LMFDB - Embedded newform 336.6.a.l.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 21)
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)

$f(q)$

$=$

$q+9.00000 q^{3} -34.0000 q^{5} +49.0000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q+9.00000 q^{3} -34.0000 q^{5} +49.0000 q^{7} +81.0000 q^{9} +340.000 q^{11} +454.000 q^{13} -306.000 q^{15} -798.000 q^{17} -892.000 q^{19} +441.000 q^{21} +3192.00 q^{23} -1969.00 q^{25} +729.000 q^{27} -8242.00 q^{29} +2496.00 q^{31} +3060.00 q^{33} -1666.00 q^{35} +9798.00 q^{37} +4086.00 q^{39} +19834.0 q^{41} +17236.0 q^{43} -2754.00 q^{45} -8928.00 q^{47} +2401.00 q^{49} -7182.00 q^{51} +150.000 q^{53} -11560.0 q^{55} -8028.00 q^{57} +42396.0 q^{59} +14758.0 q^{61} +3969.00 q^{63} -15436.0 q^{65} +1676.00 q^{67} +28728.0 q^{69} -14568.0 q^{71} +78378.0 q^{73} -17721.0 q^{75} +16660.0 q^{77} +2272.00 q^{79} +6561.00 q^{81} +37764.0 q^{83} +27132.0 q^{85} -74178.0 q^{87} -117286. q^{89} +22246.0 q^{91} +22464.0 q^{93} +30328.0 q^{95} +10002.0 q^{97} +27540.0 q^{99} +O(q^{100})$

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
$5$	−34.0000	−0.608210	−0.304105	−	0.952638i	$-0.598357\pi$
$5$	−34.0000	−0.608210	−0.304105	+	0.952638i	$0.598357\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	340.000	0.847222	0.423611	−	0.905844i	$-0.360762\pi$
$11$	340.000	0.847222	0.423611	+	0.905844i	$0.360762\pi$
$12$	0	0
$13$	454.000	0.745071	0.372535	−	0.928018i	$-0.378489\pi$
$13$	454.000	0.745071	0.372535	+	0.928018i	$0.378489\pi$
$14$	0	0
$15$	−306.000	−0.351150
$16$	0	0
$17$	−798.000	−0.669700	−0.334850	−	0.942271i	$-0.608686\pi$
$17$	−798.000	−0.669700	−0.334850	+	0.942271i	$0.608686\pi$
$18$	0	0
$19$	−892.000	−0.566867	−0.283433	−	0.958992i	$-0.591473\pi$
$19$	−892.000	−0.566867	−0.283433	+	0.958992i	$0.591473\pi$
$20$	0	0
$21$	441.000	0.218218
$22$	0	0
$23$	3192.00	1.25818	0.629091	−	0.777332i	$-0.283427\pi$
$23$	3192.00	1.25818	0.629091	+	0.777332i	$0.283427\pi$
$24$	0	0
$25$	−1969.00	−0.630080
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−8242.00	−1.81986	−0.909929	−	0.414764i	$-0.863864\pi$
$29$	−8242.00	−1.81986	−0.909929	+	0.414764i	$0.863864\pi$
$30$	0	0
$31$	2496.00	0.466488	0.233244	−	0.972418i	$-0.425066\pi$
$31$	2496.00	0.466488	0.233244	+	0.972418i	$0.425066\pi$
$32$	0	0
$33$	3060.00	0.489144
$34$	0	0
$35$	−1666.00	−0.229882
$36$	0	0
$37$	9798.00	1.17661	0.588306	−	0.808639i	$-0.299795\pi$
$37$	9798.00	1.17661	0.588306	+	0.808639i	$0.299795\pi$
$38$	0	0
$39$	4086.00	0.430167
$40$	0	0
$41$	19834.0	1.84268	0.921342	−	0.388754i	$-0.127094\pi$
$41$	19834.0	1.84268	0.921342	+	0.388754i	$0.127094\pi$
$42$	0	0
$43$	17236.0	1.42156	0.710780	−	0.703414i	$-0.248342\pi$
$43$	17236.0	1.42156	0.710780	+	0.703414i	$0.248342\pi$
$44$	0	0
$45$	−2754.00	−0.202737
$46$	0	0
$47$	−8928.00	−0.589535	−0.294767	−	0.955569i	$-0.595242\pi$
$47$	−8928.00	−0.589535	−0.294767	+	0.955569i	$0.595242\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−7182.00	−0.386652
$52$	0	0
$53$	150.000	0.00733502	0.00366751	−	0.999993i	$-0.498833\pi$
$53$	150.000	0.00733502	0.00366751	+	0.999993i	$0.498833\pi$
$54$	0	0
$55$	−11560.0	−0.515289
$56$	0	0
$57$	−8028.00	−0.327281
$58$	0	0
$59$	42396.0	1.58560	0.792802	−	0.609479i	$-0.208621\pi$
$59$	42396.0	1.58560	0.792802	+	0.609479i	$0.208621\pi$
$60$	0	0
$61$	14758.0	0.507812	0.253906	−	0.967229i	$-0.418285\pi$
$61$	14758.0	0.507812	0.253906	+	0.967229i	$0.418285\pi$
$62$	0	0
$63$	3969.00	0.125988
$64$	0	0
$65$	−15436.0	−0.453160
$66$	0	0
$67$	1676.00	0.0456128	0.0228064	−	0.999740i	$-0.492740\pi$
$67$	1676.00	0.0456128	0.0228064	+	0.999740i	$0.492740\pi$
$68$	0	0
$69$	28728.0	0.726411
$70$	0	0
$71$	−14568.0	−0.342968	−0.171484	−	0.985187i	$-0.554856\pi$
$71$	−14568.0	−0.342968	−0.171484	+	0.985187i	$0.554856\pi$
$72$	0	0
$73$	78378.0	1.72142	0.860710	−	0.509095i	$-0.170020\pi$
$73$	78378.0	1.72142	0.860710	+	0.509095i	$0.170020\pi$
$74$	0	0
$75$	−17721.0	−0.363777
$76$	0	0
$77$	16660.0	0.320220
$78$	0	0
$79$	2272.00	0.0409582	0.0204791	−	0.999790i	$-0.493481\pi$
$79$	2272.00	0.0409582	0.0204791	+	0.999790i	$0.493481\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	37764.0	0.601704	0.300852	−	0.953671i	$-0.402729\pi$
$83$	37764.0	0.601704	0.300852	+	0.953671i	$0.402729\pi$
$84$	0	0
$85$	27132.0	0.407319
$86$	0	0
$87$	−74178.0	−1.05070
$88$	0	0
$89$	−117286.	−1.56954	−0.784768	−	0.619790i	$-0.787218\pi$
$89$	−117286.	−1.56954	−0.784768	+	0.619790i	$0.787218\pi$
$90$	0	0
$91$	22246.0	0.281610
$92$	0	0
$93$	22464.0	0.269327
$94$	0	0
$95$	30328.0	0.344774
$96$	0	0
$97$	10002.0	0.107934	0.0539669	−	0.998543i	$-0.482813\pi$
$97$	10002.0	0.107934	0.0539669	+	0.998543i	$0.482813\pi$
$98$	0	0
$99$	27540.0	0.282407
$100$	0	0
$101$	−108770.	−1.06098	−0.530488	−	0.847692i	$-0.677991\pi$
$101$	−108770.	−1.06098	−0.530488	+	0.847692i	$0.677991\pi$
$102$	0	0
$103$	199192.	1.85003	0.925015	−	0.379930i	$-0.124052\pi$
$103$	199192.	1.85003	0.925015	+	0.379930i	$0.124052\pi$
$104$	0	0
$105$	−14994.0	−0.132722
$106$	0	0
$107$	79972.0	0.675272	0.337636	−	0.941277i	$-0.390373\pi$
$107$	79972.0	0.675272	0.337636	+	0.941277i	$0.390373\pi$
$108$	0	0
$109$	−46098.0	−0.371634	−0.185817	−	0.982584i	$-0.559493\pi$
$109$	−46098.0	−0.371634	−0.185817	+	0.982584i	$0.559493\pi$
$110$	0	0
$111$	88182.0	0.679317
$112$	0	0
$113$	262706.	1.93541	0.967707	−	0.252078i	$-0.0811138\pi$
$113$	262706.	1.93541	0.967707	+	0.252078i	$0.0811138\pi$
$114$	0	0
$115$	−108528.	−0.765239
$116$	0	0
$117$	36774.0	0.248357
$118$	0	0
$119$	−39102.0	−0.253123
$120$	0	0
$121$	−45451.0	−0.282215
$122$	0	0
$123$	178506.	1.06387
$124$	0	0
$125$	173196.	0.991432
$126$	0	0
$127$	−196608.	−1.08166	−0.540831	−	0.841131i	$-0.681890\pi$
$127$	−196608.	−1.08166	−0.540831	+	0.841131i	$0.681890\pi$
$128$	0	0
$129$	155124.	0.820738
$130$	0	0
$131$	77140.0	0.392737	0.196368	−	0.980530i	$-0.437085\pi$
$131$	77140.0	0.392737	0.196368	+	0.980530i	$0.437085\pi$
$132$	0	0
$133$	−43708.0	−0.214255
$134$	0	0
$135$	−24786.0	−0.117050
$136$	0	0
$137$	208170.	0.947582	0.473791	−	0.880637i	$-0.342885\pi$
$137$	208170.	0.947582	0.473791	+	0.880637i	$0.342885\pi$
$138$	0	0
$139$	275580.	1.20979	0.604896	−	0.796304i	$-0.293215\pi$
$139$	275580.	1.20979	0.604896	+	0.796304i	$0.293215\pi$
$140$	0	0
$141$	−80352.0	−0.340368
$142$	0	0
$143$	154360.	0.631240
$144$	0	0
$145$	280228.	1.10686
$146$	0	0
$147$	21609.0	0.0824786
$148$	0	0
$149$	−296106.	−1.09265	−0.546326	−	0.837573i	$-0.683974\pi$
$149$	−296106.	−1.09265	−0.546326	+	0.837573i	$0.683974\pi$
$150$	0	0
$151$	426472.	1.52212	0.761059	−	0.648683i	$-0.224680\pi$
$151$	426472.	1.52212	0.761059	+	0.648683i	$0.224680\pi$
$152$	0	0
$153$	−64638.0	−0.223233
$154$	0	0
$155$	−84864.0	−0.283723
$156$	0	0
$157$	178486.	0.577903	0.288952	−	0.957344i	$-0.406693\pi$
$157$	178486.	0.577903	0.288952	+	0.957344i	$0.406693\pi$
$158$	0	0
$159$	1350.00	0.00423488
$160$	0	0
$161$	156408.	0.475548
$162$	0	0
$163$	−252772.	−0.745178	−0.372589	−	0.927996i	$-0.621530\pi$
$163$	−252772.	−0.745178	−0.372589	+	0.927996i	$0.621530\pi$
$164$	0	0
$165$	−104040.	−0.297502
$166$	0	0
$167$	−508088.	−1.40977	−0.704884	−	0.709322i	$-0.749001\pi$
$167$	−508088.	−1.40977	−0.704884	+	0.709322i	$0.749001\pi$
$168$	0	0
$169$	−165177.	−0.444870
$170$	0	0
$171$	−72252.0	−0.188956
$172$	0	0
$173$	−221834.	−0.563525	−0.281762	−	0.959484i	$-0.590919\pi$
$173$	−221834.	−0.563525	−0.281762	+	0.959484i	$0.590919\pi$
$174$	0	0
$175$	−96481.0	−0.238148
$176$	0	0
$177$	381564.	0.915449
$178$	0	0
$179$	113564.	0.264916	0.132458	−	0.991189i	$-0.457713\pi$
$179$	113564.	0.264916	0.132458	+	0.991189i	$0.457713\pi$
$180$	0	0
$181$	663118.	1.50451	0.752254	−	0.658873i	$-0.228967\pi$
$181$	663118.	1.50451	0.752254	+	0.658873i	$0.228967\pi$
$182$	0	0
$183$	132822.	0.293185
$184$	0	0
$185$	−333132.	−0.715628
$186$	0	0
$187$	−271320.	−0.567385
$188$	0	0
$189$	35721.0	0.0727393
$190$	0	0
$191$	−505664.	−1.00295	−0.501474	−	0.865173i	$-0.667209\pi$
$191$	−505664.	−1.00295	−0.501474	+	0.865173i	$0.667209\pi$
$192$	0	0
$193$	−432382.	−0.835554	−0.417777	−	0.908550i	$-0.637191\pi$
$193$	−432382.	−0.835554	−0.417777	+	0.908550i	$0.637191\pi$
$194$	0	0
$195$	−138924.	−0.261632
$196$	0	0
$197$	−131962.	−0.242261	−0.121130	−	0.992637i	$-0.538652\pi$
$197$	−131962.	−0.242261	−0.121130	+	0.992637i	$0.538652\pi$
$198$	0	0
$199$	−298536.	−0.534397	−0.267199	−	0.963642i	$-0.586098\pi$
$199$	−298536.	−0.534397	−0.267199	+	0.963642i	$0.586098\pi$
$200$	0	0
$201$	15084.0	0.0263346
$202$	0	0
$203$	−403858.	−0.687842
$204$	0	0
$205$	−674356.	−1.12074
$206$	0	0
$207$	258552.	0.419394
$208$	0	0
$209$	−303280.	−0.480262
$210$	0	0
$211$	1.17062e6	1.81013	0.905065	−	0.425273i	$-0.139822\pi$
$211$	1.17062e6	1.81013	0.905065	+	0.425273i	$0.139822\pi$
$212$	0	0
$213$	−131112.	−0.198013
$214$	0	0
$215$	−586024.	−0.864608
$216$	0	0
$217$	122304.	0.176316
$218$	0	0
$219$	705402.	0.993863
$220$	0	0
$221$	−362292.	−0.498974
$222$	0	0
$223$	−399376.	−0.537799	−0.268899	−	0.963168i	$-0.586660\pi$
$223$	−399376.	−0.537799	−0.268899	+	0.963168i	$0.586660\pi$
$224$	0	0
$225$	−159489.	−0.210027
$226$	0	0
$227$	−707916.	−0.911837	−0.455918	−	0.890022i	$-0.650689\pi$
$227$	−707916.	−0.911837	−0.455918	+	0.890022i	$0.650689\pi$
$228$	0	0
$229$	−735778.	−0.927167	−0.463584	−	0.886053i	$-0.653437\pi$
$229$	−735778.	−0.927167	−0.463584	+	0.886053i	$0.653437\pi$
$230$	0	0
$231$	149940.	0.184879
$232$	0	0
$233$	−208758.	−0.251915	−0.125957	−	0.992036i	$-0.540200\pi$
$233$	−208758.	−0.251915	−0.125957	+	0.992036i	$0.540200\pi$
$234$	0	0
$235$	303552.	0.358561
$236$	0	0
$237$	20448.0	0.0236472
$238$	0	0
$239$	−713376.	−0.807837	−0.403919	−	0.914795i	$-0.632352\pi$
$239$	−713376.	−0.807837	−0.403919	+	0.914795i	$0.632352\pi$
$240$	0	0
$241$	−505246.	−0.560351	−0.280176	−	0.959949i	$-0.590393\pi$
$241$	−505246.	−0.560351	−0.280176	+	0.959949i	$0.590393\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	−81634.0	−0.0868872
$246$	0	0
$247$	−404968.	−0.422356
$248$	0	0
$249$	339876.	0.347394
$250$	0	0
$251$	−317108.	−0.317704	−0.158852	−	0.987302i	$-0.550779\pi$
$251$	−317108.	−0.317704	−0.158852	+	0.987302i	$0.550779\pi$
$252$	0	0
$253$	1.08528e6	1.06596
$254$	0	0
$255$	244188.	0.235166
$256$	0	0
$257$	−1.44285e6	−1.36266	−0.681329	−	0.731977i	$-0.738598\pi$
$257$	−1.44285e6	−1.36266	−0.681329	+	0.731977i	$0.738598\pi$
$258$	0	0
$259$	480102.	0.444717
$260$	0	0
$261$	−667602.	−0.606619
$262$	0	0
$263$	−271496.	−0.242033	−0.121016	−	0.992651i	$-0.538615\pi$
$263$	−271496.	−0.242033	−0.121016	+	0.992651i	$0.538615\pi$
$264$	0	0
$265$	−5100.00	−0.00446124
$266$	0	0
$267$	−1.05557e6	−0.906172
$268$	0	0
$269$	850614.	0.716724	0.358362	−	0.933583i	$-0.383335\pi$
$269$	850614.	0.716724	0.358362	+	0.933583i	$0.383335\pi$
$270$	0	0
$271$	540128.	0.446759	0.223380	−	0.974732i	$-0.428291\pi$
$271$	540128.	0.446759	0.223380	+	0.974732i	$0.428291\pi$
$272$	0	0
$273$	200214.	0.162588
$274$	0	0
$275$	−669460.	−0.533818
$276$	0	0
$277$	513574.	0.402164	0.201082	−	0.979574i	$-0.435554\pi$
$277$	513574.	0.402164	0.201082	+	0.979574i	$0.435554\pi$
$278$	0	0
$279$	202176.	0.155496
$280$	0	0
$281$	−1.35642e6	−1.02478	−0.512388	−	0.858754i	$-0.671239\pi$
$281$	−1.35642e6	−1.02478	−0.512388	+	0.858754i	$0.671239\pi$
$282$	0	0
$283$	−286756.	−0.212837	−0.106418	−	0.994321i	$-0.533938\pi$
$283$	−286756.	−0.212837	−0.106418	+	0.994321i	$0.533938\pi$
$284$	0	0
$285$	272952.	0.199055
$286$	0	0
$287$	971866.	0.696469
$288$	0	0
$289$	−783053.	−0.551501
$290$	0	0
$291$	90018.0	0.0623156
$292$	0	0
$293$	−1.70727e6	−1.16180	−0.580901	−	0.813974i	$-0.697300\pi$
$293$	−1.70727e6	−1.16180	−0.580901	+	0.813974i	$0.697300\pi$
$294$	0	0
$295$	−1.44146e6	−0.964381
$296$	0	0
$297$	247860.	0.163048
$298$	0	0
$299$	1.44917e6	0.937434
$300$	0	0
$301$	844564.	0.537299
$302$	0	0
$303$	−978930.	−0.612555
$304$	0	0
$305$	−501772.	−0.308857
$306$	0	0
$307$	546788.	0.331111	0.165555	−	0.986201i	$-0.447058\pi$
$307$	546788.	0.331111	0.165555	+	0.986201i	$0.447058\pi$
$308$	0	0
$309$	1.79273e6	1.06812
$310$	0	0
$311$	−3.23426e6	−1.89616	−0.948079	−	0.318035i	$-0.896977\pi$
$311$	−3.23426e6	−1.89616	−0.948079	+	0.318035i	$0.896977\pi$
$312$	0	0
$313$	1.81313e6	1.04609	0.523044	−	0.852306i	$-0.324796\pi$
$313$	1.81313e6	1.04609	0.523044	+	0.852306i	$0.324796\pi$
$314$	0	0
$315$	−134946.	−0.0766273
$316$	0	0
$317$	−1.27658e6	−0.713509	−0.356754	−	0.934198i	$-0.616117\pi$
$317$	−1.27658e6	−0.713509	−0.356754	+	0.934198i	$0.616117\pi$
$318$	0	0
$319$	−2.80228e6	−1.54182
$320$	0	0
$321$	719748.	0.389868
$322$	0	0
$323$	711816.	0.379631
$324$	0	0
$325$	−893926.	−0.469454
$326$	0	0
$327$	−414882.	−0.214563
$328$	0	0
$329$	−437472.	−0.222823
$330$	0	0
$331$	1.73621e6	0.871029	0.435515	−	0.900182i	$-0.356566\pi$
$331$	1.73621e6	0.871029	0.435515	+	0.900182i	$0.356566\pi$
$332$	0	0
$333$	793638.	0.392204
$334$	0	0
$335$	−56984.0	−0.0277422
$336$	0	0
$337$	2.07215e6	0.993907	0.496953	−	0.867777i	$-0.334452\pi$
$337$	2.07215e6	0.993907	0.496953	+	0.867777i	$0.334452\pi$
$338$	0	0
$339$	2.36435e6	1.11741
$340$	0	0
$341$	848640.	0.395219
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	−976752.	−0.441811
$346$	0	0
$347$	1.65146e6	0.736282	0.368141	−	0.929770i	$-0.379994\pi$
$347$	1.65146e6	0.736282	0.368141	+	0.929770i	$0.379994\pi$
$348$	0	0
$349$	1.26645e6	0.556578	0.278289	−	0.960497i	$-0.410233\pi$
$349$	1.26645e6	0.556578	0.278289	+	0.960497i	$0.410233\pi$
$350$	0	0
$351$	330966.	0.143389
$352$	0	0
$353$	573218.	0.244840	0.122420	−	0.992478i	$-0.460934\pi$
$353$	573218.	0.244840	0.122420	+	0.992478i	$0.460934\pi$
$354$	0	0
$355$	495312.	0.208597
$356$	0	0
$357$	−351918.	−0.146141
$358$	0	0
$359$	−4.46322e6	−1.82773	−0.913866	−	0.406016i	$-0.866918\pi$
$359$	−4.46322e6	−1.82773	−0.913866	+	0.406016i	$0.866918\pi$
$360$	0	0
$361$	−1.68044e6	−0.678662
$362$	0	0
$363$	−409059.	−0.162937
$364$	0	0
$365$	−2.66485e6	−1.04699
$366$	0	0
$367$	4.50797e6	1.74709	0.873546	−	0.486742i	$-0.161815\pi$
$367$	4.50797e6	1.74709	0.873546	+	0.486742i	$0.161815\pi$
$368$	0	0
$369$	1.60655e6	0.614228
$370$	0	0
$371$	7350.00	0.00277238
$372$	0	0
$373$	1.66535e6	0.619774	0.309887	−	0.950773i	$-0.399709\pi$
$373$	1.66535e6	0.619774	0.309887	+	0.950773i	$0.399709\pi$
$374$	0	0
$375$	1.55876e6	0.572403
$376$	0	0
$377$	−3.74187e6	−1.35592
$378$	0	0
$379$	2.53232e6	0.905568	0.452784	−	0.891620i	$-0.350431\pi$
$379$	2.53232e6	0.905568	0.452784	+	0.891620i	$0.350431\pi$
$380$	0	0
$381$	−1.76947e6	−0.624498
$382$	0	0
$383$	−796368.	−0.277407	−0.138703	−	0.990334i	$-0.544293\pi$
$383$	−796368.	−0.277407	−0.138703	+	0.990334i	$0.544293\pi$
$384$	0	0
$385$	−566440.	−0.194761
$386$	0	0
$387$	1.39612e6	0.473853
$388$	0	0
$389$	1.94799e6	0.652699	0.326349	−	0.945249i	$-0.394181\pi$
$389$	1.94799e6	0.652699	0.326349	+	0.945249i	$0.394181\pi$
$390$	0	0
$391$	−2.54722e6	−0.842605
$392$	0	0
$393$	694260.	0.226747
$394$	0	0
$395$	−77248.0	−0.0249112
$396$	0	0
$397$	1.08116e6	0.344281	0.172140	−	0.985072i	$-0.444932\pi$
$397$	1.08116e6	0.344281	0.172140	+	0.985072i	$0.444932\pi$
$398$	0	0
$399$	−393372.	−0.123700
$400$	0	0
$401$	2.76770e6	0.859524	0.429762	−	0.902942i	$-0.358598\pi$
$401$	2.76770e6	0.859524	0.429762	+	0.902942i	$0.358598\pi$
$402$	0	0
$403$	1.13318e6	0.347566
$404$	0	0
$405$	−223074.	−0.0675789
$406$	0	0
$407$	3.33132e6	0.996851
$408$	0	0
$409$	2.36350e6	0.698630	0.349315	−	0.937005i	$-0.386414\pi$
$409$	2.36350e6	0.698630	0.349315	+	0.937005i	$0.386414\pi$
$410$	0	0
$411$	1.87353e6	0.547087
$412$	0	0
$413$	2.07740e6	0.599302
$414$	0	0
$415$	−1.28398e6	−0.365963
$416$	0	0
$417$	2.48022e6	0.698474
$418$	0	0
$419$	2.98669e6	0.831104	0.415552	−	0.909569i	$-0.363588\pi$
$419$	2.98669e6	0.831104	0.415552	+	0.909569i	$0.363588\pi$
$420$	0	0
$421$	−3.46331e6	−0.952326	−0.476163	−	0.879357i	$-0.657973\pi$
$421$	−3.46331e6	−0.952326	−0.476163	+	0.879357i	$0.657973\pi$
$422$	0	0
$423$	−723168.	−0.196512
$424$	0	0
$425$	1.57126e6	0.421965
$426$	0	0
$427$	723142.	0.191935
$428$	0	0
$429$	1.38924e6	0.364447
$430$	0	0
$431$	−2.33693e6	−0.605971	−0.302986	−	0.952995i	$-0.597983\pi$
$431$	−2.33693e6	−0.605971	−0.302986	+	0.952995i	$0.597983\pi$
$432$	0	0
$433$	−3.50838e6	−0.899264	−0.449632	−	0.893214i	$-0.648445\pi$
$433$	−3.50838e6	−0.899264	−0.449632	+	0.893214i	$0.648445\pi$
$434$	0	0
$435$	2.52205e6	0.639044
$436$	0	0
$437$	−2.84726e6	−0.713221
$438$	0	0
$439$	−3.54833e6	−0.878744	−0.439372	−	0.898305i	$-0.644799\pi$
$439$	−3.54833e6	−0.878744	−0.439372	+	0.898305i	$0.644799\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	0	0
$443$	−1.76833e6	−0.428109	−0.214055	−	0.976822i	$-0.568667\pi$
$443$	−1.76833e6	−0.428109	−0.214055	+	0.976822i	$0.568667\pi$
$444$	0	0
$445$	3.98772e6	0.954608
$446$	0	0
$447$	−2.66495e6	−0.630842
$448$	0	0
$449$	−5.52579e6	−1.29354	−0.646768	−	0.762687i	$-0.723880\pi$
$449$	−5.52579e6	−1.29354	−0.646768	+	0.762687i	$0.723880\pi$
$450$	0	0
$451$	6.74356e6	1.56116
$452$	0	0
$453$	3.83825e6	0.878795
$454$	0	0
$455$	−756364.	−0.171278
$456$	0	0
$457$	−2.96226e6	−0.663488	−0.331744	−	0.943369i	$-0.607637\pi$
$457$	−2.96226e6	−0.663488	−0.331744	+	0.943369i	$0.607637\pi$
$458$	0	0
$459$	−581742.	−0.128884
$460$	0	0
$461$	2.11884e6	0.464350	0.232175	−	0.972674i	$-0.425416\pi$
$461$	2.11884e6	0.464350	0.232175	+	0.972674i	$0.425416\pi$
$462$	0	0
$463$	−3.19226e6	−0.692062	−0.346031	−	0.938223i	$-0.612471\pi$
$463$	−3.19226e6	−0.692062	−0.346031	+	0.938223i	$0.612471\pi$
$464$	0	0
$465$	−763776.	−0.163807
$466$	0	0
$467$	7.42621e6	1.57571	0.787853	−	0.615863i	$-0.211193\pi$
$467$	7.42621e6	1.57571	0.787853	+	0.615863i	$0.211193\pi$
$468$	0	0
$469$	82124.0	0.0172400
$470$	0	0
$471$	1.60637e6	0.333653
$472$	0	0
$473$	5.86024e6	1.20438
$474$	0	0
$475$	1.75635e6	0.357171
$476$	0	0
$477$	12150.0	0.00244501
$478$	0	0
$479$	3.39685e6	0.676453	0.338226	−	0.941065i	$-0.390173\pi$
$479$	3.39685e6	0.676453	0.338226	+	0.941065i	$0.390173\pi$
$480$	0	0
$481$	4.44829e6	0.876659
$482$	0	0
$483$	1.40767e6	0.274558
$484$	0	0
$485$	−340068.	−0.0656465
$486$	0	0
$487$	3.71382e6	0.709574	0.354787	−	0.934947i	$-0.384553\pi$
$487$	3.71382e6	0.709574	0.354787	+	0.934947i	$0.384553\pi$
$488$	0	0
$489$	−2.27495e6	−0.430229
$490$	0	0
$491$	−5.57494e6	−1.04361	−0.521803	−	0.853066i	$-0.674740\pi$
$491$	−5.57494e6	−1.04361	−0.521803	+	0.853066i	$0.674740\pi$
$492$	0	0
$493$	6.57712e6	1.21876
$494$	0	0
$495$	−936360.	−0.171763
$496$	0	0
$497$	−713832.	−0.129630
$498$	0	0
$499$	−3.92698e6	−0.706004	−0.353002	−	0.935623i	$-0.614839\pi$
$499$	−3.92698e6	−0.706004	−0.353002	+	0.935623i	$0.614839\pi$
$500$	0	0
$501$	−4.57279e6	−0.813930
$502$	0	0
$503$	−6.42079e6	−1.13154	−0.565768	−	0.824564i	$-0.691420\pi$
$503$	−6.42079e6	−1.13154	−0.565768	+	0.824564i	$0.691420\pi$
$504$	0	0
$505$	3.69818e6	0.645297
$506$	0	0
$507$	−1.48659e6	−0.256846
$508$	0	0
$509$	146278.	0.0250256	0.0125128	−	0.999922i	$-0.496017\pi$
$509$	146278.	0.0250256	0.0125128	+	0.999922i	$0.496017\pi$
$510$	0	0
$511$	3.84052e6	0.650636
$512$	0	0
$513$	−650268.	−0.109094
$514$	0	0
$515$	−6.77253e6	−1.12521
$516$	0	0
$517$	−3.03552e6	−0.499467
$518$	0	0
$519$	−1.99651e6	−0.325351
$520$	0	0
$521$	7.70937e6	1.24430	0.622149	−	0.782899i	$-0.286260\pi$
$521$	7.70937e6	1.24430	0.622149	+	0.782899i	$0.286260\pi$
$522$	0	0
$523$	569420.	0.0910287	0.0455144	−	0.998964i	$-0.485507\pi$
$523$	569420.	0.0910287	0.0455144	+	0.998964i	$0.485507\pi$
$524$	0	0
$525$	−868329.	−0.137495
$526$	0	0
$527$	−1.99181e6	−0.312407
$528$	0	0
$529$	3.75252e6	0.583021
$530$	0	0
$531$	3.43408e6	0.528535
$532$	0	0
$533$	9.00464e6	1.37293
$534$	0	0
$535$	−2.71905e6	−0.410707
$536$	0	0
$537$	1.02208e6	0.152949
$538$	0	0
$539$	816340.	0.121032
$540$	0	0
$541$	−9.44802e6	−1.38787	−0.693933	−	0.720040i	$-0.744124\pi$
$541$	−9.44802e6	−1.38787	−0.693933	+	0.720040i	$0.744124\pi$
$542$	0	0
$543$	5.96806e6	0.868628
$544$	0	0
$545$	1.56733e6	0.226032
$546$	0	0
$547$	1.35321e6	0.193374	0.0966869	−	0.995315i	$-0.469175\pi$
$547$	1.35321e6	0.193374	0.0966869	+	0.995315i	$0.469175\pi$
$548$	0	0
$549$	1.19540e6	0.169271
$550$	0	0
$551$	7.35186e6	1.03162
$552$	0	0
$553$	111328.	0.0154807
$554$	0	0
$555$	−2.99819e6	−0.413168
$556$	0	0
$557$	8.19390e6	1.11906	0.559529	−	0.828811i	$-0.310982\pi$
$557$	8.19390e6	1.11906	0.559529	+	0.828811i	$0.310982\pi$
$558$	0	0
$559$	7.82514e6	1.05916
$560$	0	0
$561$	−2.44188e6	−0.327580
$562$	0	0
$563$	1.05796e7	1.40669	0.703347	−	0.710847i	$-0.251688\pi$
$563$	1.05796e7	1.40669	0.703347	+	0.710847i	$0.251688\pi$
$564$	0	0
$565$	−8.93200e6	−1.17714
$566$	0	0
$567$	321489.	0.0419961
$568$	0	0
$569$	−1.20205e7	−1.55648	−0.778238	−	0.627969i	$-0.783886\pi$
$569$	−1.20205e7	−1.55648	−0.778238	+	0.627969i	$0.783886\pi$
$570$	0	0
$571$	2.48948e6	0.319534	0.159767	−	0.987155i	$-0.448926\pi$
$571$	2.48948e6	0.319534	0.159767	+	0.987155i	$0.448926\pi$
$572$	0	0
$573$	−4.55098e6	−0.579053
$574$	0	0
$575$	−6.28505e6	−0.792755
$576$	0	0
$577$	8.21322e6	1.02701	0.513504	−	0.858087i	$-0.328347\pi$
$577$	8.21322e6	1.02701	0.513504	+	0.858087i	$0.328347\pi$
$578$	0	0
$579$	−3.89144e6	−0.482407
$580$	0	0
$581$	1.85044e6	0.227423
$582$	0	0
$583$	51000.0	0.00621439
$584$	0	0
$585$	−1.25032e6	−0.151053
$586$	0	0
$587$	1.21827e6	0.145931	0.0729655	−	0.997334i	$-0.476754\pi$
$587$	1.21827e6	0.145931	0.0729655	+	0.997334i	$0.476754\pi$
$588$	0	0
$589$	−2.22643e6	−0.264436
$590$	0	0
$591$	−1.18766e6	−0.139869
$592$	0	0
$593$	−8.42379e6	−0.983718	−0.491859	−	0.870675i	$-0.663683\pi$
$593$	−8.42379e6	−0.983718	−0.491859	+	0.870675i	$0.663683\pi$
$594$	0	0
$595$	1.32947e6	0.153952
$596$	0	0
$597$	−2.68682e6	−0.308534
$598$	0	0
$599$	−8.21254e6	−0.935212	−0.467606	−	0.883937i	$-0.654883\pi$
$599$	−8.21254e6	−0.935212	−0.467606	+	0.883937i	$0.654883\pi$
$600$	0	0
$601$	3.25478e6	0.367566	0.183783	−	0.982967i	$-0.441166\pi$
$601$	3.25478e6	0.367566	0.183783	+	0.982967i	$0.441166\pi$
$602$	0	0
$603$	135756.	0.0152043
$604$	0	0
$605$	1.54533e6	0.171646
$606$	0	0
$607$	−7.82101e6	−0.861571	−0.430785	−	0.902454i	$-0.641763\pi$
$607$	−7.82101e6	−0.861571	−0.430785	+	0.902454i	$0.641763\pi$
$608$	0	0
$609$	−3.63472e6	−0.397126
$610$	0	0
$611$	−4.05331e6	−0.439245
$612$	0	0
$613$	−9.51670e6	−1.02290	−0.511452	−	0.859312i	$-0.670892\pi$
$613$	−9.51670e6	−1.02290	−0.511452	+	0.859312i	$0.670892\pi$
$614$	0	0
$615$	−6.06920e6	−0.647059
$616$	0	0
$617$	−7.04895e6	−0.745438	−0.372719	−	0.927944i	$-0.621574\pi$
$617$	−7.04895e6	−0.745438	−0.372719	+	0.927944i	$0.621574\pi$
$618$	0	0
$619$	6.32174e6	0.663147	0.331574	−	0.943429i	$-0.392420\pi$
$619$	6.32174e6	0.663147	0.331574	+	0.943429i	$0.392420\pi$
$620$	0	0
$621$	2.32697e6	0.242137
$622$	0	0
$623$	−5.74701e6	−0.593229
$624$	0	0
$625$	264461.	0.0270808
$626$	0	0
$627$	−2.72952e6	−0.277279
$628$	0	0
$629$	−7.81880e6	−0.787977
$630$	0	0
$631$	−8.61236e6	−0.861090	−0.430545	−	0.902569i	$-0.641679\pi$
$631$	−8.61236e6	−0.861090	−0.430545	+	0.902569i	$0.641679\pi$
$632$	0	0
$633$	1.05356e7	1.04508
$634$	0	0
$635$	6.68467e6	0.657879
$636$	0	0
$637$	1.09005e6	0.106439
$638$	0	0
$639$	−1.18001e6	−0.114323
$640$	0	0
$641$	−5.22829e6	−0.502590	−0.251295	−	0.967910i	$-0.580857\pi$
$641$	−5.22829e6	−0.502590	−0.251295	+	0.967910i	$0.580857\pi$
$642$	0	0
$643$	−1.61373e7	−1.53923	−0.769615	−	0.638508i	$-0.779552\pi$
$643$	−1.61373e7	−1.53923	−0.769615	+	0.638508i	$0.779552\pi$
$644$	0	0
$645$	−5.27422e6	−0.499182
$646$	0	0
$647$	1.58749e7	1.49090	0.745451	−	0.666560i	$-0.232234\pi$
$647$	1.58749e7	1.49090	0.745451	+	0.666560i	$0.232234\pi$
$648$	0	0
$649$	1.44146e7	1.34336
$650$	0	0
$651$	1.10074e6	0.101796
$652$	0	0
$653$	−5.94112e6	−0.545237	−0.272619	−	0.962122i	$-0.587890\pi$
$653$	−5.94112e6	−0.545237	−0.272619	+	0.962122i	$0.587890\pi$
$654$	0	0
$655$	−2.62276e6	−0.238867
$656$	0	0
$657$	6.34862e6	0.573807
$658$	0	0
$659$	7.64430e6	0.685684	0.342842	−	0.939393i	$-0.388610\pi$
$659$	7.64430e6	0.685684	0.342842	+	0.939393i	$0.388610\pi$
$660$	0	0
$661$	−7.58688e6	−0.675398	−0.337699	−	0.941254i	$-0.609649\pi$
$661$	−7.58688e6	−0.675398	−0.337699	+	0.941254i	$0.609649\pi$
$662$	0	0
$663$	−3.26063e6	−0.288083
$664$	0	0
$665$	1.48607e6	0.130312
$666$	0	0
$667$	−2.63085e7	−2.28971
$668$	0	0
$669$	−3.59438e6	−0.310498
$670$	0	0
$671$	5.01772e6	0.430229
$672$	0	0
$673$	−2.06681e7	−1.75899	−0.879494	−	0.475910i	$-0.842119\pi$
$673$	−2.06681e7	−1.75899	−0.879494	+	0.475910i	$0.842119\pi$
$674$	0	0
$675$	−1.43540e6	−0.121259
$676$	0	0
$677$	7.89541e6	0.662068	0.331034	−	0.943619i	$-0.392602\pi$
$677$	7.89541e6	0.662068	0.331034	+	0.943619i	$0.392602\pi$
$678$	0	0
$679$	490098.	0.0407951
$680$	0	0
$681$	−6.37124e6	−0.526449
$682$	0	0
$683$	1.96015e7	1.60782	0.803911	−	0.594750i	$-0.202749\pi$
$683$	1.96015e7	1.60782	0.803911	+	0.594750i	$0.202749\pi$
$684$	0	0
$685$	−7.07778e6	−0.576329
$686$	0	0
$687$	−6.62200e6	−0.535300
$688$	0	0
$689$	68100.0	0.00546511
$690$	0	0
$691$	1.72710e7	1.37601	0.688005	−	0.725706i	$-0.258487\pi$
$691$	1.72710e7	1.37601	0.688005	+	0.725706i	$0.258487\pi$
$692$	0	0
$693$	1.34946e6	0.106740
$694$	0	0
$695$	−9.36972e6	−0.735808
$696$	0	0
$697$	−1.58275e7	−1.23405
$698$	0	0
$699$	−1.87882e6	−0.145443
$700$	0	0
$701$	−5.36344e6	−0.412238	−0.206119	−	0.978527i	$-0.566083\pi$
$701$	−5.36344e6	−0.412238	−0.206119	+	0.978527i	$0.566083\pi$
$702$	0	0
$703$	−8.73982e6	−0.666982
$704$	0	0
$705$	2.73197e6	0.207015
$706$	0	0
$707$	−5.32973e6	−0.401011
$708$	0	0
$709$	−1.73733e7	−1.29798	−0.648988	−	0.760798i	$-0.724808\pi$
$709$	−1.73733e7	−1.29798	−0.648988	+	0.760798i	$0.724808\pi$
$710$	0	0
$711$	184032.	0.0136527
$712$	0	0
$713$	7.96723e6	0.586926
$714$	0	0
$715$	−5.24824e6	−0.383927
$716$	0	0
$717$	−6.42038e6	−0.466405
$718$	0	0
$719$	−424608.	−0.0306313	−0.0153157	−	0.999883i	$-0.504875\pi$
$719$	−424608.	−0.0306313	−0.0153157	+	0.999883i	$0.504875\pi$
$720$	0	0
$721$	9.76041e6	0.699246
$722$	0	0
$723$	−4.54721e6	−0.323519
$724$	0	0
$725$	1.62285e7	1.14666
$726$	0	0
$727$	−2.18290e7	−1.53179	−0.765893	−	0.642968i	$-0.777703\pi$
$727$	−2.18290e7	−1.53179	−0.765893	+	0.642968i	$0.777703\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.37543e7	−0.952020
$732$	0	0
$733$	2.17675e7	1.49640	0.748202	−	0.663470i	$-0.230917\pi$
$733$	2.17675e7	1.49640	0.748202	+	0.663470i	$0.230917\pi$
$734$	0	0
$735$	−734706.	−0.0501644
$736$	0	0
$737$	569840.	0.0386442
$738$	0	0
$739$	−6.21786e6	−0.418822	−0.209411	−	0.977828i	$-0.567155\pi$
$739$	−6.21786e6	−0.418822	−0.209411	+	0.977828i	$0.567155\pi$
$740$	0	0
$741$	−3.64471e6	−0.243847
$742$	0	0
$743$	−3.77647e6	−0.250966	−0.125483	−	0.992096i	$-0.540048\pi$
$743$	−3.77647e6	−0.250966	−0.125483	+	0.992096i	$0.540048\pi$
$744$	0	0
$745$	1.00676e7	0.664562
$746$	0	0
$747$	3.05888e6	0.200568
$748$	0	0
$749$	3.91863e6	0.255229
$750$	0	0
$751$	2.88795e6	0.186849	0.0934244	−	0.995626i	$-0.470219\pi$
$751$	2.88795e6	0.186849	0.0934244	+	0.995626i	$0.470219\pi$
$752$	0	0
$753$	−2.85397e6	−0.183427
$754$	0	0
$755$	−1.45000e7	−0.925768
$756$	0	0
$757$	1.25519e6	0.0796104	0.0398052	−	0.999207i	$-0.487326\pi$
$757$	1.25519e6	0.0796104	0.0398052	+	0.999207i	$0.487326\pi$
$758$	0	0
$759$	9.76752e6	0.615432
$760$	0	0
$761$	−1.42623e7	−0.892746	−0.446373	−	0.894847i	$-0.647284\pi$
$761$	−1.42623e7	−0.892746	−0.446373	+	0.894847i	$0.647284\pi$
$762$	0	0
$763$	−2.25880e6	−0.140465
$764$	0	0
$765$	2.19769e6	0.135773
$766$	0	0
$767$	1.92478e7	1.18139
$768$	0	0
$769$	−2.02261e7	−1.23338	−0.616689	−	0.787207i	$-0.711526\pi$
$769$	−2.02261e7	−1.23338	−0.616689	+	0.787207i	$0.711526\pi$
$770$	0	0
$771$	−1.29856e7	−0.786732
$772$	0	0
$773$	2.62288e7	1.57881	0.789406	−	0.613872i	$-0.210389\pi$
$773$	2.62288e7	1.57881	0.789406	+	0.613872i	$0.210389\pi$
$774$	0	0
$775$	−4.91462e6	−0.293925
$776$	0	0
$777$	4.32092e6	0.256758
$778$	0	0
$779$	−1.76919e7	−1.04456
$780$	0	0
$781$	−4.95312e6	−0.290570
$782$	0	0
$783$	−6.00842e6	−0.350232
$784$	0	0
$785$	−6.06852e6	−0.351487
$786$	0	0
$787$	9.92829e6	0.571397	0.285698	−	0.958320i	$-0.407774\pi$
$787$	9.92829e6	0.571397	0.285698	+	0.958320i	$0.407774\pi$
$788$	0	0
$789$	−2.44346e6	−0.139738
$790$	0	0
$791$	1.28726e7	0.731518
$792$	0	0
$793$	6.70013e6	0.378356
$794$	0	0
$795$	−45900.0	−0.00257570
$796$	0	0
$797$	1.09033e7	0.608014	0.304007	−	0.952670i	$-0.401675\pi$
$797$	1.09033e7	0.608014	0.304007	+	0.952670i	$0.401675\pi$
$798$	0	0
$799$	7.12454e6	0.394812
$800$	0	0
$801$	−9.50017e6	−0.523179
$802$	0	0
$803$	2.66485e7	1.45843
$804$	0	0
$805$	−5.31787e6	−0.289233
$806$	0	0
$807$	7.65553e6	0.413801
$808$	0	0
$809$	6.06398e6	0.325751	0.162876	−	0.986647i	$-0.447923\pi$
$809$	6.06398e6	0.325751	0.162876	+	0.986647i	$0.447923\pi$
$810$	0	0
$811$	8.59438e6	0.458841	0.229421	−	0.973327i	$-0.426317\pi$
$811$	8.59438e6	0.458841	0.229421	+	0.973327i	$0.426317\pi$
$812$	0	0
$813$	4.86115e6	0.257937
$814$	0	0
$815$	8.59425e6	0.453225
$816$	0	0
$817$	−1.53745e7	−0.805835
$818$	0	0
$819$	1.80193e6	0.0938701
$820$	0	0
$821$	−2.01396e6	−0.104278	−0.0521391	−	0.998640i	$-0.516604\pi$
$821$	−2.01396e6	−0.104278	−0.0521391	+	0.998640i	$0.516604\pi$
$822$	0	0
$823$	2.64679e7	1.36213	0.681067	−	0.732221i	$-0.261516\pi$
$823$	2.64679e7	1.36213	0.681067	+	0.732221i	$0.261516\pi$
$824$	0	0
$825$	−6.02514e6	−0.308200
$826$	0	0
$827$	3.90229e6	0.198407	0.0992033	−	0.995067i	$-0.468371\pi$
$827$	3.90229e6	0.198407	0.0992033	+	0.995067i	$0.468371\pi$
$828$	0	0
$829$	−1.95595e7	−0.988487	−0.494244	−	0.869323i	$-0.664555\pi$
$829$	−1.95595e7	−0.988487	−0.494244	+	0.869323i	$0.664555\pi$
$830$	0	0
$831$	4.62217e6	0.232190
$832$	0	0
$833$	−1.91600e6	−0.0956715
$834$	0	0
$835$	1.72750e7	0.857436
$836$	0	0
$837$	1.81958e6	0.0897756
$838$	0	0
$839$	2.45448e7	1.20380	0.601901	−	0.798570i	$-0.294410\pi$
$839$	2.45448e7	1.20380	0.601901	+	0.798570i	$0.294410\pi$
$840$	0	0
$841$	4.74194e7	2.31188
$842$	0	0
$843$	−1.22078e7	−0.591655
$844$	0	0
$845$	5.61602e6	0.270574
$846$	0	0
$847$	−2.22710e6	−0.106667
$848$	0	0
$849$	−2.58080e6	−0.122881
$850$	0	0
$851$	3.12752e7	1.48039
$852$	0	0
$853$	3.38305e7	1.59197	0.795987	−	0.605314i	$-0.206952\pi$
$853$	3.38305e7	1.59197	0.795987	+	0.605314i	$0.206952\pi$
$854$	0	0
$855$	2.45657e6	0.114925
$856$	0	0
$857$	3.18009e7	1.47907	0.739534	−	0.673120i	$-0.235046\pi$
$857$	3.18009e7	1.47907	0.739534	+	0.673120i	$0.235046\pi$
$858$	0	0
$859$	−638420.	−0.0295205	−0.0147602	−	0.999891i	$-0.504699\pi$
$859$	−638420.	−0.0295205	−0.0147602	+	0.999891i	$0.504699\pi$
$860$	0	0
$861$	8.74679e6	0.402106
$862$	0	0
$863$	4.22256e6	0.192996	0.0964981	−	0.995333i	$-0.469236\pi$
$863$	4.22256e6	0.192996	0.0964981	+	0.995333i	$0.469236\pi$
$864$	0	0
$865$	7.54236e6	0.342742
$866$	0	0
$867$	−7.04748e6	−0.318409
$868$	0	0
$869$	772480.	0.0347007
$870$	0	0
$871$	760904.	0.0339848
$872$	0	0
$873$	810162.	0.0359779
$874$	0	0
$875$	8.48660e6	0.374726
$876$	0	0
$877$	−2.45043e7	−1.07583	−0.537915	−	0.842999i	$-0.680788\pi$
$877$	−2.45043e7	−1.07583	−0.537915	+	0.842999i	$0.680788\pi$
$878$	0	0
$879$	−1.53654e7	−0.670767
$880$	0	0
$881$	−2.77630e7	−1.20511	−0.602555	−	0.798078i	$-0.705850\pi$
$881$	−2.77630e7	−1.20511	−0.602555	+	0.798078i	$0.705850\pi$
$882$	0	0
$883$	−3.30170e7	−1.42507	−0.712534	−	0.701638i	$-0.752452\pi$
$883$	−3.30170e7	−1.42507	−0.712534	+	0.701638i	$0.752452\pi$
$884$	0	0
$885$	−1.29732e7	−0.556786
$886$	0	0
$887$	−4.34462e6	−0.185414	−0.0927070	−	0.995693i	$-0.529552\pi$
$887$	−4.34462e6	−0.185414	−0.0927070	+	0.995693i	$0.529552\pi$
$888$	0	0
$889$	−9.63379e6	−0.408830
$890$	0	0
$891$	2.23074e6	0.0941358
$892$	0	0
$893$	7.96378e6	0.334188
$894$	0	0
$895$	−3.86118e6	−0.161125
$896$	0	0
$897$	1.30425e7	0.541228
$898$	0	0
$899$	−2.05720e7	−0.848942
$900$	0	0
$901$	−119700.	−0.00491227
$902$	0	0
$903$	7.60108e6	0.310210
$904$	0	0
$905$	−2.25460e7	−0.915057
$906$	0	0
$907$	−1.96499e7	−0.793128	−0.396564	−	0.918007i	$-0.629797\pi$
$907$	−1.96499e7	−0.793128	−0.396564	+	0.918007i	$0.629797\pi$
$908$	0	0
$909$	−8.81037e6	−0.353659
$910$	0	0
$911$	7.26518e6	0.290035	0.145018	−	0.989429i	$-0.453676\pi$
$911$	7.26518e6	0.290035	0.145018	+	0.989429i	$0.453676\pi$
$912$	0	0
$913$	1.28398e7	0.509777
$914$	0	0
$915$	−4.51595e6	−0.178318
$916$	0	0
$917$	3.77986e6	0.148440
$918$	0	0
$919$	−9.82532e6	−0.383758	−0.191879	−	0.981419i	$-0.561458\pi$
$919$	−9.82532e6	−0.383758	−0.191879	+	0.981419i	$0.561458\pi$
$920$	0	0
$921$	4.92109e6	0.191167
$922$	0	0
$923$	−6.61387e6	−0.255536
$924$	0	0
$925$	−1.92923e7	−0.741359
$926$	0	0
$927$	1.61346e7	0.616677
$928$	0	0
$929$	2.71152e7	1.03080	0.515399	−	0.856951i	$-0.327644\pi$
$929$	2.71152e7	1.03080	0.515399	+	0.856951i	$0.327644\pi$
$930$	0	0
$931$	−2.14169e6	−0.0809809
$932$	0	0
$933$	−2.91084e7	−1.09475
$934$	0	0
$935$	9.22488e6	0.345089
$936$	0	0
$937$	−4.53522e7	−1.68752	−0.843761	−	0.536720i	$-0.819663\pi$
$937$	−4.53522e7	−1.68752	−0.843761	+	0.536720i	$0.819663\pi$
$938$	0	0
$939$	1.63182e7	0.603959
$940$	0	0
$941$	4.65780e7	1.71477	0.857387	−	0.514672i	$-0.172086\pi$
$941$	4.65780e7	1.71477	0.857387	+	0.514672i	$0.172086\pi$
$942$	0	0
$943$	6.33101e7	2.31843
$944$	0	0
$945$	−1.21451e6	−0.0442408
$946$	0	0
$947$	−2.53799e7	−0.919632	−0.459816	−	0.888014i	$-0.652085\pi$
$947$	−2.53799e7	−0.919632	−0.459816	+	0.888014i	$0.652085\pi$
$948$	0	0
$949$	3.55836e7	1.28258
$950$	0	0
$951$	−1.14892e7	−0.411944
$952$	0	0
$953$	1.52948e7	0.545520	0.272760	−	0.962082i	$-0.412063\pi$
$953$	1.52948e7	0.545520	0.272760	+	0.962082i	$0.412063\pi$
$954$	0	0
$955$	1.71926e7	0.610004
$956$	0	0
$957$	−2.52205e7	−0.890173
$958$	0	0
$959$	1.02003e7	0.358152
$960$	0	0
$961$	−2.23991e7	−0.782389
$962$	0	0
$963$	6.47773e6	0.225091
$964$	0	0
$965$	1.47010e7	0.508192
$966$	0	0
$967$	5.71465e6	0.196527	0.0982637	−	0.995160i	$-0.468671\pi$
$967$	5.71465e6	0.196527	0.0982637	+	0.995160i	$0.468671\pi$
$968$	0	0
$969$	6.40634e6	0.219180
$970$	0	0
$971$	−1.30250e7	−0.443332	−0.221666	−	0.975123i	$-0.571149\pi$
$971$	−1.30250e7	−0.443332	−0.221666	+	0.975123i	$0.571149\pi$
$972$	0	0
$973$	1.35034e7	0.457258
$974$	0	0
$975$	−8.04533e6	−0.271039
$976$	0	0
$977$	−1.70360e7	−0.570992	−0.285496	−	0.958380i	$-0.592158\pi$
$977$	−1.70360e7	−0.570992	−0.285496	+	0.958380i	$0.592158\pi$
$978$	0	0
$979$	−3.98772e7	−1.32974
$980$	0	0
$981$	−3.73394e6	−0.123878
$982$	0	0
$983$	1.36985e7	0.452156	0.226078	−	0.974109i	$-0.427410\pi$
$983$	1.36985e7	0.452156	0.226078	+	0.974109i	$0.427410\pi$
$984$	0	0
$985$	4.48671e6	0.147346
$986$	0	0
$987$	−3.93725e6	−0.128647
$988$	0	0
$989$	5.50173e7	1.78858
$990$	0	0
$991$	3.49088e7	1.12915	0.564574	−	0.825383i	$-0.309041\pi$
$991$	3.49088e7	1.12915	0.564574	+	0.825383i	$0.309041\pi$
$992$	0	0
$993$	1.56259e7	0.502889
$994$	0	0
$995$	1.01502e7	0.325026
$996$	0	0
$997$	875662.	0.0278996	0.0139498	−	0.999903i	$-0.495559\pi$
$997$	875662.	0.0278996	0.0139498	+	0.999903i	$0.495559\pi$
$998$	0	0
$999$	7.14274e6	0.226439

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.a.l.1.1		1
3.2	odd	2		1008.6.a.t.1.1		1
4.3	odd	2		21.6.a.b.1.1	✓	1
12.11	even	2		63.6.a.c.1.1		1
20.3	even	4		525.6.d.d.274.1		2
20.7	even	4		525.6.d.d.274.2		2
20.19	odd	2		525.6.a.c.1.1		1
28.3	even	6		147.6.e.e.79.1		2
28.11	odd	6		147.6.e.f.79.1		2
28.19	even	6		147.6.e.e.67.1		2
28.23	odd	6		147.6.e.f.67.1		2
28.27	even	2		147.6.a.e.1.1		1
84.83	odd	2		441.6.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.a.b.1.1	✓	1	4.3	odd	2
63.6.a.c.1.1		1	12.11	even	2
147.6.a.e.1.1		1	28.27	even	2
147.6.e.e.67.1		2	28.19	even	6
147.6.e.e.79.1		2	28.3	even	6
147.6.e.f.67.1		2	28.23	odd	6
147.6.e.f.79.1		2	28.11	odd	6
336.6.a.l.1.1		1	1.1	even	1	trivial
441.6.a.d.1.1		1	84.83	odd	2
525.6.a.c.1.1		1	20.19	odd	2
525.6.d.d.274.1		2	20.3	even	4
525.6.d.d.274.2		2	20.7	even	4
1008.6.a.t.1.1		1	3.2	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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more