LMFDB - Embedded newform 5184.2.f.e.2591.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5184,2,Mod(2591,5184)]

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([1, 1, 1]))

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

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

chi := DirichletCharacter("5184.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5184 = 2^{6} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5184.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$41.3944484078$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8x^{14} + 49x^{12} - 104x^{10} + 160x^{8} - 104x^{6} + 49x^{4} - 8x^{2} + 1 $ x^16 - 8x^14 + 49x^12 - 104x^10 + 160x^8 - 104x^6 + 49x^4 - 8*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.6
Root			$0.670418 + 0.387066i$ of defining polynomial
Character	$\chi$	$=$	5184.2591
Dual form			5184.2.f.e.2591.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.59733 q^{5} +1.16418i q^{7} +O(q^{10})$	$q-1.59733 q^{5} +1.16418i q^{7} +1.98092i q^{11} -0.164177i q^{13} -3.57825i q^{17} +3.16418 q^{19} -1.21374 q^{23} -2.44854 q^{25} -4.98396 q^{29} -6.64469i q^{31} -1.85957i q^{35} +1.10377i q^{37} -2.02533i q^{41} +8.34477 q^{43} +2.21731 q^{47} +5.64469 q^{49} +9.65461 q^{53} -3.16418i q^{55} -0.888520i q^{59} +9.74846i q^{61} +0.262245i q^{65} +3.98359 q^{67} -12.8049 q^{71} +0.879814 q^{73} -2.30614 q^{77} -2.91179i q^{79} +13.9298i q^{83} +5.71564i q^{85} -8.41501i q^{89} +0.191131 q^{91} -5.05423 q^{95} +4.31634 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 24 q^{19} + 16 q^{25} + 24 q^{43} - 48 q^{49} + 120 q^{67} + 16 q^{73} + 168 q^{91} - 16 q^{97}+O(q^{100}) $ 16 * q + 24 * q^19 + 16 * q^25 + 24 * q^43 - 48 * q^49 + 120 * q^67 + 16 * q^73 + 168 * q^91 - 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
$5$	−1.59733	−0.714347	−0.357174	−	0.934038i	$-0.616260\pi$
$5$	−1.59733	−0.714347	−0.357174	+	0.934038i	$0.616260\pi$
$6$	0	0
$7$	1.16418i	0.440018i	0.975498	+	0.220009i	$0.0706086\pi$
$7$	1.16418i	0.440018i	−0.975498	+	0.220009i	$0.929391\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.98092i	0.597269i	0.954368	+	0.298635i	$0.0965312\pi$
$11$	1.98092i	0.597269i	−0.954368	+	0.298635i	$0.903469\pi$
$12$	0	0
$13$	− 0.164177i	− 0.0455345i	−0.999741	−	0.0227672i	$-0.992752\pi$
$13$	− 0.164177i	− 0.0455345i	0.999741	−	0.0227672i	$-0.00724766\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.57825i	− 0.867852i	−0.900948	−	0.433926i	$-0.857128\pi$
$17$	− 3.57825i	− 0.867852i	0.900948	−	0.433926i	$-0.142872\pi$
$18$	0	0
$19$	3.16418	0.725912	0.362956	−	0.931806i	$-0.381768\pi$
$19$	3.16418	0.725912	0.362956	+	0.931806i	$0.381768\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.21374	−0.253082	−0.126541	−	0.991961i	$-0.540388\pi$
$23$	−1.21374	−0.253082	−0.126541	+	0.991961i	$0.540388\pi$
$24$	0	0
$25$	−2.44854	−0.489708
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.98396	−0.925499	−0.462749	−	0.886489i	$-0.653137\pi$
$29$	−4.98396	−0.925499	−0.462749	+	0.886489i	$0.653137\pi$
$30$	0	0
$31$	− 6.64469i	− 1.19342i	−0.802456	−	0.596711i	$-0.796474\pi$
$31$	− 6.64469i	− 1.19342i	0.802456	−	0.596711i	$-0.203526\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1.85957i	− 0.314325i
$36$	0	0
$37$	1.10377i	0.181459i	0.995876	+	0.0907295i	$0.0289199\pi$
$37$	1.10377i	0.181459i	−0.995876	+	0.0907295i	$0.971080\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 2.02533i	− 0.316304i	−0.987415	−	0.158152i	$-0.949446\pi$
$41$	− 2.02533i	− 0.316304i	0.987415	−	0.158152i	$-0.0505536\pi$
$42$	0	0
$43$	8.34477	1.27257	0.636283	−	0.771456i	$-0.280471\pi$
$43$	8.34477	1.27257	0.636283	+	0.771456i	$0.280471\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.21731	0.323428	0.161714	−	0.986838i	$-0.448298\pi$
$47$	2.21731	0.323428	0.161714	+	0.986838i	$0.448298\pi$
$48$	0	0
$49$	5.64469	0.806385
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.65461	1.32616	0.663081	−	0.748548i	$-0.269248\pi$
$53$	9.65461	1.32616	0.663081	+	0.748548i	$0.269248\pi$
$54$	0	0
$55$	− 3.16418i	− 0.426658i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 0.888520i	− 0.115675i	−0.998326	−	0.0578377i	$-0.981579\pi$
$59$	− 0.888520i	− 0.115675i	0.998326	−	0.0578377i	$-0.0184206\pi$
$60$	0	0
$61$	9.74846i	1.24816i	0.781359	+	0.624081i	$0.214527\pi$
$61$	9.74846i	1.24816i	−0.781359	+	0.624081i	$0.785473\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.262245i	0.0325274i
$66$	0	0
$67$	3.98359	0.486673	0.243336	−	0.969942i	$-0.421758\pi$
$67$	3.98359	0.486673	0.243336	+	0.969942i	$0.421758\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.8049	−1.51966	−0.759828	−	0.650124i	$-0.774717\pi$
$71$	−12.8049	−1.51966	−0.759828	+	0.650124i	$0.774717\pi$
$72$	0	0
$73$	0.879814	0.102974	0.0514872	−	0.998674i	$-0.483604\pi$
$73$	0.879814	0.102974	0.0514872	+	0.998674i	$0.483604\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.30614	−0.262809
$78$	0	0
$79$	− 2.91179i	− 0.327602i	−0.986493	−	0.163801i	$-0.947625\pi$
$79$	− 2.91179i	− 0.327602i	0.986493	−	0.163801i	$-0.0523755\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.9298i	1.52899i	0.644630	+	0.764495i	$0.277012\pi$
$83$	13.9298i	1.52899i	−0.644630	+	0.764495i	$0.722988\pi$
$84$	0	0
$85$	5.71564i	0.619948i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 8.41501i	− 0.891990i	−0.895036	−	0.445995i	$-0.852850\pi$
$89$	− 8.41501i	− 0.891990i	0.895036	−	0.445995i	$-0.147150\pi$
$90$	0	0
$91$	0.191131	0.0200360
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.05423	−0.518553
$96$	0	0
$97$	4.31634	0.438258	0.219129	−	0.975696i	$-0.429678\pi$
$97$	4.31634	0.438258	0.219129	+	0.975696i	$0.429678\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.3103	−1.12542	−0.562709	−	0.826655i	$-0.690241\pi$
$101$	−11.3103	−1.12542	−0.562709	+	0.826655i	$0.690241\pi$
$102$	0	0
$103$	2.00000i	0.197066i	0.995134	+	0.0985329i	$0.0314150\pi$
$103$	2.00000i	0.197066i	−0.995134	+	0.0985329i	$0.968585\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 19.4306i	− 1.87842i	−0.343339	−	0.939211i	$-0.611558\pi$
$107$	− 19.4306i	− 1.87842i	0.343339	−	0.939211i	$-0.388442\pi$
$108$	0	0
$109$	10.7760i	1.03216i	0.856541	+	0.516079i	$0.172609\pi$
$109$	10.7760i	1.03216i	−0.856541	+	0.516079i	$0.827391\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.46677i	0.420198i	0.977680	+	0.210099i	$0.0673786\pi$
$113$	4.46677i	0.420198i	−0.977680	+	0.210099i	$0.932621\pi$
$114$	0	0
$115$	1.93874	0.180789
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.16571	0.381870
$120$	0	0
$121$	7.07597	0.643270
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.8978	1.06417
$126$	0	0
$127$	− 9.22813i	− 0.818864i	−0.912341	−	0.409432i	$-0.865727\pi$
$127$	− 9.22813i	− 0.818864i	0.912341	−	0.409432i	$-0.134273\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.6764i	0.932799i	0.884574	+	0.466399i	$0.154449\pi$
$131$	10.6764i	0.932799i	−0.884574	+	0.466399i	$0.845551\pi$
$132$	0	0
$133$	3.68366i	0.319414i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.2163i	1.72719i	0.504182	+	0.863597i	$0.331794\pi$
$137$	20.2163i	1.72719i	−0.504182	+	0.863597i	$0.668206\pi$
$138$	0	0
$139$	13.1686	1.11694	0.558472	−	0.829523i	$-0.311388\pi$
$139$	13.1686	1.11694	0.558472	+	0.829523i	$0.311388\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.325221	0.0271963
$144$	0	0
$145$	7.96103	0.661128
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.99383	0.491033	0.245517	−	0.969392i	$-0.421042\pi$
$149$	5.99383	0.491033	0.245517	+	0.969392i	$0.421042\pi$
$150$	0	0
$151$	− 2.32835i	− 0.189479i	−0.995502	−	0.0947394i	$-0.969798\pi$
$151$	− 2.32835i	− 0.189479i	0.995502	−	0.0947394i	$-0.0302018\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	10.6138i	0.852518i
$156$	0	0
$157$	9.14861i	0.730139i	0.930980	+	0.365069i	$0.118955\pi$
$157$	9.14861i	0.730139i	−0.930980	+	0.365069i	$0.881045\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1.41301i	− 0.111361i
$162$	0	0
$163$	22.1416	1.73427	0.867133	−	0.498077i	$-0.165960\pi$
$163$	22.1416	1.73427	0.867133	+	0.498077i	$0.165960\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.4440	1.65939	0.829693	−	0.558220i	$-0.188516\pi$
$167$	21.4440	1.65939	0.829693	+	0.558220i	$0.188516\pi$
$168$	0	0
$169$	12.9730	0.997927
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−16.7481	−1.27334	−0.636668	−	0.771138i	$-0.719688\pi$
$173$	−16.7481	−1.27334	−0.636668	+	0.771138i	$0.719688\pi$
$174$	0	0
$175$	− 2.85053i	− 0.215480i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 16.1146i	− 1.20446i	−0.798323	−	0.602229i	$-0.794279\pi$
$179$	− 16.1146i	− 1.20446i	0.798323	−	0.602229i	$-0.205721\pi$
$180$	0	0
$181$	16.3612i	1.21612i	0.793892	+	0.608059i	$0.208051\pi$
$181$	16.3612i	1.21612i	−0.793892	+	0.608059i	$0.791949\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 1.76309i	− 0.129625i
$186$	0	0
$187$	7.08821	0.518341
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.6326	1.34820	0.674102	−	0.738638i	$-0.264531\pi$
$191$	18.6326	1.34820	0.674102	+	0.738638i	$0.264531\pi$
$192$	0	0
$193$	−1.65671	−0.119252	−0.0596262	−	0.998221i	$-0.518991\pi$
$193$	−1.65671	−0.119252	−0.0596262	+	0.998221i	$0.518991\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.8978	0.847681	0.423840	−	0.905737i	$-0.360682\pi$
$197$	11.8978	0.847681	0.423840	+	0.905737i	$0.360682\pi$
$198$	0	0
$199$	8.32835i	0.590381i	0.955438	+	0.295191i	$0.0953832\pi$
$199$	8.32835i	0.590381i	−0.955438	+	0.295191i	$0.904617\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 5.80222i	− 0.407236i
$204$	0	0
$205$	3.23512i	0.225951i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.26797i	0.433565i
$210$	0	0
$211$	4.77187	0.328509	0.164255	−	0.986418i	$-0.447478\pi$
$211$	4.77187	0.328509	0.164255	+	0.986418i	$0.447478\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−13.3293	−0.909053
$216$	0	0
$217$	7.73560	0.525127
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.587465	−0.0395172
$222$	0	0
$223$	− 24.4415i	− 1.63673i	−0.574701	−	0.818363i	$-0.694882\pi$
$223$	− 24.4415i	− 1.63673i	0.574701	−	0.818363i	$-0.305118\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.9107i	1.05603i	0.849235	+	0.528014i	$0.177063\pi$
$227$	15.9107i	1.05603i	−0.849235	+	0.528014i	$0.822937\pi$
$228$	0	0
$229$	25.1700i	1.66328i	0.555312	+	0.831642i	$0.312599\pi$
$229$	25.1700i	1.66328i	−0.555312	+	0.831642i	$0.687401\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 6.87605i	− 0.450465i	−0.974305	−	0.225233i	$-0.927686\pi$
$233$	− 6.87605i	− 0.450465i	0.974305	−	0.225233i	$-0.0723142\pi$
$234$	0	0
$235$	−3.54177	−0.231040
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−16.8300	−1.08864	−0.544322	−	0.838876i	$-0.683213\pi$
$239$	−16.8300	−1.08864	−0.544322	+	0.838876i	$0.683213\pi$
$240$	0	0
$241$	12.1692	0.783887	0.391943	−	0.919989i	$-0.371803\pi$
$241$	12.1692	0.783887	0.391943	+	0.919989i	$0.371803\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−9.01643	−0.576039
$246$	0	0
$247$	− 0.519485i	− 0.0330540i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 22.1787i	− 1.39990i	−0.714190	−	0.699952i	$-0.753205\pi$
$251$	− 22.1787i	− 1.39990i	0.714190	−	0.699952i	$-0.246795\pi$
$252$	0	0
$253$	− 2.40432i	− 0.151158i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.86892i	0.303715i	0.988402	+	0.151857i	$0.0485254\pi$
$257$	4.86892i	0.303715i	−0.988402	+	0.151857i	$0.951475\pi$
$258$	0	0
$259$	−1.28499	−0.0798452
$260$	0	0
$261$	0	0
$262$	0	0
$263$	17.1807	1.05941	0.529705	−	0.848182i	$-0.322303\pi$
$263$	17.1807	1.05941	0.529705	+	0.848182i	$0.322303\pi$
$264$	0	0
$265$	−15.4216	−0.947340
$266$	0	0
$267$	0	0
$268$	0	0
$269$	28.1844	1.71843	0.859216	−	0.511613i	$-0.170952\pi$
$269$	28.1844	1.71843	0.859216	+	0.511613i	$0.170952\pi$
$270$	0	0
$271$	− 17.6324i	− 1.07109i	−0.844505	−	0.535547i	$-0.820105\pi$
$271$	− 17.6324i	− 1.07109i	0.844505	−	0.535547i	$-0.179895\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 4.85035i	− 0.292487i
$276$	0	0
$277$	20.0968i	1.20750i	0.797174	+	0.603749i	$0.206327\pi$
$277$	20.0968i	1.20750i	−0.797174	+	0.603749i	$0.793673\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 5.06089i	− 0.301908i	−0.988541	−	0.150954i	$-0.951766\pi$
$281$	− 5.06089i	− 0.301908i	0.988541	−	0.150954i	$-0.0482345\pi$
$282$	0	0
$283$	9.55208	0.567812	0.283906	−	0.958852i	$-0.408370\pi$
$283$	9.55208	0.567812	0.283906	+	0.958852i	$0.408370\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.35784	0.139179
$288$	0	0
$289$	4.19615	0.246832
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6.33095	−0.369858	−0.184929	−	0.982752i	$-0.559206\pi$
$293$	−6.33095	−0.369858	−0.184929	+	0.982752i	$0.559206\pi$
$294$	0	0
$295$	1.41926i	0.0826324i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.199268i	0.0115240i
$300$	0	0
$301$	9.71479i	0.559951i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 15.5715i	− 0.891622i
$306$	0	0
$307$	4.58074	0.261437	0.130718	−	0.991420i	$-0.458272\pi$
$307$	4.58074	0.261437	0.130718	+	0.991420i	$0.458272\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.0707	1.36493	0.682463	−	0.730920i	$-0.260909\pi$
$311$	24.0707	1.36493	0.682463	+	0.730920i	$0.260909\pi$
$312$	0	0
$313$	3.07597	0.173864	0.0869320	−	0.996214i	$-0.472294\pi$
$313$	3.07597	0.173864	0.0869320	+	0.996214i	$0.472294\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.37060	−0.470140	−0.235070	−	0.971978i	$-0.575532\pi$
$317$	−8.37060	−0.470140	−0.235070	+	0.971978i	$0.575532\pi$
$318$	0	0
$319$	− 9.87282i	− 0.552772i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 11.3222i	− 0.629984i
$324$	0	0
$325$	0.401994i	0.0222986i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.58134i	0.142314i
$330$	0	0
$331$	25.6445	1.40955	0.704774	−	0.709432i	$-0.251049\pi$
$331$	25.6445	1.40955	0.704774	+	0.709432i	$0.251049\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6.36310	−0.347653
$336$	0	0
$337$	10.4043	0.566759	0.283380	−	0.959008i	$-0.408544\pi$
$337$	10.4043	0.566759	0.283380	+	0.959008i	$0.408544\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	13.1626	0.712794
$342$	0	0
$343$	14.7207i	0.794841i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.971054i	0.0521289i	0.999660	+	0.0260644i	$0.00829751\pi$
$347$	0.971054i	0.0521289i	−0.999660	+	0.0260644i	$0.991702\pi$
$348$	0	0
$349$	− 9.57290i	− 0.512425i	−0.966620	−	0.256213i	$-0.917525\pi$
$349$	− 9.57290i	− 0.512425i	0.966620	−	0.256213i	$-0.0824747\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	21.3671i	1.13725i	0.822596	+	0.568627i	$0.192525\pi$
$353$	21.3671i	1.13725i	−0.822596	+	0.568627i	$0.807475\pi$
$354$	0	0
$355$	20.4536	1.08556
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−23.5074	−1.24068	−0.620338	−	0.784335i	$-0.713004\pi$
$359$	−23.5074	−1.24068	−0.620338	+	0.784335i	$0.713004\pi$
$360$	0	0
$361$	−8.98798	−0.473052
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.40535	−0.0735595
$366$	0	0
$367$	− 10.4683i	− 0.546439i	−0.961952	−	0.273220i	$-0.911911\pi$
$367$	− 10.4683i	− 0.546439i	0.961952	−	0.273220i	$-0.0880886\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	11.2397i	0.583535i
$372$	0	0
$373$	− 28.6668i	− 1.48431i	−0.670229	−	0.742154i	$-0.733804\pi$
$373$	− 28.6668i	− 1.48431i	0.670229	−	0.742154i	$-0.266196\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.818252i	0.0421421i
$378$	0	0
$379$	21.5729	1.10813	0.554063	−	0.832475i	$-0.313077\pi$
$379$	21.5729	1.10813	0.554063	+	0.832475i	$0.313077\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−22.9975	−1.17512	−0.587560	−	0.809181i	$-0.699911\pi$
$383$	−22.9975	−1.17512	−0.587560	+	0.809181i	$0.699911\pi$
$384$	0	0
$385$	3.68366	0.187737
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−19.0980	−0.968309	−0.484155	−	0.874982i	$-0.660873\pi$
$389$	−19.0980	−0.968309	−0.484155	+	0.874982i	$0.660873\pi$
$390$	0	0
$391$	4.34306i	0.219638i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.65109i	0.234022i
$396$	0	0
$397$	11.1366i	0.558930i	0.960156	+	0.279465i	$0.0901571\pi$
$397$	11.1366i	0.558930i	−0.960156	+	0.279465i	$0.909843\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 2.65161i	− 0.132415i	−0.997806	−	0.0662075i	$-0.978910\pi$
$401$	− 2.65161i	− 0.132415i	0.997806	−	0.0662075i	$-0.0210899\pi$
$402$	0	0
$403$	−1.09091	−0.0543419
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.18648	−0.108380
$408$	0	0
$409$	26.8139	1.32586	0.662931	−	0.748681i	$-0.269312\pi$
$409$	26.8139	1.32586	0.662931	+	0.748681i	$0.269312\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.03439	0.0508992
$414$	0	0
$415$	− 22.2504i	− 1.09223i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.92291i	0.0939405i	0.998896	+	0.0469703i	$0.0149566\pi$
$419$	1.92291i	0.0939405i	−0.998896	+	0.0469703i	$0.985043\pi$
$420$	0	0
$421$	− 0.272971i	− 0.0133038i	−0.999978	−	0.00665189i	$-0.997883\pi$
$421$	− 0.272971i	− 0.0133038i	0.999978	−	0.00665189i	$-0.00211738\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8.76148i	0.424994i
$426$	0	0
$427$	−11.3489	−0.549214
$428$	0	0
$429$	0	0
$430$	0	0
$431$	33.5705	1.61703	0.808517	−	0.588473i	$-0.200271\pi$
$431$	33.5705	1.61703	0.808517	+	0.588473i	$0.200271\pi$
$432$	0	0
$433$	4.11494	0.197751	0.0988756	−	0.995100i	$-0.468475\pi$
$433$	4.11494	0.197751	0.0988756	+	0.995100i	$0.468475\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.84049	−0.183716
$438$	0	0
$439$	22.2624i	1.06253i	0.847206	+	0.531264i	$0.178283\pi$
$439$	22.2624i	1.06253i	−0.847206	+	0.531264i	$0.821717\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	25.8199i	1.22674i	0.789796	+	0.613370i	$0.210186\pi$
$443$	25.8199i	1.22674i	−0.789796	+	0.613370i	$0.789814\pi$
$444$	0	0
$445$	13.4415i	0.637190i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 11.4688i	− 0.541245i	−0.962686	−	0.270622i	$-0.912771\pi$
$449$	− 11.4688i	− 0.541245i	0.962686	−	0.270622i	$-0.0872295\pi$
$450$	0	0
$451$	4.01202	0.188918
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.305299	−0.0143126
$456$	0	0
$457$	−20.5096	−0.959397	−0.479699	−	0.877433i	$-0.659254\pi$
$457$	−20.5096	−0.959397	−0.479699	+	0.877433i	$0.659254\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12.0821	0.562719	0.281359	−	0.959602i	$-0.409215\pi$
$461$	12.0821	0.562719	0.281359	+	0.959602i	$0.409215\pi$
$462$	0	0
$463$	− 21.6937i	− 1.00819i	−0.863648	−	0.504096i	$-0.831826\pi$
$463$	− 21.6937i	− 1.00819i	0.863648	−	0.504096i	$-0.168174\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.38932i	0.295662i	0.989013	+	0.147831i	$0.0472292\pi$
$467$	6.38932i	0.295662i	−0.989013	+	0.147831i	$0.952771\pi$
$468$	0	0
$469$	4.63760i	0.214144i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	16.5303i	0.760064i
$474$	0	0
$475$	−7.74761	−0.355485
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.15743	0.0528841	0.0264421	−	0.999650i	$-0.491582\pi$
$479$	1.15743	0.0528841	0.0264421	+	0.999650i	$0.491582\pi$
$480$	0	0
$481$	0.181214	0.00826264
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.89461	−0.313068
$486$	0	0
$487$	− 8.18844i	− 0.371053i	−0.982639	−	0.185527i	$-0.940601\pi$
$487$	− 8.18844i	− 0.371053i	0.982639	−	0.185527i	$-0.0593991\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2.86944i	0.129496i	0.997902	+	0.0647479i	$0.0206243\pi$
$491$	2.86944i	0.129496i	−0.997902	+	0.0647479i	$0.979376\pi$
$492$	0	0
$493$	17.8339i	0.803196i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 14.9071i	− 0.668675i
$498$	0	0
$499$	−2.07327	−0.0928124	−0.0464062	−	0.998923i	$-0.514777\pi$
$499$	−2.07327	−0.0928124	−0.0464062	+	0.998923i	$0.514777\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−39.0342	−1.74045	−0.870224	−	0.492657i	$-0.836026\pi$
$503$	−39.0342	−1.74045	−0.870224	+	0.492657i	$0.836026\pi$
$504$	0	0
$505$	18.0663	0.803939
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−4.65838	−0.206479	−0.103239	−	0.994657i	$-0.532921\pi$
$509$	−4.65838	−0.206479	−0.103239	+	0.994657i	$0.532921\pi$
$510$	0	0
$511$	1.02426i	0.0453106i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 3.19466i	− 0.140773i
$516$	0	0
$517$	4.39230i	0.193173i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 5.42555i	− 0.237698i	−0.992912	−	0.118849i	$-0.962080\pi$
$521$	− 5.42555i	− 0.237698i	0.992912	−	0.118849i	$-0.0379204\pi$
$522$	0	0
$523$	−19.8280	−0.867017	−0.433508	−	0.901149i	$-0.642725\pi$
$523$	−19.8280	−0.867017	−0.433508	+	0.901149i	$0.642725\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−23.7763	−1.03571
$528$	0	0
$529$	−21.5268	−0.935949
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−0.332513	−0.0144027
$534$	0	0
$535$	31.0370i	1.34185i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	11.1817i	0.481629i
$540$	0	0
$541$	14.5116i	0.623904i	0.950098	+	0.311952i	$0.100983\pi$
$541$	14.5116i	0.623904i	−0.950098	+	0.311952i	$0.899017\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 17.2129i	− 0.737319i
$546$	0	0
$547$	22.8295	0.976117	0.488058	−	0.872811i	$-0.337705\pi$
$547$	22.8295	0.976117	0.488058	+	0.872811i	$0.337705\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−15.7701	−0.671831
$552$	0	0
$553$	3.38984	0.144151
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10.4848	−0.444254	−0.222127	−	0.975018i	$-0.571300\pi$
$557$	−10.4848	−0.444254	−0.222127	+	0.975018i	$0.571300\pi$
$558$	0	0
$559$	− 1.37002i	− 0.0579456i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.69923i	0.155904i	0.996957	+	0.0779519i	$0.0248381\pi$
$563$	3.69923i	0.155904i	−0.996957	+	0.0779519i	$0.975162\pi$
$564$	0	0
$565$	− 7.13490i	− 0.300167i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 25.8719i	− 1.08461i	−0.840182	−	0.542304i	$-0.817552\pi$
$569$	− 25.8719i	− 1.08461i	0.840182	−	0.542304i	$-0.182448\pi$
$570$	0	0
$571$	5.21171	0.218103	0.109052	−	0.994036i	$-0.465219\pi$
$571$	5.21171	0.218103	0.109052	+	0.994036i	$0.465219\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.97189	0.123936
$576$	0	0
$577$	−24.9341	−1.03802	−0.519010	−	0.854768i	$-0.673699\pi$
$577$	−24.9341	−1.03802	−0.519010	+	0.854768i	$0.673699\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−16.2167	−0.672782
$582$	0	0
$583$	19.1250i	0.792076i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 12.0894i	− 0.498982i	−0.968377	−	0.249491i	$-0.919737\pi$
$587$	− 12.0894i	− 0.498982i	0.968377	−	0.249491i	$-0.0802633\pi$
$588$	0	0
$589$	− 21.0250i	− 0.866319i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	21.3992i	0.878760i	0.898301	+	0.439380i	$0.144802\pi$
$593$	21.3992i	0.878760i	−0.898301	+	0.439380i	$0.855198\pi$
$594$	0	0
$595$	−6.65401	−0.272788
$596$	0	0
$597$	0	0
$598$	0	0
$599$	32.8504	1.34223	0.671116	−	0.741353i	$-0.265815\pi$
$599$	32.8504	1.34223	0.671116	+	0.741353i	$0.265815\pi$
$600$	0	0
$601$	23.7086	0.967096	0.483548	−	0.875318i	$-0.339348\pi$
$601$	23.7086	0.967096	0.483548	+	0.875318i	$0.339348\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−11.3026	−0.459518
$606$	0	0
$607$	− 41.4026i	− 1.68048i	−0.542216	−	0.840239i	$-0.682414\pi$
$607$	− 41.4026i	− 1.68048i	0.542216	−	0.840239i	$-0.317586\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 0.364031i	− 0.0147271i
$612$	0	0
$613$	− 32.4820i	− 1.31194i	−0.754789	−	0.655968i	$-0.772261\pi$
$613$	− 32.4820i	− 1.31194i	0.754789	−	0.655968i	$-0.227739\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 30.1842i	− 1.21517i	−0.794254	−	0.607586i	$-0.792138\pi$
$617$	− 30.1842i	− 1.21517i	0.794254	−	0.607586i	$-0.207862\pi$
$618$	0	0
$619$	−24.8535	−0.998946	−0.499473	−	0.866329i	$-0.666473\pi$
$619$	−24.8535	−0.998946	−0.499473	+	0.866329i	$0.666473\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	9.79656	0.392491
$624$	0	0
$625$	−6.76196	−0.270478
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.94957	0.157480
$630$	0	0
$631$	31.5937i	1.25773i	0.777516	+	0.628863i	$0.216479\pi$
$631$	31.5937i	1.25773i	−0.777516	+	0.628863i	$0.783521\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	14.7404i	0.584953i
$636$	0	0
$637$	− 0.926728i	− 0.0367183i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1.48328i	− 0.0585859i	−0.999571	−	0.0292930i	$-0.990674\pi$
$641$	− 1.48328i	− 0.0585859i	0.999571	−	0.0292930i	$-0.00932558\pi$
$642$	0	0
$643$	−45.3533	−1.78856	−0.894280	−	0.447507i	$-0.852312\pi$
$643$	−45.3533	−1.78856	−0.894280	+	0.447507i	$0.852312\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.0785	−0.592796	−0.296398	−	0.955065i	$-0.595785\pi$
$647$	−15.0785	−0.592796	−0.296398	+	0.955065i	$0.595785\pi$
$648$	0	0
$649$	1.76008	0.0690894
$650$	0	0
$651$	0	0
$652$	0	0
$653$	34.3187	1.34300	0.671498	−	0.741006i	$-0.265651\pi$
$653$	34.3187	1.34300	0.671498	+	0.741006i	$0.265651\pi$
$654$	0	0
$655$	− 17.0537i	− 0.666342i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 29.5778i	− 1.15219i	−0.817383	−	0.576094i	$-0.804576\pi$
$659$	− 29.5778i	− 1.15219i	0.817383	−	0.576094i	$-0.195424\pi$
$660$	0	0
$661$	− 6.36203i	− 0.247454i	−0.992316	−	0.123727i	$-0.960515\pi$
$661$	− 6.36203i	− 0.247454i	0.992316	−	0.123727i	$-0.0394848\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 5.88402i	− 0.228173i
$666$	0	0
$667$	6.04924	0.234228
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−19.3109	−0.745489
$672$	0	0
$673$	−21.3043	−0.821221	−0.410611	−	0.911811i	$-0.634684\pi$
$673$	−21.3043	−0.821221	−0.410611	+	0.911811i	$0.634684\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16.4279	0.631375	0.315687	−	0.948863i	$-0.397765\pi$
$677$	16.4279	0.631375	0.315687	+	0.948863i	$0.397765\pi$
$678$	0	0
$679$	5.02498i	0.192841i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1.01447i	− 0.0388177i	−0.999812	−	0.0194089i	$-0.993822\pi$
$683$	− 1.01447i	− 0.0388177i	0.999812	−	0.0194089i	$-0.00617842\pi$
$684$	0	0
$685$	− 32.2921i	− 1.23382i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1.58506i	− 0.0603861i
$690$	0	0
$691$	8.91349	0.339085	0.169543	−	0.985523i	$-0.445771\pi$
$691$	8.91349	0.339085	0.169543	+	0.985523i	$0.445771\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−21.0345	−0.797886
$696$	0	0
$697$	−7.24714	−0.274505
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−29.3401	−1.10816	−0.554080	−	0.832463i	$-0.686930\pi$
$701$	−29.3401	−1.10816	−0.554080	+	0.832463i	$0.686930\pi$
$702$	0	0
$703$	3.49253i	0.131723i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 13.1672i	− 0.495203i
$708$	0	0
$709$	19.6175i	0.736751i	0.929677	+	0.368376i	$0.120086\pi$
$709$	19.6175i	0.736751i	−0.929677	+	0.368376i	$0.879914\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.06493i	0.302034i
$714$	0	0
$715$	−0.519485	−0.0194276
$716$	0	0
$717$	0	0
$718$	0	0
$719$	51.5446	1.92229	0.961145	−	0.276044i	$-0.0890235\pi$
$719$	51.5446	1.92229	0.961145	+	0.276044i	$0.0890235\pi$
$720$	0	0
$721$	−2.32835	−0.0867124
$722$	0	0
$723$	0	0
$724$	0	0
$725$	12.2034	0.453224
$726$	0	0
$727$	27.3801i	1.01547i	0.861513	+	0.507735i	$0.169517\pi$
$727$	27.3801i	1.01547i	−0.861513	+	0.507735i	$0.830483\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 29.8596i	− 1.10440i
$732$	0	0
$733$	6.21759i	0.229652i	0.993386	+	0.114826i	$0.0366310\pi$
$733$	6.21759i	0.229652i	−0.993386	+	0.114826i	$0.963369\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.89116i	0.290674i
$738$	0	0
$739$	−48.1637	−1.77173	−0.885865	−	0.463943i	$-0.846434\pi$
$739$	−48.1637	−1.77173	−0.885865	+	0.463943i	$0.846434\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	29.5198	1.08298	0.541489	−	0.840708i	$-0.317861\pi$
$743$	29.5198	1.08298	0.541489	+	0.840708i	$0.317861\pi$
$744$	0	0
$745$	−9.57411	−0.350768
$746$	0	0
$747$	0	0
$748$	0	0
$749$	22.6206	0.826539
$750$	0	0
$751$	1.01224i	0.0369373i	0.999829	+	0.0184686i	$0.00587909\pi$
$751$	1.01224i	0.0369373i	−0.999829	+	0.0184686i	$0.994121\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.71915i	0.135354i
$756$	0	0
$757$	32.3712i	1.17655i	0.808660	+	0.588276i	$0.200193\pi$
$757$	32.3712i	1.17655i	−0.808660	+	0.588276i	$0.799807\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	38.1317i	1.38227i	0.722724	+	0.691136i	$0.242890\pi$
$761$	38.1317i	1.38227i	−0.722724	+	0.691136i	$0.757110\pi$
$762$	0	0
$763$	−12.5452	−0.454167
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−0.145874	−0.00526722
$768$	0	0
$769$	−9.45626	−0.341001	−0.170501	−	0.985358i	$-0.554538\pi$
$769$	−9.45626	−0.341001	−0.170501	+	0.985358i	$0.554538\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−39.7374	−1.42925	−0.714627	−	0.699506i	$-0.753404\pi$
$773$	−39.7374	−1.42925	−0.714627	+	0.699506i	$0.753404\pi$
$774$	0	0
$775$	16.2698i	0.584428i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 6.40851i	− 0.229609i
$780$	0	0
$781$	− 25.3654i	− 0.907644i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 14.6133i	− 0.521573i
$786$	0	0
$787$	10.8010	0.385015	0.192507	−	0.981296i	$-0.438338\pi$
$787$	10.8010	0.385015	0.192507	+	0.981296i	$0.438338\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.20011	−0.184894
$792$	0	0
$793$	1.60047	0.0568345
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−13.2916	−0.470812	−0.235406	−	0.971897i	$-0.575642\pi$
$797$	−13.2916	−0.470812	−0.235406	+	0.971897i	$0.575642\pi$
$798$	0	0
$799$	− 7.93408i	− 0.280687i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.74284i	0.0615035i
$804$	0	0
$805$	2.25704i	0.0795502i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	16.5163i	0.580684i	0.956923	+	0.290342i	$0.0937690\pi$
$809$	16.5163i	0.580684i	−0.956923	+	0.290342i	$0.906231\pi$
$810$	0	0
$811$	34.5560	1.21342	0.606712	−	0.794921i	$-0.292488\pi$
$811$	34.5560	1.21342	0.606712	+	0.794921i	$0.292488\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−35.3675	−1.23887
$816$	0	0
$817$	26.4043	0.923770
$818$	0	0
$819$	0	0
$820$	0	0
$821$	46.5813	1.62570	0.812849	−	0.582475i	$-0.197915\pi$
$821$	46.5813	1.62570	0.812849	+	0.582475i	$0.197915\pi$
$822$	0	0
$823$	35.4173i	1.23457i	0.786740	+	0.617284i	$0.211767\pi$
$823$	35.4173i	1.23457i	−0.786740	+	0.617284i	$0.788233\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34.7146i	1.20714i	0.797308	+	0.603572i	$0.206257\pi$
$827$	34.7146i	1.20714i	−0.797308	+	0.603572i	$0.793743\pi$
$828$	0	0
$829$	− 11.9120i	− 0.413721i	−0.978370	−	0.206861i	$-0.933675\pi$
$829$	− 11.9120i	− 0.413721i	0.978370	−	0.206861i	$-0.0663247\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 20.1981i	− 0.699823i
$834$	0	0
$835$	−34.2531	−1.18538
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−7.50962	−0.259261	−0.129630	−	0.991562i	$-0.541379\pi$
$839$	−7.50962	−0.259261	−0.129630	+	0.991562i	$0.541379\pi$
$840$	0	0
$841$	−4.16011	−0.143452
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−20.7222	−0.712866
$846$	0	0
$847$	8.23768i	0.283050i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1.33969i	− 0.0459241i
$852$	0	0
$853$	− 47.0179i	− 1.60986i	−0.593369	−	0.804931i	$-0.702202\pi$
$853$	− 47.0179i	− 1.60986i	0.593369	−	0.804931i	$-0.297798\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 29.4641i	− 1.00648i	−0.864148	−	0.503238i	$-0.832142\pi$
$857$	− 29.4641i	− 1.00648i	0.864148	−	0.503238i	$-0.167858\pi$
$858$	0	0
$859$	−35.7888	−1.22110	−0.610549	−	0.791979i	$-0.709051\pi$
$859$	−35.7888	−1.22110	−0.610549	+	0.791979i	$0.709051\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	47.0404	1.60127	0.800636	−	0.599151i	$-0.204495\pi$
$863$	47.0404	1.60127	0.800636	+	0.599151i	$0.204495\pi$
$864$	0	0
$865$	26.7523	0.909604
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.76801	0.195666
$870$	0	0
$871$	− 0.654013i	− 0.0221604i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	13.8511i	0.468253i
$876$	0	0
$877$	− 18.4500i	− 0.623013i	−0.950244	−	0.311506i	$-0.899166\pi$
$877$	− 18.4500i	− 0.623013i	0.950244	−	0.311506i	$-0.100834\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	14.0647i	0.473852i	0.971528	+	0.236926i	$0.0761398\pi$
$881$	14.0647i	0.473852i	−0.971528	+	0.236926i	$0.923860\pi$
$882$	0	0
$883$	−18.2404	−0.613837	−0.306919	−	0.951736i	$-0.599298\pi$
$883$	−18.2404	−0.613837	−0.306919	+	0.951736i	$0.599298\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37.7362	1.26706	0.633529	−	0.773719i	$-0.281606\pi$
$887$	37.7362	1.26706	0.633529	+	0.773719i	$0.281606\pi$
$888$	0	0
$889$	10.7432	0.360314
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.01596	0.234780
$894$	0	0
$895$	25.7403i	0.860402i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	33.1169i	1.10451i
$900$	0	0
$901$	− 34.5466i	− 1.15091i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 26.1342i	− 0.868730i
$906$	0	0
$907$	−38.1744	−1.26756	−0.633781	−	0.773513i	$-0.718498\pi$
$907$	−38.1744	−1.26756	−0.633781	+	0.773513i	$0.718498\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−18.9190	−0.626813	−0.313407	−	0.949619i	$-0.601470\pi$
$911$	−18.9190	−0.626813	−0.313407	+	0.949619i	$0.601470\pi$
$912$	0	0
$913$	−27.5937	−0.913218
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−12.4292	−0.410448
$918$	0	0
$919$	− 4.03920i	− 0.133241i	−0.997778	−	0.0666204i	$-0.978778\pi$
$919$	− 4.03920i	− 0.133241i	0.997778	−	0.0666204i	$-0.0212217\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.10226i	0.0691968i
$924$	0	0
$925$	− 2.70263i	− 0.0888619i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	23.4245i	0.768534i	0.923222	+	0.384267i	$0.125546\pi$
$929$	23.4245i	0.768534i	−0.923222	+	0.384267i	$0.874454\pi$
$930$	0	0
$931$	17.8608	0.585364
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−11.3222	−0.370276
$936$	0	0
$937$	−41.0860	−1.34222	−0.671111	−	0.741357i	$-0.734182\pi$
$937$	−41.0860	−1.34222	−0.671111	+	0.741357i	$0.734182\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−24.5313	−0.799696	−0.399848	−	0.916581i	$-0.630937\pi$
$941$	−24.5313	−0.799696	−0.399848	+	0.916581i	$0.630937\pi$
$942$	0	0
$943$	2.45823i	0.0800509i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 47.0862i	− 1.53010i	−0.643973	−	0.765048i	$-0.722715\pi$
$947$	− 47.0862i	− 1.53010i	0.643973	−	0.765048i	$-0.277285\pi$
$948$	0	0
$949$	− 0.144445i	− 0.00468889i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	47.3929i	1.53521i	0.640926	+	0.767603i	$0.278551\pi$
$953$	47.3929i	1.53521i	−0.640926	+	0.767603i	$0.721449\pi$
$954$	0	0
$955$	−29.7623	−0.963086
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−23.5353	−0.759996
$960$	0	0
$961$	−13.1519	−0.424256
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.64631	0.0851876
$966$	0	0
$967$	40.0248i	1.28711i	0.765400	+	0.643555i	$0.222541\pi$
$967$	40.0248i	1.28711i	−0.765400	+	0.643555i	$0.777459\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	36.5162i	1.17186i	0.810362	+	0.585930i	$0.199271\pi$
$971$	36.5162i	1.17186i	−0.810362	+	0.585930i	$0.800729\pi$
$972$	0	0
$973$	15.3306i	0.491475i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	43.4028i	1.38858i	0.719697	+	0.694289i	$0.244281\pi$
$977$	43.4028i	1.38858i	−0.719697	+	0.694289i	$0.755719\pi$
$978$	0	0
$979$	16.6694	0.532758
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.34326	0.0747384	0.0373692	−	0.999302i	$-0.488102\pi$
$983$	2.34326	0.0747384	0.0373692	+	0.999302i	$0.488102\pi$
$984$	0	0
$985$	−19.0047	−0.605539
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−10.1284	−0.322064
$990$	0	0
$991$	− 44.6280i	− 1.41766i	−0.705382	−	0.708828i	$-0.749224\pi$
$991$	− 44.6280i	− 1.41766i	0.705382	−	0.708828i	$-0.250776\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 13.3031i	− 0.421737i
$996$	0	0
$997$	− 53.4194i	− 1.69181i	−0.533334	−	0.845904i	$-0.679061\pi$
$997$	− 53.4194i	− 1.69181i	0.533334	−	0.845904i	$-0.320939\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5184.2.f.e.2591.6	yes	16
3.2	odd	2	inner	5184.2.f.e.2591.12	yes	16
4.3	odd	2		5184.2.f.b.2591.5	✓	16
8.3	odd	2	inner	5184.2.f.e.2591.11	yes	16
8.5	even	2		5184.2.f.b.2591.12	yes	16
12.11	even	2		5184.2.f.b.2591.11	yes	16
24.5	odd	2		5184.2.f.b.2591.6	yes	16
24.11	even	2	inner	5184.2.f.e.2591.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5184.2.f.b.2591.5	✓	16	4.3	odd	2
5184.2.f.b.2591.6	yes	16	24.5	odd	2
5184.2.f.b.2591.11	yes	16	12.11	even	2
5184.2.f.b.2591.12	yes	16	8.5	even	2
5184.2.f.e.2591.5	yes	16	24.11	even	2	inner
5184.2.f.e.2591.6	yes	16	1.1	even	1	trivial
5184.2.f.e.2591.11	yes	16	8.3	odd	2	inner
5184.2.f.e.2591.12	yes	16	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5184 = 2^{6} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5184.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.59733 q^{5} +1.16418i q^{7} +O(q^{10})\)	\(q-1.59733 q^{5} +1.16418i q^{7} +1.98092i q^{11} -0.164177i q^{13} -3.57825i q^{17} +3.16418 q^{19} -1.21374 q^{23} -2.44854 q^{25} -4.98396 q^{29} -6.64469i q^{31} -1.85957i q^{35} +1.10377i q^{37} -2.02533i q^{41} +8.34477 q^{43} +2.21731 q^{47} +5.64469 q^{49} +9.65461 q^{53} -3.16418i q^{55} -0.888520i q^{59} +9.74846i q^{61} +0.262245i q^{65} +3.98359 q^{67} -12.8049 q^{71} +0.879814 q^{73} -2.30614 q^{77} -2.91179i q^{79} +13.9298i q^{83} +5.71564i q^{85} -8.41501i q^{89} +0.191131 q^{91} -5.05423 q^{95} +4.31634 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 24 q^{19} + 16 q^{25} + 24 q^{43} - 48 q^{49} + 120 q^{67} + 16 q^{73} + 168 q^{91} - 16 q^{97}+O(q^{100}) \) 16 * q + 24 * q^19 + 16 * q^25 + 24 * q^43 - 48 * q^49 + 120 * q^67 + 16 * q^73 + 168 * q^91 - 16 * q^97

Introduction

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