LMFDB - Embedded newform 5184.2.d.q.2593.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("5184.2593");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$41.3944484078$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 $ x^12 - 16x^8 - 24x^7 + 96x^5 + 304x^4 + 384x^3 + 288x^2 + 144*x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2593.4
Root			$-1.50511 + 0.403293i$ of defining polynomial
Character	$\chi$	$=$	5184.2593
Dual form			5184.2.d.q.2593.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{5} +0.990721 q^{7} +O(q^{10})$	$q-1.73205i q^{5} +0.990721 q^{7} +2.10083i q^{11} -6.36152i q^{13} +3.81681 q^{17} +2.00000i q^{19} +7.10285 q^{23} +2.00000 q^{25} -8.34296i q^{29} -2.15609 q^{31} -1.71598i q^{35} +4.62947i q^{37} +0.816810 q^{41} +2.28402i q^{43} -6.78555 q^{47} -6.01847 q^{49} +3.14681i q^{53} +3.63875 q^{55} -11.9176i q^{59} +4.87886i q^{61} -11.0185 q^{65} +13.7345i q^{67} +13.5391 q^{71} +10.0185 q^{73} +2.08134i q^{77} +9.08429 q^{79} -4.28402i q^{83} -6.61091i q^{85} -14.0185 q^{89} -6.30249i q^{91} +3.46410 q^{95} +12.4504 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 24 q^{25} - 36 q^{41} + 48 q^{49} - 12 q^{65} - 48 q^{89} + 12 q^{97}+O(q^{100}) $ 12 * q + 24 * q^25 - 36 * q^41 + 48 * q^49 - 12 * q^65 - 48 * q^89 + 12 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 1.73205i	− 0.774597i	−0.921954	−	0.387298i	$-0.873408\pi$
$5$	− 1.73205i	− 0.774597i	0.921954	−	0.387298i	$-0.126592\pi$
$6$	0	0
$7$	0.990721	0.374457	0.187229	−	0.982316i	$-0.440049\pi$
$7$	0.990721	0.374457	0.187229	+	0.982316i	$0.440049\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.10083i	0.633424i	0.948522	+	0.316712i	$0.102579\pi$
$11$	2.10083i	0.633424i	−0.948522	+	0.316712i	$0.897421\pi$
$12$	0	0
$13$	− 6.36152i	− 1.76437i	−0.470905	−	0.882184i	$-0.656073\pi$
$13$	− 6.36152i	− 1.76437i	0.470905	−	0.882184i	$-0.343927\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.81681	0.925712	0.462856	−	0.886433i	$-0.346825\pi$
$17$	3.81681	0.925712	0.462856	+	0.886433i	$0.346825\pi$
$18$	0	0
$19$	2.00000i	0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$	2.00000i	0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.10285	1.48105	0.740523	−	0.672031i	$-0.234578\pi$
$23$	7.10285	1.48105	0.740523	+	0.672031i	$0.234578\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 8.34296i	− 1.54925i	−0.632422	−	0.774624i	$-0.717939\pi$
$29$	− 8.34296i	− 1.54925i	0.632422	−	0.774624i	$-0.282061\pi$
$30$	0	0
$31$	−2.15609	−0.387245	−0.193622	−	0.981076i	$-0.562024\pi$
$31$	−2.15609	−0.387245	−0.193622	+	0.981076i	$0.562024\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1.71598i	− 0.290054i
$36$	0	0
$37$	4.62947i	0.761080i	0.924765	+	0.380540i	$0.124262\pi$
$37$	4.62947i	0.761080i	−0.924765	+	0.380540i	$0.875738\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.816810	0.127564	0.0637822	−	0.997964i	$-0.479684\pi$
$41$	0.816810	0.127564	0.0637822	+	0.997964i	$0.479684\pi$
$42$	0	0
$43$	2.28402i	0.348310i	0.984718	+	0.174155i	$0.0557193\pi$
$43$	2.28402i	0.348310i	−0.984718	+	0.174155i	$0.944281\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.78555	−0.989775	−0.494887	−	0.868957i	$-0.664791\pi$
$47$	−6.78555	−0.989775	−0.494887	+	0.868957i	$0.664791\pi$
$48$	0	0
$49$	−6.01847	−0.859782
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.14681i	0.432247i	0.976366	+	0.216124i	$0.0693414\pi$
$53$	3.14681i	0.432247i	−0.976366	+	0.216124i	$0.930659\pi$
$54$	0	0
$55$	3.63875	0.490648
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 11.9176i	− 1.55154i	−0.631013	−	0.775772i	$-0.717361\pi$
$59$	− 11.9176i	− 1.55154i	0.631013	−	0.775772i	$-0.282639\pi$
$60$	0	0
$61$	4.87886i	0.624674i	0.949971	+	0.312337i	$0.101112\pi$
$61$	4.87886i	0.624674i	−0.949971	+	0.312337i	$0.898888\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−11.0185	−1.36667
$66$	0	0
$67$	13.7345i	1.67793i	0.544185	+	0.838965i	$0.316839\pi$
$67$	13.7345i	1.67793i	−0.544185	+	0.838965i	$0.683161\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.5391	1.60680	0.803399	−	0.595442i	$-0.203023\pi$
$71$	13.5391	1.60680	0.803399	+	0.595442i	$0.203023\pi$
$72$	0	0
$73$	10.0185	1.17257	0.586287	−	0.810104i	$-0.300589\pi$
$73$	10.0185	1.17257	0.586287	+	0.810104i	$0.300589\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.08134i	0.237190i
$78$	0	0
$79$	9.08429	1.02206	0.511031	−	0.859562i	$-0.329264\pi$
$79$	9.08429	1.02206	0.511031	+	0.859562i	$0.329264\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 4.28402i	− 0.470232i	−0.971967	−	0.235116i	$-0.924453\pi$
$83$	− 4.28402i	− 0.470232i	0.971967	−	0.235116i	$-0.0755471\pi$
$84$	0	0
$85$	− 6.61091i	− 0.717054i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−14.0185	−1.48595	−0.742977	−	0.669316i	$-0.766587\pi$
$89$	−14.0185	−1.48595	−0.742977	+	0.669316i	$0.766587\pi$
$90$	0	0
$91$	− 6.30249i	− 0.660681i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.46410	0.355409
$96$	0	0
$97$	12.4504	1.26415	0.632075	−	0.774907i	$-0.282204\pi$
$97$	12.4504	1.26415	0.632075	+	0.774907i	$0.282204\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 15.9377i	− 1.58586i	−0.609310	−	0.792932i	$-0.708553\pi$
$101$	− 15.9377i	− 1.58586i	0.609310	−	0.792932i	$-0.291447\pi$
$102$	0	0
$103$	−17.1779	−1.69258	−0.846292	−	0.532719i	$-0.821170\pi$
$103$	−17.1779	−1.69258	−0.846292	+	0.532719i	$0.821170\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 16.2017i	− 1.56627i	−0.621849	−	0.783137i	$-0.713618\pi$
$107$	− 16.2017i	− 1.56627i	0.621849	−	0.783137i	$-0.286382\pi$
$108$	0	0
$109$	− 0.816078i	− 0.0781661i	−0.999236	−	0.0390830i	$-0.987556\pi$
$109$	− 0.816078i	− 0.0781661i	0.999236	−	0.0390830i	$-0.0124437\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.43196	−0.605068	−0.302534	−	0.953139i	$-0.597833\pi$
$113$	−6.43196	−0.605068	−0.302534	+	0.953139i	$0.597833\pi$
$114$	0	0
$115$	− 12.3025i	− 1.14721i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.78140	0.346640
$120$	0	0
$121$	6.58651	0.598774
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 12.1244i	− 1.08444i
$126$	0	0
$127$	16.1871	1.43638	0.718188	−	0.695849i	$-0.244972\pi$
$127$	16.1871	1.43638	0.718188	+	0.695849i	$0.244972\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	7.71598i	0.674148i	0.941478	+	0.337074i	$0.109437\pi$
$131$	7.71598i	0.674148i	−0.941478	+	0.337074i	$0.890563\pi$
$132$	0	0
$133$	1.98144i	0.171813i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.61515	−0.736042	−0.368021	−	0.929818i	$-0.619965\pi$
$137$	−8.61515	−0.736042	−0.368021	+	0.929818i	$0.619965\pi$
$138$	0	0
$139$	− 5.71598i	− 0.484823i	−0.970174	−	0.242412i	$-0.922062\pi$
$139$	− 5.71598i	− 0.484823i	0.970174	−	0.242412i	$-0.0779384\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	13.3645	1.11759
$144$	0	0
$145$	−14.4504	−1.20004
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.41476i	0.115901i	0.998319	+	0.0579507i	$0.0184566\pi$
$149$	1.41476i	0.115901i	−0.998319	+	0.0579507i	$0.981543\pi$
$150$	0	0
$151$	−3.28946	−0.267692	−0.133846	−	0.991002i	$-0.542733\pi$
$151$	−3.28946	−0.267692	−0.133846	+	0.991002i	$0.542733\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.73445i	0.299958i
$156$	0	0
$157$	6.67881i	0.533027i	0.963831	+	0.266514i	$0.0858717\pi$
$157$	6.67881i	0.533027i	−0.963831	+	0.266514i	$0.914128\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.03694	0.554589
$162$	0	0
$163$	− 7.43196i	− 0.582116i	−0.956705	−	0.291058i	$-0.905993\pi$
$163$	− 7.43196i	− 0.582116i	0.956705	−	0.291058i	$-0.0940073\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.90037	0.766113	0.383057	−	0.923725i	$-0.374871\pi$
$167$	9.90037	0.766113	0.383057	+	0.923725i	$0.374871\pi$
$168$	0	0
$169$	−27.4689	−2.11299
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 12.1244i	− 0.921798i	−0.887453	−	0.460899i	$-0.847527\pi$
$173$	− 12.1244i	− 0.921798i	0.887453	−	0.460899i	$-0.152473\pi$
$174$	0	0
$175$	1.98144	0.149783
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.00000i	0.448461i	0.974536	+	0.224231i	$0.0719869\pi$
$179$	6.00000i	0.448461i	−0.974536	+	0.224231i	$0.928013\pi$
$180$	0	0
$181$	− 4.62947i	− 0.344106i	−0.985088	−	0.172053i	$-0.944960\pi$
$181$	− 4.62947i	− 0.344106i	0.985088	−	0.172053i	$-0.0550400\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8.01847	0.589530
$186$	0	0
$187$	8.01847i	0.586369i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.1779	1.24295	0.621473	−	0.783436i	$-0.286535\pi$
$191$	17.1779	1.24295	0.621473	+	0.783436i	$0.286535\pi$
$192$	0	0
$193$	−11.5865	−0.834015	−0.417008	−	0.908903i	$-0.636921\pi$
$193$	−11.5865	−0.834015	−0.417008	+	0.908903i	$0.636921\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 17.6698i	− 1.25892i	−0.777033	−	0.629460i	$-0.783276\pi$
$197$	− 17.6698i	− 1.25892i	0.777033	−	0.629460i	$-0.216724\pi$
$198$	0	0
$199$	−16.5044	−1.16997	−0.584984	−	0.811045i	$-0.698899\pi$
$199$	−16.5044	−1.16997	−0.584984	+	0.811045i	$0.698899\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 8.26555i	− 0.580128i
$204$	0	0
$205$	− 1.41476i	− 0.0988109i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.20166	−0.290635
$210$	0	0
$211$	− 11.7345i	− 0.807833i	−0.914796	−	0.403916i	$-0.867649\pi$
$211$	− 11.7345i	− 0.807833i	0.914796	−	0.403916i	$-0.132351\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3.95604	0.269800
$216$	0	0
$217$	−2.13608	−0.145007
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 24.2807i	− 1.63330i
$222$	0	0
$223$	−1.34001	−0.0897336	−0.0448668	−	0.998993i	$-0.514286\pi$
$223$	−1.34001	−0.0897336	−0.0448668	+	0.998993i	$0.514286\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.89917i	0.258797i	0.991593	+	0.129398i	$0.0413046\pi$
$227$	3.89917i	0.258797i	−0.991593	+	0.129398i	$0.958695\pi$
$228$	0	0
$229$	2.23083i	0.147418i	0.997280	+	0.0737089i	$0.0234836\pi$
$229$	2.23083i	0.147418i	−0.997280	+	0.0737089i	$0.976516\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.26724	−0.607117	−0.303559	−	0.952813i	$-0.598175\pi$
$233$	−9.26724	−0.607117	−0.303559	+	0.952813i	$0.598175\pi$
$234$	0	0
$235$	11.7529i	0.766676i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	20.9593	1.35574	0.677871	−	0.735181i	$-0.262903\pi$
$239$	20.9593	1.35574	0.677871	+	0.735181i	$0.262903\pi$
$240$	0	0
$241$	15.0369	0.968615	0.484307	−	0.874898i	$-0.339072\pi$
$241$	15.0369	0.968615	0.484307	+	0.874898i	$0.339072\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	10.4243i	0.665984i
$246$	0	0
$247$	12.7230	0.809547
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 0.164719i	− 0.0103969i	−0.999986	−	0.00519847i	$-0.998345\pi$
$251$	− 0.164719i	− 0.0103969i	0.999986	−	0.00519847i	$-0.00165473\pi$
$252$	0	0
$253$	14.9219i	0.938130i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−26.8353	−1.67394	−0.836969	−	0.547250i	$-0.815675\pi$
$257$	−26.8353	−1.67394	−0.836969	+	0.547250i	$0.815675\pi$
$258$	0	0
$259$	4.58651i	0.284992i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	19.9754	1.23173	0.615867	−	0.787850i	$-0.288806\pi$
$263$	19.9754	1.23173	0.615867	+	0.787850i	$0.288806\pi$
$264$	0	0
$265$	5.45043	0.334817
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0.666581i	0.0406422i	0.999794	+	0.0203211i	$0.00646884\pi$
$269$	0.666581i	0.0406422i	−0.999794	+	0.0203211i	$0.993531\pi$
$270$	0	0
$271$	−17.0352	−1.03482	−0.517408	−	0.855739i	$-0.673103\pi$
$271$	−17.0352	−1.03482	−0.517408	+	0.855739i	$0.673103\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.20166i	0.253370i
$276$	0	0
$277$	− 3.71349i	− 0.223122i	−0.993758	−	0.111561i	$-0.964415\pi$
$277$	− 3.71349i	− 0.223122i	0.993758	−	0.111561i	$-0.0355851\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	27.0554	1.61399	0.806995	−	0.590558i	$-0.201092\pi$
$281$	27.0554	1.61399	0.806995	+	0.590558i	$0.201092\pi$
$282$	0	0
$283$	1.14794i	0.0682379i	0.999418	+	0.0341190i	$0.0108625\pi$
$283$	1.14794i	0.0682379i	−0.999418	+	0.0341190i	$0.989137\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.809231	0.0477674
$288$	0	0
$289$	−2.43196	−0.143056
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 26.3300i	− 1.53822i	−0.639118	−	0.769109i	$-0.720700\pi$
$293$	− 26.3300i	− 1.53822i	0.639118	−	0.769109i	$-0.279300\pi$
$294$	0	0
$295$	−20.6420	−1.20182
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 45.1849i	− 2.61311i
$300$	0	0
$301$	2.26283i	0.130427i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.45043	0.483870
$306$	0	0
$307$	7.43196i	0.424164i	0.977252	+	0.212082i	$0.0680244\pi$
$307$	7.43196i	0.424164i	−0.977252	+	0.212082i	$0.931976\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9.93236	−0.563212	−0.281606	−	0.959530i	$-0.590867\pi$
$311$	−9.93236	−0.563212	−0.281606	+	0.959530i	$0.590867\pi$
$312$	0	0
$313$	−9.03694	−0.510798	−0.255399	−	0.966836i	$-0.582207\pi$
$313$	−9.03694	−0.510798	−0.255399	+	0.966836i	$0.582207\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.69225i	0.488205i	0.969749	+	0.244103i	$0.0784934\pi$
$317$	8.69225i	0.488205i	−0.969749	+	0.244103i	$0.921507\pi$
$318$	0	0
$319$	17.5271	0.981332
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7.63362i	0.424746i
$324$	0	0
$325$	− 12.7230i	− 0.705747i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6.72259	−0.370629
$330$	0	0
$331$	− 12.8705i	− 0.707428i	−0.935354	−	0.353714i	$-0.884919\pi$
$331$	− 12.8705i	− 0.707428i	0.935354	−	0.353714i	$-0.115081\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	23.7888	1.29972
$336$	0	0
$337$	−2.43196	−0.132477	−0.0662386	−	0.997804i	$-0.521100\pi$
$337$	−2.43196	−0.132477	−0.0662386	+	0.997804i	$0.521100\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 4.52957i	− 0.245290i
$342$	0	0
$343$	−12.8977	−0.696409
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 23.3681i	− 1.25446i	−0.778833	−	0.627232i	$-0.784188\pi$
$347$	− 23.3681i	− 1.25446i	0.778833	−	0.627232i	$-0.215812\pi$
$348$	0	0
$349$	− 10.0071i	− 0.535668i	−0.963465	−	0.267834i	$-0.913692\pi$
$349$	− 10.0071i	− 0.535668i	0.963465	−	0.267834i	$-0.0863079\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.3849	−1.13820	−0.569100	−	0.822268i	$-0.692708\pi$
$353$	−21.3849	−1.13820	−0.569100	+	0.822268i	$0.692708\pi$
$354$	0	0
$355$	− 23.4504i	− 1.24462i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−23.9634	−1.26474	−0.632370	−	0.774666i	$-0.717918\pi$
$359$	−23.9634	−1.26474	−0.632370	+	0.774666i	$0.717918\pi$
$360$	0	0
$361$	15.0000	0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 17.3525i	− 0.908271i
$366$	0	0
$367$	−16.5113	−0.861882	−0.430941	−	0.902380i	$-0.641818\pi$
$367$	−16.5113	−0.861882	−0.430941	+	0.902380i	$0.641818\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.11761i	0.161858i
$372$	0	0
$373$	24.5301i	1.27012i	0.772463	+	0.635060i	$0.219025\pi$
$373$	24.5301i	1.27012i	−0.772463	+	0.635060i	$0.780975\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−53.0739	−2.73344
$378$	0	0
$379$	− 10.8824i	− 0.558991i	−0.960147	−	0.279495i	$-0.909833\pi$
$379$	− 10.8824i	− 0.558991i	0.960147	−	0.279495i	$-0.0901672\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.32145	−0.169718	−0.0848591	−	0.996393i	$-0.527044\pi$
$383$	−3.32145	−0.169718	−0.0848591	+	0.996393i	$0.527044\pi$
$384$	0	0
$385$	3.60498	0.183727
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 1.38276i	− 0.0701089i	−0.999385	−	0.0350545i	$-0.988840\pi$
$389$	− 1.38276i	− 0.0701089i	0.999385	−	0.0350545i	$-0.0111605\pi$
$390$	0	0
$391$	27.1102	1.37102
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 15.7345i	− 0.791686i
$396$	0	0
$397$	− 19.9685i	− 1.00219i	−0.865392	−	0.501096i	$-0.832930\pi$
$397$	− 19.9685i	− 1.00219i	0.865392	−	0.501096i	$-0.167070\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−0.652092	−0.0325639	−0.0162819	−	0.999867i	$-0.505183\pi$
$401$	−0.652092	−0.0325639	−0.0162819	+	0.999867i	$0.505183\pi$
$402$	0	0
$403$	13.7160i	0.683242i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.72572	−0.482086
$408$	0	0
$409$	15.0369	0.743529	0.371764	−	0.928327i	$-0.378753\pi$
$409$	15.0369	0.743529	0.371764	+	0.928327i	$0.378753\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 11.8071i	− 0.580988i
$414$	0	0
$415$	−7.42014	−0.364240
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 13.1664i	− 0.643221i	−0.946872	−	0.321611i	$-0.895776\pi$
$419$	− 13.1664i	− 0.643221i	0.946872	−	0.321611i	$-0.104224\pi$
$420$	0	0
$421$	3.88129i	0.189163i	0.995517	+	0.0945813i	$0.0301512\pi$
$421$	3.88129i	0.189163i	−0.995517	+	0.0945813i	$0.969849\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7.63362	0.370285
$426$	0	0
$427$	4.83359i	0.233914i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.81339	0.183684	0.0918422	−	0.995774i	$-0.470724\pi$
$431$	3.81339	0.183684	0.0918422	+	0.995774i	$0.470724\pi$
$432$	0	0
$433$	−10.0185	−0.481457	−0.240728	−	0.970592i	$-0.577386\pi$
$433$	−10.0185	−0.481457	−0.240728	+	0.970592i	$0.577386\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	14.2057i	0.679550i
$438$	0	0
$439$	24.4234	1.16566	0.582832	−	0.812593i	$-0.301945\pi$
$439$	24.4234	1.16566	0.582832	+	0.812593i	$0.301945\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 24.6873i	− 1.17293i	−0.809974	−	0.586466i	$-0.800519\pi$
$443$	− 24.6873i	− 1.17293i	0.809974	−	0.586466i	$-0.199481\pi$
$444$	0	0
$445$	24.2807i	1.15102i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	21.6521	1.02183	0.510913	−	0.859633i	$-0.329308\pi$
$449$	21.6521	1.02183	0.510913	+	0.859633i	$0.329308\pi$
$450$	0	0
$451$	1.71598i	0.0808023i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−10.9162	−0.511761
$456$	0	0
$457$	−22.4874	−1.05191	−0.525957	−	0.850511i	$-0.676293\pi$
$457$	−22.4874	−1.05191	−0.525957	+	0.850511i	$0.676293\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	25.0289i	1.16571i	0.812576	+	0.582856i	$0.198065\pi$
$461$	25.0289i	1.16571i	−0.812576	+	0.582856i	$0.801935\pi$
$462$	0	0
$463$	19.4766	0.905154	0.452577	−	0.891725i	$-0.350505\pi$
$463$	19.4766	0.905154	0.452577	+	0.891725i	$0.350505\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 2.40332i	− 0.111212i	−0.998453	−	0.0556062i	$-0.982291\pi$
$467$	− 2.40332i	− 0.111212i	0.998453	−	0.0556062i	$-0.0177091\pi$
$468$	0	0
$469$	13.6070i	0.628314i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.79834	−0.220628
$474$	0	0
$475$	4.00000i	0.183533i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	21.3085	0.973612	0.486806	−	0.873510i	$-0.338162\pi$
$479$	21.3085	0.973612	0.486806	+	0.873510i	$0.338162\pi$
$480$	0	0
$481$	29.4504	1.34282
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 21.5648i	− 0.979206i
$486$	0	0
$487$	40.7851	1.84815	0.924075	−	0.382210i	$-0.124837\pi$
$487$	40.7851	1.84815	0.924075	+	0.382210i	$0.124837\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	35.4235i	1.59864i	0.600906	+	0.799320i	$0.294807\pi$
$491$	35.4235i	1.59864i	−0.600906	+	0.799320i	$0.705193\pi$
$492$	0	0
$493$	− 31.8435i	− 1.43416i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13.4135	0.601677
$498$	0	0
$499$	− 4.32096i	− 0.193433i	−0.995312	−	0.0967164i	$-0.969166\pi$
$499$	− 4.32096i	− 0.193433i	0.995312	−	0.0967164i	$-0.0308340\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7.24550	0.323061	0.161530	−	0.986868i	$-0.448357\pi$
$503$	7.24550	0.323061	0.161530	+	0.986868i	$0.448357\pi$
$504$	0	0
$505$	−27.6050	−1.22841
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8.34296i	0.369795i	0.982758	+	0.184898i	$0.0591953\pi$
$509$	8.34296i	0.369795i	−0.982758	+	0.184898i	$0.940805\pi$
$510$	0	0
$511$	9.92551	0.439079
$512$	0	0
$513$	0	0
$514$	0	0
$515$	29.7529i	1.31107i
$516$	0	0
$517$	− 14.2553i	− 0.626947i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	16.9714	0.743529	0.371764	−	0.928327i	$-0.378753\pi$
$521$	16.9714	0.743529	0.371764	+	0.928327i	$0.378753\pi$
$522$	0	0
$523$	40.0554i	1.75150i	0.482764	+	0.875750i	$0.339633\pi$
$523$	40.0554i	1.75150i	−0.482764	+	0.875750i	$0.660367\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.22937	−0.358477
$528$	0	0
$529$	27.4504	1.19350
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 5.19615i	− 0.225070i
$534$	0	0
$535$	−28.0621	−1.21323
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 12.6438i	− 0.544606i
$540$	0	0
$541$	− 25.5956i	− 1.10044i	−0.835020	−	0.550220i	$-0.814544\pi$
$541$	− 25.5956i	− 1.10044i	0.835020	−	0.550220i	$-0.185456\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.41349	−0.0605472
$546$	0	0
$547$	33.7714i	1.44396i	0.691914	+	0.721980i	$0.256768\pi$
$547$	33.7714i	1.44396i	−0.691914	+	0.721980i	$0.743232\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	16.6859	0.710844
$552$	0	0
$553$	9.00000	0.382719
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 18.3364i	− 0.776937i	−0.921462	−	0.388469i	$-0.873004\pi$
$557$	− 18.3364i	− 0.776937i	0.921462	−	0.388469i	$-0.126996\pi$
$558$	0	0
$559$	14.5298	0.614546
$560$	0	0
$561$	0	0
$562$	0	0
$563$	41.3681i	1.74346i	0.489990	+	0.871728i	$0.337000\pi$
$563$	41.3681i	1.74346i	−0.489990	+	0.871728i	$0.663000\pi$
$564$	0	0
$565$	11.1405i	0.468684i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.36638	−0.0572816	−0.0286408	−	0.999590i	$-0.509118\pi$
$569$	−1.36638	−0.0572816	−0.0286408	+	0.999590i	$0.509118\pi$
$570$	0	0
$571$	17.7345i	0.742164i	0.928600	+	0.371082i	$0.121013\pi$
$571$	17.7345i	0.742164i	−0.928600	+	0.371082i	$0.878987\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	14.2057	0.592418
$576$	0	0
$577$	20.5865	0.857028	0.428514	−	0.903535i	$-0.359037\pi$
$577$	20.5865	0.857028	0.428514	+	0.903535i	$0.359037\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 4.24427i	− 0.176082i
$582$	0	0
$583$	−6.61091	−0.273796
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 14.4857i	− 0.597888i	−0.954271	−	0.298944i	$-0.903366\pi$
$587$	− 14.4857i	− 0.597888i	0.954271	−	0.298944i	$-0.0966344\pi$
$588$	0	0
$589$	− 4.31217i	− 0.177680i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4.58651	0.188345	0.0941727	−	0.995556i	$-0.469979\pi$
$593$	4.58651	0.188345	0.0941727	+	0.995556i	$0.469979\pi$
$594$	0	0
$595$	− 6.54957i	− 0.268506i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2.62288	−0.107168	−0.0535839	−	0.998563i	$-0.517064\pi$
$599$	−2.62288	−0.107168	−0.0535839	+	0.998563i	$0.517064\pi$
$600$	0	0
$601$	7.01847	0.286289	0.143145	−	0.989702i	$-0.454279\pi$
$601$	7.01847	0.286289	0.143145	+	0.989702i	$0.454279\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 11.4082i	− 0.463808i
$606$	0	0
$607$	−37.1784	−1.50902	−0.754512	−	0.656286i	$-0.772126\pi$
$607$	−37.1784	−1.50902	−0.754512	+	0.656286i	$0.772126\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	43.1664i	1.74633i
$612$	0	0
$613$	− 10.7096i	− 0.432557i	−0.976332	−	0.216278i	$-0.930608\pi$
$613$	− 10.7096i	− 0.432557i	0.976332	−	0.216278i	$-0.0693919\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	33.2201	1.33739	0.668696	−	0.743536i	$-0.266853\pi$
$617$	33.2201	1.33739	0.668696	+	0.743536i	$0.266853\pi$
$618$	0	0
$619$	− 26.3394i	− 1.05867i	−0.848413	−	0.529336i	$-0.822441\pi$
$619$	− 26.3394i	− 1.05867i	0.848413	−	0.529336i	$-0.177559\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−13.8884	−0.556427
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	17.6698i	0.704541i
$630$	0	0
$631$	−21.6327	−0.861183	−0.430592	−	0.902547i	$-0.641695\pi$
$631$	−21.6327	−0.861183	−0.430592	+	0.902547i	$0.641695\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 28.0369i	− 1.11261i
$636$	0	0
$637$	38.2866i	1.51697i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	3.76970	0.148894	0.0744471	−	0.997225i	$-0.476281\pi$
$641$	3.76970	0.148894	0.0744471	+	0.997225i	$0.476281\pi$
$642$	0	0
$643$	33.7714i	1.33181i	0.746035	+	0.665907i	$0.231955\pi$
$643$	33.7714i	1.33181i	−0.746035	+	0.665907i	$0.768045\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.7020	0.617311	0.308656	−	0.951174i	$-0.400121\pi$
$647$	15.7020	0.617311	0.308656	+	0.951174i	$0.400121\pi$
$648$	0	0
$649$	25.0369	0.982786
$650$	0	0
$651$	0	0
$652$	0	0
$653$	31.6398i	1.23816i	0.785328	+	0.619080i	$0.212494\pi$
$653$	31.6398i	1.23816i	−0.785328	+	0.619080i	$0.787506\pi$
$654$	0	0
$655$	13.3645	0.522193
$656$	0	0
$657$	0	0
$658$	0	0
$659$	36.3025i	1.41414i	0.707141	+	0.707072i	$0.249984\pi$
$659$	36.3025i	1.41414i	−0.707141	+	0.707072i	$0.750016\pi$
$660$	0	0
$661$	14.6046i	0.568052i	0.958817	+	0.284026i	$0.0916703\pi$
$661$	14.6046i	0.568052i	−0.958817	+	0.284026i	$0.908330\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.43196	0.133086
$666$	0	0
$667$	− 59.2588i	− 2.29451i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.2497	−0.395684
$672$	0	0
$673$	−3.58651	−0.138250	−0.0691249	−	0.997608i	$-0.522021\pi$
$673$	−3.58651	−0.138250	−0.0691249	+	0.997608i	$0.522021\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.44675i	− 0.0556031i	−0.999613	−	0.0278016i	$-0.991149\pi$
$677$	− 1.44675i	− 0.0556031i	0.999613	−	0.0278016i	$-0.00885065\pi$
$678$	0	0
$679$	12.3349	0.473370
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 16.9714i	− 0.649391i	−0.945819	−	0.324696i	$-0.894738\pi$
$683$	− 16.9714i	− 0.649391i	0.945819	−	0.324696i	$-0.105262\pi$
$684$	0	0
$685$	14.9219i	0.570136i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	20.0185	0.762643
$690$	0	0
$691$	− 9.75292i	− 0.371019i	−0.982643	−	0.185509i	$-0.940607\pi$
$691$	− 9.75292i	− 0.371019i	0.982643	−	0.185509i	$-0.0593935\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.90037	−0.375542
$696$	0	0
$697$	3.11761	0.118088
$698$	0	0
$699$	0	0
$700$	0	0
$701$	33.3398i	1.25923i	0.776908	+	0.629614i	$0.216787\pi$
$701$	33.3398i	1.25923i	−0.776908	+	0.629614i	$0.783213\pi$
$702$	0	0
$703$	−9.25893	−0.349207
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 15.7899i	− 0.593839i
$708$	0	0
$709$	− 51.4268i	− 1.93138i	−0.259707	−	0.965688i	$-0.583626\pi$
$709$	− 51.4268i	− 1.93138i	0.259707	−	0.965688i	$-0.416374\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−15.3143	−0.573527
$714$	0	0
$715$	− 23.1479i	− 0.865684i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−22.1178	−0.824854	−0.412427	−	0.910991i	$-0.635319\pi$
$719$	−22.1178	−0.824854	−0.412427	+	0.910991i	$0.635319\pi$
$720$	0	0
$721$	−17.0185	−0.633801
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 16.6859i	− 0.619700i
$726$	0	0
$727$	−28.5860	−1.06020	−0.530099	−	0.847936i	$-0.677845\pi$
$727$	−28.5860	−1.06020	−0.530099	+	0.847936i	$0.677845\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.71767i	0.322435i
$732$	0	0
$733$	− 45.5145i	− 1.68112i	−0.541721	−	0.840558i	$-0.682227\pi$
$733$	− 45.5145i	− 1.68112i	0.541721	−	0.840558i	$-0.317773\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−28.8538	−1.06284
$738$	0	0
$739$	19.9815i	0.735032i	0.930017	+	0.367516i	$0.119792\pi$
$739$	19.9815i	0.735032i	−0.930017	+	0.367516i	$0.880208\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	37.6772	1.38224	0.691121	−	0.722739i	$-0.257117\pi$
$743$	37.6772	1.38224	0.691121	+	0.722739i	$0.257117\pi$
$744$	0	0
$745$	2.45043	0.0897768
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 16.0513i	− 0.586503i
$750$	0	0
$751$	−12.5164	−0.456730	−0.228365	−	0.973576i	$-0.573338\pi$
$751$	−12.5164	−0.456730	−0.228365	+	0.973576i	$0.573338\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.69751i	0.207354i
$756$	0	0
$757$	12.5872i	0.457491i	0.973486	+	0.228745i	$0.0734623\pi$
$757$	12.5872i	0.457491i	−0.973486	+	0.228745i	$0.926538\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−11.8824	−0.430736	−0.215368	−	0.976533i	$-0.569095\pi$
$761$	−11.8824	−0.430736	−0.215368	+	0.976533i	$0.569095\pi$
$762$	0	0
$763$	− 0.808506i	− 0.0292699i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−75.8143	−2.73749
$768$	0	0
$769$	−45.0739	−1.62541	−0.812703	−	0.582678i	$-0.802005\pi$
$769$	−45.0739	−1.62541	−0.812703	+	0.582678i	$0.802005\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	34.0384i	1.22428i	0.790751	+	0.612138i	$0.209690\pi$
$773$	34.0384i	1.22428i	−0.790751	+	0.612138i	$0.790310\pi$
$774$	0	0
$775$	−4.31217	−0.154898
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.63362i	0.0585305i
$780$	0	0
$781$	28.4434i	1.01778i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	11.5680	0.412881
$786$	0	0
$787$	− 4.24708i	− 0.151392i	−0.997131	−	0.0756960i	$-0.975882\pi$
$787$	− 4.24708i	− 0.151392i	0.997131	−	0.0756960i	$-0.0241179\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.37228	−0.226572
$792$	0	0
$793$	31.0369	1.10215
$794$	0	0
$795$	0	0
$796$	0	0
$797$	43.0480i	1.52484i	0.647084	+	0.762419i	$0.275988\pi$
$797$	43.0480i	1.52484i	−0.647084	+	0.762419i	$0.724012\pi$
$798$	0	0
$799$	−25.8992	−0.916247
$800$	0	0
$801$	0	0
$802$	0	0
$803$	21.0471i	0.742736i
$804$	0	0
$805$	− 12.1883i	− 0.429583i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	25.2488	0.887699	0.443850	−	0.896101i	$-0.353612\pi$
$809$	25.2488	0.887699	0.443850	+	0.896101i	$0.353612\pi$
$810$	0	0
$811$	36.9378i	1.29706i	0.761188	+	0.648531i	$0.224616\pi$
$811$	36.9378i	1.29706i	−0.761188	+	0.648531i	$0.775384\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−12.8725	−0.450905
$816$	0	0
$817$	−4.56804	−0.159816
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.4416i	0.434217i	0.976148	+	0.217108i	$0.0696625\pi$
$821$	12.4416i	0.434217i	−0.976148	+	0.217108i	$0.930338\pi$
$822$	0	0
$823$	−1.52150	−0.0530361	−0.0265180	−	0.999648i	$-0.508442\pi$
$823$	−1.52150	−0.0530361	−0.0265180	+	0.999648i	$0.508442\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10.0369i	0.349019i	0.984656	+	0.174509i	$0.0558339\pi$
$827$	10.0369i	0.349019i	−0.984656	+	0.174509i	$0.944166\pi$
$828$	0	0
$829$	43.7824i	1.52063i	0.649556	+	0.760314i	$0.274955\pi$
$829$	43.7824i	1.52063i	−0.649556	+	0.760314i	$0.725045\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−22.9714	−0.795911
$834$	0	0
$835$	− 17.1479i	− 0.593429i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−42.1251	−1.45432	−0.727161	−	0.686467i	$-0.759161\pi$
$839$	−42.1251	−1.45432	−0.727161	+	0.686467i	$0.759161\pi$
$840$	0	0
$841$	−40.6050	−1.40017
$842$	0	0
$843$	0	0
$844$	0	0
$845$	47.5775i	1.63672i
$846$	0	0
$847$	6.52540	0.224215
$848$	0	0
$849$	0	0
$850$	0	0
$851$	32.8824i	1.12719i
$852$	0	0
$853$	26.2981i	0.900428i	0.892921	+	0.450214i	$0.148652\pi$
$853$	26.2981i	0.900428i	−0.892921	+	0.450214i	$0.851348\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.4874	−0.631517	−0.315758	−	0.948840i	$-0.602259\pi$
$857$	−18.4874	−0.631517	−0.315758	+	0.948840i	$0.602259\pi$
$858$	0	0
$859$	46.0673i	1.57180i	0.618357	+	0.785898i	$0.287799\pi$
$859$	46.0673i	1.57180i	−0.618357	+	0.785898i	$0.712201\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−46.0812	−1.56862	−0.784311	−	0.620368i	$-0.786983\pi$
$863$	−46.0812	−1.56862	−0.784311	+	0.620368i	$0.786983\pi$
$864$	0	0
$865$	−21.0000	−0.714021
$866$	0	0
$867$	0	0
$868$	0	0
$869$	19.0846i	0.647399i
$870$	0	0
$871$	87.3719	2.96049
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 12.0119i	− 0.406075i
$876$	0	0
$877$	− 32.9547i	− 1.11280i	−0.830915	−	0.556400i	$-0.812182\pi$
$877$	− 32.9547i	− 1.11280i	0.830915	−	0.556400i	$-0.187818\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	21.9815	0.740577	0.370288	−	0.928917i	$-0.379259\pi$
$881$	21.9815	0.740577	0.370288	+	0.928917i	$0.379259\pi$
$882$	0	0
$883$	− 10.0000i	− 0.336527i	−0.985742	−	0.168263i	$-0.946184\pi$
$883$	− 10.0000i	− 0.336527i	0.985742	−	0.168263i	$-0.0538159\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.35344	0.112598	0.0562988	−	0.998414i	$-0.482070\pi$
$887$	3.35344	0.112598	0.0562988	+	0.998414i	$0.482070\pi$
$888$	0	0
$889$	16.0369	0.537862
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 13.5711i	− 0.454140i
$894$	0	0
$895$	10.3923	0.347376
$896$	0	0
$897$	0	0
$898$	0	0
$899$	17.9881i	0.599938i
$900$	0	0
$901$	12.0108i	0.400137i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8.01847	−0.266543
$906$	0	0
$907$	10.2471i	0.340249i	0.985423	+	0.170124i	$0.0544169\pi$
$907$	10.2471i	0.340249i	−0.985423	+	0.170124i	$0.945583\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−25.4072	−0.841779	−0.420890	−	0.907112i	$-0.638282\pi$
$911$	−25.4072	−0.841779	−0.420890	+	0.907112i	$0.638282\pi$
$912$	0	0
$913$	9.00000	0.297857
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.64439i	0.252440i
$918$	0	0
$919$	44.1134	1.45517	0.727584	−	0.686019i	$-0.240643\pi$
$919$	44.1134	1.45517	0.727584	+	0.686019i	$0.240643\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 86.1293i	− 2.83498i
$924$	0	0
$925$	9.25893i	0.304432i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−35.6235	−1.16877	−0.584384	−	0.811477i	$-0.698664\pi$
$929$	−35.6235	−1.16877	−0.584384	+	0.811477i	$0.698664\pi$
$930$	0	0
$931$	− 12.0369i	− 0.394495i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	13.8884	0.454199
$936$	0	0
$937$	−15.4135	−0.503537	−0.251768	−	0.967788i	$-0.581012\pi$
$937$	−15.4135	−0.503537	−0.251768	+	0.967788i	$0.581012\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 8.72424i	− 0.284402i	−0.989838	−	0.142201i	$-0.954582\pi$
$941$	− 8.72424i	− 0.284402i	0.989838	−	0.142201i	$-0.0454180\pi$
$942$	0	0
$943$	5.80168	0.188929
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15.8992i	0.516654i	0.966058	+	0.258327i	$0.0831711\pi$
$947$	15.8992i	0.516654i	−0.966058	+	0.258327i	$0.916829\pi$
$948$	0	0
$949$	− 63.7327i	− 2.06885i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	25.6890	0.832149	0.416075	−	0.909330i	$-0.363405\pi$
$953$	25.6890	0.832149	0.416075	+	0.909330i	$0.363405\pi$
$954$	0	0
$955$	− 29.7529i	− 0.962782i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.53521	−0.275616
$960$	0	0
$961$	−26.3513	−0.850042
$962$	0	0
$963$	0	0
$964$	0	0
$965$	20.0684i	0.646026i
$966$	0	0
$967$	−30.7170	−0.987791	−0.493896	−	0.869521i	$-0.664428\pi$
$967$	−30.7170	−0.987791	−0.493896	+	0.869521i	$0.664428\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 40.0924i	− 1.28662i	−0.765604	−	0.643312i	$-0.777560\pi$
$971$	− 40.0924i	− 1.28662i	0.765604	−	0.643312i	$-0.222440\pi$
$972$	0	0
$973$	− 5.66294i	− 0.181546i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	48.8722	1.56356	0.781780	−	0.623554i	$-0.214312\pi$
$977$	48.8722	1.56356	0.781780	+	0.623554i	$0.214312\pi$
$978$	0	0
$979$	− 29.4504i	− 0.941240i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	11.2015	0.357274	0.178637	−	0.983915i	$-0.442831\pi$
$983$	11.2015	0.357274	0.178637	+	0.983915i	$0.442831\pi$
$984$	0	0
$985$	−30.6050	−0.975156
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16.2230i	0.515863i
$990$	0	0
$991$	−15.7020	−0.498792	−0.249396	−	0.968402i	$-0.580232\pi$
$991$	−15.7020	−0.498792	−0.249396	+	0.968402i	$0.580232\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	28.5865i	0.906253i
$996$	0	0
$997$	24.3623i	0.771562i	0.922590	+	0.385781i	$0.126068\pi$
$997$	24.3623i	0.771562i	−0.922590	+	0.385781i	$0.873932\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5184.2.d.q.2593.4		12
3.2	odd	2		5184.2.d.r.2593.10		12
4.3	odd	2	inner	5184.2.d.q.2593.3		12
8.3	odd	2	inner	5184.2.d.q.2593.9		12
8.5	even	2	inner	5184.2.d.q.2593.10		12
9.2	odd	6		1728.2.r.e.1441.3		12
9.4	even	3		576.2.r.e.97.5	yes	12
9.5	odd	6		1728.2.r.f.289.3		12
9.7	even	3		576.2.r.f.481.2	yes	12
12.11	even	2		5184.2.d.r.2593.9		12
24.5	odd	2		5184.2.d.r.2593.4		12
24.11	even	2		5184.2.d.r.2593.3		12
36.7	odd	6		576.2.r.f.481.5	yes	12
36.11	even	6		1728.2.r.e.1441.4		12
36.23	even	6		1728.2.r.f.289.4		12
36.31	odd	6		576.2.r.e.97.2	✓	12
72.5	odd	6		1728.2.r.e.289.3		12
72.11	even	6		1728.2.r.f.1441.4		12
72.13	even	6		576.2.r.f.97.2	yes	12
72.29	odd	6		1728.2.r.f.1441.3		12
72.43	odd	6		576.2.r.e.481.2	yes	12
72.59	even	6		1728.2.r.e.289.4		12
72.61	even	6		576.2.r.e.481.5	yes	12
72.67	odd	6		576.2.r.f.97.5	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.r.e.97.2	✓	12	36.31	odd	6
576.2.r.e.97.5	yes	12	9.4	even	3
576.2.r.e.481.2	yes	12	72.43	odd	6
576.2.r.e.481.5	yes	12	72.61	even	6
576.2.r.f.97.2	yes	12	72.13	even	6
576.2.r.f.97.5	yes	12	72.67	odd	6
576.2.r.f.481.2	yes	12	9.7	even	3
576.2.r.f.481.5	yes	12	36.7	odd	6
1728.2.r.e.289.3		12	72.5	odd	6
1728.2.r.e.289.4		12	72.59	even	6
1728.2.r.e.1441.3		12	9.2	odd	6
1728.2.r.e.1441.4		12	36.11	even	6
1728.2.r.f.289.3		12	9.5	odd	6
1728.2.r.f.289.4		12	36.23	even	6
1728.2.r.f.1441.3		12	72.29	odd	6
1728.2.r.f.1441.4		12	72.11	even	6
5184.2.d.q.2593.3		12	4.3	odd	2	inner
5184.2.d.q.2593.4		12	1.1	even	1	trivial
5184.2.d.q.2593.9		12	8.3	odd	2	inner
5184.2.d.q.2593.10		12	8.5	even	2	inner
5184.2.d.r.2593.3		12	24.11	even	2
5184.2.d.r.2593.4		12	24.5	odd	2
5184.2.d.r.2593.9		12	12.11	even	2
5184.2.d.r.2593.10		12	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.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{5} +0.990721 q^{7} +O(q^{10})\)	\(q-1.73205i q^{5} +0.990721 q^{7} +2.10083i q^{11} -6.36152i q^{13} +3.81681 q^{17} +2.00000i q^{19} +7.10285 q^{23} +2.00000 q^{25} -8.34296i q^{29} -2.15609 q^{31} -1.71598i q^{35} +4.62947i q^{37} +0.816810 q^{41} +2.28402i q^{43} -6.78555 q^{47} -6.01847 q^{49} +3.14681i q^{53} +3.63875 q^{55} -11.9176i q^{59} +4.87886i q^{61} -11.0185 q^{65} +13.7345i q^{67} +13.5391 q^{71} +10.0185 q^{73} +2.08134i q^{77} +9.08429 q^{79} -4.28402i q^{83} -6.61091i q^{85} -14.0185 q^{89} -6.30249i q^{91} +3.46410 q^{95} +12.4504 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 24 q^{25} - 36 q^{41} + 48 q^{49} - 12 q^{65} - 48 q^{89} + 12 q^{97}+O(q^{100}) \) 12 * q + 24 * q^25 - 36 * q^41 + 48 * q^49 - 12 * q^65 - 48 * q^89 + 12 * q^97

Introduction

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