LMFDB - Embedded newform 5184.2.a.cd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5184,2,Mod(1,5184)]

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, 0, 0]))

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

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

chi := DirichletCharacter("5184.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5184 = 2^{6} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5184.a (trivial)

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:	yes
Analytic conductor:	$41.3944484078$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 1 $ x^4 - 5*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2592)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.456850$ of defining polynomial
Character	$\chi$	$=$	5184.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64575 q^{5} +4.37780 q^{7} +O(q^{10})$	$q-2.64575 q^{5} +4.37780 q^{7} +3.58258 q^{11} +6.58258 q^{13} +1.73205 q^{17} +2.55040 q^{19} +7.58258 q^{23} +2.00000 q^{25} -6.10985 q^{29} +8.75560 q^{31} -11.5826 q^{35} -2.58258 q^{37} -1.82740 q^{41} -2.55040 q^{43} +8.00000 q^{47} +12.1652 q^{49} -1.82740 q^{53} -9.47860 q^{55} -8.00000 q^{59} +1.41742 q^{61} -17.4159 q^{65} -2.55040 q^{67} +0.417424 q^{71} -6.16515 q^{73} +15.6838 q^{77} +9.47860 q^{79} -15.1652 q^{83} -4.58258 q^{85} -12.3151 q^{89} +28.8172 q^{91} -6.74773 q^{95} -5.16515 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 4 q^{11} + 8 q^{13} + 12 q^{23} + 8 q^{25} - 28 q^{35} + 8 q^{37} + 32 q^{47} + 12 q^{49} - 32 q^{59} + 24 q^{61} + 20 q^{71} + 12 q^{73} - 24 q^{83} + 28 q^{95} + 16 q^{97}+O(q^{100}) $ 4 * q - 4 * q^11 + 8 * q^13 + 12 * q^23 + 8 * q^25 - 28 * q^35 + 8 * q^37 + 32 * q^47 + 12 * q^49 - 32 * q^59 + 24 * q^61 + 20 * q^71 + 12 * q^73 - 24 * q^83 + 28 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−2.64575	−1.18322	−0.591608	−	0.806226i	$-0.701507\pi$
$5$	−2.64575	−1.18322	−0.591608	+	0.806226i	$0.701507\pi$
$6$	0	0
$7$	4.37780	1.65465	0.827327	−	0.561721i	$-0.189860\pi$
$7$	4.37780	1.65465	0.827327	+	0.561721i	$0.189860\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.58258	1.08019	0.540094	−	0.841605i	$-0.318389\pi$
$11$	3.58258	1.08019	0.540094	+	0.841605i	$0.318389\pi$
$12$	0	0
$13$	6.58258	1.82568	0.912839	−	0.408320i	$-0.133885\pi$
$13$	6.58258	1.82568	0.912839	+	0.408320i	$0.133885\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.73205	0.420084	0.210042	−	0.977692i	$-0.432640\pi$
$17$	1.73205	0.420084	0.210042	+	0.977692i	$0.432640\pi$
$18$	0	0
$19$	2.55040	0.585102	0.292551	−	0.956250i	$-0.405496\pi$
$19$	2.55040	0.585102	0.292551	+	0.956250i	$0.405496\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.58258	1.58108	0.790538	−	0.612413i	$-0.209801\pi$
$23$	7.58258	1.58108	0.790538	+	0.612413i	$0.209801\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.10985	−1.13457	−0.567286	−	0.823521i	$-0.692006\pi$
$29$	−6.10985	−1.13457	−0.567286	+	0.823521i	$0.692006\pi$
$30$	0	0
$31$	8.75560	1.57255	0.786276	−	0.617875i	$-0.212006\pi$
$31$	8.75560	1.57255	0.786276	+	0.617875i	$0.212006\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−11.5826	−1.95781
$36$	0	0
$37$	−2.58258	−0.424573	−0.212286	−	0.977207i	$-0.568091\pi$
$37$	−2.58258	−0.424573	−0.212286	+	0.977207i	$0.568091\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.82740	−0.285392	−0.142696	−	0.989767i	$-0.545577\pi$
$41$	−1.82740	−0.285392	−0.142696	+	0.989767i	$0.545577\pi$
$42$	0	0
$43$	−2.55040	−0.388933	−0.194466	−	0.980909i	$-0.562297\pi$
$43$	−2.55040	−0.388933	−0.194466	+	0.980909i	$0.562297\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	12.1652	1.73788
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.82740	−0.251013	−0.125506	−	0.992093i	$-0.540056\pi$
$53$	−1.82740	−0.251013	−0.125506	+	0.992093i	$0.540056\pi$
$54$	0	0
$55$	−9.47860	−1.27809
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	1.41742	0.181483	0.0907413	−	0.995874i	$-0.471076\pi$
$61$	1.41742	0.181483	0.0907413	+	0.995874i	$0.471076\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−17.4159	−2.16017
$66$	0	0
$67$	−2.55040	−0.311581	−0.155791	−	0.987790i	$-0.549792\pi$
$67$	−2.55040	−0.311581	−0.155791	+	0.987790i	$0.549792\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0.417424	0.0495392	0.0247696	−	0.999693i	$-0.492115\pi$
$71$	0.417424	0.0495392	0.0247696	+	0.999693i	$0.492115\pi$
$72$	0	0
$73$	−6.16515	−0.721576	−0.360788	−	0.932648i	$-0.617492\pi$
$73$	−6.16515	−0.721576	−0.360788	+	0.932648i	$0.617492\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	15.6838	1.78734
$78$	0	0
$79$	9.47860	1.06643	0.533213	−	0.845981i	$-0.320984\pi$
$79$	9.47860	1.06643	0.533213	+	0.845981i	$0.320984\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−15.1652	−1.66459	−0.832296	−	0.554332i	$-0.812974\pi$
$83$	−15.1652	−1.66459	−0.832296	+	0.554332i	$0.812974\pi$
$84$	0	0
$85$	−4.58258	−0.497050
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.3151	−1.30539	−0.652697	−	0.757619i	$-0.726362\pi$
$89$	−12.3151	−1.30539	−0.652697	+	0.757619i	$0.726362\pi$
$90$	0	0
$91$	28.8172	3.02086
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.74773	−0.692302
$96$	0	0
$97$	−5.16515	−0.524442	−0.262221	−	0.965008i	$-0.584455\pi$
$97$	−5.16515	−0.524442	−0.262221	+	0.965008i	$0.584455\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.8564	−1.37876	−0.689382	−	0.724398i	$-0.742118\pi$
$101$	−13.8564	−1.37876	−0.689382	+	0.724398i	$0.742118\pi$
$102$	0	0
$103$	−12.4104	−1.22283	−0.611417	−	0.791309i	$-0.709400\pi$
$103$	−12.4104	−1.22283	−0.611417	+	0.791309i	$0.709400\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.16515	0.692681	0.346341	−	0.938109i	$-0.387424\pi$
$107$	7.16515	0.692681	0.346341	+	0.938109i	$0.387424\pi$
$108$	0	0
$109$	14.5826	1.39676	0.698379	−	0.715728i	$-0.253905\pi$
$109$	14.5826	1.39676	0.698379	+	0.715728i	$0.253905\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	19.2433	1.81025	0.905127	−	0.425142i	$-0.139776\pi$
$113$	19.2433	1.81025	0.905127	+	0.425142i	$0.139776\pi$
$114$	0	0
$115$	−20.0616	−1.87075
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.58258	0.695094
$120$	0	0
$121$	1.83485	0.166804
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.93725	0.709930
$126$	0	0
$127$	−18.2342	−1.61802	−0.809012	−	0.587792i	$-0.799997\pi$
$127$	−18.2342	−1.61802	−0.809012	+	0.587792i	$0.799997\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.58258	0.313011	0.156506	−	0.987677i	$-0.449977\pi$
$131$	3.58258	0.313011	0.156506	+	0.987677i	$0.449977\pi$
$132$	0	0
$133$	11.1652	0.968141
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.66025	−0.739895	−0.369948	−	0.929053i	$-0.620624\pi$
$137$	−8.66025	−0.739895	−0.369948	+	0.929053i	$0.620624\pi$
$138$	0	0
$139$	−8.75560	−0.742641	−0.371320	−	0.928505i	$-0.621095\pi$
$139$	−8.75560	−0.742641	−0.371320	+	0.928505i	$0.621095\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	23.5826	1.97207
$144$	0	0
$145$	16.1652	1.34244
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.00905	−0.0826647	−0.0413323	−	0.999145i	$-0.513160\pi$
$149$	−1.00905	−0.0826647	−0.0413323	+	0.999145i	$0.513160\pi$
$150$	0	0
$151$	−12.4104	−1.00994	−0.504972	−	0.863136i	$-0.668497\pi$
$151$	−12.4104	−1.00994	−0.504972	+	0.863136i	$0.668497\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−23.1652	−1.86067
$156$	0	0
$157$	20.5826	1.64267	0.821334	−	0.570447i	$-0.193230\pi$
$157$	20.5826	1.64267	0.821334	+	0.570447i	$0.193230\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	33.1950	2.61613
$162$	0	0
$163$	−3.65480	−0.286266	−0.143133	−	0.989703i	$-0.545718\pi$
$163$	−3.65480	−0.286266	−0.143133	+	0.989703i	$0.545718\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.5826	1.20582	0.602908	−	0.797811i	$-0.294009\pi$
$167$	15.5826	1.20582	0.602908	+	0.797811i	$0.294009\pi$
$168$	0	0
$169$	30.3303	2.33310
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2.45505	−0.186654	−0.0933270	−	0.995636i	$-0.529750\pi$
$173$	−2.45505	−0.186654	−0.0933270	+	0.995636i	$0.529750\pi$
$174$	0	0
$175$	8.75560	0.661861
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.83285	0.502361
$186$	0	0
$187$	6.20520	0.453769
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.58258	−0.548656	−0.274328	−	0.961636i	$-0.588455\pi$
$191$	−7.58258	−0.548656	−0.274328	+	0.961636i	$0.588455\pi$
$192$	0	0
$193$	−12.1652	−0.875667	−0.437833	−	0.899056i	$-0.644254\pi$
$193$	−12.1652	−0.875667	−0.437833	+	0.899056i	$0.644254\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.6748	1.04553	0.522767	−	0.852476i	$-0.324900\pi$
$197$	14.6748	1.04553	0.522767	+	0.852476i	$0.324900\pi$
$198$	0	0
$199$	−17.5112	−1.24134	−0.620668	−	0.784073i	$-0.713139\pi$
$199$	−17.5112	−1.24134	−0.620668	+	0.784073i	$0.713139\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−26.7477	−1.87732
$204$	0	0
$205$	4.83485	0.337680
$206$	0	0
$207$	0	0
$208$	0	0
$209$	9.13701	0.632020
$210$	0	0
$211$	−11.3060	−0.778338	−0.389169	−	0.921166i	$-0.627238\pi$
$211$	−11.3060	−0.778338	−0.389169	+	0.921166i	$0.627238\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.74773	0.460191
$216$	0	0
$217$	38.3303	2.60203
$218$	0	0
$219$	0	0
$220$	0	0
$221$	11.4014	0.766938
$222$	0	0
$223$	16.7882	1.12422	0.562111	−	0.827062i	$-0.309989\pi$
$223$	16.7882	1.12422	0.562111	+	0.827062i	$0.309989\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.7477	0.713352	0.356676	−	0.934228i	$-0.383910\pi$
$227$	10.7477	0.713352	0.356676	+	0.934228i	$0.383910\pi$
$228$	0	0
$229$	−3.74773	−0.247657	−0.123828	−	0.992304i	$-0.539517\pi$
$229$	−3.74773	−0.247657	−0.123828	+	0.992304i	$0.539517\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.6784	0.699562	0.349781	−	0.936831i	$-0.386256\pi$
$233$	10.6784	0.699562	0.349781	+	0.936831i	$0.386256\pi$
$234$	0	0
$235$	−21.1660	−1.38072
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.83485	−0.571479	−0.285739	−	0.958307i	$-0.592239\pi$
$239$	−8.83485	−0.571479	−0.285739	+	0.958307i	$0.592239\pi$
$240$	0	0
$241$	1.00000	0.0644157	0.0322078	−	0.999481i	$-0.489746\pi$
$241$	1.00000	0.0644157	0.0322078	+	0.999481i	$0.489746\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−32.1860	−2.05629
$246$	0	0
$247$	16.7882	1.06821
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4.41742	−0.278825	−0.139413	−	0.990234i	$-0.544521\pi$
$251$	−4.41742	−0.278825	−0.139413	+	0.990234i	$0.544521\pi$
$252$	0	0
$253$	27.1652	1.70786
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.55945	−0.222033	−0.111016	−	0.993819i	$-0.535411\pi$
$257$	−3.55945	−0.222033	−0.111016	+	0.993819i	$0.535411\pi$
$258$	0	0
$259$	−11.3060	−0.702521
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.33030	−0.390343	−0.195172	−	0.980769i	$-0.562526\pi$
$263$	−6.33030	−0.390343	−0.195172	+	0.980769i	$0.562526\pi$
$264$	0	0
$265$	4.83485	0.297002
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5.91915	−0.360897	−0.180449	−	0.983584i	$-0.557755\pi$
$269$	−5.91915	−0.360897	−0.180449	+	0.983584i	$0.557755\pi$
$270$	0	0
$271$	13.1334	0.797798	0.398899	−	0.916995i	$-0.369392\pi$
$271$	13.1334	0.797798	0.398899	+	0.916995i	$0.369392\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.16515	0.432075
$276$	0	0
$277$	20.3303	1.22153	0.610765	−	0.791812i	$-0.290862\pi$
$277$	20.3303	1.22153	0.610765	+	0.791812i	$0.290862\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−24.5348	−1.46362	−0.731811	−	0.681508i	$-0.761324\pi$
$281$	−24.5348	−1.46362	−0.731811	+	0.681508i	$0.761324\pi$
$282$	0	0
$283$	17.5112	1.04093	0.520467	−	0.853882i	$-0.325758\pi$
$283$	17.5112	1.04093	0.520467	+	0.853882i	$0.325758\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.00000	−0.472225
$288$	0	0
$289$	−14.0000	−0.823529
$290$	0	0
$291$	0	0
$292$	0	0
$293$	14.8655	0.868449	0.434225	−	0.900805i	$-0.357022\pi$
$293$	14.8655	0.868449	0.434225	+	0.900805i	$0.357022\pi$
$294$	0	0
$295$	21.1660	1.23233
$296$	0	0
$297$	0	0
$298$	0	0
$299$	49.9129	2.88654
$300$	0	0
$301$	−11.1652	−0.643549
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.75015	−0.214733
$306$	0	0
$307$	12.4104	0.708299	0.354150	−	0.935189i	$-0.384770\pi$
$307$	12.4104	0.708299	0.354150	+	0.935189i	$0.384770\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	15.1652	0.859937	0.429968	−	0.902844i	$-0.358525\pi$
$311$	15.1652	0.859937	0.429968	+	0.902844i	$0.358525\pi$
$312$	0	0
$313$	19.0000	1.07394	0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000	1.07394	0.536972	+	0.843600i	$0.319568\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.2759	1.53197	0.765983	−	0.642861i	$-0.222253\pi$
$317$	27.2759	1.53197	0.765983	+	0.642861i	$0.222253\pi$
$318$	0	0
$319$	−21.8890	−1.22555
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.41742	0.245792
$324$	0	0
$325$	13.1652	0.730271
$326$	0	0
$327$	0	0
$328$	0	0
$329$	35.0224	1.93085
$330$	0	0
$331$	−1.10440	−0.0607034	−0.0303517	−	0.999539i	$-0.509663\pi$
$331$	−1.10440	−0.0607034	−0.0303517	+	0.999539i	$0.509663\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.74773	0.368668
$336$	0	0
$337$	17.1652	0.935045	0.467523	−	0.883981i	$-0.345147\pi$
$337$	17.1652	0.935045	0.467523	+	0.883981i	$0.345147\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	31.3676	1.69865
$342$	0	0
$343$	22.6120	1.22093
$344$	0	0
$345$	0	0
$346$	0	0
$347$	35.5826	1.91017	0.955086	−	0.296328i	$-0.0957620\pi$
$347$	35.5826	1.91017	0.955086	+	0.296328i	$0.0957620\pi$
$348$	0	0
$349$	−1.16515	−0.0623691	−0.0311846	−	0.999514i	$-0.509928\pi$
$349$	−1.16515	−0.0623691	−0.0311846	+	0.999514i	$0.509928\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−31.3676	−1.66953	−0.834765	−	0.550607i	$-0.814396\pi$
$353$	−31.3676	−1.66953	−0.834765	+	0.550607i	$0.814396\pi$
$354$	0	0
$355$	−1.10440	−0.0586155
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−21.9129	−1.15652	−0.578259	−	0.815853i	$-0.696268\pi$
$359$	−21.9129	−1.15652	−0.578259	+	0.815853i	$0.696268\pi$
$360$	0	0
$361$	−12.4955	−0.657655
$362$	0	0
$363$	0	0
$364$	0	0
$365$	16.3115	0.853781
$366$	0	0
$367$	−26.2668	−1.37112	−0.685558	−	0.728018i	$-0.740442\pi$
$367$	−26.2668	−1.37112	−0.685558	+	0.728018i	$0.740442\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8.00000	−0.415339
$372$	0	0
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−40.2186	−2.07136
$378$	0	0
$379$	−5.10080	−0.262011	−0.131005	−	0.991382i	$-0.541820\pi$
$379$	−5.10080	−0.262011	−0.131005	+	0.991382i	$0.541820\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	15.5826	0.796232	0.398116	−	0.917335i	$-0.369664\pi$
$383$	15.5826	0.796232	0.398116	+	0.917335i	$0.369664\pi$
$384$	0	0
$385$	−41.4955	−2.11480
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−33.1950	−1.68305	−0.841527	−	0.540215i	$-0.818343\pi$
$389$	−33.1950	−1.68305	−0.841527	+	0.540215i	$0.818343\pi$
$390$	0	0
$391$	13.1334	0.664185
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−25.0780	−1.26181
$396$	0	0
$397$	−17.7477	−0.890733	−0.445366	−	0.895348i	$-0.646927\pi$
$397$	−17.7477	−0.890733	−0.445366	+	0.895348i	$0.646927\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.0953502	0.00476156	0.00238078	−	0.999997i	$-0.499242\pi$
$401$	0.0953502	0.00476156	0.00238078	+	0.999997i	$0.499242\pi$
$402$	0	0
$403$	57.6344	2.87098
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.25227	−0.458618
$408$	0	0
$409$	16.1652	0.799315	0.399658	−	0.916664i	$-0.369129\pi$
$409$	16.1652	0.799315	0.399658	+	0.916664i	$0.369129\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−35.0224	−1.72334
$414$	0	0
$415$	40.1232	1.96957
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−7.16515	−0.350041	−0.175020	−	0.984565i	$-0.555999\pi$
$419$	−7.16515	−0.350041	−0.175020	+	0.984565i	$0.555999\pi$
$420$	0	0
$421$	−0.582576	−0.0283930	−0.0141965	−	0.999899i	$-0.504519\pi$
$421$	−0.582576	−0.0283930	−0.0141965	+	0.999899i	$0.504519\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.46410	0.168034
$426$	0	0
$427$	6.20520	0.300291
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−39.1652	−1.88652	−0.943259	−	0.332057i	$-0.892258\pi$
$431$	−39.1652	−1.88652	−0.943259	+	0.332057i	$0.892258\pi$
$432$	0	0
$433$	−23.3303	−1.12118	−0.560591	−	0.828093i	$-0.689426\pi$
$433$	−23.3303	−1.12118	−0.560591	+	0.828093i	$0.689426\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	19.3386	0.925091
$438$	0	0
$439$	−21.1660	−1.01020	−0.505099	−	0.863061i	$-0.668544\pi$
$439$	−21.1660	−1.01020	−0.505099	+	0.863061i	$0.668544\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−36.6606	−1.74180	−0.870899	−	0.491462i	$-0.836463\pi$
$443$	−36.6606	−1.74180	−0.870899	+	0.491462i	$0.836463\pi$
$444$	0	0
$445$	32.5826	1.54456
$446$	0	0
$447$	0	0
$448$	0	0
$449$	29.5402	1.39409	0.697044	−	0.717028i	$-0.254498\pi$
$449$	29.5402	1.39409	0.697044	+	0.717028i	$0.254498\pi$
$450$	0	0
$451$	−6.54680	−0.308277
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−76.2432	−3.57434
$456$	0	0
$457$	8.16515	0.381950	0.190975	−	0.981595i	$-0.438835\pi$
$457$	8.16515	0.381950	0.190975	+	0.981595i	$0.438835\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−5.48220	−0.255332	−0.127666	−	0.991817i	$-0.540749\pi$
$461$	−5.48220	−0.255332	−0.127666	+	0.991817i	$0.540749\pi$
$462$	0	0
$463$	−3.65480	−0.169853	−0.0849265	−	0.996387i	$-0.527066\pi$
$463$	−3.65480	−0.169853	−0.0849265	+	0.996387i	$0.527066\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−0.834849	−0.0386322	−0.0193161	−	0.999813i	$-0.506149\pi$
$467$	−0.834849	−0.0386322	−0.0193161	+	0.999813i	$0.506149\pi$
$468$	0	0
$469$	−11.1652	−0.515559
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−9.13701	−0.420120
$474$	0	0
$475$	5.10080	0.234041
$476$	0	0
$477$	0	0
$478$	0	0
$479$	22.7477	1.03937	0.519685	−	0.854358i	$-0.326049\pi$
$479$	22.7477	1.03937	0.519685	+	0.854358i	$0.326049\pi$
$480$	0	0
$481$	−17.0000	−0.775133
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13.6657	0.620528
$486$	0	0
$487$	18.9572	0.859033	0.429517	−	0.903059i	$-0.358684\pi$
$487$	18.9572	0.859033	0.429517	+	0.903059i	$0.358684\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3.58258	0.161679	0.0808397	−	0.996727i	$-0.474240\pi$
$491$	3.58258	0.161679	0.0808397	+	0.996727i	$0.474240\pi$
$492$	0	0
$493$	−10.5826	−0.476615
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.82740	0.0819701
$498$	0	0
$499$	33.9180	1.51838	0.759189	−	0.650870i	$-0.225596\pi$
$499$	33.9180	1.51838	0.759189	+	0.650870i	$0.225596\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−15.1652	−0.676181	−0.338090	−	0.941114i	$-0.609781\pi$
$503$	−15.1652	−0.676181	−0.338090	+	0.941114i	$0.609781\pi$
$504$	0	0
$505$	36.6606	1.63138
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−33.1950	−1.47134	−0.735672	−	0.677338i	$-0.763133\pi$
$509$	−33.1950	−1.47134	−0.735672	+	0.677338i	$0.763133\pi$
$510$	0	0
$511$	−26.9898	−1.19396
$512$	0	0
$513$	0	0
$514$	0	0
$515$	32.8348	1.44688
$516$	0	0
$517$	28.6606	1.26049
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5.48220	0.240180	0.120090	−	0.992763i	$-0.461682\pi$
$521$	5.48220	0.240180	0.120090	+	0.992763i	$0.461682\pi$
$522$	0	0
$523$	−21.5076	−0.940462	−0.470231	−	0.882543i	$-0.655829\pi$
$523$	−21.5076	−0.940462	−0.470231	+	0.882543i	$0.655829\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	15.1652	0.660604
$528$	0	0
$529$	34.4955	1.49980
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−12.0290	−0.521034
$534$	0	0
$535$	−18.9572	−0.819592
$536$	0	0
$537$	0	0
$538$	0	0
$539$	43.5826	1.87723
$540$	0	0
$541$	−2.91288	−0.125234	−0.0626172	−	0.998038i	$-0.519945\pi$
$541$	−2.91288	−0.125234	−0.0626172	+	0.998038i	$0.519945\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−38.5819	−1.65267
$546$	0	0
$547$	13.8564	0.592457	0.296229	−	0.955117i	$-0.404271\pi$
$547$	13.8564	0.592457	0.296229	+	0.955117i	$0.404271\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−15.5826	−0.663840
$552$	0	0
$553$	41.4955	1.76457
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−33.8227	−1.43311	−0.716556	−	0.697529i	$-0.754283\pi$
$557$	−33.8227	−1.43311	−0.716556	+	0.697529i	$0.754283\pi$
$558$	0	0
$559$	−16.7882	−0.710066
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	−50.9129	−2.14192
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20.6893	0.867339	0.433669	−	0.901072i	$-0.357219\pi$
$569$	20.6893	0.867339	0.433669	+	0.901072i	$0.357219\pi$
$570$	0	0
$571$	21.1660	0.885770	0.442885	−	0.896578i	$-0.353955\pi$
$571$	21.1660	0.885770	0.442885	+	0.896578i	$0.353955\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	15.1652	0.632431
$576$	0	0
$577$	18.1652	0.756225	0.378113	−	0.925760i	$-0.376573\pi$
$577$	18.1652	0.756225	0.378113	+	0.925760i	$0.376573\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−66.3900	−2.75432
$582$	0	0
$583$	−6.54680	−0.271141
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4.41742	−0.182327	−0.0911633	−	0.995836i	$-0.529059\pi$
$587$	−4.41742	−0.182327	−0.0911633	+	0.995836i	$0.529059\pi$
$588$	0	0
$589$	22.3303	0.920104
$590$	0	0
$591$	0	0
$592$	0	0
$593$	12.1244	0.497888	0.248944	−	0.968518i	$-0.419917\pi$
$593$	12.1244	0.497888	0.248944	+	0.968518i	$0.419917\pi$
$594$	0	0
$595$	−20.0616	−0.822446
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−31.1652	−1.27337	−0.636687	−	0.771123i	$-0.719696\pi$
$599$	−31.1652	−1.27337	−0.636687	+	0.771123i	$0.719696\pi$
$600$	0	0
$601$	−14.4955	−0.591282	−0.295641	−	0.955299i	$-0.595533\pi$
$601$	−14.4955	−0.591282	−0.295641	+	0.955299i	$0.595533\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−4.85455	−0.197366
$606$	0	0
$607$	−30.6446	−1.24383	−0.621913	−	0.783086i	$-0.713644\pi$
$607$	−30.6446	−1.24383	−0.621913	+	0.783086i	$0.713644\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	52.6606	2.13042
$612$	0	0
$613$	−30.0000	−1.21169	−0.605844	−	0.795583i	$-0.707165\pi$
$613$	−30.0000	−1.21169	−0.605844	+	0.795583i	$0.707165\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10.6784	0.429894	0.214947	−	0.976626i	$-0.431042\pi$
$617$	10.6784	0.429894	0.214947	+	0.976626i	$0.431042\pi$
$618$	0	0
$619$	−40.1232	−1.61269	−0.806344	−	0.591447i	$-0.798557\pi$
$619$	−40.1232	−1.61269	−0.806344	+	0.591447i	$0.798557\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−53.9129	−2.15997
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.47315	−0.178356
$630$	0	0
$631$	22.6120	0.900170	0.450085	−	0.892986i	$-0.351394\pi$
$631$	22.6120	0.900170	0.450085	+	0.892986i	$0.351394\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	48.2432	1.91447
$636$	0	0
$637$	80.0780	3.17281
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−19.2433	−0.760063	−0.380032	−	0.924974i	$-0.624087\pi$
$641$	−19.2433	−0.760063	−0.380032	+	0.924974i	$0.624087\pi$
$642$	0	0
$643$	−26.2668	−1.03586	−0.517931	−	0.855422i	$-0.673298\pi$
$643$	−26.2668	−1.03586	−0.517931	+	0.855422i	$0.673298\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	13.9129	0.546972	0.273486	−	0.961876i	$-0.411823\pi$
$647$	13.9129	0.546972	0.273486	+	0.961876i	$0.411823\pi$
$648$	0	0
$649$	−28.6606	−1.12503
$650$	0	0
$651$	0	0
$652$	0	0
$653$	25.8854	1.01297	0.506487	−	0.862247i	$-0.330944\pi$
$653$	25.8854	1.01297	0.506487	+	0.862247i	$0.330944\pi$
$654$	0	0
$655$	−9.47860	−0.370360
$656$	0	0
$657$	0	0
$658$	0	0
$659$	34.7477	1.35358	0.676790	−	0.736176i	$-0.263371\pi$
$659$	34.7477	1.35358	0.676790	+	0.736176i	$0.263371\pi$
$660$	0	0
$661$	11.7477	0.456934	0.228467	−	0.973552i	$-0.426629\pi$
$661$	11.7477	0.456934	0.228467	+	0.973552i	$0.426629\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−29.5402	−1.14552
$666$	0	0
$667$	−46.3284	−1.79384
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.07803	0.196035
$672$	0	0
$673$	−21.8348	−0.841672	−0.420836	−	0.907137i	$-0.638263\pi$
$673$	−21.8348	−0.841672	−0.420836	+	0.907137i	$0.638263\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	36.8498	1.41625	0.708127	−	0.706085i	$-0.249541\pi$
$677$	36.8498	1.41625	0.708127	+	0.706085i	$0.249541\pi$
$678$	0	0
$679$	−22.6120	−0.867769
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	0	0
$685$	22.9129	0.875456
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−12.0290	−0.458269
$690$	0	0
$691$	16.4068	0.624144	0.312072	−	0.950058i	$-0.398977\pi$
$691$	16.4068	0.624144	0.312072	+	0.950058i	$0.398977\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	23.1652	0.878704
$696$	0	0
$697$	−3.16515	−0.119889
$698$	0	0
$699$	0	0
$700$	0	0
$701$	27.2759	1.03020	0.515098	−	0.857132i	$-0.327756\pi$
$701$	27.2759	1.03020	0.515098	+	0.857132i	$0.327756\pi$
$702$	0	0
$703$	−6.58660	−0.248418
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−60.6606	−2.28138
$708$	0	0
$709$	−35.7477	−1.34253	−0.671267	−	0.741216i	$-0.734250\pi$
$709$	−35.7477	−1.34253	−0.671267	+	0.741216i	$0.734250\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	66.3900	2.48633
$714$	0	0
$715$	−62.3936	−2.33339
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.08712	−0.0778365	−0.0389182	−	0.999242i	$-0.512391\pi$
$719$	−2.08712	−0.0778365	−0.0389182	+	0.999242i	$0.512391\pi$
$720$	0	0
$721$	−54.3303	−2.02337
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−12.2197	−0.453828
$726$	0	0
$727$	−0.723000	−0.0268146	−0.0134073	−	0.999910i	$-0.504268\pi$
$727$	−0.723000	−0.0268146	−0.0134073	+	0.999910i	$0.504268\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4.41742	−0.163384
$732$	0	0
$733$	−32.3303	−1.19415	−0.597073	−	0.802187i	$-0.703670\pi$
$733$	−32.3303	−1.19415	−0.597073	+	0.802187i	$0.703670\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.13701	−0.336566
$738$	0	0
$739$	−42.3320	−1.55721	−0.778604	−	0.627515i	$-0.784072\pi$
$739$	−42.3320	−1.55721	−0.778604	+	0.627515i	$0.784072\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	30.3303	1.11271	0.556355	−	0.830944i	$-0.312199\pi$
$743$	30.3303	1.11271	0.556355	+	0.830944i	$0.312199\pi$
$744$	0	0
$745$	2.66970	0.0978101
$746$	0	0
$747$	0	0
$748$	0	0
$749$	31.3676	1.14615
$750$	0	0
$751$	20.4430	0.745976	0.372988	−	0.927836i	$-0.378333\pi$
$751$	20.4430	0.745976	0.372988	+	0.927836i	$0.378333\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	32.8348	1.19498
$756$	0	0
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−22.5167	−0.816228	−0.408114	−	0.912931i	$-0.633813\pi$
$761$	−22.5167	−0.816228	−0.408114	+	0.912931i	$0.633813\pi$
$762$	0	0
$763$	63.8396	2.31115
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−52.6606	−1.90146
$768$	0	0
$769$	3.83485	0.138288	0.0691441	−	0.997607i	$-0.477973\pi$
$769$	3.83485	0.138288	0.0691441	+	0.997607i	$0.477973\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	37.6682	1.35483	0.677415	−	0.735601i	$-0.263100\pi$
$773$	37.6682	1.35483	0.677415	+	0.735601i	$0.263100\pi$
$774$	0	0
$775$	17.5112	0.629021
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.66061	−0.166984
$780$	0	0
$781$	1.49545	0.0535116
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−54.4564	−1.94363
$786$	0	0
$787$	32.4720	1.15750	0.578751	−	0.815504i	$-0.303540\pi$
$787$	32.4720	1.15750	0.578751	+	0.815504i	$0.303540\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	84.2432	2.99534
$792$	0	0
$793$	9.33030	0.331329
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.02715	0.107227	0.0536136	−	0.998562i	$-0.482926\pi$
$797$	3.02715	0.107227	0.0536136	+	0.998562i	$0.482926\pi$
$798$	0	0
$799$	13.8564	0.490204
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−22.0871	−0.779438
$804$	0	0
$805$	−87.8258	−3.09545
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−34.5457	−1.21456	−0.607280	−	0.794488i	$-0.707740\pi$
$809$	−34.5457	−1.21456	−0.607280	+	0.794488i	$0.707740\pi$
$810$	0	0
$811$	−17.5112	−0.614902	−0.307451	−	0.951564i	$-0.599476\pi$
$811$	−17.5112	−0.614902	−0.307451	+	0.951564i	$0.599476\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	9.66970	0.338715
$816$	0	0
$817$	−6.50455	−0.227565
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.6929	0.582585	0.291292	−	0.956634i	$-0.405915\pi$
$821$	16.6929	0.582585	0.291292	+	0.956634i	$0.405915\pi$
$822$	0	0
$823$	45.2240	1.57641	0.788205	−	0.615413i	$-0.211011\pi$
$823$	45.2240	1.57641	0.788205	+	0.615413i	$0.211011\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	18.7477	0.651922	0.325961	−	0.945383i	$-0.394312\pi$
$827$	18.7477	0.651922	0.325961	+	0.945383i	$0.394312\pi$
$828$	0	0
$829$	28.3303	0.983952	0.491976	−	0.870609i	$-0.336275\pi$
$829$	28.3303	0.983952	0.491976	+	0.870609i	$0.336275\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	21.0707	0.730055
$834$	0	0
$835$	−41.2276	−1.42674
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−17.2523	−0.595615	−0.297807	−	0.954626i	$-0.596255\pi$
$839$	−17.2523	−0.595615	−0.297807	+	0.954626i	$0.596255\pi$
$840$	0	0
$841$	8.33030	0.287252
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−80.2464	−2.76056
$846$	0	0
$847$	8.03260	0.276004
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−19.5826	−0.671282
$852$	0	0
$853$	−36.3303	−1.24393	−0.621963	−	0.783047i	$-0.713665\pi$
$853$	−36.3303	−1.24393	−0.621963	+	0.783047i	$0.713665\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	34.7364	1.18657	0.593286	−	0.804992i	$-0.297830\pi$
$857$	34.7364	1.18657	0.593286	+	0.804992i	$0.297830\pi$
$858$	0	0
$859$	6.54680	0.223374	0.111687	−	0.993743i	$-0.464375\pi$
$859$	6.54680	0.223374	0.111687	+	0.993743i	$0.464375\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−51.4083	−1.74996	−0.874980	−	0.484159i	$-0.839126\pi$
$863$	−51.4083	−1.74996	−0.874980	+	0.484159i	$0.839126\pi$
$864$	0	0
$865$	6.49545	0.220852
$866$	0	0
$867$	0	0
$868$	0	0
$869$	33.9578	1.15194
$870$	0	0
$871$	−16.7882	−0.568847
$872$	0	0
$873$	0	0
$874$	0	0
$875$	34.7477	1.17469
$876$	0	0
$877$	−24.0780	−0.813057	−0.406529	−	0.913638i	$-0.633261\pi$
$877$	−24.0780	−0.813057	−0.406529	+	0.913638i	$0.633261\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	33.1950	1.11837	0.559184	−	0.829043i	$-0.311114\pi$
$881$	33.1950	1.11837	0.559184	+	0.829043i	$0.311114\pi$
$882$	0	0
$883$	43.7780	1.47325	0.736624	−	0.676303i	$-0.236419\pi$
$883$	43.7780	1.47325	0.736624	+	0.676303i	$0.236419\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−16.4174	−0.551243	−0.275622	−	0.961266i	$-0.588884\pi$
$887$	−16.4174	−0.551243	−0.275622	+	0.961266i	$0.588884\pi$
$888$	0	0
$889$	−79.8258	−2.67727
$890$	0	0
$891$	0	0
$892$	0	0
$893$	20.4032	0.682767
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−53.4955	−1.78417
$900$	0	0
$901$	−3.16515	−0.105446
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.29150	0.175895
$906$	0	0
$907$	−29.9216	−0.993531	−0.496765	−	0.867885i	$-0.665479\pi$
$907$	−29.9216	−0.993531	−0.496765	+	0.867885i	$0.665479\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	31.5826	1.04638	0.523189	−	0.852217i	$-0.324742\pi$
$911$	31.5826	1.04638	0.523189	+	0.852217i	$0.324742\pi$
$912$	0	0
$913$	−54.3303	−1.79807
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15.6838	0.517925
$918$	0	0
$919$	−14.5794	−0.480930	−0.240465	−	0.970658i	$-0.577300\pi$
$919$	−14.5794	−0.480930	−0.240465	+	0.970658i	$0.577300\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.74773	0.0904425
$924$	0	0
$925$	−5.16515	−0.169829
$926$	0	0
$927$	0	0
$928$	0	0
$929$	15.5885	0.511441	0.255720	−	0.966751i	$-0.417687\pi$
$929$	15.5885	0.511441	0.255720	+	0.966751i	$0.417687\pi$
$930$	0	0
$931$	31.0260	1.01684
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−16.4174	−0.536907
$936$	0	0
$937$	−16.1652	−0.528092	−0.264046	−	0.964510i	$-0.585057\pi$
$937$	−16.1652	−0.528092	−0.264046	+	0.964510i	$0.585057\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	28.9126	0.942523	0.471261	−	0.881994i	$-0.343799\pi$
$941$	28.9126	0.942523	0.471261	+	0.881994i	$0.343799\pi$
$942$	0	0
$943$	−13.8564	−0.451227
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−17.0780	−0.554961	−0.277481	−	0.960731i	$-0.589499\pi$
$947$	−17.0780	−0.554961	−0.277481	+	0.960731i	$0.589499\pi$
$948$	0	0
$949$	−40.5826	−1.31737
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−17.0345	−0.551800	−0.275900	−	0.961186i	$-0.588976\pi$
$953$	−17.0345	−0.551800	−0.275900	+	0.961186i	$0.588976\pi$
$954$	0	0
$955$	20.0616	0.649178
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−37.9129	−1.22427
$960$	0	0
$961$	45.6606	1.47292
$962$	0	0
$963$	0	0
$964$	0	0
$965$	32.1860	1.03610
$966$	0	0
$967$	14.5794	0.468842	0.234421	−	0.972135i	$-0.424681\pi$
$967$	14.5794	0.468842	0.234421	+	0.972135i	$0.424681\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−26.7477	−0.858375	−0.429188	−	0.903215i	$-0.641200\pi$
$971$	−26.7477	−0.858375	−0.429188	+	0.903215i	$0.641200\pi$
$972$	0	0
$973$	−38.3303	−1.22881
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−29.5402	−0.945075	−0.472538	−	0.881311i	$-0.656662\pi$
$977$	−29.5402	−0.945075	−0.472538	+	0.881311i	$0.656662\pi$
$978$	0	0
$979$	−44.1196	−1.41007
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.50455	0.0798826	0.0399413	−	0.999202i	$-0.487283\pi$
$983$	2.50455	0.0798826	0.0399413	+	0.999202i	$0.487283\pi$
$984$	0	0
$985$	−38.8258	−1.23709
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−19.3386	−0.614932
$990$	0	0
$991$	−18.2342	−0.579229	−0.289614	−	0.957143i	$-0.593527\pi$
$991$	−18.2342	−0.579229	−0.289614	+	0.957143i	$0.593527\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	46.3303	1.46877
$996$	0	0
$997$	−37.7477	−1.19548	−0.597741	−	0.801689i	$-0.703935\pi$
$997$	−37.7477	−1.19548	−0.597741	+	0.801689i	$0.703935\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5184.2.a.cd.1.2		4
3.2	odd	2		5184.2.a.ce.1.4		4
4.3	odd	2		5184.2.a.ce.1.1		4
8.3	odd	2		2592.2.a.v.1.3	yes	4
8.5	even	2		2592.2.a.w.1.4	yes	4
12.11	even	2	inner	5184.2.a.cd.1.3		4
24.5	odd	2		2592.2.a.v.1.2	✓	4
24.11	even	2		2592.2.a.w.1.1	yes	4
72.5	odd	6		2592.2.i.bh.865.3		8
72.11	even	6		2592.2.i.bg.1729.4		8
72.13	even	6		2592.2.i.bg.865.1		8
72.29	odd	6		2592.2.i.bh.1729.3		8
72.43	odd	6		2592.2.i.bh.1729.2		8
72.59	even	6		2592.2.i.bg.865.4		8
72.61	even	6		2592.2.i.bg.1729.1		8
72.67	odd	6		2592.2.i.bh.865.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2592.2.a.v.1.2	✓	4	24.5	odd	2
2592.2.a.v.1.3	yes	4	8.3	odd	2
2592.2.a.w.1.1	yes	4	24.11	even	2
2592.2.a.w.1.4	yes	4	8.5	even	2
2592.2.i.bg.865.1		8	72.13	even	6
2592.2.i.bg.865.4		8	72.59	even	6
2592.2.i.bg.1729.1		8	72.61	even	6
2592.2.i.bg.1729.4		8	72.11	even	6
2592.2.i.bh.865.2		8	72.67	odd	6
2592.2.i.bh.865.3		8	72.5	odd	6
2592.2.i.bh.1729.2		8	72.43	odd	6
2592.2.i.bh.1729.3		8	72.29	odd	6
5184.2.a.cd.1.2		4	1.1	even	1	trivial
5184.2.a.cd.1.3		4	12.11	even	2	inner
5184.2.a.ce.1.1		4	4.3	odd	2
5184.2.a.ce.1.4		4	3.2	odd	2

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.a (trivial)

\(f(q)\)	\(=\)	\(q-2.64575 q^{5} +4.37780 q^{7} +O(q^{10})\)	\(q-2.64575 q^{5} +4.37780 q^{7} +3.58258 q^{11} +6.58258 q^{13} +1.73205 q^{17} +2.55040 q^{19} +7.58258 q^{23} +2.00000 q^{25} -6.10985 q^{29} +8.75560 q^{31} -11.5826 q^{35} -2.58258 q^{37} -1.82740 q^{41} -2.55040 q^{43} +8.00000 q^{47} +12.1652 q^{49} -1.82740 q^{53} -9.47860 q^{55} -8.00000 q^{59} +1.41742 q^{61} -17.4159 q^{65} -2.55040 q^{67} +0.417424 q^{71} -6.16515 q^{73} +15.6838 q^{77} +9.47860 q^{79} -15.1652 q^{83} -4.58258 q^{85} -12.3151 q^{89} +28.8172 q^{91} -6.74773 q^{95} -5.16515 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 4 q^{11} + 8 q^{13} + 12 q^{23} + 8 q^{25} - 28 q^{35} + 8 q^{37} + 32 q^{47} + 12 q^{49} - 32 q^{59} + 24 q^{61} + 20 q^{71} + 12 q^{73} - 24 q^{83} + 28 q^{95} + 16 q^{97}+O(q^{100}) \) 4 * q - 4 * q^11 + 8 * q^13 + 12 * q^23 + 8 * q^25 - 28 * q^35 + 8 * q^37 + 32 * q^47 + 12 * q^49 - 32 * q^59 + 24 * q^61 + 20 * q^71 + 12 * q^73 - 24 * q^83 + 28 * q^95 + 16 * q^97

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