LMFDB - Embedded newform 5184.2.f.e.2591.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5184,2,Mod(2591,5184)]

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([1, 1, 1]))

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

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

chi := DirichletCharacter("5184.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$41.3944484078$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8x^{14} + 49x^{12} - 104x^{10} + 160x^{8} - 104x^{6} + 49x^{4} - 8x^{2} + 1 $ x^16 - 8x^14 + 49x^12 - 104x^10 + 160x^8 - 104x^6 + 49x^4 - 8*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.9
Root			$2.04058 - 1.17813i$ of defining polynomial
Character	$\chi$	$=$	5184.2591
Dual form			5184.2.f.e.2591.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.206954 q^{5} -3.03957i q^{7} +O(q^{10})$	$q+0.206954 q^{5} -3.03957i q^{7} +5.02319i q^{11} +4.03957i q^{13} -4.81624i q^{17} -1.03957 q^{19} +5.43710 q^{23} -4.95717 q^{25} +2.90487 q^{29} +1.23898i q^{31} -0.629051i q^{35} +8.77162i q^{37} +0.979317i q^{41} +3.18555 q^{43} -3.26332 q^{47} -2.23898 q^{49} -7.91987 q^{53} +1.03957i q^{55} -14.8544i q^{59} +9.53264i q^{61} +0.836005i q^{65} +0.735311 q^{67} +2.33127 q^{71} -10.0363 q^{73} +15.2683 q^{77} +14.1929i q^{79} +4.23665i q^{83} -0.996740i q^{85} +11.6053i q^{89} +12.2786 q^{91} -0.215143 q^{95} +4.84016 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 24 q^{19} + 16 q^{25} + 24 q^{43} - 48 q^{49} + 120 q^{67} + 16 q^{73} + 168 q^{91} - 16 q^{97}+O(q^{100}) $ 16 * q + 24 * q^19 + 16 * q^25 + 24 * q^43 - 48 * q^49 + 120 * q^67 + 16 * q^73 + 168 * q^91 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0.206954	0.0925526	0.0462763	−	0.998929i	$-0.485265\pi$
$5$	0.206954	0.0925526	0.0462763	+	0.998929i	$0.485265\pi$
$6$	0	0
$7$	− 3.03957i	− 1.14885i	−0.818557	−	0.574425i	$-0.805226\pi$
$7$	− 3.03957i	− 1.14885i	0.818557	−	0.574425i	$-0.194774\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.02319i	1.51455i	0.653096	+	0.757275i	$0.273470\pi$
$11$	5.02319i	1.51455i	−0.653096	+	0.757275i	$0.726530\pi$
$12$	0	0
$13$	4.03957i	1.12037i	0.828366	+	0.560187i	$0.189271\pi$
$13$	4.03957i	1.12037i	−0.828366	+	0.560187i	$0.810729\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.81624i	− 1.16811i	−0.811714	−	0.584055i	$-0.801465\pi$
$17$	− 4.81624i	− 1.16811i	0.811714	−	0.584055i	$-0.198535\pi$
$18$	0	0
$19$	−1.03957	−0.238494	−0.119247	−	0.992865i	$-0.538048\pi$
$19$	−1.03957	−0.238494	−0.119247	+	0.992865i	$0.538048\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.43710	1.13371	0.566857	−	0.823816i	$-0.308159\pi$
$23$	5.43710	1.13371	0.566857	+	0.823816i	$0.308159\pi$
$24$	0	0
$25$	−4.95717	−0.991434
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.90487	0.539421	0.269710	−	0.962941i	$-0.413072\pi$
$29$	2.90487	0.539421	0.269710	+	0.962941i	$0.413072\pi$
$30$	0	0
$31$	1.23898i	0.222528i	0.993791	+	0.111264i	$0.0354899\pi$
$31$	1.23898i	0.222528i	−0.993791	+	0.111264i	$0.964510\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 0.629051i	− 0.106329i
$36$	0	0
$37$	8.77162i	1.44205i	0.692911	+	0.721023i	$0.256328\pi$
$37$	8.77162i	1.44205i	−0.692911	+	0.721023i	$0.743672\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.979317i	0.152944i	0.997072	+	0.0764718i	$0.0243655\pi$
$41$	0.979317i	0.152944i	−0.997072	+	0.0764718i	$0.975634\pi$
$42$	0	0
$43$	3.18555	0.485792	0.242896	−	0.970052i	$-0.421903\pi$
$43$	3.18555	0.485792	0.242896	+	0.970052i	$0.421903\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.26332	−0.476005	−0.238002	−	0.971265i	$-0.576493\pi$
$47$	−3.26332	−0.476005	−0.238002	+	0.971265i	$0.576493\pi$
$48$	0	0
$49$	−2.23898	−0.319855
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−7.91987	−1.08788	−0.543939	−	0.839125i	$-0.683068\pi$
$53$	−7.91987	−1.08788	−0.543939	+	0.839125i	$0.683068\pi$
$54$	0	0
$55$	1.03957i	0.140176i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 14.8544i	− 1.93388i	−0.254999	−	0.966941i	$-0.582075\pi$
$59$	− 14.8544i	− 1.93388i	0.254999	−	0.966941i	$-0.417925\pi$
$60$	0	0
$61$	9.53264i	1.22053i	0.792198	+	0.610265i	$0.208937\pi$
$61$	9.53264i	1.22053i	−0.792198	+	0.610265i	$0.791063\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.836005i	0.103694i
$66$	0	0
$67$	0.735311	0.0898326	0.0449163	−	0.998991i	$-0.485698\pi$
$67$	0.735311	0.0898326	0.0449163	+	0.998991i	$0.485698\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.33127	0.276671	0.138336	−	0.990385i	$-0.455825\pi$
$71$	2.33127	0.276671	0.138336	+	0.990385i	$0.455825\pi$
$72$	0	0
$73$	−10.0363	−1.17466	−0.587331	−	0.809347i	$-0.699821\pi$
$73$	−10.0363	−1.17466	−0.587331	+	0.809347i	$0.699821\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	15.2683	1.73999
$78$	0	0
$79$	14.1929i	1.59683i	0.602111	+	0.798413i	$0.294327\pi$
$79$	14.1929i	1.59683i	−0.602111	+	0.798413i	$0.705673\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.23665i	0.465032i	0.972593	+	0.232516i	$0.0746959\pi$
$83$	4.23665i	0.465032i	−0.972593	+	0.232516i	$0.925304\pi$
$84$	0	0
$85$	− 0.996740i	− 0.108112i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.6053i	1.23016i	0.788465	+	0.615079i	$0.210876\pi$
$89$	11.6053i	1.23016i	−0.788465	+	0.615079i	$0.789124\pi$
$90$	0	0
$91$	12.2786	1.28714
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.215143	−0.0220732
$96$	0	0
$97$	4.84016	0.491443	0.245722	−	0.969340i	$-0.420975\pi$
$97$	4.84016	0.491443	0.245722	+	0.969340i	$0.420975\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−17.3949	−1.73085	−0.865426	−	0.501036i	$-0.832952\pi$
$101$	−17.3949	−1.73085	−0.865426	+	0.501036i	$0.832952\pi$
$102$	0	0
$103$	2.00000i	0.197066i	0.995134	+	0.0985329i	$0.0314150\pi$
$103$	2.00000i	0.197066i	−0.995134	+	0.0985329i	$0.968585\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.4456i	1.10649i	0.833019	+	0.553244i	$0.186610\pi$
$107$	11.4456i	1.10649i	−0.833019	+	0.553244i	$0.813390\pi$
$108$	0	0
$109$	− 7.80793i	− 0.747864i	−0.927456	−	0.373932i	$-0.878009\pi$
$109$	− 7.80793i	− 0.747864i	0.927456	−	0.373932i	$-0.121991\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	19.6707i	1.85046i	0.379405	+	0.925231i	$0.376129\pi$
$113$	19.6707i	1.85046i	−0.379405	+	0.925231i	$0.623871\pi$
$114$	0	0
$115$	1.12523	0.104928
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−14.6393	−1.34198
$120$	0	0
$121$	−14.2325	−1.29386
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−2.06068	−0.184312
$126$	0	0
$127$	7.35274i	0.652450i	0.945292	+	0.326225i	$0.105777\pi$
$127$	7.35274i	0.652450i	−0.945292	+	0.326225i	$0.894223\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 11.0730i	− 0.967449i	−0.875220	−	0.483725i	$-0.839284\pi$
$131$	− 11.0730i	− 0.967449i	0.875220	−	0.483725i	$-0.160716\pi$
$132$	0	0
$133$	3.15984i	0.273993i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 16.1103i	− 1.37640i	−0.725521	−	0.688200i	$-0.758401\pi$
$137$	− 16.1103i	− 1.37640i	0.725521	−	0.688200i	$-0.241599\pi$
$138$	0	0
$139$	21.1444	1.79345	0.896723	−	0.442592i	$-0.145941\pi$
$139$	21.1444	1.79345	0.896723	+	0.442592i	$0.145941\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−20.2915	−1.69686
$144$	0	0
$145$	0.601174	0.0499248
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.3437	1.33893	0.669464	−	0.742844i	$-0.266524\pi$
$149$	16.3437	1.33893	0.669464	+	0.742844i	$0.266524\pi$
$150$	0	0
$151$	6.07914i	0.494713i	0.968924	+	0.247357i	$0.0795619\pi$
$151$	6.07914i	0.494713i	−0.968924	+	0.247357i	$0.920438\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.256412i	0.0205955i
$156$	0	0
$157$	14.3817i	1.14778i	0.818931	+	0.573892i	$0.194567\pi$
$157$	14.3817i	1.14778i	−0.818931	+	0.573892i	$0.805433\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 16.5264i	− 1.30247i
$162$	0	0
$163$	13.8263	1.08296	0.541479	−	0.840714i	$-0.317864\pi$
$163$	13.8263	1.08296	0.541479	+	0.840714i	$0.317864\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.97675	0.772024	0.386012	−	0.922494i	$-0.373852\pi$
$167$	9.97675	0.772024	0.386012	+	0.922494i	$0.373852\pi$
$168$	0	0
$169$	−3.31812	−0.255240
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−22.8401	−1.73650	−0.868252	−	0.496123i	$-0.834756\pi$
$173$	−22.8401	−1.73650	−0.868252	+	0.496123i	$0.834756\pi$
$174$	0	0
$175$	15.0677i	1.13901i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	15.4258i	1.15298i	0.817104	+	0.576491i	$0.195578\pi$
$179$	15.4258i	1.15298i	−0.817104	+	0.576491i	$0.804422\pi$
$180$	0	0
$181$	14.4502i	1.07408i	0.843557	+	0.537039i	$0.180457\pi$
$181$	14.4502i	1.07408i	−0.843557	+	0.537039i	$0.819543\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.81532i	0.133465i
$186$	0	0
$187$	24.1929	1.76916
$188$	0	0
$189$	0	0
$190$	0	0
$191$	25.4190	1.83925	0.919626	−	0.392796i	$-0.128492\pi$
$191$	25.4190	1.83925	0.919626	+	0.392796i	$0.128492\pi$
$192$	0	0
$193$	15.1583	1.09112	0.545558	−	0.838073i	$-0.316318\pi$
$193$	15.1583	1.09112	0.545558	+	0.838073i	$0.316318\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.06068	−0.146817	−0.0734085	−	0.997302i	$-0.523388\pi$
$197$	−2.06068	−0.146817	−0.0734085	+	0.997302i	$0.523388\pi$
$198$	0	0
$199$	− 0.0791389i	− 0.00561001i	−0.999996	−	0.00280500i	$-0.999107\pi$
$199$	− 0.0791389i	− 0.00561001i	0.999996	−	0.00280500i	$-0.000892861\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 8.82955i	− 0.619713i
$204$	0	0
$205$	0.202673i	0.0141553i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 5.22196i	− 0.361210i
$210$	0	0
$211$	21.3527	1.46998	0.734991	−	0.678076i	$-0.237186\pi$
$211$	21.3527	1.46998	0.734991	+	0.678076i	$0.237186\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.659262	0.0449613
$216$	0	0
$217$	3.76597	0.255651
$218$	0	0
$219$	0	0
$220$	0	0
$221$	19.4555	1.30872
$222$	0	0
$223$	− 13.4018i	− 0.897448i	−0.893670	−	0.448724i	$-0.851879\pi$
$223$	− 13.4018i	− 0.897448i	0.893670	−	0.448724i	$-0.148121\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.25984i	0.614597i	0.951613	+	0.307299i	$0.0994250\pi$
$227$	9.25984i	0.614597i	−0.951613	+	0.307299i	$0.900575\pi$
$228$	0	0
$229$	11.1717i	0.738246i	0.929381	+	0.369123i	$0.120342\pi$
$229$	11.1717i	0.738246i	−0.929381	+	0.369123i	$0.879658\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 14.1233i	− 0.925251i	−0.886554	−	0.462625i	$-0.846907\pi$
$233$	− 14.1233i	− 0.925251i	0.886554	−	0.462625i	$-0.153093\pi$
$234$	0	0
$235$	−0.675358	−0.0440555
$236$	0	0
$237$	0	0
$238$	0	0
$239$	23.2106	1.50137	0.750684	−	0.660661i	$-0.229724\pi$
$239$	23.2106	1.50137	0.750684	+	0.660661i	$0.229724\pi$
$240$	0	0
$241$	−14.5143	−0.934947	−0.467473	−	0.884007i	$-0.654836\pi$
$241$	−14.5143	−0.934947	−0.467473	+	0.884007i	$0.654836\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.463366	−0.0296034
$246$	0	0
$247$	− 4.19941i	− 0.267202i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 4.03788i	− 0.254869i	−0.991847	−	0.127434i	$-0.959326\pi$
$251$	− 4.03788i	− 0.254869i	0.991847	−	0.127434i	$-0.0406742\pi$
$252$	0	0
$253$	27.3116i	1.71707i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.77579i	0.609797i	0.952385	+	0.304898i	$0.0986225\pi$
$257$	9.77579i	0.609797i	−0.952385	+	0.304898i	$0.901377\pi$
$258$	0	0
$259$	26.6619	1.65669
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−24.5814	−1.51576	−0.757878	−	0.652396i	$-0.773764\pi$
$263$	−24.5814	−1.51576	−0.757878	+	0.652396i	$0.773764\pi$
$264$	0	0
$265$	−1.63905	−0.100686
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.02008	−0.123167	−0.0615833	−	0.998102i	$-0.519615\pi$
$269$	−2.02008	−0.123167	−0.0615833	+	0.998102i	$0.519615\pi$
$270$	0	0
$271$	28.6643i	1.74123i	0.491961	+	0.870617i	$0.336280\pi$
$271$	28.6643i	1.74123i	−0.491961	+	0.870617i	$0.663720\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 24.9008i	− 1.50158i
$276$	0	0
$277$	14.2162i	0.854169i	0.904212	+	0.427085i	$0.140459\pi$
$277$	14.2162i	0.854169i	−0.904212	+	0.427085i	$0.859541\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 7.49178i	− 0.446922i	−0.974713	−	0.223461i	$-0.928264\pi$
$281$	− 7.49178i	− 0.446922i	0.974713	−	0.223461i	$-0.0717356\pi$
$282$	0	0
$283$	−27.6159	−1.64159	−0.820796	−	0.571221i	$-0.806470\pi$
$283$	−27.6159	−1.64159	−0.820796	+	0.571221i	$0.806470\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.97670	0.175709
$288$	0	0
$289$	−6.19615	−0.364480
$290$	0	0
$291$	0	0
$292$	0	0
$293$	26.3495	1.53935	0.769677	−	0.638433i	$-0.220417\pi$
$293$	26.3495	1.53935	0.769677	+	0.638433i	$0.220417\pi$
$294$	0	0
$295$	− 3.07418i	− 0.178986i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	21.9635i	1.27018i
$300$	0	0
$301$	− 9.68270i	− 0.558102i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.97282i	0.112963i
$306$	0	0
$307$	9.07418	0.517891	0.258946	−	0.965892i	$-0.416625\pi$
$307$	9.07418	0.517891	0.258946	+	0.965892i	$0.416625\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	21.0661	1.19455	0.597274	−	0.802037i	$-0.296250\pi$
$311$	21.0661	1.19455	0.597274	+	0.802037i	$0.296250\pi$
$312$	0	0
$313$	−18.2325	−1.03056	−0.515280	−	0.857022i	$-0.672312\pi$
$313$	−18.2325	−1.03056	−0.515280	+	0.857022i	$0.672312\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.60278	0.314684	0.157342	−	0.987544i	$-0.449708\pi$
$317$	5.60278	0.314684	0.157342	+	0.987544i	$0.449708\pi$
$318$	0	0
$319$	14.5917i	0.816979i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.00681i	0.278587i
$324$	0	0
$325$	− 20.0248i	− 1.11078i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	9.91910i	0.546858i
$330$	0	0
$331$	−29.5836	−1.62606	−0.813032	−	0.582219i	$-0.802184\pi$
$331$	−29.5836	−1.62606	−0.813032	+	0.582219i	$0.802184\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.152176	0.00831424
$336$	0	0
$337$	−19.3116	−1.05197	−0.525985	−	0.850494i	$-0.676303\pi$
$337$	−19.3116	−1.05197	−0.525985	+	0.850494i	$0.676303\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−6.22365	−0.337029
$342$	0	0
$343$	− 14.4714i	− 0.781385i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 14.2254i	− 0.763659i	−0.924233	−	0.381829i	$-0.875294\pi$
$347$	− 14.2254i	− 0.763659i	0.924233	−	0.381829i	$-0.124706\pi$
$348$	0	0
$349$	12.1672i	0.651295i	0.945491	+	0.325647i	$0.105582\pi$
$349$	12.1672i	0.651295i	−0.945491	+	0.325647i	$0.894418\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 0.419903i	− 0.0223492i	−0.999938	−	0.0111746i	$-0.996443\pi$
$353$	− 0.419903i	− 0.0223492i	0.999938	−	0.0111746i	$-0.00355705\pi$
$354$	0	0
$355$	0.482466	0.0256066
$356$	0	0
$357$	0	0
$358$	0	0
$359$	14.0799	0.743107	0.371554	−	0.928411i	$-0.378825\pi$
$359$	14.0799	0.743107	0.371554	+	0.928411i	$0.378825\pi$
$360$	0	0
$361$	−17.9193	−0.943121
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.07705	−0.108718
$366$	0	0
$367$	31.6248i	1.65080i	0.564549	+	0.825400i	$0.309050\pi$
$367$	31.6248i	1.65080i	−0.564549	+	0.825400i	$0.690950\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	24.0730i	1.24981i
$372$	0	0
$373$	33.1077i	1.71425i	0.515108	+	0.857126i	$0.327752\pi$
$373$	33.1077i	1.71425i	−0.515108	+	0.857126i	$0.672248\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11.7344i	0.604353i
$378$	0	0
$379$	−0.167186	−0.00858775	−0.00429387	−	0.999991i	$-0.501367\pi$
$379$	−0.167186	−0.00858775	−0.00429387	+	0.999991i	$0.501367\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−32.7432	−1.67310	−0.836550	−	0.547890i	$-0.815431\pi$
$383$	−32.7432	−1.67310	−0.836550	+	0.547890i	$0.815431\pi$
$384$	0	0
$385$	3.15984	0.161041
$386$	0	0
$387$	0	0
$388$	0	0
$389$	15.4016	0.780893	0.390447	−	0.920626i	$-0.372321\pi$
$389$	15.4016	0.780893	0.390447	+	0.920626i	$0.372321\pi$
$390$	0	0
$391$	− 26.1864i	− 1.32430i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.93727i	0.147790i
$396$	0	0
$397$	25.3010i	1.26982i	0.772586	+	0.634910i	$0.218963\pi$
$397$	25.3010i	1.26982i	−0.772586	+	0.634910i	$0.781037\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 13.0391i	− 0.651142i	−0.945518	−	0.325571i	$-0.894443\pi$
$401$	− 13.0391i	− 0.651142i	0.945518	−	0.325571i	$-0.105557\pi$
$402$	0	0
$403$	−5.00496	−0.249315
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−44.0615	−2.18405
$408$	0	0
$409$	−7.75326	−0.383374	−0.191687	−	0.981456i	$-0.561396\pi$
$409$	−7.75326	−0.383374	−0.191687	+	0.981456i	$0.561396\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−45.1511	−2.22174
$414$	0	0
$415$	0.876791i	0.0430400i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 30.2966i	− 1.48009i	−0.672558	−	0.740044i	$-0.734805\pi$
$419$	− 30.2966i	− 1.48009i	0.672558	−	0.740044i	$-0.265195\pi$
$420$	0	0
$421$	18.7427i	0.913461i	0.889605	+	0.456731i	$0.150980\pi$
$421$	18.7427i	0.913461i	−0.889605	+	0.456731i	$0.849020\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	23.8749i	1.15810i
$426$	0	0
$427$	28.9751	1.40220
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−14.8198	−0.713846	−0.356923	−	0.934134i	$-0.616174\pi$
$431$	−14.8198	−0.713846	−0.356923	+	0.934134i	$0.616174\pi$
$432$	0	0
$433$	−9.83364	−0.472574	−0.236287	−	0.971683i	$-0.575931\pi$
$433$	−9.83364	−0.472574	−0.236287	+	0.971683i	$0.575931\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.65224	−0.270383
$438$	0	0
$439$	− 9.79609i	− 0.467542i	−0.972292	−	0.233771i	$-0.924893\pi$
$439$	− 9.79609i	− 0.467542i	0.972292	−	0.233771i	$-0.0751065\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 12.2734i	− 0.583128i	−0.956551	−	0.291564i	$-0.905824\pi$
$443$	− 12.2734i	− 0.583128i	0.956551	−	0.291564i	$-0.0941756\pi$
$444$	0	0
$445$	2.40176i	0.113854i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8.46106i	0.399302i	0.979867	+	0.199651i	$0.0639809\pi$
$449$	8.46106i	0.399302i	−0.979867	+	0.199651i	$0.936019\pi$
$450$	0	0
$451$	−4.91930	−0.231641
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.54109	0.119128
$456$	0	0
$457$	23.5127	1.09988	0.549939	−	0.835205i	$-0.314651\pi$
$457$	23.5127	1.09988	0.549939	+	0.835205i	$0.314651\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−18.7941	−0.875327	−0.437664	−	0.899139i	$-0.644194\pi$
$461$	−18.7941	−0.875327	−0.437664	+	0.899139i	$0.644194\pi$
$462$	0	0
$463$	23.7896i	1.10559i	0.833316	+	0.552797i	$0.186440\pi$
$463$	23.7896i	1.10559i	−0.833316	+	0.552797i	$0.813560\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 0.827816i	− 0.0383067i	−0.999817	−	0.0191534i	$-0.993903\pi$
$467$	− 0.827816i	− 0.0383067i	0.999817	−	0.0191534i	$-0.00609708\pi$
$468$	0	0
$469$	− 2.23503i	− 0.103204i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	16.0016i	0.735756i
$474$	0	0
$475$	5.15332	0.236451
$476$	0	0
$477$	0	0
$478$	0	0
$479$	22.9671	1.04939	0.524696	−	0.851290i	$-0.324179\pi$
$479$	22.9671	1.04939	0.524696	+	0.851290i	$0.324179\pi$
$480$	0	0
$481$	−35.4336	−1.61563
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.00169	0.0454844
$486$	0	0
$487$	− 33.4665i	− 1.51651i	−0.651957	−	0.758256i	$-0.726052\pi$
$487$	− 33.4665i	− 1.51651i	0.651957	−	0.758256i	$-0.273948\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	19.8776i	0.897065i	0.893767	+	0.448532i	$0.148053\pi$
$491$	19.8776i	0.897065i	−0.893767	+	0.448532i	$0.851947\pi$
$492$	0	0
$493$	− 13.9905i	− 0.630102i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 7.08606i	− 0.317853i
$498$	0	0
$499$	6.04452	0.270590	0.135295	−	0.990805i	$-0.456802\pi$
$499$	6.04452	0.270590	0.135295	+	0.990805i	$0.456802\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0.252024	0.0112372	0.00561859	−	0.999984i	$-0.498212\pi$
$503$	0.252024	0.0112372	0.00561859	+	0.999984i	$0.498212\pi$
$504$	0	0
$505$	−3.59993	−0.160195
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−27.1848	−1.20495	−0.602473	−	0.798139i	$-0.705818\pi$
$509$	−27.1848	−1.20495	−0.602473	+	0.798139i	$0.705818\pi$
$510$	0	0
$511$	30.5061i	1.34951i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.413908i	0.0182390i
$516$	0	0
$517$	− 16.3923i	− 0.720933i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 37.6451i	− 1.64926i	−0.565671	−	0.824631i	$-0.691383\pi$
$521$	− 37.6451i	− 1.64926i	0.565671	−	0.824631i	$-0.308617\pi$
$522$	0	0
$523$	2.20180	0.0962780	0.0481390	−	0.998841i	$-0.484671\pi$
$523$	2.20180	0.0962780	0.0481390	+	0.998841i	$0.484671\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.96723	0.259937
$528$	0	0
$529$	6.56206	0.285307
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.95602	−0.171354
$534$	0	0
$535$	2.36871i	0.102408i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 11.2468i	− 0.484436i
$540$	0	0
$541$	− 8.04196i	− 0.345751i	−0.984944	−	0.172875i	$-0.944694\pi$
$541$	− 8.04196i	− 0.345751i	0.984944	−	0.172875i	$-0.0553058\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 1.61588i	− 0.0692168i
$546$	0	0
$547$	−21.1745	−0.905357	−0.452679	−	0.891674i	$-0.649532\pi$
$547$	−21.1745	−0.905357	−0.452679	+	0.891674i	$0.649532\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.01981	−0.128648
$552$	0	0
$553$	43.1403	1.83451
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18.5871	0.787562	0.393781	−	0.919204i	$-0.371167\pi$
$557$	18.5871	0.787562	0.393781	+	0.919204i	$0.371167\pi$
$558$	0	0
$559$	12.8682i	0.544269i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	19.0085i	0.801115i	0.916272	+	0.400557i	$0.131183\pi$
$563$	19.0085i	0.801115i	−0.916272	+	0.400557i	$0.868817\pi$
$564$	0	0
$565$	4.07092i	0.171265i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.82654i	0.160417i	0.996778	+	0.0802084i	$0.0255586\pi$
$569$	3.82654i	0.160417i	−0.996778	+	0.0802084i	$0.974441\pi$
$570$	0	0
$571$	−14.6174	−0.611720	−0.305860	−	0.952076i	$-0.598944\pi$
$571$	−14.6174	−0.611720	−0.305860	+	0.952076i	$0.598944\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−26.9526	−1.12400
$576$	0	0
$577$	−1.28305	−0.0534142	−0.0267071	−	0.999643i	$-0.508502\pi$
$577$	−1.28305	−0.0534142	−0.0267071	+	0.999643i	$0.508502\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	12.8776	0.534252
$582$	0	0
$583$	− 39.7830i	− 1.64765i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 5.45348i	− 0.225089i	−0.993647	−	0.112545i	$-0.964100\pi$
$587$	− 5.45348i	− 0.225089i	0.993647	−	0.112545i	$-0.0359001\pi$
$588$	0	0
$589$	− 1.28801i	− 0.0530715i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	25.7774i	1.05855i	0.848450	+	0.529276i	$0.177536\pi$
$593$	25.7774i	1.05855i	−0.848450	+	0.529276i	$0.822464\pi$
$594$	0	0
$595$	−3.02966	−0.124204
$596$	0	0
$597$	0	0
$598$	0	0
$599$	39.6141	1.61859	0.809295	−	0.587403i	$-0.199849\pi$
$599$	39.6141	1.61859	0.809295	+	0.587403i	$0.199849\pi$
$600$	0	0
$601$	3.44785	0.140641	0.0703204	−	0.997524i	$-0.477598\pi$
$601$	3.44785	0.140641	0.0703204	+	0.997524i	$0.477598\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.94546	−0.119750
$606$	0	0
$607$	− 23.0029i	− 0.933660i	−0.884347	−	0.466830i	$-0.845396\pi$
$607$	− 23.0029i	− 0.933660i	0.884347	−	0.466830i	$-0.154604\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 13.1824i	− 0.533304i
$612$	0	0
$613$	− 6.82786i	− 0.275775i	−0.990448	−	0.137887i	$-0.955969\pi$
$613$	− 6.82786i	− 0.275775i	0.990448	−	0.137887i	$-0.0440312\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.9201i	0.882469i	0.897392	+	0.441235i	$0.145459\pi$
$617$	21.9201i	0.882469i	−0.897392	+	0.441235i	$0.854541\pi$
$618$	0	0
$619$	37.0131	1.48768	0.743841	−	0.668356i	$-0.233002\pi$
$619$	37.0131	1.48768	0.743841	+	0.668356i	$0.233002\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	35.2751	1.41327
$624$	0	0
$625$	24.3594	0.974375
$626$	0	0
$627$	0	0
$628$	0	0
$629$	42.2462	1.68447
$630$	0	0
$631$	25.2815i	1.00644i	0.864158	+	0.503220i	$0.167851\pi$
$631$	25.2815i	1.00644i	−0.864158	+	0.503220i	$0.832149\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.52168i	0.0603859i
$636$	0	0
$637$	− 9.04452i	− 0.358357i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 19.6465i	− 0.775988i	−0.921662	−	0.387994i	$-0.873168\pi$
$641$	− 19.6465i	− 0.775988i	0.921662	−	0.387994i	$-0.126832\pi$
$642$	0	0
$643$	−17.2089	−0.678652	−0.339326	−	0.940669i	$-0.610199\pi$
$643$	−17.2089	−0.678652	−0.339326	+	0.940669i	$0.610199\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	33.9988	1.33663	0.668315	−	0.743879i	$-0.267016\pi$
$647$	33.9988	1.33663	0.668315	+	0.743879i	$0.267016\pi$
$648$	0	0
$649$	74.6167	2.92896
$650$	0	0
$651$	0	0
$652$	0	0
$653$	20.5636	0.804718	0.402359	−	0.915482i	$-0.368190\pi$
$653$	20.5636	0.804718	0.402359	+	0.915482i	$0.368190\pi$
$654$	0	0
$655$	− 2.29159i	− 0.0895400i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 22.4586i	− 0.874864i	−0.899251	−	0.437432i	$-0.855888\pi$
$659$	− 22.4586i	− 0.874864i	0.899251	−	0.437432i	$-0.144112\pi$
$660$	0	0
$661$	− 17.1362i	− 0.666521i	−0.942835	−	0.333260i	$-0.891851\pi$
$661$	− 17.1362i	− 0.666521i	0.942835	−	0.333260i	$-0.108149\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.653942i	0.0253588i
$666$	0	0
$667$	15.7941	0.611549
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−47.8843	−1.84855
$672$	0	0
$673$	−30.7595	−1.18569	−0.592845	−	0.805317i	$-0.701995\pi$
$673$	−30.7595	−1.18569	−0.592845	+	0.805317i	$0.701995\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−13.3157	−0.511764	−0.255882	−	0.966708i	$-0.582366\pi$
$677$	−13.3157	−0.511764	−0.255882	+	0.966708i	$0.582366\pi$
$678$	0	0
$679$	− 14.7120i	− 0.564594i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	27.4006i	1.04846i	0.851578	+	0.524228i	$0.175646\pi$
$683$	27.4006i	1.04846i	−0.851578	+	0.524228i	$0.824354\pi$
$684$	0	0
$685$	− 3.33410i	− 0.127389i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 31.9929i	− 1.21883i
$690$	0	0
$691$	17.1790	0.653521	0.326761	−	0.945107i	$-0.394043\pi$
$691$	17.1790	0.653521	0.326761	+	0.945107i	$0.394043\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.37592	0.165988
$696$	0	0
$697$	4.71662	0.178655
$698$	0	0
$699$	0	0
$700$	0	0
$701$	42.7770	1.61567	0.807833	−	0.589412i	$-0.200640\pi$
$701$	42.7770	1.61567	0.807833	+	0.589412i	$0.200640\pi$
$702$	0	0
$703$	− 9.11871i	− 0.343919i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	52.8729i	1.98849i
$708$	0	0
$709$	− 51.9018i	− 1.94921i	−0.223928	−	0.974606i	$-0.571888\pi$
$709$	− 51.9018i	− 1.94921i	0.223928	−	0.974606i	$-0.428112\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.73647i	0.252283i
$714$	0	0
$715$	−4.19941	−0.157049
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−29.6166	−1.10451	−0.552257	−	0.833674i	$-0.686233\pi$
$719$	−29.6166	−1.10451	−0.552257	+	0.833674i	$0.686233\pi$
$720$	0	0
$721$	6.07914	0.226399
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−14.3999	−0.534800
$726$	0	0
$727$	− 31.8177i	− 1.18005i	−0.807384	−	0.590026i	$-0.799118\pi$
$727$	− 31.8177i	− 1.18005i	0.807384	−	0.590026i	$-0.200882\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 15.3424i	− 0.567458i
$732$	0	0
$733$	− 23.4062i	− 0.864527i	−0.901747	−	0.432263i	$-0.857715\pi$
$733$	− 23.4062i	− 0.864527i	0.901747	−	0.432263i	$-0.142285\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.69361i	0.136056i
$738$	0	0
$739$	14.0424	0.516558	0.258279	−	0.966070i	$-0.416845\pi$
$739$	14.0424	0.516558	0.258279	+	0.966070i	$0.416845\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−12.8612	−0.471832	−0.235916	−	0.971773i	$-0.575809\pi$
$743$	−12.8612	−0.471832	−0.235916	+	0.971773i	$0.575809\pi$
$744$	0	0
$745$	3.38239	0.123921
$746$	0	0
$747$	0	0
$748$	0	0
$749$	34.7897	1.27119
$750$	0	0
$751$	39.4254i	1.43865i	0.694673	+	0.719326i	$0.255549\pi$
$751$	39.4254i	1.43865i	−0.694673	+	0.719326i	$0.744451\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.25810i	0.0457870i
$756$	0	0
$757$	− 14.4992i	− 0.526981i	−0.964662	−	0.263491i	$-0.915126\pi$
$757$	− 14.4992i	− 0.526981i	0.964662	−	0.263491i	$-0.0848738\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 46.6306i	− 1.69036i	−0.534482	−	0.845180i	$-0.679493\pi$
$761$	− 46.6306i	− 1.69036i	0.534482	−	0.845180i	$-0.320507\pi$
$762$	0	0
$763$	−23.7327	−0.859183
$764$	0	0
$765$	0	0
$766$	0	0
$767$	60.0055	2.16667
$768$	0	0
$769$	23.7055	0.854841	0.427421	−	0.904053i	$-0.359422\pi$
$769$	23.7055	0.854841	0.427421	+	0.904053i	$0.359422\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−24.1631	−0.869085	−0.434542	−	0.900651i	$-0.643090\pi$
$773$	−24.1631	−0.869085	−0.434542	+	0.900651i	$0.643090\pi$
$774$	0	0
$775$	− 6.14185i	− 0.220622i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1.01807i	− 0.0364761i
$780$	0	0
$781$	11.7104i	0.419032i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.97635i	0.106230i
$786$	0	0
$787$	−27.5199	−0.980979	−0.490490	−	0.871447i	$-0.663182\pi$
$787$	−27.5199	−0.980979	−0.490490	+	0.871447i	$0.663182\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	59.7904	2.12590
$792$	0	0
$793$	−38.5078	−1.36745
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−12.6199	−0.447019	−0.223510	−	0.974702i	$-0.571751\pi$
$797$	−12.6199	−0.447019	−0.223510	+	0.974702i	$0.571751\pi$
$798$	0	0
$799$	15.7169i	0.556026i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 50.4143i	− 1.77908i
$804$	0	0
$805$	− 3.42021i	− 0.120547i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 15.5226i	− 0.545744i	−0.962050	−	0.272872i	$-0.912026\pi$
$809$	− 15.5226i	− 0.545744i	0.962050	−	0.272872i	$-0.0879736\pi$
$810$	0	0
$811$	−48.4347	−1.70077	−0.850386	−	0.526159i	$-0.823632\pi$
$811$	−48.4347	−1.70077	−0.850386	+	0.526159i	$0.823632\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.86141	0.100231
$816$	0	0
$817$	−3.31160	−0.115858
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.0891	0.421912	0.210956	−	0.977496i	$-0.432342\pi$
$821$	12.0891	0.421912	0.210956	+	0.977496i	$0.432342\pi$
$822$	0	0
$823$	− 5.10430i	− 0.177925i	−0.996035	−	0.0889623i	$-0.971645\pi$
$823$	− 5.10430i	− 0.177925i	0.996035	−	0.0889623i	$-0.0283551\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 6.40603i	− 0.222760i	−0.993778	−	0.111380i	$-0.964473\pi$
$827$	− 6.40603i	− 0.222760i	0.993778	−	0.111380i	$-0.0355270\pi$
$828$	0	0
$829$	− 42.1518i	− 1.46399i	−0.681309	−	0.731996i	$-0.738589\pi$
$829$	− 42.1518i	− 1.46399i	0.681309	−	0.731996i	$-0.261411\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	10.7835i	0.373625i
$834$	0	0
$835$	2.06473	0.0714528
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−52.3893	−1.80868	−0.904340	−	0.426813i	$-0.859636\pi$
$839$	−52.3893	−1.80868	−0.904340	+	0.426813i	$0.859636\pi$
$840$	0	0
$841$	−20.5617	−0.709025
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−0.686698	−0.0236231
$846$	0	0
$847$	43.2606i	1.48645i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	47.6922i	1.63487i
$852$	0	0
$853$	− 28.2920i	− 0.968698i	−0.874875	−	0.484349i	$-0.839056\pi$
$853$	− 28.2920i	− 0.968698i	0.874875	−	0.484349i	$-0.160944\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 32.5139i	− 1.11065i	−0.831632	−	0.555327i	$-0.812593\pi$
$857$	− 32.5139i	− 1.11065i	0.831632	−	0.555327i	$-0.187407\pi$
$858$	0	0
$859$	40.9453	1.39703	0.698517	−	0.715593i	$-0.253843\pi$
$859$	40.9453	1.39703	0.698517	+	0.715593i	$0.253843\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−9.23907	−0.314502	−0.157251	−	0.987559i	$-0.550263\pi$
$863$	−9.23907	−0.314502	−0.157251	+	0.987559i	$0.550263\pi$
$864$	0	0
$865$	−4.72686	−0.160718
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−71.2936	−2.41847
$870$	0	0
$871$	2.97034i	0.100646i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	6.26357i	0.211747i
$876$	0	0
$877$	1.01556i	0.0342930i	0.999853	+	0.0171465i	$0.00545816\pi$
$877$	1.01556i	0.0342930i	−0.999853	+	0.0171465i	$0.994542\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	49.9531i	1.68296i	0.540285	+	0.841482i	$0.318316\pi$
$881$	49.9531i	1.68296i	−0.540285	+	0.841482i	$0.681684\pi$
$882$	0	0
$883$	−40.0726	−1.34855	−0.674275	−	0.738480i	$-0.735544\pi$
$883$	−40.0726	−1.34855	−0.674275	+	0.738480i	$0.735544\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−29.4591	−0.989141	−0.494570	−	0.869138i	$-0.664675\pi$
$887$	−29.4591	−0.989141	−0.494570	+	0.869138i	$0.664675\pi$
$888$	0	0
$889$	22.3491	0.749566
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.39245	0.113524
$894$	0	0
$895$	3.19244i	0.106711i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.59908i	0.120036i
$900$	0	0
$901$	38.1440i	1.27076i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.99053i	0.0994087i
$906$	0	0
$907$	−36.3557	−1.20717	−0.603585	−	0.797299i	$-0.706262\pi$
$907$	−36.3557	−1.20717	−0.603585	+	0.797299i	$0.706262\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	28.3465	0.939162	0.469581	−	0.882889i	$-0.344405\pi$
$911$	28.3465	0.939162	0.469581	+	0.882889i	$0.344405\pi$
$912$	0	0
$913$	−21.2815	−0.704314
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−33.6570	−1.11145
$918$	0	0
$919$	− 58.7435i	− 1.93777i	−0.247514	−	0.968884i	$-0.579614\pi$
$919$	− 58.7435i	− 1.93777i	0.247514	−	0.968884i	$-0.420386\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9.41733i	0.309975i
$924$	0	0
$925$	− 43.4824i	− 1.42969i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	24.7981i	0.813599i	0.913517	+	0.406800i	$0.133355\pi$
$929$	24.7981i	0.813599i	−0.913517	+	0.406800i	$0.866645\pi$
$930$	0	0
$931$	2.32758	0.0762833
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.00681	0.163740
$936$	0	0
$937$	25.1819	0.822656	0.411328	−	0.911487i	$-0.365065\pi$
$937$	25.1819	0.822656	0.411328	+	0.911487i	$0.365065\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−36.6929	−1.19615	−0.598077	−	0.801439i	$-0.704068\pi$
$941$	−36.6929	−1.19615	−0.598077	+	0.801439i	$0.704068\pi$
$942$	0	0
$943$	5.32464i	0.173394i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 21.7134i	− 0.705590i	−0.935701	−	0.352795i	$-0.885231\pi$
$947$	− 21.7134i	− 0.705590i	0.935701	−	0.352795i	$-0.114769\pi$
$948$	0	0
$949$	− 40.5424i	− 1.31606i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	16.5469i	0.536005i	0.963418	+	0.268003i	$0.0863636\pi$
$953$	16.5469i	0.536005i	−0.963418	+	0.268003i	$0.913636\pi$
$954$	0	0
$955$	5.26055	0.170228
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−48.9685	−1.58128
$960$	0	0
$961$	29.4649	0.950481
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.13707	0.100986
$966$	0	0
$967$	− 27.0566i	− 0.870083i	−0.900410	−	0.435041i	$-0.856734\pi$
$967$	− 27.0566i	− 0.870083i	0.900410	−	0.435041i	$-0.143266\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 41.0968i	− 1.31886i	−0.751766	−	0.659430i	$-0.770798\pi$
$971$	− 41.0968i	− 1.31886i	0.751766	−	0.659430i	$-0.229202\pi$
$972$	0	0
$973$	− 64.2699i	− 2.06040i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 46.7497i	− 1.49566i	−0.663892	−	0.747828i	$-0.731097\pi$
$977$	− 46.7497i	− 1.49566i	0.663892	−	0.747828i	$-0.268903\pi$
$978$	0	0
$979$	−58.2956	−1.86314
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−45.5184	−1.45181	−0.725906	−	0.687794i	$-0.758579\pi$
$983$	−45.5184	−1.45181	−0.725906	+	0.687794i	$0.758579\pi$
$984$	0	0
$985$	−0.426465	−0.0135883
$986$	0	0
$987$	0	0
$988$	0	0
$989$	17.3202	0.550749
$990$	0	0
$991$	− 22.8381i	− 0.725477i	−0.931891	−	0.362739i	$-0.881842\pi$
$991$	− 22.8381i	− 0.725477i	0.931891	−	0.362739i	$-0.118158\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 0.0163781i	0 0.000519221i
$996$	0	0
$997$	43.7357i	1.38512i	0.721358	+	0.692562i	$0.243518\pi$
$997$	43.7357i	1.38512i	−0.721358	+	0.692562i	$0.756482\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5184.2.f.e.2591.9	yes	16
3.2	odd	2	inner	5184.2.f.e.2591.7	yes	16
4.3	odd	2		5184.2.f.b.2591.10	yes	16
8.3	odd	2	inner	5184.2.f.e.2591.8	yes	16
8.5	even	2		5184.2.f.b.2591.7	✓	16
12.11	even	2		5184.2.f.b.2591.8	yes	16
24.5	odd	2		5184.2.f.b.2591.9	yes	16
24.11	even	2	inner	5184.2.f.e.2591.10	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5184.2.f.b.2591.7	✓	16	8.5	even	2
5184.2.f.b.2591.8	yes	16	12.11	even	2
5184.2.f.b.2591.9	yes	16	24.5	odd	2
5184.2.f.b.2591.10	yes	16	4.3	odd	2
5184.2.f.e.2591.7	yes	16	3.2	odd	2	inner
5184.2.f.e.2591.8	yes	16	8.3	odd	2	inner
5184.2.f.e.2591.9	yes	16	1.1	even	1	trivial
5184.2.f.e.2591.10	yes	16	24.11	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.206954 q^{5} -3.03957i q^{7} +O(q^{10})\)	\(q+0.206954 q^{5} -3.03957i q^{7} +5.02319i q^{11} +4.03957i q^{13} -4.81624i q^{17} -1.03957 q^{19} +5.43710 q^{23} -4.95717 q^{25} +2.90487 q^{29} +1.23898i q^{31} -0.629051i q^{35} +8.77162i q^{37} +0.979317i q^{41} +3.18555 q^{43} -3.26332 q^{47} -2.23898 q^{49} -7.91987 q^{53} +1.03957i q^{55} -14.8544i q^{59} +9.53264i q^{61} +0.836005i q^{65} +0.735311 q^{67} +2.33127 q^{71} -10.0363 q^{73} +15.2683 q^{77} +14.1929i q^{79} +4.23665i q^{83} -0.996740i q^{85} +11.6053i q^{89} +12.2786 q^{91} -0.215143 q^{95} +4.84016 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 24 q^{19} + 16 q^{25} + 24 q^{43} - 48 q^{49} + 120 q^{67} + 16 q^{73} + 168 q^{91} - 16 q^{97}+O(q^{100}) \) 16 * q + 24 * q^19 + 16 * q^25 + 24 * q^43 - 48 * q^49 + 120 * q^67 + 16 * q^73 + 168 * q^91 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more