LMFDB - Embedded newform 336.6.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,Mod(1,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("336.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	336.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:	$53.8889634572$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	336.1

$q$ -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} -38.0000 q^{5} +49.0000 q^{7} +81.0000 q^{9} -600.000 q^{11} -674.000 q^{13} +342.000 q^{15} +78.0000 q^{17} +916.000 q^{19} -441.000 q^{21} +4604.00 q^{23} -1681.00 q^{25} -729.000 q^{27} -6810.00 q^{29} -7912.00 q^{31} +5400.00 q^{33} -1862.00 q^{35} -9274.00 q^{37} +6066.00 q^{39} -242.000 q^{41} -1116.00 q^{43} -3078.00 q^{45} +28312.0 q^{47} +2401.00 q^{49} -702.000 q^{51} +10230.0 q^{53} +22800.0 q^{55} -8244.00 q^{57} +4108.00 q^{59} +15878.0 q^{61} +3969.00 q^{63} +25612.0 q^{65} +67668.0 q^{67} -41436.0 q^{69} +67492.0 q^{71} +1106.00 q^{73} +15129.0 q^{75} -29400.0 q^{77} -84152.0 q^{79} +6561.00 q^{81} +2908.00 q^{83} -2964.00 q^{85} +61290.0 q^{87} -8322.00 q^{89} -33026.0 q^{91} +71208.0 q^{93} -34808.0 q^{95} +130810. q^{97} -48600.0 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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−38.0000	−0.679765	−0.339882	−	0.940468i	$-0.610387\pi$
$5$	−38.0000	−0.679765	−0.339882	+	0.940468i	$0.610387\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−600.000	−1.49510	−0.747549	−	0.664207i	$-0.768769\pi$
$11$	−600.000	−1.49510	−0.747549	+	0.664207i	$0.768769\pi$
$12$	0	0
$13$	−674.000	−1.10612	−0.553059	−	0.833142i	$-0.686540\pi$
$13$	−674.000	−1.10612	−0.553059	+	0.833142i	$0.686540\pi$
$14$	0	0
$15$	342.000	0.392462
$16$	0	0
$17$	78.0000	0.0654594	0.0327297	−	0.999464i	$-0.489580\pi$
$17$	78.0000	0.0654594	0.0327297	+	0.999464i	$0.489580\pi$
$18$	0	0
$19$	916.000	0.582119	0.291059	−	0.956705i	$-0.405992\pi$
$19$	916.000	0.582119	0.291059	+	0.956705i	$0.405992\pi$
$20$	0	0
$21$	−441.000	−0.218218
$22$	0	0
$23$	4604.00	1.81475	0.907373	−	0.420327i	$-0.138085\pi$
$23$	4604.00	1.81475	0.907373	+	0.420327i	$0.138085\pi$
$24$	0	0
$25$	−1681.00	−0.537920
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−6810.00	−1.50367	−0.751834	−	0.659352i	$-0.770831\pi$
$29$	−6810.00	−1.50367	−0.751834	+	0.659352i	$0.770831\pi$
$30$	0	0
$31$	−7912.00	−1.47871	−0.739353	−	0.673318i	$-0.764869\pi$
$31$	−7912.00	−1.47871	−0.739353	+	0.673318i	$0.764869\pi$
$32$	0	0
$33$	5400.00	0.863195
$34$	0	0
$35$	−1862.00	−0.256927
$36$	0	0
$37$	−9274.00	−1.11369	−0.556843	−	0.830618i	$-0.687988\pi$
$37$	−9274.00	−1.11369	−0.556843	+	0.830618i	$0.687988\pi$
$38$	0	0
$39$	6066.00	0.638618
$40$	0	0
$41$	−242.000	−0.0224831	−0.0112415	−	0.999937i	$-0.503578\pi$
$41$	−242.000	−0.0224831	−0.0112415	+	0.999937i	$0.503578\pi$
$42$	0	0
$43$	−1116.00	−0.0920435	−0.0460217	−	0.998940i	$-0.514654\pi$
$43$	−1116.00	−0.0920435	−0.0460217	+	0.998940i	$0.514654\pi$
$44$	0	0
$45$	−3078.00	−0.226588
$46$	0	0
$47$	28312.0	1.86950	0.934751	−	0.355304i	$-0.115623\pi$
$47$	28312.0	1.86950	0.934751	+	0.355304i	$0.115623\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−702.000	−0.0377930
$52$	0	0
$53$	10230.0	0.500249	0.250124	−	0.968214i	$-0.419528\pi$
$53$	10230.0	0.500249	0.250124	+	0.968214i	$0.419528\pi$
$54$	0	0
$55$	22800.0	1.01631
$56$	0	0
$57$	−8244.00	−0.336086
$58$	0	0
$59$	4108.00	0.153639	0.0768193	−	0.997045i	$-0.475524\pi$
$59$	4108.00	0.153639	0.0768193	+	0.997045i	$0.475524\pi$
$60$	0	0
$61$	15878.0	0.546350	0.273175	−	0.961964i	$-0.411926\pi$
$61$	15878.0	0.546350	0.273175	+	0.961964i	$0.411926\pi$
$62$	0	0
$63$	3969.00	0.125988
$64$	0	0
$65$	25612.0	0.751900
$66$	0	0
$67$	67668.0	1.84160	0.920802	−	0.390030i	$-0.127535\pi$
$67$	67668.0	1.84160	0.920802	+	0.390030i	$0.127535\pi$
$68$	0	0
$69$	−41436.0	−1.04774
$70$	0	0
$71$	67492.0	1.58894	0.794468	−	0.607306i	$-0.207750\pi$
$71$	67492.0	1.58894	0.794468	+	0.607306i	$0.207750\pi$
$72$	0	0
$73$	1106.00	0.0242911	0.0121456	−	0.999926i	$-0.496134\pi$
$73$	1106.00	0.0242911	0.0121456	+	0.999926i	$0.496134\pi$
$74$	0	0
$75$	15129.0	0.310568
$76$	0	0
$77$	−29400.0	−0.565094
$78$	0	0
$79$	−84152.0	−1.51704	−0.758519	−	0.651650i	$-0.774077\pi$
$79$	−84152.0	−1.51704	−0.758519	+	0.651650i	$0.774077\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	2908.00	0.0463339	0.0231670	−	0.999732i	$-0.492625\pi$
$83$	2908.00	0.0463339	0.0231670	+	0.999732i	$0.492625\pi$
$84$	0	0
$85$	−2964.00	−0.0444970
$86$	0	0
$87$	61290.0	0.868143
$88$	0	0
$89$	−8322.00	−0.111366	−0.0556830	−	0.998448i	$-0.517734\pi$
$89$	−8322.00	−0.111366	−0.0556830	+	0.998448i	$0.517734\pi$
$90$	0	0
$91$	−33026.0	−0.418073
$92$	0	0
$93$	71208.0	0.853732
$94$	0	0
$95$	−34808.0	−0.395704
$96$	0	0
$97$	130810.	1.41160	0.705800	−	0.708411i	$-0.250588\pi$
$97$	130810.	1.41160	0.705800	+	0.708411i	$0.250588\pi$
$98$	0	0
$99$	−48600.0	−0.498366
$100$	0	0
$101$	159794.	1.55868	0.779340	−	0.626601i	$-0.215554\pi$
$101$	159794.	1.55868	0.779340	+	0.626601i	$0.215554\pi$
$102$	0	0
$103$	91440.0	0.849265	0.424632	−	0.905366i	$-0.360403\pi$
$103$	91440.0	0.849265	0.424632	+	0.905366i	$0.360403\pi$
$104$	0	0
$105$	16758.0	0.148337
$106$	0	0
$107$	139480.	1.17775	0.588874	−	0.808225i	$-0.299571\pi$
$107$	139480.	1.17775	0.588874	+	0.808225i	$0.299571\pi$
$108$	0	0
$109$	141758.	1.14283	0.571415	−	0.820662i	$-0.306395\pi$
$109$	141758.	1.14283	0.571415	+	0.820662i	$0.306395\pi$
$110$	0	0
$111$	83466.0	0.642987
$112$	0	0
$113$	−81742.0	−0.602212	−0.301106	−	0.953591i	$-0.597356\pi$
$113$	−81742.0	−0.602212	−0.301106	+	0.953591i	$0.597356\pi$
$114$	0	0
$115$	−174952.	−1.23360
$116$	0	0
$117$	−54594.0	−0.368706
$118$	0	0
$119$	3822.00	0.0247413
$120$	0	0
$121$	198949.	1.23532
$122$	0	0
$123$	2178.00	0.0129806
$124$	0	0
$125$	182628.	1.04542
$126$	0	0
$127$	−90488.0	−0.497831	−0.248915	−	0.968525i	$-0.580074\pi$
$127$	−90488.0	−0.497831	−0.248915	+	0.968525i	$0.580074\pi$
$128$	0	0
$129$	10044.0	0.0531413
$130$	0	0
$131$	−187476.	−0.954481	−0.477241	−	0.878773i	$-0.658363\pi$
$131$	−187476.	−0.954481	−0.477241	+	0.878773i	$0.658363\pi$
$132$	0	0
$133$	44884.0	0.220020
$134$	0	0
$135$	27702.0	0.130821
$136$	0	0
$137$	168482.	0.766924	0.383462	−	0.923557i	$-0.374732\pi$
$137$	168482.	0.766924	0.383462	+	0.923557i	$0.374732\pi$
$138$	0	0
$139$	−668.000	−0.00293251	−0.00146625	−	0.999999i	$-0.500467\pi$
$139$	−668.000	−0.00293251	−0.00146625	+	0.999999i	$0.500467\pi$
$140$	0	0
$141$	−254808.	−1.07936
$142$	0	0
$143$	404400.	1.65375
$144$	0	0
$145$	258780.	1.02214
$146$	0	0
$147$	−21609.0	−0.0824786
$148$	0	0
$149$	−300618.	−1.10930	−0.554650	−	0.832084i	$-0.687148\pi$
$149$	−300618.	−1.10930	−0.554650	+	0.832084i	$0.687148\pi$
$150$	0	0
$151$	−359032.	−1.28142	−0.640709	−	0.767784i	$-0.721359\pi$
$151$	−359032.	−1.28142	−0.640709	+	0.767784i	$0.721359\pi$
$152$	0	0
$153$	6318.00	0.0218198
$154$	0	0
$155$	300656.	1.00517
$156$	0	0
$157$	110822.	0.358820	0.179410	−	0.983774i	$-0.442581\pi$
$157$	110822.	0.358820	0.179410	+	0.983774i	$0.442581\pi$
$158$	0	0
$159$	−92070.0	−0.288819
$160$	0	0
$161$	225596.	0.685909
$162$	0	0
$163$	−372380.	−1.09779	−0.548893	−	0.835893i	$-0.684950\pi$
$163$	−372380.	−1.09779	−0.548893	+	0.835893i	$0.684950\pi$
$164$	0	0
$165$	−205200.	−0.586769
$166$	0	0
$167$	109600.	0.304102	0.152051	−	0.988373i	$-0.451412\pi$
$167$	109600.	0.304102	0.152051	+	0.988373i	$0.451412\pi$
$168$	0	0
$169$	82983.0	0.223497
$170$	0	0
$171$	74196.0	0.194040
$172$	0	0
$173$	−388598.	−0.987155	−0.493577	−	0.869702i	$-0.664311\pi$
$173$	−388598.	−0.987155	−0.493577	+	0.869702i	$0.664311\pi$
$174$	0	0
$175$	−82369.0	−0.203315
$176$	0	0
$177$	−36972.0	−0.0887033
$178$	0	0
$179$	241656.	0.563722	0.281861	−	0.959455i	$-0.409048\pi$
$179$	241656.	0.563722	0.281861	+	0.959455i	$0.409048\pi$
$180$	0	0
$181$	190854.	0.433017	0.216508	−	0.976281i	$-0.430533\pi$
$181$	190854.	0.433017	0.216508	+	0.976281i	$0.430533\pi$
$182$	0	0
$183$	−142902.	−0.315436
$184$	0	0
$185$	352412.	0.757044
$186$	0	0
$187$	−46800.0	−0.0978683
$188$	0	0
$189$	−35721.0	−0.0727393
$190$	0	0
$191$	−253300.	−0.502402	−0.251201	−	0.967935i	$-0.580826\pi$
$191$	−253300.	−0.502402	−0.251201	+	0.967935i	$0.580826\pi$
$192$	0	0
$193$	−712302.	−1.37648	−0.688242	−	0.725482i	$-0.741617\pi$
$193$	−712302.	−1.37648	−0.688242	+	0.725482i	$0.741617\pi$
$194$	0	0
$195$	−230508.	−0.434110
$196$	0	0
$197$	−107194.	−0.196791	−0.0983954	−	0.995147i	$-0.531371\pi$
$197$	−107194.	−0.196791	−0.0983954	+	0.995147i	$0.531371\pi$
$198$	0	0
$199$	956952.	1.71300	0.856500	−	0.516147i	$-0.172634\pi$
$199$	956952.	1.71300	0.856500	+	0.516147i	$0.172634\pi$
$200$	0	0
$201$	−609012.	−1.06325
$202$	0	0
$203$	−333690.	−0.568333
$204$	0	0
$205$	9196.00	0.0152832
$206$	0	0
$207$	372924.	0.604915
$208$	0	0
$209$	−549600.	−0.870324
$210$	0	0
$211$	320700.	0.495899	0.247949	−	0.968773i	$-0.420243\pi$
$211$	320700.	0.495899	0.247949	+	0.968773i	$0.420243\pi$
$212$	0	0
$213$	−607428.	−0.917373
$214$	0	0
$215$	42408.0	0.0625679
$216$	0	0
$217$	−387688.	−0.558899
$218$	0	0
$219$	−9954.00	−0.0140245
$220$	0	0
$221$	−52572.0	−0.0724059
$222$	0	0
$223$	−1.06840e6	−1.43870	−0.719352	−	0.694645i	$-0.755561\pi$
$223$	−1.06840e6	−1.43870	−0.719352	+	0.694645i	$0.755561\pi$
$224$	0	0
$225$	−136161.	−0.179307
$226$	0	0
$227$	806820.	1.03923	0.519615	−	0.854400i	$-0.326075\pi$
$227$	806820.	1.03923	0.519615	+	0.854400i	$0.326075\pi$
$228$	0	0
$229$	1.21661e6	1.53308	0.766539	−	0.642198i	$-0.221977\pi$
$229$	1.21661e6	1.53308	0.766539	+	0.642198i	$0.221977\pi$
$230$	0	0
$231$	264600.	0.326257
$232$	0	0
$233$	214722.	0.259112	0.129556	−	0.991572i	$-0.458645\pi$
$233$	214722.	0.259112	0.129556	+	0.991572i	$0.458645\pi$
$234$	0	0
$235$	−1.07586e6	−1.27082
$236$	0	0
$237$	757368.	0.875863
$238$	0	0
$239$	615780.	0.697318	0.348659	−	0.937250i	$-0.386637\pi$
$239$	615780.	0.697318	0.348659	+	0.937250i	$0.386637\pi$
$240$	0	0
$241$	−995590.	−1.10418	−0.552088	−	0.833786i	$-0.686169\pi$
$241$	−995590.	−1.10418	−0.552088	+	0.833786i	$0.686169\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−91238.0	−0.0971092
$246$	0	0
$247$	−617384.	−0.643892
$248$	0	0
$249$	−26172.0	−0.0267509
$250$	0	0
$251$	169340.	0.169658	0.0848292	−	0.996396i	$-0.472966\pi$
$251$	169340.	0.169658	0.0848292	+	0.996396i	$0.472966\pi$
$252$	0	0
$253$	−2.76240e6	−2.71322
$254$	0	0
$255$	26676.0	0.0256904
$256$	0	0
$257$	1.67825e6	1.58498	0.792488	−	0.609887i	$-0.208785\pi$
$257$	1.67825e6	1.58498	0.792488	+	0.609887i	$0.208785\pi$
$258$	0	0
$259$	−454426.	−0.420934
$260$	0	0
$261$	−551610.	−0.501223
$262$	0	0
$263$	1.77507e6	1.58243	0.791217	−	0.611535i	$-0.209448\pi$
$263$	1.77507e6	1.58243	0.791217	+	0.611535i	$0.209448\pi$
$264$	0	0
$265$	−388740.	−0.340051
$266$	0	0
$267$	74898.0	0.0642972
$268$	0	0
$269$	−15646.0	−0.0131833	−0.00659163	−	0.999978i	$-0.502098\pi$
$269$	−15646.0	−0.0131833	−0.00659163	+	0.999978i	$0.502098\pi$
$270$	0	0
$271$	635672.	0.525787	0.262894	−	0.964825i	$-0.415323\pi$
$271$	635672.	0.525787	0.262894	+	0.964825i	$0.415323\pi$
$272$	0	0
$273$	297234.	0.241375
$274$	0	0
$275$	1.00860e6	0.804243
$276$	0	0
$277$	−92218.0	−0.0722131	−0.0361066	−	0.999348i	$-0.511496\pi$
$277$	−92218.0	−0.0722131	−0.0361066	+	0.999348i	$0.511496\pi$
$278$	0	0
$279$	−640872.	−0.492902
$280$	0	0
$281$	−1.15678e6	−0.873948	−0.436974	−	0.899474i	$-0.643950\pi$
$281$	−1.15678e6	−0.873948	−0.436974	+	0.899474i	$0.643950\pi$
$282$	0	0
$283$	−1.97380e6	−1.46500	−0.732498	−	0.680770i	$-0.761645\pi$
$283$	−1.97380e6	−1.46500	−0.732498	+	0.680770i	$0.761645\pi$
$284$	0	0
$285$	313272.	0.228460
$286$	0	0
$287$	−11858.0	−0.00849780
$288$	0	0
$289$	−1.41377e6	−0.995715
$290$	0	0
$291$	−1.17729e6	−0.814988
$292$	0	0
$293$	−113966.	−0.0775544	−0.0387772	−	0.999248i	$-0.512346\pi$
$293$	−113966.	−0.0775544	−0.0387772	+	0.999248i	$0.512346\pi$
$294$	0	0
$295$	−156104.	−0.104438
$296$	0	0
$297$	437400.	0.287732
$298$	0	0
$299$	−3.10310e6	−2.00732
$300$	0	0
$301$	−54684.0	−0.0347892
$302$	0	0
$303$	−1.43815e6	−0.899904
$304$	0	0
$305$	−603364.	−0.371390
$306$	0	0
$307$	−2.18500e6	−1.32314	−0.661571	−	0.749883i	$-0.730110\pi$
$307$	−2.18500e6	−1.32314	−0.661571	+	0.749883i	$0.730110\pi$
$308$	0	0
$309$	−822960.	−0.490323
$310$	0	0
$311$	365520.	0.214294	0.107147	−	0.994243i	$-0.465828\pi$
$311$	365520.	0.214294	0.107147	+	0.994243i	$0.465828\pi$
$312$	0	0
$313$	1.24550e6	0.718592	0.359296	−	0.933224i	$-0.383017\pi$
$313$	1.24550e6	0.718592	0.359296	+	0.933224i	$0.383017\pi$
$314$	0	0
$315$	−150822.	−0.0856423
$316$	0	0
$317$	1.79646e6	1.00408	0.502042	−	0.864843i	$-0.332582\pi$
$317$	1.79646e6	1.00408	0.502042	+	0.864843i	$0.332582\pi$
$318$	0	0
$319$	4.08600e6	2.24813
$320$	0	0
$321$	−1.25532e6	−0.679973
$322$	0	0
$323$	71448.0	0.0381052
$324$	0	0
$325$	1.13299e6	0.595003
$326$	0	0
$327$	−1.27582e6	−0.659813
$328$	0	0
$329$	1.38729e6	0.706605
$330$	0	0
$331$	−1.55840e6	−0.781822	−0.390911	−	0.920428i	$-0.627840\pi$
$331$	−1.55840e6	−0.781822	−0.390911	+	0.920428i	$0.627840\pi$
$332$	0	0
$333$	−751194.	−0.371229
$334$	0	0
$335$	−2.57138e6	−1.25186
$336$	0	0
$337$	−1.33472e6	−0.640199	−0.320099	−	0.947384i	$-0.603716\pi$
$337$	−1.33472e6	−0.640199	−0.320099	+	0.947384i	$0.603716\pi$
$338$	0	0
$339$	735678.	0.347687
$340$	0	0
$341$	4.74720e6	2.21081
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	1.57457e6	0.712219
$346$	0	0
$347$	−3.56086e6	−1.58757	−0.793783	−	0.608201i	$-0.791891\pi$
$347$	−3.56086e6	−1.58757	−0.793783	+	0.608201i	$0.791891\pi$
$348$	0	0
$349$	−1.60451e6	−0.705144	−0.352572	−	0.935785i	$-0.614693\pi$
$349$	−1.60451e6	−0.705144	−0.352572	+	0.935785i	$0.614693\pi$
$350$	0	0
$351$	491346.	0.212873
$352$	0	0
$353$	−384290.	−0.164143	−0.0820715	−	0.996626i	$-0.526154\pi$
$353$	−384290.	−0.164143	−0.0820715	+	0.996626i	$0.526154\pi$
$354$	0	0
$355$	−2.56470e6	−1.08010
$356$	0	0
$357$	−34398.0	−0.0142844
$358$	0	0
$359$	500532.	0.204973	0.102486	−	0.994734i	$-0.467320\pi$
$359$	500532.	0.204973	0.102486	+	0.994734i	$0.467320\pi$
$360$	0	0
$361$	−1.63704e6	−0.661138
$362$	0	0
$363$	−1.79054e6	−0.713210
$364$	0	0
$365$	−42028.0	−0.0165123
$366$	0	0
$367$	4.13210e6	1.60142	0.800710	−	0.599052i	$-0.204456\pi$
$367$	4.13210e6	1.60142	0.800710	+	0.599052i	$0.204456\pi$
$368$	0	0
$369$	−19602.0	−0.00749436
$370$	0	0
$371$	501270.	0.189076
$372$	0	0
$373$	3.41287e6	1.27013	0.635064	−	0.772459i	$-0.280974\pi$
$373$	3.41287e6	1.27013	0.635064	+	0.772459i	$0.280974\pi$
$374$	0	0
$375$	−1.64365e6	−0.603576
$376$	0	0
$377$	4.58994e6	1.66324
$378$	0	0
$379$	3.16959e6	1.13346	0.566728	−	0.823905i	$-0.308209\pi$
$379$	3.16959e6	1.13346	0.566728	+	0.823905i	$0.308209\pi$
$380$	0	0
$381$	814392.	0.287423
$382$	0	0
$383$	5.07848e6	1.76904	0.884518	−	0.466506i	$-0.154487\pi$
$383$	5.07848e6	1.76904	0.884518	+	0.466506i	$0.154487\pi$
$384$	0	0
$385$	1.11720e6	0.384131
$386$	0	0
$387$	−90396.0	−0.0306812
$388$	0	0
$389$	1.57224e6	0.526798	0.263399	−	0.964687i	$-0.415156\pi$
$389$	1.57224e6	0.526798	0.263399	+	0.964687i	$0.415156\pi$
$390$	0	0
$391$	359112.	0.118792
$392$	0	0
$393$	1.68728e6	0.551070
$394$	0	0
$395$	3.19778e6	1.03123
$396$	0	0
$397$	−3.00804e6	−0.957872	−0.478936	−	0.877850i	$-0.658977\pi$
$397$	−3.00804e6	−0.957872	−0.478936	+	0.877850i	$0.658977\pi$
$398$	0	0
$399$	−403956.	−0.127029
$400$	0	0
$401$	−54406.0	−0.0168961	−0.00844804	−	0.999964i	$-0.502689\pi$
$401$	−54406.0	−0.0168961	−0.00844804	+	0.999964i	$0.502689\pi$
$402$	0	0
$403$	5.33269e6	1.63562
$404$	0	0
$405$	−249318.	−0.0755294
$406$	0	0
$407$	5.56440e6	1.66507
$408$	0	0
$409$	−2.07030e6	−0.611963	−0.305982	−	0.952037i	$-0.598985\pi$
$409$	−2.07030e6	−0.611963	−0.305982	+	0.952037i	$0.598985\pi$
$410$	0	0
$411$	−1.51634e6	−0.442784
$412$	0	0
$413$	201292.	0.0580699
$414$	0	0
$415$	−110504.	−0.0314962
$416$	0	0
$417$	6012.00	0.00169309
$418$	0	0
$419$	−1.48062e6	−0.412011	−0.206005	−	0.978551i	$-0.566046\pi$
$419$	−1.48062e6	−0.412011	−0.206005	+	0.978551i	$0.566046\pi$
$420$	0	0
$421$	−2.22283e6	−0.611224	−0.305612	−	0.952156i	$-0.598861\pi$
$421$	−2.22283e6	−0.611224	−0.305612	+	0.952156i	$0.598861\pi$
$422$	0	0
$423$	2.29327e6	0.623167
$424$	0	0
$425$	−131118.	−0.0352119
$426$	0	0
$427$	778022.	0.206501
$428$	0	0
$429$	−3.63960e6	−0.954796
$430$	0	0
$431$	1.96259e6	0.508904	0.254452	−	0.967085i	$-0.418105\pi$
$431$	1.96259e6	0.508904	0.254452	+	0.967085i	$0.418105\pi$
$432$	0	0
$433$	1.92503e6	0.493420	0.246710	−	0.969089i	$-0.420650\pi$
$433$	1.92503e6	0.493420	0.246710	+	0.969089i	$0.420650\pi$
$434$	0	0
$435$	−2.32902e6	−0.590133
$436$	0	0
$437$	4.21726e6	1.05640
$438$	0	0
$439$	−1.68838e6	−0.418127	−0.209063	−	0.977902i	$-0.567041\pi$
$439$	−1.68838e6	−0.418127	−0.209063	+	0.977902i	$0.567041\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	0	0
$443$	−7.37277e6	−1.78493	−0.892465	−	0.451116i	$-0.851026\pi$
$443$	−7.37277e6	−1.78493	−0.892465	+	0.451116i	$0.851026\pi$
$444$	0	0
$445$	316236.	0.0757027
$446$	0	0
$447$	2.70556e6	0.640455
$448$	0	0
$449$	−4.61309e6	−1.07988	−0.539940	−	0.841703i	$-0.681553\pi$
$449$	−4.61309e6	−1.07988	−0.539940	+	0.841703i	$0.681553\pi$
$450$	0	0
$451$	145200.	0.0336144
$452$	0	0
$453$	3.23129e6	0.739827
$454$	0	0
$455$	1.25499e6	0.284191
$456$	0	0
$457$	−5.75663e6	−1.28937	−0.644685	−	0.764448i	$-0.723012\pi$
$457$	−5.75663e6	−1.28937	−0.644685	+	0.764448i	$0.723012\pi$
$458$	0	0
$459$	−56862.0	−0.0125977
$460$	0	0
$461$	5.67782e6	1.24431	0.622156	−	0.782893i	$-0.286257\pi$
$461$	5.67782e6	1.24431	0.622156	+	0.782893i	$0.286257\pi$
$462$	0	0
$463$	8.06452e6	1.74834	0.874170	−	0.485619i	$-0.161406\pi$
$463$	8.06452e6	1.74834	0.874170	+	0.485619i	$0.161406\pi$
$464$	0	0
$465$	−2.70590e6	−0.580337
$466$	0	0
$467$	6.92248e6	1.46882	0.734412	−	0.678704i	$-0.237458\pi$
$467$	6.92248e6	1.46882	0.734412	+	0.678704i	$0.237458\pi$
$468$	0	0
$469$	3.31573e6	0.696061
$470$	0	0
$471$	−997398.	−0.207165
$472$	0	0
$473$	669600.	0.137614
$474$	0	0
$475$	−1.53980e6	−0.313133
$476$	0	0
$477$	828630.	0.166750
$478$	0	0
$479$	8.33297e6	1.65944	0.829719	−	0.558182i	$-0.188501\pi$
$479$	8.33297e6	1.65944	0.829719	+	0.558182i	$0.188501\pi$
$480$	0	0
$481$	6.25068e6	1.23187
$482$	0	0
$483$	−2.03036e6	−0.396010
$484$	0	0
$485$	−4.97078e6	−0.959556
$486$	0	0
$487$	−496792.	−0.0949188	−0.0474594	−	0.998873i	$-0.515112\pi$
$487$	−496792.	−0.0949188	−0.0474594	+	0.998873i	$0.515112\pi$
$488$	0	0
$489$	3.35142e6	0.633807
$490$	0	0
$491$	760704.	0.142401	0.0712003	−	0.997462i	$-0.477317\pi$
$491$	760704.	0.142401	0.0712003	+	0.997462i	$0.477317\pi$
$492$	0	0
$493$	−531180.	−0.0984293
$494$	0	0
$495$	1.84680e6	0.338772
$496$	0	0
$497$	3.30711e6	0.600561
$498$	0	0
$499$	−2.19192e6	−0.394071	−0.197035	−	0.980396i	$-0.563131\pi$
$499$	−2.19192e6	−0.394071	−0.197035	+	0.980396i	$0.563131\pi$
$500$	0	0
$501$	−986400.	−0.175573
$502$	0	0
$503$	790680.	0.139342	0.0696708	−	0.997570i	$-0.477805\pi$
$503$	790680.	0.139342	0.0696708	+	0.997570i	$0.477805\pi$
$504$	0	0
$505$	−6.07217e6	−1.05954
$506$	0	0
$507$	−746847.	−0.129036
$508$	0	0
$509$	2.75272e6	0.470943	0.235471	−	0.971881i	$-0.424337\pi$
$509$	2.75272e6	0.470943	0.235471	+	0.971881i	$0.424337\pi$
$510$	0	0
$511$	54194.0	0.00918119
$512$	0	0
$513$	−667764.	−0.112029
$514$	0	0
$515$	−3.47472e6	−0.577300
$516$	0	0
$517$	−1.69872e7	−2.79509
$518$	0	0
$519$	3.49738e6	0.569934
$520$	0	0
$521$	−6.27911e6	−1.01345	−0.506727	−	0.862107i	$-0.669145\pi$
$521$	−6.27911e6	−1.01345	−0.506727	+	0.862107i	$0.669145\pi$
$522$	0	0
$523$	5.60610e6	0.896203	0.448102	−	0.893983i	$-0.352100\pi$
$523$	5.60610e6	0.896203	0.448102	+	0.893983i	$0.352100\pi$
$524$	0	0
$525$	741321.	0.117384
$526$	0	0
$527$	−617136.	−0.0967953
$528$	0	0
$529$	1.47605e7	2.29330
$530$	0	0
$531$	332748.	0.0512129
$532$	0	0
$533$	163108.	0.0248689
$534$	0	0
$535$	−5.30024e6	−0.800592
$536$	0	0
$537$	−2.17490e6	−0.325465
$538$	0	0
$539$	−1.44060e6	−0.213585
$540$	0	0
$541$	3.31128e6	0.486410	0.243205	−	0.969975i	$-0.421801\pi$
$541$	3.31128e6	0.486410	0.243205	+	0.969975i	$0.421801\pi$
$542$	0	0
$543$	−1.71769e6	−0.250002
$544$	0	0
$545$	−5.38680e6	−0.776855
$546$	0	0
$547$	−5.66566e6	−0.809622	−0.404811	−	0.914400i	$-0.632663\pi$
$547$	−5.66566e6	−0.809622	−0.404811	+	0.914400i	$0.632663\pi$
$548$	0	0
$549$	1.28612e6	0.182117
$550$	0	0
$551$	−6.23796e6	−0.875313
$552$	0	0
$553$	−4.12345e6	−0.573387
$554$	0	0
$555$	−3.17171e6	−0.437080
$556$	0	0
$557$	−6.36880e6	−0.869801	−0.434900	−	0.900479i	$-0.643216\pi$
$557$	−6.36880e6	−0.869801	−0.434900	+	0.900479i	$0.643216\pi$
$558$	0	0
$559$	752184.	0.101811
$560$	0	0
$561$	421200.	0.0565043
$562$	0	0
$563$	−181404.	−0.0241199	−0.0120600	−	0.999927i	$-0.503839\pi$
$563$	−181404.	−0.0241199	−0.0120600	+	0.999927i	$0.503839\pi$
$564$	0	0
$565$	3.10620e6	0.409362
$566$	0	0
$567$	321489.	0.0419961
$568$	0	0
$569$	−5.02851e6	−0.651116	−0.325558	−	0.945522i	$-0.605552\pi$
$569$	−5.02851e6	−0.651116	−0.325558	+	0.945522i	$0.605552\pi$
$570$	0	0
$571$	7.57430e6	0.972192	0.486096	−	0.873905i	$-0.338420\pi$
$571$	7.57430e6	0.972192	0.486096	+	0.873905i	$0.338420\pi$
$572$	0	0
$573$	2.27970e6	0.290062
$574$	0	0
$575$	−7.73932e6	−0.976188
$576$	0	0
$577$	1.52011e7	1.90080	0.950400	−	0.311029i	$-0.100674\pi$
$577$	1.52011e7	1.90080	0.950400	+	0.311029i	$0.100674\pi$
$578$	0	0
$579$	6.41072e6	0.794713
$580$	0	0
$581$	142492.	0.0175126
$582$	0	0
$583$	−6.13800e6	−0.747921
$584$	0	0
$585$	2.07457e6	0.250633
$586$	0	0
$587$	−4.64015e6	−0.555823	−0.277912	−	0.960607i	$-0.589642\pi$
$587$	−4.64015e6	−0.555823	−0.277912	+	0.960607i	$0.589642\pi$
$588$	0	0
$589$	−7.24739e6	−0.860783
$590$	0	0
$591$	964746.	0.113617
$592$	0	0
$593$	−4.18658e6	−0.488903	−0.244451	−	0.969662i	$-0.578608\pi$
$593$	−4.18658e6	−0.488903	−0.244451	+	0.969662i	$0.578608\pi$
$594$	0	0
$595$	−145236.	−0.0168183
$596$	0	0
$597$	−8.61257e6	−0.989001
$598$	0	0
$599$	1.02901e7	1.17180	0.585898	−	0.810385i	$-0.300742\pi$
$599$	1.02901e7	1.17180	0.585898	+	0.810385i	$0.300742\pi$
$600$	0	0
$601$	1.43512e7	1.62070	0.810349	−	0.585947i	$-0.199277\pi$
$601$	1.43512e7	1.62070	0.810349	+	0.585947i	$0.199277\pi$
$602$	0	0
$603$	5.48111e6	0.613868
$604$	0	0
$605$	−7.56006e6	−0.839725
$606$	0	0
$607$	−1.09870e7	−1.21034	−0.605169	−	0.796097i	$-0.706894\pi$
$607$	−1.09870e7	−1.21034	−0.605169	+	0.796097i	$0.706894\pi$
$608$	0	0
$609$	3.00321e6	0.328127
$610$	0	0
$611$	−1.90823e7	−2.06789
$612$	0	0
$613$	3.00637e6	0.323141	0.161570	−	0.986861i	$-0.448344\pi$
$613$	3.00637e6	0.323141	0.161570	+	0.986861i	$0.448344\pi$
$614$	0	0
$615$	−82764.0	−0.00882376
$616$	0	0
$617$	1.82040e7	1.92510	0.962550	−	0.271105i	$-0.0873890\pi$
$617$	1.82040e7	1.92510	0.962550	+	0.271105i	$0.0873890\pi$
$618$	0	0
$619$	−6.45803e6	−0.677444	−0.338722	−	0.940887i	$-0.609995\pi$
$619$	−6.45803e6	−0.677444	−0.338722	+	0.940887i	$0.609995\pi$
$620$	0	0
$621$	−3.35632e6	−0.349248
$622$	0	0
$623$	−407778.	−0.0420924
$624$	0	0
$625$	−1.68674e6	−0.172722
$626$	0	0
$627$	4.94640e6	0.502482
$628$	0	0
$629$	−723372.	−0.0729013
$630$	0	0
$631$	−3.59598e6	−0.359538	−0.179769	−	0.983709i	$-0.557535\pi$
$631$	−3.59598e6	−0.359538	−0.179769	+	0.983709i	$0.557535\pi$
$632$	0	0
$633$	−2.88630e6	−0.286307
$634$	0	0
$635$	3.43854e6	0.338408
$636$	0	0
$637$	−1.61827e6	−0.158017
$638$	0	0
$639$	5.46685e6	0.529645
$640$	0	0
$641$	6.54433e6	0.629101	0.314550	−	0.949241i	$-0.398146\pi$
$641$	6.54433e6	0.629101	0.314550	+	0.949241i	$0.398146\pi$
$642$	0	0
$643$	−1.48417e7	−1.41565	−0.707827	−	0.706386i	$-0.750324\pi$
$643$	−1.48417e7	−1.41565	−0.707827	+	0.706386i	$0.750324\pi$
$644$	0	0
$645$	−381672.	−0.0361236
$646$	0	0
$647$	−4.34043e6	−0.407636	−0.203818	−	0.979009i	$-0.565335\pi$
$647$	−4.34043e6	−0.407636	−0.203818	+	0.979009i	$0.565335\pi$
$648$	0	0
$649$	−2.46480e6	−0.229705
$650$	0	0
$651$	3.48919e6	0.322680
$652$	0	0
$653$	5.31207e6	0.487507	0.243753	−	0.969837i	$-0.421621\pi$
$653$	5.31207e6	0.487507	0.243753	+	0.969837i	$0.421621\pi$
$654$	0	0
$655$	7.12409e6	0.648823
$656$	0	0
$657$	89586.0	0.00809705
$658$	0	0
$659$	2.78371e6	0.249696	0.124848	−	0.992176i	$-0.460156\pi$
$659$	2.78371e6	0.249696	0.124848	+	0.992176i	$0.460156\pi$
$660$	0	0
$661$	2.01522e7	1.79398	0.896992	−	0.442047i	$-0.145747\pi$
$661$	2.01522e7	1.79398	0.896992	+	0.442047i	$0.145747\pi$
$662$	0	0
$663$	473148.	0.0418036
$664$	0	0
$665$	−1.70559e6	−0.149562
$666$	0	0
$667$	−3.13532e7	−2.72878
$668$	0	0
$669$	9.61560e6	0.830637
$670$	0	0
$671$	−9.52680e6	−0.816847
$672$	0	0
$673$	−1.26277e7	−1.07470	−0.537349	−	0.843360i	$-0.680574\pi$
$673$	−1.26277e7	−1.07470	−0.537349	+	0.843360i	$0.680574\pi$
$674$	0	0
$675$	1.22545e6	0.103523
$676$	0	0
$677$	−6.94541e6	−0.582406	−0.291203	−	0.956661i	$-0.594056\pi$
$677$	−6.94541e6	−0.582406	−0.291203	+	0.956661i	$0.594056\pi$
$678$	0	0
$679$	6.40969e6	0.533535
$680$	0	0
$681$	−7.26138e6	−0.600000
$682$	0	0
$683$	278144.	0.0228149	0.0114074	−	0.999935i	$-0.496369\pi$
$683$	278144.	0.0228149	0.0114074	+	0.999935i	$0.496369\pi$
$684$	0	0
$685$	−6.40232e6	−0.521328
$686$	0	0
$687$	−1.09495e7	−0.885123
$688$	0	0
$689$	−6.89502e6	−0.553334
$690$	0	0
$691$	6.38656e6	0.508829	0.254414	−	0.967095i	$-0.418117\pi$
$691$	6.38656e6	0.508829	0.254414	+	0.967095i	$0.418117\pi$
$692$	0	0
$693$	−2.38140e6	−0.188365
$694$	0	0
$695$	25384.0	0.00199342
$696$	0	0
$697$	−18876.0	−0.00147173
$698$	0	0
$699$	−1.93250e6	−0.149598
$700$	0	0
$701$	−1.26602e7	−0.973075	−0.486537	−	0.873660i	$-0.661740\pi$
$701$	−1.26602e7	−0.973075	−0.486537	+	0.873660i	$0.661740\pi$
$702$	0	0
$703$	−8.49498e6	−0.648297
$704$	0	0
$705$	9.68270e6	0.733709
$706$	0	0
$707$	7.82991e6	0.589126
$708$	0	0
$709$	−1.29990e7	−0.971169	−0.485585	−	0.874190i	$-0.661393\pi$
$709$	−1.29990e7	−0.971169	−0.485585	+	0.874190i	$0.661393\pi$
$710$	0	0
$711$	−6.81631e6	−0.505680
$712$	0	0
$713$	−3.64268e7	−2.68348
$714$	0	0
$715$	−1.53672e7	−1.12416
$716$	0	0
$717$	−5.54202e6	−0.402597
$718$	0	0
$719$	−1.27912e6	−0.0922761	−0.0461380	−	0.998935i	$-0.514691\pi$
$719$	−1.27912e6	−0.0922761	−0.0461380	+	0.998935i	$0.514691\pi$
$720$	0	0
$721$	4.48056e6	0.320992
$722$	0	0
$723$	8.96031e6	0.637496
$724$	0	0
$725$	1.14476e7	0.808853
$726$	0	0
$727$	2.01247e7	1.41219	0.706097	−	0.708115i	$-0.250454\pi$
$727$	2.01247e7	1.41219	0.706097	+	0.708115i	$0.250454\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−87048.0	−0.00602512
$732$	0	0
$733$	2.59171e7	1.78167	0.890835	−	0.454328i	$-0.150121\pi$
$733$	2.59171e7	1.78167	0.890835	+	0.454328i	$0.150121\pi$
$734$	0	0
$735$	821142.	0.0560660
$736$	0	0
$737$	−4.06008e7	−2.75338
$738$	0	0
$739$	−2.57638e6	−0.173540	−0.0867698	−	0.996228i	$-0.527654\pi$
$739$	−2.57638e6	−0.173540	−0.0867698	+	0.996228i	$0.527654\pi$
$740$	0	0
$741$	5.55646e6	0.371751
$742$	0	0
$743$	−2.36046e7	−1.56865	−0.784323	−	0.620353i	$-0.786990\pi$
$743$	−2.36046e7	−1.56865	−0.784323	+	0.620353i	$0.786990\pi$
$744$	0	0
$745$	1.14235e7	0.754063
$746$	0	0
$747$	235548.	0.0154446
$748$	0	0
$749$	6.83452e6	0.445147
$750$	0	0
$751$	9.32237e6	0.603151	0.301576	−	0.953442i	$-0.402487\pi$
$751$	9.32237e6	0.603151	0.301576	+	0.953442i	$0.402487\pi$
$752$	0	0
$753$	−1.52406e6	−0.0979523
$754$	0	0
$755$	1.36432e7	0.871063
$756$	0	0
$757$	1.90696e7	1.20949	0.604744	−	0.796420i	$-0.293276\pi$
$757$	1.90696e7	1.20949	0.604744	+	0.796420i	$0.293276\pi$
$758$	0	0
$759$	2.48616e7	1.56648
$760$	0	0
$761$	−6.87530e6	−0.430358	−0.215179	−	0.976575i	$-0.569033\pi$
$761$	−6.87530e6	−0.430358	−0.215179	+	0.976575i	$0.569033\pi$
$762$	0	0
$763$	6.94614e6	0.431949
$764$	0	0
$765$	−240084.	−0.0148323
$766$	0	0
$767$	−2.76879e6	−0.169942
$768$	0	0
$769$	1.90337e7	1.16067	0.580334	−	0.814379i	$-0.302922\pi$
$769$	1.90337e7	1.16067	0.580334	+	0.814379i	$0.302922\pi$
$770$	0	0
$771$	−1.51042e7	−0.915087
$772$	0	0
$773$	1.64278e7	0.988851	0.494426	−	0.869220i	$-0.335378\pi$
$773$	1.64278e7	0.988851	0.494426	+	0.869220i	$0.335378\pi$
$774$	0	0
$775$	1.33001e7	0.795426
$776$	0	0
$777$	4.08983e6	0.243026
$778$	0	0
$779$	−221672.	−0.0130878
$780$	0	0
$781$	−4.04952e7	−2.37561
$782$	0	0
$783$	4.96449e6	0.289381
$784$	0	0
$785$	−4.21124e6	−0.243913
$786$	0	0
$787$	4.19153e6	0.241233	0.120616	−	0.992699i	$-0.461513\pi$
$787$	4.19153e6	0.241233	0.120616	+	0.992699i	$0.461513\pi$
$788$	0	0
$789$	−1.59756e7	−0.913619
$790$	0	0
$791$	−4.00536e6	−0.227615
$792$	0	0
$793$	−1.07018e7	−0.604328
$794$	0	0
$795$	3.49866e6	0.196329
$796$	0	0
$797$	−1.28046e7	−0.714035	−0.357018	−	0.934098i	$-0.616206\pi$
$797$	−1.28046e7	−0.714035	−0.357018	+	0.934098i	$0.616206\pi$
$798$	0	0
$799$	2.20834e6	0.122377
$800$	0	0
$801$	−674082.	−0.0371220
$802$	0	0
$803$	−663600.	−0.0363176
$804$	0	0
$805$	−8.57265e6	−0.466257
$806$	0	0
$807$	140814.	0.00761136
$808$	0	0
$809$	−1.33669e7	−0.718057	−0.359029	−	0.933327i	$-0.616892\pi$
$809$	−1.33669e7	−0.718057	−0.359029	+	0.933327i	$0.616892\pi$
$810$	0	0
$811$	3.51226e7	1.87514	0.937571	−	0.347794i	$-0.113069\pi$
$811$	3.51226e7	1.87514	0.937571	+	0.347794i	$0.113069\pi$
$812$	0	0
$813$	−5.72105e6	−0.303563
$814$	0	0
$815$	1.41504e7	0.746236
$816$	0	0
$817$	−1.02226e6	−0.0535802
$818$	0	0
$819$	−2.67511e6	−0.139358
$820$	0	0
$821$	−2.66290e7	−1.37878	−0.689392	−	0.724389i	$-0.742122\pi$
$821$	−2.66290e7	−1.37878	−0.689392	+	0.724389i	$0.742122\pi$
$822$	0	0
$823$	1.11835e7	0.575545	0.287773	−	0.957699i	$-0.407085\pi$
$823$	1.11835e7	0.575545	0.287773	+	0.957699i	$0.407085\pi$
$824$	0	0
$825$	−9.07740e6	−0.464330
$826$	0	0
$827$	2.56108e6	0.130215	0.0651073	−	0.997878i	$-0.479261\pi$
$827$	2.56108e6	0.130215	0.0651073	+	0.997878i	$0.479261\pi$
$828$	0	0
$829$	2.27188e7	1.14815	0.574076	−	0.818802i	$-0.305362\pi$
$829$	2.27188e7	1.14815	0.574076	+	0.818802i	$0.305362\pi$
$830$	0	0
$831$	829962.	0.0416923
$832$	0	0
$833$	187278.	0.00935135
$834$	0	0
$835$	−4.16480e6	−0.206718
$836$	0	0
$837$	5.76785e6	0.284577
$838$	0	0
$839$	−3.94627e7	−1.93545	−0.967725	−	0.252007i	$-0.918909\pi$
$839$	−3.94627e7	−1.93545	−0.967725	+	0.252007i	$0.918909\pi$
$840$	0	0
$841$	2.58650e7	1.26102
$842$	0	0
$843$	1.04110e7	0.504574
$844$	0	0
$845$	−3.15335e6	−0.151926
$846$	0	0
$847$	9.74850e6	0.466906
$848$	0	0
$849$	1.77642e7	0.845815
$850$	0	0
$851$	−4.26975e7	−2.02106
$852$	0	0
$853$	−2.87657e7	−1.35364	−0.676818	−	0.736150i	$-0.736642\pi$
$853$	−2.87657e7	−1.35364	−0.676818	+	0.736150i	$0.736642\pi$
$854$	0	0
$855$	−2.81945e6	−0.131901
$856$	0	0
$857$	−1.40799e7	−0.654860	−0.327430	−	0.944875i	$-0.606183\pi$
$857$	−1.40799e7	−0.654860	−0.327430	+	0.944875i	$0.606183\pi$
$858$	0	0
$859$	3.10876e7	1.43749	0.718744	−	0.695275i	$-0.244718\pi$
$859$	3.10876e7	1.43749	0.718744	+	0.695275i	$0.244718\pi$
$860$	0	0
$861$	106722.	0.00490621
$862$	0	0
$863$	9.66226e6	0.441623	0.220812	−	0.975316i	$-0.429129\pi$
$863$	9.66226e6	0.441623	0.220812	+	0.975316i	$0.429129\pi$
$864$	0	0
$865$	1.47667e7	0.671033
$866$	0	0
$867$	1.27240e7	0.574876
$868$	0	0
$869$	5.04912e7	2.26812
$870$	0	0
$871$	−4.56082e7	−2.03703
$872$	0	0
$873$	1.05956e7	0.470533
$874$	0	0
$875$	8.94877e6	0.395133
$876$	0	0
$877$	−1.98937e7	−0.873409	−0.436704	−	0.899605i	$-0.643854\pi$
$877$	−1.98937e7	−0.873409	−0.436704	+	0.899605i	$0.643854\pi$
$878$	0	0
$879$	1.02569e6	0.0447760
$880$	0	0
$881$	−2.03248e6	−0.0882240	−0.0441120	−	0.999027i	$-0.514046\pi$
$881$	−2.03248e6	−0.0882240	−0.0441120	+	0.999027i	$0.514046\pi$
$882$	0	0
$883$	3.69655e7	1.59549	0.797747	−	0.602993i	$-0.206025\pi$
$883$	3.69655e7	1.59549	0.797747	+	0.602993i	$0.206025\pi$
$884$	0	0
$885$	1.40494e6	0.0602974
$886$	0	0
$887$	2.75837e7	1.17718	0.588591	−	0.808431i	$-0.299683\pi$
$887$	2.75837e7	1.17718	0.588591	+	0.808431i	$0.299683\pi$
$888$	0	0
$889$	−4.43391e6	−0.188162
$890$	0	0
$891$	−3.93660e6	−0.166122
$892$	0	0
$893$	2.59338e7	1.08827
$894$	0	0
$895$	−9.18293e6	−0.383198
$896$	0	0
$897$	2.79279e7	1.15893
$898$	0	0
$899$	5.38807e7	2.22348
$900$	0	0
$901$	797940.	0.0327460
$902$	0	0
$903$	492156.	0.0200855
$904$	0	0
$905$	−7.25245e6	−0.294350
$906$	0	0
$907$	1.73426e7	0.699997	0.349998	−	0.936750i	$-0.386182\pi$
$907$	1.73426e7	0.699997	0.349998	+	0.936750i	$0.386182\pi$
$908$	0	0
$909$	1.29433e7	0.519560
$910$	0	0
$911$	−3.98913e7	−1.59251	−0.796255	−	0.604961i	$-0.793189\pi$
$911$	−3.98913e7	−1.59251	−0.796255	+	0.604961i	$0.793189\pi$
$912$	0	0
$913$	−1.74480e6	−0.0692738
$914$	0	0
$915$	5.43028e6	0.214422
$916$	0	0
$917$	−9.18632e6	−0.360760
$918$	0	0
$919$	−4.01774e7	−1.56925	−0.784626	−	0.619970i	$-0.787145\pi$
$919$	−4.01774e7	−1.56925	−0.784626	+	0.619970i	$0.787145\pi$
$920$	0	0
$921$	1.96650e7	0.763916
$922$	0	0
$923$	−4.54896e7	−1.75755
$924$	0	0
$925$	1.55896e7	0.599074
$926$	0	0
$927$	7.40664e6	0.283088
$928$	0	0
$929$	3.03389e7	1.15335	0.576675	−	0.816974i	$-0.304350\pi$
$929$	3.03389e7	1.15335	0.576675	+	0.816974i	$0.304350\pi$
$930$	0	0
$931$	2.19932e6	0.0831598
$932$	0	0
$933$	−3.28968e6	−0.123723
$934$	0	0
$935$	1.77840e6	0.0665274
$936$	0	0
$937$	−4.32143e7	−1.60797	−0.803986	−	0.594648i	$-0.797291\pi$
$937$	−4.32143e7	−1.60797	−0.803986	+	0.594648i	$0.797291\pi$
$938$	0	0
$939$	−1.12095e7	−0.414879
$940$	0	0
$941$	4.39045e7	1.61635	0.808174	−	0.588944i	$-0.200456\pi$
$941$	4.39045e7	1.61635	0.808174	+	0.588944i	$0.200456\pi$
$942$	0	0
$943$	−1.11417e6	−0.0408011
$944$	0	0
$945$	1.35740e6	0.0494456
$946$	0	0
$947$	4.80030e7	1.73937	0.869687	−	0.493603i	$-0.164320\pi$
$947$	4.80030e7	1.73937	0.869687	+	0.493603i	$0.164320\pi$
$948$	0	0
$949$	−745444.	−0.0268689
$950$	0	0
$951$	−1.61682e7	−0.579708
$952$	0	0
$953$	−2.26010e6	−0.0806113	−0.0403056	−	0.999187i	$-0.512833\pi$
$953$	−2.26010e6	−0.0806113	−0.0403056	+	0.999187i	$0.512833\pi$
$954$	0	0
$955$	9.62540e6	0.341515
$956$	0	0
$957$	−3.67740e7	−1.29796
$958$	0	0
$959$	8.25562e6	0.289870
$960$	0	0
$961$	3.39706e7	1.18657
$962$	0	0
$963$	1.12979e7	0.392583
$964$	0	0
$965$	2.70675e7	0.935685
$966$	0	0
$967$	−2.43715e6	−0.0838140	−0.0419070	−	0.999122i	$-0.513343\pi$
$967$	−2.43715e6	−0.0838140	−0.0419070	+	0.999122i	$0.513343\pi$
$968$	0	0
$969$	−643032.	−0.0220000
$970$	0	0
$971$	4.15303e7	1.41357	0.706784	−	0.707429i	$-0.250145\pi$
$971$	4.15303e7	1.41357	0.706784	+	0.707429i	$0.250145\pi$
$972$	0	0
$973$	−32732.0	−0.00110838
$974$	0	0
$975$	−1.01969e7	−0.343525
$976$	0	0
$977$	3.59151e6	0.120376	0.0601882	−	0.998187i	$-0.480830\pi$
$977$	3.59151e6	0.120376	0.0601882	+	0.998187i	$0.480830\pi$
$978$	0	0
$979$	4.99320e6	0.166503
$980$	0	0
$981$	1.14824e7	0.380943
$982$	0	0
$983$	−3.84268e7	−1.26838	−0.634191	−	0.773176i	$-0.718667\pi$
$983$	−3.84268e7	−1.26838	−0.634191	+	0.773176i	$0.718667\pi$
$984$	0	0
$985$	4.07337e6	0.133771
$986$	0	0
$987$	−1.24856e7	−0.407959
$988$	0	0
$989$	−5.13806e6	−0.167035
$990$	0	0
$991$	−7.96056e6	−0.257489	−0.128745	−	0.991678i	$-0.541095\pi$
$991$	−7.96056e6	−0.257489	−0.128745	+	0.991678i	$0.541095\pi$
$992$	0	0
$993$	1.40256e7	0.451385
$994$	0	0
$995$	−3.63642e7	−1.16444
$996$	0	0
$997$	2.67858e7	0.853427	0.426713	−	0.904387i	$-0.359671\pi$
$997$	2.67858e7	0.853427	0.426713	+	0.904387i	$0.359671\pi$
$998$	0	0
$999$	6.76075e6	0.214329

Display

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a_n

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a_n

instead) (See

a_n

instead) Display

a_n

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n

up to: 50 250 1000 (See only

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a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.a.d.1.1		1
3.2	odd	2		1008.6.a.v.1.1		1
4.3	odd	2		168.6.a.e.1.1	✓	1
12.11	even	2		504.6.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.6.a.e.1.1	✓	1	4.3	odd	2
336.6.a.d.1.1		1	1.1	even	1	trivial
504.6.a.g.1.1		1	12.11	even	2
1008.6.a.v.1.1		1	3.2	odd	2

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

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Newspace parameters

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Embedding invariants

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Twists

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

Label	336.6.a.d.1.1

Level	$336$
Weight	$6$
Character	336.1
Self dual	yes
Analytic conductor	$53.889$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$