LMFDB - Embedded newform 864.2.c.a.863.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(863,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("864.863");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	864.c (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:	$6.89907473464$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			863.2
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	864.863
Dual form			864.2.c.a.863.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.14626i q^{5} +3.44949i q^{7} +O(q^{10})$	$q-3.14626i q^{5} +3.44949i q^{7} -4.56048 q^{11} -6.89898 q^{13} +3.46410i q^{17} +4.89898i q^{19} +2.82843 q^{23} -4.89898 q^{25} +2.19275i q^{29} -2.55051i q^{31} +10.8530 q^{35} -4.89898 q^{37} +4.09978i q^{41} +2.89898i q^{43} -2.19275 q^{47} -4.89898 q^{49} -12.9029i q^{53} +14.3485i q^{55} +2.19275 q^{59} +4.00000 q^{61} +21.7060i q^{65} +14.8990i q^{67} -13.2207 q^{71} -7.89898 q^{73} -15.7313i q^{77} +2.00000i q^{79} -12.4101 q^{83} +10.8990 q^{85} -5.02118i q^{89} -23.7980i q^{91} +15.4135 q^{95} -5.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 16 q^{13} + 32 q^{61} - 24 q^{73} + 48 q^{85} - 40 q^{97}+O(q^{100}) $ 8 * q - 16 * q^13 + 32 * q^61 - 24 * q^73 + 48 * q^85 - 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	− 3.14626i	− 1.40705i	−0.710669	−	0.703526i	$-0.751608\pi$
$5$	− 3.14626i	− 1.40705i	0.710669	−	0.703526i	$-0.248392\pi$
$6$	0	0
$7$	3.44949i	1.30378i	0.758312	+	0.651892i	$0.226025\pi$
$7$	3.44949i	1.30378i	−0.758312	+	0.651892i	$0.773975\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.56048	−1.37504	−0.687518	−	0.726167i	$-0.741300\pi$
$11$	−4.56048	−1.37504	−0.687518	+	0.726167i	$0.741300\pi$
$12$	0	0
$13$	−6.89898	−1.91343	−0.956716	−	0.291022i	$-0.906005\pi$
$13$	−6.89898	−1.91343	−0.956716	+	0.291022i	$0.906005\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.46410i	0.840168i	0.907485	+	0.420084i	$0.137999\pi$
$17$	3.46410i	0.840168i	−0.907485	+	0.420084i	$0.862001\pi$
$18$	0	0
$19$	4.89898i	1.12390i	0.827170	+	0.561951i	$0.189949\pi$
$19$	4.89898i	1.12390i	−0.827170	+	0.561951i	$0.810051\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	−4.89898	−0.979796
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.19275i	0.407184i	0.979056	+	0.203592i	$0.0652616\pi$
$29$	2.19275i	0.407184i	−0.979056	+	0.203592i	$0.934738\pi$
$30$	0	0
$31$	− 2.55051i	− 0.458085i	−0.973416	−	0.229043i	$-0.926440\pi$
$31$	− 2.55051i	− 0.458085i	0.973416	−	0.229043i	$-0.0735595\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	10.8530	1.83449
$36$	0	0
$37$	−4.89898	−0.805387	−0.402694	−	0.915335i	$-0.631926\pi$
$37$	−4.89898	−0.805387	−0.402694	+	0.915335i	$0.631926\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.09978i	0.640277i	0.947371	+	0.320139i	$0.103730\pi$
$41$	4.09978i	0.640277i	−0.947371	+	0.320139i	$0.896270\pi$
$42$	0	0
$43$	2.89898i	0.442090i	0.975264	+	0.221045i	$0.0709468\pi$
$43$	2.89898i	0.442090i	−0.975264	+	0.221045i	$0.929053\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.19275	−0.319846	−0.159923	−	0.987130i	$-0.551125\pi$
$47$	−2.19275	−0.319846	−0.159923	+	0.987130i	$0.551125\pi$
$48$	0	0
$49$	−4.89898	−0.699854
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 12.9029i	− 1.77235i	−0.463352	−	0.886174i	$-0.653353\pi$
$53$	− 12.9029i	− 1.77235i	0.463352	−	0.886174i	$-0.346647\pi$
$54$	0	0
$55$	14.3485i	1.93475i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.19275	0.285472	0.142736	−	0.989761i	$-0.454410\pi$
$59$	2.19275	0.285472	0.142736	+	0.989761i	$0.454410\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	21.7060i	2.69230i
$66$	0	0
$67$	14.8990i	1.82020i	0.414389	+	0.910100i	$0.363995\pi$
$67$	14.8990i	1.82020i	−0.414389	+	0.910100i	$0.636005\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.2207	−1.56901	−0.784506	−	0.620121i	$-0.787083\pi$
$71$	−13.2207	−1.56901	−0.784506	+	0.620121i	$0.787083\pi$
$72$	0	0
$73$	−7.89898	−0.924506	−0.462253	−	0.886748i	$-0.652959\pi$
$73$	−7.89898	−0.924506	−0.462253	+	0.886748i	$0.652959\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 15.7313i	− 1.79275i
$78$	0	0
$79$	2.00000i	0.225018i	0.993651	+	0.112509i	$0.0358886\pi$
$79$	2.00000i	0.225018i	−0.993651	+	0.112509i	$0.964111\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−12.4101	−1.36218	−0.681092	−	0.732198i	$-0.738495\pi$
$83$	−12.4101	−1.36218	−0.681092	+	0.732198i	$0.738495\pi$
$84$	0	0
$85$	10.8990	1.18216
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 5.02118i	− 0.532244i	−0.963939	−	0.266122i	$-0.914258\pi$
$89$	− 5.02118i	− 0.532244i	0.963939	−	0.266122i	$-0.0857424\pi$
$90$	0	0
$91$	− 23.7980i	− 2.49470i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	15.4135	1.58139
$96$	0	0
$97$	−5.00000	−0.507673	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000	−0.507673	−0.253837	+	0.967247i	$0.581693\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.58919i	− 0.158130i	−0.996869	−	0.0790650i	$-0.974807\pi$
$101$	− 1.58919i	− 0.158130i	0.996869	−	0.0790650i	$-0.0251935\pi$
$102$	0	0
$103$	− 10.0000i	− 0.985329i	−0.870219	−	0.492665i	$-0.836023\pi$
$103$	− 10.0000i	− 0.985329i	0.870219	−	0.492665i	$-0.163977\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.66025	−0.837218	−0.418609	−	0.908166i	$-0.637482\pi$
$107$	−8.66025	−0.837218	−0.418609	+	0.908166i	$0.637482\pi$
$108$	0	0
$109$	4.89898	0.469237	0.234619	−	0.972088i	$-0.424616\pi$
$109$	4.89898	0.469237	0.234619	+	0.972088i	$0.424616\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 14.1421i	− 1.33038i	−0.746674	−	0.665190i	$-0.768350\pi$
$113$	− 14.1421i	− 1.33038i	0.746674	−	0.665190i	$-0.231650\pi$
$114$	0	0
$115$	− 8.89898i	− 0.829834i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−11.9494	−1.09540
$120$	0	0
$121$	9.79796	0.890724
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 0.317837i	− 0.0284282i
$126$	0	0
$127$	− 9.24745i	− 0.820578i	−0.911955	−	0.410289i	$-0.865428\pi$
$127$	− 9.24745i	− 0.820578i	0.911955	−	0.410289i	$-0.134572\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.3171	1.25089	0.625446	−	0.780268i	$-0.284917\pi$
$131$	14.3171	1.25089	0.625446	+	0.780268i	$0.284917\pi$
$132$	0	0
$133$	−16.8990	−1.46533
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.6062i	1.50420i	0.659048	+	0.752101i	$0.270960\pi$
$137$	17.6062i	1.50420i	−0.659048	+	0.752101i	$0.729040\pi$
$138$	0	0
$139$	20.0000i	1.69638i	0.529694	+	0.848189i	$0.322307\pi$
$139$	20.0000i	1.69638i	−0.529694	+	0.848189i	$0.677693\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	31.4626	2.63104
$144$	0	0
$145$	6.89898	0.572929
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.61037i	0.541542i	0.962644	+	0.270771i	$0.0872787\pi$
$149$	6.61037i	0.541542i	−0.962644	+	0.270771i	$0.912721\pi$
$150$	0	0
$151$	− 7.24745i	− 0.589789i	−0.955530	−	0.294895i	$-0.904715\pi$
$151$	− 7.24745i	− 0.589789i	0.955530	−	0.294895i	$-0.0952845\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−8.02458	−0.644550
$156$	0	0
$157$	5.79796	0.462728	0.231364	−	0.972867i	$-0.425681\pi$
$157$	5.79796	0.462728	0.231364	+	0.972867i	$0.425681\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	9.75663i	0.768930i
$162$	0	0
$163$	− 6.00000i	− 0.469956i	−0.972001	−	0.234978i	$-0.924498\pi$
$163$	− 6.00000i	− 0.469956i	0.972001	−	0.234978i	$-0.0755019\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−11.6637	−0.902561	−0.451280	−	0.892382i	$-0.649033\pi$
$167$	−11.6637	−0.902561	−0.451280	+	0.892382i	$0.649033\pi$
$168$	0	0
$169$	34.5959	2.66122
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.80312i	0.669289i	0.942344	+	0.334644i	$0.108616\pi$
$173$	8.80312i	0.669289i	−0.942344	+	0.334644i	$0.891384\pi$
$174$	0	0
$175$	− 16.8990i	− 1.27744i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.66025	0.647298	0.323649	−	0.946177i	$-0.395090\pi$
$179$	8.66025	0.647298	0.323649	+	0.946177i	$0.395090\pi$
$180$	0	0
$181$	9.79796	0.728277	0.364138	−	0.931345i	$-0.381364\pi$
$181$	9.79796	0.728277	0.364138	+	0.931345i	$0.381364\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	15.4135i	1.13322i
$186$	0	0
$187$	− 15.7980i	− 1.15526i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.6780	−0.772635	−0.386318	−	0.922366i	$-0.626253\pi$
$191$	−10.6780	−0.772635	−0.386318	+	0.922366i	$0.626253\pi$
$192$	0	0
$193$	11.8990	0.856507	0.428254	−	0.903659i	$-0.359129\pi$
$193$	11.8990	0.856507	0.428254	+	0.903659i	$0.359129\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1.87492i	− 0.133582i	−0.997767	−	0.0667911i	$-0.978724\pi$
$197$	− 1.87492i	− 0.133582i	0.997767	−	0.0667911i	$-0.0212761\pi$
$198$	0	0
$199$	5.65153i	0.400626i	0.979732	+	0.200313i	$0.0641960\pi$
$199$	5.65153i	0.400626i	−0.979732	+	0.200313i	$0.935804\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.56388	−0.530880
$204$	0	0
$205$	12.8990	0.900904
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 22.3417i	− 1.54541i
$210$	0	0
$211$	− 20.4949i	− 1.41093i	−0.708746	−	0.705463i	$-0.750739\pi$
$211$	− 20.4949i	− 1.41093i	0.708746	−	0.705463i	$-0.249261\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.12096	0.622044
$216$	0	0
$217$	8.79796	0.597244
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 23.8988i	− 1.60760i
$222$	0	0
$223$	19.7980i	1.32577i	0.748721	+	0.662885i	$0.230668\pi$
$223$	19.7980i	1.32577i	−0.748721	+	0.662885i	$0.769332\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	22.9774	1.52506	0.762531	−	0.646952i	$-0.223957\pi$
$227$	22.9774	1.52506	0.762531	+	0.646952i	$0.223957\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 7.56388i	− 0.495526i	−0.968821	−	0.247763i	$-0.920305\pi$
$233$	− 7.56388i	− 0.495526i	0.968821	−	0.247763i	$-0.0796954\pi$
$234$	0	0
$235$	6.89898i	0.450040i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−7.21393	−0.466630	−0.233315	−	0.972401i	$-0.574957\pi$
$239$	−7.21393	−0.466630	−0.233315	+	0.972401i	$0.574957\pi$
$240$	0	0
$241$	−0.202041	−0.0130146	−0.00650730	−	0.999979i	$-0.502071\pi$
$241$	−0.202041	−0.0130146	−0.00650730	+	0.999979i	$0.502071\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	15.4135i	0.984731i
$246$	0	0
$247$	− 33.7980i	− 2.15051i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−17.3205	−1.09326	−0.546630	−	0.837374i	$-0.684090\pi$
$251$	−17.3205	−1.09326	−0.546630	+	0.837374i	$0.684090\pi$
$252$	0	0
$253$	−12.8990	−0.810952
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 22.9774i	− 1.43329i	−0.697439	−	0.716644i	$-0.745677\pi$
$257$	− 22.9774i	− 1.43329i	0.697439	−	0.716644i	$-0.254323\pi$
$258$	0	0
$259$	− 16.8990i	− 1.05005i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−30.5412	−1.88325	−0.941627	−	0.336659i	$-0.890703\pi$
$263$	−30.5412	−1.88325	−0.941627	+	0.336659i	$0.890703\pi$
$264$	0	0
$265$	−40.5959	−2.49379
$266$	0	0
$267$	0	0
$268$	0	0
$269$	10.3923i	0.633630i	0.948487	+	0.316815i	$0.102613\pi$
$269$	10.3923i	0.633630i	−0.948487	+	0.316815i	$0.897387\pi$
$270$	0	0
$271$	17.4495i	1.05998i	0.848004	+	0.529991i	$0.177805\pi$
$271$	17.4495i	1.05998i	−0.848004	+	0.529991i	$0.822195\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	22.3417	1.34725
$276$	0	0
$277$	5.79796	0.348366	0.174183	−	0.984713i	$-0.444272\pi$
$277$	5.79796	0.348366	0.174183	+	0.984713i	$0.444272\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 21.7060i	− 1.29487i	−0.762120	−	0.647436i	$-0.775841\pi$
$281$	− 21.7060i	− 1.29487i	0.762120	−	0.647436i	$-0.224159\pi$
$282$	0	0
$283$	12.8990i	0.766765i	0.923590	+	0.383382i	$0.125241\pi$
$283$	12.8990i	0.766765i	−0.923590	+	0.383382i	$0.874759\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−14.1421	−0.834784
$288$	0	0
$289$	5.00000	0.294118
$290$	0	0
$291$	0	0
$292$	0	0
$293$	28.6342i	1.67283i	0.548098	+	0.836414i	$0.315352\pi$
$293$	28.6342i	1.67283i	−0.548098	+	0.836414i	$0.684648\pi$
$294$	0	0
$295$	− 6.89898i	− 0.401674i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−19.5133	−1.12848
$300$	0	0
$301$	−10.0000	−0.576390
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 12.5851i	− 0.720618i
$306$	0	0
$307$	− 18.0000i	− 1.02731i	−0.857996	−	0.513657i	$-0.828290\pi$
$307$	− 18.0000i	− 1.02731i	0.857996	−	0.513657i	$-0.171710\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.75663	0.553248	0.276624	−	0.960978i	$-0.410784\pi$
$311$	9.75663	0.553248	0.276624	+	0.960978i	$0.410784\pi$
$312$	0	0
$313$	−19.4949	−1.10192	−0.550958	−	0.834533i	$-0.685738\pi$
$313$	−19.4949	−1.10192	−0.550958	+	0.834533i	$0.685738\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.4456i	0.867511i	0.901031	+	0.433755i	$0.142812\pi$
$317$	15.4456i	0.867511i	−0.901031	+	0.433755i	$0.857188\pi$
$318$	0	0
$319$	− 10.0000i	− 0.559893i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−16.9706	−0.944267
$324$	0	0
$325$	33.7980	1.87477
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 7.56388i	− 0.417010i
$330$	0	0
$331$	28.6969i	1.57733i	0.614825	+	0.788663i	$0.289226\pi$
$331$	28.6969i	1.57733i	−0.614825	+	0.788663i	$0.710774\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	46.8761	2.56112
$336$	0	0
$337$	23.7980	1.29636	0.648179	−	0.761488i	$-0.275531\pi$
$337$	23.7980	1.29636	0.648179	+	0.761488i	$0.275531\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	11.6315i	0.629884i
$342$	0	0
$343$	7.24745i	0.391325i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−22.8024	−1.22410	−0.612048	−	0.790820i	$-0.709654\pi$
$347$	−22.8024	−1.22410	−0.612048	+	0.790820i	$0.709654\pi$
$348$	0	0
$349$	15.1010	0.808339	0.404170	−	0.914684i	$-0.367561\pi$
$349$	15.1010	0.808339	0.404170	+	0.914684i	$0.367561\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.1421i	0.752710i	0.926476	+	0.376355i	$0.122823\pi$
$353$	14.1421i	0.752710i	−0.926476	+	0.376355i	$0.877177\pi$
$354$	0	0
$355$	41.5959i	2.20768i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−0.349945	−0.0184694	−0.00923470	−	0.999957i	$-0.502940\pi$
$359$	−0.349945	−0.0184694	−0.00923470	+	0.999957i	$0.502940\pi$
$360$	0	0
$361$	−5.00000	−0.263158
$362$	0	0
$363$	0	0
$364$	0	0
$365$	24.8523i	1.30083i
$366$	0	0
$367$	13.4495i	0.702058i	0.936365	+	0.351029i	$0.114168\pi$
$367$	13.4495i	0.702058i	−0.936365	+	0.351029i	$0.885832\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	44.5084	2.31076
$372$	0	0
$373$	−14.4949	−0.750517	−0.375259	−	0.926920i	$-0.622446\pi$
$373$	−14.4949	−0.750517	−0.375259	+	0.926920i	$0.622446\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 15.1278i	− 0.779119i
$378$	0	0
$379$	− 8.00000i	− 0.410932i	−0.978664	−	0.205466i	$-0.934129\pi$
$379$	− 8.00000i	− 0.410932i	0.978664	−	0.205466i	$-0.0658711\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	24.5344	1.25365	0.626826	−	0.779160i	$-0.284354\pi$
$383$	24.5344	1.25365	0.626826	+	0.779160i	$0.284354\pi$
$384$	0	0
$385$	−49.4949	−2.52249
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 21.6739i	− 1.09891i	−0.835523	−	0.549455i	$-0.814835\pi$
$389$	− 21.6739i	− 1.09891i	0.835523	−	0.549455i	$-0.185165\pi$
$390$	0	0
$391$	9.79796i	0.495504i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6.29253	0.316611
$396$	0	0
$397$	−20.6969	−1.03875	−0.519375	−	0.854547i	$-0.673835\pi$
$397$	−20.6969	−1.03875	−0.519375	+	0.854547i	$0.673835\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	22.6274i	1.12996i	0.825105	+	0.564980i	$0.191116\pi$
$401$	22.6274i	1.12996i	−0.825105	+	0.564980i	$0.808884\pi$
$402$	0	0
$403$	17.5959i	0.876515i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	22.3417	1.10744
$408$	0	0
$409$	−7.00000	−0.346128	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000	−0.346128	−0.173064	+	0.984911i	$0.555367\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.56388i	0.372194i
$414$	0	0
$415$	39.0454i	1.91666i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−19.1633	−0.936189	−0.468095	−	0.883678i	$-0.655059\pi$
$419$	−19.1633	−0.936189	−0.468095	+	0.883678i	$0.655059\pi$
$420$	0	0
$421$	−10.2020	−0.497217	−0.248609	−	0.968604i	$-0.579973\pi$
$421$	−10.2020	−0.497217	−0.248609	+	0.968604i	$0.579973\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 16.9706i	− 0.823193i
$426$	0	0
$427$	13.7980i	0.667730i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6.57826	−0.316864	−0.158432	−	0.987370i	$-0.550644\pi$
$431$	−6.57826	−0.316864	−0.158432	+	0.987370i	$0.550644\pi$
$432$	0	0
$433$	−26.3939	−1.26841	−0.634204	−	0.773165i	$-0.718672\pi$
$433$	−26.3939	−1.26841	−0.634204	+	0.773165i	$0.718672\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	13.8564i	0.662842i
$438$	0	0
$439$	5.24745i	0.250447i	0.992129	+	0.125224i	$0.0399648\pi$
$439$	5.24745i	0.250447i	−0.992129	+	0.125224i	$0.960035\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6.00680	−0.285392	−0.142696	−	0.989767i	$-0.545577\pi$
$443$	−6.00680	−0.285392	−0.142696	+	0.989767i	$0.545577\pi$
$444$	0	0
$445$	−15.7980	−0.748895
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 5.37113i	− 0.253479i	−0.991936	−	0.126740i	$-0.959549\pi$
$449$	− 5.37113i	− 0.253479i	0.991936	−	0.126740i	$-0.0404512\pi$
$450$	0	0
$451$	− 18.6969i	− 0.880404i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−74.8747	−3.51018
$456$	0	0
$457$	25.0000	1.16945	0.584725	−	0.811231i	$-0.301202\pi$
$457$	25.0000	1.16945	0.584725	+	0.811231i	$0.301202\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 12.2672i	− 0.571341i	−0.958328	−	0.285671i	$-0.907784\pi$
$461$	− 12.2672i	− 0.571341i	0.958328	−	0.285671i	$-0.0922164\pi$
$462$	0	0
$463$	17.0454i	0.792167i	0.918214	+	0.396084i	$0.129631\pi$
$463$	17.0454i	0.792167i	−0.918214	+	0.396084i	$0.870369\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.1455	0.793401	0.396700	−	0.917948i	$-0.370155\pi$
$467$	17.1455	0.793401	0.396700	+	0.917948i	$0.370155\pi$
$468$	0	0
$469$	−51.3939	−2.37315
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 13.2207i	− 0.607890i
$474$	0	0
$475$	− 24.0000i	− 1.10120i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	19.1633	0.875594	0.437797	−	0.899074i	$-0.355759\pi$
$479$	19.1633	0.875594	0.437797	+	0.899074i	$0.355759\pi$
$480$	0	0
$481$	33.7980	1.54105
$482$	0	0
$483$	0	0
$484$	0	0
$485$	15.7313i	0.714323i
$486$	0	0
$487$	− 15.7980i	− 0.715874i	−0.933746	−	0.357937i	$-0.883480\pi$
$487$	− 15.7980i	− 0.715874i	0.933746	−	0.357937i	$-0.116520\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.00340	−0.135542	−0.0677708	−	0.997701i	$-0.521589\pi$
$491$	−3.00340	−0.135542	−0.0677708	+	0.997701i	$0.521589\pi$
$492$	0	0
$493$	−7.59592	−0.342103
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 45.6048i	− 2.04565i
$498$	0	0
$499$	− 5.10102i	− 0.228353i	−0.993460	−	0.114177i	$-0.963577\pi$
$499$	− 5.10102i	− 0.228353i	0.993460	−	0.114177i	$-0.0364229\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−31.1127	−1.38725	−0.693623	−	0.720338i	$-0.743987\pi$
$503$	−31.1127	−1.38725	−0.693623	+	0.720338i	$0.743987\pi$
$504$	0	0
$505$	−5.00000	−0.222497
$506$	0	0
$507$	0	0
$508$	0	0
$509$	44.0156i	1.95096i	0.220095	+	0.975478i	$0.429363\pi$
$509$	44.0156i	1.95096i	−0.220095	+	0.975478i	$0.570637\pi$
$510$	0	0
$511$	− 27.2474i	− 1.20536i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−31.4626	−1.38641
$516$	0	0
$517$	10.0000	0.439799
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 0.921404i	− 0.0403674i	−0.999796	−	0.0201837i	$-0.993575\pi$
$521$	− 0.921404i	− 0.0403674i	0.999796	−	0.0201837i	$-0.00642511\pi$
$522$	0	0
$523$	31.3939i	1.37276i	0.727244	+	0.686379i	$0.240801\pi$
$523$	31.3939i	1.37276i	−0.727244	+	0.686379i	$0.759199\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.83523	0.384869
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 28.2843i	− 1.22513i
$534$	0	0
$535$	27.2474i	1.17801i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	22.3417	0.962325
$540$	0	0
$541$	−20.0000	−0.859867	−0.429934	−	0.902861i	$-0.641463\pi$
$541$	−20.0000	−0.859867	−0.429934	+	0.902861i	$0.641463\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 15.4135i	− 0.660241i
$546$	0	0
$547$	4.20204i	0.179666i	0.995957	+	0.0898332i	$0.0286334\pi$
$547$	4.20204i	0.179666i	−0.995957	+	0.0898332i	$0.971367\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−10.7423	−0.457635
$552$	0	0
$553$	−6.89898	−0.293374
$554$	0	0
$555$	0	0
$556$	0	0
$557$	29.8735i	1.26578i	0.774242	+	0.632890i	$0.218131\pi$
$557$	29.8735i	1.26578i	−0.774242	+	0.632890i	$0.781869\pi$
$558$	0	0
$559$	− 20.0000i	− 0.845910i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5.48188	−0.231034	−0.115517	−	0.993306i	$-0.536852\pi$
$563$	−5.48188	−0.231034	−0.115517	+	0.993306i	$0.536852\pi$
$564$	0	0
$565$	−44.4949	−1.87191
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 19.5133i	− 0.818038i	−0.912526	−	0.409019i	$-0.865871\pi$
$569$	− 19.5133i	− 0.818038i	0.912526	−	0.409019i	$-0.134129\pi$
$570$	0	0
$571$	0.202041i	0.00845515i	0.999991	+	0.00422758i	$0.00134568\pi$
$571$	0.202041i	0.00845515i	−0.999991	+	0.00422758i	$0.998654\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−13.8564	−0.577852
$576$	0	0
$577$	15.7980	0.657678	0.328839	−	0.944386i	$-0.393343\pi$
$577$	15.7980	0.657678	0.328839	+	0.944386i	$0.393343\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 42.8085i	− 1.77599i
$582$	0	0
$583$	58.8434i	2.43704i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	33.4804	1.38188	0.690942	−	0.722910i	$-0.257196\pi$
$587$	33.4804	1.38188	0.690942	+	0.722910i	$0.257196\pi$
$588$	0	0
$589$	12.4949	0.514843
$590$	0	0
$591$	0	0
$592$	0	0
$593$	42.7764i	1.75661i	0.478097	+	0.878307i	$0.341327\pi$
$593$	42.7764i	1.75661i	−0.478097	+	0.878307i	$0.658673\pi$
$594$	0	0
$595$	37.5959i	1.54128i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	36.8338	1.50499	0.752493	−	0.658600i	$-0.228851\pi$
$599$	36.8338	1.50499	0.752493	+	0.658600i	$0.228851\pi$
$600$	0	0
$601$	39.6969	1.61927	0.809636	−	0.586932i	$-0.199665\pi$
$601$	39.6969	1.61927	0.809636	+	0.586932i	$0.199665\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 30.8270i	− 1.25329i
$606$	0	0
$607$	4.20204i	0.170556i	0.996357	+	0.0852778i	$0.0271778\pi$
$607$	4.20204i	0.170556i	−0.996357	+	0.0852778i	$0.972822\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	15.1278	0.612003
$612$	0	0
$613$	30.0000	1.21169	0.605844	−	0.795583i	$-0.292835\pi$
$613$	30.0000	1.21169	0.605844	+	0.795583i	$0.292835\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.285729i	0.0115030i	0.999983	+	0.00575151i	$0.00183077\pi$
$617$	0.285729i	0.0115030i	−0.999983	+	0.00575151i	$0.998169\pi$
$618$	0	0
$619$	− 17.5959i	− 0.707240i	−0.935389	−	0.353620i	$-0.884951\pi$
$619$	− 17.5959i	− 0.707240i	0.935389	−	0.353620i	$-0.115049\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	17.3205	0.693932
$624$	0	0
$625$	−25.4949	−1.01980
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 16.9706i	− 0.676661i
$630$	0	0
$631$	− 27.7423i	− 1.10441i	−0.833710	−	0.552203i	$-0.813787\pi$
$631$	− 27.7423i	− 1.10441i	0.833710	−	0.552203i	$-0.186213\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−29.0949	−1.15460
$636$	0	0
$637$	33.7980	1.33912
$638$	0	0
$639$	0	0
$640$	0	0
$641$	20.4989i	0.809657i	0.914393	+	0.404829i	$0.132669\pi$
$641$	20.4989i	0.809657i	−0.914393	+	0.404829i	$0.867331\pi$
$642$	0	0
$643$	21.3939i	0.843692i	0.906667	+	0.421846i	$0.138618\pi$
$643$	21.3939i	0.843692i	−0.906667	+	0.421846i	$0.861382\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20.7846	−0.817127	−0.408564	−	0.912730i	$-0.633970\pi$
$647$	−20.7846	−0.817127	−0.408564	+	0.912730i	$0.633970\pi$
$648$	0	0
$649$	−10.0000	−0.392534
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 1.23924i	− 0.0484952i	−0.999706	−	0.0242476i	$-0.992281\pi$
$653$	− 1.23924i	− 0.0484952i	0.999706	−	0.0242476i	$-0.00771901\pi$
$654$	0	0
$655$	− 45.0454i	− 1.76007i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	17.4313	0.679026	0.339513	−	0.940601i	$-0.389738\pi$
$659$	17.4313	0.679026	0.339513	+	0.940601i	$0.389738\pi$
$660$	0	0
$661$	44.4949	1.73065	0.865325	−	0.501210i	$-0.167112\pi$
$661$	44.4949	1.73065	0.865325	+	0.501210i	$0.167112\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	53.1687i	2.06179i
$666$	0	0
$667$	6.20204i	0.240144i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−18.2419	−0.704221
$672$	0	0
$673$	2.59592	0.100065	0.0500326	−	0.998748i	$-0.484067\pi$
$673$	2.59592	0.100065	0.0500326	+	0.998748i	$0.484067\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	45.6048i	1.75273i	0.481644	+	0.876367i	$0.340040\pi$
$677$	45.6048i	1.75273i	−0.481644	+	0.876367i	$0.659960\pi$
$678$	0	0
$679$	− 17.2474i	− 0.661896i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−10.3923	−0.397650	−0.198825	−	0.980035i	$-0.563713\pi$
$683$	−10.3923	−0.397650	−0.198825	+	0.980035i	$0.563713\pi$
$684$	0	0
$685$	55.3939	2.11649
$686$	0	0
$687$	0	0
$688$	0	0
$689$	89.0168i	3.39127i
$690$	0	0
$691$	4.00000i	0.152167i	0.997101	+	0.0760836i	$0.0242416\pi$
$691$	4.00000i	0.152167i	−0.997101	+	0.0760836i	$0.975758\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	62.9253	2.38689
$696$	0	0
$697$	−14.2020	−0.537941
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 13.2528i	− 0.500553i	−0.968174	−	0.250276i	$-0.919478\pi$
$701$	− 13.2528i	− 0.500553i	0.968174	−	0.250276i	$-0.0805215\pi$
$702$	0	0
$703$	− 24.0000i	− 0.905177i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.48188	0.206167
$708$	0	0
$709$	−40.0000	−1.50223	−0.751116	−	0.660171i	$-0.770484\pi$
$709$	−40.0000	−1.50223	−0.751116	+	0.660171i	$0.770484\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 7.21393i	− 0.270164i
$714$	0	0
$715$	− 98.9898i	− 3.70201i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6.92820	0.258378	0.129189	−	0.991620i	$-0.458763\pi$
$719$	6.92820	0.258378	0.129189	+	0.991620i	$0.458763\pi$
$720$	0	0
$721$	34.4949	1.28466
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 10.7423i	− 0.398957i
$726$	0	0
$727$	17.2474i	0.639672i	0.947473	+	0.319836i	$0.103628\pi$
$727$	17.2474i	0.639672i	−0.947473	+	0.319836i	$0.896372\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−10.0424	−0.371430
$732$	0	0
$733$	−44.4949	−1.64346	−0.821728	−	0.569880i	$-0.806990\pi$
$733$	−44.4949	−1.64346	−0.821728	+	0.569880i	$0.806990\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 67.9465i	− 2.50284i
$738$	0	0
$739$	− 39.3939i	− 1.44913i	−0.689208	−	0.724564i	$-0.742041\pi$
$739$	− 39.3939i	− 1.44913i	0.689208	−	0.724564i	$-0.257959\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−3.17837	−0.116603	−0.0583016	−	0.998299i	$-0.518569\pi$
$743$	−3.17837	−0.116603	−0.0583016	+	0.998299i	$0.518569\pi$
$744$	0	0
$745$	20.7980	0.761978
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 29.8735i	− 1.09155i
$750$	0	0
$751$	− 25.7423i	− 0.939352i	−0.882839	−	0.469676i	$-0.844371\pi$
$751$	− 25.7423i	− 0.939352i	0.882839	−	0.469676i	$-0.155629\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−22.8024	−0.829864
$756$	0	0
$757$	19.3939	0.704882	0.352441	−	0.935834i	$-0.385352\pi$
$757$	19.3939	0.704882	0.352441	+	0.935834i	$0.385352\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	30.2555i	1.09676i	0.836229	+	0.548381i	$0.184756\pi$
$761$	30.2555i	1.09676i	−0.836229	+	0.548381i	$0.815244\pi$
$762$	0	0
$763$	16.8990i	0.611784i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−15.1278	−0.546232
$768$	0	0
$769$	−37.4949	−1.35210	−0.676050	−	0.736855i	$-0.736310\pi$
$769$	−37.4949	−1.35210	−0.676050	+	0.736855i	$0.736310\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 0.349945i	− 0.0125867i	−0.999980	−	0.00629333i	$-0.997997\pi$
$773$	− 0.349945i	− 0.0125867i	0.999980	−	0.00629333i	$-0.00200324\pi$
$774$	0	0
$775$	12.4949i	0.448830i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−20.0847	−0.719610
$780$	0	0
$781$	60.2929	2.15745
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 18.2419i	− 0.651082i
$786$	0	0
$787$	− 43.1918i	− 1.53962i	−0.638271	−	0.769811i	$-0.720350\pi$
$787$	− 43.1918i	− 1.53962i	0.638271	−	0.769811i	$-0.279650\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	48.7832	1.73453
$792$	0	0
$793$	−27.5959	−0.979960
$794$	0	0
$795$	0	0
$796$	0	0
$797$	10.0745i	0.356856i	0.983953	+	0.178428i	$0.0571012\pi$
$797$	10.0745i	0.356856i	−0.983953	+	0.178428i	$0.942899\pi$
$798$	0	0
$799$	− 7.59592i	− 0.268724i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	36.0231	1.27123
$804$	0	0
$805$	30.6969	1.08192
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 5.02118i	− 0.176535i	−0.996097	−	0.0882676i	$-0.971867\pi$
$809$	− 5.02118i	− 0.176535i	0.996097	−	0.0882676i	$-0.0281331\pi$
$810$	0	0
$811$	− 24.4949i	− 0.860132i	−0.902797	−	0.430066i	$-0.858490\pi$
$811$	− 24.4949i	− 0.860132i	0.902797	−	0.430066i	$-0.141510\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−18.8776	−0.661253
$816$	0	0
$817$	−14.2020	−0.496867
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 11.6637i	− 0.407064i	−0.979068	−	0.203532i	$-0.934758\pi$
$821$	− 11.6637i	− 0.407064i	0.979068	−	0.203532i	$-0.0652421\pi$
$822$	0	0
$823$	44.3485i	1.54589i	0.634473	+	0.772945i	$0.281217\pi$
$823$	44.3485i	1.54589i	−0.634473	+	0.772945i	$0.718783\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−51.2616	−1.78254	−0.891271	−	0.453471i	$-0.850185\pi$
$827$	−51.2616	−1.78254	−0.891271	+	0.453471i	$0.850185\pi$
$828$	0	0
$829$	1.59592	0.0554285	0.0277143	−	0.999616i	$-0.491177\pi$
$829$	1.59592	0.0554285	0.0277143	+	0.999616i	$0.491177\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 16.9706i	− 0.587995i
$834$	0	0
$835$	36.6969i	1.26995i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−31.4626	−1.08621	−0.543106	−	0.839664i	$-0.682752\pi$
$839$	−31.4626	−1.08621	−0.543106	+	0.839664i	$0.682752\pi$
$840$	0	0
$841$	24.1918	0.834201
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 108.848i	− 3.74448i
$846$	0	0
$847$	33.7980i	1.16131i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−13.8564	−0.474991
$852$	0	0
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	22.9774i	0.784892i	0.919775	+	0.392446i	$0.128371\pi$
$857$	22.9774i	0.784892i	−0.919775	+	0.392446i	$0.871629\pi$
$858$	0	0
$859$	− 0.404082i	− 0.0137871i	−0.999976	−	0.00689355i	$-0.997806\pi$
$859$	− 0.404082i	− 0.0137871i	0.999976	−	0.00689355i	$-0.00219430\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−6.92820	−0.235839	−0.117919	−	0.993023i	$-0.537622\pi$
$863$	−6.92820	−0.235839	−0.117919	+	0.993023i	$0.537622\pi$
$864$	0	0
$865$	27.6969	0.941724
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 9.12096i	− 0.309407i
$870$	0	0
$871$	− 102.788i	− 3.48283i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.09638	0.0370643
$876$	0	0
$877$	−35.7980	−1.20881	−0.604406	−	0.796677i	$-0.706589\pi$
$877$	−35.7980	−1.20881	−0.604406	+	0.796677i	$0.706589\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	41.2193i	1.38871i	0.719631	+	0.694356i	$0.244311\pi$
$881$	41.2193i	1.38871i	−0.719631	+	0.694356i	$0.755689\pi$
$882$	0	0
$883$	43.1010i	1.45046i	0.688504	+	0.725232i	$0.258268\pi$
$883$	43.1010i	1.45046i	−0.688504	+	0.725232i	$0.741732\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	17.0348	0.571972	0.285986	−	0.958234i	$-0.407679\pi$
$887$	17.0348	0.571972	0.285986	+	0.958234i	$0.407679\pi$
$888$	0	0
$889$	31.8990	1.06986
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 10.7423i	− 0.359476i
$894$	0	0
$895$	− 27.2474i	− 0.910782i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.59264	0.186525
$900$	0	0
$901$	44.6969	1.48907
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 30.8270i	− 1.02472i
$906$	0	0
$907$	− 44.4949i	− 1.47743i	−0.674019	−	0.738714i	$-0.735433\pi$
$907$	− 44.4949i	− 1.47743i	0.674019	−	0.738714i	$-0.264567\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−21.7060	−0.719152	−0.359576	−	0.933116i	$-0.617079\pi$
$911$	−21.7060	−0.719152	−0.359576	+	0.933116i	$0.617079\pi$
$912$	0	0
$913$	56.5959	1.87305
$914$	0	0
$915$	0	0
$916$	0	0
$917$	49.3867i	1.63089i
$918$	0	0
$919$	− 27.2474i	− 0.898810i	−0.893328	−	0.449405i	$-0.851636\pi$
$919$	− 27.2474i	− 0.898810i	0.893328	−	0.449405i	$-0.148364\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	91.2096	3.00220
$924$	0	0
$925$	24.0000	0.789115
$926$	0	0
$927$	0	0
$928$	0	0
$929$	26.4415i	0.867516i	0.901029	+	0.433758i	$0.142813\pi$
$929$	26.4415i	0.867516i	−0.901029	+	0.433758i	$0.857187\pi$
$930$	0	0
$931$	− 24.0000i	− 0.786568i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−49.7046	−1.62551
$936$	0	0
$937$	−15.0000	−0.490029	−0.245014	−	0.969519i	$-0.578793\pi$
$937$	−15.0000	−0.490029	−0.245014	+	0.969519i	$0.578793\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 10.4244i	− 0.339826i	−0.985459	−	0.169913i	$-0.945651\pi$
$941$	− 10.4244i	− 0.339826i	0.985459	−	0.169913i	$-0.0543487\pi$
$942$	0	0
$943$	11.5959i	0.377615i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−33.4804	−1.08797	−0.543984	−	0.839096i	$-0.683085\pi$
$947$	−33.4804	−1.08797	−0.543984	+	0.839096i	$0.683085\pi$
$948$	0	0
$949$	54.4949	1.76898
$950$	0	0
$951$	0	0
$952$	0	0
$953$	44.6834i	1.44744i	0.690096	+	0.723718i	$0.257568\pi$
$953$	44.6834i	1.44744i	−0.690096	+	0.723718i	$0.742432\pi$
$954$	0	0
$955$	33.5959i	1.08714i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−60.7325	−1.96116
$960$	0	0
$961$	24.4949	0.790158
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 37.4373i	− 1.20515i
$966$	0	0
$967$	24.7526i	0.795988i	0.917388	+	0.397994i	$0.130294\pi$
$967$	24.7526i	0.795988i	−0.917388	+	0.397994i	$0.869706\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−8.94598	−0.287090	−0.143545	−	0.989644i	$-0.545850\pi$
$971$	−8.94598	−0.287090	−0.143545	+	0.989644i	$0.545850\pi$
$972$	0	0
$973$	−68.9898	−2.21171
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.17837i	0.101685i	0.998707	+	0.0508426i	$0.0161907\pi$
$977$	3.17837i	0.101685i	−0.998707	+	0.0508426i	$0.983809\pi$
$978$	0	0
$979$	22.8990i	0.731855i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4.03556	−0.128714	−0.0643572	−	0.997927i	$-0.520500\pi$
$983$	−4.03556	−0.128714	−0.0643572	+	0.997927i	$0.520500\pi$
$984$	0	0
$985$	−5.89898	−0.187957
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.19955i	0.260731i
$990$	0	0
$991$	− 33.2474i	− 1.05614i	−0.849201	−	0.528070i	$-0.822916\pi$
$991$	− 33.2474i	− 1.05614i	0.849201	−	0.528070i	$-0.177084\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	17.7812	0.563702
$996$	0	0
$997$	−31.3031	−0.991378	−0.495689	−	0.868500i	$-0.665084\pi$
$997$	−31.3031	−0.991378	−0.495689	+	0.868500i	$0.665084\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.c.a.863.2	yes	8
3.2	odd	2	inner	864.2.c.a.863.8	yes	8
4.3	odd	2	inner	864.2.c.a.863.1	✓	8
8.3	odd	2		1728.2.c.g.1727.7		8
8.5	even	2		1728.2.c.g.1727.8		8
9.2	odd	6		2592.2.s.a.863.2		8
9.4	even	3		2592.2.s.a.1727.1		8
9.5	odd	6		2592.2.s.h.1727.3		8
9.7	even	3		2592.2.s.h.863.4		8
12.11	even	2	inner	864.2.c.a.863.7	yes	8
24.5	odd	2		1728.2.c.g.1727.2		8
24.11	even	2		1728.2.c.g.1727.1		8
36.7	odd	6		2592.2.s.h.863.3		8
36.11	even	6		2592.2.s.a.863.1		8
36.23	even	6		2592.2.s.h.1727.4		8
36.31	odd	6		2592.2.s.a.1727.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.c.a.863.1	✓	8	4.3	odd	2	inner
864.2.c.a.863.2	yes	8	1.1	even	1	trivial
864.2.c.a.863.7	yes	8	12.11	even	2	inner
864.2.c.a.863.8	yes	8	3.2	odd	2	inner
1728.2.c.g.1727.1		8	24.11	even	2
1728.2.c.g.1727.2		8	24.5	odd	2
1728.2.c.g.1727.7		8	8.3	odd	2
1728.2.c.g.1727.8		8	8.5	even	2
2592.2.s.a.863.1		8	36.11	even	6
2592.2.s.a.863.2		8	9.2	odd	6
2592.2.s.a.1727.1		8	9.4	even	3
2592.2.s.a.1727.2		8	36.31	odd	6
2592.2.s.h.863.3		8	36.7	odd	6
2592.2.s.h.863.4		8	9.7	even	3
2592.2.s.h.1727.3		8	9.5	odd	6
2592.2.s.h.1727.4		8	36.23	even	6

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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	864.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.14626i q^{5} +3.44949i q^{7} +O(q^{10})\)	\(q-3.14626i q^{5} +3.44949i q^{7} -4.56048 q^{11} -6.89898 q^{13} +3.46410i q^{17} +4.89898i q^{19} +2.82843 q^{23} -4.89898 q^{25} +2.19275i q^{29} -2.55051i q^{31} +10.8530 q^{35} -4.89898 q^{37} +4.09978i q^{41} +2.89898i q^{43} -2.19275 q^{47} -4.89898 q^{49} -12.9029i q^{53} +14.3485i q^{55} +2.19275 q^{59} +4.00000 q^{61} +21.7060i q^{65} +14.8990i q^{67} -13.2207 q^{71} -7.89898 q^{73} -15.7313i q^{77} +2.00000i q^{79} -12.4101 q^{83} +10.8990 q^{85} -5.02118i q^{89} -23.7980i q^{91} +15.4135 q^{95} -5.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 16 q^{13} + 32 q^{61} - 24 q^{73} + 48 q^{85} - 40 q^{97}+O(q^{100}) \) 8 * q - 16 * q^13 + 32 * q^61 - 24 * q^73 + 48 * q^85 - 40 * q^97

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