LMFDB - Embedded newform 288.6.a.l.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [288,6,Mod(1,288)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("288.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	288.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-92,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$46.1905401061$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	no (minimal twist has level 32)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	288.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-46.0000 q^{5} +166.277 q^{7} +83.1384 q^{11} -42.0000 q^{13} -962.000 q^{17} +2078.46 q^{19} -3159.26 q^{23} -1009.00 q^{25} +2554.00 q^{29} -1995.32 q^{31} -7648.74 q^{35} +11950.0 q^{37} +5078.00 q^{41} +12553.9 q^{43} +12304.5 q^{47} +10841.0 q^{49} +19714.0 q^{53} -3824.37 q^{55} -8895.81 q^{59} +29318.0 q^{61} +1932.00 q^{65} -16877.1 q^{67} +80976.8 q^{71} +37914.0 q^{73} +13824.0 q^{77} -88791.9 q^{79} -39324.5 q^{83} +44252.0 q^{85} -13930.0 q^{89} -6983.63 q^{91} -95609.2 q^{95} +163602. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 92 q^{5} - 84 q^{13} - 1924 q^{17} - 2018 q^{25} + 5108 q^{29} + 23900 q^{37} + 10156 q^{41} + 21682 q^{49} + 39428 q^{53} + 58636 q^{61} + 3864 q^{65} + 75828 q^{73} + 27648 q^{77} + 88504 q^{85}+ \cdots + 327204 q^{97}+O(q^{100}) $ 2 * q - 92 * q^5 - 84 * q^13 - 1924 * q^17 - 2018 * q^25 + 5108 * q^29 + 23900 * q^37 + 10156 * q^41 + 21682 * q^49 + 39428 * q^53 + 58636 * q^61 + 3864 * q^65 + 75828 * q^73 + 27648 * q^77 + 88504 * q^85 - 27860 * q^89 + 327204 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−46.0000	−0.822873	−0.411437	−	0.911438i	$-0.634973\pi$
$5$	−46.0000	−0.822873	−0.411437	+	0.911438i	$0.634973\pi$
$6$	0	0
$7$	166.277	1.28259	0.641293	−	0.767296i	$-0.278398\pi$
$7$	166.277	1.28259	0.641293	+	0.767296i	$0.278398\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	83.1384	0.207167	0.103583	−	0.994621i	$-0.466969\pi$
$11$	83.1384	0.207167	0.103583	+	0.994621i	$0.466969\pi$
$12$	0	0
$13$	−42.0000	−0.0689272	−0.0344636	−	0.999406i	$-0.510972\pi$
$13$	−42.0000	−0.0689272	−0.0344636	+	0.999406i	$0.510972\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−962.000	−0.807333	−0.403667	−	0.914906i	$-0.632264\pi$
$17$	−962.000	−0.807333	−0.403667	+	0.914906i	$0.632264\pi$
$18$	0	0
$19$	2078.46	1.32086	0.660432	−	0.750886i	$-0.270373\pi$
$19$	2078.46	1.32086	0.660432	+	0.750886i	$0.270373\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3159.26	−1.24528	−0.622638	−	0.782510i	$-0.713939\pi$
$23$	−3159.26	−1.24528	−0.622638	+	0.782510i	$0.713939\pi$
$24$	0	0
$25$	−1009.00	−0.322880
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2554.00	0.563931	0.281965	−	0.959425i	$-0.409014\pi$
$29$	2554.00	0.563931	0.281965	+	0.959425i	$0.409014\pi$
$30$	0	0
$31$	−1995.32	−0.372914	−0.186457	−	0.982463i	$-0.559701\pi$
$31$	−1995.32	−0.372914	−0.186457	+	0.982463i	$0.559701\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−7648.74	−1.05541
$36$	0	0
$37$	11950.0	1.43504	0.717519	−	0.696539i	$-0.245278\pi$
$37$	11950.0	1.43504	0.717519	+	0.696539i	$0.245278\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5078.00	0.471773	0.235886	−	0.971781i	$-0.424201\pi$
$41$	5078.00	0.471773	0.235886	+	0.971781i	$0.424201\pi$
$42$	0	0
$43$	12553.9	1.03540	0.517699	−	0.855563i	$-0.326789\pi$
$43$	12553.9	1.03540	0.517699	+	0.855563i	$0.326789\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12304.5	0.812492	0.406246	−	0.913764i	$-0.366838\pi$
$47$	12304.5	0.812492	0.406246	+	0.913764i	$0.366838\pi$
$48$	0	0
$49$	10841.0	0.645029
$50$	0	0
$51$	0	0
$52$	0	0
$53$	19714.0	0.964018	0.482009	−	0.876166i	$-0.339907\pi$
$53$	19714.0	0.964018	0.482009	+	0.876166i	$0.339907\pi$
$54$	0	0
$55$	−3824.37	−0.170472
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8895.81	−0.332702	−0.166351	−	0.986067i	$-0.553199\pi$
$59$	−8895.81	−0.332702	−0.166351	+	0.986067i	$0.553199\pi$
$60$	0	0
$61$	29318.0	1.00881	0.504405	−	0.863467i	$-0.331712\pi$
$61$	29318.0	1.00881	0.504405	+	0.863467i	$0.331712\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1932.00	0.0567184
$66$	0	0
$67$	−16877.1	−0.459315	−0.229658	−	0.973271i	$-0.573761\pi$
$67$	−16877.1	−0.459315	−0.229658	+	0.973271i	$0.573761\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	80976.8	1.90640	0.953202	−	0.302334i	$-0.0977658\pi$
$71$	80976.8	1.90640	0.953202	+	0.302334i	$0.0977658\pi$
$72$	0	0
$73$	37914.0	0.832707	0.416354	−	0.909203i	$-0.363308\pi$
$73$	37914.0	0.832707	0.416354	+	0.909203i	$0.363308\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	13824.0	0.265709
$78$	0	0
$79$	−88791.9	−1.60068	−0.800342	−	0.599544i	$-0.795349\pi$
$79$	−88791.9	−1.60068	−0.800342	+	0.599544i	$0.795349\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−39324.5	−0.626567	−0.313284	−	0.949660i	$-0.601429\pi$
$83$	−39324.5	−0.626567	−0.313284	+	0.949660i	$0.601429\pi$
$84$	0	0
$85$	44252.0	0.664333
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−13930.0	−0.186413	−0.0932065	−	0.995647i	$-0.529712\pi$
$89$	−13930.0	−0.186413	−0.0932065	+	0.995647i	$0.529712\pi$
$90$	0	0
$91$	−6983.63	−0.0884052
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−95609.2	−1.08690
$96$	0	0
$97$	163602.	1.76547	0.882733	−	0.469875i	$-0.155701\pi$
$97$	163602.	1.76547	0.882733	+	0.469875i	$0.155701\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	148562.	1.44912	0.724560	−	0.689212i	$-0.242043\pi$
$101$	148562.	1.44912	0.724560	+	0.689212i	$0.242043\pi$
$102$	0	0
$103$	100598.	0.934317	0.467158	−	0.884174i	$-0.345278\pi$
$103$	100598.	0.934317	0.467158	+	0.884174i	$0.345278\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	194960.	1.64621	0.823105	−	0.567889i	$-0.192240\pi$
$107$	194960.	1.64621	0.823105	+	0.567889i	$0.192240\pi$
$108$	0	0
$109$	123222.	0.993395	0.496698	−	0.867924i	$-0.334546\pi$
$109$	123222.	0.993395	0.496698	+	0.867924i	$0.334546\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	51278.0	0.377777	0.188888	−	0.981999i	$-0.439512\pi$
$113$	51278.0	0.377777	0.188888	+	0.981999i	$0.439512\pi$
$114$	0	0
$115$	145326.	1.02470
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−159958.	−1.03547
$120$	0	0
$121$	−154139.	−0.957082
$122$	0	0
$123$	0	0
$124$	0	0
$125$	190164.	1.08856
$126$	0	0
$127$	148984.	0.819654	0.409827	−	0.912163i	$-0.365589\pi$
$127$	148984.	0.819654	0.409827	+	0.912163i	$0.365589\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−221231.	−1.12634	−0.563169	−	0.826342i	$-0.690418\pi$
$131$	−221231.	−1.12634	−0.563169	+	0.826342i	$0.690418\pi$
$132$	0	0
$133$	345600.	1.69412
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−241034.	−1.09718	−0.548589	−	0.836092i	$-0.684835\pi$
$137$	−241034.	−1.09718	−0.548589	+	0.836092i	$0.684835\pi$
$138$	0	0
$139$	−169020.	−0.741997	−0.370999	−	0.928633i	$-0.620985\pi$
$139$	−169020.	−0.741997	−0.370999	+	0.928633i	$0.620985\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3491.81	−0.0142794
$144$	0	0
$145$	−117484.	−0.464044
$146$	0	0
$147$	0	0
$148$	0	0
$149$	103810.	0.383066	0.191533	−	0.981486i	$-0.438654\pi$
$149$	103810.	0.383066	0.191533	+	0.981486i	$0.438654\pi$
$150$	0	0
$151$	101595.	0.362602	0.181301	−	0.983428i	$-0.441969\pi$
$151$	101595.	0.362602	0.181301	+	0.983428i	$0.441969\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	91784.8	0.306861
$156$	0	0
$157$	−234458.	−0.759130	−0.379565	−	0.925165i	$-0.623926\pi$
$157$	−234458.	−0.759130	−0.379565	+	0.925165i	$0.623926\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−525312.	−1.59718
$162$	0	0
$163$	−388672.	−1.14581	−0.572907	−	0.819620i	$-0.694185\pi$
$163$	−388672.	−1.14581	−0.572907	+	0.819620i	$0.694185\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−67342.1	−0.186851	−0.0934256	−	0.995626i	$-0.529782\pi$
$167$	−67342.1	−0.186851	−0.0934256	+	0.995626i	$0.529782\pi$
$168$	0	0
$169$	−369529.	−0.995249
$170$	0	0
$171$	0	0
$172$	0	0
$173$	206282.	0.524018	0.262009	−	0.965065i	$-0.415615\pi$
$173$	206282.	0.524018	0.262009	+	0.965065i	$0.415615\pi$
$174$	0	0
$175$	−167773.	−0.414122
$176$	0	0
$177$	0	0
$178$	0	0
$179$	198119.	0.462161	0.231081	−	0.972935i	$-0.425774\pi$
$179$	198119.	0.462161	0.231081	+	0.972935i	$0.425774\pi$
$180$	0	0
$181$	−19746.0	−0.0448005	−0.0224002	−	0.999749i	$-0.507131\pi$
$181$	−19746.0	−0.0448005	−0.0224002	+	0.999749i	$0.507131\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−549700.	−1.18085
$186$	0	0
$187$	−79979.2	−0.167253
$188$	0	0
$189$	0	0
$190$	0	0
$191$	701023.	1.39043	0.695215	−	0.718802i	$-0.255309\pi$
$191$	701023.	1.39043	0.695215	+	0.718802i	$0.255309\pi$
$192$	0	0
$193$	−628622.	−1.21478	−0.607388	−	0.794405i	$-0.707783\pi$
$193$	−628622.	−1.21478	−0.607388	+	0.794405i	$0.707783\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	334034.	0.613232	0.306616	−	0.951833i	$-0.400803\pi$
$197$	334034.	0.613232	0.306616	+	0.951833i	$0.400803\pi$
$198$	0	0
$199$	318088.	0.569396	0.284698	−	0.958617i	$-0.408107\pi$
$199$	318088.	0.569396	0.284698	+	0.958617i	$0.408107\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	424671.	0.723290
$204$	0	0
$205$	−233588.	−0.388209
$206$	0	0
$207$	0	0
$208$	0	0
$209$	172800.	0.273639
$210$	0	0
$211$	736357.	1.13863	0.569315	−	0.822120i	$-0.307209\pi$
$211$	736357.	1.13863	0.569315	+	0.822120i	$0.307209\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−577480.	−0.852002
$216$	0	0
$217$	−331776.	−0.478295
$218$	0	0
$219$	0	0
$220$	0	0
$221$	40404.0	0.0556472
$222$	0	0
$223$	−1.31226e6	−1.76708	−0.883541	−	0.468354i	$-0.844847\pi$
$223$	−1.31226e6	−1.76708	−0.883541	+	0.468354i	$0.844847\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−344276.	−0.443448	−0.221724	−	0.975109i	$-0.571168\pi$
$227$	−344276.	−0.443448	−0.221724	+	0.975109i	$0.571168\pi$
$228$	0	0
$229$	−664626.	−0.837507	−0.418754	−	0.908100i	$-0.637533\pi$
$229$	−664626.	−0.837507	−0.418754	+	0.908100i	$0.637533\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.09433e6	−1.32056	−0.660281	−	0.751019i	$-0.729563\pi$
$233$	−1.09433e6	−1.32056	−0.660281	+	0.751019i	$0.729563\pi$
$234$	0	0
$235$	−566006.	−0.668577
$236$	0	0
$237$	0	0
$238$	0	0
$239$	596269.	0.675223	0.337612	−	0.941285i	$-0.390381\pi$
$239$	596269.	0.675223	0.337612	+	0.941285i	$0.390381\pi$
$240$	0	0
$241$	−64734.0	−0.0717943	−0.0358971	−	0.999355i	$-0.511429\pi$
$241$	−64734.0	−0.0717943	−0.0358971	+	0.999355i	$0.511429\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−498686.	−0.530777
$246$	0	0
$247$	−87295.4	−0.0910435
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.29521e6	−1.29765	−0.648824	−	0.760938i	$-0.724739\pi$
$251$	−1.29521e6	−1.29765	−0.648824	+	0.760938i	$0.724739\pi$
$252$	0	0
$253$	−262656.	−0.257980
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−52226.0	−0.0493235	−0.0246618	−	0.999696i	$-0.507851\pi$
$257$	−52226.0	−0.0493235	−0.0246618	+	0.999696i	$0.507851\pi$
$258$	0	0
$259$	1.98701e6	1.84056
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.17342e6	−1.04607	−0.523037	−	0.852310i	$-0.675201\pi$
$263$	−1.17342e6	−1.04607	−0.523037	+	0.852310i	$0.675201\pi$
$264$	0	0
$265$	−906844.	−0.793264
$266$	0	0
$267$	0	0
$268$	0	0
$269$	359498.	0.302912	0.151456	−	0.988464i	$-0.451604\pi$
$269$	359498.	0.302912	0.151456	+	0.988464i	$0.451604\pi$
$270$	0	0
$271$	1.00331e6	0.829877	0.414939	−	0.909849i	$-0.363803\pi$
$271$	1.00331e6	0.829877	0.414939	+	0.909849i	$0.363803\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−83886.7	−0.0668900
$276$	0	0
$277$	−638370.	−0.499888	−0.249944	−	0.968260i	$-0.580412\pi$
$277$	−638370.	−0.499888	−0.249944	+	0.968260i	$0.580412\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	112790.	0.0852128	0.0426064	−	0.999092i	$-0.486434\pi$
$281$	112790.	0.0852128	0.0426064	+	0.999092i	$0.486434\pi$
$282$	0	0
$283$	−487939.	−0.362160	−0.181080	−	0.983468i	$-0.557959\pi$
$283$	−487939.	−0.362160	−0.181080	+	0.983468i	$0.557959\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	844354.	0.605090
$288$	0	0
$289$	−494413.	−0.348213
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−990350.	−0.673938	−0.336969	−	0.941516i	$-0.609402\pi$
$293$	−990350.	−0.673938	−0.336969	+	0.941516i	$0.609402\pi$
$294$	0	0
$295$	409207.	0.273772
$296$	0	0
$297$	0	0
$298$	0	0
$299$	132689.	0.0858335
$300$	0	0
$301$	2.08742e6	1.32799
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.34863e6	−0.830123
$306$	0	0
$307$	340618.	0.206263	0.103132	−	0.994668i	$-0.467114\pi$
$307$	340618.	0.206263	0.103132	+	0.994668i	$0.467114\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.41418e6	0.829097	0.414548	−	0.910027i	$-0.363940\pi$
$311$	1.41418e6	0.829097	0.414548	+	0.910027i	$0.363940\pi$
$312$	0	0
$313$	940154.	0.542423	0.271212	−	0.962520i	$-0.412576\pi$
$313$	940154.	0.542423	0.271212	+	0.962520i	$0.412576\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.42825e6	1.35720	0.678602	−	0.734506i	$-0.262586\pi$
$317$	2.42825e6	1.35720	0.678602	+	0.734506i	$0.262586\pi$
$318$	0	0
$319$	212336.	0.116828
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.99948e6	−1.06638
$324$	0	0
$325$	42378.0	0.0222552
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.04595e6	1.04209
$330$	0	0
$331$	−762296.	−0.382432	−0.191216	−	0.981548i	$-0.561243\pi$
$331$	−762296.	−0.382432	−0.191216	+	0.981548i	$0.561243\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	776347.	0.377958
$336$	0	0
$337$	2.97506e6	1.42699	0.713495	−	0.700661i	$-0.247111\pi$
$337$	2.97506e6	1.42699	0.713495	+	0.700661i	$0.247111\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−165888.	−0.0772554
$342$	0	0
$343$	−992008.	−0.455281
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.91159e6	1.29810	0.649048	−	0.760748i	$-0.275167\pi$
$347$	2.91159e6	1.29810	0.649048	+	0.760748i	$0.275167\pi$
$348$	0	0
$349$	1.60641e6	0.705979	0.352989	−	0.935627i	$-0.385165\pi$
$349$	1.60641e6	0.705979	0.352989	+	0.935627i	$0.385165\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4.22819e6	1.80600	0.903000	−	0.429641i	$-0.141360\pi$
$353$	4.22819e6	1.80600	0.903000	+	0.429641i	$0.141360\pi$
$354$	0	0
$355$	−3.72493e6	−1.56873
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.07464e6	−0.849583	−0.424792	−	0.905291i	$-0.639653\pi$
$359$	−2.07464e6	−0.849583	−0.424792	+	0.905291i	$0.639653\pi$
$360$	0	0
$361$	1.84390e6	0.744680
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.74404e6	−0.685212
$366$	0	0
$367$	−2.33087e6	−0.903343	−0.451672	−	0.892184i	$-0.649172\pi$
$367$	−2.33087e6	−0.903343	−0.451672	+	0.892184i	$0.649172\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.27798e6	1.23644
$372$	0	0
$373$	1.50630e6	0.560583	0.280292	−	0.959915i	$-0.409569\pi$
$373$	1.50630e6	0.560583	0.280292	+	0.959915i	$0.409569\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−107268.	−0.0388702
$378$	0	0
$379$	−843772.	−0.301736	−0.150868	−	0.988554i	$-0.548207\pi$
$379$	−843772.	−0.301736	−0.150868	+	0.988554i	$0.548207\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.58360e6	−1.24831	−0.624155	−	0.781300i	$-0.714557\pi$
$383$	−3.58360e6	−1.24831	−0.624155	+	0.781300i	$0.714557\pi$
$384$	0	0
$385$	−635904.	−0.218645
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−786446.	−0.263509	−0.131754	−	0.991282i	$-0.542061\pi$
$389$	−786446.	−0.263509	−0.131754	+	0.991282i	$0.542061\pi$
$390$	0	0
$391$	3.03921e6	1.00535
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.08443e6	1.31716
$396$	0	0
$397$	1.16889e6	0.372217	0.186108	−	0.982529i	$-0.440412\pi$
$397$	1.16889e6	0.372217	0.186108	+	0.982529i	$0.440412\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.58400e6	1.42359	0.711793	−	0.702390i	$-0.247884\pi$
$401$	4.58400e6	1.42359	0.711793	+	0.702390i	$0.247884\pi$
$402$	0	0
$403$	83803.5	0.0257039
$404$	0	0
$405$	0	0
$406$	0	0
$407$	993504.	0.297292
$408$	0	0
$409$	4.31356e6	1.27505	0.637526	−	0.770429i	$-0.279958\pi$
$409$	4.31356e6	1.27505	0.637526	+	0.770429i	$0.279958\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.47917e6	−0.426719
$414$	0	0
$415$	1.80893e6	0.515585
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3.63257e6	−1.01083	−0.505416	−	0.862876i	$-0.668661\pi$
$419$	−3.63257e6	−1.01083	−0.505416	+	0.862876i	$0.668661\pi$
$420$	0	0
$421$	−5.01688e6	−1.37952	−0.689761	−	0.724037i	$-0.742284\pi$
$421$	−5.01688e6	−1.37952	−0.689761	+	0.724037i	$0.742284\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	970658.	0.260672
$426$	0	0
$427$	4.87491e6	1.29389
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−189888.	−0.0492385	−0.0246192	−	0.999697i	$-0.507837\pi$
$431$	−189888.	−0.0492385	−0.0246192	+	0.999697i	$0.507837\pi$
$432$	0	0
$433$	5.21687e6	1.33718	0.668590	−	0.743631i	$-0.266898\pi$
$433$	5.21687e6	1.33718	0.668590	+	0.743631i	$0.266898\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.56640e6	−1.64484
$438$	0	0
$439$	3.36661e6	0.833741	0.416871	−	0.908966i	$-0.363127\pi$
$439$	3.36661e6	0.833741	0.416871	+	0.908966i	$0.363127\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.42654e6	−1.07166	−0.535828	−	0.844327i	$-0.680000\pi$
$443$	−4.42654e6	−1.07166	−0.535828	+	0.844327i	$0.680000\pi$
$444$	0	0
$445$	640780.	0.153394
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−443506.	−0.103821	−0.0519103	−	0.998652i	$-0.516531\pi$
$449$	−443506.	−0.103821	−0.0519103	+	0.998652i	$0.516531\pi$
$450$	0	0
$451$	422177.	0.0977357
$452$	0	0
$453$	0	0
$454$	0	0
$455$	321247.	0.0727462
$456$	0	0
$457$	590538.	0.132269	0.0661344	−	0.997811i	$-0.478933\pi$
$457$	590538.	0.132269	0.0661344	+	0.997811i	$0.478933\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4.79463e6	1.05076	0.525380	−	0.850868i	$-0.323923\pi$
$461$	4.79463e6	1.05076	0.525380	+	0.850868i	$0.323923\pi$
$462$	0	0
$463$	−1.40038e6	−0.303595	−0.151798	−	0.988412i	$-0.548506\pi$
$463$	−1.40038e6	−0.303595	−0.151798	+	0.988412i	$0.548506\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.31681e6	0.703766	0.351883	−	0.936044i	$-0.385542\pi$
$467$	3.31681e6	0.703766	0.351883	+	0.936044i	$0.385542\pi$
$468$	0	0
$469$	−2.80627e6	−0.589112
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.04371e6	0.214500
$474$	0	0
$475$	−2.09717e6	−0.426480
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−5.00360e6	−0.996424	−0.498212	−	0.867055i	$-0.666010\pi$
$479$	−5.00360e6	−0.996424	−0.498212	+	0.867055i	$0.666010\pi$
$480$	0	0
$481$	−501900.	−0.0989133
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.52569e6	−1.45275
$486$	0	0
$487$	−5.90133e6	−1.12753	−0.563764	−	0.825936i	$-0.690647\pi$
$487$	−5.90133e6	−1.12753	−0.563764	+	0.825936i	$0.690647\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−8.15480e6	−1.52654	−0.763272	−	0.646077i	$-0.776409\pi$
$491$	−8.15480e6	−1.52654	−0.763272	+	0.646077i	$0.776409\pi$
$492$	0	0
$493$	−2.45695e6	−0.455280
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.34646e7	2.44513
$498$	0	0
$499$	470979.	0.0846741	0.0423370	−	0.999103i	$-0.486520\pi$
$499$	470979.	0.0846741	0.0423370	+	0.999103i	$0.486520\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9.23452e6	1.62740	0.813700	−	0.581285i	$-0.197450\pi$
$503$	9.23452e6	1.62740	0.813700	+	0.581285i	$0.197450\pi$
$504$	0	0
$505$	−6.83385e6	−1.19244
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.95283e6	0.505177	0.252588	−	0.967574i	$-0.418718\pi$
$509$	2.95283e6	0.505177	0.252588	+	0.967574i	$0.418718\pi$
$510$	0	0
$511$	6.30422e6	1.06802
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.62749e6	−0.768824
$516$	0	0
$517$	1.02298e6	0.168321
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.04570e7	−1.68776	−0.843882	−	0.536529i	$-0.819735\pi$
$521$	−1.04570e7	−1.68776	−0.843882	+	0.536529i	$0.819735\pi$
$522$	0	0
$523$	5.77571e6	0.923318	0.461659	−	0.887058i	$-0.347254\pi$
$523$	5.77571e6	0.923318	0.461659	+	0.887058i	$0.347254\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.91950e6	0.301066
$528$	0	0
$529$	3.54459e6	0.550714
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−213276.	−0.0325180
$534$	0	0
$535$	−8.96814e6	−1.35462
$536$	0	0
$537$	0	0
$538$	0	0
$539$	901304.	0.133629
$540$	0	0
$541$	−3.86409e6	−0.567615	−0.283808	−	0.958881i	$-0.591598\pi$
$541$	−3.86409e6	−0.567615	−0.283808	+	0.958881i	$0.591598\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.66821e6	−0.817438
$546$	0	0
$547$	−8.49633e6	−1.21412	−0.607062	−	0.794654i	$-0.707652\pi$
$547$	−8.49633e6	−1.21412	−0.607062	+	0.794654i	$0.707652\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.30839e6	0.744876
$552$	0	0
$553$	−1.47640e7	−2.05302
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6.33253e6	−0.864848	−0.432424	−	0.901671i	$-0.642342\pi$
$557$	−6.33253e6	−0.864848	−0.432424	+	0.901671i	$0.642342\pi$
$558$	0	0
$559$	−527264.	−0.0713672
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.47605e6	1.12700	0.563498	−	0.826117i	$-0.309455\pi$
$563$	8.47605e6	1.12700	0.563498	+	0.826117i	$0.309455\pi$
$564$	0	0
$565$	−2.35879e6	−0.310862
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.69279e6	−0.219191	−0.109596	−	0.993976i	$-0.534956\pi$
$569$	−1.69279e6	−0.219191	−0.109596	+	0.993976i	$0.534956\pi$
$570$	0	0
$571$	4.50901e6	0.578750	0.289375	−	0.957216i	$-0.406552\pi$
$571$	4.50901e6	0.578750	0.289375	+	0.957216i	$0.406552\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.18769e6	0.402075
$576$	0	0
$577$	−4.07577e6	−0.509648	−0.254824	−	0.966987i	$-0.582018\pi$
$577$	−4.07577e6	−0.509648	−0.254824	+	0.966987i	$0.582018\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6.53875e6	−0.803627
$582$	0	0
$583$	1.63899e6	0.199712
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.15563e6	0.976928	0.488464	−	0.872584i	$-0.337557\pi$
$587$	8.15563e6	0.976928	0.488464	+	0.872584i	$0.337557\pi$
$588$	0	0
$589$	−4.14720e6	−0.492569
$590$	0	0
$591$	0	0
$592$	0	0
$593$	528622.	0.0617317	0.0308659	−	0.999524i	$-0.490174\pi$
$593$	528622.	0.0617317	0.0308659	+	0.999524i	$0.490174\pi$
$594$	0	0
$595$	7.35808e6	0.852064
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4.84381e6	−0.551595	−0.275797	−	0.961216i	$-0.588942\pi$
$599$	−4.84381e6	−0.551595	−0.275797	+	0.961216i	$0.588942\pi$
$600$	0	0
$601$	−2.11804e6	−0.239193	−0.119596	−	0.992823i	$-0.538160\pi$
$601$	−2.11804e6	−0.239193	−0.119596	+	0.992823i	$0.538160\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.09039e6	0.787557
$606$	0	0
$607$	−6.93109e6	−0.763536	−0.381768	−	0.924258i	$-0.624685\pi$
$607$	−6.93109e6	−0.763536	−0.381768	+	0.924258i	$0.624685\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−516789.	−0.0560028
$612$	0	0
$613$	−3.21599e6	−0.345671	−0.172836	−	0.984951i	$-0.555293\pi$
$613$	−3.21599e6	−0.345671	−0.172836	+	0.984951i	$0.555293\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.46326e7	−1.54742	−0.773709	−	0.633541i	$-0.781601\pi$
$617$	−1.46326e7	−1.54742	−0.773709	+	0.633541i	$0.781601\pi$
$618$	0	0
$619$	−2.42706e6	−0.254597	−0.127299	−	0.991864i	$-0.540631\pi$
$619$	−2.42706e6	−0.254597	−0.127299	+	0.991864i	$0.540631\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.31624e6	−0.239091
$624$	0	0
$625$	−5.59442e6	−0.572869
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.14959e7	−1.15855
$630$	0	0
$631$	−1.62235e7	−1.62207	−0.811036	−	0.584996i	$-0.801096\pi$
$631$	−1.62235e7	−1.62207	−0.811036	+	0.584996i	$0.801096\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6.85327e6	−0.674471
$636$	0	0
$637$	−455322.	−0.0444601
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.22399e7	1.17662	0.588308	−	0.808637i	$-0.299794\pi$
$641$	1.22399e7	1.17662	0.588308	+	0.808637i	$0.299794\pi$
$642$	0	0
$643$	−2.40461e6	−0.229360	−0.114680	−	0.993402i	$-0.536584\pi$
$643$	−2.40461e6	−0.229360	−0.114680	+	0.993402i	$0.536584\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.39515e7	−1.31026	−0.655132	−	0.755514i	$-0.727387\pi$
$647$	−1.39515e7	−1.31026	−0.655132	+	0.755514i	$0.727387\pi$
$648$	0	0
$649$	−739584.	−0.0689248
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.59391e7	1.46279	0.731395	−	0.681954i	$-0.238870\pi$
$653$	1.59391e7	1.46279	0.731395	+	0.681954i	$0.238870\pi$
$654$	0	0
$655$	1.01766e7	0.926833
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3.36470e6	0.301809	0.150905	−	0.988548i	$-0.451781\pi$
$659$	3.36470e6	0.301809	0.150905	+	0.988548i	$0.451781\pi$
$660$	0	0
$661$	−8.51194e6	−0.757748	−0.378874	−	0.925448i	$-0.623689\pi$
$661$	−8.51194e6	−0.757748	−0.378874	+	0.925448i	$0.623689\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.58976e7	−1.39405
$666$	0	0
$667$	−8.06875e6	−0.702250
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.43745e6	0.208992
$672$	0	0
$673$	1.15169e7	0.980161	0.490080	−	0.871677i	$-0.336967\pi$
$673$	1.15169e7	0.980161	0.490080	+	0.871677i	$0.336967\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.32692e7	1.95124	0.975618	−	0.219475i	$-0.0704346\pi$
$677$	2.32692e7	1.95124	0.975618	+	0.219475i	$0.0704346\pi$
$678$	0	0
$679$	2.72032e7	2.26436
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.90399e7	−1.56176	−0.780880	−	0.624682i	$-0.785229\pi$
$683$	−1.90399e7	−1.56176	−0.780880	+	0.624682i	$0.785229\pi$
$684$	0	0
$685$	1.10876e7	0.902838
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−827988.	−0.0664471
$690$	0	0
$691$	−7.15780e6	−0.570275	−0.285138	−	0.958487i	$-0.592039\pi$
$691$	−7.15780e6	−0.570275	−0.285138	+	0.958487i	$0.592039\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.77494e6	0.610569
$696$	0	0
$697$	−4.88504e6	−0.380878
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−8.77684e6	−0.674595	−0.337297	−	0.941398i	$-0.609513\pi$
$701$	−8.77684e6	−0.674595	−0.337297	+	0.941398i	$0.609513\pi$
$702$	0	0
$703$	2.48376e7	1.89549
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.47024e7	1.85862
$708$	0	0
$709$	8.08481e6	0.604024	0.302012	−	0.953304i	$-0.402342\pi$
$709$	8.08481e6	0.604024	0.302012	+	0.953304i	$0.402342\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.30374e6	0.464381
$714$	0	0
$715$	160623.	0.0117502
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.42855e7	−1.03056	−0.515280	−	0.857022i	$-0.672312\pi$
$719$	−1.42855e7	−1.03056	−0.515280	+	0.857022i	$0.672312\pi$
$720$	0	0
$721$	1.67270e7	1.19834
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.57699e6	−0.182082
$726$	0	0
$727$	−1.17227e7	−0.822605	−0.411302	−	0.911499i	$-0.634926\pi$
$727$	−1.17227e7	−0.822605	−0.411302	+	0.911499i	$0.634926\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.20769e7	−0.835912
$732$	0	0
$733$	1.81661e7	1.24882	0.624412	−	0.781095i	$-0.285339\pi$
$733$	1.81661e7	1.24882	0.624412	+	0.781095i	$0.285339\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.40314e6	−0.0951549
$738$	0	0
$739$	1.27312e7	0.857550	0.428775	−	0.903411i	$-0.358945\pi$
$739$	1.27312e7	0.857550	0.428775	+	0.903411i	$0.358945\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.82988e6	−0.121605	−0.0608023	−	0.998150i	$-0.519366\pi$
$743$	−1.82988e6	−0.121605	−0.0608023	+	0.998150i	$0.519366\pi$
$744$	0	0
$745$	−4.77526e6	−0.315215
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.24173e7	2.11141
$750$	0	0
$751$	−5.64577e6	−0.365278	−0.182639	−	0.983180i	$-0.558464\pi$
$751$	−5.64577e6	−0.365278	−0.182639	+	0.983180i	$0.558464\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4.67338e6	−0.298376
$756$	0	0
$757$	139230.	0.00883066	0.00441533	−	0.999990i	$-0.498595\pi$
$757$	139230.	0.00883066	0.00441533	+	0.999990i	$0.498595\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.36251e7	−0.852863	−0.426432	−	0.904520i	$-0.640230\pi$
$761$	−1.36251e7	−0.852863	−0.426432	+	0.904520i	$0.640230\pi$
$762$	0	0
$763$	2.04890e7	1.27412
$764$	0	0
$765$	0	0
$766$	0	0
$767$	373624.	0.0229322
$768$	0	0
$769$	−1.06840e7	−0.651504	−0.325752	−	0.945455i	$-0.605618\pi$
$769$	−1.06840e7	−0.651504	−0.325752	+	0.945455i	$0.605618\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−2.22367e7	−1.33851	−0.669255	−	0.743033i	$-0.733386\pi$
$773$	−2.22367e7	−1.33851	−0.669255	+	0.743033i	$0.733386\pi$
$774$	0	0
$775$	2.01328e6	0.120407
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.05544e7	0.623148
$780$	0	0
$781$	6.73229e6	0.394944
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.07851e7	0.624667
$786$	0	0
$787$	−2.81333e7	−1.61914	−0.809569	−	0.587025i	$-0.800299\pi$
$787$	−2.81333e7	−1.61914	−0.809569	+	0.587025i	$0.800299\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8.52635e6	0.484531
$792$	0	0
$793$	−1.23136e6	−0.0695346
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.43450e7	1.35758	0.678789	−	0.734333i	$-0.262505\pi$
$797$	2.43450e7	1.35758	0.678789	+	0.734333i	$0.262505\pi$
$798$	0	0
$799$	−1.18369e7	−0.655951
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3.15211e6	0.172509
$804$	0	0
$805$	2.41644e7	1.31427
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−2.30810e7	−1.23989	−0.619945	−	0.784645i	$-0.712845\pi$
$809$	−2.30810e7	−1.23989	−0.619945	+	0.784645i	$0.712845\pi$
$810$	0	0
$811$	−3.33586e7	−1.78096	−0.890482	−	0.455019i	$-0.849632\pi$
$811$	−3.33586e7	−1.78096	−0.890482	+	0.455019i	$0.849632\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.78789e7	0.942860
$816$	0	0
$817$	2.60928e7	1.36762
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.03134e7	1.05178	0.525890	−	0.850553i	$-0.323732\pi$
$821$	2.03134e7	1.05178	0.525890	+	0.850553i	$0.323732\pi$
$822$	0	0
$823$	−6.90531e6	−0.355372	−0.177686	−	0.984087i	$-0.556861\pi$
$823$	−6.90531e6	−0.355372	−0.177686	+	0.984087i	$0.556861\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.12498e7	1.58885	0.794427	−	0.607360i	$-0.207771\pi$
$827$	3.12498e7	1.58885	0.794427	+	0.607360i	$0.207771\pi$
$828$	0	0
$829$	2.32015e7	1.17254	0.586272	−	0.810114i	$-0.300595\pi$
$829$	2.32015e7	1.17254	0.586272	+	0.810114i	$0.300595\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.04290e7	−0.520753
$834$	0	0
$835$	3.09774e6	0.153755
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2.91651e7	−1.43041	−0.715203	−	0.698917i	$-0.753666\pi$
$839$	−2.91651e7	−1.43041	−0.715203	+	0.698917i	$0.753666\pi$
$840$	0	0
$841$	−1.39882e7	−0.681982
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.69983e7	0.818964
$846$	0	0
$847$	−2.56298e7	−1.22754
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3.77532e7	−1.78702
$852$	0	0
$853$	−9.23146e6	−0.434408	−0.217204	−	0.976126i	$-0.569694\pi$
$853$	−9.23146e6	−0.434408	−0.217204	+	0.976126i	$0.569694\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.34077e6	0.341421	0.170710	−	0.985321i	$-0.445394\pi$
$857$	7.34077e6	0.341421	0.170710	+	0.985321i	$0.445394\pi$
$858$	0	0
$859$	1.57274e7	0.727233	0.363616	−	0.931549i	$-0.381542\pi$
$859$	1.57274e7	0.727233	0.363616	+	0.931549i	$0.381542\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−5.72990e6	−0.261891	−0.130945	−	0.991390i	$-0.541801\pi$
$863$	−5.72990e6	−0.261891	−0.130945	+	0.991390i	$0.541801\pi$
$864$	0	0
$865$	−9.48897e6	−0.431200
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−7.38202e6	−0.331608
$870$	0	0
$871$	708838.	0.0316593
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.16199e7	1.39618
$876$	0	0
$877$	3.57123e7	1.56790	0.783950	−	0.620823i	$-0.213202\pi$
$877$	3.57123e7	1.56790	0.783950	+	0.620823i	$0.213202\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.77000e6	0.0768304	0.0384152	−	0.999262i	$-0.487769\pi$
$881$	1.77000e6	0.0768304	0.0384152	+	0.999262i	$0.487769\pi$
$882$	0	0
$883$	2.70394e7	1.16706	0.583532	−	0.812090i	$-0.301670\pi$
$883$	2.70394e7	1.16706	0.583532	+	0.812090i	$0.301670\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.07893e7	1.31399	0.656994	−	0.753896i	$-0.271828\pi$
$887$	3.07893e7	1.31399	0.656994	+	0.753896i	$0.271828\pi$
$888$	0	0
$889$	2.47726e7	1.05128
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.55744e7	1.07319
$894$	0	0
$895$	−9.11347e6	−0.380300
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−5.09605e6	−0.210298
$900$	0	0
$901$	−1.89649e7	−0.778284
$902$	0	0
$903$	0	0
$904$	0	0
$905$	908316.	0.0368651
$906$	0	0
$907$	1.94729e7	0.785983	0.392992	−	0.919542i	$-0.371440\pi$
$907$	1.94729e7	0.785983	0.392992	+	0.919542i	$0.371440\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	2.44863e7	0.977522	0.488761	−	0.872418i	$-0.337449\pi$
$911$	2.44863e7	0.977522	0.488761	+	0.872418i	$0.337449\pi$
$912$	0	0
$913$	−3.26938e6	−0.129804
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.67857e7	−1.44463
$918$	0	0
$919$	2.05656e7	0.803254	0.401627	−	0.915803i	$-0.368445\pi$
$919$	2.05656e7	0.803254	0.401627	+	0.915803i	$0.368445\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3.40103e6	−0.131403
$924$	0	0
$925$	−1.20576e7	−0.463345
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.50171e7	1.71135	0.855674	−	0.517514i	$-0.173143\pi$
$929$	4.50171e7	1.71135	0.855674	+	0.517514i	$0.173143\pi$
$930$	0	0
$931$	2.25326e7	0.851995
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3.67904e6	0.137628
$936$	0	0
$937$	2.31151e7	0.860097	0.430048	−	0.902806i	$-0.358496\pi$
$937$	2.31151e7	0.860097	0.430048	+	0.902806i	$0.358496\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	664042.	0.0244468	0.0122234	−	0.999925i	$-0.496109\pi$
$941$	664042.	0.0244468	0.0122234	+	0.999925i	$0.496109\pi$
$942$	0	0
$943$	−1.60427e7	−0.587488
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−5.17375e7	−1.87469	−0.937347	−	0.348398i	$-0.886726\pi$
$947$	−5.17375e7	−1.87469	−0.937347	+	0.348398i	$0.886726\pi$
$948$	0	0
$949$	−1.59239e6	−0.0573962
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.89030e7	−1.03089	−0.515444	−	0.856923i	$-0.672373\pi$
$953$	−2.89030e7	−1.03089	−0.515444	+	0.856923i	$0.672373\pi$
$954$	0	0
$955$	−3.22471e7	−1.14415
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.00784e7	−1.40723
$960$	0	0
$961$	−2.46478e7	−0.860935
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.89166e7	0.999607
$966$	0	0
$967$	2.36554e7	0.813512	0.406756	−	0.913537i	$-0.366660\pi$
$967$	2.36554e7	0.813512	0.406756	+	0.913537i	$0.366660\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−3.15207e7	−1.07287	−0.536436	−	0.843941i	$-0.680230\pi$
$971$	−3.15207e7	−1.07287	−0.536436	+	0.843941i	$0.680230\pi$
$972$	0	0
$973$	−2.81042e7	−0.951676
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.82778e6	−0.161812	−0.0809060	−	0.996722i	$-0.525781\pi$
$977$	−4.82778e6	−0.161812	−0.0809060	+	0.996722i	$0.525781\pi$
$978$	0	0
$979$	−1.15812e6	−0.0386186
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.81641e7	0.929636	0.464818	−	0.885406i	$-0.346120\pi$
$983$	2.81641e7	0.929636	0.464818	+	0.885406i	$0.346120\pi$
$984$	0	0
$985$	−1.53656e7	−0.504612
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.96611e7	−1.28936
$990$	0	0
$991$	2.95301e7	0.955171	0.477585	−	0.878585i	$-0.341512\pi$
$991$	2.95301e7	0.955171	0.477585	+	0.878585i	$0.341512\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.46320e7	−0.468540
$996$	0	0
$997$	−5.16306e7	−1.64501	−0.822507	−	0.568756i	$-0.807425\pi$
$997$	−5.16306e7	−1.64501	−0.822507	+	0.568756i	$0.807425\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.6.a.l.1.2		2
3.2	odd	2		32.6.a.d.1.1	✓	2
4.3	odd	2	inner	288.6.a.l.1.1		2
8.3	odd	2		576.6.a.bp.1.1		2
8.5	even	2		576.6.a.bp.1.2		2
12.11	even	2		32.6.a.d.1.2	yes	2
15.2	even	4		800.6.c.d.449.3		4
15.8	even	4		800.6.c.d.449.1		4
15.14	odd	2		800.6.a.k.1.2		2
24.5	odd	2		64.6.a.h.1.2		2
24.11	even	2		64.6.a.h.1.1		2
48.5	odd	4		256.6.b.l.129.4		4
48.11	even	4		256.6.b.l.129.2		4
48.29	odd	4		256.6.b.l.129.1		4
48.35	even	4		256.6.b.l.129.3		4
60.23	odd	4		800.6.c.d.449.4		4
60.47	odd	4		800.6.c.d.449.2		4
60.59	even	2		800.6.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.6.a.d.1.1	✓	2	3.2	odd	2
32.6.a.d.1.2	yes	2	12.11	even	2
64.6.a.h.1.1		2	24.11	even	2
64.6.a.h.1.2		2	24.5	odd	2
256.6.b.l.129.1		4	48.29	odd	4
256.6.b.l.129.2		4	48.11	even	4
256.6.b.l.129.3		4	48.35	even	4
256.6.b.l.129.4		4	48.5	odd	4
288.6.a.l.1.1		2	4.3	odd	2	inner
288.6.a.l.1.2		2	1.1	even	1	trivial
576.6.a.bp.1.1		2	8.3	odd	2
576.6.a.bp.1.2		2	8.5	even	2
800.6.a.k.1.1		2	60.59	even	2
800.6.a.k.1.2		2	15.14	odd	2
800.6.c.d.449.1		4	15.8	even	4
800.6.c.d.449.2		4	60.47	odd	4
800.6.c.d.449.3		4	15.2	even	4
800.6.c.d.449.4		4	60.23	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	288.a (trivial)

\(f(q)\)	\(=\)	\(q-46.0000 q^{5} +166.277 q^{7} +83.1384 q^{11} -42.0000 q^{13} -962.000 q^{17} +2078.46 q^{19} -3159.26 q^{23} -1009.00 q^{25} +2554.00 q^{29} -1995.32 q^{31} -7648.74 q^{35} +11950.0 q^{37} +5078.00 q^{41} +12553.9 q^{43} +12304.5 q^{47} +10841.0 q^{49} +19714.0 q^{53} -3824.37 q^{55} -8895.81 q^{59} +29318.0 q^{61} +1932.00 q^{65} -16877.1 q^{67} +80976.8 q^{71} +37914.0 q^{73} +13824.0 q^{77} -88791.9 q^{79} -39324.5 q^{83} +44252.0 q^{85} -13930.0 q^{89} -6983.63 q^{91} -95609.2 q^{95} +163602. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 92 q^{5} - 84 q^{13} - 1924 q^{17} - 2018 q^{25} + 5108 q^{29} + 23900 q^{37} + 10156 q^{41} + 21682 q^{49} + 39428 q^{53} + 58636 q^{61} + 3864 q^{65} + 75828 q^{73} + 27648 q^{77} + 88504 q^{85}+ \cdots + 327204 q^{97}+O(q^{100}) \) 2 * q - 92 * q^5 - 84 * q^13 - 1924 * q^17 - 2018 * q^25 + 5108 * q^29 + 23900 * q^37 + 10156 * q^41 + 21682 * q^49 + 39428 * q^53 + 58636 * q^61 + 3864 * q^65 + 75828 * q^73 + 27648 * q^77 + 88504 * q^85 - 27860 * q^89 + 327204 * q^97

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