LMFDB - Embedded newform 5184.2.d.r.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			$-0.403293 - 1.50511i$ of defining polynomial
Character	$\chi$	$=$	5184.2593
Dual form			5184.2.d.r.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	$0.343927\pi$
$13$	6.36152i	1.76437i	−0.470905	+	0.882184i	$0.656073\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.81681	−0.925712	−0.462856	−	0.886433i	$-0.653175\pi$
$17$	−3.81681	−0.925712	−0.462856	+	0.886433i	$0.653175\pi$
$18$	0	0
$19$	− 2.00000i	− 0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$	− 2.00000i	− 0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.10285	−1.48105	−0.740523	−	0.672031i	$-0.765422\pi$
$23$	−7.10285	−1.48105	−0.740523	+	0.672031i	$0.765422\pi$
$24$	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	$-0.875738\pi$
$37$	− 4.62947i	− 0.761080i	0.924765	−	0.380540i	$-0.124262\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.816810	−0.127564	−0.0637822	−	0.997964i	$-0.520316\pi$
$41$	−0.816810	−0.127564	−0.0637822	+	0.997964i	$0.520316\pi$
$42$	0	0
$43$	− 2.28402i	− 0.348310i	−0.984718	−	0.174155i	$-0.944281\pi$
$43$	− 2.28402i	− 0.348310i	0.984718	−	0.174155i	$-0.0557193\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.78555	0.989775	0.494887	−	0.868957i	$-0.335209\pi$
$47$	6.78555	0.989775	0.494887	+	0.868957i	$0.335209\pi$
$48$	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	$-0.898888\pi$
$61$	− 4.87886i	− 0.624674i	0.949971	−	0.312337i	$-0.101112\pi$
$62$	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	$-0.683161\pi$
$67$	− 13.7345i	− 1.67793i	0.544185	−	0.838965i	$-0.316839\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.5391	−1.60680	−0.803399	−	0.595442i	$-0.796977\pi$
$71$	−13.5391	−1.60680	−0.803399	+	0.595442i	$0.796977\pi$
$72$	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	$-0.233413\pi$
$89$	14.0185	1.48595	0.742977	+	0.669316i	$0.233413\pi$
$90$	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	$0.0124437\pi$
$109$	0.816078i	0.0781661i	−0.999236	+	0.0390830i	$0.987556\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.43196	0.605068	0.302534	−	0.953139i	$-0.402167\pi$
$113$	6.43196	0.605068	0.302534	+	0.953139i	$0.402167\pi$
$114$	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	$-0.380035\pi$
$137$	8.61515	0.736042	0.368021	+	0.929818i	$0.380035\pi$
$138$	0	0
$139$	5.71598i	0.484823i	0.970174	+	0.242412i	$0.0779384\pi$
$139$	5.71598i	0.484823i	−0.970174	+	0.242412i	$0.922062\pi$
$140$	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	$-0.914128\pi$
$157$	− 6.67881i	− 0.533027i	0.963831	−	0.266514i	$-0.0858717\pi$
$158$	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	$0.0940073\pi$
$163$	7.43196i	0.582116i	−0.956705	+	0.291058i	$0.905993\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.90037	−0.766113	−0.383057	−	0.923725i	$-0.625129\pi$
$167$	−9.90037	−0.766113	−0.383057	+	0.923725i	$0.625129\pi$
$168$	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	$0.0550400\pi$
$181$	4.62947i	0.344106i	−0.985088	+	0.172053i	$0.944960\pi$
$182$	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	$-0.713465\pi$
$191$	−17.1779	−1.24295	−0.621473	+	0.783436i	$0.713465\pi$
$192$	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	$0.132351\pi$
$211$	11.7345i	0.807833i	−0.914796	+	0.403916i	$0.867649\pi$
$212$	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	$-0.976516\pi$
$229$	− 2.23083i	− 0.147418i	0.997280	−	0.0737089i	$-0.0234836\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.26724	0.607117	0.303559	−	0.952813i	$-0.401825\pi$
$233$	9.26724	0.607117	0.303559	+	0.952813i	$0.401825\pi$
$234$	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	$-0.737097\pi$
$239$	−20.9593	−1.35574	−0.677871	+	0.735181i	$0.737097\pi$
$240$	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	$-0.184325\pi$
$257$	26.8353	1.67394	0.836969	+	0.547250i	$0.184325\pi$
$258$	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	$-0.711194\pi$
$263$	−19.9754	−1.23173	−0.615867	+	0.787850i	$0.711194\pi$
$264$	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	$0.0355851\pi$
$277$	3.71349i	0.223122i	−0.993758	+	0.111561i	$0.964415\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−27.0554	−1.61399	−0.806995	−	0.590558i	$-0.798908\pi$
$281$	−27.0554	−1.61399	−0.806995	+	0.590558i	$0.798908\pi$
$282$	0	0
$283$	− 1.14794i	− 0.0682379i	−0.999418	−	0.0341190i	$-0.989137\pi$
$283$	− 1.14794i	− 0.0682379i	0.999418	−	0.0341190i	$-0.0108625\pi$
$284$	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	$-0.931976\pi$
$307$	− 7.43196i	− 0.424164i	0.977252	−	0.212082i	$-0.0680244\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.93236	0.563212	0.281606	−	0.959530i	$-0.409133\pi$
$311$	9.93236	0.563212	0.281606	+	0.959530i	$0.409133\pi$
$312$	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	$0.115081\pi$
$331$	12.8705i	0.707428i	−0.935354	+	0.353714i	$0.884919\pi$
$332$	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	$0.0863079\pi$
$349$	10.0071i	0.535668i	−0.963465	+	0.267834i	$0.913692\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	21.3849	1.13820	0.569100	−	0.822268i	$-0.307292\pi$
$353$	21.3849	1.13820	0.569100	+	0.822268i	$0.307292\pi$
$354$	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	$-0.282082\pi$
$359$	23.9634	1.26474	0.632370	+	0.774666i	$0.282082\pi$
$360$	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	$-0.780975\pi$
$373$	− 24.5301i	− 1.27012i	0.772463	−	0.635060i	$-0.219025\pi$
$374$	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	$0.0901672\pi$
$379$	10.8824i	0.558991i	−0.960147	+	0.279495i	$0.909833\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.32145	0.169718	0.0848591	−	0.996393i	$-0.472956\pi$
$383$	3.32145	0.169718	0.0848591	+	0.996393i	$0.472956\pi$
$384$	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	$0.167070\pi$
$397$	19.9685i	1.00219i	−0.865392	+	0.501096i	$0.832930\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.652092	0.0325639	0.0162819	−	0.999867i	$-0.494817\pi$
$401$	0.652092	0.0325639	0.0162819	+	0.999867i	$0.494817\pi$
$402$	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	$-0.969849\pi$
$421$	− 3.88129i	− 0.189163i	0.995517	−	0.0945813i	$-0.0301512\pi$
$422$	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	$-0.529276\pi$
$431$	−3.81339	−0.183684	−0.0918422	+	0.995774i	$0.529276\pi$
$432$	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	$-0.670692\pi$
$449$	−21.6521	−1.02183	−0.510913	+	0.859633i	$0.670692\pi$
$450$	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	$-0.661838\pi$
$479$	−21.3085	−0.973612	−0.486806	+	0.873510i	$0.661838\pi$
$480$	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	$0.0308340\pi$
$499$	4.32096i	0.193433i	−0.995312	+	0.0967164i	$0.969166\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−7.24550	−0.323061	−0.161530	−	0.986868i	$-0.551643\pi$
$503$	−7.24550	−0.323061	−0.161530	+	0.986868i	$0.551643\pi$
$504$	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	$-0.621247\pi$
$521$	−16.9714	−0.743529	−0.371764	+	0.928327i	$0.621247\pi$
$522$	0	0
$523$	− 40.0554i	− 1.75150i	−0.482764	−	0.875750i	$-0.660367\pi$
$523$	− 40.0554i	− 1.75150i	0.482764	−	0.875750i	$-0.339633\pi$
$524$	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	$0.185456\pi$
$541$	25.5956i	1.10044i	−0.835020	+	0.550220i	$0.814544\pi$
$542$	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	$-0.743232\pi$
$547$	− 33.7714i	− 1.44396i	0.691914	−	0.721980i	$-0.256768\pi$
$548$	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	$-0.490882\pi$
$569$	1.36638	0.0572816	0.0286408	+	0.999590i	$0.490882\pi$
$570$	0	0
$571$	− 17.7345i	− 0.742164i	−0.928600	−	0.371082i	$-0.878987\pi$
$571$	− 17.7345i	− 0.742164i	0.928600	−	0.371082i	$-0.121013\pi$
$572$	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	$-0.530021\pi$
$593$	−4.58651	−0.188345	−0.0941727	+	0.995556i	$0.530021\pi$
$594$	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	$-0.482936\pi$
$599$	2.62288	0.107168	0.0535839	+	0.998563i	$0.482936\pi$
$600$	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	$0.0693919\pi$
$613$	10.7096i	0.432557i	−0.976332	+	0.216278i	$0.930608\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−33.2201	−1.33739	−0.668696	−	0.743536i	$-0.733147\pi$
$617$	−33.2201	−1.33739	−0.668696	+	0.743536i	$0.733147\pi$
$618$	0	0
$619$	26.3394i	1.05867i	0.848413	+	0.529336i	$0.177559\pi$
$619$	26.3394i	1.05867i	−0.848413	+	0.529336i	$0.822441\pi$
$620$	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	$-0.523719\pi$
$641$	−3.76970	−0.148894	−0.0744471	+	0.997225i	$0.523719\pi$
$642$	0	0
$643$	− 33.7714i	− 1.33181i	−0.746035	−	0.665907i	$-0.768045\pi$
$643$	− 33.7714i	− 1.33181i	0.746035	−	0.665907i	$-0.231955\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.7020	−0.617311	−0.308656	−	0.951174i	$-0.599879\pi$
$647$	−15.7020	−0.617311	−0.308656	+	0.951174i	$0.599879\pi$
$648$	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	$-0.908330\pi$
$661$	− 14.6046i	− 0.568052i	0.958817	−	0.284026i	$-0.0916703\pi$
$662$	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	$0.0593935\pi$
$691$	9.75292i	0.371019i	−0.982643	+	0.185509i	$0.940607\pi$
$692$	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	$0.416374\pi$
$709$	51.4268i	1.93138i	−0.259707	+	0.965688i	$0.583626\pi$
$710$	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	$-0.364681\pi$
$719$	22.1178	0.824854	0.412427	+	0.910991i	$0.364681\pi$
$720$	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	$0.317773\pi$
$733$	45.5145i	1.68112i	−0.541721	+	0.840558i	$0.682227\pi$
$734$	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	$-0.880208\pi$
$739$	− 19.9815i	− 0.735032i	0.930017	−	0.367516i	$-0.119792\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−37.6772	−1.38224	−0.691121	−	0.722739i	$-0.742883\pi$
$743$	−37.6772	−1.38224	−0.691121	+	0.722739i	$0.742883\pi$
$744$	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	$-0.926538\pi$
$757$	− 12.5872i	− 0.457491i	0.973486	−	0.228745i	$-0.0734623\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	11.8824	0.430736	0.215368	−	0.976533i	$-0.430905\pi$
$761$	11.8824	0.430736	0.215368	+	0.976533i	$0.430905\pi$
$762$	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	$0.0241179\pi$
$787$	4.24708i	0.151392i	−0.997131	+	0.0756960i	$0.975882\pi$
$788$	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	$-0.646388\pi$
$809$	−25.2488	−0.887699	−0.443850	+	0.896101i	$0.646388\pi$
$810$	0	0
$811$	− 36.9378i	− 1.29706i	−0.761188	−	0.648531i	$-0.775384\pi$
$811$	− 36.9378i	− 1.29706i	0.761188	−	0.648531i	$-0.224616\pi$
$812$	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	$-0.725045\pi$
$829$	− 43.7824i	− 1.52063i	0.649556	−	0.760314i	$-0.274955\pi$
$830$	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	$-0.240839\pi$
$839$	42.1251	1.45432	0.727161	+	0.686467i	$0.240839\pi$
$840$	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	$-0.851348\pi$
$853$	− 26.2981i	− 0.900428i	0.892921	−	0.450214i	$-0.148652\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18.4874	0.631517	0.315758	−	0.948840i	$-0.397741\pi$
$857$	18.4874	0.631517	0.315758	+	0.948840i	$0.397741\pi$
$858$	0	0
$859$	− 46.0673i	− 1.57180i	−0.618357	−	0.785898i	$-0.712201\pi$
$859$	− 46.0673i	− 1.57180i	0.618357	−	0.785898i	$-0.287799\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	46.0812	1.56862	0.784311	−	0.620368i	$-0.213017\pi$
$863$	46.0812	1.56862	0.784311	+	0.620368i	$0.213017\pi$
$864$	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	$0.187818\pi$
$877$	32.9547i	1.11280i	−0.830915	+	0.556400i	$0.812182\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−21.9815	−0.740577	−0.370288	−	0.928917i	$-0.620741\pi$
$881$	−21.9815	−0.740577	−0.370288	+	0.928917i	$0.620741\pi$
$882$	0	0
$883$	10.0000i	0.336527i	0.985742	+	0.168263i	$0.0538159\pi$
$883$	10.0000i	0.336527i	−0.985742	+	0.168263i	$0.946184\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3.35344	−0.112598	−0.0562988	−	0.998414i	$-0.517930\pi$
$887$	−3.35344	−0.112598	−0.0562988	+	0.998414i	$0.517930\pi$
$888$	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	$-0.945583\pi$
$907$	− 10.2471i	− 0.340249i	0.985423	−	0.170124i	$-0.0544169\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	25.4072	0.841779	0.420890	−	0.907112i	$-0.361718\pi$
$911$	25.4072	0.841779	0.420890	+	0.907112i	$0.361718\pi$
$912$	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	$-0.301336\pi$
$929$	35.6235	1.16877	0.584384	+	0.811477i	$0.301336\pi$
$930$	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	$-0.636595\pi$
$953$	−25.6890	−0.832149	−0.416075	+	0.909330i	$0.636595\pi$
$954$	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	$-0.785688\pi$
$977$	−48.8722	−1.56356	−0.781780	+	0.623554i	$0.785688\pi$
$978$	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	$-0.557169\pi$
$983$	−11.2015	−0.357274	−0.178637	+	0.983915i	$0.557169\pi$
$984$	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	$-0.873932\pi$
$997$	− 24.3623i	− 0.771562i	0.922590	−	0.385781i	$-0.126068\pi$
$998$	0	0
$999$	0	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

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

L-functions

Modular forms

Varieties

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Representations

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Database

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