LMFDB - Embedded newform 1008.3.d.d.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,3,Mod(449,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1008.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.d (of order $2$, degree $1$, not 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:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$-2.57794i$ of defining polynomial
Character	$\chi$	$=$	1008.449
Dual form			1008.3.d.d.449.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.15587i q^{5} +2.64575 q^{7} +O(q^{10})$	$q-5.15587i q^{5} +2.64575 q^{7} -14.5544i q^{11} +16.5830 q^{13} -9.98823i q^{17} -32.4575 q^{19} -9.72202i q^{23} -1.58301 q^{25} +40.0102i q^{29} +32.7085 q^{31} -13.6412i q^{35} -42.5830 q^{37} -22.7735i q^{41} +11.7490 q^{43} -39.4205i q^{47} +7.00000 q^{49} -49.1425i q^{53} -75.0405 q^{55} +28.4617i q^{59} -112.539 q^{61} -85.4998i q^{65} -68.0810 q^{67} -69.1190i q^{71} +41.0405 q^{73} -38.5073i q^{77} +22.0000 q^{79} -13.9647i q^{83} -51.4980 q^{85} +110.632i q^{89} +43.8745 q^{91} +167.347i q^{95} +45.2915 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 24 q^{13} - 24 q^{19} + 36 q^{25} + 152 q^{31} - 128 q^{37} - 80 q^{43} + 28 q^{49} - 152 q^{55} - 48 q^{61} + 24 q^{67} + 16 q^{73} + 88 q^{79} + 48 q^{85} + 112 q^{91} + 160 q^{97}+O(q^{100}) $ 4 * q + 24 * q^13 - 24 * q^19 + 36 * q^25 + 152 * q^31 - 128 * q^37 - 80 * q^43 + 28 * q^49 - 152 * q^55 - 48 * q^61 + 24 * q^67 + 16 * q^73 + 88 * q^79 + 48 * q^85 + 112 * q^91 + 160 * q^97

Display 100 coefficients 10 coefficients

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
$5$	− 5.15587i	− 1.03117i	−0.856837	−	0.515587i	$-0.827574\pi$
$5$	− 5.15587i	− 1.03117i	0.856837	−	0.515587i	$-0.172426\pi$
$6$	0	0
$7$	2.64575	0.377964
$7$	2.64575	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 14.5544i	− 1.32313i	−0.749890	−	0.661563i	$-0.769893\pi$
$11$	− 14.5544i	− 1.32313i	0.749890	−	0.661563i	$-0.230107\pi$
$12$	0	0
$13$	16.5830	1.27562	0.637808	−	0.770195i	$-0.279841\pi$
$13$	16.5830	1.27562	0.637808	+	0.770195i	$0.279841\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 9.98823i	− 0.587543i	−0.955876	−	0.293772i	$-0.905090\pi$
$17$	− 9.98823i	− 0.587543i	0.955876	−	0.293772i	$-0.0949105\pi$
$18$	0	0
$19$	−32.4575	−1.70829	−0.854145	−	0.520035i	$-0.825919\pi$
$19$	−32.4575	−1.70829	−0.854145	+	0.520035i	$0.825919\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 9.72202i	− 0.422697i	−0.977411	−	0.211348i	$-0.932215\pi$
$23$	− 9.72202i	− 0.422697i	0.977411	−	0.211348i	$-0.0677854\pi$
$24$	0	0
$25$	−1.58301	−0.0633202
$26$	0	0
$27$	0	0
$28$	0	0
$29$	40.0102i	1.37966i	0.723970	+	0.689831i	$0.242315\pi$
$29$	40.0102i	1.37966i	−0.723970	+	0.689831i	$0.757685\pi$
$30$	0	0
$31$	32.7085	1.05511	0.527556	−	0.849520i	$-0.323108\pi$
$31$	32.7085	1.05511	0.527556	+	0.849520i	$0.323108\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 13.6412i	− 0.389747i
$36$	0	0
$37$	−42.5830	−1.15089	−0.575446	−	0.817840i	$-0.695172\pi$
$37$	−42.5830	−1.15089	−0.575446	+	0.817840i	$0.695172\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 22.7735i	− 0.555450i	−0.960661	−	0.277725i	$-0.910420\pi$
$41$	− 22.7735i	− 0.555450i	0.960661	−	0.277725i	$-0.0895804\pi$
$42$	0	0
$43$	11.7490	0.273233	0.136616	−	0.990624i	$-0.456377\pi$
$43$	11.7490	0.273233	0.136616	+	0.990624i	$0.456377\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 39.4205i	− 0.838734i	−0.907817	−	0.419367i	$-0.862252\pi$
$47$	− 39.4205i	− 0.838734i	0.907817	−	0.419367i	$-0.137748\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 49.1425i	− 0.927218i	−0.886040	−	0.463609i	$-0.846554\pi$
$53$	− 49.1425i	− 0.927218i	0.886040	−	0.463609i	$-0.153446\pi$
$54$	0	0
$55$	−75.0405	−1.36437
$56$	0	0
$57$	0	0
$58$	0	0
$59$	28.4617i	0.482402i	0.970475	+	0.241201i	$0.0775414\pi$
$59$	28.4617i	0.482402i	−0.970475	+	0.241201i	$0.922459\pi$
$60$	0	0
$61$	−112.539	−1.84489	−0.922447	−	0.386123i	$-0.873814\pi$
$61$	−112.539	−1.84489	−0.922447	+	0.386123i	$0.873814\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 85.4998i	− 1.31538i
$66$	0	0
$67$	−68.0810	−1.01613	−0.508067	−	0.861317i	$-0.669640\pi$
$67$	−68.0810	−1.01613	−0.508067	+	0.861317i	$0.669640\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 69.1190i	− 0.973507i	−0.873539	−	0.486753i	$-0.838181\pi$
$71$	− 69.1190i	− 0.973507i	0.873539	−	0.486753i	$-0.161819\pi$
$72$	0	0
$73$	41.0405	0.562199	0.281099	−	0.959679i	$-0.409301\pi$
$73$	41.0405	0.562199	0.281099	+	0.959679i	$0.409301\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 38.5073i	− 0.500095i
$78$	0	0
$79$	22.0000	0.278481	0.139241	−	0.990259i	$-0.455534\pi$
$79$	22.0000	0.278481	0.139241	+	0.990259i	$0.455534\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 13.9647i	− 0.168249i	−0.996455	−	0.0841245i	$-0.973191\pi$
$83$	− 13.9647i	− 0.168249i	0.996455	−	0.0841245i	$-0.0268093\pi$
$84$	0	0
$85$	−51.4980	−0.605859
$86$	0	0
$87$	0	0
$88$	0	0
$89$	110.632i	1.24306i	0.783391	+	0.621529i	$0.213488\pi$
$89$	110.632i	1.24306i	−0.783391	+	0.621529i	$0.786512\pi$
$90$	0	0
$91$	43.8745	0.482137
$92$	0	0
$93$	0	0
$94$	0	0
$95$	167.347i	1.76154i
$96$	0	0
$97$	45.2915	0.466923	0.233461	−	0.972366i	$-0.424995\pi$
$97$	45.2915	0.466923	0.233461	+	0.972366i	$0.424995\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 165.958i	− 1.64315i	−0.570099	−	0.821576i	$-0.693095\pi$
$101$	− 165.958i	− 1.64315i	0.570099	−	0.821576i	$-0.306905\pi$
$102$	0	0
$103$	−94.1255	−0.913840	−0.456920	−	0.889508i	$-0.651047\pi$
$103$	−94.1255	−0.913840	−0.456920	+	0.889508i	$0.651047\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 93.3954i	− 0.872854i	−0.899740	−	0.436427i	$-0.856244\pi$
$107$	− 93.3954i	− 0.872854i	0.899740	−	0.436427i	$-0.143756\pi$
$108$	0	0
$109$	−86.3320	−0.792037	−0.396018	−	0.918243i	$-0.629608\pi$
$109$	−86.3320	−0.792037	−0.396018	+	0.918243i	$0.629608\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 120.678i	− 1.06794i	−0.845502	−	0.533972i	$-0.820699\pi$
$113$	− 120.678i	− 1.06794i	0.845502	−	0.533972i	$-0.179301\pi$
$114$	0	0
$115$	−50.1255	−0.435874
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 26.4264i	− 0.222070i
$120$	0	0
$121$	−90.8301	−0.750662
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 120.735i	− 0.965880i
$126$	0	0
$127$	−78.4131	−0.617426	−0.308713	−	0.951155i	$-0.599898\pi$
$127$	−78.4131	−0.617426	−0.308713	+	0.951155i	$0.599898\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 132.112i	− 1.00849i	−0.863562	−	0.504243i	$-0.831772\pi$
$131$	− 132.112i	− 1.00849i	0.863562	−	0.504243i	$-0.168228\pi$
$132$	0	0
$133$	−85.8745	−0.645673
$134$	0	0
$135$	0	0
$136$	0	0
$137$	110.481i	0.806427i	0.915106	+	0.403214i	$0.132107\pi$
$137$	110.481i	0.806427i	−0.915106	+	0.403214i	$0.867893\pi$
$138$	0	0
$139$	154.834	1.11391	0.556957	−	0.830541i	$-0.311969\pi$
$139$	154.834	1.11391	0.556957	+	0.830541i	$0.311969\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 241.355i	− 1.68780i
$144$	0	0
$145$	206.288	1.42267
$146$	0	0
$147$	0	0
$148$	0	0
$149$	77.0719i	0.517261i	0.965976	+	0.258630i	$0.0832712\pi$
$149$	77.0719i	0.517261i	−0.965976	+	0.258630i	$0.916729\pi$
$150$	0	0
$151$	68.2510	0.451993	0.225997	−	0.974128i	$-0.427436\pi$
$151$	68.2510	0.451993	0.225997	+	0.974128i	$0.427436\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 168.641i	− 1.08801i
$156$	0	0
$157$	−56.7895	−0.361717	−0.180858	−	0.983509i	$-0.557888\pi$
$157$	−56.7895	−0.361717	−0.180858	+	0.983509i	$0.557888\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 25.7221i	− 0.159764i
$162$	0	0
$163$	43.4170	0.266362	0.133181	−	0.991092i	$-0.457481\pi$
$163$	43.4170	0.266362	0.133181	+	0.991092i	$0.457481\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 6.77342i	− 0.0405594i	−0.999794	−	0.0202797i	$-0.993544\pi$
$167$	− 6.77342i	− 0.0405594i	0.999794	−	0.0202797i	$-0.00645568\pi$
$168$	0	0
$169$	105.996	0.627196
$170$	0	0
$171$	0	0
$172$	0	0
$173$	39.6294i	0.229072i	0.993419	+	0.114536i	$0.0365381\pi$
$173$	39.6294i	0.229072i	−0.993419	+	0.114536i	$0.963462\pi$
$174$	0	0
$175$	−4.18824	−0.0239328
$176$	0	0
$177$	0	0
$178$	0	0
$179$	85.9750i	0.480307i	0.970735	+	0.240153i	$0.0771978\pi$
$179$	85.9750i	0.480307i	−0.970735	+	0.240153i	$0.922802\pi$
$180$	0	0
$181$	339.830	1.87751	0.938757	−	0.344580i	$-0.111979\pi$
$181$	339.830	1.87751	0.938757	+	0.344580i	$0.111979\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	219.552i	1.18677i
$186$	0	0
$187$	−145.373	−0.777393
$188$	0	0
$189$	0	0
$190$	0	0
$191$	207.472i	1.08624i	0.839656	+	0.543119i	$0.182757\pi$
$191$	207.472i	1.08624i	−0.839656	+	0.543119i	$0.817243\pi$
$192$	0	0
$193$	−223.498	−1.15802	−0.579010	−	0.815320i	$-0.696561\pi$
$193$	−223.498	−1.15802	−0.579010	+	0.815320i	$0.696561\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 174.063i	− 0.883568i	−0.897121	−	0.441784i	$-0.854346\pi$
$197$	− 174.063i	− 0.883568i	0.897121	−	0.441784i	$-0.145654\pi$
$198$	0	0
$199$	42.5830	0.213985	0.106992	−	0.994260i	$-0.465878\pi$
$199$	42.5830	0.213985	0.106992	+	0.994260i	$0.465878\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	105.857i	0.521464i
$204$	0	0
$205$	−117.417	−0.572766
$206$	0	0
$207$	0	0
$208$	0	0
$209$	472.399i	2.26028i
$210$	0	0
$211$	−67.8379	−0.321507	−0.160753	−	0.986995i	$-0.551392\pi$
$211$	−67.8379	−0.321507	−0.160753	+	0.986995i	$0.551392\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 60.5764i	− 0.281751i
$216$	0	0
$217$	86.5385	0.398795
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 165.635i	− 0.749479i
$222$	0	0
$223$	203.911	0.914400	0.457200	−	0.889364i	$-0.348852\pi$
$223$	203.911	0.914400	0.457200	+	0.889364i	$0.348852\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	422.667i	1.86197i	0.365060	+	0.930984i	$0.381049\pi$
$227$	422.667i	1.86197i	−0.365060	+	0.930984i	$0.618951\pi$
$228$	0	0
$229$	38.7530	0.169227	0.0846134	−	0.996414i	$-0.473034\pi$
$229$	38.7530	0.169227	0.0846134	+	0.996414i	$0.473034\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 185.439i	− 0.795877i	−0.917412	−	0.397939i	$-0.869726\pi$
$233$	− 185.439i	− 0.795877i	0.917412	−	0.397939i	$-0.130274\pi$
$234$	0	0
$235$	−203.247	−0.864881
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 227.866i	− 0.953414i	−0.879062	−	0.476707i	$-0.841830\pi$
$239$	− 227.866i	− 0.953414i	0.879062	−	0.476707i	$-0.158170\pi$
$240$	0	0
$241$	43.6235	0.181010	0.0905052	−	0.995896i	$-0.471152\pi$
$241$	43.6235	0.181010	0.0905052	+	0.995896i	$0.471152\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 36.0911i	− 0.147311i
$246$	0	0
$247$	−538.243	−2.17912
$248$	0	0
$249$	0	0
$250$	0	0
$251$	248.661i	0.990682i	0.868699	+	0.495341i	$0.164957\pi$
$251$	248.661i	0.990682i	−0.868699	+	0.495341i	$0.835043\pi$
$252$	0	0
$253$	−141.498	−0.559281
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.08514i	0.00422232i	0.999998	+	0.00211116i	$0.000672003\pi$
$257$	1.08514i	0.00422232i	−0.999998	+	0.00211116i	$0.999328\pi$
$258$	0	0
$259$	−112.664	−0.434996
$260$	0	0
$261$	0	0
$262$	0	0
$263$	145.601i	0.553616i	0.960925	+	0.276808i	$0.0892767\pi$
$263$	145.601i	0.553616i	−0.960925	+	0.276808i	$0.910723\pi$
$264$	0	0
$265$	−253.373	−0.956123
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 84.4147i	− 0.313809i	−0.987614	−	0.156905i	$-0.949848\pi$
$269$	− 84.4147i	− 0.313809i	0.987614	−	0.156905i	$-0.0501515\pi$
$270$	0	0
$271$	317.535	1.17171	0.585857	−	0.810414i	$-0.300758\pi$
$271$	317.535	1.17171	0.585857	+	0.810414i	$0.300758\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	23.0397i	0.0837806i
$276$	0	0
$277$	37.8379	0.136599	0.0682995	−	0.997665i	$-0.478243\pi$
$277$	37.8379	0.136599	0.0682995	+	0.997665i	$0.478243\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	348.183i	1.23909i	0.784963	+	0.619543i	$0.212682\pi$
$281$	348.183i	1.23909i	−0.784963	+	0.619543i	$0.787318\pi$
$282$	0	0
$283$	375.542	1.32701	0.663503	−	0.748174i	$-0.269069\pi$
$283$	375.542	1.32701	0.663503	+	0.748174i	$0.269069\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 60.2529i	− 0.209940i
$288$	0	0
$289$	189.235	0.654793
$290$	0	0
$291$	0	0
$292$	0	0
$293$	308.153i	1.05172i	0.850573	+	0.525858i	$0.176256\pi$
$293$	308.153i	1.05172i	−0.850573	+	0.525858i	$0.823744\pi$
$294$	0	0
$295$	146.745	0.497441
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 161.220i	− 0.539198i
$300$	0	0
$301$	31.0850	0.103272
$302$	0	0
$303$	0	0
$304$	0	0
$305$	580.234i	1.90241i
$306$	0	0
$307$	400.029	1.30303	0.651513	−	0.758638i	$-0.274135\pi$
$307$	400.029	1.30303	0.651513	+	0.758638i	$0.274135\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	304.938i	0.980507i	0.871580	+	0.490254i	$0.163096\pi$
$311$	304.938i	0.980507i	−0.871580	+	0.490254i	$0.836904\pi$
$312$	0	0
$313$	88.7451	0.283531	0.141765	−	0.989900i	$-0.454722\pi$
$313$	88.7451	0.283531	0.141765	+	0.989900i	$0.454722\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 539.159i	− 1.70082i	−0.526122	−	0.850409i	$-0.676355\pi$
$317$	− 539.159i	− 1.70082i	0.526122	−	0.850409i	$-0.323645\pi$
$318$	0	0
$319$	582.324	1.82547
$320$	0	0
$321$	0	0
$322$	0	0
$323$	324.193i	1.00369i
$324$	0	0
$325$	−26.2510	−0.0807723
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 104.297i	− 0.317012i
$330$	0	0
$331$	420.988	1.27187	0.635934	−	0.771744i	$-0.280615\pi$
$331$	420.988	1.27187	0.635934	+	0.771744i	$0.280615\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	351.017i	1.04781i
$336$	0	0
$337$	441.077	1.30883	0.654417	−	0.756134i	$-0.272914\pi$
$337$	441.077	1.30883	0.654417	+	0.756134i	$0.272914\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 476.052i	− 1.39605i
$342$	0	0
$343$	18.5203	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 152.678i	− 0.439994i	−0.975501	−	0.219997i	$-0.929395\pi$
$347$	− 152.678i	− 0.439994i	0.975501	−	0.219997i	$-0.0706047\pi$
$348$	0	0
$349$	36.4654	0.104485	0.0522427	−	0.998634i	$-0.483363\pi$
$349$	36.4654	0.104485	0.0522427	+	0.998634i	$0.483363\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	255.758i	0.724527i	0.932076	+	0.362264i	$0.117996\pi$
$353$	255.758i	0.724527i	−0.932076	+	0.362264i	$0.882004\pi$
$354$	0	0
$355$	−356.369	−1.00386
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 110.251i	− 0.307107i	−0.988140	−	0.153553i	$-0.950928\pi$
$359$	− 110.251i	− 0.307107i	0.988140	−	0.153553i	$-0.0490717\pi$
$360$	0	0
$361$	692.490	1.91826
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 211.600i	− 0.579725i
$366$	0	0
$367$	92.5751	0.252248	0.126124	−	0.992014i	$-0.459746\pi$
$367$	92.5751	0.252248	0.126124	+	0.992014i	$0.459746\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 130.019i	− 0.350455i
$372$	0	0
$373$	−390.988	−1.04823	−0.524113	−	0.851649i	$-0.675603\pi$
$373$	−390.988	−1.04823	−0.524113	+	0.851649i	$0.675603\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	663.490i	1.75992i
$378$	0	0
$379$	−387.158	−1.02153	−0.510763	−	0.859722i	$-0.670637\pi$
$379$	−387.158	−1.02153	−0.510763	+	0.859722i	$0.670637\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	128.688i	0.336000i	0.985787	+	0.168000i	$0.0537308\pi$
$383$	128.688i	0.336000i	−0.985787	+	0.168000i	$0.946269\pi$
$384$	0	0
$385$	−198.539	−0.515685
$386$	0	0
$387$	0	0
$388$	0	0
$389$	63.1072i	0.162229i	0.996705	+	0.0811146i	$0.0258480\pi$
$389$	63.1072i	0.162229i	−0.996705	+	0.0811146i	$0.974152\pi$
$390$	0	0
$391$	−97.1058	−0.248352
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 113.429i	− 0.287162i
$396$	0	0
$397$	−105.218	−0.265034	−0.132517	−	0.991181i	$-0.542306\pi$
$397$	−105.218	−0.265034	−0.132517	+	0.991181i	$0.542306\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	486.650i	1.21359i	0.794858	+	0.606796i	$0.207545\pi$
$401$	486.650i	1.21359i	−0.794858	+	0.606796i	$0.792455\pi$
$402$	0	0
$403$	542.405	1.34592
$404$	0	0
$405$	0	0
$406$	0	0
$407$	619.769i	1.52277i
$408$	0	0
$409$	78.8706	0.192838	0.0964188	−	0.995341i	$-0.469261\pi$
$409$	78.8706	0.192838	0.0964188	+	0.995341i	$0.469261\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	75.3027i	0.182331i
$414$	0	0
$415$	−72.0000	−0.173494
$416$	0	0
$417$	0	0
$418$	0	0
$419$	55.5148i	0.132494i	0.997803	+	0.0662468i	$0.0211025\pi$
$419$	55.5148i	0.132494i	−0.997803	+	0.0662468i	$0.978898\pi$
$420$	0	0
$421$	−315.004	−0.748228	−0.374114	−	0.927383i	$-0.622053\pi$
$421$	−315.004	−0.748228	−0.374114	+	0.927383i	$0.622053\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	15.8114i	0.0372033i
$426$	0	0
$427$	−297.749	−0.697304
$428$	0	0
$429$	0	0
$430$	0	0
$431$	43.3193i	0.100509i	0.998736	+	0.0502545i	$0.0160032\pi$
$431$	43.3193i	0.100509i	−0.998736	+	0.0502545i	$0.983997\pi$
$432$	0	0
$433$	496.583	1.14684	0.573421	−	0.819261i	$-0.305616\pi$
$433$	496.583	1.14684	0.573421	+	0.819261i	$0.305616\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	315.553i	0.722088i
$438$	0	0
$439$	−229.166	−0.522018	−0.261009	−	0.965336i	$-0.584055\pi$
$439$	−229.166	−0.522018	−0.261009	+	0.965336i	$0.584055\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 841.623i	− 1.89983i	−0.312513	−	0.949914i	$-0.601171\pi$
$443$	− 841.623i	− 1.89983i	0.312513	−	0.949914i	$-0.398829\pi$
$444$	0	0
$445$	570.405	1.28181
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 114.207i	− 0.254360i	−0.991880	−	0.127180i	$-0.959407\pi$
$449$	− 114.207i	− 0.254360i	0.991880	−	0.127180i	$-0.0405925\pi$
$450$	0	0
$451$	−331.454	−0.734930
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 226.211i	− 0.497168i
$456$	0	0
$457$	807.320	1.76657	0.883283	−	0.468841i	$-0.155328\pi$
$457$	807.320	1.76657	0.883283	+	0.468841i	$0.155328\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 291.640i	− 0.632626i	−0.948655	−	0.316313i	$-0.897555\pi$
$461$	− 291.640i	− 0.632626i	0.948655	−	0.316313i	$-0.102445\pi$
$462$	0	0
$463$	−742.073	−1.60275	−0.801375	−	0.598162i	$-0.795898\pi$
$463$	−742.073	−1.60275	−0.801375	+	0.598162i	$0.795898\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	793.013i	1.69810i	0.528311	+	0.849051i	$0.322825\pi$
$467$	793.013i	1.69810i	−0.528311	+	0.849051i	$0.677175\pi$
$468$	0	0
$469$	−180.125	−0.384063
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 171.000i	− 0.361522i
$474$	0	0
$475$	51.3804	0.108169
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 464.332i	− 0.969377i	−0.874687	−	0.484689i	$-0.838933\pi$
$479$	− 464.332i	− 0.969377i	0.874687	−	0.484689i	$-0.161067\pi$
$480$	0	0
$481$	−706.154	−1.46810
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 233.517i	− 0.481479i
$486$	0	0
$487$	406.405	0.834508	0.417254	−	0.908790i	$-0.362993\pi$
$487$	406.405	0.834508	0.417254	+	0.908790i	$0.362993\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 964.414i	− 1.96418i	−0.188403	−	0.982092i	$-0.560331\pi$
$491$	− 964.414i	− 1.96418i	0.188403	−	0.982092i	$-0.439669\pi$
$492$	0	0
$493$	399.631	0.810611
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 182.872i	− 0.367951i
$498$	0	0
$499$	−127.085	−0.254679	−0.127340	−	0.991859i	$-0.540644\pi$
$499$	−127.085	−0.254679	−0.127340	+	0.991859i	$0.540644\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	911.275i	1.81168i	0.423620	+	0.905840i	$0.360759\pi$
$503$	911.275i	1.81168i	−0.423620	+	0.905840i	$0.639241\pi$
$504$	0	0
$505$	−855.660	−1.69438
$506$	0	0
$507$	0	0
$508$	0	0
$509$	275.620i	0.541493i	0.962651	+	0.270747i	$0.0872705\pi$
$509$	275.620i	0.541493i	−0.962651	+	0.270747i	$0.912729\pi$
$510$	0	0
$511$	108.583	0.212491
$512$	0	0
$513$	0	0
$514$	0	0
$515$	485.299i	0.942328i
$516$	0	0
$517$	−573.741	−1.10975
$518$	0	0
$519$	0	0
$520$	0	0
$521$	240.688i	0.461973i	0.972957	+	0.230987i	$0.0741954\pi$
$521$	240.688i	0.461973i	−0.972957	+	0.230987i	$0.925805\pi$
$522$	0	0
$523$	522.154	0.998383	0.499191	−	0.866492i	$-0.333630\pi$
$523$	522.154	0.998383	0.499191	+	0.866492i	$0.333630\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 326.700i	− 0.619924i
$528$	0	0
$529$	434.482	0.821328
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 377.652i	− 0.708541i
$534$	0	0
$535$	−481.535	−0.900065
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 101.881i	− 0.189018i
$540$	0	0
$541$	−522.988	−0.966706	−0.483353	−	0.875425i	$-0.660581\pi$
$541$	−522.988	−0.966706	−0.483353	+	0.875425i	$0.660581\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	445.117i	0.816728i
$546$	0	0
$547$	929.158	1.69864	0.849322	−	0.527875i	$-0.177011\pi$
$547$	929.158	1.69864	0.849322	+	0.527875i	$0.177011\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 1298.63i	− 2.35686i
$552$	0	0
$553$	58.2065	0.105256
$554$	0	0
$555$	0	0
$556$	0	0
$557$	454.913i	0.816720i	0.912821	+	0.408360i	$0.133899\pi$
$557$	454.913i	0.816720i	−0.912821	+	0.408360i	$0.866101\pi$
$558$	0	0
$559$	194.834	0.348540
$560$	0	0
$561$	0	0
$562$	0	0
$563$	881.330i	1.56542i	0.622388	+	0.782709i	$0.286163\pi$
$563$	881.330i	1.56542i	−0.622388	+	0.782709i	$0.713837\pi$
$564$	0	0
$565$	−622.199	−1.10124
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 250.963i	− 0.441059i	−0.975380	−	0.220530i	$-0.929221\pi$
$569$	− 250.963i	− 0.441059i	0.975380	−	0.220530i	$-0.0707786\pi$
$570$	0	0
$571$	−664.478	−1.16371	−0.581855	−	0.813293i	$-0.697673\pi$
$571$	−664.478	−1.16371	−0.581855	+	0.813293i	$0.697673\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	15.3900i	0.0267652i
$576$	0	0
$577$	90.0889	0.156133	0.0780666	−	0.996948i	$-0.475125\pi$
$577$	90.0889	0.156133	0.0780666	+	0.996948i	$0.475125\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 36.9470i	− 0.0635921i
$582$	0	0
$583$	−715.239	−1.22683
$584$	0	0
$585$	0	0
$586$	0	0
$587$	638.108i	1.08707i	0.839388	+	0.543533i	$0.182914\pi$
$587$	638.108i	1.08707i	−0.839388	+	0.543533i	$0.817086\pi$
$588$	0	0
$589$	−1061.64	−1.80244
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 854.257i	− 1.44057i	−0.693679	−	0.720284i	$-0.744011\pi$
$593$	− 854.257i	− 1.44057i	0.693679	−	0.720284i	$-0.255989\pi$
$594$	0	0
$595$	−136.251	−0.228993
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 705.138i	− 1.17719i	−0.808427	−	0.588596i	$-0.799681\pi$
$599$	− 705.138i	− 1.17719i	0.808427	−	0.588596i	$-0.200319\pi$
$600$	0	0
$601$	1145.39	1.90581	0.952906	−	0.303265i	$-0.0980767\pi$
$601$	1145.39	1.90581	0.952906	+	0.303265i	$0.0980767\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	468.308i	0.774063i
$606$	0	0
$607$	555.247	0.914740	0.457370	−	0.889277i	$-0.348792\pi$
$607$	555.247	0.914740	0.457370	+	0.889277i	$0.348792\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 653.710i	− 1.06990i
$612$	0	0
$613$	59.0928	0.0963994	0.0481997	−	0.998838i	$-0.484652\pi$
$613$	59.0928	0.0963994	0.0481997	+	0.998838i	$0.484652\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 761.374i	− 1.23399i	−0.786966	−	0.616997i	$-0.788349\pi$
$617$	− 761.374i	− 1.23399i	0.786966	−	0.616997i	$-0.211651\pi$
$618$	0	0
$619$	−717.563	−1.15923	−0.579615	−	0.814890i	$-0.696797\pi$
$619$	−717.563	−1.15923	−0.579615	+	0.814890i	$0.696797\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	292.705i	0.469832i
$624$	0	0
$625$	−662.069	−1.05931
$626$	0	0
$627$	0	0
$628$	0	0
$629$	425.329i	0.676199i
$630$	0	0
$631$	417.733	0.662018	0.331009	−	0.943628i	$-0.392611\pi$
$631$	417.733	0.662018	0.331009	+	0.943628i	$0.392611\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	404.288i	0.636673i
$636$	0	0
$637$	116.081	0.182231
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 676.144i	− 1.05483i	−0.849609	−	0.527413i	$-0.823162\pi$
$641$	− 676.144i	− 1.05483i	0.849609	−	0.527413i	$-0.176838\pi$
$642$	0	0
$643$	−243.616	−0.378873	−0.189437	−	0.981893i	$-0.560666\pi$
$643$	−243.616	−0.378873	−0.189437	+	0.981893i	$0.560666\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 657.896i	− 1.01684i	−0.861109	−	0.508420i	$-0.830230\pi$
$647$	− 657.896i	− 1.01684i	0.861109	−	0.508420i	$-0.169770\pi$
$648$	0	0
$649$	414.243	0.638279
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 969.246i	− 1.48430i	−0.670235	−	0.742149i	$-0.733807\pi$
$653$	− 969.246i	− 1.48430i	0.670235	−	0.742149i	$-0.266193\pi$
$654$	0	0
$655$	−681.150	−1.03992
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 735.574i	− 1.11620i	−0.829775	−	0.558099i	$-0.811531\pi$
$659$	− 735.574i	− 1.11620i	0.829775	−	0.558099i	$-0.188469\pi$
$660$	0	0
$661$	−818.280	−1.23794	−0.618971	−	0.785414i	$-0.712450\pi$
$661$	−818.280	−1.23794	−0.618971	+	0.785414i	$0.712450\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	442.758i	0.665801i
$666$	0	0
$667$	388.980	0.583179
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1637.93i	2.44103i
$672$	0	0
$673$	−314.170	−0.466820	−0.233410	−	0.972378i	$-0.574988\pi$
$673$	−314.170	−0.466820	−0.233410	+	0.972378i	$0.574988\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1292.67i	− 1.90942i	−0.297545	−	0.954708i	$-0.596168\pi$
$677$	− 1292.67i	− 1.90942i	0.297545	−	0.954708i	$-0.403832\pi$
$678$	0	0
$679$	119.830	0.176480
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 363.974i	− 0.532905i	−0.963848	−	0.266453i	$-0.914148\pi$
$683$	− 363.974i	− 0.532905i	0.963848	−	0.266453i	$-0.0858516\pi$
$684$	0	0
$685$	569.624	0.831567
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 814.931i	− 1.18277i
$690$	0	0
$691$	−302.243	−0.437400	−0.218700	−	0.975792i	$-0.570182\pi$
$691$	−302.243	−0.437400	−0.218700	+	0.975792i	$0.570182\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 798.304i	− 1.14864i
$696$	0	0
$697$	−227.467	−0.326351
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 720.700i	− 1.02810i	−0.857759	−	0.514051i	$-0.828144\pi$
$701$	− 720.700i	− 1.02810i	0.857759	−	0.514051i	$-0.171856\pi$
$702$	0	0
$703$	1382.14	1.96606
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 439.085i	− 0.621053i
$708$	0	0
$709$	214.154	0.302051	0.151026	−	0.988530i	$-0.451742\pi$
$709$	214.154	0.302051	0.151026	+	0.988530i	$0.451742\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 317.993i	− 0.445993i
$714$	0	0
$715$	−1244.40	−1.74042
$716$	0	0
$717$	0	0
$718$	0	0
$719$	142.464i	0.198142i	0.995080	+	0.0990709i	$0.0315871\pi$
$719$	142.464i	0.198142i	−0.995080	+	0.0990709i	$0.968413\pi$
$720$	0	0
$721$	−249.033	−0.345399
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 63.3364i	− 0.0873605i
$726$	0	0
$727$	601.875	0.827888	0.413944	−	0.910302i	$-0.364151\pi$
$727$	601.875	0.827888	0.413944	+	0.910302i	$0.364151\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 117.352i	− 0.160536i
$732$	0	0
$733$	269.660	0.367886	0.183943	−	0.982937i	$-0.441114\pi$
$733$	269.660	0.367886	0.183943	+	0.982937i	$0.441114\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	990.877i	1.34447i
$738$	0	0
$739$	457.069	0.618497	0.309248	−	0.950981i	$-0.399923\pi$
$739$	457.069	0.618497	0.309248	+	0.950981i	$0.399923\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 639.385i	− 0.860545i	−0.902699	−	0.430273i	$-0.858417\pi$
$743$	− 639.385i	− 0.860545i	0.902699	−	0.430273i	$-0.141583\pi$
$744$	0	0
$745$	397.373	0.533386
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 247.101i	− 0.329908i
$750$	0	0
$751$	−798.818	−1.06367	−0.531836	−	0.846847i	$-0.678498\pi$
$751$	−798.818	−1.06367	−0.531836	+	0.846847i	$0.678498\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 351.893i	− 0.466084i
$756$	0	0
$757$	1373.47	1.81437	0.907183	−	0.420737i	$-0.138229\pi$
$757$	1373.47	1.81437	0.907183	+	0.420737i	$0.138229\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 334.902i	− 0.440082i	−0.975491	−	0.220041i	$-0.929381\pi$
$761$	− 334.902i	− 0.440082i	0.975491	−	0.220041i	$-0.0706191\pi$
$762$	0	0
$763$	−228.413	−0.299362
$764$	0	0
$765$	0	0
$766$	0	0
$767$	471.981i	0.615360i
$768$	0	0
$769$	823.004	1.07023	0.535113	−	0.844780i	$-0.320269\pi$
$769$	823.004	1.07023	0.535113	+	0.844780i	$0.320269\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 452.517i	− 0.585403i	−0.956204	−	0.292702i	$-0.905446\pi$
$773$	− 452.517i	− 0.585403i	0.956204	−	0.292702i	$-0.0945543\pi$
$774$	0	0
$775$	−51.7777	−0.0668100
$776$	0	0
$777$	0	0
$778$	0	0
$779$	739.170i	0.948870i
$780$	0	0
$781$	−1005.98	−1.28807
$782$	0	0
$783$	0	0
$784$	0	0
$785$	292.800i	0.372993i
$786$	0	0
$787$	317.182	0.403026	0.201513	−	0.979486i	$-0.435414\pi$
$787$	317.182	0.403026	0.201513	+	0.979486i	$0.435414\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 319.283i	− 0.403645i
$792$	0	0
$793$	−1866.23	−2.35338
$794$	0	0
$795$	0	0
$796$	0	0
$797$	944.475i	1.18504i	0.805557	+	0.592519i	$0.201866\pi$
$797$	944.475i	1.18504i	−0.805557	+	0.592519i	$0.798134\pi$
$798$	0	0
$799$	−393.741	−0.492792
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 597.319i	− 0.743860i
$804$	0	0
$805$	−132.620	−0.164745
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1432.21i	1.77035i	0.465262	+	0.885173i	$0.345960\pi$
$809$	1432.21i	1.77035i	−0.465262	+	0.885173i	$0.654040\pi$
$810$	0	0
$811$	296.251	0.365291	0.182645	−	0.983179i	$-0.441534\pi$
$811$	296.251	0.365291	0.182645	+	0.983179i	$0.441534\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 223.852i	− 0.274666i
$816$	0	0
$817$	−381.344	−0.466761
$818$	0	0
$819$	0	0
$820$	0	0
$821$	372.042i	0.453157i	0.973993	+	0.226578i	$0.0727539\pi$
$821$	372.042i	0.453157i	−0.973993	+	0.226578i	$0.927246\pi$
$822$	0	0
$823$	−800.138	−0.972222	−0.486111	−	0.873897i	$-0.661585\pi$
$823$	−800.138	−0.972222	−0.486111	+	0.873897i	$0.661585\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 902.085i	− 1.09079i	−0.838178	−	0.545396i	$-0.816379\pi$
$827$	− 902.085i	− 1.09079i	0.838178	−	0.545396i	$-0.183621\pi$
$828$	0	0
$829$	331.976	0.400454	0.200227	−	0.979750i	$-0.435832\pi$
$829$	331.976	0.400454	0.200227	+	0.979750i	$0.435832\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 69.9176i	− 0.0839347i
$834$	0	0
$835$	−34.9229	−0.0418238
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1234.10i	− 1.47092i	−0.677569	−	0.735459i	$-0.736967\pi$
$839$	− 1234.10i	− 1.47092i	0.677569	−	0.735459i	$-0.263033\pi$
$840$	0	0
$841$	−759.818	−0.903470
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 546.502i	− 0.646748i
$846$	0	0
$847$	−240.314	−0.283723
$848$	0	0
$849$	0	0
$850$	0	0
$851$	413.993i	0.486478i
$852$	0	0
$853$	−159.624	−0.187132	−0.0935660	−	0.995613i	$-0.529827\pi$
$853$	−159.624	−0.187132	−0.0935660	+	0.995613i	$0.529827\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	455.867i	0.531933i	0.963982	+	0.265967i	$0.0856911\pi$
$857$	455.867i	0.531933i	−0.963982	+	0.265967i	$0.914309\pi$
$858$	0	0
$859$	−511.114	−0.595010	−0.297505	−	0.954720i	$-0.596155\pi$
$859$	−511.114	−0.595010	−0.297505	+	0.954720i	$0.596155\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 331.098i	− 0.383659i	−0.981428	−	0.191830i	$-0.938558\pi$
$863$	− 331.098i	− 0.383659i	0.981428	−	0.191830i	$-0.0614421\pi$
$864$	0	0
$865$	204.324	0.236213
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 320.196i	− 0.368465i
$870$	0	0
$871$	−1128.99	−1.29620
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 319.435i	− 0.365068i
$876$	0	0
$877$	207.563	0.236674	0.118337	−	0.992973i	$-0.462244\pi$
$877$	207.563	0.236674	0.118337	+	0.992973i	$0.462244\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 363.176i	− 0.412231i	−0.978528	−	0.206116i	$-0.933918\pi$
$881$	− 363.176i	− 0.412231i	0.978528	−	0.206116i	$-0.0660823\pi$
$882$	0	0
$883$	173.725	0.196744	0.0983722	−	0.995150i	$-0.468636\pi$
$883$	173.725	0.196744	0.0983722	+	0.995150i	$0.468636\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 562.772i	− 0.634467i	−0.948348	−	0.317233i	$-0.897246\pi$
$887$	− 562.772i	− 0.634467i	0.948348	−	0.317233i	$-0.102754\pi$
$888$	0	0
$889$	−207.461	−0.233365
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1279.49i	1.43280i
$894$	0	0
$895$	443.276	0.495280
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1308.67i	1.45570i
$900$	0	0
$901$	−490.847	−0.544780
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1752.12i	− 1.93604i
$906$	0	0
$907$	−91.2470	−0.100603	−0.0503016	−	0.998734i	$-0.516018\pi$
$907$	−91.2470	−0.100603	−0.0503016	+	0.998734i	$0.516018\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1421.78i	1.56068i	0.625353	+	0.780342i	$0.284955\pi$
$911$	1421.78i	1.56068i	−0.625353	+	0.780342i	$0.715045\pi$
$912$	0	0
$913$	−203.247	−0.222615
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 349.534i	− 0.381172i
$918$	0	0
$919$	−1406.73	−1.53072	−0.765359	−	0.643604i	$-0.777438\pi$
$919$	−1406.73	−1.53072	−0.765359	+	0.643604i	$0.777438\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1146.20i	− 1.24182i
$924$	0	0
$925$	67.4091	0.0728747
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1476.46i	1.58930i	0.607068	+	0.794650i	$0.292345\pi$
$929$	1476.46i	1.58930i	−0.607068	+	0.794650i	$0.707655\pi$
$930$	0	0
$931$	−227.203	−0.244041
$932$	0	0
$933$	0	0
$934$	0	0
$935$	749.522i	0.801628i
$936$	0	0
$937$	−742.324	−0.792235	−0.396117	−	0.918200i	$-0.629643\pi$
$937$	−742.324	−0.792235	−0.396117	+	0.918200i	$0.629643\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	120.944i	0.128527i	0.997933	+	0.0642635i	$0.0204698\pi$
$941$	120.944i	0.128527i	−0.997933	+	0.0642635i	$0.979530\pi$
$942$	0	0
$943$	−221.404	−0.234787
$944$	0	0
$945$	0	0
$946$	0	0
$947$	675.308i	0.713103i	0.934276	+	0.356551i	$0.116047\pi$
$947$	675.308i	0.713103i	−0.934276	+	0.356551i	$0.883953\pi$
$948$	0	0
$949$	680.575	0.717150
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 361.197i	− 0.379011i	−0.981880	−	0.189505i	$-0.939312\pi$
$953$	− 361.197i	− 0.379011i	0.981880	−	0.189505i	$-0.0606885\pi$
$954$	0	0
$955$	1069.70	1.12010
$956$	0	0
$957$	0	0
$958$	0	0
$959$	292.304i	0.304801i
$960$	0	0
$961$	108.846	0.113263
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1152.33i	1.19412i
$966$	0	0
$967$	579.433	0.599207	0.299603	−	0.954064i	$-0.403146\pi$
$967$	579.433	0.599207	0.299603	+	0.954064i	$0.403146\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 230.700i	− 0.237590i	−0.992919	−	0.118795i	$-0.962097\pi$
$971$	− 230.700i	− 0.237590i	0.992919	−	0.118795i	$-0.0379031\pi$
$972$	0	0
$973$	409.652	0.421020
$974$	0	0
$975$	0	0
$976$	0	0
$977$	384.483i	0.393534i	0.980450	+	0.196767i	$0.0630443\pi$
$977$	384.483i	0.393534i	−0.980450	+	0.196767i	$0.936956\pi$
$978$	0	0
$979$	1610.18	1.64472
$980$	0	0
$981$	0	0
$982$	0	0
$983$	19.5992i	0.0199382i	0.999950	+	0.00996910i	$0.00317331\pi$
$983$	19.5992i	0.0199382i	−0.999950	+	0.00996910i	$0.996827\pi$
$984$	0	0
$985$	−897.446	−0.911112
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 114.224i	− 0.115495i
$990$	0	0
$991$	672.235	0.678340	0.339170	−	0.940725i	$-0.389854\pi$
$991$	672.235	0.678340	0.339170	+	0.940725i	$0.389854\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 219.552i	− 0.220656i
$996$	0	0
$997$	1181.20	1.18476	0.592378	−	0.805660i	$-0.298189\pi$
$997$	1181.20	1.18476	0.592378	+	0.805660i	$0.298189\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.d.d.449.1		4
3.2	odd	2	inner	1008.3.d.d.449.4		4
4.3	odd	2		63.3.b.a.8.1	✓	4
8.3	odd	2		4032.3.d.b.449.4		4
8.5	even	2		4032.3.d.c.449.4		4
12.11	even	2		63.3.b.a.8.4	yes	4
20.3	even	4		1575.3.f.a.449.2		8
20.7	even	4		1575.3.f.a.449.7		8
20.19	odd	2		1575.3.c.a.701.4		4
24.5	odd	2		4032.3.d.c.449.1		4
24.11	even	2		4032.3.d.b.449.1		4
28.3	even	6		441.3.q.a.422.4		8
28.11	odd	6		441.3.q.b.422.4		8
28.19	even	6		441.3.q.a.116.1		8
28.23	odd	6		441.3.q.b.116.1		8
28.27	even	2		441.3.b.b.197.1		4
36.7	odd	6		567.3.r.a.134.1		8
36.11	even	6		567.3.r.a.134.4		8
36.23	even	6		567.3.r.a.512.1		8
36.31	odd	6		567.3.r.a.512.4		8
60.23	odd	4		1575.3.f.a.449.8		8
60.47	odd	4		1575.3.f.a.449.1		8
60.59	even	2		1575.3.c.a.701.1		4
84.11	even	6		441.3.q.b.422.1		8
84.23	even	6		441.3.q.b.116.4		8
84.47	odd	6		441.3.q.a.116.4		8
84.59	odd	6		441.3.q.a.422.1		8
84.83	odd	2		441.3.b.b.197.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.3.b.a.8.1	✓	4	4.3	odd	2
63.3.b.a.8.4	yes	4	12.11	even	2
441.3.b.b.197.1		4	28.27	even	2
441.3.b.b.197.4		4	84.83	odd	2
441.3.q.a.116.1		8	28.19	even	6
441.3.q.a.116.4		8	84.47	odd	6
441.3.q.a.422.1		8	84.59	odd	6
441.3.q.a.422.4		8	28.3	even	6
441.3.q.b.116.1		8	28.23	odd	6
441.3.q.b.116.4		8	84.23	even	6
441.3.q.b.422.1		8	84.11	even	6
441.3.q.b.422.4		8	28.11	odd	6
567.3.r.a.134.1		8	36.7	odd	6
567.3.r.a.134.4		8	36.11	even	6
567.3.r.a.512.1		8	36.23	even	6
567.3.r.a.512.4		8	36.31	odd	6
1008.3.d.d.449.1		4	1.1	even	1	trivial
1008.3.d.d.449.4		4	3.2	odd	2	inner
1575.3.c.a.701.1		4	60.59	even	2
1575.3.c.a.701.4		4	20.19	odd	2
1575.3.f.a.449.1		8	60.47	odd	4
1575.3.f.a.449.2		8	20.3	even	4
1575.3.f.a.449.7		8	20.7	even	4
1575.3.f.a.449.8		8	60.23	odd	4
4032.3.d.b.449.1		4	24.11	even	2
4032.3.d.b.449.4		4	8.3	odd	2
4032.3.d.c.449.1		4	24.5	odd	2
4032.3.d.c.449.4		4	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.15587i q^{5} +2.64575 q^{7} +O(q^{10})\)	\(q-5.15587i q^{5} +2.64575 q^{7} -14.5544i q^{11} +16.5830 q^{13} -9.98823i q^{17} -32.4575 q^{19} -9.72202i q^{23} -1.58301 q^{25} +40.0102i q^{29} +32.7085 q^{31} -13.6412i q^{35} -42.5830 q^{37} -22.7735i q^{41} +11.7490 q^{43} -39.4205i q^{47} +7.00000 q^{49} -49.1425i q^{53} -75.0405 q^{55} +28.4617i q^{59} -112.539 q^{61} -85.4998i q^{65} -68.0810 q^{67} -69.1190i q^{71} +41.0405 q^{73} -38.5073i q^{77} +22.0000 q^{79} -13.9647i q^{83} -51.4980 q^{85} +110.632i q^{89} +43.8745 q^{91} +167.347i q^{95} +45.2915 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 24 q^{13} - 24 q^{19} + 36 q^{25} + 152 q^{31} - 128 q^{37} - 80 q^{43} + 28 q^{49} - 152 q^{55} - 48 q^{61} + 24 q^{67} + 16 q^{73} + 88 q^{79} + 48 q^{85} + 112 q^{91} + 160 q^{97}+O(q^{100}) \) 4 * q + 24 * q^13 - 24 * q^19 + 36 * q^25 + 152 * q^31 - 128 * q^37 - 80 * q^43 + 28 * q^49 - 152 * q^55 - 48 * q^61 + 24 * q^67 + 16 * q^73 + 88 * q^79 + 48 * q^85 + 112 * q^91 + 160 * q^97

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