LMFDB - Embedded newform 864.4.c.b.863.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,4,Mod(863,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("864.863");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	864.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$50.9776502450$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			863.18
Character	$\chi$	$=$	864.863
Dual form			864.4.c.b.863.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.99651i q^{5} +35.3962i q^{7} +O(q^{10})$	$q+8.99651i q^{5} +35.3962i q^{7} +2.88651 q^{11} +47.2668 q^{13} +3.19790i q^{17} +92.6761i q^{19} +182.495 q^{23} +44.0628 q^{25} +161.054i q^{29} +11.7511i q^{31} -318.442 q^{35} +71.1977 q^{37} +111.322i q^{41} +30.0027i q^{43} +523.230 q^{47} -909.889 q^{49} -683.907i q^{53} +25.9685i q^{55} -146.873 q^{59} -717.089 q^{61} +425.237i q^{65} -228.684i q^{67} -1042.11 q^{71} -676.843 q^{73} +102.171i q^{77} +865.762i q^{79} +334.416 q^{83} -28.7699 q^{85} -598.589i q^{89} +1673.07i q^{91} -833.762 q^{95} +1034.49 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 144 q^{13} - 768 q^{25} + 1536 q^{37} - 2304 q^{49} + 2784 q^{61} - 1416 q^{73} - 816 q^{85} + 648 q^{97}+O(q^{100}) $ 24 * q + 144 * q^13 - 768 * q^25 + 1536 * q^37 - 2304 * q^49 + 2784 * q^61 - 1416 * q^73 - 816 * q^85 + 648 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	8.99651i	0.804673i	0.915492	+	0.402336i	$0.131802\pi$
$5$	8.99651i	0.804673i	−0.915492	+	0.402336i	$0.868198\pi$
$6$	0	0
$7$	35.3962i	1.91121i	0.294645	+	0.955607i	$0.404799\pi$
$7$	35.3962i	1.91121i	−0.294645	+	0.955607i	$0.595201\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.88651	0.0791195	0.0395598	−	0.999217i	$-0.487404\pi$
$11$	2.88651	0.0791195	0.0395598	+	0.999217i	$0.487404\pi$
$12$	0	0
$13$	47.2668	1.00842	0.504210	−	0.863581i	$-0.331784\pi$
$13$	47.2668	1.00842	0.504210	+	0.863581i	$0.331784\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.19790i	0.0456238i	0.999740	+	0.0228119i	$0.00726188\pi$
$17$	3.19790i	0.0456238i	−0.999740	+	0.0228119i	$0.992738\pi$
$18$	0	0
$19$	92.6761i	1.11902i	0.828824	+	0.559509i	$0.189010\pi$
$19$	92.6761i	1.11902i	−0.828824	+	0.559509i	$0.810990\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	182.495	1.65447	0.827235	−	0.561855i	$-0.189912\pi$
$23$	182.495	1.65447	0.827235	+	0.561855i	$0.189912\pi$
$24$	0	0
$25$	44.0628	0.352502
$26$	0	0
$27$	0	0
$28$	0	0
$29$	161.054i	1.03128i	0.856806	+	0.515639i	$0.172445\pi$
$29$	161.054i	1.03128i	−0.856806	+	0.515639i	$0.827555\pi$
$30$	0	0
$31$	11.7511i	0.0680825i	0.999420	+	0.0340412i	$0.0108378\pi$
$31$	11.7511i	0.0680825i	−0.999420	+	0.0340412i	$0.989162\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−318.442	−1.53790
$36$	0	0
$37$	71.1977	0.316347	0.158173	−	0.987411i	$-0.449440\pi$
$37$	71.1977	0.316347	0.158173	+	0.987411i	$0.449440\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	111.322i	0.424040i	0.977265	+	0.212020i	$0.0680042\pi$
$41$	111.322i	0.424040i	−0.977265	+	0.212020i	$0.931996\pi$
$42$	0	0
$43$	30.0027i	0.106404i	0.998584	+	0.0532019i	$0.0169427\pi$
$43$	30.0027i	0.106404i	−0.998584	+	0.0532019i	$0.983057\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	523.230	1.62385	0.811924	−	0.583763i	$-0.198420\pi$
$47$	523.230	1.62385	0.811924	+	0.583763i	$0.198420\pi$
$48$	0	0
$49$	−909.889	−2.65274
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 683.907i	− 1.77249i	−0.463219	−	0.886244i	$-0.653306\pi$
$53$	− 683.907i	− 1.77249i	0.463219	−	0.886244i	$-0.346694\pi$
$54$	0	0
$55$	25.9685i	0.0636653i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−146.873	−0.324089	−0.162044	−	0.986783i	$-0.551809\pi$
$59$	−146.873	−0.324089	−0.162044	+	0.986783i	$0.551809\pi$
$60$	0	0
$61$	−717.089	−1.50515	−0.752573	−	0.658509i	$-0.771187\pi$
$61$	−717.089	−1.50515	−0.752573	+	0.658509i	$0.771187\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	425.237i	0.811448i
$66$	0	0
$67$	− 228.684i	− 0.416988i	−0.978024	−	0.208494i	$-0.933144\pi$
$67$	− 228.684i	− 0.416988i	0.978024	−	0.208494i	$-0.0668562\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1042.11	−1.74191	−0.870955	−	0.491363i	$-0.836499\pi$
$71$	−1042.11	−1.74191	−0.870955	+	0.491363i	$0.836499\pi$
$72$	0	0
$73$	−676.843	−1.08518	−0.542592	−	0.839996i	$-0.682557\pi$
$73$	−676.843	−1.08518	−0.542592	+	0.839996i	$0.682557\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	102.171i	0.151214i
$78$	0	0
$79$	865.762i	1.23298i	0.787361	+	0.616492i	$0.211447\pi$
$79$	865.762i	1.23298i	−0.787361	+	0.616492i	$0.788553\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	334.416	0.442251	0.221126	−	0.975245i	$-0.429027\pi$
$83$	334.416	0.442251	0.221126	+	0.975245i	$0.429027\pi$
$84$	0	0
$85$	−28.7699	−0.0367122
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 598.589i	− 0.712924i	−0.934310	−	0.356462i	$-0.883983\pi$
$89$	− 598.589i	− 0.712924i	0.934310	−	0.356462i	$-0.116017\pi$
$90$	0	0
$91$	1673.07i	1.92731i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−833.762	−0.900443
$96$	0	0
$97$	1034.49	1.08286	0.541428	−	0.840747i	$-0.317884\pi$
$97$	1034.49	1.08286	0.541428	+	0.840747i	$0.317884\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1563.41i	− 1.54025i	−0.637892	−	0.770126i	$-0.720194\pi$
$101$	− 1563.41i	− 1.54025i	0.637892	−	0.770126i	$-0.279806\pi$
$102$	0	0
$103$	769.270i	0.735907i	0.929844	+	0.367954i	$0.119941\pi$
$103$	769.270i	0.735907i	−0.929844	+	0.367954i	$0.880059\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1577.71	1.42545	0.712725	−	0.701443i	$-0.247461\pi$
$107$	1577.71	1.42545	0.712725	+	0.701443i	$0.247461\pi$
$108$	0	0
$109$	371.530	0.326478	0.163239	−	0.986587i	$-0.447806\pi$
$109$	371.530	0.326478	0.163239	+	0.986587i	$0.447806\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1713.42i	1.42642i	0.700952	+	0.713208i	$0.252759\pi$
$113$	1713.42i	1.42642i	−0.700952	+	0.713208i	$0.747241\pi$
$114$	0	0
$115$	1641.82i	1.33131i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−113.193	−0.0871967
$120$	0	0
$121$	−1322.67	−0.993740
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1520.98i	1.08832i
$126$	0	0
$127$	2035.62i	1.42230i	0.703040	+	0.711150i	$0.251826\pi$
$127$	2035.62i	1.42230i	−0.703040	+	0.711150i	$0.748174\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−31.6824	−0.0211305	−0.0105653	−	0.999944i	$-0.503363\pi$
$131$	−31.6824	−0.0211305	−0.0105653	+	0.999944i	$0.503363\pi$
$132$	0	0
$133$	−3280.38	−2.13868
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1032.16i	− 0.643676i	−0.946795	−	0.321838i	$-0.895699\pi$
$137$	− 1032.16i	− 0.643676i	0.946795	−	0.321838i	$-0.104301\pi$
$138$	0	0
$139$	1315.52i	0.802743i	0.915915	+	0.401371i	$0.131466\pi$
$139$	1315.52i	0.802743i	−0.915915	+	0.401371i	$0.868534\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	136.436	0.0797857
$144$	0	0
$145$	−1448.93	−0.829841
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 1824.84i	− 1.00334i	−0.865060	−	0.501668i	$-0.832720\pi$
$149$	− 1824.84i	− 1.00334i	0.865060	−	0.501668i	$-0.167280\pi$
$150$	0	0
$151$	− 247.339i	− 0.133299i	−0.997776	−	0.0666497i	$-0.978769\pi$
$151$	− 247.339i	− 0.133299i	0.997776	−	0.0666497i	$-0.0212310\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−105.719	−0.0547841
$156$	0	0
$157$	−1889.16	−0.960329	−0.480164	−	0.877179i	$-0.659423\pi$
$157$	−1889.16	−0.960329	−0.480164	+	0.877179i	$0.659423\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6459.62i	3.16205i
$162$	0	0
$163$	− 1966.48i	− 0.944950i	−0.881344	−	0.472475i	$-0.843361\pi$
$163$	− 1966.48i	− 0.944950i	0.881344	−	0.472475i	$-0.156639\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	443.537	0.205520	0.102760	−	0.994706i	$-0.467233\pi$
$167$	443.537	0.205520	0.102760	+	0.994706i	$0.467233\pi$
$168$	0	0
$169$	37.1537	0.0169111
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 3105.57i	− 1.36481i	−0.730975	−	0.682404i	$-0.760934\pi$
$173$	− 3105.57i	− 1.36481i	0.730975	−	0.682404i	$-0.239066\pi$
$174$	0	0
$175$	1559.65i	0.673707i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1973.24	0.823947	0.411974	−	0.911196i	$-0.364840\pi$
$179$	1973.24	0.823947	0.411974	+	0.911196i	$0.364840\pi$
$180$	0	0
$181$	−3126.34	−1.28386	−0.641932	−	0.766762i	$-0.721867\pi$
$181$	−3126.34	−1.28386	−0.641932	+	0.766762i	$0.721867\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	640.531i	0.254555i
$186$	0	0
$187$	9.23075i	0.00360973i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3230.96	1.22400	0.612000	−	0.790857i	$-0.290365\pi$
$191$	3230.96	1.22400	0.612000	+	0.790857i	$0.290365\pi$
$192$	0	0
$193$	−707.651	−0.263927	−0.131963	−	0.991255i	$-0.542128\pi$
$193$	−707.651	−0.263927	−0.131963	+	0.991255i	$0.542128\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2826.44i	− 1.02221i	−0.859518	−	0.511106i	$-0.829236\pi$
$197$	− 2826.44i	− 1.02221i	0.859518	−	0.511106i	$-0.170764\pi$
$198$	0	0
$199$	− 2702.47i	− 0.962678i	−0.876534	−	0.481339i	$-0.840151\pi$
$199$	− 2702.47i	− 0.962678i	0.876534	−	0.481339i	$-0.159849\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5700.71	−1.97099
$204$	0	0
$205$	−1001.51	−0.341213
$206$	0	0
$207$	0	0
$208$	0	0
$209$	267.510i	0.0885362i
$210$	0	0
$211$	− 231.147i	− 0.0754161i	−0.999289	−	0.0377080i	$-0.987994\pi$
$211$	− 231.147i	− 0.0754161i	0.999289	−	0.0377080i	$-0.0120057\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−269.919	−0.0856202
$216$	0	0
$217$	−415.943	−0.130120
$218$	0	0
$219$	0	0
$220$	0	0
$221$	151.154i	0.0460079i
$222$	0	0
$223$	− 784.305i	− 0.235520i	−0.993042	−	0.117760i	$-0.962429\pi$
$223$	− 784.305i	− 0.235520i	0.993042	−	0.117760i	$-0.0375714\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1070.17	−0.312907	−0.156453	−	0.987685i	$-0.550006\pi$
$227$	−1070.17	−0.312907	−0.156453	+	0.987685i	$0.550006\pi$
$228$	0	0
$229$	4827.56	1.39308	0.696538	−	0.717520i	$-0.254723\pi$
$229$	4827.56	1.39308	0.696538	+	0.717520i	$0.254723\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2689.22i	0.756124i	0.925780	+	0.378062i	$0.123409\pi$
$233$	2689.22i	0.756124i	−0.925780	+	0.378062i	$0.876591\pi$
$234$	0	0
$235$	4707.24i	1.30667i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1031.51	−0.279175	−0.139587	−	0.990210i	$-0.544578\pi$
$239$	−1031.51	−0.279175	−0.139587	+	0.990210i	$0.544578\pi$
$240$	0	0
$241$	−2351.47	−0.628513	−0.314256	−	0.949338i	$-0.601755\pi$
$241$	−2351.47	−0.628513	−0.314256	+	0.949338i	$0.601755\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 8185.83i	− 2.13459i
$246$	0	0
$247$	4380.51i	1.12844i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5723.75	−1.43936	−0.719681	−	0.694305i	$-0.755712\pi$
$251$	−5723.75	−1.43936	−0.719681	+	0.694305i	$0.755712\pi$
$252$	0	0
$253$	526.773	0.130901
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5348.44i	1.29816i	0.760721	+	0.649079i	$0.224846\pi$
$257$	5348.44i	1.29816i	−0.760721	+	0.649079i	$0.775154\pi$
$258$	0	0
$259$	2520.13i	0.604606i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7061.50	1.65563	0.827815	−	0.561000i	$-0.189583\pi$
$263$	7061.50	1.65563	0.827815	+	0.561000i	$0.189583\pi$
$264$	0	0
$265$	6152.78	1.42627
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 2946.25i	− 0.667792i	−0.942610	−	0.333896i	$-0.891637\pi$
$269$	− 2946.25i	− 0.667792i	0.942610	−	0.333896i	$-0.108363\pi$
$270$	0	0
$271$	2048.60i	0.459201i	0.973285	+	0.229600i	$0.0737419\pi$
$271$	2048.60i	0.459201i	−0.973285	+	0.229600i	$0.926258\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	127.187	0.0278898
$276$	0	0
$277$	8653.43	1.87702	0.938509	−	0.345254i	$-0.112207\pi$
$277$	8653.43	1.87702	0.938509	+	0.345254i	$0.112207\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 2228.69i	− 0.473141i	−0.971614	−	0.236570i	$-0.923977\pi$
$281$	− 2228.69i	− 0.473141i	0.971614	−	0.236570i	$-0.0760233\pi$
$282$	0	0
$283$	− 3226.05i	− 0.677628i	−0.940853	−	0.338814i	$-0.889974\pi$
$283$	− 3226.05i	− 0.677628i	0.940853	−	0.338814i	$-0.110026\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3940.39	−0.810431
$288$	0	0
$289$	4902.77	0.997918
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3044.49i	0.607034i	0.952826	+	0.303517i	$0.0981610\pi$
$293$	3044.49i	0.607034i	−0.952826	+	0.303517i	$0.901839\pi$
$294$	0	0
$295$	− 1321.34i	− 0.260785i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8625.96	1.66840
$300$	0	0
$301$	−1061.98	−0.203360
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 6451.30i	− 1.21115i
$306$	0	0
$307$	− 7394.12i	− 1.37461i	−0.726370	−	0.687304i	$-0.758794\pi$
$307$	− 7394.12i	− 1.37461i	0.726370	−	0.687304i	$-0.241206\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8469.46	−1.54424	−0.772121	−	0.635476i	$-0.780804\pi$
$311$	−8469.46	−1.54424	−0.772121	+	0.635476i	$0.780804\pi$
$312$	0	0
$313$	−6394.80	−1.15481	−0.577405	−	0.816458i	$-0.695934\pi$
$313$	−6394.80	−1.15481	−0.577405	+	0.816458i	$0.695934\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 3234.14i	− 0.573021i	−0.958077	−	0.286510i	$-0.907505\pi$
$317$	− 3234.14i	− 0.573021i	0.958077	−	0.286510i	$-0.0924953\pi$
$318$	0	0
$319$	464.885i	0.0815942i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−296.369	−0.0510538
$324$	0	0
$325$	2082.71	0.355470
$326$	0	0
$327$	0	0
$328$	0	0
$329$	18520.3i	3.10352i
$330$	0	0
$331$	− 7913.16i	− 1.31404i	−0.753874	−	0.657019i	$-0.771817\pi$
$331$	− 7913.16i	− 1.31404i	0.753874	−	0.657019i	$-0.228183\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2057.36	0.335539
$336$	0	0
$337$	638.612	0.103227	0.0516134	−	0.998667i	$-0.483564\pi$
$337$	638.612	0.103227	0.0516134	+	0.998667i	$0.483564\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	33.9196i	0.00538665i
$342$	0	0
$343$	− 20065.7i	− 3.15873i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2900.91	0.448786	0.224393	−	0.974499i	$-0.427960\pi$
$347$	2900.91	0.448786	0.224393	+	0.974499i	$0.427960\pi$
$348$	0	0
$349$	11177.2	1.71432	0.857162	−	0.515047i	$-0.172225\pi$
$349$	11177.2	1.71432	0.857162	+	0.515047i	$0.172225\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4077.91i	0.614859i	0.951571	+	0.307430i	$0.0994688\pi$
$353$	4077.91i	0.614859i	−0.951571	+	0.307430i	$0.900531\pi$
$354$	0	0
$355$	− 9375.35i	− 1.40167i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10771.0	−1.58349	−0.791744	−	0.610853i	$-0.790827\pi$
$359$	−10771.0	−1.58349	−0.791744	+	0.610853i	$0.790827\pi$
$360$	0	0
$361$	−1729.86	−0.252202
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 6089.22i	− 0.873218i
$366$	0	0
$367$	− 11402.5i	− 1.62181i	−0.585178	−	0.810905i	$-0.698975\pi$
$367$	− 11402.5i	− 1.62181i	0.585178	−	0.810905i	$-0.301025\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	24207.7	3.38760
$372$	0	0
$373$	−9787.33	−1.35863	−0.679315	−	0.733847i	$-0.737723\pi$
$373$	−9787.33	−1.35863	−0.679315	+	0.733847i	$0.737723\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7612.53i	1.03996i
$378$	0	0
$379$	14341.9i	1.94378i	0.235438	+	0.971889i	$0.424348\pi$
$379$	14341.9i	1.94378i	−0.235438	+	0.971889i	$0.575652\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4395.34	0.586400	0.293200	−	0.956051i	$-0.405280\pi$
$383$	4395.34	0.586400	0.293200	+	0.956051i	$0.405280\pi$
$384$	0	0
$385$	−919.185	−0.121678
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 5527.39i	− 0.720436i	−0.932868	−	0.360218i	$-0.882702\pi$
$389$	− 5527.39i	− 0.720436i	0.932868	−	0.360218i	$-0.117298\pi$
$390$	0	0
$391$	583.600i	0.0754832i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7788.84	−0.992149
$396$	0	0
$397$	2814.85	0.355852	0.177926	−	0.984044i	$-0.443061\pi$
$397$	2814.85	0.355852	0.177926	+	0.984044i	$0.443061\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 9331.30i	− 1.16205i	−0.813885	−	0.581026i	$-0.802651\pi$
$401$	− 9331.30i	− 1.16205i	0.813885	−	0.581026i	$-0.197349\pi$
$402$	0	0
$403$	555.437i	0.0686557i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	205.513	0.0250292
$408$	0	0
$409$	12861.5	1.55492	0.777458	−	0.628935i	$-0.216509\pi$
$409$	12861.5	1.55492	0.777458	+	0.628935i	$0.216509\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 5198.74i	− 0.619403i
$414$	0	0
$415$	3008.57i	0.355868i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−6006.40	−0.700314	−0.350157	−	0.936691i	$-0.613872\pi$
$419$	−6006.40	−0.700314	−0.350157	+	0.936691i	$0.613872\pi$
$420$	0	0
$421$	3455.89	0.400071	0.200035	−	0.979789i	$-0.435894\pi$
$421$	3455.89	0.400071	0.200035	+	0.979789i	$0.435894\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	140.908i	0.0160825i
$426$	0	0
$427$	− 25382.2i	− 2.87665i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11022.5	1.23187	0.615935	−	0.787797i	$-0.288779\pi$
$431$	11022.5	1.23187	0.615935	+	0.787797i	$0.288779\pi$
$432$	0	0
$433$	11314.1	1.25571	0.627856	−	0.778330i	$-0.283933\pi$
$433$	11314.1	1.25571	0.627856	+	0.778330i	$0.283933\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	16912.9i	1.85138i
$438$	0	0
$439$	14000.0i	1.52205i	0.648720	+	0.761027i	$0.275305\pi$
$439$	14000.0i	1.52205i	−0.648720	+	0.761027i	$0.724695\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9289.05	−0.996244	−0.498122	−	0.867107i	$-0.665977\pi$
$443$	−9289.05	−0.996244	−0.498122	+	0.867107i	$0.665977\pi$
$444$	0	0
$445$	5385.21	0.573671
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14592.2i	1.53374i	0.641804	+	0.766869i	$0.278186\pi$
$449$	14592.2i	1.53374i	−0.641804	+	0.766869i	$0.721814\pi$
$450$	0	0
$451$	321.333i	0.0335498i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−15051.8	−1.55085
$456$	0	0
$457$	6034.71	0.617707	0.308853	−	0.951110i	$-0.400055\pi$
$457$	6034.71	0.617707	0.308853	+	0.951110i	$0.400055\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 14113.8i	− 1.42591i	−0.701211	−	0.712953i	$-0.747357\pi$
$461$	− 14113.8i	− 1.42591i	0.701211	−	0.712953i	$-0.252643\pi$
$462$	0	0
$463$	615.069i	0.0617380i	0.999523	+	0.0308690i	$0.00982746\pi$
$463$	615.069i	0.0617380i	−0.999523	+	0.0308690i	$0.990173\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7925.30	−0.785309	−0.392655	−	0.919686i	$-0.628443\pi$
$467$	−7925.30	−0.785309	−0.392655	+	0.919686i	$0.628443\pi$
$468$	0	0
$469$	8094.54	0.796954
$470$	0	0
$471$	0	0
$472$	0	0
$473$	86.6029i	0.00841862i
$474$	0	0
$475$	4083.56i	0.394456i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	12221.7	1.16581	0.582904	−	0.812541i	$-0.301916\pi$
$479$	12221.7	1.16581	0.582904	+	0.812541i	$0.301916\pi$
$480$	0	0
$481$	3365.29	0.319010
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9306.84i	0.871344i
$486$	0	0
$487$	− 4058.84i	− 0.377667i	−0.982009	−	0.188833i	$-0.939529\pi$
$487$	− 4058.84i	− 0.377667i	0.982009	−	0.188833i	$-0.0604706\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4906.51	−0.450973	−0.225486	−	0.974246i	$-0.572397\pi$
$491$	−4906.51	−0.450973	−0.225486	+	0.974246i	$0.572397\pi$
$492$	0	0
$493$	−515.035	−0.0470508
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 36886.7i	− 3.32916i
$498$	0	0
$499$	7905.92i	0.709254i	0.935008	+	0.354627i	$0.115392\pi$
$499$	7905.92i	0.709254i	−0.935008	+	0.354627i	$0.884608\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	16309.7	1.44575	0.722876	−	0.690977i	$-0.242820\pi$
$503$	16309.7	1.44575	0.722876	+	0.690977i	$0.242820\pi$
$504$	0	0
$505$	14065.3	1.23940
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 10047.1i	− 0.874915i	−0.899239	−	0.437458i	$-0.855879\pi$
$509$	− 10047.1i	− 0.874915i	0.899239	−	0.437458i	$-0.144121\pi$
$510$	0	0
$511$	− 23957.6i	− 2.07402i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6920.75	−0.592164
$516$	0	0
$517$	1510.31	0.128478
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1449.81i	0.121914i	0.998140	+	0.0609572i	$0.0194153\pi$
$521$	1449.81i	0.121914i	−0.998140	+	0.0609572i	$0.980585\pi$
$522$	0	0
$523$	191.955i	0.0160489i	0.999968	+	0.00802447i	$0.00255430\pi$
$523$	191.955i	0.0160489i	−0.999968	+	0.00802447i	$0.997446\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−37.5788	−0.00310618
$528$	0	0
$529$	21137.4	1.73727
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5261.86i	0.427610i
$534$	0	0
$535$	14193.9i	1.14702i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2626.40	−0.209883
$540$	0	0
$541$	−3901.48	−0.310051	−0.155026	−	0.987910i	$-0.549546\pi$
$541$	−3901.48	−0.310051	−0.155026	+	0.987910i	$0.549546\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3342.47i	0.262708i
$546$	0	0
$547$	15716.7i	1.22851i	0.789107	+	0.614256i	$0.210544\pi$
$547$	15716.7i	1.22851i	−0.789107	+	0.614256i	$0.789456\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−14925.9	−1.15402
$552$	0	0
$553$	−30644.7	−2.35650
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20798.3i	1.58214i	0.611725	+	0.791071i	$0.290476\pi$
$557$	20798.3i	1.58214i	−0.611725	+	0.791071i	$0.709524\pi$
$558$	0	0
$559$	1418.13i	0.107300i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−11103.8	−0.831203	−0.415602	−	0.909547i	$-0.636429\pi$
$563$	−11103.8	−0.831203	−0.415602	+	0.909547i	$0.636429\pi$
$564$	0	0
$565$	−15414.8	−1.14780
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 14825.2i	− 1.09228i	−0.837694	−	0.546139i	$-0.816097\pi$
$569$	− 14825.2i	− 1.09228i	0.837694	−	0.546139i	$-0.183903\pi$
$570$	0	0
$571$	− 14831.3i	− 1.08699i	−0.839412	−	0.543495i	$-0.817101\pi$
$571$	− 14831.3i	− 1.08699i	0.839412	−	0.543495i	$-0.182899\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8041.23	0.583204
$576$	0	0
$577$	−25829.0	−1.86356	−0.931780	−	0.363024i	$-0.881744\pi$
$577$	−25829.0	−1.86356	−0.931780	+	0.363024i	$0.881744\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	11837.0i	0.845237i
$582$	0	0
$583$	− 1974.10i	− 0.140238i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4780.76	0.336155	0.168078	−	0.985774i	$-0.446244\pi$
$587$	4780.76	0.336155	0.168078	+	0.985774i	$0.446244\pi$
$588$	0	0
$589$	−1089.04	−0.0761856
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 4513.09i	− 0.312530i	−0.987715	−	0.156265i	$-0.950055\pi$
$593$	− 4513.09i	− 0.312530i	0.987715	−	0.156265i	$-0.0499455\pi$
$594$	0	0
$595$	− 1018.34i	− 0.0701648i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4646.83	−0.316968	−0.158484	−	0.987362i	$-0.550661\pi$
$599$	−4646.83	−0.316968	−0.158484	+	0.987362i	$0.550661\pi$
$600$	0	0
$601$	1610.99	0.109341	0.0546703	−	0.998504i	$-0.482589\pi$
$601$	1610.99	0.109341	0.0546703	+	0.998504i	$0.482589\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 11899.4i	− 0.799635i
$606$	0	0
$607$	10227.7i	0.683906i	0.939717	+	0.341953i	$0.111088\pi$
$607$	10227.7i	0.683906i	−0.939717	+	0.341953i	$0.888912\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	24731.4	1.63752
$612$	0	0
$613$	−11957.6	−0.787869	−0.393934	−	0.919139i	$-0.628886\pi$
$613$	−11957.6	−0.787869	−0.393934	+	0.919139i	$0.628886\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11950.8i	0.779777i	0.920862	+	0.389889i	$0.127486\pi$
$617$	11950.8i	0.779777i	−0.920862	+	0.389889i	$0.872514\pi$
$618$	0	0
$619$	− 5595.35i	− 0.363322i	−0.983361	−	0.181661i	$-0.941853\pi$
$619$	− 5595.35i	− 0.363322i	0.983361	−	0.181661i	$-0.0581473\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	21187.7	1.36255
$624$	0	0
$625$	−8175.63	−0.523240
$626$	0	0
$627$	0	0
$628$	0	0
$629$	227.683i	0.0144329i
$630$	0	0
$631$	− 19082.2i	− 1.20389i	−0.798539	−	0.601943i	$-0.794393\pi$
$631$	− 19082.2i	− 1.20389i	0.798539	−	0.601943i	$-0.205607\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−18313.5	−1.14449
$636$	0	0
$637$	−43007.6	−2.67507
$638$	0	0
$639$	0	0
$640$	0	0
$641$	27622.7i	1.70208i	0.525105	+	0.851038i	$0.324026\pi$
$641$	27622.7i	1.70208i	−0.525105	+	0.851038i	$0.675974\pi$
$642$	0	0
$643$	− 19288.6i	− 1.18300i	−0.806306	−	0.591498i	$-0.798537\pi$
$643$	− 19288.6i	− 1.18300i	0.806306	−	0.591498i	$-0.201463\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−11188.6	−0.679861	−0.339931	−	0.940450i	$-0.610404\pi$
$647$	−11188.6	−0.679861	−0.339931	+	0.940450i	$0.610404\pi$
$648$	0	0
$649$	−423.950	−0.0256417
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14406.5i	0.863356i	0.902028	+	0.431678i	$0.142078\pi$
$653$	14406.5i	0.863356i	−0.902028	+	0.431678i	$0.857922\pi$
$654$	0	0
$655$	− 285.031i	− 0.0170032i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	27480.4	1.62441	0.812205	−	0.583373i	$-0.198267\pi$
$659$	27480.4	1.62441	0.812205	+	0.583373i	$0.198267\pi$
$660$	0	0
$661$	24268.4	1.42804	0.714019	−	0.700127i	$-0.246873\pi$
$661$	24268.4	1.42804	0.714019	+	0.700127i	$0.246873\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 29512.0i	− 1.72094i
$666$	0	0
$667$	29391.6i	1.70622i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2069.88	−0.119086
$672$	0	0
$673$	−15316.2	−0.877261	−0.438631	−	0.898667i	$-0.644536\pi$
$673$	−15316.2	−0.877261	−0.438631	+	0.898667i	$0.644536\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3781.94i	0.214700i	0.994221	+	0.107350i	$0.0342365\pi$
$677$	3781.94i	0.214700i	−0.994221	+	0.107350i	$0.965763\pi$
$678$	0	0
$679$	36617.1i	2.06957i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−24496.7	−1.37238	−0.686192	−	0.727420i	$-0.740719\pi$
$683$	−24496.7	−1.37238	−0.686192	+	0.727420i	$0.740719\pi$
$684$	0	0
$685$	9285.87	0.517948
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 32326.1i	− 1.78741i
$690$	0	0
$691$	21646.8i	1.19172i	0.803087	+	0.595862i	$0.203190\pi$
$691$	21646.8i	1.19172i	−0.803087	+	0.595862i	$0.796810\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11835.1	−0.645945
$696$	0	0
$697$	−355.998	−0.0193463
$698$	0	0
$699$	0	0
$700$	0	0
$701$	22628.9i	1.21923i	0.792696	+	0.609617i	$0.208677\pi$
$701$	22628.9i	1.21923i	−0.792696	+	0.609617i	$0.791323\pi$
$702$	0	0
$703$	6598.32i	0.353998i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	55338.8	2.94375
$708$	0	0
$709$	14610.9	0.773940	0.386970	−	0.922092i	$-0.373522\pi$
$709$	14610.9	0.773940	0.386970	+	0.922092i	$0.373522\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2144.51i	0.112640i
$714$	0	0
$715$	1227.45i	0.0642014i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−12211.2	−0.633381	−0.316690	−	0.948529i	$-0.602572\pi$
$719$	−12211.2	−0.633381	−0.316690	+	0.948529i	$0.602572\pi$
$720$	0	0
$721$	−27229.2	−1.40648
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7096.50i	0.363528i
$726$	0	0
$727$	13197.1i	0.673250i	0.941639	+	0.336625i	$0.109285\pi$
$727$	13197.1i	0.673250i	−0.941639	+	0.336625i	$0.890715\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−95.9454	−0.00485454
$732$	0	0
$733$	22565.2	1.13706	0.568531	−	0.822662i	$-0.307512\pi$
$733$	22565.2	1.13706	0.568531	+	0.822662i	$0.307512\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 660.098i	− 0.0329919i
$738$	0	0
$739$	25340.2i	1.26137i	0.776038	+	0.630686i	$0.217226\pi$
$739$	25340.2i	1.26137i	−0.776038	+	0.630686i	$0.782774\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	4341.02	0.214343	0.107171	−	0.994241i	$-0.465821\pi$
$743$	4341.02	0.214343	0.107171	+	0.994241i	$0.465821\pi$
$744$	0	0
$745$	16417.2	0.807356
$746$	0	0
$747$	0	0
$748$	0	0
$749$	55845.0i	2.72434i
$750$	0	0
$751$	30349.5i	1.47466i	0.675534	+	0.737328i	$0.263913\pi$
$751$	30349.5i	1.47466i	−0.675534	+	0.737328i	$0.736087\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2225.19	0.107262
$756$	0	0
$757$	−44.5144	−0.00213726	−0.00106863	−	0.999999i	$-0.500340\pi$
$757$	−44.5144	−0.00213726	−0.00106863	+	0.999999i	$0.500340\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 12698.4i	− 0.604885i	−0.953168	−	0.302442i	$-0.902198\pi$
$761$	− 12698.4i	− 0.604885i	0.953168	−	0.302442i	$-0.0978020\pi$
$762$	0	0
$763$	13150.7i	0.623969i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6942.22	−0.326817
$768$	0	0
$769$	−3620.32	−0.169769	−0.0848843	−	0.996391i	$-0.527052\pi$
$769$	−3620.32	−0.169769	−0.0848843	+	0.996391i	$0.527052\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 30688.4i	− 1.42792i	−0.700184	−	0.713962i	$-0.746899\pi$
$773$	− 30688.4i	− 1.42792i	0.700184	−	0.713962i	$-0.253101\pi$
$774$	0	0
$775$	517.785i	0.0239992i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−10316.9	−0.474509
$780$	0	0
$781$	−3008.06	−0.137819
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 16995.9i	− 0.772750i
$786$	0	0
$787$	26627.7i	1.20607i	0.797715	+	0.603035i	$0.206042\pi$
$787$	26627.7i	1.20607i	−0.797715	+	0.603035i	$0.793958\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−60648.6	−2.72619
$792$	0	0
$793$	−33894.5	−1.51782
$794$	0	0
$795$	0	0
$796$	0	0
$797$	11706.4i	0.520279i	0.965571	+	0.260139i	$0.0837685\pi$
$797$	11706.4i	0.520279i	−0.965571	+	0.260139i	$0.916232\pi$
$798$	0	0
$799$	1673.23i	0.0740861i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1953.71	−0.0858593
$804$	0	0
$805$	−58114.1	−2.54441
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 34745.1i	− 1.50998i	−0.655736	−	0.754990i	$-0.727642\pi$
$809$	− 34745.1i	− 1.50998i	0.655736	−	0.754990i	$-0.272358\pi$
$810$	0	0
$811$	5538.24i	0.239795i	0.992786	+	0.119898i	$0.0382567\pi$
$811$	5538.24i	0.239795i	−0.992786	+	0.119898i	$0.961743\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	17691.5	0.760375
$816$	0	0
$817$	−2780.53	−0.119068
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1271.12i	0.0540347i	0.999635	+	0.0270174i	$0.00860094\pi$
$821$	1271.12i	0.0540347i	−0.999635	+	0.0270174i	$0.991399\pi$
$822$	0	0
$823$	36457.6i	1.54414i	0.635535	+	0.772072i	$0.280780\pi$
$823$	36457.6i	1.54414i	−0.635535	+	0.772072i	$0.719220\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10229.2	0.430115	0.215058	−	0.976601i	$-0.431006\pi$
$827$	10229.2	0.430115	0.215058	+	0.976601i	$0.431006\pi$
$828$	0	0
$829$	−11008.2	−0.461196	−0.230598	−	0.973049i	$-0.574068\pi$
$829$	−11008.2	−0.461196	−0.230598	+	0.973049i	$0.574068\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 2909.73i	− 0.121028i
$834$	0	0
$835$	3990.29i	0.165377i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	9498.63	0.390857	0.195429	−	0.980718i	$-0.437390\pi$
$839$	9498.63	0.390857	0.195429	+	0.980718i	$0.437390\pi$
$840$	0	0
$841$	−1549.53	−0.0635338
$842$	0	0
$843$	0	0
$844$	0	0
$845$	334.254i	0.0136079i
$846$	0	0
$847$	− 46817.4i	− 1.89925i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	12993.2	0.523386
$852$	0	0
$853$	−18750.8	−0.752655	−0.376328	−	0.926487i	$-0.622813\pi$
$853$	−18750.8	−0.752655	−0.376328	+	0.926487i	$0.622813\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 37740.6i	− 1.50431i	−0.658985	−	0.752156i	$-0.729014\pi$
$857$	− 37740.6i	− 1.50431i	0.658985	−	0.752156i	$-0.270986\pi$
$858$	0	0
$859$	− 21464.7i	− 0.852579i	−0.904587	−	0.426290i	$-0.859820\pi$
$859$	− 21464.7i	− 0.852579i	0.904587	−	0.426290i	$-0.140180\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	29001.8	1.14396	0.571978	−	0.820269i	$-0.306176\pi$
$863$	29001.8	1.14396	0.571978	+	0.820269i	$0.306176\pi$
$864$	0	0
$865$	27939.3	1.09822
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2499.03i	0.0975532i
$870$	0	0
$871$	− 10809.2i	− 0.420499i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−53836.7	−2.08001
$876$	0	0
$877$	15055.6	0.579695	0.289848	−	0.957073i	$-0.406395\pi$
$877$	15055.6	0.579695	0.289848	+	0.957073i	$0.406395\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 6679.13i	− 0.255421i	−0.991812	−	0.127710i	$-0.959237\pi$
$881$	− 6679.13i	− 0.255421i	0.991812	−	0.127710i	$-0.0407628\pi$
$882$	0	0
$883$	14960.3i	0.570165i	0.958503	+	0.285083i	$0.0920210\pi$
$883$	14960.3i	0.570165i	−0.958503	+	0.285083i	$0.907979\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	13974.6	0.528997	0.264498	−	0.964386i	$-0.414794\pi$
$887$	13974.6	0.528997	0.264498	+	0.964386i	$0.414794\pi$
$888$	0	0
$889$	−72053.2	−2.71832
$890$	0	0
$891$	0	0
$892$	0	0
$893$	48490.9i	1.81712i
$894$	0	0
$895$	17752.2i	0.663008i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1892.56	−0.0702119
$900$	0	0
$901$	2187.06	0.0808675
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 28126.2i	− 1.03309i
$906$	0	0
$907$	− 15103.3i	− 0.552919i	−0.961026	−	0.276459i	$-0.910839\pi$
$907$	− 15103.3i	− 0.552919i	0.961026	−	0.276459i	$-0.0891611\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	7158.99	0.260360	0.130180	−	0.991490i	$-0.458444\pi$
$911$	7158.99	0.260360	0.130180	+	0.991490i	$0.458444\pi$
$912$	0	0
$913$	965.293	0.0349907
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1121.43i	− 0.0403850i
$918$	0	0
$919$	− 6718.51i	− 0.241157i	−0.992704	−	0.120578i	$-0.961525\pi$
$919$	− 6718.51i	− 0.241157i	0.992704	−	0.120578i	$-0.0384749\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−49257.2	−1.75658
$924$	0	0
$925$	3137.17	0.111513
$926$	0	0
$927$	0	0
$928$	0	0
$929$	52525.0i	1.85499i	0.373832	+	0.927497i	$0.378044\pi$
$929$	52525.0i	1.85499i	−0.373832	+	0.927497i	$0.621956\pi$
$930$	0	0
$931$	− 84324.9i	− 2.96846i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−83.0446	−0.00290465
$936$	0	0
$937$	14330.5	0.499635	0.249818	−	0.968293i	$-0.419629\pi$
$937$	14330.5	0.499635	0.249818	+	0.968293i	$0.419629\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 8295.24i	− 0.287372i	−0.989623	−	0.143686i	$-0.954104\pi$
$941$	− 8295.24i	− 0.287372i	0.989623	−	0.143686i	$-0.0458956\pi$
$942$	0	0
$943$	20315.8i	0.701562i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−39543.2	−1.35690	−0.678449	−	0.734647i	$-0.737348\pi$
$947$	−39543.2	−1.35690	−0.678449	+	0.734647i	$0.737348\pi$
$948$	0	0
$949$	−31992.2	−1.09432
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 2750.61i	− 0.0934954i	−0.998907	−	0.0467477i	$-0.985114\pi$
$953$	− 2750.61i	− 0.0934954i	0.998907	−	0.0467477i	$-0.0148857\pi$
$954$	0	0
$955$	29067.4i	0.984920i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	36534.6	1.23020
$960$	0	0
$961$	29652.9	0.995365
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 6366.39i	− 0.212375i
$966$	0	0
$967$	− 13315.8i	− 0.442821i	−0.975181	−	0.221411i	$-0.928934\pi$
$967$	− 13315.8i	− 0.442821i	0.975181	−	0.221411i	$-0.0710661\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	13075.9	0.432158	0.216079	−	0.976376i	$-0.430673\pi$
$971$	13075.9	0.432158	0.216079	+	0.976376i	$0.430673\pi$
$972$	0	0
$973$	−46564.5	−1.53421
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4325.21i	0.141633i	0.997489	+	0.0708167i	$0.0225605\pi$
$977$	4325.21i	0.141633i	−0.997489	+	0.0708167i	$0.977439\pi$
$978$	0	0
$979$	− 1727.83i	− 0.0564062i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	4345.41	0.140994	0.0704970	−	0.997512i	$-0.477541\pi$
$983$	4345.41	0.140994	0.0704970	+	0.997512i	$0.477541\pi$
$984$	0	0
$985$	25428.1	0.822546
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5475.34i	0.176042i
$990$	0	0
$991$	− 15332.7i	− 0.491484i	−0.969335	−	0.245742i	$-0.920968\pi$
$991$	− 15332.7i	− 0.491484i	0.969335	−	0.245742i	$-0.0790316\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	24312.8	0.774641
$996$	0	0
$997$	58134.3	1.84667	0.923335	−	0.383995i	$-0.125452\pi$
$997$	58134.3	1.84667	0.923335	+	0.383995i	$0.125452\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.4.c.b.863.18	yes	24
3.2	odd	2	inner	864.4.c.b.863.8	yes	24
4.3	odd	2	inner	864.4.c.b.863.17	yes	24
8.3	odd	2		1728.4.c.k.1727.7		24
8.5	even	2		1728.4.c.k.1727.8		24
12.11	even	2	inner	864.4.c.b.863.7	✓	24
24.5	odd	2		1728.4.c.k.1727.18		24
24.11	even	2		1728.4.c.k.1727.17		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.4.c.b.863.7	✓	24	12.11	even	2	inner
864.4.c.b.863.8	yes	24	3.2	odd	2	inner
864.4.c.b.863.17	yes	24	4.3	odd	2	inner
864.4.c.b.863.18	yes	24	1.1	even	1	trivial
1728.4.c.k.1727.7		24	8.3	odd	2
1728.4.c.k.1727.8		24	8.5	even	2
1728.4.c.k.1727.17		24	24.11	even	2
1728.4.c.k.1727.18		24	24.5	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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	864.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+8.99651i q^{5} +35.3962i q^{7} +O(q^{10})\)	\(q+8.99651i q^{5} +35.3962i q^{7} +2.88651 q^{11} +47.2668 q^{13} +3.19790i q^{17} +92.6761i q^{19} +182.495 q^{23} +44.0628 q^{25} +161.054i q^{29} +11.7511i q^{31} -318.442 q^{35} +71.1977 q^{37} +111.322i q^{41} +30.0027i q^{43} +523.230 q^{47} -909.889 q^{49} -683.907i q^{53} +25.9685i q^{55} -146.873 q^{59} -717.089 q^{61} +425.237i q^{65} -228.684i q^{67} -1042.11 q^{71} -676.843 q^{73} +102.171i q^{77} +865.762i q^{79} +334.416 q^{83} -28.7699 q^{85} -598.589i q^{89} +1673.07i q^{91} -833.762 q^{95} +1034.49 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 144 q^{13} - 768 q^{25} + 1536 q^{37} - 2304 q^{49} + 2784 q^{61} - 1416 q^{73} - 816 q^{85} + 648 q^{97}+O(q^{100}) \) 24 * q + 144 * q^13 - 768 * q^25 + 1536 * q^37 - 2304 * q^49 + 2784 * q^61 - 1416 * q^73 - 816 * q^85 + 648 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more