LMFDB - Embedded newform 1728.4.d.f.865.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,4,Mod(865,1728)]

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([0, 1, 0]))

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

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

chi := DirichletCharacter("1728.865");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.d (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:	8.0.592240896.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 7x^{6} + 40x^{4} - 63x^{2} + 81 $ x^8 - 7x^6 + 40x^4 - 63*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			865.5
Root			$1.12824 + 0.651388i$ of defining polynomial
Character	$\chi$	$=$	1728.865
Dual form			1728.4.d.f.865.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+12.4900i q^{5} -12.1244 q^{7} +O(q^{10})$	$q+12.4900i q^{5} -12.1244 q^{7} -21.6333i q^{11} -15.5885i q^{13} -64.8999 q^{17} +49.0000i q^{19} +62.4500 q^{23} -31.0000 q^{25} -24.9800i q^{29} -24.2487 q^{31} -151.433i q^{35} -102.191i q^{37} -346.133 q^{41} +260.000i q^{43} +362.210 q^{47} -196.000 q^{49} -574.540i q^{53} +270.200 q^{55} -324.500i q^{59} -174.937i q^{61} +194.700 q^{65} -241.000i q^{67} -249.800 q^{71} +353.000 q^{73} +262.290i q^{77} +5.19615 q^{79} -1038.40i q^{83} -810.600i q^{85} -800.432 q^{89} +189.000i q^{91} -612.010 q^{95} +1111.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 248 q^{25} - 1568 q^{49} + 2824 q^{73} + 8888 q^{97}+O(q^{100}) $ 8 * q - 248 * q^25 - 1568 * q^49 + 2824 * q^73 + 8888 * 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
$5$	12.4900i	1.11714i	0.829458	+	0.558570i	$0.188650\pi$
$5$	12.4900i	1.11714i	−0.829458	+	0.558570i	$0.811350\pi$
$6$	0	0
$7$	−12.1244	−0.654654	−0.327327	−	0.944911i	$-0.606148\pi$
$7$	−12.1244	−0.654654	−0.327327	+	0.944911i	$0.606148\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 21.6333i	− 0.592972i	−0.955037	−	0.296486i	$-0.904185\pi$
$11$	− 21.6333i	− 0.592972i	0.955037	−	0.296486i	$-0.0958147\pi$
$12$	0	0
$13$	− 15.5885i	− 0.332574i	−0.986077	−	0.166287i	$-0.946822\pi$
$13$	− 15.5885i	− 0.332574i	0.986077	−	0.166287i	$-0.0531778\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−64.8999	−0.925914	−0.462957	−	0.886381i	$-0.653212\pi$
$17$	−64.8999	−0.925914	−0.462957	+	0.886381i	$0.653212\pi$
$18$	0	0
$19$	49.0000i	0.591651i	0.955242	+	0.295826i	$0.0955947\pi$
$19$	49.0000i	0.591651i	−0.955242	+	0.295826i	$0.904405\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	62.4500	0.566162	0.283081	−	0.959096i	$-0.408644\pi$
$23$	62.4500	0.566162	0.283081	+	0.959096i	$0.408644\pi$
$24$	0	0
$25$	−31.0000	−0.248000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 24.9800i	− 0.159954i	−0.996797	−	0.0799770i	$-0.974515\pi$
$29$	− 24.9800i	− 0.159954i	0.996797	−	0.0799770i	$-0.0254847\pi$
$30$	0	0
$31$	−24.2487	−0.140490	−0.0702451	−	0.997530i	$-0.522378\pi$
$31$	−24.2487	−0.140490	−0.0702451	+	0.997530i	$0.522378\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 151.433i	− 0.731339i
$36$	0	0
$37$	− 102.191i	− 0.454057i	−0.973888	−	0.227028i	$-0.927099\pi$
$37$	− 102.191i	− 0.454057i	0.973888	−	0.227028i	$-0.0729010\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−346.133	−1.31846	−0.659230	−	0.751941i	$-0.729118\pi$
$41$	−346.133	−1.31846	−0.659230	+	0.751941i	$0.729118\pi$
$42$	0	0
$43$	260.000i	0.922084i	0.887378	+	0.461042i	$0.152524\pi$
$43$	260.000i	0.922084i	−0.887378	+	0.461042i	$0.847476\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	362.210	1.12412	0.562061	−	0.827096i	$-0.310009\pi$
$47$	362.210	1.12412	0.562061	+	0.827096i	$0.310009\pi$
$48$	0	0
$49$	−196.000	−0.571429
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 574.540i	− 1.48904i	−0.667600	−	0.744520i	$-0.732678\pi$
$53$	− 574.540i	− 1.48904i	0.667600	−	0.744520i	$-0.267322\pi$
$54$	0	0
$55$	270.200	0.662432
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 324.500i	− 0.716038i	−0.933714	−	0.358019i	$-0.883452\pi$
$59$	− 324.500i	− 0.716038i	0.933714	−	0.358019i	$-0.116548\pi$
$60$	0	0
$61$	− 174.937i	− 0.367187i	−0.983002	−	0.183593i	$-0.941227\pi$
$61$	− 174.937i	− 0.367187i	0.983002	−	0.183593i	$-0.0587730\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	194.700	0.371531
$66$	0	0
$67$	− 241.000i	− 0.439445i	−0.975562	−	0.219723i	$-0.929485\pi$
$67$	− 241.000i	− 0.439445i	0.975562	−	0.219723i	$-0.0705152\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−249.800	−0.417546	−0.208773	−	0.977964i	$-0.566947\pi$
$71$	−249.800	−0.417546	−0.208773	+	0.977964i	$0.566947\pi$
$72$	0	0
$73$	353.000	0.565966	0.282983	−	0.959125i	$-0.408676\pi$
$73$	353.000	0.565966	0.282983	+	0.959125i	$0.408676\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	262.290i	0.388191i
$78$	0	0
$79$	5.19615	0.00740016	0.00370008	−	0.999993i	$-0.498822\pi$
$79$	5.19615	0.00740016	0.00370008	+	0.999993i	$0.498822\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 1038.40i	− 1.37324i	−0.727016	−	0.686621i	$-0.759093\pi$
$83$	− 1038.40i	− 1.37324i	0.727016	−	0.686621i	$-0.240907\pi$
$84$	0	0
$85$	− 810.600i	− 1.03438i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−800.432	−0.953322	−0.476661	−	0.879087i	$-0.658153\pi$
$89$	−800.432	−0.953322	−0.476661	+	0.879087i	$0.658153\pi$
$90$	0	0
$91$	189.000i	0.217721i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−612.010	−0.660957
$96$	0	0
$97$	1111.00	1.16294	0.581469	−	0.813569i	$-0.302478\pi$
$97$	1111.00	1.16294	0.581469	+	0.813569i	$0.302478\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	24.9800i	0.0246099i	0.999924	+	0.0123050i	$0.00391689\pi$
$101$	24.9800i	0.0246099i	−0.999924	+	0.0123050i	$0.996083\pi$
$102$	0	0
$103$	1557.11	1.48958	0.744791	−	0.667298i	$-0.232549\pi$
$103$	1557.11	1.48958	0.744791	+	0.667298i	$0.232549\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1752.30i	1.58319i	0.611048	+	0.791594i	$0.290748\pi$
$107$	1752.30i	1.58319i	−0.611048	+	0.791594i	$0.709252\pi$
$108$	0	0
$109$	69.2820i	0.0608809i	0.999537	+	0.0304404i	$0.00969099\pi$
$109$	69.2820i	0.0608809i	−0.999537	+	0.0304404i	$0.990309\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	281.233	0.234125	0.117063	−	0.993125i	$-0.462652\pi$
$113$	281.233	0.234125	0.117063	+	0.993125i	$0.462652\pi$
$114$	0	0
$115$	780.000i	0.632482i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	786.870	0.606153
$120$	0	0
$121$	863.000	0.648385
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1174.06i	0.840089i
$126$	0	0
$127$	1125.83	0.786626	0.393313	−	0.919405i	$-0.371329\pi$
$127$	1125.83	0.786626	0.393313	+	0.919405i	$0.371329\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1341.27i	0.894556i	0.894395	+	0.447278i	$0.147607\pi$
$131$	1341.27i	0.894556i	−0.894395	+	0.447278i	$0.852393\pi$
$132$	0	0
$133$	− 594.093i	− 0.387327i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3093.56	1.92920	0.964602	−	0.263710i	$-0.0849463\pi$
$137$	3093.56	1.92920	0.964602	+	0.263710i	$0.0849463\pi$
$138$	0	0
$139$	− 233.000i	− 0.142178i	−0.997470	−	0.0710892i	$-0.977352\pi$
$139$	− 233.000i	− 0.142178i	0.997470	−	0.0710892i	$-0.0226475\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−337.230	−0.197207
$144$	0	0
$145$	312.000	0.178691
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 1199.04i	− 0.659256i	−0.944111	−	0.329628i	$-0.893077\pi$
$149$	− 1199.04i	− 0.659256i	0.944111	−	0.329628i	$-0.106923\pi$
$150$	0	0
$151$	2149.48	1.15842	0.579211	−	0.815177i	$-0.303361\pi$
$151$	2149.48	1.15842	0.579211	+	0.815177i	$0.303361\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 302.866i	− 0.156947i
$156$	0	0
$157$	− 3062.27i	− 1.55666i	−0.627856	−	0.778329i	$-0.716067\pi$
$157$	− 3062.27i	− 1.55666i	0.627856	−	0.778329i	$-0.283933\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−757.166	−0.370640
$162$	0	0
$163$	2095.00i	1.00671i	0.864081	+	0.503353i	$0.167900\pi$
$163$	2095.00i	1.00671i	−0.864081	+	0.503353i	$0.832100\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−287.270	−0.133111	−0.0665557	−	0.997783i	$-0.521201\pi$
$167$	−287.270	−0.133111	−0.0665557	+	0.997783i	$0.521201\pi$
$168$	0	0
$169$	1954.00	0.889395
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4246.60i	1.86626i	0.359540	+	0.933130i	$0.382934\pi$
$173$	4246.60i	1.86626i	−0.359540	+	0.933130i	$0.617066\pi$
$174$	0	0
$175$	375.855	0.162354
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2682.53i	1.12012i	0.828452	+	0.560061i	$0.189222\pi$
$179$	2682.53i	1.12012i	−0.828452	+	0.560061i	$0.810778\pi$
$180$	0	0
$181$	− 2277.65i	− 0.935338i	−0.883904	−	0.467669i	$-0.845094\pi$
$181$	− 2277.65i	− 0.935338i	0.883904	−	0.467669i	$-0.154906\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1276.37	0.507244
$186$	0	0
$187$	1404.00i	0.549041i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−736.910	−0.279167	−0.139584	−	0.990210i	$-0.544576\pi$
$191$	−736.910	−0.279167	−0.139584	+	0.990210i	$0.544576\pi$
$192$	0	0
$193$	−37.0000	−0.0137996	−0.00689979	−	0.999976i	$-0.502196\pi$
$193$	−37.0000	−0.0137996	−0.00689979	+	0.999976i	$0.502196\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3035.07i	− 1.09766i	−0.835933	−	0.548832i	$-0.815073\pi$
$197$	− 3035.07i	− 1.09766i	0.835933	−	0.548832i	$-0.184927\pi$
$198$	0	0
$199$	4286.83	1.52706	0.763530	−	0.645772i	$-0.223464\pi$
$199$	4286.83	1.52706	0.763530	+	0.645772i	$0.223464\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	302.866i	0.104715i
$204$	0	0
$205$	− 4323.20i	− 1.47290i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1060.03	0.350832
$210$	0	0
$211$	1829.00i	0.596747i	0.954449	+	0.298373i	$0.0964440\pi$
$211$	1829.00i	0.596747i	−0.954449	+	0.298373i	$0.903556\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3247.40	−1.03010
$216$	0	0
$217$	294.000	0.0919724
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1011.69i	0.307935i
$222$	0	0
$223$	1687.02	0.506597	0.253298	−	0.967388i	$-0.418485\pi$
$223$	1687.02	0.506597	0.253298	+	0.967388i	$0.418485\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3461.33i	− 1.01206i	−0.862517	−	0.506028i	$-0.831113\pi$
$227$	− 3461.33i	− 1.01206i	0.862517	−	0.506028i	$-0.168887\pi$
$228$	0	0
$229$	− 4579.54i	− 1.32151i	−0.750604	−	0.660753i	$-0.770237\pi$
$229$	− 4579.54i	− 1.32151i	0.750604	−	0.660753i	$-0.229763\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2293.13	−0.644755	−0.322378	−	0.946611i	$-0.604482\pi$
$233$	−2293.13	−0.644755	−0.322378	+	0.946611i	$0.604482\pi$
$234$	0	0
$235$	4524.00i	1.25580i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	724.420	0.196062	0.0980310	−	0.995183i	$-0.468746\pi$
$239$	724.420	0.196062	0.0980310	+	0.995183i	$0.468746\pi$
$240$	0	0
$241$	−379.000	−0.101301	−0.0506505	−	0.998716i	$-0.516129\pi$
$241$	−379.000	−0.101301	−0.0506505	+	0.998716i	$0.516129\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 2448.04i	− 0.638365i
$246$	0	0
$247$	763.834	0.196768
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 3201.73i	− 0.805145i	−0.915388	−	0.402572i	$-0.868116\pi$
$251$	− 3201.73i	− 0.805145i	0.915388	−	0.402572i	$-0.131884\pi$
$252$	0	0
$253$	− 1351.00i	− 0.335718i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2293.13	−0.556582	−0.278291	−	0.960497i	$-0.589768\pi$
$257$	−2293.13	−0.556582	−0.278291	+	0.960497i	$0.589768\pi$
$258$	0	0
$259$	1239.00i	0.297250i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	824.340	0.193274	0.0966368	−	0.995320i	$-0.469191\pi$
$263$	824.340	0.193274	0.0966368	+	0.995320i	$0.469191\pi$
$264$	0	0
$265$	7176.00	1.66346
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 262.290i	− 0.0594502i	−0.999558	−	0.0297251i	$-0.990537\pi$
$269$	− 262.290i	− 0.0594502i	0.999558	−	0.0297251i	$-0.00946318\pi$
$270$	0	0
$271$	8197.80	1.83757	0.918784	−	0.394762i	$-0.129173\pi$
$271$	8197.80	1.83757	0.918784	+	0.394762i	$0.129173\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	670.633i	0.147057i
$276$	0	0
$277$	− 2438.73i	− 0.528985i	−0.964388	−	0.264493i	$-0.914796\pi$
$277$	− 2438.73i	− 0.528985i	0.964388	−	0.264493i	$-0.0852045\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2639.26	−0.560303	−0.280152	−	0.959956i	$-0.590385\pi$
$281$	−2639.26	−0.560303	−0.280152	+	0.959956i	$0.590385\pi$
$282$	0	0
$283$	3292.00i	0.691481i	0.938330	+	0.345740i	$0.112372\pi$
$283$	3292.00i	0.691481i	−0.938330	+	0.345740i	$0.887628\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4196.64	0.863135
$288$	0	0
$289$	−701.000	−0.142683
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 3085.03i	− 0.615117i	−0.951529	−	0.307559i	$-0.900488\pi$
$293$	− 3085.03i	− 0.615117i	0.951529	−	0.307559i	$-0.0995120\pi$
$294$	0	0
$295$	4053.00	0.799914
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 973.499i	− 0.188291i
$300$	0	0
$301$	− 3152.33i	− 0.603646i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2184.96	0.410199
$306$	0	0
$307$	2372.00i	0.440968i	0.975391	+	0.220484i	$0.0707637\pi$
$307$	2372.00i	0.440968i	−0.975391	+	0.220484i	$0.929236\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8705.53	1.58728	0.793642	−	0.608385i	$-0.208182\pi$
$311$	8705.53	1.58728	0.793642	+	0.608385i	$0.208182\pi$
$312$	0	0
$313$	3337.00	0.602615	0.301307	−	0.953527i	$-0.402577\pi$
$313$	3337.00	0.602615	0.301307	+	0.953527i	$0.402577\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1798.56i	− 0.318666i	−0.987225	−	0.159333i	$-0.949066\pi$
$317$	− 1798.56i	− 0.318666i	0.987225	−	0.159333i	$-0.0509344\pi$
$318$	0	0
$319$	−540.400	−0.0948482
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 3180.10i	− 0.547818i
$324$	0	0
$325$	483.242i	0.0824783i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4391.56	−0.735911
$330$	0	0
$331$	− 2057.00i	− 0.341580i	−0.985307	−	0.170790i	$-0.945368\pi$
$331$	− 2057.00i	− 0.341580i	0.985307	−	0.170790i	$-0.0546320\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3010.09	0.490922
$336$	0	0
$337$	6271.00	1.01366	0.506830	−	0.862046i	$-0.330817\pi$
$337$	6271.00	1.01366	0.506830	+	0.862046i	$0.330817\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	524.580i	0.0833067i
$342$	0	0
$343$	6535.03	1.02874
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 5581.39i	− 0.863472i	−0.902000	−	0.431736i	$-0.857901\pi$
$347$	− 5581.39i	− 0.863472i	0.902000	−	0.431736i	$-0.142099\pi$
$348$	0	0
$349$	− 11034.9i	− 1.69251i	−0.532782	−	0.846253i	$-0.678853\pi$
$349$	− 11034.9i	− 1.69251i	0.532782	−	0.846253i	$-0.321147\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8436.99	−1.27211	−0.636056	−	0.771643i	$-0.719435\pi$
$353$	−8436.99	−1.27211	−0.636056	+	0.771643i	$0.719435\pi$
$354$	0	0
$355$	− 3120.00i	− 0.466457i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4259.09	−0.626145	−0.313073	−	0.949729i	$-0.601358\pi$
$359$	−4259.09	−0.626145	−0.313073	+	0.949729i	$0.601358\pi$
$360$	0	0
$361$	4458.00	0.649949
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4408.97i	0.632263i
$366$	0	0
$367$	5901.10	0.839332	0.419666	−	0.907679i	$-0.362147\pi$
$367$	5901.10	0.839332	0.419666	+	0.907679i	$0.362147\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6965.93i	0.974805i
$372$	0	0
$373$	− 8651.59i	− 1.20097i	−0.799635	−	0.600486i	$-0.794974\pi$
$373$	− 8651.59i	− 1.20097i	0.799635	−	0.600486i	$-0.205026\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−389.400	−0.0531965
$378$	0	0
$379$	335.000i	0.0454032i	0.999742	+	0.0227016i	$0.00722676\pi$
$379$	335.000i	0.0454032i	−0.999742	+	0.0227016i	$0.992773\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−14413.5	−1.92296	−0.961479	−	0.274877i	$-0.911363\pi$
$383$	−14413.5	−1.92296	−0.961479	+	0.274877i	$0.911363\pi$
$384$	0	0
$385$	−3276.00	−0.433663
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 1211.53i	− 0.157910i	−0.996878	−	0.0789550i	$-0.974842\pi$
$389$	− 1211.53i	− 0.157910i	0.996878	−	0.0789550i	$-0.0251583\pi$
$390$	0	0
$391$	−4053.00	−0.524217
$392$	0	0
$393$	0	0
$394$	0	0
$395$	64.8999i	0.00826701i
$396$	0	0
$397$	13856.4i	1.75172i	0.482565	+	0.875860i	$0.339705\pi$
$397$	13856.4i	1.75172i	−0.482565	+	0.875860i	$0.660295\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6360.19	0.792052	0.396026	−	0.918239i	$-0.370389\pi$
$401$	6360.19	0.792052	0.396026	+	0.918239i	$0.370389\pi$
$402$	0	0
$403$	378.000i	0.0467234i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2210.73	−0.269243
$408$	0	0
$409$	9389.00	1.13510	0.567550	−	0.823339i	$-0.307891\pi$
$409$	9389.00	1.13510	0.567550	+	0.823339i	$0.307891\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3934.35i	0.468757i
$414$	0	0
$415$	12969.6	1.53410
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 2358.03i	− 0.274934i	−0.990506	−	0.137467i	$-0.956104\pi$
$419$	− 2358.03i	− 0.274934i	0.990506	−	0.137467i	$-0.0438961\pi$
$420$	0	0
$421$	1945.09i	0.225173i	0.993642	+	0.112587i	$0.0359136\pi$
$421$	1945.09i	0.225173i	−0.993642	+	0.112587i	$0.964086\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2011.90	0.229627
$426$	0	0
$427$	2121.00i	0.240380i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16424.3	1.83558	0.917788	−	0.397072i	$-0.129974\pi$
$431$	16424.3	1.83558	0.917788	+	0.397072i	$0.129974\pi$
$432$	0	0
$433$	−15662.0	−1.73826	−0.869131	−	0.494581i	$-0.835321\pi$
$433$	−15662.0	−1.73826	−0.869131	+	0.494581i	$0.835321\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3060.05i	0.334970i
$438$	0	0
$439$	−5996.36	−0.651915	−0.325957	−	0.945384i	$-0.605687\pi$
$439$	−5996.36	−0.651915	−0.325957	+	0.945384i	$0.605687\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7831.26i	0.839897i	0.907548	+	0.419948i	$0.137952\pi$
$443$	7831.26i	0.839897i	−0.907548	+	0.419948i	$0.862048\pi$
$444$	0	0
$445$	− 9997.40i	− 1.06499i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11573.8	1.21649	0.608243	−	0.793751i	$-0.291875\pi$
$449$	11573.8	1.21649	0.608243	+	0.793751i	$0.291875\pi$
$450$	0	0
$451$	7488.00i	0.781810i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2360.61	−0.243224
$456$	0	0
$457$	5066.00	0.518550	0.259275	−	0.965803i	$-0.416516\pi$
$457$	5066.00	0.518550	0.259275	+	0.965803i	$0.416516\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 13376.8i	− 1.35145i	−0.737153	−	0.675726i	$-0.763830\pi$
$461$	− 13376.8i	− 1.35145i	0.737153	−	0.675726i	$-0.236170\pi$
$462$	0	0
$463$	−6157.44	−0.618057	−0.309029	−	0.951053i	$-0.600004\pi$
$463$	−6157.44	−0.618057	−0.309029	+	0.951053i	$0.600004\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6122.23i	− 0.606644i	−0.952888	−	0.303322i	$-0.901904\pi$
$467$	− 6122.23i	− 0.606644i	0.952888	−	0.303322i	$-0.0980958\pi$
$468$	0	0
$469$	2921.97i	0.287684i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5624.66	0.546770
$474$	0	0
$475$	− 1519.00i	− 0.146729i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−17610.9	−1.67988	−0.839940	−	0.542680i	$-0.817410\pi$
$479$	−17610.9	−1.67988	−0.839940	+	0.542680i	$0.817410\pi$
$480$	0	0
$481$	−1593.00	−0.151007
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13876.4i	1.29916i
$486$	0	0
$487$	4442.71	0.413385	0.206692	−	0.978406i	$-0.433730\pi$
$487$	4442.71	0.413385	0.206692	+	0.978406i	$0.433730\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9367.22i	0.860971i	0.902597	+	0.430486i	$0.141658\pi$
$491$	9367.22i	0.860971i	−0.902597	+	0.430486i	$0.858342\pi$
$492$	0	0
$493$	1621.20i	0.148104i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3028.66	0.273348
$498$	0	0
$499$	− 6088.00i	− 0.546165i	−0.961991	−	0.273082i	$-0.911957\pi$
$499$	− 6088.00i	− 0.546165i	0.961991	−	0.273082i	$-0.0880432\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3159.97	−0.280111	−0.140056	−	0.990144i	$-0.544728\pi$
$503$	−3159.97	−0.280111	−0.140056	+	0.990144i	$0.544728\pi$
$504$	0	0
$505$	−312.000	−0.0274927
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 5632.99i	− 0.490526i	−0.969457	−	0.245263i	$-0.921126\pi$
$509$	− 5632.99i	− 0.490526i	0.969457	−	0.245263i	$-0.0788743\pi$
$510$	0	0
$511$	−4279.90	−0.370512
$512$	0	0
$513$	0	0
$514$	0	0
$515$	19448.3i	1.66407i
$516$	0	0
$517$	− 7835.80i	− 0.666573i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	11963.2	1.00598	0.502992	−	0.864291i	$-0.332232\pi$
$521$	11963.2	1.00598	0.502992	+	0.864291i	$0.332232\pi$
$522$	0	0
$523$	12611.0i	1.05438i	0.849748	+	0.527190i	$0.176754\pi$
$523$	12611.0i	1.05438i	−0.849748	+	0.527190i	$0.823246\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1573.74	0.130082
$528$	0	0
$529$	−8267.00	−0.679461
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5395.68i	0.438485i
$534$	0	0
$535$	−21886.2	−1.76864
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4240.13i	0.338841i
$540$	0	0
$541$	14873.1i	1.18197i	0.806683	+	0.590985i	$0.201261\pi$
$541$	14873.1i	1.18197i	−0.806683	+	0.590985i	$0.798739\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−865.332	−0.0680124
$546$	0	0
$547$	4291.00i	0.335411i	0.985837	+	0.167706i	$0.0536358\pi$
$547$	4291.00i	0.335411i	−0.985837	+	0.167706i	$0.946364\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1224.02	0.0946370
$552$	0	0
$553$	−63.0000	−0.00484454
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 8056.05i	− 0.612829i	−0.951898	−	0.306414i	$-0.900871\pi$
$557$	− 8056.05i	− 0.612829i	0.951898	−	0.306414i	$-0.0991293\pi$
$558$	0	0
$559$	4053.00	0.306661
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 10124.4i	− 0.757890i	−0.925419	−	0.378945i	$-0.876287\pi$
$563$	− 10124.4i	− 0.757890i	0.925419	−	0.378945i	$-0.123713\pi$
$564$	0	0
$565$	3512.60i	0.261551i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7723.09	−0.569014	−0.284507	−	0.958674i	$-0.591830\pi$
$569$	−7723.09	−0.569014	−0.284507	+	0.958674i	$0.591830\pi$
$570$	0	0
$571$	− 21143.0i	− 1.54957i	−0.632222	−	0.774787i	$-0.717857\pi$
$571$	− 21143.0i	− 1.54957i	0.632222	−	0.774787i	$-0.282143\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1935.95	−0.140408
$576$	0	0
$577$	−20459.0	−1.47612	−0.738058	−	0.674737i	$-0.764257\pi$
$577$	−20459.0	−1.47612	−0.738058	+	0.674737i	$0.764257\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	12589.9i	0.898998i
$582$	0	0
$583$	−12429.2	−0.882958
$584$	0	0
$585$	0	0
$586$	0	0
$587$	281.233i	0.0197747i	0.999951	+	0.00988733i	$0.00314729\pi$
$587$	281.233i	0.0197747i	−0.999951	+	0.00988733i	$0.996853\pi$
$588$	0	0
$589$	− 1188.19i	− 0.0831212i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−10297.5	−0.713096	−0.356548	−	0.934277i	$-0.616046\pi$
$593$	−10297.5	−0.713096	−0.356548	+	0.934277i	$0.616046\pi$
$594$	0	0
$595$	9828.00i	0.677158i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	17261.2	1.17742	0.588708	−	0.808346i	$-0.299637\pi$
$599$	17261.2	1.17742	0.588708	+	0.808346i	$0.299637\pi$
$600$	0	0
$601$	22646.0	1.53702	0.768510	−	0.639837i	$-0.220998\pi$
$601$	22646.0	1.53702	0.768510	+	0.639837i	$0.220998\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10778.9i	0.724336i
$606$	0	0
$607$	−18510.4	−1.23775	−0.618876	−	0.785489i	$-0.712412\pi$
$607$	−18510.4	−1.23775	−0.618876	+	0.785489i	$0.712412\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 5646.29i	− 0.373854i
$612$	0	0
$613$	22501.1i	1.48256i	0.671196	+	0.741280i	$0.265781\pi$
$613$	22501.1i	1.48256i	−0.671196	+	0.741280i	$0.734219\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4651.16	0.303482	0.151741	−	0.988420i	$-0.451512\pi$
$617$	4651.16	0.303482	0.151741	+	0.988420i	$0.451512\pi$
$618$	0	0
$619$	− 13067.0i	− 0.848477i	−0.905551	−	0.424238i	$-0.860542\pi$
$619$	− 13067.0i	− 0.848477i	0.905551	−	0.424238i	$-0.139458\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	9704.73	0.624096
$624$	0	0
$625$	−18539.0	−1.18650
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6632.19i	0.420418i
$630$	0	0
$631$	18250.6	1.15142	0.575710	−	0.817654i	$-0.304726\pi$
$631$	18250.6	1.15142	0.575710	+	0.817654i	$0.304726\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	14061.6i	0.878771i
$636$	0	0
$637$	3055.34i	0.190042i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−20335.3	−1.25304	−0.626518	−	0.779407i	$-0.715521\pi$
$641$	−20335.3	−1.25304	−0.626518	+	0.779407i	$0.715521\pi$
$642$	0	0
$643$	− 23612.0i	− 1.44816i	−0.689716	−	0.724080i	$-0.742265\pi$
$643$	− 23612.0i	− 1.44816i	0.689716	−	0.724080i	$-0.257735\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−26029.2	−1.58163	−0.790813	−	0.612058i	$-0.790342\pi$
$647$	−26029.2	−1.58163	−0.790813	+	0.612058i	$0.790342\pi$
$648$	0	0
$649$	−7020.00	−0.424590
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 10341.7i	− 0.619759i	−0.950776	−	0.309880i	$-0.899711\pi$
$653$	− 10341.7i	− 0.619759i	0.950776	−	0.309880i	$-0.100289\pi$
$654$	0	0
$655$	−16752.4	−0.999344
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 23580.3i	− 1.39387i	−0.717136	−	0.696933i	$-0.754547\pi$
$659$	− 23580.3i	− 1.39387i	0.717136	−	0.696933i	$-0.245453\pi$
$660$	0	0
$661$	− 3230.27i	− 0.190080i	−0.995473	−	0.0950402i	$-0.969702\pi$
$661$	− 3230.27i	− 0.190080i	0.995473	−	0.0950402i	$-0.0302980\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7420.22	0.432698
$666$	0	0
$667$	− 1560.00i	− 0.0905599i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3784.47	−0.217731
$672$	0	0
$673$	−21911.0	−1.25499	−0.627494	−	0.778621i	$-0.715919\pi$
$673$	−21911.0	−1.25499	−0.627494	+	0.778621i	$0.715919\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1711.13i	− 0.0971404i	−0.998820	−	0.0485702i	$-0.984534\pi$
$677$	− 1711.13i	− 0.0971404i	0.998820	−	0.0485702i	$-0.0154665\pi$
$678$	0	0
$679$	−13470.2	−0.761321
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 22888.0i	− 1.28226i	−0.767431	−	0.641132i	$-0.778465\pi$
$683$	− 22888.0i	− 1.28226i	0.767431	−	0.641132i	$-0.221535\pi$
$684$	0	0
$685$	38638.6i	2.15519i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−8956.19	−0.495216
$690$	0	0
$691$	− 7796.00i	− 0.429195i	−0.976703	−	0.214598i	$-0.931156\pi$
$691$	− 7796.00i	− 0.429195i	0.976703	−	0.214598i	$-0.0688440\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2910.17	0.158833
$696$	0	0
$697$	22464.0	1.22078
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 21994.9i	− 1.18507i	−0.805544	−	0.592536i	$-0.798127\pi$
$701$	− 21994.9i	− 1.18507i	0.805544	−	0.592536i	$-0.201873\pi$
$702$	0	0
$703$	5007.36	0.268643
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 302.866i	− 0.0161110i
$708$	0	0
$709$	28043.6i	1.48547i	0.669583	+	0.742737i	$0.266473\pi$
$709$	28043.6i	1.48547i	−0.669583	+	0.742737i	$0.733527\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1514.33	−0.0795402
$714$	0	0
$715$	− 4212.00i	− 0.220308i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	34622.3	1.79582	0.897909	−	0.440182i	$-0.145086\pi$
$719$	34622.3	1.79582	0.897909	+	0.440182i	$0.145086\pi$
$720$	0	0
$721$	−18879.0	−0.975160
$722$	0	0
$723$	0	0
$724$	0	0
$725$	774.380i	0.0396686i
$726$	0	0
$727$	−12834.5	−0.654753	−0.327376	−	0.944894i	$-0.606164\pi$
$727$	−12834.5	−0.654753	−0.327376	+	0.944894i	$0.606164\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 16874.0i	− 0.853771i
$732$	0	0
$733$	− 17223.5i	− 0.867892i	−0.900939	−	0.433946i	$-0.857121\pi$
$733$	− 17223.5i	− 0.867892i	0.900939	−	0.433946i	$-0.142879\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5213.63	−0.260579
$738$	0	0
$739$	− 9964.00i	− 0.495983i	−0.968762	−	0.247992i	$-0.920229\pi$
$739$	− 9964.00i	− 0.495983i	0.968762	−	0.247992i	$-0.0797706\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17323.6	0.855373	0.427687	−	0.903927i	$-0.359329\pi$
$743$	17323.6	0.855373	0.427687	+	0.903927i	$0.359329\pi$
$744$	0	0
$745$	14976.0	0.736481
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 21245.5i	− 1.03644i
$750$	0	0
$751$	−24028.7	−1.16754	−0.583769	−	0.811920i	$-0.698423\pi$
$751$	−24028.7	−1.16754	−0.583769	+	0.811920i	$0.698423\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	26846.9i	1.29412i
$756$	0	0
$757$	38408.2i	1.84408i	0.387091	+	0.922041i	$0.373480\pi$
$757$	38408.2i	1.84408i	−0.387091	+	0.922041i	$0.626520\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−23169.3	−1.10366	−0.551830	−	0.833957i	$-0.686070\pi$
$761$	−23169.3	−1.10366	−0.551830	+	0.833957i	$0.686070\pi$
$762$	0	0
$763$	− 840.000i	− 0.0398559i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5058.45	−0.238136
$768$	0	0
$769$	−8833.00	−0.414208	−0.207104	−	0.978319i	$-0.566404\pi$
$769$	−8833.00	−0.414208	−0.207104	+	0.978319i	$0.566404\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 12490.0i	− 0.581156i	−0.956851	−	0.290578i	$-0.906152\pi$
$773$	− 12490.0i	− 0.581156i	0.956851	−	0.290578i	$-0.0938476\pi$
$774$	0	0
$775$	751.710	0.0348416
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 16960.5i	− 0.780069i
$780$	0	0
$781$	5404.00i	0.247593i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	38247.7	1.73900
$786$	0	0
$787$	− 39791.0i	− 1.80228i	−0.433526	−	0.901141i	$-0.642731\pi$
$787$	− 39791.0i	− 1.80228i	0.433526	−	0.901141i	$-0.357269\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3409.77	−0.153271
$792$	0	0
$793$	−2727.00	−0.122117
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 28052.5i	− 1.24676i	−0.781917	−	0.623382i	$-0.785758\pi$
$797$	− 28052.5i	− 1.24676i	0.781917	−	0.623382i	$-0.214242\pi$
$798$	0	0
$799$	−23507.4	−1.04084
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 7636.56i	− 0.335602i
$804$	0	0
$805$	− 9457.00i	− 0.414056i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−37815.0	−1.64339	−0.821697	−	0.569925i	$-0.806972\pi$
$809$	−37815.0	−1.64339	−0.821697	+	0.569925i	$0.806972\pi$
$810$	0	0
$811$	1460.00i	0.0632152i	0.999500	+	0.0316076i	$0.0100627\pi$
$811$	1460.00i	0.0632152i	−0.999500	+	0.0316076i	$0.989937\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−26166.5	−1.12463
$816$	0	0
$817$	−12740.0	−0.545552
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28714.5i	1.22064i	0.792156	+	0.610319i	$0.208959\pi$
$821$	28714.5i	1.22064i	−0.792156	+	0.610319i	$0.791041\pi$
$822$	0	0
$823$	−34424.5	−1.45804	−0.729018	−	0.684495i	$-0.760023\pi$
$823$	−34424.5	−1.45804	−0.729018	+	0.684495i	$0.760023\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13131.4i	0.552145i	0.961137	+	0.276073i	$0.0890330\pi$
$827$	13131.4i	0.552145i	−0.961137	+	0.276073i	$0.910967\pi$
$828$	0	0
$829$	31982.3i	1.33992i	0.742398	+	0.669959i	$0.233688\pi$
$829$	31982.3i	1.33992i	−0.742398	+	0.669959i	$0.766312\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	12720.4	0.529094
$834$	0	0
$835$	− 3588.00i	− 0.148704i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1673.66	0.0688691	0.0344345	−	0.999407i	$-0.489037\pi$
$839$	1673.66	0.0688691	0.0344345	+	0.999407i	$0.489037\pi$
$840$	0	0
$841$	23765.0	0.974415
$842$	0	0
$843$	0	0
$844$	0	0
$845$	24405.5i	0.993578i
$846$	0	0
$847$	−10463.3	−0.424467
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 6381.83i	− 0.257070i
$852$	0	0
$853$	− 6663.20i	− 0.267460i	−0.991018	−	0.133730i	$-0.957304\pi$
$853$	− 6663.20i	− 0.267460i	0.991018	−	0.133730i	$-0.0426955\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3980.53	−0.158661	−0.0793304	−	0.996848i	$-0.525278\pi$
$857$	−3980.53	−0.158661	−0.0793304	+	0.996848i	$0.525278\pi$
$858$	0	0
$859$	− 511.000i	− 0.0202970i	−0.999949	−	0.0101485i	$-0.996770\pi$
$859$	− 511.000i	− 0.0202970i	0.999949	−	0.0101485i	$-0.00323042\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−21570.2	−0.850821	−0.425411	−	0.905000i	$-0.639870\pi$
$863$	−21570.2	−0.850821	−0.425411	+	0.905000i	$0.639870\pi$
$864$	0	0
$865$	−53040.0	−2.08487
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 112.410i	− 0.00438809i
$870$	0	0
$871$	−3756.82	−0.146148
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 14234.7i	− 0.549967i
$876$	0	0
$877$	11474.8i	0.441822i	0.975294	+	0.220911i	$0.0709030\pi$
$877$	11474.8i	0.441822i	−0.975294	+	0.220911i	$0.929097\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−28707.4	−1.09782	−0.548909	−	0.835882i	$-0.684956\pi$
$881$	−28707.4	−1.09782	−0.548909	+	0.835882i	$0.684956\pi$
$882$	0	0
$883$	32443.0i	1.23646i	0.785997	+	0.618230i	$0.212150\pi$
$883$	32443.0i	1.23646i	−0.785997	+	0.618230i	$0.787850\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14238.6	0.538991	0.269496	−	0.963002i	$-0.413143\pi$
$887$	14238.6	0.538991	0.269496	+	0.963002i	$0.413143\pi$
$888$	0	0
$889$	−13650.0	−0.514968
$890$	0	0
$891$	0	0
$892$	0	0
$893$	17748.3i	0.665088i
$894$	0	0
$895$	−33504.8	−1.25133
$896$	0	0
$897$	0	0
$898$	0	0
$899$	605.733i	0.0224720i
$900$	0	0
$901$	37287.6i	1.37872i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	28447.8	1.04490
$906$	0	0
$907$	− 18013.0i	− 0.659440i	−0.944079	−	0.329720i	$-0.893046\pi$
$907$	− 18013.0i	− 0.659440i	0.944079	−	0.329720i	$-0.106954\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	26029.2	0.946635	0.473317	−	0.880892i	$-0.343056\pi$
$911$	26029.2	0.946635	0.473317	+	0.880892i	$0.343056\pi$
$912$	0	0
$913$	−22464.0	−0.814293
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 16262.0i	− 0.585625i
$918$	0	0
$919$	43782.8	1.57156	0.785778	−	0.618508i	$-0.212263\pi$
$919$	43782.8	1.57156	0.785778	+	0.618508i	$0.212263\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3894.00i	0.138865i
$924$	0	0
$925$	3167.92i	0.112606i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	50405.6	1.78014	0.890072	−	0.455819i	$-0.150654\pi$
$929$	50405.6	1.78014	0.890072	+	0.455819i	$0.150654\pi$
$930$	0	0
$931$	− 9604.00i	− 0.338086i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−17536.0	−0.613355
$936$	0	0
$937$	11449.0	0.399170	0.199585	−	0.979880i	$-0.436041\pi$
$937$	11449.0	0.399170	0.199585	+	0.979880i	$0.436041\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	43927.3i	1.52177i	0.648884	+	0.760887i	$0.275236\pi$
$941$	43927.3i	1.52177i	−0.648884	+	0.760887i	$0.724764\pi$
$942$	0	0
$943$	−21616.0	−0.746462
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15208.2i	0.521859i	0.965358	+	0.260930i	$0.0840290\pi$
$947$	15208.2i	0.521859i	−0.965358	+	0.260930i	$0.915971\pi$
$948$	0	0
$949$	− 5502.73i	− 0.188226i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	21828.0	0.741950	0.370975	−	0.928643i	$-0.379024\pi$
$953$	21828.0	0.741950	0.370975	+	0.928643i	$0.379024\pi$
$954$	0	0
$955$	− 9204.00i	− 0.311869i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−37507.5	−1.26296
$960$	0	0
$961$	−29203.0	−0.980262
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 462.130i	− 0.0154160i
$966$	0	0
$967$	−31351.9	−1.04261	−0.521307	−	0.853369i	$-0.674555\pi$
$967$	−31351.9	−1.04261	−0.521307	+	0.853369i	$0.674555\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	41038.4i	1.35632i	0.734915	+	0.678159i	$0.237222\pi$
$971$	41038.4i	1.35632i	−0.734915	+	0.678159i	$0.762778\pi$
$972$	0	0
$973$	2824.97i	0.0930776i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−31325.0	−1.02577	−0.512885	−	0.858458i	$-0.671423\pi$
$977$	−31325.0	−1.02577	−0.512885	+	0.858458i	$0.671423\pi$
$978$	0	0
$979$	17316.0i	0.565293i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	29189.1	0.947089	0.473544	−	0.880770i	$-0.342974\pi$
$983$	29189.1	0.947089	0.473544	+	0.880770i	$0.342974\pi$
$984$	0	0
$985$	37908.0	1.22624
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16237.0i	0.522049i
$990$	0	0
$991$	17831.5	0.571579	0.285790	−	0.958292i	$-0.407744\pi$
$991$	17831.5	0.571579	0.285790	+	0.958292i	$0.407744\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	53542.4i	1.70594i
$996$	0	0
$997$	− 20535.2i	− 0.652313i	−0.945316	−	0.326157i	$-0.894246\pi$
$997$	− 20535.2i	− 0.652313i	0.945316	−	0.326157i	$-0.105754\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.d.f.865.5	yes	8
3.2	odd	2	inner	1728.4.d.f.865.1	✓	8
4.3	odd	2	inner	1728.4.d.f.865.7	yes	8
8.3	odd	2	inner	1728.4.d.f.865.4	yes	8
8.5	even	2	inner	1728.4.d.f.865.2	yes	8
12.11	even	2	inner	1728.4.d.f.865.3	yes	8
24.5	odd	2	inner	1728.4.d.f.865.6	yes	8
24.11	even	2	inner	1728.4.d.f.865.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.4.d.f.865.1	✓	8	3.2	odd	2	inner
1728.4.d.f.865.2	yes	8	8.5	even	2	inner
1728.4.d.f.865.3	yes	8	12.11	even	2	inner
1728.4.d.f.865.4	yes	8	8.3	odd	2	inner
1728.4.d.f.865.5	yes	8	1.1	even	1	trivial
1728.4.d.f.865.6	yes	8	24.5	odd	2	inner
1728.4.d.f.865.7	yes	8	4.3	odd	2	inner
1728.4.d.f.865.8	yes	8	24.11	even	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.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+12.4900i q^{5} -12.1244 q^{7} +O(q^{10})\)	\(q+12.4900i q^{5} -12.1244 q^{7} -21.6333i q^{11} -15.5885i q^{13} -64.8999 q^{17} +49.0000i q^{19} +62.4500 q^{23} -31.0000 q^{25} -24.9800i q^{29} -24.2487 q^{31} -151.433i q^{35} -102.191i q^{37} -346.133 q^{41} +260.000i q^{43} +362.210 q^{47} -196.000 q^{49} -574.540i q^{53} +270.200 q^{55} -324.500i q^{59} -174.937i q^{61} +194.700 q^{65} -241.000i q^{67} -249.800 q^{71} +353.000 q^{73} +262.290i q^{77} +5.19615 q^{79} -1038.40i q^{83} -810.600i q^{85} -800.432 q^{89} +189.000i q^{91} -612.010 q^{95} +1111.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 248 q^{25} - 1568 q^{49} + 2824 q^{73} + 8888 q^{97}+O(q^{100}) \) 8 * q - 248 * q^25 - 1568 * q^49 + 2824 * q^73 + 8888 * q^97

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