LMFDB - Embedded newform 240.8.a.d.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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
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)

$f(q)$

$=$

$q-27.0000 q^{3} -125.000 q^{5} +540.000 q^{7} +729.000 q^{9} +O(q^{10})$

$q-27.0000 q^{3} -125.000 q^{5} +540.000 q^{7} +729.000 q^{9} -3584.00 q^{11} +5994.00 q^{13} +3375.00 q^{15} -24666.0 q^{17} +31276.0 q^{19} -14580.0 q^{21} -5376.00 q^{23} +15625.0 q^{25} -19683.0 q^{27} -194846. q^{29} +43592.0 q^{31} +96768.0 q^{33} -67500.0 q^{35} -244358. q^{37} -161838. q^{39} -73686.0 q^{41} +440268. q^{43} -91125.0 q^{45} -465920. q^{47} -531943. q^{49} +665982. q^{51} +47154.0 q^{53} +448000. q^{55} -844452. q^{57} +2.28902e6 q^{59} +1.60648e6 q^{61} +393660. q^{63} -749250. q^{65} +3.65323e6 q^{67} +145152. q^{69} +1.99283e6 q^{71} -4.03707e6 q^{73} -421875. q^{75} -1.93536e6 q^{77} +1.94247e6 q^{79} +531441. q^{81} -1.10567e6 q^{83} +3.08325e6 q^{85} +5.26084e6 q^{87} +14626.0 q^{89} +3.23676e6 q^{91} -1.17698e6 q^{93} -3.90950e6 q^{95} +9.36787e6 q^{97} -2.61274e6 q^{99} +O(q^{100})$

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
$7$	540.000	0.595046	0.297523	−	0.954715i	$-0.403839\pi$
$7$	540.000	0.595046	0.297523	+	0.954715i	$0.403839\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−3584.00	−0.811883	−0.405942	−	0.913899i	$-0.633056\pi$
$11$	−3584.00	−0.811883	−0.405942	+	0.913899i	$0.633056\pi$
$12$	0	0
$13$	5994.00	0.756685	0.378342	−	0.925666i	$-0.376494\pi$
$13$	5994.00	0.756685	0.378342	+	0.925666i	$0.376494\pi$
$14$	0	0
$15$	3375.00	0.258199
$16$	0	0
$17$	−24666.0	−1.21766	−0.608832	−	0.793299i	$-0.708362\pi$
$17$	−24666.0	−1.21766	−0.608832	+	0.793299i	$0.708362\pi$
$18$	0	0
$19$	31276.0	1.04610	0.523050	−	0.852302i	$-0.324794\pi$
$19$	31276.0	1.04610	0.523050	+	0.852302i	$0.324794\pi$
$20$	0	0
$21$	−14580.0	−0.343550
$22$	0	0
$23$	−5376.00	−0.0921323	−0.0460661	−	0.998938i	$-0.514668\pi$
$23$	−5376.00	−0.0921323	−0.0460661	+	0.998938i	$0.514668\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	−194846.	−1.48354	−0.741769	−	0.670656i	$-0.766013\pi$
$29$	−194846.	−1.48354	−0.741769	+	0.670656i	$0.766013\pi$
$30$	0	0
$31$	43592.0	0.262809	0.131405	−	0.991329i	$-0.458051\pi$
$31$	43592.0	0.262809	0.131405	+	0.991329i	$0.458051\pi$
$32$	0	0
$33$	96768.0	0.468741
$34$	0	0
$35$	−67500.0	−0.266113
$36$	0	0
$37$	−244358.	−0.793086	−0.396543	−	0.918016i	$-0.629790\pi$
$37$	−244358.	−0.793086	−0.396543	+	0.918016i	$0.629790\pi$
$38$	0	0
$39$	−161838.	−0.436872
$40$	0	0
$41$	−73686.0	−0.166971	−0.0834856	−	0.996509i	$-0.526605\pi$
$41$	−73686.0	−0.166971	−0.0834856	+	0.996509i	$0.526605\pi$
$42$	0	0
$43$	440268.	0.844457	0.422228	−	0.906489i	$-0.361248\pi$
$43$	440268.	0.844457	0.422228	+	0.906489i	$0.361248\pi$
$44$	0	0
$45$	−91125.0	−0.149071
$46$	0	0
$47$	−465920.	−0.654589	−0.327295	−	0.944922i	$-0.606137\pi$
$47$	−465920.	−0.654589	−0.327295	+	0.944922i	$0.606137\pi$
$48$	0	0
$49$	−531943.	−0.645920
$50$	0	0
$51$	665982.	0.703019
$52$	0	0
$53$	47154.0	0.0435064	0.0217532	−	0.999763i	$-0.493075\pi$
$53$	47154.0	0.0435064	0.0217532	+	0.999763i	$0.493075\pi$
$54$	0	0
$55$	448000.	0.363085
$56$	0	0
$57$	−844452.	−0.603967
$58$	0	0
$59$	2.28902e6	1.45100	0.725501	−	0.688221i	$-0.241608\pi$
$59$	2.28902e6	1.45100	0.725501	+	0.688221i	$0.241608\pi$
$60$	0	0
$61$	1.60648e6	0.906192	0.453096	−	0.891462i	$-0.350319\pi$
$61$	1.60648e6	0.906192	0.453096	+	0.891462i	$0.350319\pi$
$62$	0	0
$63$	393660.	0.198349
$64$	0	0
$65$	−749250.	−0.338400
$66$	0	0
$67$	3.65323e6	1.48394	0.741968	−	0.670436i	$-0.233893\pi$
$67$	3.65323e6	1.48394	0.741968	+	0.670436i	$0.233893\pi$
$68$	0	0
$69$	145152.	0.0531926
$70$	0	0
$71$	1.99283e6	0.660795	0.330397	−	0.943842i	$-0.392817\pi$
$71$	1.99283e6	0.660795	0.330397	+	0.943842i	$0.392817\pi$
$72$	0	0
$73$	−4.03707e6	−1.21461	−0.607304	−	0.794469i	$-0.707749\pi$
$73$	−4.03707e6	−1.21461	−0.607304	+	0.794469i	$0.707749\pi$
$74$	0	0
$75$	−421875.	−0.115470
$76$	0	0
$77$	−1.93536e6	−0.483108
$78$	0	0
$79$	1.94247e6	0.443261	0.221631	−	0.975131i	$-0.428862\pi$
$79$	1.94247e6	0.443261	0.221631	+	0.975131i	$0.428862\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−1.10567e6	−0.212252	−0.106126	−	0.994353i	$-0.533845\pi$
$83$	−1.10567e6	−0.212252	−0.106126	+	0.994353i	$0.533845\pi$
$84$	0	0
$85$	3.08325e6	0.544556
$86$	0	0
$87$	5.26084e6	0.856521
$88$	0	0
$89$	14626.0	0.00219918	0.00109959	−	0.999999i	$-0.499650\pi$
$89$	14626.0	0.00219918	0.00109959	+	0.999999i	$0.499650\pi$
$90$	0	0
$91$	3.23676e6	0.450262
$92$	0	0
$93$	−1.17698e6	−0.151733
$94$	0	0
$95$	−3.90950e6	−0.467831
$96$	0	0
$97$	9.36787e6	1.04217	0.521087	−	0.853504i	$-0.325527\pi$
$97$	9.36787e6	1.04217	0.521087	+	0.853504i	$0.325527\pi$
$98$	0	0
$99$	−2.61274e6	−0.270628
$100$	0	0
$101$	1.46520e7	1.41506	0.707528	−	0.706686i	$-0.249811\pi$
$101$	1.46520e7	1.41506	0.707528	+	0.706686i	$0.249811\pi$
$102$	0	0
$103$	4.00105e6	0.360781	0.180391	−	0.983595i	$-0.442264\pi$
$103$	4.00105e6	0.360781	0.180391	+	0.983595i	$0.442264\pi$
$104$	0	0
$105$	1.82250e6	0.153640
$106$	0	0
$107$	1.72613e7	1.36217	0.681083	−	0.732206i	$-0.261509\pi$
$107$	1.72613e7	1.36217	0.681083	+	0.732206i	$0.261509\pi$
$108$	0	0
$109$	6.67294e6	0.493543	0.246771	−	0.969074i	$-0.420630\pi$
$109$	6.67294e6	0.493543	0.246771	+	0.969074i	$0.420630\pi$
$110$	0	0
$111$	6.59767e6	0.457889
$112$	0	0
$113$	2.51498e7	1.63968	0.819841	−	0.572592i	$-0.194062\pi$
$113$	2.51498e7	1.63968	0.819841	+	0.572592i	$0.194062\pi$
$114$	0	0
$115$	672000.	0.0412028
$116$	0	0
$117$	4.36963e6	0.252228
$118$	0	0
$119$	−1.33196e7	−0.724566
$120$	0	0
$121$	−6.64212e6	−0.340846
$122$	0	0
$123$	1.98952e6	0.0964008
$124$	0	0
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	−3.65508e6	−0.158338	−0.0791689	−	0.996861i	$-0.525227\pi$
$127$	−3.65508e6	−0.158338	−0.0791689	+	0.996861i	$0.525227\pi$
$128$	0	0
$129$	−1.18872e7	−0.487547
$130$	0	0
$131$	−3.25654e7	−1.26563	−0.632815	−	0.774303i	$-0.718101\pi$
$131$	−3.25654e7	−1.26563	−0.632815	+	0.774303i	$0.718101\pi$
$132$	0	0
$133$	1.68890e7	0.622478
$134$	0	0
$135$	2.46038e6	0.0860663
$136$	0	0
$137$	4.40777e7	1.46453	0.732263	−	0.681022i	$-0.238464\pi$
$137$	4.40777e7	1.46453	0.732263	+	0.681022i	$0.238464\pi$
$138$	0	0
$139$	5.02559e7	1.58721	0.793607	−	0.608430i	$-0.208200\pi$
$139$	5.02559e7	1.58721	0.793607	+	0.608430i	$0.208200\pi$
$140$	0	0
$141$	1.25798e7	0.377927
$142$	0	0
$143$	−2.14825e7	−0.614340
$144$	0	0
$145$	2.43557e7	0.663458
$146$	0	0
$147$	1.43625e7	0.372922
$148$	0	0
$149$	5.40505e7	1.33859	0.669296	−	0.742996i	$-0.266596\pi$
$149$	5.40505e7	1.33859	0.669296	+	0.742996i	$0.266596\pi$
$150$	0	0
$151$	−3.56658e6	−0.0843011	−0.0421505	−	0.999111i	$-0.513421\pi$
$151$	−3.56658e6	−0.0843011	−0.0421505	+	0.999111i	$0.513421\pi$
$152$	0	0
$153$	−1.79815e7	−0.405888
$154$	0	0
$155$	−5.44900e6	−0.117532
$156$	0	0
$157$	−2.07073e7	−0.427045	−0.213523	−	0.976938i	$-0.568494\pi$
$157$	−2.07073e7	−0.427045	−0.213523	+	0.976938i	$0.568494\pi$
$158$	0	0
$159$	−1.27316e6	−0.0251184
$160$	0	0
$161$	−2.90304e6	−0.0548230
$162$	0	0
$163$	−1.61059e7	−0.291291	−0.145646	−	0.989337i	$-0.546526\pi$
$163$	−1.61059e7	−0.291291	−0.145646	+	0.989337i	$0.546526\pi$
$164$	0	0
$165$	−1.20960e7	−0.209627
$166$	0	0
$167$	−569728.	−0.00946586	−0.00473293	−	0.999989i	$-0.501507\pi$
$167$	−569728.	−0.00946586	−0.00473293	+	0.999989i	$0.501507\pi$
$168$	0	0
$169$	−2.68205e7	−0.427428
$170$	0	0
$171$	2.28002e7	0.348700
$172$	0	0
$173$	3.81223e7	0.559780	0.279890	−	0.960032i	$-0.409702\pi$
$173$	3.81223e7	0.559780	0.279890	+	0.960032i	$0.409702\pi$
$174$	0	0
$175$	8.43750e6	0.119009
$176$	0	0
$177$	−6.18036e7	−0.837737
$178$	0	0
$179$	−5.43251e7	−0.707970	−0.353985	−	0.935251i	$-0.615174\pi$
$179$	−5.43251e7	−0.707970	−0.353985	+	0.935251i	$0.615174\pi$
$180$	0	0
$181$	−3.40094e7	−0.426308	−0.213154	−	0.977019i	$-0.568374\pi$
$181$	−3.40094e7	−0.426308	−0.213154	+	0.977019i	$0.568374\pi$
$182$	0	0
$183$	−4.33749e7	−0.523190
$184$	0	0
$185$	3.05448e7	0.354679
$186$	0	0
$187$	8.84029e7	0.988601
$188$	0	0
$189$	−1.06288e7	−0.114517
$190$	0	0
$191$	9.78493e7	1.01611	0.508055	−	0.861324i	$-0.330365\pi$
$191$	9.78493e7	1.01611	0.508055	+	0.861324i	$0.330365\pi$
$192$	0	0
$193$	1.27215e8	1.27377	0.636883	−	0.770961i	$-0.280224\pi$
$193$	1.27215e8	1.27377	0.636883	+	0.770961i	$0.280224\pi$
$194$	0	0
$195$	2.02298e7	0.195375
$196$	0	0
$197$	8.31465e7	0.774840	0.387420	−	0.921903i	$-0.373366\pi$
$197$	8.31465e7	0.774840	0.387420	+	0.921903i	$0.373366\pi$
$198$	0	0
$199$	−6.73202e6	−0.0605564	−0.0302782	−	0.999542i	$-0.509639\pi$
$199$	−6.73202e6	−0.0605564	−0.0302782	+	0.999542i	$0.509639\pi$
$200$	0	0
$201$	−9.86372e7	−0.856750
$202$	0	0
$203$	−1.05217e8	−0.882773
$204$	0	0
$205$	9.21075e6	0.0746718
$206$	0	0
$207$	−3.91910e6	−0.0307108
$208$	0	0
$209$	−1.12093e8	−0.849312
$210$	0	0
$211$	2.32363e8	1.70286	0.851428	−	0.524472i	$-0.175737\pi$
$211$	2.32363e8	1.70286	0.851428	+	0.524472i	$0.175737\pi$
$212$	0	0
$213$	−5.38065e7	−0.381510
$214$	0	0
$215$	−5.50335e7	−0.377653
$216$	0	0
$217$	2.35397e7	0.156384
$218$	0	0
$219$	1.09001e8	0.701254
$220$	0	0
$221$	−1.47848e8	−0.921388
$222$	0	0
$223$	1.64917e8	0.995861	0.497931	−	0.867217i	$-0.334093\pi$
$223$	1.64917e8	0.995861	0.497931	+	0.867217i	$0.334093\pi$
$224$	0	0
$225$	1.13906e7	0.0666667
$226$	0	0
$227$	2.45397e7	0.139245	0.0696224	−	0.997573i	$-0.477821\pi$
$227$	2.45397e7	0.139245	0.0696224	+	0.997573i	$0.477821\pi$
$228$	0	0
$229$	−1.47416e8	−0.811183	−0.405592	−	0.914054i	$-0.632934\pi$
$229$	−1.47416e8	−0.811183	−0.405592	+	0.914054i	$0.632934\pi$
$230$	0	0
$231$	5.22547e7	0.278923
$232$	0	0
$233$	1.80364e7	0.0934121	0.0467060	−	0.998909i	$-0.485128\pi$
$233$	1.80364e7	0.0934121	0.0467060	+	0.998909i	$0.485128\pi$
$234$	0	0
$235$	5.82400e7	0.292741
$236$	0	0
$237$	−5.24467e7	−0.255917
$238$	0	0
$239$	−3.06434e8	−1.45192	−0.725962	−	0.687734i	$-0.758605\pi$
$239$	−3.06434e8	−1.45192	−0.725962	+	0.687734i	$0.758605\pi$
$240$	0	0
$241$	−4.20718e8	−1.93611	−0.968057	−	0.250729i	$-0.919330\pi$
$241$	−4.20718e8	−1.93611	−0.968057	+	0.250729i	$0.919330\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	6.64929e7	0.288864
$246$	0	0
$247$	1.87468e8	0.791569
$248$	0	0
$249$	2.98530e7	0.122544
$250$	0	0
$251$	−1.60387e8	−0.640193	−0.320097	−	0.947385i	$-0.603715\pi$
$251$	−1.60387e8	−0.640193	−0.320097	+	0.947385i	$0.603715\pi$
$252$	0	0
$253$	1.92676e7	0.0748007
$254$	0	0
$255$	−8.32478e7	−0.314400
$256$	0	0
$257$	−1.15916e8	−0.425970	−0.212985	−	0.977056i	$-0.568318\pi$
$257$	−1.15916e8	−0.425970	−0.212985	+	0.977056i	$0.568318\pi$
$258$	0	0
$259$	−1.31953e8	−0.471923
$260$	0	0
$261$	−1.42043e8	−0.494512
$262$	0	0
$263$	−4.29419e8	−1.45558	−0.727790	−	0.685800i	$-0.759453\pi$
$263$	−4.29419e8	−1.45558	−0.727790	+	0.685800i	$0.759453\pi$
$264$	0	0
$265$	−5.89425e6	−0.0194566
$266$	0	0
$267$	−394902.	−0.00126970
$268$	0	0
$269$	−2.43599e8	−0.763030	−0.381515	−	0.924363i	$-0.624598\pi$
$269$	−2.43599e8	−0.763030	−0.381515	+	0.924363i	$0.624598\pi$
$270$	0	0
$271$	−1.63200e8	−0.498111	−0.249056	−	0.968489i	$-0.580120\pi$
$271$	−1.63200e8	−0.498111	−0.249056	+	0.968489i	$0.580120\pi$
$272$	0	0
$273$	−8.73925e7	−0.259959
$274$	0	0
$275$	−5.60000e7	−0.162377
$276$	0	0
$277$	1.69043e8	0.477879	0.238940	−	0.971034i	$-0.423200\pi$
$277$	1.69043e8	0.477879	0.238940	+	0.971034i	$0.423200\pi$
$278$	0	0
$279$	3.17786e7	0.0876031
$280$	0	0
$281$	−4.40644e8	−1.18472	−0.592360	−	0.805673i	$-0.701804\pi$
$281$	−4.40644e8	−1.18472	−0.592360	+	0.805673i	$0.701804\pi$
$282$	0	0
$283$	−7.37145e8	−1.93330	−0.966652	−	0.256092i	$-0.917565\pi$
$283$	−7.37145e8	−1.93330	−0.966652	+	0.256092i	$0.917565\pi$
$284$	0	0
$285$	1.05557e8	0.270102
$286$	0	0
$287$	−3.97904e7	−0.0993555
$288$	0	0
$289$	1.98073e8	0.482706
$290$	0	0
$291$	−2.52933e8	−0.601699
$292$	0	0
$293$	6.95918e8	1.61630	0.808149	−	0.588978i	$-0.200470\pi$
$293$	6.95918e8	1.61630	0.808149	+	0.588978i	$0.200470\pi$
$294$	0	0
$295$	−2.86128e8	−0.648908
$296$	0	0
$297$	7.05439e7	0.156247
$298$	0	0
$299$	−3.22237e7	−0.0697151
$300$	0	0
$301$	2.37745e8	0.502491
$302$	0	0
$303$	−3.95605e8	−0.816982
$304$	0	0
$305$	−2.00810e8	−0.405262
$306$	0	0
$307$	3.73398e8	0.736526	0.368263	−	0.929722i	$-0.379953\pi$
$307$	3.73398e8	0.736526	0.368263	+	0.929722i	$0.379953\pi$
$308$	0	0
$309$	−1.08028e8	−0.208297
$310$	0	0
$311$	3.22066e8	0.607132	0.303566	−	0.952810i	$-0.401823\pi$
$311$	3.22066e8	0.607132	0.303566	+	0.952810i	$0.401823\pi$
$312$	0	0
$313$	6.28126e8	1.15782	0.578911	−	0.815391i	$-0.303478\pi$
$313$	6.28126e8	1.15782	0.578911	+	0.815391i	$0.303478\pi$
$314$	0	0
$315$	−4.92075e7	−0.0887042
$316$	0	0
$317$	−1.79960e8	−0.317299	−0.158649	−	0.987335i	$-0.550714\pi$
$317$	−1.79960e8	−0.317299	−0.158649	+	0.987335i	$0.550714\pi$
$318$	0	0
$319$	6.98328e8	1.20446
$320$	0	0
$321$	−4.66055e8	−0.786447
$322$	0	0
$323$	−7.71454e8	−1.27380
$324$	0	0
$325$	9.36562e7	0.151337
$326$	0	0
$327$	−1.80169e8	−0.284947
$328$	0	0
$329$	−2.51597e8	−0.389511
$330$	0	0
$331$	5.17622e8	0.784539	0.392270	−	0.919850i	$-0.371690\pi$
$331$	5.17622e8	0.784539	0.392270	+	0.919850i	$0.371690\pi$
$332$	0	0
$333$	−1.78137e8	−0.264362
$334$	0	0
$335$	−4.56654e8	−0.663636
$336$	0	0
$337$	−2.06359e8	−0.293710	−0.146855	−	0.989158i	$-0.546915\pi$
$337$	−2.06359e8	−0.293710	−0.146855	+	0.989158i	$0.546915\pi$
$338$	0	0
$339$	−6.79044e8	−0.946671
$340$	0	0
$341$	−1.56234e8	−0.213371
$342$	0	0
$343$	−7.31962e8	−0.979398
$344$	0	0
$345$	−1.81440e7	−0.0237885
$346$	0	0
$347$	8.56243e7	0.110013	0.0550065	−	0.998486i	$-0.482482\pi$
$347$	8.56243e7	0.110013	0.0550065	+	0.998486i	$0.482482\pi$
$348$	0	0
$349$	1.44884e9	1.82445	0.912226	−	0.409688i	$-0.134362\pi$
$349$	1.44884e9	1.82445	0.912226	+	0.409688i	$0.134362\pi$
$350$	0	0
$351$	−1.17980e8	−0.145624
$352$	0	0
$353$	−6.63738e8	−0.803129	−0.401564	−	0.915831i	$-0.631533\pi$
$353$	−6.63738e8	−0.803129	−0.401564	+	0.915831i	$0.631533\pi$
$354$	0	0
$355$	−2.49104e8	−0.295516
$356$	0	0
$357$	3.59630e8	0.418329
$358$	0	0
$359$	8.11116e8	0.925236	0.462618	−	0.886558i	$-0.346910\pi$
$359$	8.11116e8	0.925236	0.462618	+	0.886558i	$0.346910\pi$
$360$	0	0
$361$	8.43164e7	0.0943272
$362$	0	0
$363$	1.79337e8	0.196787
$364$	0	0
$365$	5.04634e8	0.543189
$366$	0	0
$367$	1.25253e9	1.32268	0.661342	−	0.750084i	$-0.269987\pi$
$367$	1.25253e9	1.32268	0.661342	+	0.750084i	$0.269987\pi$
$368$	0	0
$369$	−5.37171e7	−0.0556571
$370$	0	0
$371$	2.54632e7	0.0258883
$372$	0	0
$373$	1.61555e9	1.61190	0.805950	−	0.591983i	$-0.201655\pi$
$373$	1.61555e9	1.61190	0.805950	+	0.591983i	$0.201655\pi$
$374$	0	0
$375$	5.27344e7	0.0516398
$376$	0	0
$377$	−1.16791e9	−1.12257
$378$	0	0
$379$	8.94292e8	0.843805	0.421903	−	0.906641i	$-0.361362\pi$
$379$	8.94292e8	0.843805	0.421903	+	0.906641i	$0.361362\pi$
$380$	0	0
$381$	9.86873e7	0.0914163
$382$	0	0
$383$	1.86145e9	1.69299	0.846496	−	0.532396i	$-0.178708\pi$
$383$	1.86145e9	1.69299	0.846496	+	0.532396i	$0.178708\pi$
$384$	0	0
$385$	2.41920e8	0.216052
$386$	0	0
$387$	3.20955e8	0.281486
$388$	0	0
$389$	−2.69370e8	−0.232020	−0.116010	−	0.993248i	$-0.537010\pi$
$389$	−2.69370e8	−0.232020	−0.116010	+	0.993248i	$0.537010\pi$
$390$	0	0
$391$	1.32604e8	0.112186
$392$	0	0
$393$	8.79265e8	0.730711
$394$	0	0
$395$	−2.42809e8	−0.198233
$396$	0	0
$397$	1.33926e9	1.07423	0.537116	−	0.843509i	$-0.319514\pi$
$397$	1.33926e9	1.07423	0.537116	+	0.843509i	$0.319514\pi$
$398$	0	0
$399$	−4.56004e8	−0.359388
$400$	0	0
$401$	−1.46336e9	−1.13331	−0.566653	−	0.823957i	$-0.691762\pi$
$401$	−1.46336e9	−1.13331	−0.566653	+	0.823957i	$0.691762\pi$
$402$	0	0
$403$	2.61290e8	0.198864
$404$	0	0
$405$	−6.64301e7	−0.0496904
$406$	0	0
$407$	8.75779e8	0.643894
$408$	0	0
$409$	−9.30881e8	−0.672763	−0.336382	−	0.941726i	$-0.609203\pi$
$409$	−9.30881e8	−0.672763	−0.336382	+	0.941726i	$0.609203\pi$
$410$	0	0
$411$	−1.19010e9	−0.845545
$412$	0	0
$413$	1.23607e9	0.863414
$414$	0	0
$415$	1.38208e8	0.0949219
$416$	0	0
$417$	−1.35691e9	−0.916379
$418$	0	0
$419$	2.16338e7	0.0143676	0.00718379	−	0.999974i	$-0.497713\pi$
$419$	2.16338e7	0.0143676	0.00718379	+	0.999974i	$0.497713\pi$
$420$	0	0
$421$	−3.29561e8	−0.215253	−0.107626	−	0.994191i	$-0.534325\pi$
$421$	−3.29561e8	−0.215253	−0.107626	+	0.994191i	$0.534325\pi$
$422$	0	0
$423$	−3.39656e8	−0.218196
$424$	0	0
$425$	−3.85406e8	−0.243533
$426$	0	0
$427$	8.67498e8	0.539226
$428$	0	0
$429$	5.80027e8	0.354689
$430$	0	0
$431$	1.61775e9	0.973285	0.486643	−	0.873601i	$-0.338221\pi$
$431$	1.61775e9	0.973285	0.486643	+	0.873601i	$0.338221\pi$
$432$	0	0
$433$	−3.13091e8	−0.185337	−0.0926687	−	0.995697i	$-0.529540\pi$
$433$	−3.13091e8	−0.185337	−0.0926687	+	0.995697i	$0.529540\pi$
$434$	0	0
$435$	−6.57605e8	−0.383048
$436$	0	0
$437$	−1.68140e8	−0.0963797
$438$	0	0
$439$	3.41644e9	1.92729	0.963646	−	0.267181i	$-0.0860922\pi$
$439$	3.41644e9	1.92729	0.963646	+	0.267181i	$0.0860922\pi$
$440$	0	0
$441$	−3.87786e8	−0.215307
$442$	0	0
$443$	1.41875e9	0.775340	0.387670	−	0.921798i	$-0.373280\pi$
$443$	1.41875e9	0.775340	0.387670	+	0.921798i	$0.373280\pi$
$444$	0	0
$445$	−1.82825e6	−0.000983503 0
$446$	0	0
$447$	−1.45936e9	−0.772836
$448$	0	0
$449$	6.05549e8	0.315709	0.157854	−	0.987462i	$-0.449542\pi$
$449$	6.05549e8	0.315709	0.157854	+	0.987462i	$0.449542\pi$
$450$	0	0
$451$	2.64091e8	0.135561
$452$	0	0
$453$	9.62978e7	0.0486713
$454$	0	0
$455$	−4.04595e8	−0.201363
$456$	0	0
$457$	2.29699e9	1.12578	0.562890	−	0.826532i	$-0.309690\pi$
$457$	2.29699e9	1.12578	0.562890	+	0.826532i	$0.309690\pi$
$458$	0	0
$459$	4.85501e8	0.234340
$460$	0	0
$461$	−4.03718e9	−1.91922	−0.959609	−	0.281337i	$-0.909222\pi$
$461$	−4.03718e9	−1.91922	−0.959609	+	0.281337i	$0.909222\pi$
$462$	0	0
$463$	−7.00365e8	−0.327937	−0.163969	−	0.986466i	$-0.552430\pi$
$463$	−7.00365e8	−0.327937	−0.163969	+	0.986466i	$0.552430\pi$
$464$	0	0
$465$	1.47123e8	0.0678571
$466$	0	0
$467$	4.08634e9	1.85663	0.928316	−	0.371793i	$-0.121257\pi$
$467$	4.08634e9	1.85663	0.928316	+	0.371793i	$0.121257\pi$
$468$	0	0
$469$	1.97274e9	0.883010
$470$	0	0
$471$	5.59096e8	0.246555
$472$	0	0
$473$	−1.57792e9	−0.685601
$474$	0	0
$475$	4.88687e8	0.209220
$476$	0	0
$477$	3.43753e7	0.0145021
$478$	0	0
$479$	1.33732e8	0.0555982	0.0277991	−	0.999614i	$-0.491150\pi$
$479$	1.33732e8	0.0555982	0.0277991	+	0.999614i	$0.491150\pi$
$480$	0	0
$481$	−1.46468e9	−0.600116
$482$	0	0
$483$	7.83821e7	0.0316520
$484$	0	0
$485$	−1.17098e9	−0.466074
$486$	0	0
$487$	−6.73233e8	−0.264128	−0.132064	−	0.991241i	$-0.542160\pi$
$487$	−6.73233e8	−0.264128	−0.132064	+	0.991241i	$0.542160\pi$
$488$	0	0
$489$	4.34859e8	0.168177
$490$	0	0
$491$	−3.32002e9	−1.26577	−0.632886	−	0.774245i	$-0.718130\pi$
$491$	−3.32002e9	−1.26577	−0.632886	+	0.774245i	$0.718130\pi$
$492$	0	0
$493$	4.80607e9	1.80645
$494$	0	0
$495$	3.26592e8	0.121028
$496$	0	0
$497$	1.07613e9	0.393203
$498$	0	0
$499$	−3.01858e8	−0.108755	−0.0543776	−	0.998520i	$-0.517317\pi$
$499$	−3.01858e8	−0.108755	−0.0543776	+	0.998520i	$0.517317\pi$
$500$	0	0
$501$	1.53827e7	0.00546511
$502$	0	0
$503$	−3.37577e9	−1.18273	−0.591363	−	0.806405i	$-0.701410\pi$
$503$	−3.37577e9	−1.18273	−0.591363	+	0.806405i	$0.701410\pi$
$504$	0	0
$505$	−1.83151e9	−0.632832
$506$	0	0
$507$	7.24153e8	0.246776
$508$	0	0
$509$	4.50975e9	1.51580	0.757898	−	0.652373i	$-0.226227\pi$
$509$	4.50975e9	1.51580	0.757898	+	0.652373i	$0.226227\pi$
$510$	0	0
$511$	−2.18002e9	−0.722748
$512$	0	0
$513$	−6.15606e8	−0.201322
$514$	0	0
$515$	−5.00132e8	−0.161346
$516$	0	0
$517$	1.66986e9	0.531450
$518$	0	0
$519$	−1.02930e9	−0.323189
$520$	0	0
$521$	−5.68799e7	−0.0176209	−0.00881043	−	0.999961i	$-0.502804\pi$
$521$	−5.68799e7	−0.0176209	−0.00881043	+	0.999961i	$0.502804\pi$
$522$	0	0
$523$	5.69608e9	1.74108	0.870542	−	0.492094i	$-0.163768\pi$
$523$	5.69608e9	1.74108	0.870542	+	0.492094i	$0.163768\pi$
$524$	0	0
$525$	−2.27812e8	−0.0687100
$526$	0	0
$527$	−1.07524e9	−0.320014
$528$	0	0
$529$	−3.37592e9	−0.991512
$530$	0	0
$531$	1.66870e9	0.483668
$532$	0	0
$533$	−4.41674e8	−0.126345
$534$	0	0
$535$	−2.15766e9	−0.609179
$536$	0	0
$537$	1.46678e9	0.408747
$538$	0	0
$539$	1.90648e9	0.524412
$540$	0	0
$541$	5.67986e9	1.54222	0.771111	−	0.636701i	$-0.219701\pi$
$541$	5.67986e9	1.54222	0.771111	+	0.636701i	$0.219701\pi$
$542$	0	0
$543$	9.18253e8	0.246129
$544$	0	0
$545$	−8.34118e8	−0.220719
$546$	0	0
$547$	−4.04334e9	−1.05629	−0.528146	−	0.849153i	$-0.677113\pi$
$547$	−4.04334e9	−1.05629	−0.528146	+	0.849153i	$0.677113\pi$
$548$	0	0
$549$	1.17112e9	0.302064
$550$	0	0
$551$	−6.09400e9	−1.55193
$552$	0	0
$553$	1.04893e9	0.263761
$554$	0	0
$555$	−8.24708e8	−0.204774
$556$	0	0
$557$	7.35076e9	1.80235	0.901175	−	0.433456i	$-0.142706\pi$
$557$	7.35076e9	1.80235	0.901175	+	0.433456i	$0.142706\pi$
$558$	0	0
$559$	2.63897e9	0.638988
$560$	0	0
$561$	−2.38688e9	−0.570769
$562$	0	0
$563$	−2.95680e8	−0.0698301	−0.0349151	−	0.999390i	$-0.511116\pi$
$563$	−2.95680e8	−0.0698301	−0.0349151	+	0.999390i	$0.511116\pi$
$564$	0	0
$565$	−3.14372e9	−0.733288
$566$	0	0
$567$	2.86978e8	0.0661162
$568$	0	0
$569$	9.95363e8	0.226511	0.113255	−	0.993566i	$-0.463872\pi$
$569$	9.95363e8	0.226511	0.113255	+	0.993566i	$0.463872\pi$
$570$	0	0
$571$	−6.41612e9	−1.44227	−0.721134	−	0.692795i	$-0.756379\pi$
$571$	−6.41612e9	−1.44227	−0.721134	+	0.692795i	$0.756379\pi$
$572$	0	0
$573$	−2.64193e9	−0.586652
$574$	0	0
$575$	−8.40000e7	−0.0184265
$576$	0	0
$577$	1.59733e9	0.346163	0.173081	−	0.984908i	$-0.444628\pi$
$577$	1.59733e9	0.346163	0.173081	+	0.984908i	$0.444628\pi$
$578$	0	0
$579$	−3.43482e9	−0.735409
$580$	0	0
$581$	−5.97061e8	−0.126300
$582$	0	0
$583$	−1.69000e8	−0.0353221
$584$	0	0
$585$	−5.46203e8	−0.112800
$586$	0	0
$587$	−4.67543e9	−0.954087	−0.477043	−	0.878880i	$-0.658292\pi$
$587$	−4.67543e9	−0.954087	−0.477043	+	0.878880i	$0.658292\pi$
$588$	0	0
$589$	1.36338e9	0.274925
$590$	0	0
$591$	−2.24496e9	−0.447354
$592$	0	0
$593$	−6.43350e9	−1.26694	−0.633469	−	0.773768i	$-0.718370\pi$
$593$	−6.43350e9	−1.26694	−0.633469	+	0.773768i	$0.718370\pi$
$594$	0	0
$595$	1.66495e9	0.324036
$596$	0	0
$597$	1.81765e8	0.0349622
$598$	0	0
$599$	−5.31872e9	−1.01115	−0.505573	−	0.862784i	$-0.668719\pi$
$599$	−5.31872e9	−1.01115	−0.505573	+	0.862784i	$0.668719\pi$
$600$	0	0
$601$	−8.95968e9	−1.68357	−0.841786	−	0.539811i	$-0.818496\pi$
$601$	−8.95968e9	−1.68357	−0.841786	+	0.539811i	$0.818496\pi$
$602$	0	0
$603$	2.66320e9	0.494645
$604$	0	0
$605$	8.30264e8	0.152431
$606$	0	0
$607$	8.65725e9	1.57116	0.785579	−	0.618762i	$-0.212365\pi$
$607$	8.65725e9	1.57116	0.785579	+	0.618762i	$0.212365\pi$
$608$	0	0
$609$	2.84085e9	0.509669
$610$	0	0
$611$	−2.79272e9	−0.495318
$612$	0	0
$613$	−1.66931e9	−0.292703	−0.146351	−	0.989233i	$-0.546753\pi$
$613$	−1.66931e9	−0.292703	−0.146351	+	0.989233i	$0.546753\pi$
$614$	0	0
$615$	−2.48690e8	−0.0431118
$616$	0	0
$617$	1.00732e9	0.172651	0.0863255	−	0.996267i	$-0.472487\pi$
$617$	1.00732e9	0.172651	0.0863255	+	0.996267i	$0.472487\pi$
$618$	0	0
$619$	−2.91303e9	−0.493660	−0.246830	−	0.969059i	$-0.579389\pi$
$619$	−2.91303e9	−0.493660	−0.246830	+	0.969059i	$0.579389\pi$
$620$	0	0
$621$	1.05816e8	0.0177309
$622$	0	0
$623$	7.89804e6	0.00130861
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	0	0
$627$	3.02652e9	0.490350
$628$	0	0
$629$	6.02733e9	0.965713
$630$	0	0
$631$	−4.06191e9	−0.643617	−0.321809	−	0.946805i	$-0.604291\pi$
$631$	−4.06191e9	−0.643617	−0.321809	+	0.946805i	$0.604291\pi$
$632$	0	0
$633$	−6.27379e9	−0.983144
$634$	0	0
$635$	4.56886e8	0.0708108
$636$	0	0
$637$	−3.18847e9	−0.488758
$638$	0	0
$639$	1.45277e9	0.220265
$640$	0	0
$641$	4.55612e7	0.00683269	0.00341635	−	0.999994i	$-0.498913\pi$
$641$	4.55612e7	0.00683269	0.00341635	+	0.999994i	$0.498913\pi$
$642$	0	0
$643$	−2.86005e9	−0.424263	−0.212131	−	0.977241i	$-0.568040\pi$
$643$	−2.86005e9	−0.424263	−0.212131	+	0.977241i	$0.568040\pi$
$644$	0	0
$645$	1.48590e9	0.218038
$646$	0	0
$647$	8.57059e7	0.0124407	0.00622037	−	0.999981i	$-0.498020\pi$
$647$	8.57059e7	0.0124407	0.00622037	+	0.999981i	$0.498020\pi$
$648$	0	0
$649$	−8.20386e9	−1.17805
$650$	0	0
$651$	−6.35571e8	−0.0902882
$652$	0	0
$653$	−9.45177e9	−1.32836	−0.664182	−	0.747571i	$-0.731220\pi$
$653$	−9.45177e9	−1.32836	−0.664182	+	0.747571i	$0.731220\pi$
$654$	0	0
$655$	4.07067e9	0.566007
$656$	0	0
$657$	−2.94302e9	−0.404869
$658$	0	0
$659$	6.09489e9	0.829597	0.414798	−	0.909913i	$-0.363852\pi$
$659$	6.09489e9	0.829597	0.414798	+	0.909913i	$0.363852\pi$
$660$	0	0
$661$	−5.32644e9	−0.717352	−0.358676	−	0.933462i	$-0.616772\pi$
$661$	−5.32644e9	−0.717352	−0.358676	+	0.933462i	$0.616772\pi$
$662$	0	0
$663$	3.99190e9	0.531964
$664$	0	0
$665$	−2.11113e9	−0.278381
$666$	0	0
$667$	1.04749e9	0.136682
$668$	0	0
$669$	−4.45276e9	−0.574961
$670$	0	0
$671$	−5.75762e9	−0.735722
$672$	0	0
$673$	−5.98275e9	−0.756569	−0.378284	−	0.925689i	$-0.623486\pi$
$673$	−5.98275e9	−0.756569	−0.378284	+	0.925689i	$0.623486\pi$
$674$	0	0
$675$	−3.07547e8	−0.0384900
$676$	0	0
$677$	1.24611e10	1.54347	0.771733	−	0.635946i	$-0.219390\pi$
$677$	1.24611e10	1.54347	0.771733	+	0.635946i	$0.219390\pi$
$678$	0	0
$679$	5.05865e9	0.620141
$680$	0	0
$681$	−6.62572e8	−0.0803930
$682$	0	0
$683$	5.51389e9	0.662195	0.331098	−	0.943597i	$-0.392581\pi$
$683$	5.51389e9	0.662195	0.331098	+	0.943597i	$0.392581\pi$
$684$	0	0
$685$	−5.50971e9	−0.654956
$686$	0	0
$687$	3.98022e9	0.468337
$688$	0	0
$689$	2.82641e8	0.0329206
$690$	0	0
$691$	−1.52984e10	−1.76389	−0.881946	−	0.471350i	$-0.843767\pi$
$691$	−1.52984e10	−1.76389	−0.881946	+	0.471350i	$0.843767\pi$
$692$	0	0
$693$	−1.41088e9	−0.161036
$694$	0	0
$695$	−6.28199e9	−0.709824
$696$	0	0
$697$	1.81754e9	0.203315
$698$	0	0
$699$	−4.86981e8	−0.0539315
$700$	0	0
$701$	1.44144e10	1.58046	0.790232	−	0.612808i	$-0.209960\pi$
$701$	1.44144e10	1.58046	0.790232	+	0.612808i	$0.209960\pi$
$702$	0	0
$703$	−7.64254e9	−0.829648
$704$	0	0
$705$	−1.57248e9	−0.169014
$706$	0	0
$707$	7.91210e9	0.842023
$708$	0	0
$709$	−1.54497e10	−1.62802	−0.814009	−	0.580853i	$-0.802719\pi$
$709$	−1.54497e10	−1.62802	−0.814009	+	0.580853i	$0.802719\pi$
$710$	0	0
$711$	1.41606e9	0.147754
$712$	0	0
$713$	−2.34351e8	−0.0242132
$714$	0	0
$715$	2.68531e9	0.274741
$716$	0	0
$717$	8.27372e9	0.838269
$718$	0	0
$719$	−1.12386e10	−1.12761	−0.563806	−	0.825907i	$-0.690664\pi$
$719$	−1.12386e10	−1.12761	−0.563806	+	0.825907i	$0.690664\pi$
$720$	0	0
$721$	2.16057e9	0.214681
$722$	0	0
$723$	1.13594e10	1.11782
$724$	0	0
$725$	−3.04447e9	−0.296707
$726$	0	0
$727$	5.48758e9	0.529676	0.264838	−	0.964293i	$-0.414681\pi$
$727$	5.48758e9	0.529676	0.264838	+	0.964293i	$0.414681\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−1.08597e10	−1.02826
$732$	0	0
$733$	−7.65914e9	−0.718316	−0.359158	−	0.933277i	$-0.616936\pi$
$733$	−7.65914e9	−0.718316	−0.359158	+	0.933277i	$0.616936\pi$
$734$	0	0
$735$	−1.79531e9	−0.166776
$736$	0	0
$737$	−1.30932e10	−1.20478
$738$	0	0
$739$	8.88152e9	0.809528	0.404764	−	0.914421i	$-0.367354\pi$
$739$	8.88152e9	0.809528	0.404764	+	0.914421i	$0.367354\pi$
$740$	0	0
$741$	−5.06165e9	−0.457012
$742$	0	0
$743$	1.46556e10	1.31082	0.655409	−	0.755274i	$-0.272496\pi$
$743$	1.46556e10	1.31082	0.655409	+	0.755274i	$0.272496\pi$
$744$	0	0
$745$	−6.75632e9	−0.598636
$746$	0	0
$747$	−8.06032e8	−0.0707506
$748$	0	0
$749$	9.32109e9	0.810551
$750$	0	0
$751$	−2.03087e10	−1.74961	−0.874805	−	0.484475i	$-0.839011\pi$
$751$	−2.03087e10	−1.74961	−0.874805	+	0.484475i	$0.839011\pi$
$752$	0	0
$753$	4.33045e9	0.369616
$754$	0	0
$755$	4.45823e8	0.0377006
$756$	0	0
$757$	1.26420e10	1.05921	0.529603	−	0.848246i	$-0.322341\pi$
$757$	1.26420e10	1.05921	0.529603	+	0.848246i	$0.322341\pi$
$758$	0	0
$759$	−5.20225e8	−0.0431862
$760$	0	0
$761$	−3.15079e9	−0.259163	−0.129582	−	0.991569i	$-0.541363\pi$
$761$	−3.15079e9	−0.259163	−0.129582	+	0.991569i	$0.541363\pi$
$762$	0	0
$763$	3.60339e9	0.293681
$764$	0	0
$765$	2.24769e9	0.181519
$766$	0	0
$767$	1.37204e10	1.09795
$768$	0	0
$769$	−9.85574e9	−0.781532	−0.390766	−	0.920490i	$-0.627790\pi$
$769$	−9.85574e9	−0.781532	−0.390766	+	0.920490i	$0.627790\pi$
$770$	0	0
$771$	3.12974e9	0.245934
$772$	0	0
$773$	−4.51998e9	−0.351972	−0.175986	−	0.984393i	$-0.556311\pi$
$773$	−4.51998e9	−0.351972	−0.175986	+	0.984393i	$0.556311\pi$
$774$	0	0
$775$	6.81125e8	0.0525619
$776$	0	0
$777$	3.56274e9	0.272465
$778$	0	0
$779$	−2.30460e9	−0.174669
$780$	0	0
$781$	−7.14231e9	−0.536488
$782$	0	0
$783$	3.83515e9	0.285507
$784$	0	0
$785$	2.58841e9	0.190980
$786$	0	0
$787$	−2.14337e9	−0.156742	−0.0783712	−	0.996924i	$-0.524972\pi$
$787$	−2.14337e9	−0.156742	−0.0783712	+	0.996924i	$0.524972\pi$
$788$	0	0
$789$	1.15943e10	0.840380
$790$	0	0
$791$	1.35809e10	0.975686
$792$	0	0
$793$	9.62923e9	0.685702
$794$	0	0
$795$	1.59145e8	0.0112333
$796$	0	0
$797$	1.74444e10	1.22054	0.610270	−	0.792193i	$-0.291061\pi$
$797$	1.74444e10	1.22054	0.610270	+	0.792193i	$0.291061\pi$
$798$	0	0
$799$	1.14924e10	0.797070
$800$	0	0
$801$	1.06624e7	0.000733060 0
$802$	0	0
$803$	1.44689e10	0.986120
$804$	0	0
$805$	3.62880e8	0.0245176
$806$	0	0
$807$	6.57716e9	0.440536
$808$	0	0
$809$	1.77807e10	1.18067	0.590334	−	0.807159i	$-0.298996\pi$
$809$	1.77807e10	1.18067	0.590334	+	0.807159i	$0.298996\pi$
$810$	0	0
$811$	−2.38186e10	−1.56799	−0.783995	−	0.620767i	$-0.786821\pi$
$811$	−2.38186e10	−1.56799	−0.783995	+	0.620767i	$0.786821\pi$
$812$	0	0
$813$	4.40639e9	0.287585
$814$	0	0
$815$	2.01323e9	0.130269
$816$	0	0
$817$	1.37698e10	0.883387
$818$	0	0
$819$	2.35960e9	0.150087
$820$	0	0
$821$	−2.17123e10	−1.36932	−0.684659	−	0.728863i	$-0.740049\pi$
$821$	−2.17123e10	−1.36932	−0.684659	+	0.728863i	$0.740049\pi$
$822$	0	0
$823$	−5.61245e9	−0.350957	−0.175478	−	0.984483i	$-0.556147\pi$
$823$	−5.61245e9	−0.350957	−0.175478	+	0.984483i	$0.556147\pi$
$824$	0	0
$825$	1.51200e9	0.0937482
$826$	0	0
$827$	−8.18286e9	−0.503079	−0.251539	−	0.967847i	$-0.580937\pi$
$827$	−8.18286e9	−0.503079	−0.251539	+	0.967847i	$0.580937\pi$
$828$	0	0
$829$	1.12415e10	0.685301	0.342651	−	0.939463i	$-0.388675\pi$
$829$	1.12415e10	0.685301	0.342651	+	0.939463i	$0.388675\pi$
$830$	0	0
$831$	−4.56416e9	−0.275904
$832$	0	0
$833$	1.31209e10	0.786514
$834$	0	0
$835$	7.12160e7	0.00423326
$836$	0	0
$837$	−8.58021e8	−0.0505777
$838$	0	0
$839$	−6.13514e9	−0.358639	−0.179320	−	0.983791i	$-0.557390\pi$
$839$	−6.13514e9	−0.358639	−0.179320	+	0.983791i	$0.557390\pi$
$840$	0	0
$841$	2.07151e10	1.20088
$842$	0	0
$843$	1.18974e10	0.683998
$844$	0	0
$845$	3.35256e9	0.191152
$846$	0	0
$847$	−3.58674e9	−0.202819
$848$	0	0
$849$	1.99029e10	1.11619
$850$	0	0
$851$	1.31367e9	0.0730689
$852$	0	0
$853$	2.19770e10	1.21240	0.606200	−	0.795312i	$-0.292693\pi$
$853$	2.19770e10	1.21240	0.606200	+	0.795312i	$0.292693\pi$
$854$	0	0
$855$	−2.85003e9	−0.155944
$856$	0	0
$857$	3.11557e10	1.69085	0.845424	−	0.534096i	$-0.179348\pi$
$857$	3.11557e10	1.69085	0.845424	+	0.534096i	$0.179348\pi$
$858$	0	0
$859$	2.38146e10	1.28194	0.640970	−	0.767566i	$-0.278532\pi$
$859$	2.38146e10	1.28194	0.640970	+	0.767566i	$0.278532\pi$
$860$	0	0
$861$	1.07434e9	0.0573629
$862$	0	0
$863$	−5.48563e9	−0.290528	−0.145264	−	0.989393i	$-0.546403\pi$
$863$	−5.48563e9	−0.290528	−0.145264	+	0.989393i	$0.546403\pi$
$864$	0	0
$865$	−4.76529e9	−0.250341
$866$	0	0
$867$	−5.34797e9	−0.278690
$868$	0	0
$869$	−6.96182e9	−0.359877
$870$	0	0
$871$	2.18974e10	1.12287
$872$	0	0
$873$	6.82918e9	0.347391
$874$	0	0
$875$	−1.05469e9	−0.0532225
$876$	0	0
$877$	−2.32316e10	−1.16300	−0.581501	−	0.813545i	$-0.697534\pi$
$877$	−2.32316e10	−1.16300	−0.581501	+	0.813545i	$0.697534\pi$
$878$	0	0
$879$	−1.87898e10	−0.933170
$880$	0	0
$881$	9.50127e9	0.468129	0.234065	−	0.972221i	$-0.424797\pi$
$881$	9.50127e9	0.468129	0.234065	+	0.972221i	$0.424797\pi$
$882$	0	0
$883$	−3.56277e10	−1.74151	−0.870754	−	0.491719i	$-0.836369\pi$
$883$	−3.56277e10	−1.74151	−0.870754	+	0.491719i	$0.836369\pi$
$884$	0	0
$885$	7.72546e9	0.374647
$886$	0	0
$887$	2.18377e10	1.05069	0.525345	−	0.850889i	$-0.323936\pi$
$887$	2.18377e10	1.05069	0.525345	+	0.850889i	$0.323936\pi$
$888$	0	0
$889$	−1.97375e9	−0.0942182
$890$	0	0
$891$	−1.90468e9	−0.0902093
$892$	0	0
$893$	−1.45721e10	−0.684766
$894$	0	0
$895$	6.79064e9	0.316614
$896$	0	0
$897$	8.70041e8	0.0402500
$898$	0	0
$899$	−8.49373e9	−0.389888
$900$	0	0
$901$	−1.16310e9	−0.0529762
$902$	0	0
$903$	−6.41911e9	−0.290113
$904$	0	0
$905$	4.25117e9	0.190651
$906$	0	0
$907$	−2.40770e10	−1.07146	−0.535731	−	0.844389i	$-0.679964\pi$
$907$	−2.40770e10	−1.07146	−0.535731	+	0.844389i	$0.679964\pi$
$908$	0	0
$909$	1.06813e10	0.471685
$910$	0	0
$911$	1.93946e10	0.849899	0.424949	−	0.905217i	$-0.360292\pi$
$911$	1.93946e10	0.849899	0.424949	+	0.905217i	$0.360292\pi$
$912$	0	0
$913$	3.96271e9	0.172324
$914$	0	0
$915$	5.42186e9	0.233978
$916$	0	0
$917$	−1.75853e10	−0.753108
$918$	0	0
$919$	−8.23845e9	−0.350140	−0.175070	−	0.984556i	$-0.556015\pi$
$919$	−8.23845e9	−0.350140	−0.175070	+	0.984556i	$0.556015\pi$
$920$	0	0
$921$	−1.00818e10	−0.425234
$922$	0	0
$923$	1.19450e10	0.500013
$924$	0	0
$925$	−3.81809e9	−0.158617
$926$	0	0
$927$	2.91677e9	0.120260
$928$	0	0
$929$	−4.77268e9	−0.195302	−0.0976512	−	0.995221i	$-0.531133\pi$
$929$	−4.77268e9	−0.195302	−0.0976512	+	0.995221i	$0.531133\pi$
$930$	0	0
$931$	−1.66370e10	−0.675698
$932$	0	0
$933$	−8.69577e9	−0.350528
$934$	0	0
$935$	−1.10504e10	−0.442116
$936$	0	0
$937$	3.00758e10	1.19434	0.597171	−	0.802114i	$-0.296291\pi$
$937$	3.00758e10	1.19434	0.597171	+	0.802114i	$0.296291\pi$
$938$	0	0
$939$	−1.69594e10	−0.668469
$940$	0	0
$941$	1.31601e8	0.00514869	0.00257435	−	0.999997i	$-0.499181\pi$
$941$	1.31601e8	0.00514869	0.00257435	+	0.999997i	$0.499181\pi$
$942$	0	0
$943$	3.96136e8	0.0153834
$944$	0	0
$945$	1.32860e9	0.0512134
$946$	0	0
$947$	3.71764e10	1.42247	0.711234	−	0.702955i	$-0.248137\pi$
$947$	3.71764e10	1.42247	0.711234	+	0.702955i	$0.248137\pi$
$948$	0	0
$949$	−2.41982e10	−0.919076
$950$	0	0
$951$	4.85892e9	0.183193
$952$	0	0
$953$	−1.65884e10	−0.620840	−0.310420	−	0.950600i	$-0.600470\pi$
$953$	−1.65884e10	−0.620840	−0.310420	+	0.950600i	$0.600470\pi$
$954$	0	0
$955$	−1.22312e10	−0.454419
$956$	0	0
$957$	−1.88549e10	−0.695395
$958$	0	0
$959$	2.38020e10	0.871461
$960$	0	0
$961$	−2.56124e10	−0.930931
$962$	0	0
$963$	1.25835e10	0.454055
$964$	0	0
$965$	−1.59019e10	−0.569645
$966$	0	0
$967$	−2.95263e10	−1.05007	−0.525033	−	0.851082i	$-0.675947\pi$
$967$	−2.95263e10	−1.05007	−0.525033	+	0.851082i	$0.675947\pi$
$968$	0	0
$969$	2.08293e10	0.735429
$970$	0	0
$971$	3.85110e10	1.34995	0.674975	−	0.737841i	$-0.264155\pi$
$971$	3.85110e10	1.34995	0.674975	+	0.737841i	$0.264155\pi$
$972$	0	0
$973$	2.71382e10	0.944466
$974$	0	0
$975$	−2.52872e9	−0.0873744
$976$	0	0
$977$	−8.65181e9	−0.296808	−0.148404	−	0.988927i	$-0.547414\pi$
$977$	−8.65181e9	−0.296808	−0.148404	+	0.988927i	$0.547414\pi$
$978$	0	0
$979$	−5.24196e7	−0.00178548
$980$	0	0
$981$	4.86457e9	0.164514
$982$	0	0
$983$	−2.83575e10	−0.952205	−0.476102	−	0.879390i	$-0.657951\pi$
$983$	−2.83575e10	−0.952205	−0.476102	+	0.879390i	$0.657951\pi$
$984$	0	0
$985$	−1.03933e10	−0.346519
$986$	0	0
$987$	6.79311e9	0.224884
$988$	0	0
$989$	−2.36688e9	−0.0778017
$990$	0	0
$991$	5.25975e10	1.71675	0.858375	−	0.513023i	$-0.171474\pi$
$991$	5.25975e10	1.71675	0.858375	+	0.513023i	$0.171474\pi$
$992$	0	0
$993$	−1.39758e10	−0.452954
$994$	0	0
$995$	8.41503e8	0.0270816
$996$	0	0
$997$	−3.11108e10	−0.994210	−0.497105	−	0.867690i	$-0.665604\pi$
$997$	−3.11108e10	−0.994210	−0.497105	+	0.867690i	$0.665604\pi$
$998$	0	0
$999$	4.80970e9	0.152630

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.8.a.d.1.1		1
4.3	odd	2		120.8.a.a.1.1	✓	1
12.11	even	2		360.8.a.c.1.1		1
20.3	even	4		600.8.f.c.49.2		2
20.7	even	4		600.8.f.c.49.1		2
20.19	odd	2		600.8.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.8.a.a.1.1	✓	1	4.3	odd	2
240.8.a.d.1.1		1	1.1	even	1	trivial
360.8.a.c.1.1		1	12.11	even	2
600.8.a.c.1.1		1	20.19	odd	2
600.8.f.c.49.1		2	20.7	even	4
600.8.f.c.49.2		2	20.3	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:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	240.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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