.00	−0.366937
$409$	3670.00	0.443691	0.221846	−	0.975082i	$-0.428792\pi$
$409$	3670.00	0.443691	0.221846	+	0.975082i	$0.428792\pi$
$410$	−504.000	−0.0607092
$411$	−2178.00	−0.261394
$412$	−512.000	−0.0612243
$413$	0	0
$414$	−3024.00	−0.358989
$415$	−2952.00	−0.349176
$416$	1216.00	0.143316
$417$	−1140.00	−0.133875
$418$	480.000	0.0561664
$419$	9660.00	1.12631	0.563153	−	0.826353i	$-0.309588\pi$
$419$	9660.00	1.12631	0.563153	+	0.826353i	$0.309588\pi$
$420$	0	0
$421$	8462.00	0.979602	0.489801	−	0.871834i	$-0.337069\pi$
$421$	8462.00	0.979602	0.489801	+	0.871834i	$0.337069\pi$
$422$	−6664.00	−0.768717
$423$	864.000	0.0993123
$424$	−1584.00	−0.181429
$425$	−11214.0	−1.27990
$426$	−4752.00	−0.540458
$427$	0	0
$428$	−5904.00	−0.666777
$429$	−1368.00	−0.153957
$430$	−624.000	−0.0699813
$431$	9792.00	1.09435	0.547174	−	0.837019i	$-0.315704\pi$
$431$	9792.00	1.09435	0.547174	+	0.837019i	$0.315704\pi$
$432$	432.000	0.0481125
$433$	7342.00	0.814859	0.407430	−	0.913237i	$-0.366425\pi$
$433$	7342.00	0.814859	0.407430	+	0.913237i	$0.366425\pi$
$434$	0	0
$435$	−540.000	−0.0595196
$436$	4760.00	0.522850
$437$	−3360.00	−0.367805
$438$	1308.00	0.142691
$439$	−10640.0	−1.15676	−0.578382	−	0.815766i	$-0.696316\pi$
$439$	−10640.0	−1.15676	−0.578382	+	0.815766i	$0.696316\pi$
$440$	576.000	0.0624085
$441$	0	0
$442$	9576.00	1.03051
$443$	−17412.0	−1.86742	−0.933712	−	0.358024i	$-0.883451\pi$
$443$	−17412.0	−1.86742	−0.933712	+	0.358024i	$0.883451\pi$
$444$	3048.00	0.325792
$445$	4860.00	0.517722
$446$	5296.00	0.562271
$447$	4770.00	0.504728
$448$	0	0
$449$	−1710.00	−0.179732	−0.0898662	−	0.995954i	$-0.528644\pi$
$449$	−1710.00	−0.179732	−0.0898662	+	0.995954i	$0.528644\pi$
$450$	1602.00	0.167820
$451$	−504.000	−0.0526218
$452$	−1848.00	−0.192307
$453$	7296.00	0.756724
$454$	4488.00	0.463948
$455$	0	0
$456$	480.000	0.0492940
$457$	−646.000	−0.0661239	−0.0330619	−	0.999453i	$-0.510526\pi$
$457$	−646.000	−0.0661239	−0.0330619	+	0.999453i	$0.510526\pi$
$458$	−11300.0	−1.15287
$459$	3402.00	0.345952
$460$	−4032.00	−0.408680
$461$	6018.00	0.607996	0.303998	−	0.952673i	$-0.401678\pi$
$461$	6018.00	0.607996	0.303998	+	0.952673i	$0.401678\pi$
$462$	0	0
$463$	−6712.00	−0.673722	−0.336861	−	0.941554i	$-0.609365\pi$
$463$	−6712.00	−0.673722	−0.336861	+	0.941554i	$0.609365\pi$
$464$	480.000	0.0480247
$465$	−1584.00	−0.157970
$466$	−9396.00	−0.934037
$467$	−5364.00	−0.531512	−0.265756	−	0.964040i	$-0.585622\pi$
$467$	−5364.00	−0.531512	−0.265756	+	0.964040i	$0.585622\pi$
$468$	−1368.00	−0.135119
$469$	0	0
$470$	1152.00	0.113059
$471$	−1842.00	−0.180201
$472$	−5280.00	−0.514898
$473$	−624.000	−0.0606587
$474$	3120.00	0.302334
$475$	1780.00	0.171941
$476$	0	0
$477$	1782.00	0.171053
$478$	2400.00	0.229652
$479$	−9840.00	−0.938624	−0.469312	−	0.883032i	$-0.655498\pi$
$479$	−9840.00	−0.938624	−0.469312	+	0.883032i	$0.655498\pi$
$480$	576.000	0.0547723
$481$	−9652.00	−0.914955
$482$	−1436.00	−0.135701
$483$	0	0
$484$	−4748.00	−0.445905
$485$	6924.00	0.648253
$486$	−486.000	−0.0453609
$487$	1424.00	0.132500	0.0662501	−	0.997803i	$-0.478896\pi$
$487$	1424.00	0.132500	0.0662501	+	0.997803i	$0.478896\pi$
$488$	−4304.00	−0.399248
$489$	−5556.00	−0.513806
$490$	0	0
$491$	−4548.00	−0.418021	−0.209011	−	0.977913i	$-0.567024\pi$
$491$	−4548.00	−0.418021	−0.209011	+	0.977913i	$0.567024\pi$
$492$	−504.000	−0.0461831
$493$	3780.00	0.345320
$494$	−1520.00	−0.138437
$495$	−648.000	−0.0588393
$496$	1408.00	0.127462
$497$	0	0
$498$	−2952.00	−0.265627
$499$	6500.00	0.583126	0.291563	−	0.956552i	$-0.405825\pi$
$499$	6500.00	0.583126	0.291563	+	0.956552i	$0.405825\pi$
$500$	5136.00	0.459378
$501$	6408.00	0.571434
$502$	12024.0	1.06904
$503$	−12168.0	−1.07862	−0.539308	−	0.842108i	$-0.681314\pi$
$503$	−12168.0	−1.07862	−0.539308	+	0.842108i	$0.681314\pi$
$504$	0	0
$505$	−3708.00	−0.326740
$506$	−4032.00	−0.354238
$507$	−2259.00	−0.197881
$508$	−10144.0	−0.885959
$509$	21090.0	1.83654	0.918269	−	0.395957i	$-0.129587\pi$
$509$	21090.0	1.83654	0.918269	+	0.395957i	$0.129587\pi$
$510$	4536.00	0.393838
$511$	0	0
$512$	−512.000	−0.0441942
$513$	−540.000	−0.0464748
$514$	−4092.00	−0.351149
$515$	768.000	0.0657129
$516$	−624.000	−0.0532366
$517$	1152.00	0.0979979
$518$	0	0
$519$	−5274.00	−0.446056
$520$	−1824.00	−0.153822
$521$	5238.00	0.440462	0.220231	−	0.975448i	$-0.429319\pi$
$521$	5238.00	0.440462	0.220231	+	0.975448i	$0.429319\pi$
$522$	−540.000	−0.0452781
$523$	−8588.00	−0.718025	−0.359012	−	0.933333i	$-0.616886\pi$
$523$	−8588.00	−0.718025	−0.359012	+	0.933333i	$0.616886\pi$
$524$	−9168.00	−0.764324
$525$	0	0
$526$	12144.0	1.00666
$527$	11088.0	0.916510
$528$	576.000	0.0474757
$529$	16057.0	1.31972
$530$	2376.00	0.194730
$531$	5940.00	0.485450
$532$	0	0
$533$	1596.00	0.129701
$534$	4860.00	0.393844
$535$	8856.00	0.715660
$536$	−7072.00	−0.569895
$537$	−1620.00	−0.130183
$538$	−13860.0	−1.11068
$539$	0	0
$540$	−648.000	−0.0516398
$541$	3062.00	0.243338	0.121669	−	0.992571i	$-0.461175\pi$
$541$	3062.00	0.243338	0.121669	+	0.992571i	$0.461175\pi$
$542$	2704.00	0.214293
$543$	−5946.00	−0.469921
$544$	−4032.00	−0.317777
$545$	−7140.00	−0.561182
$546$	0	0
$547$	−8476.00	−0.662537	−0.331268	−	0.943537i	$-0.607477\pi$
$547$	−8476.00	−0.662537	−0.331268	+	0.943537i	$0.607477\pi$
$548$	−2904.00	−0.226374
$549$	4842.00	0.376414
$550$	2136.00	0.165599
$551$	−600.000	−0.0463899
$552$	−4032.00	−0.310894
$553$	0	0
$554$	2372.00	0.181907
$555$	−4572.00	−0.349677
$556$	−1520.00	−0.115939
$557$	−12546.0	−0.954383	−0.477191	−	0.878799i	$-0.658345\pi$
$557$	−12546.0	−0.954383	−0.477191	+	0.878799i	$0.658345\pi$
$558$	−1584.00	−0.120172
$559$	1976.00	0.149510
$560$	0	0
$561$	4536.00	0.341373
$562$	−4884.00	−0.366582
$563$	12.0000	0.000898294 0	0.000449147	−	1.00000i	$-0.499857\pi$
$563$	12.0000	0.000898294 0	0.000449147		1.00000i	$0.499857\pi$
$564$	1152.00	0.0860070
$565$	2772.00	0.206405
$566$	5656.00	0.420034
$567$	0	0
$568$	−6336.00	−0.468050
$569$	19290.0	1.42123	0.710614	−	0.703582i	$-0.248417\pi$
$569$	19290.0	1.42123	0.710614	+	0.703582i	$0.248417\pi$
$570$	−720.000	−0.0529079
$571$	−12148.0	−0.890329	−0.445165	−	0.895449i	$-0.646855\pi$
$571$	−12148.0	−0.890329	−0.445165	+	0.895449i	$0.646855\pi$
$572$	−1824.00	−0.133331
$573$	−8064.00	−0.587920
$574$	0	0
$575$	−14952.0	−1.08442
$576$	576.000	0.0416667
$577$	10366.0	0.747907	0.373953	−	0.927447i	$-0.378002\pi$
$577$	10366.0	0.747907	0.373953	+	0.927447i	$0.378002\pi$
$578$	−21926.0	−1.57786
$579$	−6906.00	−0.495688
$580$	−720.000	−0.0515455
$581$	0	0
$582$	6924.00	0.493143
$583$	2376.00	0.168789
$584$	1744.00	0.123574
$585$	2052.00	0.145025
$586$	9516.00	0.670823
$587$	−7644.00	−0.537482	−0.268741	−	0.963213i	$-0.586607\pi$
$587$	−7644.00	−0.537482	−0.268741	+	0.963213i	$0.586607\pi$
$588$	0	0
$589$	−1760.00	−0.123123
$590$	7920.00	0.552646
$591$	13122.0	0.913311
$592$	4064.00	0.282144
$593$	−8658.00	−0.599564	−0.299782	−	0.954008i	$-0.596914\pi$
$593$	−8658.00	−0.599564	−0.299782	+	0.954008i	$0.596914\pi$
$594$	−648.000	−0.0447605
$595$	0	0
$596$	6360.00	0.437107
$597$	4800.00	0.329064
$598$	12768.0	0.873114
$599$	25800.0	1.75987	0.879933	−	0.475098i	$-0.157587\pi$
$599$	25800.0	1.75987	0.879933	+	0.475098i	$0.157587\pi$
$600$	2136.00	0.145336
$601$	−16202.0	−1.09966	−0.549828	−	0.835278i	$-0.685307\pi$
$601$	−16202.0	−1.09966	−0.549828	+	0.835278i	$0.685307\pi$
$602$	0	0
$603$	7956.00	0.537302
$604$	9728.00	0.655342
$605$	7122.00	0.478596
$606$	−3708.00	−0.248560
$607$	24136.0	1.61392	0.806960	−	0.590605i	$-0.201111\pi$
$607$	24136.0	1.61392	0.806960	+	0.590605i	$0.201111\pi$
$608$	640.000	0.0426898
$609$	0	0
$610$	6456.00	0.428518
$611$	−3648.00	−0.241542
$612$	4536.00	0.299603
$613$	−4642.00	−0.305854	−0.152927	−	0.988237i	$-0.548870\pi$
$613$	−4642.00	−0.305854	−0.152927	+	0.988237i	$0.548870\pi$
$614$	−16952.0	−1.11421
$615$	756.000	0.0495689
$616$	0	0
$617$	−6726.00	−0.438863	−0.219432	−	0.975628i	$-0.570420\pi$
$617$	−6726.00	−0.438863	−0.219432	+	0.975628i	$0.570420\pi$
$618$	768.000	0.0499895
$619$	21220.0	1.37787	0.688937	−	0.724821i	$-0.258078\pi$
$619$	21220.0	1.37787	0.688937	+	0.724821i	$0.258078\pi$
$620$	−2112.00	−0.136806
$621$	4536.00	0.293113
$622$	9264.00	0.597191
$623$	0	0
$624$	−1824.00	−0.117017
$625$	3421.00	0.218944
$626$	−9644.00	−0.615738
$627$	−720.000	−0.0458597
$628$	−2456.00	−0.156059
$629$	32004.0	2.02875
$630$	0	0
$631$	29792.0	1.87956	0.939779	−	0.341783i	$-0.111031\pi$
$631$	29792.0	1.87956	0.939779	+	0.341783i	$0.111031\pi$
$632$	4160.00	0.261829
$633$	9996.00	0.627655
$634$	6852.00	0.429223
$635$	15216.0	0.950911
$636$	2376.00	0.148136
$637$	0	0
$638$	−720.000	−0.0446788
$639$	7128.00	0.441282
$640$	768.000	0.0474342
$641$	−10158.0	−0.625923	−0.312962	−	0.949766i	$-0.601321\pi$
$641$	−10158.0	−0.625923	−0.312962	+	0.949766i	$0.601321\pi$
$642$	8856.00	0.544421
$643$	−29828.0	−1.82940	−0.914698	−	0.404138i	$-0.867571\pi$
$643$	−29828.0	−1.82940	−0.914698	+	0.404138i	$0.867571\pi$
$644$	0	0
$645$	936.000	0.0571395
$646$	5040.00	0.306960
$647$	−1944.00	−0.118124	−0.0590622	−	0.998254i	$-0.518811\pi$
$647$	−1944.00	−0.118124	−0.0590622	+	0.998254i	$0.518811\pi$
$648$	−648.000	−0.0392837
$649$	7920.00	0.479025
$650$	−6764.00	−0.408163
$651$	0	0
$652$	−7408.00	−0.444969
$653$	26718.0	1.60116	0.800579	−	0.599227i	$-0.204525\pi$
$653$	26718.0	1.60116	0.800579	+	0.599227i	$0.204525\pi$
$654$	−7140.00	−0.426905
$655$	13752.0	0.820359
$656$	−672.000	−0.0399957
$657$	−1962.00	−0.116507
$658$	0	0
$659$	4260.00	0.251815	0.125907	−	0.992042i	$-0.459816\pi$
$659$	4260.00	0.251815	0.125907	+	0.992042i	$0.459816\pi$
$660$	−864.000	−0.0509563
$661$	−22862.0	−1.34528	−0.672639	−	0.739971i	$-0.734839\pi$
$661$	−22862.0	−1.34528	−0.672639	+	0.739971i	$0.734839\pi$
$662$	5576.00	0.327368
$663$	−14364.0	−0.841405
$664$	−3936.00	−0.230040
$665$	0	0
$666$	−4572.00	−0.266008
$667$	5040.00	0.292578
$668$	8544.00	0.494876
$669$	−7944.00	−0.459092
$670$	10608.0	0.611676
$671$	6456.00	0.371432
$672$	0	0
$673$	−32542.0	−1.86390	−0.931948	−	0.362592i	$-0.881892\pi$
$673$	−32542.0	−1.86390	−0.931948	+	0.362592i	$0.881892\pi$
$674$	−868.000	−0.0496055
$675$	−2403.00	−0.137024
$676$	−3012.00	−0.171370
$677$	−14214.0	−0.806925	−0.403463	−	0.914996i	$-0.632193\pi$
$677$	−14214.0	−0.806925	−0.403463	+	0.914996i	$0.632193\pi$
$678$	2772.00	0.157018
$679$	0	0
$680$	6048.00	0.341074
$681$	−6732.00	−0.378812
$682$	−2112.00	−0.118582
$683$	−7092.00	−0.397317	−0.198659	−	0.980069i	$-0.563659\pi$
$683$	−7092.00	−0.397317	−0.198659	+	0.980069i	$0.563659\pi$
$684$	−720.000	−0.0402484
$685$	4356.00	0.242970
$686$	0	0
$687$	16950.0	0.941314
$688$	−832.000	−0.0461042
$689$	−7524.00	−0.416026
$690$	6048.00	0.333686
$691$	13228.0	0.728244	0.364122	−	0.931351i	$-0.381369\pi$
$691$	13228.0	0.728244	0.364122	+	0.931351i	$0.381369\pi$
$692$	−7032.00	−0.386296
$693$	0	0
$694$	−13368.0	−0.731185
$695$	2280.00	0.124439
$696$	−720.000	−0.0392120
$697$	−5292.00	−0.287588
$698$	5260.00	0.285235
$699$	14094.0	0.762638
$700$	0	0
$701$	28062.0	1.51196	0.755982	−	0.654592i	$-0.227160\pi$
$701$	28062.0	1.51196	0.755982	+	0.654592i	$0.227160\pi$
$702$	2052.00	0.110324
$703$	−5080.00	−0.272540
$704$	768.000	0.0411152
$705$	−1728.00	−0.0923124
$706$	−14844.0	−0.791305
$707$	0	0
$708$	7920.00	0.420412
$709$	−27250.0	−1.44343	−0.721717	−	0.692188i	$-0.756647\pi$
$709$	−27250.0	−1.44343	−0.721717	+	0.692188i	$0.756647\pi$
$710$	9504.00	0.502364
$711$	−4680.00	−0.246855
$712$	6480.00	0.341079
$713$	14784.0	0.776529
$714$	0	0
$715$	2736.00	0.143106
$716$	−2160.00	−0.112742
$717$	−3600.00	−0.187510
$718$	20880.0	1.08529
$719$	14400.0	0.746912	0.373456	−	0.927648i	$-0.378173\pi$
$719$	14400.0	0.746912	0.373456	+	0.927648i	$0.378173\pi$
$720$	−864.000	−0.0447214
$721$	0	0
$722$	12918.0	0.665870
$723$	2154.00	0.110800
$724$	−7928.00	−0.406964
$725$	−2670.00	−0.136774
$726$	7122.00	0.364080
$727$	−17984.0	−0.917455	−0.458727	−	0.888577i	$-0.651695\pi$
$727$	−17984.0	−0.917455	−0.458727	+	0.888577i	$0.651695\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−2616.00	−0.132634
$731$	−6552.00	−0.331511
$732$	6456.00	0.325984
$733$	−16598.0	−0.836373	−0.418186	−	0.908361i	$-0.637334\pi$
$733$	−16598.0	−0.836373	−0.418186	+	0.908361i	$0.637334\pi$
$734$	20848.0	1.04838
$735$	0	0
$736$	−5376.00	−0.269242
$737$	10608.0	0.530191
$738$	756.000	0.0377083
$739$	1460.00	0.0726752	0.0363376	−	0.999340i	$-0.488431\pi$
$739$	1460.00	0.0726752	0.0363376	+	0.999340i	$0.488431\pi$
$740$	−6096.00	−0.302829
$741$	2280.00	0.113034
$742$	0	0
$743$	−30072.0	−1.48484	−0.742419	−	0.669936i	$-0.766322\pi$
$743$	−30072.0	−1.48484	−0.742419	+	0.669936i	$0.766322\pi$
$744$	−2112.00	−0.104072
$745$	−9540.00	−0.469152
$746$	−6556.00	−0.321759
$747$	4428.00	0.216884
$748$	6048.00	0.295637
$749$	0	0
$750$	−7704.00	−0.375080
$751$	−18088.0	−0.878882	−0.439441	−	0.898271i	$-0.644823\pi$
$751$	−18088.0	−0.878882	−0.439441	+	0.898271i	$0.644823\pi$
$752$	1536.00	0.0744843
$753$	−18036.0	−0.872866
$754$	2280.00	0.110123
$755$	−14592.0	−0.703387
$756$	0	0
$757$	24734.0	1.18755	0.593773	−	0.804633i	$-0.297638\pi$
$757$	24734.0	1.18755	0.593773	+	0.804633i	$0.297638\pi$
$758$	−12280.0	−0.588430
$759$	6048.00	0.289234
$760$	−960.000	−0.0458196
$761$	22278.0	1.06120	0.530602	−	0.847621i	$-0.321966\pi$
$761$	22278.0	1.06120	0.530602	+	0.847621i	$0.321966\pi$
$762$	15216.0	0.723383
$763$	0	0
$764$	−10752.0	−0.509154
$765$	−6804.00	−0.321568
$766$	−6144.00	−0.289806
$767$	−25080.0	−1.18069
$768$	768.000	0.0360844
$769$	−16130.0	−0.756388	−0.378194	−	0.925726i	$-0.623455\pi$
$769$	−16130.0	−0.756388	−0.378194	+	0.925726i	$0.623455\pi$
$770$	0	0
$771$	6138.00	0.286712
$772$	−9208.00	−0.429279
$773$	−29718.0	−1.38277	−0.691386	−	0.722486i	$-0.742999\pi$
$773$	−29718.0	−1.38277	−0.691386	+	0.722486i	$0.742999\pi$
$774$	936.000	0.0434675
$775$	−7832.00	−0.363011
$776$	9232.00	0.427074
$777$	0	0
$778$	−12300.0	−0.566808
$779$	840.000	0.0386343
$780$	2736.00	0.125596
$781$	9504.00	0.435442
$782$	−42336.0	−1.93597
$783$	810.000	0.0369694
$784$	0	0
$785$	3684.00	0.167500
$786$	13752.0	0.624068
$787$	−9524.00	−0.431377	−0.215689	−	0.976462i	$-0.569200\pi$
$787$	−9524.00	−0.431377	−0.215689	+	0.976462i	$0.569200\pi$
$788$	17496.0	0.790951
$789$	−18216.0	−0.821935
$790$	−6240.00	−0.281024
$791$	0	0
$792$	−864.000	−0.0387638
$793$	−20444.0	−0.915495
$794$	−212.000	−0.00947556
$795$	−3564.00	−0.158996
$796$	6400.00	0.284977
$797$	33906.0	1.50692	0.753458	−	0.657496i	$-0.228384\pi$
$797$	33906.0	1.50692	0.753458	+	0.657496i	$0.228384\pi$
$798$	0	0
$799$	12096.0	0.535577
$800$	2848.00	0.125865
$801$	−7290.00	−0.321572
$802$	3516.00	0.154806
$803$	−2616.00	−0.114965
$804$	10608.0	0.465318
$805$	0	0
$806$	6688.00	0.292276
$807$	20790.0	0.906868
$808$	−4944.00	−0.215259
$809$	−630.000	−0.0273790	−0.0136895	−	0.999906i	$-0.504358\pi$
$809$	−630.000	−0.0273790	−0.0136895	+	0.999906i	$0.504358\pi$
$810$	972.000	0.0421637
$811$	20788.0	0.900081	0.450040	−	0.893008i	$-0.351410\pi$
$811$	20788.0	0.900081	0.450040	+	0.893008i	$0.351410\pi$
$812$	0	0
$813$	−4056.00	−0.174969
$814$	−6096.00	−0.262487
$815$	11112.0	0.477591
$816$	6048.00	0.259464
$817$	1040.00	0.0445349
$818$	−7340.00	−0.313737
$819$	0	0
$820$	1008.00	0.0429279
$821$	−43098.0	−1.83207	−0.916036	−	0.401097i	$-0.868629\pi$
$821$	−43098.0	−1.83207	−0.916036	+	0.401097i	$0.868629\pi$
$822$	4356.00	0.184833
$823$	−14272.0	−0.604484	−0.302242	−	0.953231i	$-0.597735\pi$
$823$	−14272.0	−0.604484	−0.302242	+	0.953231i	$0.597735\pi$
$824$	1024.00	0.0432921
$825$	−3204.00	−0.135211
$826$	0	0
$827$	13644.0	0.573698	0.286849	−	0.957976i	$-0.407392\pi$
$827$	13644.0	0.573698	0.286849	+	0.957976i	$0.407392\pi$
$828$	6048.00	0.253844
$829$	2410.00	0.100968	0.0504842	−	0.998725i	$-0.483924\pi$
$829$	2410.00	0.100968	0.0504842	+	0.998725i	$0.483924\pi$
$830$	5904.00	0.246905
$831$	−3558.00	−0.148527
$832$	−2432.00	−0.101339
$833$	0	0
$834$	2280.00	0.0946642
$835$	−12816.0	−0.531157
$836$	−960.000	−0.0397157
$837$	2376.00	0.0981202
$838$	−19320.0	−0.796418
$839$	−23160.0	−0.953006	−0.476503	−	0.879173i	$-0.658096\pi$
$839$	−23160.0	−0.953006	−0.476503	+	0.879173i	$0.658096\pi$
$840$	0	0
$841$	−23489.0	−0.963098
$842$	−16924.0	−0.692684
$843$	7326.00	0.299313
$844$	13328.0	0.543565
$845$	4518.00	0.183934
$846$	−1728.00	−0.0702244
$847$	0	0
$848$	3168.00	0.128290
$849$	−8484.00	−0.342957
$850$	22428.0	0.905028
$851$	42672.0	1.71889
$852$	9504.00	0.382162
$853$	−32078.0	−1.28761	−0.643804	−	0.765190i	$-0.722645\pi$
$853$	−32078.0	−1.28761	−0.643804	+	0.765190i	$0.722645\pi$
$854$	0	0
$855$	1080.00	0.0431991
$856$	11808.0	0.471483
$857$	14406.0	0.574212	0.287106	−	0.957899i	$-0.407307\pi$
$857$	14406.0	0.574212	0.287106	+	0.957899i	$0.407307\pi$
$858$	2736.00	0.108864
$859$	−30620.0	−1.21623	−0.608115	−	0.793849i	$-0.708074\pi$
$859$	−30620.0	−1.21623	−0.608115	+	0.793849i	$0.708074\pi$
$860$	1248.00	0.0494842
$861$	0	0
$862$	−19584.0	−0.773821
$863$	17568.0	0.692957	0.346478	−	0.938058i	$-0.387377\pi$
$863$	17568.0	0.692957	0.346478	+	0.938058i	$0.387377\pi$
$864$	−864.000	−0.0340207
$865$	10548.0	0.414616
$866$	−14684.0	−0.576192
$867$	32889.0	1.28831
$868$	0	0
$869$	−6240.00	−0.243587
$870$	1080.00	0.0420867
$871$	−33592.0	−1.30680
$872$	−9520.00	−0.369711
$873$	−10386.0	−0.402649
$874$	6720.00	0.260077
$875$	0	0
$876$	−2616.00	−0.100898
$877$	−21706.0	−0.835758	−0.417879	−	0.908503i	$-0.637226\pi$
$877$	−21706.0	−0.835758	−0.417879	+	0.908503i	$0.637226\pi$
$878$	21280.0	0.817956
$879$	−14274.0	−0.547725
$880$	−1152.00	−0.0441294
$881$	14958.0	0.572018	0.286009	−	0.958227i	$-0.407671\pi$
$881$	14958.0	0.572018	0.286009	+	0.958227i	$0.407671\pi$
$882$	0	0
$883$	−32812.0	−1.25052	−0.625261	−	0.780415i	$-0.715008\pi$
$883$	−32812.0	−1.25052	−0.625261	+	0.780415i	$0.715008\pi$
$884$	−19152.0	−0.728678
$885$	−11880.0	−0.451234
$886$	34824.0	1.32047
$887$	38856.0	1.47086	0.735432	−	0.677598i	$-0.236979\pi$
$887$	38856.0	1.47086	0.735432	+	0.677598i	$0.236979\pi$
$888$	−6096.00	−0.230370
$889$	0	0
$890$	−9720.00	−0.366084
$891$	972.000	0.0365468
$892$	−10592.0	−0.397586
$893$	−1920.00	−0.0719489
$894$	−9540.00	−0.356896
$895$	3240.00	0.121007
$896$	0	0
$897$	−19152.0	−0.712895
$898$	3420.00	0.127090
$899$	2640.00	0.0979410
$900$	−3204.00	−0.118667
$901$	24948.0	0.922462
$902$	1008.00	0.0372092
$903$	0	0
$904$	3696.00	0.135981
$905$	11892.0	0.436799
$906$	−14592.0	−0.535085
$907$	−28276.0	−1.03516	−0.517579	−	0.855635i	$-0.673167\pi$
$907$	−28276.0	−1.03516	−0.517579	+	0.855635i	$0.673167\pi$
$908$	−8976.00	−0.328061
$909$	5562.00	0.202948
$910$	0	0
$911$	8112.00	0.295019	0.147510	−	0.989061i	$-0.452874\pi$
$911$	8112.00	0.295019	0.147510	+	0.989061i	$0.452874\pi$
$912$	−960.000	−0.0348561
$913$	5904.00	0.214013
$914$	1292.00	0.0467566
$915$	−9684.00	−0.349883
$916$	22600.0	0.815202
$917$	0	0
$918$	−6804.00	−0.244625
$919$	−26080.0	−0.936126	−0.468063	−	0.883695i	$-0.655048\pi$
$919$	−26080.0	−0.936126	−0.468063	+	0.883695i	$0.655048\pi$
$920$	8064.00	0.288981
$921$	25428.0	0.909751
$922$	−12036.0	−0.429918
$923$	−30096.0	−1.07326
$924$	0	0
$925$	−22606.0	−0.803547
$926$	13424.0	0.476393
$927$	−1152.00	−0.0408162
$928$	−960.000	−0.0339586
$929$	−49170.0	−1.73651	−0.868254	−	0.496120i	$-0.834757\pi$
$929$	−49170.0	−1.73651	−0.868254	+	0.496120i	$0.834757\pi$
$930$	3168.00	0.111702
$931$	0	0
$932$	18792.0	0.660464
$933$	−13896.0	−0.487604
$934$	10728.0	0.375836
$935$	−9072.00	−0.317311
$936$	2736.00	0.0955438
$937$	−48314.0	−1.68447	−0.842236	−	0.539110i	$-0.818761\pi$
$937$	−48314.0	−1.68447	−0.842236	+	0.539110i	$0.818761\pi$
$938$	0	0
$939$	14466.0	0.502748
$940$	−2304.00	−0.0799449
$941$	−34782.0	−1.20495	−0.602477	−	0.798137i	$-0.705819\pi$
$941$	−34782.0	−1.20495	−0.602477	+	0.798137i	$0.705819\pi$
$942$	3684.00	0.127422
$943$	−7056.00	−0.243664
$944$	10560.0	0.364088
$945$	0	0
$946$	1248.00	0.0428922
$947$	−25116.0	−0.861838	−0.430919	−	0.902391i	$-0.641810\pi$
$947$	−25116.0	−0.861838	−0.430919	+	0.902391i	$0.641810\pi$
$948$	−6240.00	−0.213782
$949$	8284.00	0.283361
$950$	−3560.00	−0.121581
$951$	−10278.0	−0.350460
$952$	0	0
$953$	−15462.0	−0.525565	−0.262782	−	0.964855i	$-0.584640\pi$
$953$	−15462.0	−0.525565	−0.262782	+	0.964855i	$0.584640\pi$
$954$	−3564.00	−0.120953
$955$	16128.0	0.546481
$956$	−4800.00	−0.162388
$957$	1080.00	0.0364801
$958$	19680.0	0.663708
$959$	0	0
$960$	−1152.00	−0.0387298
$961$	−22047.0	−0.740056
$962$	19304.0	0.646971
$963$	−13284.0	−0.444518
$964$	2872.00	0.0959553
$965$	13812.0	0.460750
$966$	0	0
$967$	−736.000	−0.0244759	−0.0122379	−	0.999925i	$-0.503896\pi$
$967$	−736.000	−0.0244759	−0.0122379	+	0.999925i	$0.503896\pi$
$968$	9496.00	0.315303
$969$	−7560.00	−0.250632
$970$	−13848.0	−0.458384
$971$	29268.0	0.967307	0.483653	−	0.875260i	$-0.339310\pi$
$971$	29268.0	0.967307	0.483653	+	0.875260i	$0.339310\pi$
$972$	972.000	0.0320750
$973$	0	0
$974$	−2848.00	−0.0936918
$975$	10146.0	0.333264
$976$	8608.00	0.282311
$977$	16674.0	0.546007	0.273003	−	0.962013i	$-0.411983\pi$
$977$	16674.0	0.546007	0.273003	+	0.962013i	$0.411983\pi$
$978$	11112.0	0.363316
$979$	−9720.00	−0.317316
$980$	0	0
$981$	10710.0	0.348567
$982$	9096.00	0.295586
$983$	31272.0	1.01467	0.507336	−	0.861749i	$-0.330630\pi$
$983$	31272.0	1.01467	0.507336	+	0.861749i	$0.330630\pi$
$984$	1008.00	0.0326564
$985$	−26244.0	−0.848937
$986$	−7560.00	−0.244178
$987$	0	0
$988$	3040.00	0.0978900
$989$	−8736.00	−0.280878
$990$	1296.00	0.0416056
$991$	−15928.0	−0.510565	−0.255282	−	0.966867i	$-0.582168\pi$
$991$	−15928.0	−0.510565	−0.255282	+	0.966867i	$0.582168\pi$
$992$	−2816.00	−0.0901291
$993$	−8364.00	−0.267295
$994$	0	0
$995$	−9600.00	−0.305870
$996$	5904.00	0.187827
$997$	−42014.0	−1.33460	−0.667300	−	0.744789i	$-0.732550\pi$
$997$	−42014.0	−1.33460	−0.667300	+	0.744789i	$0.732550\pi$
$998$	−13000.0	−0.412332
$999$	6858.00	0.217195

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	294.4.a.e.1.1		1
3.2	odd	2		882.4.a.n.1.1		1
4.3	odd	2		2352.4.a.e.1.1		1
7.2	even	3		294.4.e.g.67.1		2
7.3	odd	6		294.4.e.h.79.1		2
7.4	even	3		294.4.e.g.79.1		2
7.5	odd	6		294.4.e.h.67.1		2
7.6	odd	2		6.4.a.a.1.1	✓	1
21.2	odd	6		882.4.g.f.361.1		2
21.5	even	6		882.4.g.i.361.1		2
21.11	odd	6		882.4.g.f.667.1		2
21.17	even	6		882.4.g.i.667.1		2
21.20	even	2		18.4.a.a.1.1		1
28.27	even	2		48.4.a.c.1.1		1
35.13	even	4		150.4.c.d.49.2		2
35.27	even	4		150.4.c.d.49.1		2
35.34	odd	2		150.4.a.i.1.1		1
56.13	odd	2		192.4.a.i.1.1		1
56.27	even	2		192.4.a.c.1.1		1
63.13	odd	6		162.4.c.f.55.1		2
63.20	even	6		162.4.c.c.109.1		2
63.34	odd	6		162.4.c.f.109.1		2
63.41	even	6		162.4.c.c.55.1		2
77.76	even	2		726.4.a.f.1.1		1
84.83	odd	2		144.4.a.c.1.1		1
91.34	even	4		1014.4.b.d.337.1		2
91.83	even	4		1014.4.b.d.337.2		2
91.90	odd	2		1014.4.a.g.1.1		1
105.62	odd	4		450.4.c.e.199.2		2
105.83	odd	4		450.4.c.e.199.1		2
105.104	even	2		450.4.a.h.1.1		1
112.13	odd	4		768.4.d.n.385.1		2
112.27	even	4		768.4.d.c.385.1		2
112.69	odd	4		768.4.d.n.385.2		2
112.83	even	4		768.4.d.c.385.2		2
119.118	odd	2		1734.4.a.d.1.1		1
133.132	even	2		2166.4.a.i.1.1		1
140.27	odd	4		1200.4.f.j.49.1		2
140.83	odd	4		1200.4.f.j.49.2		2
140.139	even	2		1200.4.a.b.1.1		1
168.83	odd	2		576.4.a.r.1.1		1
168.125	even	2		576.4.a.q.1.1		1
231.230	odd	2		2178.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.4.a.a.1.1	✓	1	7.6	odd	2
18.4.a.a.1.1		1	21.20	even	2
48.4.a.c.1.1		1	28.27	even	2
144.4.a.c.1.1		1	84.83	odd	2
150.4.a.i.1.1		1	35.34	odd	2
150.4.c.d.49.1		2	35.27	even	4
150.4.c.d.49.2		2	35.13	even	4
162.4.c.c.55.1		2	63.41	even	6
162.4.c.c.109.1		2	63.20	even	6
162.4.c.f.55.1		2	63.13	odd	6
162.4.c.f.109.1		2	63.34	odd	6
192.4.a.c.1.1		1	56.27	even	2
192.4.a.i.1.1		1	56.13	odd	2
294.4.a.e.1.1		1	1.1	even	1	trivial
294.4.e.g.67.1		2	7.2	even	3
294.4.e.g.79.1		2	7.4	even	3
294.4.e.h.67.1		2	7.5	odd	6
294.4.e.h.79.1		2	7.3	odd	6
450.4.a.h.1.1		1	105.104	even	2
450.4.c.e.199.1		2	105.83	odd	4
450.4.c.e.199.2		2	105.62	odd	4
576.4.a.q.1.1		1	168.125	even	2
576.4.a.r.1.1		1	168.83	odd	2
726.4.a.f.1.1		1	77.76	even	2
768.4.d.c.385.1		2	112.27	even	4
768.4.d.c.385.2		2	112.83	even	4
768.4.d.n.385.1		2	112.13	odd	4
768.4.d.n.385.2		2	112.69	odd	4
882.4.a.n.1.1		1	3.2	odd	2
882.4.g.f.361.1		2	21.2	odd	6
882.4.g.f.667.1		2	21.11	odd	6
882.4.g.i.361.1		2	21.5	even	6
882.4.g.i.667.1		2	21.17	even	6
1014.4.a.g.1.1		1	91.90	odd	2
1014.4.b.d.337.1		2	91.34	even	4
1014.4.b.d.337.2		2	91.83	even	4
1200.4.a.b.1.1		1	140.139	even	2
1200.4.f.j.49.1		2	140.27	odd	4
1200.4.f.j.49.2		2	140.83	odd	4
1734.4.a.d.1.1		1	119.118	odd	2
2166.4.a.i.1.1		1	133.132	even	2
2178.4.a.e.1.1		1	231.230	odd	2
2352.4.a.e.1.1		1	4.3	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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	294.4.a.e.1.1

Level	$294$
Weight	$4$
Character	294.1
Self dual	yes
Analytic conductor	$17.347$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$