LMFDB - Embedded newform 1008.3.o.b.1007.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1008,3,Mod(1007,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1008.1007"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.o (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1007.12
Character	$\chi$	$=$	1008.1007
Dual form			1008.3.o.b.1007.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.718749 q^{5} +(-6.52255 + 2.54092i) q^{7} -13.1805 q^{11} -12.0430i q^{13} +17.0788 q^{17} +6.05336 q^{19} +10.8926 q^{23} -24.4834 q^{25} +8.20529i q^{29} +36.6031 q^{31} +(4.68808 - 1.82629i) q^{35} +45.0874 q^{37} +62.4228 q^{41} +12.0100i q^{43} +87.3347i q^{47} +(36.0874 - 33.1466i) q^{49} +74.7970i q^{53} +9.47344 q^{55} +22.0297i q^{59} -57.2996i q^{61} +8.65587i q^{65} -47.6026i q^{67} -66.4226 q^{71} -36.0414i q^{73} +(85.9703 - 33.4905i) q^{77} +46.6882i q^{79} +38.2957i q^{83} -12.2753 q^{85} +126.425 q^{89} +(30.6003 + 78.5509i) q^{91} -4.35085 q^{95} -126.730i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 264 q^{25} - 96 q^{37} - 312 q^{49} - 96 q^{85}+O(q^{100}) $ 24 * q + 264 * q^25 - 96 * q^37 - 312 * q^49 - 96 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$5$	−0.718749			−0.143750			−0.0718749	−	0.997414i	$-0.522898\pi$
$5$	−0.718749			−0.143750			−0.0718749	+	0.997414i	$0.522898\pi$
$6$		0			0
$7$	−6.52255	+	2.54092i	−0.931793	+	0.362989i
$7$	−6.52255	+	2.54092i	−0.931793	+	0.362989i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−13.1805			−1.19822			−0.599112	−	0.800665i	$-0.704480\pi$
$11$	−13.1805			−1.19822			−0.599112	+	0.800665i	$0.704480\pi$
$12$		0			0
$13$		−	12.0430i		−	0.926383i	−0.886258	−	0.463191i	$-0.846704\pi$
$13$		−	12.0430i		−	0.926383i	0.886258	−	0.463191i	$-0.153296\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	17.0788			1.00463			0.502317	−	0.864684i	$-0.332481\pi$
$17$	17.0788			1.00463			0.502317	+	0.864684i	$0.332481\pi$
$18$		0			0
$19$	6.05336			0.318598			0.159299	−	0.987230i	$-0.449077\pi$
$19$	6.05336			0.318598			0.159299	+	0.987230i	$0.449077\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	10.8926			0.473593			0.236796	−	0.971559i	$-0.423903\pi$
$23$	10.8926			0.473593			0.236796	+	0.971559i	$0.423903\pi$
$24$		0			0
$25$	−24.4834			−0.979336
$26$		0			0
$27$		0			0
$28$		0			0
$29$			8.20529i			0.282941i	0.989942	+	0.141470i	$0.0451830\pi$
$29$			8.20529i			0.282941i	−0.989942	+	0.141470i	$0.954817\pi$
$30$		0			0
$31$	36.6031			1.18075			0.590373	−	0.807131i	$-0.298981\pi$
$31$	36.6031			1.18075			0.590373	+	0.807131i	$0.298981\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.68808	−	1.82629i	0.133945	−	0.0521796i
$36$		0			0
$37$	45.0874			1.21858			0.609290	−	0.792948i	$-0.291455\pi$
$37$	45.0874			1.21858			0.609290	+	0.792948i	$0.291455\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	62.4228			1.52251			0.761253	−	0.648454i	$-0.224584\pi$
$41$	62.4228			1.52251			0.761253	+	0.648454i	$0.224584\pi$
$42$		0			0
$43$			12.0100i			0.279303i	0.990201	+	0.139652i	$0.0445983\pi$
$43$			12.0100i			0.279303i	−0.990201	+	0.139652i	$0.955402\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			87.3347i			1.85818i	0.369848	+	0.929092i	$0.379410\pi$
$47$			87.3347i			1.85818i	−0.369848	+	0.929092i	$0.620590\pi$
$48$		0			0
$49$	36.0874	−	33.1466i	0.736478	−	0.676461i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			74.7970i			1.41126i	0.708579	+	0.705632i	$0.249337\pi$
$53$			74.7970i			1.41126i	−0.708579	+	0.705632i	$0.750663\pi$
$54$		0			0
$55$	9.47344			0.172244
$56$		0			0
$57$		0			0
$58$		0			0
$59$			22.0297i			0.373384i	0.982418	+	0.186692i	$0.0597767\pi$
$59$			22.0297i			0.373384i	−0.982418	+	0.186692i	$0.940223\pi$
$60$		0			0
$61$		−	57.2996i		−	0.939338i	−0.882843	−	0.469669i	$-0.844373\pi$
$61$		−	57.2996i		−	0.939338i	0.882843	−	0.469669i	$-0.155627\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			8.65587i			0.133167i
$66$		0			0
$67$		−	47.6026i		−	0.710487i	−0.934774	−	0.355244i	$-0.884398\pi$
$67$		−	47.6026i		−	0.710487i	0.934774	−	0.355244i	$-0.115602\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−66.4226			−0.935530			−0.467765	−	0.883853i	$-0.654941\pi$
$71$	−66.4226			−0.935530			−0.467765	+	0.883853i	$0.654941\pi$
$72$		0			0
$73$		−	36.0414i		−	0.493718i	−0.969051	−	0.246859i	$-0.920602\pi$
$73$		−	36.0414i		−	0.493718i	0.969051	−	0.246859i	$-0.0793984\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	85.9703	−	33.4905i	1.11650	−	0.434942i
$78$		0			0
$79$			46.6882i			0.590990i	0.955344	+	0.295495i	$0.0954845\pi$
$79$			46.6882i			0.590990i	−0.955344	+	0.295495i	$0.904515\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			38.2957i			0.461394i	0.973026	+	0.230697i	$0.0741005\pi$
$83$			38.2957i			0.461394i	−0.973026	+	0.230697i	$0.925899\pi$
$84$		0			0
$85$	−12.2753			−0.144416
$86$		0			0
$87$		0			0
$88$		0			0
$89$	126.425			1.42051			0.710255	−	0.703945i	$-0.248580\pi$
$89$	126.425			1.42051			0.710255	+	0.703945i	$0.248580\pi$
$90$		0			0
$91$	30.6003	+	78.5509i	0.336267	+	0.863197i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.35085			−0.0457984
$96$		0			0
$97$		−	126.730i		−	1.30649i	−0.757146	−	0.653246i	$-0.773407\pi$
$97$		−	126.730i		−	1.30649i	0.757146	−	0.653246i	$-0.226593\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	129.877			1.28591			0.642955	−	0.765904i	$-0.277708\pi$
$101$	129.877			1.28591			0.642955	+	0.765904i	$0.277708\pi$
$102$		0			0
$103$	107.831			1.04691			0.523454	−	0.852054i	$-0.324643\pi$
$103$	107.831			1.04691			0.523454	+	0.852054i	$0.324643\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	31.6555			0.295845			0.147923	−	0.988999i	$-0.452741\pi$
$107$	31.6555			0.295845			0.147923	+	0.988999i	$0.452741\pi$
$108$		0			0
$109$	37.6040			0.344991			0.172496	−	0.985010i	$-0.444817\pi$
$109$	37.6040			0.344991			0.172496	+	0.985010i	$0.444817\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			61.7890i			0.546806i	0.961900	+	0.273403i	$0.0881492\pi$
$113$			61.7890i			0.546806i	−0.961900	+	0.273403i	$0.911851\pi$
$114$		0			0
$115$	−7.82907			−0.0680788
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−111.397	+	43.3958i	−0.936111	+	0.364671i
$120$		0			0
$121$	52.7247			0.435741
$122$		0			0
$123$		0			0
$124$		0			0
$125$	35.5661			0.284529
$126$		0			0
$127$			28.1504i			0.221656i	0.993840	+	0.110828i	$0.0353503\pi$
$127$			28.1504i			0.221656i	−0.993840	+	0.110828i	$0.964650\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	114.344i		−	0.872855i	−0.899739	−	0.436427i	$-0.856244\pi$
$131$		−	114.344i		−	0.872855i	0.899739	−	0.436427i	$-0.143756\pi$
$132$		0			0
$133$	−39.4834	+	15.3811i	−0.296868	+	0.115648i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	4.07205i		−	0.0297230i	−0.999890	−	0.0148615i	$-0.995269\pi$
$137$		−	4.07205i		−	0.0297230i	0.999890	−	0.0148615i	$-0.00473073\pi$
$138$		0			0
$139$	−9.02009			−0.0648927			−0.0324464	−	0.999473i	$-0.510330\pi$
$139$	−9.02009			−0.0648927			−0.0324464	+	0.999473i	$0.510330\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			158.732i			1.11001i
$144$		0			0
$145$		−	5.89754i		−	0.0406727i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			184.375i			1.23742i	0.785621	+	0.618708i	$0.212344\pi$
$149$			184.375i			1.23742i	−0.785621	+	0.618708i	$0.787656\pi$
$150$		0			0
$151$			113.229i			0.749861i	0.927053	+	0.374931i	$0.122334\pi$
$151$			113.229i			0.749861i	−0.927053	+	0.374931i	$0.877666\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−26.3084			−0.169732
$156$		0			0
$157$		−	201.949i		−	1.28630i	−0.765739	−	0.643151i	$-0.777627\pi$
$157$		−	201.949i		−	1.28630i	0.765739	−	0.643151i	$-0.222373\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−71.0478	+	27.6773i	−0.441291	+	0.171909i
$162$		0			0
$163$		−	307.256i		−	1.88501i	−0.334198	−	0.942503i	$-0.608466\pi$
$163$		−	307.256i		−	1.88501i	0.334198	−	0.942503i	$-0.391534\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			217.945i			1.30506i	0.757764	+	0.652529i	$0.226292\pi$
$167$			217.945i			1.30506i	−0.757764	+	0.652529i	$0.773708\pi$
$168$		0			0
$169$	23.9668			0.141815
$170$		0			0
$171$		0			0
$172$		0			0
$173$	17.6552			0.102053			0.0510266	−	0.998697i	$-0.483751\pi$
$173$	17.6552			0.102053			0.0510266	+	0.998697i	$0.483751\pi$
$174$		0			0
$175$	159.694	−	62.2104i	0.912539	−	0.355488i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	71.5186			0.399545			0.199773	−	0.979842i	$-0.435980\pi$
$179$	71.5186			0.399545			0.199773	+	0.979842i	$0.435980\pi$
$180$		0			0
$181$			162.482i			0.897693i	0.893609	+	0.448846i	$0.148165\pi$
$181$			162.482i			0.897693i	−0.893609	+	0.448846i	$0.851835\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−32.4065			−0.175170
$186$		0			0
$187$	−225.106			−1.20378
$188$		0			0
$189$		0			0
$190$		0			0
$191$	248.036			1.29862			0.649309	−	0.760525i	$-0.275058\pi$
$191$	248.036			1.29862			0.649309	+	0.760525i	$0.275058\pi$
$192$		0			0
$193$	179.833			0.931778			0.465889	−	0.884843i	$-0.345735\pi$
$193$	179.833			0.931778			0.465889	+	0.884843i	$0.345735\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	276.983i		−	1.40600i	−0.711188	−	0.703001i	$-0.751843\pi$
$197$		−	276.983i		−	1.40600i	0.711188	−	0.703001i	$-0.248157\pi$
$198$		0			0
$199$	−4.12608			−0.0207341			−0.0103670	−	0.999946i	$-0.503300\pi$
$199$	−4.12608			−0.0207341			−0.0103670	+	0.999946i	$0.503300\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−20.8490	−	53.5194i	−0.102704	−	0.263642i
$204$		0			0
$205$	−44.8663			−0.218860
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−79.7862			−0.381752
$210$		0			0
$211$			289.252i			1.37086i	0.728138	+	0.685431i	$0.240386\pi$
$211$			289.252i			1.37086i	−0.728138	+	0.685431i	$0.759614\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	8.63221i		−	0.0401498i
$216$		0			0
$217$	−238.746	+	93.0056i	−1.10021	+	0.428597i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	205.679i		−	0.930675i
$222$		0			0
$223$	141.619			0.635065			0.317532	−	0.948247i	$-0.397146\pi$
$223$	141.619			0.635065			0.317532	+	0.948247i	$0.397146\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	366.717i		−	1.61549i	−0.589529	−	0.807747i	$-0.700686\pi$
$227$		−	366.717i		−	1.61549i	0.589529	−	0.807747i	$-0.299314\pi$
$228$		0			0
$229$		−	48.2856i		−	0.210854i	−0.994427	−	0.105427i	$-0.966379\pi$
$229$		−	48.2856i		−	0.210854i	0.994427	−	0.105427i	$-0.0336210\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			199.063i			0.854347i	0.904170	+	0.427174i	$0.140491\pi$
$233$			199.063i			0.854347i	−0.904170	+	0.427174i	$0.859509\pi$
$234$		0			0
$235$		−	62.7717i		−	0.267114i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−369.790			−1.54724			−0.773619	−	0.633651i	$-0.781556\pi$
$239$	−369.790			−1.54724			−0.773619	+	0.633651i	$0.781556\pi$
$240$		0			0
$241$			446.214i			1.85151i	0.378123	+	0.925756i	$0.376570\pi$
$241$			446.214i			1.85151i	−0.378123	+	0.925756i	$0.623430\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−25.9378	+	23.8241i	−0.105869	+	0.0972412i
$246$		0			0
$247$		−	72.9005i		−	0.295144i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	397.708i		−	1.58449i	−0.610201	−	0.792247i	$-0.708911\pi$
$251$		−	397.708i		−	1.58449i	0.610201	−	0.792247i	$-0.291089\pi$
$252$		0			0
$253$	−143.570			−0.567470
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−160.583			−0.624836			−0.312418	−	0.949945i	$-0.601139\pi$
$257$	−160.583			−0.624836			−0.312418	+	0.949945i	$0.601139\pi$
$258$		0			0
$259$	−294.085	+	114.564i	−1.13546	+	0.442331i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	161.172			0.612822			0.306411	−	0.951899i	$-0.400872\pi$
$263$	161.172			0.612822			0.306411	+	0.951899i	$0.400872\pi$
$264$		0			0
$265$		−	53.7602i		−	0.202869i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−293.506			−1.09110			−0.545550	−	0.838078i	$-0.683679\pi$
$269$	−293.506			−1.09110			−0.545550	+	0.838078i	$0.683679\pi$
$270$		0			0
$271$	−9.10088			−0.0335826			−0.0167913	−	0.999859i	$-0.505345\pi$
$271$	−9.10088			−0.0335826			−0.0167913	+	0.999859i	$0.505345\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	322.703			1.17346
$276$		0			0
$277$	58.0341			0.209509			0.104755	−	0.994498i	$-0.466594\pi$
$277$	58.0341			0.209509			0.104755	+	0.994498i	$0.466594\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	237.740i		−	0.846050i	−0.906118	−	0.423025i	$-0.860968\pi$
$281$		−	237.740i		−	0.846050i	0.906118	−	0.423025i	$-0.139032\pi$
$282$		0			0
$283$	−118.869			−0.420030			−0.210015	−	0.977698i	$-0.567351\pi$
$283$	−118.869			−0.420030			−0.210015	+	0.977698i	$0.567351\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−407.156	+	158.611i	−1.41866	+	0.552653i
$288$		0			0
$289$	2.68439			0.00928853
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−343.134			−1.17110			−0.585552	−	0.810635i	$-0.699122\pi$
$293$	−343.134			−1.17110			−0.585552	+	0.810635i	$0.699122\pi$
$294$		0			0
$295$		−	15.8338i		−	0.0536739i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	131.180i		−	0.438728i
$300$		0			0
$301$	−30.5166	−	78.3362i	−0.101384	−	0.260253i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			41.1840i			0.135030i
$306$		0			0
$307$	−438.690			−1.42896			−0.714479	−	0.699657i	$-0.753336\pi$
$307$	−438.690			−1.42896			−0.714479	+	0.699657i	$0.753336\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	81.3566i		−	0.261597i	−0.991409	−	0.130798i	$-0.958246\pi$
$311$		−	81.3566i		−	0.261597i	0.991409	−	0.130798i	$-0.0417541\pi$
$312$		0			0
$313$			240.509i			0.768401i	0.923250	+	0.384200i	$0.125523\pi$
$313$			240.509i			0.768401i	−0.923250	+	0.384200i	$0.874477\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	36.9841i		−	0.116669i	−0.998297	−	0.0583346i	$-0.981421\pi$
$317$		−	36.9841i		−	0.116669i	0.998297	−	0.0583346i	$-0.0185790\pi$
$318$		0			0
$319$		−	108.149i		−	0.339027i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	103.384			0.320074
$324$		0			0
$325$			294.853i			0.907240i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−221.911	−	569.645i	−0.674500	−	1.73144i
$330$		0			0
$331$		−	606.081i		−	1.83106i	−0.402250	−	0.915530i	$-0.631772\pi$
$331$		−	606.081i		−	1.83106i	0.402250	−	0.915530i	$-0.368228\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			34.2143i			0.102132i
$336$		0			0
$337$	−193.692			−0.574755			−0.287377	−	0.957817i	$-0.592783\pi$
$337$	−193.692			−0.574755			−0.287377	+	0.957817i	$0.592783\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−482.446			−1.41480
$342$		0			0
$343$	−151.159	+	307.896i	−0.440697	+	0.897656i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	404.557			1.16587			0.582935	−	0.812518i	$-0.301904\pi$
$347$	404.557			1.16587			0.582935	+	0.812518i	$0.301904\pi$
$348$		0			0
$349$			316.549i			0.907016i	0.891252	+	0.453508i	$0.149828\pi$
$349$			316.549i			0.907016i	−0.891252	+	0.453508i	$0.850172\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−273.644			−0.775195			−0.387598	−	0.921829i	$-0.626695\pi$
$353$	−273.644			−0.775195			−0.387598	+	0.921829i	$0.626695\pi$
$354$		0			0
$355$	47.7412			0.134482
$356$		0			0
$357$		0			0
$358$		0			0
$359$	628.441			1.75053			0.875267	−	0.483641i	$-0.160686\pi$
$359$	628.441			1.75053			0.875267	+	0.483641i	$0.160686\pi$
$360$		0			0
$361$	−324.357			−0.898495
$362$		0			0
$363$		0			0
$364$		0			0
$365$			25.9047i			0.0709718i
$366$		0			0
$367$	363.759			0.991169			0.495584	−	0.868560i	$-0.334954\pi$
$367$	363.759			0.991169			0.495584	+	0.868560i	$0.334954\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−190.053	−	487.867i	−0.512273	−	1.31501i
$372$		0			0
$373$	358.591			0.961370			0.480685	−	0.876893i	$-0.340388\pi$
$373$	358.591			0.961370			0.480685	+	0.876893i	$0.340388\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	98.8160			0.262112
$378$		0			0
$379$			20.2357i			0.0533924i	0.999644	+	0.0266962i	$0.00849867\pi$
$379$			20.2357i			0.0533924i	−0.999644	+	0.0266962i	$0.991501\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			362.193i			0.945674i	0.881150	+	0.472837i	$0.156770\pi$
$383$			362.193i			0.945674i	−0.881150	+	0.472837i	$0.843230\pi$
$384$		0			0
$385$	−61.7911	+	24.0713i	−0.160496	+	0.0625228i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			401.899i			1.03316i	0.856239	+	0.516580i	$0.172795\pi$
$389$			401.899i			1.03316i	−0.856239	+	0.516580i	$0.827205\pi$
$390$		0			0
$391$	186.033			0.475787
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	33.5571i		−	0.0849546i
$396$		0			0
$397$			600.636i			1.51294i	0.654030	+	0.756469i	$0.273077\pi$
$397$			600.636i			1.51294i	−0.654030	+	0.756469i	$0.726923\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			320.267i			0.798672i	0.916805	+	0.399336i	$0.130759\pi$
$401$			320.267i			0.798672i	−0.916805	+	0.399336i	$0.869241\pi$
$402$		0			0
$403$		−	440.810i		−	1.09382i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−594.273			−1.46013
$408$		0			0
$409$		−	536.109i		−	1.31078i	−0.755291	−	0.655390i	$-0.772504\pi$
$409$		−	536.109i		−	1.31078i	0.755291	−	0.655390i	$-0.227496\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−55.9757	−	143.690i	−0.135534	−	0.347917i
$414$		0			0
$415$		−	27.5250i		−	0.0663252i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	572.718i		−	1.36687i	−0.730012	−	0.683435i	$-0.760485\pi$
$419$		−	572.718i		−	1.36687i	0.730012	−	0.683435i	$-0.239515\pi$
$420$		0			0
$421$	−525.282			−1.24770			−0.623851	−	0.781543i	$-0.714433\pi$
$421$	−525.282			−1.24770			−0.623851	+	0.781543i	$0.714433\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−418.146			−0.983874
$426$		0			0
$427$	145.594	+	373.740i	0.340969	+	0.875269i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−319.218			−0.740645			−0.370323	−	0.928903i	$-0.620753\pi$
$431$	−319.218			−0.740645			−0.370323	+	0.928903i	$0.620753\pi$
$432$		0			0
$433$			415.694i			0.960032i	0.877260	+	0.480016i	$0.159369\pi$
$433$			415.694i			0.960032i	−0.877260	+	0.480016i	$0.840631\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	65.9371			0.150886
$438$		0			0
$439$	5.25771			0.0119766			0.00598828	−	0.999982i	$-0.498094\pi$
$439$	5.25771			0.0119766			0.00598828	+	0.999982i	$0.498094\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	847.180			1.91237			0.956186	−	0.292761i	$-0.0945741\pi$
$443$	847.180			1.91237			0.956186	+	0.292761i	$0.0945741\pi$
$444$		0			0
$445$	−90.8681			−0.204198
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	445.707i		−	0.992667i	−0.868132	−	0.496333i	$-0.834679\pi$
$449$		−	445.707i		−	0.992667i	0.868132	−	0.496333i	$-0.165321\pi$
$450$		0			0
$451$	−822.761			−1.82430
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−21.9939	−	56.4584i	−0.0483382	−	0.124084i
$456$		0			0
$457$	719.184			1.57371			0.786853	−	0.617140i	$-0.211709\pi$
$457$	719.184			1.57371			0.786853	+	0.617140i	$0.211709\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−126.659			−0.274749			−0.137375	−	0.990519i	$-0.543866\pi$
$461$	−126.659			−0.274749			−0.137375	+	0.990519i	$0.543866\pi$
$462$		0			0
$463$			711.905i			1.53759i	0.639494	+	0.768796i	$0.279144\pi$
$463$			711.905i			1.53759i	−0.639494	+	0.768796i	$0.720856\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	486.913i		−	1.04264i	−0.853361	−	0.521320i	$-0.825440\pi$
$467$		−	486.913i		−	1.04264i	0.853361	−	0.521320i	$-0.174560\pi$
$468$		0			0
$469$	120.955	+	310.491i	0.257899	+	0.662027i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	158.298i		−	0.334668i
$474$		0			0
$475$	−148.207			−0.312015
$476$		0			0
$477$		0			0
$478$		0			0
$479$			708.637i			1.47941i	0.672932	+	0.739704i	$0.265035\pi$
$479$			708.637i			1.47941i	−0.672932	+	0.739704i	$0.734965\pi$
$480$		0			0
$481$		−	542.987i		−	1.12887i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			91.0869i			0.187808i
$486$		0			0
$487$		−	309.923i		−	0.636392i	−0.948025	−	0.318196i	$-0.896923\pi$
$487$		−	309.923i		−	0.636392i	0.948025	−	0.318196i	$-0.103077\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−905.482			−1.84416			−0.922080	−	0.387000i	$-0.873511\pi$
$491$	−905.482			−1.84416			−0.922080	+	0.387000i	$0.873511\pi$
$492$		0			0
$493$			140.136i			0.284252i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	433.245	−	168.775i	0.871721	−	0.339587i
$498$		0			0
$499$		−	484.957i		−	0.971857i	−0.873998	−	0.485929i	$-0.838481\pi$
$499$		−	484.957i		−	0.971857i	0.873998	−	0.485929i	$-0.161519\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			170.145i			0.338261i	0.985594	+	0.169130i	$0.0540959\pi$
$503$			170.145i			0.338261i	−0.985594	+	0.169130i	$0.945904\pi$
$504$		0			0
$505$	−93.3488			−0.184849
$506$		0			0
$507$		0			0
$508$		0			0
$509$	740.516			1.45484			0.727422	−	0.686190i	$-0.240718\pi$
$509$	740.516			1.45484			0.727422	+	0.686190i	$0.240718\pi$
$510$		0			0
$511$	91.5784	+	235.082i	0.179214	+	0.460043i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−77.5038			−0.150493
$516$		0			0
$517$		−	1151.11i		−	2.22652i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	132.788			0.254871			0.127436	−	0.991847i	$-0.459325\pi$
$521$	132.788			0.254871			0.127436	+	0.991847i	$0.459325\pi$
$522$		0			0
$523$	28.0680			0.0536674			0.0268337	−	0.999640i	$-0.491458\pi$
$523$	28.0680			0.0536674			0.0268337	+	0.999640i	$0.491458\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	625.136			1.18622
$528$		0			0
$529$	−410.351			−0.775710
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	751.756i		−	1.41042i
$534$		0			0
$535$	−22.7523			−0.0425277
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−475.649	+	436.888i	−0.882466	+	0.810552i
$540$		0			0
$541$	142.826			0.264004			0.132002	−	0.991249i	$-0.457860\pi$
$541$	142.826			0.264004			0.132002	+	0.991249i	$0.457860\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−27.0279			−0.0495924
$546$		0			0
$547$			464.249i			0.848718i	0.905494	+	0.424359i	$0.139501\pi$
$547$			464.249i			0.848718i	−0.905494	+	0.424359i	$0.860499\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			49.6696i			0.0901444i
$552$		0			0
$553$	−118.631	−	304.526i	−0.214523	−	0.550680i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			755.433i			1.35625i	0.734945	+	0.678127i	$0.237208\pi$
$557$			755.433i			1.35625i	−0.734945	+	0.678127i	$0.762792\pi$
$558$		0			0
$559$	144.637			0.258742
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	426.385i		−	0.757345i	−0.925531	−	0.378673i	$-0.876381\pi$
$563$		−	426.385i		−	0.757345i	0.925531	−	0.378673i	$-0.123619\pi$
$564$		0			0
$565$		−	44.4108i		−	0.0786032i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			761.417i			1.33817i	0.743187	+	0.669083i	$0.233313\pi$
$569$			761.417i			1.33817i	−0.743187	+	0.669083i	$0.766687\pi$
$570$		0			0
$571$			405.274i			0.709762i	0.934911	+	0.354881i	$0.115479\pi$
$571$			405.274i			0.709762i	−0.934911	+	0.354881i	$0.884521\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−266.689			−0.463806
$576$		0			0
$577$			103.329i			0.179080i	0.995983	+	0.0895401i	$0.0285397\pi$
$577$			103.329i			0.179080i	−0.995983	+	0.0895401i	$0.971460\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−97.3063	−	249.786i	−0.167481	−	0.429923i
$582$		0			0
$583$		−	985.859i		−	1.69101i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			748.072i			1.27440i	0.770699	+	0.637199i	$0.219907\pi$
$587$			748.072i			1.27440i	−0.770699	+	0.637199i	$0.780093\pi$
$588$		0			0
$589$	221.572			0.376183
$590$		0			0
$591$		0			0
$592$		0			0
$593$	708.242			1.19434			0.597169	−	0.802116i	$-0.296292\pi$
$593$	708.242			1.19434			0.597169	+	0.802116i	$0.296292\pi$
$594$		0			0
$595$	80.0666	−	31.1907i	0.134566	−	0.0524213i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	414.088			0.691298			0.345649	−	0.938364i	$-0.387659\pi$
$599$	414.088			0.691298			0.345649	+	0.938364i	$0.387659\pi$
$600$		0			0
$601$			682.465i			1.13555i	0.823184	+	0.567775i	$0.192196\pi$
$601$			682.465i			1.13555i	−0.823184	+	0.567775i	$0.807804\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−37.8958			−0.0626377
$606$		0			0
$607$	929.956			1.53205			0.766027	−	0.642809i	$-0.222231\pi$
$607$	929.956			1.53205			0.766027	+	0.642809i	$0.222231\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1051.77			1.72139
$612$		0			0
$613$	596.834			0.973628			0.486814	−	0.873506i	$-0.338159\pi$
$613$	596.834			0.973628			0.486814	+	0.873506i	$0.338159\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			1025.43i			1.66197i	0.556298	+	0.830983i	$0.312221\pi$
$617$			1025.43i			1.66197i	−0.556298	+	0.830983i	$0.687779\pi$
$618$		0			0
$619$	562.181			0.908209			0.454104	−	0.890948i	$-0.349959\pi$
$619$	562.181			0.908209			0.454104	+	0.890948i	$0.349959\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−824.616	+	321.237i	−1.32362	+	0.515629i
$624$		0			0
$625$	586.522			0.938435
$626$		0			0
$627$		0			0
$628$		0			0
$629$	770.038			1.22423
$630$		0			0
$631$			231.394i			0.366710i	0.983047	+	0.183355i	$0.0586957\pi$
$631$			231.394i			0.366710i	−0.983047	+	0.183355i	$0.941304\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	20.2330i		−	0.0318631i
$636$		0			0
$637$	−399.184	−	434.600i	−0.626662	−	0.682260i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	133.202i		−	0.207803i	−0.994588	−	0.103902i	$-0.966867\pi$
$641$		−	133.202i		−	0.207803i	0.994588	−	0.103902i	$-0.0331327\pi$
$642$		0			0
$643$	773.102			1.20234			0.601168	−	0.799123i	$-0.294702\pi$
$643$	773.102			1.20234			0.601168	+	0.799123i	$0.294702\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			58.7839i			0.0908561i	0.998968	+	0.0454281i	$0.0144652\pi$
$647$			58.7839i			0.0908561i	−0.998968	+	0.0454281i	$0.985535\pi$
$648$		0			0
$649$		−	290.361i		−	0.447398i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			1019.61i			1.56142i	0.624893	+	0.780711i	$0.285143\pi$
$653$			1019.61i			1.56142i	−0.624893	+	0.780711i	$0.714857\pi$
$654$		0			0
$655$			82.1846i			0.125473i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−222.681			−0.337908			−0.168954	−	0.985624i	$-0.554039\pi$
$659$	−222.681			−0.337908			−0.168954	+	0.985624i	$0.554039\pi$
$660$		0			0
$661$		−	936.819i		−	1.41727i	−0.705573	−	0.708637i	$-0.749310\pi$
$661$		−	936.819i		−	1.41727i	0.705573	−	0.708637i	$-0.250690\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	28.3786	−	11.0552i	0.0426747	−	0.0166243i
$666$		0			0
$667$			89.3771i			0.133999i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			755.236i			1.12554i
$672$		0			0
$673$	180.866			0.268746			0.134373	−	0.990931i	$-0.457098\pi$
$673$	180.866			0.268746			0.134373	+	0.990931i	$0.457098\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	714.110			1.05482			0.527408	−	0.849612i	$-0.323164\pi$
$677$	714.110			1.05482			0.527408	+	0.849612i	$0.323164\pi$
$678$		0			0
$679$	322.010	+	826.602i	0.474242	+	1.21738i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	418.174			0.612260			0.306130	−	0.951990i	$-0.400966\pi$
$683$	418.174			0.612260			0.306130	+	0.951990i	$0.400966\pi$
$684$		0			0
$685$			2.92678i			0.00427267i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	900.778			1.30737
$690$		0			0
$691$	−1196.06			−1.73091			−0.865453	−	0.500991i	$-0.832969\pi$
$691$	−1196.06			−1.73091			−0.865453	+	0.500991i	$0.832969\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.48318			0.00932831
$696$		0			0
$697$	1066.10			1.52956
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1102.31i		−	1.57248i	−0.617923	−	0.786238i	$-0.712026\pi$
$701$		−	1102.31i		−	1.57248i	0.617923	−	0.786238i	$-0.287974\pi$
$702$		0			0
$703$	272.931			0.388237
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−847.129	+	330.007i	−1.19820	+	0.466771i
$708$		0			0
$709$	−130.721			−0.184374			−0.0921871	−	0.995742i	$-0.529386\pi$
$709$	−130.721			−0.184374			−0.0921871	+	0.995742i	$0.529386\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	398.704			0.559192
$714$		0			0
$715$		−	114.088i		−	0.159564i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			572.947i			0.796866i	0.917197	+	0.398433i	$0.130446\pi$
$719$			572.947i			0.796866i	−0.917197	+	0.398433i	$0.869554\pi$
$720$		0			0
$721$	−703.337	+	273.991i	−0.975502	+	0.380016i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	200.893i		−	0.277094i
$726$		0			0
$727$	23.1626			0.0318606			0.0159303	−	0.999873i	$-0.494929\pi$
$727$	23.1626			0.0318606			0.0159303	+	0.999873i	$0.494929\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			205.117i			0.280598i
$732$		0			0
$733$		−	1288.22i		−	1.75746i	−0.477318	−	0.878730i	$-0.658391\pi$
$733$		−	1288.22i		−	1.75746i	0.477318	−	0.878730i	$-0.341609\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			627.425i			0.851323i
$738$		0			0
$739$		−	654.581i		−	0.885765i	−0.896580	−	0.442883i	$-0.853956\pi$
$739$		−	654.581i		−	0.885765i	0.896580	−	0.442883i	$-0.146044\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−765.496			−1.03028			−0.515139	−	0.857107i	$-0.672260\pi$
$743$	−765.496			−1.03028			−0.515139	+	0.857107i	$0.672260\pi$
$744$		0			0
$745$		−	132.519i		−	0.177878i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−206.474	+	80.4341i	−0.275667	+	0.107389i
$750$		0			0
$751$		−	1055.73i		−	1.40577i	−0.711305	−	0.702884i	$-0.751895\pi$
$751$		−	1055.73i		−	1.40577i	0.711305	−	0.702884i	$-0.248105\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	81.3833i		−	0.107792i
$756$		0			0
$757$	−974.372			−1.28715			−0.643574	−	0.765384i	$-0.722549\pi$
$757$	−974.372			−1.28715			−0.643574	+	0.765384i	$0.722549\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−545.838			−0.717264			−0.358632	−	0.933479i	$-0.616757\pi$
$761$	−545.838			−0.717264			−0.358632	+	0.933479i	$0.616757\pi$
$762$		0			0
$763$	−245.274	+	95.5489i	−0.321460	+	0.125228i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	265.303			0.345897
$768$		0			0
$769$		−	1399.09i		−	1.81936i	−0.415306	−	0.909682i	$-0.636325\pi$
$769$		−	1399.09i		−	1.81936i	0.415306	−	0.909682i	$-0.363675\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	852.651			1.10304			0.551520	−	0.834161i	$-0.314048\pi$
$773$	852.651			1.10304			0.551520	+	0.834161i	$0.314048\pi$
$774$		0			0
$775$	−896.168			−1.15635
$776$		0			0
$777$		0			0
$778$		0			0
$779$	377.868			0.485068
$780$		0			0
$781$	875.481			1.12097
$782$		0			0
$783$		0			0
$784$		0			0
$785$			145.151i			0.184906i
$786$		0			0
$787$	902.486			1.14674			0.573371	−	0.819296i	$-0.305635\pi$
$787$	902.486			1.14674			0.573371	+	0.819296i	$0.305635\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−157.001	−	403.022i	−0.198484	−	0.509510i
$792$		0			0
$793$	−690.058			−0.870187
$794$		0			0
$795$		0			0
$796$		0			0
$797$	884.614			1.10993			0.554965	−	0.831874i	$-0.312732\pi$
$797$	884.614			1.10993			0.554965	+	0.831874i	$0.312732\pi$
$798$		0			0
$799$			1491.57i			1.86679i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			475.042i			0.591585i
$804$		0			0
$805$	51.0655	−	19.8930i	0.0634354	−	0.0247119i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	107.405i		−	0.132762i	−0.997794	−	0.0663812i	$-0.978855\pi$
$809$		−	107.405i		−	0.132762i	0.997794	−	0.0663812i	$-0.0211453\pi$
$810$		0			0
$811$	215.158			0.265299			0.132650	−	0.991163i	$-0.457651\pi$
$811$	215.158			0.265299			0.132650	+	0.991163i	$0.457651\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			220.840i			0.270969i
$816$		0			0
$817$			72.7012i			0.0889856i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			781.594i			0.952002i	0.879445	+	0.476001i	$0.157914\pi$
$821$			781.594i			0.952002i	−0.879445	+	0.476001i	$0.842086\pi$
$822$		0			0
$823$			701.058i			0.851833i	0.904763	+	0.425916i	$0.140048\pi$
$823$			701.058i			0.851833i	−0.904763	+	0.425916i	$0.859952\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1524.87			−1.84386			−0.921930	−	0.387356i	$-0.873388\pi$
$827$	−1524.87			−1.84386			−0.921930	+	0.387356i	$0.873388\pi$
$828$		0			0
$829$			789.295i			0.952104i	0.879417	+	0.476052i	$0.157933\pi$
$829$			789.295i			0.952104i	−0.879417	+	0.476052i	$0.842067\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	616.329	−	566.103i	0.739891	−	0.679596i
$834$		0			0
$835$		−	156.647i		−	0.187602i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1184.60i		−	1.41192i	−0.708250	−	0.705962i	$-0.750515\pi$
$839$		−	1184.60i		−	1.41192i	0.708250	−	0.705962i	$-0.249485\pi$
$840$		0			0
$841$	773.673			0.919944
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−17.2261			−0.0203859
$846$		0			0
$847$	−343.899	+	133.969i	−0.406021	+	0.158169i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	491.121			0.577110
$852$		0			0
$853$			158.721i			0.186074i	0.995663	+	0.0930371i	$0.0296575\pi$
$853$			158.721i			0.186074i	−0.995663	+	0.0930371i	$0.970342\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	819.923			0.956737			0.478368	−	0.878159i	$-0.341228\pi$
$857$	819.923			0.956737			0.478368	+	0.878159i	$0.341228\pi$
$858$		0			0
$859$	1566.30			1.82340			0.911700	−	0.410856i	$-0.134770\pi$
$859$	1566.30			1.82340			0.911700	+	0.410856i	$0.134770\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	355.853			0.412344			0.206172	−	0.978516i	$-0.433899\pi$
$863$	355.853			0.412344			0.206172	+	0.978516i	$0.433899\pi$
$864$		0			0
$865$	−12.6897			−0.0146701
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	615.372i		−	0.708138i
$870$		0			0
$871$	−573.277			−0.658183
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−231.982	+	90.3708i	−0.265122	+	0.103281i
$876$		0			0
$877$	1272.14			1.45056			0.725279	−	0.688456i	$-0.241711\pi$
$877$	1272.14			1.45056			0.725279	+	0.688456i	$0.241711\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1195.13			−1.35656			−0.678281	−	0.734803i	$-0.737275\pi$
$881$	−1195.13			−1.35656			−0.678281	+	0.734803i	$0.737275\pi$
$882$		0			0
$883$			502.807i			0.569431i	0.958612	+	0.284715i	$0.0918991\pi$
$883$			502.807i			0.569431i	−0.958612	+	0.284715i	$0.908101\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	703.859i		−	0.793527i	−0.917921	−	0.396764i	$-0.870133\pi$
$887$		−	703.859i		−	0.793527i	0.917921	−	0.396764i	$-0.129867\pi$
$888$		0			0
$889$	−71.5279	−	183.612i	−0.0804588	−	0.206538i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			528.669i			0.592014i
$894$		0			0
$895$	−51.4039			−0.0574345
$896$		0			0
$897$		0			0
$898$		0			0
$899$			300.339i			0.334081i
$900$		0			0
$901$			1277.44i			1.41780i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	116.784i		−	0.129043i
$906$		0			0
$907$			834.516i			0.920084i	0.887897	+	0.460042i	$0.152166\pi$
$907$			834.516i			0.920084i	−0.887897	+	0.460042i	$0.847834\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	391.022			0.429223			0.214611	−	0.976700i	$-0.431151\pi$
$911$	391.022			0.429223			0.214611	+	0.976700i	$0.431151\pi$
$912$		0			0
$913$		−	504.755i		−	0.552853i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	290.539	+	745.815i	0.316837	+	0.813320i
$918$		0			0
$919$		−	1552.19i		−	1.68899i	−0.535560	−	0.844497i	$-0.679900\pi$
$919$		−	1552.19i		−	1.68899i	0.535560	−	0.844497i	$-0.320100\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			799.926i			0.866658i
$924$		0			0
$925$	−1103.89			−1.19340
$926$		0			0
$927$		0			0
$928$		0			0
$929$	196.405			0.211415			0.105708	−	0.994397i	$-0.466289\pi$
$929$	196.405			0.211415			0.105708	+	0.994397i	$0.466289\pi$
$930$		0			0
$931$	218.450	−	200.649i	0.234641	−	0.215519i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	161.795			0.173043
$936$		0			0
$937$			529.809i			0.565431i	0.959204	+	0.282716i	$0.0912353\pi$
$937$			529.809i			0.565431i	−0.959204	+	0.282716i	$0.908765\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1744.64			−1.85403			−0.927016	−	0.375021i	$-0.877635\pi$
$941$	−1744.64			−1.85403			−0.927016	+	0.375021i	$0.877635\pi$
$942$		0			0
$943$	679.948			0.721048
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1635.75			1.72730			0.863651	−	0.504090i	$-0.168172\pi$
$947$	1635.75			1.72730			0.863651	+	0.504090i	$0.168172\pi$
$948$		0			0
$949$	−434.046			−0.457372
$950$		0			0
$951$		0			0
$952$		0			0
$953$			206.977i			0.217185i	0.994086	+	0.108592i	$0.0346343\pi$
$953$			206.977i			0.217185i	−0.994086	+	0.108592i	$0.965366\pi$
$954$		0			0
$955$	−178.276			−0.186676
$956$		0			0
$957$		0			0
$958$		0			0
$959$	10.3468	+	26.5602i	0.0107891	+	0.0276957i
$960$		0			0
$961$	378.787			0.394159
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−129.255			−0.133943
$966$		0			0
$967$		−	444.327i		−	0.459491i	−0.973251	−	0.229745i	$-0.926211\pi$
$967$		−	444.327i		−	0.459491i	0.973251	−	0.229745i	$-0.0737893\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1130.03i			1.16378i	0.813268	+	0.581890i	$0.197686\pi$
$971$			1130.03i			1.16378i	−0.813268	+	0.581890i	$0.802314\pi$
$972$		0			0
$973$	58.8340	−	22.9193i	0.0604666	−	0.0235553i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	622.036i		−	0.636680i	−0.947977	−	0.318340i	$-0.896875\pi$
$977$		−	622.036i		−	0.636680i	0.947977	−	0.318340i	$-0.103125\pi$
$978$		0			0
$979$	−1666.35			−1.70209
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1354.99i		−	1.37842i	−0.724560	−	0.689212i	$-0.757957\pi$
$983$		−	1354.99i		−	1.37842i	0.724560	−	0.689212i	$-0.242043\pi$
$984$		0			0
$985$			199.081i			0.202113i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			130.821i			0.132276i
$990$		0			0
$991$		−	1642.45i		−	1.65736i	−0.559720	−	0.828682i	$-0.689091\pi$
$991$		−	1642.45i		−	1.65736i	0.559720	−	0.828682i	$-0.310909\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2.96561			0.00298052
$996$		0			0
$997$			70.4772i			0.0706893i	0.999375	+	0.0353446i	$0.0112529\pi$
$997$			70.4772i			0.0706893i	−0.999375	+	0.0353446i	$0.988747\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.o.b.1007.12	yes	24
3.2	odd	2	inner	1008.3.o.b.1007.16	yes	24
4.3	odd	2	inner	1008.3.o.b.1007.11	yes	24
7.6	odd	2	inner	1008.3.o.b.1007.14	yes	24
12.11	even	2	inner	1008.3.o.b.1007.15	yes	24
21.20	even	2	inner	1008.3.o.b.1007.10	yes	24
28.27	even	2	inner	1008.3.o.b.1007.13	yes	24
84.83	odd	2	inner	1008.3.o.b.1007.9	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.3.o.b.1007.9	✓	24	84.83	odd	2	inner
1008.3.o.b.1007.10	yes	24	21.20	even	2	inner
1008.3.o.b.1007.11	yes	24	4.3	odd	2	inner
1008.3.o.b.1007.12	yes	24	1.1	even	1	trivial
1008.3.o.b.1007.13	yes	24	28.27	even	2	inner
1008.3.o.b.1007.14	yes	24	7.6	odd	2	inner
1008.3.o.b.1007.15	yes	24	12.11	even	2	inner
1008.3.o.b.1007.16	yes	24	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.718749 q^{5} +(-6.52255 + 2.54092i) q^{7} -13.1805 q^{11} -12.0430i q^{13} +17.0788 q^{17} +6.05336 q^{19} +10.8926 q^{23} -24.4834 q^{25} +8.20529i q^{29} +36.6031 q^{31} +(4.68808 - 1.82629i) q^{35} +45.0874 q^{37} +62.4228 q^{41} +12.0100i q^{43} +87.3347i q^{47} +(36.0874 - 33.1466i) q^{49} +74.7970i q^{53} +9.47344 q^{55} +22.0297i q^{59} -57.2996i q^{61} +8.65587i q^{65} -47.6026i q^{67} -66.4226 q^{71} -36.0414i q^{73} +(85.9703 - 33.4905i) q^{77} +46.6882i q^{79} +38.2957i q^{83} -12.2753 q^{85} +126.425 q^{89} +(30.6003 + 78.5509i) q^{91} -4.35085 q^{95} -126.730i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 264 q^{25} - 96 q^{37} - 312 q^{49} - 96 q^{85}+O(q^{100}) \) 24 * q + 264 * q^25 - 96 * q^37 - 312 * q^49 - 96 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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