LMFDB - Embedded newform 1728.4.f.e.863.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,4,Mod(863,1728)]

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, 1, 1]))

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

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

chi := DirichletCharacter("1728.863");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$101.955300490$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 $ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			863.1
Root			$0.535233 - 0.309017i$ of defining polynomial
Character	$\chi$	$=$	1728.863
Dual form			1728.4.f.e.863.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-15.4919 q^{5} -13.0000i q^{7} +O(q^{10})$	$q-15.4919 q^{5} -13.0000i q^{7} +15.4919i q^{11} -22.5167i q^{13} +26.8328i q^{17} +81.4064 q^{19} +26.8328 q^{23} +115.000 q^{25} -185.903 q^{29} +172.000i q^{31} +201.395i q^{35} +32.9090i q^{37} +268.328i q^{41} -197.454 q^{43} +295.161 q^{47} +174.000 q^{49} -526.726 q^{53} -240.000i q^{55} +697.137i q^{59} -469.386i q^{61} +348.827i q^{65} -19.0526 q^{67} +590.322 q^{71} +391.000 q^{73} +201.395 q^{77} -413.000i q^{79} +123.935i q^{83} -415.692i q^{85} -1314.81i q^{89} -292.717 q^{91} -1261.14 q^{95} -427.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 920 q^{25} + 1392 q^{49} + 3128 q^{73} - 3416 q^{97}+O(q^{100}) $ 8 * q + 920 * q^25 + 1392 * q^49 + 3128 * q^73 - 3416 * 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
$5$	−15.4919	−1.38564	−0.692820	−	0.721110i	$-0.743632\pi$
$5$	−15.4919	−1.38564	−0.692820	+	0.721110i	$0.743632\pi$
$6$	0	0
$7$	− 13.0000i	− 0.701934i	−0.936388	−	0.350967i	$-0.885853\pi$
$7$	− 13.0000i	− 0.701934i	0.936388	−	0.350967i	$-0.114147\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	15.4919i	0.424636i	0.977201	+	0.212318i	$0.0681012\pi$
$11$	15.4919i	0.424636i	−0.977201	+	0.212318i	$0.931899\pi$
$12$	0	0
$13$	− 22.5167i	− 0.480384i	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	− 22.5167i	− 0.480384i	0.970725	−	0.240192i	$-0.0772105\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	26.8328i	0.382818i	0.981510	+	0.191409i	$0.0613058\pi$
$17$	26.8328i	0.382818i	−0.981510	+	0.191409i	$0.938694\pi$
$18$	0	0
$19$	81.4064	0.982942	0.491471	−	0.870894i	$-0.336459\pi$
$19$	81.4064	0.982942	0.491471	+	0.870894i	$0.336459\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	26.8328	0.243262	0.121631	−	0.992575i	$-0.461188\pi$
$23$	26.8328	0.243262	0.121631	+	0.992575i	$0.461188\pi$
$24$	0	0
$25$	115.000	0.920000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−185.903	−1.19039	−0.595196	−	0.803581i	$-0.702926\pi$
$29$	−185.903	−1.19039	−0.595196	+	0.803581i	$0.702926\pi$
$30$	0	0
$31$	172.000i	0.996520i	0.867028	+	0.498260i	$0.166027\pi$
$31$	172.000i	0.996520i	−0.867028	+	0.498260i	$0.833973\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	201.395i	0.972628i
$36$	0	0
$37$	32.9090i	0.146222i	0.997324	+	0.0731108i	$0.0232927\pi$
$37$	32.9090i	0.146222i	−0.997324	+	0.0731108i	$0.976707\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	268.328i	1.02209i	0.859553	+	0.511047i	$0.170742\pi$
$41$	268.328i	1.02209i	−0.859553	+	0.511047i	$0.829258\pi$
$42$	0	0
$43$	−197.454	−0.700266	−0.350133	−	0.936700i	$-0.613864\pi$
$43$	−197.454	−0.700266	−0.350133	+	0.936700i	$0.613864\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	295.161	0.916035	0.458018	−	0.888943i	$-0.348560\pi$
$47$	295.161	0.916035	0.458018	+	0.888943i	$0.348560\pi$
$48$	0	0
$49$	174.000	0.507289
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−526.726	−1.36512	−0.682560	−	0.730830i	$-0.739133\pi$
$53$	−526.726	−1.36512	−0.682560	+	0.730830i	$0.739133\pi$
$54$	0	0
$55$	− 240.000i	− 0.588393i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	697.137i	1.53830i	0.639070	+	0.769148i	$0.279319\pi$
$59$	697.137i	1.53830i	−0.639070	+	0.769148i	$0.720681\pi$
$60$	0	0
$61$	− 469.386i	− 0.985224i	−0.870249	−	0.492612i	$-0.836042\pi$
$61$	− 469.386i	− 0.985224i	0.870249	−	0.492612i	$-0.163958\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	348.827i	0.665640i
$66$	0	0
$67$	−19.0526	−0.0347409	−0.0173705	−	0.999849i	$-0.505529\pi$
$67$	−19.0526	−0.0347409	−0.0173705	+	0.999849i	$0.505529\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	590.322	0.986737	0.493368	−	0.869820i	$-0.335765\pi$
$71$	590.322	0.986737	0.493368	+	0.869820i	$0.335765\pi$
$72$	0	0
$73$	391.000	0.626892	0.313446	−	0.949606i	$-0.398517\pi$
$73$	391.000	0.626892	0.313446	+	0.949606i	$0.398517\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	201.395	0.298066
$78$	0	0
$79$	− 413.000i	− 0.588179i	−0.955778	−	0.294089i	$-0.904984\pi$
$79$	− 413.000i	− 0.588179i	0.955778	−	0.294089i	$-0.0950164\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	123.935i	0.163900i	0.996636	+	0.0819499i	$0.0261148\pi$
$83$	123.935i	0.163900i	−0.996636	+	0.0819499i	$0.973885\pi$
$84$	0	0
$85$	− 415.692i	− 0.530449i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 1314.81i	− 1.56595i	−0.622054	−	0.782974i	$-0.713702\pi$
$89$	− 1314.81i	− 1.56595i	0.622054	−	0.782974i	$-0.286298\pi$
$90$	0	0
$91$	−292.717	−0.337198
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1261.14	−1.36200
$96$	0	0
$97$	−427.000	−0.446962	−0.223481	−	0.974708i	$-0.571742\pi$
$97$	−427.000	−0.446962	−0.223481	+	0.974708i	$0.571742\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	30.9839	0.0305249	0.0152624	−	0.999884i	$-0.495142\pi$
$101$	30.9839	0.0305249	0.0152624	+	0.999884i	$0.495142\pi$
$102$	0	0
$103$	− 1151.00i	− 1.10108i	−0.834808	−	0.550541i	$-0.814422\pi$
$103$	− 1151.00i	− 1.10108i	0.834808	−	0.550541i	$-0.185578\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 945.008i	− 0.853807i	−0.904297	−	0.426904i	$-0.859604\pi$
$107$	− 945.008i	− 0.853807i	0.904297	−	0.426904i	$-0.140396\pi$
$108$	0	0
$109$	1143.15i	1.00453i	0.864712	+	0.502267i	$0.167501\pi$
$109$	1143.15i	1.00453i	−0.864712	+	0.502267i	$0.832499\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2280.79i	1.89875i	0.314147	+	0.949374i	$0.398282\pi$
$113$	2280.79i	1.89875i	−0.314147	+	0.949374i	$0.601718\pi$
$114$	0	0
$115$	−415.692	−0.337074
$116$	0	0
$117$	0	0
$118$	0	0
$119$	348.827	0.268713
$120$	0	0
$121$	1091.00	0.819684
$122$	0	0
$123$	0	0
$124$	0	0
$125$	154.919	0.110851
$126$	0	0
$127$	− 2260.00i	− 1.57908i	−0.613702	−	0.789538i	$-0.710320\pi$
$127$	− 2260.00i	− 1.57908i	0.613702	−	0.789538i	$-0.289680\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 495.742i	− 0.330635i	−0.986240	−	0.165317i	$-0.947135\pi$
$131$	− 495.742i	− 0.330635i	0.986240	−	0.165317i	$-0.0528649\pi$
$132$	0	0
$133$	− 1058.28i	− 0.689961i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2924.78i	− 1.82395i	−0.410251	−	0.911973i	$-0.634559\pi$
$137$	− 2924.78i	− 1.82395i	0.410251	−	0.911973i	$-0.365441\pi$
$138$	0	0
$139$	−2939.29	−1.79358	−0.896789	−	0.442458i	$-0.854106\pi$
$139$	−2939.29	−1.79358	−0.896789	+	0.442458i	$0.854106\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	348.827	0.203988
$144$	0	0
$145$	2880.00	1.64946
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2943.47	1.61838	0.809189	−	0.587549i	$-0.199907\pi$
$149$	2943.47	1.61838	0.809189	+	0.587549i	$0.199907\pi$
$150$	0	0
$151$	− 1039.00i	− 0.559951i	−0.960007	−	0.279976i	$-0.909674\pi$
$151$	− 1039.00i	− 0.559951i	0.960007	−	0.279976i	$-0.0903264\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 2664.61i	− 1.38082i
$156$	0	0
$157$	− 852.169i	− 0.433188i	−0.976262	−	0.216594i	$-0.930505\pi$
$157$	− 852.169i	− 0.433188i	0.976262	−	0.216594i	$-0.0694948\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 348.827i	− 0.170754i
$162$	0	0
$163$	635.663	0.305454	0.152727	−	0.988268i	$-0.451195\pi$
$163$	635.663	0.305454	0.152727	+	0.988268i	$0.451195\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3944.42	−1.82772	−0.913858	−	0.406033i	$-0.866912\pi$
$167$	−3944.42	−1.82772	−0.913858	+	0.406033i	$0.866912\pi$
$168$	0	0
$169$	1690.00	0.769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	898.532	0.394879	0.197440	−	0.980315i	$-0.436737\pi$
$173$	898.532	0.394879	0.197440	+	0.980315i	$0.436737\pi$
$174$	0	0
$175$	− 1495.00i	− 0.645779i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 2943.47i	− 1.22908i	−0.788886	−	0.614539i	$-0.789342\pi$
$179$	− 2943.47i	− 1.22908i	0.788886	−	0.614539i	$-0.210658\pi$
$180$	0	0
$181$	− 3448.51i	− 1.41617i	−0.706129	−	0.708083i	$-0.749560\pi$
$181$	− 3448.51i	− 1.41617i	0.706129	−	0.708083i	$-0.250440\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 509.823i	− 0.202611i
$186$	0	0
$187$	−415.692	−0.162558
$188$	0	0
$189$	0	0
$190$	0	0
$191$	402.492	0.152478	0.0762390	−	0.997090i	$-0.475709\pi$
$191$	402.492	0.152478	0.0762390	+	0.997090i	$0.475709\pi$
$192$	0	0
$193$	1751.00	0.653056	0.326528	−	0.945188i	$-0.394121\pi$
$193$	1751.00	0.653056	0.326528	+	0.945188i	$0.394121\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	108.444	0.0392197	0.0196099	−	0.999808i	$-0.493758\pi$
$197$	108.444	0.0392197	0.0196099	+	0.999808i	$0.493758\pi$
$198$	0	0
$199$	− 743.000i	− 0.264673i	−0.991205	−	0.132336i	$-0.957752\pi$
$199$	− 743.000i	− 0.264673i	0.991205	−	0.132336i	$-0.0422479\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2416.74i	0.835576i
$204$	0	0
$205$	− 4156.92i	− 1.41625i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1261.14i	0.417393i
$210$	0	0
$211$	−781.155	−0.254867	−0.127433	−	0.991847i	$-0.540674\pi$
$211$	−781.155	−0.254867	−0.127433	+	0.991847i	$0.540674\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3058.94	0.970316
$216$	0	0
$217$	2236.00	0.699491
$218$	0	0
$219$	0	0
$220$	0	0
$221$	604.185	0.183900
$222$	0	0
$223$	− 4660.00i	− 1.39936i	−0.714458	−	0.699679i	$-0.753327\pi$
$223$	− 4660.00i	− 1.39936i	0.714458	−	0.699679i	$-0.246673\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2881.50i	0.842519i	0.906940	+	0.421260i	$0.138412\pi$
$227$	2881.50i	0.842519i	−0.906940	+	0.421260i	$0.861588\pi$
$228$	0	0
$229$	935.307i	0.269899i	0.990852	+	0.134949i	$0.0430872\pi$
$229$	935.307i	0.269899i	−0.990852	+	0.134949i	$0.956913\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 3810.26i	− 1.07132i	−0.844433	−	0.535662i	$-0.820062\pi$
$233$	− 3810.26i	− 1.07132i	0.844433	−	0.535662i	$-0.179938\pi$
$234$	0	0
$235$	−4572.61	−1.26930
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−751.319	−0.203342	−0.101671	−	0.994818i	$-0.532419\pi$
$239$	−751.319	−0.203342	−0.101671	+	0.994818i	$0.532419\pi$
$240$	0	0
$241$	6751.00	1.80444	0.902220	−	0.431276i	$-0.141936\pi$
$241$	6751.00	1.80444	0.902220	+	0.431276i	$0.141936\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2695.60	−0.702920
$246$	0	0
$247$	− 1833.00i	− 0.472190i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2695.60i	0.677867i	0.940810	+	0.338933i	$0.110066\pi$
$251$	2695.60i	0.677867i	−0.940810	+	0.338933i	$0.889934\pi$
$252$	0	0
$253$	415.692i	0.103298i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5420.23i	1.31558i	0.753200	+	0.657791i	$0.228509\pi$
$257$	5420.23i	1.31558i	−0.753200	+	0.657791i	$0.771491\pi$
$258$	0	0
$259$	427.817	0.102638
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−107.331	−0.0251647	−0.0125824	−	0.999921i	$-0.504005\pi$
$263$	−107.331	−0.0251647	−0.0125824	+	0.999921i	$0.504005\pi$
$264$	0	0
$265$	8160.00	1.89157
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2370.27	−0.537240	−0.268620	−	0.963246i	$-0.586568\pi$
$269$	−2370.27	−0.537240	−0.268620	+	0.963246i	$0.586568\pi$
$270$	0	0
$271$	− 31.0000i	− 0.00694877i	−0.999994	−	0.00347438i	$-0.998894\pi$
$271$	− 31.0000i	− 0.00694877i	0.999994	−	0.00347438i	$-0.00110593\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1781.57i	0.390665i
$276$	0	0
$277$	− 6630.29i	− 1.43818i	−0.694918	−	0.719089i	$-0.744559\pi$
$277$	− 6630.29i	− 1.43818i	0.694918	−	0.719089i	$-0.255441\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 321.994i	− 0.0683578i	−0.999416	−	0.0341789i	$-0.989118\pi$
$281$	− 321.994i	− 0.0683578i	0.999416	−	0.0341789i	$-0.0108816\pi$
$282$	0	0
$283$	−2296.70	−0.482419	−0.241210	−	0.970473i	$-0.577544\pi$
$283$	−2296.70	−0.482419	−0.241210	+	0.970473i	$0.577544\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3488.27	0.717442
$288$	0	0
$289$	4193.00	0.853450
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3764.54	−0.750603	−0.375302	−	0.926903i	$-0.622461\pi$
$293$	−3764.54	−0.750603	−0.375302	+	0.926903i	$0.622461\pi$
$294$	0	0
$295$	− 10800.0i	− 2.13153i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 604.185i	− 0.116859i
$300$	0	0
$301$	2566.90i	0.491540i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7271.69i	1.36517i
$306$	0	0
$307$	−7970.90	−1.48183	−0.740917	−	0.671596i	$-0.765609\pi$
$307$	−7970.90	−1.48183	−0.740917	+	0.671596i	$0.765609\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5178.73	0.944241	0.472121	−	0.881534i	$-0.343489\pi$
$311$	5178.73	0.944241	0.472121	+	0.881534i	$0.343489\pi$
$312$	0	0
$313$	−4811.00	−0.868798	−0.434399	−	0.900721i	$-0.643039\pi$
$313$	−4811.00	−0.868798	−0.434399	+	0.900721i	$0.643039\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3470.19	−0.614844	−0.307422	−	0.951573i	$-0.599466\pi$
$317$	−3470.19	−0.614844	−0.307422	+	0.951573i	$0.599466\pi$
$318$	0	0
$319$	− 2880.00i	− 0.505483i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2184.36i	0.376289i
$324$	0	0
$325$	− 2589.42i	− 0.441954i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 3837.09i	− 0.642996i
$330$	0	0
$331$	9406.77	1.56206	0.781031	−	0.624492i	$-0.214694\pi$
$331$	9406.77	1.56206	0.781031	+	0.624492i	$0.214694\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	295.161	0.0481384
$336$	0	0
$337$	413.000	0.0667583	0.0333791	−	0.999443i	$-0.489373\pi$
$337$	413.000	0.0667583	0.0333791	+	0.999443i	$0.489373\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2664.61	−0.423158
$342$	0	0
$343$	− 6721.00i	− 1.05802i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1425.26i	0.220495i	0.993904	+	0.110248i	$0.0351644\pi$
$347$	1425.26i	0.220495i	−0.993904	+	0.110248i	$0.964836\pi$
$348$	0	0
$349$	1945.09i	0.298334i	0.988812	+	0.149167i	$0.0476591\pi$
$349$	1945.09i	0.298334i	−0.988812	+	0.149167i	$0.952341\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4400.58i	0.663511i	0.943365	+	0.331755i	$0.107641\pi$
$353$	4400.58i	0.663511i	−0.943365	+	0.331755i	$0.892359\pi$
$354$	0	0
$355$	−9145.23	−1.36726
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9042.66	−1.32940	−0.664698	−	0.747112i	$-0.731440\pi$
$359$	−9042.66	−1.32940	−0.664698	+	0.747112i	$0.731440\pi$
$360$	0	0
$361$	−232.000	−0.0338242
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6057.35	−0.868647
$366$	0	0
$367$	− 3983.00i	− 0.566515i	−0.959044	−	0.283257i	$-0.908585\pi$
$367$	− 3983.00i	− 0.566515i	0.959044	−	0.283257i	$-0.0914151\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6847.43i	0.958224i
$372$	0	0
$373$	− 6576.60i	− 0.912931i	−0.889741	−	0.456466i	$-0.849115\pi$
$373$	− 6576.60i	− 0.912931i	0.889741	−	0.456466i	$-0.150885\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4185.92i	0.571846i
$378$	0	0
$379$	−6212.87	−0.842041	−0.421020	−	0.907051i	$-0.638328\pi$
$379$	−6212.87	−0.842041	−0.421020	+	0.907051i	$0.638328\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	9069.49	1.21000	0.604999	−	0.796226i	$-0.293173\pi$
$383$	9069.49	1.21000	0.604999	+	0.796226i	$0.293173\pi$
$384$	0	0
$385$	−3120.00	−0.413013
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3082.89	0.401823	0.200911	−	0.979609i	$-0.435610\pi$
$389$	3082.89	0.401823	0.200911	+	0.979609i	$0.435610\pi$
$390$	0	0
$391$	720.000i	0.0931252i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6398.17i	0.815005i
$396$	0	0
$397$	− 852.169i	− 0.107731i	−0.998548	−	0.0538654i	$-0.982846\pi$
$397$	− 852.169i	− 0.107731i	0.998548	−	0.0538654i	$-0.0171542\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 12879.8i	− 1.60395i	−0.597357	−	0.801975i	$-0.703783\pi$
$401$	− 12879.8i	− 1.60395i	0.597357	−	0.801975i	$-0.296217\pi$
$402$	0	0
$403$	3872.87	0.478713
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−509.823	−0.0620909
$408$	0	0
$409$	14143.0	1.70984	0.854922	−	0.518756i	$-0.173605\pi$
$409$	14143.0	1.70984	0.854922	+	0.518756i	$0.173605\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9062.78	1.07978
$414$	0	0
$415$	− 1920.00i	− 0.227106i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 15011.7i	− 1.75028i	−0.483867	−	0.875142i	$-0.660768\pi$
$419$	− 15011.7i	− 1.75028i	0.483867	−	0.875142i	$-0.339232\pi$
$420$	0	0
$421$	− 13283.1i	− 1.53772i	−0.639419	−	0.768858i	$-0.720825\pi$
$421$	− 13283.1i	− 1.53772i	0.639419	−	0.768858i	$-0.279175\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3085.77i	0.352193i
$426$	0	0
$427$	−6102.01	−0.691563
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−14570.2	−1.62836	−0.814180	−	0.580613i	$-0.802813\pi$
$431$	−14570.2	−1.62836	−0.814180	+	0.580613i	$0.802813\pi$
$432$	0	0
$433$	2350.00	0.260817	0.130409	−	0.991460i	$-0.458371\pi$
$433$	2350.00	0.260817	0.130409	+	0.991460i	$0.458371\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2184.36	0.239113
$438$	0	0
$439$	− 1484.00i	− 0.161338i	−0.996741	−	0.0806691i	$-0.974294\pi$
$439$	− 1484.00i	− 0.161338i	0.996741	−	0.0806691i	$-0.0257057\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 13849.8i	− 1.48538i	−0.669635	−	0.742690i	$-0.733550\pi$
$443$	− 13849.8i	− 1.48538i	0.669635	−	0.742690i	$-0.266450\pi$
$444$	0	0
$445$	20368.9i	2.16984i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 6627.71i	− 0.696616i	−0.937380	−	0.348308i	$-0.886756\pi$
$449$	− 6627.71i	− 0.696616i	0.937380	−	0.348308i	$-0.113244\pi$
$450$	0	0
$451$	−4156.92	−0.434017
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4534.75	0.467236
$456$	0	0
$457$	−1430.00	−0.146373	−0.0731866	−	0.997318i	$-0.523317\pi$
$457$	−1430.00	−0.146373	−0.0731866	+	0.997318i	$0.523317\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14856.8	1.50097	0.750486	−	0.660886i	$-0.229819\pi$
$461$	14856.8	1.50097	0.750486	+	0.660886i	$0.229819\pi$
$462$	0	0
$463$	1499.00i	0.150463i	0.997166	+	0.0752316i	$0.0239696\pi$
$463$	1499.00i	0.150463i	−0.997166	+	0.0752316i	$0.976030\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6429.15i	− 0.637057i	−0.947913	−	0.318529i	$-0.896811\pi$
$467$	− 6429.15i	− 0.637057i	0.947913	−	0.318529i	$-0.103189\pi$
$468$	0	0
$469$	247.683i	0.0243858i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 3058.94i	− 0.297358i
$474$	0	0
$475$	9361.73	0.904307
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1019.65	−0.0972628	−0.0486314	−	0.998817i	$-0.515486\pi$
$479$	−1019.65	−0.0972628	−0.0486314	+	0.998817i	$0.515486\pi$
$480$	0	0
$481$	741.000	0.0702426
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6615.06	0.619328
$486$	0	0
$487$	15227.0i	1.41684i	0.705791	+	0.708420i	$0.250592\pi$
$487$	15227.0i	1.41684i	−0.705791	+	0.708420i	$0.749408\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 4260.28i	− 0.391576i	−0.980646	−	0.195788i	$-0.937274\pi$
$491$	− 4260.28i	− 0.391576i	0.980646	−	0.195788i	$-0.0627264\pi$
$492$	0	0
$493$	− 4988.31i	− 0.455704i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 7674.19i	− 0.692624i
$498$	0	0
$499$	−1402.96	−0.125862	−0.0629310	−	0.998018i	$-0.520045\pi$
$499$	−1402.96	−0.125862	−0.0629310	+	0.998018i	$0.520045\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3032.11	0.268777	0.134389	−	0.990929i	$-0.457093\pi$
$503$	3032.11	0.268777	0.134389	+	0.990929i	$0.457093\pi$
$504$	0	0
$505$	−480.000	−0.0422965
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1595.67	−0.138952	−0.0694762	−	0.997584i	$-0.522133\pi$
$509$	−1595.67	−0.138952	−0.0694762	+	0.997584i	$0.522133\pi$
$510$	0	0
$511$	− 5083.00i	− 0.440037i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	17831.2i	1.52570i
$516$	0	0
$517$	4572.61i	0.388981i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10974.6i	0.922854i	0.887178	+	0.461427i	$0.152662\pi$
$521$	10974.6i	0.922854i	−0.887178	+	0.461427i	$0.847338\pi$
$522$	0	0
$523$	13116.8	1.09667	0.548335	−	0.836259i	$-0.315262\pi$
$523$	13116.8	1.09667	0.548335	+	0.836259i	$0.315262\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4615.24	−0.381486
$528$	0	0
$529$	−11447.0	−0.940824
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6041.85	0.490998
$534$	0	0
$535$	14640.0i	1.18307i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2695.60i	0.215413i
$540$	0	0
$541$	11741.6i	0.933105i	0.884494	+	0.466552i	$0.154504\pi$
$541$	11741.6i	0.933105i	−0.884494	+	0.466552i	$0.845496\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 17709.7i	− 1.39192i
$546$	0	0
$547$	20630.5	1.61260	0.806302	−	0.591504i	$-0.201466\pi$
$547$	20630.5	1.61260	0.806302	+	0.591504i	$0.201466\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−15133.7	−1.17009
$552$	0	0
$553$	−5369.00	−0.412863
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2029.44	0.154381	0.0771905	−	0.997016i	$-0.475405\pi$
$557$	2029.44	0.154381	0.0771905	+	0.997016i	$0.475405\pi$
$558$	0	0
$559$	4446.00i	0.336397i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 21038.0i	− 1.57486i	−0.616402	−	0.787432i	$-0.711410\pi$
$563$	− 21038.0i	− 1.57486i	0.616402	−	0.787432i	$-0.288590\pi$
$564$	0	0
$565$	− 35333.8i	− 2.63098i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	17307.2i	1.27514i	0.770393	+	0.637570i	$0.220060\pi$
$569$	17307.2i	1.27514i	−0.770393	+	0.637570i	$0.779940\pi$
$570$	0	0
$571$	−3119.42	−0.228623	−0.114312	−	0.993445i	$-0.536466\pi$
$571$	−3119.42	−0.228623	−0.114312	+	0.993445i	$0.536466\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3085.77	0.223801
$576$	0	0
$577$	14077.0	1.01566	0.507828	−	0.861459i	$-0.330449\pi$
$577$	14077.0	1.01566	0.507828	+	0.861459i	$0.330449\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1611.16	0.115047
$582$	0	0
$583$	− 8160.00i	− 0.579679i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 24461.8i	− 1.72001i	−0.510287	−	0.860004i	$-0.670461\pi$
$587$	− 24461.8i	− 1.72001i	0.510287	−	0.860004i	$-0.329539\pi$
$588$	0	0
$589$	14001.9i	0.979522i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 3434.60i	− 0.237845i	−0.992904	−	0.118923i	$-0.962056\pi$
$593$	− 3434.60i	− 0.237845i	0.992904	−	0.118923i	$-0.0379440\pi$
$594$	0	0
$595$	−5404.00	−0.372340
$596$	0	0
$597$	0	0
$598$	0	0
$599$	19105.0	1.30318	0.651592	−	0.758569i	$-0.274101\pi$
$599$	19105.0	1.30318	0.651592	+	0.758569i	$0.274101\pi$
$600$	0	0
$601$	−10342.0	−0.701928	−0.350964	−	0.936389i	$-0.614146\pi$
$601$	−10342.0	−0.701928	−0.350964	+	0.936389i	$0.614146\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−16901.7	−1.13579
$606$	0	0
$607$	20627.0i	1.37928i	0.724151	+	0.689641i	$0.242232\pi$
$607$	20627.0i	1.37928i	−0.724151	+	0.689641i	$0.757768\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 6646.04i	− 0.440049i
$612$	0	0
$613$	20148.9i	1.32758i	0.747918	+	0.663791i	$0.231054\pi$
$613$	20148.9i	1.32758i	−0.747918	+	0.663791i	$0.768946\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3783.43i	0.246864i	0.992353	+	0.123432i	$0.0393901\pi$
$617$	3783.43i	0.246864i	−0.992353	+	0.123432i	$0.960610\pi$
$618$	0	0
$619$	16667.5	1.08227	0.541134	−	0.840936i	$-0.317995\pi$
$619$	16667.5	1.08227	0.541134	+	0.840936i	$0.317995\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−17092.5	−1.09919
$624$	0	0
$625$	−16775.0	−1.07360
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−883.040	−0.0559763
$630$	0	0
$631$	13289.0i	0.838394i	0.907895	+	0.419197i	$0.137688\pi$
$631$	13289.0i	0.838394i	−0.907895	+	0.419197i	$0.862312\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	35011.8i	2.18803i
$636$	0	0
$637$	− 3917.90i	− 0.243694i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 17226.7i	− 1.06149i	−0.847533	−	0.530743i	$-0.821913\pi$
$641$	− 17226.7i	− 1.06149i	0.847533	−	0.530743i	$-0.178087\pi$
$642$	0	0
$643$	−25450.8	−1.56093	−0.780466	−	0.625198i	$-0.785018\pi$
$643$	−25450.8	−1.56093	−0.780466	+	0.625198i	$0.785018\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15831.4	−0.961971	−0.480985	−	0.876729i	$-0.659721\pi$
$647$	−15831.4	−0.961971	−0.480985	+	0.876729i	$0.659721\pi$
$648$	0	0
$649$	−10800.0	−0.653216
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5701.03	−0.341652	−0.170826	−	0.985301i	$-0.554644\pi$
$653$	−5701.03	−0.341652	−0.170826	+	0.985301i	$0.554644\pi$
$654$	0	0
$655$	7680.00i	0.458141i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	25499.7i	1.50733i	0.657261	+	0.753663i	$0.271715\pi$
$659$	25499.7i	1.50733i	−0.657261	+	0.753663i	$0.728285\pi$
$660$	0	0
$661$	4851.47i	0.285477i	0.989760	+	0.142739i	$0.0455908\pi$
$661$	4851.47i	0.285477i	−0.989760	+	0.142739i	$0.954409\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	16394.9i	0.956038i
$666$	0	0
$667$	−4988.31	−0.289577
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7271.69	0.418362
$672$	0	0
$673$	−30659.0	−1.75604	−0.878022	−	0.478620i	$-0.841137\pi$
$673$	−30659.0	−1.75604	−0.878022	+	0.478620i	$0.841137\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8690.97	−0.493384	−0.246692	−	0.969094i	$-0.579344\pi$
$677$	−8690.97	−0.493384	−0.246692	+	0.969094i	$0.579344\pi$
$678$	0	0
$679$	5551.00i	0.313738i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	11433.0i	0.640517i	0.947330	+	0.320259i	$0.103770\pi$
$683$	11433.0i	0.640517i	−0.947330	+	0.320259i	$0.896230\pi$
$684$	0	0
$685$	45310.4i	2.52733i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	11860.1i	0.655782i
$690$	0	0
$691$	18030.6	0.992646	0.496323	−	0.868138i	$-0.334683\pi$
$691$	18030.6	0.992646	0.496323	+	0.868138i	$0.334683\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	45535.3	2.48525
$696$	0	0
$697$	−7200.00	−0.391276
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6336.20	−0.341391	−0.170695	−	0.985324i	$-0.554601\pi$
$701$	−6336.20	−0.341391	−0.170695	+	0.985324i	$0.554601\pi$
$702$	0	0
$703$	2679.00i	0.143727i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 402.790i	− 0.0214264i
$708$	0	0
$709$	− 18728.7i	− 0.992059i	−0.868306	−	0.496029i	$-0.834791\pi$
$709$	− 18728.7i	− 0.992059i	0.868306	−	0.496029i	$-0.165209\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	4615.24i	0.242416i
$714$	0	0
$715$	−5404.00	−0.282655
$716$	0	0
$717$	0	0
$718$	0	0
$719$	22861.6	1.18580	0.592901	−	0.805275i	$-0.297982\pi$
$719$	22861.6	1.18580	0.592901	+	0.805275i	$0.297982\pi$
$720$	0	0
$721$	−14963.0	−0.772887
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−21378.9	−1.09516
$726$	0	0
$727$	− 1180.00i	− 0.0601978i	−0.999547	−	0.0300989i	$-0.990418\pi$
$727$	− 1180.00i	− 0.0601978i	0.999547	−	0.0300989i	$-0.00958222\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 5298.24i	− 0.268075i
$732$	0	0
$733$	39220.6i	1.97632i	0.153419	+	0.988161i	$0.450972\pi$
$733$	39220.6i	1.97632i	−0.153419	+	0.988161i	$0.549028\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 295.161i	− 0.0147522i
$738$	0	0
$739$	28526.9	1.42000	0.709999	−	0.704203i	$-0.248695\pi$
$739$	28526.9	1.42000	0.709999	+	0.704203i	$0.248695\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−37324.4	−1.84294	−0.921468	−	0.388455i	$-0.873009\pi$
$743$	−37324.4	−1.84294	−0.921468	+	0.388455i	$0.873009\pi$
$744$	0	0
$745$	−45600.0	−2.24249
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−12285.1	−0.599316
$750$	0	0
$751$	31819.0i	1.54606i	0.634369	+	0.773030i	$0.281260\pi$
$751$	31819.0i	1.54606i	−0.634369	+	0.773030i	$0.718740\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	16096.1i	0.775891i
$756$	0	0
$757$	− 28248.0i	− 1.35626i	−0.734940	−	0.678132i	$-0.762790\pi$
$757$	− 28248.0i	− 1.35626i	0.734940	−	0.678132i	$-0.237210\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	17951.2i	0.855097i	0.903992	+	0.427548i	$0.140623\pi$
$761$	17951.2i	0.855097i	−0.903992	+	0.427548i	$0.859377\pi$
$762$	0	0
$763$	14861.0	0.705117
$764$	0	0
$765$	0	0
$766$	0	0
$767$	15697.2	0.738974
$768$	0	0
$769$	15503.0	0.726986	0.363493	−	0.931597i	$-0.381584\pi$
$769$	15503.0	0.726986	0.363493	+	0.931597i	$0.381584\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−4275.77	−0.198951	−0.0994753	−	0.995040i	$-0.531716\pi$
$773$	−4275.77	−0.198951	−0.0994753	+	0.995040i	$0.531716\pi$
$774$	0	0
$775$	19780.0i	0.916798i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	21843.6i	1.00466i
$780$	0	0
$781$	9145.23i	0.419004i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	13201.7i	0.600243i
$786$	0	0
$787$	3864.21	0.175024	0.0875121	−	0.996163i	$-0.472108\pi$
$787$	3864.21	0.175024	0.0875121	+	0.996163i	$0.472108\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	29650.3	1.33280
$792$	0	0
$793$	−10569.0	−0.473287
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27296.8	1.21318	0.606588	−	0.795016i	$-0.292538\pi$
$797$	27296.8	1.21318	0.606588	+	0.795016i	$0.292538\pi$
$798$	0	0
$799$	7920.00i	0.350675i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6057.35i	0.266201i
$804$	0	0
$805$	5404.00i	0.236604i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 10464.8i	− 0.454787i	−0.973803	−	0.227394i	$-0.926980\pi$
$809$	− 10464.8i	− 0.454787i	0.973803	−	0.227394i	$-0.0730204\pi$
$810$	0	0
$811$	−10340.3	−0.447717	−0.223859	−	0.974622i	$-0.571865\pi$
$811$	−10340.3	−0.447717	−0.223859	+	0.974622i	$0.571865\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−9847.64	−0.423249
$816$	0	0
$817$	−16074.0	−0.688321
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6088.33	0.258811	0.129406	−	0.991592i	$-0.458693\pi$
$821$	6088.33	0.258811	0.129406	+	0.991592i	$0.458693\pi$
$822$	0	0
$823$	− 5719.00i	− 0.242226i	−0.992639	−	0.121113i	$-0.961354\pi$
$823$	− 5719.00i	− 0.242226i	0.992639	−	0.121113i	$-0.0386463\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 14268.1i	− 0.599939i	−0.953949	−	0.299969i	$-0.903024\pi$
$827$	− 14268.1i	− 0.599939i	0.953949	−	0.299969i	$-0.0969765\pi$
$828$	0	0
$829$	33381.8i	1.39855i	0.714853	+	0.699275i	$0.246494\pi$
$829$	33381.8i	1.39855i	−0.714853	+	0.699275i	$0.753506\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4668.91i	0.194199i
$834$	0	0
$835$	61106.8	2.53256
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−43308.2	−1.78208	−0.891039	−	0.453926i	$-0.850023\pi$
$839$	−43308.2	−1.78208	−0.891039	+	0.453926i	$0.850023\pi$
$840$	0	0
$841$	10171.0	0.417032
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−26181.4	−1.06588
$846$	0	0
$847$	− 14183.0i	− 0.575364i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	883.040i	0.0355702i
$852$	0	0
$853$	− 29855.4i	− 1.19839i	−0.800603	−	0.599196i	$-0.795487\pi$
$853$	− 29855.4i	− 1.19839i	0.800603	−	0.599196i	$-0.204513\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 27101.1i	− 1.08023i	−0.841591	−	0.540115i	$-0.818381\pi$
$857$	− 27101.1i	− 1.08023i	0.841591	−	0.540115i	$-0.181619\pi$
$858$	0	0
$859$	−12171.1	−0.483438	−0.241719	−	0.970346i	$-0.577711\pi$
$859$	−12171.1	−0.483438	−0.241719	+	0.970346i	$0.577711\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7379.02	0.291060	0.145530	−	0.989354i	$-0.453511\pi$
$863$	7379.02	0.291060	0.145530	+	0.989354i	$0.453511\pi$
$864$	0	0
$865$	−13920.0	−0.547161
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6398.17	0.249762
$870$	0	0
$871$	429.000i	0.0166890i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 2013.95i	− 0.0778103i
$876$	0	0
$877$	38276.6i	1.47378i	0.676010	+	0.736892i	$0.263707\pi$
$877$	38276.6i	1.47378i	−0.676010	+	0.736892i	$0.736293\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 46340.3i	− 1.77213i	−0.463564	−	0.886063i	$-0.653430\pi$
$881$	− 46340.3i	− 1.77213i	0.463564	−	0.886063i	$-0.346570\pi$
$882$	0	0
$883$	−24347.4	−0.927924	−0.463962	−	0.885855i	$-0.653573\pi$
$883$	−24347.4	−0.927924	−0.463962	+	0.885855i	$0.653573\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−5044.57	−0.190958	−0.0954792	−	0.995431i	$-0.530438\pi$
$887$	−5044.57	−0.190958	−0.0954792	+	0.995431i	$0.530438\pi$
$888$	0	0
$889$	−29380.0	−1.10841
$890$	0	0
$891$	0	0
$892$	0	0
$893$	24028.0	0.900410
$894$	0	0
$895$	45600.0i	1.70306i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 31975.4i	− 1.18625i
$900$	0	0
$901$	− 14133.5i	− 0.522593i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	53424.1i	1.96230i
$906$	0	0
$907$	14738.0	0.539546	0.269773	−	0.962924i	$-0.413051\pi$
$907$	14738.0	0.539546	0.269773	+	0.962924i	$0.413051\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	16690.0	0.606987	0.303493	−	0.952834i	$-0.401847\pi$
$911$	16690.0	0.606987	0.303493	+	0.952834i	$0.401847\pi$
$912$	0	0
$913$	−1920.00	−0.0695977
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6444.64	−0.232084
$918$	0	0
$919$	45476.0i	1.63233i	0.577816	+	0.816167i	$0.303905\pi$
$919$	45476.0i	1.63233i	−0.577816	+	0.816167i	$0.696095\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 13292.1i	− 0.474013i
$924$	0	0
$925$	3784.53i	0.134524i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 8371.84i	− 0.295663i	−0.989013	−	0.147832i	$-0.952771\pi$
$929$	− 8371.84i	− 0.295663i	0.989013	−	0.147832i	$-0.0472294\pi$
$930$	0	0
$931$	14164.7	0.498636
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6439.88	0.225248
$936$	0	0
$937$	10453.0	0.364445	0.182222	−	0.983257i	$-0.441671\pi$
$937$	10453.0	0.364445	0.182222	+	0.983257i	$0.441671\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	32703.5	1.13295	0.566473	−	0.824080i	$-0.308307\pi$
$941$	32703.5	1.13295	0.566473	+	0.824080i	$0.308307\pi$
$942$	0	0
$943$	7200.00i	0.248637i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1192.88i	0.0409328i	0.999791	+	0.0204664i	$0.00651511\pi$
$947$	1192.88i	0.0409328i	−0.999791	+	0.0204664i	$0.993485\pi$
$948$	0	0
$949$	− 8804.01i	− 0.301149i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 24444.7i	− 0.830893i	−0.909618	−	0.415447i	$-0.863625\pi$
$953$	− 24444.7i	− 0.830893i	0.909618	−	0.415447i	$-0.136375\pi$
$954$	0	0
$955$	−6235.38	−0.211280
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−38022.1	−1.28029
$960$	0	0
$961$	207.000	0.00694841
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−27126.4	−0.904900
$966$	0	0
$967$	45643.0i	1.51787i	0.651167	+	0.758935i	$0.274280\pi$
$967$	45643.0i	1.51787i	−0.651167	+	0.758935i	$0.725720\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 15972.2i	− 0.527880i	−0.964539	−	0.263940i	$-0.914978\pi$
$971$	− 15972.2i	− 0.527880i	0.964539	−	0.263940i	$-0.0850221\pi$
$972$	0	0
$973$	38210.8i	1.25897i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	11377.1i	0.372555i	0.982497	+	0.186277i	$0.0596423\pi$
$977$	11377.1i	0.372555i	−0.982497	+	0.186277i	$0.940358\pi$
$978$	0	0
$979$	20368.9	0.664958
$980$	0	0
$981$	0	0
$982$	0	0
$983$	18434.1	0.598126	0.299063	−	0.954233i	$-0.403326\pi$
$983$	18434.1	0.598126	0.299063	+	0.954233i	$0.403326\pi$
$984$	0	0
$985$	−1680.00	−0.0543444
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5298.24	−0.170348
$990$	0	0
$991$	40081.0i	1.28478i	0.766379	+	0.642389i	$0.222057\pi$
$991$	40081.0i	1.28478i	−0.766379	+	0.642389i	$0.777943\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	11510.5i	0.366741i
$996$	0	0
$997$	− 31821.2i	− 1.01082i	−0.862879	−	0.505411i	$-0.831341\pi$
$997$	− 31821.2i	− 1.01082i	0.862879	−	0.505411i	$-0.168659\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.f.e.863.1	✓	8
3.2	odd	2	inner	1728.4.f.e.863.5	yes	8
4.3	odd	2	inner	1728.4.f.e.863.3	yes	8
8.3	odd	2	inner	1728.4.f.e.863.8	yes	8
8.5	even	2	inner	1728.4.f.e.863.6	yes	8
12.11	even	2	inner	1728.4.f.e.863.7	yes	8
24.5	odd	2	inner	1728.4.f.e.863.2	yes	8
24.11	even	2	inner	1728.4.f.e.863.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.4.f.e.863.1	✓	8	1.1	even	1	trivial
1728.4.f.e.863.2	yes	8	24.5	odd	2	inner
1728.4.f.e.863.3	yes	8	4.3	odd	2	inner
1728.4.f.e.863.4	yes	8	24.11	even	2	inner
1728.4.f.e.863.5	yes	8	3.2	odd	2	inner
1728.4.f.e.863.6	yes	8	8.5	even	2	inner
1728.4.f.e.863.7	yes	8	12.11	even	2	inner
1728.4.f.e.863.8	yes	8	8.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-15.4919 q^{5} -13.0000i q^{7} +O(q^{10})\)	\(q-15.4919 q^{5} -13.0000i q^{7} +15.4919i q^{11} -22.5167i q^{13} +26.8328i q^{17} +81.4064 q^{19} +26.8328 q^{23} +115.000 q^{25} -185.903 q^{29} +172.000i q^{31} +201.395i q^{35} +32.9090i q^{37} +268.328i q^{41} -197.454 q^{43} +295.161 q^{47} +174.000 q^{49} -526.726 q^{53} -240.000i q^{55} +697.137i q^{59} -469.386i q^{61} +348.827i q^{65} -19.0526 q^{67} +590.322 q^{71} +391.000 q^{73} +201.395 q^{77} -413.000i q^{79} +123.935i q^{83} -415.692i q^{85} -1314.81i q^{89} -292.717 q^{91} -1261.14 q^{95} -427.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 920 q^{25} + 1392 q^{49} + 3128 q^{73} - 3416 q^{97}+O(q^{100}) \) 8 * q + 920 * q^25 + 1392 * q^49 + 3128 * q^73 - 3416 * q^97

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