LMFDB - Embedded newform 294.8.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [294,8,Mod(1,294)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(294, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("294.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$N$	$=$	$294 = 2 \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	294.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$91.8411974923$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 42)
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-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +290.000 q^{5} +216.000 q^{6} -512.000 q^{8} +729.000 q^{9} -2320.00 q^{10} +1130.00 q^{11} -1728.00 q^{12} +7563.00 q^{13} -7830.00 q^{15} +4096.00 q^{16} -14504.0 q^{17} -5832.00 q^{18} -25043.0 q^{19} +18560.0 q^{20} -9040.00 q^{22} -6664.00 q^{23} +13824.0 q^{24} +5975.00 q^{25} -60504.0 q^{26} -19683.0 q^{27} +6820.00 q^{29} +62640.0 q^{30} -176079. q^{31} -32768.0 q^{32} -30510.0 q^{33} +116032. q^{34} +46656.0 q^{36} -132985. q^{37} +200344. q^{38} -204201. q^{39} -148480. q^{40} -661206. q^{41} +147095. q^{43} +72320.0 q^{44} +211410. q^{45} +53312.0 q^{46} +899634. q^{47} -110592. q^{48} -47800.0 q^{50} +391608. q^{51} +484032. q^{52} -1.30513e6 q^{53} +157464. q^{54} +327700. q^{55} +676161. q^{57} -54560.0 q^{58} +2.38156e6 q^{59} -501120. q^{60} +2.62510e6 q^{61} +1.40863e6 q^{62} +262144. q^{64} +2.19327e6 q^{65} +244080. q^{66} +3.97339e6 q^{67} -928256. q^{68} +179928. q^{69} -4.29176e6 q^{71} -373248. q^{72} -5.93029e6 q^{73} +1.06388e6 q^{74} -161325. q^{75} -1.60275e6 q^{76} +1.63361e6 q^{78} +6.28369e6 q^{79} +1.18784e6 q^{80} +531441. q^{81} +5.28965e6 q^{82} -4.25225e6 q^{83} -4.20616e6 q^{85} -1.17676e6 q^{86} -184140. q^{87} -578560. q^{88} +1.48080e6 q^{89} -1.69128e6 q^{90} -426496. q^{92} +4.75413e6 q^{93} -7.19707e6 q^{94} -7.26247e6 q^{95} +884736. q^{96} -9.40743e6 q^{97} +823770. 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

)

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

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	64.0000	0.500000
$5$	290.000	1.03754	0.518768	−	0.854915i	$-0.326391\pi$
$5$	290.000	1.03754	0.518768	+	0.854915i	$0.326391\pi$
$6$	216.000	0.408248
$7$	0	0
$7$	0	0
$8$	−512.000	−0.353553
$9$	729.000	0.333333
$10$	−2320.00	−0.733648
$11$	1130.00	0.255979	0.127989	−	0.991776i	$-0.459148\pi$
$11$	1130.00	0.255979	0.127989	+	0.991776i	$0.459148\pi$
$12$	−1728.00	−0.288675
$13$	7563.00	0.954756	0.477378	−	0.878698i	$-0.341587\pi$
$13$	7563.00	0.954756	0.477378	+	0.878698i	$0.341587\pi$
$14$	0	0
$15$	−7830.00	−0.599021
$16$	4096.00	0.250000
$17$	−14504.0	−0.716006	−0.358003	−	0.933720i	$-0.616542\pi$
$17$	−14504.0	−0.716006	−0.358003	+	0.933720i	$0.616542\pi$
$18$	−5832.00	−0.235702
$19$	−25043.0	−0.837623	−0.418812	−	0.908073i	$-0.637553\pi$
$19$	−25043.0	−0.837623	−0.418812	+	0.908073i	$0.637553\pi$
$20$	18560.0	0.518768
$21$	0	0
$22$	−9040.00	−0.181004
$23$	−6664.00	−0.114206	−0.0571028	−	0.998368i	$-0.518186\pi$
$23$	−6664.00	−0.114206	−0.0571028	+	0.998368i	$0.518186\pi$
$24$	13824.0	0.204124
$25$	5975.00	0.0764800
$26$	−60504.0	−0.675114
$27$	−19683.0	−0.192450
$28$	0	0
$29$	6820.00	0.0519268	0.0259634	−	0.999663i	$-0.491735\pi$
$29$	6820.00	0.0519268	0.0259634	+	0.999663i	$0.491735\pi$
$30$	62640.0	0.423572
$31$	−176079.	−1.06155	−0.530776	−	0.847512i	$-0.678100\pi$
$31$	−176079.	−1.06155	−0.530776	+	0.847512i	$0.678100\pi$
$32$	−32768.0	−0.176777
$33$	−30510.0	−0.147789
$34$	116032.	0.506293
$35$	0	0
$36$	46656.0	0.166667
$37$	−132985.	−0.431615	−0.215808	−	0.976436i	$-0.569238\pi$
$37$	−132985.	−0.431615	−0.215808	+	0.976436i	$0.569238\pi$
$38$	200344.	0.592289
$39$	−204201.	−0.551229
$40$	−148480.	−0.366824
$41$	−661206.	−1.49828	−0.749141	−	0.662411i	$-0.769533\pi$
$41$	−661206.	−1.49828	−0.749141	+	0.662411i	$0.769533\pi$
$42$	0	0
$43$	147095.	0.282136	0.141068	−	0.990000i	$-0.454946\pi$
$43$	147095.	0.282136	0.141068	+	0.990000i	$0.454946\pi$
$44$	72320.0	0.127989
$45$	211410.	0.345845
$46$	53312.0	0.0807556
$47$	899634.	1.26393	0.631965	−	0.774997i	$-0.282248\pi$
$47$	899634.	1.26393	0.631965	+	0.774997i	$0.282248\pi$
$48$	−110592.	−0.144338
$49$	0	0
$50$	−47800.0	−0.0540795
$51$	391608.	0.413386
$52$	484032.	0.477378
$53$	−1.30513e6	−1.20417	−0.602087	−	0.798431i	$-0.705664\pi$
$53$	−1.30513e6	−1.20417	−0.602087	+	0.798431i	$0.705664\pi$
$54$	157464.	0.136083
$55$	327700.	0.265587
$56$	0	0
$57$	676161.	0.483602
$58$	−54560.0	−0.0367178
$59$	2.38156e6	1.50966	0.754832	−	0.655918i	$-0.227718\pi$
$59$	2.38156e6	1.50966	0.754832	+	0.655918i	$0.227718\pi$
$60$	−501120.	−0.299511
$61$	2.62510e6	1.48078	0.740392	−	0.672175i	$-0.234640\pi$
$61$	2.62510e6	1.48078	0.740392	+	0.672175i	$0.234640\pi$
$62$	1.40863e6	0.750631
$63$	0	0
$64$	262144.	0.125000
$65$	2.19327e6	0.990593
$66$	244080.	0.104503
$67$	3.97339e6	1.61399	0.806993	−	0.590561i	$-0.201094\pi$
$67$	3.97339e6	1.61399	0.806993	+	0.590561i	$0.201094\pi$
$68$	−928256.	−0.358003
$69$	179928.	0.0659367
$70$	0	0
$71$	−4.29176e6	−1.42309	−0.711543	−	0.702642i	$-0.752004\pi$
$71$	−4.29176e6	−1.42309	−0.711543	+	0.702642i	$0.752004\pi$
$72$	−373248.	−0.117851
$73$	−5.93029e6	−1.78421	−0.892105	−	0.451827i	$-0.850772\pi$
$73$	−5.93029e6	−1.78421	−0.892105	+	0.451827i	$0.850772\pi$
$74$	1.06388e6	0.305198
$75$	−161325.	−0.0441557
$76$	−1.60275e6	−0.418812
$77$	0	0
$78$	1.63361e6	0.389777
$79$	6.28369e6	1.43390	0.716952	−	0.697123i	$-0.245537\pi$
$79$	6.28369e6	1.43390	0.716952	+	0.697123i	$0.245537\pi$
$80$	1.18784e6	0.259384
$81$	531441.	0.111111
$82$	5.28965e6	1.05944
$83$	−4.25225e6	−0.816292	−0.408146	−	0.912917i	$-0.633825\pi$
$83$	−4.25225e6	−0.816292	−0.408146	+	0.912917i	$0.633825\pi$
$84$	0	0
$85$	−4.20616e6	−0.742882
$86$	−1.17676e6	−0.199500
$87$	−184140.	−0.0299799
$88$	−578560.	−0.0905022
$89$	1.48080e6	0.222654	0.111327	−	0.993784i	$-0.464490\pi$
$89$	1.48080e6	0.222654	0.111327	+	0.993784i	$0.464490\pi$
$90$	−1.69128e6	−0.244549
$91$	0	0
$92$	−426496.	−0.0571028
$93$	4.75413e6	0.612888
$94$	−7.19707e6	−0.893734
$95$	−7.26247e6	−0.869064
$96$	884736.	0.102062
$97$	−9.40743e6	−1.04657	−0.523287	−	0.852156i	$-0.675295\pi$
$97$	−9.40743e6	−1.04657	−0.523287	+	0.852156i	$0.675295\pi$
$98$	0	0
$99$	823770.	0.0853263
$100$	382400.	0.0382400
$101$	1.08475e7	1.04762	0.523809	−	0.851835i	$-0.324510\pi$
$101$	1.08475e7	1.04762	0.523809	+	0.851835i	$0.324510\pi$
$102$	−3.13286e6	−0.292308
$103$	202105.	0.0182241	0.00911206	−	0.999958i	$-0.497099\pi$
$103$	202105.	0.0182241	0.00911206	+	0.999958i	$0.497099\pi$
$104$	−3.87226e6	−0.337557
$105$	0	0
$106$	1.04411e7	0.851479
$107$	−1.28697e7	−1.01560	−0.507802	−	0.861474i	$-0.669542\pi$
$107$	−1.28697e7	−1.01560	−0.507802	+	0.861474i	$0.669542\pi$
$108$	−1.25971e6	−0.0962250
$109$	−2.18508e7	−1.61612	−0.808061	−	0.589098i	$-0.799483\pi$
$109$	−2.18508e7	−1.61612	−0.808061	+	0.589098i	$0.799483\pi$
$110$	−2.62160e6	−0.187798
$111$	3.59060e6	0.249193
$112$	0	0
$113$	4.61360e6	0.300792	0.150396	−	0.988626i	$-0.451945\pi$
$113$	4.61360e6	0.300792	0.150396	+	0.988626i	$0.451945\pi$
$114$	−5.40929e6	−0.341958
$115$	−1.93256e6	−0.118492
$116$	436480.	0.0259634
$117$	5.51343e6	0.318252
$118$	−1.90525e7	−1.06749
$119$	0	0
$120$	4.00896e6	0.211786
$121$	−1.82103e7	−0.934475
$122$	−2.10008e7	−1.04707
$123$	1.78526e7	0.865033
$124$	−1.12691e7	−0.530776
$125$	−2.09235e7	−0.958185
$126$	0	0
$127$	−3.70924e7	−1.60684	−0.803419	−	0.595415i	$-0.796988\pi$
$127$	−3.70924e7	−1.60684	−0.803419	+	0.595415i	$0.796988\pi$
$128$	−2.09715e6	−0.0883883
$129$	−3.97156e6	−0.162891
$130$	−1.75462e7	−0.700455
$131$	3.98348e7	1.54815	0.774076	−	0.633093i	$-0.218215\pi$
$131$	3.98348e7	1.54815	0.774076	+	0.633093i	$0.218215\pi$
$132$	−1.95264e6	−0.0738947
$133$	0	0
$134$	−3.17871e7	−1.14126
$135$	−5.70807e6	−0.199674
$136$	7.42605e6	0.253146
$137$	−2.57447e7	−0.855392	−0.427696	−	0.903923i	$-0.640675\pi$
$137$	−2.57447e7	−0.855392	−0.427696	+	0.903923i	$0.640675\pi$
$138$	−1.43942e6	−0.0466243
$139$	1.29485e7	0.408948	0.204474	−	0.978872i	$-0.434452\pi$
$139$	1.29485e7	0.408948	0.204474	+	0.978872i	$0.434452\pi$
$140$	0	0
$141$	−2.42901e7	−0.729731
$142$	3.43341e7	1.00627
$143$	8.54619e6	0.244397
$144$	2.98598e6	0.0833333
$145$	1.97780e6	0.0538759
$146$	4.74423e7	1.26163
$147$	0	0
$148$	−8.51104e6	−0.215808
$149$	−3.13156e7	−0.775549	−0.387775	−	0.921754i	$-0.626756\pi$
$149$	−3.13156e7	−0.775549	−0.387775	+	0.921754i	$0.626756\pi$
$150$	1.29060e6	0.0312228
$151$	−4.12328e7	−0.974592	−0.487296	−	0.873237i	$-0.662017\pi$
$151$	−4.12328e7	−0.974592	−0.487296	+	0.873237i	$0.662017\pi$
$152$	1.28220e7	0.296145
$153$	−1.05734e7	−0.238669
$154$	0	0
$155$	−5.10629e7	−1.10140
$156$	−1.30689e7	−0.275614
$157$	−5.53231e6	−0.114093	−0.0570464	−	0.998372i	$-0.518168\pi$
$157$	−5.53231e6	−0.114093	−0.0570464	+	0.998372i	$0.518168\pi$
$158$	−5.02695e7	−1.01392
$159$	3.52386e7	0.695230
$160$	−9.50272e6	−0.183412
$161$	0	0
$162$	−4.25153e6	−0.0785674
$163$	−3.35309e7	−0.606441	−0.303220	−	0.952920i	$-0.598062\pi$
$163$	−3.35309e7	−0.606441	−0.303220	+	0.952920i	$0.598062\pi$
$164$	−4.23172e7	−0.749141
$165$	−8.84790e6	−0.153337
$166$	3.40180e7	0.577206
$167$	5.23132e7	0.869167	0.434584	−	0.900632i	$-0.356896\pi$
$167$	5.23132e7	0.869167	0.434584	+	0.900632i	$0.356896\pi$
$168$	0	0
$169$	−5.54955e6	−0.0884411
$170$	3.36493e7	0.525297
$171$	−1.82563e7	−0.279208
$172$	9.41408e6	0.141068
$173$	8.20425e7	1.20470	0.602348	−	0.798234i	$-0.294232\pi$
$173$	8.20425e7	1.20470	0.602348	+	0.798234i	$0.294232\pi$
$174$	1.47312e6	0.0211990
$175$	0	0
$176$	4.62848e6	0.0639947
$177$	−6.43022e7	−0.871605
$178$	−1.18464e7	−0.157440
$179$	−1.14630e8	−1.49387	−0.746934	−	0.664898i	$-0.768475\pi$
$179$	−1.14630e8	−1.49387	−0.746934	+	0.664898i	$0.768475\pi$
$180$	1.35302e7	0.172923
$181$	−5.29000e7	−0.663102	−0.331551	−	0.943437i	$-0.607572\pi$
$181$	−5.29000e7	−0.663102	−0.331551	+	0.943437i	$0.607572\pi$
$182$	0	0
$183$	−7.08778e7	−0.854931
$184$	3.41197e6	0.0403778
$185$	−3.85656e7	−0.447816
$186$	−3.80331e7	−0.433377
$187$	−1.63895e7	−0.183282
$188$	5.75766e7	0.631965
$189$	0	0
$190$	5.80998e7	0.614521
$191$	1.02847e8	1.06801	0.534006	−	0.845480i	$-0.320686\pi$
$191$	1.02847e8	1.06801	0.534006	+	0.845480i	$0.320686\pi$
$192$	−7.07789e6	−0.0721688
$193$	−1.75242e8	−1.75464	−0.877318	−	0.479910i	$-0.840669\pi$
$193$	−1.75242e8	−1.75464	−0.877318	+	0.479910i	$0.840669\pi$
$194$	7.52595e7	0.740040
$195$	−5.92183e7	−0.571919
$196$	0	0
$197$	1.57220e8	1.46513	0.732563	−	0.680700i	$-0.238324\pi$
$197$	1.57220e8	1.46513	0.732563	+	0.680700i	$0.238324\pi$
$198$	−6.59016e6	−0.0603348
$199$	−1.10748e8	−0.996212	−0.498106	−	0.867116i	$-0.665971\pi$
$199$	−1.10748e8	−0.996212	−0.498106	+	0.867116i	$0.665971\pi$
$200$	−3.05920e6	−0.0270398
$201$	−1.07282e8	−0.931835
$202$	−8.67797e7	−0.740778
$203$	0	0
$204$	2.50629e7	0.206693
$205$	−1.91750e8	−1.55452
$206$	−1.61684e6	−0.0128864
$207$	−4.85806e6	−0.0380685
$208$	3.09780e7	0.238689
$209$	−2.82986e7	−0.214414
$210$	0	0
$211$	−1.46673e8	−1.07488	−0.537440	−	0.843302i	$-0.680609\pi$
$211$	−1.46673e8	−1.07488	−0.537440	+	0.843302i	$0.680609\pi$
$212$	−8.35284e7	−0.602087
$213$	1.15878e8	0.821620
$214$	1.02957e8	0.718140
$215$	4.26576e7	0.292726
$216$	1.00777e7	0.0680414
$217$	0	0
$218$	1.74806e8	1.14277
$219$	1.60118e8	1.03011
$220$	2.09728e7	0.132794
$221$	−1.09694e8	−0.683611
$222$	−2.87248e7	−0.176206
$223$	5.90964e7	0.356857	0.178428	−	0.983953i	$-0.442899\pi$
$223$	5.90964e7	0.356857	0.178428	+	0.983953i	$0.442899\pi$
$224$	0	0
$225$	4.35578e6	0.0254933
$226$	−3.69088e7	−0.212692
$227$	4.29052e7	0.243455	0.121728	−	0.992564i	$-0.461157\pi$
$227$	4.29052e7	0.243455	0.121728	+	0.992564i	$0.461157\pi$
$228$	4.32743e7	0.241801
$229$	−2.76252e8	−1.52013	−0.760066	−	0.649845i	$-0.774834\pi$
$229$	−2.76252e8	−1.52013	−0.760066	+	0.649845i	$0.774834\pi$
$230$	1.54605e7	0.0837868
$231$	0	0
$232$	−3.49184e6	−0.0183589
$233$	1.53341e8	0.794169	0.397085	−	0.917782i	$-0.370022\pi$
$233$	1.53341e8	0.794169	0.397085	+	0.917782i	$0.370022\pi$
$234$	−4.41074e7	−0.225038
$235$	2.60894e8	1.31137
$236$	1.52420e8	0.754832
$237$	−1.69660e8	−0.827864
$238$	0	0
$239$	−3.45622e8	−1.63760	−0.818801	−	0.574078i	$-0.805361\pi$
$239$	−3.45622e8	−1.63760	−0.818801	+	0.574078i	$0.805361\pi$
$240$	−3.20717e7	−0.149755
$241$	−5.40667e7	−0.248812	−0.124406	−	0.992231i	$-0.539702\pi$
$241$	−5.40667e7	−0.248812	−0.124406	+	0.992231i	$0.539702\pi$
$242$	1.45682e8	0.660773
$243$	−1.43489e7	−0.0641500
$244$	1.68007e8	0.740392
$245$	0	0
$246$	−1.42820e8	−0.611671
$247$	−1.89400e8	−0.799726
$248$	9.01524e7	0.375316
$249$	1.14811e8	0.471286
$250$	1.67388e8	0.677539
$251$	9.40011e7	0.375210	0.187605	−	0.982245i	$-0.439927\pi$
$251$	9.40011e7	0.375210	0.187605	+	0.982245i	$0.439927\pi$
$252$	0	0
$253$	−7.53032e6	−0.0292342
$254$	2.96739e8	1.13621
$255$	1.13566e8	0.428903
$256$	1.67772e7	0.0625000
$257$	−6.98001e7	−0.256502	−0.128251	−	0.991742i	$-0.540936\pi$
$257$	−6.98001e7	−0.256502	−0.128251	+	0.991742i	$0.540936\pi$
$258$	3.17725e7	0.115181
$259$	0	0
$260$	1.40369e8	0.495297
$261$	4.97178e6	0.0173089
$262$	−3.18679e8	−1.09471
$263$	7.13209e7	0.241753	0.120876	−	0.992668i	$-0.461430\pi$
$263$	7.13209e7	0.241753	0.120876	+	0.992668i	$0.461430\pi$
$264$	1.56211e7	0.0522515
$265$	−3.78488e8	−1.24937
$266$	0	0
$267$	−3.99816e7	−0.128550
$268$	2.54297e8	0.806993
$269$	2.17505e8	0.681297	0.340648	−	0.940191i	$-0.389353\pi$
$269$	2.17505e8	0.681297	0.340648	+	0.940191i	$0.389353\pi$
$270$	4.56646e7	0.141191
$271$	−1.51037e7	−0.0460990	−0.0230495	−	0.999734i	$-0.507338\pi$
$271$	−1.51037e7	−0.0460990	−0.0230495	+	0.999734i	$0.507338\pi$
$272$	−5.94084e7	−0.179001
$273$	0	0
$274$	2.05957e8	0.604853
$275$	6.75175e6	0.0195773
$276$	1.15154e7	0.0329683
$277$	1.63805e8	0.463072	0.231536	−	0.972826i	$-0.425625\pi$
$277$	1.63805e8	0.463072	0.231536	+	0.972826i	$0.425625\pi$
$278$	−1.03588e8	−0.289170
$279$	−1.28362e8	−0.353851
$280$	0	0
$281$	−4.21200e8	−1.13244	−0.566221	−	0.824253i	$-0.691595\pi$
$281$	−4.21200e8	−1.13244	−0.566221	+	0.824253i	$0.691595\pi$
$282$	1.94321e8	0.515998
$283$	6.72003e8	1.76246	0.881229	−	0.472689i	$-0.156717\pi$
$283$	6.72003e8	1.76246	0.881229	+	0.472689i	$0.156717\pi$
$284$	−2.74673e8	−0.711543
$285$	1.96087e8	0.501754
$286$	−6.83695e7	−0.172815
$287$	0	0
$288$	−2.38879e7	−0.0589256
$289$	−1.99973e8	−0.487336
$290$	−1.58224e7	−0.0380960
$291$	2.54001e8	0.604240
$292$	−3.79539e8	−0.892105
$293$	−771192.	−0.00179112	−0.000895562	−	1.00000i	$-0.500285\pi$
$293$	−771192.	−0.00179112	−0.000895562		1.00000i	$0.500285\pi$
$294$	0	0
$295$	6.90654e8	1.56633
$296$	6.80883e7	0.152599
$297$	−2.22418e7	−0.0492631
$298$	2.50525e8	0.548396
$299$	−5.03998e7	−0.109039
$300$	−1.03248e7	−0.0220779
$301$	0	0
$302$	3.29862e8	0.689141
$303$	−2.92881e8	−0.604843
$304$	−1.02576e8	−0.209406
$305$	7.61280e8	1.53637
$306$	8.45873e7	0.168764
$307$	−2.79387e8	−0.551089	−0.275544	−	0.961288i	$-0.588858\pi$
$307$	−2.79387e8	−0.551089	−0.275544	+	0.961288i	$0.588858\pi$
$308$	0	0
$309$	−5.45684e6	−0.0105217
$310$	4.08503e8	0.778807
$311$	7.21550e8	1.36021	0.680103	−	0.733116i	$-0.261935\pi$
$311$	7.21550e8	1.36021	0.680103	+	0.733116i	$0.261935\pi$
$312$	1.04551e8	0.194889
$313$	−9.91405e8	−1.82745	−0.913726	−	0.406330i	$-0.866808\pi$
$313$	−9.91405e8	−1.82745	−0.913726	+	0.406330i	$0.866808\pi$
$314$	4.42585e7	0.0806758
$315$	0	0
$316$	4.02156e8	0.716952
$317$	−1.20414e8	−0.212310	−0.106155	−	0.994350i	$-0.533854\pi$
$317$	−1.20414e8	−0.212310	−0.106155	+	0.994350i	$0.533854\pi$
$318$	−2.81909e8	−0.491602
$319$	7.70660e6	0.0132922
$320$	7.60218e7	0.129692
$321$	3.47481e8	0.586359
$322$	0	0
$323$	3.63224e8	0.599743
$324$	3.40122e7	0.0555556
$325$	4.51889e7	0.0730197
$326$	2.68247e8	0.428818
$327$	5.89971e8	0.933069
$328$	3.38537e8	0.529722
$329$	0	0
$330$	7.07832e7	0.108425
$331$	1.89531e8	0.287265	0.143632	−	0.989631i	$-0.454122\pi$
$331$	1.89531e8	0.287265	0.143632	+	0.989631i	$0.454122\pi$
$332$	−2.72144e8	−0.408146
$333$	−9.69461e7	−0.143872
$334$	−4.18505e8	−0.614594
$335$	1.15228e9	1.67457
$336$	0	0
$337$	9.06750e8	1.29057	0.645287	−	0.763940i	$-0.276738\pi$
$337$	9.06750e8	1.29057	0.645287	+	0.763940i	$0.276738\pi$
$338$	4.43964e7	0.0625373
$339$	−1.24567e8	−0.173662
$340$	−2.69194e8	−0.371441
$341$	−1.98969e8	−0.271735
$342$	1.46051e8	0.197430
$343$	0	0
$344$	−7.53126e7	−0.0997501
$345$	5.21791e7	0.0684116
$346$	−6.56340e8	−0.851849
$347$	−3.45121e8	−0.443423	−0.221712	−	0.975112i	$-0.571164\pi$
$347$	−3.45121e8	−0.443423	−0.221712	+	0.975112i	$0.571164\pi$
$348$	−1.17850e7	−0.0149900
$349$	6.01221e8	0.757086	0.378543	−	0.925584i	$-0.376425\pi$
$349$	6.01221e8	0.757086	0.378543	+	0.925584i	$0.376425\pi$
$350$	0	0
$351$	−1.48863e8	−0.183743
$352$	−3.70278e7	−0.0452511
$353$	−2.91547e8	−0.352774	−0.176387	−	0.984321i	$-0.556441\pi$
$353$	−2.91547e8	−0.352774	−0.176387	+	0.984321i	$0.556441\pi$
$354$	5.14418e8	0.616318
$355$	−1.24461e9	−1.47650
$356$	9.47712e7	0.111327
$357$	0	0
$358$	9.17039e8	1.05632
$359$	−4.75589e8	−0.542502	−0.271251	−	0.962509i	$-0.587437\pi$
$359$	−4.75589e8	−0.542502	−0.271251	+	0.962509i	$0.587437\pi$
$360$	−1.08242e8	−0.122275
$361$	−2.66720e8	−0.298387
$362$	4.23200e8	0.468884
$363$	4.91677e8	0.539519
$364$	0	0
$365$	−1.71978e9	−1.85118
$366$	5.67022e8	0.604528
$367$	−1.65079e8	−0.174326	−0.0871629	−	0.996194i	$-0.527780\pi$
$367$	−1.65079e8	−0.174326	−0.0871629	+	0.996194i	$0.527780\pi$
$368$	−2.72957e7	−0.0285514
$369$	−4.82019e8	−0.499427
$370$	3.08525e8	0.316654
$371$	0	0
$372$	3.04265e8	0.306444
$373$	4.66848e8	0.465795	0.232897	−	0.972501i	$-0.425179\pi$
$373$	4.66848e8	0.465795	0.232897	+	0.972501i	$0.425179\pi$
$374$	1.31116e8	0.129600
$375$	5.64934e8	0.553208
$376$	−4.60613e8	−0.446867
$377$	5.15797e7	0.0495774
$378$	0	0
$379$	−2.77160e8	−0.261513	−0.130756	−	0.991415i	$-0.541741\pi$
$379$	−2.77160e8	−0.261513	−0.130756	+	0.991415i	$0.541741\pi$
$380$	−4.64798e8	−0.434532
$381$	1.00149e9	0.927708
$382$	−8.22779e8	−0.755199
$383$	7.97883e6	0.00725677	0.00362839	−	0.999993i	$-0.498845\pi$
$383$	7.97883e6	0.00725677	0.00362839	+	0.999993i	$0.498845\pi$
$384$	5.66231e7	0.0510310
$385$	0	0
$386$	1.40193e9	1.24071
$387$	1.07232e8	0.0940453
$388$	−6.02076e8	−0.523287
$389$	1.72774e9	1.48817	0.744087	−	0.668082i	$-0.232885\pi$
$389$	1.72774e9	1.48817	0.744087	+	0.668082i	$0.232885\pi$
$390$	4.73746e8	0.404408
$391$	9.66547e7	0.0817719
$392$	0	0
$393$	−1.07554e9	−0.893826
$394$	−1.25776e9	−1.03600
$395$	1.82227e9	1.48773
$396$	5.27213e7	0.0426631
$397$	−7.92498e8	−0.635669	−0.317835	−	0.948146i	$-0.602956\pi$
$397$	−7.92498e8	−0.635669	−0.317835	+	0.948146i	$0.602956\pi$
$398$	8.85988e8	0.704429
$399$	0	0
$400$	2.44736e7	0.0191200
$401$	−6.95185e8	−0.538387	−0.269194	−	0.963086i	$-0.586757\pi$
$401$	−6.95185e8	−0.538387	−0.269194	+	0.963086i	$0.586757\pi$
$402$	8.58253e8	0.658907
$403$	−1.33169e9	−1.01352
$404$	6.94238e8	0.523809
$405$	1.54118e8	0.115282
$406$	0	0
$407$	−1.50273e8	−0.110484
$408$	−2.00503e8	−0.146154
$409$	−1.96607e9	−1.42091	−0.710456	−	0.703741i	$-0.751511\pi$
$409$	−1.96607e9	−1.42091	−0.710456	+	0.703741i	$0.751511\pi$
$410$	1.53400e9	1.09921
$411$	6.95106e8	0.493861
$412$	1.29347e7	0.00911206
$413$	0	0
$414$	3.88644e7	0.0269185
$415$	−1.23315e9	−0.846932
$416$	−2.47824e8	−0.168779
$417$	−3.49609e8	−0.236106
$418$	2.26389e8	0.151613
$419$	−2.20900e9	−1.46706	−0.733529	−	0.679658i	$-0.762128\pi$
$419$	−2.20900e9	−1.46706	−0.733529	+	0.679658i	$0.762128\pi$
$420$	0	0
$421$	−4.59955e8	−0.300419	−0.150210	−	0.988654i	$-0.547995\pi$
$421$	−4.59955e8	−0.300419	−0.150210	+	0.988654i	$0.547995\pi$
$422$	1.17338e9	0.760056
$423$	6.55833e8	0.421310
$424$	6.68228e8	0.425739
$425$	−8.66614e7	−0.0547601
$426$	−9.27021e8	−0.580973
$427$	0	0
$428$	−8.23658e8	−0.507802
$429$	−2.30747e8	−0.141103
$430$	−3.41260e8	−0.206989
$431$	−2.15517e9	−1.29662	−0.648308	−	0.761378i	$-0.724523\pi$
$431$	−2.15517e9	−1.29662	−0.648308	+	0.761378i	$0.724523\pi$
$432$	−8.06216e7	−0.0481125
$433$	−1.78833e9	−1.05862	−0.529311	−	0.848428i	$-0.677550\pi$
$433$	−1.78833e9	−1.05862	−0.529311	+	0.848428i	$0.677550\pi$
$434$	0	0
$435$	−5.34006e7	−0.0311053
$436$	−1.39845e9	−0.808061
$437$	1.66887e8	0.0956613
$438$	−1.28094e9	−0.728401
$439$	−1.52100e8	−0.0858035	−0.0429017	−	0.999079i	$-0.513660\pi$
$439$	−1.52100e8	−0.0858035	−0.0429017	+	0.999079i	$0.513660\pi$
$440$	−1.67782e8	−0.0938992
$441$	0	0
$442$	8.77550e8	0.483386
$443$	2.69618e9	1.47345	0.736727	−	0.676190i	$-0.236370\pi$
$443$	2.69618e9	1.47345	0.736727	+	0.676190i	$0.236370\pi$
$444$	2.29798e8	0.124597
$445$	4.29432e8	0.231012
$446$	−4.72771e8	−0.252336
$447$	8.45522e8	0.447764
$448$	0	0
$449$	−5.37859e8	−0.280418	−0.140209	−	0.990122i	$-0.544777\pi$
$449$	−5.37859e8	−0.280418	−0.140209	+	0.990122i	$0.544777\pi$
$450$	−3.48462e7	−0.0180265
$451$	−7.47163e8	−0.383528
$452$	2.95271e8	0.150396
$453$	1.11328e9	0.562681
$454$	−3.43241e8	−0.172149
$455$	0	0
$456$	−3.46194e8	−0.170979
$457$	−5.38793e8	−0.264068	−0.132034	−	0.991245i	$-0.542151\pi$
$457$	−5.38793e8	−0.264068	−0.132034	+	0.991245i	$0.542151\pi$
$458$	2.21002e9	1.07490
$459$	2.85482e8	0.137795
$460$	−1.23684e8	−0.0592462
$461$	−5.24391e6	−0.00249288	−0.00124644	−	0.999999i	$-0.500397\pi$
$461$	−5.24391e6	−0.00249288	−0.00124644	+	0.999999i	$0.500397\pi$
$462$	0	0
$463$	2.18426e8	0.102275	0.0511376	−	0.998692i	$-0.483715\pi$
$463$	2.18426e8	0.102275	0.0511376	+	0.998692i	$0.483715\pi$
$464$	2.79347e7	0.0129817
$465$	1.37870e9	0.635893
$466$	−1.22673e9	−0.561562
$467$	3.49442e8	0.158769	0.0793845	−	0.996844i	$-0.474705\pi$
$467$	3.49442e8	0.158769	0.0793845	+	0.996844i	$0.474705\pi$
$468$	3.52859e8	0.159126
$469$	0	0
$470$	−2.08715e9	−0.927281
$471$	1.49372e8	0.0658715
$472$	−1.21936e9	−0.533747
$473$	1.66217e8	0.0722208
$474$	1.35728e9	0.585389
$475$	−1.49632e8	−0.0640614
$476$	0	0
$477$	−9.51441e8	−0.401391
$478$	2.76497e9	1.15796
$479$	−4.05022e9	−1.68385	−0.841927	−	0.539591i	$-0.818579\pi$
$479$	−4.05022e9	−1.68385	−0.841927	+	0.539591i	$0.818579\pi$
$480$	2.56573e8	0.105893
$481$	−1.00577e9	−0.412087
$482$	4.32534e8	0.175936
$483$	0	0
$484$	−1.16546e9	−0.467237
$485$	−2.72816e9	−1.08586
$486$	1.14791e8	0.0453609
$487$	9.26634e8	0.363544	0.181772	−	0.983341i	$-0.441817\pi$
$487$	9.26634e8	0.363544	0.181772	+	0.983341i	$0.441817\pi$
$488$	−1.34405e9	−0.523536
$489$	9.05334e8	0.350129
$490$	0	0
$491$	2.65906e9	1.01378	0.506889	−	0.862011i	$-0.330795\pi$
$491$	2.65906e9	1.01378	0.506889	+	0.862011i	$0.330795\pi$
$492$	1.14256e9	0.432516
$493$	−9.89173e7	−0.0371799
$494$	1.51520e9	0.565492
$495$	2.38893e8	0.0885290
$496$	−7.21220e8	−0.265388
$497$	0	0
$498$	−9.18486e8	−0.333250
$499$	−3.16496e9	−1.14029	−0.570147	−	0.821543i	$-0.693114\pi$
$499$	−3.16496e9	−1.14029	−0.570147	+	0.821543i	$0.693114\pi$
$500$	−1.33910e9	−0.479092
$501$	−1.41246e9	−0.501814
$502$	−7.52009e8	−0.265314
$503$	−4.12431e9	−1.44499	−0.722493	−	0.691379i	$-0.757004\pi$
$503$	−4.12431e9	−1.44499	−0.722493	+	0.691379i	$0.757004\pi$
$504$	0	0
$505$	3.14576e9	1.08694
$506$	6.02426e7	0.0206717
$507$	1.49838e8	0.0510615
$508$	−2.37391e9	−0.803419
$509$	5.55132e9	1.86588	0.932940	−	0.360031i	$-0.117234\pi$
$509$	5.55132e9	1.86588	0.932940	+	0.360031i	$0.117234\pi$
$510$	−9.08531e8	−0.303280
$511$	0	0
$512$	−1.34218e8	−0.0441942
$513$	4.92921e8	0.161201
$514$	5.58401e8	0.181374
$515$	5.86104e7	0.0189082
$516$	−2.54180e8	−0.0814456
$517$	1.01659e9	0.323540
$518$	0	0
$519$	−2.21515e9	−0.695532
$520$	−1.12295e9	−0.350228
$521$	−2.14145e9	−0.663400	−0.331700	−	0.943385i	$-0.607622\pi$
$521$	−2.14145e9	−0.663400	−0.331700	+	0.943385i	$0.607622\pi$
$522$	−3.97742e7	−0.0122393
$523$	−1.96040e9	−0.599224	−0.299612	−	0.954061i	$-0.596857\pi$
$523$	−1.96040e9	−0.599224	−0.299612	+	0.954061i	$0.596857\pi$
$524$	2.54943e9	0.774076
$525$	0	0
$526$	−5.70567e8	−0.170945
$527$	2.55385e9	0.760078
$528$	−1.24969e8	−0.0369474
$529$	−3.36042e9	−0.986957
$530$	3.02791e9	0.883440
$531$	1.73616e9	0.503221
$532$	0	0
$533$	−5.00070e9	−1.43049
$534$	3.19853e8	0.0908983
$535$	−3.73220e9	−1.05372
$536$	−2.03438e9	−0.570630
$537$	3.09501e9	0.862485
$538$	−1.74004e9	−0.481749
$539$	0	0
$540$	−3.65316e8	−0.0998369
$541$	4.03245e9	1.09491	0.547455	−	0.836835i	$-0.315597\pi$
$541$	4.03245e9	1.09491	0.547455	+	0.836835i	$0.315597\pi$
$542$	1.20830e8	0.0325969
$543$	1.42830e9	0.382842
$544$	4.75267e8	0.126573
$545$	−6.33673e9	−1.67678
$546$	0	0
$547$	4.81827e9	1.25874	0.629369	−	0.777107i	$-0.283313\pi$
$547$	4.81827e9	1.25874	0.629369	+	0.777107i	$0.283313\pi$
$548$	−1.64766e9	−0.427696
$549$	1.91370e9	0.493595
$550$	−5.40140e7	−0.0138432
$551$	−1.70793e8	−0.0434951
$552$	−9.21231e7	−0.0233121
$553$	0	0
$554$	−1.31044e9	−0.327442
$555$	1.04127e9	0.258547
$556$	8.28704e8	0.204474
$557$	−2.47986e9	−0.608043	−0.304022	−	0.952665i	$-0.598330\pi$
$557$	−2.47986e9	−0.608043	−0.304022	+	0.952665i	$0.598330\pi$
$558$	1.02689e9	0.250210
$559$	1.11248e9	0.269371
$560$	0	0
$561$	4.42517e8	0.105818
$562$	3.36960e9	0.800758
$563$	4.70161e9	1.11037	0.555184	−	0.831728i	$-0.312648\pi$
$563$	4.70161e9	1.11037	0.555184	+	0.831728i	$0.312648\pi$
$564$	−1.55457e9	−0.364865
$565$	1.33794e9	0.312082
$566$	−5.37603e9	−1.24625
$567$	0	0
$568$	2.19738e9	0.503137
$569$	1.41800e9	0.322688	0.161344	−	0.986898i	$-0.448417\pi$
$569$	1.41800e9	0.322688	0.161344	+	0.986898i	$0.448417\pi$
$570$	−1.56869e9	−0.354794
$571$	−8.49814e9	−1.91028	−0.955141	−	0.296152i	$-0.904297\pi$
$571$	−8.49814e9	−1.91028	−0.955141	+	0.296152i	$0.904297\pi$
$572$	5.46956e8	0.122199
$573$	−2.77688e9	−0.616617
$574$	0	0
$575$	−3.98174e7	−0.00873445
$576$	1.91103e8	0.0416667
$577$	5.31460e9	1.15174	0.575871	−	0.817541i	$-0.304663\pi$
$577$	5.31460e9	1.15174	0.575871	+	0.817541i	$0.304663\pi$
$578$	1.59978e9	0.344598
$579$	4.73153e9	1.01304
$580$	1.26579e8	0.0269379
$581$	0	0
$582$	−2.03201e9	−0.427262
$583$	−1.47480e9	−0.308243
$584$	3.03631e9	0.630814
$585$	1.59889e9	0.330198
$586$	6.16954e6	0.00126652
$587$	6.24032e9	1.27343	0.636713	−	0.771101i	$-0.280294\pi$
$587$	6.24032e9	1.27343	0.636713	+	0.771101i	$0.280294\pi$
$588$	0	0
$589$	4.40955e9	0.889181
$590$	−5.52523e9	−1.10756
$591$	−4.24493e9	−0.845890
$592$	−5.44707e8	−0.107904
$593$	6.71126e9	1.32164	0.660820	−	0.750545i	$-0.270209\pi$
$593$	6.71126e9	1.32164	0.660820	+	0.750545i	$0.270209\pi$
$594$	1.77934e8	0.0348343
$595$	0	0
$596$	−2.00420e9	−0.387775
$597$	2.99021e9	0.575163
$598$	4.03199e8	0.0771019
$599$	5.85019e9	1.11218	0.556091	−	0.831121i	$-0.312300\pi$
$599$	5.85019e9	1.11218	0.556091	+	0.831121i	$0.312300\pi$
$600$	8.25984e7	0.0156114
$601$	1.23472e9	0.232010	0.116005	−	0.993249i	$-0.462991\pi$
$601$	1.23472e9	0.232010	0.116005	+	0.993249i	$0.462991\pi$
$602$	0	0
$603$	2.89660e9	0.537995
$604$	−2.63890e9	−0.487296
$605$	−5.28098e9	−0.969551
$606$	2.34305e9	0.427689
$607$	−7.48797e8	−0.135895	−0.0679475	−	0.997689i	$-0.521645\pi$
$607$	−7.48797e8	−0.135895	−0.0679475	+	0.997689i	$0.521645\pi$
$608$	8.20609e8	0.148072
$609$	0	0
$610$	−6.09024e9	−1.08637
$611$	6.80393e9	1.20675
$612$	−6.76699e8	−0.119334
$613$	−9.73268e9	−1.70656	−0.853279	−	0.521455i	$-0.825389\pi$
$613$	−9.73268e9	−1.70656	−0.853279	+	0.521455i	$0.825389\pi$
$614$	2.23509e9	0.389678
$615$	5.17724e9	0.897502
$616$	0	0
$617$	−9.09192e9	−1.55832	−0.779161	−	0.626823i	$-0.784355\pi$
$617$	−9.09192e9	−1.55832	−0.779161	+	0.626823i	$0.784355\pi$
$618$	4.36547e7	0.00743997
$619$	−2.54761e9	−0.431733	−0.215866	−	0.976423i	$-0.569258\pi$
$619$	−2.54761e9	−0.431733	−0.215866	+	0.976423i	$0.569258\pi$
$620$	−3.26803e9	−0.550699
$621$	1.31168e8	0.0219789
$622$	−5.77240e9	−0.961811
$623$	0	0
$624$	−8.36407e8	−0.137807
$625$	−6.53461e9	−1.07063
$626$	7.93124e9	1.29220
$627$	7.64062e8	0.123792
$628$	−3.54068e8	−0.0570464
$629$	1.92881e9	0.309039
$630$	0	0
$631$	−3.43067e9	−0.543595	−0.271798	−	0.962354i	$-0.587618\pi$
$631$	−3.43067e9	−0.543595	−0.271798	+	0.962354i	$0.587618\pi$
$632$	−3.21725e9	−0.506961
$633$	3.96016e9	0.620583
$634$	9.63314e8	0.150126
$635$	−1.07568e10	−1.66715
$636$	2.25527e9	0.347615
$637$	0	0
$638$	−6.16528e7	−0.00939897
$639$	−3.12869e9	−0.474362
$640$	−6.08174e8	−0.0917061
$641$	4.41127e9	0.661546	0.330773	−	0.943710i	$-0.392691\pi$
$641$	4.41127e9	0.661546	0.330773	+	0.943710i	$0.392691\pi$
$642$	−2.77985e9	−0.414618
$643$	5.39024e9	0.799595	0.399797	−	0.916604i	$-0.369081\pi$
$643$	5.39024e9	0.799595	0.399797	+	0.916604i	$0.369081\pi$
$644$	0	0
$645$	−1.15175e9	−0.169005
$646$	−2.90579e9	−0.424082
$647$	−2.52488e9	−0.366501	−0.183251	−	0.983066i	$-0.558662\pi$
$647$	−2.52488e9	−0.366501	−0.183251	+	0.983066i	$0.558662\pi$
$648$	−2.72098e8	−0.0392837
$649$	2.69117e9	0.386442
$650$	−3.61511e8	−0.0516327
$651$	0	0
$652$	−2.14598e9	−0.303220
$653$	−5.91356e8	−0.0831099	−0.0415550	−	0.999136i	$-0.513231\pi$
$653$	−5.91356e8	−0.0831099	−0.0415550	+	0.999136i	$0.513231\pi$
$654$	−4.71977e9	−0.659779
$655$	1.15521e10	1.60626
$656$	−2.70830e9	−0.374570
$657$	−4.32318e9	−0.594737
$658$	0	0
$659$	−7.80244e9	−1.06202	−0.531009	−	0.847366i	$-0.678187\pi$
$659$	−7.80244e9	−1.06202	−0.531009	+	0.847366i	$0.678187\pi$
$660$	−5.66266e8	−0.0766684
$661$	−1.12132e10	−1.51017	−0.755086	−	0.655626i	$-0.772405\pi$
$661$	−1.12132e10	−1.51017	−0.755086	+	0.655626i	$0.772405\pi$
$662$	−1.51625e9	−0.203127
$663$	2.96173e9	0.394683
$664$	2.17715e9	0.288603
$665$	0	0
$666$	7.75569e8	0.101733
$667$	−4.54485e7	−0.00593033
$668$	3.34804e9	0.434584
$669$	−1.59560e9	−0.206031
$670$	−9.21827e9	−1.18410
$671$	2.96637e9	0.379049
$672$	0	0
$673$	1.09343e10	1.38273	0.691365	−	0.722506i	$-0.257010\pi$
$673$	1.09343e10	1.38273	0.691365	+	0.722506i	$0.257010\pi$
$674$	−7.25400e9	−0.912573
$675$	−1.17606e8	−0.0147186
$676$	−3.55171e8	−0.0442206
$677$	−6.17913e9	−0.765362	−0.382681	−	0.923881i	$-0.624999\pi$
$677$	−6.17913e9	−0.765362	−0.382681	+	0.923881i	$0.624999\pi$
$678$	9.96538e8	0.122798
$679$	0	0
$680$	2.15355e9	0.262648
$681$	−1.15844e9	−0.140559
$682$	1.59175e9	0.192146
$683$	−1.11149e10	−1.33486	−0.667428	−	0.744674i	$-0.732605\pi$
$683$	−1.11149e10	−1.33486	−0.667428	+	0.744674i	$0.732605\pi$
$684$	−1.16841e9	−0.139604
$685$	−7.46595e9	−0.887499
$686$	0	0
$687$	7.45881e9	0.877649
$688$	6.02501e8	0.0705340
$689$	−9.87071e9	−1.14969
$690$	−4.17433e8	−0.0483743
$691$	−1.09613e10	−1.26383	−0.631916	−	0.775037i	$-0.717731\pi$
$691$	−1.09613e10	−1.26383	−0.631916	+	0.775037i	$0.717731\pi$
$692$	5.25072e9	0.602348
$693$	0	0
$694$	2.76097e9	0.313548
$695$	3.75506e9	0.424298
$696$	9.42797e7	0.0105995
$697$	9.59013e9	1.07278
$698$	−4.80977e9	−0.535341
$699$	−4.14021e9	−0.458514
$700$	0	0
$701$	1.27383e10	1.39669	0.698344	−	0.715762i	$-0.253920\pi$
$701$	1.27383e10	1.39669	0.698344	+	0.715762i	$0.253920\pi$
$702$	1.19090e9	0.129926
$703$	3.33034e9	0.361531
$704$	2.96223e8	0.0319974
$705$	−7.04413e9	−0.757122
$706$	2.33238e9	0.249449
$707$	0	0
$708$	−4.11534e9	−0.435802
$709$	9.08191e8	0.0957007	0.0478504	−	0.998855i	$-0.484763\pi$
$709$	9.08191e8	0.0957007	0.0478504	+	0.998855i	$0.484763\pi$
$710$	9.95689e9	1.04405
$711$	4.58081e9	0.477968
$712$	−7.58170e8	−0.0787202
$713$	1.17339e9	0.121235
$714$	0	0
$715$	2.47840e9	0.253571
$716$	−7.33631e9	−0.746934
$717$	9.33178e9	0.945469
$718$	3.80471e9	0.383607
$719$	1.18602e10	1.18999	0.594994	−	0.803730i	$-0.297155\pi$
$719$	1.18602e10	1.18999	0.594994	+	0.803730i	$0.297155\pi$
$720$	8.65935e8	0.0864613
$721$	0	0
$722$	2.13376e9	0.210992
$723$	1.45980e9	0.143651
$724$	−3.38560e9	−0.331551
$725$	4.07495e7	0.00397136
$726$	−3.93342e9	−0.381498
$727$	8.01213e9	0.773352	0.386676	−	0.922216i	$-0.373623\pi$
$727$	8.01213e9	0.773352	0.386676	+	0.922216i	$0.373623\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	1.37583e10	1.30898
$731$	−2.13347e9	−0.202011
$732$	−4.53618e9	−0.427466
$733$	9.80967e9	0.920005	0.460003	−	0.887918i	$-0.347848\pi$
$733$	9.80967e9	0.920005	0.460003	+	0.887918i	$0.347848\pi$
$734$	1.32064e9	0.123267
$735$	0	0
$736$	2.18366e8	0.0201889
$737$	4.48993e9	0.413146
$738$	3.85615e9	0.353148
$739$	1.59250e10	1.45152	0.725762	−	0.687945i	$-0.241487\pi$
$739$	1.59250e10	1.45152	0.725762	+	0.687945i	$0.241487\pi$
$740$	−2.46820e9	−0.223908
$741$	5.11381e9	0.461722
$742$	0	0
$743$	1.85101e10	1.65558	0.827788	−	0.561042i	$-0.189599\pi$
$743$	1.85101e10	1.65558	0.827788	+	0.561042i	$0.189599\pi$
$744$	−2.43412e9	−0.216689
$745$	−9.08154e9	−0.804660
$746$	−3.73479e9	−0.329367
$747$	−3.09989e9	−0.272097
$748$	−1.04893e9	−0.0916412
$749$	0	0
$750$	−4.51948e9	−0.391177
$751$	−1.53951e10	−1.32631	−0.663154	−	0.748483i	$-0.730782\pi$
$751$	−1.53951e10	−1.32631	−0.663154	+	0.748483i	$0.730782\pi$
$752$	3.68490e9	0.315983
$753$	−2.53803e9	−0.216628
$754$	−4.12637e8	−0.0350565
$755$	−1.19575e10	−1.01117
$756$	0	0
$757$	5.14693e9	0.431233	0.215617	−	0.976478i	$-0.430824\pi$
$757$	5.14693e9	0.431233	0.215617	+	0.976478i	$0.430824\pi$
$758$	2.21728e9	0.184917
$759$	2.03319e8	0.0168784
$760$	3.71838e9	0.307261
$761$	2.00227e9	0.164694	0.0823469	−	0.996604i	$-0.473758\pi$
$761$	2.00227e9	0.164694	0.0823469	+	0.996604i	$0.473758\pi$
$762$	−8.01196e9	−0.655988
$763$	0	0
$764$	6.58223e9	0.534006
$765$	−3.06629e9	−0.247627
$766$	−6.38306e7	−0.00513131
$767$	1.80118e10	1.44136
$768$	−4.52985e8	−0.0360844
$769$	9.62216e9	0.763010	0.381505	−	0.924367i	$-0.375406\pi$
$769$	9.62216e9	0.763010	0.381505	+	0.924367i	$0.375406\pi$
$770$	0	0
$771$	1.88460e9	0.148091
$772$	−1.12155e10	−0.877318
$773$	9.13035e8	0.0710983	0.0355491	−	0.999368i	$-0.488682\pi$
$773$	9.13035e8	0.0710983	0.0355491	+	0.999368i	$0.488682\pi$
$774$	−8.57858e8	−0.0665001
$775$	−1.05207e9	−0.0811876
$776$	4.81661e9	0.370020
$777$	0	0
$778$	−1.38219e10	−1.05230
$779$	1.65586e10	1.25500
$780$	−3.78997e9	−0.285960
$781$	−4.84969e9	−0.364280
$782$	−7.73237e8	−0.0578215
$783$	−1.34238e8	−0.00999331
$784$	0	0
$785$	−1.60437e9	−0.118375
$786$	8.60433e9	0.632030
$787$	1.62505e10	1.18838	0.594189	−	0.804325i	$-0.297473\pi$
$787$	1.62505e10	1.18838	0.594189	+	0.804325i	$0.297473\pi$
$788$	1.00621e10	0.732563
$789$	−1.92566e9	−0.139576
$790$	−1.45782e10	−1.05198
$791$	0	0
$792$	−4.21770e8	−0.0301674
$793$	1.98536e10	1.41379
$794$	6.33998e9	0.449486
$795$	1.02192e10	0.721325
$796$	−7.08790e9	−0.498106
$797$	2.00600e10	1.40355	0.701774	−	0.712400i	$-0.252392\pi$
$797$	2.00600e10	1.40355	0.701774	+	0.712400i	$0.252392\pi$
$798$	0	0
$799$	−1.30483e10	−0.904982
$800$	−1.95789e8	−0.0135199
$801$	1.07950e9	0.0742182
$802$	5.56148e9	0.380697
$803$	−6.70123e9	−0.456720
$804$	−6.86602e9	−0.465918
$805$	0	0
$806$	1.06535e10	0.716670
$807$	−5.87263e9	−0.393347
$808$	−5.55390e9	−0.370389
$809$	−1.78197e10	−1.18326	−0.591630	−	0.806209i	$-0.701515\pi$
$809$	−1.78197e10	−1.18326	−0.591630	+	0.806209i	$0.701515\pi$
$810$	−1.23294e9	−0.0815165
$811$	−4.72659e9	−0.311154	−0.155577	−	0.987824i	$-0.549724\pi$
$811$	−4.72659e9	−0.311154	−0.155577	+	0.987824i	$0.549724\pi$
$812$	0	0
$813$	4.07801e8	0.0266153
$814$	1.20218e9	0.0781242
$815$	−9.72396e9	−0.629204
$816$	1.60403e9	0.103347
$817$	−3.68370e9	−0.236324
$818$	1.57286e10	1.00474
$819$	0	0
$820$	−1.22720e10	−0.777260
$821$	2.00031e10	1.26153	0.630764	−	0.775975i	$-0.282742\pi$
$821$	2.00031e10	1.26153	0.630764	+	0.775975i	$0.282742\pi$
$822$	−5.56084e9	−0.349212
$823$	1.53836e10	0.961962	0.480981	−	0.876731i	$-0.340281\pi$
$823$	1.53836e10	0.961962	0.480981	+	0.876731i	$0.340281\pi$
$824$	−1.03478e8	−0.00644320
$825$	−1.82297e8	−0.0113029
$826$	0	0
$827$	1.42246e10	0.874520	0.437260	−	0.899335i	$-0.355949\pi$
$827$	1.42246e10	0.874520	0.437260	+	0.899335i	$0.355949\pi$
$828$	−3.10916e8	−0.0190343
$829$	2.17248e10	1.32439	0.662194	−	0.749332i	$-0.269625\pi$
$829$	2.17248e10	1.32439	0.662194	+	0.749332i	$0.269625\pi$
$830$	9.86522e9	0.598871
$831$	−4.42274e9	−0.267355
$832$	1.98260e9	0.119344
$833$	0	0
$834$	2.79688e9	0.166952
$835$	1.51708e10	0.901792
$836$	−1.81111e9	−0.107207
$837$	3.46576e9	0.204296
$838$	1.76720e10	1.03737
$839$	1.54623e10	0.903871	0.451935	−	0.892051i	$-0.350734\pi$
$839$	1.54623e10	0.903871	0.451935	+	0.892051i	$0.350734\pi$
$840$	0	0
$841$	−1.72034e10	−0.997304
$842$	3.67964e9	0.212429
$843$	1.13724e10	0.653816
$844$	−9.38705e9	−0.537440
$845$	−1.60937e9	−0.0917608
$846$	−5.24667e9	−0.297911
$847$	0	0
$848$	−5.34582e9	−0.301043
$849$	−1.81441e10	−1.01756
$850$	6.93291e8	0.0387213
$851$	8.86212e8	0.0492929
$852$	7.41616e9	0.410810
$853$	1.80110e10	0.993608	0.496804	−	0.867863i	$-0.334507\pi$
$853$	1.80110e10	0.993608	0.496804	+	0.867863i	$0.334507\pi$
$854$	0	0
$855$	−5.29434e9	−0.289688
$856$	6.58927e9	0.359070
$857$	1.71715e8	0.00931916	0.00465958	−	0.999989i	$-0.498517\pi$
$857$	1.71715e8	0.00931916	0.00465958	+	0.999989i	$0.498517\pi$
$858$	1.84598e9	0.0997748
$859$	−1.37786e9	−0.0741700	−0.0370850	−	0.999312i	$-0.511807\pi$
$859$	−1.37786e9	−0.0741700	−0.0370850	+	0.999312i	$0.511807\pi$
$860$	2.73008e9	0.146363
$861$	0	0
$862$	1.72414e10	0.916847
$863$	9.44605e7	0.00500279	0.00250140	−	0.999997i	$-0.499204\pi$
$863$	9.44605e7	0.00500279	0.00250140	+	0.999997i	$0.499204\pi$
$864$	6.44973e8	0.0340207
$865$	2.37923e10	1.24992
$866$	1.43067e10	0.748559
$867$	5.39926e9	0.281363
$868$	0	0
$869$	7.10057e9	0.367049
$870$	4.27205e8	0.0219947
$871$	3.00508e10	1.54096
$872$	1.11876e10	0.571386
$873$	−6.85802e9	−0.348858
$874$	−1.33509e9	−0.0676428
$875$	0	0
$876$	1.02475e10	0.515057
$877$	−1.37376e10	−0.687722	−0.343861	−	0.939021i	$-0.611735\pi$
$877$	−1.37376e10	−0.687722	−0.343861	+	0.939021i	$0.611735\pi$
$878$	1.21680e9	0.0606722
$879$	2.08222e7	0.00103411
$880$	1.34226e9	0.0663968
$881$	−1.09605e10	−0.540026	−0.270013	−	0.962857i	$-0.587028\pi$
$881$	−1.09605e10	−0.540026	−0.270013	+	0.962857i	$0.587028\pi$
$882$	0	0
$883$	7.24742e9	0.354259	0.177129	−	0.984188i	$-0.443319\pi$
$883$	7.24742e9	0.354259	0.177129	+	0.984188i	$0.443319\pi$
$884$	−7.02040e9	−0.341805
$885$	−1.86476e10	−0.904321
$886$	−2.15695e10	−1.04189
$887$	−1.12154e10	−0.539613	−0.269807	−	0.962915i	$-0.586960\pi$
$887$	−1.12154e10	−0.539613	−0.269807	+	0.962915i	$0.586960\pi$
$888$	−1.83838e9	−0.0881031
$889$	0	0
$890$	−3.43546e9	−0.163350
$891$	6.00528e8	0.0284421
$892$	3.78217e9	0.178428
$893$	−2.25295e10	−1.05870
$894$	−6.76418e9	−0.316617
$895$	−3.32427e10	−1.54994
$896$	0	0
$897$	1.36080e9	0.0629534
$898$	4.30287e9	0.198285
$899$	−1.20086e9	−0.0551230
$900$	2.78770e8	0.0127467
$901$	1.89296e10	0.862195
$902$	5.97730e9	0.271195
$903$	0	0
$904$	−2.36216e9	−0.106346
$905$	−1.53410e10	−0.687992
$906$	−8.90627e9	−0.397876
$907$	−6.12250e9	−0.272460	−0.136230	−	0.990677i	$-0.543499\pi$
$907$	−6.12250e9	−0.272460	−0.136230	+	0.990677i	$0.543499\pi$
$908$	2.74593e9	0.121728
$909$	7.90780e9	0.349206
$910$	0	0
$911$	1.72053e10	0.753960	0.376980	−	0.926221i	$-0.376963\pi$
$911$	1.72053e10	0.753960	0.376980	+	0.926221i	$0.376963\pi$
$912$	2.76956e9	0.120901
$913$	−4.80504e9	−0.208953
$914$	4.31035e9	0.186724
$915$	−2.05545e10	−0.887021
$916$	−1.76801e10	−0.760066
$917$	0	0
$918$	−2.28386e9	−0.0974361
$919$	2.48590e10	1.05652	0.528261	−	0.849082i	$-0.322844\pi$
$919$	2.48590e10	1.05652	0.528261	+	0.849082i	$0.322844\pi$
$920$	9.89471e8	0.0418934
$921$	7.54344e9	0.318171
$922$	4.19513e7	0.00176273
$923$	−3.24586e10	−1.35870
$924$	0	0
$925$	−7.94585e8	−0.0330099
$926$	−1.74741e9	−0.0723195
$927$	1.47335e8	0.00607471
$928$	−2.23478e8	−0.00917944
$929$	−1.75978e10	−0.720115	−0.360058	−	0.932930i	$-0.617243\pi$
$929$	−1.75978e10	−0.720115	−0.360058	+	0.932930i	$0.617243\pi$
$930$	−1.10296e10	−0.449644
$931$	0	0
$932$	9.81383e9	0.397085
$933$	−1.94818e10	−0.785316
$934$	−2.79553e9	−0.112267
$935$	−4.75296e9	−0.190162
$936$	−2.82287e9	−0.112519
$937$	−1.80228e10	−0.715703	−0.357852	−	0.933778i	$-0.616491\pi$
$937$	−1.80228e10	−0.715703	−0.357852	+	0.933778i	$0.616491\pi$
$938$	0	0
$939$	2.67679e10	1.05508
$940$	1.66972e10	0.655687
$941$	−2.96116e10	−1.15850	−0.579252	−	0.815149i	$-0.696655\pi$
$941$	−2.96116e10	−1.15850	−0.579252	+	0.815149i	$0.696655\pi$
$942$	−1.19498e9	−0.0465782
$943$	4.40628e9	0.171112
$944$	9.75489e9	0.377416
$945$	0	0
$946$	−1.32974e9	−0.0510678
$947$	−2.44524e10	−0.935612	−0.467806	−	0.883831i	$-0.654955\pi$
$947$	−2.44524e10	−0.935612	−0.467806	+	0.883831i	$0.654955\pi$
$948$	−1.08582e10	−0.413932
$949$	−4.48508e10	−1.70349
$950$	1.19706e9	0.0452983
$951$	3.25119e9	0.122577
$952$	0	0
$953$	−3.27114e10	−1.22426	−0.612130	−	0.790757i	$-0.709687\pi$
$953$	−3.27114e10	−1.22426	−0.612130	+	0.790757i	$0.709687\pi$
$954$	7.61153e9	0.283826
$955$	2.98257e10	1.10810
$956$	−2.21198e10	−0.818801
$957$	−2.08078e8	−0.00767423
$958$	3.24018e10	1.19066
$959$	0	0
$960$	−2.05259e9	−0.0748777
$961$	3.49120e9	0.126895
$962$	8.04612e9	0.291390
$963$	−9.38199e9	−0.338534
$964$	−3.46027e9	−0.124406
$965$	−5.08201e10	−1.82050
$966$	0	0
$967$	−5.41923e9	−0.192728	−0.0963640	−	0.995346i	$-0.530721\pi$
$967$	−5.41923e9	−0.192728	−0.0963640	+	0.995346i	$0.530721\pi$
$968$	9.32366e9	0.330387
$969$	−9.80704e9	−0.346262
$970$	2.18252e10	0.767818
$971$	−8.82543e9	−0.309363	−0.154681	−	0.987964i	$-0.549435\pi$
$971$	−8.82543e9	−0.309363	−0.154681	+	0.987964i	$0.549435\pi$
$972$	−9.18330e8	−0.0320750
$973$	0	0
$974$	−7.41307e9	−0.257064
$975$	−1.22010e9	−0.0421580
$976$	1.07524e10	0.370196
$977$	−3.26822e10	−1.12119	−0.560597	−	0.828089i	$-0.689428\pi$
$977$	−3.26822e10	−1.12119	−0.560597	+	0.828089i	$0.689428\pi$
$978$	−7.24267e9	−0.247578
$979$	1.67330e9	0.0569948
$980$	0	0
$981$	−1.59292e10	−0.538708
$982$	−2.12725e10	−0.716849
$983$	−3.56345e10	−1.19656	−0.598279	−	0.801288i	$-0.704148\pi$
$983$	−3.56345e10	−1.19656	−0.598279	+	0.801288i	$0.704148\pi$
$984$	−9.14051e9	−0.305835
$985$	4.55937e10	1.52012
$986$	7.91338e8	0.0262901
$987$	0	0
$988$	−1.21216e10	−0.399863
$989$	−9.80241e8	−0.0322215
$990$	−1.91115e9	−0.0625995
$991$	5.12314e10	1.67216	0.836080	−	0.548607i	$-0.184842\pi$
$991$	5.12314e10	1.67216	0.836080	+	0.548607i	$0.184842\pi$
$992$	5.76976e9	0.187658
$993$	−5.11734e9	−0.165852
$994$	0	0
$995$	−3.21171e10	−1.03361
$996$	7.34789e9	0.235643
$997$	−2.04673e10	−0.654076	−0.327038	−	0.945011i	$-0.606051\pi$
$997$	−2.04673e10	−0.654076	−0.327038	+	0.945011i	$0.606051\pi$
$998$	2.53197e10	0.806309
$999$	2.61754e9	0.0830644

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	294.8.a.e.1.1		1
7.2	even	3		294.8.e.l.67.1		2
7.3	odd	6		42.8.e.b.37.1	yes	2
7.4	even	3		294.8.e.l.79.1		2
7.5	odd	6		42.8.e.b.25.1	✓	2
7.6	odd	2		294.8.a.f.1.1		1
21.5	even	6		126.8.g.a.109.1		2
21.17	even	6		126.8.g.a.37.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.8.e.b.25.1	✓	2	7.5	odd	6
42.8.e.b.37.1	yes	2	7.3	odd	6
126.8.g.a.37.1		2	21.17	even	6
126.8.g.a.109.1		2	21.5	even	6
294.8.a.e.1.1		1	1.1	even	1	trivial
294.8.a.f.1.1		1	7.6	odd	2
294.8.e.l.67.1		2	7.2	even	3
294.8.e.l.79.1		2	7.4	even	3

Overview	Random
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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

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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	294.8.a.e.1.1

Level	$294$
Weight	$8$
Character	294.1
Self dual	yes
Analytic conductor	$91.841$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$