LMFDB - Embedded newform 240.8.a.e.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 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)

$f(q)$

$=$

$q-27.0000 q^{3} +125.000 q^{5} +232.000 q^{7} +729.000 q^{9} +60.0000 q^{11} +14054.0 q^{13} -3375.00 q^{15} +13938.0 q^{17} -53564.0 q^{19} -6264.00 q^{21} +27000.0 q^{23} +15625.0 q^{25} -19683.0 q^{27} -88554.0 q^{29} +120352. q^{31} -1620.00 q^{33} +29000.0 q^{35} +469646. q^{37} -379458. q^{39} -510246. q^{41} -145412. q^{43} +91125.0 q^{45} -304560. q^{47} -769719. q^{49} -376326. q^{51} +94398.0 q^{53} +7500.00 q^{55} +1.44623e6 q^{57} +1.88574e6 q^{59} -271690. q^{61} +169128. q^{63} +1.75675e6 q^{65} -1.07467e6 q^{67} -729000. q^{69} +2.50548e6 q^{71} -2.95457e6 q^{73} -421875. q^{75} +13920.0 q^{77} +7.35477e6 q^{79} +531441. q^{81} +3.84676e6 q^{83} +1.74225e6 q^{85} +2.39096e6 q^{87} +5.68516e6 q^{89} +3.26053e6 q^{91} -3.24950e6 q^{93} -6.69550e6 q^{95} +7.98135e6 q^{97} +43740.0 q^{99} +O(q^{100})$

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
$7$	232.000	0.255649	0.127825	−	0.991797i	$-0.459201\pi$
$7$	232.000	0.255649	0.127825	+	0.991797i	$0.459201\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	60.0000	0.0135918	0.00679590	−	0.999977i	$-0.497837\pi$
$11$	60.0000	0.0135918	0.00679590	+	0.999977i	$0.497837\pi$
$12$	0	0
$13$	14054.0	1.77418	0.887091	−	0.461594i	$-0.152722\pi$
$13$	14054.0	1.77418	0.887091	+	0.461594i	$0.152722\pi$
$14$	0	0
$15$	−3375.00	−0.258199
$16$	0	0
$17$	13938.0	0.688065	0.344032	−	0.938958i	$-0.388207\pi$
$17$	13938.0	0.688065	0.344032	+	0.938958i	$0.388207\pi$
$18$	0	0
$19$	−53564.0	−1.79158	−0.895788	−	0.444481i	$-0.853388\pi$
$19$	−53564.0	−1.79158	−0.895788	+	0.444481i	$0.853388\pi$
$20$	0	0
$21$	−6264.00	−0.147599
$22$	0	0
$23$	27000.0	0.462718	0.231359	−	0.972868i	$-0.425683\pi$
$23$	27000.0	0.462718	0.231359	+	0.972868i	$0.425683\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	−88554.0	−0.674241	−0.337121	−	0.941461i	$-0.609453\pi$
$29$	−88554.0	−0.674241	−0.337121	+	0.941461i	$0.609453\pi$
$30$	0	0
$31$	120352.	0.725583	0.362792	−	0.931870i	$-0.381824\pi$
$31$	120352.	0.725583	0.362792	+	0.931870i	$0.381824\pi$
$32$	0	0
$33$	−1620.00	−0.00784723
$34$	0	0
$35$	29000.0	0.114330
$36$	0	0
$37$	469646.	1.52428	0.762140	−	0.647413i	$-0.224149\pi$
$37$	469646.	1.52428	0.762140	+	0.647413i	$0.224149\pi$
$38$	0	0
$39$	−379458.	−1.02432
$40$	0	0
$41$	−510246.	−1.15621	−0.578104	−	0.815963i	$-0.696207\pi$
$41$	−510246.	−1.15621	−0.578104	+	0.815963i	$0.696207\pi$
$42$	0	0
$43$	−145412.	−0.278908	−0.139454	−	0.990229i	$-0.544535\pi$
$43$	−145412.	−0.278908	−0.139454	+	0.990229i	$0.544535\pi$
$44$	0	0
$45$	91125.0	0.149071
$46$	0	0
$47$	−304560.	−0.427888	−0.213944	−	0.976846i	$-0.568631\pi$
$47$	−304560.	−0.427888	−0.213944	+	0.976846i	$0.568631\pi$
$48$	0	0
$49$	−769719.	−0.934643
$50$	0	0
$51$	−376326.	−0.397254
$52$	0	0
$53$	94398.0	0.0870958	0.0435479	−	0.999051i	$-0.486134\pi$
$53$	94398.0	0.0870958	0.0435479	+	0.999051i	$0.486134\pi$
$54$	0	0
$55$	7500.00	0.00607844
$56$	0	0
$57$	1.44623e6	1.03437
$58$	0	0
$59$	1.88574e6	1.19536	0.597681	−	0.801734i	$-0.296089\pi$
$59$	1.88574e6	1.19536	0.597681	+	0.801734i	$0.296089\pi$
$60$	0	0
$61$	−271690.	−0.153257	−0.0766283	−	0.997060i	$-0.524415\pi$
$61$	−271690.	−0.153257	−0.0766283	+	0.997060i	$0.524415\pi$
$62$	0	0
$63$	169128.	0.0852165
$64$	0	0
$65$	1.75675e6	0.793438
$66$	0	0
$67$	−1.07467e6	−0.436528	−0.218264	−	0.975890i	$-0.570039\pi$
$67$	−1.07467e6	−0.436528	−0.218264	+	0.975890i	$0.570039\pi$
$68$	0	0
$69$	−729000.	−0.267150
$70$	0	0
$71$	2.50548e6	0.830781	0.415391	−	0.909643i	$-0.363645\pi$
$71$	2.50548e6	0.830781	0.415391	+	0.909643i	$0.363645\pi$
$72$	0	0
$73$	−2.95457e6	−0.888922	−0.444461	−	0.895798i	$-0.646605\pi$
$73$	−2.95457e6	−0.888922	−0.444461	+	0.895798i	$0.646605\pi$
$74$	0	0
$75$	−421875.	−0.115470
$76$	0	0
$77$	13920.0	0.00347474
$78$	0	0
$79$	7.35477e6	1.67832	0.839159	−	0.543886i	$-0.183048\pi$
$79$	7.35477e6	1.67832	0.839159	+	0.543886i	$0.183048\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	3.84676e6	0.738450	0.369225	−	0.929340i	$-0.379623\pi$
$83$	3.84676e6	0.738450	0.369225	+	0.929340i	$0.379623\pi$
$84$	0	0
$85$	1.74225e6	0.307712
$86$	0	0
$87$	2.39096e6	0.389273
$88$	0	0
$89$	5.68516e6	0.854826	0.427413	−	0.904056i	$-0.359425\pi$
$89$	5.68516e6	0.854826	0.427413	+	0.904056i	$0.359425\pi$
$90$	0	0
$91$	3.26053e6	0.453569
$92$	0	0
$93$	−3.24950e6	−0.418916
$94$	0	0
$95$	−6.69550e6	−0.801217
$96$	0	0
$97$	7.98135e6	0.887923	0.443961	−	0.896046i	$-0.353573\pi$
$97$	7.98135e6	0.887923	0.443961	+	0.896046i	$0.353573\pi$
$98$	0	0
$99$	43740.0	0.00453060
$100$	0	0
$101$	1.54628e7	1.49336	0.746678	−	0.665186i	$-0.231648\pi$
$101$	1.54628e7	1.49336	0.746678	+	0.665186i	$0.231648\pi$
$102$	0	0
$103$	2.09024e7	1.88480	0.942402	−	0.334483i	$-0.108562\pi$
$103$	2.09024e7	1.88480	0.942402	+	0.334483i	$0.108562\pi$
$104$	0	0
$105$	−783000.	−0.0660084
$106$	0	0
$107$	−2.05645e7	−1.62284	−0.811418	−	0.584466i	$-0.801304\pi$
$107$	−2.05645e7	−1.62284	−0.811418	+	0.584466i	$0.801304\pi$
$108$	0	0
$109$	1.39850e7	1.03436	0.517179	−	0.855877i	$-0.326982\pi$
$109$	1.39850e7	1.03436	0.517179	+	0.855877i	$0.326982\pi$
$110$	0	0
$111$	−1.26804e7	−0.880043
$112$	0	0
$113$	2.64438e7	1.72405	0.862024	−	0.506868i	$-0.169197\pi$
$113$	2.64438e7	1.72405	0.862024	+	0.506868i	$0.169197\pi$
$114$	0	0
$115$	3.37500e6	0.206934
$116$	0	0
$117$	1.02454e7	0.591394
$118$	0	0
$119$	3.23362e6	0.175903
$120$	0	0
$121$	−1.94836e7	−0.999815
$122$	0	0
$123$	1.37766e7	0.667537
$124$	0	0
$125$	1.95312e6	0.0894427
$126$	0	0
$127$	9.34000e6	0.404607	0.202304	−	0.979323i	$-0.435157\pi$
$127$	9.34000e6	0.404607	0.202304	+	0.979323i	$0.435157\pi$
$128$	0	0
$129$	3.92612e6	0.161027
$130$	0	0
$131$	4.46795e7	1.73644	0.868219	−	0.496182i	$-0.165265\pi$
$131$	4.46795e7	1.73644	0.868219	+	0.496182i	$0.165265\pi$
$132$	0	0
$133$	−1.24268e7	−0.458016
$134$	0	0
$135$	−2.46038e6	−0.0860663
$136$	0	0
$137$	−5.69929e7	−1.89365	−0.946823	−	0.321755i	$-0.895727\pi$
$137$	−5.69929e7	−1.89365	−0.946823	+	0.321755i	$0.895727\pi$
$138$	0	0
$139$	−3.06822e7	−0.969026	−0.484513	−	0.874784i	$-0.661003\pi$
$139$	−3.06822e7	−0.969026	−0.484513	+	0.874784i	$0.661003\pi$
$140$	0	0
$141$	8.22312e6	0.247041
$142$	0	0
$143$	843240.	0.0241143
$144$	0	0
$145$	−1.10692e7	−0.301530
$146$	0	0
$147$	2.07824e7	0.539617
$148$	0	0
$149$	2.83628e7	0.702422	0.351211	−	0.936296i	$-0.385770\pi$
$149$	2.83628e7	0.702422	0.351211	+	0.936296i	$0.385770\pi$
$150$	0	0
$151$	3.45653e7	0.816997	0.408499	−	0.912759i	$-0.366052\pi$
$151$	3.45653e7	0.816997	0.408499	+	0.912759i	$0.366052\pi$
$152$	0	0
$153$	1.01608e7	0.229355
$154$	0	0
$155$	1.50440e7	0.324491
$156$	0	0
$157$	5.00660e7	1.03251	0.516255	−	0.856435i	$-0.327326\pi$
$157$	5.00660e7	1.03251	0.516255	+	0.856435i	$0.327326\pi$
$158$	0	0
$159$	−2.54875e6	−0.0502848
$160$	0	0
$161$	6.26400e6	0.118294
$162$	0	0
$163$	1.63359e6	0.0295451	0.0147726	−	0.999891i	$-0.495298\pi$
$163$	1.63359e6	0.0295451	0.0147726	+	0.999891i	$0.495298\pi$
$164$	0	0
$165$	−202500.	−0.00350939
$166$	0	0
$167$	9.94818e7	1.65286	0.826430	−	0.563040i	$-0.190368\pi$
$167$	9.94818e7	1.65286	0.826430	+	0.563040i	$0.190368\pi$
$168$	0	0
$169$	1.34766e8	2.14772
$170$	0	0
$171$	−3.90482e7	−0.597192
$172$	0	0
$173$	−1.00775e8	−1.47976	−0.739881	−	0.672738i	$-0.765118\pi$
$173$	−1.00775e8	−1.47976	−0.739881	+	0.672738i	$0.765118\pi$
$174$	0	0
$175$	3.62500e6	0.0511299
$176$	0	0
$177$	−5.09150e7	−0.690143
$178$	0	0
$179$	−8.59163e7	−1.11967	−0.559835	−	0.828604i	$-0.689135\pi$
$179$	−8.59163e7	−1.11967	−0.559835	+	0.828604i	$0.689135\pi$
$180$	0	0
$181$	−7.34951e6	−0.0921262	−0.0460631	−	0.998939i	$-0.514668\pi$
$181$	−7.34951e6	−0.0921262	−0.0460631	+	0.998939i	$0.514668\pi$
$182$	0	0
$183$	7.33563e6	0.0884827
$184$	0	0
$185$	5.87058e7	0.681678
$186$	0	0
$187$	836280.	0.00935203
$188$	0	0
$189$	−4.56646e6	−0.0491998
$190$	0	0
$191$	−2.55529e7	−0.265353	−0.132676	−	0.991159i	$-0.542357\pi$
$191$	−2.55529e7	−0.265353	−0.132676	+	0.991159i	$0.542357\pi$
$192$	0	0
$193$	1.36815e8	1.36988	0.684940	−	0.728600i	$-0.259828\pi$
$193$	1.36815e8	1.36988	0.684940	+	0.728600i	$0.259828\pi$
$194$	0	0
$195$	−4.74322e7	−0.458092
$196$	0	0
$197$	−1.38053e8	−1.28651	−0.643255	−	0.765652i	$-0.722417\pi$
$197$	−1.38053e8	−1.28651	−0.643255	+	0.765652i	$0.722417\pi$
$198$	0	0
$199$	−6.20415e6	−0.0558080	−0.0279040	−	0.999611i	$-0.508883\pi$
$199$	−6.20415e6	−0.0558080	−0.0279040	+	0.999611i	$0.508883\pi$
$200$	0	0
$201$	2.90160e7	0.252030
$202$	0	0
$203$	−2.05445e7	−0.172369
$204$	0	0
$205$	−6.37808e7	−0.517072
$206$	0	0
$207$	1.96830e7	0.154239
$208$	0	0
$209$	−3.21384e6	−0.0243507
$210$	0	0
$211$	1.65042e8	1.20950	0.604750	−	0.796416i	$-0.293273\pi$
$211$	1.65042e8	1.20950	0.604750	+	0.796416i	$0.293273\pi$
$212$	0	0
$213$	−6.76480e7	−0.479652
$214$	0	0
$215$	−1.81765e7	−0.124731
$216$	0	0
$217$	2.79217e7	0.185495
$218$	0	0
$219$	7.97733e7	0.513219
$220$	0	0
$221$	1.95885e8	1.22075
$222$	0	0
$223$	5.43117e7	0.327964	0.163982	−	0.986463i	$-0.447566\pi$
$223$	5.43117e7	0.327964	0.163982	+	0.986463i	$0.447566\pi$
$224$	0	0
$225$	1.13906e7	0.0666667
$226$	0	0
$227$	−1.33928e8	−0.759942	−0.379971	−	0.924998i	$-0.624066\pi$
$227$	−1.33928e8	−0.759942	−0.379971	+	0.924998i	$0.624066\pi$
$228$	0	0
$229$	2.09843e7	0.115470	0.0577351	−	0.998332i	$-0.481612\pi$
$229$	2.09843e7	0.115470	0.0577351	+	0.998332i	$0.481612\pi$
$230$	0	0
$231$	−375840.	−0.00200614
$232$	0	0
$233$	1.21685e7	0.0630217	0.0315109	−	0.999503i	$-0.489968\pi$
$233$	1.21685e7	0.0630217	0.0315109	+	0.999503i	$0.489968\pi$
$234$	0	0
$235$	−3.80700e7	−0.191357
$236$	0	0
$237$	−1.98579e8	−0.968977
$238$	0	0
$239$	4.69235e7	0.222330	0.111165	−	0.993802i	$-0.464542\pi$
$239$	4.69235e7	0.222330	0.111165	+	0.993802i	$0.464542\pi$
$240$	0	0
$241$	−2.29327e8	−1.05535	−0.527674	−	0.849447i	$-0.676936\pi$
$241$	−2.29327e8	−1.05535	−0.527674	+	0.849447i	$0.676936\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	−9.62149e7	−0.417985
$246$	0	0
$247$	−7.52788e8	−3.17858
$248$	0	0
$249$	−1.03862e8	−0.426345
$250$	0	0
$251$	−4.50379e8	−1.79771	−0.898857	−	0.438242i	$-0.855601\pi$
$251$	−4.50379e8	−1.79771	−0.898857	+	0.438242i	$0.855601\pi$
$252$	0	0
$253$	1.62000e6	0.00628917
$254$	0	0
$255$	−4.70408e7	−0.177658
$256$	0	0
$257$	6.65252e7	0.244467	0.122233	−	0.992501i	$-0.460994\pi$
$257$	6.65252e7	0.244467	0.122233	+	0.992501i	$0.460994\pi$
$258$	0	0
$259$	1.08958e8	0.389681
$260$	0	0
$261$	−6.45559e7	−0.224747
$262$	0	0
$263$	1.10723e8	0.375312	0.187656	−	0.982235i	$-0.439911\pi$
$263$	1.10723e8	0.375312	0.187656	+	0.982235i	$0.439911\pi$
$264$	0	0
$265$	1.17998e7	0.0389504
$266$	0	0
$267$	−1.53499e8	−0.493534
$268$	0	0
$269$	−1.31054e8	−0.410504	−0.205252	−	0.978709i	$-0.565801\pi$
$269$	−1.31054e8	−0.410504	−0.205252	+	0.978709i	$0.565801\pi$
$270$	0	0
$271$	5.66890e8	1.73024	0.865119	−	0.501566i	$-0.167243\pi$
$271$	5.66890e8	1.73024	0.865119	+	0.501566i	$0.167243\pi$
$272$	0	0
$273$	−8.80343e7	−0.261868
$274$	0	0
$275$	937500.	0.00271836
$276$	0	0
$277$	1.30574e8	0.369129	0.184565	−	0.982820i	$-0.440912\pi$
$277$	1.30574e8	0.369129	0.184565	+	0.982820i	$0.440912\pi$
$278$	0	0
$279$	8.77366e7	0.241861
$280$	0	0
$281$	−2.70368e8	−0.726914	−0.363457	−	0.931611i	$-0.618404\pi$
$281$	−2.70368e8	−0.726914	−0.363457	+	0.931611i	$0.618404\pi$
$282$	0	0
$283$	9.11074e7	0.238947	0.119473	−	0.992837i	$-0.461879\pi$
$283$	9.11074e7	0.238947	0.119473	+	0.992837i	$0.461879\pi$
$284$	0	0
$285$	1.80778e8	0.462583
$286$	0	0
$287$	−1.18377e8	−0.295584
$288$	0	0
$289$	−2.16071e8	−0.526567
$290$	0	0
$291$	−2.15496e8	−0.512642
$292$	0	0
$293$	−6.22671e8	−1.44618	−0.723090	−	0.690754i	$-0.757279\pi$
$293$	−6.22671e8	−1.44618	−0.723090	+	0.690754i	$0.757279\pi$
$294$	0	0
$295$	2.35717e8	0.534583
$296$	0	0
$297$	−1.18098e6	−0.00261574
$298$	0	0
$299$	3.79458e8	0.820946
$300$	0	0
$301$	−3.37356e7	−0.0713026
$302$	0	0
$303$	−4.17496e8	−0.862189
$304$	0	0
$305$	−3.39612e7	−0.0685384
$306$	0	0
$307$	1.13282e8	0.223447	0.111724	−	0.993739i	$-0.464363\pi$
$307$	1.13282e8	0.223447	0.111724	+	0.993739i	$0.464363\pi$
$308$	0	0
$309$	−5.64365e8	−1.08819
$310$	0	0
$311$	−3.78344e8	−0.713222	−0.356611	−	0.934253i	$-0.616068\pi$
$311$	−3.78344e8	−0.713222	−0.356611	+	0.934253i	$0.616068\pi$
$312$	0	0
$313$	−1.11400e8	−0.205344	−0.102672	−	0.994715i	$-0.532739\pi$
$313$	−1.11400e8	−0.205344	−0.102672	+	0.994715i	$0.532739\pi$
$314$	0	0
$315$	2.11410e7	0.0381100
$316$	0	0
$317$	−6.70098e8	−1.18149	−0.590746	−	0.806857i	$-0.701167\pi$
$317$	−6.70098e8	−1.18149	−0.590746	+	0.806857i	$0.701167\pi$
$318$	0	0
$319$	−5.31324e6	−0.00916415
$320$	0	0
$321$	5.55241e8	0.936945
$322$	0	0
$323$	−7.46575e8	−1.23272
$324$	0	0
$325$	2.19594e8	0.354836
$326$	0	0
$327$	−3.77596e8	−0.597187
$328$	0	0
$329$	−7.06579e7	−0.109389
$330$	0	0
$331$	5.44096e8	0.824665	0.412332	−	0.911033i	$-0.364714\pi$
$331$	5.44096e8	0.824665	0.412332	+	0.911033i	$0.364714\pi$
$332$	0	0
$333$	3.42372e8	0.508093
$334$	0	0
$335$	−1.34334e8	−0.195221
$336$	0	0
$337$	9.62374e8	1.36974	0.684872	−	0.728664i	$-0.259858\pi$
$337$	9.62374e8	1.36974	0.684872	+	0.728664i	$0.259858\pi$
$338$	0	0
$339$	−7.13982e8	−0.995379
$340$	0	0
$341$	7.22112e6	0.00986198
$342$	0	0
$343$	−3.69637e8	−0.494590
$344$	0	0
$345$	−9.11250e7	−0.119473
$346$	0	0
$347$	5.92829e8	0.761686	0.380843	−	0.924640i	$-0.375634\pi$
$347$	5.92829e8	0.761686	0.380843	+	0.924640i	$0.375634\pi$
$348$	0	0
$349$	−9.73268e8	−1.22558	−0.612792	−	0.790244i	$-0.709954\pi$
$349$	−9.73268e8	−1.22558	−0.612792	+	0.790244i	$0.709954\pi$
$350$	0	0
$351$	−2.76625e8	−0.341442
$352$	0	0
$353$	1.59127e9	1.92545	0.962723	−	0.270491i	$-0.0871859\pi$
$353$	1.59127e9	1.92545	0.962723	+	0.270491i	$0.0871859\pi$
$354$	0	0
$355$	3.13185e8	0.371537
$356$	0	0
$357$	−8.73076e7	−0.101558
$358$	0	0
$359$	−8.37928e8	−0.955820	−0.477910	−	0.878409i	$-0.658606\pi$
$359$	−8.37928e8	−0.955820	−0.477910	+	0.878409i	$0.658606\pi$
$360$	0	0
$361$	1.97523e9	2.20975
$362$	0	0
$363$	5.26056e8	0.577244
$364$	0	0
$365$	−3.69321e8	−0.397538
$366$	0	0
$367$	1.14448e9	1.20858	0.604290	−	0.796765i	$-0.293457\pi$
$367$	1.14448e9	1.20858	0.604290	+	0.796765i	$0.293457\pi$
$368$	0	0
$369$	−3.71969e8	−0.385403
$370$	0	0
$371$	2.19003e7	0.0222660
$372$	0	0
$373$	1.36673e9	1.36365	0.681825	−	0.731516i	$-0.261187\pi$
$373$	1.36673e9	1.36365	0.681825	+	0.731516i	$0.261187\pi$
$374$	0	0
$375$	−5.27344e7	−0.0516398
$376$	0	0
$377$	−1.24454e9	−1.19623
$378$	0	0
$379$	4.91619e8	0.463865	0.231933	−	0.972732i	$-0.425495\pi$
$379$	4.91619e8	0.463865	0.231933	+	0.972732i	$0.425495\pi$
$380$	0	0
$381$	−2.52180e8	−0.233600
$382$	0	0
$383$	−5.32012e8	−0.483867	−0.241933	−	0.970293i	$-0.577782\pi$
$383$	−5.32012e8	−0.483867	−0.241933	+	0.970293i	$0.577782\pi$
$384$	0	0
$385$	1.74000e6	0.00155395
$386$	0	0
$387$	−1.06005e8	−0.0929693
$388$	0	0
$389$	6.83513e8	0.588739	0.294370	−	0.955692i	$-0.404890\pi$
$389$	6.83513e8	0.588739	0.294370	+	0.955692i	$0.404890\pi$
$390$	0	0
$391$	3.76326e8	0.318380
$392$	0	0
$393$	−1.20635e9	−1.00253
$394$	0	0
$395$	9.19346e8	0.750566
$396$	0	0
$397$	−1.46842e9	−1.17784	−0.588918	−	0.808193i	$-0.700446\pi$
$397$	−1.46842e9	−1.17784	−0.588918	+	0.808193i	$0.700446\pi$
$398$	0	0
$399$	3.35525e8	0.264435
$400$	0	0
$401$	7.08292e8	0.548538	0.274269	−	0.961653i	$-0.411564\pi$
$401$	7.08292e8	0.548538	0.274269	+	0.961653i	$0.411564\pi$
$402$	0	0
$403$	1.69143e9	1.28732
$404$	0	0
$405$	6.64301e7	0.0496904
$406$	0	0
$407$	2.81788e7	0.0207177
$408$	0	0
$409$	2.13603e9	1.54375	0.771873	−	0.635777i	$-0.219320\pi$
$409$	2.13603e9	1.54375	0.771873	+	0.635777i	$0.219320\pi$
$410$	0	0
$411$	1.53881e9	1.09330
$412$	0	0
$413$	4.37492e8	0.305594
$414$	0	0
$415$	4.80844e8	0.330245
$416$	0	0
$417$	8.28421e8	0.559467
$418$	0	0
$419$	−1.22130e9	−0.811098	−0.405549	−	0.914073i	$-0.632920\pi$
$419$	−1.22130e9	−0.811098	−0.405549	+	0.914073i	$0.632920\pi$
$420$	0	0
$421$	−1.17754e8	−0.0769110	−0.0384555	−	0.999260i	$-0.512244\pi$
$421$	−1.17754e8	−0.0769110	−0.0384555	+	0.999260i	$0.512244\pi$
$422$	0	0
$423$	−2.22024e8	−0.142629
$424$	0	0
$425$	2.17781e8	0.137613
$426$	0	0
$427$	−6.30321e7	−0.0391800
$428$	0	0
$429$	−2.27675e7	−0.0139224
$430$	0	0
$431$	2.92749e9	1.76127	0.880634	−	0.473797i	$-0.157117\pi$
$431$	2.92749e9	1.76127	0.880634	+	0.473797i	$0.157117\pi$
$432$	0	0
$433$	−9.22066e8	−0.545826	−0.272913	−	0.962039i	$-0.587987\pi$
$433$	−9.22066e8	−0.545826	−0.272913	+	0.962039i	$0.587987\pi$
$434$	0	0
$435$	2.98870e8	0.174088
$436$	0	0
$437$	−1.44623e9	−0.828995
$438$	0	0
$439$	2.69431e9	1.51992	0.759962	−	0.649967i	$-0.225217\pi$
$439$	2.69431e9	1.51992	0.759962	+	0.649967i	$0.225217\pi$
$440$	0	0
$441$	−5.61125e8	−0.311548
$442$	0	0
$443$	−1.50928e9	−0.824815	−0.412407	−	0.911000i	$-0.635312\pi$
$443$	−1.50928e9	−0.824815	−0.412407	+	0.911000i	$0.635312\pi$
$444$	0	0
$445$	7.10645e8	0.382290
$446$	0	0
$447$	−7.65797e8	−0.405543
$448$	0	0
$449$	7.94655e8	0.414302	0.207151	−	0.978309i	$-0.433581\pi$
$449$	7.94655e8	0.414302	0.207151	+	0.978309i	$0.433581\pi$
$450$	0	0
$451$	−3.06148e7	−0.0157149
$452$	0	0
$453$	−9.33262e8	−0.471694
$454$	0	0
$455$	4.07566e8	0.202842
$456$	0	0
$457$	1.82270e9	0.893324	0.446662	−	0.894703i	$-0.352613\pi$
$457$	1.82270e9	0.893324	0.446662	+	0.894703i	$0.352613\pi$
$458$	0	0
$459$	−2.74342e8	−0.132418
$460$	0	0
$461$	−1.79582e8	−0.0853711	−0.0426855	−	0.999089i	$-0.513591\pi$
$461$	−1.79582e8	−0.0853711	−0.0426855	+	0.999089i	$0.513591\pi$
$462$	0	0
$463$	−3.95891e8	−0.185371	−0.0926855	−	0.995695i	$-0.529545\pi$
$463$	−3.95891e8	−0.185371	−0.0926855	+	0.995695i	$0.529545\pi$
$464$	0	0
$465$	−4.06188e8	−0.187345
$466$	0	0
$467$	−5.70281e8	−0.259107	−0.129554	−	0.991572i	$-0.541354\pi$
$467$	−5.70281e8	−0.259107	−0.129554	+	0.991572i	$0.541354\pi$
$468$	0	0
$469$	−2.49323e8	−0.111598
$470$	0	0
$471$	−1.35178e9	−0.596120
$472$	0	0
$473$	−8.72472e6	−0.00379086
$474$	0	0
$475$	−8.36938e8	−0.358315
$476$	0	0
$477$	6.88161e7	0.0290319
$478$	0	0
$479$	−4.36823e9	−1.81607	−0.908033	−	0.418900i	$-0.862416\pi$
$479$	−4.36823e9	−1.81607	−0.908033	+	0.418900i	$0.862416\pi$
$480$	0	0
$481$	6.60040e9	2.70435
$482$	0	0
$483$	−1.69128e8	−0.0682968
$484$	0	0
$485$	9.97668e8	0.397091
$486$	0	0
$487$	−3.29328e8	−0.129204	−0.0646022	−	0.997911i	$-0.520578\pi$
$487$	−3.29328e8	−0.129204	−0.0646022	+	0.997911i	$0.520578\pi$
$488$	0	0
$489$	−4.41069e7	−0.0170579
$490$	0	0
$491$	2.54666e9	0.970924	0.485462	−	0.874258i	$-0.338651\pi$
$491$	2.54666e9	0.970924	0.485462	+	0.874258i	$0.338651\pi$
$492$	0	0
$493$	−1.23427e9	−0.463921
$494$	0	0
$495$	5.46750e6	0.00202615
$496$	0	0
$497$	5.81271e8	0.212389
$498$	0	0
$499$	−1.42495e9	−0.513391	−0.256696	−	0.966492i	$-0.582634\pi$
$499$	−1.42495e9	−0.513391	−0.256696	+	0.966492i	$0.582634\pi$
$500$	0	0
$501$	−2.68601e9	−0.954279
$502$	0	0
$503$	2.18397e9	0.765172	0.382586	−	0.923920i	$-0.375034\pi$
$503$	2.18397e9	0.765172	0.382586	+	0.923920i	$0.375034\pi$
$504$	0	0
$505$	1.93285e9	0.667849
$506$	0	0
$507$	−3.63869e9	−1.23999
$508$	0	0
$509$	−5.25623e9	−1.76670	−0.883349	−	0.468715i	$-0.844717\pi$
$509$	−5.25623e9	−1.76670	−0.883349	+	0.468715i	$0.844717\pi$
$510$	0	0
$511$	−6.85459e8	−0.227252
$512$	0	0
$513$	1.05430e9	0.344789
$514$	0	0
$515$	2.61280e9	0.842910
$516$	0	0
$517$	−1.82736e7	−0.00581577
$518$	0	0
$519$	2.72093e9	0.854340
$520$	0	0
$521$	3.19481e9	0.989720	0.494860	−	0.868973i	$-0.335219\pi$
$521$	3.19481e9	0.989720	0.494860	+	0.868973i	$0.335219\pi$
$522$	0	0
$523$	1.92219e9	0.587545	0.293772	−	0.955875i	$-0.405089\pi$
$523$	1.92219e9	0.587545	0.293772	+	0.955875i	$0.405089\pi$
$524$	0	0
$525$	−9.78750e7	−0.0295199
$526$	0	0
$527$	1.67747e9	0.499248
$528$	0	0
$529$	−2.67583e9	−0.785892
$530$	0	0
$531$	1.37470e9	0.398454
$532$	0	0
$533$	−7.17100e9	−2.05132
$534$	0	0
$535$	−2.57056e9	−0.725754
$536$	0	0
$537$	2.31974e9	0.646442
$538$	0	0
$539$	−4.61831e7	−0.0127035
$540$	0	0
$541$	1.62352e9	0.440826	0.220413	−	0.975407i	$-0.429259\pi$
$541$	1.62352e9	0.440826	0.220413	+	0.975407i	$0.429259\pi$
$542$	0	0
$543$	1.98437e8	0.0531891
$544$	0	0
$545$	1.74813e9	0.462579
$546$	0	0
$547$	8.76824e8	0.229064	0.114532	−	0.993420i	$-0.463463\pi$
$547$	8.76824e8	0.229064	0.114532	+	0.993420i	$0.463463\pi$
$548$	0	0
$549$	−1.98062e8	−0.0510855
$550$	0	0
$551$	4.74331e9	1.20795
$552$	0	0
$553$	1.70631e9	0.429061
$554$	0	0
$555$	−1.58506e9	−0.393567
$556$	0	0
$557$	−2.29689e8	−0.0563179	−0.0281590	−	0.999603i	$-0.508964\pi$
$557$	−2.29689e8	−0.0563179	−0.0281590	+	0.999603i	$0.508964\pi$
$558$	0	0
$559$	−2.04362e9	−0.494833
$560$	0	0
$561$	−2.25796e7	−0.00539940
$562$	0	0
$563$	−7.86151e8	−0.185664	−0.0928318	−	0.995682i	$-0.529592\pi$
$563$	−7.86151e8	−0.185664	−0.0928318	+	0.995682i	$0.529592\pi$
$564$	0	0
$565$	3.30547e9	0.771017
$566$	0	0
$567$	1.23294e8	0.0284055
$568$	0	0
$569$	3.30088e9	0.751166	0.375583	−	0.926789i	$-0.377442\pi$
$569$	3.30088e9	0.751166	0.375583	+	0.926789i	$0.377442\pi$
$570$	0	0
$571$	1.54033e9	0.346248	0.173124	−	0.984900i	$-0.444614\pi$
$571$	1.54033e9	0.346248	0.173124	+	0.984900i	$0.444614\pi$
$572$	0	0
$573$	6.89928e8	0.153201
$574$	0	0
$575$	4.21875e8	0.0925436
$576$	0	0
$577$	−3.79769e9	−0.823008	−0.411504	−	0.911408i	$-0.634996\pi$
$577$	−3.79769e9	−0.823008	−0.411504	+	0.911408i	$0.634996\pi$
$578$	0	0
$579$	−3.69400e9	−0.790900
$580$	0	0
$581$	8.92447e8	0.188784
$582$	0	0
$583$	5.66388e6	0.00118379
$584$	0	0
$585$	1.28067e9	0.264479
$586$	0	0
$587$	−8.29676e9	−1.69307	−0.846536	−	0.532332i	$-0.821316\pi$
$587$	−8.29676e9	−1.69307	−0.846536	+	0.532332i	$0.821316\pi$
$588$	0	0
$589$	−6.44653e9	−1.29994
$590$	0	0
$591$	3.72743e9	0.742767
$592$	0	0
$593$	4.93196e9	0.971243	0.485621	−	0.874169i	$-0.338593\pi$
$593$	4.93196e9	0.971243	0.485621	+	0.874169i	$0.338593\pi$
$594$	0	0
$595$	4.04202e8	0.0786664
$596$	0	0
$597$	1.67512e8	0.0322208
$598$	0	0
$599$	−6.51152e9	−1.23791	−0.618954	−	0.785427i	$-0.712443\pi$
$599$	−6.51152e9	−1.23791	−0.618954	+	0.785427i	$0.712443\pi$
$600$	0	0
$601$	−4.17460e9	−0.784430	−0.392215	−	0.919874i	$-0.628291\pi$
$601$	−4.17460e9	−0.784430	−0.392215	+	0.919874i	$0.628291\pi$
$602$	0	0
$603$	−7.83433e8	−0.145509
$604$	0	0
$605$	−2.43545e9	−0.447131
$606$	0	0
$607$	−5.46979e9	−0.992682	−0.496341	−	0.868128i	$-0.665323\pi$
$607$	−5.46979e9	−0.992682	−0.496341	+	0.868128i	$0.665323\pi$
$608$	0	0
$609$	5.54702e8	0.0995175
$610$	0	0
$611$	−4.28029e9	−0.759152
$612$	0	0
$613$	−7.92764e9	−1.39006	−0.695028	−	0.718982i	$-0.744608\pi$
$613$	−7.92764e9	−1.39006	−0.695028	+	0.718982i	$0.744608\pi$
$614$	0	0
$615$	1.72208e9	0.298532
$616$	0	0
$617$	−9.70277e9	−1.66302	−0.831510	−	0.555510i	$-0.812523\pi$
$617$	−9.70277e9	−1.66302	−0.831510	+	0.555510i	$0.812523\pi$
$618$	0	0
$619$	−5.34840e8	−0.0906373	−0.0453186	−	0.998973i	$-0.514430\pi$
$619$	−5.34840e8	−0.0906373	−0.0453186	+	0.998973i	$0.514430\pi$
$620$	0	0
$621$	−5.31441e8	−0.0890501
$622$	0	0
$623$	1.31896e9	0.218536
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	0	0
$627$	8.67737e7	0.0140589
$628$	0	0
$629$	6.54593e9	1.04880
$630$	0	0
$631$	8.98145e9	1.42313	0.711563	−	0.702622i	$-0.247987\pi$
$631$	8.98145e9	1.42313	0.711563	+	0.702622i	$0.247987\pi$
$632$	0	0
$633$	−4.45613e9	−0.698305
$634$	0	0
$635$	1.16750e9	0.180946
$636$	0	0
$637$	−1.08176e10	−1.65823
$638$	0	0
$639$	1.82649e9	0.276927
$640$	0	0
$641$	6.81488e8	0.102201	0.0511005	−	0.998694i	$-0.483727\pi$
$641$	6.81488e8	0.102201	0.0511005	+	0.998694i	$0.483727\pi$
$642$	0	0
$643$	3.68007e9	0.545906	0.272953	−	0.962027i	$-0.412000\pi$
$643$	3.68007e9	0.545906	0.272953	+	0.962027i	$0.412000\pi$
$644$	0	0
$645$	4.90765e8	0.0720137
$646$	0	0
$647$	−9.03269e9	−1.31115	−0.655574	−	0.755131i	$-0.727573\pi$
$647$	−9.03269e9	−1.31115	−0.655574	+	0.755131i	$0.727573\pi$
$648$	0	0
$649$	1.13144e8	0.0162471
$650$	0	0
$651$	−7.53885e8	−0.107096
$652$	0	0
$653$	2.90477e9	0.408240	0.204120	−	0.978946i	$-0.434567\pi$
$653$	2.90477e9	0.408240	0.204120	+	0.978946i	$0.434567\pi$
$654$	0	0
$655$	5.58494e9	0.776558
$656$	0	0
$657$	−2.15388e9	−0.296307
$658$	0	0
$659$	2.88225e8	0.0392313	0.0196157	−	0.999808i	$-0.493756\pi$
$659$	2.88225e8	0.0392313	0.0196157	+	0.999808i	$0.493756\pi$
$660$	0	0
$661$	4.04326e9	0.544536	0.272268	−	0.962221i	$-0.412226\pi$
$661$	4.04326e9	0.544536	0.272268	+	0.962221i	$0.412226\pi$
$662$	0	0
$663$	−5.28889e9	−0.704802
$664$	0	0
$665$	−1.55336e9	−0.204831
$666$	0	0
$667$	−2.39096e9	−0.311983
$668$	0	0
$669$	−1.46642e9	−0.189350
$670$	0	0
$671$	−1.63014e7	−0.00208303
$672$	0	0
$673$	6.38900e9	0.807942	0.403971	−	0.914772i	$-0.367630\pi$
$673$	6.38900e9	0.807942	0.403971	+	0.914772i	$0.367630\pi$
$674$	0	0
$675$	−3.07547e8	−0.0384900
$676$	0	0
$677$	−1.08256e10	−1.34089	−0.670445	−	0.741960i	$-0.733897\pi$
$677$	−1.08256e10	−1.34089	−0.670445	+	0.741960i	$0.733897\pi$
$678$	0	0
$679$	1.85167e9	0.226997
$680$	0	0
$681$	3.61605e9	0.438753
$682$	0	0
$683$	−8.81962e8	−0.105920	−0.0529599	−	0.998597i	$-0.516866\pi$
$683$	−8.81962e8	−0.105920	−0.0529599	+	0.998597i	$0.516866\pi$
$684$	0	0
$685$	−7.12411e9	−0.846864
$686$	0	0
$687$	−5.66575e8	−0.0666667
$688$	0	0
$689$	1.32667e9	0.154524
$690$	0	0
$691$	−5.36883e9	−0.619022	−0.309511	−	0.950896i	$-0.600165\pi$
$691$	−5.36883e9	−0.619022	−0.309511	+	0.950896i	$0.600165\pi$
$692$	0	0
$693$	1.01477e7	0.00115825
$694$	0	0
$695$	−3.83528e9	−0.433361
$696$	0	0
$697$	−7.11181e9	−0.795546
$698$	0	0
$699$	−3.28549e8	−0.0363856
$700$	0	0
$701$	4.37341e9	0.479521	0.239760	−	0.970832i	$-0.422931\pi$
$701$	4.37341e9	0.479521	0.239760	+	0.970832i	$0.422931\pi$
$702$	0	0
$703$	−2.51561e10	−2.73086
$704$	0	0
$705$	1.02789e9	0.110480
$706$	0	0
$707$	3.58737e9	0.381776
$708$	0	0
$709$	4.60383e9	0.485129	0.242565	−	0.970135i	$-0.422011\pi$
$709$	4.60383e9	0.485129	0.242565	+	0.970135i	$0.422011\pi$
$710$	0	0
$711$	5.36163e9	0.559439
$712$	0	0
$713$	3.24950e9	0.335740
$714$	0	0
$715$	1.05405e8	0.0107843
$716$	0	0
$717$	−1.26693e9	−0.128362
$718$	0	0
$719$	−5.78324e9	−0.580257	−0.290129	−	0.956988i	$-0.593698\pi$
$719$	−5.78324e9	−0.580257	−0.290129	+	0.956988i	$0.593698\pi$
$720$	0	0
$721$	4.84936e9	0.481849
$722$	0	0
$723$	6.19183e9	0.609305
$724$	0	0
$725$	−1.38366e9	−0.134848
$726$	0	0
$727$	−1.42871e10	−1.37903	−0.689514	−	0.724272i	$-0.742176\pi$
$727$	−1.42871e10	−1.37903	−0.689514	+	0.724272i	$0.742176\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−2.02675e9	−0.191907
$732$	0	0
$733$	−1.18896e10	−1.11507	−0.557535	−	0.830153i	$-0.688253\pi$
$733$	−1.18896e10	−1.11507	−0.557535	+	0.830153i	$0.688253\pi$
$734$	0	0
$735$	2.59780e9	0.241324
$736$	0	0
$737$	−6.44801e7	−0.00593321
$738$	0	0
$739$	−1.41993e10	−1.29423	−0.647113	−	0.762394i	$-0.724024\pi$
$739$	−1.41993e10	−1.29423	−0.647113	+	0.762394i	$0.724024\pi$
$740$	0	0
$741$	2.03253e10	1.83516
$742$	0	0
$743$	8.29539e9	0.741953	0.370976	−	0.928642i	$-0.379023\pi$
$743$	8.29539e9	0.741953	0.370976	+	0.928642i	$0.379023\pi$
$744$	0	0
$745$	3.54536e9	0.314132
$746$	0	0
$747$	2.80429e9	0.246150
$748$	0	0
$749$	−4.77096e9	−0.414877
$750$	0	0
$751$	1.07334e10	0.924693	0.462347	−	0.886699i	$-0.347008\pi$
$751$	1.07334e10	0.924693	0.462347	+	0.886699i	$0.347008\pi$
$752$	0	0
$753$	1.21602e10	1.03791
$754$	0	0
$755$	4.32066e9	0.365372
$756$	0	0
$757$	1.28389e9	0.107570	0.0537850	−	0.998553i	$-0.482871\pi$
$757$	1.28389e9	0.107570	0.0537850	+	0.998553i	$0.482871\pi$
$758$	0	0
$759$	−4.37400e7	−0.00363105
$760$	0	0
$761$	1.13991e10	0.937616	0.468808	−	0.883300i	$-0.344684\pi$
$761$	1.13991e10	0.937616	0.468808	+	0.883300i	$0.344684\pi$
$762$	0	0
$763$	3.24453e9	0.264433
$764$	0	0
$765$	1.27010e9	0.102571
$766$	0	0
$767$	2.65022e10	2.12079
$768$	0	0
$769$	−1.26388e10	−1.00222	−0.501112	−	0.865383i	$-0.667075\pi$
$769$	−1.26388e10	−1.00222	−0.501112	+	0.865383i	$0.667075\pi$
$770$	0	0
$771$	−1.79618e9	−0.141143
$772$	0	0
$773$	1.78935e10	1.39337	0.696687	−	0.717375i	$-0.254657\pi$
$773$	1.78935e10	1.39337	0.696687	+	0.717375i	$0.254657\pi$
$774$	0	0
$775$	1.88050e9	0.145117
$776$	0	0
$777$	−2.94186e9	−0.224983
$778$	0	0
$779$	2.73308e10	2.07144
$780$	0	0
$781$	1.50329e8	0.0112918
$782$	0	0
$783$	1.74301e9	0.129758
$784$	0	0
$785$	6.25825e9	0.461752
$786$	0	0
$787$	−1.65085e10	−1.20725	−0.603623	−	0.797270i	$-0.706277\pi$
$787$	−1.65085e10	−1.20725	−0.603623	+	0.797270i	$0.706277\pi$
$788$	0	0
$789$	−2.98952e9	−0.216686
$790$	0	0
$791$	6.13496e9	0.440752
$792$	0	0
$793$	−3.81833e9	−0.271905
$794$	0	0
$795$	−3.18593e8	−0.0224880
$796$	0	0
$797$	−1.82052e9	−0.127377	−0.0636886	−	0.997970i	$-0.520286\pi$
$797$	−1.82052e9	−0.127377	−0.0636886	+	0.997970i	$0.520286\pi$
$798$	0	0
$799$	−4.24496e9	−0.294415
$800$	0	0
$801$	4.14448e9	0.284942
$802$	0	0
$803$	−1.77274e8	−0.0120820
$804$	0	0
$805$	7.83000e8	0.0529025
$806$	0	0
$807$	3.53846e9	0.237004
$808$	0	0
$809$	−2.17445e10	−1.44388	−0.721938	−	0.691958i	$-0.756748\pi$
$809$	−2.17445e10	−1.44388	−0.721938	+	0.691958i	$0.756748\pi$
$810$	0	0
$811$	1.93869e10	1.27625	0.638126	−	0.769932i	$-0.279710\pi$
$811$	1.93869e10	1.27625	0.638126	+	0.769932i	$0.279710\pi$
$812$	0	0
$813$	−1.53060e10	−0.998954
$814$	0	0
$815$	2.04198e8	0.0132130
$816$	0	0
$817$	7.78885e9	0.499685
$818$	0	0
$819$	2.37692e9	0.151190
$820$	0	0
$821$	−1.16565e10	−0.735134	−0.367567	−	0.929997i	$-0.619809\pi$
$821$	−1.16565e10	−0.735134	−0.367567	+	0.929997i	$0.619809\pi$
$822$	0	0
$823$	4.64054e9	0.290181	0.145091	−	0.989418i	$-0.453653\pi$
$823$	4.64054e9	0.290181	0.145091	+	0.989418i	$0.453653\pi$
$824$	0	0
$825$	−2.53125e7	−0.00156945
$826$	0	0
$827$	−1.00604e10	−0.618506	−0.309253	−	0.950980i	$-0.600079\pi$
$827$	−1.00604e10	−0.618506	−0.309253	+	0.950980i	$0.600079\pi$
$828$	0	0
$829$	1.56256e10	0.952566	0.476283	−	0.879292i	$-0.341984\pi$
$829$	1.56256e10	0.952566	0.476283	+	0.879292i	$0.341984\pi$
$830$	0	0
$831$	−3.52551e9	−0.213117
$832$	0	0
$833$	−1.07283e10	−0.643095
$834$	0	0
$835$	1.24352e10	0.739181
$836$	0	0
$837$	−2.36889e9	−0.139639
$838$	0	0
$839$	2.83557e10	1.65757	0.828787	−	0.559564i	$-0.189032\pi$
$839$	2.83557e10	1.65757	0.828787	+	0.559564i	$0.189032\pi$
$840$	0	0
$841$	−9.40807e9	−0.545399
$842$	0	0
$843$	7.29994e9	0.419684
$844$	0	0
$845$	1.68458e10	0.960491
$846$	0	0
$847$	−4.52019e9	−0.255602
$848$	0	0
$849$	−2.45990e9	−0.137956
$850$	0	0
$851$	1.26804e10	0.705311
$852$	0	0
$853$	1.52072e10	0.838933	0.419466	−	0.907771i	$-0.362217\pi$
$853$	1.52072e10	0.838933	0.419466	+	0.907771i	$0.362217\pi$
$854$	0	0
$855$	−4.88102e9	−0.267072
$856$	0	0
$857$	2.69776e10	1.46410	0.732049	−	0.681251i	$-0.238564\pi$
$857$	2.69776e10	1.46410	0.732049	+	0.681251i	$0.238564\pi$
$858$	0	0
$859$	1.59964e10	0.861086	0.430543	−	0.902570i	$-0.358322\pi$
$859$	1.59964e10	0.861086	0.430543	+	0.902570i	$0.358322\pi$
$860$	0	0
$861$	3.19618e9	0.170656
$862$	0	0
$863$	−2.22647e10	−1.17918	−0.589588	−	0.807704i	$-0.700710\pi$
$863$	−2.22647e10	−1.17918	−0.589588	+	0.807704i	$0.700710\pi$
$864$	0	0
$865$	−1.25969e10	−0.661769
$866$	0	0
$867$	5.83391e9	0.304014
$868$	0	0
$869$	4.41286e8	0.0228114
$870$	0	0
$871$	−1.51034e10	−0.774481
$872$	0	0
$873$	5.81840e9	0.295974
$874$	0	0
$875$	4.53125e8	0.0228660
$876$	0	0
$877$	2.08359e10	1.04307	0.521534	−	0.853230i	$-0.325360\pi$
$877$	2.08359e10	1.04307	0.521534	+	0.853230i	$0.325360\pi$
$878$	0	0
$879$	1.68121e10	0.834952
$880$	0	0
$881$	2.41363e10	1.18920	0.594601	−	0.804021i	$-0.297310\pi$
$881$	2.41363e10	1.18920	0.594601	+	0.804021i	$0.297310\pi$
$882$	0	0
$883$	−8.17127e9	−0.399417	−0.199709	−	0.979855i	$-0.564000\pi$
$883$	−8.17127e9	−0.399417	−0.199709	+	0.979855i	$0.564000\pi$
$884$	0	0
$885$	−6.36437e9	−0.308641
$886$	0	0
$887$	−2.64726e10	−1.27369	−0.636846	−	0.770991i	$-0.719761\pi$
$887$	−2.64726e10	−1.27369	−0.636846	+	0.770991i	$0.719761\pi$
$888$	0	0
$889$	2.16688e9	0.103438
$890$	0	0
$891$	3.18865e7	0.00151020
$892$	0	0
$893$	1.63135e10	0.766595
$894$	0	0
$895$	−1.07395e10	−0.500732
$896$	0	0
$897$	−1.02454e10	−0.473973
$898$	0	0
$899$	−1.06577e10	−0.489218
$900$	0	0
$901$	1.31572e9	0.0599275
$902$	0	0
$903$	9.10861e8	0.0411666
$904$	0	0
$905$	−9.18688e8	−0.0412001
$906$	0	0
$907$	−3.84880e10	−1.71277	−0.856387	−	0.516335i	$-0.827296\pi$
$907$	−3.84880e10	−1.71277	−0.856387	+	0.516335i	$0.827296\pi$
$908$	0	0
$909$	1.12724e10	0.497785
$910$	0	0
$911$	8.73430e9	0.382749	0.191374	−	0.981517i	$-0.438706\pi$
$911$	8.73430e9	0.382749	0.191374	+	0.981517i	$0.438706\pi$
$912$	0	0
$913$	2.30805e8	0.0100369
$914$	0	0
$915$	9.16954e8	0.0395707
$916$	0	0
$917$	1.03657e10	0.443919
$918$	0	0
$919$	−1.91695e10	−0.814717	−0.407358	−	0.913268i	$-0.633550\pi$
$919$	−1.91695e10	−0.814717	−0.407358	+	0.913268i	$0.633550\pi$
$920$	0	0
$921$	−3.05860e9	−0.129007
$922$	0	0
$923$	3.52120e10	1.47396
$924$	0	0
$925$	7.33822e9	0.304856
$926$	0	0
$927$	1.52379e10	0.628268
$928$	0	0
$929$	9.51352e9	0.389302	0.194651	−	0.980873i	$-0.437643\pi$
$929$	9.51352e9	0.389302	0.194651	+	0.980873i	$0.437643\pi$
$930$	0	0
$931$	4.12292e10	1.67449
$932$	0	0
$933$	1.02153e10	0.411779
$934$	0	0
$935$	1.04535e8	0.00418236
$936$	0	0
$937$	−2.09871e10	−0.833422	−0.416711	−	0.909039i	$-0.636817\pi$
$937$	−2.09871e10	−0.833422	−0.416711	+	0.909039i	$0.636817\pi$
$938$	0	0
$939$	3.00781e9	0.118555
$940$	0	0
$941$	1.92302e9	0.0752349	0.0376175	−	0.999292i	$-0.488023\pi$
$941$	1.92302e9	0.0752349	0.0376175	+	0.999292i	$0.488023\pi$
$942$	0	0
$943$	−1.37766e10	−0.534998
$944$	0	0
$945$	−5.70807e8	−0.0220028
$946$	0	0
$947$	−1.52483e10	−0.583440	−0.291720	−	0.956504i	$-0.594228\pi$
$947$	−1.52483e10	−0.583440	−0.291720	+	0.956504i	$0.594228\pi$
$948$	0	0
$949$	−4.15235e10	−1.57711
$950$	0	0
$951$	1.80927e10	0.682135
$952$	0	0
$953$	−1.60051e10	−0.599011	−0.299505	−	0.954095i	$-0.596822\pi$
$953$	−1.60051e10	−0.599011	−0.299505	+	0.954095i	$0.596822\pi$
$954$	0	0
$955$	−3.19411e9	−0.118669
$956$	0	0
$957$	1.43457e8	0.00529092
$958$	0	0
$959$	−1.32224e10	−0.484110
$960$	0	0
$961$	−1.30280e10	−0.473529
$962$	0	0
$963$	−1.49915e10	−0.540945
$964$	0	0
$965$	1.71018e10	0.612629
$966$	0	0
$967$	3.05254e10	1.08560	0.542798	−	0.839863i	$-0.317365\pi$
$967$	3.05254e10	1.08560	0.542798	+	0.839863i	$0.317365\pi$
$968$	0	0
$969$	2.01575e10	0.711712
$970$	0	0
$971$	−3.85732e10	−1.35213	−0.676066	−	0.736841i	$-0.736316\pi$
$971$	−3.85732e10	−1.35213	−0.676066	+	0.736841i	$0.736316\pi$
$972$	0	0
$973$	−7.11828e9	−0.247731
$974$	0	0
$975$	−5.92903e9	−0.204865
$976$	0	0
$977$	−1.90805e10	−0.654575	−0.327287	−	0.944925i	$-0.606134\pi$
$977$	−1.90805e10	−0.654575	−0.327287	+	0.944925i	$0.606134\pi$
$978$	0	0
$979$	3.41110e8	0.0116186
$980$	0	0
$981$	1.01951e10	0.344786
$982$	0	0
$983$	4.88330e10	1.63974	0.819872	−	0.572546i	$-0.194044\pi$
$983$	4.88330e10	1.63974	0.819872	+	0.572546i	$0.194044\pi$
$984$	0	0
$985$	−1.72566e10	−0.575345
$986$	0	0
$987$	1.90776e9	0.0631560
$988$	0	0
$989$	−3.92612e9	−0.129056
$990$	0	0
$991$	−3.83732e10	−1.25248	−0.626239	−	0.779631i	$-0.715407\pi$
$991$	−3.83732e10	−1.25248	−0.626239	+	0.779631i	$0.715407\pi$
$992$	0	0
$993$	−1.46906e10	−0.476120
$994$	0	0
$995$	−7.75519e8	−0.0249581
$996$	0	0
$997$	−2.22521e10	−0.711111	−0.355555	−	0.934655i	$-0.615708\pi$
$997$	−2.22521e10	−0.711111	−0.355555	+	0.934655i	$0.615708\pi$
$998$	0	0
$999$	−9.24404e9	−0.293348

Twists

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

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

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