LMFDB - Embedded newform 150.8.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,8,Mod(1,150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, 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("150.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	150.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:	$46.8577538226$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	150.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -216.000 q^{6} -512.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})$

\(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -216.000 q^{6} -512.000 q^{7} -512.000 q^{8} +729.000 q^{9} +5460.00 q^{11} +1728.00 q^{12} -10166.0 q^{13} +4096.00 q^{14} +4096.00 q^{16} +9918.00 q^{17} -5832.00 q^{18} -12436.0 q^{19} -13824.0 q^{21} -43680.0 q^{22} -33600.0 q^{23} -13824.0 q^{24} +81328.0 q^{26} +19683.0 q^{27} -32768.0 q^{28} -187914. q^{29} -42592.0 q^{31} -32768.0 q^{32} +147420. q^{33} -79344.0 q^{34} +46656.0 q^{36} +544066. q^{37} +99488.0 q^{38} -274482. q^{39} +374394. q^{41} +110592. q^{42} +540532. q^{43} +349440. q^{44} +268800. q^{46} -1.33836e6 q^{47} +110592. q^{48} -561399. q^{49} +267786. q^{51} -650624. q^{52} -1.30822e6 q^{53} -157464. q^{54} +262144. q^{56} -335772. q^{57} +1.50331e6 q^{58} +262740. q^{59} -976330. q^{61} +340736. q^{62} -373248. q^{63} +262144. q^{64} -1.17936e6 q^{66} -3.55917e6 q^{67} +634752. q^{68} -907200. q^{69} -2.67372e6 q^{71} -373248. q^{72} +3.03213e6 q^{73} -4.35253e6 q^{74} -795904. q^{76} -2.79552e6 q^{77} +2.19586e6 q^{78} -5.47581e6 q^{79} +531441. q^{81} -2.99515e6 q^{82} -2.23156e6 q^{83} -884736. q^{84} -4.32426e6 q^{86} -5.07368e6 q^{87} -2.79552e6 q^{88} -1.00507e7 q^{89} +5.20499e6 q^{91} -2.15040e6 q^{92} -1.14998e6 q^{93} +1.07069e7 q^{94} -884736. q^{96} -5.72755e6 q^{97} +4.49119e6 q^{98} +3.98034e6 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
$5$	0	0
$5$	0	0
$6$	−216.000	−0.408248
$7$	−512.000	−0.564192	−0.282096	−	0.959386i	$-0.591030\pi$
$7$	−512.000	−0.564192	−0.282096	+	0.959386i	$0.591030\pi$
$8$	−512.000	−0.353553
$9$	729.000	0.333333
$10$	0	0
$11$	5460.00	1.23685	0.618427	−	0.785842i	$-0.287770\pi$
$11$	5460.00	1.23685	0.618427	+	0.785842i	$0.287770\pi$
$12$	1728.00	0.288675
$13$	−10166.0	−1.28336	−0.641680	−	0.766973i	$-0.721762\pi$
$13$	−10166.0	−1.28336	−0.641680	+	0.766973i	$0.721762\pi$
$14$	4096.00	0.398944
$15$	0	0
$16$	4096.00	0.250000
$17$	9918.00	0.489613	0.244806	−	0.969572i	$-0.421276\pi$
$17$	9918.00	0.489613	0.244806	+	0.969572i	$0.421276\pi$
$18$	−5832.00	−0.235702
$19$	−12436.0	−0.415952	−0.207976	−	0.978134i	$-0.566688\pi$
$19$	−12436.0	−0.415952	−0.207976	+	0.978134i	$0.566688\pi$
$20$	0	0
$21$	−13824.0	−0.325736
$22$	−43680.0	−0.874587
$23$	−33600.0	−0.575827	−0.287913	−	0.957656i	$-0.592962\pi$
$23$	−33600.0	−0.575827	−0.287913	+	0.957656i	$0.592962\pi$
$24$	−13824.0	−0.204124
$25$	0	0
$26$	81328.0	0.907472
$27$	19683.0	0.192450
$28$	−32768.0	−0.282096
$29$	−187914.	−1.43076	−0.715379	−	0.698737i	$-0.753746\pi$
$29$	−187914.	−1.43076	−0.715379	+	0.698737i	$0.753746\pi$
$30$	0	0
$31$	−42592.0	−0.256781	−0.128390	−	0.991724i	$-0.540981\pi$
$31$	−42592.0	−0.256781	−0.128390	+	0.991724i	$0.540981\pi$
$32$	−32768.0	−0.176777
$33$	147420.	0.714098
$34$	−79344.0	−0.346209
$35$	0	0
$36$	46656.0	0.166667
$37$	544066.	1.76582	0.882908	−	0.469546i	$-0.155582\pi$
$37$	544066.	1.76582	0.882908	+	0.469546i	$0.155582\pi$
$38$	99488.0	0.294122
$39$	−274482.	−0.740948
$40$	0	0
$41$	374394.	0.848370	0.424185	−	0.905576i	$-0.360561\pi$
$41$	374394.	0.848370	0.424185	+	0.905576i	$0.360561\pi$
$42$	110592.	0.230330
$43$	540532.	1.03677	0.518384	−	0.855148i	$-0.326534\pi$
$43$	540532.	1.03677	0.518384	+	0.855148i	$0.326534\pi$
$44$	349440.	0.618427
$45$	0	0
$46$	268800.	0.407171
$47$	−1.33836e6	−1.88031	−0.940157	−	0.340741i	$-0.889322\pi$
$47$	−1.33836e6	−1.88031	−0.940157	+	0.340741i	$0.889322\pi$
$48$	110592.	0.144338
$49$	−561399.	−0.681688
$50$	0	0
$51$	267786.	0.282678
$52$	−650624.	−0.641680
$53$	−1.30822e6	−1.20702	−0.603512	−	0.797354i	$-0.706232\pi$
$53$	−1.30822e6	−1.20702	−0.603512	+	0.797354i	$0.706232\pi$
$54$	−157464.	−0.136083
$55$	0	0
$56$	262144.	0.199472
$57$	−335772.	−0.240150
$58$	1.50331e6	1.01170
$59$	262740.	0.166550	0.0832749	−	0.996527i	$-0.473462\pi$
$59$	262740.	0.166550	0.0832749	+	0.996527i	$0.473462\pi$
$60$	0	0
$61$	−976330.	−0.550734	−0.275367	−	0.961339i	$-0.588799\pi$
$61$	−976330.	−0.550734	−0.275367	+	0.961339i	$0.588799\pi$
$62$	340736.	0.181571
$63$	−373248.	−0.188064
$64$	262144.	0.125000
$65$	0	0
$66$	−1.17936e6	−0.504943
$67$	−3.55917e6	−1.44573	−0.722865	−	0.690989i	$-0.757175\pi$
$67$	−3.55917e6	−1.44573	−0.722865	+	0.690989i	$0.757175\pi$
$68$	634752.	0.244806
$69$	−907200.	−0.332454
$70$	0	0
$71$	−2.67372e6	−0.886567	−0.443284	−	0.896381i	$-0.646187\pi$
$71$	−2.67372e6	−0.886567	−0.443284	+	0.896381i	$0.646187\pi$
$72$	−373248.	−0.117851
$73$	3.03213e6	0.912259	0.456130	−	0.889913i	$-0.349235\pi$
$73$	3.03213e6	0.912259	0.456130	+	0.889913i	$0.349235\pi$
$74$	−4.35253e6	−1.24862
$75$	0	0
$76$	−795904.	−0.207976
$77$	−2.79552e6	−0.697823
$78$	2.19586e6	0.523929
$79$	−5.47581e6	−1.24955	−0.624775	−	0.780805i	$-0.714809\pi$
$79$	−5.47581e6	−1.24955	−0.624775	+	0.780805i	$0.714809\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	−2.99515e6	−0.599888
$83$	−2.23156e6	−0.428385	−0.214193	−	0.976791i	$-0.568712\pi$
$83$	−2.23156e6	−0.428385	−0.214193	+	0.976791i	$0.568712\pi$
$84$	−884736.	−0.162868
$85$	0	0
$86$	−4.32426e6	−0.733106
$87$	−5.07368e6	−0.826048
$88$	−2.79552e6	−0.437294
$89$	−1.00507e7	−1.51123	−0.755615	−	0.655016i	$-0.772662\pi$
$89$	−1.00507e7	−1.51123	−0.755615	+	0.655016i	$0.772662\pi$
$90$	0	0
$91$	5.20499e6	0.724061
$92$	−2.15040e6	−0.287913
$93$	−1.14998e6	−0.148252
$94$	1.07069e7	1.32958
$95$	0	0
$96$	−884736.	−0.102062
$97$	−5.72755e6	−0.637189	−0.318594	−	0.947891i	$-0.603211\pi$
$97$	−5.72755e6	−0.637189	−0.318594	+	0.947891i	$0.603211\pi$
$98$	4.49119e6	0.482026
$99$	3.98034e6	0.412284
$100$	0	0
$101$	−1.33358e7	−1.28793	−0.643966	−	0.765054i	$-0.722712\pi$
$101$	−1.33358e7	−1.28793	−0.643966	+	0.765054i	$0.722712\pi$
$102$	−2.14229e6	−0.199884
$103$	2.71019e6	0.244382	0.122191	−	0.992507i	$-0.461008\pi$
$103$	2.71019e6	0.244382	0.122191	+	0.992507i	$0.461008\pi$
$104$	5.20499e6	0.453736
$105$	0	0
$106$	1.04658e7	0.853495
$107$	−1.13195e7	−0.893274	−0.446637	−	0.894715i	$-0.647378\pi$
$107$	−1.13195e7	−0.893274	−0.446637	+	0.894715i	$0.647378\pi$
$108$	1.25971e6	0.0962250
$109$	−2.19732e7	−1.62518	−0.812589	−	0.582838i	$-0.801942\pi$
$109$	−2.19732e7	−1.62518	−0.812589	+	0.582838i	$0.801942\pi$
$110$	0	0
$111$	1.46898e7	1.01949
$112$	−2.09715e6	−0.141048
$113$	−3.68359e6	−0.240158	−0.120079	−	0.992764i	$-0.538315\pi$
$113$	−3.68359e6	−0.240158	−0.120079	+	0.992764i	$0.538315\pi$
$114$	2.68618e6	0.169812
$115$	0	0
$116$	−1.20265e7	−0.715379
$117$	−7.41101e6	−0.427787
$118$	−2.10192e6	−0.117769
$119$	−5.07802e6	−0.276236
$120$	0	0
$121$	1.03244e7	0.529806
$122$	7.81064e6	0.389428
$123$	1.01086e7	0.489807
$124$	−2.72589e6	−0.128390
$125$	0	0
$126$	2.98598e6	0.132981
$127$	2.67953e7	1.16077	0.580385	−	0.814342i	$-0.302902\pi$
$127$	2.67953e7	1.16077	0.580385	+	0.814342i	$0.302902\pi$
$128$	−2.09715e6	−0.0883883
$129$	1.45944e7	0.598579
$130$	0	0
$131$	1.48085e7	0.575523	0.287762	−	0.957702i	$-0.407089\pi$
$131$	1.48085e7	0.575523	0.287762	+	0.957702i	$0.407089\pi$
$132$	9.43488e6	0.357049
$133$	6.36723e6	0.234677
$134$	2.84734e7	1.02229
$135$	0	0
$136$	−5.07802e6	−0.173104
$137$	5.76532e7	1.91559	0.957793	−	0.287458i	$-0.0928103\pi$
$137$	5.76532e7	1.91559	0.957793	+	0.287458i	$0.0928103\pi$
$138$	7.25760e6	0.235080
$139$	−8.37800e6	−0.264599	−0.132300	−	0.991210i	$-0.542236\pi$
$139$	−8.37800e6	−0.264599	−0.132300	+	0.991210i	$0.542236\pi$
$140$	0	0
$141$	−3.61357e7	−1.08560
$142$	2.13898e7	0.626898
$143$	−5.55064e7	−1.58733
$144$	2.98598e6	0.0833333
$145$	0	0
$146$	−2.42571e7	−0.645065
$147$	−1.51578e7	−0.393572
$148$	3.48202e7	0.882908
$149$	5.56477e6	0.137815	0.0689073	−	0.997623i	$-0.478049\pi$
$149$	5.56477e6	0.137815	0.0689073	+	0.997623i	$0.478049\pi$
$150$	0	0
$151$	6.62933e7	1.56693	0.783466	−	0.621434i	$-0.213450\pi$
$151$	6.62933e7	1.56693	0.783466	+	0.621434i	$0.213450\pi$
$152$	6.36723e6	0.147061
$153$	7.23022e6	0.163204
$154$	2.23642e7	0.493435
$155$	0	0
$156$	−1.75668e7	−0.370474
$157$	6.42791e7	1.32563	0.662813	−	0.748785i	$-0.269363\pi$
$157$	6.42791e7	1.32563	0.662813	+	0.748785i	$0.269363\pi$
$158$	4.38065e7	0.883565
$159$	−3.53220e7	−0.696876
$160$	0	0
$161$	1.72032e7	0.324877
$162$	−4.25153e6	−0.0785674
$163$	−8.48552e7	−1.53469	−0.767347	−	0.641232i	$-0.778424\pi$
$163$	−8.48552e7	−1.53469	−0.767347	+	0.641232i	$0.778424\pi$
$164$	2.39612e7	0.424185
$165$	0	0
$166$	1.78524e7	0.302914
$167$	−6.42144e7	−1.06690	−0.533451	−	0.845831i	$-0.679105\pi$
$167$	−6.42144e7	−1.06690	−0.533451	+	0.845831i	$0.679105\pi$
$168$	7.07789e6	0.115165
$169$	4.05990e7	0.647012
$170$	0	0
$171$	−9.06584e6	−0.138651
$172$	3.45940e7	0.518384
$173$	−1.10058e8	−1.61607	−0.808036	−	0.589134i	$-0.799469\pi$
$173$	−1.10058e8	−1.61607	−0.808036	+	0.589134i	$0.799469\pi$
$174$	4.05894e7	0.584104
$175$	0	0
$176$	2.23642e7	0.309213
$177$	7.09398e6	0.0961576
$178$	8.04054e7	1.06860
$179$	8.68778e7	1.13220	0.566100	−	0.824337i	$-0.308452\pi$
$179$	8.68778e7	1.13220	0.566100	+	0.824337i	$0.308452\pi$
$180$	0	0
$181$	1.29730e8	1.62617	0.813083	−	0.582148i	$-0.197787\pi$
$181$	1.29730e8	1.62617	0.813083	+	0.582148i	$0.197787\pi$
$182$	−4.16399e7	−0.511988
$183$	−2.63609e7	−0.317967
$184$	1.72032e7	0.203586
$185$	0	0
$186$	9.19987e6	0.104830
$187$	5.41523e7	0.605579
$188$	−8.56550e7	−0.940157
$189$	−1.00777e7	−0.108579
$190$	0	0
$191$	−1.42024e8	−1.47484	−0.737418	−	0.675437i	$-0.763955\pi$
$191$	−1.42024e8	−1.47484	−0.737418	+	0.675437i	$0.763955\pi$
$192$	7.07789e6	0.0721688
$193$	2.58910e7	0.259238	0.129619	−	0.991564i	$-0.458625\pi$
$193$	2.58910e7	0.259238	0.129619	+	0.991564i	$0.458625\pi$
$194$	4.58204e7	0.450561
$195$	0	0
$196$	−3.59295e7	−0.340844
$197$	3.59130e7	0.334673	0.167336	−	0.985900i	$-0.446483\pi$
$197$	3.59130e7	0.334673	0.167336	+	0.985900i	$0.446483\pi$
$198$	−3.18427e7	−0.291529
$199$	−1.75453e8	−1.57824	−0.789122	−	0.614237i	$-0.789464\pi$
$199$	−1.75453e8	−1.57824	−0.789122	+	0.614237i	$0.789464\pi$
$200$	0	0
$201$	−9.60976e7	−0.834693
$202$	1.06686e8	0.910706
$203$	9.62120e7	0.807222
$204$	1.71383e7	0.141339
$205$	0	0
$206$	−2.16815e7	−0.172804
$207$	−2.44944e7	−0.191942
$208$	−4.16399e7	−0.320840
$209$	−6.79006e7	−0.514472
$210$	0	0
$211$	−9.18337e7	−0.672998	−0.336499	−	0.941684i	$-0.609243\pi$
$211$	−9.18337e7	−0.672998	−0.336499	+	0.941684i	$0.609243\pi$
$212$	−8.37262e7	−0.603512
$213$	−7.21904e7	−0.511860
$214$	9.05561e7	0.631640
$215$	0	0
$216$	−1.00777e7	−0.0680414
$217$	2.18071e7	0.144873
$218$	1.75786e8	1.14917
$219$	8.18676e7	0.526693
$220$	0	0
$221$	−1.00826e8	−0.628349
$222$	−1.17518e8	−0.720892
$223$	−1.15780e7	−0.0699146	−0.0349573	−	0.999389i	$-0.511130\pi$
$223$	−1.15780e7	−0.0699146	−0.0349573	+	0.999389i	$0.511130\pi$
$224$	1.67772e7	0.0997360
$225$	0	0
$226$	2.94687e7	0.169817
$227$	2.99769e7	0.170097	0.0850485	−	0.996377i	$-0.472895\pi$
$227$	2.99769e7	0.170097	0.0850485	+	0.996377i	$0.472895\pi$
$228$	−2.14894e7	−0.120075
$229$	1.19721e8	0.658787	0.329394	−	0.944193i	$-0.393156\pi$
$229$	1.19721e8	0.658787	0.329394	+	0.944193i	$0.393156\pi$
$230$	0	0
$231$	−7.54790e7	−0.402888
$232$	9.62120e7	0.505849
$233$	1.27607e8	0.660890	0.330445	−	0.943825i	$-0.392801\pi$
$233$	1.27607e8	0.660890	0.330445	+	0.943825i	$0.392801\pi$
$234$	5.92881e7	0.302491
$235$	0	0
$236$	1.68154e7	0.0832749
$237$	−1.47847e8	−0.721428
$238$	4.06241e7	0.195328
$239$	−1.64949e8	−0.781549	−0.390775	−	0.920486i	$-0.627793\pi$
$239$	−1.64949e8	−0.781549	−0.390775	+	0.920486i	$0.627793\pi$
$240$	0	0
$241$	3.98628e8	1.83446	0.917230	−	0.398357i	$-0.130420\pi$
$241$	3.98628e8	1.83446	0.917230	+	0.398357i	$0.130420\pi$
$242$	−8.25954e7	−0.374630
$243$	1.43489e7	0.0641500
$244$	−6.24851e7	−0.275367
$245$	0	0
$246$	−8.08691e7	−0.346346
$247$	1.26424e8	0.533816
$248$	2.18071e7	0.0907856
$249$	−6.02520e7	−0.247328
$250$	0	0
$251$	2.42512e8	0.967999	0.484000	−	0.875068i	$-0.339184\pi$
$251$	2.42512e8	0.967999	0.484000	+	0.875068i	$0.339184\pi$
$252$	−2.38879e7	−0.0940320
$253$	−1.83456e8	−0.712213
$254$	−2.14363e8	−0.820788
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	−3.22956e8	−1.18680	−0.593399	−	0.804908i	$-0.702214\pi$
$257$	−3.22956e8	−1.18680	−0.593399	+	0.804908i	$0.702214\pi$
$258$	−1.16755e8	−0.423259
$259$	−2.78562e8	−0.996259
$260$	0	0
$261$	−1.36989e8	−0.476919
$262$	−1.18468e8	−0.406956
$263$	3.37426e8	1.14376	0.571879	−	0.820338i	$-0.306215\pi$
$263$	3.37426e8	1.14376	0.571879	+	0.820338i	$0.306215\pi$
$264$	−7.54790e7	−0.252472
$265$	0	0
$266$	−5.09379e7	−0.165941
$267$	−2.71368e8	−0.872509
$268$	−2.27787e8	−0.722865
$269$	2.55886e8	0.801517	0.400759	−	0.916184i	$-0.368747\pi$
$269$	2.55886e8	0.801517	0.400759	+	0.916184i	$0.368747\pi$
$270$	0	0
$271$	4.85679e8	1.48237	0.741185	−	0.671301i	$-0.234264\pi$
$271$	4.85679e8	1.48237	0.741185	+	0.671301i	$0.234264\pi$
$272$	4.06241e7	0.122403
$273$	1.40535e8	0.418037
$274$	−4.61226e8	−1.35452
$275$	0	0
$276$	−5.80608e7	−0.166227
$277$	−2.92732e8	−0.827545	−0.413772	−	0.910380i	$-0.635789\pi$
$277$	−2.92732e8	−0.827545	−0.413772	+	0.910380i	$0.635789\pi$
$278$	6.70240e7	0.187100
$279$	−3.10496e7	−0.0855935
$280$	0	0
$281$	−1.86225e8	−0.500685	−0.250343	−	0.968157i	$-0.580543\pi$
$281$	−1.86225e8	−0.500685	−0.250343	+	0.968157i	$0.580543\pi$
$282$	2.89086e8	0.767635
$283$	6.58251e8	1.72639	0.863196	−	0.504870i	$-0.168459\pi$
$283$	6.58251e8	1.72639	0.863196	+	0.504870i	$0.168459\pi$
$284$	−1.71118e8	−0.443284
$285$	0	0
$286$	4.44051e8	1.12241
$287$	−1.91690e8	−0.478644
$288$	−2.38879e7	−0.0589256
$289$	−3.11972e8	−0.760279
$290$	0	0
$291$	−1.54644e8	−0.367881
$292$	1.94057e8	0.456130
$293$	−1.61426e8	−0.374918	−0.187459	−	0.982272i	$-0.560025\pi$
$293$	−1.61426e8	−0.374918	−0.187459	+	0.982272i	$0.560025\pi$
$294$	1.21262e8	0.278298
$295$	0	0
$296$	−2.78562e8	−0.624310
$297$	1.07469e8	0.238033
$298$	−4.45181e7	−0.0974496
$299$	3.41578e8	0.738993
$300$	0	0
$301$	−2.76752e8	−0.584936
$302$	−5.30346e8	−1.10799
$303$	−3.60066e8	−0.743588
$304$	−5.09379e7	−0.103988
$305$	0	0
$306$	−5.78418e7	−0.115403
$307$	7.15488e8	1.41130	0.705648	−	0.708562i	$-0.250656\pi$
$307$	7.15488e8	1.41130	0.705648	+	0.708562i	$0.250656\pi$
$308$	−1.78913e8	−0.348911
$309$	7.31752e7	0.141094
$310$	0	0
$311$	−6.86692e7	−0.129449	−0.0647247	−	0.997903i	$-0.520617\pi$
$311$	−6.86692e7	−0.129449	−0.0647247	+	0.997903i	$0.520617\pi$
$312$	1.40535e8	0.261965
$313$	−9.27771e7	−0.171015	−0.0855077	−	0.996338i	$-0.527251\pi$
$313$	−9.27771e7	−0.171015	−0.0855077	+	0.996338i	$0.527251\pi$
$314$	−5.14233e8	−0.937359
$315$	0	0
$316$	−3.50452e8	−0.624775
$317$	−3.90749e8	−0.688954	−0.344477	−	0.938795i	$-0.611944\pi$
$317$	−3.90749e8	−0.688954	−0.344477	+	0.938795i	$0.611944\pi$
$318$	2.82576e8	0.492765
$319$	−1.02601e9	−1.76964
$320$	0	0
$321$	−3.05627e8	−0.515732
$322$	−1.37626e8	−0.229723
$323$	−1.23340e8	−0.203655
$324$	3.40122e7	0.0555556
$325$	0	0
$326$	6.78842e8	1.08519
$327$	−5.93277e8	−0.938297
$328$	−1.91690e8	−0.299944
$329$	6.85240e8	1.06086
$330$	0	0
$331$	−4.80269e8	−0.727925	−0.363962	−	0.931414i	$-0.618576\pi$
$331$	−4.80269e8	−0.727925	−0.363962	+	0.931414i	$0.618576\pi$
$332$	−1.42820e8	−0.214193
$333$	3.96624e8	0.588605
$334$	5.13715e8	0.754414
$335$	0	0
$336$	−5.66231e7	−0.0814341
$337$	2.30504e8	0.328075	0.164038	−	0.986454i	$-0.447548\pi$
$337$	2.30504e8	0.328075	0.164038	+	0.986454i	$0.447548\pi$
$338$	−3.24792e8	−0.457507
$339$	−9.94568e7	−0.138655
$340$	0	0
$341$	−2.32552e8	−0.317600
$342$	7.25268e7	0.0980408
$343$	7.09090e8	0.948794
$344$	−2.76752e8	−0.366553
$345$	0	0
$346$	8.80464e8	1.14273
$347$	8.36727e8	1.07505	0.537527	−	0.843246i	$-0.319359\pi$
$347$	8.36727e8	1.07505	0.537527	+	0.843246i	$0.319359\pi$
$348$	−3.24715e8	−0.413024
$349$	1.51659e9	1.90976	0.954881	−	0.296988i	$-0.0959820\pi$
$349$	1.51659e9	1.90976	0.954881	+	0.296988i	$0.0959820\pi$
$350$	0	0
$351$	−2.00097e8	−0.246983
$352$	−1.78913e8	−0.218647
$353$	−2.33016e7	−0.0281952	−0.0140976	−	0.999901i	$-0.504488\pi$
$353$	−2.33016e7	−0.0281952	−0.0140976	+	0.999901i	$0.504488\pi$
$354$	−5.67518e7	−0.0679937
$355$	0	0
$356$	−6.43243e8	−0.755615
$357$	−1.37106e8	−0.159485
$358$	−6.95022e8	−0.800586
$359$	1.63162e9	1.86118	0.930590	−	0.366063i	$-0.119295\pi$
$359$	1.63162e9	1.86118	0.930590	+	0.366063i	$0.119295\pi$
$360$	0	0
$361$	−7.39218e8	−0.826984
$362$	−1.03784e9	−1.14987
$363$	2.78760e8	0.305884
$364$	3.33119e8	0.362031
$365$	0	0
$366$	2.10887e8	0.224836
$367$	−3.47807e8	−0.367289	−0.183644	−	0.982993i	$-0.558789\pi$
$367$	−3.47807e8	−0.367289	−0.183644	+	0.982993i	$0.558789\pi$
$368$	−1.37626e8	−0.143957
$369$	2.72933e8	0.282790
$370$	0	0
$371$	6.69810e8	0.680993
$372$	−7.35990e7	−0.0741262
$373$	−5.00769e7	−0.0499639	−0.0249820	−	0.999688i	$-0.507953\pi$
$373$	−5.00769e7	−0.0499639	−0.0249820	+	0.999688i	$0.507953\pi$
$374$	−4.33218e8	−0.428209
$375$	0	0
$376$	6.85240e8	0.664791
$377$	1.91033e9	1.83618
$378$	8.06216e7	0.0767768
$379$	1.40154e9	1.32241	0.661207	−	0.750204i	$-0.270045\pi$
$379$	1.40154e9	1.32241	0.661207	+	0.750204i	$0.270045\pi$
$380$	0	0
$381$	7.23474e8	0.670170
$382$	1.13619e9	1.04287
$383$	−1.06830e9	−0.971623	−0.485811	−	0.874064i	$-0.661476\pi$
$383$	−1.06830e9	−0.971623	−0.485811	+	0.874064i	$0.661476\pi$
$384$	−5.66231e7	−0.0510310
$385$	0	0
$386$	−2.07128e8	−0.183309
$387$	3.94048e8	0.345590
$388$	−3.66563e8	−0.318594
$389$	−2.13713e8	−0.184080	−0.0920401	−	0.995755i	$-0.529339\pi$
$389$	−2.13713e8	−0.184080	−0.0920401	+	0.995755i	$0.529339\pi$
$390$	0	0
$391$	−3.33245e8	−0.281932
$392$	2.87436e8	0.241013
$393$	3.99831e8	0.332278
$394$	−2.87304e8	−0.236649
$395$	0	0
$396$	2.54742e8	0.206142
$397$	−9.69088e8	−0.777314	−0.388657	−	0.921383i	$-0.627061\pi$
$397$	−9.69088e8	−0.777314	−0.388657	+	0.921383i	$0.627061\pi$
$398$	1.40362e9	1.11599
$399$	1.71915e8	0.135491
$400$	0	0
$401$	−2.21767e9	−1.71748	−0.858738	−	0.512415i	$-0.828751\pi$
$401$	−2.21767e9	−1.71748	−0.858738	+	0.512415i	$0.828751\pi$
$402$	7.68781e8	0.590217
$403$	4.32990e8	0.329542
$404$	−8.53489e8	−0.643966
$405$	0	0
$406$	−7.69696e8	−0.570792
$407$	2.97060e9	2.18406
$408$	−1.37106e8	−0.0999418
$409$	−1.04837e9	−0.757674	−0.378837	−	0.925463i	$-0.623676\pi$
$409$	−1.04837e9	−0.757674	−0.378837	+	0.925463i	$0.623676\pi$
$410$	0	0
$411$	1.55664e9	1.10596
$412$	1.73452e8	0.122191
$413$	−1.34523e8	−0.0939661
$414$	1.95955e8	0.135724
$415$	0	0
$416$	3.33119e8	0.226868
$417$	−2.26206e8	−0.152766
$418$	5.43204e8	0.363786
$419$	−2.21021e8	−0.146786	−0.0733930	−	0.997303i	$-0.523383\pi$
$419$	−2.21021e8	−0.146786	−0.0733930	+	0.997303i	$0.523383\pi$
$420$	0	0
$421$	1.51339e9	0.988472	0.494236	−	0.869328i	$-0.335448\pi$
$421$	1.51339e9	0.988472	0.494236	+	0.869328i	$0.335448\pi$
$422$	7.34670e8	0.475881
$423$	−9.75664e8	−0.626771
$424$	6.69810e8	0.426747
$425$	0	0
$426$	5.77524e8	0.361940
$427$	4.99881e8	0.310720
$428$	−7.24449e8	−0.446637
$429$	−1.49867e9	−0.916444
$430$	0	0
$431$	1.46801e9	0.883201	0.441601	−	0.897212i	$-0.354411\pi$
$431$	1.46801e9	0.883201	0.441601	+	0.897212i	$0.354411\pi$
$432$	8.06216e7	0.0481125
$433$	−3.97508e8	−0.235309	−0.117654	−	0.993055i	$-0.537537\pi$
$433$	−3.97508e8	−0.235309	−0.117654	+	0.993055i	$0.537537\pi$
$434$	−1.74457e8	−0.102441
$435$	0	0
$436$	−1.40629e9	−0.812589
$437$	4.17850e8	0.239516
$438$	−6.54941e8	−0.372428
$439$	1.22071e9	0.688629	0.344315	−	0.938854i	$-0.388111\pi$
$439$	1.22071e9	0.688629	0.344315	+	0.938854i	$0.388111\pi$
$440$	0	0
$441$	−4.09260e8	−0.227229
$442$	8.06611e8	0.444310
$443$	−3.24551e9	−1.77366	−0.886830	−	0.462095i	$-0.847098\pi$
$443$	−3.24551e9	−1.77366	−0.886830	+	0.462095i	$0.847098\pi$
$444$	9.40146e8	0.509747
$445$	0	0
$446$	9.26243e7	0.0494371
$447$	1.50249e8	0.0795672
$448$	−1.34218e8	−0.0705240
$449$	−7.04211e8	−0.367147	−0.183574	−	0.983006i	$-0.558767\pi$
$449$	−7.04211e8	−0.367147	−0.183574	+	0.983006i	$0.558767\pi$
$450$	0	0
$451$	2.04419e9	1.04931
$452$	−2.35750e8	−0.120079
$453$	1.78992e9	0.904669
$454$	−2.39815e8	−0.120277
$455$	0	0
$456$	1.71915e8	0.0849058
$457$	−3.70970e9	−1.81816	−0.909080	−	0.416621i	$-0.863214\pi$
$457$	−3.70970e9	−1.81816	−0.909080	+	0.416621i	$0.863214\pi$
$458$	−9.57766e8	−0.465833
$459$	1.95216e8	0.0942261
$460$	0	0
$461$	1.12514e9	0.534878	0.267439	−	0.963575i	$-0.413823\pi$
$461$	1.12514e9	0.534878	0.267439	+	0.963575i	$0.413823\pi$
$462$	6.03832e8	0.284885
$463$	7.64328e7	0.0357887	0.0178944	−	0.999840i	$-0.494304\pi$
$463$	7.64328e7	0.0357887	0.0178944	+	0.999840i	$0.494304\pi$
$464$	−7.69696e8	−0.357689
$465$	0	0
$466$	−1.02086e9	−0.467320
$467$	6.41328e7	0.0291388	0.0145694	−	0.999894i	$-0.495362\pi$
$467$	6.41328e7	0.0291388	0.0145694	+	0.999894i	$0.495362\pi$
$468$	−4.74305e8	−0.213893
$469$	1.82230e9	0.815669
$470$	0	0
$471$	1.73553e9	0.765350
$472$	−1.34523e8	−0.0588843
$473$	2.95130e9	1.28233
$474$	1.18277e9	0.510126
$475$	0	0
$476$	−3.24993e8	−0.138118
$477$	−9.53694e8	−0.402341
$478$	1.31959e9	0.552639
$479$	1.01386e9	0.421505	0.210753	−	0.977539i	$-0.432409\pi$
$479$	1.01386e9	0.421505	0.210753	+	0.977539i	$0.432409\pi$
$480$	0	0
$481$	−5.53097e9	−2.26618
$482$	−3.18903e9	−1.29716
$483$	4.64486e8	0.187568
$484$	6.60763e8	0.264903
$485$	0	0
$486$	−1.14791e8	−0.0453609
$487$	2.79009e8	0.109463	0.0547315	−	0.998501i	$-0.482570\pi$
$487$	2.79009e8	0.109463	0.0547315	+	0.998501i	$0.482570\pi$
$488$	4.99881e8	0.194714
$489$	−2.29109e9	−0.886056
$490$	0	0
$491$	1.77339e9	0.676112	0.338056	−	0.941126i	$-0.390231\pi$
$491$	1.77339e9	0.676112	0.338056	+	0.941126i	$0.390231\pi$
$492$	6.46953e8	0.244903
$493$	−1.86373e9	−0.700518
$494$	−1.01140e9	−0.377465
$495$	0	0
$496$	−1.74457e8	−0.0641951
$497$	1.36894e9	0.500194
$498$	4.82016e8	0.174888
$499$	−4.66125e9	−1.67939	−0.839693	−	0.543061i	$-0.817265\pi$
$499$	−4.66125e9	−1.67939	−0.839693	+	0.543061i	$0.817265\pi$
$500$	0	0
$501$	−1.73379e9	−0.615977
$502$	−1.94010e9	−0.684479
$503$	2.59549e9	0.909352	0.454676	−	0.890657i	$-0.349755\pi$
$503$	2.59549e9	0.909352	0.454676	+	0.890657i	$0.349755\pi$
$504$	1.91103e8	0.0664906
$505$	0	0
$506$	1.46765e9	0.503611
$507$	1.09617e9	0.373553
$508$	1.71490e9	0.580385
$509$	9.18100e8	0.308587	0.154293	−	0.988025i	$-0.450690\pi$
$509$	9.18100e8	0.308587	0.154293	+	0.988025i	$0.450690\pi$
$510$	0	0
$511$	−1.55245e9	−0.514689
$512$	−1.34218e8	−0.0441942
$513$	−2.44778e8	−0.0800500
$514$	2.58365e9	0.839193
$515$	0	0
$516$	9.34039e8	0.299289
$517$	−7.30745e9	−2.32567
$518$	2.22849e9	0.704462
$519$	−2.97157e9	−0.933039
$520$	0	0
$521$	−5.35323e8	−0.165838	−0.0829189	−	0.996556i	$-0.526424\pi$
$521$	−5.35323e8	−0.165838	−0.0829189	+	0.996556i	$0.526424\pi$
$522$	1.09591e9	0.337233
$523$	−5.20020e8	−0.158951	−0.0794756	−	0.996837i	$-0.525325\pi$
$523$	−5.20020e8	−0.158951	−0.0794756	+	0.996837i	$0.525325\pi$
$524$	9.47747e8	0.287762
$525$	0	0
$526$	−2.69941e9	−0.808758
$527$	−4.22427e8	−0.125723
$528$	6.03832e8	0.178524
$529$	−2.27587e9	−0.668424
$530$	0	0
$531$	1.91537e8	0.0555166
$532$	4.07503e8	0.117338
$533$	−3.80609e9	−1.08876
$534$	2.17095e9	0.616957
$535$	0	0
$536$	1.82230e9	0.511143
$537$	2.34570e9	0.653676
$538$	−2.04708e9	−0.566758
$539$	−3.06524e9	−0.843148
$540$	0	0
$541$	−3.93186e9	−1.06760	−0.533798	−	0.845612i	$-0.679236\pi$
$541$	−3.93186e9	−1.06760	−0.533798	+	0.845612i	$0.679236\pi$
$542$	−3.88543e9	−1.04819
$543$	3.50271e9	0.938867
$544$	−3.24993e8	−0.0865522
$545$	0	0
$546$	−1.12428e9	−0.295597
$547$	−3.11516e9	−0.813814	−0.406907	−	0.913470i	$-0.633393\pi$
$547$	−3.11516e9	−0.813814	−0.406907	+	0.913470i	$0.633393\pi$
$548$	3.68981e9	0.957793
$549$	−7.11745e8	−0.183578
$550$	0	0
$551$	2.33690e9	0.595126
$552$	4.64486e8	0.117540
$553$	2.80361e9	0.704986
$554$	2.34186e9	0.585163
$555$	0	0
$556$	−5.36192e8	−0.132300
$557$	−4.14733e9	−1.01689	−0.508447	−	0.861093i	$-0.669780\pi$
$557$	−4.14733e9	−1.01689	−0.508447	+	0.861093i	$0.669780\pi$
$558$	2.48397e8	0.0605238
$559$	−5.49505e9	−1.33055
$560$	0	0
$561$	1.46211e9	0.349631
$562$	1.48980e9	0.354038
$563$	5.15380e8	0.121716	0.0608580	−	0.998146i	$-0.480616\pi$
$563$	5.15380e8	0.121716	0.0608580	+	0.998146i	$0.480616\pi$
$564$	−2.31269e9	−0.542800
$565$	0	0
$566$	−5.26601e9	−1.22074
$567$	−2.72098e8	−0.0626880
$568$	1.36894e9	0.313449
$569$	−4.82868e8	−0.109884	−0.0549421	−	0.998490i	$-0.517497\pi$
$569$	−4.82868e8	−0.109884	−0.0549421	+	0.998490i	$0.517497\pi$
$570$	0	0
$571$	−2.23974e9	−0.503466	−0.251733	−	0.967797i	$-0.581001\pi$
$571$	−2.23974e9	−0.503466	−0.251733	+	0.967797i	$0.581001\pi$
$572$	−3.55241e9	−0.793664
$573$	−3.83463e9	−0.851496
$574$	1.53352e9	0.338452
$575$	0	0
$576$	1.91103e8	0.0416667
$577$	−6.93164e9	−1.50218	−0.751088	−	0.660202i	$-0.770471\pi$
$577$	−6.93164e9	−1.50218	−0.751088	+	0.660202i	$0.770471\pi$
$578$	2.49578e9	0.537599
$579$	6.99057e8	0.149671
$580$	0	0
$581$	1.14256e9	0.241691
$582$	1.23715e9	0.260131
$583$	−7.14289e9	−1.49291
$584$	−1.55245e9	−0.322532
$585$	0	0
$586$	1.29141e9	0.265107
$587$	4.17248e9	0.851454	0.425727	−	0.904852i	$-0.360018\pi$
$587$	4.17248e9	0.851454	0.425727	+	0.904852i	$0.360018\pi$
$588$	−9.70097e8	−0.196786
$589$	5.29674e8	0.106808
$590$	0	0
$591$	9.69652e8	0.193223
$592$	2.22849e9	0.441454
$593$	−6.27101e9	−1.23494	−0.617470	−	0.786594i	$-0.711842\pi$
$593$	−6.27101e9	−1.23494	−0.617470	+	0.786594i	$0.711842\pi$
$594$	−8.59753e8	−0.168314
$595$	0	0
$596$	3.56145e8	0.0689073
$597$	−4.73722e9	−0.911199
$598$	−2.73262e9	−0.522547
$599$	−2.10159e9	−0.399535	−0.199767	−	0.979843i	$-0.564019\pi$
$599$	−2.10159e9	−0.399535	−0.199767	+	0.979843i	$0.564019\pi$
$600$	0	0
$601$	−2.41110e9	−0.453059	−0.226529	−	0.974004i	$-0.572738\pi$
$601$	−2.41110e9	−0.453059	−0.226529	+	0.974004i	$0.572738\pi$
$602$	2.21402e9	0.413612
$603$	−2.59464e9	−0.481910
$604$	4.24277e9	0.783466
$605$	0	0
$606$	2.88052e9	0.525796
$607$	5.94932e9	1.07971	0.539855	−	0.841758i	$-0.318479\pi$
$607$	5.94932e9	1.07971	0.539855	+	0.841758i	$0.318479\pi$
$608$	4.07503e8	0.0735306
$609$	2.59772e9	0.466050
$610$	0	0
$611$	1.36058e10	2.41312
$612$	4.62734e8	0.0816022
$613$	−5.45618e9	−0.956702	−0.478351	−	0.878169i	$-0.658765\pi$
$613$	−5.45618e9	−0.956702	−0.478351	+	0.878169i	$0.658765\pi$
$614$	−5.72391e9	−0.997937
$615$	0	0
$616$	1.43131e9	0.246718
$617$	4.11503e9	0.705301	0.352651	−	0.935755i	$-0.385280\pi$
$617$	4.11503e9	0.705301	0.352651	+	0.935755i	$0.385280\pi$
$618$	−5.85401e8	−0.0997687
$619$	2.78751e9	0.472388	0.236194	−	0.971706i	$-0.424100\pi$
$619$	2.78751e9	0.472388	0.236194	+	0.971706i	$0.424100\pi$
$620$	0	0
$621$	−6.61349e8	−0.110818
$622$	5.49353e8	0.0915346
$623$	5.14595e9	0.852623
$624$	−1.12428e9	−0.185237
$625$	0	0
$626$	7.42217e8	0.120926
$627$	−1.83332e9	−0.297030
$628$	4.11386e9	0.662813
$629$	5.39605e9	0.864567
$630$	0	0
$631$	4.87155e8	0.0771906	0.0385953	−	0.999255i	$-0.487712\pi$
$631$	4.87155e8	0.0771906	0.0385953	+	0.999255i	$0.487712\pi$
$632$	2.80361e9	0.441782
$633$	−2.47951e9	−0.388555
$634$	3.12599e9	0.487164
$635$	0	0
$636$	−2.26061e9	−0.348438
$637$	5.70718e9	0.874850
$638$	8.20808e9	1.25132
$639$	−1.94914e9	−0.295522
$640$	0	0
$641$	−2.39918e9	−0.359798	−0.179899	−	0.983685i	$-0.557577\pi$
$641$	−2.39918e9	−0.359798	−0.179899	+	0.983685i	$0.557577\pi$
$642$	2.44502e9	0.364678
$643$	1.02984e10	1.52767	0.763836	−	0.645410i	$-0.223313\pi$
$643$	1.02984e10	1.52767	0.763836	+	0.645410i	$0.223313\pi$
$644$	1.10100e9	0.162438
$645$	0	0
$646$	9.86722e8	0.144006
$647$	8.49896e9	1.23367	0.616837	−	0.787091i	$-0.288414\pi$
$647$	8.49896e9	1.23367	0.616837	+	0.787091i	$0.288414\pi$
$648$	−2.72098e8	−0.0392837
$649$	1.43456e9	0.205998
$650$	0	0
$651$	5.88792e8	0.0836427
$652$	−5.43073e9	−0.767347
$653$	3.78543e9	0.532009	0.266004	−	0.963972i	$-0.414296\pi$
$653$	3.78543e9	0.532009	0.266004	+	0.963972i	$0.414296\pi$
$654$	4.74621e9	0.663476
$655$	0	0
$656$	1.53352e9	0.212093
$657$	2.21043e9	0.304086
$658$	−5.48192e9	−0.750140
$659$	−7.69061e9	−1.04680	−0.523398	−	0.852089i	$-0.675336\pi$
$659$	−7.69061e9	−1.04680	−0.523398	+	0.852089i	$0.675336\pi$
$660$	0	0
$661$	5.78185e9	0.778685	0.389343	−	0.921093i	$-0.372702\pi$
$661$	5.78185e9	0.778685	0.389343	+	0.921093i	$0.372702\pi$
$662$	3.84215e9	0.514720
$663$	−2.72231e9	−0.362778
$664$	1.14256e9	0.151457
$665$	0	0
$666$	−3.17299e9	−0.416207
$667$	6.31391e9	0.823869
$668$	−4.10972e9	−0.533451
$669$	−3.12607e8	−0.0403652
$670$	0	0
$671$	−5.33076e9	−0.681178
$672$	4.52985e8	0.0575826
$673$	1.07125e10	1.35468	0.677340	−	0.735670i	$-0.263133\pi$
$673$	1.07125e10	1.35468	0.677340	+	0.735670i	$0.263133\pi$
$674$	−1.84403e9	−0.231984
$675$	0	0
$676$	2.59834e9	0.323506
$677$	−1.24253e8	−0.0153903	−0.00769516	−	0.999970i	$-0.502449\pi$
$677$	−1.24253e8	−0.0153903	−0.00769516	+	0.999970i	$0.502449\pi$
$678$	7.95655e8	0.0980439
$679$	2.93251e9	0.359497
$680$	0	0
$681$	8.09377e8	0.0982055
$682$	1.86042e9	0.224577
$683$	5.48973e8	0.0659294	0.0329647	−	0.999457i	$-0.489505\pi$
$683$	5.48973e8	0.0659294	0.0329647	+	0.999457i	$0.489505\pi$
$684$	−5.80214e8	−0.0693253
$685$	0	0
$686$	−5.67272e9	−0.670899
$687$	3.23246e9	0.380351
$688$	2.21402e9	0.259192
$689$	1.32994e10	1.54905
$690$	0	0
$691$	3.86163e9	0.445243	0.222622	−	0.974905i	$-0.428539\pi$
$691$	3.86163e9	0.445243	0.222622	+	0.974905i	$0.428539\pi$
$692$	−7.04372e9	−0.808036
$693$	−2.03793e9	−0.232608
$694$	−6.69381e9	−0.760178
$695$	0	0
$696$	2.59772e9	0.292052
$697$	3.71324e9	0.415373
$698$	−1.21327e10	−1.35041
$699$	3.44539e9	0.381565
$700$	0	0
$701$	−1.24129e10	−1.36100	−0.680501	−	0.732748i	$-0.738238\pi$
$701$	−1.24129e10	−1.36100	−0.680501	+	0.732748i	$0.738238\pi$
$702$	1.60078e9	0.174643
$703$	−6.76600e9	−0.734495
$704$	1.43131e9	0.154607
$705$	0	0
$706$	1.86413e8	0.0199370
$707$	6.82791e9	0.726641
$708$	4.54015e8	0.0480788
$709$	8.68970e9	0.915678	0.457839	−	0.889035i	$-0.348624\pi$
$709$	8.68970e9	0.915678	0.457839	+	0.889035i	$0.348624\pi$
$710$	0	0
$711$	−3.99186e9	−0.416516
$712$	5.14595e9	0.534300
$713$	1.43109e9	0.147861
$714$	1.09685e9	0.112773
$715$	0	0
$716$	5.56018e9	0.566100
$717$	−4.45362e9	−0.451228
$718$	−1.30530e10	−1.31605
$719$	1.34874e10	1.35325	0.676625	−	0.736328i	$-0.263442\pi$
$719$	1.34874e10	1.35325	0.676625	+	0.736328i	$0.263442\pi$
$720$	0	0
$721$	−1.38762e9	−0.137879
$722$	5.91374e9	0.584766
$723$	1.07630e10	1.05913
$724$	8.30271e9	0.813083
$725$	0	0
$726$	−2.23008e9	−0.216293
$727$	4.09956e9	0.395700	0.197850	−	0.980232i	$-0.436604\pi$
$727$	4.09956e9	0.395700	0.197850	+	0.980232i	$0.436604\pi$
$728$	−2.66496e9	−0.255994
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	5.36100e9	0.507615
$732$	−1.68710e9	−0.158983
$733$	−9.50716e9	−0.891635	−0.445818	−	0.895124i	$-0.647087\pi$
$733$	−9.50716e9	−0.891635	−0.445818	+	0.895124i	$0.647087\pi$
$734$	2.78246e9	0.259712
$735$	0	0
$736$	1.10100e9	0.101793
$737$	−1.94331e10	−1.78816
$738$	−2.18347e9	−0.199963
$739$	2.75490e9	0.251102	0.125551	−	0.992087i	$-0.459930\pi$
$739$	2.75490e9	0.251102	0.125551	+	0.992087i	$0.459930\pi$
$740$	0	0
$741$	3.41346e9	0.308199
$742$	−5.35848e9	−0.481535
$743$	−7.88700e9	−0.705425	−0.352712	−	0.935732i	$-0.614741\pi$
$743$	−7.88700e9	−0.705425	−0.352712	+	0.935732i	$0.614741\pi$
$744$	5.88792e8	0.0524151
$745$	0	0
$746$	4.00615e8	0.0353298
$747$	−1.62680e9	−0.142795
$748$	3.46575e9	0.302790
$749$	5.79559e9	0.503978
$750$	0	0
$751$	1.54231e10	1.32871	0.664356	−	0.747416i	$-0.268706\pi$
$751$	1.54231e10	1.32871	0.664356	+	0.747416i	$0.268706\pi$
$752$	−5.48192e9	−0.470079
$753$	6.54782e9	0.558875
$754$	−1.52827e10	−1.29837
$755$	0	0
$756$	−6.44973e8	−0.0542894
$757$	2.28786e10	1.91687	0.958436	−	0.285307i	$-0.0920956\pi$
$757$	2.28786e10	1.91687	0.958436	+	0.285307i	$0.0920956\pi$
$758$	−1.12123e10	−0.935087
$759$	−4.95331e9	−0.411197
$760$	0	0
$761$	−2.06723e10	−1.70037	−0.850183	−	0.526488i	$-0.823509\pi$
$761$	−2.06723e10	−1.70037	−0.850183	+	0.526488i	$0.823509\pi$
$762$	−5.78779e9	−0.473882
$763$	1.12503e10	0.916912
$764$	−9.08950e9	−0.737418
$765$	0	0
$766$	8.54640e9	0.687041
$767$	−2.67101e9	−0.213743
$768$	4.52985e8	0.0360844
$769$	−7.74699e9	−0.614315	−0.307157	−	0.951659i	$-0.599378\pi$
$769$	−7.74699e9	−0.614315	−0.307157	+	0.951659i	$0.599378\pi$
$770$	0	0
$771$	−8.71980e9	−0.685198
$772$	1.65702e9	0.129619
$773$	−1.55352e10	−1.20973	−0.604866	−	0.796327i	$-0.706773\pi$
$773$	−1.55352e10	−1.20973	−0.604866	+	0.796327i	$0.706773\pi$
$774$	−3.15238e9	−0.244369
$775$	0	0
$776$	2.93251e9	0.225280
$777$	−7.52117e9	−0.575191
$778$	1.70970e9	0.130164
$779$	−4.65596e9	−0.352881
$780$	0	0
$781$	−1.45985e10	−1.09655
$782$	2.66596e9	0.199356
$783$	−3.69871e9	−0.275349
$784$	−2.29949e9	−0.170422
$785$	0	0
$786$	−3.19864e9	−0.234956
$787$	−1.21211e10	−0.886399	−0.443199	−	0.896423i	$-0.646157\pi$
$787$	−1.21211e10	−0.886399	−0.443199	+	0.896423i	$0.646157\pi$
$788$	2.29843e9	0.167336
$789$	9.11051e9	0.660349
$790$	0	0
$791$	1.88600e9	0.135495
$792$	−2.03793e9	−0.145765
$793$	9.92537e9	0.706790
$794$	7.75270e9	0.549644
$795$	0	0
$796$	−1.12290e10	−0.789122
$797$	−8.20343e9	−0.573973	−0.286987	−	0.957935i	$-0.592654\pi$
$797$	−8.20343e9	−0.573973	−0.286987	+	0.957935i	$0.592654\pi$
$798$	−1.37532e9	−0.0958064
$799$	−1.32739e10	−0.920626
$800$	0	0
$801$	−7.32694e9	−0.503743
$802$	1.77413e10	1.21444
$803$	1.65555e10	1.12833
$804$	−6.15025e9	−0.417346
$805$	0	0
$806$	−3.46392e9	−0.233021
$807$	6.90891e9	0.462756
$808$	6.82791e9	0.455353
$809$	9.91739e9	0.658533	0.329267	−	0.944237i	$-0.393199\pi$
$809$	9.91739e9	0.658533	0.329267	+	0.944237i	$0.393199\pi$
$810$	0	0
$811$	2.13417e10	1.40494	0.702469	−	0.711715i	$-0.252081\pi$
$811$	2.13417e10	1.40494	0.702469	+	0.711715i	$0.252081\pi$
$812$	6.15757e9	0.403611
$813$	1.31133e10	0.855847
$814$	−2.37648e10	−1.54436
$815$	0	0
$816$	1.09685e9	0.0706695
$817$	−6.72206e9	−0.431246
$818$	8.38695e9	0.535756
$819$	3.79444e9	0.241354
$820$	0	0
$821$	1.81759e9	0.114629	0.0573145	−	0.998356i	$-0.481746\pi$
$821$	1.81759e9	0.114629	0.0573145	+	0.998356i	$0.481746\pi$
$822$	−1.24531e10	−0.782035
$823$	−9.68482e9	−0.605609	−0.302804	−	0.953053i	$-0.597923\pi$
$823$	−9.68482e9	−0.605609	−0.302804	+	0.953053i	$0.597923\pi$
$824$	−1.38762e9	−0.0864022
$825$	0	0
$826$	1.07618e9	0.0664440
$827$	−1.22370e10	−0.752326	−0.376163	−	0.926553i	$-0.622757\pi$
$827$	−1.22370e10	−0.752326	−0.376163	+	0.926553i	$0.622757\pi$
$828$	−1.56764e9	−0.0959711
$829$	−1.30073e9	−0.0792951	−0.0396476	−	0.999214i	$-0.512624\pi$
$829$	−1.30073e9	−0.0792951	−0.0396476	+	0.999214i	$0.512624\pi$
$830$	0	0
$831$	−7.90377e9	−0.477783
$832$	−2.66496e9	−0.160420
$833$	−5.56796e9	−0.333763
$834$	1.80965e9	0.108022
$835$	0	0
$836$	−4.34564e9	−0.257236
$837$	−8.38338e8	−0.0494174
$838$	1.76817e9	0.103793
$839$	−1.65720e10	−0.968743	−0.484372	−	0.874862i	$-0.660952\pi$
$839$	−1.65720e10	−0.968743	−0.484372	+	0.874862i	$0.660952\pi$
$840$	0	0
$841$	1.80618e10	1.04707
$842$	−1.21071e10	−0.698956
$843$	−5.02806e9	−0.289071
$844$	−5.87736e9	−0.336499
$845$	0	0
$846$	7.80532e9	0.443194
$847$	−5.28611e9	−0.298912
$848$	−5.35848e9	−0.301756
$849$	1.77728e10	0.996732
$850$	0	0
$851$	−1.82806e10	−1.01680
$852$	−4.62019e9	−0.255930
$853$	2.87728e10	1.58730	0.793652	−	0.608372i	$-0.208177\pi$
$853$	2.87728e10	1.58730	0.793652	+	0.608372i	$0.208177\pi$
$854$	−3.99905e9	−0.219712
$855$	0	0
$856$	5.79559e9	0.315820
$857$	2.85737e9	0.155072	0.0775361	−	0.996990i	$-0.475295\pi$
$857$	2.85737e9	0.155072	0.0775361	+	0.996990i	$0.475295\pi$
$858$	1.19894e10	0.648024
$859$	1.38088e10	0.743327	0.371663	−	0.928368i	$-0.378788\pi$
$859$	1.38088e10	0.743327	0.371663	+	0.928368i	$0.378788\pi$
$860$	0	0
$861$	−5.17562e9	−0.276345
$862$	−1.17441e10	−0.624518
$863$	2.72992e9	0.144581	0.0722905	−	0.997384i	$-0.476969\pi$
$863$	2.72992e9	0.144581	0.0722905	+	0.997384i	$0.476969\pi$
$864$	−6.44973e8	−0.0340207
$865$	0	0
$866$	3.18006e9	0.166388
$867$	−8.42324e9	−0.438947
$868$	1.39565e9	0.0724367
$869$	−2.98979e10	−1.54551
$870$	0	0
$871$	3.61825e10	1.85539
$872$	1.12503e10	0.574587
$873$	−4.17539e9	−0.212396
$874$	−3.34280e9	−0.169364
$875$	0	0
$876$	5.23953e9	0.263347
$877$	−1.52412e10	−0.762992	−0.381496	−	0.924370i	$-0.624591\pi$
$877$	−1.52412e10	−0.762992	−0.381496	+	0.924370i	$0.624591\pi$
$878$	−9.76565e9	−0.486935
$879$	−4.35850e9	−0.216459
$880$	0	0
$881$	−1.89703e10	−0.934672	−0.467336	−	0.884080i	$-0.654786\pi$
$881$	−1.89703e10	−0.934672	−0.467336	+	0.884080i	$0.654786\pi$
$882$	3.27408e9	0.160675
$883$	−1.03565e10	−0.506232	−0.253116	−	0.967436i	$-0.581455\pi$
$883$	−1.03565e10	−0.506232	−0.253116	+	0.967436i	$0.581455\pi$
$884$	−6.45289e9	−0.314175
$885$	0	0
$886$	2.59641e10	1.25417
$887$	4.04503e9	0.194621	0.0973103	−	0.995254i	$-0.468976\pi$
$887$	4.04503e9	0.194621	0.0973103	+	0.995254i	$0.468976\pi$
$888$	−7.52117e9	−0.360446
$889$	−1.37192e10	−0.654897
$890$	0	0
$891$	2.90167e9	0.137428
$892$	−7.40995e8	−0.0349573
$893$	1.66438e10	0.782120
$894$	−1.20199e9	−0.0562625
$895$	0	0
$896$	1.07374e9	0.0498680
$897$	9.22260e9	0.426658
$898$	5.63369e9	0.259612
$899$	8.00363e9	0.367391
$900$	0	0
$901$	−1.29749e10	−0.590975
$902$	−1.63535e10	−0.741974
$903$	−7.47231e9	−0.337713
$904$	1.88600e9	0.0849085
$905$	0	0
$906$	−1.43194e10	−0.639697
$907$	3.30880e10	1.47247	0.736233	−	0.676728i	$-0.236603\pi$
$907$	3.30880e10	1.47247	0.736233	+	0.676728i	$0.236603\pi$
$908$	1.91852e9	0.0850485
$909$	−9.72177e9	−0.429311
$910$	0	0
$911$	−2.13526e10	−0.935700	−0.467850	−	0.883808i	$-0.654971\pi$
$911$	−2.13526e10	−0.935700	−0.467850	+	0.883808i	$0.654971\pi$
$912$	−1.37532e9	−0.0600375
$913$	−1.21843e10	−0.529850
$914$	2.96776e10	1.28563
$915$	0	0
$916$	7.66213e9	0.329394
$917$	−7.58197e9	−0.324705
$918$	−1.56173e9	−0.0666279
$919$	4.49811e10	1.91173	0.955863	−	0.293813i	$-0.0949242\pi$
$919$	4.49811e10	1.91173	0.955863	+	0.293813i	$0.0949242\pi$
$920$	0	0
$921$	1.93182e10	0.814813
$922$	−9.00115e9	−0.378216
$923$	2.71810e10	1.13778
$924$	−4.83066e9	−0.201444
$925$	0	0
$926$	−6.11462e8	−0.0253065
$927$	1.97573e9	0.0814608
$928$	6.15757e9	0.252925
$929$	−2.15112e10	−0.880258	−0.440129	−	0.897935i	$-0.645067\pi$
$929$	−2.15112e10	−0.880258	−0.440129	+	0.897935i	$0.645067\pi$
$930$	0	0
$931$	6.98156e9	0.283549
$932$	8.16685e9	0.330445
$933$	−1.85407e9	−0.0747377
$934$	−5.13062e8	−0.0206042
$935$	0	0
$936$	3.79444e9	0.151245
$937$	3.31276e10	1.31553	0.657766	−	0.753222i	$-0.271502\pi$
$937$	3.31276e10	1.31553	0.657766	+	0.753222i	$0.271502\pi$
$938$	−1.45784e10	−0.576765
$939$	−2.50498e9	−0.0987358
$940$	0	0
$941$	1.55361e10	0.607824	0.303912	−	0.952700i	$-0.401707\pi$
$941$	1.55361e10	0.607824	0.303912	+	0.952700i	$0.401707\pi$
$942$	−1.38843e10	−0.541184
$943$	−1.25796e10	−0.488514
$944$	1.07618e9	0.0416375
$945$	0	0
$946$	−2.36104e10	−0.906745
$947$	−5.05218e10	−1.93310	−0.966549	−	0.256482i	$-0.917436\pi$
$947$	−5.05218e10	−1.93310	−0.966549	+	0.256482i	$0.917436\pi$
$948$	−9.46220e9	−0.360714
$949$	−3.08247e10	−1.17076
$950$	0	0
$951$	−1.05502e10	−0.397768
$952$	2.59994e9	0.0976640
$953$	7.93237e9	0.296878	0.148439	−	0.988922i	$-0.452575\pi$
$953$	7.93237e9	0.296878	0.148439	+	0.988922i	$0.452575\pi$
$954$	7.62955e9	0.284498
$955$	0	0
$956$	−1.05567e10	−0.390775
$957$	−2.77023e10	−1.02170
$958$	−8.11087e9	−0.298049
$959$	−2.95185e10	−1.08076
$960$	0	0
$961$	−2.56985e10	−0.934064
$962$	4.42478e10	1.60243
$963$	−8.25193e9	−0.297758
$964$	2.55122e10	0.917230
$965$	0	0
$966$	−3.71589e9	−0.132630
$967$	−2.50000e10	−0.889095	−0.444547	−	0.895755i	$-0.646635\pi$
$967$	−2.50000e10	−0.889095	−0.444547	+	0.895755i	$0.646635\pi$
$968$	−5.28611e9	−0.187315
$969$	−3.33019e9	−0.117581
$970$	0	0
$971$	2.23544e10	0.783603	0.391801	−	0.920050i	$-0.371852\pi$
$971$	2.23544e10	0.783603	0.391801	+	0.920050i	$0.371852\pi$
$972$	9.18330e8	0.0320750
$973$	4.28953e9	0.149285
$974$	−2.23208e9	−0.0774020
$975$	0	0
$976$	−3.99905e9	−0.137684
$977$	1.84311e9	0.0632297	0.0316148	−	0.999500i	$-0.489935\pi$
$977$	1.84311e9	0.0632297	0.0316148	+	0.999500i	$0.489935\pi$
$978$	1.83287e10	0.626536
$979$	−5.48767e10	−1.86917
$980$	0	0
$981$	−1.60185e10	−0.541726
$982$	−1.41871e10	−0.478083
$983$	−1.25816e10	−0.422473	−0.211236	−	0.977435i	$-0.567749\pi$
$983$	−1.25816e10	−0.422473	−0.211236	+	0.977435i	$0.567749\pi$
$984$	−5.17562e9	−0.173173
$985$	0	0
$986$	1.49098e10	0.495341
$987$	1.85015e10	0.612487
$988$	8.09116e9	0.266908
$989$	−1.81619e10	−0.596999
$990$	0	0
$991$	−2.30746e10	−0.753142	−0.376571	−	0.926388i	$-0.622897\pi$
$991$	−2.30746e10	−0.753142	−0.376571	+	0.926388i	$0.622897\pi$
$992$	1.39565e9	0.0453928
$993$	−1.29673e10	−0.420267
$994$	−1.09516e10	−0.353691
$995$	0	0
$996$	−3.85613e9	−0.123664
$997$	5.91619e9	0.189064	0.0945320	−	0.995522i	$-0.469865\pi$
$997$	5.91619e9	0.189064	0.0945320	+	0.995522i	$0.469865\pi$
$998$	3.72900e10	1.18751
$999$	1.07089e10	0.339832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.8.a.g.1.1		1
3.2	odd	2		450.8.a.r.1.1		1
5.2	odd	4		150.8.c.e.49.1		2
5.3	odd	4		150.8.c.e.49.2		2
5.4	even	2		30.8.a.e.1.1	✓	1
15.2	even	4		450.8.c.c.199.2		2
15.8	even	4		450.8.c.c.199.1		2
15.14	odd	2		90.8.a.b.1.1		1
20.19	odd	2		240.8.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.8.a.e.1.1	✓	1	5.4	even	2
90.8.a.b.1.1		1	15.14	odd	2
150.8.a.g.1.1		1	1.1	even	1	trivial
150.8.c.e.49.1		2	5.2	odd	4
150.8.c.e.49.2		2	5.3	odd	4
240.8.a.l.1.1		1	20.19	odd	2
450.8.a.r.1.1		1	3.2	odd	2
450.8.c.c.199.1		2	15.8	even	4
450.8.c.c.199.2		2	15.2	even	4

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	150.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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