LMFDB - Embedded newform 48.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,6,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	48.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:	$7.69842335102$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	48.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-9.00000 q^{3} +6.00000 q^{5} +40.0000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q-9.00000 q^{3} +6.00000 q^{5} +40.0000 q^{7} +81.0000 q^{9} +564.000 q^{11} +638.000 q^{13} -54.0000 q^{15} +882.000 q^{17} +556.000 q^{19} -360.000 q^{21} +840.000 q^{23} -3089.00 q^{25} -729.000 q^{27} +4638.00 q^{29} -4400.00 q^{31} -5076.00 q^{33} +240.000 q^{35} -2410.00 q^{37} -5742.00 q^{39} -6870.00 q^{41} -9644.00 q^{43} +486.000 q^{45} +18672.0 q^{47} -15207.0 q^{49} -7938.00 q^{51} +33750.0 q^{53} +3384.00 q^{55} -5004.00 q^{57} +18084.0 q^{59} +39758.0 q^{61} +3240.00 q^{63} +3828.00 q^{65} +23068.0 q^{67} -7560.00 q^{69} +4248.00 q^{71} -41110.0 q^{73} +27801.0 q^{75} +22560.0 q^{77} -21920.0 q^{79} +6561.00 q^{81} -82452.0 q^{83} +5292.00 q^{85} -41742.0 q^{87} -94086.0 q^{89} +25520.0 q^{91} +39600.0 q^{93} +3336.00 q^{95} +49442.0 q^{97} +45684.0 q^{99} +O(q^{100})$

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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	6.00000	0.107331	0.0536656	−	0.998559i	$-0.482909\pi$
$5$	6.00000	0.107331	0.0536656	+	0.998559i	$0.482909\pi$
$6$	0	0
$7$	40.0000	0.308542	0.154271	−	0.988029i	$-0.450697\pi$
$7$	40.0000	0.308542	0.154271	+	0.988029i	$0.450697\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	564.000	1.40539	0.702696	−	0.711490i	$-0.251979\pi$
$11$	564.000	1.40539	0.702696	+	0.711490i	$0.251979\pi$
$12$	0	0
$13$	638.000	1.04704	0.523519	−	0.852014i	$-0.324619\pi$
$13$	638.000	1.04704	0.523519	+	0.852014i	$0.324619\pi$
$14$	0	0
$15$	−54.0000	−0.0619677
$16$	0	0
$17$	882.000	0.740195	0.370098	−	0.928993i	$-0.379324\pi$
$17$	882.000	0.740195	0.370098	+	0.928993i	$0.379324\pi$
$18$	0	0
$19$	556.000	0.353338	0.176669	−	0.984270i	$-0.443468\pi$
$19$	556.000	0.353338	0.176669	+	0.984270i	$0.443468\pi$
$20$	0	0
$21$	−360.000	−0.178137
$22$	0	0
$23$	840.000	0.331100	0.165550	−	0.986201i	$-0.447060\pi$
$23$	840.000	0.331100	0.165550	+	0.986201i	$0.447060\pi$
$24$	0	0
$25$	−3089.00	−0.988480
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	4638.00	1.02408	0.512042	−	0.858960i	$-0.328889\pi$
$29$	4638.00	1.02408	0.512042	+	0.858960i	$0.328889\pi$
$30$	0	0
$31$	−4400.00	−0.822334	−0.411167	−	0.911560i	$-0.634879\pi$
$31$	−4400.00	−0.822334	−0.411167	+	0.911560i	$0.634879\pi$
$32$	0	0
$33$	−5076.00	−0.811403
$34$	0	0
$35$	240.000	0.0331162
$36$	0	0
$37$	−2410.00	−0.289409	−0.144705	−	0.989475i	$-0.546223\pi$
$37$	−2410.00	−0.289409	−0.144705	+	0.989475i	$0.546223\pi$
$38$	0	0
$39$	−5742.00	−0.604507
$40$	0	0
$41$	−6870.00	−0.638259	−0.319130	−	0.947711i	$-0.603391\pi$
$41$	−6870.00	−0.638259	−0.319130	+	0.947711i	$0.603391\pi$
$42$	0	0
$43$	−9644.00	−0.795401	−0.397700	−	0.917515i	$-0.630192\pi$
$43$	−9644.00	−0.795401	−0.397700	+	0.917515i	$0.630192\pi$
$44$	0	0
$45$	486.000	0.0357771
$46$	0	0
$47$	18672.0	1.23295	0.616476	−	0.787374i	$-0.288560\pi$
$47$	18672.0	1.23295	0.616476	+	0.787374i	$0.288560\pi$
$48$	0	0
$49$	−15207.0	−0.904802
$50$	0	0
$51$	−7938.00	−0.427352
$52$	0	0
$53$	33750.0	1.65038	0.825190	−	0.564855i	$-0.191068\pi$
$53$	33750.0	1.65038	0.825190	+	0.564855i	$0.191068\pi$
$54$	0	0
$55$	3384.00	0.150842
$56$	0	0
$57$	−5004.00	−0.204000
$58$	0	0
$59$	18084.0	0.676339	0.338170	−	0.941085i	$-0.390192\pi$
$59$	18084.0	0.676339	0.338170	+	0.941085i	$0.390192\pi$
$60$	0	0
$61$	39758.0	1.36804	0.684022	−	0.729462i	$-0.260229\pi$
$61$	39758.0	1.36804	0.684022	+	0.729462i	$0.260229\pi$
$62$	0	0
$63$	3240.00	0.102847
$64$	0	0
$65$	3828.00	0.112380
$66$	0	0
$67$	23068.0	0.627802	0.313901	−	0.949456i	$-0.398364\pi$
$67$	23068.0	0.627802	0.313901	+	0.949456i	$0.398364\pi$
$68$	0	0
$69$	−7560.00	−0.191161
$70$	0	0
$71$	4248.00	0.100009	0.0500044	−	0.998749i	$-0.484076\pi$
$71$	4248.00	0.100009	0.0500044	+	0.998749i	$0.484076\pi$
$72$	0	0
$73$	−41110.0	−0.902901	−0.451451	−	0.892296i	$-0.649093\pi$
$73$	−41110.0	−0.902901	−0.451451	+	0.892296i	$0.649093\pi$
$74$	0	0
$75$	27801.0	0.570699
$76$	0	0
$77$	22560.0	0.433623
$78$	0	0
$79$	−21920.0	−0.395160	−0.197580	−	0.980287i	$-0.563308\pi$
$79$	−21920.0	−0.395160	−0.197580	+	0.980287i	$0.563308\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−82452.0	−1.31373	−0.656865	−	0.754008i	$-0.728118\pi$
$83$	−82452.0	−1.31373	−0.656865	+	0.754008i	$0.728118\pi$
$84$	0	0
$85$	5292.00	0.0794461
$86$	0	0
$87$	−41742.0	−0.591255
$88$	0	0
$89$	−94086.0	−1.25907	−0.629535	−	0.776972i	$-0.716755\pi$
$89$	−94086.0	−1.25907	−0.629535	+	0.776972i	$0.716755\pi$
$90$	0	0
$91$	25520.0	0.323056
$92$	0	0
$93$	39600.0	0.474775
$94$	0	0
$95$	3336.00	0.0379243
$96$	0	0
$97$	49442.0	0.533540	0.266770	−	0.963760i	$-0.414044\pi$
$97$	49442.0	0.533540	0.266770	+	0.963760i	$0.414044\pi$
$98$	0	0
$99$	45684.0	0.468464
$100$	0	0
$101$	−143034.	−1.39520	−0.697599	−	0.716488i	$-0.745748\pi$
$101$	−143034.	−1.39520	−0.697599	+	0.716488i	$0.745748\pi$
$102$	0	0
$103$	−53144.0	−0.493584	−0.246792	−	0.969068i	$-0.579376\pi$
$103$	−53144.0	−0.493584	−0.246792	+	0.969068i	$0.579376\pi$
$104$	0	0
$105$	−2160.00	−0.0191197
$106$	0	0
$107$	−90828.0	−0.766938	−0.383469	−	0.923554i	$-0.625271\pi$
$107$	−90828.0	−0.766938	−0.383469	+	0.923554i	$0.625271\pi$
$108$	0	0
$109$	−61666.0	−0.497141	−0.248570	−	0.968614i	$-0.579961\pi$
$109$	−61666.0	−0.497141	−0.248570	+	0.968614i	$0.579961\pi$
$110$	0	0
$111$	21690.0	0.167091
$112$	0	0
$113$	10482.0	0.0772232	0.0386116	−	0.999254i	$-0.487706\pi$
$113$	10482.0	0.0772232	0.0386116	+	0.999254i	$0.487706\pi$
$114$	0	0
$115$	5040.00	0.0355374
$116$	0	0
$117$	51678.0	0.349013
$118$	0	0
$119$	35280.0	0.228382
$120$	0	0
$121$	157045.	0.975126
$122$	0	0
$123$	61830.0	0.368499
$124$	0	0
$125$	−37284.0	−0.213426
$126$	0	0
$127$	171088.	0.941261	0.470631	−	0.882330i	$-0.344026\pi$
$127$	171088.	0.941261	0.470631	+	0.882330i	$0.344026\pi$
$128$	0	0
$129$	86796.0	0.459225
$130$	0	0
$131$	−258468.	−1.31592	−0.657959	−	0.753054i	$-0.728580\pi$
$131$	−258468.	−1.31592	−0.657959	+	0.753054i	$0.728580\pi$
$132$	0	0
$133$	22240.0	0.109020
$134$	0	0
$135$	−4374.00	−0.0206559
$136$	0	0
$137$	300234.	1.36665	0.683327	−	0.730113i	$-0.260532\pi$
$137$	300234.	1.36665	0.683327	+	0.730113i	$0.260532\pi$
$138$	0	0
$139$	350164.	1.53721	0.768607	−	0.639721i	$-0.220950\pi$
$139$	350164.	1.53721	0.768607	+	0.639721i	$0.220950\pi$
$140$	0	0
$141$	−168048.	−0.711845
$142$	0	0
$143$	359832.	1.47150
$144$	0	0
$145$	27828.0	0.109916
$146$	0	0
$147$	136863.	0.522387
$148$	0	0
$149$	−105258.	−0.388409	−0.194205	−	0.980961i	$-0.562213\pi$
$149$	−105258.	−0.388409	−0.194205	+	0.980961i	$0.562213\pi$
$150$	0	0
$151$	−396392.	−1.41476	−0.707380	−	0.706834i	$-0.750123\pi$
$151$	−396392.	−1.41476	−0.707380	+	0.706834i	$0.750123\pi$
$152$	0	0
$153$	71442.0	0.246732
$154$	0	0
$155$	−26400.0	−0.0882622
$156$	0	0
$157$	−137746.	−0.445995	−0.222997	−	0.974819i	$-0.571584\pi$
$157$	−137746.	−0.445995	−0.222997	+	0.974819i	$0.571584\pi$
$158$	0	0
$159$	−303750.	−0.952848
$160$	0	0
$161$	33600.0	0.102159
$162$	0	0
$163$	−352676.	−1.03970	−0.519849	−	0.854258i	$-0.674012\pi$
$163$	−352676.	−1.03970	−0.519849	+	0.854258i	$0.674012\pi$
$164$	0	0
$165$	−30456.0	−0.0870889
$166$	0	0
$167$	217560.	0.603654	0.301827	−	0.953363i	$-0.402404\pi$
$167$	217560.	0.603654	0.301827	+	0.953363i	$0.402404\pi$
$168$	0	0
$169$	35751.0	0.0962878
$170$	0	0
$171$	45036.0	0.117779
$172$	0	0
$173$	−163698.	−0.415842	−0.207921	−	0.978146i	$-0.566670\pi$
$173$	−163698.	−0.415842	−0.207921	+	0.978146i	$0.566670\pi$
$174$	0	0
$175$	−123560.	−0.304988
$176$	0	0
$177$	−162756.	−0.390485
$178$	0	0
$179$	−358740.	−0.836849	−0.418425	−	0.908252i	$-0.637418\pi$
$179$	−358740.	−0.836849	−0.418425	+	0.908252i	$0.637418\pi$
$180$	0	0
$181$	−507130.	−1.15060	−0.575298	−	0.817944i	$-0.695114\pi$
$181$	−507130.	−1.15060	−0.575298	+	0.817944i	$0.695114\pi$
$182$	0	0
$183$	−357822.	−0.789840
$184$	0	0
$185$	−14460.0	−0.0310627
$186$	0	0
$187$	497448.	1.04026
$188$	0	0
$189$	−29160.0	−0.0593790
$190$	0	0
$191$	648384.	1.28602	0.643012	−	0.765856i	$-0.277685\pi$
$191$	648384.	1.28602	0.643012	+	0.765856i	$0.277685\pi$
$192$	0	0
$193$	−27838.0	−0.0537954	−0.0268977	−	0.999638i	$-0.508563\pi$
$193$	−27838.0	−0.0537954	−0.0268977	+	0.999638i	$0.508563\pi$
$194$	0	0
$195$	−34452.0	−0.0648826
$196$	0	0
$197$	611046.	1.12178	0.560891	−	0.827890i	$-0.310459\pi$
$197$	611046.	1.12178	0.560891	+	0.827890i	$0.310459\pi$
$198$	0	0
$199$	−879032.	−1.57352	−0.786760	−	0.617260i	$-0.788243\pi$
$199$	−879032.	−1.57352	−0.786760	+	0.617260i	$0.788243\pi$
$200$	0	0
$201$	−207612.	−0.362462
$202$	0	0
$203$	185520.	0.315973
$204$	0	0
$205$	−41220.0	−0.0685052
$206$	0	0
$207$	68040.0	0.110367
$208$	0	0
$209$	313584.	0.496579
$210$	0	0
$211$	−48500.0	−0.0749956	−0.0374978	−	0.999297i	$-0.511939\pi$
$211$	−48500.0	−0.0749956	−0.0374978	+	0.999297i	$0.511939\pi$
$212$	0	0
$213$	−38232.0	−0.0577402
$214$	0	0
$215$	−57864.0	−0.0853714
$216$	0	0
$217$	−176000.	−0.253725
$218$	0	0
$219$	369990.	0.521290
$220$	0	0
$221$	562716.	0.775012
$222$	0	0
$223$	999472.	1.34589	0.672943	−	0.739694i	$-0.265030\pi$
$223$	999472.	1.34589	0.672943	+	0.739694i	$0.265030\pi$
$224$	0	0
$225$	−250209.	−0.329493
$226$	0	0
$227$	−606180.	−0.780795	−0.390397	−	0.920646i	$-0.627662\pi$
$227$	−606180.	−0.780795	−0.390397	+	0.920646i	$0.627662\pi$
$228$	0	0
$229$	1.35993e6	1.71367	0.856834	−	0.515593i	$-0.172428\pi$
$229$	1.35993e6	1.71367	0.856834	+	0.515593i	$0.172428\pi$
$230$	0	0
$231$	−203040.	−0.250352
$232$	0	0
$233$	−392886.	−0.474107	−0.237054	−	0.971497i	$-0.576182\pi$
$233$	−392886.	−0.474107	−0.237054	+	0.971497i	$0.576182\pi$
$234$	0	0
$235$	112032.	0.132334
$236$	0	0
$237$	197280.	0.228146
$238$	0	0
$239$	1.32514e6	1.50060	0.750301	−	0.661096i	$-0.229908\pi$
$239$	1.32514e6	1.50060	0.750301	+	0.661096i	$0.229908\pi$
$240$	0	0
$241$	−990094.	−1.09808	−0.549040	−	0.835796i	$-0.685006\pi$
$241$	−990094.	−1.09808	−0.549040	+	0.835796i	$0.685006\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−91242.0	−0.0971135
$246$	0	0
$247$	354728.	0.369959
$248$	0	0
$249$	742068.	0.758482
$250$	0	0
$251$	−147132.	−0.147409	−0.0737043	−	0.997280i	$-0.523482\pi$
$251$	−147132.	−0.147409	−0.0737043	+	0.997280i	$0.523482\pi$
$252$	0	0
$253$	473760.	0.465326
$254$	0	0
$255$	−47628.0	−0.0458682
$256$	0	0
$257$	−483582.	−0.456707	−0.228353	−	0.973578i	$-0.573334\pi$
$257$	−483582.	−0.456707	−0.228353	+	0.973578i	$0.573334\pi$
$258$	0	0
$259$	−96400.0	−0.0892951
$260$	0	0
$261$	375678.	0.341361
$262$	0	0
$263$	−813576.	−0.725285	−0.362643	−	0.931928i	$-0.618125\pi$
$263$	−813576.	−0.725285	−0.362643	+	0.931928i	$0.618125\pi$
$264$	0	0
$265$	202500.	0.177137
$266$	0	0
$267$	846774.	0.726925
$268$	0	0
$269$	−461106.	−0.388526	−0.194263	−	0.980949i	$-0.562232\pi$
$269$	−461106.	−0.388526	−0.194263	+	0.980949i	$0.562232\pi$
$270$	0	0
$271$	−1.67514e6	−1.38556	−0.692782	−	0.721147i	$-0.743615\pi$
$271$	−1.67514e6	−1.38556	−0.692782	+	0.721147i	$0.743615\pi$
$272$	0	0
$273$	−229680.	−0.186516
$274$	0	0
$275$	−1.74220e6	−1.38920
$276$	0	0
$277$	401126.	0.314110	0.157055	−	0.987590i	$-0.449800\pi$
$277$	401126.	0.314110	0.157055	+	0.987590i	$0.449800\pi$
$278$	0	0
$279$	−356400.	−0.274111
$280$	0	0
$281$	−2.30977e6	−1.74503	−0.872514	−	0.488590i	$-0.837511\pi$
$281$	−2.30977e6	−1.74503	−0.872514	+	0.488590i	$0.837511\pi$
$282$	0	0
$283$	1.12877e6	0.837800	0.418900	−	0.908032i	$-0.362416\pi$
$283$	1.12877e6	0.837800	0.418900	+	0.908032i	$0.362416\pi$
$284$	0	0
$285$	−30024.0	−0.0218956
$286$	0	0
$287$	−274800.	−0.196930
$288$	0	0
$289$	−641933.	−0.452111
$290$	0	0
$291$	−444978.	−0.308039
$292$	0	0
$293$	−938874.	−0.638908	−0.319454	−	0.947602i	$-0.603499\pi$
$293$	−938874.	−0.638908	−0.319454	+	0.947602i	$0.603499\pi$
$294$	0	0
$295$	108504.	0.0725923
$296$	0	0
$297$	−411156.	−0.270468
$298$	0	0
$299$	535920.	0.346675
$300$	0	0
$301$	−385760.	−0.245415
$302$	0	0
$303$	1.28731e6	0.805518
$304$	0	0
$305$	238548.	0.146834
$306$	0	0
$307$	−692948.	−0.419619	−0.209809	−	0.977742i	$-0.567284\pi$
$307$	−692948.	−0.419619	−0.209809	+	0.977742i	$0.567284\pi$
$308$	0	0
$309$	478296.	0.284971
$310$	0	0
$311$	−2.94310e6	−1.72545	−0.862727	−	0.505670i	$-0.831245\pi$
$311$	−2.94310e6	−1.72545	−0.862727	+	0.505670i	$0.831245\pi$
$312$	0	0
$313$	885146.	0.510686	0.255343	−	0.966851i	$-0.417812\pi$
$313$	885146.	0.510686	0.255343	+	0.966851i	$0.417812\pi$
$314$	0	0
$315$	19440.0	0.0110387
$316$	0	0
$317$	2.50880e6	1.40222	0.701112	−	0.713051i	$-0.252687\pi$
$317$	2.50880e6	1.40222	0.701112	+	0.713051i	$0.252687\pi$
$318$	0	0
$319$	2.61583e6	1.43924
$320$	0	0
$321$	817452.	0.442792
$322$	0	0
$323$	490392.	0.261539
$324$	0	0
$325$	−1.97078e6	−1.03498
$326$	0	0
$327$	554994.	0.287024
$328$	0	0
$329$	746880.	0.380418
$330$	0	0
$331$	216148.	0.108438	0.0542190	−	0.998529i	$-0.482733\pi$
$331$	216148.	0.108438	0.0542190	+	0.998529i	$0.482733\pi$
$332$	0	0
$333$	−195210.	−0.0964698
$334$	0	0
$335$	138408.	0.0673828
$336$	0	0
$337$	3.25263e6	1.56012	0.780062	−	0.625702i	$-0.215187\pi$
$337$	3.25263e6	1.56012	0.780062	+	0.625702i	$0.215187\pi$
$338$	0	0
$339$	−94338.0	−0.0445849
$340$	0	0
$341$	−2.48160e6	−1.15570
$342$	0	0
$343$	−1.28056e6	−0.587712
$344$	0	0
$345$	−45360.0	−0.0205175
$346$	0	0
$347$	2.93207e6	1.30723	0.653613	−	0.756829i	$-0.273253\pi$
$347$	2.93207e6	1.30723	0.653613	+	0.756829i	$0.273253\pi$
$348$	0	0
$349$	905198.	0.397814	0.198907	−	0.980018i	$-0.436261\pi$
$349$	905198.	0.397814	0.198907	+	0.980018i	$0.436261\pi$
$350$	0	0
$351$	−465102.	−0.201502
$352$	0	0
$353$	1.91786e6	0.819181	0.409590	−	0.912270i	$-0.365672\pi$
$353$	1.91786e6	0.819181	0.409590	+	0.912270i	$0.365672\pi$
$354$	0	0
$355$	25488.0	0.0107341
$356$	0	0
$357$	−317520.	−0.131856
$358$	0	0
$359$	2.43698e6	0.997968	0.498984	−	0.866611i	$-0.333707\pi$
$359$	2.43698e6	0.997968	0.498984	+	0.866611i	$0.333707\pi$
$360$	0	0
$361$	−2.16696e6	−0.875152
$362$	0	0
$363$	−1.41340e6	−0.562989
$364$	0	0
$365$	−246660.	−0.0969095
$366$	0	0
$367$	984064.	0.381380	0.190690	−	0.981650i	$-0.438927\pi$
$367$	984064.	0.381380	0.190690	+	0.981650i	$0.438927\pi$
$368$	0	0
$369$	−556470.	−0.212753
$370$	0	0
$371$	1.35000e6	0.509212
$372$	0	0
$373$	1.70365e6	0.634029	0.317015	−	0.948421i	$-0.397320\pi$
$373$	1.70365e6	0.634029	0.317015	+	0.948421i	$0.397320\pi$
$374$	0	0
$375$	335556.	0.123222
$376$	0	0
$377$	2.95904e6	1.07225
$378$	0	0
$379$	−2.75654e6	−0.985749	−0.492874	−	0.870100i	$-0.664054\pi$
$379$	−2.75654e6	−0.985749	−0.492874	+	0.870100i	$0.664054\pi$
$380$	0	0
$381$	−1.53979e6	−0.543438
$382$	0	0
$383$	−456576.	−0.159044	−0.0795218	−	0.996833i	$-0.525339\pi$
$383$	−456576.	−0.159044	−0.0795218	+	0.996833i	$0.525339\pi$
$384$	0	0
$385$	135360.	0.0465413
$386$	0	0
$387$	−781164.	−0.265134
$388$	0	0
$389$	−2.00639e6	−0.672268	−0.336134	−	0.941814i	$-0.609119\pi$
$389$	−2.00639e6	−0.672268	−0.336134	+	0.941814i	$0.609119\pi$
$390$	0	0
$391$	740880.	0.245079
$392$	0	0
$393$	2.32621e6	0.759745
$394$	0	0
$395$	−131520.	−0.0424130
$396$	0	0
$397$	−5.77040e6	−1.83751	−0.918755	−	0.394828i	$-0.870804\pi$
$397$	−5.77040e6	−1.83751	−0.918755	+	0.394828i	$0.870804\pi$
$398$	0	0
$399$	−200160.	−0.0629427
$400$	0	0
$401$	3.00626e6	0.933610	0.466805	−	0.884360i	$-0.345405\pi$
$401$	3.00626e6	0.933610	0.466805	+	0.884360i	$0.345405\pi$
$402$	0	0
$403$	−2.80720e6	−0.861015
$404$	0	0
$405$	39366.0	0.0119257
$406$	0	0
$407$	−1.35924e6	−0.406734
$408$	0	0
$409$	1.53363e6	0.453327	0.226663	−	0.973973i	$-0.427218\pi$
$409$	1.53363e6	0.453327	0.226663	+	0.973973i	$0.427218\pi$
$410$	0	0
$411$	−2.70211e6	−0.789038
$412$	0	0
$413$	723360.	0.208679
$414$	0	0
$415$	−494712.	−0.141004
$416$	0	0
$417$	−3.15148e6	−0.887511
$418$	0	0
$419$	3.87376e6	1.07795	0.538973	−	0.842323i	$-0.318812\pi$
$419$	3.87376e6	1.07795	0.538973	+	0.842323i	$0.318812\pi$
$420$	0	0
$421$	−1.33307e6	−0.366561	−0.183281	−	0.983061i	$-0.558672\pi$
$421$	−1.33307e6	−0.366561	−0.183281	+	0.983061i	$0.558672\pi$
$422$	0	0
$423$	1.51243e6	0.410984
$424$	0	0
$425$	−2.72450e6	−0.731668
$426$	0	0
$427$	1.59032e6	0.422099
$428$	0	0
$429$	−3.23849e6	−0.849570
$430$	0	0
$431$	−6.45192e6	−1.67300	−0.836500	−	0.547967i	$-0.815402\pi$
$431$	−6.45192e6	−1.67300	−0.836500	+	0.547967i	$0.815402\pi$
$432$	0	0
$433$	−4.16577e6	−1.06777	−0.533883	−	0.845558i	$-0.679268\pi$
$433$	−4.16577e6	−1.06777	−0.533883	+	0.845558i	$0.679268\pi$
$434$	0	0
$435$	−250452.	−0.0634602
$436$	0	0
$437$	467040.	0.116990
$438$	0	0
$439$	−792680.	−0.196307	−0.0981537	−	0.995171i	$-0.531294\pi$
$439$	−792680.	−0.196307	−0.0981537	+	0.995171i	$0.531294\pi$
$440$	0	0
$441$	−1.23177e6	−0.301601
$442$	0	0
$443$	1.39981e6	0.338891	0.169446	−	0.985540i	$-0.445802\pi$
$443$	1.39981e6	0.338891	0.169446	+	0.985540i	$0.445802\pi$
$444$	0	0
$445$	−564516.	−0.135138
$446$	0	0
$447$	947322.	0.224248
$448$	0	0
$449$	2.99248e6	0.700512	0.350256	−	0.936654i	$-0.386095\pi$
$449$	2.99248e6	0.700512	0.350256	+	0.936654i	$0.386095\pi$
$450$	0	0
$451$	−3.87468e6	−0.897004
$452$	0	0
$453$	3.56753e6	0.816812
$454$	0	0
$455$	153120.	0.0346740
$456$	0	0
$457$	6.29969e6	1.41101	0.705503	−	0.708707i	$-0.250721\pi$
$457$	6.29969e6	1.41101	0.705503	+	0.708707i	$0.250721\pi$
$458$	0	0
$459$	−642978.	−0.142451
$460$	0	0
$461$	3.40318e6	0.745818	0.372909	−	0.927868i	$-0.378360\pi$
$461$	3.40318e6	0.745818	0.372909	+	0.927868i	$0.378360\pi$
$462$	0	0
$463$	2.23034e6	0.483524	0.241762	−	0.970336i	$-0.422275\pi$
$463$	2.23034e6	0.483524	0.241762	+	0.970336i	$0.422275\pi$
$464$	0	0
$465$	237600.	0.0509582
$466$	0	0
$467$	6.51409e6	1.38217	0.691085	−	0.722773i	$-0.257133\pi$
$467$	6.51409e6	1.38217	0.691085	+	0.722773i	$0.257133\pi$
$468$	0	0
$469$	922720.	0.193704
$470$	0	0
$471$	1.23971e6	0.257495
$472$	0	0
$473$	−5.43922e6	−1.11785
$474$	0	0
$475$	−1.71748e6	−0.349268
$476$	0	0
$477$	2.73375e6	0.550127
$478$	0	0
$479$	−2.39232e6	−0.476410	−0.238205	−	0.971215i	$-0.576559\pi$
$479$	−2.39232e6	−0.476410	−0.238205	+	0.971215i	$0.576559\pi$
$480$	0	0
$481$	−1.53758e6	−0.303023
$482$	0	0
$483$	−302400.	−0.0589812
$484$	0	0
$485$	296652.	0.0572655
$486$	0	0
$487$	6.13089e6	1.17139	0.585694	−	0.810532i	$-0.300822\pi$
$487$	6.13089e6	1.17139	0.585694	+	0.810532i	$0.300822\pi$
$488$	0	0
$489$	3.17408e6	0.600269
$490$	0	0
$491$	1.23589e6	0.231354	0.115677	−	0.993287i	$-0.463096\pi$
$491$	1.23589e6	0.231354	0.115677	+	0.993287i	$0.463096\pi$
$492$	0	0
$493$	4.09072e6	0.758022
$494$	0	0
$495$	274104.	0.0502808
$496$	0	0
$497$	169920.	0.0308570
$498$	0	0
$499$	9.85496e6	1.77175	0.885877	−	0.463921i	$-0.153558\pi$
$499$	9.85496e6	1.77175	0.885877	+	0.463921i	$0.153558\pi$
$500$	0	0
$501$	−1.95804e6	−0.348520
$502$	0	0
$503$	−1.16777e6	−0.205796	−0.102898	−	0.994692i	$-0.532812\pi$
$503$	−1.16777e6	−0.205796	−0.102898	+	0.994692i	$0.532812\pi$
$504$	0	0
$505$	−858204.	−0.149748
$506$	0	0
$507$	−321759.	−0.0555918
$508$	0	0
$509$	1.04941e6	0.179535	0.0897675	−	0.995963i	$-0.471388\pi$
$509$	1.04941e6	0.179535	0.0897675	+	0.995963i	$0.471388\pi$
$510$	0	0
$511$	−1.64440e6	−0.278583
$512$	0	0
$513$	−405324.	−0.0680000
$514$	0	0
$515$	−318864.	−0.0529770
$516$	0	0
$517$	1.05310e7	1.73278
$518$	0	0
$519$	1.47328e6	0.240086
$520$	0	0
$521$	−9.61407e6	−1.55172	−0.775859	−	0.630906i	$-0.782683\pi$
$521$	−9.61407e6	−1.55172	−0.775859	+	0.630906i	$0.782683\pi$
$522$	0	0
$523$	−6.96148e6	−1.11288	−0.556439	−	0.830888i	$-0.687833\pi$
$523$	−6.96148e6	−1.11288	−0.556439	+	0.830888i	$0.687833\pi$
$524$	0	0
$525$	1.11204e6	0.176085
$526$	0	0
$527$	−3.88080e6	−0.608688
$528$	0	0
$529$	−5.73074e6	−0.890373
$530$	0	0
$531$	1.46480e6	0.225446
$532$	0	0
$533$	−4.38306e6	−0.668281
$534$	0	0
$535$	−544968.	−0.0823164
$536$	0	0
$537$	3.22866e6	0.483155
$538$	0	0
$539$	−8.57675e6	−1.27160
$540$	0	0
$541$	−712690.	−0.104691	−0.0523453	−	0.998629i	$-0.516670\pi$
$541$	−712690.	−0.104691	−0.0523453	+	0.998629i	$0.516670\pi$
$542$	0	0
$543$	4.56417e6	0.664297
$544$	0	0
$545$	−369996.	−0.0533588
$546$	0	0
$547$	3.62614e6	0.518175	0.259087	−	0.965854i	$-0.416578\pi$
$547$	3.62614e6	0.518175	0.259087	+	0.965854i	$0.416578\pi$
$548$	0	0
$549$	3.22040e6	0.456015
$550$	0	0
$551$	2.57873e6	0.361848
$552$	0	0
$553$	−876800.	−0.121924
$554$	0	0
$555$	130140.	0.0179340
$556$	0	0
$557$	4.84846e6	0.662165	0.331082	−	0.943602i	$-0.392586\pi$
$557$	4.84846e6	0.662165	0.331082	+	0.943602i	$0.392586\pi$
$558$	0	0
$559$	−6.15287e6	−0.832815
$560$	0	0
$561$	−4.47703e6	−0.600597
$562$	0	0
$563$	−8.50405e6	−1.13072	−0.565360	−	0.824844i	$-0.691263\pi$
$563$	−8.50405e6	−1.13072	−0.565360	+	0.824844i	$0.691263\pi$
$564$	0	0
$565$	62892.0	0.00828847
$566$	0	0
$567$	262440.	0.0342825
$568$	0	0
$569$	362874.	0.0469867	0.0234934	−	0.999724i	$-0.492521\pi$
$569$	362874.	0.0469867	0.0234934	+	0.999724i	$0.492521\pi$
$570$	0	0
$571$	−4.11024e6	−0.527566	−0.263783	−	0.964582i	$-0.584970\pi$
$571$	−4.11024e6	−0.527566	−0.263783	+	0.964582i	$0.584970\pi$
$572$	0	0
$573$	−5.83546e6	−0.742486
$574$	0	0
$575$	−2.59476e6	−0.327286
$576$	0	0
$577$	−7.87680e6	−0.984941	−0.492470	−	0.870329i	$-0.663906\pi$
$577$	−7.87680e6	−0.984941	−0.492470	+	0.870329i	$0.663906\pi$
$578$	0	0
$579$	250542.	0.0310588
$580$	0	0
$581$	−3.29808e6	−0.405341
$582$	0	0
$583$	1.90350e7	2.31943
$584$	0	0
$585$	310068.	0.0374600
$586$	0	0
$587$	−603948.	−0.0723443	−0.0361721	−	0.999346i	$-0.511516\pi$
$587$	−603948.	−0.0723443	−0.0361721	+	0.999346i	$0.511516\pi$
$588$	0	0
$589$	−2.44640e6	−0.290562
$590$	0	0
$591$	−5.49941e6	−0.647661
$592$	0	0
$593$	−5.39077e6	−0.629526	−0.314763	−	0.949170i	$-0.601925\pi$
$593$	−5.39077e6	−0.629526	−0.314763	+	0.949170i	$0.601925\pi$
$594$	0	0
$595$	211680.	0.0245125
$596$	0	0
$597$	7.91129e6	0.908472
$598$	0	0
$599$	−4.27999e6	−0.487389	−0.243695	−	0.969852i	$-0.578359\pi$
$599$	−4.27999e6	−0.487389	−0.243695	+	0.969852i	$0.578359\pi$
$600$	0	0
$601$	1.02483e6	0.115735	0.0578674	−	0.998324i	$-0.481570\pi$
$601$	1.02483e6	0.115735	0.0578674	+	0.998324i	$0.481570\pi$
$602$	0	0
$603$	1.86851e6	0.209267
$604$	0	0
$605$	942270.	0.104661
$606$	0	0
$607$	−1.24342e7	−1.36976	−0.684882	−	0.728654i	$-0.740146\pi$
$607$	−1.24342e7	−1.36976	−0.684882	+	0.728654i	$0.740146\pi$
$608$	0	0
$609$	−1.66968e6	−0.182427
$610$	0	0
$611$	1.19127e7	1.29095
$612$	0	0
$613$	4.21506e6	0.453057	0.226528	−	0.974005i	$-0.427262\pi$
$613$	4.21506e6	0.453057	0.226528	+	0.974005i	$0.427262\pi$
$614$	0	0
$615$	370980.	0.0395515
$616$	0	0
$617$	−4.40665e6	−0.466010	−0.233005	−	0.972476i	$-0.574856\pi$
$617$	−4.40665e6	−0.466010	−0.233005	+	0.972476i	$0.574856\pi$
$618$	0	0
$619$	−4.80168e6	−0.503693	−0.251847	−	0.967767i	$-0.581038\pi$
$619$	−4.80168e6	−0.503693	−0.251847	+	0.967767i	$0.581038\pi$
$620$	0	0
$621$	−612360.	−0.0637203
$622$	0	0
$623$	−3.76344e6	−0.388477
$624$	0	0
$625$	9.42942e6	0.965573
$626$	0	0
$627$	−2.82226e6	−0.286700
$628$	0	0
$629$	−2.12562e6	−0.214220
$630$	0	0
$631$	−8.30727e6	−0.830587	−0.415293	−	0.909688i	$-0.636321\pi$
$631$	−8.30727e6	−0.830587	−0.415293	+	0.909688i	$0.636321\pi$
$632$	0	0
$633$	436500.	0.0432987
$634$	0	0
$635$	1.02653e6	0.101027
$636$	0	0
$637$	−9.70207e6	−0.947361
$638$	0	0
$639$	344088.	0.0333363
$640$	0	0
$641$	1.76956e7	1.70107	0.850534	−	0.525921i	$-0.176279\pi$
$641$	1.76956e7	1.70107	0.850534	+	0.525921i	$0.176279\pi$
$642$	0	0
$643$	1.28394e7	1.22466	0.612330	−	0.790602i	$-0.290232\pi$
$643$	1.28394e7	1.22466	0.612330	+	0.790602i	$0.290232\pi$
$644$	0	0
$645$	520776.	0.0492892
$646$	0	0
$647$	2.08468e7	1.95785	0.978924	−	0.204226i	$-0.0654678\pi$
$647$	2.08468e7	1.95785	0.978924	+	0.204226i	$0.0654678\pi$
$648$	0	0
$649$	1.01994e7	0.950521
$650$	0	0
$651$	1.58400e6	0.146488
$652$	0	0
$653$	1.29632e7	1.18968	0.594841	−	0.803843i	$-0.297215\pi$
$653$	1.29632e7	1.18968	0.594841	+	0.803843i	$0.297215\pi$
$654$	0	0
$655$	−1.55081e6	−0.141239
$656$	0	0
$657$	−3.32991e6	−0.300967
$658$	0	0
$659$	5.66862e6	0.508468	0.254234	−	0.967143i	$-0.418177\pi$
$659$	5.66862e6	0.508468	0.254234	+	0.967143i	$0.418177\pi$
$660$	0	0
$661$	−3.11430e6	−0.277240	−0.138620	−	0.990346i	$-0.544267\pi$
$661$	−3.11430e6	−0.277240	−0.138620	+	0.990346i	$0.544267\pi$
$662$	0	0
$663$	−5.06444e6	−0.447454
$664$	0	0
$665$	133440.	0.0117012
$666$	0	0
$667$	3.89592e6	0.339075
$668$	0	0
$669$	−8.99525e6	−0.777048
$670$	0	0
$671$	2.24235e7	1.92264
$672$	0	0
$673$	105890.	0.00901192	0.00450596	−	0.999990i	$-0.498566\pi$
$673$	105890.	0.00901192	0.00450596	+	0.999990i	$0.498566\pi$
$674$	0	0
$675$	2.25188e6	0.190233
$676$	0	0
$677$	−1.60910e7	−1.34931	−0.674656	−	0.738132i	$-0.735708\pi$
$677$	−1.60910e7	−1.34931	−0.674656	+	0.738132i	$0.735708\pi$
$678$	0	0
$679$	1.97768e6	0.164620
$680$	0	0
$681$	5.45562e6	0.450792
$682$	0	0
$683$	−1.60780e7	−1.31880	−0.659402	−	0.751791i	$-0.729190\pi$
$683$	−1.60780e7	−1.31880	−0.659402	+	0.751791i	$0.729190\pi$
$684$	0	0
$685$	1.80140e6	0.146685
$686$	0	0
$687$	−1.22393e7	−0.989386
$688$	0	0
$689$	2.15325e7	1.72801
$690$	0	0
$691$	165964.	0.0132227	0.00661133	−	0.999978i	$-0.497896\pi$
$691$	165964.	0.0132227	0.00661133	+	0.999978i	$0.497896\pi$
$692$	0	0
$693$	1.82736e6	0.144541
$694$	0	0
$695$	2.10098e6	0.164991
$696$	0	0
$697$	−6.05934e6	−0.472436
$698$	0	0
$699$	3.53597e6	0.273726
$700$	0	0
$701$	1.77248e7	1.36234	0.681171	−	0.732124i	$-0.261471\pi$
$701$	1.77248e7	1.36234	0.681171	+	0.732124i	$0.261471\pi$
$702$	0	0
$703$	−1.33996e6	−0.102259
$704$	0	0
$705$	−1.00829e6	−0.0764032
$706$	0	0
$707$	−5.72136e6	−0.430478
$708$	0	0
$709$	−1.06023e7	−0.792112	−0.396056	−	0.918226i	$-0.629621\pi$
$709$	−1.06023e7	−0.792112	−0.396056	+	0.918226i	$0.629621\pi$
$710$	0	0
$711$	−1.77552e6	−0.131720
$712$	0	0
$713$	−3.69600e6	−0.272275
$714$	0	0
$715$	2.15899e6	0.157938
$716$	0	0
$717$	−1.19262e7	−0.866373
$718$	0	0
$719$	−9.03211e6	−0.651579	−0.325790	−	0.945442i	$-0.605630\pi$
$719$	−9.03211e6	−0.651579	−0.325790	+	0.945442i	$0.605630\pi$
$720$	0	0
$721$	−2.12576e6	−0.152292
$722$	0	0
$723$	8.91085e6	0.633977
$724$	0	0
$725$	−1.43268e7	−1.01229
$726$	0	0
$727$	−1.87575e7	−1.31625	−0.658127	−	0.752907i	$-0.728651\pi$
$727$	−1.87575e7	−1.31625	−0.658127	+	0.752907i	$0.728651\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−8.50601e6	−0.588752
$732$	0	0
$733$	−1.17773e7	−0.809626	−0.404813	−	0.914399i	$-0.632663\pi$
$733$	−1.17773e7	−0.809626	−0.404813	+	0.914399i	$0.632663\pi$
$734$	0	0
$735$	821178.	0.0560685
$736$	0	0
$737$	1.30104e7	0.882308
$738$	0	0
$739$	−5.88948e6	−0.396703	−0.198352	−	0.980131i	$-0.563559\pi$
$739$	−5.88948e6	−0.396703	−0.198352	+	0.980131i	$0.563559\pi$
$740$	0	0
$741$	−3.19255e6	−0.213596
$742$	0	0
$743$	1.00476e7	0.667712	0.333856	−	0.942624i	$-0.391650\pi$
$743$	1.00476e7	0.667712	0.333856	+	0.942624i	$0.391650\pi$
$744$	0	0
$745$	−631548.	−0.0416884
$746$	0	0
$747$	−6.67861e6	−0.437910
$748$	0	0
$749$	−3.63312e6	−0.236633
$750$	0	0
$751$	−4.81530e6	−0.311547	−0.155773	−	0.987793i	$-0.549787\pi$
$751$	−4.81530e6	−0.311547	−0.155773	+	0.987793i	$0.549787\pi$
$752$	0	0
$753$	1.32419e6	0.0851064
$754$	0	0
$755$	−2.37835e6	−0.151848
$756$	0	0
$757$	3.12973e6	0.198503	0.0992516	−	0.995062i	$-0.468355\pi$
$757$	3.12973e6	0.198503	0.0992516	+	0.995062i	$0.468355\pi$
$758$	0	0
$759$	−4.26384e6	−0.268656
$760$	0	0
$761$	−1.17773e7	−0.737197	−0.368599	−	0.929589i	$-0.620162\pi$
$761$	−1.17773e7	−0.737197	−0.368599	+	0.929589i	$0.620162\pi$
$762$	0	0
$763$	−2.46664e6	−0.153389
$764$	0	0
$765$	428652.	0.0264820
$766$	0	0
$767$	1.15376e7	0.708152
$768$	0	0
$769$	−1.49376e6	−0.0910887	−0.0455443	−	0.998962i	$-0.514502\pi$
$769$	−1.49376e6	−0.0910887	−0.0455443	+	0.998962i	$0.514502\pi$
$770$	0	0
$771$	4.35224e6	0.263680
$772$	0	0
$773$	−2.25125e7	−1.35511	−0.677555	−	0.735472i	$-0.736960\pi$
$773$	−2.25125e7	−1.35511	−0.677555	+	0.735472i	$0.736960\pi$
$774$	0	0
$775$	1.35916e7	0.812861
$776$	0	0
$777$	867600.	0.0515545
$778$	0	0
$779$	−3.81972e6	−0.225521
$780$	0	0
$781$	2.39587e6	0.140552
$782$	0	0
$783$	−3.38110e6	−0.197085
$784$	0	0
$785$	−826476.	−0.0478692
$786$	0	0
$787$	−1.19547e7	−0.688022	−0.344011	−	0.938966i	$-0.611786\pi$
$787$	−1.19547e7	−0.688022	−0.344011	+	0.938966i	$0.611786\pi$
$788$	0	0
$789$	7.32218e6	0.418744
$790$	0	0
$791$	419280.	0.0238266
$792$	0	0
$793$	2.53656e7	1.43239
$794$	0	0
$795$	−1.82250e6	−0.102270
$796$	0	0
$797$	540798.	0.0301571	0.0150785	−	0.999886i	$-0.495200\pi$
$797$	540798.	0.0301571	0.0150785	+	0.999886i	$0.495200\pi$
$798$	0	0
$799$	1.64687e7	0.912625
$800$	0	0
$801$	−7.62097e6	−0.419690
$802$	0	0
$803$	−2.31860e7	−1.26893
$804$	0	0
$805$	201600.	0.0109648
$806$	0	0
$807$	4.14995e6	0.224316
$808$	0	0
$809$	−6.14223e6	−0.329955	−0.164978	−	0.986297i	$-0.552755\pi$
$809$	−6.14223e6	−0.329955	−0.164978	+	0.986297i	$0.552755\pi$
$810$	0	0
$811$	3.16734e7	1.69100	0.845499	−	0.533977i	$-0.179303\pi$
$811$	3.16734e7	1.69100	0.845499	+	0.533977i	$0.179303\pi$
$812$	0	0
$813$	1.50762e7	0.799956
$814$	0	0
$815$	−2.11606e6	−0.111592
$816$	0	0
$817$	−5.36206e6	−0.281046
$818$	0	0
$819$	2.06712e6	0.107685
$820$	0	0
$821$	2.66175e7	1.37819	0.689095	−	0.724671i	$-0.258008\pi$
$821$	2.66175e7	1.37819	0.689095	+	0.724671i	$0.258008\pi$
$822$	0	0
$823$	−3.62817e7	−1.86719	−0.933593	−	0.358335i	$-0.883345\pi$
$823$	−3.62817e7	−1.86719	−0.933593	+	0.358335i	$0.883345\pi$
$824$	0	0
$825$	1.56798e7	0.802056
$826$	0	0
$827$	−1.09033e6	−0.0554364	−0.0277182	−	0.999616i	$-0.508824\pi$
$827$	−1.09033e6	−0.0554364	−0.0277182	+	0.999616i	$0.508824\pi$
$828$	0	0
$829$	−1.03016e7	−0.520620	−0.260310	−	0.965525i	$-0.583825\pi$
$829$	−1.03016e7	−0.520620	−0.260310	+	0.965525i	$0.583825\pi$
$830$	0	0
$831$	−3.61013e6	−0.181351
$832$	0	0
$833$	−1.34126e7	−0.669730
$834$	0	0
$835$	1.30536e6	0.0647909
$836$	0	0
$837$	3.20760e6	0.158258
$838$	0	0
$839$	1.96134e7	0.961940	0.480970	−	0.876737i	$-0.340285\pi$
$839$	1.96134e7	0.961940	0.480970	+	0.876737i	$0.340285\pi$
$840$	0	0
$841$	999895.	0.0487489
$842$	0	0
$843$	2.07879e7	1.00749
$844$	0	0
$845$	214506.	0.0103347
$846$	0	0
$847$	6.28180e6	0.300868
$848$	0	0
$849$	−1.01589e7	−0.483704
$850$	0	0
$851$	−2.02440e6	−0.0958236
$852$	0	0
$853$	3.27565e7	1.54143	0.770717	−	0.637178i	$-0.219898\pi$
$853$	3.27565e7	1.54143	0.770717	+	0.637178i	$0.219898\pi$
$854$	0	0
$855$	270216.	0.0126414
$856$	0	0
$857$	−2.57953e7	−1.19974	−0.599872	−	0.800096i	$-0.704782\pi$
$857$	−2.57953e7	−1.19974	−0.599872	+	0.800096i	$0.704782\pi$
$858$	0	0
$859$	1.98548e7	0.918085	0.459043	−	0.888414i	$-0.348193\pi$
$859$	1.98548e7	0.918085	0.459043	+	0.888414i	$0.348193\pi$
$860$	0	0
$861$	2.47320e6	0.113698
$862$	0	0
$863$	673056.	0.0307627	0.0153813	−	0.999882i	$-0.495104\pi$
$863$	673056.	0.0307627	0.0153813	+	0.999882i	$0.495104\pi$
$864$	0	0
$865$	−982188.	−0.0446328
$866$	0	0
$867$	5.77740e6	0.261026
$868$	0	0
$869$	−1.23629e7	−0.555354
$870$	0	0
$871$	1.47174e7	0.657333
$872$	0	0
$873$	4.00480e6	0.177847
$874$	0	0
$875$	−1.49136e6	−0.0658510
$876$	0	0
$877$	5.32115e6	0.233618	0.116809	−	0.993154i	$-0.462733\pi$
$877$	5.32115e6	0.233618	0.116809	+	0.993154i	$0.462733\pi$
$878$	0	0
$879$	8.44987e6	0.368874
$880$	0	0
$881$	2.78891e7	1.21058	0.605291	−	0.796004i	$-0.293057\pi$
$881$	2.78891e7	1.21058	0.605291	+	0.796004i	$0.293057\pi$
$882$	0	0
$883$	2.83786e7	1.22487	0.612435	−	0.790521i	$-0.290190\pi$
$883$	2.83786e7	1.22487	0.612435	+	0.790521i	$0.290190\pi$
$884$	0	0
$885$	−976536.	−0.0419112
$886$	0	0
$887$	−4.22678e7	−1.80385	−0.901925	−	0.431893i	$-0.857846\pi$
$887$	−4.22678e7	−1.80385	−0.901925	+	0.431893i	$0.857846\pi$
$888$	0	0
$889$	6.84352e6	0.290419
$890$	0	0
$891$	3.70040e6	0.156155
$892$	0	0
$893$	1.03816e7	0.435649
$894$	0	0
$895$	−2.15244e6	−0.0898201
$896$	0	0
$897$	−4.82328e6	−0.200153
$898$	0	0
$899$	−2.04072e7	−0.842140
$900$	0	0
$901$	2.97675e7	1.22160
$902$	0	0
$903$	3.47184e6	0.141690
$904$	0	0
$905$	−3.04278e6	−0.123495
$906$	0	0
$907$	−3.19526e7	−1.28970	−0.644849	−	0.764310i	$-0.723080\pi$
$907$	−3.19526e7	−1.28970	−0.644849	+	0.764310i	$0.723080\pi$
$908$	0	0
$909$	−1.15858e7	−0.465066
$910$	0	0
$911$	1.16429e7	0.464800	0.232400	−	0.972620i	$-0.425342\pi$
$911$	1.16429e7	0.464800	0.232400	+	0.972620i	$0.425342\pi$
$912$	0	0
$913$	−4.65029e7	−1.84630
$914$	0	0
$915$	−2.14693e6	−0.0847746
$916$	0	0
$917$	−1.03387e7	−0.406016
$918$	0	0
$919$	−1.39844e6	−0.0546204	−0.0273102	−	0.999627i	$-0.508694\pi$
$919$	−1.39844e6	−0.0546204	−0.0273102	+	0.999627i	$0.508694\pi$
$920$	0	0
$921$	6.23653e6	0.242267
$922$	0	0
$923$	2.71022e6	0.104713
$924$	0	0
$925$	7.44449e6	0.286075
$926$	0	0
$927$	−4.30466e6	−0.164528
$928$	0	0
$929$	−1.66792e7	−0.634067	−0.317033	−	0.948414i	$-0.602687\pi$
$929$	−1.66792e7	−0.634067	−0.317033	+	0.948414i	$0.602687\pi$
$930$	0	0
$931$	−8.45509e6	−0.319701
$932$	0	0
$933$	2.64879e7	0.996191
$934$	0	0
$935$	2.98469e6	0.111653
$936$	0	0
$937$	−2.47956e7	−0.922625	−0.461312	−	0.887238i	$-0.652621\pi$
$937$	−2.47956e7	−0.922625	−0.461312	+	0.887238i	$0.652621\pi$
$938$	0	0
$939$	−7.96631e6	−0.294845
$940$	0	0
$941$	2.79574e7	1.02925	0.514627	−	0.857414i	$-0.327930\pi$
$941$	2.79574e7	1.02925	0.514627	+	0.857414i	$0.327930\pi$
$942$	0	0
$943$	−5.77080e6	−0.211328
$944$	0	0
$945$	−174960.	−0.00637322
$946$	0	0
$947$	−7.64936e6	−0.277173	−0.138586	−	0.990350i	$-0.544256\pi$
$947$	−7.64936e6	−0.277173	−0.138586	+	0.990350i	$0.544256\pi$
$948$	0	0
$949$	−2.62282e7	−0.945372
$950$	0	0
$951$	−2.25792e7	−0.809575
$952$	0	0
$953$	−4.62179e7	−1.64846	−0.824228	−	0.566257i	$-0.808391\pi$
$953$	−4.62179e7	−1.64846	−0.824228	+	0.566257i	$0.808391\pi$
$954$	0	0
$955$	3.89030e6	0.138031
$956$	0	0
$957$	−2.35425e7	−0.830945
$958$	0	0
$959$	1.20094e7	0.421671
$960$	0	0
$961$	−9.26915e6	−0.323766
$962$	0	0
$963$	−7.35707e6	−0.255646
$964$	0	0
$965$	−167028.	−0.00577392
$966$	0	0
$967$	−2.08557e7	−0.717229	−0.358615	−	0.933486i	$-0.616751\pi$
$967$	−2.08557e7	−0.717229	−0.358615	+	0.933486i	$0.616751\pi$
$968$	0	0
$969$	−4.41353e6	−0.151000
$970$	0	0
$971$	4.58152e7	1.55941	0.779707	−	0.626144i	$-0.215368\pi$
$971$	4.58152e7	1.55941	0.779707	+	0.626144i	$0.215368\pi$
$972$	0	0
$973$	1.40066e7	0.474296
$974$	0	0
$975$	1.77370e7	0.597544
$976$	0	0
$977$	−1.09544e6	−0.0367157	−0.0183578	−	0.999831i	$-0.505844\pi$
$977$	−1.09544e6	−0.0367157	−0.0183578	+	0.999831i	$0.505844\pi$
$978$	0	0
$979$	−5.30645e7	−1.76949
$980$	0	0
$981$	−4.99495e6	−0.165714
$982$	0	0
$983$	−5.25817e7	−1.73561	−0.867803	−	0.496909i	$-0.834468\pi$
$983$	−5.25817e7	−1.73561	−0.867803	+	0.496909i	$0.834468\pi$
$984$	0	0
$985$	3.66628e6	0.120402
$986$	0	0
$987$	−6.72192e6	−0.219634
$988$	0	0
$989$	−8.10096e6	−0.263358
$990$	0	0
$991$	4.90389e7	1.58620	0.793098	−	0.609094i	$-0.208467\pi$
$991$	4.90389e7	1.58620	0.793098	+	0.609094i	$0.208467\pi$
$992$	0	0
$993$	−1.94533e6	−0.0626067
$994$	0	0
$995$	−5.27419e6	−0.168888
$996$	0	0
$997$	3.05461e6	0.0973237	0.0486618	−	0.998815i	$-0.484504\pi$
$997$	3.05461e6	0.0973237	0.0486618	+	0.998815i	$0.484504\pi$
$998$	0	0
$999$	1.75689e6	0.0556969

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.6.a.a.1.1		1
3.2	odd	2		144.6.a.f.1.1		1
4.3	odd	2		3.6.a.a.1.1	✓	1
8.3	odd	2		192.6.a.d.1.1		1
8.5	even	2		192.6.a.l.1.1		1
12.11	even	2		9.6.a.a.1.1		1
16.3	odd	4		768.6.d.k.385.2		2
16.5	even	4		768.6.d.h.385.2		2
16.11	odd	4		768.6.d.k.385.1		2
16.13	even	4		768.6.d.h.385.1		2
20.3	even	4		75.6.b.b.49.2		2
20.7	even	4		75.6.b.b.49.1		2
20.19	odd	2		75.6.a.e.1.1		1
24.5	odd	2		576.6.a.t.1.1		1
24.11	even	2		576.6.a.s.1.1		1
28.3	even	6		147.6.e.k.79.1		2
28.11	odd	6		147.6.e.h.79.1		2
28.19	even	6		147.6.e.k.67.1		2
28.23	odd	6		147.6.e.h.67.1		2
28.27	even	2		147.6.a.a.1.1		1
36.7	odd	6		81.6.c.c.28.1		2
36.11	even	6		81.6.c.a.28.1		2
36.23	even	6		81.6.c.a.55.1		2
36.31	odd	6		81.6.c.c.55.1		2
44.43	even	2		363.6.a.d.1.1		1
52.51	odd	2		507.6.a.b.1.1		1
60.23	odd	4		225.6.b.b.199.1		2
60.47	odd	4		225.6.b.b.199.2		2
60.59	even	2		225.6.a.a.1.1		1
68.67	odd	2		867.6.a.a.1.1		1
76.75	even	2		1083.6.a.c.1.1		1
84.83	odd	2		441.6.a.i.1.1		1
132.131	odd	2		1089.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.6.a.a.1.1	✓	1	4.3	odd	2
9.6.a.a.1.1		1	12.11	even	2
48.6.a.a.1.1		1	1.1	even	1	trivial
75.6.a.e.1.1		1	20.19	odd	2
75.6.b.b.49.1		2	20.7	even	4
75.6.b.b.49.2		2	20.3	even	4
81.6.c.a.28.1		2	36.11	even	6
81.6.c.a.55.1		2	36.23	even	6
81.6.c.c.28.1		2	36.7	odd	6
81.6.c.c.55.1		2	36.31	odd	6
144.6.a.f.1.1		1	3.2	odd	2
147.6.a.a.1.1		1	28.27	even	2
147.6.e.h.67.1		2	28.23	odd	6
147.6.e.h.79.1		2	28.11	odd	6
147.6.e.k.67.1		2	28.19	even	6
147.6.e.k.79.1		2	28.3	even	6
192.6.a.d.1.1		1	8.3	odd	2
192.6.a.l.1.1		1	8.5	even	2
225.6.a.a.1.1		1	60.59	even	2
225.6.b.b.199.1		2	60.23	odd	4
225.6.b.b.199.2		2	60.47	odd	4
363.6.a.d.1.1		1	44.43	even	2
441.6.a.i.1.1		1	84.83	odd	2
507.6.a.b.1.1		1	52.51	odd	2
576.6.a.s.1.1		1	24.11	even	2
576.6.a.t.1.1		1	24.5	odd	2
768.6.d.h.385.1		2	16.13	even	4
768.6.d.h.385.2		2	16.5	even	4
768.6.d.k.385.1		2	16.11	odd	4
768.6.d.k.385.2		2	16.3	odd	4
867.6.a.a.1.1		1	68.67	odd	2
1083.6.a.c.1.1		1	76.75	even	2
1089.6.a.b.1.1		1	132.131	odd	2

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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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