LMFDB - Embedded newform 240.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,8,Mod(1,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("240.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	240.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:	$74.9724061162$
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	$\chi$	$=$	240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-27.0000 q^{3} -125.000 q^{5} -1604.00 q^{7} +729.000 q^{9} +O(q^{10})$

$q-27.0000 q^{3} -125.000 q^{5} -1604.00 q^{7} +729.000 q^{9} +2208.00 q^{11} +5738.00 q^{13} +3375.00 q^{15} +15654.0 q^{17} +19660.0 q^{19} +43308.0 q^{21} +28512.0 q^{23} +15625.0 q^{25} -19683.0 q^{27} -140190. q^{29} +291208. q^{31} -59616.0 q^{33} +200500. q^{35} -135046. q^{37} -154926. q^{39} -804438. q^{41} -721268. q^{43} -91125.0 q^{45} +802656. q^{47} +1.74927e6 q^{49} -422658. q^{51} +274098. q^{53} -276000. q^{55} -530820. q^{57} -1.96944e6 q^{59} +3.17934e6 q^{61} -1.16932e6 q^{63} -717250. q^{65} +1.36376e6 q^{67} -769824. q^{69} +4.38989e6 q^{71} -4.27886e6 q^{73} -421875. q^{75} -3.54163e6 q^{77} -3.85196e6 q^{79} +531441. q^{81} -8.53223e6 q^{83} -1.95675e6 q^{85} +3.78513e6 q^{87} +3.73341e6 q^{89} -9.20375e6 q^{91} -7.86262e6 q^{93} -2.45750e6 q^{95} -1.56862e7 q^{97} +1.60963e6 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	0	0
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	0	0
$7$	−1604.00	−1.76751	−0.883754	−	0.467952i	$-0.844992\pi$
$7$	−1604.00	−1.76751	−0.883754	+	0.467952i	$0.844992\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	2208.00	0.500178	0.250089	−	0.968223i	$-0.419540\pi$
$11$	2208.00	0.500178	0.250089	+	0.968223i	$0.419540\pi$
$12$	0	0
$13$	5738.00	0.724367	0.362184	−	0.932107i	$-0.382031\pi$
$13$	5738.00	0.724367	0.362184	+	0.932107i	$0.382031\pi$
$14$	0	0
$15$	3375.00	0.258199
$16$	0	0
$17$	15654.0	0.772777	0.386388	−	0.922336i	$-0.373722\pi$
$17$	15654.0	0.772777	0.386388	+	0.922336i	$0.373722\pi$
$18$	0	0
$19$	19660.0	0.657576	0.328788	−	0.944404i	$-0.393360\pi$
$19$	19660.0	0.657576	0.328788	+	0.944404i	$0.393360\pi$
$20$	0	0
$21$	43308.0	1.02047
$22$	0	0
$23$	28512.0	0.488630	0.244315	−	0.969696i	$-0.421437\pi$
$23$	28512.0	0.488630	0.244315	+	0.969696i	$0.421437\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	−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	$-0.158995\pi$
$31$	291208.	1.75565	0.877824	+	0.478984i	$0.158995\pi$
$32$	0	0
$33$	−59616.0	−0.288778
$34$	0	0
$35$	200500.	0.790453
$36$	0	0
$37$	−135046.	−0.438304	−0.219152	−	0.975691i	$-0.570329\pi$
$37$	−135046.	−0.438304	−0.219152	+	0.975691i	$0.570329\pi$
$38$	0	0
$39$	−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	$0.865014\pi$
$42$	0	0
$43$	−721268.	−1.38343	−0.691715	−	0.722171i	$-0.743144\pi$
$43$	−721268.	−1.38343	−0.691715	+	0.722171i	$0.743144\pi$
$44$	0	0
$45$	−91125.0	−0.149071
$46$	0	0
$47$	802656.	1.12768	0.563841	−	0.825883i	$-0.309323\pi$
$47$	802656.	1.12768	0.563841	+	0.825883i	$0.309323\pi$
$48$	0	0
$49$	1.74927e6	2.12408
$50$	0	0
$51$	−422658.	−0.446163
$52$	0	0
$53$	274098.	0.252895	0.126448	−	0.991973i	$-0.459642\pi$
$53$	274098.	0.252895	0.126448	+	0.991973i	$0.459642\pi$
$54$	0	0
$55$	−276000.	−0.223686
$56$	0	0
$57$	−530820.	−0.379652
$58$	0	0
$59$	−1.96944e6	−1.24842	−0.624210	−	0.781257i	$-0.714579\pi$
$59$	−1.96944e6	−1.24842	−0.624210	+	0.781257i	$0.714579\pi$
$60$	0	0
$61$	3.17934e6	1.79342	0.896712	−	0.442615i	$-0.145949\pi$
$61$	3.17934e6	1.79342	0.896712	+	0.442615i	$0.145949\pi$
$62$	0	0
$63$	−1.16932e6	−0.589169
$64$	0	0
$65$	−717250.	−0.323947
$66$	0	0
$67$	1.36376e6	0.553955	0.276978	−	0.960876i	$-0.410667\pi$
$67$	1.36376e6	0.553955	0.276978	+	0.960876i	$0.410667\pi$
$68$	0	0
$69$	−769824.	−0.282111
$70$	0	0
$71$	4.38989e6	1.45562	0.727812	−	0.685777i	$-0.240537\pi$
$71$	4.38989e6	1.45562	0.727812	+	0.685777i	$0.240537\pi$
$72$	0	0
$73$	−4.27886e6	−1.28735	−0.643677	−	0.765297i	$-0.722592\pi$
$73$	−4.27886e6	−1.28735	−0.643677	+	0.765297i	$0.722592\pi$
$74$	0	0
$75$	−421875.	−0.115470
$76$	0	0
$77$	−3.54163e6	−0.884068
$78$	0	0
$79$	−3.85196e6	−0.878996	−0.439498	−	0.898244i	$-0.644844\pi$
$79$	−3.85196e6	−0.878996	−0.439498	+	0.898244i	$0.644844\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−8.53223e6	−1.63791	−0.818953	−	0.573860i	$-0.805445\pi$
$83$	−8.53223e6	−1.63791	−0.818953	+	0.573860i	$0.805445\pi$
$84$	0	0
$85$	−1.95675e6	−0.345596
$86$	0	0
$87$	3.78513e6	0.616259
$88$	0	0
$89$	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
$92$	0	0
$93$	−7.86262e6	−1.01362
$94$	0	0
$95$	−2.45750e6	−0.294077
$96$	0	0
$97$	−1.56862e7	−1.74509	−0.872543	−	0.488537i	$-0.837531\pi$
$97$	−1.56862e7	−1.74509	−0.872543	+	0.488537i	$0.837531\pi$
$98$	0	0
$99$	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	$0.450807\pi$
$102$	0	0
$103$	−1.01905e7	−0.918891	−0.459445	−	0.888206i	$-0.651952\pi$
$103$	−1.01905e7	−0.918891	−0.459445	+	0.888206i	$0.651952\pi$
$104$	0	0
$105$	−5.41350e6	−0.456368
$106$	0	0
$107$	3.96964e6	0.313262	0.156631	−	0.987657i	$-0.449937\pi$
$107$	3.96964e6	0.313262	0.156631	+	0.987657i	$0.449937\pi$
$108$	0	0
$109$	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
$112$	0	0
$113$	−1.14854e7	−0.748807	−0.374404	−	0.927266i	$-0.622153\pi$
$113$	−1.14854e7	−0.748807	−0.374404	+	0.927266i	$0.622153\pi$
$114$	0	0
$115$	−3.56400e6	−0.218522
$116$	0	0
$117$	4.18300e6	0.241456
$118$	0	0
$119$	−2.51090e7	−1.36589
$120$	0	0
$121$	−1.46119e7	−0.749822
$122$	0	0
$123$	2.17198e7	1.05242
$124$	0	0
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	−2.37634e7	−1.02943	−0.514714	−	0.857362i	$-0.672102\pi$
$127$	−2.37634e7	−1.02943	−0.514714	+	0.857362i	$0.672102\pi$
$128$	0	0
$129$	1.94742e7	0.798723
$130$	0	0
$131$	3.37491e7	1.31163	0.655817	−	0.754920i	$-0.272324\pi$
$131$	3.37491e7	1.31163	0.655817	+	0.754920i	$0.272324\pi$
$132$	0	0
$133$	−3.15346e7	−1.16227
$134$	0	0
$135$	2.46038e6	0.0860663
$136$	0	0
$137$	−6.40267e6	−0.212735	−0.106367	−	0.994327i	$-0.533922\pi$
$137$	−6.40267e6	−0.212735	−0.106367	+	0.994327i	$0.533922\pi$
$138$	0	0
$139$	4.09246e6	0.129251	0.0646253	−	0.997910i	$-0.479415\pi$
$139$	4.09246e6	0.129251	0.0646253	+	0.997910i	$0.479415\pi$
$140$	0	0
$141$	−2.16717e7	−0.651068
$142$	0	0
$143$	1.26695e7	0.362313
$144$	0	0
$145$	1.75237e7	0.477352
$146$	0	0
$147$	−4.72304e7	−1.22634
$148$	0	0
$149$	−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	$-0.775789\pi$
$151$	−6.44781e7	−1.52403	−0.762014	+	0.647561i	$0.775789\pi$
$152$	0	0
$153$	1.14118e7	0.257592
$154$	0	0
$155$	−3.64010e7	−0.785150
$156$	0	0
$157$	2.70913e7	0.558703	0.279351	−	0.960189i	$-0.409881\pi$
$157$	2.70913e7	0.558703	0.279351	+	0.960189i	$0.409881\pi$
$158$	0	0
$159$	−7.40065e6	−0.146009
$160$	0	0
$161$	−4.57332e7	−0.863657
$162$	0	0
$163$	6.82802e7	1.23492	0.617459	−	0.786603i	$-0.288162\pi$
$163$	6.82802e7	1.23492	0.617459	+	0.786603i	$0.288162\pi$
$164$	0	0
$165$	7.45200e6	0.129145
$166$	0	0
$167$	−7.17321e7	−1.19181	−0.595903	−	0.803056i	$-0.703206\pi$
$167$	−7.17321e7	−1.19181	−0.595903	+	0.803056i	$0.703206\pi$
$168$	0	0
$169$	−2.98239e7	−0.475292
$170$	0	0
$171$	1.43321e7	0.219192
$172$	0	0
$173$	5.63879e7	0.827989	0.413994	−	0.910279i	$-0.364133\pi$
$173$	5.63879e7	0.827989	0.413994	+	0.910279i	$0.364133\pi$
$174$	0	0
$175$	−2.50625e7	−0.353501
$176$	0	0
$177$	5.31749e7	0.720776
$178$	0	0
$179$	−4.10039e7	−0.534367	−0.267183	−	0.963646i	$-0.586093\pi$
$179$	−4.10039e7	−0.534367	−0.267183	+	0.963646i	$0.586093\pi$
$180$	0	0
$181$	2.44312e7	0.306246	0.153123	−	0.988207i	$-0.451067\pi$
$181$	2.44312e7	0.306246	0.153123	+	0.988207i	$0.451067\pi$
$182$	0	0
$183$	−8.58422e7	−1.03543
$184$	0	0
$185$	1.68808e7	0.196016
$186$	0	0
$187$	3.45640e7	0.386526
$188$	0	0
$189$	3.15715e7	0.340157
$190$	0	0
$191$	4.72840e7	0.491018	0.245509	−	0.969394i	$-0.421045\pi$
$191$	4.72840e7	0.491018	0.245509	+	0.969394i	$0.421045\pi$
$192$	0	0
$193$	−2.50714e7	−0.251031	−0.125516	−	0.992092i	$-0.540059\pi$
$193$	−2.50714e7	−0.251031	−0.125516	+	0.992092i	$0.540059\pi$
$194$	0	0
$195$	1.93658e7	0.187031
$196$	0	0
$197$	−8.42077e7	−0.784730	−0.392365	−	0.919810i	$-0.628343\pi$
$197$	−8.42077e7	−0.784730	−0.392365	+	0.919810i	$0.628343\pi$
$198$	0	0
$199$	8.87650e7	0.798465	0.399233	−	0.916850i	$-0.369277\pi$
$199$	8.87650e7	0.798465	0.399233	+	0.916850i	$0.369277\pi$
$200$	0	0
$201$	−3.68214e7	−0.319826
$202$	0	0
$203$	2.24865e8	1.88662
$204$	0	0
$205$	1.00555e8	0.815200
$206$	0	0
$207$	2.07852e7	0.162877
$208$	0	0
$209$	4.34093e7	0.328905
$210$	0	0
$211$	−9.13398e7	−0.669378	−0.334689	−	0.942329i	$-0.608631\pi$
$211$	−9.13398e7	−0.669378	−0.334689	+	0.942329i	$0.608631\pi$
$212$	0	0
$213$	−1.18527e8	−0.840405
$214$	0	0
$215$	9.01585e7	0.618689
$216$	0	0
$217$	−4.67098e8	−3.10312
$218$	0	0
$219$	1.15529e8	0.743255
$220$	0	0
$221$	8.98227e7	0.559774
$222$	0	0
$223$	9.49239e7	0.573203	0.286602	−	0.958050i	$-0.407474\pi$
$223$	9.49239e7	0.573203	0.286602	+	0.958050i	$0.407474\pi$
$224$	0	0
$225$	1.13906e7	0.0666667
$226$	0	0
$227$	−2.98928e8	−1.69620	−0.848098	−	0.529839i	$-0.822252\pi$
$227$	−2.98928e8	−1.69620	−0.848098	+	0.529839i	$0.822252\pi$
$228$	0	0
$229$	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
$232$	0	0
$233$	2.27345e8	1.17744	0.588720	−	0.808337i	$-0.299632\pi$
$233$	2.27345e8	1.17744	0.588720	+	0.808337i	$0.299632\pi$
$234$	0	0
$235$	−1.00332e8	−0.504315
$236$	0	0
$237$	1.04003e8	0.507489
$238$	0	0
$239$	−2.90118e8	−1.37462	−0.687309	−	0.726365i	$-0.741208\pi$
$239$	−2.90118e8	−1.37462	−0.687309	+	0.726365i	$0.741208\pi$
$240$	0	0
$241$	−3.05573e7	−0.140623	−0.0703113	−	0.997525i	$-0.522399\pi$
$241$	−3.05573e7	−0.140623	−0.0703113	+	0.997525i	$0.522399\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	−2.18659e8	−0.949918
$246$	0	0
$247$	1.12809e8	0.476326
$248$	0	0
$249$	2.30370e8	0.945646
$250$	0	0
$251$	2.21212e8	0.882980	0.441490	−	0.897266i	$-0.354450\pi$
$251$	2.21212e8	0.882980	0.441490	+	0.897266i	$0.354450\pi$
$252$	0	0
$253$	6.29545e7	0.244402
$254$	0	0
$255$	5.28322e7	0.199530
$256$	0	0
$257$	4.55108e8	1.67243	0.836217	−	0.548399i	$-0.184763\pi$
$257$	4.55108e8	1.67243	0.836217	+	0.548399i	$0.184763\pi$
$258$	0	0
$259$	2.16614e8	0.774706
$260$	0	0
$261$	−1.02199e8	−0.355797
$262$	0	0
$263$	8.36484e7	0.283539	0.141769	−	0.989900i	$-0.454721\pi$
$263$	8.36484e7	0.283539	0.141769	+	0.989900i	$0.454721\pi$
$264$	0	0
$265$	−3.42622e7	−0.113098
$266$	0	0
$267$	−1.00802e8	−0.324101
$268$	0	0
$269$	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	$-0.670389\pi$
$271$	−3.34251e8	−1.02019	−0.510094	+	0.860119i	$0.670389\pi$
$272$	0	0
$273$	2.48501e8	0.739196
$274$	0	0
$275$	3.45000e7	0.100036
$276$	0	0
$277$	−1.79379e8	−0.507099	−0.253549	−	0.967322i	$-0.581598\pi$
$277$	−1.79379e8	−0.507099	−0.253549	+	0.967322i	$0.581598\pi$
$278$	0	0
$279$	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	$0.340964\pi$
$282$	0	0
$283$	−6.33056e8	−1.66031	−0.830156	−	0.557531i	$-0.811749\pi$
$283$	−6.33056e8	−1.66031	−0.830156	+	0.557531i	$0.811749\pi$
$284$	0	0
$285$	6.63525e7	0.169785
$286$	0	0
$287$	1.29032e9	3.22189
$288$	0	0
$289$	−1.65291e8	−0.402816
$290$	0	0
$291$	4.23528e8	1.00753
$292$	0	0
$293$	3.81234e8	0.885432	0.442716	−	0.896662i	$-0.354015\pi$
$293$	3.81234e8	0.885432	0.442716	+	0.896662i	$0.354015\pi$
$294$	0	0
$295$	2.46180e8	0.558310
$296$	0	0
$297$	−4.34601e7	−0.0962593
$298$	0	0
$299$	1.63602e8	0.353948
$300$	0	0
$301$	1.15691e9	2.44522
$302$	0	0
$303$	−8.60685e7	−0.177744
$304$	0	0
$305$	−3.97418e8	−0.802043
$306$	0	0
$307$	−3.54930e8	−0.700096	−0.350048	−	0.936732i	$-0.613835\pi$
$307$	−3.54930e8	−0.700096	−0.350048	+	0.936732i	$0.613835\pi$
$308$	0	0
$309$	2.75143e8	0.530522
$310$	0	0
$311$	7.50592e8	1.41496	0.707478	−	0.706736i	$-0.249833\pi$
$311$	7.50592e8	1.41496	0.707478	+	0.706736i	$0.249833\pi$
$312$	0	0
$313$	−7.09319e8	−1.30748	−0.653742	−	0.756718i	$-0.726802\pi$
$313$	−7.09319e8	−1.30748	−0.653742	+	0.756718i	$0.726802\pi$
$314$	0	0
$315$	1.46164e8	0.263484
$316$	0	0
$317$	−5.00147e8	−0.881841	−0.440921	−	0.897546i	$-0.645348\pi$
$317$	−5.00147e8	−0.881841	−0.440921	+	0.897546i	$0.645348\pi$
$318$	0	0
$319$	−3.09540e8	−0.533886
$320$	0	0
$321$	−1.07180e8	−0.180862
$322$	0	0
$323$	3.07758e8	0.508159
$324$	0	0
$325$	8.96563e7	0.144873
$326$	0	0
$327$	−2.46580e8	−0.389978
$328$	0	0
$329$	−1.28746e9	−1.99319
$330$	0	0
$331$	2.07604e8	0.314657	0.157328	−	0.987546i	$-0.449712\pi$
$331$	2.07604e8	0.314657	0.157328	+	0.987546i	$0.449712\pi$
$332$	0	0
$333$	−9.84485e7	−0.146101
$334$	0	0
$335$	−1.70470e8	−0.247736
$336$	0	0
$337$	1.86712e8	0.265746	0.132873	−	0.991133i	$-0.457580\pi$
$337$	1.86712e8	0.265746	0.132873	+	0.991133i	$0.457580\pi$
$338$	0	0
$339$	3.10105e8	0.432324
$340$	0	0
$341$	6.42987e8	0.878137
$342$	0	0
$343$	−1.48487e9	−1.98682
$344$	0	0
$345$	9.62280e7	0.126164
$346$	0	0
$347$	−5.19597e8	−0.667596	−0.333798	−	0.942645i	$-0.608330\pi$
$347$	−5.19597e8	−0.667596	−0.333798	+	0.942645i	$0.608330\pi$
$348$	0	0
$349$	−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
$352$	0	0
$353$	−1.73844e8	−0.210353	−0.105177	−	0.994454i	$-0.533541\pi$
$353$	−1.73844e8	−0.210353	−0.105177	+	0.994454i	$0.533541\pi$
$354$	0	0
$355$	−5.48736e8	−0.650975
$356$	0	0
$357$	6.77943e8	0.788596
$358$	0	0
$359$	−6.61102e8	−0.754115	−0.377058	−	0.926190i	$-0.623064\pi$
$359$	−6.61102e8	−0.754115	−0.377058	+	0.926190i	$0.623064\pi$
$360$	0	0
$361$	−5.07356e8	−0.567594
$362$	0	0
$363$	3.94521e8	0.432910
$364$	0	0
$365$	5.34858e8	0.575723
$366$	0	0
$367$	−3.36278e8	−0.355114	−0.177557	−	0.984111i	$-0.556819\pi$
$367$	−3.36278e8	−0.355114	−0.177557	+	0.984111i	$0.556819\pi$
$368$	0	0
$369$	−5.86435e8	−0.607614
$370$	0	0
$371$	−4.39653e8	−0.446994
$372$	0	0
$373$	−2.08973e8	−0.208502	−0.104251	−	0.994551i	$-0.533245\pi$
$373$	−2.08973e8	−0.208502	−0.104251	+	0.994551i	$0.533245\pi$
$374$	0	0
$375$	5.27344e7	0.0516398
$376$	0	0
$377$	−8.04410e8	−0.773184
$378$	0	0
$379$	1.34843e9	1.27230	0.636151	−	0.771564i	$-0.280525\pi$
$379$	1.34843e9	1.27230	0.636151	+	0.771564i	$0.280525\pi$
$380$	0	0
$381$	6.41612e8	0.594340
$382$	0	0
$383$	−8.28746e8	−0.753747	−0.376874	−	0.926265i	$-0.623001\pi$
$383$	−8.28746e8	−0.753747	−0.376874	+	0.926265i	$0.623001\pi$
$384$	0	0
$385$	4.42704e8	0.395367
$386$	0	0
$387$	−5.25804e8	−0.461143
$388$	0	0
$389$	−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
$392$	0	0
$393$	−9.11225e8	−0.757272
$394$	0	0
$395$	4.81495e8	0.393099
$396$	0	0
$397$	2.97473e8	0.238605	0.119303	−	0.992858i	$-0.461934\pi$
$397$	2.97473e8	0.238605	0.119303	+	0.992858i	$0.461934\pi$
$398$	0	0
$399$	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	$0.547996\pi$
$402$	0	0
$403$	1.67095e9	1.27173
$404$	0	0
$405$	−6.64301e7	−0.0496904
$406$	0	0
$407$	−2.98182e8	−0.219230
$408$	0	0
$409$	−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
$412$	0	0
$413$	3.15898e9	2.20659
$414$	0	0
$415$	1.06653e9	0.732494
$416$	0	0
$417$	−1.10496e8	−0.0746229
$418$	0	0
$419$	−8.22926e8	−0.546527	−0.273264	−	0.961939i	$-0.588103\pi$
$419$	−8.22926e8	−0.546527	−0.273264	+	0.961939i	$0.588103\pi$
$420$	0	0
$421$	−2.10690e9	−1.37612	−0.688061	−	0.725653i	$-0.741538\pi$
$421$	−2.10690e9	−1.37612	−0.688061	+	0.725653i	$0.741538\pi$
$422$	0	0
$423$	5.85136e8	0.375894
$424$	0	0
$425$	2.44594e8	0.154555
$426$	0	0
$427$	−5.09966e9	−3.16989
$428$	0	0
$429$	−3.42077e8	−0.209181
$430$	0	0
$431$	2.66148e8	0.160123	0.0800614	−	0.996790i	$-0.474488\pi$
$431$	2.66148e8	0.160123	0.0800614	+	0.996790i	$0.474488\pi$
$432$	0	0
$433$	−2.09405e9	−1.23959	−0.619797	−	0.784762i	$-0.712785\pi$
$433$	−2.09405e9	−1.23959	−0.619797	+	0.784762i	$0.712785\pi$
$434$	0	0
$435$	−4.73141e8	−0.275599
$436$	0	0
$437$	5.60546e8	0.321311
$438$	0	0
$439$	−2.89321e9	−1.63213	−0.816065	−	0.577960i	$-0.803849\pi$
$439$	−2.89321e9	−1.63213	−0.816065	+	0.577960i	$0.803849\pi$
$440$	0	0
$441$	1.27522e9	0.708027
$442$	0	0
$443$	−8.70595e8	−0.475777	−0.237888	−	0.971292i	$-0.576455\pi$
$443$	−8.70595e8	−0.475777	−0.237888	+	0.971292i	$0.576455\pi$
$444$	0	0
$445$	−4.66676e8	−0.251047
$446$	0	0
$447$	1.05526e9	0.558836
$448$	0	0
$449$	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
$452$	0	0
$453$	1.74091e9	0.879898
$454$	0	0
$455$	1.15047e9	0.572578
$456$	0	0
$457$	−1.73811e9	−0.851863	−0.425932	−	0.904755i	$-0.640054\pi$
$457$	−1.73811e9	−0.851863	−0.425932	+	0.904755i	$0.640054\pi$
$458$	0	0
$459$	−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	$0.428453\pi$
$462$	0	0
$463$	−8.40528e8	−0.393567	−0.196783	−	0.980447i	$-0.563050\pi$
$463$	−8.40528e8	−0.393567	−0.196783	+	0.980447i	$0.563050\pi$
$464$	0	0
$465$	9.82827e8	0.453306
$466$	0	0
$467$	−1.06616e6	−0.000484412 0	−0.000242206	−	1.00000i	$-0.500077\pi$
$467$	−1.06616e6	−0.000484412 0	−0.000242206		1.00000i	$0.500077\pi$
$468$	0	0
$469$	−2.18746e9	−0.979120
$470$	0	0
$471$	−7.31464e8	−0.322567
$472$	0	0
$473$	−1.59256e9	−0.691961
$474$	0	0
$475$	3.07188e8	0.131515
$476$	0	0
$477$	1.99817e8	0.0842983
$478$	0	0
$479$	−2.22754e9	−0.926086	−0.463043	−	0.886336i	$-0.653243\pi$
$479$	−2.22754e9	−0.926086	−0.463043	+	0.886336i	$0.653243\pi$
$480$	0	0
$481$	−7.74894e8	−0.317493
$482$	0	0
$483$	1.23480e9	0.498633
$484$	0	0
$485$	1.96078e9	0.780426
$486$	0	0
$487$	−3.23635e8	−0.126971	−0.0634855	−	0.997983i	$-0.520222\pi$
$487$	−3.23635e8	−0.126971	−0.0634855	+	0.997983i	$0.520222\pi$
$488$	0	0
$489$	−1.84356e9	−0.712980
$490$	0	0
$491$	1.82808e9	0.696963	0.348482	−	0.937316i	$-0.386697\pi$
$491$	1.82808e9	0.696963	0.348482	+	0.937316i	$0.386697\pi$
$492$	0	0
$493$	−2.19453e9	−0.824856
$494$	0	0
$495$	−2.01204e8	−0.0745621
$496$	0	0
$497$	−7.04138e9	−2.57283
$498$	0	0
$499$	−3.29577e9	−1.18742	−0.593711	−	0.804678i	$-0.702338\pi$
$499$	−3.29577e9	−1.18742	−0.593711	+	0.804678i	$0.702338\pi$
$500$	0	0
$501$	1.93677e9	0.688090
$502$	0	0
$503$	2.38389e9	0.835214	0.417607	−	0.908628i	$-0.362869\pi$
$503$	2.38389e9	0.835214	0.417607	+	0.908628i	$0.362869\pi$
$504$	0	0
$505$	−3.98465e8	−0.137680
$506$	0	0
$507$	8.05245e8	0.274410
$508$	0	0
$509$	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
$512$	0	0
$513$	−3.86968e8	−0.126551
$514$	0	0
$515$	1.27381e9	0.410940
$516$	0	0
$517$	1.77226e9	0.564042
$518$	0	0
$519$	−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	$0.442256\pi$
$522$	0	0
$523$	−1.03455e9	−0.316225	−0.158113	−	0.987421i	$-0.550541\pi$
$523$	−1.03455e9	−0.316225	−0.158113	+	0.987421i	$0.550541\pi$
$524$	0	0
$525$	6.76688e8	0.204094
$526$	0	0
$527$	4.55857e9	1.35672
$528$	0	0
$529$	−2.59189e9	−0.761241
$530$	0	0
$531$	−1.43572e9	−0.416140
$532$	0	0
$533$	−4.61587e9	−1.32041
$534$	0	0
$535$	−4.96204e8	−0.140095
$536$	0	0
$537$	1.10710e9	0.308517
$538$	0	0
$539$	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	$0.532342\pi$
$542$	0	0
$543$	−6.59643e8	−0.176811
$544$	0	0
$545$	−1.14157e9	−0.302076
$546$	0	0
$547$	2.81842e9	0.736292	0.368146	−	0.929768i	$-0.379993\pi$
$547$	2.81842e9	0.736292	0.368146	+	0.929768i	$0.379993\pi$
$548$	0	0
$549$	2.31774e9	0.597808
$550$	0	0
$551$	−2.75614e9	−0.701891
$552$	0	0
$553$	6.17854e9	1.55363
$554$	0	0
$555$	−4.55780e8	−0.113170
$556$	0	0
$557$	−5.76438e9	−1.41338	−0.706691	−	0.707523i	$-0.749813\pi$
$557$	−5.76438e9	−1.41338	−0.706691	+	0.707523i	$0.749813\pi$
$558$	0	0
$559$	−4.13864e9	−1.00211
$560$	0	0
$561$	−9.33229e8	−0.223161
$562$	0	0
$563$	−5.39704e9	−1.27461	−0.637304	−	0.770612i	$-0.719950\pi$
$563$	−5.39704e9	−1.27461	−0.637304	+	0.770612i	$0.719950\pi$
$564$	0	0
$565$	1.43567e9	0.334877
$566$	0	0
$567$	−8.52431e8	−0.196390
$568$	0	0
$569$	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	$-0.306129\pi$
$571$	5.09011e9	1.14420	0.572099	+	0.820185i	$0.306129\pi$
$572$	0	0
$573$	−1.27667e9	−0.283489
$574$	0	0
$575$	4.45500e8	0.0977260
$576$	0	0
$577$	−1.92429e9	−0.417019	−0.208509	−	0.978020i	$-0.566861\pi$
$577$	−1.92429e9	−0.417019	−0.208509	+	0.978020i	$0.566861\pi$
$578$	0	0
$579$	6.76927e8	0.144933
$580$	0	0
$581$	1.36857e10	2.89501
$582$	0	0
$583$	6.05208e8	0.126493
$584$	0	0
$585$	−5.22875e8	−0.107982
$586$	0	0
$587$	1.23738e9	0.252504	0.126252	−	0.991998i	$-0.459705\pi$
$587$	1.23738e9	0.252504	0.126252	+	0.991998i	$0.459705\pi$
$588$	0	0
$589$	5.72515e9	1.15447
$590$	0	0
$591$	2.27361e9	0.453064
$592$	0	0
$593$	−6.46283e9	−1.27272	−0.636358	−	0.771394i	$-0.719560\pi$
$593$	−6.46283e9	−1.27272	−0.636358	+	0.771394i	$0.719560\pi$
$594$	0	0
$595$	3.13863e9	0.610844
$596$	0	0
$597$	−2.39666e9	−0.460994
$598$	0	0
$599$	−5.38102e9	−1.02299	−0.511494	−	0.859287i	$-0.670908\pi$
$599$	−5.38102e9	−1.02299	−0.511494	+	0.859287i	$0.670908\pi$
$600$	0	0
$601$	4.49380e9	0.844410	0.422205	−	0.906500i	$-0.361256\pi$
$601$	4.49380e9	0.844410	0.422205	+	0.906500i	$0.361256\pi$
$602$	0	0
$603$	9.94178e8	0.184652
$604$	0	0
$605$	1.82649e9	0.335331
$606$	0	0
$607$	−8.84951e9	−1.60605	−0.803025	−	0.595946i	$-0.796777\pi$
$607$	−8.84951e9	−1.60605	−0.803025	+	0.595946i	$0.796777\pi$
$608$	0	0
$609$	−6.07135e9	−1.08924
$610$	0	0
$611$	4.60564e9	0.816856
$612$	0	0
$613$	5.51640e9	0.967262	0.483631	−	0.875272i	$-0.339318\pi$
$613$	5.51640e9	0.967262	0.483631	+	0.875272i	$0.339318\pi$
$614$	0	0
$615$	−2.71498e9	−0.470656
$616$	0	0
$617$	1.92241e9	0.329494	0.164747	−	0.986336i	$-0.447319\pi$
$617$	1.92241e9	0.329494	0.164747	+	0.986336i	$0.447319\pi$
$618$	0	0
$619$	−7.24498e9	−1.22778	−0.613889	−	0.789392i	$-0.710396\pi$
$619$	−7.24498e9	−1.22778	−0.613889	+	0.789392i	$0.710396\pi$
$620$	0	0
$621$	−5.61202e8	−0.0940369
$622$	0	0
$623$	−5.98839e9	−0.992206
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	0	0
$627$	−1.17205e9	−0.189893
$628$	0	0
$629$	−2.11401e9	−0.338711
$630$	0	0
$631$	1.16278e10	1.84244	0.921221	−	0.389040i	$-0.127193\pi$
$631$	1.16278e10	1.84244	0.921221	+	0.389040i	$0.127193\pi$
$632$	0	0
$633$	2.46617e9	0.386465
$634$	0	0
$635$	2.97043e9	0.460374
$636$	0	0
$637$	1.00373e10	1.53862
$638$	0	0
$639$	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	$0.571299\pi$
$642$	0	0
$643$	1.06783e10	1.58403	0.792015	−	0.610502i	$-0.209032\pi$
$643$	1.06783e10	1.58403	0.792015	+	0.610502i	$0.209032\pi$
$644$	0	0
$645$	−2.43428e9	−0.357200
$646$	0	0
$647$	−1.09787e10	−1.59363	−0.796813	−	0.604225i	$-0.793483\pi$
$647$	−1.09787e10	−1.59363	−0.796813	+	0.604225i	$0.793483\pi$
$648$	0	0
$649$	−4.34852e9	−0.624432
$650$	0	0
$651$	1.26116e10	1.79159
$652$	0	0
$653$	2.19211e9	0.308082	0.154041	−	0.988064i	$-0.450771\pi$
$653$	2.19211e9	0.308082	0.154041	+	0.988064i	$0.450771\pi$
$654$	0	0
$655$	−4.21864e9	−0.586580
$656$	0	0
$657$	−3.11929e9	−0.429118
$658$	0	0
$659$	4.73747e9	0.644833	0.322417	−	0.946598i	$-0.395505\pi$
$659$	4.73747e9	0.644833	0.322417	+	0.946598i	$0.395505\pi$
$660$	0	0
$661$	1.10047e10	1.48209	0.741044	−	0.671457i	$-0.234331\pi$
$661$	1.10047e10	1.48209	0.741044	+	0.671457i	$0.234331\pi$
$662$	0	0
$663$	−2.42521e9	−0.323186
$664$	0	0
$665$	3.94183e9	0.519783
$666$	0	0
$667$	−3.99710e9	−0.521560
$668$	0	0
$669$	−2.56295e9	−0.330939
$670$	0	0
$671$	7.01999e9	0.897031
$672$	0	0
$673$	−1.24627e10	−1.57601	−0.788006	−	0.615668i	$-0.788886\pi$
$673$	−1.24627e10	−1.57601	−0.788006	+	0.615668i	$0.788886\pi$
$674$	0	0
$675$	−3.07547e8	−0.0384900
$676$	0	0
$677$	−6.81622e8	−0.0844274	−0.0422137	−	0.999109i	$-0.513441\pi$
$677$	−6.81622e8	−0.0844274	−0.0422137	+	0.999109i	$0.513441\pi$
$678$	0	0
$679$	2.51607e10	3.08445
$680$	0	0
$681$	8.07106e9	0.979300
$682$	0	0
$683$	1.31911e10	1.58419	0.792097	−	0.610395i	$-0.208989\pi$
$683$	1.31911e10	1.58419	0.792097	+	0.610395i	$0.208989\pi$
$684$	0	0
$685$	8.00333e8	0.0951380
$686$	0	0
$687$	−1.22133e9	−0.143710
$688$	0	0
$689$	1.57277e9	0.183189
$690$	0	0
$691$	2.60166e9	0.299970	0.149985	−	0.988688i	$-0.452077\pi$
$691$	2.60166e9	0.299970	0.149985	+	0.988688i	$0.452077\pi$
$692$	0	0
$693$	−2.58185e9	−0.294689
$694$	0	0
$695$	−5.11558e8	−0.0578026
$696$	0	0
$697$	−1.25927e10	−1.40865
$698$	0	0
$699$	−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	$0.554532\pi$
$702$	0	0
$703$	−2.65500e9	−0.288218
$704$	0	0
$705$	2.70896e9	0.291166
$706$	0	0
$707$	−5.11311e9	−0.544148
$708$	0	0
$709$	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
$712$	0	0
$713$	8.30292e9	0.857862
$714$	0	0
$715$	−1.58369e9	−0.162031
$716$	0	0
$717$	7.83319e9	0.793636
$718$	0	0
$719$	1.12740e10	1.13117	0.565585	−	0.824690i	$-0.308650\pi$
$719$	1.12740e10	1.13117	0.565585	+	0.824690i	$0.308650\pi$
$720$	0	0
$721$	1.63455e10	1.62415
$722$	0	0
$723$	8.25047e8	0.0811885
$724$	0	0
$725$	−2.19047e9	−0.213478
$726$	0	0
$727$	−2.86921e9	−0.276943	−0.138472	−	0.990366i	$-0.544219\pi$
$727$	−2.86921e9	−0.276943	−0.138472	+	0.990366i	$0.544219\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−1.12907e10	−1.06908
$732$	0	0
$733$	5.48654e9	0.514559	0.257279	−	0.966337i	$-0.417174\pi$
$733$	5.48654e9	0.514559	0.257279	+	0.966337i	$0.417174\pi$
$734$	0	0
$735$	5.90380e9	0.548436
$736$	0	0
$737$	3.01117e9	0.277076
$738$	0	0
$739$	−3.88260e9	−0.353889	−0.176944	−	0.984221i	$-0.556621\pi$
$739$	−3.88260e9	−0.353889	−0.176944	+	0.984221i	$0.556621\pi$
$740$	0	0
$741$	−3.04585e9	−0.275007
$742$	0	0
$743$	−2.02325e9	−0.180962	−0.0904812	−	0.995898i	$-0.528841\pi$
$743$	−2.02325e9	−0.180962	−0.0904812	+	0.995898i	$0.528841\pi$
$744$	0	0
$745$	4.88547e9	0.432872
$746$	0	0
$747$	−6.21999e9	−0.545969
$748$	0	0
$749$	−6.36730e9	−0.553693
$750$	0	0
$751$	−1.05797e10	−0.911454	−0.455727	−	0.890120i	$-0.650621\pi$
$751$	−1.05797e10	−0.911454	−0.455727	+	0.890120i	$0.650621\pi$
$752$	0	0
$753$	−5.97272e9	−0.509789
$754$	0	0
$755$	8.05976e9	0.681566
$756$	0	0
$757$	−1.16388e10	−0.975157	−0.487578	−	0.873079i	$-0.662120\pi$
$757$	−1.16388e10	−0.975157	−0.487578	+	0.873079i	$0.662120\pi$
$758$	0	0
$759$	−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	$0.800664\pi$
$762$	0	0
$763$	−1.46487e10	−1.19388
$764$	0	0
$765$	−1.42647e9	−0.115199
$766$	0	0
$767$	−1.13006e10	−0.904315
$768$	0	0
$769$	−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
$772$	0	0
$773$	−2.19754e9	−0.171123	−0.0855614	−	0.996333i	$-0.527268\pi$
$773$	−2.19754e9	−0.171123	−0.0855614	+	0.996333i	$0.527268\pi$
$774$	0	0
$775$	4.55013e9	0.351130
$776$	0	0
$777$	−5.84857e9	−0.447277
$778$	0	0
$779$	−1.58153e10	−1.19866
$780$	0	0
$781$	9.69287e9	0.728071
$782$	0	0
$783$	2.75936e9	0.205420
$784$	0	0
$785$	−3.38641e9	−0.249859
$786$	0	0
$787$	9.39302e9	0.686900	0.343450	−	0.939171i	$-0.388404\pi$
$787$	9.39302e9	0.686900	0.343450	+	0.939171i	$0.388404\pi$
$788$	0	0
$789$	−2.25851e9	−0.163701
$790$	0	0
$791$	1.84225e10	1.32352
$792$	0	0
$793$	1.82431e10	1.29910
$794$	0	0
$795$	9.25081e8	0.0652972
$796$	0	0
$797$	−2.46054e10	−1.72157	−0.860787	−	0.508966i	$-0.830028\pi$
$797$	−2.46054e10	−1.72157	−0.860787	+	0.508966i	$0.830028\pi$
$798$	0	0
$799$	1.25648e10	0.871447
$800$	0	0
$801$	2.72166e9	0.187120
$802$	0	0
$803$	−9.44773e9	−0.643907
$804$	0	0
$805$	5.71666e9	0.386239
$806$	0	0
$807$	−1.45546e10	−0.974860
$808$	0	0
$809$	−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	$-0.733644\pi$
$811$	−2.03509e10	−1.33971	−0.669856	+	0.742491i	$0.733644\pi$
$812$	0	0
$813$	9.02478e9	0.589006
$814$	0	0
$815$	−8.53502e9	−0.552272
$816$	0	0
$817$	−1.41801e10	−0.909710
$818$	0	0
$819$	−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	$0.459462\pi$
$822$	0	0
$823$	1.78044e10	1.11334	0.556671	−	0.830733i	$-0.312078\pi$
$823$	1.78044e10	1.11334	0.556671	+	0.830733i	$0.312078\pi$
$824$	0	0
$825$	−9.31500e8	−0.0577556
$826$	0	0
$827$	−5.63276e9	−0.346300	−0.173150	−	0.984895i	$-0.555395\pi$
$827$	−5.63276e9	−0.346300	−0.173150	+	0.984895i	$0.555395\pi$
$828$	0	0
$829$	−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
$832$	0	0
$833$	2.73831e10	1.64144
$834$	0	0
$835$	8.96651e9	0.532992
$836$	0	0
$837$	−5.73185e9	−0.337875
$838$	0	0
$839$	−8.12392e9	−0.474896	−0.237448	−	0.971400i	$-0.576311\pi$
$839$	−8.12392e9	−0.474896	−0.237448	+	0.971400i	$0.576311\pi$
$840$	0	0
$841$	2.40336e9	0.139326
$842$	0	0
$843$	−9.62255e9	−0.553214
$844$	0	0
$845$	3.72798e9	0.212557
$846$	0	0
$847$	2.34375e10	1.32532
$848$	0	0
$849$	1.70925e10	0.958582
$850$	0	0
$851$	−3.85043e9	−0.214169
$852$	0	0
$853$	−1.63287e10	−0.900805	−0.450403	−	0.892826i	$-0.648720\pi$
$853$	−1.63287e10	−0.900805	−0.450403	+	0.892826i	$0.648720\pi$
$854$	0	0
$855$	−1.79152e9	−0.0980256
$856$	0	0
$857$	2.27799e10	1.23629	0.618143	−	0.786066i	$-0.287885\pi$
$857$	2.27799e10	1.23629	0.618143	+	0.786066i	$0.287885\pi$
$858$	0	0
$859$	−1.14488e9	−0.0616289	−0.0308144	−	0.999525i	$-0.509810\pi$
$859$	−1.14488e9	−0.0616289	−0.0308144	+	0.999525i	$0.509810\pi$
$860$	0	0
$861$	−3.48386e10	−1.86016
$862$	0	0
$863$	−8.63678e9	−0.457419	−0.228709	−	0.973495i	$-0.573451\pi$
$863$	−8.63678e9	−0.457419	−0.228709	+	0.973495i	$0.573451\pi$
$864$	0	0
$865$	−7.04849e9	−0.370288
$866$	0	0
$867$	4.46286e9	0.232566
$868$	0	0
$869$	−8.50513e9	−0.439655
$870$	0	0
$871$	7.82523e9	0.401267
$872$	0	0
$873$	−1.14352e10	−0.581695
$874$	0	0
$875$	3.13281e9	0.158091
$876$	0	0
$877$	9.18002e8	0.0459563	0.0229782	−	0.999736i	$-0.492685\pi$
$877$	9.18002e8	0.0459563	0.0229782	+	0.999736i	$0.492685\pi$
$878$	0	0
$879$	−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	$0.548890\pi$
$882$	0	0
$883$	−2.46069e9	−0.120280	−0.0601402	−	0.998190i	$-0.519155\pi$
$883$	−2.46069e9	−0.120280	−0.0601402	+	0.998190i	$0.519155\pi$
$884$	0	0
$885$	−6.64686e9	−0.322341
$886$	0	0
$887$	−3.89923e10	−1.87606	−0.938028	−	0.346558i	$-0.887350\pi$
$887$	−3.89923e10	−1.87606	−0.938028	+	0.346558i	$0.887350\pi$
$888$	0	0
$889$	3.81165e10	1.81952
$890$	0	0
$891$	1.17342e9	0.0555753
$892$	0	0
$893$	1.57802e10	0.741537
$894$	0	0
$895$	5.12548e9	0.238976
$896$	0	0
$897$	−4.41725e9	−0.204352
$898$	0	0
$899$	−4.08244e10	−1.87396
$900$	0	0
$901$	4.29073e9	0.195431
$902$	0	0
$903$	−3.12367e10	−1.41175
$904$	0	0
$905$	−3.05390e9	−0.136957
$906$	0	0
$907$	2.48805e8	0.0110722	0.00553611	−	0.999985i	$-0.498238\pi$
$907$	2.48805e8	0.0110722	0.00553611	+	0.999985i	$0.498238\pi$
$908$	0	0
$909$	2.32385e9	0.102621
$910$	0	0
$911$	5.81723e9	0.254919	0.127459	−	0.991844i	$-0.459318\pi$
$911$	5.81723e9	0.254919	0.127459	+	0.991844i	$0.459318\pi$
$912$	0	0
$913$	−1.88392e10	−0.819245
$914$	0	0
$915$	1.07303e10	0.463060
$916$	0	0
$917$	−5.41335e10	−2.31832
$918$	0	0
$919$	−7.31515e9	−0.310899	−0.155449	−	0.987844i	$-0.549683\pi$
$919$	−7.31515e9	−0.310899	−0.155449	+	0.987844i	$0.549683\pi$
$920$	0	0
$921$	9.58310e9	0.404201
$922$	0	0
$923$	2.51892e10	1.05441
$924$	0	0
$925$	−2.11009e9	−0.0876609
$926$	0	0
$927$	−7.42885e9	−0.306297
$928$	0	0
$929$	−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
$932$	0	0
$933$	−2.02660e10	−0.816925
$934$	0	0
$935$	−4.32050e9	−0.172860
$936$	0	0
$937$	−2.84087e10	−1.12814	−0.564070	−	0.825727i	$-0.690765\pi$
$937$	−2.84087e10	−1.12814	−0.564070	+	0.825727i	$0.690765\pi$
$938$	0	0
$939$	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	$0.290998\pi$
$942$	0	0
$943$	−2.29361e10	−0.890696
$944$	0	0
$945$	−3.94644e9	−0.152123
$946$	0	0
$947$	1.52962e10	0.585272	0.292636	−	0.956224i	$-0.405468\pi$
$947$	1.52962e10	0.585272	0.292636	+	0.956224i	$0.405468\pi$
$948$	0	0
$949$	−2.45521e10	−0.932518
$950$	0	0
$951$	1.35040e10	0.509131
$952$	0	0
$953$	−1.23407e10	−0.461864	−0.230932	−	0.972970i	$-0.574177\pi$
$953$	−1.23407e10	−0.461864	−0.230932	+	0.972970i	$0.574177\pi$
$954$	0	0
$955$	−5.91050e9	−0.219590
$956$	0	0
$957$	8.35757e9	0.308239
$958$	0	0
$959$	1.02699e10	0.376011
$960$	0	0
$961$	5.72895e10	2.08230
$962$	0	0
$963$	2.89386e9	0.104421
$964$	0	0
$965$	3.13392e9	0.112265
$966$	0	0
$967$	3.88311e10	1.38098	0.690489	−	0.723343i	$-0.257395\pi$
$967$	3.88311e10	1.38098	0.690489	+	0.723343i	$0.257395\pi$
$968$	0	0
$969$	−8.30946e9	−0.293386
$970$	0	0
$971$	−8.36183e9	−0.293112	−0.146556	−	0.989202i	$-0.546819\pi$
$971$	−8.36183e9	−0.293112	−0.146556	+	0.989202i	$0.546819\pi$
$972$	0	0
$973$	−6.56431e9	−0.228451
$974$	0	0
$975$	−2.42072e9	−0.0836427
$976$	0	0
$977$	5.42304e10	1.86042	0.930212	−	0.367023i	$-0.119623\pi$
$977$	5.42304e10	1.86042	0.930212	+	0.367023i	$0.119623\pi$
$978$	0	0
$979$	8.24337e9	0.280779
$980$	0	0
$981$	6.65766e9	0.225154
$982$	0	0
$983$	2.72599e10	0.915351	0.457675	−	0.889119i	$-0.348682\pi$
$983$	2.72599e10	0.915351	0.457675	+	0.889119i	$0.348682\pi$
$984$	0	0
$985$	1.05260e10	0.350942
$986$	0	0
$987$	3.47614e10	1.15077
$988$	0	0
$989$	−2.05648e10	−0.675985
$990$	0	0
$991$	2.77316e10	0.905142	0.452571	−	0.891728i	$-0.350507\pi$
$991$	2.77316e10	0.905142	0.452571	+	0.891728i	$0.350507\pi$
$992$	0	0
$993$	−5.60529e9	−0.181667
$994$	0	0
$995$	−1.10956e10	−0.357084
$996$	0	0
$997$	3.92804e10	1.25529	0.627643	−	0.778502i	$-0.284020\pi$
$997$	3.92804e10	1.25529	0.627643	+	0.778502i	$0.284020\pi$
$998$	0	0
$999$	2.65811e9	0.0843517

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.8.a.a.1.1		1
4.3	odd	2		30.8.a.f.1.1	✓	1
12.11	even	2		90.8.a.e.1.1		1
20.3	even	4		150.8.c.h.49.1		2
20.7	even	4		150.8.c.h.49.2		2
20.19	odd	2		150.8.a.a.1.1		1
60.23	odd	4		450.8.c.m.199.2		2
60.47	odd	4		450.8.c.m.199.1		2
60.59	even	2		450.8.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.8.a.f.1.1	✓	1	4.3	odd	2
90.8.a.e.1.1		1	12.11	even	2
150.8.a.a.1.1		1	20.19	odd	2
150.8.c.h.49.1		2	20.3	even	4
150.8.c.h.49.2		2	20.7	even	4
240.8.a.a.1.1		1	1.1	even	1	trivial
450.8.a.n.1.1		1	60.59	even	2
450.8.c.m.199.1		2	60.47	odd	4
450.8.c.m.199.2		2	60.23	odd	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:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	240.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more