LMFDB - Embedded newform 5184.2.d.n.2593.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5184,2,Mod(2593,5184)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5184, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5184.2593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5184 = 2^{6} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5184.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:	$41.3944484078$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2593.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	5184.2593
Dual form			5184.2.d.n.2593.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{5} +1.26795 q^{7} +O(q^{10})$	$q-1.73205i q^{5} +1.26795 q^{7} -1.26795i q^{11} -3.00000i q^{13} +4.26795 q^{17} -4.19615i q^{19} -1.26795 q^{23} +2.00000 q^{25} +4.26795i q^{29} -3.46410 q^{31} -2.19615i q^{35} -0.464102i q^{37} +3.46410 q^{41} -6.19615i q^{43} +12.9282 q^{47} -5.39230 q^{49} +0.928203i q^{53} -2.19615 q^{55} -9.46410i q^{59} -6.46410i q^{61} -5.19615 q^{65} +4.19615i q^{67} -4.73205 q^{71} +5.00000 q^{73} -1.60770i q^{77} -14.1962 q^{79} +10.3923i q^{83} -7.39230i q^{85} -0.803848 q^{89} -3.80385i q^{91} -7.26795 q^{95} +4.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{7}+O(q^{10}) $ 4 * q + 12 * q^7	$ 4 q + 12 q^{7} + 24 q^{17} - 12 q^{23} + 8 q^{25} + 24 q^{47} + 20 q^{49} + 12 q^{55} - 12 q^{71} + 20 q^{73} - 36 q^{79} - 24 q^{89} - 36 q^{95} + 16 q^{97}+O(q^{100}) $ 4 * q + 12 * q^7 + 24 * q^17 - 12 * q^23 + 8 * q^25 + 24 * q^47 + 20 * q^49 + 12 * q^55 - 12 * q^71 + 20 * q^73 - 36 * q^79 - 24 * q^89 - 36 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$-1$	$1$	$1$

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
$5$	− 1.73205i	− 0.774597i	−0.921954	−	0.387298i	$-0.873408\pi$
$5$	− 1.73205i	− 0.774597i	0.921954	−	0.387298i	$-0.126592\pi$
$6$	0	0
$7$	1.26795	0.479240	0.239620	−	0.970867i	$-0.422977\pi$
$7$	1.26795	0.479240	0.239620	+	0.970867i	$0.422977\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.26795i	− 0.382301i	−0.981561	−	0.191151i	$-0.938778\pi$
$11$	− 1.26795i	− 0.382301i	0.981561	−	0.191151i	$-0.0612219\pi$
$12$	0	0
$13$	− 3.00000i	− 0.832050i	−0.909353	−	0.416025i	$-0.863423\pi$
$13$	− 3.00000i	− 0.832050i	0.909353	−	0.416025i	$-0.136577\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.26795	1.03513	0.517565	−	0.855644i	$-0.326839\pi$
$17$	4.26795	1.03513	0.517565	+	0.855644i	$0.326839\pi$
$18$	0	0
$19$	− 4.19615i	− 0.962663i	−0.876539	−	0.481332i	$-0.840153\pi$
$19$	− 4.19615i	− 0.962663i	0.876539	−	0.481332i	$-0.159847\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.26795	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$23$	−1.26795	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.26795i	0.792538i	0.918134	+	0.396269i	$0.129695\pi$
$29$	4.26795i	0.792538i	−0.918134	+	0.396269i	$0.870305\pi$
$30$	0	0
$31$	−3.46410	−0.622171	−0.311086	−	0.950382i	$-0.600693\pi$
$31$	−3.46410	−0.622171	−0.311086	+	0.950382i	$0.600693\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 2.19615i	− 0.371218i
$36$	0	0
$37$	− 0.464102i	− 0.0762978i	−0.999272	−	0.0381489i	$-0.987854\pi$
$37$	− 0.464102i	− 0.0762978i	0.999272	−	0.0381489i	$-0.0121461\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	0	0
$43$	− 6.19615i	− 0.944904i	−0.881356	−	0.472452i	$-0.843369\pi$
$43$	− 6.19615i	− 0.944904i	0.881356	−	0.472452i	$-0.156631\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.9282	1.88577	0.942886	−	0.333115i	$-0.108100\pi$
$47$	12.9282	1.88577	0.942886	+	0.333115i	$0.108100\pi$
$48$	0	0
$49$	−5.39230	−0.770329
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.928203i	0.127499i	0.997966	+	0.0637493i	$0.0203058\pi$
$53$	0.928203i	0.127499i	−0.997966	+	0.0637493i	$0.979694\pi$
$54$	0	0
$55$	−2.19615	−0.296129
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 9.46410i	− 1.23212i	−0.787699	−	0.616061i	$-0.788728\pi$
$59$	− 9.46410i	− 1.23212i	0.787699	−	0.616061i	$-0.211272\pi$
$60$	0	0
$61$	− 6.46410i	− 0.827643i	−0.910358	−	0.413822i	$-0.864194\pi$
$61$	− 6.46410i	− 0.827643i	0.910358	−	0.413822i	$-0.135806\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.19615	−0.644503
$66$	0	0
$67$	4.19615i	0.512642i	0.966592	+	0.256321i	$0.0825104\pi$
$67$	4.19615i	0.512642i	−0.966592	+	0.256321i	$0.917490\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.73205	−0.561591	−0.280796	−	0.959768i	$-0.590598\pi$
$71$	−4.73205	−0.561591	−0.280796	+	0.959768i	$0.590598\pi$
$72$	0	0
$73$	5.00000	0.585206	0.292603	−	0.956234i	$-0.405479\pi$
$73$	5.00000	0.585206	0.292603	+	0.956234i	$0.405479\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.60770i	− 0.183214i
$78$	0	0
$79$	−14.1962	−1.59719	−0.798596	−	0.601867i	$-0.794424\pi$
$79$	−14.1962	−1.59719	−0.798596	+	0.601867i	$0.794424\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.3923i	1.14070i	0.821401	+	0.570352i	$0.193193\pi$
$83$	10.3923i	1.14070i	−0.821401	+	0.570352i	$0.806807\pi$
$84$	0	0
$85$	− 7.39230i	− 0.801808i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.803848	−0.0852077	−0.0426038	−	0.999092i	$-0.513565\pi$
$89$	−0.803848	−0.0852077	−0.0426038	+	0.999092i	$0.513565\pi$
$90$	0	0
$91$	− 3.80385i	− 0.398752i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.26795	−0.745676
$96$	0	0
$97$	4.00000	0.406138	0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000	0.406138	0.203069	+	0.979164i	$0.434908\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.9282i	1.88343i	0.336416	+	0.941713i	$0.390785\pi$
$101$	18.9282i	1.88343i	−0.336416	+	0.941713i	$0.609215\pi$
$102$	0	0
$103$	−12.9282	−1.27385	−0.636927	−	0.770924i	$-0.719795\pi$
$103$	−12.9282	−1.27385	−0.636927	+	0.770924i	$0.719795\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.928203i	0.0897328i	0.998993	+	0.0448664i	$0.0142862\pi$
$107$	0.928203i	0.0897328i	−0.998993	+	0.0448664i	$0.985714\pi$
$108$	0	0
$109$	− 12.4641i	− 1.19384i	−0.802299	−	0.596922i	$-0.796390\pi$
$109$	− 12.4641i	− 1.19384i	0.802299	−	0.596922i	$-0.203610\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11.1962	1.05325	0.526623	−	0.850099i	$-0.323458\pi$
$113$	11.1962	1.05325	0.526623	+	0.850099i	$0.323458\pi$
$114$	0	0
$115$	2.19615i	0.204792i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.41154	0.496075
$120$	0	0
$121$	9.39230	0.853846
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 12.1244i	− 1.08444i
$126$	0	0
$127$	−9.12436	−0.809656	−0.404828	−	0.914393i	$-0.632669\pi$
$127$	−9.12436	−0.809656	−0.404828	+	0.914393i	$0.632669\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 2.19615i	− 0.191879i	−0.995387	−	0.0959394i	$-0.969415\pi$
$131$	− 2.19615i	− 0.191879i	0.995387	−	0.0959394i	$-0.0305855\pi$
$132$	0	0
$133$	− 5.32051i	− 0.461347i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.1244	−1.03585	−0.517927	−	0.855425i	$-0.673296\pi$
$137$	−12.1244	−1.03585	−0.517927	+	0.855425i	$0.673296\pi$
$138$	0	0
$139$	− 14.0000i	− 1.18746i	−0.804663	−	0.593732i	$-0.797654\pi$
$139$	− 14.0000i	− 1.18746i	0.804663	−	0.593732i	$-0.202346\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.80385	−0.318094
$144$	0	0
$145$	7.39230	0.613898
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 21.5885i	− 1.76860i	−0.466924	−	0.884298i	$-0.654638\pi$
$149$	− 21.5885i	− 1.76860i	0.466924	−	0.884298i	$-0.345362\pi$
$150$	0	0
$151$	5.07180	0.412737	0.206368	−	0.978474i	$-0.433835\pi$
$151$	5.07180	0.412737	0.206368	+	0.978474i	$0.433835\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.00000i	0.481932i
$156$	0	0
$157$	− 9.92820i	− 0.792357i	−0.918174	−	0.396178i	$-0.870336\pi$
$157$	− 9.92820i	− 0.792357i	0.918174	−	0.396178i	$-0.129664\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.60770	−0.126704
$162$	0	0
$163$	8.39230i	0.657336i	0.944446	+	0.328668i	$0.106600\pi$
$163$	8.39230i	0.657336i	−0.944446	+	0.328668i	$0.893400\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.1244	−1.63465	−0.817326	−	0.576176i	$-0.804544\pi$
$167$	−21.1244	−1.63465	−0.817326	+	0.576176i	$0.804544\pi$
$168$	0	0
$169$	4.00000	0.307692
$170$	0	0
$171$	0	0
$172$	0	0
$173$	16.2679i	1.23683i	0.785852	+	0.618415i	$0.212225\pi$
$173$	16.2679i	1.23683i	−0.785852	+	0.618415i	$0.787775\pi$
$174$	0	0
$175$	2.53590	0.191696
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 22.3923i	− 1.67368i	−0.547448	−	0.836840i	$-0.684401\pi$
$179$	− 22.3923i	− 1.67368i	0.547448	−	0.836840i	$-0.315599\pi$
$180$	0	0
$181$	− 11.3205i	− 0.841447i	−0.907189	−	0.420723i	$-0.861776\pi$
$181$	− 11.3205i	− 0.841447i	0.907189	−	0.420723i	$-0.138224\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.803848	−0.0591000
$186$	0	0
$187$	− 5.41154i	− 0.395731i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.2679	1.39418	0.697090	−	0.716984i	$-0.254478\pi$
$191$	19.2679	1.39418	0.697090	+	0.716984i	$0.254478\pi$
$192$	0	0
$193$	25.7846	1.85602	0.928008	−	0.372559i	$-0.121520\pi$
$193$	25.7846	1.85602	0.928008	+	0.372559i	$0.121520\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 11.1962i	− 0.797693i	−0.917018	−	0.398846i	$-0.869411\pi$
$197$	− 11.1962i	− 0.797693i	0.917018	−	0.398846i	$-0.130589\pi$
$198$	0	0
$199$	−18.0000	−1.27599	−0.637993	−	0.770042i	$-0.720235\pi$
$199$	−18.0000	−1.27599	−0.637993	+	0.770042i	$0.720235\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.41154i	0.379816i
$204$	0	0
$205$	− 6.00000i	− 0.419058i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.32051	−0.368027
$210$	0	0
$211$	− 10.5885i	− 0.728939i	−0.931215	−	0.364470i	$-0.881250\pi$
$211$	− 10.5885i	− 0.728939i	0.931215	−	0.364470i	$-0.118750\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−10.7321	−0.731920
$216$	0	0
$217$	−4.39230	−0.298169
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 12.8038i	− 0.861280i
$222$	0	0
$223$	−21.1244	−1.41459	−0.707296	−	0.706918i	$-0.750085\pi$
$223$	−21.1244	−1.41459	−0.707296	+	0.706918i	$0.750085\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	28.0526i	1.86191i	0.365129	+	0.930957i	$0.381025\pi$
$227$	28.0526i	1.86191i	−0.365129	+	0.930957i	$0.618975\pi$
$228$	0	0
$229$	8.07180i	0.533399i	0.963780	+	0.266700i	$0.0859332\pi$
$229$	8.07180i	0.533399i	−0.963780	+	0.266700i	$0.914067\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.73205	0.506543	0.253272	−	0.967395i	$-0.418493\pi$
$233$	7.73205	0.506543	0.253272	+	0.967395i	$0.418493\pi$
$234$	0	0
$235$	− 22.3923i	− 1.46071i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.53590	−0.552141	−0.276071	−	0.961137i	$-0.589032\pi$
$239$	−8.53590	−0.552141	−0.276071	+	0.961137i	$0.589032\pi$
$240$	0	0
$241$	9.39230	0.605012	0.302506	−	0.953148i	$-0.402177\pi$
$241$	9.39230	0.605012	0.302506	+	0.953148i	$0.402177\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	9.33975i	0.596694i
$246$	0	0
$247$	−12.5885	−0.800984
$248$	0	0
$249$	0	0
$250$	0	0
$251$	27.1244i	1.71207i	0.516914	+	0.856037i	$0.327081\pi$
$251$	27.1244i	1.71207i	−0.516914	+	0.856037i	$0.672919\pi$
$252$	0	0
$253$	1.60770i	0.101075i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.2679	−0.640497	−0.320249	−	0.947334i	$-0.603766\pi$
$257$	−10.2679	−0.640497	−0.320249	+	0.947334i	$0.603766\pi$
$258$	0	0
$259$	− 0.588457i	− 0.0365650i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	26.7846	1.65161	0.825805	−	0.563956i	$-0.190721\pi$
$263$	26.7846	1.65161	0.825805	+	0.563956i	$0.190721\pi$
$264$	0	0
$265$	1.60770	0.0987599
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 0.803848i	− 0.0490115i	−0.999700	−	0.0245057i	$-0.992199\pi$
$269$	− 0.803848i	− 0.0490115i	0.999700	−	0.0245057i	$-0.00780120\pi$
$270$	0	0
$271$	−4.73205	−0.287452	−0.143726	−	0.989618i	$-0.545908\pi$
$271$	−4.73205	−0.287452	−0.143726	+	0.989618i	$0.545908\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 2.53590i	− 0.152920i
$276$	0	0
$277$	− 18.9282i	− 1.13729i	−0.822585	−	0.568643i	$-0.807469\pi$
$277$	− 18.9282i	− 1.13729i	0.822585	−	0.568643i	$-0.192531\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−17.1962	−1.02584	−0.512918	−	0.858437i	$-0.671436\pi$
$281$	−17.1962	−1.02584	−0.512918	+	0.858437i	$0.671436\pi$
$282$	0	0
$283$	− 14.0000i	− 0.832214i	−0.909316	−	0.416107i	$-0.863394\pi$
$283$	− 14.0000i	− 0.832214i	0.909316	−	0.416107i	$-0.136606\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.39230	0.259270
$288$	0	0
$289$	1.21539	0.0714935
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 4.26795i	− 0.249336i	−0.992198	−	0.124668i	$-0.960213\pi$
$293$	− 4.26795i	− 0.249336i	0.992198	−	0.124668i	$-0.0397866\pi$
$294$	0	0
$295$	−16.3923	−0.954397
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.80385i	0.219982i
$300$	0	0
$301$	− 7.85641i	− 0.452836i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−11.1962	−0.641090
$306$	0	0
$307$	18.3923i	1.04970i	0.851193	+	0.524852i	$0.175879\pi$
$307$	18.3923i	1.04970i	−0.851193	+	0.524852i	$0.824121\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−14.5359	−0.824255	−0.412128	−	0.911126i	$-0.635214\pi$
$311$	−14.5359	−0.824255	−0.412128	+	0.911126i	$0.635214\pi$
$312$	0	0
$313$	−3.39230	−0.191744	−0.0958722	−	0.995394i	$-0.530564\pi$
$313$	−3.39230	−0.191744	−0.0958722	+	0.995394i	$0.530564\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.8038i	0.719136i	0.933119	+	0.359568i	$0.117076\pi$
$317$	12.8038i	0.719136i	−0.933119	+	0.359568i	$0.882924\pi$
$318$	0	0
$319$	5.41154	0.302988
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 17.9090i	− 0.996481i
$324$	0	0
$325$	− 6.00000i	− 0.332820i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	16.3923	0.903737
$330$	0	0
$331$	− 16.1962i	− 0.890221i	−0.895476	−	0.445111i	$-0.853164\pi$
$331$	− 16.1962i	− 0.890221i	0.895476	−	0.445111i	$-0.146836\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7.26795	0.397090
$336$	0	0
$337$	−8.00000	−0.435788	−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000	−0.435788	−0.217894	+	0.975972i	$0.569919\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.39230i	0.237857i
$342$	0	0
$343$	−15.7128	−0.848412
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 23.6603i	− 1.27015i	−0.772451	−	0.635074i	$-0.780969\pi$
$347$	− 23.6603i	− 1.27015i	0.772451	−	0.635074i	$-0.219031\pi$
$348$	0	0
$349$	− 19.8564i	− 1.06289i	−0.847093	−	0.531445i	$-0.821649\pi$
$349$	− 19.8564i	− 1.06289i	0.847093	−	0.531445i	$-0.178351\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−17.0718	−0.908640	−0.454320	−	0.890839i	$-0.650118\pi$
$353$	−17.0718	−0.908640	−0.454320	+	0.890839i	$0.650118\pi$
$354$	0	0
$355$	8.19615i	0.435007i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−19.5167	−1.03005	−0.515025	−	0.857175i	$-0.672217\pi$
$359$	−19.5167	−1.03005	−0.515025	+	0.857175i	$0.672217\pi$
$360$	0	0
$361$	1.39230	0.0732792
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 8.66025i	− 0.453298i
$366$	0	0
$367$	9.46410	0.494022	0.247011	−	0.969013i	$-0.420552\pi$
$367$	9.46410	0.494022	0.247011	+	0.969013i	$0.420552\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.17691i	0.0611024i
$372$	0	0
$373$	− 24.0000i	− 1.24267i	−0.783544	−	0.621336i	$-0.786590\pi$
$373$	− 24.0000i	− 1.24267i	0.783544	−	0.621336i	$-0.213410\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.8038	0.659432
$378$	0	0
$379$	− 14.3923i	− 0.739283i	−0.929174	−	0.369642i	$-0.879480\pi$
$379$	− 14.3923i	− 0.739283i	0.929174	−	0.369642i	$-0.120520\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	16.0526	0.820247	0.410124	−	0.912030i	$-0.365486\pi$
$383$	16.0526	0.820247	0.410124	+	0.912030i	$0.365486\pi$
$384$	0	0
$385$	−2.78461	−0.141917
$386$	0	0
$387$	0	0
$388$	0	0
$389$	12.9282i	0.655486i	0.944767	+	0.327743i	$0.106288\pi$
$389$	12.9282i	0.655486i	−0.944767	+	0.327743i	$0.893712\pi$
$390$	0	0
$391$	−5.41154	−0.273673
$392$	0	0
$393$	0	0
$394$	0	0
$395$	24.5885i	1.23718i
$396$	0	0
$397$	− 33.9282i	− 1.70281i	−0.524511	−	0.851404i	$-0.675752\pi$
$397$	− 33.9282i	− 1.70281i	0.524511	−	0.851404i	$-0.324248\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	25.0526	1.25107	0.625533	−	0.780198i	$-0.284882\pi$
$401$	25.0526	1.25107	0.625533	+	0.780198i	$0.284882\pi$
$402$	0	0
$403$	10.3923i	0.517678i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.588457	−0.0291687
$408$	0	0
$409$	−3.78461	−0.187137	−0.0935685	−	0.995613i	$-0.529827\pi$
$409$	−3.78461	−0.187137	−0.0935685	+	0.995613i	$0.529827\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 12.0000i	− 0.590481i
$414$	0	0
$415$	18.0000	0.883585
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.39230i	0.214578i	0.994228	+	0.107289i	$0.0342170\pi$
$419$	4.39230i	0.214578i	−0.994228	+	0.107289i	$0.965783\pi$
$420$	0	0
$421$	20.3205i	0.990361i	0.868790	+	0.495180i	$0.164898\pi$
$421$	20.3205i	0.990361i	−0.868790	+	0.495180i	$0.835102\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8.53590	0.414052
$426$	0	0
$427$	− 8.19615i	− 0.396640i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.32051	−0.256280	−0.128140	−	0.991756i	$-0.540901\pi$
$431$	−5.32051	−0.256280	−0.128140	+	0.991756i	$0.540901\pi$
$432$	0	0
$433$	−5.00000	−0.240285	−0.120142	−	0.992757i	$-0.538335\pi$
$433$	−5.00000	−0.240285	−0.120142	+	0.992757i	$0.538335\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.32051i	0.254514i
$438$	0	0
$439$	1.60770	0.0767311	0.0383656	−	0.999264i	$-0.487785\pi$
$439$	1.60770	0.0767311	0.0383656	+	0.999264i	$0.487785\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 24.9282i	− 1.18437i	−0.805800	−	0.592187i	$-0.798265\pi$
$443$	− 24.9282i	− 1.18437i	0.805800	−	0.592187i	$-0.201735\pi$
$444$	0	0
$445$	1.39230i	0.0660016i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	24.9282	1.17643	0.588217	−	0.808703i	$-0.299830\pi$
$449$	24.9282	1.17643	0.588217	+	0.808703i	$0.299830\pi$
$450$	0	0
$451$	− 4.39230i	− 0.206826i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.58846	−0.308872
$456$	0	0
$457$	9.39230	0.439353	0.219677	−	0.975573i	$-0.429500\pi$
$457$	9.39230	0.439353	0.219677	+	0.975573i	$0.429500\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10.3923i	0.484018i	0.970274	+	0.242009i	$0.0778063\pi$
$461$	10.3923i	0.484018i	−0.970274	+	0.242009i	$0.922194\pi$
$462$	0	0
$463$	−28.3923	−1.31950	−0.659751	−	0.751484i	$-0.729338\pi$
$463$	−28.3923	−1.31950	−0.659751	+	0.751484i	$0.729338\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 27.7128i	− 1.28240i	−0.767375	−	0.641198i	$-0.778438\pi$
$467$	− 27.7128i	− 1.28240i	0.767375	−	0.641198i	$-0.221562\pi$
$468$	0	0
$469$	5.32051i	0.245678i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−7.85641	−0.361238
$474$	0	0
$475$	− 8.39230i	− 0.385065i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−6.58846	−0.301034	−0.150517	−	0.988607i	$-0.548094\pi$
$479$	−6.58846	−0.301034	−0.150517	+	0.988607i	$0.548094\pi$
$480$	0	0
$481$	−1.39230	−0.0634836
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 6.92820i	− 0.314594i
$486$	0	0
$487$	10.3923	0.470920	0.235460	−	0.971884i	$-0.424340\pi$
$487$	10.3923	0.470920	0.235460	+	0.971884i	$0.424340\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 27.1244i	− 1.22411i	−0.790817	−	0.612053i	$-0.790344\pi$
$491$	− 27.1244i	− 1.22411i	0.790817	−	0.612053i	$-0.209656\pi$
$492$	0	0
$493$	18.2154i	0.820380i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.00000	−0.269137
$498$	0	0
$499$	34.1962i	1.53083i	0.643537	+	0.765415i	$0.277466\pi$
$499$	34.1962i	1.53083i	−0.643537	+	0.765415i	$0.722534\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	11.3205	0.504757	0.252378	−	0.967629i	$-0.418787\pi$
$503$	11.3205	0.504757	0.252378	+	0.967629i	$0.418787\pi$
$504$	0	0
$505$	32.7846	1.45890
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 12.2487i	− 0.542915i	−0.962450	−	0.271457i	$-0.912494\pi$
$509$	− 12.2487i	− 0.542915i	0.962450	−	0.271457i	$-0.0875056\pi$
$510$	0	0
$511$	6.33975	0.280454
$512$	0	0
$513$	0	0
$514$	0	0
$515$	22.3923i	0.986723i
$516$	0	0
$517$	− 16.3923i	− 0.720933i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−12.9282	−0.566395	−0.283197	−	0.959062i	$-0.591395\pi$
$521$	−12.9282	−0.566395	−0.283197	+	0.959062i	$0.591395\pi$
$522$	0	0
$523$	− 9.41154i	− 0.411538i	−0.978601	−	0.205769i	$-0.934030\pi$
$523$	− 9.41154i	− 0.411538i	0.978601	−	0.205769i	$-0.0659696\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−14.7846	−0.644028
$528$	0	0
$529$	−21.3923	−0.930100
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 10.3923i	− 0.450141i
$534$	0	0
$535$	1.60770	0.0695067
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.83717i	0.294498i
$540$	0	0
$541$	45.9282i	1.97461i	0.158843	+	0.987304i	$0.449224\pi$
$541$	45.9282i	1.97461i	−0.158843	+	0.987304i	$0.550776\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−21.5885	−0.924748
$546$	0	0
$547$	5.60770i	0.239768i	0.992788	+	0.119884i	$0.0382522\pi$
$547$	5.60770i	0.239768i	−0.992788	+	0.119884i	$0.961748\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	17.9090	0.762948
$552$	0	0
$553$	−18.0000	−0.765438
$554$	0	0
$555$	0	0
$556$	0	0
$557$	36.3731i	1.54118i	0.637333	+	0.770588i	$0.280037\pi$
$557$	36.3731i	1.54118i	−0.637333	+	0.770588i	$0.719963\pi$
$558$	0	0
$559$	−18.5885	−0.786208
$560$	0	0
$561$	0	0
$562$	0	0
$563$	19.8564i	0.836848i	0.908252	+	0.418424i	$0.137417\pi$
$563$	19.8564i	0.836848i	−0.908252	+	0.418424i	$0.862583\pi$
$564$	0	0
$565$	− 19.3923i	− 0.815840i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	21.3397	0.894609	0.447304	−	0.894382i	$-0.352384\pi$
$569$	21.3397	0.894609	0.447304	+	0.894382i	$0.352384\pi$
$570$	0	0
$571$	− 15.6077i	− 0.653162i	−0.945169	−	0.326581i	$-0.894103\pi$
$571$	− 15.6077i	− 0.653162i	0.945169	−	0.326581i	$-0.105897\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.53590	−0.105754
$576$	0	0
$577$	5.39230	0.224485	0.112242	−	0.993681i	$-0.464197\pi$
$577$	5.39230	0.224485	0.112242	+	0.993681i	$0.464197\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	13.1769i	0.546671i
$582$	0	0
$583$	1.17691	0.0487428
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 38.4449i	− 1.58679i	−0.608708	−	0.793395i	$-0.708312\pi$
$587$	− 38.4449i	− 1.58679i	0.608708	−	0.793395i	$-0.291688\pi$
$588$	0	0
$589$	14.5359i	0.598941i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0.803848	0.0330101	0.0165050	−	0.999864i	$-0.494746\pi$
$593$	0.803848	0.0330101	0.0165050	+	0.999864i	$0.494746\pi$
$594$	0	0
$595$	− 9.37307i	− 0.384258i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	21.7128	0.887161	0.443581	−	0.896234i	$-0.353708\pi$
$599$	21.7128	0.887161	0.443581	+	0.896234i	$0.353708\pi$
$600$	0	0
$601$	24.1769	0.986197	0.493098	−	0.869974i	$-0.335864\pi$
$601$	24.1769	0.986197	0.493098	+	0.869974i	$0.335864\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 16.2679i	− 0.661386i
$606$	0	0
$607$	14.4449	0.586299	0.293149	−	0.956067i	$-0.405297\pi$
$607$	14.4449	0.586299	0.293149	+	0.956067i	$0.405297\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 38.7846i	− 1.56906i
$612$	0	0
$613$	− 18.2487i	− 0.737059i	−0.929616	−	0.368529i	$-0.879861\pi$
$613$	− 18.2487i	− 0.737059i	0.929616	−	0.368529i	$-0.120139\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.73205	0.311281	0.155640	−	0.987814i	$-0.450256\pi$
$617$	7.73205	0.311281	0.155640	+	0.987814i	$0.450256\pi$
$618$	0	0
$619$	− 36.3923i	− 1.46273i	−0.681986	−	0.731365i	$-0.738884\pi$
$619$	− 36.3923i	− 1.46273i	0.681986	−	0.731365i	$-0.261116\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.01924	−0.0408349
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1.98076i	− 0.0789782i
$630$	0	0
$631$	27.7128	1.10323	0.551615	−	0.834099i	$-0.314012\pi$
$631$	27.7128	1.10323	0.551615	+	0.834099i	$0.314012\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	15.8038i	0.627157i
$636$	0	0
$637$	16.1769i	0.640953i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	36.1244	1.42683	0.713413	−	0.700744i	$-0.247148\pi$
$641$	36.1244	1.42683	0.713413	+	0.700744i	$0.247148\pi$
$642$	0	0
$643$	18.7846i	0.740793i	0.928874	+	0.370396i	$0.120778\pi$
$643$	18.7846i	0.740793i	−0.928874	+	0.370396i	$0.879222\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.4449	1.27554	0.637770	−	0.770227i	$-0.279857\pi$
$647$	32.4449	1.27554	0.637770	+	0.770227i	$0.279857\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	0	0
$652$	0	0
$653$	30.0000i	1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$	30.0000i	1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$	0	0
$655$	−3.80385	−0.148629
$656$	0	0
$657$	0	0
$658$	0	0
$659$	33.8038i	1.31681i	0.752663	+	0.658405i	$0.228769\pi$
$659$	33.8038i	1.31681i	−0.752663	+	0.658405i	$0.771231\pi$
$660$	0	0
$661$	36.7128i	1.42796i	0.700165	+	0.713981i	$0.253110\pi$
$661$	36.7128i	1.42796i	−0.700165	+	0.713981i	$0.746890\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−9.21539	−0.357358
$666$	0	0
$667$	− 5.41154i	− 0.209536i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8.19615	−0.316409
$672$	0	0
$673$	38.1769	1.47161	0.735806	−	0.677192i	$-0.236804\pi$
$673$	38.1769	1.47161	0.735806	+	0.677192i	$0.236804\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 8.53590i	− 0.328061i	−0.986455	−	0.164031i	$-0.947550\pi$
$677$	− 8.53590i	− 0.328061i	0.986455	−	0.164031i	$-0.0524496\pi$
$678$	0	0
$679$	5.07180	0.194638
$680$	0	0
$681$	0	0
$682$	0	0
$683$	28.6410i	1.09592i	0.836505	+	0.547959i	$0.184595\pi$
$683$	28.6410i	1.09592i	−0.836505	+	0.547959i	$0.815405\pi$
$684$	0	0
$685$	21.0000i	0.802369i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.78461	0.106085
$690$	0	0
$691$	44.9808i	1.71115i	0.517680	+	0.855574i	$0.326796\pi$
$691$	44.9808i	1.71115i	−0.517680	+	0.855574i	$0.673204\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−24.2487	−0.919806
$696$	0	0
$697$	14.7846	0.560007
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 3.58846i	− 0.135534i	−0.997701	−	0.0677671i	$-0.978413\pi$
$701$	− 3.58846i	− 0.135534i	0.997701	−	0.0677671i	$-0.0215875\pi$
$702$	0	0
$703$	−1.94744	−0.0734491
$704$	0	0
$705$	0	0
$706$	0	0
$707$	24.0000i	0.902613i
$708$	0	0
$709$	− 0.215390i	− 0.00808915i	−0.999992	−	0.00404458i	$-0.998713\pi$
$709$	− 0.215390i	− 0.00808915i	0.999992	−	0.00404458i	$-0.00128743\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	4.39230	0.164493
$714$	0	0
$715$	6.58846i	0.246394i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	12.5885	0.469470	0.234735	−	0.972059i	$-0.424578\pi$
$719$	12.5885	0.469470	0.234735	+	0.972059i	$0.424578\pi$
$720$	0	0
$721$	−16.3923	−0.610481
$722$	0	0
$723$	0	0
$724$	0	0
$725$	8.53590i	0.317015i
$726$	0	0
$727$	−12.5885	−0.466880	−0.233440	−	0.972371i	$-0.574998\pi$
$727$	−12.5885	−0.466880	−0.233440	+	0.972371i	$0.574998\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 26.4449i	− 0.978099i
$732$	0	0
$733$	− 18.2487i	− 0.674032i	−0.941499	−	0.337016i	$-0.890582\pi$
$733$	− 18.2487i	− 0.674032i	0.941499	−	0.337016i	$-0.109418\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.32051	0.195983
$738$	0	0
$739$	38.0000i	1.39785i	0.715194	+	0.698926i	$0.246338\pi$
$739$	38.0000i	1.39785i	−0.715194	+	0.698926i	$0.753662\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	39.4641	1.44780	0.723899	−	0.689906i	$-0.242348\pi$
$743$	39.4641	1.44780	0.723899	+	0.689906i	$0.242348\pi$
$744$	0	0
$745$	−37.3923	−1.36995
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.17691i	0.0430035i
$750$	0	0
$751$	10.0526	0.366823	0.183412	−	0.983036i	$-0.441286\pi$
$751$	10.0526	0.366823	0.183412	+	0.983036i	$0.441286\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 8.78461i	− 0.319705i
$756$	0	0
$757$	30.2487i	1.09941i	0.835359	+	0.549704i	$0.185260\pi$
$757$	30.2487i	1.09941i	−0.835359	+	0.549704i	$0.814740\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.41154	0.0874184	0.0437092	−	0.999044i	$-0.486083\pi$
$761$	2.41154	0.0874184	0.0437092	+	0.999044i	$0.486083\pi$
$762$	0	0
$763$	− 15.8038i	− 0.572138i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−28.3923	−1.02519
$768$	0	0
$769$	−11.3923	−0.410817	−0.205409	−	0.978676i	$-0.565852\pi$
$769$	−11.3923	−0.410817	−0.205409	+	0.978676i	$0.565852\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 2.66025i	− 0.0956827i	−0.998855	−	0.0478413i	$-0.984766\pi$
$773$	− 2.66025i	− 0.0956827i	0.998855	−	0.0478413i	$-0.0152342\pi$
$774$	0	0
$775$	−6.92820	−0.248868
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 14.5359i	− 0.520803i
$780$	0	0
$781$	6.00000i	0.214697i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−17.1962	−0.613757
$786$	0	0
$787$	18.1962i	0.648623i	0.945950	+	0.324311i	$0.105133\pi$
$787$	18.1962i	0.648623i	−0.945950	+	0.324311i	$0.894867\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	14.1962	0.504757
$792$	0	0
$793$	−19.3923	−0.688641
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 10.5167i	− 0.372519i	−0.982501	−	0.186260i	$-0.940363\pi$
$797$	− 10.5167i	− 0.372519i	0.982501	−	0.186260i	$-0.0596366\pi$
$798$	0	0
$799$	55.1769	1.95202
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 6.33975i	− 0.223725i
$804$	0	0
$805$	2.78461i	0.0981446i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−18.1244	−0.637218	−0.318609	−	0.947886i	$-0.603216\pi$
$809$	−18.1244	−0.637218	−0.318609	+	0.947886i	$0.603216\pi$
$810$	0	0
$811$	16.7846i	0.589387i	0.955592	+	0.294694i	$0.0952176\pi$
$811$	16.7846i	0.589387i	−0.955592	+	0.294694i	$0.904782\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	14.5359	0.509170
$816$	0	0
$817$	−26.0000	−0.909625
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 24.1244i	− 0.841946i	−0.907073	−	0.420973i	$-0.861689\pi$
$821$	− 24.1244i	− 0.841946i	0.907073	−	0.420973i	$-0.138311\pi$
$822$	0	0
$823$	50.7846	1.77024	0.885120	−	0.465363i	$-0.154076\pi$
$823$	50.7846	1.77024	0.885120	+	0.465363i	$0.154076\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33.3731i	1.16050i	0.814440	+	0.580248i	$0.197044\pi$
$827$	33.3731i	1.16050i	−0.814440	+	0.580248i	$0.802956\pi$
$828$	0	0
$829$	− 16.3923i	− 0.569328i	−0.958627	−	0.284664i	$-0.908118\pi$
$829$	− 16.3923i	− 0.569328i	0.958627	−	0.284664i	$-0.0918821\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−23.0141	−0.797391
$834$	0	0
$835$	36.5885i	1.26620i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−46.0526	−1.58991	−0.794955	−	0.606668i	$-0.792506\pi$
$839$	−46.0526	−1.58991	−0.794955	+	0.606668i	$0.792506\pi$
$840$	0	0
$841$	10.7846	0.371883
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 6.92820i	− 0.238337i
$846$	0	0
$847$	11.9090	0.409197
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0.588457i	0.0201721i
$852$	0	0
$853$	25.8564i	0.885306i	0.896693	+	0.442653i	$0.145963\pi$
$853$	25.8564i	0.885306i	−0.896693	+	0.442653i	$0.854037\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.98076	0.0676615	0.0338308	−	0.999428i	$-0.489229\pi$
$857$	1.98076	0.0676615	0.0338308	+	0.999428i	$0.489229\pi$
$858$	0	0
$859$	− 23.1769i	− 0.790786i	−0.918512	−	0.395393i	$-0.870608\pi$
$859$	− 23.1769i	− 0.790786i	0.918512	−	0.395393i	$-0.129392\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−31.5167	−1.07284	−0.536420	−	0.843951i	$-0.680224\pi$
$863$	−31.5167	−1.07284	−0.536420	+	0.843951i	$0.680224\pi$
$864$	0	0
$865$	28.1769	0.958044
$866$	0	0
$867$	0	0
$868$	0	0
$869$	18.0000i	0.610608i
$870$	0	0
$871$	12.5885	0.426544
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 15.3731i	− 0.519705i
$876$	0	0
$877$	45.4974i	1.53634i	0.640247	+	0.768169i	$0.278832\pi$
$877$	45.4974i	1.53634i	−0.640247	+	0.768169i	$0.721168\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−47.5692	−1.60265	−0.801324	−	0.598231i	$-0.795871\pi$
$881$	−47.5692	−1.60265	−0.801324	+	0.598231i	$0.795871\pi$
$882$	0	0
$883$	27.5692i	0.927778i	0.885893	+	0.463889i	$0.153546\pi$
$883$	27.5692i	0.927778i	−0.885893	+	0.463889i	$0.846454\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28.9808	0.973079	0.486539	−	0.873659i	$-0.338259\pi$
$887$	28.9808	0.973079	0.486539	+	0.873659i	$0.338259\pi$
$888$	0	0
$889$	−11.5692	−0.388019
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 54.2487i	− 1.81536i
$894$	0	0
$895$	−38.7846	−1.29643
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 14.7846i	− 0.493094i
$900$	0	0
$901$	3.96152i	0.131978i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−19.6077	−0.651782
$906$	0	0
$907$	45.1769i	1.50007i	0.661395	+	0.750037i	$0.269965\pi$
$907$	45.1769i	1.50007i	−0.661395	+	0.750037i	$0.730035\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−34.0526	−1.12821	−0.564106	−	0.825703i	$-0.690779\pi$
$911$	−34.0526	−1.12821	−0.564106	+	0.825703i	$0.690779\pi$
$912$	0	0
$913$	13.1769	0.436092
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 2.78461i	− 0.0919559i
$918$	0	0
$919$	42.5885	1.40486	0.702432	−	0.711751i	$-0.252098\pi$
$919$	42.5885	1.40486	0.702432	+	0.711751i	$0.252098\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	14.1962i	0.467272i
$924$	0	0
$925$	− 0.928203i	− 0.0305191i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	35.1962	1.15475	0.577374	−	0.816480i	$-0.304077\pi$
$929$	35.1962	1.15475	0.577374	+	0.816480i	$0.304077\pi$
$930$	0	0
$931$	22.6269i	0.741568i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−9.37307	−0.306532
$936$	0	0
$937$	−30.1769	−0.985837	−0.492918	−	0.870076i	$-0.664070\pi$
$937$	−30.1769	−0.985837	−0.492918	+	0.870076i	$0.664070\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	8.66025i	0.282316i	0.989987	+	0.141158i	$0.0450826\pi$
$941$	8.66025i	0.282316i	−0.989987	+	0.141158i	$0.954917\pi$
$942$	0	0
$943$	−4.39230	−0.143033
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.05256i	0.131690i	0.997830	+	0.0658452i	$0.0209744\pi$
$947$	4.05256i	0.131690i	−0.997830	+	0.0658452i	$0.979026\pi$
$948$	0	0
$949$	− 15.0000i	− 0.486921i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	19.7321	0.639184	0.319592	−	0.947555i	$-0.396454\pi$
$953$	19.7321	0.639184	0.319592	+	0.947555i	$0.396454\pi$
$954$	0	0
$955$	− 33.3731i	− 1.07993i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−15.3731	−0.496422
$960$	0	0
$961$	−19.0000	−0.612903
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 44.6603i	− 1.43766i
$966$	0	0
$967$	23.9090	0.768860	0.384430	−	0.923154i	$-0.374398\pi$
$967$	23.9090	0.768860	0.384430	+	0.923154i	$0.374398\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	58.9808i	1.89278i	0.323022	+	0.946391i	$0.395301\pi$
$971$	58.9808i	1.89278i	−0.323022	+	0.946391i	$0.604699\pi$
$972$	0	0
$973$	− 17.7513i	− 0.569080i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	28.6410	0.916307	0.458154	−	0.888873i	$-0.348511\pi$
$977$	28.6410	0.916307	0.458154	+	0.888873i	$0.348511\pi$
$978$	0	0
$979$	1.01924i	0.0325750i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−31.6077	−1.00813	−0.504064	−	0.863666i	$-0.668163\pi$
$983$	−31.6077	−1.00813	−0.504064	+	0.863666i	$0.668163\pi$
$984$	0	0
$985$	−19.3923	−0.617890
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7.85641i	0.249819i
$990$	0	0
$991$	−27.1244	−0.861634	−0.430817	−	0.902439i	$-0.641774\pi$
$991$	−27.1244	−0.861634	−0.430817	+	0.902439i	$0.641774\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	31.1769i	0.988375i
$996$	0	0
$997$	10.6077i	0.335949i	0.985791	+	0.167975i	$0.0537226\pi$
$997$	10.6077i	0.335949i	−0.985791	+	0.167975i	$0.946277\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5184.2.d.n.2593.1	yes	4
3.2	odd	2		5184.2.d.m.2593.3	yes	4
4.3	odd	2		5184.2.d.b.2593.2	yes	4
8.3	odd	2		5184.2.d.b.2593.4	yes	4
8.5	even	2	inner	5184.2.d.n.2593.3	yes	4
12.11	even	2		5184.2.d.a.2593.4	yes	4
24.5	odd	2		5184.2.d.m.2593.1	yes	4
24.11	even	2		5184.2.d.a.2593.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5184.2.d.a.2593.2	✓	4	24.11	even	2
5184.2.d.a.2593.4	yes	4	12.11	even	2
5184.2.d.b.2593.2	yes	4	4.3	odd	2
5184.2.d.b.2593.4	yes	4	8.3	odd	2
5184.2.d.m.2593.1	yes	4	24.5	odd	2
5184.2.d.m.2593.3	yes	4	3.2	odd	2
5184.2.d.n.2593.1	yes	4	1.1	even	1	trivial
5184.2.d.n.2593.3	yes	4	8.5	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:	\( N \)	\(=\)	\( 5184 = 2^{6} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5184.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{5} +1.26795 q^{7} +O(q^{10})\)	\(q-1.73205i q^{5} +1.26795 q^{7} -1.26795i q^{11} -3.00000i q^{13} +4.26795 q^{17} -4.19615i q^{19} -1.26795 q^{23} +2.00000 q^{25} +4.26795i q^{29} -3.46410 q^{31} -2.19615i q^{35} -0.464102i q^{37} +3.46410 q^{41} -6.19615i q^{43} +12.9282 q^{47} -5.39230 q^{49} +0.928203i q^{53} -2.19615 q^{55} -9.46410i q^{59} -6.46410i q^{61} -5.19615 q^{65} +4.19615i q^{67} -4.73205 q^{71} +5.00000 q^{73} -1.60770i q^{77} -14.1962 q^{79} +10.3923i q^{83} -7.39230i q^{85} -0.803848 q^{89} -3.80385i q^{91} -7.26795 q^{95} +4.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{7}+O(q^{10}) \) 4 * q + 12 * q^7	\( 4 q + 12 q^{7} + 24 q^{17} - 12 q^{23} + 8 q^{25} + 24 q^{47} + 20 q^{49} + 12 q^{55} - 12 q^{71} + 20 q^{73} - 36 q^{79} - 24 q^{89} - 36 q^{95} + 16 q^{97}+O(q^{100}) \) 4 * q + 12 * q^7 + 24 * q^17 - 12 * q^23 + 8 * q^25 + 24 * q^47 + 20 * q^49 + 12 * q^55 - 12 * q^71 + 20 * q^73 - 36 * q^79 - 24 * q^89 - 36 * q^95 + 16 * q^97

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