LMFDB - Embedded newform 1728.3.g.n.703.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(703,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1728.703");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.g (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:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.22581504.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 $ x^8 - 4x^7 + 5x^6 + 2x^5 - 11x^4 + 4x^3 + 20x^2 - 32*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 864)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.2
Root			$1.20036 - 0.747754i$ of defining polynomial
Character	$\chi$	$=$	1728.703
Dual form			1728.3.g.n.703.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.18059 q^{5} +2.58429i q^{7} +O(q^{10})$	$q-4.18059 q^{5} +2.58429i q^{7} -6.38876i q^{11} -11.5849 q^{13} +7.58491 q^{17} -0.0311239i q^{19} +7.52976i q^{23} -7.52266 q^{25} +13.5609 q^{29} +29.7852i q^{31} -10.8038i q^{35} -57.4290 q^{37} +27.1757 q^{41} -77.8454i q^{43} -43.8370i q^{47} +42.3215 q^{49} +87.7199 q^{53} +26.7088i q^{55} +67.8299i q^{59} -31.2943 q^{61} +48.4318 q^{65} -23.6302i q^{67} +50.8625i q^{71} +70.9868 q^{73} +16.5104 q^{77} +67.3835i q^{79} -8.55335i q^{83} -31.7094 q^{85} -5.22274 q^{89} -29.9387i q^{91} +0.130116i q^{95} +120.354 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5}+O(q^{10}) $ 8 * q + 8 * q^5	$ 8 q + 8 q^{5} + 16 q^{13} - 48 q^{17} + 48 q^{25} - 32 q^{29} + 96 q^{37} + 128 q^{41} + 168 q^{53} - 32 q^{61} + 112 q^{65} + 24 q^{73} - 440 q^{77} - 144 q^{85} - 624 q^{89} - 136 q^{97}+O(q^{100}) $ 8 * q + 8 * q^5 + 16 * q^13 - 48 * q^17 + 48 * q^25 - 32 * q^29 + 96 * q^37 + 128 * q^41 + 168 * q^53 - 32 * q^61 + 112 * q^65 + 24 * q^73 - 440 * q^77 - 144 * q^85 - 624 * q^89 - 136 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	−4.18059	−0.836118	−0.418059	−	0.908420i	$-0.637289\pi$
$5$	−4.18059	−0.836118	−0.418059	+	0.908420i	$0.637289\pi$
$6$	0	0
$7$	2.58429i	0.369184i	0.982815	+	0.184592i	$0.0590964\pi$
$7$	2.58429i	0.369184i	−0.982815	+	0.184592i	$0.940904\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 6.38876i	− 0.580796i	−0.956906	−	0.290398i	$-0.906212\pi$
$11$	− 6.38876i	− 0.580796i	0.956906	−	0.290398i	$-0.0937877\pi$
$12$	0	0
$13$	−11.5849	−0.891147	−0.445573	−	0.895245i	$-0.647000\pi$
$13$	−11.5849	−0.891147	−0.445573	+	0.895245i	$0.647000\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.58491	0.446171	0.223086	−	0.974799i	$-0.428387\pi$
$17$	7.58491	0.446171	0.223086	+	0.974799i	$0.428387\pi$
$18$	0	0
$19$	− 0.0311239i	− 0.00163810i	−1.00000		0.000819049i	$-0.999739\pi$
$19$	− 0.0311239i	− 0.00163810i	1.00000		0.000819049i	$-0.000260711\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.52976i	0.327381i	0.986512	+	0.163690i	$0.0523398\pi$
$23$	7.52976i	0.327381i	−0.986512	+	0.163690i	$0.947660\pi$
$24$	0	0
$25$	−7.52266	−0.300907
$26$	0	0
$27$	0	0
$28$	0	0
$29$	13.5609	0.467617	0.233808	−	0.972283i	$-0.424881\pi$
$29$	13.5609	0.467617	0.233808	+	0.972283i	$0.424881\pi$
$30$	0	0
$31$	29.7852i	0.960814i	0.877046	+	0.480407i	$0.159511\pi$
$31$	29.7852i	0.960814i	−0.877046	+	0.480407i	$0.840489\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 10.8038i	− 0.308681i
$36$	0	0
$37$	−57.4290	−1.55214	−0.776068	−	0.630649i	$-0.782789\pi$
$37$	−57.4290	−1.55214	−0.776068	+	0.630649i	$0.782789\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	27.1757	0.662821	0.331411	−	0.943487i	$-0.392475\pi$
$41$	27.1757	0.662821	0.331411	+	0.943487i	$0.392475\pi$
$42$	0	0
$43$	− 77.8454i	− 1.81036i	−0.425031	−	0.905179i	$-0.639737\pi$
$43$	− 77.8454i	− 1.81036i	0.425031	−	0.905179i	$-0.360263\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 43.8370i	− 0.932703i	−0.884600	−	0.466351i	$-0.845568\pi$
$47$	− 43.8370i	− 0.932703i	0.884600	−	0.466351i	$-0.154432\pi$
$48$	0	0
$49$	42.3215	0.863703
$50$	0	0
$51$	0	0
$52$	0	0
$53$	87.7199	1.65509	0.827547	−	0.561397i	$-0.189736\pi$
$53$	87.7199	1.65509	0.827547	+	0.561397i	$0.189736\pi$
$54$	0	0
$55$	26.7088i	0.485614i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	67.8299i	1.14966i	0.818273	+	0.574830i	$0.194932\pi$
$59$	67.8299i	1.14966i	−0.818273	+	0.574830i	$0.805068\pi$
$60$	0	0
$61$	−31.2943	−0.513022	−0.256511	−	0.966541i	$-0.582573\pi$
$61$	−31.2943	−0.513022	−0.256511	+	0.966541i	$0.582573\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	48.4318	0.745104
$66$	0	0
$67$	− 23.6302i	− 0.352690i	−0.984328	−	0.176345i	$-0.943573\pi$
$67$	− 23.6302i	− 0.352690i	0.984328	−	0.176345i	$-0.0564274\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	50.8625i	0.716374i	0.933650	+	0.358187i	$0.116605\pi$
$71$	50.8625i	0.716374i	−0.933650	+	0.358187i	$0.883395\pi$
$72$	0	0
$73$	70.9868	0.972422	0.486211	−	0.873841i	$-0.338379\pi$
$73$	70.9868	0.972422	0.486211	+	0.873841i	$0.338379\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	16.5104	0.214421
$78$	0	0
$79$	67.3835i	0.852955i	0.904498	+	0.426478i	$0.140246\pi$
$79$	67.3835i	0.852955i	−0.904498	+	0.426478i	$0.859754\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 8.55335i	− 0.103052i	−0.998672	−	0.0515262i	$-0.983591\pi$
$83$	− 8.55335i	− 0.103052i	0.998672	−	0.0515262i	$-0.0164086\pi$
$84$	0	0
$85$	−31.7094	−0.373052
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.22274	−0.0586825	−0.0293412	−	0.999569i	$-0.509341\pi$
$89$	−5.22274	−0.0586825	−0.0293412	+	0.999569i	$0.509341\pi$
$90$	0	0
$91$	− 29.9387i	− 0.328997i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.130116i	0.00136964i
$96$	0	0
$97$	120.354	1.24076	0.620379	−	0.784302i	$-0.286979\pi$
$97$	120.354	1.24076	0.620379	+	0.784302i	$0.286979\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	47.1221	0.466555	0.233278	−	0.972410i	$-0.425055\pi$
$101$	47.1221	0.466555	0.233278	+	0.972410i	$0.425055\pi$
$102$	0	0
$103$	30.7664i	0.298703i	0.988784	+	0.149352i	$0.0477186\pi$
$103$	30.7664i	0.298703i	−0.988784	+	0.149352i	$0.952281\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 149.818i	− 1.40017i	−0.714061	−	0.700083i	$-0.753146\pi$
$107$	− 149.818i	− 1.40017i	0.714061	−	0.700083i	$-0.246854\pi$
$108$	0	0
$109$	−12.0441	−0.110496	−0.0552480	−	0.998473i	$-0.517595\pi$
$109$	−12.0441	−0.110496	−0.0552480	+	0.998473i	$0.517595\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	212.750	1.88274	0.941371	−	0.337373i	$-0.109538\pi$
$113$	212.750	1.88274	0.941371	+	0.337373i	$0.109538\pi$
$114$	0	0
$115$	− 31.4788i	− 0.273729i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	19.6016i	0.164719i
$120$	0	0
$121$	80.1838	0.662676
$122$	0	0
$123$	0	0
$124$	0	0
$125$	135.964	1.08771
$126$	0	0
$127$	192.982i	1.51954i	0.650190	+	0.759771i	$0.274689\pi$
$127$	192.982i	1.51954i	−0.650190	+	0.759771i	$0.725311\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	189.604i	1.44736i	0.690135	+	0.723680i	$0.257551\pi$
$131$	189.604i	1.44736i	−0.690135	+	0.723680i	$0.742449\pi$
$132$	0	0
$133$	0.0804330	0.000604760 0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	196.740	1.43606	0.718031	−	0.696011i	$-0.245044\pi$
$137$	196.740	1.43606	0.718031	+	0.696011i	$0.245044\pi$
$138$	0	0
$139$	− 104.557i	− 0.752211i	−0.926577	−	0.376105i	$-0.877263\pi$
$139$	− 104.557i	− 0.752211i	0.926577	−	0.376105i	$-0.122737\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	74.0132i	0.517575i
$144$	0	0
$145$	−56.6925	−0.390983
$146$	0	0
$147$	0	0
$148$	0	0
$149$	51.2196	0.343755	0.171878	−	0.985118i	$-0.445017\pi$
$149$	51.2196	0.343755	0.171878	+	0.985118i	$0.445017\pi$
$150$	0	0
$151$	− 226.393i	− 1.49929i	−0.661839	−	0.749646i	$-0.730224\pi$
$151$	− 226.393i	− 1.49929i	0.661839	−	0.749646i	$-0.269776\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 124.520i	− 0.803354i
$156$	0	0
$157$	43.1698	0.274967	0.137484	−	0.990504i	$-0.456099\pi$
$157$	43.1698	0.274967	0.137484	+	0.990504i	$0.456099\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−19.4591	−0.120864
$162$	0	0
$163$	137.264i	0.842113i	0.907034	+	0.421057i	$0.138341\pi$
$163$	137.264i	0.842113i	−0.907034	+	0.421057i	$0.861659\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 143.149i	− 0.857177i	−0.903500	−	0.428588i	$-0.859011\pi$
$167$	− 143.149i	− 0.857177i	0.903500	−	0.428588i	$-0.140989\pi$
$168$	0	0
$169$	−34.7898	−0.205857
$170$	0	0
$171$	0	0
$172$	0	0
$173$	188.031	1.08688	0.543442	−	0.839446i	$-0.317121\pi$
$173$	188.031	1.08688	0.543442	+	0.839446i	$0.317121\pi$
$174$	0	0
$175$	− 19.4407i	− 0.111090i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 75.8284i	− 0.423622i	−0.977311	−	0.211811i	$-0.932064\pi$
$179$	− 75.8284i	− 0.423622i	0.977311	−	0.211811i	$-0.0679362\pi$
$180$	0	0
$181$	153.743	0.849406	0.424703	−	0.905333i	$-0.360378\pi$
$181$	153.743	0.849406	0.424703	+	0.905333i	$0.360378\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	240.087	1.29777
$186$	0	0
$187$	− 48.4582i	− 0.259135i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 201.836i	− 1.05673i	−0.849017	−	0.528366i	$-0.822805\pi$
$191$	− 201.836i	− 1.05673i	0.849017	−	0.528366i	$-0.177195\pi$
$192$	0	0
$193$	−142.854	−0.740175	−0.370088	−	0.928997i	$-0.620672\pi$
$193$	−142.854	−0.740175	−0.370088	+	0.928997i	$0.620672\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	388.500	1.97208	0.986041	−	0.166504i	$-0.0532479\pi$
$197$	388.500	1.97208	0.986041	+	0.166504i	$0.0532479\pi$
$198$	0	0
$199$	19.9376i	0.100189i	0.998744	+	0.0500945i	$0.0159523\pi$
$199$	19.9376i	0.100189i	−0.998744	+	0.0500945i	$0.984048\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	35.0452i	0.172637i
$204$	0	0
$205$	−113.610	−0.554197
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.198843	−0.000951401 0
$210$	0	0
$211$	174.438i	0.826720i	0.910568	+	0.413360i	$0.135645\pi$
$211$	174.438i	0.826720i	−0.910568	+	0.413360i	$0.864355\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	325.440i	1.51367i
$216$	0	0
$217$	−76.9736	−0.354717
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−87.8705	−0.397604
$222$	0	0
$223$	− 354.898i	− 1.59147i	−0.605643	−	0.795736i	$-0.707084\pi$
$223$	− 354.898i	− 1.59147i	0.605643	−	0.795736i	$-0.292916\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	360.716i	1.58906i	0.607227	+	0.794528i	$0.292282\pi$
$227$	360.716i	1.58906i	−0.607227	+	0.794528i	$0.707718\pi$
$228$	0	0
$229$	−45.4706	−0.198562	−0.0992809	−	0.995059i	$-0.531654\pi$
$229$	−45.4706	−0.198562	−0.0992809	+	0.995059i	$0.531654\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−294.123	−1.26233	−0.631166	−	0.775648i	$-0.717423\pi$
$233$	−294.123	−1.26233	−0.631166	+	0.775648i	$0.717423\pi$
$234$	0	0
$235$	183.265i	0.779850i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 295.342i	− 1.23574i	−0.786279	−	0.617871i	$-0.787995\pi$
$239$	− 295.342i	− 1.23574i	0.786279	−	0.617871i	$-0.212005\pi$
$240$	0	0
$241$	−200.546	−0.832142	−0.416071	−	0.909332i	$-0.636593\pi$
$241$	−200.546	−0.832142	−0.416071	+	0.909332i	$0.636593\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−176.929	−0.722158
$246$	0	0
$247$	0.360567i	0.00145979i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 272.988i	− 1.08760i	−0.839214	−	0.543801i	$-0.816985\pi$
$251$	− 272.988i	− 1.08760i	0.839214	−	0.543801i	$-0.183015\pi$
$252$	0	0
$253$	48.1058	0.190141
$254$	0	0
$255$	0	0
$256$	0	0
$257$	211.739	0.823885	0.411943	−	0.911210i	$-0.364850\pi$
$257$	211.739	0.823885	0.411943	+	0.911210i	$0.364850\pi$
$258$	0	0
$259$	− 148.413i	− 0.573024i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	189.044i	0.718799i	0.933184	+	0.359400i	$0.117019\pi$
$263$	189.044i	0.718799i	−0.933184	+	0.359400i	$0.882981\pi$
$264$	0	0
$265$	−366.721	−1.38385
$266$	0	0
$267$	0	0
$268$	0	0
$269$	139.727	0.519432	0.259716	−	0.965685i	$-0.416371\pi$
$269$	139.727	0.519432	0.259716	+	0.965685i	$0.416371\pi$
$270$	0	0
$271$	4.24913i	0.0156794i	0.999969	+	0.00783972i	$0.00249549\pi$
$271$	4.24913i	0.0156794i	−0.999969	+	0.00783972i	$0.997505\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	48.0605i	0.174765i
$276$	0	0
$277$	230.546	0.832297	0.416149	−	0.909297i	$-0.363380\pi$
$277$	230.546	0.832297	0.416149	+	0.909297i	$0.363380\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	247.125	0.879449	0.439725	−	0.898133i	$-0.355076\pi$
$281$	247.125	0.879449	0.439725	+	0.898133i	$0.355076\pi$
$282$	0	0
$283$	542.670i	1.91756i	0.284149	+	0.958780i	$0.408289\pi$
$283$	542.670i	1.91756i	−0.284149	+	0.958780i	$0.591711\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	70.2297i	0.244703i
$288$	0	0
$289$	−231.469	−0.800931
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−245.662	−0.838437	−0.419218	−	0.907885i	$-0.637696\pi$
$293$	−245.662	−0.838437	−0.419218	+	0.907885i	$0.637696\pi$
$294$	0	0
$295$	− 283.569i	− 0.961251i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 87.2316i	− 0.291744i
$300$	0	0
$301$	201.175	0.668355
$302$	0	0
$303$	0	0
$304$	0	0
$305$	130.829	0.428947
$306$	0	0
$307$	481.064i	1.56698i	0.621403	+	0.783491i	$0.286563\pi$
$307$	481.064i	1.56698i	−0.621403	+	0.783491i	$0.713437\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	71.8405i	0.230998i	0.993308	+	0.115499i	$0.0368468\pi$
$311$	71.8405i	0.230998i	−0.993308	+	0.115499i	$0.963153\pi$
$312$	0	0
$313$	−152.089	−0.485907	−0.242953	−	0.970038i	$-0.578116\pi$
$313$	−152.089	−0.485907	−0.242953	+	0.970038i	$0.578116\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	183.035	0.577396	0.288698	−	0.957420i	$-0.406778\pi$
$317$	183.035	0.577396	0.288698	+	0.957420i	$0.406778\pi$
$318$	0	0
$319$	− 86.6372i	− 0.271590i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 0.236072i	0 0.000730872i
$324$	0	0
$325$	87.1494	0.268152
$326$	0	0
$327$	0	0
$328$	0	0
$329$	113.287	0.344339
$330$	0	0
$331$	− 546.449i	− 1.65090i	−0.564473	−	0.825452i	$-0.690921\pi$
$331$	− 546.449i	− 1.65090i	0.564473	−	0.825452i	$-0.309079\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	98.7883i	0.294890i
$336$	0	0
$337$	176.360	0.523324	0.261662	−	0.965160i	$-0.415729\pi$
$337$	176.360	0.523324	0.261662	+	0.965160i	$0.415729\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	190.291	0.558037
$342$	0	0
$343$	236.001i	0.688049i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 113.338i	− 0.326622i	−0.986575	−	0.163311i	$-0.947783\pi$
$347$	− 113.338i	− 0.326622i	0.986575	−	0.163311i	$-0.0522174\pi$
$348$	0	0
$349$	408.318	1.16996	0.584982	−	0.811046i	$-0.301102\pi$
$349$	408.318	1.16996	0.584982	+	0.811046i	$0.301102\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−374.570	−1.06111	−0.530553	−	0.847652i	$-0.678016\pi$
$353$	−374.570	−1.06111	−0.530553	+	0.847652i	$0.678016\pi$
$354$	0	0
$355$	− 212.635i	− 0.598973i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	336.934i	0.938534i	0.883056	+	0.469267i	$0.155482\pi$
$359$	336.934i	0.938534i	−0.883056	+	0.469267i	$0.844518\pi$
$360$	0	0
$361$	360.999	0.999997
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−296.767	−0.813060
$366$	0	0
$367$	− 430.222i	− 1.17227i	−0.810214	−	0.586134i	$-0.800649\pi$
$367$	− 430.222i	− 1.17227i	0.810214	−	0.586134i	$-0.199351\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	226.694i	0.611034i
$372$	0	0
$373$	62.2557	0.166905	0.0834527	−	0.996512i	$-0.473405\pi$
$373$	62.2557	0.166905	0.0834527	+	0.996512i	$0.473405\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−157.102	−0.416715
$378$	0	0
$379$	657.970i	1.73607i	0.496505	+	0.868034i	$0.334616\pi$
$379$	657.970i	1.73607i	−0.496505	+	0.868034i	$0.665384\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 726.694i	− 1.89737i	−0.316219	−	0.948686i	$-0.602413\pi$
$383$	− 726.694i	− 1.89737i	0.316219	−	0.948686i	$-0.397587\pi$
$384$	0	0
$385$	−69.0232	−0.179281
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−363.139	−0.933521	−0.466760	−	0.884384i	$-0.654579\pi$
$389$	−363.139	−0.933521	−0.466760	+	0.884384i	$0.654579\pi$
$390$	0	0
$391$	57.1125i	0.146068i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 281.703i	− 0.713171i
$396$	0	0
$397$	−69.9305	−0.176147	−0.0880737	−	0.996114i	$-0.528071\pi$
$397$	−69.9305	−0.176147	−0.0880737	+	0.996114i	$0.528071\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−371.449	−0.926307	−0.463153	−	0.886278i	$-0.653282\pi$
$401$	−371.449	−0.926307	−0.463153	+	0.886278i	$0.653282\pi$
$402$	0	0
$403$	− 345.059i	− 0.856226i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	366.900i	0.901475i
$408$	0	0
$409$	33.3496	0.0815394	0.0407697	−	0.999169i	$-0.487019\pi$
$409$	33.3496	0.0815394	0.0407697	+	0.999169i	$0.487019\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−175.292	−0.424436
$414$	0	0
$415$	35.7581i	0.0861640i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	221.058i	0.527585i	0.964579	+	0.263792i	$0.0849734\pi$
$419$	221.058i	0.527585i	−0.964579	+	0.263792i	$0.915027\pi$
$420$	0	0
$421$	−199.813	−0.474615	−0.237307	−	0.971435i	$-0.576265\pi$
$421$	−199.813	−0.474615	−0.237307	+	0.971435i	$0.576265\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−57.0587	−0.134256
$426$	0	0
$427$	− 80.8735i	− 0.189399i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 575.274i	− 1.33474i	−0.744725	−	0.667371i	$-0.767419\pi$
$431$	− 575.274i	− 1.33474i	0.744725	−	0.667371i	$-0.232581\pi$
$432$	0	0
$433$	90.1509	0.208201	0.104100	−	0.994567i	$-0.466804\pi$
$433$	90.1509	0.208201	0.104100	+	0.994567i	$0.466804\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.234355	0.000536282 0
$438$	0	0
$439$	− 153.475i	− 0.349602i	−0.984604	−	0.174801i	$-0.944072\pi$
$439$	− 153.475i	− 0.349602i	0.984604	−	0.174801i	$-0.0559282\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	89.5187i	0.202074i	0.994883	+	0.101037i	$0.0322160\pi$
$443$	89.5187i	0.202074i	−0.994883	+	0.101037i	$0.967784\pi$
$444$	0	0
$445$	21.8341	0.0490655
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−487.678	−1.08614	−0.543071	−	0.839687i	$-0.682739\pi$
$449$	−487.678	−1.08614	−0.543071	+	0.839687i	$0.682739\pi$
$450$	0	0
$451$	− 173.619i	− 0.384964i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	125.162i	0.275080i
$456$	0	0
$457$	77.2928	0.169131	0.0845654	−	0.996418i	$-0.473050\pi$
$457$	77.2928	0.169131	0.0845654	+	0.996418i	$0.473050\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−201.504	−0.437101	−0.218551	−	0.975826i	$-0.570133\pi$
$461$	−201.504	−0.437101	−0.218551	+	0.975826i	$0.570133\pi$
$462$	0	0
$463$	− 82.5940i	− 0.178389i	−0.996014	−	0.0891944i	$-0.971571\pi$
$463$	− 82.5940i	− 0.178389i	0.996014	−	0.0891944i	$-0.0284292\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	444.443i	0.951698i	0.879527	+	0.475849i	$0.157859\pi$
$467$	444.443i	0.951698i	−0.879527	+	0.475849i	$0.842141\pi$
$468$	0	0
$469$	61.0673	0.130207
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−497.335	−1.05145
$474$	0	0
$475$	0.234134i	0 0.000492914i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 556.519i	− 1.16184i	−0.813962	−	0.580918i	$-0.802694\pi$
$479$	− 556.519i	− 1.16184i	0.813962	−	0.580918i	$-0.197306\pi$
$480$	0	0
$481$	665.310	1.38318
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−503.149	−1.03742
$486$	0	0
$487$	705.234i	1.44812i	0.689737	+	0.724060i	$0.257726\pi$
$487$	705.234i	1.44812i	−0.689737	+	0.724060i	$0.742274\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 605.015i	− 1.23221i	−0.787664	−	0.616105i	$-0.788710\pi$
$491$	− 605.015i	− 1.23221i	0.787664	−	0.616105i	$-0.211290\pi$
$492$	0	0
$493$	102.858	0.208637
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−131.443	−0.264474
$498$	0	0
$499$	− 192.748i	− 0.386268i	−0.981172	−	0.193134i	$-0.938135\pi$
$499$	− 192.748i	− 0.386268i	0.981172	−	0.193134i	$-0.0618652\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	558.416i	1.11017i	0.831793	+	0.555085i	$0.187314\pi$
$503$	558.416i	1.11017i	−0.831793	+	0.555085i	$0.812686\pi$
$504$	0	0
$505$	−196.998	−0.390095
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−126.597	−0.248718	−0.124359	−	0.992237i	$-0.539687\pi$
$509$	−126.597	−0.248718	−0.124359	+	0.992237i	$0.539687\pi$
$510$	0	0
$511$	183.450i	0.359003i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 128.622i	− 0.249751i
$516$	0	0
$517$	−280.064	−0.541710
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−590.984	−1.13433	−0.567163	−	0.823605i	$-0.691959\pi$
$521$	−590.984	−1.13433	−0.567163	+	0.823605i	$0.691959\pi$
$522$	0	0
$523$	50.5498i	0.0966535i	0.998832	+	0.0483268i	$0.0153889\pi$
$523$	50.5498i	0.0966535i	−0.998832	+	0.0483268i	$0.984611\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	225.918i	0.428688i
$528$	0	0
$529$	472.303	0.892822
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−314.828	−0.590671
$534$	0	0
$535$	626.327i	1.17070i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 270.382i	− 0.501636i
$540$	0	0
$541$	−909.128	−1.68046	−0.840229	−	0.542231i	$-0.817580\pi$
$541$	−909.128	−1.68046	−0.840229	+	0.542231i	$0.817580\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	50.3513	0.0923877
$546$	0	0
$547$	− 731.866i	− 1.33796i	−0.743279	−	0.668981i	$-0.766731\pi$
$547$	− 731.866i	− 1.33796i	0.743279	−	0.668981i	$-0.233269\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 0.422067i	0 0.000766002i
$552$	0	0
$553$	−174.138	−0.314897
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−617.363	−1.10837	−0.554186	−	0.832393i	$-0.686970\pi$
$557$	−617.363	−1.10837	−0.554186	+	0.832393i	$0.686970\pi$
$558$	0	0
$559$	901.832i	1.61329i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	829.105i	1.47266i	0.676625	+	0.736328i	$0.263442\pi$
$563$	829.105i	1.47266i	−0.676625	+	0.736328i	$0.736558\pi$
$564$	0	0
$565$	−889.420	−1.57419
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−287.157	−0.504670	−0.252335	−	0.967640i	$-0.581198\pi$
$569$	−287.157	−0.504670	−0.252335	+	0.967640i	$0.581198\pi$
$570$	0	0
$571$	− 161.987i	− 0.283691i	−0.989889	−	0.141845i	$-0.954696\pi$
$571$	− 161.987i	− 0.283691i	0.989889	−	0.141845i	$-0.0453036\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 56.6438i	− 0.0985110i
$576$	0	0
$577$	−197.092	−0.341581	−0.170790	−	0.985307i	$-0.554632\pi$
$577$	−197.092	−0.341581	−0.170790	+	0.985307i	$0.554632\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	22.1043	0.0380453
$582$	0	0
$583$	− 560.421i	− 0.961272i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 757.516i	− 1.29049i	−0.763977	−	0.645244i	$-0.776756\pi$
$587$	− 757.516i	− 1.29049i	0.763977	−	0.645244i	$-0.223244\pi$
$588$	0	0
$589$	0.927032	0.00157391
$590$	0	0
$591$	0	0
$592$	0	0
$593$	767.673	1.29456	0.647279	−	0.762253i	$-0.275907\pi$
$593$	767.673	1.29456	0.647279	+	0.762253i	$0.275907\pi$
$594$	0	0
$595$	− 81.9462i	− 0.137725i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1043.77i	1.74252i	0.490821	+	0.871260i	$0.336697\pi$
$599$	1043.77i	1.74252i	−0.490821	+	0.871260i	$0.663303\pi$
$600$	0	0
$601$	697.990	1.16138	0.580691	−	0.814124i	$-0.302783\pi$
$601$	697.990	1.16138	0.580691	+	0.814124i	$0.302783\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−335.215	−0.554075
$606$	0	0
$607$	67.5726i	0.111322i	0.998450	+	0.0556611i	$0.0177267\pi$
$607$	67.5726i	0.111322i	−0.998450	+	0.0556611i	$0.982273\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	507.848i	0.831175i
$612$	0	0
$613$	627.420	1.02352	0.511762	−	0.859127i	$-0.328993\pi$
$613$	627.420	1.02352	0.511762	+	0.859127i	$0.328993\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−98.1234	−0.159033	−0.0795165	−	0.996834i	$-0.525338\pi$
$617$	−98.1234	−0.159033	−0.0795165	+	0.996834i	$0.525338\pi$
$618$	0	0
$619$	− 622.064i	− 1.00495i	−0.864592	−	0.502475i	$-0.832423\pi$
$619$	− 622.064i	− 1.00495i	0.864592	−	0.502475i	$-0.167577\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 13.4971i	− 0.0216646i
$624$	0	0
$625$	−380.343	−0.608549
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−435.594	−0.692518
$630$	0	0
$631$	583.009i	0.923944i	0.886895	+	0.461972i	$0.152858\pi$
$631$	583.009i	0.923944i	−0.886895	+	0.461972i	$0.847142\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 806.779i	− 1.27052i
$636$	0	0
$637$	−490.290	−0.769687
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−642.626	−1.00254	−0.501268	−	0.865292i	$-0.667133\pi$
$641$	−642.626	−1.00254	−0.501268	+	0.865292i	$0.667133\pi$
$642$	0	0
$643$	224.989i	0.349904i	0.984577	+	0.174952i	$0.0559771\pi$
$643$	224.989i	0.349904i	−0.984577	+	0.174952i	$0.944023\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1234.87i	− 1.90861i	−0.298840	−	0.954303i	$-0.596600\pi$
$647$	− 1234.87i	− 1.90861i	0.298840	−	0.954303i	$-0.403400\pi$
$648$	0	0
$649$	433.349	0.667718
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−54.3250	−0.0831929	−0.0415965	−	0.999134i	$-0.513244\pi$
$653$	−54.3250	−0.0831929	−0.0415965	+	0.999134i	$0.513244\pi$
$654$	0	0
$655$	− 792.658i	− 1.21016i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	592.696i	0.899387i	0.893183	+	0.449694i	$0.148467\pi$
$659$	592.696i	0.899387i	−0.893183	+	0.449694i	$0.851533\pi$
$660$	0	0
$661$	−817.609	−1.23693	−0.618464	−	0.785813i	$-0.712245\pi$
$661$	−817.609	−1.23693	−0.618464	+	0.785813i	$0.712245\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.336258	−0.000505650 0
$666$	0	0
$667$	102.110i	0.153089i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	199.932i	0.297961i
$672$	0	0
$673$	−294.801	−0.438040	−0.219020	−	0.975720i	$-0.570286\pi$
$673$	−294.801	−0.438040	−0.219020	+	0.975720i	$0.570286\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	238.903	0.352884	0.176442	−	0.984311i	$-0.443541\pi$
$677$	238.903	0.352884	0.176442	+	0.984311i	$0.443541\pi$
$678$	0	0
$679$	311.028i	0.458068i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	358.601i	0.525038i	0.964927	+	0.262519i	$0.0845532\pi$
$683$	358.601i	0.525038i	−0.964927	+	0.262519i	$0.915447\pi$
$684$	0	0
$685$	−822.491	−1.20072
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1016.23	−1.47493
$690$	0	0
$691$	− 590.750i	− 0.854921i	−0.904034	−	0.427460i	$-0.859408\pi$
$691$	− 590.750i	− 0.854921i	0.904034	−	0.427460i	$-0.140592\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	437.111i	0.628937i
$696$	0	0
$697$	206.125	0.295732
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1122.36	−1.60108	−0.800539	−	0.599281i	$-0.795453\pi$
$701$	−1122.36	−1.60108	−0.800539	+	0.599281i	$0.795453\pi$
$702$	0	0
$703$	1.78741i	0.00254255i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	121.777i	0.172245i
$708$	0	0
$709$	−209.168	−0.295018	−0.147509	−	0.989061i	$-0.547126\pi$
$709$	−209.168	−0.295018	−0.147509	+	0.989061i	$0.547126\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−224.276	−0.314552
$714$	0	0
$715$	− 309.419i	− 0.432754i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 601.883i	− 0.837111i	−0.908191	−	0.418555i	$-0.862537\pi$
$719$	− 601.883i	− 0.837111i	0.908191	−	0.418555i	$-0.137463\pi$
$720$	0	0
$721$	−79.5093	−0.110276
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−102.014	−0.140709
$726$	0	0
$727$	131.548i	0.180947i	0.995899	+	0.0904734i	$0.0288380\pi$
$727$	131.548i	0.180947i	−0.995899	+	0.0904734i	$0.971162\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 590.450i	− 0.807729i
$732$	0	0
$733$	401.891	0.548283	0.274141	−	0.961689i	$-0.411606\pi$
$733$	401.891	0.548283	0.274141	+	0.961689i	$0.411606\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−150.968	−0.204841
$738$	0	0
$739$	214.088i	0.289699i	0.989454	+	0.144850i	$0.0462698\pi$
$739$	214.088i	0.289699i	−0.989454	+	0.144850i	$0.953730\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	891.968i	1.20050i	0.799814	+	0.600248i	$0.204931\pi$
$743$	891.968i	1.20050i	−0.799814	+	0.600248i	$0.795069\pi$
$744$	0	0
$745$	−214.128	−0.287420
$746$	0	0
$747$	0	0
$748$	0	0
$749$	387.172	0.516919
$750$	0	0
$751$	879.020i	1.17047i	0.810865	+	0.585233i	$0.198997\pi$
$751$	879.020i	1.17047i	−0.810865	+	0.585233i	$0.801003\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	946.457i	1.25358i
$756$	0	0
$757$	510.218	0.673999	0.337000	−	0.941505i	$-0.390588\pi$
$757$	510.218	0.673999	0.337000	+	0.941505i	$0.390588\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−926.923	−1.21803	−0.609016	−	0.793158i	$-0.708436\pi$
$761$	−926.923	−1.21803	−0.609016	+	0.793158i	$0.708436\pi$
$762$	0	0
$763$	− 31.1253i	− 0.0407933i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 785.804i	− 1.02452i
$768$	0	0
$769$	890.596	1.15812	0.579061	−	0.815284i	$-0.303419\pi$
$769$	890.596	1.15812	0.579061	+	0.815284i	$0.303419\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	175.401	0.226910	0.113455	−	0.993543i	$-0.463808\pi$
$773$	175.401	0.226910	0.113455	+	0.993543i	$0.463808\pi$
$774$	0	0
$775$	− 224.064i	− 0.289115i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 0.845812i	− 0.00108577i
$780$	0	0
$781$	324.949	0.416067
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−180.475	−0.229905
$786$	0	0
$787$	461.631i	0.586571i	0.956025	+	0.293286i	$0.0947487\pi$
$787$	461.631i	0.586571i	−0.956025	+	0.293286i	$0.905251\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	549.807i	0.695078i
$792$	0	0
$793$	362.542	0.457178
$794$	0	0
$795$	0	0
$796$	0	0
$797$	792.157	0.993923	0.496961	−	0.867773i	$-0.334449\pi$
$797$	792.157	0.993923	0.496961	+	0.867773i	$0.334449\pi$
$798$	0	0
$799$	− 332.500i	− 0.416145i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 453.518i	− 0.564779i
$804$	0	0
$805$	81.3503	0.101056
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1078.34	1.33292	0.666462	−	0.745539i	$-0.267808\pi$
$809$	1078.34	1.33292	0.666462	+	0.745539i	$0.267808\pi$
$810$	0	0
$811$	− 606.040i	− 0.747275i	−0.927575	−	0.373638i	$-0.878110\pi$
$811$	− 606.040i	− 0.747275i	0.927575	−	0.373638i	$-0.121890\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 573.846i	− 0.704106i
$816$	0	0
$817$	−2.42285	−0.00296554
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−326.498	−0.397683	−0.198842	−	0.980032i	$-0.563718\pi$
$821$	−326.498	−0.397683	−0.198842	+	0.980032i	$0.563718\pi$
$822$	0	0
$823$	− 1114.44i	− 1.35412i	−0.735928	−	0.677060i	$-0.763254\pi$
$823$	− 1114.44i	− 1.35412i	0.735928	−	0.677060i	$-0.236746\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 381.985i	− 0.461892i	−0.972967	−	0.230946i	$-0.925818\pi$
$827$	− 381.985i	− 0.461892i	0.972967	−	0.230946i	$-0.0741821\pi$
$828$	0	0
$829$	238.484	0.287677	0.143838	−	0.989601i	$-0.454055\pi$
$829$	238.484	0.287677	0.143838	+	0.989601i	$0.454055\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	321.004	0.385360
$834$	0	0
$835$	598.445i	0.716701i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 478.703i	− 0.570563i	−0.958444	−	0.285282i	$-0.907913\pi$
$839$	− 478.703i	− 0.570563i	0.958444	−	0.285282i	$-0.0920872\pi$
$840$	0	0
$841$	−657.103	−0.781335
$842$	0	0
$843$	0	0
$844$	0	0
$845$	145.442	0.172121
$846$	0	0
$847$	207.218i	0.244649i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 432.427i	− 0.508139i
$852$	0	0
$853$	1425.70	1.67140	0.835700	−	0.549186i	$-0.185062\pi$
$853$	1425.70	1.67140	0.835700	+	0.549186i	$0.185062\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.4474	0.0133575	0.00667876	−	0.999978i	$-0.497874\pi$
$857$	11.4474	0.0133575	0.00667876	+	0.999978i	$0.497874\pi$
$858$	0	0
$859$	221.720i	0.258114i	0.991637	+	0.129057i	$0.0411951\pi$
$859$	221.720i	0.258114i	−0.991637	+	0.129057i	$0.958805\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1675.77i	− 1.94180i	−0.239487	−	0.970900i	$-0.576979\pi$
$863$	− 1675.77i	− 1.94180i	0.239487	−	0.970900i	$-0.423021\pi$
$864$	0	0
$865$	−786.081	−0.908764
$866$	0	0
$867$	0	0
$868$	0	0
$869$	430.497	0.495393
$870$	0	0
$871$	273.754i	0.314299i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	351.370i	0.401566i
$876$	0	0
$877$	748.650	0.853649	0.426825	−	0.904334i	$-0.359632\pi$
$877$	748.650	0.853649	0.426825	+	0.904334i	$0.359632\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	705.252	0.800513	0.400256	−	0.916403i	$-0.368921\pi$
$881$	705.252	0.800513	0.400256	+	0.916403i	$0.368921\pi$
$882$	0	0
$883$	1657.97i	1.87765i	0.344392	+	0.938826i	$0.388085\pi$
$883$	1657.97i	1.87765i	−0.344392	+	0.938826i	$0.611915\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1407.84i	1.58719i	0.608444	+	0.793597i	$0.291794\pi$
$887$	1407.84i	1.58719i	−0.608444	+	0.793597i	$0.708206\pi$
$888$	0	0
$889$	−498.721	−0.560991
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.36438	−0.00152786
$894$	0	0
$895$	317.007i	0.354198i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	403.914i	0.449293i
$900$	0	0
$901$	665.348	0.738455
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−642.735	−0.710204
$906$	0	0
$907$	− 238.955i	− 0.263457i	−0.991286	−	0.131728i	$-0.957947\pi$
$907$	− 238.955i	− 0.263457i	0.991286	−	0.131728i	$-0.0420527\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	659.103i	0.723494i	0.932276	+	0.361747i	$0.117820\pi$
$911$	659.103i	0.723494i	−0.932276	+	0.361747i	$0.882180\pi$
$912$	0	0
$913$	−54.6453	−0.0598525
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−489.992	−0.534342
$918$	0	0
$919$	− 1171.93i	− 1.27522i	−0.770357	−	0.637612i	$-0.779922\pi$
$919$	− 1171.93i	− 1.27522i	0.770357	−	0.637612i	$-0.220078\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 589.238i	− 0.638394i
$924$	0	0
$925$	432.019	0.467048
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1676.14	1.80424	0.902121	−	0.431482i	$-0.142009\pi$
$929$	1676.14	1.80424	0.902121	+	0.431482i	$0.142009\pi$
$930$	0	0
$931$	− 1.31721i	− 0.00141483i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	202.584i	0.216667i
$936$	0	0
$937$	1791.65	1.91212	0.956058	−	0.293176i	$-0.0947123\pi$
$937$	1791.65	1.91212	0.956058	+	0.293176i	$0.0947123\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	269.577	0.286479	0.143239	−	0.989688i	$-0.454248\pi$
$941$	269.577	0.286479	0.143239	+	0.989688i	$0.454248\pi$
$942$	0	0
$943$	204.626i	0.216995i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 37.8401i	− 0.0399579i	−0.999800	−	0.0199789i	$-0.993640\pi$
$947$	− 37.8401i	− 0.0399579i	0.999800	−	0.0199789i	$-0.00635992\pi$
$948$	0	0
$949$	−822.376	−0.866571
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1574.25	1.65188	0.825942	−	0.563755i	$-0.190644\pi$
$953$	1574.25	1.65188	0.825942	+	0.563755i	$0.190644\pi$
$954$	0	0
$955$	843.793i	0.883553i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	508.434i	0.530171i
$960$	0	0
$961$	73.8399	0.0768365
$962$	0	0
$963$	0	0
$964$	0	0
$965$	597.213	0.618874
$966$	0	0
$967$	− 1105.96i	− 1.14371i	−0.820356	−	0.571853i	$-0.806225\pi$
$967$	− 1105.96i	− 1.14371i	0.820356	−	0.571853i	$-0.193775\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	390.576i	0.402241i	0.979567	+	0.201120i	$0.0644583\pi$
$971$	390.576i	0.402241i	−0.979567	+	0.201120i	$0.935542\pi$
$972$	0	0
$973$	270.206	0.277704
$974$	0	0
$975$	0	0
$976$	0	0
$977$	125.557	0.128513	0.0642564	−	0.997933i	$-0.479532\pi$
$977$	125.557	0.128513	0.0642564	+	0.997933i	$0.479532\pi$
$978$	0	0
$979$	33.3668i	0.0340826i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1264.46i	1.28633i	0.765729	+	0.643164i	$0.222379\pi$
$983$	1264.46i	1.28633i	−0.765729	+	0.643164i	$0.777621\pi$
$984$	0	0
$985$	−1624.16	−1.64889
$986$	0	0
$987$	0	0
$988$	0	0
$989$	586.157	0.592676
$990$	0	0
$991$	− 264.800i	− 0.267205i	−0.991035	−	0.133602i	$-0.957346\pi$
$991$	− 264.800i	− 0.267205i	0.991035	−	0.133602i	$-0.0426545\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 83.3511i	− 0.0837699i
$996$	0	0
$997$	166.052	0.166552	0.0832758	−	0.996527i	$-0.473462\pi$
$997$	166.052	0.166552	0.0832758	+	0.996527i	$0.473462\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.g.n.703.2		8
3.2	odd	2		1728.3.g.k.703.8		8
4.3	odd	2	inner	1728.3.g.n.703.1		8
8.3	odd	2		864.3.g.a.703.7	✓	8
8.5	even	2		864.3.g.a.703.8	yes	8
12.11	even	2		1728.3.g.k.703.7		8
24.5	odd	2		864.3.g.c.703.2	yes	8
24.11	even	2		864.3.g.c.703.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.3.g.a.703.7	✓	8	8.3	odd	2
864.3.g.a.703.8	yes	8	8.5	even	2
864.3.g.c.703.1	yes	8	24.11	even	2
864.3.g.c.703.2	yes	8	24.5	odd	2
1728.3.g.k.703.7		8	12.11	even	2
1728.3.g.k.703.8		8	3.2	odd	2
1728.3.g.n.703.1		8	4.3	odd	2	inner
1728.3.g.n.703.2		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.18059 q^{5} +2.58429i q^{7} +O(q^{10})\)	\(q-4.18059 q^{5} +2.58429i q^{7} -6.38876i q^{11} -11.5849 q^{13} +7.58491 q^{17} -0.0311239i q^{19} +7.52976i q^{23} -7.52266 q^{25} +13.5609 q^{29} +29.7852i q^{31} -10.8038i q^{35} -57.4290 q^{37} +27.1757 q^{41} -77.8454i q^{43} -43.8370i q^{47} +42.3215 q^{49} +87.7199 q^{53} +26.7088i q^{55} +67.8299i q^{59} -31.2943 q^{61} +48.4318 q^{65} -23.6302i q^{67} +50.8625i q^{71} +70.9868 q^{73} +16.5104 q^{77} +67.3835i q^{79} -8.55335i q^{83} -31.7094 q^{85} -5.22274 q^{89} -29.9387i q^{91} +0.130116i q^{95} +120.354 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5}+O(q^{10}) \) 8 * q + 8 * q^5	\( 8 q + 8 q^{5} + 16 q^{13} - 48 q^{17} + 48 q^{25} - 32 q^{29} + 96 q^{37} + 128 q^{41} + 168 q^{53} - 32 q^{61} + 112 q^{65} + 24 q^{73} - 440 q^{77} - 144 q^{85} - 624 q^{89} - 136 q^{97}+O(q^{100}) \) 8 * q + 8 * q^5 + 16 * q^13 - 48 * q^17 + 48 * q^25 - 32 * q^29 + 96 * q^37 + 128 * q^41 + 168 * q^53 - 32 * q^61 + 112 * q^65 + 24 * q^73 - 440 * q^77 - 144 * q^85 - 624 * q^89 - 136 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more