LMFDB - Embedded newform 9216.2.a.be.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9216,2,Mod(1,9216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9216, 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("9216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9216 = 2^{10} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9216.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:	$73.5901305028$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 2304)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.93185$ of defining polynomial
Character	$\chi$	$=$	9216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.46410 q^{5} -1.41421 q^{7} +O(q^{10})$	$q+3.46410 q^{5} -1.41421 q^{7} +4.89898 q^{11} -1.41421 q^{13} -4.89898 q^{17} -6.00000 q^{19} -6.92820 q^{23} +7.00000 q^{25} -3.46410 q^{29} +1.41421 q^{31} -4.89898 q^{35} +9.89949 q^{37} -4.89898 q^{41} -6.00000 q^{43} -6.92820 q^{47} -5.00000 q^{49} +3.46410 q^{53} +16.9706 q^{55} +9.79796 q^{59} +7.07107 q^{61} -4.89898 q^{65} +8.00000 q^{67} -13.8564 q^{71} -12.0000 q^{73} -6.92820 q^{77} -15.5563 q^{79} -14.6969 q^{83} -16.9706 q^{85} -9.79796 q^{89} +2.00000 q^{91} -20.7846 q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 24 q^{19} + 28 q^{25} - 24 q^{43} - 20 q^{49} + 32 q^{67} - 48 q^{73} + 8 q^{91}+O(q^{100}) $ 4 * q - 24 * q^19 + 28 * q^25 - 24 * q^43 - 20 * q^49 + 32 * q^67 - 48 * q^73 + 8 * q^91

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
$7$	−1.41421	−0.534522	−0.267261	−	0.963624i	$-0.586119\pi$
$7$	−1.41421	−0.534522	−0.267261	+	0.963624i	$0.586119\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.89898	1.47710	0.738549	−	0.674200i	$-0.235511\pi$
$11$	4.89898	1.47710	0.738549	+	0.674200i	$0.235511\pi$
$12$	0	0
$13$	−1.41421	−0.392232	−0.196116	−	0.980581i	$-0.562833\pi$
$13$	−1.41421	−0.392232	−0.196116	+	0.980581i	$0.562833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.89898	−1.18818	−0.594089	−	0.804400i	$-0.702487\pi$
$17$	−4.89898	−1.18818	−0.594089	+	0.804400i	$0.702487\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.92820	−1.44463	−0.722315	−	0.691564i	$-0.756922\pi$
$23$	−6.92820	−1.44463	−0.722315	+	0.691564i	$0.756922\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	1.41421	0.254000	0.127000	−	0.991903i	$-0.459465\pi$
$31$	1.41421	0.254000	0.127000	+	0.991903i	$0.459465\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.89898	−0.828079
$36$	0	0
$37$	9.89949	1.62747	0.813733	−	0.581238i	$-0.197432\pi$
$37$	9.89949	1.62747	0.813733	+	0.581238i	$0.197432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.89898	−0.765092	−0.382546	−	0.923936i	$-0.624953\pi$
$41$	−4.89898	−0.765092	−0.382546	+	0.923936i	$0.624953\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.46410	0.475831	0.237915	−	0.971286i	$-0.423536\pi$
$53$	3.46410	0.475831	0.237915	+	0.971286i	$0.423536\pi$
$54$	0	0
$55$	16.9706	2.28831
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.79796	1.27559	0.637793	−	0.770208i	$-0.279848\pi$
$59$	9.79796	1.27559	0.637793	+	0.770208i	$0.279848\pi$
$60$	0	0
$61$	7.07107	0.905357	0.452679	−	0.891674i	$-0.350468\pi$
$61$	7.07107	0.905357	0.452679	+	0.891674i	$0.350468\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.89898	−0.607644
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.8564	−1.64445	−0.822226	−	0.569160i	$-0.807268\pi$
$71$	−13.8564	−1.64445	−0.822226	+	0.569160i	$0.807268\pi$
$72$	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
$77$	−6.92820	−0.789542
$78$	0	0
$79$	−15.5563	−1.75023	−0.875113	−	0.483919i	$-0.839213\pi$
$79$	−15.5563	−1.75023	−0.875113	+	0.483919i	$0.839213\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−14.6969	−1.61320	−0.806599	−	0.591099i	$-0.798694\pi$
$83$	−14.6969	−1.61320	−0.806599	+	0.591099i	$0.798694\pi$
$84$	0	0
$85$	−16.9706	−1.84072
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.79796	−1.03858	−0.519291	−	0.854598i	$-0.673804\pi$
$89$	−9.79796	−1.03858	−0.519291	+	0.854598i	$0.673804\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−20.7846	−2.13246
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	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
$103$	−7.07107	−0.696733	−0.348367	−	0.937358i	$-0.613264\pi$
$103$	−7.07107	−0.696733	−0.348367	+	0.937358i	$0.613264\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	1.41421	0.135457	0.0677285	−	0.997704i	$-0.478425\pi$
$109$	1.41421	0.135457	0.0677285	+	0.997704i	$0.478425\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	19.5959	1.84343	0.921714	−	0.387869i	$-0.126789\pi$
$113$	19.5959	1.84343	0.921714	+	0.387869i	$0.126789\pi$
$114$	0	0
$115$	−24.0000	−2.23801
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.92820	0.635107
$120$	0	0
$121$	13.0000	1.18182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.92820	0.619677
$126$	0	0
$127$	−9.89949	−0.878438	−0.439219	−	0.898380i	$-0.644745\pi$
$127$	−9.89949	−0.878438	−0.439219	+	0.898380i	$0.644745\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−9.79796	−0.856052	−0.428026	−	0.903767i	$-0.640791\pi$
$131$	−9.79796	−0.856052	−0.428026	+	0.903767i	$0.640791\pi$
$132$	0	0
$133$	8.48528	0.735767
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.6969	1.25564	0.627822	−	0.778357i	$-0.283947\pi$
$137$	14.6969	1.25564	0.627822	+	0.778357i	$0.283947\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.92820	−0.579365
$144$	0	0
$145$	−12.0000	−0.996546
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.46410	−0.283790	−0.141895	−	0.989882i	$-0.545320\pi$
$149$	−3.46410	−0.283790	−0.141895	+	0.989882i	$0.545320\pi$
$150$	0	0
$151$	15.5563	1.26596	0.632979	−	0.774169i	$-0.281832\pi$
$151$	15.5563	1.26596	0.632979	+	0.774169i	$0.281832\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.89898	0.393496
$156$	0	0
$157$	9.89949	0.790066	0.395033	−	0.918667i	$-0.370733\pi$
$157$	9.89949	0.790066	0.395033	+	0.918667i	$0.370733\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	9.79796	0.772187
$162$	0	0
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.92820	0.536120	0.268060	−	0.963402i	$-0.413617\pi$
$167$	6.92820	0.536120	0.268060	+	0.963402i	$0.413617\pi$
$168$	0	0
$169$	−11.0000	−0.846154
$170$	0	0
$171$	0	0
$172$	0	0
$173$	17.3205	1.31685	0.658427	−	0.752645i	$-0.271222\pi$
$173$	17.3205	1.31685	0.658427	+	0.752645i	$0.271222\pi$
$174$	0	0
$175$	−9.89949	−0.748331
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	1.41421	0.105118	0.0525588	−	0.998618i	$-0.483262\pi$
$181$	1.41421	0.105118	0.0525588	+	0.998618i	$0.483262\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	34.2929	2.52126
$186$	0	0
$187$	−24.0000	−1.75505
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.7846	1.50392	0.751961	−	0.659208i	$-0.229108\pi$
$191$	20.7846	1.50392	0.751961	+	0.659208i	$0.229108\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.46410	0.246807	0.123404	−	0.992357i	$-0.460619\pi$
$197$	3.46410	0.246807	0.123404	+	0.992357i	$0.460619\pi$
$198$	0	0
$199$	7.07107	0.501255	0.250627	−	0.968084i	$-0.419363\pi$
$199$	7.07107	0.501255	0.250627	+	0.968084i	$0.419363\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.89898	0.343841
$204$	0	0
$205$	−16.9706	−1.18528
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−29.3939	−2.03322
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−20.7846	−1.41750
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.92820	0.466041
$222$	0	0
$223$	−7.07107	−0.473514	−0.236757	−	0.971569i	$-0.576084\pi$
$223$	−7.07107	−0.473514	−0.236757	+	0.971569i	$0.576084\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.89898	0.325157	0.162578	−	0.986696i	$-0.448019\pi$
$227$	4.89898	0.325157	0.162578	+	0.986696i	$0.448019\pi$
$228$	0	0
$229$	−1.41421	−0.0934539	−0.0467269	−	0.998908i	$-0.514879\pi$
$229$	−1.41421	−0.0934539	−0.0467269	+	0.998908i	$0.514879\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.79796	−0.641886	−0.320943	−	0.947099i	$-0.604000\pi$
$233$	−9.79796	−0.641886	−0.320943	+	0.947099i	$0.604000\pi$
$234$	0	0
$235$	−24.0000	−1.56559
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.92820	0.448148	0.224074	−	0.974572i	$-0.428064\pi$
$239$	6.92820	0.448148	0.224074	+	0.974572i	$0.428064\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−17.3205	−1.10657
$246$	0	0
$247$	8.48528	0.539906
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.4949	−1.54610	−0.773052	−	0.634343i	$-0.781271\pi$
$251$	−24.4949	−1.54610	−0.773052	+	0.634343i	$0.781271\pi$
$252$	0	0
$253$	−33.9411	−2.13386
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.79796	0.611180	0.305590	−	0.952163i	$-0.401146\pi$
$257$	9.79796	0.611180	0.305590	+	0.952163i	$0.401146\pi$
$258$	0	0
$259$	−14.0000	−0.869918
$260$	0	0
$261$	0	0
$262$	0	0
$263$	13.8564	0.854423	0.427211	−	0.904152i	$-0.359496\pi$
$263$	13.8564	0.854423	0.427211	+	0.904152i	$0.359496\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	0	0
$267$	0	0
$268$	0	0
$269$	10.3923	0.633630	0.316815	−	0.948487i	$-0.397387\pi$
$269$	10.3923	0.633630	0.316815	+	0.948487i	$0.397387\pi$
$270$	0	0
$271$	−7.07107	−0.429537	−0.214768	−	0.976665i	$-0.568900\pi$
$271$	−7.07107	−0.429537	−0.214768	+	0.976665i	$0.568900\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	34.2929	2.06794
$276$	0	0
$277$	−1.41421	−0.0849719	−0.0424859	−	0.999097i	$-0.513528\pi$
$277$	−1.41421	−0.0849719	−0.0424859	+	0.999097i	$0.513528\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−29.3939	−1.75349	−0.876746	−	0.480954i	$-0.840290\pi$
$281$	−29.3939	−1.75349	−0.876746	+	0.480954i	$0.840290\pi$
$282$	0	0
$283$	−8.00000	−0.475551	−0.237775	−	0.971320i	$-0.576418\pi$
$283$	−8.00000	−0.475551	−0.237775	+	0.971320i	$0.576418\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.92820	0.408959
$288$	0	0
$289$	7.00000	0.411765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	17.3205	1.01187	0.505937	−	0.862570i	$-0.331147\pi$
$293$	17.3205	1.01187	0.505937	+	0.862570i	$0.331147\pi$
$294$	0	0
$295$	33.9411	1.97613
$296$	0	0
$297$	0	0
$298$	0	0
$299$	9.79796	0.566631
$300$	0	0
$301$	8.48528	0.489083
$302$	0	0
$303$	0	0
$304$	0	0
$305$	24.4949	1.40257
$306$	0	0
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.8564	0.785725	0.392862	−	0.919597i	$-0.371485\pi$
$311$	13.8564	0.785725	0.392862	+	0.919597i	$0.371485\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.3923	−0.583690	−0.291845	−	0.956466i	$-0.594269\pi$
$317$	−10.3923	−0.583690	−0.291845	+	0.956466i	$0.594269\pi$
$318$	0	0
$319$	−16.9706	−0.950169
$320$	0	0
$321$	0	0
$322$	0	0
$323$	29.3939	1.63552
$324$	0	0
$325$	−9.89949	−0.549125
$326$	0	0
$327$	0	0
$328$	0	0
$329$	9.79796	0.540179
$330$	0	0
$331$	16.0000	0.879440	0.439720	−	0.898135i	$-0.355078\pi$
$331$	16.0000	0.879440	0.439720	+	0.898135i	$0.355078\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	27.7128	1.51411
$336$	0	0
$337$	12.0000	0.653682	0.326841	−	0.945079i	$-0.394016\pi$
$337$	12.0000	0.653682	0.326841	+	0.945079i	$0.394016\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.92820	0.375183
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.89898	−0.262991	−0.131495	−	0.991317i	$-0.541978\pi$
$347$	−4.89898	−0.262991	−0.131495	+	0.991317i	$0.541978\pi$
$348$	0	0
$349$	26.8701	1.43832	0.719161	−	0.694844i	$-0.244527\pi$
$349$	26.8701	1.43832	0.719161	+	0.694844i	$0.244527\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9.79796	−0.521493	−0.260746	−	0.965407i	$-0.583969\pi$
$353$	−9.79796	−0.521493	−0.260746	+	0.965407i	$0.583969\pi$
$354$	0	0
$355$	−48.0000	−2.54758
$356$	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
$361$	17.0000	0.894737
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−41.5692	−2.17583
$366$	0	0
$367$	−32.5269	−1.69789	−0.848945	−	0.528480i	$-0.822762\pi$
$367$	−32.5269	−1.69789	−0.848945	+	0.528480i	$0.822762\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4.89898	−0.254342
$372$	0	0
$373$	26.8701	1.39128	0.695639	−	0.718391i	$-0.255121\pi$
$373$	26.8701	1.39128	0.695639	+	0.718391i	$0.255121\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.89898	0.252310
$378$	0	0
$379$	−6.00000	−0.308199	−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000	−0.308199	−0.154100	+	0.988055i	$0.549248\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6.92820	0.354015	0.177007	−	0.984210i	$-0.443358\pi$
$383$	6.92820	0.354015	0.177007	+	0.984210i	$0.443358\pi$
$384$	0	0
$385$	−24.0000	−1.22315
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−24.2487	−1.22946	−0.614729	−	0.788738i	$-0.710735\pi$
$389$	−24.2487	−1.22946	−0.614729	+	0.788738i	$0.710735\pi$
$390$	0	0
$391$	33.9411	1.71648
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−53.8888	−2.71144
$396$	0	0
$397$	9.89949	0.496841	0.248421	−	0.968652i	$-0.420088\pi$
$397$	9.89949	0.496841	0.248421	+	0.968652i	$0.420088\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.6969	−0.733930	−0.366965	−	0.930235i	$-0.619603\pi$
$401$	−14.6969	−0.733930	−0.366965	+	0.930235i	$0.619603\pi$
$402$	0	0
$403$	−2.00000	−0.0996271
$404$	0	0
$405$	0	0
$406$	0	0
$407$	48.4974	2.40393
$408$	0	0
$409$	−24.0000	−1.18672	−0.593362	−	0.804936i	$-0.702200\pi$
$409$	−24.0000	−1.18672	−0.593362	+	0.804936i	$0.702200\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−13.8564	−0.681829
$414$	0	0
$415$	−50.9117	−2.49916
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14.6969	0.717992	0.358996	−	0.933339i	$-0.383119\pi$
$419$	14.6969	0.717992	0.358996	+	0.933339i	$0.383119\pi$
$420$	0	0
$421$	−35.3553	−1.72311	−0.861557	−	0.507661i	$-0.830510\pi$
$421$	−35.3553	−1.72311	−0.861557	+	0.507661i	$0.830510\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−34.2929	−1.66345
$426$	0	0
$427$	−10.0000	−0.483934
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.92820	0.333720	0.166860	−	0.985981i	$-0.446637\pi$
$431$	6.92820	0.333720	0.166860	+	0.985981i	$0.446637\pi$
$432$	0	0
$433$	−24.0000	−1.15337	−0.576683	−	0.816968i	$-0.695653\pi$
$433$	−24.0000	−1.15337	−0.576683	+	0.816968i	$0.695653\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	41.5692	1.98853
$438$	0	0
$439$	−26.8701	−1.28244	−0.641219	−	0.767358i	$-0.721571\pi$
$439$	−26.8701	−1.28244	−0.641219	+	0.767358i	$0.721571\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14.6969	0.698273	0.349136	−	0.937072i	$-0.386475\pi$
$443$	14.6969	0.698273	0.349136	+	0.937072i	$0.386475\pi$
$444$	0	0
$445$	−33.9411	−1.60896
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−34.2929	−1.61838	−0.809190	−	0.587547i	$-0.800094\pi$
$449$	−34.2929	−1.61838	−0.809190	+	0.587547i	$0.800094\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.92820	0.324799
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3.46410	−0.161339	−0.0806696	−	0.996741i	$-0.525706\pi$
$461$	−3.46410	−0.161339	−0.0806696	+	0.996741i	$0.525706\pi$
$462$	0	0
$463$	24.0416	1.11731	0.558655	−	0.829400i	$-0.311318\pi$
$463$	24.0416	1.11731	0.558655	+	0.829400i	$0.311318\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.89898	0.226698	0.113349	−	0.993555i	$-0.463842\pi$
$467$	4.89898	0.226698	0.113349	+	0.993555i	$0.463842\pi$
$468$	0	0
$469$	−11.3137	−0.522419
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−29.3939	−1.35153
$474$	0	0
$475$	−42.0000	−1.92709
$476$	0	0
$477$	0	0
$478$	0	0
$479$	27.7128	1.26623	0.633115	−	0.774057i	$-0.281776\pi$
$479$	27.7128	1.26623	0.633115	+	0.774057i	$0.281776\pi$
$480$	0	0
$481$	−14.0000	−0.638345
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	35.3553	1.60210	0.801052	−	0.598595i	$-0.204274\pi$
$487$	35.3553	1.60210	0.801052	+	0.598595i	$0.204274\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−29.3939	−1.32653	−0.663264	−	0.748386i	$-0.730829\pi$
$491$	−29.3939	−1.32653	−0.663264	+	0.748386i	$0.730829\pi$
$492$	0	0
$493$	16.9706	0.764316
$494$	0	0
$495$	0	0
$496$	0	0
$497$	19.5959	0.878997
$498$	0	0
$499$	−8.00000	−0.358129	−0.179065	−	0.983837i	$-0.557307\pi$
$499$	−8.00000	−0.358129	−0.179065	+	0.983837i	$0.557307\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−36.0000	−1.60198
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−10.3923	−0.460631	−0.230315	−	0.973116i	$-0.573976\pi$
$509$	−10.3923	−0.460631	−0.230315	+	0.973116i	$0.573976\pi$
$510$	0	0
$511$	16.9706	0.750733
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−24.4949	−1.07937
$516$	0	0
$517$	−33.9411	−1.49273
$518$	0	0
$519$	0	0
$520$	0	0
$521$	14.6969	0.643885	0.321942	−	0.946759i	$-0.395664\pi$
$521$	14.6969	0.643885	0.321942	+	0.946759i	$0.395664\pi$
$522$	0	0
$523$	−30.0000	−1.31181	−0.655904	−	0.754844i	$-0.727712\pi$
$523$	−30.0000	−1.31181	−0.655904	+	0.754844i	$0.727712\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.92820	−0.301797
$528$	0	0
$529$	25.0000	1.08696
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.92820	0.300094
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−24.4949	−1.05507
$540$	0	0
$541$	15.5563	0.668820	0.334410	−	0.942428i	$-0.391463\pi$
$541$	15.5563	0.668820	0.334410	+	0.942428i	$0.391463\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4.89898	0.209849
$546$	0	0
$547$	−18.0000	−0.769624	−0.384812	−	0.922995i	$-0.625734\pi$
$547$	−18.0000	−0.769624	−0.384812	+	0.922995i	$0.625734\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	20.7846	0.885454
$552$	0	0
$553$	22.0000	0.935535
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.3923	0.440336	0.220168	−	0.975462i	$-0.429339\pi$
$557$	10.3923	0.440336	0.220168	+	0.975462i	$0.429339\pi$
$558$	0	0
$559$	8.48528	0.358889
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−4.89898	−0.206467	−0.103234	−	0.994657i	$-0.532919\pi$
$563$	−4.89898	−0.206467	−0.103234	+	0.994657i	$0.532919\pi$
$564$	0	0
$565$	67.8823	2.85583
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.6969	0.616128	0.308064	−	0.951366i	$-0.400319\pi$
$569$	14.6969	0.616128	0.308064	+	0.951366i	$0.400319\pi$
$570$	0	0
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−48.4974	−2.02248
$576$	0	0
$577$	12.0000	0.499567	0.249783	−	0.968302i	$-0.419641\pi$
$577$	12.0000	0.499567	0.249783	+	0.968302i	$0.419641\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	20.7846	0.862291
$582$	0	0
$583$	16.9706	0.702849
$584$	0	0
$585$	0	0
$586$	0	0
$587$	39.1918	1.61762	0.808810	−	0.588070i	$-0.200112\pi$
$587$	39.1918	1.61762	0.808810	+	0.588070i	$0.200112\pi$
$588$	0	0
$589$	−8.48528	−0.349630
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−39.1918	−1.60942	−0.804708	−	0.593671i	$-0.797678\pi$
$593$	−39.1918	−1.60942	−0.804708	+	0.593671i	$0.797678\pi$
$594$	0	0
$595$	24.0000	0.983904
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6.92820	0.283079	0.141539	−	0.989933i	$-0.454795\pi$
$599$	6.92820	0.283079	0.141539	+	0.989933i	$0.454795\pi$
$600$	0	0
$601$	12.0000	0.489490	0.244745	−	0.969587i	$-0.421296\pi$
$601$	12.0000	0.489490	0.244745	+	0.969587i	$0.421296\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	45.0333	1.83086
$606$	0	0
$607$	−9.89949	−0.401808	−0.200904	−	0.979611i	$-0.564388\pi$
$607$	−9.89949	−0.401808	−0.200904	+	0.979611i	$0.564388\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.79796	0.396383
$612$	0	0
$613$	26.8701	1.08527	0.542636	−	0.839968i	$-0.317426\pi$
$613$	26.8701	1.08527	0.542636	+	0.839968i	$0.317426\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19.5959	0.788902	0.394451	−	0.918917i	$-0.370935\pi$
$617$	19.5959	0.788902	0.394451	+	0.918917i	$0.370935\pi$
$618$	0	0
$619$	16.0000	0.643094	0.321547	−	0.946894i	$-0.395797\pi$
$619$	16.0000	0.643094	0.321547	+	0.946894i	$0.395797\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	13.8564	0.555145
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−48.4974	−1.93372
$630$	0	0
$631$	1.41421	0.0562990	0.0281495	−	0.999604i	$-0.491039\pi$
$631$	1.41421	0.0562990	0.0281495	+	0.999604i	$0.491039\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−34.2929	−1.36087
$636$	0	0
$637$	7.07107	0.280166
$638$	0	0
$639$	0	0
$640$	0	0
$641$	14.6969	0.580494	0.290247	−	0.956952i	$-0.406263\pi$
$641$	14.6969	0.580494	0.290247	+	0.956952i	$0.406263\pi$
$642$	0	0
$643$	−42.0000	−1.65632	−0.828159	−	0.560493i	$-0.810612\pi$
$643$	−42.0000	−1.65632	−0.828159	+	0.560493i	$0.810612\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.92820	0.272376	0.136188	−	0.990683i	$-0.456515\pi$
$647$	6.92820	0.272376	0.136188	+	0.990683i	$0.456515\pi$
$648$	0	0
$649$	48.0000	1.88416
$650$	0	0
$651$	0	0
$652$	0	0
$653$	38.1051	1.49117	0.745584	−	0.666411i	$-0.232171\pi$
$653$	38.1051	1.49117	0.745584	+	0.666411i	$0.232171\pi$
$654$	0	0
$655$	−33.9411	−1.32619
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−9.79796	−0.381674	−0.190837	−	0.981622i	$-0.561120\pi$
$659$	−9.79796	−0.381674	−0.190837	+	0.981622i	$0.561120\pi$
$660$	0	0
$661$	−26.8701	−1.04512	−0.522562	−	0.852601i	$-0.675024\pi$
$661$	−26.8701	−1.04512	−0.522562	+	0.852601i	$0.675024\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	29.3939	1.13985
$666$	0	0
$667$	24.0000	0.929284
$668$	0	0
$669$	0	0
$670$	0	0
$671$	34.6410	1.33730
$672$	0	0
$673$	−24.0000	−0.925132	−0.462566	−	0.886585i	$-0.653071\pi$
$673$	−24.0000	−0.925132	−0.462566	+	0.886585i	$0.653071\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.1051	1.46450	0.732249	−	0.681037i	$-0.238471\pi$
$677$	38.1051	1.46450	0.732249	+	0.681037i	$0.238471\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24.4949	0.937271	0.468636	−	0.883392i	$-0.344746\pi$
$683$	24.4949	0.937271	0.468636	+	0.883392i	$0.344746\pi$
$684$	0	0
$685$	50.9117	1.94524
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4.89898	−0.186636
$690$	0	0
$691$	−6.00000	−0.228251	−0.114125	−	0.993466i	$-0.536407\pi$
$691$	−6.00000	−0.228251	−0.114125	+	0.993466i	$0.536407\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−55.4256	−2.10241
$696$	0	0
$697$	24.0000	0.909065
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−3.46410	−0.130837	−0.0654187	−	0.997858i	$-0.520838\pi$
$701$	−3.46410	−0.130837	−0.0654187	+	0.997858i	$0.520838\pi$
$702$	0	0
$703$	−59.3970	−2.24020
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.6969	0.552735
$708$	0	0
$709$	1.41421	0.0531119	0.0265560	−	0.999647i	$-0.491546\pi$
$709$	1.41421	0.0531119	0.0265560	+	0.999647i	$0.491546\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−9.79796	−0.366936
$714$	0	0
$715$	−24.0000	−0.897549
$716$	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
$721$	10.0000	0.372419
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−24.2487	−0.900575
$726$	0	0
$727$	−15.5563	−0.576953	−0.288477	−	0.957487i	$-0.593149\pi$
$727$	−15.5563	−0.576953	−0.288477	+	0.957487i	$0.593149\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	29.3939	1.08717
$732$	0	0
$733$	−15.5563	−0.574587	−0.287293	−	0.957843i	$-0.592755\pi$
$733$	−15.5563	−0.574587	−0.287293	+	0.957843i	$0.592755\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	39.1918	1.44365
$738$	0	0
$739$	8.00000	0.294285	0.147142	−	0.989115i	$-0.452992\pi$
$739$	8.00000	0.294285	0.147142	+	0.989115i	$0.452992\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−34.6410	−1.27086	−0.635428	−	0.772160i	$-0.719176\pi$
$743$	−34.6410	−1.27086	−0.635428	+	0.772160i	$0.719176\pi$
$744$	0	0
$745$	−12.0000	−0.439646
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−18.3848	−0.670870	−0.335435	−	0.942063i	$-0.608883\pi$
$751$	−18.3848	−0.670870	−0.335435	+	0.942063i	$0.608883\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	53.8888	1.96121
$756$	0	0
$757$	35.3553	1.28501	0.642506	−	0.766281i	$-0.277895\pi$
$757$	35.3553	1.28501	0.642506	+	0.766281i	$0.277895\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.89898	0.177588	0.0887939	−	0.996050i	$-0.471699\pi$
$761$	4.89898	0.177588	0.0887939	+	0.996050i	$0.471699\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−13.8564	−0.500326
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−38.1051	−1.37055	−0.685273	−	0.728286i	$-0.740317\pi$
$773$	−38.1051	−1.37055	−0.685273	+	0.728286i	$0.740317\pi$
$774$	0	0
$775$	9.89949	0.355600
$776$	0	0
$777$	0	0
$778$	0	0
$779$	29.3939	1.05314
$780$	0	0
$781$	−67.8823	−2.42902
$782$	0	0
$783$	0	0
$784$	0	0
$785$	34.2929	1.22396
$786$	0	0
$787$	−18.0000	−0.641631	−0.320815	−	0.947142i	$-0.603957\pi$
$787$	−18.0000	−0.641631	−0.320815	+	0.947142i	$0.603957\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−27.7128	−0.985354
$792$	0	0
$793$	−10.0000	−0.355110
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−51.9615	−1.84057	−0.920286	−	0.391247i	$-0.872044\pi$
$797$	−51.9615	−1.84057	−0.920286	+	0.391247i	$0.872044\pi$
$798$	0	0
$799$	33.9411	1.20075
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−58.7878	−2.07457
$804$	0	0
$805$	33.9411	1.19627
$806$	0	0
$807$	0	0
$808$	0	0
$809$	4.89898	0.172239	0.0861195	−	0.996285i	$-0.472553\pi$
$809$	4.89898	0.172239	0.0861195	+	0.996285i	$0.472553\pi$
$810$	0	0
$811$	6.00000	0.210688	0.105344	−	0.994436i	$-0.466406\pi$
$811$	6.00000	0.210688	0.105344	+	0.994436i	$0.466406\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−20.7846	−0.728053
$816$	0	0
$817$	36.0000	1.25948
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10.3923	0.362694	0.181347	−	0.983419i	$-0.441954\pi$
$821$	10.3923	0.362694	0.181347	+	0.983419i	$0.441954\pi$
$822$	0	0
$823$	35.3553	1.23241	0.616205	−	0.787586i	$-0.288669\pi$
$823$	35.3553	1.23241	0.616205	+	0.787586i	$0.288669\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	48.9898	1.70354	0.851771	−	0.523914i	$-0.175529\pi$
$827$	48.9898	1.70354	0.851771	+	0.523914i	$0.175529\pi$
$828$	0	0
$829$	−32.5269	−1.12971	−0.564853	−	0.825191i	$-0.691067\pi$
$829$	−32.5269	−1.12971	−0.564853	+	0.825191i	$0.691067\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	24.4949	0.848698
$834$	0	0
$835$	24.0000	0.830554
$836$	0	0
$837$	0	0
$838$	0	0
$839$	13.8564	0.478376	0.239188	−	0.970973i	$-0.423119\pi$
$839$	13.8564	0.478376	0.239188	+	0.970973i	$0.423119\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−38.1051	−1.31086
$846$	0	0
$847$	−18.3848	−0.631708
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−68.5857	−2.35109
$852$	0	0
$853$	−41.0122	−1.40423	−0.702115	−	0.712063i	$-0.747761\pi$
$853$	−41.0122	−1.40423	−0.702115	+	0.712063i	$0.747761\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	24.4949	0.836730	0.418365	−	0.908279i	$-0.362603\pi$
$857$	24.4949	0.836730	0.418365	+	0.908279i	$0.362603\pi$
$858$	0	0
$859$	−18.0000	−0.614152	−0.307076	−	0.951685i	$-0.599351\pi$
$859$	−18.0000	−0.614152	−0.307076	+	0.951685i	$0.599351\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−6.92820	−0.235839	−0.117919	−	0.993023i	$-0.537622\pi$
$863$	−6.92820	−0.235839	−0.117919	+	0.993023i	$0.537622\pi$
$864$	0	0
$865$	60.0000	2.04006
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−76.2102	−2.58526
$870$	0	0
$871$	−11.3137	−0.383350
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−9.79796	−0.331231
$876$	0	0
$877$	7.07107	0.238773	0.119386	−	0.992848i	$-0.461907\pi$
$877$	7.07107	0.238773	0.119386	+	0.992848i	$0.461907\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	9.79796	0.330102	0.165051	−	0.986285i	$-0.447221\pi$
$881$	9.79796	0.330102	0.165051	+	0.986285i	$0.447221\pi$
$882$	0	0
$883$	18.0000	0.605748	0.302874	−	0.953031i	$-0.402054\pi$
$883$	18.0000	0.605748	0.302874	+	0.953031i	$0.402054\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−27.7128	−0.930505	−0.465253	−	0.885178i	$-0.654037\pi$
$887$	−27.7128	−0.930505	−0.465253	+	0.885178i	$0.654037\pi$
$888$	0	0
$889$	14.0000	0.469545
$890$	0	0
$891$	0	0
$892$	0	0
$893$	41.5692	1.39106
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4.89898	−0.163390
$900$	0	0
$901$	−16.9706	−0.565371
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.89898	0.162848
$906$	0	0
$907$	30.0000	0.996134	0.498067	−	0.867139i	$-0.334043\pi$
$907$	30.0000	0.996134	0.498067	+	0.867139i	$0.334043\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	34.6410	1.14771	0.573854	−	0.818958i	$-0.305448\pi$
$911$	34.6410	1.14771	0.573854	+	0.818958i	$0.305448\pi$
$912$	0	0
$913$	−72.0000	−2.38285
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13.8564	0.457579
$918$	0	0
$919$	35.3553	1.16627	0.583133	−	0.812377i	$-0.301827\pi$
$919$	35.3553	1.16627	0.583133	+	0.812377i	$0.301827\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	19.5959	0.645007
$924$	0	0
$925$	69.2965	2.27845
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−24.4949	−0.803652	−0.401826	−	0.915716i	$-0.631624\pi$
$929$	−24.4949	−0.803652	−0.401826	+	0.915716i	$0.631624\pi$
$930$	0	0
$931$	30.0000	0.983210
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−83.1384	−2.71892
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	51.9615	1.69390	0.846949	−	0.531675i	$-0.178437\pi$
$941$	51.9615	1.69390	0.846949	+	0.531675i	$0.178437\pi$
$942$	0	0
$943$	33.9411	1.10528
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−39.1918	−1.27356	−0.636782	−	0.771044i	$-0.719735\pi$
$947$	−39.1918	−1.27356	−0.636782	+	0.771044i	$0.719735\pi$
$948$	0	0
$949$	16.9706	0.550888
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−44.0908	−1.42824	−0.714121	−	0.700022i	$-0.753173\pi$
$953$	−44.0908	−1.42824	−0.714121	+	0.700022i	$0.753173\pi$
$954$	0	0
$955$	72.0000	2.32987
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−20.7846	−0.671170
$960$	0	0
$961$	−29.0000	−0.935484
$962$	0	0
$963$	0	0
$964$	0	0
$965$	34.6410	1.11513
$966$	0	0
$967$	−24.0416	−0.773127	−0.386563	−	0.922263i	$-0.626338\pi$
$967$	−24.0416	−0.773127	−0.386563	+	0.922263i	$0.626338\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	14.6969	0.471647	0.235824	−	0.971796i	$-0.424221\pi$
$971$	14.6969	0.471647	0.235824	+	0.971796i	$0.424221\pi$
$972$	0	0
$973$	22.6274	0.725402
$974$	0	0
$975$	0	0
$976$	0	0
$977$	53.8888	1.72405	0.862027	−	0.506862i	$-0.169195\pi$
$977$	53.8888	1.72405	0.862027	+	0.506862i	$0.169195\pi$
$978$	0	0
$979$	−48.0000	−1.53409
$980$	0	0
$981$	0	0
$982$	0	0
$983$	55.4256	1.76780	0.883901	−	0.467673i	$-0.154908\pi$
$983$	55.4256	1.76780	0.883901	+	0.467673i	$0.154908\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	0	0
$987$	0	0
$988$	0	0
$989$	41.5692	1.32182
$990$	0	0
$991$	15.5563	0.494164	0.247082	−	0.968995i	$-0.420528\pi$
$991$	15.5563	0.494164	0.247082	+	0.968995i	$0.420528\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	24.4949	0.776540
$996$	0	0
$997$	9.89949	0.313520	0.156760	−	0.987637i	$-0.449895\pi$
$997$	9.89949	0.313520	0.156760	+	0.987637i	$0.449895\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9216.2.a.be.1.3		4
3.2	odd	2	inner	9216.2.a.be.1.1		4
4.3	odd	2		9216.2.a.bh.1.4		4
8.3	odd	2	inner	9216.2.a.be.1.2		4
8.5	even	2		9216.2.a.bh.1.1		4
12.11	even	2		9216.2.a.bh.1.2		4
24.5	odd	2		9216.2.a.bh.1.3		4
24.11	even	2	inner	9216.2.a.be.1.4		4
32.3	odd	8		2304.2.k.i.577.1	yes	8
32.5	even	8		2304.2.k.h.1729.3	yes	8
32.11	odd	8		2304.2.k.i.1729.2	yes	8
32.13	even	8		2304.2.k.h.577.4	yes	8
32.19	odd	8		2304.2.k.h.577.3	yes	8
32.21	even	8		2304.2.k.i.1729.1	yes	8
32.27	odd	8		2304.2.k.h.1729.4	yes	8
32.29	even	8		2304.2.k.i.577.2	yes	8
96.5	odd	8		2304.2.k.h.1729.1	yes	8
96.11	even	8		2304.2.k.i.1729.4	yes	8
96.29	odd	8		2304.2.k.i.577.4	yes	8
96.35	even	8		2304.2.k.i.577.3	yes	8
96.53	odd	8		2304.2.k.i.1729.3	yes	8
96.59	even	8		2304.2.k.h.1729.2	yes	8
96.77	odd	8		2304.2.k.h.577.2	yes	8
96.83	even	8		2304.2.k.h.577.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2304.2.k.h.577.1	✓	8	96.83	even	8
2304.2.k.h.577.2	yes	8	96.77	odd	8
2304.2.k.h.577.3	yes	8	32.19	odd	8
2304.2.k.h.577.4	yes	8	32.13	even	8
2304.2.k.h.1729.1	yes	8	96.5	odd	8
2304.2.k.h.1729.2	yes	8	96.59	even	8
2304.2.k.h.1729.3	yes	8	32.5	even	8
2304.2.k.h.1729.4	yes	8	32.27	odd	8
2304.2.k.i.577.1	yes	8	32.3	odd	8
2304.2.k.i.577.2	yes	8	32.29	even	8
2304.2.k.i.577.3	yes	8	96.35	even	8
2304.2.k.i.577.4	yes	8	96.29	odd	8
2304.2.k.i.1729.1	yes	8	32.21	even	8
2304.2.k.i.1729.2	yes	8	32.11	odd	8
2304.2.k.i.1729.3	yes	8	96.53	odd	8
2304.2.k.i.1729.4	yes	8	96.11	even	8
9216.2.a.be.1.1		4	3.2	odd	2	inner
9216.2.a.be.1.2		4	8.3	odd	2	inner
9216.2.a.be.1.3		4	1.1	even	1	trivial
9216.2.a.be.1.4		4	24.11	even	2	inner
9216.2.a.bh.1.1		4	8.5	even	2
9216.2.a.bh.1.2		4	12.11	even	2
9216.2.a.bh.1.3		4	24.5	odd	2
9216.2.a.bh.1.4		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9216 = 2^{10} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9216.a (trivial)

\(f(q)\)	\(=\)	\(q+3.46410 q^{5} -1.41421 q^{7} +O(q^{10})\)	\(q+3.46410 q^{5} -1.41421 q^{7} +4.89898 q^{11} -1.41421 q^{13} -4.89898 q^{17} -6.00000 q^{19} -6.92820 q^{23} +7.00000 q^{25} -3.46410 q^{29} +1.41421 q^{31} -4.89898 q^{35} +9.89949 q^{37} -4.89898 q^{41} -6.00000 q^{43} -6.92820 q^{47} -5.00000 q^{49} +3.46410 q^{53} +16.9706 q^{55} +9.79796 q^{59} +7.07107 q^{61} -4.89898 q^{65} +8.00000 q^{67} -13.8564 q^{71} -12.0000 q^{73} -6.92820 q^{77} -15.5563 q^{79} -14.6969 q^{83} -16.9706 q^{85} -9.79796 q^{89} +2.00000 q^{91} -20.7846 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 24 q^{19} + 28 q^{25} - 24 q^{43} - 20 q^{49} + 32 q^{67} - 48 q^{73} + 8 q^{91}+O(q^{100}) \) 4 * q - 24 * q^19 + 28 * q^25 - 24 * q^43 - 20 * q^49 + 32 * q^67 - 48 * q^73 + 8 * q^91

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