LMFDB - Embedded newform 150.8.a.a.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:	$0$
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)

$f(q)$

$=$

$q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +216.000 q^{6} -1604.00 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})$

\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +216.000 q^{6} -1604.00 q^{7} -512.000 q^{8} +729.000 q^{9} -2208.00 q^{11} -1728.00 q^{12} -5738.00 q^{13} +12832.0 q^{14} +4096.00 q^{16} -15654.0 q^{17} -5832.00 q^{18} -19660.0 q^{19} +43308.0 q^{21} +17664.0 q^{22} +28512.0 q^{23} +13824.0 q^{24} +45904.0 q^{26} -19683.0 q^{27} -102656. q^{28} -140190. q^{29} -291208. q^{31} -32768.0 q^{32} +59616.0 q^{33} +125232. q^{34} +46656.0 q^{36} +135046. q^{37} +157280. q^{38} +154926. q^{39} -804438. q^{41} -346464. q^{42} -721268. q^{43} -141312. q^{44} -228096. q^{46} +802656. q^{47} -110592. q^{48} +1.74927e6 q^{49} +422658. q^{51} -367232. q^{52} -274098. q^{53} +157464. q^{54} +821248. q^{56} +530820. q^{57} +1.12152e6 q^{58} +1.96944e6 q^{59} +3.17934e6 q^{61} +2.32966e6 q^{62} -1.16932e6 q^{63} +262144. q^{64} -476928. q^{66} +1.36376e6 q^{67} -1.00186e6 q^{68} -769824. q^{69} -4.38989e6 q^{71} -373248. q^{72} +4.27886e6 q^{73} -1.08037e6 q^{74} -1.25824e6 q^{76} +3.54163e6 q^{77} -1.23941e6 q^{78} +3.85196e6 q^{79} +531441. q^{81} +6.43550e6 q^{82} -8.53223e6 q^{83} +2.77171e6 q^{84} +5.77014e6 q^{86} +3.78513e6 q^{87} +1.13050e6 q^{88} +3.73341e6 q^{89} +9.20375e6 q^{91} +1.82477e6 q^{92} +7.86262e6 q^{93} -6.42125e6 q^{94} +884736. q^{96} +1.56862e7 q^{97} -1.39942e7 q^{98} -1.60963e6 q^{99} +O(q^{100})\)

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
$7$	−1604.00	−1.76751	−0.883754	−	0.467952i	$-0.844992\pi$
$7$	−1604.00	−1.76751	−0.883754	+	0.467952i	$0.844992\pi$
$8$	−512.000	−0.353553
$9$	729.000	0.333333
$10$	0	0
$11$	−2208.00	−0.500178	−0.250089	−	0.968223i	$-0.580460\pi$
$11$	−2208.00	−0.500178	−0.250089	+	0.968223i	$0.580460\pi$
$12$	−1728.00	−0.288675
$13$	−5738.00	−0.724367	−0.362184	−	0.932107i	$-0.617969\pi$
$13$	−5738.00	−0.724367	−0.362184	+	0.932107i	$0.617969\pi$
$14$	12832.0	1.24982
$15$	0	0
$16$	4096.00	0.250000
$17$	−15654.0	−0.772777	−0.386388	−	0.922336i	$-0.626278\pi$
$17$	−15654.0	−0.772777	−0.386388	+	0.922336i	$0.626278\pi$
$18$	−5832.00	−0.235702
$19$	−19660.0	−0.657576	−0.328788	−	0.944404i	$-0.606640\pi$
$19$	−19660.0	−0.657576	−0.328788	+	0.944404i	$0.606640\pi$
$20$	0	0
$21$	43308.0	1.02047
$22$	17664.0	0.353679
$23$	28512.0	0.488630	0.244315	−	0.969696i	$-0.421437\pi$
$23$	28512.0	0.488630	0.244315	+	0.969696i	$0.421437\pi$
$24$	13824.0	0.204124
$25$	0	0
$26$	45904.0	0.512205
$27$	−19683.0	−0.192450
$28$	−102656.	−0.883754
$29$	−140190.	−1.06739	−0.533696	−	0.845676i	$-0.679197\pi$
$29$	−140190.	−1.06739	−0.533696	+	0.845676i	$0.679197\pi$
$30$	0	0
$31$	−291208.	−1.75565	−0.877824	−	0.478984i	$-0.841005\pi$
$31$	−291208.	−1.75565	−0.877824	+	0.478984i	$0.841005\pi$
$32$	−32768.0	−0.176777
$33$	59616.0	0.288778
$34$	125232.	0.546436
$35$	0	0
$36$	46656.0	0.166667
$37$	135046.	0.438304	0.219152	−	0.975691i	$-0.429671\pi$
$37$	135046.	0.438304	0.219152	+	0.975691i	$0.429671\pi$
$38$	157280.	0.464976
$39$	154926.	0.418214
$40$	0	0
$41$	−804438.	−1.82284	−0.911421	−	0.411475i	$-0.865014\pi$
$41$	−804438.	−1.82284	−0.911421	+	0.411475i	$0.865014\pi$
$42$	−346464.	−0.721582
$43$	−721268.	−1.38343	−0.691715	−	0.722171i	$-0.743144\pi$
$43$	−721268.	−1.38343	−0.691715	+	0.722171i	$0.743144\pi$
$44$	−141312.	−0.250089
$45$	0	0
$46$	−228096.	−0.345514
$47$	802656.	1.12768	0.563841	−	0.825883i	$-0.309323\pi$
$47$	802656.	1.12768	0.563841	+	0.825883i	$0.309323\pi$
$48$	−110592.	−0.144338
$49$	1.74927e6	2.12408
$50$	0	0
$51$	422658.	0.446163
$52$	−367232.	−0.362184
$53$	−274098.	−0.252895	−0.126448	−	0.991973i	$-0.540358\pi$
$53$	−274098.	−0.252895	−0.126448	+	0.991973i	$0.540358\pi$
$54$	157464.	0.136083
$55$	0	0
$56$	821248.	0.624908
$57$	530820.	0.379652
$58$	1.12152e6	0.754760
$59$	1.96944e6	1.24842	0.624210	−	0.781257i	$-0.285421\pi$
$59$	1.96944e6	1.24842	0.624210	+	0.781257i	$0.285421\pi$
$60$	0	0
$61$	3.17934e6	1.79342	0.896712	−	0.442615i	$-0.145949\pi$
$61$	3.17934e6	1.79342	0.896712	+	0.442615i	$0.145949\pi$
$62$	2.32966e6	1.24143
$63$	−1.16932e6	−0.589169
$64$	262144.	0.125000
$65$	0	0
$66$	−476928.	−0.204197
$67$	1.36376e6	0.553955	0.276978	−	0.960876i	$-0.410667\pi$
$67$	1.36376e6	0.553955	0.276978	+	0.960876i	$0.410667\pi$
$68$	−1.00186e6	−0.386388
$69$	−769824.	−0.282111
$70$	0	0
$71$	−4.38989e6	−1.45562	−0.727812	−	0.685777i	$-0.759463\pi$
$71$	−4.38989e6	−1.45562	−0.727812	+	0.685777i	$0.759463\pi$
$72$	−373248.	−0.117851
$73$	4.27886e6	1.28735	0.643677	−	0.765297i	$-0.277408\pi$
$73$	4.27886e6	1.28735	0.643677	+	0.765297i	$0.277408\pi$
$74$	−1.08037e6	−0.309928
$75$	0	0
$76$	−1.25824e6	−0.328788
$77$	3.54163e6	0.884068
$78$	−1.23941e6	−0.295722
$79$	3.85196e6	0.878996	0.439498	−	0.898244i	$-0.355156\pi$
$79$	3.85196e6	0.878996	0.439498	+	0.898244i	$0.355156\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	6.43550e6	1.28894
$83$	−8.53223e6	−1.63791	−0.818953	−	0.573860i	$-0.805445\pi$
$83$	−8.53223e6	−1.63791	−0.818953	+	0.573860i	$0.805445\pi$
$84$	2.77171e6	0.510235
$85$	0	0
$86$	5.77014e6	0.978232
$87$	3.78513e6	0.616259
$88$	1.13050e6	0.176840
$89$	3.73341e6	0.561359	0.280679	−	0.959802i	$-0.409440\pi$
$89$	3.73341e6	0.561359	0.280679	+	0.959802i	$0.409440\pi$
$90$	0	0
$91$	9.20375e6	1.28032
$92$	1.82477e6	0.244315
$93$	7.86262e6	1.01362
$94$	−6.42125e6	−0.797392
$95$	0	0
$96$	884736.	0.102062
$97$	1.56862e7	1.74509	0.872543	−	0.488537i	$-0.162469\pi$
$97$	1.56862e7	1.74509	0.872543	+	0.488537i	$0.162469\pi$
$98$	−1.39942e7	−1.50195
$99$	−1.60963e6	−0.166726
$100$	0	0
$101$	3.18772e6	0.307862	0.153931	−	0.988082i	$-0.450807\pi$
$101$	3.18772e6	0.307862	0.153931	+	0.988082i	$0.450807\pi$
$102$	−3.38126e6	−0.315485
$103$	−1.01905e7	−0.918891	−0.459445	−	0.888206i	$-0.651952\pi$
$103$	−1.01905e7	−0.918891	−0.459445	+	0.888206i	$0.651952\pi$
$104$	2.93786e6	0.256102
$105$	0	0
$106$	2.19278e6	0.178824
$107$	3.96964e6	0.313262	0.156631	−	0.987657i	$-0.449937\pi$
$107$	3.96964e6	0.313262	0.156631	+	0.987657i	$0.449937\pi$
$108$	−1.25971e6	−0.0962250
$109$	9.13259e6	0.675462	0.337731	−	0.941243i	$-0.390341\pi$
$109$	9.13259e6	0.675462	0.337731	+	0.941243i	$0.390341\pi$
$110$	0	0
$111$	−3.64624e6	−0.253055
$112$	−6.56998e6	−0.441877
$113$	1.14854e7	0.748807	0.374404	−	0.927266i	$-0.377847\pi$
$113$	1.14854e7	0.748807	0.374404	+	0.927266i	$0.377847\pi$
$114$	−4.24656e6	−0.268454
$115$	0	0
$116$	−8.97216e6	−0.533696
$117$	−4.18300e6	−0.241456
$118$	−1.57555e7	−0.882766
$119$	2.51090e7	1.36589
$120$	0	0
$121$	−1.46119e7	−0.749822
$122$	−2.54347e7	−1.26814
$123$	2.17198e7	1.05242
$124$	−1.86373e7	−0.877824
$125$	0	0
$126$	9.35453e6	0.416605
$127$	−2.37634e7	−1.02943	−0.514714	−	0.857362i	$-0.672102\pi$
$127$	−2.37634e7	−1.02943	−0.514714	+	0.857362i	$0.672102\pi$
$128$	−2.09715e6	−0.0883883
$129$	1.94742e7	0.798723
$130$	0	0
$131$	−3.37491e7	−1.31163	−0.655817	−	0.754920i	$-0.727676\pi$
$131$	−3.37491e7	−1.31163	−0.655817	+	0.754920i	$0.727676\pi$
$132$	3.81542e6	0.144389
$133$	3.15346e7	1.16227
$134$	−1.09100e7	−0.391706
$135$	0	0
$136$	8.01485e6	0.273218
$137$	6.40267e6	0.212735	0.106367	−	0.994327i	$-0.466078\pi$
$137$	6.40267e6	0.212735	0.106367	+	0.994327i	$0.466078\pi$
$138$	6.15859e6	0.199482
$139$	−4.09246e6	−0.129251	−0.0646253	−	0.997910i	$-0.520585\pi$
$139$	−4.09246e6	−0.129251	−0.0646253	+	0.997910i	$0.520585\pi$
$140$	0	0
$141$	−2.16717e7	−0.651068
$142$	3.51191e7	1.02928
$143$	1.26695e7	0.362313
$144$	2.98598e6	0.0833333
$145$	0	0
$146$	−3.42309e7	−0.910297
$147$	−4.72304e7	−1.22634
$148$	8.64294e6	0.219152
$149$	−3.90838e7	−0.967932	−0.483966	−	0.875087i	$-0.660804\pi$
$149$	−3.90838e7	−0.967932	−0.483966	+	0.875087i	$0.660804\pi$
$150$	0	0
$151$	6.44781e7	1.52403	0.762014	−	0.647561i	$-0.224211\pi$
$151$	6.44781e7	1.52403	0.762014	+	0.647561i	$0.224211\pi$
$152$	1.00659e7	0.232488
$153$	−1.14118e7	−0.257592
$154$	−2.83331e7	−0.625131
$155$	0	0
$156$	9.91526e6	0.209107
$157$	−2.70913e7	−0.558703	−0.279351	−	0.960189i	$-0.590119\pi$
$157$	−2.70913e7	−0.558703	−0.279351	+	0.960189i	$0.590119\pi$
$158$	−3.08157e7	−0.621544
$159$	7.40065e6	0.146009
$160$	0	0
$161$	−4.57332e7	−0.863657
$162$	−4.25153e6	−0.0785674
$163$	6.82802e7	1.23492	0.617459	−	0.786603i	$-0.288162\pi$
$163$	6.82802e7	1.23492	0.617459	+	0.786603i	$0.288162\pi$
$164$	−5.14840e7	−0.911421
$165$	0	0
$166$	6.82578e7	1.15817
$167$	−7.17321e7	−1.19181	−0.595903	−	0.803056i	$-0.703206\pi$
$167$	−7.17321e7	−1.19181	−0.595903	+	0.803056i	$0.703206\pi$
$168$	−2.21737e7	−0.360791
$169$	−2.98239e7	−0.475292
$170$	0	0
$171$	−1.43321e7	−0.219192
$172$	−4.61612e7	−0.691715
$173$	−5.63879e7	−0.827989	−0.413994	−	0.910279i	$-0.635867\pi$
$173$	−5.63879e7	−0.827989	−0.413994	+	0.910279i	$0.635867\pi$
$174$	−3.02810e7	−0.435761
$175$	0	0
$176$	−9.04397e6	−0.125045
$177$	−5.31749e7	−0.720776
$178$	−2.98673e7	−0.396941
$179$	4.10039e7	0.534367	0.267183	−	0.963646i	$-0.413907\pi$
$179$	4.10039e7	0.534367	0.267183	+	0.963646i	$0.413907\pi$
$180$	0	0
$181$	2.44312e7	0.306246	0.153123	−	0.988207i	$-0.451067\pi$
$181$	2.44312e7	0.306246	0.153123	+	0.988207i	$0.451067\pi$
$182$	−7.36300e7	−0.905326
$183$	−8.58422e7	−1.03543
$184$	−1.45981e7	−0.172757
$185$	0	0
$186$	−6.29009e7	−0.716740
$187$	3.45640e7	0.386526
$188$	5.13700e7	0.563841
$189$	3.15715e7	0.340157
$190$	0	0
$191$	−4.72840e7	−0.491018	−0.245509	−	0.969394i	$-0.578955\pi$
$191$	−4.72840e7	−0.491018	−0.245509	+	0.969394i	$0.578955\pi$
$192$	−7.07789e6	−0.0721688
$193$	2.50714e7	0.251031	0.125516	−	0.992092i	$-0.459941\pi$
$193$	2.50714e7	0.251031	0.125516	+	0.992092i	$0.459941\pi$
$194$	−1.25490e8	−1.23396
$195$	0	0
$196$	1.11953e8	1.06204
$197$	8.42077e7	0.784730	0.392365	−	0.919810i	$-0.371657\pi$
$197$	8.42077e7	0.784730	0.392365	+	0.919810i	$0.371657\pi$
$198$	1.28771e7	0.117893
$199$	−8.87650e7	−0.798465	−0.399233	−	0.916850i	$-0.630723\pi$
$199$	−8.87650e7	−0.798465	−0.399233	+	0.916850i	$0.630723\pi$
$200$	0	0
$201$	−3.68214e7	−0.319826
$202$	−2.55018e7	−0.217691
$203$	2.24865e8	1.88662
$204$	2.70501e7	0.223081
$205$	0	0
$206$	8.15237e7	0.649754
$207$	2.07852e7	0.162877
$208$	−2.35028e7	−0.181092
$209$	4.34093e7	0.328905
$210$	0	0
$211$	9.13398e7	0.669378	0.334689	−	0.942329i	$-0.391369\pi$
$211$	9.13398e7	0.669378	0.334689	+	0.942329i	$0.391369\pi$
$212$	−1.75423e7	−0.126448
$213$	1.18527e8	0.840405
$214$	−3.17571e7	−0.221510
$215$	0	0
$216$	1.00777e7	0.0680414
$217$	4.67098e8	3.10312
$218$	−7.30607e7	−0.477624
$219$	−1.15529e8	−0.743255
$220$	0	0
$221$	8.98227e7	0.559774
$222$	2.91699e7	0.178937
$223$	9.49239e7	0.573203	0.286602	−	0.958050i	$-0.407474\pi$
$223$	9.49239e7	0.573203	0.286602	+	0.958050i	$0.407474\pi$
$224$	5.25599e7	0.312454
$225$	0	0
$226$	−9.18829e7	−0.529487
$227$	−2.98928e8	−1.69620	−0.848098	−	0.529839i	$-0.822252\pi$
$227$	−2.98928e8	−1.69620	−0.848098	+	0.529839i	$0.822252\pi$
$228$	3.39725e7	0.189826
$229$	4.52346e7	0.248912	0.124456	−	0.992225i	$-0.460281\pi$
$229$	4.52346e7	0.248912	0.124456	+	0.992225i	$0.460281\pi$
$230$	0	0
$231$	−9.56241e7	−0.510417
$232$	7.17773e7	0.377380
$233$	−2.27345e8	−1.17744	−0.588720	−	0.808337i	$-0.700368\pi$
$233$	−2.27345e8	−1.17744	−0.588720	+	0.808337i	$0.700368\pi$
$234$	3.34640e7	0.170735
$235$	0	0
$236$	1.26044e8	0.624210
$237$	−1.04003e8	−0.507489
$238$	−2.00872e8	−0.965829
$239$	2.90118e8	1.37462	0.687309	−	0.726365i	$-0.258792\pi$
$239$	2.90118e8	1.37462	0.687309	+	0.726365i	$0.258792\pi$
$240$	0	0
$241$	−3.05573e7	−0.140623	−0.0703113	−	0.997525i	$-0.522399\pi$
$241$	−3.05573e7	−0.140623	−0.0703113	+	0.997525i	$0.522399\pi$
$242$	1.16895e8	0.530204
$243$	−1.43489e7	−0.0641500
$244$	2.03478e8	0.896712
$245$	0	0
$246$	−1.73759e8	−0.744172
$247$	1.12809e8	0.476326
$248$	1.49098e8	0.620715
$249$	2.30370e8	0.945646
$250$	0	0
$251$	−2.21212e8	−0.882980	−0.441490	−	0.897266i	$-0.645550\pi$
$251$	−2.21212e8	−0.882980	−0.441490	+	0.897266i	$0.645550\pi$
$252$	−7.48362e7	−0.294585
$253$	−6.29545e7	−0.244402
$254$	1.90107e8	0.727915
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	−4.55108e8	−1.67243	−0.836217	−	0.548399i	$-0.815237\pi$
$257$	−4.55108e8	−1.67243	−0.836217	+	0.548399i	$0.815237\pi$
$258$	−1.55794e8	−0.564783
$259$	−2.16614e8	−0.774706
$260$	0	0
$261$	−1.02199e8	−0.355797
$262$	2.69993e8	0.927465
$263$	8.36484e7	0.283539	0.141769	−	0.989900i	$-0.454721\pi$
$263$	8.36484e7	0.283539	0.141769	+	0.989900i	$0.454721\pi$
$264$	−3.05234e7	−0.102098
$265$	0	0
$266$	−2.52277e8	−0.821849
$267$	−1.00802e8	−0.324101
$268$	8.72804e7	0.276978
$269$	5.39059e8	1.68851	0.844254	−	0.535943i	$-0.180044\pi$
$269$	5.39059e8	1.68851	0.844254	+	0.535943i	$0.180044\pi$
$270$	0	0
$271$	3.34251e8	1.02019	0.510094	−	0.860119i	$-0.329611\pi$
$271$	3.34251e8	1.02019	0.510094	+	0.860119i	$0.329611\pi$
$272$	−6.41188e7	−0.193194
$273$	−2.48501e8	−0.739196
$274$	−5.12213e7	−0.150426
$275$	0	0
$276$	−4.92687e7	−0.141055
$277$	1.79379e8	0.507099	0.253549	−	0.967322i	$-0.418402\pi$
$277$	1.79379e8	0.507099	0.253549	+	0.967322i	$0.418402\pi$
$278$	3.27397e7	0.0913940
$279$	−2.12291e8	−0.585216
$280$	0	0
$281$	3.56391e8	0.958195	0.479098	−	0.877762i	$-0.340964\pi$
$281$	3.56391e8	0.958195	0.479098	+	0.877762i	$0.340964\pi$
$282$	1.73374e8	0.460375
$283$	−6.33056e8	−1.66031	−0.830156	−	0.557531i	$-0.811749\pi$
$283$	−6.33056e8	−1.66031	−0.830156	+	0.557531i	$0.811749\pi$
$284$	−2.80953e8	−0.727812
$285$	0	0
$286$	−1.01356e8	−0.256194
$287$	1.29032e9	3.22189
$288$	−2.38879e7	−0.0589256
$289$	−1.65291e8	−0.402816
$290$	0	0
$291$	−4.23528e8	−1.00753
$292$	2.73847e8	0.643677
$293$	−3.81234e8	−0.885432	−0.442716	−	0.896662i	$-0.645985\pi$
$293$	−3.81234e8	−0.885432	−0.442716	+	0.896662i	$0.645985\pi$
$294$	3.77843e8	0.867153
$295$	0	0
$296$	−6.91436e7	−0.154964
$297$	4.34601e7	0.0962593
$298$	3.12670e8	0.684431
$299$	−1.63602e8	−0.353948
$300$	0	0
$301$	1.15691e9	2.44522
$302$	−5.15825e8	−1.07765
$303$	−8.60685e7	−0.177744
$304$	−8.05274e7	−0.164394
$305$	0	0
$306$	9.12941e7	0.182145
$307$	−3.54930e8	−0.700096	−0.350048	−	0.936732i	$-0.613835\pi$
$307$	−3.54930e8	−0.700096	−0.350048	+	0.936732i	$0.613835\pi$
$308$	2.26664e8	0.442034
$309$	2.75143e8	0.530522
$310$	0	0
$311$	−7.50592e8	−1.41496	−0.707478	−	0.706736i	$-0.750167\pi$
$311$	−7.50592e8	−1.41496	−0.707478	+	0.706736i	$0.750167\pi$
$312$	−7.93221e7	−0.147861
$313$	7.09319e8	1.30748	0.653742	−	0.756718i	$-0.273198\pi$
$313$	7.09319e8	1.30748	0.653742	+	0.756718i	$0.273198\pi$
$314$	2.16730e8	0.395062
$315$	0	0
$316$	2.46525e8	0.439498
$317$	5.00147e8	0.881841	0.440921	−	0.897546i	$-0.354652\pi$
$317$	5.00147e8	0.881841	0.440921	+	0.897546i	$0.354652\pi$
$318$	−5.92052e7	−0.103244
$319$	3.09540e8	0.533886
$320$	0	0
$321$	−1.07180e8	−0.180862
$322$	3.65866e8	0.610698
$323$	3.07758e8	0.508159
$324$	3.40122e7	0.0555556
$325$	0	0
$326$	−5.46241e8	−0.873219
$327$	−2.46580e8	−0.389978
$328$	4.11872e8	0.644472
$329$	−1.28746e9	−1.99319
$330$	0	0
$331$	−2.07604e8	−0.314657	−0.157328	−	0.987546i	$-0.550288\pi$
$331$	−2.07604e8	−0.314657	−0.157328	+	0.987546i	$0.550288\pi$
$332$	−5.46063e8	−0.818953
$333$	9.84485e7	0.146101
$334$	5.73857e8	0.842734
$335$	0	0
$336$	1.77390e8	0.255118
$337$	−1.86712e8	−0.265746	−0.132873	−	0.991133i	$-0.542420\pi$
$337$	−1.86712e8	−0.265746	−0.132873	+	0.991133i	$0.542420\pi$
$338$	2.38591e8	0.336082
$339$	−3.10105e8	−0.432324
$340$	0	0
$341$	6.42987e8	0.878137
$342$	1.14657e8	0.154992
$343$	−1.48487e9	−1.98682
$344$	3.69289e8	0.489116
$345$	0	0
$346$	4.51103e8	0.585477
$347$	−5.19597e8	−0.667596	−0.333798	−	0.942645i	$-0.608330\pi$
$347$	−5.19597e8	−0.667596	−0.333798	+	0.942645i	$0.608330\pi$
$348$	2.42248e8	0.308130
$349$	−7.42950e8	−0.935558	−0.467779	−	0.883845i	$-0.654946\pi$
$349$	−7.42950e8	−0.935558	−0.467779	+	0.883845i	$0.654946\pi$
$350$	0	0
$351$	1.12941e8	0.139405
$352$	7.23517e7	0.0884198
$353$	1.73844e8	0.210353	0.105177	−	0.994454i	$-0.466459\pi$
$353$	1.73844e8	0.210353	0.105177	+	0.994454i	$0.466459\pi$
$354$	4.25399e8	0.509665
$355$	0	0
$356$	2.38938e8	0.280679
$357$	−6.77943e8	−0.788596
$358$	−3.28031e8	−0.377854
$359$	6.61102e8	0.754115	0.377058	−	0.926190i	$-0.376936\pi$
$359$	6.61102e8	0.754115	0.377058	+	0.926190i	$0.376936\pi$
$360$	0	0
$361$	−5.07356e8	−0.567594
$362$	−1.95450e8	−0.216549
$363$	3.94521e8	0.432910
$364$	5.89040e8	0.640162
$365$	0	0
$366$	6.86738e8	0.732162
$367$	−3.36278e8	−0.355114	−0.177557	−	0.984111i	$-0.556819\pi$
$367$	−3.36278e8	−0.355114	−0.177557	+	0.984111i	$0.556819\pi$
$368$	1.16785e8	0.122158
$369$	−5.86435e8	−0.607614
$370$	0	0
$371$	4.39653e8	0.446994
$372$	5.03207e8	0.506812
$373$	2.08973e8	0.208502	0.104251	−	0.994551i	$-0.466755\pi$
$373$	2.08973e8	0.208502	0.104251	+	0.994551i	$0.466755\pi$
$374$	−2.76512e8	−0.273315
$375$	0	0
$376$	−4.10960e8	−0.398696
$377$	8.04410e8	0.773184
$378$	−2.52572e8	−0.240527
$379$	−1.34843e9	−1.27230	−0.636151	−	0.771564i	$-0.719475\pi$
$379$	−1.34843e9	−1.27230	−0.636151	+	0.771564i	$0.719475\pi$
$380$	0	0
$381$	6.41612e8	0.594340
$382$	3.78272e8	0.347202
$383$	−8.28746e8	−0.753747	−0.376874	−	0.926265i	$-0.623001\pi$
$383$	−8.28746e8	−0.753747	−0.376874	+	0.926265i	$0.623001\pi$
$384$	5.66231e7	0.0510310
$385$	0	0
$386$	−2.00571e8	−0.177506
$387$	−5.25804e8	−0.461143
$388$	1.00392e9	0.872543
$389$	−8.83784e8	−0.761242	−0.380621	−	0.924731i	$-0.624290\pi$
$389$	−8.83784e8	−0.761242	−0.380621	+	0.924731i	$0.624290\pi$
$390$	0	0
$391$	−4.46327e8	−0.377602
$392$	−8.95628e8	−0.750976
$393$	9.11225e8	0.757272
$394$	−6.73662e8	−0.554888
$395$	0	0
$396$	−1.03016e8	−0.0833630
$397$	−2.97473e8	−0.238605	−0.119303	−	0.992858i	$-0.538066\pi$
$397$	−2.97473e8	−0.238605	−0.119303	+	0.992858i	$0.538066\pi$
$398$	7.10120e8	0.564600
$399$	−8.51435e8	−0.671037
$400$	0	0
$401$	−3.87923e8	−0.300428	−0.150214	−	0.988654i	$-0.547996\pi$
$401$	−3.87923e8	−0.300428	−0.150214	+	0.988654i	$0.547996\pi$
$402$	2.94571e8	0.226151
$403$	1.67095e9	1.27173
$404$	2.04014e8	0.153931
$405$	0	0
$406$	−1.79892e9	−1.33404
$407$	−2.98182e8	−0.219230
$408$	−2.16401e8	−0.157742
$409$	−8.73693e8	−0.631433	−0.315716	−	0.948854i	$-0.602245\pi$
$409$	−8.73693e8	−0.631433	−0.315716	+	0.948854i	$0.602245\pi$
$410$	0	0
$411$	−1.72872e8	−0.122823
$412$	−6.52190e8	−0.459445
$413$	−3.15898e9	−2.20659
$414$	−1.66282e8	−0.115171
$415$	0	0
$416$	1.88023e8	0.128051
$417$	1.10496e8	0.0746229
$418$	−3.47274e8	−0.232571
$419$	8.22926e8	0.546527	0.273264	−	0.961939i	$-0.411897\pi$
$419$	8.22926e8	0.546527	0.273264	+	0.961939i	$0.411897\pi$
$420$	0	0
$421$	−2.10690e9	−1.37612	−0.688061	−	0.725653i	$-0.741538\pi$
$421$	−2.10690e9	−1.37612	−0.688061	+	0.725653i	$0.741538\pi$
$422$	−7.30718e8	−0.473322
$423$	5.85136e8	0.375894
$424$	1.40338e8	0.0894119
$425$	0	0
$426$	−9.48216e8	−0.594256
$427$	−5.09966e9	−3.16989
$428$	2.54057e8	0.156631
$429$	−3.42077e8	−0.209181
$430$	0	0
$431$	−2.66148e8	−0.160123	−0.0800614	−	0.996790i	$-0.525512\pi$
$431$	−2.66148e8	−0.160123	−0.0800614	+	0.996790i	$0.525512\pi$
$432$	−8.06216e7	−0.0481125
$433$	2.09405e9	1.23959	0.619797	−	0.784762i	$-0.287215\pi$
$433$	2.09405e9	1.23959	0.619797	+	0.784762i	$0.287215\pi$
$434$	−3.73678e9	−2.19424
$435$	0	0
$436$	5.84486e8	0.337731
$437$	−5.60546e8	−0.321311
$438$	9.24234e8	0.525560
$439$	2.89321e9	1.63213	0.816065	−	0.577960i	$-0.196151\pi$
$439$	2.89321e9	1.63213	0.816065	+	0.577960i	$0.196151\pi$
$440$	0	0
$441$	1.27522e9	0.708027
$442$	−7.18581e8	−0.395820
$443$	−8.70595e8	−0.475777	−0.237888	−	0.971292i	$-0.576455\pi$
$443$	−8.70595e8	−0.475777	−0.237888	+	0.971292i	$0.576455\pi$
$444$	−2.33359e8	−0.126528
$445$	0	0
$446$	−7.59391e8	−0.405316
$447$	1.05526e9	0.558836
$448$	−4.20479e8	−0.220938
$449$	1.20917e9	0.630412	0.315206	−	0.949023i	$-0.397926\pi$
$449$	1.20917e9	0.630412	0.315206	+	0.949023i	$0.397926\pi$
$450$	0	0
$451$	1.77620e9	0.911746
$452$	7.35063e8	0.374404
$453$	−1.74091e9	−0.879898
$454$	2.39142e9	1.19939
$455$	0	0
$456$	−2.71780e8	−0.134227
$457$	1.73811e9	0.851863	0.425932	−	0.904755i	$-0.359946\pi$
$457$	1.73811e9	0.851863	0.425932	+	0.904755i	$0.359946\pi$
$458$	−3.61877e8	−0.176008
$459$	3.08118e8	0.148721
$460$	0	0
$461$	9.37700e8	0.445770	0.222885	−	0.974845i	$-0.428453\pi$
$461$	9.37700e8	0.445770	0.222885	+	0.974845i	$0.428453\pi$
$462$	7.64993e8	0.360919
$463$	−8.40528e8	−0.393567	−0.196783	−	0.980447i	$-0.563050\pi$
$463$	−8.40528e8	−0.393567	−0.196783	+	0.980447i	$0.563050\pi$
$464$	−5.74218e8	−0.266848
$465$	0	0
$466$	1.81876e9	0.832576
$467$	−1.06616e6	−0.000484412 0	−0.000242206	−	1.00000i	$-0.500077\pi$
$467$	−1.06616e6	−0.000484412 0	−0.000242206		1.00000i	$0.500077\pi$
$468$	−2.67712e8	−0.120728
$469$	−2.18746e9	−0.979120
$470$	0	0
$471$	7.31464e8	0.322567
$472$	−1.00835e9	−0.441383
$473$	1.59256e9	0.691961
$474$	8.32023e8	0.358849
$475$	0	0
$476$	1.60698e9	0.682944
$477$	−1.99817e8	−0.0842983
$478$	−2.32094e9	−0.972002
$479$	2.22754e9	0.926086	0.463043	−	0.886336i	$-0.346757\pi$
$479$	2.22754e9	0.926086	0.463043	+	0.886336i	$0.346757\pi$
$480$	0	0
$481$	−7.74894e8	−0.317493
$482$	2.44458e8	0.0994352
$483$	1.23480e9	0.498633
$484$	−9.35162e8	−0.374911
$485$	0	0
$486$	1.14791e8	0.0453609
$487$	−3.23635e8	−0.126971	−0.0634855	−	0.997983i	$-0.520222\pi$
$487$	−3.23635e8	−0.126971	−0.0634855	+	0.997983i	$0.520222\pi$
$488$	−1.62782e9	−0.634071
$489$	−1.84356e9	−0.712980
$490$	0	0
$491$	−1.82808e9	−0.696963	−0.348482	−	0.937316i	$-0.613303\pi$
$491$	−1.82808e9	−0.696963	−0.348482	+	0.937316i	$0.613303\pi$
$492$	1.39007e9	0.526209
$493$	2.19453e9	0.824856
$494$	−9.02473e8	−0.336814
$495$	0	0
$496$	−1.19279e9	−0.438912
$497$	7.04138e9	2.57283
$498$	−1.84296e9	−0.668673
$499$	3.29577e9	1.18742	0.593711	−	0.804678i	$-0.297662\pi$
$499$	3.29577e9	1.18742	0.593711	+	0.804678i	$0.297662\pi$
$500$	0	0
$501$	1.93677e9	0.688090
$502$	1.76970e9	0.624361
$503$	2.38389e9	0.835214	0.417607	−	0.908628i	$-0.362869\pi$
$503$	2.38389e9	0.835214	0.417607	+	0.908628i	$0.362869\pi$
$504$	5.98690e8	0.208303
$505$	0	0
$506$	5.03636e8	0.172818
$507$	8.05245e8	0.274410
$508$	−1.52086e9	−0.514714
$509$	5.41262e9	1.81926	0.909631	−	0.415417i	$-0.136364\pi$
$509$	5.41262e9	1.81926	0.909631	+	0.415417i	$0.136364\pi$
$510$	0	0
$511$	−6.86329e9	−2.27541
$512$	−1.34218e8	−0.0441942
$513$	3.86968e8	0.126551
$514$	3.64087e9	1.18259
$515$	0	0
$516$	1.24635e9	0.399362
$517$	−1.77226e9	−0.564042
$518$	1.73291e9	0.547800
$519$	1.52247e9	0.478040
$520$	0	0
$521$	1.16474e9	0.360827	0.180413	−	0.983591i	$-0.442256\pi$
$521$	1.16474e9	0.360827	0.180413	+	0.983591i	$0.442256\pi$
$522$	8.17588e8	0.251587
$523$	−1.03455e9	−0.316225	−0.158113	−	0.987421i	$-0.550541\pi$
$523$	−1.03455e9	−0.316225	−0.158113	+	0.987421i	$0.550541\pi$
$524$	−2.15994e9	−0.655817
$525$	0	0
$526$	−6.69187e8	−0.200492
$527$	4.55857e9	1.35672
$528$	2.44187e8	0.0721945
$529$	−2.59189e9	−0.761241
$530$	0	0
$531$	1.43572e9	0.416140
$532$	2.01822e9	0.581135
$533$	4.61587e9	1.32041
$534$	8.06417e8	0.229174
$535$	0	0
$536$	−6.98243e8	−0.195853
$537$	−1.10710e9	−0.308517
$538$	−4.31247e9	−1.19396
$539$	−3.86239e9	−1.06242
$540$	0	0
$541$	−7.47113e8	−0.202860	−0.101430	−	0.994843i	$-0.532342\pi$
$541$	−7.47113e8	−0.202860	−0.101430	+	0.994843i	$0.532342\pi$
$542$	−2.67401e9	−0.721382
$543$	−6.59643e8	−0.176811
$544$	5.12950e8	0.136609
$545$	0	0
$546$	1.98801e9	0.522690
$547$	2.81842e9	0.736292	0.368146	−	0.929768i	$-0.379993\pi$
$547$	2.81842e9	0.736292	0.368146	+	0.929768i	$0.379993\pi$
$548$	4.09771e8	0.106367
$549$	2.31774e9	0.597808
$550$	0	0
$551$	2.75614e9	0.701891
$552$	3.94150e8	0.0997412
$553$	−6.17854e9	−1.55363
$554$	−1.43503e9	−0.358573
$555$	0	0
$556$	−2.61917e8	−0.0646253
$557$	5.76438e9	1.41338	0.706691	−	0.707523i	$-0.250187\pi$
$557$	5.76438e9	1.41338	0.706691	+	0.707523i	$0.250187\pi$
$558$	1.69833e9	0.413810
$559$	4.13864e9	1.00211
$560$	0	0
$561$	−9.33229e8	−0.223161
$562$	−2.85112e9	−0.677546
$563$	−5.39704e9	−1.27461	−0.637304	−	0.770612i	$-0.719950\pi$
$563$	−5.39704e9	−1.27461	−0.637304	+	0.770612i	$0.719950\pi$
$564$	−1.38699e9	−0.325534
$565$	0	0
$566$	5.06445e9	1.17402
$567$	−8.52431e8	−0.196390
$568$	2.24762e9	0.514641
$569$	3.64521e9	0.829524	0.414762	−	0.909930i	$-0.363865\pi$
$569$	3.64521e9	0.829524	0.414762	+	0.909930i	$0.363865\pi$
$570$	0	0
$571$	−5.09011e9	−1.14420	−0.572099	−	0.820185i	$-0.693871\pi$
$571$	−5.09011e9	−1.14420	−0.572099	+	0.820185i	$0.693871\pi$
$572$	8.10848e8	0.181156
$573$	1.27667e9	0.283489
$574$	−1.03225e10	−2.27822
$575$	0	0
$576$	1.91103e8	0.0416667
$577$	1.92429e9	0.417019	0.208509	−	0.978020i	$-0.433139\pi$
$577$	1.92429e9	0.417019	0.208509	+	0.978020i	$0.433139\pi$
$578$	1.32233e9	0.284834
$579$	−6.76927e8	−0.144933
$580$	0	0
$581$	1.36857e10	2.89501
$582$	3.38822e9	0.712428
$583$	6.05208e8	0.126493
$584$	−2.19078e9	−0.455149
$585$	0	0
$586$	3.04987e9	0.626095
$587$	1.23738e9	0.252504	0.126252	−	0.991998i	$-0.459705\pi$
$587$	1.23738e9	0.252504	0.126252	+	0.991998i	$0.459705\pi$
$588$	−3.02274e9	−0.613170
$589$	5.72515e9	1.15447
$590$	0	0
$591$	−2.27361e9	−0.453064
$592$	5.53148e8	0.109576
$593$	6.46283e9	1.27272	0.636358	−	0.771394i	$-0.280440\pi$
$593$	6.46283e9	1.27272	0.636358	+	0.771394i	$0.280440\pi$
$594$	−3.47681e8	−0.0680656
$595$	0	0
$596$	−2.50136e9	−0.483966
$597$	2.39666e9	0.460994
$598$	1.30881e9	0.250279
$599$	5.38102e9	1.02299	0.511494	−	0.859287i	$-0.329092\pi$
$599$	5.38102e9	1.02299	0.511494	+	0.859287i	$0.329092\pi$
$600$	0	0
$601$	4.49380e9	0.844410	0.422205	−	0.906500i	$-0.361256\pi$
$601$	4.49380e9	0.844410	0.422205	+	0.906500i	$0.361256\pi$
$602$	−9.25531e9	−1.72903
$603$	9.94178e8	0.184652
$604$	4.12660e9	0.762014
$605$	0	0
$606$	6.88548e8	0.125684
$607$	−8.84951e9	−1.60605	−0.803025	−	0.595946i	$-0.796777\pi$
$607$	−8.84951e9	−1.60605	−0.803025	+	0.595946i	$0.796777\pi$
$608$	6.44219e8	0.116244
$609$	−6.07135e9	−1.08924
$610$	0	0
$611$	−4.60564e9	−0.816856
$612$	−7.30353e8	−0.128796
$613$	−5.51640e9	−0.967262	−0.483631	−	0.875272i	$-0.660682\pi$
$613$	−5.51640e9	−0.967262	−0.483631	+	0.875272i	$0.660682\pi$
$614$	2.83944e9	0.495043
$615$	0	0
$616$	−1.81332e9	−0.312565
$617$	−1.92241e9	−0.329494	−0.164747	−	0.986336i	$-0.552681\pi$
$617$	−1.92241e9	−0.329494	−0.164747	+	0.986336i	$0.552681\pi$
$618$	−2.20114e9	−0.375136
$619$	7.24498e9	1.22778	0.613889	−	0.789392i	$-0.289604\pi$
$619$	7.24498e9	1.22778	0.613889	+	0.789392i	$0.289604\pi$
$620$	0	0
$621$	−5.61202e8	−0.0940369
$622$	6.00474e9	1.00052
$623$	−5.98839e9	−0.992206
$624$	6.34577e8	0.104553
$625$	0	0
$626$	−5.67455e9	−0.924530
$627$	−1.17205e9	−0.189893
$628$	−1.73384e9	−0.279351
$629$	−2.11401e9	−0.338711
$630$	0	0
$631$	−1.16278e10	−1.84244	−0.921221	−	0.389040i	$-0.872807\pi$
$631$	−1.16278e10	−1.84244	−0.921221	+	0.389040i	$0.872807\pi$
$632$	−1.97220e9	−0.310772
$633$	−2.46617e9	−0.386465
$634$	−4.00118e9	−0.623556
$635$	0	0
$636$	4.73641e8	0.0730045
$637$	−1.00373e10	−1.53862
$638$	−2.47632e9	−0.377515
$639$	−3.20023e9	−0.485208
$640$	0	0
$641$	−2.96229e9	−0.444246	−0.222123	−	0.975019i	$-0.571299\pi$
$641$	−2.96229e9	−0.444246	−0.222123	+	0.975019i	$0.571299\pi$
$642$	8.57441e8	0.127889
$643$	1.06783e10	1.58403	0.792015	−	0.610502i	$-0.209032\pi$
$643$	1.06783e10	1.58403	0.792015	+	0.610502i	$0.209032\pi$
$644$	−2.92693e9	−0.431829
$645$	0	0
$646$	−2.46206e9	−0.359323
$647$	−1.09787e10	−1.59363	−0.796813	−	0.604225i	$-0.793483\pi$
$647$	−1.09787e10	−1.59363	−0.796813	+	0.604225i	$0.793483\pi$
$648$	−2.72098e8	−0.0392837
$649$	−4.34852e9	−0.624432
$650$	0	0
$651$	−1.26116e10	−1.79159
$652$	4.36993e9	0.617459
$653$	−2.19211e9	−0.308082	−0.154041	−	0.988064i	$-0.549229\pi$
$653$	−2.19211e9	−0.308082	−0.154041	+	0.988064i	$0.549229\pi$
$654$	1.97264e9	0.275756
$655$	0	0
$656$	−3.29498e9	−0.455711
$657$	3.11929e9	0.429118
$658$	1.02997e10	1.40940
$659$	−4.73747e9	−0.644833	−0.322417	−	0.946598i	$-0.604495\pi$
$659$	−4.73747e9	−0.644833	−0.322417	+	0.946598i	$0.604495\pi$
$660$	0	0
$661$	1.10047e10	1.48209	0.741044	−	0.671457i	$-0.234331\pi$
$661$	1.10047e10	1.48209	0.741044	+	0.671457i	$0.234331\pi$
$662$	1.66083e9	0.222496
$663$	−2.42521e9	−0.323186
$664$	4.36850e9	0.579087
$665$	0	0
$666$	−7.87588e8	−0.103309
$667$	−3.99710e9	−0.521560
$668$	−4.59085e9	−0.595903
$669$	−2.56295e9	−0.330939
$670$	0	0
$671$	−7.01999e9	−0.897031
$672$	−1.41912e9	−0.180395
$673$	1.24627e10	1.57601	0.788006	−	0.615668i	$-0.211114\pi$
$673$	1.24627e10	1.57601	0.788006	+	0.615668i	$0.211114\pi$
$674$	1.49369e9	0.187911
$675$	0	0
$676$	−1.90873e9	−0.237646
$677$	6.81622e8	0.0844274	0.0422137	−	0.999109i	$-0.486559\pi$
$677$	6.81622e8	0.0844274	0.0422137	+	0.999109i	$0.486559\pi$
$678$	2.48084e9	0.305699
$679$	−2.51607e10	−3.08445
$680$	0	0
$681$	8.07106e9	0.979300
$682$	−5.14390e9	−0.620936
$683$	1.31911e10	1.58419	0.792097	−	0.610395i	$-0.208989\pi$
$683$	1.31911e10	1.58419	0.792097	+	0.610395i	$0.208989\pi$
$684$	−9.17257e8	−0.109596
$685$	0	0
$686$	1.18790e10	1.40490
$687$	−1.22133e9	−0.143710
$688$	−2.95431e9	−0.345857
$689$	1.57277e9	0.183189
$690$	0	0
$691$	−2.60166e9	−0.299970	−0.149985	−	0.988688i	$-0.547923\pi$
$691$	−2.60166e9	−0.299970	−0.149985	+	0.988688i	$0.547923\pi$
$692$	−3.60883e9	−0.413994
$693$	2.58185e9	0.294689
$694$	4.15678e9	0.472061
$695$	0	0
$696$	−1.93799e9	−0.217881
$697$	1.25927e10	1.40865
$698$	5.94360e9	0.661539
$699$	6.13831e9	0.679796
$700$	0	0
$701$	−3.10968e9	−0.340960	−0.170480	−	0.985361i	$-0.554532\pi$
$701$	−3.10968e9	−0.340960	−0.170480	+	0.985361i	$0.554532\pi$
$702$	−9.03528e8	−0.0985739
$703$	−2.65500e9	−0.288218
$704$	−5.78814e8	−0.0625223
$705$	0	0
$706$	−1.39075e9	−0.148742
$707$	−5.11311e9	−0.544148
$708$	−3.40319e9	−0.360388
$709$	5.04862e9	0.531999	0.265999	−	0.963973i	$-0.414298\pi$
$709$	5.04862e9	0.531999	0.265999	+	0.963973i	$0.414298\pi$
$710$	0	0
$711$	2.80808e9	0.292999
$712$	−1.91151e9	−0.198470
$713$	−8.30292e9	−0.857862
$714$	5.42355e9	0.557622
$715$	0	0
$716$	2.62425e9	0.267183
$717$	−7.83319e9	−0.793636
$718$	−5.28881e9	−0.533240
$719$	−1.12740e10	−1.13117	−0.565585	−	0.824690i	$-0.691350\pi$
$719$	−1.12740e10	−1.13117	−0.565585	+	0.824690i	$0.691350\pi$
$720$	0	0
$721$	1.63455e10	1.62415
$722$	4.05885e9	0.401349
$723$	8.25047e8	0.0811885
$724$	1.56360e9	0.153123
$725$	0	0
$726$	−3.15617e9	−0.306113
$727$	−2.86921e9	−0.276943	−0.138472	−	0.990366i	$-0.544219\pi$
$727$	−2.86921e9	−0.276943	−0.138472	+	0.990366i	$0.544219\pi$
$728$	−4.71232e9	−0.452663
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	1.12907e10	1.06908
$732$	−5.49390e9	−0.517717
$733$	−5.48654e9	−0.514559	−0.257279	−	0.966337i	$-0.582826\pi$
$733$	−5.48654e9	−0.514559	−0.257279	+	0.966337i	$0.582826\pi$
$734$	2.69023e9	0.251103
$735$	0	0
$736$	−9.34281e8	−0.0863784
$737$	−3.01117e9	−0.277076
$738$	4.69148e9	0.429648
$739$	3.88260e9	0.353889	0.176944	−	0.984221i	$-0.443379\pi$
$739$	3.88260e9	0.353889	0.176944	+	0.984221i	$0.443379\pi$
$740$	0	0
$741$	−3.04585e9	−0.275007
$742$	−3.51723e9	−0.316072
$743$	−2.02325e9	−0.180962	−0.0904812	−	0.995898i	$-0.528841\pi$
$743$	−2.02325e9	−0.180962	−0.0904812	+	0.995898i	$0.528841\pi$
$744$	−4.02566e9	−0.358370
$745$	0	0
$746$	−1.67179e9	−0.147433
$747$	−6.21999e9	−0.545969
$748$	2.21210e9	0.193263
$749$	−6.36730e9	−0.553693
$750$	0	0
$751$	1.05797e10	0.911454	0.455727	−	0.890120i	$-0.349379\pi$
$751$	1.05797e10	0.911454	0.455727	+	0.890120i	$0.349379\pi$
$752$	3.28768e9	0.281921
$753$	5.97272e9	0.509789
$754$	−6.43528e9	−0.546724
$755$	0	0
$756$	2.02058e9	0.170078
$757$	1.16388e10	0.975157	0.487578	−	0.873079i	$-0.337880\pi$
$757$	1.16388e10	0.975157	0.487578	+	0.873079i	$0.337880\pi$
$758$	1.07874e10	0.899654
$759$	1.69977e9	0.141106
$760$	0	0
$761$	−1.97011e10	−1.62048	−0.810241	−	0.586097i	$-0.800664\pi$
$761$	−1.97011e10	−1.62048	−0.810241	+	0.586097i	$0.800664\pi$
$762$	−5.13290e9	−0.420262
$763$	−1.46487e10	−1.19388
$764$	−3.02618e9	−0.245509
$765$	0	0
$766$	6.62997e9	0.532980
$767$	−1.13006e10	−0.904315
$768$	−4.52985e8	−0.0360844
$769$	−1.59579e10	−1.26542	−0.632710	−	0.774389i	$-0.718057\pi$
$769$	−1.59579e10	−1.26542	−0.632710	+	0.774389i	$0.718057\pi$
$770$	0	0
$771$	1.22879e10	0.965580
$772$	1.60457e9	0.125516
$773$	2.19754e9	0.171123	0.0855614	−	0.996333i	$-0.472732\pi$
$773$	2.19754e9	0.171123	0.0855614	+	0.996333i	$0.472732\pi$
$774$	4.20643e9	0.326077
$775$	0	0
$776$	−8.03134e9	−0.616981
$777$	5.84857e9	0.447277
$778$	7.07028e9	0.538279
$779$	1.58153e10	1.19866
$780$	0	0
$781$	9.69287e9	0.728071
$782$	3.57061e9	0.267005
$783$	2.75936e9	0.205420
$784$	7.16502e9	0.531021
$785$	0	0
$786$	−7.28980e9	−0.535472
$787$	9.39302e9	0.686900	0.343450	−	0.939171i	$-0.388404\pi$
$787$	9.39302e9	0.686900	0.343450	+	0.939171i	$0.388404\pi$
$788$	5.38929e9	0.392365
$789$	−2.25851e9	−0.163701
$790$	0	0
$791$	−1.84225e10	−1.32352
$792$	8.24132e8	0.0589466
$793$	−1.82431e10	−1.29910
$794$	2.37978e9	0.168720
$795$	0	0
$796$	−5.68096e9	−0.399233
$797$	2.46054e10	1.72157	0.860787	−	0.508966i	$-0.169972\pi$
$797$	2.46054e10	1.72157	0.860787	+	0.508966i	$0.169972\pi$
$798$	6.81148e9	0.474495
$799$	−1.25648e10	−0.871447
$800$	0	0
$801$	2.72166e9	0.187120
$802$	3.10339e9	0.212435
$803$	−9.44773e9	−0.643907
$804$	−2.35657e9	−0.159913
$805$	0	0
$806$	−1.33676e10	−0.899252
$807$	−1.45546e10	−0.974860
$808$	−1.63211e9	−0.108846
$809$	−2.09320e10	−1.38993	−0.694963	−	0.719046i	$-0.744579\pi$
$809$	−2.09320e10	−1.38993	−0.694963	+	0.719046i	$0.744579\pi$
$810$	0	0
$811$	2.03509e10	1.33971	0.669856	−	0.742491i	$-0.266356\pi$
$811$	2.03509e10	1.33971	0.669856	+	0.742491i	$0.266356\pi$
$812$	1.43913e10	0.943312
$813$	−9.02478e9	−0.589006
$814$	2.38545e9	0.155019
$815$	0	0
$816$	1.73121e9	0.111541
$817$	1.41801e10	0.909710
$818$	6.98954e9	0.446491
$819$	6.70954e9	0.426775
$820$	0	0
$821$	4.02784e9	0.254022	0.127011	−	0.991901i	$-0.459462\pi$
$821$	4.02784e9	0.254022	0.127011	+	0.991901i	$0.459462\pi$
$822$	1.38298e9	0.0868487
$823$	1.78044e10	1.11334	0.556671	−	0.830733i	$-0.312078\pi$
$823$	1.78044e10	1.11334	0.556671	+	0.830733i	$0.312078\pi$
$824$	5.21752e9	0.324877
$825$	0	0
$826$	2.52719e10	1.56030
$827$	−5.63276e9	−0.346300	−0.173150	−	0.984895i	$-0.555395\pi$
$827$	−5.63276e9	−0.346300	−0.173150	+	0.984895i	$0.555395\pi$
$828$	1.33026e9	0.0814384
$829$	−1.27818e9	−0.0779204	−0.0389602	−	0.999241i	$-0.512405\pi$
$829$	−1.27818e9	−0.0779204	−0.0389602	+	0.999241i	$0.512405\pi$
$830$	0	0
$831$	−4.84323e9	−0.292774
$832$	−1.50418e9	−0.0905459
$833$	−2.73831e10	−1.64144
$834$	−8.83971e8	−0.0527663
$835$	0	0
$836$	2.77819e9	0.164453
$837$	5.73185e9	0.337875
$838$	−6.58341e9	−0.386453
$839$	8.12392e9	0.474896	0.237448	−	0.971400i	$-0.423689\pi$
$839$	8.12392e9	0.474896	0.237448	+	0.971400i	$0.423689\pi$
$840$	0	0
$841$	2.40336e9	0.139326
$842$	1.68552e10	0.973065
$843$	−9.62255e9	−0.553214
$844$	5.84575e9	0.334689
$845$	0	0
$846$	−4.68109e9	−0.265797
$847$	2.34375e10	1.32532
$848$	−1.12271e9	−0.0632238
$849$	1.70925e10	0.958582
$850$	0	0
$851$	3.85043e9	0.214169
$852$	7.58573e9	0.420202
$853$	1.63287e10	0.900805	0.450403	−	0.892826i	$-0.351280\pi$
$853$	1.63287e10	0.900805	0.450403	+	0.892826i	$0.351280\pi$
$854$	4.07973e10	2.24145
$855$	0	0
$856$	−2.03245e9	−0.110755
$857$	−2.27799e10	−1.23629	−0.618143	−	0.786066i	$-0.712115\pi$
$857$	−2.27799e10	−1.23629	−0.618143	+	0.786066i	$0.712115\pi$
$858$	2.73661e9	0.147914
$859$	1.14488e9	0.0616289	0.0308144	−	0.999525i	$-0.490190\pi$
$859$	1.14488e9	0.0616289	0.0308144	+	0.999525i	$0.490190\pi$
$860$	0	0
$861$	−3.48386e10	−1.86016
$862$	2.12919e9	0.113224
$863$	−8.63678e9	−0.457419	−0.228709	−	0.973495i	$-0.573451\pi$
$863$	−8.63678e9	−0.457419	−0.228709	+	0.973495i	$0.573451\pi$
$864$	6.44973e8	0.0340207
$865$	0	0
$866$	−1.67524e10	−0.876525
$867$	4.46286e9	0.232566
$868$	2.98942e10	1.55156
$869$	−8.50513e9	−0.439655
$870$	0	0
$871$	−7.82523e9	−0.401267
$872$	−4.67589e9	−0.238812
$873$	1.14352e10	0.581695
$874$	4.48437e9	0.227201
$875$	0	0
$876$	−7.39387e9	−0.371627
$877$	−9.18002e8	−0.0459563	−0.0229782	−	0.999736i	$-0.507315\pi$
$877$	−9.18002e8	−0.0459563	−0.0229782	+	0.999736i	$0.507315\pi$
$878$	−2.31457e10	−1.15409
$879$	1.02933e10	0.511204
$880$	0	0
$881$	−6.21021e9	−0.305978	−0.152989	−	0.988228i	$-0.548890\pi$
$881$	−6.21021e9	−0.305978	−0.152989	+	0.988228i	$0.548890\pi$
$882$	−1.02018e10	−0.500651
$883$	−2.46069e9	−0.120280	−0.0601402	−	0.998190i	$-0.519155\pi$
$883$	−2.46069e9	−0.120280	−0.0601402	+	0.998190i	$0.519155\pi$
$884$	5.74865e9	0.279887
$885$	0	0
$886$	6.96476e9	0.336425
$887$	−3.89923e10	−1.87606	−0.938028	−	0.346558i	$-0.887350\pi$
$887$	−3.89923e10	−1.87606	−0.938028	+	0.346558i	$0.887350\pi$
$888$	1.86688e9	0.0894685
$889$	3.81165e10	1.81952
$890$	0	0
$891$	−1.17342e9	−0.0555753
$892$	6.07513e9	0.286602
$893$	−1.57802e10	−0.741537
$894$	−8.44210e9	−0.395156
$895$	0	0
$896$	3.36383e9	0.156227
$897$	4.41725e9	0.204352
$898$	−9.67334e9	−0.445768
$899$	4.08244e10	1.87396
$900$	0	0
$901$	4.29073e9	0.195431
$902$	−1.42096e10	−0.644702
$903$	−3.12367e10	−1.41175
$904$	−5.88051e9	−0.264743
$905$	0	0
$906$	1.39273e10	0.622182
$907$	2.48805e8	0.0110722	0.00553611	−	0.999985i	$-0.498238\pi$
$907$	2.48805e8	0.0110722	0.00553611	+	0.999985i	$0.498238\pi$
$908$	−1.91314e10	−0.848098
$909$	2.32385e9	0.102621
$910$	0	0
$911$	−5.81723e9	−0.254919	−0.127459	−	0.991844i	$-0.540682\pi$
$911$	−5.81723e9	−0.254919	−0.127459	+	0.991844i	$0.540682\pi$
$912$	2.17424e9	0.0949129
$913$	1.88392e10	0.819245
$914$	−1.39049e10	−0.602358
$915$	0	0
$916$	2.89501e9	0.124456
$917$	5.41335e10	2.31832
$918$	−2.46494e9	−0.105162
$919$	7.31515e9	0.310899	0.155449	−	0.987844i	$-0.450317\pi$
$919$	7.31515e9	0.310899	0.155449	+	0.987844i	$0.450317\pi$
$920$	0	0
$921$	9.58310e9	0.404201
$922$	−7.50160e9	−0.315207
$923$	2.51892e10	1.05441
$924$	−6.11994e9	−0.255209
$925$	0	0
$926$	6.72422e9	0.278294
$927$	−7.42885e9	−0.306297
$928$	4.59375e9	0.188690
$929$	−9.37937e9	−0.383812	−0.191906	−	0.981413i	$-0.561467\pi$
$929$	−9.37937e9	−0.383812	−0.191906	+	0.981413i	$0.561467\pi$
$930$	0	0
$931$	−3.43907e10	−1.39675
$932$	−1.45501e10	−0.588720
$933$	2.02660e10	0.816925
$934$	8.52931e6	0.000342531 0
$935$	0	0
$936$	2.14170e9	0.0853675
$937$	2.84087e10	1.12814	0.564070	−	0.825727i	$-0.309235\pi$
$937$	2.84087e10	1.12814	0.564070	+	0.825727i	$0.309235\pi$
$938$	1.74997e10	0.692343
$939$	−1.91516e10	−0.754876
$940$	0	0
$941$	3.12052e10	1.22085	0.610426	−	0.792073i	$-0.290998\pi$
$941$	3.12052e10	1.22085	0.610426	+	0.792073i	$0.290998\pi$
$942$	−5.85172e9	−0.228089
$943$	−2.29361e10	−0.890696
$944$	8.06683e9	0.312105
$945$	0	0
$946$	−1.27405e10	−0.489290
$947$	1.52962e10	0.585272	0.292636	−	0.956224i	$-0.405468\pi$
$947$	1.52962e10	0.585272	0.292636	+	0.956224i	$0.405468\pi$
$948$	−6.65619e9	−0.253744
$949$	−2.45521e10	−0.932518
$950$	0	0
$951$	−1.35040e10	−0.509131
$952$	−1.28558e10	−0.482915
$953$	1.23407e10	0.461864	0.230932	−	0.972970i	$-0.425823\pi$
$953$	1.23407e10	0.461864	0.230932	+	0.972970i	$0.425823\pi$
$954$	1.59854e9	0.0596079
$955$	0	0
$956$	1.85676e10	0.687309
$957$	−8.35757e9	−0.308239
$958$	−1.78203e10	−0.654842
$959$	−1.02699e10	−0.376011
$960$	0	0
$961$	5.72895e10	2.08230
$962$	6.19915e9	0.224502
$963$	2.89386e9	0.104421
$964$	−1.95567e9	−0.0703113
$965$	0	0
$966$	−9.87838e9	−0.352587
$967$	3.88311e10	1.38098	0.690489	−	0.723343i	$-0.257395\pi$
$967$	3.88311e10	1.38098	0.690489	+	0.723343i	$0.257395\pi$
$968$	7.48130e9	0.265102
$969$	−8.30946e9	−0.293386
$970$	0	0
$971$	8.36183e9	0.293112	0.146556	−	0.989202i	$-0.453181\pi$
$971$	8.36183e9	0.293112	0.146556	+	0.989202i	$0.453181\pi$
$972$	−9.18330e8	−0.0320750
$973$	6.56431e9	0.228451
$974$	2.58908e9	0.0897821
$975$	0	0
$976$	1.30226e10	0.448356
$977$	−5.42304e10	−1.86042	−0.930212	−	0.367023i	$-0.880377\pi$
$977$	−5.42304e10	−1.86042	−0.930212	+	0.367023i	$0.880377\pi$
$978$	1.47485e10	0.504153
$979$	−8.24337e9	−0.280779
$980$	0	0
$981$	6.65766e9	0.225154
$982$	1.46246e10	0.492827
$983$	2.72599e10	0.915351	0.457675	−	0.889119i	$-0.348682\pi$
$983$	2.72599e10	0.915351	0.457675	+	0.889119i	$0.348682\pi$
$984$	−1.11206e10	−0.372086
$985$	0	0
$986$	−1.75563e10	−0.583261
$987$	3.47614e10	1.15077
$988$	7.21978e9	0.238163
$989$	−2.05648e10	−0.675985
$990$	0	0
$991$	−2.77316e10	−0.905142	−0.452571	−	0.891728i	$-0.649493\pi$
$991$	−2.77316e10	−0.905142	−0.452571	+	0.891728i	$0.649493\pi$
$992$	9.54230e9	0.310358
$993$	5.60529e9	0.181667
$994$	−5.63310e10	−1.81926
$995$	0	0
$996$	1.47437e10	0.472823
$997$	−3.92804e10	−1.25529	−0.627643	−	0.778502i	$-0.715980\pi$
$997$	−3.92804e10	−1.25529	−0.627643	+	0.778502i	$0.715980\pi$
$998$	−2.63662e10	−0.839634
$999$	−2.65811e9	−0.0843517

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.8.a.f.1.1	✓	1	5.4	even	2
90.8.a.e.1.1		1	15.14	odd	2
150.8.a.a.1.1		1	1.1	even	1	trivial
150.8.c.h.49.1		2	5.2	odd	4
150.8.c.h.49.2		2	5.3	odd	4
240.8.a.a.1.1		1	20.19	odd	2
450.8.a.n.1.1		1	3.2	odd	2
450.8.c.m.199.1		2	15.8	even	4
450.8.c.m.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

Related objects

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