LMFDB - Embedded newform 56.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [56,6,Mod(1,56)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(56, 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("56.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 56 = 2^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	56.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:	$8.98149390953$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	56.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-6.00000 q^{3} +4.00000 q^{5} +49.0000 q^{7} -207.000 q^{9} +O(q^{10})$

$q-6.00000 q^{3} +4.00000 q^{5} +49.0000 q^{7} -207.000 q^{9} -240.000 q^{11} -744.000 q^{13} -24.0000 q^{15} -1042.00 q^{17} -986.000 q^{19} -294.000 q^{21} +184.000 q^{23} -3109.00 q^{25} +2700.00 q^{27} -734.000 q^{29} +5140.00 q^{31} +1440.00 q^{33} +196.000 q^{35} -6054.00 q^{37} +4464.00 q^{39} +7598.00 q^{41} +13016.0 q^{43} -828.000 q^{45} +14668.0 q^{47} +2401.00 q^{49} +6252.00 q^{51} -14522.0 q^{53} -960.000 q^{55} +5916.00 q^{57} -13362.0 q^{59} +9676.00 q^{61} -10143.0 q^{63} -2976.00 q^{65} -62124.0 q^{67} -1104.00 q^{69} -2112.00 q^{71} -28910.0 q^{73} +18654.0 q^{75} -11760.0 q^{77} -101768. q^{79} +34101.0 q^{81} -23922.0 q^{83} -4168.00 q^{85} +4404.00 q^{87} +141674. q^{89} -36456.0 q^{91} -30840.0 q^{93} -3944.00 q^{95} +99982.0 q^{97} +49680.0 q^{99} +O(q^{100})$

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
$3$	−6.00000	−0.384900	−0.192450	−	0.981307i	$-0.561643\pi$
$3$	−6.00000	−0.384900	−0.192450	+	0.981307i	$0.561643\pi$
$4$	0	0
$5$	4.00000	0.0715542	0.0357771	−	0.999360i	$-0.488609\pi$
$5$	4.00000	0.0715542	0.0357771	+	0.999360i	$0.488609\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	−207.000	−0.851852
$10$	0	0
$11$	−240.000	−0.598039	−0.299020	−	0.954247i	$-0.596660\pi$
$11$	−240.000	−0.598039	−0.299020	+	0.954247i	$0.596660\pi$
$12$	0	0
$13$	−744.000	−1.22100	−0.610498	−	0.792017i	$-0.709031\pi$
$13$	−744.000	−1.22100	−0.610498	+	0.792017i	$0.709031\pi$
$14$	0	0
$15$	−24.0000	−0.0275412
$16$	0	0
$17$	−1042.00	−0.874471	−0.437236	−	0.899347i	$-0.644042\pi$
$17$	−1042.00	−0.874471	−0.437236	+	0.899347i	$0.644042\pi$
$18$	0	0
$19$	−986.000	−0.626604	−0.313302	−	0.949654i	$-0.601435\pi$
$19$	−986.000	−0.626604	−0.313302	+	0.949654i	$0.601435\pi$
$20$	0	0
$21$	−294.000	−0.145479
$22$	0	0
$23$	184.000	0.0725268	0.0362634	−	0.999342i	$-0.488454\pi$
$23$	184.000	0.0725268	0.0362634	+	0.999342i	$0.488454\pi$
$24$	0	0
$25$	−3109.00	−0.994880
$26$	0	0
$27$	2700.00	0.712778
$28$	0	0
$29$	−734.000	−0.162069	−0.0810347	−	0.996711i	$-0.525822\pi$
$29$	−734.000	−0.162069	−0.0810347	+	0.996711i	$0.525822\pi$
$30$	0	0
$31$	5140.00	0.960636	0.480318	−	0.877094i	$-0.340521\pi$
$31$	5140.00	0.960636	0.480318	+	0.877094i	$0.340521\pi$
$32$	0	0
$33$	1440.00	0.230185
$34$	0	0
$35$	196.000	0.0270449
$36$	0	0
$37$	−6054.00	−0.727006	−0.363503	−	0.931593i	$-0.618419\pi$
$37$	−6054.00	−0.727006	−0.363503	+	0.931593i	$0.618419\pi$
$38$	0	0
$39$	4464.00	0.469962
$40$	0	0
$41$	7598.00	0.705894	0.352947	−	0.935643i	$-0.385180\pi$
$41$	7598.00	0.705894	0.352947	+	0.935643i	$0.385180\pi$
$42$	0	0
$43$	13016.0	1.07351	0.536755	−	0.843738i	$-0.319650\pi$
$43$	13016.0	1.07351	0.536755	+	0.843738i	$0.319650\pi$
$44$	0	0
$45$	−828.000	−0.0609536
$46$	0	0
$47$	14668.0	0.968559	0.484280	−	0.874913i	$-0.339082\pi$
$47$	14668.0	0.968559	0.484280	+	0.874913i	$0.339082\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	6252.00	0.336584
$52$	0	0
$53$	−14522.0	−0.710128	−0.355064	−	0.934842i	$-0.615541\pi$
$53$	−14522.0	−0.710128	−0.355064	+	0.934842i	$0.615541\pi$
$54$	0	0
$55$	−960.000	−0.0427922
$56$	0	0
$57$	5916.00	0.241180
$58$	0	0
$59$	−13362.0	−0.499737	−0.249868	−	0.968280i	$-0.580387\pi$
$59$	−13362.0	−0.499737	−0.249868	+	0.968280i	$0.580387\pi$
$60$	0	0
$61$	9676.00	0.332944	0.166472	−	0.986046i	$-0.446762\pi$
$61$	9676.00	0.332944	0.166472	+	0.986046i	$0.446762\pi$
$62$	0	0
$63$	−10143.0	−0.321970
$64$	0	0
$65$	−2976.00	−0.0873674
$66$	0	0
$67$	−62124.0	−1.69072	−0.845361	−	0.534195i	$-0.820615\pi$
$67$	−62124.0	−1.69072	−0.845361	+	0.534195i	$0.820615\pi$
$68$	0	0
$69$	−1104.00	−0.0279156
$70$	0	0
$71$	−2112.00	−0.0497219	−0.0248610	−	0.999691i	$-0.507914\pi$
$71$	−2112.00	−0.0497219	−0.0248610	+	0.999691i	$0.507914\pi$
$72$	0	0
$73$	−28910.0	−0.634952	−0.317476	−	0.948266i	$-0.602835\pi$
$73$	−28910.0	−0.634952	−0.317476	+	0.948266i	$0.602835\pi$
$74$	0	0
$75$	18654.0	0.382929
$76$	0	0
$77$	−11760.0	−0.226038
$78$	0	0
$79$	−101768.	−1.83461	−0.917304	−	0.398186i	$-0.869640\pi$
$79$	−101768.	−1.83461	−0.917304	+	0.398186i	$0.869640\pi$
$80$	0	0
$81$	34101.0	0.577503
$82$	0	0
$83$	−23922.0	−0.381156	−0.190578	−	0.981672i	$-0.561036\pi$
$83$	−23922.0	−0.381156	−0.190578	+	0.981672i	$0.561036\pi$
$84$	0	0
$85$	−4168.00	−0.0625721
$86$	0	0
$87$	4404.00	0.0623805
$88$	0	0
$89$	141674.	1.89590	0.947949	−	0.318421i	$-0.103153\pi$
$89$	141674.	1.89590	0.947949	+	0.318421i	$0.103153\pi$
$90$	0	0
$91$	−36456.0	−0.461493
$92$	0	0
$93$	−30840.0	−0.369749
$94$	0	0
$95$	−3944.00	−0.0448361
$96$	0	0
$97$	99982.0	1.07893	0.539464	−	0.842009i	$-0.318627\pi$
$97$	99982.0	1.07893	0.539464	+	0.842009i	$0.318627\pi$
$98$	0	0
$99$	49680.0	0.509441
$100$	0	0
$101$	−108684.	−1.06014	−0.530069	−	0.847955i	$-0.677834\pi$
$101$	−108684.	−1.06014	−0.530069	+	0.847955i	$0.677834\pi$
$102$	0	0
$103$	87396.0	0.811706	0.405853	−	0.913938i	$-0.366975\pi$
$103$	87396.0	0.811706	0.405853	+	0.913938i	$0.366975\pi$
$104$	0	0
$105$	−1176.00	−0.0104096
$106$	0	0
$107$	71892.0	0.607045	0.303523	−	0.952824i	$-0.401837\pi$
$107$	71892.0	0.607045	0.303523	+	0.952824i	$0.401837\pi$
$108$	0	0
$109$	−118166.	−0.952634	−0.476317	−	0.879274i	$-0.658029\pi$
$109$	−118166.	−0.952634	−0.476317	+	0.879274i	$0.658029\pi$
$110$	0	0
$111$	36324.0	0.279825
$112$	0	0
$113$	252774.	1.86224	0.931121	−	0.364709i	$-0.118832\pi$
$113$	252774.	1.86224	0.931121	+	0.364709i	$0.118832\pi$
$114$	0	0
$115$	736.000	0.00518959
$116$	0	0
$117$	154008.	1.04011
$118$	0	0
$119$	−51058.0	−0.330519
$120$	0	0
$121$	−103451.	−0.642349
$122$	0	0
$123$	−45588.0	−0.271699
$124$	0	0
$125$	−24936.0	−0.142742
$126$	0	0
$127$	−3592.00	−0.0197618	−0.00988091	−	0.999951i	$-0.503145\pi$
$127$	−3592.00	−0.0197618	−0.00988091	+	0.999951i	$0.503145\pi$
$128$	0	0
$129$	−78096.0	−0.413194
$130$	0	0
$131$	364534.	1.85592	0.927961	−	0.372677i	$-0.121560\pi$
$131$	364534.	1.85592	0.927961	+	0.372677i	$0.121560\pi$
$132$	0	0
$133$	−48314.0	−0.236834
$134$	0	0
$135$	10800.0	0.0510022
$136$	0	0
$137$	−77246.0	−0.351621	−0.175810	−	0.984424i	$-0.556255\pi$
$137$	−77246.0	−0.351621	−0.175810	+	0.984424i	$0.556255\pi$
$138$	0	0
$139$	−122742.	−0.538835	−0.269418	−	0.963023i	$-0.586831\pi$
$139$	−122742.	−0.538835	−0.269418	+	0.963023i	$0.586831\pi$
$140$	0	0
$141$	−88008.0	−0.372799
$142$	0	0
$143$	178560.	0.730204
$144$	0	0
$145$	−2936.00	−0.0115967
$146$	0	0
$147$	−14406.0	−0.0549857
$148$	0	0
$149$	−469234.	−1.73151	−0.865753	−	0.500472i	$-0.833160\pi$
$149$	−469234.	−1.73151	−0.865753	+	0.500472i	$0.833160\pi$
$150$	0	0
$151$	−411584.	−1.46898	−0.734490	−	0.678619i	$-0.762579\pi$
$151$	−411584.	−1.46898	−0.734490	+	0.678619i	$0.762579\pi$
$152$	0	0
$153$	215694.	0.744920
$154$	0	0
$155$	20560.0	0.0687375
$156$	0	0
$157$	−574632.	−1.86055	−0.930274	−	0.366867i	$-0.880431\pi$
$157$	−574632.	−1.86055	−0.930274	+	0.366867i	$0.880431\pi$
$158$	0	0
$159$	87132.0	0.273328
$160$	0	0
$161$	9016.00	0.0274125
$162$	0	0
$163$	−264704.	−0.780354	−0.390177	−	0.920740i	$-0.627586\pi$
$163$	−264704.	−0.780354	−0.390177	+	0.920740i	$0.627586\pi$
$164$	0	0
$165$	5760.00	0.0164707
$166$	0	0
$167$	343356.	0.952694	0.476347	−	0.879257i	$-0.341961\pi$
$167$	343356.	0.952694	0.476347	+	0.879257i	$0.341961\pi$
$168$	0	0
$169$	182243.	0.490833
$170$	0	0
$171$	204102.	0.533773
$172$	0	0
$173$	−303296.	−0.770462	−0.385231	−	0.922820i	$-0.625878\pi$
$173$	−303296.	−0.770462	−0.385231	+	0.922820i	$0.625878\pi$
$174$	0	0
$175$	−152341.	−0.376029
$176$	0	0
$177$	80172.0	0.192349
$178$	0	0
$179$	−362756.	−0.846218	−0.423109	−	0.906079i	$-0.639061\pi$
$179$	−362756.	−0.846218	−0.423109	+	0.906079i	$0.639061\pi$
$180$	0	0
$181$	−146560.	−0.332521	−0.166260	−	0.986082i	$-0.553169\pi$
$181$	−146560.	−0.332521	−0.166260	+	0.986082i	$0.553169\pi$
$182$	0	0
$183$	−58056.0	−0.128150
$184$	0	0
$185$	−24216.0	−0.0520203
$186$	0	0
$187$	250080.	0.522968
$188$	0	0
$189$	132300.	0.269405
$190$	0	0
$191$	552536.	1.09592	0.547958	−	0.836506i	$-0.315405\pi$
$191$	552536.	1.09592	0.547958	+	0.836506i	$0.315405\pi$
$192$	0	0
$193$	305358.	0.590087	0.295043	−	0.955484i	$-0.404666\pi$
$193$	305358.	0.590087	0.295043	+	0.955484i	$0.404666\pi$
$194$	0	0
$195$	17856.0	0.0336277
$196$	0	0
$197$	743838.	1.36557	0.682783	−	0.730621i	$-0.260769\pi$
$197$	743838.	1.36557	0.682783	+	0.730621i	$0.260769\pi$
$198$	0	0
$199$	286220.	0.512351	0.256175	−	0.966630i	$-0.417538\pi$
$199$	286220.	0.512351	0.256175	+	0.966630i	$0.417538\pi$
$200$	0	0
$201$	372744.	0.650760
$202$	0	0
$203$	−35966.0	−0.0612565
$204$	0	0
$205$	30392.0	0.0505097
$206$	0	0
$207$	−38088.0	−0.0617820
$208$	0	0
$209$	236640.	0.374733
$210$	0	0
$211$	−895372.	−1.38451	−0.692257	−	0.721651i	$-0.743384\pi$
$211$	−895372.	−1.38451	−0.692257	+	0.721651i	$0.743384\pi$
$212$	0	0
$213$	12672.0	0.0191380
$214$	0	0
$215$	52064.0	0.0768142
$216$	0	0
$217$	251860.	0.363086
$218$	0	0
$219$	173460.	0.244393
$220$	0	0
$221$	775248.	1.06773
$222$	0	0
$223$	−1.18812e6	−1.59992	−0.799960	−	0.600054i	$-0.795146\pi$
$223$	−1.18812e6	−1.59992	−0.799960	+	0.600054i	$0.795146\pi$
$224$	0	0
$225$	643563.	0.847490
$226$	0	0
$227$	808822.	1.04181	0.520905	−	0.853615i	$-0.325595\pi$
$227$	808822.	1.04181	0.520905	+	0.853615i	$0.325595\pi$
$228$	0	0
$229$	−344344.	−0.433914	−0.216957	−	0.976181i	$-0.569613\pi$
$229$	−344344.	−0.433914	−0.216957	+	0.976181i	$0.569613\pi$
$230$	0	0
$231$	70560.0	0.0870019
$232$	0	0
$233$	−334022.	−0.403074	−0.201537	−	0.979481i	$-0.564594\pi$
$233$	−334022.	−0.403074	−0.201537	+	0.979481i	$0.564594\pi$
$234$	0	0
$235$	58672.0	0.0693045
$236$	0	0
$237$	610608.	0.706141
$238$	0	0
$239$	−954272.	−1.08063	−0.540316	−	0.841463i	$-0.681695\pi$
$239$	−954272.	−1.08063	−0.540316	+	0.841463i	$0.681695\pi$
$240$	0	0
$241$	−272882.	−0.302644	−0.151322	−	0.988485i	$-0.548353\pi$
$241$	−272882.	−0.302644	−0.151322	+	0.988485i	$0.548353\pi$
$242$	0	0
$243$	−860706.	−0.935059
$244$	0	0
$245$	9604.00	0.0102220
$246$	0	0
$247$	733584.	0.765081
$248$	0	0
$249$	143532.	0.146707
$250$	0	0
$251$	43754.0	0.0438363	0.0219181	−	0.999760i	$-0.493023\pi$
$251$	43754.0	0.0438363	0.0219181	+	0.999760i	$0.493023\pi$
$252$	0	0
$253$	−44160.0	−0.0433738
$254$	0	0
$255$	25008.0	0.0240840
$256$	0	0
$257$	−1.73201e6	−1.63576	−0.817878	−	0.575391i	$-0.804850\pi$
$257$	−1.73201e6	−1.63576	−0.817878	+	0.575391i	$0.804850\pi$
$258$	0	0
$259$	−296646.	−0.274783
$260$	0	0
$261$	151938.	0.138059
$262$	0	0
$263$	613248.	0.546697	0.273349	−	0.961915i	$-0.411869\pi$
$263$	613248.	0.546697	0.273349	+	0.961915i	$0.411869\pi$
$264$	0	0
$265$	−58088.0	−0.0508126
$266$	0	0
$267$	−850044.	−0.729732
$268$	0	0
$269$	−2.01360e6	−1.69665	−0.848325	−	0.529475i	$-0.822389\pi$
$269$	−2.01360e6	−1.69665	−0.848325	+	0.529475i	$0.822389\pi$
$270$	0	0
$271$	−1.22138e6	−1.01024	−0.505122	−	0.863048i	$-0.668552\pi$
$271$	−1.22138e6	−1.01024	−0.505122	+	0.863048i	$0.668552\pi$
$272$	0	0
$273$	218736.	0.177629
$274$	0	0
$275$	746160.	0.594977
$276$	0	0
$277$	2.11821e6	1.65871	0.829355	−	0.558722i	$-0.188708\pi$
$277$	2.11821e6	1.65871	0.829355	+	0.558722i	$0.188708\pi$
$278$	0	0
$279$	−1.06398e6	−0.818320
$280$	0	0
$281$	−1.64516e6	−1.24292	−0.621458	−	0.783447i	$-0.713459\pi$
$281$	−1.64516e6	−1.24292	−0.621458	+	0.783447i	$0.713459\pi$
$282$	0	0
$283$	1.66393e6	1.23501	0.617504	−	0.786567i	$-0.288144\pi$
$283$	1.66393e6	1.23501	0.617504	+	0.786567i	$0.288144\pi$
$284$	0	0
$285$	23664.0	0.0172574
$286$	0	0
$287$	372302.	0.266803
$288$	0	0
$289$	−334093.	−0.235300
$290$	0	0
$291$	−599892.	−0.415280
$292$	0	0
$293$	−1.15732e6	−0.787559	−0.393779	−	0.919205i	$-0.628833\pi$
$293$	−1.15732e6	−0.787559	−0.393779	+	0.919205i	$0.628833\pi$
$294$	0	0
$295$	−53448.0	−0.0357583
$296$	0	0
$297$	−648000.	−0.426269
$298$	0	0
$299$	−136896.	−0.0885549
$300$	0	0
$301$	637784.	0.405749
$302$	0	0
$303$	652104.	0.408047
$304$	0	0
$305$	38704.0	0.0238235
$306$	0	0
$307$	344998.	0.208915	0.104458	−	0.994529i	$-0.466689\pi$
$307$	344998.	0.208915	0.104458	+	0.994529i	$0.466689\pi$
$308$	0	0
$309$	−524376.	−0.312426
$310$	0	0
$311$	3.28798e6	1.92765	0.963824	−	0.266540i	$-0.0858804\pi$
$311$	3.28798e6	1.92765	0.963824	+	0.266540i	$0.0858804\pi$
$312$	0	0
$313$	−2.21063e6	−1.27542	−0.637712	−	0.770275i	$-0.720119\pi$
$313$	−2.21063e6	−1.27542	−0.637712	+	0.770275i	$0.720119\pi$
$314$	0	0
$315$	−40572.0	−0.0230383
$316$	0	0
$317$	1.19631e6	0.668645	0.334322	−	0.942459i	$-0.391493\pi$
$317$	1.19631e6	0.668645	0.334322	+	0.942459i	$0.391493\pi$
$318$	0	0
$319$	176160.	0.0969238
$320$	0	0
$321$	−431352.	−0.233652
$322$	0	0
$323$	1.02741e6	0.547947
$324$	0	0
$325$	2.31310e6	1.21475
$326$	0	0
$327$	708996.	0.366669
$328$	0	0
$329$	718732.	0.366081
$330$	0	0
$331$	2.12828e6	1.06772	0.533862	−	0.845572i	$-0.320740\pi$
$331$	2.12828e6	1.06772	0.533862	+	0.845572i	$0.320740\pi$
$332$	0	0
$333$	1.25318e6	0.619302
$334$	0	0
$335$	−248496.	−0.120978
$336$	0	0
$337$	1.89841e6	0.910576	0.455288	−	0.890344i	$-0.349536\pi$
$337$	1.89841e6	0.910576	0.455288	+	0.890344i	$0.349536\pi$
$338$	0	0
$339$	−1.51664e6	−0.716778
$340$	0	0
$341$	−1.23360e6	−0.574498
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	−4416.00	−0.00199747
$346$	0	0
$347$	2.17548e6	0.969910	0.484955	−	0.874539i	$-0.338836\pi$
$347$	2.17548e6	0.969910	0.484955	+	0.874539i	$0.338836\pi$
$348$	0	0
$349$	2.12950e6	0.935869	0.467934	−	0.883763i	$-0.344998\pi$
$349$	2.12950e6	0.935869	0.467934	+	0.883763i	$0.344998\pi$
$350$	0	0
$351$	−2.00880e6	−0.870300
$352$	0	0
$353$	−2.54144e6	−1.08553	−0.542766	−	0.839884i	$-0.682623\pi$
$353$	−2.54144e6	−1.08553	−0.542766	+	0.839884i	$0.682623\pi$
$354$	0	0
$355$	−8448.00	−0.00355781
$356$	0	0
$357$	306348.	0.127217
$358$	0	0
$359$	−479280.	−0.196270	−0.0981348	−	0.995173i	$-0.531288\pi$
$359$	−479280.	−0.196270	−0.0981348	+	0.995173i	$0.531288\pi$
$360$	0	0
$361$	−1.50390e6	−0.607368
$362$	0	0
$363$	620706.	0.247240
$364$	0	0
$365$	−115640.	−0.0454335
$366$	0	0
$367$	1.89390e6	0.733991	0.366996	−	0.930223i	$-0.380386\pi$
$367$	1.89390e6	0.733991	0.366996	+	0.930223i	$0.380386\pi$
$368$	0	0
$369$	−1.57279e6	−0.601317
$370$	0	0
$371$	−711578.	−0.268403
$372$	0	0
$373$	1.56683e6	0.583109	0.291555	−	0.956554i	$-0.405828\pi$
$373$	1.56683e6	0.583109	0.291555	+	0.956554i	$0.405828\pi$
$374$	0	0
$375$	149616.	0.0549414
$376$	0	0
$377$	546096.	0.197886
$378$	0	0
$379$	−57360.0	−0.0205121	−0.0102561	−	0.999947i	$-0.503265\pi$
$379$	−57360.0	−0.0205121	−0.0102561	+	0.999947i	$0.503265\pi$
$380$	0	0
$381$	21552.0	0.00760633
$382$	0	0
$383$	−4.41239e6	−1.53701	−0.768505	−	0.639844i	$-0.778999\pi$
$383$	−4.41239e6	−1.53701	−0.768505	+	0.639844i	$0.778999\pi$
$384$	0	0
$385$	−47040.0	−0.0161739
$386$	0	0
$387$	−2.69431e6	−0.914472
$388$	0	0
$389$	−872470.	−0.292332	−0.146166	−	0.989260i	$-0.546693\pi$
$389$	−872470.	−0.292332	−0.146166	+	0.989260i	$0.546693\pi$
$390$	0	0
$391$	−191728.	−0.0634225
$392$	0	0
$393$	−2.18720e6	−0.714345
$394$	0	0
$395$	−407072.	−0.131274
$396$	0	0
$397$	3.63170e6	1.15647	0.578233	−	0.815871i	$-0.303742\pi$
$397$	3.63170e6	1.15647	0.578233	+	0.815871i	$0.303742\pi$
$398$	0	0
$399$	289884.	0.0911574
$400$	0	0
$401$	−3.58423e6	−1.11310	−0.556550	−	0.830814i	$-0.687875\pi$
$401$	−3.58423e6	−1.11310	−0.556550	+	0.830814i	$0.687875\pi$
$402$	0	0
$403$	−3.82416e6	−1.17293
$404$	0	0
$405$	136404.	0.0413228
$406$	0	0
$407$	1.45296e6	0.434778
$408$	0	0
$409$	2.18309e6	0.645304	0.322652	−	0.946518i	$-0.395426\pi$
$409$	2.18309e6	0.645304	0.322652	+	0.946518i	$0.395426\pi$
$410$	0	0
$411$	463476.	0.135339
$412$	0	0
$413$	−654738.	−0.188883
$414$	0	0
$415$	−95688.0	−0.0272733
$416$	0	0
$417$	736452.	0.207398
$418$	0	0
$419$	−4.91137e6	−1.36668	−0.683342	−	0.730099i	$-0.739474\pi$
$419$	−4.91137e6	−1.36668	−0.683342	+	0.730099i	$0.739474\pi$
$420$	0	0
$421$	693766.	0.190769	0.0953845	−	0.995441i	$-0.469592\pi$
$421$	693766.	0.190769	0.0953845	+	0.995441i	$0.469592\pi$
$422$	0	0
$423$	−3.03628e6	−0.825069
$424$	0	0
$425$	3.23958e6	0.869994
$426$	0	0
$427$	474124.	0.125841
$428$	0	0
$429$	−1.07136e6	−0.281056
$430$	0	0
$431$	−1.25035e6	−0.324219	−0.162110	−	0.986773i	$-0.551830\pi$
$431$	−1.25035e6	−0.324219	−0.162110	+	0.986773i	$0.551830\pi$
$432$	0	0
$433$	157750.	0.0404343	0.0202171	−	0.999796i	$-0.493564\pi$
$433$	157750.	0.0404343	0.0202171	+	0.999796i	$0.493564\pi$
$434$	0	0
$435$	17616.0	0.00446359
$436$	0	0
$437$	−181424.	−0.0454455
$438$	0	0
$439$	263736.	0.0653143	0.0326571	−	0.999467i	$-0.489603\pi$
$439$	263736.	0.0653143	0.0326571	+	0.999467i	$0.489603\pi$
$440$	0	0
$441$	−497007.	−0.121693
$442$	0	0
$443$	−1.50410e6	−0.364139	−0.182070	−	0.983286i	$-0.558280\pi$
$443$	−1.50410e6	−0.364139	−0.182070	+	0.983286i	$0.558280\pi$
$444$	0	0
$445$	566696.	0.135659
$446$	0	0
$447$	2.81540e6	0.666457
$448$	0	0
$449$	2.11128e6	0.494231	0.247116	−	0.968986i	$-0.420517\pi$
$449$	2.11128e6	0.494231	0.247116	+	0.968986i	$0.420517\pi$
$450$	0	0
$451$	−1.82352e6	−0.422152
$452$	0	0
$453$	2.46950e6	0.565411
$454$	0	0
$455$	−145824.	−0.0330218
$456$	0	0
$457$	3.99938e6	0.895782	0.447891	−	0.894088i	$-0.352175\pi$
$457$	3.99938e6	0.895782	0.447891	+	0.894088i	$0.352175\pi$
$458$	0	0
$459$	−2.81340e6	−0.623304
$460$	0	0
$461$	−2.24090e6	−0.491101	−0.245551	−	0.969384i	$-0.578969\pi$
$461$	−2.24090e6	−0.491101	−0.245551	+	0.969384i	$0.578969\pi$
$462$	0	0
$463$	−1.47304e6	−0.319346	−0.159673	−	0.987170i	$-0.551044\pi$
$463$	−1.47304e6	−0.319346	−0.159673	+	0.987170i	$0.551044\pi$
$464$	0	0
$465$	−123360.	−0.0264571
$466$	0	0
$467$	8.50472e6	1.80454	0.902272	−	0.431166i	$-0.141898\pi$
$467$	8.50472e6	1.80454	0.902272	+	0.431166i	$0.141898\pi$
$468$	0	0
$469$	−3.04408e6	−0.639033
$470$	0	0
$471$	3.44779e6	0.716125
$472$	0	0
$473$	−3.12384e6	−0.642001
$474$	0	0
$475$	3.06547e6	0.623395
$476$	0	0
$477$	3.00605e6	0.604924
$478$	0	0
$479$	−6.56984e6	−1.30833	−0.654163	−	0.756354i	$-0.726979\pi$
$479$	−6.56984e6	−1.30833	−0.654163	+	0.756354i	$0.726979\pi$
$480$	0	0
$481$	4.50418e6	0.887672
$482$	0	0
$483$	−54096.0	−0.0105511
$484$	0	0
$485$	399928.	0.0772018
$486$	0	0
$487$	7.71038e6	1.47317	0.736585	−	0.676344i	$-0.236437\pi$
$487$	7.71038e6	1.47317	0.736585	+	0.676344i	$0.236437\pi$
$488$	0	0
$489$	1.58822e6	0.300358
$490$	0	0
$491$	8.72147e6	1.63262	0.816311	−	0.577612i	$-0.196015\pi$
$491$	8.72147e6	1.63262	0.816311	+	0.577612i	$0.196015\pi$
$492$	0	0
$493$	764828.	0.141725
$494$	0	0
$495$	198720.	0.0364526
$496$	0	0
$497$	−103488.	−0.0187931
$498$	0	0
$499$	−7.87430e6	−1.41567	−0.707833	−	0.706380i	$-0.750327\pi$
$499$	−7.87430e6	−1.41567	−0.707833	+	0.706380i	$0.750327\pi$
$500$	0	0
$501$	−2.06014e6	−0.366692
$502$	0	0
$503$	8.68726e6	1.53096	0.765479	−	0.643461i	$-0.222502\pi$
$503$	8.68726e6	1.53096	0.765479	+	0.643461i	$0.222502\pi$
$504$	0	0
$505$	−434736.	−0.0758573
$506$	0	0
$507$	−1.09346e6	−0.188922
$508$	0	0
$509$	−1.34131e6	−0.229475	−0.114737	−	0.993396i	$-0.536603\pi$
$509$	−1.34131e6	−0.229475	−0.114737	+	0.993396i	$0.536603\pi$
$510$	0	0
$511$	−1.41659e6	−0.239989
$512$	0	0
$513$	−2.66220e6	−0.446629
$514$	0	0
$515$	349584.	0.0580809
$516$	0	0
$517$	−3.52032e6	−0.579236
$518$	0	0
$519$	1.81978e6	0.296551
$520$	0	0
$521$	6.00185e6	0.968704	0.484352	−	0.874873i	$-0.339055\pi$
$521$	6.00185e6	0.968704	0.484352	+	0.874873i	$0.339055\pi$
$522$	0	0
$523$	1.19109e7	1.90410	0.952048	−	0.305950i	$-0.0989740\pi$
$523$	1.19109e7	1.90410	0.952048	+	0.305950i	$0.0989740\pi$
$524$	0	0
$525$	914046.	0.144734
$526$	0	0
$527$	−5.35588e6	−0.840048
$528$	0	0
$529$	−6.40249e6	−0.994740
$530$	0	0
$531$	2.76593e6	0.425702
$532$	0	0
$533$	−5.65291e6	−0.861895
$534$	0	0
$535$	287568.	0.0434366
$536$	0	0
$537$	2.17654e6	0.325709
$538$	0	0
$539$	−576240.	−0.0854341
$540$	0	0
$541$	−7.20703e6	−1.05868	−0.529338	−	0.848411i	$-0.677560\pi$
$541$	−7.20703e6	−1.05868	−0.529338	+	0.848411i	$0.677560\pi$
$542$	0	0
$543$	879360.	0.127987
$544$	0	0
$545$	−472664.	−0.0681650
$546$	0	0
$547$	1.65172e6	0.236030	0.118015	−	0.993012i	$-0.462347\pi$
$547$	1.65172e6	0.236030	0.118015	+	0.993012i	$0.462347\pi$
$548$	0	0
$549$	−2.00293e6	−0.283619
$550$	0	0
$551$	723724.	0.101553
$552$	0	0
$553$	−4.98663e6	−0.693417
$554$	0	0
$555$	145296.	0.0200226
$556$	0	0
$557$	−7.58171e6	−1.03545	−0.517725	−	0.855547i	$-0.673221\pi$
$557$	−7.58171e6	−1.03545	−0.517725	+	0.855547i	$0.673221\pi$
$558$	0	0
$559$	−9.68390e6	−1.31075
$560$	0	0
$561$	−1.50048e6	−0.201290
$562$	0	0
$563$	−1.26568e7	−1.68288	−0.841440	−	0.540351i	$-0.818292\pi$
$563$	−1.26568e7	−1.68288	−0.841440	+	0.540351i	$0.818292\pi$
$564$	0	0
$565$	1.01110e6	0.133251
$566$	0	0
$567$	1.67095e6	0.218276
$568$	0	0
$569$	4.57445e6	0.592323	0.296162	−	0.955138i	$-0.404293\pi$
$569$	4.57445e6	0.592323	0.296162	+	0.955138i	$0.404293\pi$
$570$	0	0
$571$	−5.77802e6	−0.741632	−0.370816	−	0.928706i	$-0.620922\pi$
$571$	−5.77802e6	−0.741632	−0.370816	+	0.928706i	$0.620922\pi$
$572$	0	0
$573$	−3.31522e6	−0.421818
$574$	0	0
$575$	−572056.	−0.0721554
$576$	0	0
$577$	5.46520e6	0.683387	0.341693	−	0.939811i	$-0.389000\pi$
$577$	5.46520e6	0.683387	0.341693	+	0.939811i	$0.389000\pi$
$578$	0	0
$579$	−1.83215e6	−0.227125
$580$	0	0
$581$	−1.17218e6	−0.144063
$582$	0	0
$583$	3.48528e6	0.424684
$584$	0	0
$585$	616032.	0.0744241
$586$	0	0
$587$	9.89386e6	1.18514	0.592571	−	0.805518i	$-0.298113\pi$
$587$	9.89386e6	1.18514	0.592571	+	0.805518i	$0.298113\pi$
$588$	0	0
$589$	−5.06804e6	−0.601938
$590$	0	0
$591$	−4.46303e6	−0.525607
$592$	0	0
$593$	6.12686e6	0.715486	0.357743	−	0.933820i	$-0.383546\pi$
$593$	6.12686e6	0.715486	0.357743	+	0.933820i	$0.383546\pi$
$594$	0	0
$595$	−204232.	−0.0236500
$596$	0	0
$597$	−1.71732e6	−0.197204
$598$	0	0
$599$	299432.	0.0340982	0.0170491	−	0.999855i	$-0.494573\pi$
$599$	299432.	0.0340982	0.0170491	+	0.999855i	$0.494573\pi$
$600$	0	0
$601$	−4.98133e6	−0.562548	−0.281274	−	0.959628i	$-0.590757\pi$
$601$	−4.98133e6	−0.562548	−0.281274	+	0.959628i	$0.590757\pi$
$602$	0	0
$603$	1.28597e7	1.44025
$604$	0	0
$605$	−413804.	−0.0459628
$606$	0	0
$607$	−1.10694e7	−1.21942	−0.609709	−	0.792625i	$-0.708714\pi$
$607$	−1.10694e7	−1.21942	−0.609709	+	0.792625i	$0.708714\pi$
$608$	0	0
$609$	215796.	0.0235776
$610$	0	0
$611$	−1.09130e7	−1.18261
$612$	0	0
$613$	−1.37829e7	−1.48146	−0.740729	−	0.671804i	$-0.765520\pi$
$613$	−1.37829e7	−1.48146	−0.740729	+	0.671804i	$0.765520\pi$
$614$	0	0
$615$	−182352.	−0.0194412
$616$	0	0
$617$	1.11450e7	1.17861	0.589303	−	0.807912i	$-0.299403\pi$
$617$	1.11450e7	1.17861	0.589303	+	0.807912i	$0.299403\pi$
$618$	0	0
$619$	−3.00722e6	−0.315456	−0.157728	−	0.987483i	$-0.550417\pi$
$619$	−3.00722e6	−0.315456	−0.157728	+	0.987483i	$0.550417\pi$
$620$	0	0
$621$	496800.	0.0516955
$622$	0	0
$623$	6.94203e6	0.716582
$624$	0	0
$625$	9.61588e6	0.984666
$626$	0	0
$627$	−1.41984e6	−0.144235
$628$	0	0
$629$	6.30827e6	0.635746
$630$	0	0
$631$	−570304.	−0.0570208	−0.0285104	−	0.999593i	$-0.509076\pi$
$631$	−570304.	−0.0570208	−0.0285104	+	0.999593i	$0.509076\pi$
$632$	0	0
$633$	5.37223e6	0.532900
$634$	0	0
$635$	−14368.0	−0.00141404
$636$	0	0
$637$	−1.78634e6	−0.174428
$638$	0	0
$639$	437184.	0.0423557
$640$	0	0
$641$	1.37359e7	1.32042	0.660212	−	0.751080i	$-0.270467\pi$
$641$	1.37359e7	1.32042	0.660212	+	0.751080i	$0.270467\pi$
$642$	0	0
$643$	2.58692e6	0.246749	0.123375	−	0.992360i	$-0.460628\pi$
$643$	2.58692e6	0.246749	0.123375	+	0.992360i	$0.460628\pi$
$644$	0	0
$645$	−312384.	−0.0295658
$646$	0	0
$647$	6.52446e6	0.612751	0.306375	−	0.951911i	$-0.400884\pi$
$647$	6.52446e6	0.612751	0.306375	+	0.951911i	$0.400884\pi$
$648$	0	0
$649$	3.20688e6	0.298862
$650$	0	0
$651$	−1.51116e6	−0.139752
$652$	0	0
$653$	3.75793e6	0.344878	0.172439	−	0.985020i	$-0.444835\pi$
$653$	3.75793e6	0.344878	0.172439	+	0.985020i	$0.444835\pi$
$654$	0	0
$655$	1.45814e6	0.132799
$656$	0	0
$657$	5.98437e6	0.540885
$658$	0	0
$659$	−6.97436e6	−0.625591	−0.312796	−	0.949820i	$-0.601266\pi$
$659$	−6.97436e6	−0.625591	−0.312796	+	0.949820i	$0.601266\pi$
$660$	0	0
$661$	1.17059e7	1.04208	0.521042	−	0.853531i	$-0.325543\pi$
$661$	1.17059e7	1.04208	0.521042	+	0.853531i	$0.325543\pi$
$662$	0	0
$663$	−4.65149e6	−0.410968
$664$	0	0
$665$	−193256.	−0.0169465
$666$	0	0
$667$	−135056.	−0.0117544
$668$	0	0
$669$	7.12872e6	0.615809
$670$	0	0
$671$	−2.32224e6	−0.199114
$672$	0	0
$673$	−1.82825e7	−1.55596	−0.777980	−	0.628289i	$-0.783755\pi$
$673$	−1.82825e7	−1.55596	−0.777980	+	0.628289i	$0.783755\pi$
$674$	0	0
$675$	−8.39430e6	−0.709129
$676$	0	0
$677$	2.05661e6	0.172457	0.0862283	−	0.996275i	$-0.472519\pi$
$677$	2.05661e6	0.172457	0.0862283	+	0.996275i	$0.472519\pi$
$678$	0	0
$679$	4.89912e6	0.407796
$680$	0	0
$681$	−4.85293e6	−0.400993
$682$	0	0
$683$	1.24913e7	1.02461	0.512303	−	0.858805i	$-0.328792\pi$
$683$	1.24913e7	1.02461	0.512303	+	0.858805i	$0.328792\pi$
$684$	0	0
$685$	−308984.	−0.0251599
$686$	0	0
$687$	2.06606e6	0.167014
$688$	0	0
$689$	1.08044e7	0.867064
$690$	0	0
$691$	176630.	0.0140724	0.00703622	−	0.999975i	$-0.497760\pi$
$691$	176630.	0.0140724	0.00703622	+	0.999975i	$0.497760\pi$
$692$	0	0
$693$	2.43432e6	0.192550
$694$	0	0
$695$	−490968.	−0.0385559
$696$	0	0
$697$	−7.91712e6	−0.617284
$698$	0	0
$699$	2.00413e6	0.155143
$700$	0	0
$701$	4.03111e6	0.309835	0.154917	−	0.987927i	$-0.450489\pi$
$701$	4.03111e6	0.309835	0.154917	+	0.987927i	$0.450489\pi$
$702$	0	0
$703$	5.96924e6	0.455545
$704$	0	0
$705$	−352032.	−0.0266753
$706$	0	0
$707$	−5.32552e6	−0.400694
$708$	0	0
$709$	1.41839e7	1.05969	0.529847	−	0.848093i	$-0.322249\pi$
$709$	1.41839e7	1.05969	0.529847	+	0.848093i	$0.322249\pi$
$710$	0	0
$711$	2.10660e7	1.56282
$712$	0	0
$713$	945760.	0.0696718
$714$	0	0
$715$	714240.	0.0522491
$716$	0	0
$717$	5.72563e6	0.415935
$718$	0	0
$719$	−2.46272e7	−1.77661	−0.888306	−	0.459253i	$-0.848117\pi$
$719$	−2.46272e7	−1.77661	−0.888306	+	0.459253i	$0.848117\pi$
$720$	0	0
$721$	4.28240e6	0.306796
$722$	0	0
$723$	1.63729e6	0.116488
$724$	0	0
$725$	2.28201e6	0.161240
$726$	0	0
$727$	−1.30482e7	−0.915615	−0.457808	−	0.889051i	$-0.651365\pi$
$727$	−1.30482e7	−0.915615	−0.457808	+	0.889051i	$0.651365\pi$
$728$	0	0
$729$	−3.12231e6	−0.217599
$730$	0	0
$731$	−1.35627e7	−0.938754
$732$	0	0
$733$	−2.08870e7	−1.43587	−0.717936	−	0.696109i	$-0.754913\pi$
$733$	−2.08870e7	−1.43587	−0.717936	+	0.696109i	$0.754913\pi$
$734$	0	0
$735$	−57624.0	−0.00393446
$736$	0	0
$737$	1.49098e7	1.01112
$738$	0	0
$739$	−1.47615e7	−0.994303	−0.497151	−	0.867664i	$-0.665621\pi$
$739$	−1.47615e7	−0.994303	−0.497151	+	0.867664i	$0.665621\pi$
$740$	0	0
$741$	−4.40150e6	−0.294480
$742$	0	0
$743$	−4.44570e6	−0.295439	−0.147719	−	0.989029i	$-0.547193\pi$
$743$	−4.44570e6	−0.295439	−0.147719	+	0.989029i	$0.547193\pi$
$744$	0	0
$745$	−1.87694e6	−0.123896
$746$	0	0
$747$	4.95185e6	0.324688
$748$	0	0
$749$	3.52271e6	0.229442
$750$	0	0
$751$	1.19094e7	0.770528	0.385264	−	0.922806i	$-0.374110\pi$
$751$	1.19094e7	0.770528	0.385264	+	0.922806i	$0.374110\pi$
$752$	0	0
$753$	−262524.	−0.0168726
$754$	0	0
$755$	−1.64634e6	−0.105112
$756$	0	0
$757$	−2.55035e7	−1.61756	−0.808781	−	0.588110i	$-0.799872\pi$
$757$	−2.55035e7	−1.61756	−0.808781	+	0.588110i	$0.799872\pi$
$758$	0	0
$759$	264960.	0.0166946
$760$	0	0
$761$	−1.46925e7	−0.919675	−0.459837	−	0.888003i	$-0.652092\pi$
$761$	−1.46925e7	−0.919675	−0.459837	+	0.888003i	$0.652092\pi$
$762$	0	0
$763$	−5.79013e6	−0.360062
$764$	0	0
$765$	862776.	0.0533021
$766$	0	0
$767$	9.94133e6	0.610177
$768$	0	0
$769$	−1.92779e7	−1.17556	−0.587780	−	0.809021i	$-0.699998\pi$
$769$	−1.92779e7	−1.17556	−0.587780	+	0.809021i	$0.699998\pi$
$770$	0	0
$771$	1.03921e7	0.629603
$772$	0	0
$773$	−8.56584e6	−0.515610	−0.257805	−	0.966197i	$-0.582999\pi$
$773$	−8.56584e6	−0.515610	−0.257805	+	0.966197i	$0.582999\pi$
$774$	0	0
$775$	−1.59803e7	−0.955718
$776$	0	0
$777$	1.77988e6	0.105764
$778$	0	0
$779$	−7.49163e6	−0.442316
$780$	0	0
$781$	506880.	0.0297357
$782$	0	0
$783$	−1.98180e6	−0.115520
$784$	0	0
$785$	−2.29853e6	−0.133130
$786$	0	0
$787$	1.89027e7	1.08789	0.543947	−	0.839119i	$-0.316929\pi$
$787$	1.89027e7	1.08789	0.543947	+	0.839119i	$0.316929\pi$
$788$	0	0
$789$	−3.67949e6	−0.210424
$790$	0	0
$791$	1.23859e7	0.703862
$792$	0	0
$793$	−7.19894e6	−0.406524
$794$	0	0
$795$	348528.	0.0195578
$796$	0	0
$797$	−1.71259e7	−0.955010	−0.477505	−	0.878629i	$-0.658459\pi$
$797$	−1.71259e7	−0.955010	−0.477505	+	0.878629i	$0.658459\pi$
$798$	0	0
$799$	−1.52841e7	−0.846977
$800$	0	0
$801$	−2.93265e7	−1.61502
$802$	0	0
$803$	6.93840e6	0.379726
$804$	0	0
$805$	36064.0	0.00196148
$806$	0	0
$807$	1.20816e7	0.653041
$808$	0	0
$809$	2.84511e7	1.52837	0.764185	−	0.644997i	$-0.223142\pi$
$809$	2.84511e7	1.52837	0.764185	+	0.644997i	$0.223142\pi$
$810$	0	0
$811$	6.55604e6	0.350017	0.175009	−	0.984567i	$-0.444005\pi$
$811$	6.55604e6	0.350017	0.175009	+	0.984567i	$0.444005\pi$
$812$	0	0
$813$	7.32826e6	0.388843
$814$	0	0
$815$	−1.05882e6	−0.0558376
$816$	0	0
$817$	−1.28338e7	−0.672666
$818$	0	0
$819$	7.54639e6	0.393124
$820$	0	0
$821$	−3.14356e6	−0.162766	−0.0813831	−	0.996683i	$-0.525934\pi$
$821$	−3.14356e6	−0.162766	−0.0813831	+	0.996683i	$0.525934\pi$
$822$	0	0
$823$	1.62191e7	0.834694	0.417347	−	0.908747i	$-0.362960\pi$
$823$	1.62191e7	0.834694	0.417347	+	0.908747i	$0.362960\pi$
$824$	0	0
$825$	−4.47696e6	−0.229007
$826$	0	0
$827$	−4.74707e6	−0.241358	−0.120679	−	0.992692i	$-0.538507\pi$
$827$	−4.74707e6	−0.241358	−0.120679	+	0.992692i	$0.538507\pi$
$828$	0	0
$829$	3.47333e7	1.75533	0.877666	−	0.479272i	$-0.159099\pi$
$829$	3.47333e7	1.75533	0.877666	+	0.479272i	$0.159099\pi$
$830$	0	0
$831$	−1.27093e7	−0.638438
$832$	0	0
$833$	−2.50184e6	−0.124924
$834$	0	0
$835$	1.37342e6	0.0681692
$836$	0	0
$837$	1.38780e7	0.684720
$838$	0	0
$839$	−6.10552e6	−0.299445	−0.149723	−	0.988728i	$-0.547838\pi$
$839$	−6.10552e6	−0.299445	−0.149723	+	0.988728i	$0.547838\pi$
$840$	0	0
$841$	−1.99724e7	−0.973734
$842$	0	0
$843$	9.87095e6	0.478399
$844$	0	0
$845$	728972.	0.0351212
$846$	0	0
$847$	−5.06910e6	−0.242785
$848$	0	0
$849$	−9.98360e6	−0.475355
$850$	0	0
$851$	−1.11394e6	−0.0527274
$852$	0	0
$853$	2.75613e7	1.29696	0.648481	−	0.761231i	$-0.275405\pi$
$853$	2.75613e7	1.29696	0.648481	+	0.761231i	$0.275405\pi$
$854$	0	0
$855$	816408.	0.0381937
$856$	0	0
$857$	−1.82100e7	−0.846950	−0.423475	−	0.905908i	$-0.639190\pi$
$857$	−1.82100e7	−0.846950	−0.423475	+	0.905908i	$0.639190\pi$
$858$	0	0
$859$	3.35920e7	1.55329	0.776647	−	0.629936i	$-0.216919\pi$
$859$	3.35920e7	1.55329	0.776647	+	0.629936i	$0.216919\pi$
$860$	0	0
$861$	−2.23381e6	−0.102693
$862$	0	0
$863$	3.26084e7	1.49040	0.745199	−	0.666843i	$-0.232355\pi$
$863$	3.26084e7	1.49040	0.745199	+	0.666843i	$0.232355\pi$
$864$	0	0
$865$	−1.21318e6	−0.0551298
$866$	0	0
$867$	2.00456e6	0.0905672
$868$	0	0
$869$	2.44243e7	1.09717
$870$	0	0
$871$	4.62203e7	2.06437
$872$	0	0
$873$	−2.06963e7	−0.919087
$874$	0	0
$875$	−1.22186e6	−0.0539514
$876$	0	0
$877$	1.33352e7	0.585463	0.292732	−	0.956195i	$-0.405436\pi$
$877$	1.33352e7	0.585463	0.292732	+	0.956195i	$0.405436\pi$
$878$	0	0
$879$	6.94390e6	0.303131
$880$	0	0
$881$	1.43194e7	0.621564	0.310782	−	0.950481i	$-0.399409\pi$
$881$	1.43194e7	0.621564	0.310782	+	0.950481i	$0.399409\pi$
$882$	0	0
$883$	4.01556e6	0.173318	0.0866592	−	0.996238i	$-0.472381\pi$
$883$	4.01556e6	0.173318	0.0866592	+	0.996238i	$0.472381\pi$
$884$	0	0
$885$	320688.	0.0137634
$886$	0	0
$887$	−4.29049e7	−1.83104	−0.915520	−	0.402272i	$-0.868221\pi$
$887$	−4.29049e7	−1.83104	−0.915520	+	0.402272i	$0.868221\pi$
$888$	0	0
$889$	−176008.	−0.00746927
$890$	0	0
$891$	−8.18424e6	−0.345370
$892$	0	0
$893$	−1.44626e7	−0.606903
$894$	0	0
$895$	−1.45102e6	−0.0605504
$896$	0	0
$897$	821376.	0.0340848
$898$	0	0
$899$	−3.77276e6	−0.155690
$900$	0	0
$901$	1.51319e7	0.620987
$902$	0	0
$903$	−3.82670e6	−0.156173
$904$	0	0
$905$	−586240.	−0.0237933
$906$	0	0
$907$	−3.04706e7	−1.22988	−0.614940	−	0.788574i	$-0.710820\pi$
$907$	−3.04706e7	−1.22988	−0.614940	+	0.788574i	$0.710820\pi$
$908$	0	0
$909$	2.24976e7	0.903080
$910$	0	0
$911$	−2.75748e7	−1.10082	−0.550411	−	0.834894i	$-0.685529\pi$
$911$	−2.75748e7	−1.10082	−0.550411	+	0.834894i	$0.685529\pi$
$912$	0	0
$913$	5.74128e6	0.227946
$914$	0	0
$915$	−232224.	−0.00916968
$916$	0	0
$917$	1.78622e7	0.701473
$918$	0	0
$919$	3.33346e7	1.30199	0.650993	−	0.759084i	$-0.274353\pi$
$919$	3.33346e7	1.30199	0.650993	+	0.759084i	$0.274353\pi$
$920$	0	0
$921$	−2.06999e6	−0.0804116
$922$	0	0
$923$	1.57133e6	0.0607103
$924$	0	0
$925$	1.88219e7	0.723284
$926$	0	0
$927$	−1.80910e7	−0.691453
$928$	0	0
$929$	2.08624e7	0.793096	0.396548	−	0.918014i	$-0.370208\pi$
$929$	2.08624e7	0.793096	0.396548	+	0.918014i	$0.370208\pi$
$930$	0	0
$931$	−2.36739e6	−0.0895148
$932$	0	0
$933$	−1.97279e7	−0.741952
$934$	0	0
$935$	1.00032e6	0.0374205
$936$	0	0
$937$	−1.66618e7	−0.619975	−0.309987	−	0.950741i	$-0.600325\pi$
$937$	−1.66618e7	−0.619975	−0.309987	+	0.950741i	$0.600325\pi$
$938$	0	0
$939$	1.32638e7	0.490911
$940$	0	0
$941$	−3.36202e7	−1.23773	−0.618865	−	0.785497i	$-0.712407\pi$
$941$	−3.36202e7	−1.23773	−0.618865	+	0.785497i	$0.712407\pi$
$942$	0	0
$943$	1.39803e6	0.0511962
$944$	0	0
$945$	529200.	0.0192770
$946$	0	0
$947$	−4.25335e7	−1.54119	−0.770595	−	0.637325i	$-0.780041\pi$
$947$	−4.25335e7	−1.54119	−0.770595	+	0.637325i	$0.780041\pi$
$948$	0	0
$949$	2.15090e7	0.775274
$950$	0	0
$951$	−7.17786e6	−0.257362
$952$	0	0
$953$	3.20613e7	1.14353	0.571767	−	0.820416i	$-0.306258\pi$
$953$	3.20613e7	1.14353	0.571767	+	0.820416i	$0.306258\pi$
$954$	0	0
$955$	2.21014e6	0.0784173
$956$	0	0
$957$	−1.05696e6	−0.0373060
$958$	0	0
$959$	−3.78505e6	−0.132900
$960$	0	0
$961$	−2.20955e6	−0.0771784
$962$	0	0
$963$	−1.48816e7	−0.517113
$964$	0	0
$965$	1.22143e6	0.0422232
$966$	0	0
$967$	−5.65115e7	−1.94344	−0.971719	−	0.236139i	$-0.924118\pi$
$967$	−5.65115e7	−1.94344	−0.971719	+	0.236139i	$0.924118\pi$
$968$	0	0
$969$	−6.16447e6	−0.210905
$970$	0	0
$971$	4.41580e7	1.50301	0.751504	−	0.659729i	$-0.229329\pi$
$971$	4.41580e7	1.50301	0.751504	+	0.659729i	$0.229329\pi$
$972$	0	0
$973$	−6.01436e6	−0.203661
$974$	0	0
$975$	−1.38786e7	−0.467556
$976$	0	0
$977$	−2.69053e7	−0.901782	−0.450891	−	0.892579i	$-0.648894\pi$
$977$	−2.69053e7	−0.901782	−0.450891	+	0.892579i	$0.648894\pi$
$978$	0	0
$979$	−3.40018e7	−1.13382
$980$	0	0
$981$	2.44604e7	0.811503
$982$	0	0
$983$	8.68688e6	0.286735	0.143367	−	0.989670i	$-0.454207\pi$
$983$	8.68688e6	0.286735	0.143367	+	0.989670i	$0.454207\pi$
$984$	0	0
$985$	2.97535e6	0.0977120
$986$	0	0
$987$	−4.31239e6	−0.140905
$988$	0	0
$989$	2.39494e6	0.0778582
$990$	0	0
$991$	−1.54909e7	−0.501063	−0.250532	−	0.968108i	$-0.580605\pi$
$991$	−1.54909e7	−0.501063	−0.250532	+	0.968108i	$0.580605\pi$
$992$	0	0
$993$	−1.27697e7	−0.410967
$994$	0	0
$995$	1.14488e6	0.0366608
$996$	0	0
$997$	−4.47588e6	−0.142607	−0.0713034	−	0.997455i	$-0.522716\pi$
$997$	−4.47588e6	−0.142607	−0.0713034	+	0.997455i	$0.522716\pi$
$998$	0	0
$999$	−1.63458e7	−0.518194

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	56.6.a.a.1.1	✓	1
3.2	odd	2		504.6.a.e.1.1		1
4.3	odd	2		112.6.a.f.1.1		1
7.2	even	3		392.6.i.d.361.1		2
7.3	odd	6		392.6.i.c.177.1		2
7.4	even	3		392.6.i.d.177.1		2
7.5	odd	6		392.6.i.c.361.1		2
7.6	odd	2		392.6.a.c.1.1		1
8.3	odd	2		448.6.a.g.1.1		1
8.5	even	2		448.6.a.j.1.1		1
12.11	even	2		1008.6.a.p.1.1		1
28.27	even	2		784.6.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.6.a.a.1.1	✓	1	1.1	even	1	trivial
112.6.a.f.1.1		1	4.3	odd	2
392.6.a.c.1.1		1	7.6	odd	2
392.6.i.c.177.1		2	7.3	odd	6
392.6.i.c.361.1		2	7.5	odd	6
392.6.i.d.177.1		2	7.4	even	3
392.6.i.d.361.1		2	7.2	even	3
448.6.a.g.1.1		1	8.3	odd	2
448.6.a.j.1.1		1	8.5	even	2
504.6.a.e.1.1		1	3.2	odd	2
784.6.a.e.1.1		1	28.27	even	2
1008.6.a.p.1.1		1	12.11	even	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:	\( N \)	\(=\)	\( 56 = 2^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	56.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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