LMFDB - Embedded newform 9576.2.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9576.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:	$76.4647449756$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3192)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	9576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.46410 q^{5} -1.00000 q^{7} +O(q^{10})$	$q+3.46410 q^{5} -1.00000 q^{7} -2.00000 q^{13} -3.46410 q^{17} -1.00000 q^{19} -4.00000 q^{23} +7.00000 q^{25} +7.46410 q^{29} -2.92820 q^{31} -3.46410 q^{35} -2.00000 q^{37} -6.00000 q^{41} -6.92820 q^{43} -2.53590 q^{47} +1.00000 q^{49} -3.46410 q^{53} -4.00000 q^{59} +6.00000 q^{61} -6.92820 q^{65} +1.07180 q^{67} -6.53590 q^{71} +12.9282 q^{73} -1.46410 q^{83} -12.0000 q^{85} -11.8564 q^{89} +2.00000 q^{91} -3.46410 q^{95} +10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^7	$ 2 q - 2 q^{7} - 4 q^{13} - 2 q^{19} - 8 q^{23} + 14 q^{25} + 8 q^{29} + 8 q^{31} - 4 q^{37} - 12 q^{41} - 12 q^{47} + 2 q^{49} - 8 q^{59} + 12 q^{61} + 16 q^{67} - 20 q^{71} + 12 q^{73} + 4 q^{83} - 24 q^{85} + 4 q^{89} + 4 q^{91} + 20 q^{97}+O(q^{100}) $ 2 * q - 2 * q^7 - 4 * q^13 - 2 * q^19 - 8 * q^23 + 14 * q^25 + 8 * q^29 + 8 * q^31 - 4 * q^37 - 12 * q^41 - 12 * q^47 + 2 * q^49 - 8 * q^59 + 12 * q^61 + 16 * q^67 - 20 * q^71 + 12 * q^73 + 4 * q^83 - 24 * q^85 + 4 * q^89 + 4 * q^91 + 20 * 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
$5$	3.46410	1.54919	0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410	1.54919	0.774597	+	0.632456i	$0.217953\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.46410	1.38605	0.693024	−	0.720914i	$-0.256278\pi$
$29$	7.46410	1.38605	0.693024	+	0.720914i	$0.256278\pi$
$30$	0	0
$31$	−2.92820	−0.525921	−0.262960	−	0.964807i	$-0.584699\pi$
$31$	−2.92820	−0.525921	−0.262960	+	0.964807i	$0.584699\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.46410	−0.585540
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−6.92820	−1.05654	−0.528271	−	0.849076i	$-0.677159\pi$
$43$	−6.92820	−1.05654	−0.528271	+	0.849076i	$0.677159\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.53590	−0.369899	−0.184949	−	0.982748i	$-0.559212\pi$
$47$	−2.53590	−0.369899	−0.184949	+	0.982748i	$0.559212\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.46410	−0.475831	−0.237915	−	0.971286i	$-0.576464\pi$
$53$	−3.46410	−0.475831	−0.237915	+	0.971286i	$0.576464\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.92820	−0.859338
$66$	0	0
$67$	1.07180	0.130941	0.0654704	−	0.997855i	$-0.479145\pi$
$67$	1.07180	0.130941	0.0654704	+	0.997855i	$0.479145\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.53590	−0.775668	−0.387834	−	0.921729i	$-0.626777\pi$
$71$	−6.53590	−0.775668	−0.387834	+	0.921729i	$0.626777\pi$
$72$	0	0
$73$	12.9282	1.51313	0.756566	−	0.653917i	$-0.226876\pi$
$73$	12.9282	1.51313	0.756566	+	0.653917i	$0.226876\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.46410	−0.160706	−0.0803530	−	0.996766i	$-0.525605\pi$
$83$	−1.46410	−0.160706	−0.0803530	+	0.996766i	$0.525605\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−11.8564	−1.25678	−0.628388	−	0.777900i	$-0.716285\pi$
$89$	−11.8564	−1.25678	−0.628388	+	0.777900i	$0.716285\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.46410	−0.355409
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.3923	−1.03407	−0.517036	−	0.855963i	$-0.672965\pi$
$101$	−10.3923	−1.03407	−0.517036	+	0.855963i	$0.672965\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.39230	−0.811315	−0.405657	−	0.914025i	$-0.632957\pi$
$107$	−8.39230	−0.811315	−0.405657	+	0.914025i	$0.632957\pi$
$108$	0	0
$109$	0.928203	0.0889057	0.0444529	−	0.999011i	$-0.485846\pi$
$109$	0.928203	0.0889057	0.0444529	+	0.999011i	$0.485846\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.53590	−0.426701	−0.213351	−	0.976976i	$-0.568438\pi$
$113$	−4.53590	−0.426701	−0.213351	+	0.976976i	$0.568438\pi$
$114$	0	0
$115$	−13.8564	−1.29212
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.46410	0.317554
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.92820	0.619677
$126$	0	0
$127$	−5.07180	−0.450049	−0.225025	−	0.974353i	$-0.572246\pi$
$127$	−5.07180	−0.450049	−0.225025	+	0.974353i	$0.572246\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.3923	−1.08272	−0.541360	−	0.840791i	$-0.682091\pi$
$131$	−12.3923	−1.08272	−0.541360	+	0.840791i	$0.682091\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	6.92820	0.587643	0.293821	−	0.955860i	$-0.405073\pi$
$139$	6.92820	0.587643	0.293821	+	0.955860i	$0.405073\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	25.8564	2.14726
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.92820	−0.731427	−0.365713	−	0.930727i	$-0.619175\pi$
$149$	−8.92820	−0.731427	−0.365713	+	0.930727i	$0.619175\pi$
$150$	0	0
$151$	10.9282	0.889325	0.444662	−	0.895698i	$-0.353324\pi$
$151$	10.9282	0.889325	0.444662	+	0.895698i	$0.353324\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−10.1436	−0.814753
$156$	0	0
$157$	−15.8564	−1.26548	−0.632740	−	0.774365i	$-0.718070\pi$
$157$	−15.8564	−1.26548	−0.632740	+	0.774365i	$0.718070\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	−1.07180	−0.0839496	−0.0419748	−	0.999119i	$-0.513365\pi$
$163$	−1.07180	−0.0839496	−0.0419748	+	0.999119i	$0.513365\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.9282	−0.845650	−0.422825	−	0.906211i	$-0.638961\pi$
$167$	−10.9282	−0.845650	−0.422825	+	0.906211i	$0.638961\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	−7.00000	−0.529150
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−13.4641	−1.00635	−0.503177	−	0.864183i	$-0.667836\pi$
$179$	−13.4641	−1.00635	−0.503177	+	0.864183i	$0.667836\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.92820	−0.509372
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−9.85641	−0.713185	−0.356592	−	0.934260i	$-0.616061\pi$
$191$	−9.85641	−0.713185	−0.356592	+	0.934260i	$0.616061\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.85641	0.559746	0.279873	−	0.960037i	$-0.409708\pi$
$197$	7.85641	0.559746	0.279873	+	0.960037i	$0.409708\pi$
$198$	0	0
$199$	−18.9282	−1.34178	−0.670892	−	0.741555i	$-0.734089\pi$
$199$	−18.9282	−1.34178	−0.670892	+	0.741555i	$0.734089\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.46410	−0.523877
$204$	0	0
$205$	−20.7846	−1.45166
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−1.85641	−0.127800	−0.0639001	−	0.997956i	$-0.520354\pi$
$211$	−1.85641	−0.127800	−0.0639001	+	0.997956i	$0.520354\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−24.0000	−1.63679
$216$	0	0
$217$	2.92820	0.198779
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.92820	0.466041
$222$	0	0
$223$	16.7846	1.12398	0.561990	−	0.827144i	$-0.310036\pi$
$223$	16.7846	1.12398	0.561990	+	0.827144i	$0.310036\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6.92820	−0.459841	−0.229920	−	0.973209i	$-0.573847\pi$
$227$	−6.92820	−0.459841	−0.229920	+	0.973209i	$0.573847\pi$
$228$	0	0
$229$	22.7846	1.50565	0.752825	−	0.658221i	$-0.228691\pi$
$229$	22.7846	1.50565	0.752825	+	0.658221i	$0.228691\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.92820	0.584906	0.292453	−	0.956280i	$-0.405528\pi$
$233$	8.92820	0.584906	0.292453	+	0.956280i	$0.405528\pi$
$234$	0	0
$235$	−8.78461	−0.573045
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.92820	−0.448148	−0.224074	−	0.974572i	$-0.571936\pi$
$239$	−6.92820	−0.448148	−0.224074	+	0.974572i	$0.571936\pi$
$240$	0	0
$241$	−0.143594	−0.00924967	−0.00462484	−	0.999989i	$-0.501472\pi$
$241$	−0.143594	−0.00924967	−0.00462484	+	0.999989i	$0.501472\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.46410	0.221313
$246$	0	0
$247$	2.00000	0.127257
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−12.3923	−0.782195	−0.391098	−	0.920349i	$-0.627905\pi$
$251$	−12.3923	−0.782195	−0.391098	+	0.920349i	$0.627905\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.7846	1.17175	0.585876	−	0.810401i	$-0.300751\pi$
$257$	18.7846	1.17175	0.585876	+	0.810401i	$0.300751\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	0	0
$262$	0	0
$263$	22.9282	1.41381	0.706907	−	0.707307i	$-0.250090\pi$
$263$	22.9282	1.41381	0.706907	+	0.707307i	$0.250090\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	0	0
$268$	0	0
$269$	16.9282	1.03213	0.516065	−	0.856549i	$-0.327396\pi$
$269$	16.9282	1.03213	0.516065	+	0.856549i	$0.327396\pi$
$270$	0	0
$271$	−2.14359	−0.130214	−0.0651070	−	0.997878i	$-0.520739\pi$
$271$	−2.14359	−0.130214	−0.0651070	+	0.997878i	$0.520739\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−23.8564	−1.43339	−0.716696	−	0.697385i	$-0.754347\pi$
$277$	−23.8564	−1.43339	−0.716696	+	0.697385i	$0.754347\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	17.3205	1.03325	0.516627	−	0.856210i	$-0.327187\pi$
$281$	17.3205	1.03325	0.516627	+	0.856210i	$0.327187\pi$
$282$	0	0
$283$	−22.9282	−1.36294	−0.681470	−	0.731846i	$-0.738659\pi$
$283$	−22.9282	−1.36294	−0.681470	+	0.731846i	$0.738659\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	0	0
$292$	0	0
$293$	22.0000	1.28525	0.642627	−	0.766179i	$-0.277845\pi$
$293$	22.0000	1.28525	0.642627	+	0.766179i	$0.277845\pi$
$294$	0	0
$295$	−13.8564	−0.806751
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.00000	0.462652
$300$	0	0
$301$	6.92820	0.399335
$302$	0	0
$303$	0	0
$304$	0	0
$305$	20.7846	1.19012
$306$	0	0
$307$	14.9282	0.851998	0.425999	−	0.904724i	$-0.359923\pi$
$307$	14.9282	0.851998	0.425999	+	0.904724i	$0.359923\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−10.5359	−0.597436	−0.298718	−	0.954341i	$-0.596559\pi$
$311$	−10.5359	−0.597436	−0.298718	+	0.954341i	$0.596559\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.535898	−0.0300991	−0.0150495	−	0.999887i	$-0.504791\pi$
$317$	−0.535898	−0.0300991	−0.0150495	+	0.999887i	$0.504791\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.46410	0.192748
$324$	0	0
$325$	−14.0000	−0.776580
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.53590	0.139809
$330$	0	0
$331$	1.07180	0.0589113	0.0294556	−	0.999566i	$-0.490623\pi$
$331$	1.07180	0.0589113	0.0294556	+	0.999566i	$0.490623\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.71281	0.202853
$336$	0	0
$337$	7.07180	0.385225	0.192613	−	0.981275i	$-0.438304\pi$
$337$	7.07180	0.385225	0.192613	+	0.981275i	$0.438304\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.78461	0.471583	0.235791	−	0.971804i	$-0.424232\pi$
$347$	8.78461	0.471583	0.235791	+	0.971804i	$0.424232\pi$
$348$	0	0
$349$	−15.8564	−0.848774	−0.424387	−	0.905481i	$-0.639510\pi$
$349$	−15.8564	−0.848774	−0.424387	+	0.905481i	$0.639510\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.5359	0.667219	0.333609	−	0.942711i	$-0.391733\pi$
$353$	12.5359	0.667219	0.333609	+	0.942711i	$0.391733\pi$
$354$	0	0
$355$	−22.6410	−1.20166
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	44.7846	2.34413
$366$	0	0
$367$	−10.9282	−0.570448	−0.285224	−	0.958461i	$-0.592068\pi$
$367$	−10.9282	−0.570448	−0.285224	+	0.958461i	$0.592068\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.46410	0.179847
$372$	0	0
$373$	19.8564	1.02813	0.514063	−	0.857753i	$-0.328140\pi$
$373$	19.8564	1.02813	0.514063	+	0.857753i	$0.328140\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−14.9282	−0.768842
$378$	0	0
$379$	12.7846	0.656701	0.328351	−	0.944556i	$-0.393507\pi$
$379$	12.7846	0.656701	0.328351	+	0.944556i	$0.393507\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.07180	−0.259157	−0.129578	−	0.991569i	$-0.541362\pi$
$383$	−5.07180	−0.259157	−0.129578	+	0.991569i	$0.541362\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	20.9282	1.06110	0.530551	−	0.847653i	$-0.321985\pi$
$389$	20.9282	1.06110	0.530551	+	0.847653i	$0.321985\pi$
$390$	0	0
$391$	13.8564	0.700749
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	27.8564	1.39807	0.699036	−	0.715086i	$-0.253612\pi$
$397$	27.8564	1.39807	0.699036	+	0.715086i	$0.253612\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.3923	0.718717	0.359359	−	0.933200i	$-0.382995\pi$
$401$	14.3923	0.718717	0.359359	+	0.933200i	$0.382995\pi$
$402$	0	0
$403$	5.85641	0.291728
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.00000	0.196827
$414$	0	0
$415$	−5.07180	−0.248965
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−9.46410	−0.462352	−0.231176	−	0.972912i	$-0.574257\pi$
$419$	−9.46410	−0.462352	−0.231176	+	0.972912i	$0.574257\pi$
$420$	0	0
$421$	−28.9282	−1.40987	−0.704937	−	0.709270i	$-0.749025\pi$
$421$	−28.9282	−1.40987	−0.704937	+	0.709270i	$0.749025\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−24.2487	−1.17624
$426$	0	0
$427$	−6.00000	−0.290360
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−30.5359	−1.47086	−0.735431	−	0.677599i	$-0.763020\pi$
$431$	−30.5359	−1.47086	−0.735431	+	0.677599i	$0.763020\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.00000	0.191346
$438$	0	0
$439$	−40.7846	−1.94654	−0.973272	−	0.229657i	$-0.926240\pi$
$439$	−40.7846	−1.94654	−0.973272	+	0.229657i	$0.926240\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−24.7846	−1.17755	−0.588776	−	0.808296i	$-0.700390\pi$
$443$	−24.7846	−1.17755	−0.588776	+	0.808296i	$0.700390\pi$
$444$	0	0
$445$	−41.0718	−1.94699
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0.535898	0.0252906	0.0126453	−	0.999920i	$-0.495975\pi$
$449$	0.535898	0.0252906	0.0126453	+	0.999920i	$0.495975\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.92820	0.324799
$456$	0	0
$457$	23.8564	1.11596	0.557978	−	0.829856i	$-0.311577\pi$
$457$	23.8564	1.11596	0.557978	+	0.829856i	$0.311577\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	19.4641	0.906534	0.453267	−	0.891375i	$-0.350258\pi$
$461$	19.4641	0.906534	0.453267	+	0.891375i	$0.350258\pi$
$462$	0	0
$463$	35.7128	1.65972	0.829858	−	0.557975i	$-0.188422\pi$
$463$	35.7128	1.65972	0.829858	+	0.557975i	$0.188422\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−30.5359	−1.41303	−0.706516	−	0.707697i	$-0.749734\pi$
$467$	−30.5359	−1.41303	−0.706516	+	0.707697i	$0.749734\pi$
$468$	0	0
$469$	−1.07180	−0.0494910
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−7.00000	−0.321182
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−29.4641	−1.34625	−0.673125	−	0.739529i	$-0.735048\pi$
$479$	−29.4641	−1.34625	−0.673125	+	0.739529i	$0.735048\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	34.6410	1.57297
$486$	0	0
$487$	27.7128	1.25579	0.627894	−	0.778299i	$-0.283917\pi$
$487$	27.7128	1.25579	0.627894	+	0.778299i	$0.283917\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−18.9282	−0.854218	−0.427109	−	0.904200i	$-0.640468\pi$
$491$	−18.9282	−0.854218	−0.427109	+	0.904200i	$0.640468\pi$
$492$	0	0
$493$	−25.8564	−1.16451
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.53590	0.293175
$498$	0	0
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	25.1769	1.12258	0.561292	−	0.827618i	$-0.310305\pi$
$503$	25.1769	1.12258	0.561292	+	0.827618i	$0.310305\pi$
$504$	0	0
$505$	−36.0000	−1.60198
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−42.7846	−1.89639	−0.948197	−	0.317682i	$-0.897095\pi$
$509$	−42.7846	−1.89639	−0.948197	+	0.317682i	$0.897095\pi$
$510$	0	0
$511$	−12.9282	−0.571910
$512$	0	0
$513$	0	0
$514$	0	0
$515$	27.7128	1.22117
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14.7846	−0.647726	−0.323863	−	0.946104i	$-0.604982\pi$
$521$	−14.7846	−0.647726	−0.323863	+	0.946104i	$0.604982\pi$
$522$	0	0
$523$	9.07180	0.396682	0.198341	−	0.980133i	$-0.436445\pi$
$523$	9.07180	0.396682	0.198341	+	0.980133i	$0.436445\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.1436	0.441862
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	12.0000	0.519778
$534$	0	0
$535$	−29.0718	−1.25688
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	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
$545$	3.21539	0.137732
$546$	0	0
$547$	−15.7128	−0.671831	−0.335916	−	0.941892i	$-0.609046\pi$
$547$	−15.7128	−0.671831	−0.335916	+	0.941892i	$0.609046\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7.46410	−0.317981
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−22.7846	−0.965415	−0.482707	−	0.875782i	$-0.660347\pi$
$557$	−22.7846	−0.965415	−0.482707	+	0.875782i	$0.660347\pi$
$558$	0	0
$559$	13.8564	0.586064
$560$	0	0
$561$	0	0
$562$	0	0
$563$	14.9282	0.629149	0.314574	−	0.949233i	$-0.398138\pi$
$563$	14.9282	0.629149	0.314574	+	0.949233i	$0.398138\pi$
$564$	0	0
$565$	−15.7128	−0.661043
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20.5359	−0.860910	−0.430455	−	0.902612i	$-0.641647\pi$
$569$	−20.5359	−0.860910	−0.430455	+	0.902612i	$0.641647\pi$
$570$	0	0
$571$	15.7128	0.657561	0.328780	−	0.944406i	$-0.393362\pi$
$571$	15.7128	0.657561	0.328780	+	0.944406i	$0.393362\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−28.0000	−1.16768
$576$	0	0
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.46410	0.0607412
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12.3923	−0.511485	−0.255743	−	0.966745i	$-0.582320\pi$
$587$	−12.3923	−0.511485	−0.255743	+	0.966745i	$0.582320\pi$
$588$	0	0
$589$	2.92820	0.120655
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−8.53590	−0.350527	−0.175264	−	0.984522i	$-0.556078\pi$
$593$	−8.53590	−0.350527	−0.175264	+	0.984522i	$0.556078\pi$
$594$	0	0
$595$	12.0000	0.491952
$596$	0	0
$597$	0	0
$598$	0	0
$599$	36.3923	1.48695	0.743475	−	0.668764i	$-0.233176\pi$
$599$	36.3923	1.48695	0.743475	+	0.668764i	$0.233176\pi$
$600$	0	0
$601$	−17.7128	−0.722521	−0.361260	−	0.932465i	$-0.617653\pi$
$601$	−17.7128	−0.722521	−0.361260	+	0.932465i	$0.617653\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−38.1051	−1.54919
$606$	0	0
$607$	35.7128	1.44954	0.724769	−	0.688992i	$-0.241946\pi$
$607$	35.7128	1.44954	0.724769	+	0.688992i	$0.241946\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.07180	0.205183
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	0	0
$619$	−25.0718	−1.00772	−0.503860	−	0.863785i	$-0.668087\pi$
$619$	−25.0718	−1.00772	−0.503860	+	0.863785i	$0.668087\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	11.8564	0.475017
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.92820	0.276246
$630$	0	0
$631$	40.7846	1.62361	0.811805	−	0.583929i	$-0.198485\pi$
$631$	40.7846	1.62361	0.811805	+	0.583929i	$0.198485\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−17.5692	−0.697213
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−19.1769	−0.757443	−0.378721	−	0.925511i	$-0.623636\pi$
$641$	−19.1769	−0.757443	−0.378721	+	0.925511i	$0.623636\pi$
$642$	0	0
$643$	44.7846	1.76613	0.883066	−	0.469248i	$-0.155475\pi$
$643$	44.7846	1.76613	0.883066	+	0.469248i	$0.155475\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.24871	−0.245662	−0.122831	−	0.992428i	$-0.539197\pi$
$647$	−6.24871	−0.245662	−0.122831	+	0.992428i	$0.539197\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.71281	−0.380092	−0.190046	−	0.981775i	$-0.560864\pi$
$653$	−9.71281	−0.380092	−0.190046	+	0.981775i	$0.560864\pi$
$654$	0	0
$655$	−42.9282	−1.67734
$656$	0	0
$657$	0	0
$658$	0	0
$659$	44.1051	1.71809	0.859046	−	0.511899i	$-0.171058\pi$
$659$	44.1051	1.71809	0.859046	+	0.511899i	$0.171058\pi$
$660$	0	0
$661$	−34.0000	−1.32245	−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000	−1.32245	−0.661223	+	0.750189i	$0.729962\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.46410	0.134332
$666$	0	0
$667$	−29.8564	−1.15604
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−42.7846	−1.64435	−0.822173	−	0.569238i	$-0.807238\pi$
$677$	−42.7846	−1.64435	−0.822173	+	0.569238i	$0.807238\pi$
$678$	0	0
$679$	−10.0000	−0.383765
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−15.6077	−0.597212	−0.298606	−	0.954376i	$-0.596522\pi$
$683$	−15.6077	−0.597212	−0.298606	+	0.954376i	$0.596522\pi$
$684$	0	0
$685$	−6.92820	−0.264713
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6.92820	0.263944
$690$	0	0
$691$	−17.8564	−0.679290	−0.339645	−	0.940554i	$-0.610307\pi$
$691$	−17.8564	−0.679290	−0.339645	+	0.940554i	$0.610307\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	24.0000	0.910372
$696$	0	0
$697$	20.7846	0.787273
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−19.8564	−0.749966	−0.374983	−	0.927032i	$-0.622351\pi$
$701$	−19.8564	−0.749966	−0.374983	+	0.927032i	$0.622351\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.3923	0.390843
$708$	0	0
$709$	−2.00000	−0.0751116	−0.0375558	−	0.999295i	$-0.511957\pi$
$709$	−2.00000	−0.0751116	−0.0375558	+	0.999295i	$0.511957\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	11.7128	0.438648
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	22.2487	0.829737	0.414868	−	0.909881i	$-0.363828\pi$
$719$	22.2487	0.829737	0.414868	+	0.909881i	$0.363828\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	52.2487	1.94047
$726$	0	0
$727$	−21.8564	−0.810609	−0.405305	−	0.914182i	$-0.632835\pi$
$727$	−21.8564	−0.810609	−0.405305	+	0.914182i	$0.632835\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	24.0000	0.887672
$732$	0	0
$733$	−18.7846	−0.693825	−0.346913	−	0.937897i	$-0.612770\pi$
$733$	−18.7846	−0.693825	−0.346913	+	0.937897i	$0.612770\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−11.2154	−0.412565	−0.206282	−	0.978492i	$-0.566137\pi$
$739$	−11.2154	−0.412565	−0.206282	+	0.978492i	$0.566137\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−28.3923	−1.04161	−0.520806	−	0.853675i	$-0.674369\pi$
$743$	−28.3923	−1.04161	−0.520806	+	0.853675i	$0.674369\pi$
$744$	0	0
$745$	−30.9282	−1.13312
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8.39230	0.306648
$750$	0	0
$751$	18.9282	0.690700	0.345350	−	0.938474i	$-0.387760\pi$
$751$	18.9282	0.690700	0.345350	+	0.938474i	$0.387760\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	37.8564	1.37774
$756$	0	0
$757$	39.5692	1.43817	0.719084	−	0.694923i	$-0.244562\pi$
$757$	39.5692	1.43817	0.719084	+	0.694923i	$0.244562\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	51.1769	1.85516	0.927581	−	0.373622i	$-0.121884\pi$
$761$	51.1769	1.85516	0.927581	+	0.373622i	$0.121884\pi$
$762$	0	0
$763$	−0.928203	−0.0336032
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	−31.5692	−1.13842	−0.569208	−	0.822194i	$-0.692750\pi$
$769$	−31.5692	−1.13842	−0.569208	+	0.822194i	$0.692750\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−15.8564	−0.570315	−0.285158	−	0.958481i	$-0.592046\pi$
$773$	−15.8564	−0.570315	−0.285158	+	0.958481i	$0.592046\pi$
$774$	0	0
$775$	−20.4974	−0.736289
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.00000	0.214972
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−54.9282	−1.96047
$786$	0	0
$787$	−4.78461	−0.170553	−0.0852765	−	0.996357i	$-0.527177\pi$
$787$	−4.78461	−0.170553	−0.0852765	+	0.996357i	$0.527177\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.53590	0.161278
$792$	0	0
$793$	−12.0000	−0.426132
$794$	0	0
$795$	0	0
$796$	0	0
$797$	41.7128	1.47754	0.738772	−	0.673956i	$-0.235406\pi$
$797$	41.7128	1.47754	0.738772	+	0.673956i	$0.235406\pi$
$798$	0	0
$799$	8.78461	0.310777
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	13.8564	0.488374
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−15.8564	−0.557482	−0.278741	−	0.960366i	$-0.589917\pi$
$809$	−15.8564	−0.557482	−0.278741	+	0.960366i	$0.589917\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.71281	−0.130054
$816$	0	0
$817$	6.92820	0.242387
$818$	0	0
$819$	0	0
$820$	0	0
$821$	44.9282	1.56801	0.784003	−	0.620758i	$-0.213175\pi$
$821$	44.9282	1.56801	0.784003	+	0.620758i	$0.213175\pi$
$822$	0	0
$823$	−2.92820	−0.102071	−0.0510354	−	0.998697i	$-0.516252\pi$
$823$	−2.92820	−0.102071	−0.0510354	+	0.998697i	$0.516252\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19.3205	0.671840	0.335920	−	0.941891i	$-0.390953\pi$
$827$	19.3205	0.671840	0.335920	+	0.941891i	$0.390953\pi$
$828$	0	0
$829$	−47.8564	−1.66212	−0.831061	−	0.556182i	$-0.812266\pi$
$829$	−47.8564	−1.66212	−0.831061	+	0.556182i	$0.812266\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.46410	−0.120024
$834$	0	0
$835$	−37.8564	−1.31007
$836$	0	0
$837$	0	0
$838$	0	0
$839$	37.8564	1.30695	0.653474	−	0.756949i	$-0.273311\pi$
$839$	37.8564	1.30695	0.653474	+	0.756949i	$0.273311\pi$
$840$	0	0
$841$	26.7128	0.921131
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−31.1769	−1.07252
$846$	0	0
$847$	11.0000	0.377964
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.00000	0.274236
$852$	0	0
$853$	47.5692	1.62874	0.814370	−	0.580347i	$-0.197083\pi$
$853$	47.5692	1.62874	0.814370	+	0.580347i	$0.197083\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.14359	0.141542	0.0707712	−	0.997493i	$-0.477454\pi$
$857$	4.14359	0.141542	0.0707712	+	0.997493i	$0.477454\pi$
$858$	0	0
$859$	31.7128	1.08203	0.541014	−	0.841014i	$-0.318041\pi$
$859$	31.7128	1.08203	0.541014	+	0.841014i	$0.318041\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−7.32051	−0.249193	−0.124596	−	0.992207i	$-0.539764\pi$
$863$	−7.32051	−0.249193	−0.124596	+	0.992207i	$0.539764\pi$
$864$	0	0
$865$	−6.92820	−0.235566
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−2.14359	−0.0726329
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−6.92820	−0.234216
$876$	0	0
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−41.3205	−1.39212	−0.696062	−	0.717982i	$-0.745066\pi$
$881$	−41.3205	−1.39212	−0.696062	+	0.717982i	$0.745066\pi$
$882$	0	0
$883$	−33.8564	−1.13936	−0.569679	−	0.821867i	$-0.692933\pi$
$883$	−33.8564	−1.13936	−0.569679	+	0.821867i	$0.692933\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22.6410	0.760211	0.380105	−	0.924943i	$-0.375888\pi$
$887$	22.6410	0.760211	0.380105	+	0.924943i	$0.375888\pi$
$888$	0	0
$889$	5.07180	0.170103
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.53590	0.0848606
$894$	0	0
$895$	−46.6410	−1.55904
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−21.8564	−0.728952
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.92820	−0.230301
$906$	0	0
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−57.4641	−1.90387	−0.951935	−	0.306299i	$-0.900909\pi$
$911$	−57.4641	−1.90387	−0.951935	+	0.306299i	$0.900909\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12.3923	0.409230
$918$	0	0
$919$	−51.7128	−1.70585	−0.852924	−	0.522035i	$-0.825173\pi$
$919$	−51.7128	−1.70585	−0.852924	+	0.522035i	$0.825173\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	13.0718	0.430263
$924$	0	0
$925$	−14.0000	−0.460317
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−27.4641	−0.901068	−0.450534	−	0.892759i	$-0.648766\pi$
$929$	−27.4641	−0.901068	−0.450534	+	0.892759i	$0.648766\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−1.71281	−0.0559552	−0.0279776	−	0.999609i	$-0.508907\pi$
$937$	−1.71281	−0.0559552	−0.0279776	+	0.999609i	$0.508907\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−9.21539	−0.300413	−0.150207	−	0.988655i	$-0.547994\pi$
$941$	−9.21539	−0.300413	−0.150207	+	0.988655i	$0.547994\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−29.8564	−0.970203	−0.485101	−	0.874458i	$-0.661217\pi$
$947$	−29.8564	−0.970203	−0.485101	+	0.874458i	$0.661217\pi$
$948$	0	0
$949$	−25.8564	−0.839334
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.60770	−0.0520784	−0.0260392	−	0.999661i	$-0.508289\pi$
$953$	−1.60770	−0.0520784	−0.0260392	+	0.999661i	$0.508289\pi$
$954$	0	0
$955$	−34.1436	−1.10486
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.00000	0.0645834
$960$	0	0
$961$	−22.4256	−0.723407
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−48.4974	−1.56119
$966$	0	0
$967$	2.92820	0.0941647	0.0470823	−	0.998891i	$-0.485008\pi$
$967$	2.92820	0.0941647	0.0470823	+	0.998891i	$0.485008\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−33.8564	−1.08650	−0.543252	−	0.839570i	$-0.682807\pi$
$971$	−33.8564	−1.08650	−0.543252	+	0.839570i	$0.682807\pi$
$972$	0	0
$973$	−6.92820	−0.222108
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−19.1769	−0.613524	−0.306762	−	0.951786i	$-0.599246\pi$
$977$	−19.1769	−0.613524	−0.306762	+	0.951786i	$0.599246\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	40.0000	1.27580	0.637901	−	0.770118i	$-0.279803\pi$
$983$	40.0000	1.27580	0.637901	+	0.770118i	$0.279803\pi$
$984$	0	0
$985$	27.2154	0.867154
$986$	0	0
$987$	0	0
$988$	0	0
$989$	27.7128	0.881216
$990$	0	0
$991$	25.5692	0.812233	0.406117	−	0.913821i	$-0.366883\pi$
$991$	25.5692	0.812233	0.406117	+	0.913821i	$0.366883\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−65.5692	−2.07868
$996$	0	0
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.bk.1.2		2
3.2	odd	2		3192.2.a.t.1.1	✓	2
12.11	even	2		6384.2.a.bk.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.t.1.1	✓	2	3.2	odd	2
6384.2.a.bk.1.1		2	12.11	even	2
9576.2.a.bk.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9576.a (trivial)

\(f(q)\)	\(=\)	\(q+3.46410 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+3.46410 q^{5} -1.00000 q^{7} -2.00000 q^{13} -3.46410 q^{17} -1.00000 q^{19} -4.00000 q^{23} +7.00000 q^{25} +7.46410 q^{29} -2.92820 q^{31} -3.46410 q^{35} -2.00000 q^{37} -6.00000 q^{41} -6.92820 q^{43} -2.53590 q^{47} +1.00000 q^{49} -3.46410 q^{53} -4.00000 q^{59} +6.00000 q^{61} -6.92820 q^{65} +1.07180 q^{67} -6.53590 q^{71} +12.9282 q^{73} -1.46410 q^{83} -12.0000 q^{85} -11.8564 q^{89} +2.00000 q^{91} -3.46410 q^{95} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^7	\( 2 q - 2 q^{7} - 4 q^{13} - 2 q^{19} - 8 q^{23} + 14 q^{25} + 8 q^{29} + 8 q^{31} - 4 q^{37} - 12 q^{41} - 12 q^{47} + 2 q^{49} - 8 q^{59} + 12 q^{61} + 16 q^{67} - 20 q^{71} + 12 q^{73} + 4 q^{83} - 24 q^{85} + 4 q^{89} + 4 q^{91} + 20 q^{97}+O(q^{100}) \) 2 * q - 2 * q^7 - 4 * q^13 - 2 * q^19 - 8 * q^23 + 14 * q^25 + 8 * q^29 + 8 * q^31 - 4 * q^37 - 12 * q^41 - 12 * q^47 + 2 * q^49 - 8 * q^59 + 12 * q^61 + 16 * q^67 - 20 * q^71 + 12 * q^73 + 4 * q^83 - 24 * q^85 + 4 * q^89 + 4 * q^91 + 20 * q^97

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