LMFDB - Embedded newform 240.8.a.l.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} -512.000 q^{7} +729.000 q^{9} -5460.00 q^{11} +10166.0 q^{13} +3375.00 q^{15} -9918.00 q^{17} +12436.0 q^{19} -13824.0 q^{21} -33600.0 q^{23} +15625.0 q^{25} +19683.0 q^{27} -187914. q^{29} +42592.0 q^{31} -147420. q^{33} -64000.0 q^{35} -544066. q^{37} +274482. q^{39} +374394. q^{41} +540532. q^{43} +91125.0 q^{45} -1.33836e6 q^{47} -561399. q^{49} -267786. q^{51} +1.30822e6 q^{53} -682500. q^{55} +335772. q^{57} -262740. q^{59} -976330. q^{61} -373248. q^{63} +1.27075e6 q^{65} -3.55917e6 q^{67} -907200. q^{69} +2.67372e6 q^{71} -3.03213e6 q^{73} +421875. q^{75} +2.79552e6 q^{77} +5.47581e6 q^{79} +531441. q^{81} -2.23156e6 q^{83} -1.23975e6 q^{85} -5.07368e6 q^{87} -1.00507e7 q^{89} -5.20499e6 q^{91} +1.14998e6 q^{93} +1.55450e6 q^{95} +5.72755e6 q^{97} -3.98034e6 q^{99} +O(q^{100})

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	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
$7$	−512.000	−0.564192	−0.282096	−	0.959386i	$-0.591030\pi$
$7$	−512.000	−0.564192	−0.282096	+	0.959386i	$0.591030\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−5460.00	−1.23685	−0.618427	−	0.785842i	$-0.712230\pi$
$11$	−5460.00	−1.23685	−0.618427	+	0.785842i	$0.712230\pi$
$12$	0	0
$13$	10166.0	1.28336	0.641680	−	0.766973i	$-0.278238\pi$
$13$	10166.0	1.28336	0.641680	+	0.766973i	$0.278238\pi$
$14$	0	0
$15$	3375.00	0.258199
$16$	0	0
$17$	−9918.00	−0.489613	−0.244806	−	0.969572i	$-0.578724\pi$
$17$	−9918.00	−0.489613	−0.244806	+	0.969572i	$0.578724\pi$
$18$	0	0
$19$	12436.0	0.415952	0.207976	−	0.978134i	$-0.433312\pi$
$19$	12436.0	0.415952	0.207976	+	0.978134i	$0.433312\pi$
$20$	0	0
$21$	−13824.0	−0.325736
$22$	0	0
$23$	−33600.0	−0.575827	−0.287913	−	0.957656i	$-0.592962\pi$
$23$	−33600.0	−0.575827	−0.287913	+	0.957656i	$0.592962\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	−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	$-0.459019\pi$
$31$	42592.0	0.256781	0.128390	+	0.991724i	$0.459019\pi$
$32$	0	0
$33$	−147420.	−0.714098
$34$	0	0
$35$	−64000.0	−0.252314
$36$	0	0
$37$	−544066.	−1.76582	−0.882908	−	0.469546i	$-0.844418\pi$
$37$	−544066.	−1.76582	−0.882908	+	0.469546i	$0.844418\pi$
$38$	0	0
$39$	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	$0.360561\pi$
$42$	0	0
$43$	540532.	1.03677	0.518384	−	0.855148i	$-0.326534\pi$
$43$	540532.	1.03677	0.518384	+	0.855148i	$0.326534\pi$
$44$	0	0
$45$	91125.0	0.149071
$46$	0	0
$47$	−1.33836e6	−1.88031	−0.940157	−	0.340741i	$-0.889322\pi$
$47$	−1.33836e6	−1.88031	−0.940157	+	0.340741i	$0.889322\pi$
$48$	0	0
$49$	−561399.	−0.681688
$50$	0	0
$51$	−267786.	−0.282678
$52$	0	0
$53$	1.30822e6	1.20702	0.603512	−	0.797354i	$-0.293768\pi$
$53$	1.30822e6	1.20702	0.603512	+	0.797354i	$0.293768\pi$
$54$	0	0
$55$	−682500.	−0.553138
$56$	0	0
$57$	335772.	0.240150
$58$	0	0
$59$	−262740.	−0.166550	−0.0832749	−	0.996527i	$-0.526538\pi$
$59$	−262740.	−0.166550	−0.0832749	+	0.996527i	$0.526538\pi$
$60$	0	0
$61$	−976330.	−0.550734	−0.275367	−	0.961339i	$-0.588799\pi$
$61$	−976330.	−0.550734	−0.275367	+	0.961339i	$0.588799\pi$
$62$	0	0
$63$	−373248.	−0.188064
$64$	0	0
$65$	1.27075e6	0.573936
$66$	0	0
$67$	−3.55917e6	−1.44573	−0.722865	−	0.690989i	$-0.757175\pi$
$67$	−3.55917e6	−1.44573	−0.722865	+	0.690989i	$0.757175\pi$
$68$	0	0
$69$	−907200.	−0.332454
$70$	0	0
$71$	2.67372e6	0.886567	0.443284	−	0.896381i	$-0.353813\pi$
$71$	2.67372e6	0.886567	0.443284	+	0.896381i	$0.353813\pi$
$72$	0	0
$73$	−3.03213e6	−0.912259	−0.456130	−	0.889913i	$-0.650765\pi$
$73$	−3.03213e6	−0.912259	−0.456130	+	0.889913i	$0.650765\pi$
$74$	0	0
$75$	421875.	0.115470
$76$	0	0
$77$	2.79552e6	0.697823
$78$	0	0
$79$	5.47581e6	1.24955	0.624775	−	0.780805i	$-0.285191\pi$
$79$	5.47581e6	1.24955	0.624775	+	0.780805i	$0.285191\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−2.23156e6	−0.428385	−0.214193	−	0.976791i	$-0.568712\pi$
$83$	−2.23156e6	−0.428385	−0.214193	+	0.976791i	$0.568712\pi$
$84$	0	0
$85$	−1.23975e6	−0.218962
$86$	0	0
$87$	−5.07368e6	−0.826048
$88$	0	0
$89$	−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
$92$	0	0
$93$	1.14998e6	0.148252
$94$	0	0
$95$	1.55450e6	0.186019
$96$	0	0
$97$	5.72755e6	0.637189	0.318594	−	0.947891i	$-0.396789\pi$
$97$	5.72755e6	0.637189	0.318594	+	0.947891i	$0.396789\pi$
$98$	0	0
$99$	−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	$0.722712\pi$
$102$	0	0
$103$	2.71019e6	0.244382	0.122191	−	0.992507i	$-0.461008\pi$
$103$	2.71019e6	0.244382	0.122191	+	0.992507i	$0.461008\pi$
$104$	0	0
$105$	−1.72800e6	−0.145674
$106$	0	0
$107$	−1.13195e7	−0.893274	−0.446637	−	0.894715i	$-0.647378\pi$
$107$	−1.13195e7	−0.893274	−0.446637	+	0.894715i	$0.647378\pi$
$108$	0	0
$109$	−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
$112$	0	0
$113$	3.68359e6	0.240158	0.120079	−	0.992764i	$-0.461685\pi$
$113$	3.68359e6	0.240158	0.120079	+	0.992764i	$0.461685\pi$
$114$	0	0
$115$	−4.20000e6	−0.257518
$116$	0	0
$117$	7.41101e6	0.427787
$118$	0	0
$119$	5.07802e6	0.276236
$120$	0	0
$121$	1.03244e7	0.529806
$122$	0	0
$123$	1.01086e7	0.489807
$124$	0	0
$125$	1.95312e6	0.0894427
$126$	0	0
$127$	2.67953e7	1.16077	0.580385	−	0.814342i	$-0.302902\pi$
$127$	2.67953e7	1.16077	0.580385	+	0.814342i	$0.302902\pi$
$128$	0	0
$129$	1.45944e7	0.598579
$130$	0	0
$131$	−1.48085e7	−0.575523	−0.287762	−	0.957702i	$-0.592911\pi$
$131$	−1.48085e7	−0.575523	−0.287762	+	0.957702i	$0.592911\pi$
$132$	0	0
$133$	−6.36723e6	−0.234677
$134$	0	0
$135$	2.46038e6	0.0860663
$136$	0	0
$137$	−5.76532e7	−1.91559	−0.957793	−	0.287458i	$-0.907190\pi$
$137$	−5.76532e7	−1.91559	−0.957793	+	0.287458i	$0.907190\pi$
$138$	0	0
$139$	8.37800e6	0.264599	0.132300	−	0.991210i	$-0.457764\pi$
$139$	8.37800e6	0.264599	0.132300	+	0.991210i	$0.457764\pi$
$140$	0	0
$141$	−3.61357e7	−1.08560
$142$	0	0
$143$	−5.55064e7	−1.58733
$144$	0	0
$145$	−2.34892e7	−0.639854
$146$	0	0
$147$	−1.51578e7	−0.393572
$148$	0	0
$149$	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	$-0.786550\pi$
$151$	−6.62933e7	−1.56693	−0.783466	+	0.621434i	$0.786550\pi$
$152$	0	0
$153$	−7.23022e6	−0.163204
$154$	0	0
$155$	5.32400e6	0.114836
$156$	0	0
$157$	−6.42791e7	−1.32563	−0.662813	−	0.748785i	$-0.730637\pi$
$157$	−6.42791e7	−1.32563	−0.662813	+	0.748785i	$0.730637\pi$
$158$	0	0
$159$	3.53220e7	0.696876
$160$	0	0
$161$	1.72032e7	0.324877
$162$	0	0
$163$	−8.48552e7	−1.53469	−0.767347	−	0.641232i	$-0.778424\pi$
$163$	−8.48552e7	−1.53469	−0.767347	+	0.641232i	$0.778424\pi$
$164$	0	0
$165$	−1.84275e7	−0.319354
$166$	0	0
$167$	−6.42144e7	−1.06690	−0.533451	−	0.845831i	$-0.679105\pi$
$167$	−6.42144e7	−1.06690	−0.533451	+	0.845831i	$0.679105\pi$
$168$	0	0
$169$	4.05990e7	0.647012
$170$	0	0
$171$	9.06584e6	0.138651
$172$	0	0
$173$	1.10058e8	1.61607	0.808036	−	0.589134i	$-0.200531\pi$
$173$	1.10058e8	1.61607	0.808036	+	0.589134i	$0.200531\pi$
$174$	0	0
$175$	−8.00000e6	−0.112838
$176$	0	0
$177$	−7.09398e6	−0.0961576
$178$	0	0
$179$	−8.68778e7	−1.13220	−0.566100	−	0.824337i	$-0.691548\pi$
$179$	−8.68778e7	−1.13220	−0.566100	+	0.824337i	$0.691548\pi$
$180$	0	0
$181$	1.29730e8	1.62617	0.813083	−	0.582148i	$-0.197787\pi$
$181$	1.29730e8	1.62617	0.813083	+	0.582148i	$0.197787\pi$
$182$	0	0
$183$	−2.63609e7	−0.317967
$184$	0	0
$185$	−6.80082e7	−0.789697
$186$	0	0
$187$	5.41523e7	0.605579
$188$	0	0
$189$	−1.00777e7	−0.108579
$190$	0	0
$191$	1.42024e8	1.47484	0.737418	−	0.675437i	$-0.236045\pi$
$191$	1.42024e8	1.47484	0.737418	+	0.675437i	$0.236045\pi$
$192$	0	0
$193$	−2.58910e7	−0.259238	−0.129619	−	0.991564i	$-0.541375\pi$
$193$	−2.58910e7	−0.259238	−0.129619	+	0.991564i	$0.541375\pi$
$194$	0	0
$195$	3.43102e7	0.331362
$196$	0	0
$197$	−3.59130e7	−0.334673	−0.167336	−	0.985900i	$-0.553517\pi$
$197$	−3.59130e7	−0.334673	−0.167336	+	0.985900i	$0.553517\pi$
$198$	0	0
$199$	1.75453e8	1.57824	0.789122	−	0.614237i	$-0.210536\pi$
$199$	1.75453e8	1.57824	0.789122	+	0.614237i	$0.210536\pi$
$200$	0	0
$201$	−9.60976e7	−0.834693
$202$	0	0
$203$	9.62120e7	0.807222
$204$	0	0
$205$	4.67992e7	0.379403
$206$	0	0
$207$	−2.44944e7	−0.191942
$208$	0	0
$209$	−6.79006e7	−0.514472
$210$	0	0
$211$	9.18337e7	0.672998	0.336499	−	0.941684i	$-0.390757\pi$
$211$	9.18337e7	0.672998	0.336499	+	0.941684i	$0.390757\pi$
$212$	0	0
$213$	7.21904e7	0.511860
$214$	0	0
$215$	6.75665e7	0.463657
$216$	0	0
$217$	−2.18071e7	−0.144873
$218$	0	0
$219$	−8.18676e7	−0.526693
$220$	0	0
$221$	−1.00826e8	−0.628349
$222$	0	0
$223$	−1.15780e7	−0.0699146	−0.0349573	−	0.999389i	$-0.511130\pi$
$223$	−1.15780e7	−0.0699146	−0.0349573	+	0.999389i	$0.511130\pi$
$224$	0	0
$225$	1.13906e7	0.0666667
$226$	0	0
$227$	2.99769e7	0.170097	0.0850485	−	0.996377i	$-0.472895\pi$
$227$	2.99769e7	0.170097	0.0850485	+	0.996377i	$0.472895\pi$
$228$	0	0
$229$	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
$232$	0	0
$233$	−1.27607e8	−0.660890	−0.330445	−	0.943825i	$-0.607199\pi$
$233$	−1.27607e8	−0.660890	−0.330445	+	0.943825i	$0.607199\pi$
$234$	0	0
$235$	−1.67295e8	−0.840902
$236$	0	0
$237$	1.47847e8	0.721428
$238$	0	0
$239$	1.64949e8	0.781549	0.390775	−	0.920486i	$-0.372207\pi$
$239$	1.64949e8	0.781549	0.390775	+	0.920486i	$0.372207\pi$
$240$	0	0
$241$	3.98628e8	1.83446	0.917230	−	0.398357i	$-0.130420\pi$
$241$	3.98628e8	1.83446	0.917230	+	0.398357i	$0.130420\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	−7.01749e7	−0.304860
$246$	0	0
$247$	1.26424e8	0.533816
$248$	0	0
$249$	−6.02520e7	−0.247328
$250$	0	0
$251$	−2.42512e8	−0.967999	−0.484000	−	0.875068i	$-0.660816\pi$
$251$	−2.42512e8	−0.967999	−0.484000	+	0.875068i	$0.660816\pi$
$252$	0	0
$253$	1.83456e8	0.712213
$254$	0	0
$255$	−3.34732e7	−0.126418
$256$	0	0
$257$	3.22956e8	1.18680	0.593399	−	0.804908i	$-0.297786\pi$
$257$	3.22956e8	1.18680	0.593399	+	0.804908i	$0.297786\pi$
$258$	0	0
$259$	2.78562e8	0.996259
$260$	0	0
$261$	−1.36989e8	−0.476919
$262$	0	0
$263$	3.37426e8	1.14376	0.571879	−	0.820338i	$-0.306215\pi$
$263$	3.37426e8	1.14376	0.571879	+	0.820338i	$0.306215\pi$
$264$	0	0
$265$	1.63528e8	0.539798
$266$	0	0
$267$	−2.71368e8	−0.872509
$268$	0	0
$269$	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	$-0.765736\pi$
$271$	−4.85679e8	−1.48237	−0.741185	+	0.671301i	$0.765736\pi$
$272$	0	0
$273$	−1.40535e8	−0.418037
$274$	0	0
$275$	−8.53125e7	−0.247371
$276$	0	0
$277$	2.92732e8	0.827545	0.413772	−	0.910380i	$-0.364211\pi$
$277$	2.92732e8	0.827545	0.413772	+	0.910380i	$0.364211\pi$
$278$	0	0
$279$	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	$0.580543\pi$
$282$	0	0
$283$	6.58251e8	1.72639	0.863196	−	0.504870i	$-0.168459\pi$
$283$	6.58251e8	1.72639	0.863196	+	0.504870i	$0.168459\pi$
$284$	0	0
$285$	4.19715e7	0.107398
$286$	0	0
$287$	−1.91690e8	−0.478644
$288$	0	0
$289$	−3.11972e8	−0.760279
$290$	0	0
$291$	1.54644e8	0.367881
$292$	0	0
$293$	1.61426e8	0.374918	0.187459	−	0.982272i	$-0.439975\pi$
$293$	1.61426e8	0.374918	0.187459	+	0.982272i	$0.439975\pi$
$294$	0	0
$295$	−3.28425e7	−0.0744833
$296$	0	0
$297$	−1.07469e8	−0.238033
$298$	0	0
$299$	−3.41578e8	−0.738993
$300$	0	0
$301$	−2.76752e8	−0.584936
$302$	0	0
$303$	−3.60066e8	−0.743588
$304$	0	0
$305$	−1.22041e8	−0.246296
$306$	0	0
$307$	7.15488e8	1.41130	0.705648	−	0.708562i	$-0.250656\pi$
$307$	7.15488e8	1.41130	0.705648	+	0.708562i	$0.250656\pi$
$308$	0	0
$309$	7.31752e7	0.141094
$310$	0	0
$311$	6.86692e7	0.129449	0.0647247	−	0.997903i	$-0.479383\pi$
$311$	6.86692e7	0.129449	0.0647247	+	0.997903i	$0.479383\pi$
$312$	0	0
$313$	9.27771e7	0.171015	0.0855077	−	0.996338i	$-0.472749\pi$
$313$	9.27771e7	0.171015	0.0855077	+	0.996338i	$0.472749\pi$
$314$	0	0
$315$	−4.66560e7	−0.0841048
$316$	0	0
$317$	3.90749e8	0.688954	0.344477	−	0.938795i	$-0.388056\pi$
$317$	3.90749e8	0.688954	0.344477	+	0.938795i	$0.388056\pi$
$318$	0	0
$319$	1.02601e9	1.76964
$320$	0	0
$321$	−3.05627e8	−0.515732
$322$	0	0
$323$	−1.23340e8	−0.203655
$324$	0	0
$325$	1.58844e8	0.256672
$326$	0	0
$327$	−5.93277e8	−0.938297
$328$	0	0
$329$	6.85240e8	1.06086
$330$	0	0
$331$	4.80269e8	0.727925	0.363962	−	0.931414i	$-0.381424\pi$
$331$	4.80269e8	0.727925	0.363962	+	0.931414i	$0.381424\pi$
$332$	0	0
$333$	−3.96624e8	−0.588605
$334$	0	0
$335$	−4.44896e8	−0.646550
$336$	0	0
$337$	−2.30504e8	−0.328075	−0.164038	−	0.986454i	$-0.552452\pi$
$337$	−2.30504e8	−0.328075	−0.164038	+	0.986454i	$0.552452\pi$
$338$	0	0
$339$	9.94568e7	0.138655
$340$	0	0
$341$	−2.32552e8	−0.317600
$342$	0	0
$343$	7.09090e8	0.948794
$344$	0	0
$345$	−1.13400e8	−0.148678
$346$	0	0
$347$	8.36727e8	1.07505	0.537527	−	0.843246i	$-0.319359\pi$
$347$	8.36727e8	1.07505	0.537527	+	0.843246i	$0.319359\pi$
$348$	0	0
$349$	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
$352$	0	0
$353$	2.33016e7	0.0281952	0.0140976	−	0.999901i	$-0.495512\pi$
$353$	2.33016e7	0.0281952	0.0140976	+	0.999901i	$0.495512\pi$
$354$	0	0
$355$	3.34215e8	0.396485
$356$	0	0
$357$	1.37106e8	0.159485
$358$	0	0
$359$	−1.63162e9	−1.86118	−0.930590	−	0.366063i	$-0.880705\pi$
$359$	−1.63162e9	−1.86118	−0.930590	+	0.366063i	$0.880705\pi$
$360$	0	0
$361$	−7.39218e8	−0.826984
$362$	0	0
$363$	2.78760e8	0.305884
$364$	0	0
$365$	−3.79017e8	−0.407975
$366$	0	0
$367$	−3.47807e8	−0.367289	−0.183644	−	0.982993i	$-0.558789\pi$
$367$	−3.47807e8	−0.367289	−0.183644	+	0.982993i	$0.558789\pi$
$368$	0	0
$369$	2.72933e8	0.282790
$370$	0	0
$371$	−6.69810e8	−0.680993
$372$	0	0
$373$	5.00769e7	0.0499639	0.0249820	−	0.999688i	$-0.492047\pi$
$373$	5.00769e7	0.0499639	0.0249820	+	0.999688i	$0.492047\pi$
$374$	0	0
$375$	5.27344e7	0.0516398
$376$	0	0
$377$	−1.91033e9	−1.83618
$378$	0	0
$379$	−1.40154e9	−1.32241	−0.661207	−	0.750204i	$-0.729955\pi$
$379$	−1.40154e9	−1.32241	−0.661207	+	0.750204i	$0.729955\pi$
$380$	0	0
$381$	7.23474e8	0.670170
$382$	0	0
$383$	−1.06830e9	−0.971623	−0.485811	−	0.874064i	$-0.661476\pi$
$383$	−1.06830e9	−0.971623	−0.485811	+	0.874064i	$0.661476\pi$
$384$	0	0
$385$	3.49440e8	0.312076
$386$	0	0
$387$	3.94048e8	0.345590
$388$	0	0
$389$	−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
$392$	0	0
$393$	−3.99831e8	−0.332278
$394$	0	0
$395$	6.84476e8	0.558815
$396$	0	0
$397$	9.69088e8	0.777314	0.388657	−	0.921383i	$-0.372939\pi$
$397$	9.69088e8	0.777314	0.388657	+	0.921383i	$0.372939\pi$
$398$	0	0
$399$	−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	$0.828751\pi$
$402$	0	0
$403$	4.32990e8	0.329542
$404$	0	0
$405$	6.64301e7	0.0496904
$406$	0	0
$407$	2.97060e9	2.18406
$408$	0	0
$409$	−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
$412$	0	0
$413$	1.34523e8	0.0939661
$414$	0	0
$415$	−2.78944e8	−0.191580
$416$	0	0
$417$	2.26206e8	0.152766
$418$	0	0
$419$	2.21021e8	0.146786	0.0733930	−	0.997303i	$-0.476617\pi$
$419$	2.21021e8	0.146786	0.0733930	+	0.997303i	$0.476617\pi$
$420$	0	0
$421$	1.51339e9	0.988472	0.494236	−	0.869328i	$-0.335448\pi$
$421$	1.51339e9	0.988472	0.494236	+	0.869328i	$0.335448\pi$
$422$	0	0
$423$	−9.75664e8	−0.626771
$424$	0	0
$425$	−1.54969e8	−0.0979226
$426$	0	0
$427$	4.99881e8	0.310720
$428$	0	0
$429$	−1.49867e9	−0.916444
$430$	0	0
$431$	−1.46801e9	−0.883201	−0.441601	−	0.897212i	$-0.645589\pi$
$431$	−1.46801e9	−0.883201	−0.441601	+	0.897212i	$0.645589\pi$
$432$	0	0
$433$	3.97508e8	0.235309	0.117654	−	0.993055i	$-0.462463\pi$
$433$	3.97508e8	0.235309	0.117654	+	0.993055i	$0.462463\pi$
$434$	0	0
$435$	−6.34210e8	−0.369420
$436$	0	0
$437$	−4.17850e8	−0.239516
$438$	0	0
$439$	−1.22071e9	−0.688629	−0.344315	−	0.938854i	$-0.611889\pi$
$439$	−1.22071e9	−0.688629	−0.344315	+	0.938854i	$0.611889\pi$
$440$	0	0
$441$	−4.09260e8	−0.227229
$442$	0	0
$443$	−3.24551e9	−1.77366	−0.886830	−	0.462095i	$-0.847098\pi$
$443$	−3.24551e9	−1.77366	−0.886830	+	0.462095i	$0.847098\pi$
$444$	0	0
$445$	−1.25633e9	−0.675842
$446$	0	0
$447$	1.50249e8	0.0795672
$448$	0	0
$449$	−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
$452$	0	0
$453$	−1.78992e9	−0.904669
$454$	0	0
$455$	−6.50624e8	−0.323810
$456$	0	0
$457$	3.70970e9	1.81816	0.909080	−	0.416621i	$-0.136786\pi$
$457$	3.70970e9	1.81816	0.909080	+	0.416621i	$0.136786\pi$
$458$	0	0
$459$	−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	$0.413823\pi$
$462$	0	0
$463$	7.64328e7	0.0357887	0.0178944	−	0.999840i	$-0.494304\pi$
$463$	7.64328e7	0.0357887	0.0178944	+	0.999840i	$0.494304\pi$
$464$	0	0
$465$	1.43748e8	0.0663004
$466$	0	0
$467$	6.41328e7	0.0291388	0.0145694	−	0.999894i	$-0.495362\pi$
$467$	6.41328e7	0.0291388	0.0145694	+	0.999894i	$0.495362\pi$
$468$	0	0
$469$	1.82230e9	0.815669
$470$	0	0
$471$	−1.73553e9	−0.765350
$472$	0	0
$473$	−2.95130e9	−1.28233
$474$	0	0
$475$	1.94312e8	0.0831904
$476$	0	0
$477$	9.53694e8	0.402341
$478$	0	0
$479$	−1.01386e9	−0.421505	−0.210753	−	0.977539i	$-0.567591\pi$
$479$	−1.01386e9	−0.421505	−0.210753	+	0.977539i	$0.567591\pi$
$480$	0	0
$481$	−5.53097e9	−2.26618
$482$	0	0
$483$	4.64486e8	0.187568
$484$	0	0
$485$	7.15944e8	0.284959
$486$	0	0
$487$	2.79009e8	0.109463	0.0547315	−	0.998501i	$-0.482570\pi$
$487$	2.79009e8	0.109463	0.0547315	+	0.998501i	$0.482570\pi$
$488$	0	0
$489$	−2.29109e9	−0.886056
$490$	0	0
$491$	−1.77339e9	−0.676112	−0.338056	−	0.941126i	$-0.609769\pi$
$491$	−1.77339e9	−0.676112	−0.338056	+	0.941126i	$0.609769\pi$
$492$	0	0
$493$	1.86373e9	0.700518
$494$	0	0
$495$	−4.97542e8	−0.184379
$496$	0	0
$497$	−1.36894e9	−0.500194
$498$	0	0
$499$	4.66125e9	1.67939	0.839693	−	0.543061i	$-0.182735\pi$
$499$	4.66125e9	1.67939	0.839693	+	0.543061i	$0.182735\pi$
$500$	0	0
$501$	−1.73379e9	−0.615977
$502$	0	0
$503$	2.59549e9	0.909352	0.454676	−	0.890657i	$-0.349755\pi$
$503$	2.59549e9	0.909352	0.454676	+	0.890657i	$0.349755\pi$
$504$	0	0
$505$	−1.66697e9	−0.575981
$506$	0	0
$507$	1.09617e9	0.373553
$508$	0	0
$509$	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
$512$	0	0
$513$	2.44778e8	0.0800500
$514$	0	0
$515$	3.38774e8	0.109291
$516$	0	0
$517$	7.30745e9	2.32567
$518$	0	0
$519$	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	$0.526424\pi$
$522$	0	0
$523$	−5.20020e8	−0.158951	−0.0794756	−	0.996837i	$-0.525325\pi$
$523$	−5.20020e8	−0.158951	−0.0794756	+	0.996837i	$0.525325\pi$
$524$	0	0
$525$	−2.16000e8	−0.0651473
$526$	0	0
$527$	−4.22427e8	−0.125723
$528$	0	0
$529$	−2.27587e9	−0.668424
$530$	0	0
$531$	−1.91537e8	−0.0555166
$532$	0	0
$533$	3.80609e9	1.08876
$534$	0	0
$535$	−1.41494e9	−0.399484
$536$	0	0
$537$	−2.34570e9	−0.653676
$538$	0	0
$539$	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	$0.679236\pi$
$542$	0	0
$543$	3.50271e9	0.938867
$544$	0	0
$545$	−2.74665e9	−0.726801
$546$	0	0
$547$	−3.11516e9	−0.813814	−0.406907	−	0.913470i	$-0.633393\pi$
$547$	−3.11516e9	−0.813814	−0.406907	+	0.913470i	$0.633393\pi$
$548$	0	0
$549$	−7.11745e8	−0.183578
$550$	0	0
$551$	−2.33690e9	−0.595126
$552$	0	0
$553$	−2.80361e9	−0.704986
$554$	0	0
$555$	−1.83622e9	−0.455932
$556$	0	0
$557$	4.14733e9	1.01689	0.508447	−	0.861093i	$-0.330220\pi$
$557$	4.14733e9	1.01689	0.508447	+	0.861093i	$0.330220\pi$
$558$	0	0
$559$	5.49505e9	1.33055
$560$	0	0
$561$	1.46211e9	0.349631
$562$	0	0
$563$	5.15380e8	0.121716	0.0608580	−	0.998146i	$-0.480616\pi$
$563$	5.15380e8	0.121716	0.0608580	+	0.998146i	$0.480616\pi$
$564$	0	0
$565$	4.60448e8	0.107402
$566$	0	0
$567$	−2.72098e8	−0.0626880
$568$	0	0
$569$	−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	$-0.418999\pi$
$571$	2.23974e9	0.503466	0.251733	+	0.967797i	$0.418999\pi$
$572$	0	0
$573$	3.83463e9	0.851496
$574$	0	0
$575$	−5.25000e8	−0.115165
$576$	0	0
$577$	6.93164e9	1.50218	0.751088	−	0.660202i	$-0.229529\pi$
$577$	6.93164e9	1.50218	0.751088	+	0.660202i	$0.229529\pi$
$578$	0	0
$579$	−6.99057e8	−0.149671
$580$	0	0
$581$	1.14256e9	0.241691
$582$	0	0
$583$	−7.14289e9	−1.49291
$584$	0	0
$585$	9.26377e8	0.191312
$586$	0	0
$587$	4.17248e9	0.851454	0.425727	−	0.904852i	$-0.360018\pi$
$587$	4.17248e9	0.851454	0.425727	+	0.904852i	$0.360018\pi$
$588$	0	0
$589$	5.29674e8	0.106808
$590$	0	0
$591$	−9.69652e8	−0.193223
$592$	0	0
$593$	6.27101e9	1.23494	0.617470	−	0.786594i	$-0.288158\pi$
$593$	6.27101e9	1.23494	0.617470	+	0.786594i	$0.288158\pi$
$594$	0	0
$595$	6.34752e8	0.123536
$596$	0	0
$597$	4.73722e9	0.911199
$598$	0	0
$599$	2.10159e9	0.399535	0.199767	−	0.979843i	$-0.435981\pi$
$599$	2.10159e9	0.399535	0.199767	+	0.979843i	$0.435981\pi$
$600$	0	0
$601$	−2.41110e9	−0.453059	−0.226529	−	0.974004i	$-0.572738\pi$
$601$	−2.41110e9	−0.453059	−0.226529	+	0.974004i	$0.572738\pi$
$602$	0	0
$603$	−2.59464e9	−0.481910
$604$	0	0
$605$	1.29055e9	0.236937
$606$	0	0
$607$	5.94932e9	1.07971	0.539855	−	0.841758i	$-0.318479\pi$
$607$	5.94932e9	1.07971	0.539855	+	0.841758i	$0.318479\pi$
$608$	0	0
$609$	2.59772e9	0.466050
$610$	0	0
$611$	−1.36058e10	−2.41312
$612$	0	0
$613$	5.45618e9	0.956702	0.478351	−	0.878169i	$-0.341235\pi$
$613$	5.45618e9	0.956702	0.478351	+	0.878169i	$0.341235\pi$
$614$	0	0
$615$	1.26358e9	0.219048
$616$	0	0
$617$	−4.11503e9	−0.705301	−0.352651	−	0.935755i	$-0.614720\pi$
$617$	−4.11503e9	−0.705301	−0.352651	+	0.935755i	$0.614720\pi$
$618$	0	0
$619$	−2.78751e9	−0.472388	−0.236194	−	0.971706i	$-0.575900\pi$
$619$	−2.78751e9	−0.472388	−0.236194	+	0.971706i	$0.575900\pi$
$620$	0	0
$621$	−6.61349e8	−0.110818
$622$	0	0
$623$	5.14595e9	0.852623
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	0	0
$627$	−1.83332e9	−0.297030
$628$	0	0
$629$	5.39605e9	0.864567
$630$	0	0
$631$	−4.87155e8	−0.0771906	−0.0385953	−	0.999255i	$-0.512288\pi$
$631$	−4.87155e8	−0.0771906	−0.0385953	+	0.999255i	$0.512288\pi$
$632$	0	0
$633$	2.47951e9	0.388555
$634$	0	0
$635$	3.34942e9	0.519112
$636$	0	0
$637$	−5.70718e9	−0.874850
$638$	0	0
$639$	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	$0.557577\pi$
$642$	0	0
$643$	1.02984e10	1.52767	0.763836	−	0.645410i	$-0.223313\pi$
$643$	1.02984e10	1.52767	0.763836	+	0.645410i	$0.223313\pi$
$644$	0	0
$645$	1.82430e9	0.267692
$646$	0	0
$647$	8.49896e9	1.23367	0.616837	−	0.787091i	$-0.288414\pi$
$647$	8.49896e9	1.23367	0.616837	+	0.787091i	$0.288414\pi$
$648$	0	0
$649$	1.43456e9	0.205998
$650$	0	0
$651$	−5.88792e8	−0.0836427
$652$	0	0
$653$	−3.78543e9	−0.532009	−0.266004	−	0.963972i	$-0.585704\pi$
$653$	−3.78543e9	−0.532009	−0.266004	+	0.963972i	$0.585704\pi$
$654$	0	0
$655$	−1.85107e9	−0.257382
$656$	0	0
$657$	−2.21043e9	−0.304086
$658$	0	0
$659$	7.69061e9	1.04680	0.523398	−	0.852089i	$-0.324664\pi$
$659$	7.69061e9	1.04680	0.523398	+	0.852089i	$0.324664\pi$
$660$	0	0
$661$	5.78185e9	0.778685	0.389343	−	0.921093i	$-0.372702\pi$
$661$	5.78185e9	0.778685	0.389343	+	0.921093i	$0.372702\pi$
$662$	0	0
$663$	−2.72231e9	−0.362778
$664$	0	0
$665$	−7.95904e8	−0.104951
$666$	0	0
$667$	6.31391e9	0.823869
$668$	0	0
$669$	−3.12607e8	−0.0403652
$670$	0	0
$671$	5.33076e9	0.681178
$672$	0	0
$673$	−1.07125e10	−1.35468	−0.677340	−	0.735670i	$-0.736867\pi$
$673$	−1.07125e10	−1.35468	−0.677340	+	0.735670i	$0.736867\pi$
$674$	0	0
$675$	3.07547e8	0.0384900
$676$	0	0
$677$	1.24253e8	0.0153903	0.00769516	−	0.999970i	$-0.497551\pi$
$677$	1.24253e8	0.0153903	0.00769516	+	0.999970i	$0.497551\pi$
$678$	0	0
$679$	−2.93251e9	−0.359497
$680$	0	0
$681$	8.09377e8	0.0982055
$682$	0	0
$683$	5.48973e8	0.0659294	0.0329647	−	0.999457i	$-0.489505\pi$
$683$	5.48973e8	0.0659294	0.0329647	+	0.999457i	$0.489505\pi$
$684$	0	0
$685$	−7.20665e9	−0.856676
$686$	0	0
$687$	3.23246e9	0.380351
$688$	0	0
$689$	1.32994e10	1.54905
$690$	0	0
$691$	−3.86163e9	−0.445243	−0.222622	−	0.974905i	$-0.571461\pi$
$691$	−3.86163e9	−0.445243	−0.222622	+	0.974905i	$0.571461\pi$
$692$	0	0
$693$	2.03793e9	0.232608
$694$	0	0
$695$	1.04725e9	0.118332
$696$	0	0
$697$	−3.71324e9	−0.415373
$698$	0	0
$699$	−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	$0.738238\pi$
$702$	0	0
$703$	−6.76600e9	−0.734495
$704$	0	0
$705$	−4.51697e9	−0.485495
$706$	0	0
$707$	6.82791e9	0.726641
$708$	0	0
$709$	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
$712$	0	0
$713$	−1.43109e9	−0.147861
$714$	0	0
$715$	−6.93830e9	−0.709875
$716$	0	0
$717$	4.45362e9	0.451228
$718$	0	0
$719$	−1.34874e10	−1.35325	−0.676625	−	0.736328i	$-0.736558\pi$
$719$	−1.34874e10	−1.35325	−0.676625	+	0.736328i	$0.736558\pi$
$720$	0	0
$721$	−1.38762e9	−0.137879
$722$	0	0
$723$	1.07630e10	1.05913
$724$	0	0
$725$	−2.93616e9	−0.286152
$726$	0	0
$727$	4.09956e9	0.395700	0.197850	−	0.980232i	$-0.436604\pi$
$727$	4.09956e9	0.395700	0.197850	+	0.980232i	$0.436604\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−5.36100e9	−0.507615
$732$	0	0
$733$	9.50716e9	0.891635	0.445818	−	0.895124i	$-0.352913\pi$
$733$	9.50716e9	0.891635	0.445818	+	0.895124i	$0.352913\pi$
$734$	0	0
$735$	−1.89472e9	−0.176011
$736$	0	0
$737$	1.94331e10	1.78816
$738$	0	0
$739$	−2.75490e9	−0.251102	−0.125551	−	0.992087i	$-0.540070\pi$
$739$	−2.75490e9	−0.251102	−0.125551	+	0.992087i	$0.540070\pi$
$740$	0	0
$741$	3.41346e9	0.308199
$742$	0	0
$743$	−7.88700e9	−0.705425	−0.352712	−	0.935732i	$-0.614741\pi$
$743$	−7.88700e9	−0.705425	−0.352712	+	0.935732i	$0.614741\pi$
$744$	0	0
$745$	6.95596e8	0.0616325
$746$	0	0
$747$	−1.62680e9	−0.142795
$748$	0	0
$749$	5.79559e9	0.503978
$750$	0	0
$751$	−1.54231e10	−1.32871	−0.664356	−	0.747416i	$-0.731294\pi$
$751$	−1.54231e10	−1.32871	−0.664356	+	0.747416i	$0.731294\pi$
$752$	0	0
$753$	−6.54782e9	−0.558875
$754$	0	0
$755$	−8.28666e9	−0.700753
$756$	0	0
$757$	−2.28786e10	−1.91687	−0.958436	−	0.285307i	$-0.907904\pi$
$757$	−2.28786e10	−1.91687	−0.958436	+	0.285307i	$0.907904\pi$
$758$	0	0
$759$	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	$0.823509\pi$
$762$	0	0
$763$	1.12503e10	0.916912
$764$	0	0
$765$	−9.03778e8	−0.0729872
$766$	0	0
$767$	−2.67101e9	−0.213743
$768$	0	0
$769$	−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
$772$	0	0
$773$	1.55352e10	1.20973	0.604866	−	0.796327i	$-0.293227\pi$
$773$	1.55352e10	1.20973	0.604866	+	0.796327i	$0.293227\pi$
$774$	0	0
$775$	6.65500e8	0.0513561
$776$	0	0
$777$	7.52117e9	0.575191
$778$	0	0
$779$	4.65596e9	0.352881
$780$	0	0
$781$	−1.45985e10	−1.09655
$782$	0	0
$783$	−3.69871e9	−0.275349
$784$	0	0
$785$	−8.03488e9	−0.592838
$786$	0	0
$787$	−1.21211e10	−0.886399	−0.443199	−	0.896423i	$-0.646157\pi$
$787$	−1.21211e10	−0.886399	−0.443199	+	0.896423i	$0.646157\pi$
$788$	0	0
$789$	9.11051e9	0.660349
$790$	0	0
$791$	−1.88600e9	−0.135495
$792$	0	0
$793$	−9.92537e9	−0.706790
$794$	0	0
$795$	4.41525e9	0.311652
$796$	0	0
$797$	8.20343e9	0.573973	0.286987	−	0.957935i	$-0.407346\pi$
$797$	8.20343e9	0.573973	0.286987	+	0.957935i	$0.407346\pi$
$798$	0	0
$799$	1.32739e10	0.920626
$800$	0	0
$801$	−7.32694e9	−0.503743
$802$	0	0
$803$	1.65555e10	1.12833
$804$	0	0
$805$	2.15040e9	0.145289
$806$	0	0
$807$	6.90891e9	0.462756
$808$	0	0
$809$	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	$-0.747919\pi$
$811$	−2.13417e10	−1.40494	−0.702469	+	0.711715i	$0.747919\pi$
$812$	0	0
$813$	−1.31133e10	−0.855847
$814$	0	0
$815$	−1.06069e10	−0.686336
$816$	0	0
$817$	6.72206e9	0.431246
$818$	0	0
$819$	−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	$0.481746\pi$
$822$	0	0
$823$	−9.68482e9	−0.605609	−0.302804	−	0.953053i	$-0.597923\pi$
$823$	−9.68482e9	−0.605609	−0.302804	+	0.953053i	$0.597923\pi$
$824$	0	0
$825$	−2.30344e9	−0.142820
$826$	0	0
$827$	−1.22370e10	−0.752326	−0.376163	−	0.926553i	$-0.622757\pi$
$827$	−1.22370e10	−0.752326	−0.376163	+	0.926553i	$0.622757\pi$
$828$	0	0
$829$	−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
$832$	0	0
$833$	5.56796e9	0.333763
$834$	0	0
$835$	−8.02680e9	−0.477133
$836$	0	0
$837$	8.38338e8	0.0494174
$838$	0	0
$839$	1.65720e10	0.968743	0.484372	−	0.874862i	$-0.339048\pi$
$839$	1.65720e10	0.968743	0.484372	+	0.874862i	$0.339048\pi$
$840$	0	0
$841$	1.80618e10	1.04707
$842$	0	0
$843$	−5.02806e9	−0.289071
$844$	0	0
$845$	5.07488e9	0.289353
$846$	0	0
$847$	−5.28611e9	−0.298912
$848$	0	0
$849$	1.77728e10	0.996732
$850$	0	0
$851$	1.82806e10	1.01680
$852$	0	0
$853$	−2.87728e10	−1.58730	−0.793652	−	0.608372i	$-0.791823\pi$
$853$	−2.87728e10	−1.58730	−0.793652	+	0.608372i	$0.791823\pi$
$854$	0	0
$855$	1.13323e9	0.0620064
$856$	0	0
$857$	−2.85737e9	−0.155072	−0.0775361	−	0.996990i	$-0.524705\pi$
$857$	−2.85737e9	−0.155072	−0.0775361	+	0.996990i	$0.524705\pi$
$858$	0	0
$859$	−1.38088e10	−0.743327	−0.371663	−	0.928368i	$-0.621212\pi$
$859$	−1.38088e10	−0.743327	−0.371663	+	0.928368i	$0.621212\pi$
$860$	0	0
$861$	−5.17562e9	−0.276345
$862$	0	0
$863$	2.72992e9	0.144581	0.0722905	−	0.997384i	$-0.476969\pi$
$863$	2.72992e9	0.144581	0.0722905	+	0.997384i	$0.476969\pi$
$864$	0	0
$865$	1.37573e10	0.722729
$866$	0	0
$867$	−8.42324e9	−0.438947
$868$	0	0
$869$	−2.98979e10	−1.54551
$870$	0	0
$871$	−3.61825e10	−1.85539
$872$	0	0
$873$	4.17539e9	0.212396
$874$	0	0
$875$	−1.00000e9	−0.0504629
$876$	0	0
$877$	1.52412e10	0.762992	0.381496	−	0.924370i	$-0.375409\pi$
$877$	1.52412e10	0.762992	0.381496	+	0.924370i	$0.375409\pi$
$878$	0	0
$879$	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	$0.654786\pi$
$882$	0	0
$883$	−1.03565e10	−0.506232	−0.253116	−	0.967436i	$-0.581455\pi$
$883$	−1.03565e10	−0.506232	−0.253116	+	0.967436i	$0.581455\pi$
$884$	0	0
$885$	−8.86748e8	−0.0430030
$886$	0	0
$887$	4.04503e9	0.194621	0.0973103	−	0.995254i	$-0.468976\pi$
$887$	4.04503e9	0.194621	0.0973103	+	0.995254i	$0.468976\pi$
$888$	0	0
$889$	−1.37192e10	−0.654897
$890$	0	0
$891$	−2.90167e9	−0.137428
$892$	0	0
$893$	−1.66438e10	−0.782120
$894$	0	0
$895$	−1.08597e10	−0.506335
$896$	0	0
$897$	−9.22260e9	−0.426658
$898$	0	0
$899$	−8.00363e9	−0.367391
$900$	0	0
$901$	−1.29749e10	−0.590975
$902$	0	0
$903$	−7.47231e9	−0.337713
$904$	0	0
$905$	1.62162e10	0.727244
$906$	0	0
$907$	3.30880e10	1.47247	0.736233	−	0.676728i	$-0.236603\pi$
$907$	3.30880e10	1.47247	0.736233	+	0.676728i	$0.236603\pi$
$908$	0	0
$909$	−9.72177e9	−0.429311
$910$	0	0
$911$	2.13526e10	0.935700	0.467850	−	0.883808i	$-0.345029\pi$
$911$	2.13526e10	0.935700	0.467850	+	0.883808i	$0.345029\pi$
$912$	0	0
$913$	1.21843e10	0.529850
$914$	0	0
$915$	−3.29511e9	−0.142199
$916$	0	0
$917$	7.58197e9	0.324705
$918$	0	0
$919$	−4.49811e10	−1.91173	−0.955863	−	0.293813i	$-0.905076\pi$
$919$	−4.49811e10	−1.91173	−0.955863	+	0.293813i	$0.905076\pi$
$920$	0	0
$921$	1.93182e10	0.814813
$922$	0	0
$923$	2.71810e10	1.13778
$924$	0	0
$925$	−8.50103e9	−0.353163
$926$	0	0
$927$	1.97573e9	0.0814608
$928$	0	0
$929$	−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
$932$	0	0
$933$	1.85407e9	0.0747377
$934$	0	0
$935$	6.76903e9	0.270823
$936$	0	0
$937$	−3.31276e10	−1.31553	−0.657766	−	0.753222i	$-0.728498\pi$
$937$	−3.31276e10	−1.31553	−0.657766	+	0.753222i	$0.728498\pi$
$938$	0	0
$939$	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	$0.401707\pi$
$942$	0	0
$943$	−1.25796e10	−0.488514
$944$	0	0
$945$	−1.25971e9	−0.0485579
$946$	0	0
$947$	−5.05218e10	−1.93310	−0.966549	−	0.256482i	$-0.917436\pi$
$947$	−5.05218e10	−1.93310	−0.966549	+	0.256482i	$0.917436\pi$
$948$	0	0
$949$	−3.08247e10	−1.17076
$950$	0	0
$951$	1.05502e10	0.397768
$952$	0	0
$953$	−7.93237e9	−0.296878	−0.148439	−	0.988922i	$-0.547425\pi$
$953$	−7.93237e9	−0.296878	−0.148439	+	0.988922i	$0.547425\pi$
$954$	0	0
$955$	1.77529e10	0.659566
$956$	0	0
$957$	2.77023e10	1.02170
$958$	0	0
$959$	2.95185e10	1.08076
$960$	0	0
$961$	−2.56985e10	−0.934064
$962$	0	0
$963$	−8.25193e9	−0.297758
$964$	0	0
$965$	−3.23638e9	−0.115935
$966$	0	0
$967$	−2.50000e10	−0.889095	−0.444547	−	0.895755i	$-0.646635\pi$
$967$	−2.50000e10	−0.889095	−0.444547	+	0.895755i	$0.646635\pi$
$968$	0	0
$969$	−3.33019e9	−0.117581
$970$	0	0
$971$	−2.23544e10	−0.783603	−0.391801	−	0.920050i	$-0.628148\pi$
$971$	−2.23544e10	−0.783603	−0.391801	+	0.920050i	$0.628148\pi$
$972$	0	0
$973$	−4.28953e9	−0.149285
$974$	0	0
$975$	4.28878e9	0.148190
$976$	0	0
$977$	−1.84311e9	−0.0632297	−0.0316148	−	0.999500i	$-0.510065\pi$
$977$	−1.84311e9	−0.0632297	−0.0316148	+	0.999500i	$0.510065\pi$
$978$	0	0
$979$	5.48767e10	1.86917
$980$	0	0
$981$	−1.60185e10	−0.541726
$982$	0	0
$983$	−1.25816e10	−0.422473	−0.211236	−	0.977435i	$-0.567749\pi$
$983$	−1.25816e10	−0.422473	−0.211236	+	0.977435i	$0.567749\pi$
$984$	0	0
$985$	−4.48913e9	−0.149670
$986$	0	0
$987$	1.85015e10	0.612487
$988$	0	0
$989$	−1.81619e10	−0.596999
$990$	0	0
$991$	2.30746e10	0.753142	0.376571	−	0.926388i	$-0.377103\pi$
$991$	2.30746e10	0.753142	0.376571	+	0.926388i	$0.377103\pi$
$992$	0	0
$993$	1.29673e10	0.420267
$994$	0	0
$995$	2.19316e10	0.705812
$996$	0	0
$997$	−5.91619e9	−0.189064	−0.0945320	−	0.995522i	$-0.530135\pi$
$997$	−5.91619e9	−0.189064	−0.0945320	+	0.995522i	$0.530135\pi$
$998$	0	0
$999$	−1.07089e10	−0.339832

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.8.a.e.1.1	✓	1	4.3	odd	2
90.8.a.b.1.1		1	12.11	even	2
150.8.a.g.1.1		1	20.19	odd	2
150.8.c.e.49.1		2	20.3	even	4
150.8.c.e.49.2		2	20.7	even	4
240.8.a.l.1.1		1	1.1	even	1	trivial
450.8.a.r.1.1		1	60.59	even	2
450.8.c.c.199.1		2	60.47	odd	4
450.8.c.c.199.2		2	60.23	odd	4

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

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

Label	240.8.a.l.1.1

Level	$240$
Weight	$8$
Character	240.1
Self dual	yes
Analytic conductor	$74.972$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$