LMFDB - Embedded newform 5184.2.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("5184.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$41.3944484078$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 288)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	5184.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +1.73205 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +1.73205 q^{7} -1.73205 q^{11} +3.00000 q^{13} -4.00000 q^{17} +6.92820 q^{19} -8.66025 q^{23} -4.00000 q^{25} +1.00000 q^{29} -5.19615 q^{31} -1.73205 q^{35} +8.00000 q^{37} -5.00000 q^{41} -8.66025 q^{43} +12.1244 q^{47} -4.00000 q^{49} -8.00000 q^{53} +1.73205 q^{55} -1.73205 q^{59} +7.00000 q^{61} -3.00000 q^{65} -8.66025 q^{67} +3.46410 q^{71} -12.0000 q^{73} -3.00000 q^{77} +5.19615 q^{79} +8.66025 q^{83} +4.00000 q^{85} +4.00000 q^{89} +5.19615 q^{91} -6.92820 q^{95} -3.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^5	$ 2 q - 2 q^{5} + 6 q^{13} - 8 q^{17} - 8 q^{25} + 2 q^{29} + 16 q^{37} - 10 q^{41} - 8 q^{49} - 16 q^{53} + 14 q^{61} - 6 q^{65} - 24 q^{73} - 6 q^{77} + 8 q^{85} + 8 q^{89} - 6 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 6 * q^13 - 8 * q^17 - 8 * q^25 + 2 * q^29 + 16 * q^37 - 10 * q^41 - 8 * q^49 - 16 * q^53 + 14 * q^61 - 6 * q^65 - 24 * q^73 - 6 * q^77 + 8 * q^85 + 8 * q^89 - 6 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	1.73205	0.654654	0.327327	−	0.944911i	$-0.393852\pi$
$7$	1.73205	0.654654	0.327327	+	0.944911i	$0.393852\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.73205	−0.522233	−0.261116	−	0.965307i	$-0.584091\pi$
$11$	−1.73205	−0.522233	−0.261116	+	0.965307i	$0.584091\pi$
$12$	0	0
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	6.92820	1.58944	0.794719	−	0.606977i	$-0.207618\pi$
$19$	6.92820	1.58944	0.794719	+	0.606977i	$0.207618\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.66025	−1.80579	−0.902894	−	0.429863i	$-0.858562\pi$
$23$	−8.66025	−1.80579	−0.902894	+	0.429863i	$0.858562\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	−5.19615	−0.933257	−0.466628	−	0.884454i	$-0.654531\pi$
$31$	−5.19615	−0.933257	−0.466628	+	0.884454i	$0.654531\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.73205	−0.292770
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	0	0
$43$	−8.66025	−1.32068	−0.660338	−	0.750968i	$-0.729587\pi$
$43$	−8.66025	−1.32068	−0.660338	+	0.750968i	$0.729587\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.1244	1.76852	0.884260	−	0.466996i	$-0.154664\pi$
$47$	12.1244	1.76852	0.884260	+	0.466996i	$0.154664\pi$
$48$	0	0
$49$	−4.00000	−0.571429
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.00000	−1.09888	−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000	−1.09888	−0.549442	+	0.835532i	$0.685160\pi$
$54$	0	0
$55$	1.73205	0.233550
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.73205	−0.225494	−0.112747	−	0.993624i	$-0.535965\pi$
$59$	−1.73205	−0.225494	−0.112747	+	0.993624i	$0.535965\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.00000	−0.372104
$66$	0	0
$67$	−8.66025	−1.05802	−0.529009	−	0.848616i	$-0.677436\pi$
$67$	−8.66025	−1.05802	−0.529009	+	0.848616i	$0.677436\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.46410	0.411113	0.205557	−	0.978645i	$-0.434100\pi$
$71$	3.46410	0.411113	0.205557	+	0.978645i	$0.434100\pi$
$72$	0	0
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.00000	−0.341882
$78$	0	0
$79$	5.19615	0.584613	0.292306	−	0.956325i	$-0.405577\pi$
$79$	5.19615	0.584613	0.292306	+	0.956325i	$0.405577\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.66025	0.950586	0.475293	−	0.879827i	$-0.342342\pi$
$83$	8.66025	0.950586	0.475293	+	0.879827i	$0.342342\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	5.19615	0.544705
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.92820	−0.710819
$96$	0	0
$97$	−3.00000	−0.304604	−0.152302	−	0.988334i	$-0.548669\pi$
$97$	−3.00000	−0.304604	−0.152302	+	0.988334i	$0.548669\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.0000	−1.29355	−0.646774	−	0.762682i	$-0.723882\pi$
$101$	−13.0000	−1.29355	−0.646774	+	0.762682i	$0.723882\pi$
$102$	0	0
$103$	−1.73205	−0.170664	−0.0853320	−	0.996353i	$-0.527195\pi$
$103$	−1.73205	−0.170664	−0.0853320	+	0.996353i	$0.527195\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.92820	0.669775	0.334887	−	0.942258i	$-0.391302\pi$
$107$	6.92820	0.669775	0.334887	+	0.942258i	$0.391302\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	0	0
$115$	8.66025	0.807573
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.92820	−0.635107
$120$	0	0
$121$	−8.00000	−0.727273
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	13.8564	1.22956	0.614779	−	0.788700i	$-0.289245\pi$
$127$	13.8564	1.22956	0.614779	+	0.788700i	$0.289245\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.5885	−1.36197	−0.680985	−	0.732297i	$-0.738448\pi$
$131$	−15.5885	−1.36197	−0.680985	+	0.732297i	$0.738448\pi$
$132$	0	0
$133$	12.0000	1.04053
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.0000	−1.11066	−0.555332	−	0.831628i	$-0.687409\pi$
$137$	−13.0000	−1.11066	−0.555332	+	0.831628i	$0.687409\pi$
$138$	0	0
$139$	1.73205	0.146911	0.0734553	−	0.997299i	$-0.476597\pi$
$139$	1.73205	0.146911	0.0734553	+	0.997299i	$0.476597\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.19615	−0.434524
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−11.0000	−0.901155	−0.450578	−	0.892737i	$-0.648782\pi$
$149$	−11.0000	−0.901155	−0.450578	+	0.892737i	$0.648782\pi$
$150$	0	0
$151$	−22.5167	−1.83238	−0.916190	−	0.400744i	$-0.868752\pi$
$151$	−22.5167	−1.83238	−0.916190	+	0.400744i	$0.868752\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.19615	0.417365
$156$	0	0
$157$	17.0000	1.35675	0.678374	−	0.734717i	$-0.262685\pi$
$157$	17.0000	1.35675	0.678374	+	0.734717i	$0.262685\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−15.0000	−1.18217
$162$	0	0
$163$	−6.92820	−0.542659	−0.271329	−	0.962487i	$-0.587463\pi$
$163$	−6.92820	−0.542659	−0.271329	+	0.962487i	$0.587463\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.5885	1.20627	0.603136	−	0.797639i	$-0.293918\pi$
$167$	15.5885	1.20627	0.603136	+	0.797639i	$0.293918\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−13.0000	−0.988372	−0.494186	−	0.869356i	$-0.664534\pi$
$173$	−13.0000	−0.988372	−0.494186	+	0.869356i	$0.664534\pi$
$174$	0	0
$175$	−6.92820	−0.523723
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−6.92820	−0.517838	−0.258919	−	0.965899i	$-0.583366\pi$
$179$	−6.92820	−0.517838	−0.258919	+	0.965899i	$0.583366\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−8.00000	−0.588172
$186$	0	0
$187$	6.92820	0.506640
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.73205	−0.125327	−0.0626634	−	0.998035i	$-0.519959\pi$
$191$	−1.73205	−0.125327	−0.0626634	+	0.998035i	$0.519959\pi$
$192$	0	0
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	0	0
$199$	3.46410	0.245564	0.122782	−	0.992434i	$-0.460818\pi$
$199$	3.46410	0.245564	0.122782	+	0.992434i	$0.460818\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.73205	0.121566
$204$	0	0
$205$	5.00000	0.349215
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−8.66025	−0.596196	−0.298098	−	0.954535i	$-0.596352\pi$
$211$	−8.66025	−0.596196	−0.298098	+	0.954535i	$0.596352\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.66025	0.590624
$216$	0	0
$217$	−9.00000	−0.610960
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	1.73205	0.115987	0.0579934	−	0.998317i	$-0.481530\pi$
$223$	1.73205	0.115987	0.0579934	+	0.998317i	$0.481530\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.1244	0.804722	0.402361	−	0.915481i	$-0.368190\pi$
$227$	12.1244	0.804722	0.402361	+	0.915481i	$0.368190\pi$
$228$	0	0
$229$	−15.0000	−0.991228	−0.495614	−	0.868543i	$-0.665057\pi$
$229$	−15.0000	−0.991228	−0.495614	+	0.868543i	$0.665057\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	−12.1244	−0.790906
$236$	0	0
$237$	0	0
$238$	0	0
$239$	19.0526	1.23241	0.616204	−	0.787587i	$-0.288670\pi$
$239$	19.0526	1.23241	0.616204	+	0.787587i	$0.288670\pi$
$240$	0	0
$241$	−9.00000	−0.579741	−0.289870	−	0.957066i	$-0.593612\pi$
$241$	−9.00000	−0.579741	−0.289870	+	0.957066i	$0.593612\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.00000	0.255551
$246$	0	0
$247$	20.7846	1.32249
$248$	0	0
$249$	0	0
$250$	0	0
$251$	20.7846	1.31191	0.655956	−	0.754799i	$-0.272265\pi$
$251$	20.7846	1.31191	0.655956	+	0.754799i	$0.272265\pi$
$252$	0	0
$253$	15.0000	0.943042
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−31.0000	−1.93373	−0.966863	−	0.255294i	$-0.917828\pi$
$257$	−31.0000	−1.93373	−0.966863	+	0.255294i	$0.917828\pi$
$258$	0	0
$259$	13.8564	0.860995
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−19.0526	−1.17483	−0.587416	−	0.809285i	$-0.699855\pi$
$263$	−19.0526	−1.17483	−0.587416	+	0.809285i	$0.699855\pi$
$264$	0	0
$265$	8.00000	0.491436
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−16.0000	−0.975537	−0.487769	−	0.872973i	$-0.662189\pi$
$269$	−16.0000	−0.975537	−0.487769	+	0.872973i	$0.662189\pi$
$270$	0	0
$271$	−13.8564	−0.841717	−0.420858	−	0.907126i	$-0.638271\pi$
$271$	−13.8564	−0.841717	−0.420858	+	0.907126i	$0.638271\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.92820	0.417786
$276$	0	0
$277$	−27.0000	−1.62227	−0.811136	−	0.584857i	$-0.801151\pi$
$277$	−27.0000	−1.62227	−0.811136	+	0.584857i	$0.801151\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	19.0000	1.13344	0.566722	−	0.823909i	$-0.308211\pi$
$281$	19.0000	1.13344	0.566722	+	0.823909i	$0.308211\pi$
$282$	0	0
$283$	29.4449	1.75032	0.875158	−	0.483838i	$-0.160758\pi$
$283$	29.4449	1.75032	0.875158	+	0.483838i	$0.160758\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.66025	−0.511199
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.00000	0.408944	0.204472	−	0.978872i	$-0.434452\pi$
$293$	7.00000	0.408944	0.204472	+	0.978872i	$0.434452\pi$
$294$	0	0
$295$	1.73205	0.100844
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−25.9808	−1.50251
$300$	0	0
$301$	−15.0000	−0.864586
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−7.00000	−0.400819
$306$	0	0
$307$	−6.92820	−0.395413	−0.197707	−	0.980261i	$-0.563349\pi$
$307$	−6.92820	−0.395413	−0.197707	+	0.980261i	$0.563349\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5.19615	0.294647	0.147323	−	0.989088i	$-0.452934\pi$
$311$	5.19615	0.294647	0.147323	+	0.989088i	$0.452934\pi$
$312$	0	0
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.0000	0.954815	0.477408	−	0.878682i	$-0.341577\pi$
$317$	17.0000	0.954815	0.477408	+	0.878682i	$0.341577\pi$
$318$	0	0
$319$	−1.73205	−0.0969762
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−27.7128	−1.54198
$324$	0	0
$325$	−12.0000	−0.665640
$326$	0	0
$327$	0	0
$328$	0	0
$329$	21.0000	1.15777
$330$	0	0
$331$	−12.1244	−0.666415	−0.333207	−	0.942854i	$-0.608131\pi$
$331$	−12.1244	−0.666415	−0.333207	+	0.942854i	$0.608131\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.66025	0.473160
$336$	0	0
$337$	15.0000	0.817102	0.408551	−	0.912735i	$-0.366034\pi$
$337$	15.0000	0.817102	0.408551	+	0.912735i	$0.366034\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	9.00000	0.487377
$342$	0	0
$343$	−19.0526	−1.02874
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.5167	1.20876	0.604379	−	0.796697i	$-0.293421\pi$
$347$	22.5167	1.20876	0.604379	+	0.796697i	$0.293421\pi$
$348$	0	0
$349$	−11.0000	−0.588817	−0.294408	−	0.955680i	$-0.595123\pi$
$349$	−11.0000	−0.588817	−0.294408	+	0.955680i	$0.595123\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−29.0000	−1.54351	−0.771757	−	0.635917i	$-0.780622\pi$
$353$	−29.0000	−1.54351	−0.771757	+	0.635917i	$0.780622\pi$
$354$	0	0
$355$	−3.46410	−0.183855
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	29.0000	1.52632
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12.0000	0.628109
$366$	0	0
$367$	32.9090	1.71783	0.858917	−	0.512115i	$-0.171138\pi$
$367$	32.9090	1.71783	0.858917	+	0.512115i	$0.171138\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−13.8564	−0.719389
$372$	0	0
$373$	19.0000	0.983783	0.491891	−	0.870657i	$-0.336306\pi$
$373$	19.0000	0.983783	0.491891	+	0.870657i	$0.336306\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.00000	0.154508
$378$	0	0
$379$	−10.3923	−0.533817	−0.266908	−	0.963722i	$-0.586002\pi$
$379$	−10.3923	−0.533817	−0.266908	+	0.963722i	$0.586002\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−25.9808	−1.32755	−0.663777	−	0.747930i	$-0.731048\pi$
$383$	−25.9808	−1.32755	−0.663777	+	0.747930i	$0.731048\pi$
$384$	0	0
$385$	3.00000	0.152894
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−29.0000	−1.47036	−0.735179	−	0.677873i	$-0.762902\pi$
$389$	−29.0000	−1.47036	−0.735179	+	0.677873i	$0.762902\pi$
$390$	0	0
$391$	34.6410	1.75187
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−5.19615	−0.261447
$396$	0	0
$397$	−8.00000	−0.401508	−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000	−0.401508	−0.200754	+	0.979642i	$0.564339\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−13.0000	−0.649189	−0.324595	−	0.945853i	$-0.605228\pi$
$401$	−13.0000	−0.649189	−0.324595	+	0.945853i	$0.605228\pi$
$402$	0	0
$403$	−15.5885	−0.776516
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−13.8564	−0.686837
$408$	0	0
$409$	−9.00000	−0.445021	−0.222511	−	0.974930i	$-0.571425\pi$
$409$	−9.00000	−0.445021	−0.222511	+	0.974930i	$0.571425\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.00000	−0.147620
$414$	0	0
$415$	−8.66025	−0.425115
$416$	0	0
$417$	0	0
$418$	0	0
$419$	22.5167	1.10001	0.550005	−	0.835161i	$-0.314626\pi$
$419$	22.5167	1.10001	0.550005	+	0.835161i	$0.314626\pi$
$420$	0	0
$421$	15.0000	0.731055	0.365528	−	0.930800i	$-0.380889\pi$
$421$	15.0000	0.731055	0.365528	+	0.930800i	$0.380889\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	16.0000	0.776114
$426$	0	0
$427$	12.1244	0.586739
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−27.7128	−1.33488	−0.667440	−	0.744664i	$-0.732610\pi$
$431$	−27.7128	−1.33488	−0.667440	+	0.744664i	$0.732610\pi$
$432$	0	0
$433$	4.00000	0.192228	0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000	0.192228	0.0961139	+	0.995370i	$0.469359\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−60.0000	−2.87019
$438$	0	0
$439$	5.19615	0.247999	0.123999	−	0.992282i	$-0.460428\pi$
$439$	5.19615	0.247999	0.123999	+	0.992282i	$0.460428\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−32.9090	−1.56355	−0.781776	−	0.623559i	$-0.785686\pi$
$443$	−32.9090	−1.56355	−0.781776	+	0.623559i	$0.785686\pi$
$444$	0	0
$445$	−4.00000	−0.189618
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	0	0
$451$	8.66025	0.407795
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.19615	−0.243599
$456$	0	0
$457$	21.0000	0.982339	0.491169	−	0.871064i	$-0.336570\pi$
$457$	21.0000	0.982339	0.491169	+	0.871064i	$0.336570\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.00000	0.0465746	0.0232873	−	0.999729i	$-0.492587\pi$
$461$	1.00000	0.0465746	0.0232873	+	0.999729i	$0.492587\pi$
$462$	0	0
$463$	−32.9090	−1.52941	−0.764705	−	0.644381i	$-0.777115\pi$
$463$	−32.9090	−1.52941	−0.764705	+	0.644381i	$0.777115\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.92820	−0.320599	−0.160300	−	0.987068i	$-0.551246\pi$
$467$	−6.92820	−0.320599	−0.160300	+	0.987068i	$0.551246\pi$
$468$	0	0
$469$	−15.0000	−0.692636
$470$	0	0
$471$	0	0
$472$	0	0
$473$	15.0000	0.689701
$474$	0	0
$475$	−27.7128	−1.27155
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8.66025	0.395697	0.197849	−	0.980233i	$-0.436605\pi$
$479$	8.66025	0.395697	0.197849	+	0.980233i	$0.436605\pi$
$480$	0	0
$481$	24.0000	1.09431
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.00000	0.136223
$486$	0	0
$487$	−27.7128	−1.25579	−0.627894	−	0.778299i	$-0.716083\pi$
$487$	−27.7128	−1.25579	−0.627894	+	0.778299i	$0.716083\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1.73205	−0.0781664	−0.0390832	−	0.999236i	$-0.512444\pi$
$491$	−1.73205	−0.0781664	−0.0390832	+	0.999236i	$0.512444\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.00000	0.269137
$498$	0	0
$499$	1.73205	0.0775372	0.0387686	−	0.999248i	$-0.487656\pi$
$499$	1.73205	0.0775372	0.0387686	+	0.999248i	$0.487656\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	24.2487	1.08120	0.540598	−	0.841281i	$-0.318198\pi$
$503$	24.2487	1.08120	0.540598	+	0.841281i	$0.318198\pi$
$504$	0	0
$505$	13.0000	0.578492
$506$	0	0
$507$	0	0
$508$	0	0
$509$	37.0000	1.64000	0.819998	−	0.572366i	$-0.193974\pi$
$509$	37.0000	1.64000	0.819998	+	0.572366i	$0.193974\pi$
$510$	0	0
$511$	−20.7846	−0.919457
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.73205	0.0763233
$516$	0	0
$517$	−21.0000	−0.923579
$518$	0	0
$519$	0	0
$520$	0	0
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	0	0
$523$	10.3923	0.454424	0.227212	−	0.973845i	$-0.427039\pi$
$523$	10.3923	0.454424	0.227212	+	0.973845i	$0.427039\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	20.7846	0.905392
$528$	0	0
$529$	52.0000	2.26087
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−15.0000	−0.649722
$534$	0	0
$535$	−6.92820	−0.299532
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.92820	0.298419
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	15.5885	0.666514	0.333257	−	0.942836i	$-0.391852\pi$
$547$	15.5885	0.666514	0.333257	+	0.942836i	$0.391852\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.92820	0.295151
$552$	0	0
$553$	9.00000	0.382719
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.0000	0.677942	0.338971	−	0.940797i	$-0.389921\pi$
$557$	16.0000	0.677942	0.338971	+	0.940797i	$0.389921\pi$
$558$	0	0
$559$	−25.9808	−1.09887
$560$	0	0
$561$	0	0
$562$	0	0
$563$	22.5167	0.948964	0.474482	−	0.880265i	$-0.342635\pi$
$563$	22.5167	0.948964	0.474482	+	0.880265i	$0.342635\pi$
$564$	0	0
$565$	−1.00000	−0.0420703
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−31.0000	−1.29959	−0.649794	−	0.760111i	$-0.725145\pi$
$569$	−31.0000	−1.29959	−0.649794	+	0.760111i	$0.725145\pi$
$570$	0	0
$571$	−8.66025	−0.362420	−0.181210	−	0.983444i	$-0.558001\pi$
$571$	−8.66025	−0.362420	−0.181210	+	0.983444i	$0.558001\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	34.6410	1.44463
$576$	0	0
$577$	−4.00000	−0.166522	−0.0832611	−	0.996528i	$-0.526534\pi$
$577$	−4.00000	−0.166522	−0.0832611	+	0.996528i	$0.526534\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	15.0000	0.622305
$582$	0	0
$583$	13.8564	0.573874
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.19615	−0.214468	−0.107234	−	0.994234i	$-0.534199\pi$
$587$	−5.19615	−0.214468	−0.107234	+	0.994234i	$0.534199\pi$
$588$	0	0
$589$	−36.0000	−1.48335
$590$	0	0
$591$	0	0
$592$	0	0
$593$	20.0000	0.821302	0.410651	−	0.911793i	$-0.365302\pi$
$593$	20.0000	0.821302	0.410651	+	0.911793i	$0.365302\pi$
$594$	0	0
$595$	6.92820	0.284029
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.73205	0.0707697	0.0353848	−	0.999374i	$-0.488734\pi$
$599$	1.73205	0.0707697	0.0353848	+	0.999374i	$0.488734\pi$
$600$	0	0
$601$	−11.0000	−0.448699	−0.224350	−	0.974509i	$-0.572026\pi$
$601$	−11.0000	−0.448699	−0.224350	+	0.974509i	$0.572026\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.00000	0.325246
$606$	0	0
$607$	12.1244	0.492112	0.246056	−	0.969256i	$-0.420865\pi$
$607$	12.1244	0.492112	0.246056	+	0.969256i	$0.420865\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	36.3731	1.47150
$612$	0	0
$613$	−8.00000	−0.323117	−0.161558	−	0.986863i	$-0.551652\pi$
$613$	−8.00000	−0.323117	−0.161558	+	0.986863i	$0.551652\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5.00000	−0.201292	−0.100646	−	0.994922i	$-0.532091\pi$
$617$	−5.00000	−0.201292	−0.100646	+	0.994922i	$0.532091\pi$
$618$	0	0
$619$	15.5885	0.626553	0.313276	−	0.949662i	$-0.398573\pi$
$619$	15.5885	0.626553	0.313276	+	0.949662i	$0.398573\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.92820	0.277573
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−32.0000	−1.27592
$630$	0	0
$631$	41.5692	1.65484	0.827422	−	0.561580i	$-0.189806\pi$
$631$	41.5692	1.65484	0.827422	+	0.561580i	$0.189806\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−13.8564	−0.549875
$636$	0	0
$637$	−12.0000	−0.475457
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−7.00000	−0.276483	−0.138242	−	0.990399i	$-0.544145\pi$
$641$	−7.00000	−0.276483	−0.138242	+	0.990399i	$0.544145\pi$
$642$	0	0
$643$	5.19615	0.204916	0.102458	−	0.994737i	$-0.467329\pi$
$643$	5.19615	0.204916	0.102458	+	0.994737i	$0.467329\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13.8564	−0.544752	−0.272376	−	0.962191i	$-0.587809\pi$
$647$	−13.8564	−0.544752	−0.272376	+	0.962191i	$0.587809\pi$
$648$	0	0
$649$	3.00000	0.117760
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.00000	−0.0391330	−0.0195665	−	0.999809i	$-0.506229\pi$
$653$	−1.00000	−0.0391330	−0.0195665	+	0.999809i	$0.506229\pi$
$654$	0	0
$655$	15.5885	0.609091
$656$	0	0
$657$	0	0
$658$	0	0
$659$	39.8372	1.55184	0.775918	−	0.630834i	$-0.217287\pi$
$659$	39.8372	1.55184	0.775918	+	0.630834i	$0.217287\pi$
$660$	0	0
$661$	17.0000	0.661223	0.330612	−	0.943767i	$-0.392745\pi$
$661$	17.0000	0.661223	0.330612	+	0.943767i	$0.392745\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−12.0000	−0.465340
$666$	0	0
$667$	−8.66025	−0.335326
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−12.1244	−0.468056
$672$	0	0
$673$	5.00000	0.192736	0.0963679	−	0.995346i	$-0.469277\pi$
$673$	5.00000	0.192736	0.0963679	+	0.995346i	$0.469277\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	11.0000	0.422764	0.211382	−	0.977403i	$-0.432204\pi$
$677$	11.0000	0.422764	0.211382	+	0.977403i	$0.432204\pi$
$678$	0	0
$679$	−5.19615	−0.199410
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−34.6410	−1.32550	−0.662751	−	0.748840i	$-0.730611\pi$
$683$	−34.6410	−1.32550	−0.662751	+	0.748840i	$0.730611\pi$
$684$	0	0
$685$	13.0000	0.496704
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	19.0526	0.724793	0.362397	−	0.932024i	$-0.381959\pi$
$691$	19.0526	0.724793	0.362397	+	0.932024i	$0.381959\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.73205	−0.0657004
$696$	0	0
$697$	20.0000	0.757554
$698$	0	0
$699$	0	0
$700$	0	0
$701$	16.0000	0.604312	0.302156	−	0.953259i	$-0.402294\pi$
$701$	16.0000	0.604312	0.302156	+	0.953259i	$0.402294\pi$
$702$	0	0
$703$	55.4256	2.09042
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−22.5167	−0.846826
$708$	0	0
$709$	−27.0000	−1.01401	−0.507003	−	0.861944i	$-0.669247\pi$
$709$	−27.0000	−1.01401	−0.507003	+	0.861944i	$0.669247\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	45.0000	1.68526
$714$	0	0
$715$	5.19615	0.194325
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−41.5692	−1.55027	−0.775135	−	0.631795i	$-0.782318\pi$
$719$	−41.5692	−1.55027	−0.775135	+	0.631795i	$0.782318\pi$
$720$	0	0
$721$	−3.00000	−0.111726
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.00000	−0.148556
$726$	0	0
$727$	−22.5167	−0.835097	−0.417548	−	0.908655i	$-0.637111\pi$
$727$	−22.5167	−0.835097	−0.417548	+	0.908655i	$0.637111\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	34.6410	1.28124
$732$	0	0
$733$	−21.0000	−0.775653	−0.387826	−	0.921732i	$-0.626774\pi$
$733$	−21.0000	−0.775653	−0.387826	+	0.921732i	$0.626774\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	15.0000	0.552532
$738$	0	0
$739$	−3.46410	−0.127429	−0.0637145	−	0.997968i	$-0.520295\pi$
$739$	−3.46410	−0.127429	−0.0637145	+	0.997968i	$0.520295\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.73205	0.0635428	0.0317714	−	0.999495i	$-0.489885\pi$
$743$	1.73205	0.0635428	0.0317714	+	0.999495i	$0.489885\pi$
$744$	0	0
$745$	11.0000	0.403009
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	25.9808	0.948051	0.474026	−	0.880511i	$-0.342800\pi$
$751$	25.9808	0.948051	0.474026	+	0.880511i	$0.342800\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	22.5167	0.819465
$756$	0	0
$757$	−6.00000	−0.218074	−0.109037	−	0.994038i	$-0.534777\pi$
$757$	−6.00000	−0.218074	−0.109037	+	0.994038i	$0.534777\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	43.0000	1.55875	0.779374	−	0.626559i	$-0.215537\pi$
$761$	43.0000	1.55875	0.779374	+	0.626559i	$0.215537\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.19615	−0.187622
$768$	0	0
$769$	−1.00000	−0.0360609	−0.0180305	−	0.999837i	$-0.505740\pi$
$769$	−1.00000	−0.0360609	−0.0180305	+	0.999837i	$0.505740\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	40.0000	1.43870	0.719350	−	0.694648i	$-0.244440\pi$
$773$	40.0000	1.43870	0.719350	+	0.694648i	$0.244440\pi$
$774$	0	0
$775$	20.7846	0.746605
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−34.6410	−1.24114
$780$	0	0
$781$	−6.00000	−0.214697
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−17.0000	−0.606756
$786$	0	0
$787$	−25.9808	−0.926114	−0.463057	−	0.886328i	$-0.653248\pi$
$787$	−25.9808	−0.926114	−0.463057	+	0.886328i	$0.653248\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.73205	0.0615846
$792$	0	0
$793$	21.0000	0.745732
$794$	0	0
$795$	0	0
$796$	0	0
$797$	29.0000	1.02723	0.513616	−	0.858020i	$-0.328305\pi$
$797$	29.0000	1.02723	0.513616	+	0.858020i	$0.328305\pi$
$798$	0	0
$799$	−48.4974	−1.71572
$800$	0	0
$801$	0	0
$802$	0	0
$803$	20.7846	0.733473
$804$	0	0
$805$	15.0000	0.528681
$806$	0	0
$807$	0	0
$808$	0	0
$809$	4.00000	0.140633	0.0703163	−	0.997525i	$-0.477599\pi$
$809$	4.00000	0.140633	0.0703163	+	0.997525i	$0.477599\pi$
$810$	0	0
$811$	20.7846	0.729846	0.364923	−	0.931038i	$-0.381095\pi$
$811$	20.7846	0.729846	0.364923	+	0.931038i	$0.381095\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6.92820	0.242684
$816$	0	0
$817$	−60.0000	−2.09913
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−7.00000	−0.244302	−0.122151	−	0.992512i	$-0.538979\pi$
$821$	−7.00000	−0.244302	−0.122151	+	0.992512i	$0.538979\pi$
$822$	0	0
$823$	−46.7654	−1.63014	−0.815069	−	0.579364i	$-0.803301\pi$
$823$	−46.7654	−1.63014	−0.815069	+	0.579364i	$0.803301\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−6.92820	−0.240917	−0.120459	−	0.992718i	$-0.538437\pi$
$827$	−6.92820	−0.240917	−0.120459	+	0.992718i	$0.538437\pi$
$828$	0	0
$829$	−24.0000	−0.833554	−0.416777	−	0.909009i	$-0.636840\pi$
$829$	−24.0000	−0.833554	−0.416777	+	0.909009i	$0.636840\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	16.0000	0.554367
$834$	0	0
$835$	−15.5885	−0.539461
$836$	0	0
$837$	0	0
$838$	0	0
$839$	25.9808	0.896956	0.448478	−	0.893794i	$-0.351966\pi$
$839$	25.9808	0.896956	0.448478	+	0.893794i	$0.351966\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4.00000	0.137604
$846$	0	0
$847$	−13.8564	−0.476112
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−69.2820	−2.37496
$852$	0	0
$853$	23.0000	0.787505	0.393753	−	0.919216i	$-0.371177\pi$
$853$	23.0000	0.787505	0.393753	+	0.919216i	$0.371177\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	43.0000	1.46885	0.734426	−	0.678689i	$-0.237451\pi$
$857$	43.0000	1.46885	0.734426	+	0.678689i	$0.237451\pi$
$858$	0	0
$859$	−36.3731	−1.24103	−0.620517	−	0.784193i	$-0.713077\pi$
$859$	−36.3731	−1.24103	−0.620517	+	0.784193i	$0.713077\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	55.4256	1.88671	0.943355	−	0.331785i	$-0.107651\pi$
$863$	55.4256	1.88671	0.943355	+	0.331785i	$0.107651\pi$
$864$	0	0
$865$	13.0000	0.442013
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−9.00000	−0.305304
$870$	0	0
$871$	−25.9808	−0.880325
$872$	0	0
$873$	0	0
$874$	0	0
$875$	15.5885	0.526986
$876$	0	0
$877$	43.0000	1.45201	0.726003	−	0.687691i	$-0.241376\pi$
$877$	43.0000	1.45201	0.726003	+	0.687691i	$0.241376\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−52.0000	−1.75192	−0.875962	−	0.482380i	$-0.839773\pi$
$881$	−52.0000	−1.75192	−0.875962	+	0.482380i	$0.839773\pi$
$882$	0	0
$883$	−20.7846	−0.699458	−0.349729	−	0.936851i	$-0.613726\pi$
$883$	−20.7846	−0.699458	−0.349729	+	0.936851i	$0.613726\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	5.19615	0.174470	0.0872349	−	0.996188i	$-0.472197\pi$
$887$	5.19615	0.174470	0.0872349	+	0.996188i	$0.472197\pi$
$888$	0	0
$889$	24.0000	0.804934
$890$	0	0
$891$	0	0
$892$	0	0
$893$	84.0000	2.81095
$894$	0	0
$895$	6.92820	0.231584
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−5.19615	−0.173301
$900$	0	0
$901$	32.0000	1.06607
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	32.9090	1.09272	0.546362	−	0.837549i	$-0.316012\pi$
$907$	32.9090	1.09272	0.546362	+	0.837549i	$0.316012\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−32.9090	−1.09032	−0.545161	−	0.838331i	$-0.683532\pi$
$911$	−32.9090	−1.09032	−0.545161	+	0.838331i	$0.683532\pi$
$912$	0	0
$913$	−15.0000	−0.496428
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−27.0000	−0.891619
$918$	0	0
$919$	13.8564	0.457081	0.228540	−	0.973534i	$-0.426605\pi$
$919$	13.8564	0.457081	0.228540	+	0.973534i	$0.426605\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	10.3923	0.342067
$924$	0	0
$925$	−32.0000	−1.05215
$926$	0	0
$927$	0	0
$928$	0	0
$929$	35.0000	1.14831	0.574156	−	0.818746i	$-0.305330\pi$
$929$	35.0000	1.14831	0.574156	+	0.818746i	$0.305330\pi$
$930$	0	0
$931$	−27.7128	−0.908251
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−6.92820	−0.226576
$936$	0	0
$937$	4.00000	0.130674	0.0653372	−	0.997863i	$-0.479188\pi$
$937$	4.00000	0.130674	0.0653372	+	0.997863i	$0.479188\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	7.00000	0.228193	0.114097	−	0.993470i	$-0.463603\pi$
$941$	7.00000	0.228193	0.114097	+	0.993470i	$0.463603\pi$
$942$	0	0
$943$	43.3013	1.41008
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−15.5885	−0.506557	−0.253278	−	0.967393i	$-0.581509\pi$
$947$	−15.5885	−0.506557	−0.253278	+	0.967393i	$0.581509\pi$
$948$	0	0
$949$	−36.0000	−1.16861
$950$	0	0
$951$	0	0
$952$	0	0
$953$	44.0000	1.42530	0.712650	−	0.701520i	$-0.247495\pi$
$953$	44.0000	1.42530	0.712650	+	0.701520i	$0.247495\pi$
$954$	0	0
$955$	1.73205	0.0560478
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−22.5167	−0.727101
$960$	0	0
$961$	−4.00000	−0.129032
$962$	0	0
$963$	0	0
$964$	0	0
$965$	11.0000	0.354103
$966$	0	0
$967$	−32.9090	−1.05828	−0.529140	−	0.848534i	$-0.677486\pi$
$967$	−32.9090	−1.05828	−0.529140	+	0.848534i	$0.677486\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	38.1051	1.22285	0.611426	−	0.791302i	$-0.290596\pi$
$971$	38.1051	1.22285	0.611426	+	0.791302i	$0.290596\pi$
$972$	0	0
$973$	3.00000	0.0961756
$974$	0	0
$975$	0	0
$976$	0	0
$977$	25.0000	0.799821	0.399910	−	0.916554i	$-0.369041\pi$
$977$	25.0000	0.799821	0.399910	+	0.916554i	$0.369041\pi$
$978$	0	0
$979$	−6.92820	−0.221426
$980$	0	0
$981$	0	0
$982$	0	0
$983$	8.66025	0.276219	0.138110	−	0.990417i	$-0.455897\pi$
$983$	8.66025	0.276219	0.138110	+	0.990417i	$0.455897\pi$
$984$	0	0
$985$	22.0000	0.700978
$986$	0	0
$987$	0	0
$988$	0	0
$989$	75.0000	2.38486
$990$	0	0
$991$	55.4256	1.76065	0.880327	−	0.474368i	$-0.157323\pi$
$991$	55.4256	1.76065	0.880327	+	0.474368i	$0.157323\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3.46410	−0.109819
$996$	0	0
$997$	13.0000	0.411714	0.205857	−	0.978582i	$-0.434002\pi$
$997$	13.0000	0.411714	0.205857	+	0.978582i	$0.434002\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5184.2.a.bl.1.2		2
3.2	odd	2		5184.2.a.bx.1.2		2
4.3	odd	2	inner	5184.2.a.bl.1.1		2
8.3	odd	2		2592.2.a.p.1.1		2
8.5	even	2		2592.2.a.p.1.2		2
9.2	odd	6		576.2.i.k.193.1		4
9.4	even	3		1728.2.i.l.1153.1		4
9.5	odd	6		576.2.i.k.385.1		4
9.7	even	3		1728.2.i.l.577.1		4
12.11	even	2		5184.2.a.bx.1.1		2
24.5	odd	2		2592.2.a.l.1.2		2
24.11	even	2		2592.2.a.l.1.1		2
36.7	odd	6		1728.2.i.l.577.2		4
36.11	even	6		576.2.i.k.193.2		4
36.23	even	6		576.2.i.k.385.2		4
36.31	odd	6		1728.2.i.l.1153.2		4
72.5	odd	6		288.2.i.d.97.2	yes	4
72.11	even	6		288.2.i.d.193.1	yes	4
72.13	even	6		864.2.i.d.289.1		4
72.29	odd	6		288.2.i.d.193.2	yes	4
72.43	odd	6		864.2.i.d.577.2		4
72.59	even	6		288.2.i.d.97.1	✓	4
72.61	even	6		864.2.i.d.577.1		4
72.67	odd	6		864.2.i.d.289.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.2.i.d.97.1	✓	4	72.59	even	6
288.2.i.d.97.2	yes	4	72.5	odd	6
288.2.i.d.193.1	yes	4	72.11	even	6
288.2.i.d.193.2	yes	4	72.29	odd	6
576.2.i.k.193.1		4	9.2	odd	6
576.2.i.k.193.2		4	36.11	even	6
576.2.i.k.385.1		4	9.5	odd	6
576.2.i.k.385.2		4	36.23	even	6
864.2.i.d.289.1		4	72.13	even	6
864.2.i.d.289.2		4	72.67	odd	6
864.2.i.d.577.1		4	72.61	even	6
864.2.i.d.577.2		4	72.43	odd	6
1728.2.i.l.577.1		4	9.7	even	3
1728.2.i.l.577.2		4	36.7	odd	6
1728.2.i.l.1153.1		4	9.4	even	3
1728.2.i.l.1153.2		4	36.31	odd	6
2592.2.a.l.1.1		2	24.11	even	2
2592.2.a.l.1.2		2	24.5	odd	2
2592.2.a.p.1.1		2	8.3	odd	2
2592.2.a.p.1.2		2	8.5	even	2
5184.2.a.bl.1.1		2	4.3	odd	2	inner
5184.2.a.bl.1.2		2	1.1	even	1	trivial
5184.2.a.bx.1.1		2	12.11	even	2
5184.2.a.bx.1.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +1.73205 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +1.73205 q^{7} -1.73205 q^{11} +3.00000 q^{13} -4.00000 q^{17} +6.92820 q^{19} -8.66025 q^{23} -4.00000 q^{25} +1.00000 q^{29} -5.19615 q^{31} -1.73205 q^{35} +8.00000 q^{37} -5.00000 q^{41} -8.66025 q^{43} +12.1244 q^{47} -4.00000 q^{49} -8.00000 q^{53} +1.73205 q^{55} -1.73205 q^{59} +7.00000 q^{61} -3.00000 q^{65} -8.66025 q^{67} +3.46410 q^{71} -12.0000 q^{73} -3.00000 q^{77} +5.19615 q^{79} +8.66025 q^{83} +4.00000 q^{85} +4.00000 q^{89} +5.19615 q^{91} -6.92820 q^{95} -3.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^5	\( 2 q - 2 q^{5} + 6 q^{13} - 8 q^{17} - 8 q^{25} + 2 q^{29} + 16 q^{37} - 10 q^{41} - 8 q^{49} - 16 q^{53} + 14 q^{61} - 6 q^{65} - 24 q^{73} - 6 q^{77} + 8 q^{85} + 8 q^{89} - 6 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 6 * q^13 - 8 * q^17 - 8 * q^25 + 2 * q^29 + 16 * q^37 - 10 * q^41 - 8 * q^49 - 16 * q^53 + 14 * q^61 - 6 * q^65 - 24 * q^73 - 6 * q^77 + 8 * q^85 + 8 * q^89 - 6 * q^97

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