LMFDB - Embedded newform 5184.2.f.b.2591.15

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			2591.15
Root			$-1.11871 - 0.645885i$ of defining polynomial
Character	$\chi$	$=$	5184.2591
Dual form			5184.2.f.b.2591.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.04682 q^{5} -3.89623i q^{7} +O(q^{10})$	$q+4.04682 q^{5} -3.89623i q^{7} +0.468572i q^{11} -4.89623i q^{13} -3.57825i q^{17} +1.89623 q^{19} -8.56221 q^{23} +11.3767 q^{25} -2.36451 q^{29} +7.18059i q^{31} -15.7673i q^{35} -6.16418i q^{37} -5.13116i q^{41} +10.5409 q^{43} -9.37380 q^{47} -8.18059 q^{49} +5.52280 q^{53} +1.89623i q^{55} +6.26797i q^{59} -0.983586i q^{61} -19.8141i q^{65} -12.7485 q^{67} +3.02890 q^{71} +4.58429 q^{73} +1.82566 q^{77} -11.6767i q^{79} +5.66616i q^{83} -14.4805i q^{85} +3.17610i q^{89} -19.0768 q^{91} +7.67369 q^{95} +0.611865 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	4.04682	1.80979	0.904896	−	0.425632i	$-0.139948\pi$
$5$	4.04682	1.80979	0.904896	+	0.425632i	$0.139948\pi$
$6$	0	0
$7$	− 3.89623i	− 1.47264i	−0.676636	−	0.736318i	$-0.736563\pi$
$7$	− 3.89623i	− 1.47264i	0.676636	−	0.736318i	$-0.263437\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.468572i	0.141280i	0.997502	+	0.0706400i	$0.0225041\pi$
$11$	0.468572i	0.141280i	−0.997502	+	0.0706400i	$0.977496\pi$
$12$	0	0
$13$	− 4.89623i	− 1.35797i	−0.734152	−	0.678985i	$-0.762420\pi$
$13$	− 4.89623i	− 1.35797i	0.734152	−	0.678985i	$-0.237580\pi$
$14$	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
$19$	1.89623	0.435024	0.217512	−	0.976058i	$-0.430206\pi$
$19$	1.89623	0.435024	0.217512	+	0.976058i	$0.430206\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.56221	−1.78534	−0.892672	−	0.450707i	$-0.851172\pi$
$23$	−8.56221	−1.78534	−0.892672	+	0.450707i	$0.851172\pi$
$24$	0	0
$25$	11.3767	2.27535
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.36451	−0.439078	−0.219539	−	0.975604i	$-0.570455\pi$
$29$	−2.36451	−0.439078	−0.219539	+	0.975604i	$0.570455\pi$
$30$	0	0
$31$	7.18059i	1.28967i	0.764321	+	0.644836i	$0.223074\pi$
$31$	7.18059i	1.28967i	−0.764321	+	0.644836i	$0.776926\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 15.7673i	− 2.66516i
$36$	0	0
$37$	− 6.16418i	− 1.01338i	−0.862127	−	0.506692i	$-0.830868\pi$
$37$	− 6.16418i	− 1.01338i	0.862127	−	0.506692i	$-0.169132\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 5.13116i	− 0.801353i	−0.916220	−	0.400676i	$-0.868775\pi$
$41$	− 5.13116i	− 0.801353i	0.916220	−	0.400676i	$-0.131225\pi$
$42$	0	0
$43$	10.5409	1.60748	0.803738	−	0.594984i	$-0.202842\pi$
$43$	10.5409	1.60748	0.803738	+	0.594984i	$0.202842\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.37380	−1.36731	−0.683655	−	0.729806i	$-0.739611\pi$
$47$	−9.37380	−1.36731	−0.683655	+	0.729806i	$0.739611\pi$
$48$	0	0
$49$	−8.18059	−1.16866
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.52280	0.758616	0.379308	−	0.925271i	$-0.376162\pi$
$53$	5.52280	0.758616	0.379308	+	0.925271i	$0.376162\pi$
$54$	0	0
$55$	1.89623i	0.255687i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.26797i	0.816021i	0.912977	+	0.408010i	$0.133777\pi$
$59$	6.26797i	0.816021i	−0.912977	+	0.408010i	$0.866223\pi$
$60$	0	0
$61$	− 0.983586i	− 0.125935i	−0.998016	−	0.0629677i	$-0.979943\pi$
$61$	− 0.983586i	− 0.125935i	0.998016	−	0.0629677i	$-0.0200565\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 19.8141i	− 2.45764i
$66$	0	0
$67$	−12.7485	−1.55747	−0.778736	−	0.627351i	$-0.784139\pi$
$67$	−12.7485	−1.55747	−0.778736	+	0.627351i	$0.784139\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.02890	0.359464	0.179732	−	0.983716i	$-0.442477\pi$
$71$	3.02890	0.359464	0.179732	+	0.983716i	$0.442477\pi$
$72$	0	0
$73$	4.58429	0.536550	0.268275	−	0.963342i	$-0.413546\pi$
$73$	4.58429	0.536550	0.268275	+	0.963342i	$0.413546\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.82566	0.208054
$78$	0	0
$79$	− 11.6767i	− 1.31373i	−0.754009	−	0.656864i	$-0.771883\pi$
$79$	− 11.6767i	− 1.31373i	0.754009	−	0.656864i	$-0.228117\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.66616i	0.621941i	0.950419	+	0.310971i	$0.100654\pi$
$83$	5.66616i	0.621941i	−0.950419	+	0.310971i	$0.899346\pi$
$84$	0	0
$85$	− 14.4805i	− 1.57063i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.17610i	0.336666i	0.985730	+	0.168333i	$0.0538383\pi$
$89$	3.17610i	0.336666i	−0.985730	+	0.168333i	$0.946162\pi$
$90$	0	0
$91$	−19.0768	−1.99979
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.67369	0.787304
$96$	0	0
$97$	0.611865	0.0621255	0.0310627	−	0.999517i	$-0.490111\pi$
$97$	0.611865	0.0621255	0.0310627	+	0.999517i	$0.490111\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−8.28561	−0.824449	−0.412225	−	0.911082i	$-0.635248\pi$
$101$	−8.28561	−0.824449	−0.412225	+	0.911082i	$0.635248\pi$
$102$	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
$107$	4.25315i	0.411167i	0.978640	+	0.205584i	$0.0659092\pi$
$107$	4.25315i	0.411167i	−0.978640	+	0.205584i	$0.934091\pi$
$108$	0	0
$109$	− 9.42011i	− 0.902283i	−0.892452	−	0.451141i	$-0.851017\pi$
$109$	− 9.42011i	− 0.902283i	0.892452	−	0.451141i	$-0.148983\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.84622i	0.926254i	0.886292	+	0.463127i	$0.153273\pi$
$113$	9.84622i	0.926254i	−0.886292	+	0.463127i	$0.846727\pi$
$114$	0	0
$115$	−34.6497	−3.23110
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−13.9417	−1.27803
$120$	0	0
$121$	10.7804	0.980040
$122$	0	0
$123$	0	0
$124$	0	0
$125$	25.8055	2.30812
$126$	0	0
$127$	− 14.2885i	− 1.26790i	−0.773373	−	0.633951i	$-0.781432\pi$
$127$	− 14.2885i	− 1.26790i	0.773373	−	0.633951i	$-0.218568\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 17.5444i	− 1.53286i	−0.642329	−	0.766429i	$-0.722032\pi$
$131$	− 17.5444i	− 1.53286i	0.642329	−	0.766429i	$-0.277968\pi$
$132$	0	0
$133$	− 7.38814i	− 0.640633i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.7310i	1.00225i	0.865375	+	0.501124i	$0.167080\pi$
$137$	11.7310i	1.00225i	−0.865375	+	0.501124i	$0.832920\pi$
$138$	0	0
$139$	−5.75963	−0.488525	−0.244263	−	0.969709i	$-0.578546\pi$
$139$	−5.75963	−0.488525	−0.244263	+	0.969709i	$0.578546\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.29424	0.191854
$144$	0	0
$145$	−9.56873	−0.794639
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−19.2022	−1.57311	−0.786554	−	0.617522i	$-0.788137\pi$
$149$	−19.2022	−1.57311	−0.786554	+	0.617522i	$0.788137\pi$
$150$	0	0
$151$	7.79246i	0.634141i	0.948402	+	0.317071i	$0.102699\pi$
$151$	7.79246i	0.634141i	−0.948402	+	0.317071i	$0.897301\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	29.0585i	2.33404i
$156$	0	0
$157$	9.73707i	0.777103i	0.921427	+	0.388551i	$0.127024\pi$
$157$	9.73707i	0.777103i	−0.921427	+	0.388551i	$0.872976\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	33.3603i	2.62916i
$162$	0	0
$163$	9.21342	0.721651	0.360825	−	0.932633i	$-0.382495\pi$
$163$	9.21342	0.721651	0.360825	+	0.932633i	$0.382495\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.88386	0.610072	0.305036	−	0.952341i	$-0.401332\pi$
$167$	7.88386	0.610072	0.305036	+	0.952341i	$0.401332\pi$
$168$	0	0
$169$	−10.9730	−0.844080
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−20.4747	−1.55666	−0.778331	−	0.627854i	$-0.783933\pi$
$173$	−20.4747	−1.55666	−0.778331	+	0.627854i	$0.783933\pi$
$174$	0	0
$175$	− 44.3264i	− 3.35076i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 19.1392i	− 1.43053i	−0.698851	−	0.715267i	$-0.746305\pi$
$179$	− 19.1392i	− 1.43053i	0.698851	−	0.715267i	$-0.253695\pi$
$180$	0	0
$181$	11.2894i	0.839133i	0.907725	+	0.419567i	$0.137818\pi$
$181$	11.2894i	0.839133i	−0.907725	+	0.419567i	$0.862182\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 24.9453i	− 1.83402i
$186$	0	0
$187$	1.67667	0.122610
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.7694	1.43046	0.715231	−	0.698889i	$-0.246322\pi$
$191$	19.7694	1.43046	0.715231	+	0.698889i	$0.246322\pi$
$192$	0	0
$193$	18.5849	1.33777	0.668886	−	0.743365i	$-0.266772\pi$
$193$	18.5849	1.33777	0.668886	+	0.743365i	$0.266772\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	25.8055	1.83857	0.919284	−	0.393596i	$-0.128769\pi$
$197$	25.8055	1.83857	0.919284	+	0.393596i	$0.128769\pi$
$198$	0	0
$199$	− 1.79246i	− 0.127064i	−0.997980	−	0.0635319i	$-0.979764\pi$
$199$	− 1.79246i	− 0.127064i	0.997980	−	0.0635319i	$-0.0202365\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.21265i	0.646601i
$204$	0	0
$205$	− 20.7649i	− 1.45028i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.888520i	0.0614602i
$210$	0	0
$211$	0.288533	0.0198634	0.00993170	−	0.999951i	$-0.496839\pi$
$211$	0.288533	0.0198634	0.00993170	+	0.999951i	$0.496839\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	42.6572	2.90920
$216$	0	0
$217$	27.9772	1.89922
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−17.5199	−1.17852
$222$	0	0
$223$	1.85309i	0.124092i	0.998073	+	0.0620460i	$0.0197625\pi$
$223$	1.85309i	0.124092i	−0.998073	+	0.0620460i	$0.980237\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.13473i	0.407176i	0.979057	+	0.203588i	$0.0652603\pi$
$227$	6.13473i	0.407176i	−0.979057	+	0.203588i	$0.934740\pi$
$228$	0	0
$229$	21.3662i	1.41192i	0.708253	+	0.705959i	$0.249484\pi$
$229$	21.3662i	1.41192i	−0.708253	+	0.705959i	$0.750516\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.0325i	0.919303i	0.888099	+	0.459651i	$0.152026\pi$
$233$	14.0325i	0.919303i	−0.888099	+	0.459651i	$0.847974\pi$
$234$	0	0
$235$	−37.9341	−2.47455
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.35219	0.410889	0.205445	−	0.978669i	$-0.434136\pi$
$239$	6.35219	0.410889	0.205445	+	0.978669i	$0.434136\pi$
$240$	0	0
$241$	−11.7769	−0.758616	−0.379308	−	0.925270i	$-0.623838\pi$
$241$	−11.7769	−0.758616	−0.379308	+	0.925270i	$0.623838\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−33.1054	−2.11502
$246$	0	0
$247$	− 9.28436i	− 0.590750i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 5.24621i	− 0.331138i	−0.986198	−	0.165569i	$-0.947054\pi$
$251$	− 5.24621i	− 0.331138i	0.986198	−	0.165569i	$-0.0529460\pi$
$252$	0	0
$253$	− 4.01202i	− 0.252233i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21.8395i	1.36231i	0.732140	+	0.681155i	$0.238522\pi$
$257$	21.8395i	1.36231i	−0.732140	+	0.681155i	$0.761478\pi$
$258$	0	0
$259$	−24.0170	−1.49235
$260$	0	0
$261$	0	0
$262$	0	0
$263$	9.52766	0.587501	0.293750	−	0.955882i	$-0.405097\pi$
$263$	9.52766	0.587501	0.293750	+	0.955882i	$0.405097\pi$
$264$	0	0
$265$	22.3498	1.37294
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−15.4565	−0.942397	−0.471198	−	0.882027i	$-0.656178\pi$
$269$	−15.4565	−0.942397	−0.471198	+	0.882027i	$0.656178\pi$
$270$	0	0
$271$	− 16.2765i	− 0.988728i	−0.869255	−	0.494364i	$-0.835401\pi$
$271$	− 16.2765i	− 0.988728i	0.869255	−	0.494364i	$-0.164599\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.33083i	0.321461i
$276$	0	0
$277$	− 12.6878i	− 0.762338i	−0.924505	−	0.381169i	$-0.875522\pi$
$277$	− 12.6878i	− 0.762338i	0.924505	−	0.381169i	$-0.124478\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 7.33452i	− 0.437541i	−0.975776	−	0.218770i	$-0.929795\pi$
$281$	− 7.33452i	− 0.437541i	0.975776	−	0.218770i	$-0.0702045\pi$
$282$	0	0
$283$	−6.84022	−0.406609	−0.203304	−	0.979116i	$-0.565168\pi$
$283$	−6.84022	−0.406609	−0.203304	+	0.979116i	$0.565168\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−19.9922	−1.18010
$288$	0	0
$289$	4.19615	0.246832
$290$	0	0
$291$	0	0
$292$	0	0
$293$	22.9969	1.34349	0.671747	−	0.740781i	$-0.265544\pi$
$293$	22.9969	1.34349	0.671747	+	0.740781i	$0.265544\pi$
$294$	0	0
$295$	25.3654i	1.47683i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	41.9225i	2.42444i
$300$	0	0
$301$	− 41.0698i	− 2.36723i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 3.98040i	− 0.227917i
$306$	0	0
$307$	19.3654	1.10524	0.552619	−	0.833434i	$-0.313628\pi$
$307$	19.3654	1.10524	0.552619	+	0.833434i	$0.313628\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−16.9142	−0.959119	−0.479559	−	0.877509i	$-0.659204\pi$
$311$	−16.9142	−0.959119	−0.479559	+	0.877509i	$0.659204\pi$
$312$	0	0
$313$	6.78044	0.383253	0.191627	−	0.981468i	$-0.438624\pi$
$313$	6.78044	0.383253	0.191627	+	0.981468i	$0.438624\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.77583	−0.492900	−0.246450	−	0.969156i	$-0.579264\pi$
$317$	−8.77583	−0.492900	−0.246450	+	0.969156i	$0.579264\pi$
$318$	0	0
$319$	− 1.10794i	− 0.0620328i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 6.78517i	− 0.377537i
$324$	0	0
$325$	− 55.7031i	− 3.08985i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	36.5225i	2.01355i
$330$	0	0
$331$	−17.8722	−0.982345	−0.491172	−	0.871062i	$-0.663431\pi$
$331$	−17.8722	−0.982345	−0.491172	+	0.871062i	$0.663431\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−51.5907	−2.81870
$336$	0	0
$337$	3.98798	0.217239	0.108620	−	0.994083i	$-0.465357\pi$
$337$	3.98798	0.217239	0.108620	+	0.994083i	$0.465357\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.36463	−0.182205
$342$	0	0
$343$	4.59985i	0.248369i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.0353i	1.18292i	0.806335	+	0.591458i	$0.201448\pi$
$347$	22.0353i	1.18292i	−0.806335	+	0.591458i	$0.798552\pi$
$348$	0	0
$349$	− 4.25239i	− 0.227625i	−0.993502	−	0.113813i	$-0.963694\pi$
$349$	− 4.25239i	− 0.227625i	0.993502	−	0.113813i	$-0.0363063\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 1.81516i	− 0.0966112i	−0.998833	−	0.0483056i	$-0.984618\pi$
$353$	− 1.81516i	− 0.0966112i	0.998833	−	0.0483056i	$-0.0153821\pi$
$354$	0	0
$355$	12.2574	0.650556
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.8880	1.10243	0.551213	−	0.834365i	$-0.314165\pi$
$359$	20.8880	1.10243	0.551213	+	0.834365i	$0.314165\pi$
$360$	0	0
$361$	−15.4043	−0.810754
$362$	0	0
$363$	0	0
$364$	0	0
$365$	18.5518	0.971045
$366$	0	0
$367$	− 14.1727i	− 0.739811i	−0.929069	−	0.369906i	$-0.879390\pi$
$367$	− 14.1727i	− 0.739811i	0.929069	−	0.369906i	$-0.120610\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 21.5181i	− 1.11716i
$372$	0	0
$373$	− 29.3462i	− 1.51949i	−0.650221	−	0.759745i	$-0.725324\pi$
$373$	− 29.3462i	− 1.51949i	0.650221	−	0.759745i	$-0.274676\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11.5772i	0.596254i
$378$	0	0
$379$	−7.74761	−0.397968	−0.198984	−	0.980003i	$-0.563764\pi$
$379$	−7.74761	−0.397968	−0.198984	+	0.980003i	$0.563764\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	36.0948	1.84436	0.922180	−	0.386762i	$-0.126406\pi$
$383$	36.0948	1.84436	0.922180	+	0.386762i	$0.126406\pi$
$384$	0	0
$385$	7.38814	0.376534
$386$	0	0
$387$	0	0
$388$	0	0
$389$	29.3765	1.48945	0.744723	−	0.667373i	$-0.232581\pi$
$389$	29.3765	1.48945	0.744723	+	0.667373i	$0.232581\pi$
$390$	0	0
$391$	30.6377i	1.54941i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 47.2534i	− 2.37757i
$396$	0	0
$397$	1.33275i	0.0668889i	0.999441	+	0.0334444i	$0.0106477\pi$
$397$	1.33275i	0.0668889i	−0.999441	+	0.0334444i	$0.989352\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 31.2133i	− 1.55872i	−0.626578	−	0.779358i	$-0.715545\pi$
$401$	− 31.2133i	− 1.55872i	0.626578	−	0.779358i	$-0.284455\pi$
$402$	0	0
$403$	35.1578	1.75134
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.88836	0.143171
$408$	0	0
$409$	−10.9575	−0.541813	−0.270906	−	0.962606i	$-0.587323\pi$
$409$	−10.9575	−0.541813	−0.270906	+	0.962606i	$0.587323\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	24.4214	1.20170
$414$	0	0
$415$	22.9299i	1.12558i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	18.1535i	0.886855i	0.896310	+	0.443428i	$0.146238\pi$
$419$	18.1535i	0.886855i	−0.896310	+	0.443428i	$0.853762\pi$
$420$	0	0
$421$	− 18.6127i	− 0.907128i	−0.891224	−	0.453564i	$-0.850152\pi$
$421$	− 18.6127i	− 0.907128i	0.891224	−	0.453564i	$-0.149848\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 40.7088i	− 1.97467i
$426$	0	0
$427$	−3.83228	−0.185457
$428$	0	0
$429$	0	0
$430$	0	0
$431$	25.0852	1.20831	0.604156	−	0.796866i	$-0.293510\pi$
$431$	25.0852	1.20831	0.604156	+	0.796866i	$0.293510\pi$
$432$	0	0
$433$	25.3492	1.21820	0.609102	−	0.793092i	$-0.291530\pi$
$433$	25.3492	1.21820	0.609102	+	0.793092i	$0.291530\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−16.2359	−0.776668
$438$	0	0
$439$	− 29.3342i	− 1.40005i	−0.714120	−	0.700023i	$-0.753173\pi$
$439$	− 29.3342i	− 1.40005i	0.714120	−	0.700023i	$-0.246827\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 20.4404i	− 0.971154i	−0.874194	−	0.485577i	$-0.838610\pi$
$443$	− 20.4404i	− 0.971154i	0.874194	−	0.485577i	$-0.161390\pi$
$444$	0	0
$445$	12.8531i	0.609295i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 40.0304i	− 1.88915i	−0.328291	−	0.944577i	$-0.606473\pi$
$449$	− 40.0304i	− 1.88915i	0.328291	−	0.944577i	$-0.393527\pi$
$450$	0	0
$451$	2.40432	0.113215
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−77.2004	−3.61921
$456$	0	0
$457$	19.9737	0.934329	0.467164	−	0.884170i	$-0.345276\pi$
$457$	19.9737	0.934329	0.467164	+	0.884170i	$0.345276\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−11.6016	−0.540341	−0.270171	−	0.962812i	$-0.587080\pi$
$461$	−11.6016	−0.540341	−0.270171	+	0.962812i	$0.587080\pi$
$462$	0	0
$463$	12.3732i	0.575031i	0.957776	+	0.287516i	$0.0928293\pi$
$463$	12.3732i	0.575031i	−0.957776	+	0.287516i	$0.907171\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 16.1873i	− 0.749058i	−0.927215	−	0.374529i	$-0.877804\pi$
$467$	− 16.1873i	− 0.749058i	0.927215	−	0.374529i	$-0.122196\pi$
$468$	0	0
$469$	49.6709i	2.29359i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.93918i	0.227104i
$474$	0	0
$475$	21.5729	0.989832
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.46203	0.0668020	0.0334010	−	0.999442i	$-0.489366\pi$
$479$	1.46203	0.0668020	0.0334010	+	0.999442i	$0.489366\pi$
$480$	0	0
$481$	−30.1812	−1.37614
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.47611	0.112434
$486$	0	0
$487$	15.7577i	0.714048i	0.934095	+	0.357024i	$0.116208\pi$
$487$	15.7577i	0.714048i	−0.934095	+	0.357024i	$0.883792\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 5.79940i	− 0.261723i	−0.991401	−	0.130862i	$-0.958226\pi$
$491$	− 5.79940i	− 0.261723i	0.991401	−	0.130862i	$-0.0417744\pi$
$492$	0	0
$493$	8.46078i	0.381055i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 11.8013i	− 0.529360i
$498$	0	0
$499$	−37.0540	−1.65877	−0.829383	−	0.558680i	$-0.811308\pi$
$499$	−37.0540	−1.65877	−0.829383	+	0.558680i	$0.811308\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.98721	−0.0886055	−0.0443027	−	0.999018i	$-0.514107\pi$
$503$	−1.98721	−0.0886055	−0.0443027	+	0.999018i	$0.514107\pi$
$504$	0	0
$505$	−33.5304	−1.49208
$506$	0	0
$507$	0	0
$508$	0	0
$509$	19.8358	0.879206	0.439603	−	0.898192i	$-0.355119\pi$
$509$	19.8358	0.879206	0.439603	+	0.898192i	$0.355119\pi$
$510$	0	0
$511$	− 17.8614i	− 0.790143i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.09364i	0.356648i
$516$	0	0
$517$	− 4.39230i	− 0.193173i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 10.8050i	− 0.473376i	−0.971586	−	0.236688i	$-0.923938\pi$
$521$	− 10.8050i	− 0.473376i	0.971586	−	0.236688i	$-0.0760619\pi$
$522$	0	0
$523$	35.0092	1.53085	0.765423	−	0.643528i	$-0.222530\pi$
$523$	35.0092	1.53085	0.765423	+	0.643528i	$0.222530\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	25.6939	1.11924
$528$	0	0
$529$	50.3114	2.18745
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−25.1233	−1.08821
$534$	0	0
$535$	17.2117i	0.744127i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 3.83320i	− 0.165108i
$540$	0	0
$541$	− 33.3973i	− 1.43586i	−0.696114	−	0.717932i	$-0.745089\pi$
$541$	− 33.3973i	− 1.43586i	0.696114	−	0.717932i	$-0.254911\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 38.1215i	− 1.63294i
$546$	0	0
$547$	1.11664	0.0477441	0.0238720	−	0.999715i	$-0.492401\pi$
$547$	1.11664	0.0477441	0.0238720	+	0.999715i	$0.492401\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4.48364	−0.191010
$552$	0	0
$553$	−45.4950	−1.93464
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.55480	0.320107	0.160054	−	0.987108i	$-0.448833\pi$
$557$	7.55480	0.320107	0.160054	+	0.987108i	$0.448833\pi$
$558$	0	0
$559$	− 51.6107i	− 2.18290i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 43.3715i	− 1.82789i	−0.405836	−	0.913946i	$-0.633019\pi$
$563$	− 43.3715i	− 1.82789i	0.405836	−	0.913946i	$-0.366981\pi$
$564$	0	0
$565$	39.8459i	1.67633i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	25.8719i	1.08461i	0.840182	+	0.542304i	$0.182448\pi$
$569$	25.8719i	1.08461i	−0.840182	+	0.542304i	$0.817552\pi$
$570$	0	0
$571$	−19.0370	−0.796674	−0.398337	−	0.917239i	$-0.630412\pi$
$571$	−19.0370	−0.796674	−0.398337	+	0.917239i	$0.630412\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−97.4101	−4.06228
$576$	0	0
$577$	16.5418	0.688643	0.344321	−	0.938852i	$-0.388109\pi$
$577$	16.5418	0.688643	0.344321	+	0.938852i	$0.388109\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	22.0766	0.915893
$582$	0	0
$583$	2.58783i	0.107177i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 15.8160i	− 0.652794i	−0.945233	−	0.326397i	$-0.894165\pi$
$587$	− 15.8160i	− 0.652794i	0.945233	−	0.326397i	$-0.105835\pi$
$588$	0	0
$589$	13.6160i	0.561039i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	26.7787i	1.09967i	0.835274	+	0.549834i	$0.185309\pi$
$593$	26.7787i	1.09967i	−0.835274	+	0.549834i	$0.814691\pi$
$594$	0	0
$595$	−56.4194	−2.31297
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−16.6199	−0.679069	−0.339534	−	0.940594i	$-0.610269\pi$
$599$	−16.6199	−0.679069	−0.339534	+	0.940594i	$0.610269\pi$
$600$	0	0
$601$	20.0042	0.815987	0.407994	−	0.912985i	$-0.366229\pi$
$601$	20.0042	0.815987	0.407994	+	0.912985i	$0.366229\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	43.6265	1.77367
$606$	0	0
$607$	2.42181i	0.0982984i	0.998791	+	0.0491492i	$0.0156510\pi$
$607$	2.42181i	0.0982984i	−0.998791	+	0.0491492i	$0.984349\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	45.8963i	1.85676i
$612$	0	0
$613$	− 15.4102i	− 0.622412i	−0.950343	−	0.311206i	$-0.899267\pi$
$613$	− 15.4102i	− 0.622412i	0.950343	−	0.311206i	$-0.100733\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 7.00200i	− 0.281890i	−0.990017	−	0.140945i	$-0.954986\pi$
$617$	− 7.00200i	− 0.281890i	0.990017	−	0.140945i	$-0.0450141\pi$
$618$	0	0
$619$	−11.9253	−0.479317	−0.239659	−	0.970857i	$-0.577036\pi$
$619$	−11.9253	−0.479317	−0.239659	+	0.970857i	$0.577036\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.3748	0.495786
$624$	0	0
$625$	47.5466	1.90186
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−22.0569	−0.879468
$630$	0	0
$631$	6.65500i	0.264932i	0.991188	+	0.132466i	$0.0422895\pi$
$631$	6.65500i	0.264932i	−0.991188	+	0.132466i	$0.957711\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 57.8231i	− 2.29464i
$636$	0	0
$637$	40.0540i	1.58700i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	38.6695i	1.52735i	0.645599	+	0.763677i	$0.276608\pi$
$641$	38.6695i	1.52735i	−0.645599	+	0.763677i	$0.723392\pi$
$642$	0	0
$643$	27.8236	1.09725	0.548627	−	0.836067i	$-0.315151\pi$
$643$	27.8236	1.09725	0.548627	+	0.836067i	$0.315151\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.30252	0.208464	0.104232	−	0.994553i	$-0.466762\pi$
$647$	5.30252	0.208464	0.104232	+	0.994553i	$0.466762\pi$
$648$	0	0
$649$	−2.93700	−0.115287
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−24.0403	−0.940770	−0.470385	−	0.882461i	$-0.655885\pi$
$653$	−24.0403	−0.940770	−0.470385	+	0.882461i	$0.655885\pi$
$654$	0	0
$655$	− 70.9989i	− 2.77416i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	32.5078i	1.26632i	0.774019	+	0.633162i	$0.218243\pi$
$659$	32.5078i	1.26632i	−0.774019	+	0.633162i	$0.781757\pi$
$660$	0	0
$661$	− 43.8787i	− 1.70668i	−0.521352	−	0.853342i	$-0.674572\pi$
$661$	− 43.8787i	− 1.70668i	0.521352	−	0.853342i	$-0.325428\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 29.8984i	− 1.15941i
$666$	0	0
$667$	20.2454	0.783905
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0.460881	0.0177921
$672$	0	0
$673$	−24.0162	−0.925756	−0.462878	−	0.886422i	$-0.653183\pi$
$673$	−24.0162	−0.925756	−0.462878	+	0.886422i	$0.653183\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18.3455	0.705073	0.352537	−	0.935798i	$-0.385319\pi$
$677$	18.3455	0.705073	0.352537	+	0.935798i	$0.385319\pi$
$678$	0	0
$679$	− 2.38397i	− 0.0914882i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	50.4847i	1.93174i	0.259018	+	0.965872i	$0.416601\pi$
$683$	50.4847i	1.93174i	−0.259018	+	0.965872i	$0.583399\pi$
$684$	0	0
$685$	47.4733i	1.81386i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 27.0409i	− 1.03018i
$690$	0	0
$691$	27.5020	1.04622	0.523112	−	0.852264i	$-0.324771\pi$
$691$	27.5020	1.04622	0.523112	+	0.852264i	$0.324771\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−23.3082	−0.884129
$696$	0	0
$697$	−18.3606	−0.695456
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.33376	0.239223	0.119611	−	0.992821i	$-0.461835\pi$
$701$	6.33376	0.239223	0.119611	+	0.992821i	$0.461835\pi$
$702$	0	0
$703$	− 11.6887i	− 0.440847i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	32.2826i	1.21411i
$708$	0	0
$709$	12.1008i	0.454457i	0.973841	+	0.227228i	$0.0729664\pi$
$709$	12.1008i	0.454457i	−0.973841	+	0.227228i	$0.927034\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 61.4817i	− 2.30251i
$714$	0	0
$715$	9.28436	0.347215
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−9.82136	−0.366275	−0.183137	−	0.983087i	$-0.558625\pi$
$719$	−9.82136	−0.366275	−0.183137	+	0.983087i	$0.558625\pi$
$720$	0	0
$721$	7.79246	0.290206
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−26.9004	−0.999055
$726$	0	0
$727$	39.8494i	1.47793i	0.673742	+	0.738966i	$0.264686\pi$
$727$	39.8494i	1.47793i	−0.673742	+	0.738966i	$0.735314\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 37.7180i	− 1.39505i
$732$	0	0
$733$	21.4330i	0.791645i	0.918327	+	0.395823i	$0.129540\pi$
$733$	21.4330i	0.791645i	−0.918327	+	0.395823i	$0.870460\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5.97358i	− 0.220040i
$738$	0	0
$739$	−27.3791	−1.00716	−0.503578	−	0.863950i	$-0.667983\pi$
$739$	−27.3791	−1.00716	−0.503578	+	0.863950i	$0.667983\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	14.8229	0.543799	0.271900	−	0.962326i	$-0.412348\pi$
$743$	14.8229	0.543799	0.271900	+	0.962326i	$0.412348\pi$
$744$	0	0
$745$	−77.7079	−2.84700
$746$	0	0
$747$	0	0
$748$	0	0
$749$	16.5712	0.605499
$750$	0	0
$751$	− 11.4571i	− 0.418076i	−0.977908	−	0.209038i	$-0.932967\pi$
$751$	− 11.4571i	− 0.418076i	0.977908	−	0.209038i	$-0.0670332\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	31.5347i	1.14766i
$756$	0	0
$757$	33.0507i	1.20125i	0.799531	+	0.600624i	$0.205081\pi$
$757$	33.0507i	1.20125i	−0.799531	+	0.600624i	$0.794919\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	8.12860i	0.294662i	0.989087	+	0.147331i	$0.0470682\pi$
$761$	8.12860i	0.294662i	−0.989087	+	0.147331i	$0.952932\pi$
$762$	0	0
$763$	−36.7029	−1.32873
$764$	0	0
$765$	0	0
$766$	0	0
$767$	30.6894	1.10813
$768$	0	0
$769$	−19.5771	−0.705967	−0.352984	−	0.935630i	$-0.614833\pi$
$769$	−19.5771	−0.705967	−0.352984	+	0.935630i	$0.614833\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	37.7684	1.35843	0.679217	−	0.733938i	$-0.262320\pi$
$773$	37.7684	1.35843	0.679217	+	0.733938i	$0.262320\pi$
$774$	0	0
$775$	81.6917i	2.93445i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 9.72985i	− 0.348608i
$780$	0	0
$781$	1.41926i	0.0507851i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	39.4042i	1.40640i
$786$	0	0
$787$	−2.03615	−0.0725807	−0.0362904	−	0.999341i	$-0.511554\pi$
$787$	−2.03615	−0.0725807	−0.0362904	+	0.999341i	$0.511554\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	38.3631	1.36404
$792$	0	0
$793$	−4.81586	−0.171016
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−33.2487	−1.17773	−0.588865	−	0.808231i	$-0.700425\pi$
$797$	−33.2487	−1.17773	−0.588865	+	0.808231i	$0.700425\pi$
$798$	0	0
$799$	33.5418i	1.18662i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.14807i	0.0758038i
$804$	0	0
$805$	135.003i	4.75824i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	17.3485i	0.609942i	0.952362	+	0.304971i	$0.0986468\pi$
$809$	17.3485i	0.609942i	−0.952362	+	0.304971i	$0.901353\pi$
$810$	0	0
$811$	40.9868	1.43924	0.719620	−	0.694368i	$-0.244316\pi$
$811$	40.9868	1.43924	0.719620	+	0.694368i	$0.244316\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	37.2850	1.30604
$816$	0	0
$817$	19.9880	0.699291
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.85801	0.169546	0.0847729	−	0.996400i	$-0.472984\pi$
$821$	4.85801	0.169546	0.0847729	+	0.996400i	$0.472984\pi$
$822$	0	0
$823$	28.0083i	0.976309i	0.872757	+	0.488155i	$0.162330\pi$
$823$	28.0083i	0.976309i	−0.872757	+	0.488155i	$0.837670\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.46917i	0.120635i	0.998179	+	0.0603174i	$0.0192113\pi$
$827$	3.46917i	0.120635i	−0.998179	+	0.0603174i	$0.980789\pi$
$828$	0	0
$829$	14.6239i	0.507908i	0.967216	+	0.253954i	$0.0817312\pi$
$829$	14.6239i	0.507908i	−0.967216	+	0.253954i	$0.918269\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	29.2722i	1.01422i
$834$	0	0
$835$	31.9046	1.10410
$836$	0	0
$837$	0	0
$838$	0	0
$839$	15.3680	0.530562	0.265281	−	0.964171i	$-0.414535\pi$
$839$	15.3680	0.530562	0.265281	+	0.964171i	$0.414535\pi$
$840$	0	0
$841$	−23.4091	−0.807211
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−44.4059	−1.52761
$846$	0	0
$847$	− 42.0030i	− 1.44324i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	52.7790i	1.80924i
$852$	0	0
$853$	− 0.874295i	− 0.0299353i	−0.999888	−	0.0149676i	$-0.995235\pi$
$853$	− 0.874295i	− 0.0299353i	0.999888	−	0.0149676i	$-0.00476453\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	34.7031i	1.18543i	0.805411	+	0.592717i	$0.201945\pi$
$857$	34.7031i	1.18543i	−0.805411	+	0.592717i	$0.798055\pi$
$858$	0	0
$859$	39.4933	1.34749	0.673746	−	0.738963i	$-0.264684\pi$
$859$	39.4933	1.34749	0.673746	+	0.738963i	$0.264684\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−28.1903	−0.959611	−0.479805	−	0.877375i	$-0.659293\pi$
$863$	−28.1903	−0.959611	−0.479805	+	0.877375i	$0.659293\pi$
$864$	0	0
$865$	−82.8574	−2.81724
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.47136	0.185603
$870$	0	0
$871$	62.4194i	2.11500i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 100.544i	− 3.39901i
$876$	0	0
$877$	− 34.5026i	− 1.16507i	−0.812806	−	0.582535i	$-0.802061\pi$
$877$	− 34.5026i	− 1.16507i	0.812806	−	0.582535i	$-0.197939\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	21.7178i	0.731690i	0.930676	+	0.365845i	$0.119220\pi$
$881$	21.7178i	0.731690i	−0.930676	+	0.365845i	$0.880780\pi$
$882$	0	0
$883$	10.8314	0.364506	0.182253	−	0.983252i	$-0.441661\pi$
$883$	10.8314	0.364506	0.182253	+	0.983252i	$0.441661\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.1435	0.374164	0.187082	−	0.982344i	$-0.440097\pi$
$887$	11.1435	0.374164	0.187082	+	0.982344i	$0.440097\pi$
$888$	0	0
$889$	−55.6714	−1.86716
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−17.7749	−0.594813
$894$	0	0
$895$	− 77.4531i	− 2.58897i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 16.9785i	− 0.566266i
$900$	0	0
$901$	− 19.7620i	− 0.658366i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	45.6861i	1.51866i
$906$	0	0
$907$	−10.7103	−0.355631	−0.177816	−	0.984064i	$-0.556903\pi$
$907$	−10.7103	−0.355631	−0.177816	+	0.984064i	$0.556903\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	21.5384	0.713600	0.356800	−	0.934181i	$-0.383868\pi$
$911$	21.5384	0.713600	0.356800	+	0.934181i	$0.383868\pi$
$912$	0	0
$913$	−2.65500	−0.0878678
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−68.3569	−2.25734
$918$	0	0
$919$	− 15.5159i	− 0.511824i	−0.966700	−	0.255912i	$-0.917624\pi$
$919$	− 15.5159i	− 0.511824i	0.966700	−	0.255912i	$-0.0823757\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 14.8302i	− 0.488141i
$924$	0	0
$925$	− 70.1283i	− 2.30580i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	31.9098i	1.04693i	0.852048	+	0.523464i	$0.175360\pi$
$929$	31.9098i	1.04693i	−0.852048	+	0.523464i	$0.824640\pi$
$930$	0	0
$931$	−15.5123	−0.508394
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.78517	0.221899
$936$	0	0
$937$	−7.01911	−0.229304	−0.114652	−	0.993406i	$-0.536575\pi$
$937$	−7.01911	−0.229304	−0.114652	+	0.993406i	$0.536575\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−11.7306	−0.382407	−0.191204	−	0.981550i	$-0.561239\pi$
$941$	−11.7306	−0.382407	−0.191204	+	0.981550i	$0.561239\pi$
$942$	0	0
$943$	43.9341i	1.43069i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.92535i	0.192548i	0.995355	+	0.0962741i	$0.0306925\pi$
$947$	5.92535i	0.192548i	−0.995355	+	0.0962741i	$0.969307\pi$
$948$	0	0
$949$	− 22.4457i	− 0.728619i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 8.28906i	− 0.268509i	−0.990947	−	0.134255i	$-0.957136\pi$
$953$	− 8.28906i	− 0.268509i	0.990947	−	0.134255i	$-0.0428640\pi$
$954$	0	0
$955$	80.0030	2.58884
$956$	0	0
$957$	0	0
$958$	0	0
$959$	45.7067	1.47595
$960$	0	0
$961$	−20.5609	−0.663254
$962$	0	0
$963$	0	0
$964$	0	0
$965$	75.2098	2.42109
$966$	0	0
$967$	38.6688i	1.24351i	0.783214	+	0.621753i	$0.213579\pi$
$967$	38.6688i	1.24351i	−0.783214	+	0.621753i	$0.786421\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	36.9214i	1.18486i	0.805620	+	0.592432i	$0.201832\pi$
$971$	36.9214i	1.18486i	−0.805620	+	0.592432i	$0.798168\pi$
$972$	0	0
$973$	22.4408i	0.719420i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	41.9613i	1.34246i	0.741248	+	0.671231i	$0.234234\pi$
$977$	41.9613i	1.34246i	−0.741248	+	0.671231i	$0.765766\pi$
$978$	0	0
$979$	−1.48823	−0.0475641
$980$	0	0
$981$	0	0
$982$	0	0
$983$	34.8430	1.11132	0.555659	−	0.831410i	$-0.312466\pi$
$983$	34.8430	1.11132	0.555659	+	0.831410i	$0.312466\pi$
$984$	0	0
$985$	104.430	3.32743
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−90.2536	−2.86990
$990$	0	0
$991$	36.9678i	1.17432i	0.809471	+	0.587160i	$0.199754\pi$
$991$	36.9678i	1.17432i	−0.809471	+	0.587160i	$0.800246\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 7.25374i	− 0.229959i
$996$	0	0
$997$	− 9.65401i	− 0.305746i	−0.988246	−	0.152873i	$-0.951148\pi$
$997$	− 9.65401i	− 0.305746i	0.988246	−	0.152873i	$-0.0488525\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5184.2.f.b.2591.15	yes	16
3.2	odd	2	inner	5184.2.f.b.2591.1	✓	16
4.3	odd	2		5184.2.f.e.2591.16	yes	16
8.3	odd	2	inner	5184.2.f.b.2591.2	yes	16
8.5	even	2		5184.2.f.e.2591.1	yes	16
12.11	even	2		5184.2.f.e.2591.2	yes	16
24.5	odd	2		5184.2.f.e.2591.15	yes	16
24.11	even	2	inner	5184.2.f.b.2591.16	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5184.2.f.b.2591.1	✓	16	3.2	odd	2	inner
5184.2.f.b.2591.2	yes	16	8.3	odd	2	inner
5184.2.f.b.2591.15	yes	16	1.1	even	1	trivial
5184.2.f.b.2591.16	yes	16	24.11	even	2	inner
5184.2.f.e.2591.1	yes	16	8.5	even	2
5184.2.f.e.2591.2	yes	16	12.11	even	2
5184.2.f.e.2591.15	yes	16	24.5	odd	2
5184.2.f.e.2591.16	yes	16	4.3	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.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.04682 q^{5} -3.89623i q^{7} +O(q^{10})\)	\(q+4.04682 q^{5} -3.89623i q^{7} +0.468572i q^{11} -4.89623i q^{13} -3.57825i q^{17} +1.89623 q^{19} -8.56221 q^{23} +11.3767 q^{25} -2.36451 q^{29} +7.18059i q^{31} -15.7673i q^{35} -6.16418i q^{37} -5.13116i q^{41} +10.5409 q^{43} -9.37380 q^{47} -8.18059 q^{49} +5.52280 q^{53} +1.89623i q^{55} +6.26797i q^{59} -0.983586i q^{61} -19.8141i q^{65} -12.7485 q^{67} +3.02890 q^{71} +4.58429 q^{73} +1.82566 q^{77} -11.6767i q^{79} +5.66616i q^{83} -14.4805i q^{85} +3.17610i q^{89} -19.0768 q^{91} +7.67369 q^{95} +0.611865 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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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