LMFDB - Embedded newform 4320.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4320,2,Mod(1,4320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4320.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.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:	$34.4953736732$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	4320.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +1.64575 q^{7} +1.64575 q^{11} +3.64575 q^{13} +4.64575 q^{17} +2.64575 q^{19} +4.29150 q^{23} +1.00000 q^{25} +5.64575 q^{29} -2.64575 q^{31} +1.64575 q^{35} -5.29150 q^{37} +0.354249 q^{41} -6.93725 q^{43} -5.29150 q^{47} -4.29150 q^{49} -3.93725 q^{53} +1.64575 q^{55} -3.64575 q^{59} +7.58301 q^{61} +3.64575 q^{65} +6.00000 q^{67} +2.35425 q^{71} -0.937254 q^{73} +2.70850 q^{77} -7.35425 q^{79} -5.00000 q^{83} +4.64575 q^{85} +8.35425 q^{89} +6.00000 q^{91} +2.64575 q^{95} -0.708497 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{7} - 2 q^{11} + 2 q^{13} + 4 q^{17} - 2 q^{23} + 2 q^{25} + 6 q^{29} - 2 q^{35} + 6 q^{41} + 2 q^{43} + 2 q^{49} + 8 q^{53} - 2 q^{55} - 2 q^{59} - 6 q^{61} + 2 q^{65} + 12 q^{67} + 10 q^{71}+ \cdots - 12 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^7 - 2 * q^11 + 2 * q^13 + 4 * q^17 - 2 * q^23 + 2 * q^25 + 6 * q^29 - 2 * q^35 + 6 * q^41 + 2 * q^43 + 2 * q^49 + 8 * q^53 - 2 * q^55 - 2 * q^59 - 6 * q^61 + 2 * q^65 + 12 * q^67 + 10 * q^71 + 14 * q^73 + 16 * q^77 - 20 * q^79 - 10 * q^83 + 4 * q^85 + 22 * q^89 + 12 * q^91 - 12 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.64575	0.622036	0.311018	−	0.950404i	$-0.399330\pi$
$7$	1.64575	0.622036	0.311018	+	0.950404i	$0.399330\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.64575	0.496213	0.248106	−	0.968733i	$-0.420192\pi$
$11$	1.64575	0.496213	0.248106	+	0.968733i	$0.420192\pi$
$12$	0	0
$13$	3.64575	1.01115	0.505575	−	0.862783i	$-0.331280\pi$
$13$	3.64575	1.01115	0.505575	+	0.862783i	$0.331280\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.64575	1.12676	0.563380	−	0.826198i	$-0.309501\pi$
$17$	4.64575	1.12676	0.563380	+	0.826198i	$0.309501\pi$
$18$	0	0
$19$	2.64575	0.606977	0.303488	−	0.952835i	$-0.401849\pi$
$19$	2.64575	0.606977	0.303488	+	0.952835i	$0.401849\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.29150	0.894840	0.447420	−	0.894324i	$-0.352343\pi$
$23$	4.29150	0.894840	0.447420	+	0.894324i	$0.352343\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.64575	1.04839	0.524195	−	0.851598i	$-0.324366\pi$
$29$	5.64575	1.04839	0.524195	+	0.851598i	$0.324366\pi$
$30$	0	0
$31$	−2.64575	−0.475191	−0.237595	−	0.971364i	$-0.576359\pi$
$31$	−2.64575	−0.475191	−0.237595	+	0.971364i	$0.576359\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.64575	0.278183
$36$	0	0
$37$	−5.29150	−0.869918	−0.434959	−	0.900450i	$-0.643237\pi$
$37$	−5.29150	−0.869918	−0.434959	+	0.900450i	$0.643237\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.354249	0.0553244	0.0276622	−	0.999617i	$-0.491194\pi$
$41$	0.354249	0.0553244	0.0276622	+	0.999617i	$0.491194\pi$
$42$	0	0
$43$	−6.93725	−1.05792	−0.528961	−	0.848646i	$-0.677418\pi$
$43$	−6.93725	−1.05792	−0.528961	+	0.848646i	$0.677418\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.29150	−0.771845	−0.385922	−	0.922531i	$-0.626117\pi$
$47$	−5.29150	−0.771845	−0.385922	+	0.922531i	$0.626117\pi$
$48$	0	0
$49$	−4.29150	−0.613072
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.93725	−0.540823	−0.270412	−	0.962745i	$-0.587160\pi$
$53$	−3.93725	−0.540823	−0.270412	+	0.962745i	$0.587160\pi$
$54$	0	0
$55$	1.64575	0.221913
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.64575	−0.474636	−0.237318	−	0.971432i	$-0.576268\pi$
$59$	−3.64575	−0.474636	−0.237318	+	0.971432i	$0.576268\pi$
$60$	0	0
$61$	7.58301	0.970904	0.485452	−	0.874263i	$-0.338655\pi$
$61$	7.58301	0.970904	0.485452	+	0.874263i	$0.338655\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.64575	0.452200
$66$	0	0
$67$	6.00000	0.733017	0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000	0.733017	0.366508	+	0.930415i	$0.380553\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.35425	0.279398	0.139699	−	0.990194i	$-0.455387\pi$
$71$	2.35425	0.279398	0.139699	+	0.990194i	$0.455387\pi$
$72$	0	0
$73$	−0.937254	−0.109697	−0.0548486	−	0.998495i	$-0.517468\pi$
$73$	−0.937254	−0.109697	−0.0548486	+	0.998495i	$0.517468\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.70850	0.308662
$78$	0	0
$79$	−7.35425	−0.827418	−0.413709	−	0.910409i	$-0.635767\pi$
$79$	−7.35425	−0.827418	−0.413709	+	0.910409i	$0.635767\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.00000	−0.548821	−0.274411	−	0.961613i	$-0.588483\pi$
$83$	−5.00000	−0.548821	−0.274411	+	0.961613i	$0.588483\pi$
$84$	0	0
$85$	4.64575	0.503902
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.35425	0.885549	0.442774	−	0.896633i	$-0.353994\pi$
$89$	8.35425	0.885549	0.442774	+	0.896633i	$0.353994\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.64575	0.271448
$96$	0	0
$97$	−0.708497	−0.0719370	−0.0359685	−	0.999353i	$-0.511452\pi$
$97$	−0.708497	−0.0719370	−0.0359685	+	0.999353i	$0.511452\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.2915	1.12355	0.561773	−	0.827291i	$-0.310119\pi$
$101$	11.2915	1.12355	0.561773	+	0.827291i	$0.310119\pi$
$102$	0	0
$103$	0.708497	0.0698103	0.0349052	−	0.999391i	$-0.488887\pi$
$103$	0.708497	0.0698103	0.0349052	+	0.999391i	$0.488887\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	0	0
$109$	2.29150	0.219486	0.109743	−	0.993960i	$-0.464997\pi$
$109$	2.29150	0.219486	0.109743	+	0.993960i	$0.464997\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.5830	1.37185	0.685927	−	0.727670i	$-0.259397\pi$
$113$	14.5830	1.37185	0.685927	+	0.727670i	$0.259397\pi$
$114$	0	0
$115$	4.29150	0.400185
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.64575	0.700885
$120$	0	0
$121$	−8.29150	−0.753773
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−20.5830	−1.82645	−0.913223	−	0.407460i	$-0.866415\pi$
$127$	−20.5830	−1.82645	−0.913223	+	0.407460i	$0.866415\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−19.8745	−1.73644	−0.868222	−	0.496176i	$-0.834737\pi$
$131$	−19.8745	−1.73644	−0.868222	+	0.496176i	$0.834737\pi$
$132$	0	0
$133$	4.35425	0.377561
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.06275	0.176232	0.0881161	−	0.996110i	$-0.471915\pi$
$137$	2.06275	0.176232	0.0881161	+	0.996110i	$0.471915\pi$
$138$	0	0
$139$	22.5830	1.91547	0.957733	−	0.287659i	$-0.0928771\pi$
$139$	22.5830	1.91547	0.957733	+	0.287659i	$0.0928771\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.00000	0.501745
$144$	0	0
$145$	5.64575	0.468854
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18.9373	1.55140	0.775700	−	0.631102i	$-0.217397\pi$
$149$	18.9373	1.55140	0.775700	+	0.631102i	$0.217397\pi$
$150$	0	0
$151$	−17.2915	−1.40716	−0.703581	−	0.710615i	$-0.748417\pi$
$151$	−17.2915	−1.40716	−0.703581	+	0.710615i	$0.748417\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.64575	−0.212512
$156$	0	0
$157$	−12.9373	−1.03251	−0.516253	−	0.856436i	$-0.672673\pi$
$157$	−12.9373	−1.03251	−0.516253	+	0.856436i	$0.672673\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.06275	0.556622
$162$	0	0
$163$	−1.29150	−0.101158	−0.0505791	−	0.998720i	$-0.516107\pi$
$163$	−1.29150	−0.101158	−0.0505791	+	0.998720i	$0.516107\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.2915	−0.951145	−0.475573	−	0.879677i	$-0.657759\pi$
$167$	−12.2915	−0.951145	−0.475573	+	0.879677i	$0.657759\pi$
$168$	0	0
$169$	0.291503	0.0224233
$170$	0	0
$171$	0	0
$172$	0	0
$173$	11.3542	0.863248	0.431624	−	0.902054i	$-0.357941\pi$
$173$	11.3542	0.863248	0.431624	+	0.902054i	$0.357941\pi$
$174$	0	0
$175$	1.64575	0.124407
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−7.29150	−0.544992	−0.272496	−	0.962157i	$-0.587849\pi$
$179$	−7.29150	−0.544992	−0.272496	+	0.962157i	$0.587849\pi$
$180$	0	0
$181$	−16.2915	−1.21094	−0.605469	−	0.795869i	$-0.707014\pi$
$181$	−16.2915	−1.21094	−0.605469	+	0.795869i	$0.707014\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.29150	−0.389039
$186$	0	0
$187$	7.64575	0.559113
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.70850	0.195980	0.0979900	−	0.995187i	$-0.468759\pi$
$191$	2.70850	0.195980	0.0979900	+	0.995187i	$0.468759\pi$
$192$	0	0
$193$	21.5203	1.54906	0.774531	−	0.632536i	$-0.217986\pi$
$193$	21.5203	1.54906	0.774531	+	0.632536i	$0.217986\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.8118	1.55402	0.777012	−	0.629486i	$-0.216734\pi$
$197$	21.8118	1.55402	0.777012	+	0.629486i	$0.216734\pi$
$198$	0	0
$199$	5.87451	0.416433	0.208216	−	0.978083i	$-0.433234\pi$
$199$	5.87451	0.416433	0.208216	+	0.978083i	$0.433234\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.29150	0.652136
$204$	0	0
$205$	0.354249	0.0247418
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.35425	0.301190
$210$	0	0
$211$	15.3542	1.05703	0.528515	−	0.848924i	$-0.322749\pi$
$211$	15.3542	1.05703	0.528515	+	0.848924i	$0.322749\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.93725	−0.473117
$216$	0	0
$217$	−4.35425	−0.295586
$218$	0	0
$219$	0	0
$220$	0	0
$221$	16.9373	1.13932
$222$	0	0
$223$	−14.7085	−0.984954	−0.492477	−	0.870326i	$-0.663908\pi$
$223$	−14.7085	−0.984954	−0.492477	+	0.870326i	$0.663908\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.8745	1.25275	0.626373	−	0.779524i	$-0.284539\pi$
$227$	18.8745	1.25275	0.626373	+	0.779524i	$0.284539\pi$
$228$	0	0
$229$	−9.58301	−0.633263	−0.316631	−	0.948549i	$-0.602552\pi$
$229$	−9.58301	−0.633263	−0.316631	+	0.948549i	$0.602552\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	24.5830	1.61049	0.805243	−	0.592945i	$-0.202035\pi$
$233$	24.5830	1.61049	0.805243	+	0.592945i	$0.202035\pi$
$234$	0	0
$235$	−5.29150	−0.345180
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.70850	−0.175198	−0.0875991	−	0.996156i	$-0.527919\pi$
$239$	−2.70850	−0.175198	−0.0875991	+	0.996156i	$0.527919\pi$
$240$	0	0
$241$	−22.2915	−1.43592	−0.717961	−	0.696083i	$-0.754924\pi$
$241$	−22.2915	−1.43592	−0.717961	+	0.696083i	$0.754924\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−4.29150	−0.274174
$246$	0	0
$247$	9.64575	0.613744
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3.29150	0.207758	0.103879	−	0.994590i	$-0.466875\pi$
$251$	3.29150	0.207758	0.103879	+	0.994590i	$0.466875\pi$
$252$	0	0
$253$	7.06275	0.444031
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.22876	−0.326161	−0.163080	−	0.986613i	$-0.552143\pi$
$257$	−5.22876	−0.326161	−0.163080	+	0.986613i	$0.552143\pi$
$258$	0	0
$259$	−8.70850	−0.541120
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−21.2915	−1.31289	−0.656445	−	0.754374i	$-0.727941\pi$
$263$	−21.2915	−1.31289	−0.656445	+	0.754374i	$0.727941\pi$
$264$	0	0
$265$	−3.93725	−0.241864
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−15.8745	−0.967886	−0.483943	−	0.875100i	$-0.660796\pi$
$269$	−15.8745	−0.967886	−0.483943	+	0.875100i	$0.660796\pi$
$270$	0	0
$271$	−14.6458	−0.889666	−0.444833	−	0.895614i	$-0.646737\pi$
$271$	−14.6458	−0.889666	−0.444833	+	0.895614i	$0.646737\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.64575	0.0992425
$276$	0	0
$277$	−1.64575	−0.0988836	−0.0494418	−	0.998777i	$-0.515744\pi$
$277$	−1.64575	−0.0988836	−0.0494418	+	0.998777i	$0.515744\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.35425	0.379063	0.189531	−	0.981875i	$-0.439303\pi$
$281$	6.35425	0.379063	0.189531	+	0.981875i	$0.439303\pi$
$282$	0	0
$283$	31.6458	1.88114	0.940572	−	0.339593i	$-0.110289\pi$
$283$	31.6458	1.88114	0.940572	+	0.339593i	$0.110289\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.583005	0.0344137
$288$	0	0
$289$	4.58301	0.269589
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.77124	0.161898	0.0809489	−	0.996718i	$-0.474205\pi$
$293$	2.77124	0.161898	0.0809489	+	0.996718i	$0.474205\pi$
$294$	0	0
$295$	−3.64575	−0.212264
$296$	0	0
$297$	0	0
$298$	0	0
$299$	15.6458	0.904817
$300$	0	0
$301$	−11.4170	−0.658065
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7.58301	0.434202
$306$	0	0
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	14.2288	0.806839	0.403419	−	0.915015i	$-0.367822\pi$
$311$	14.2288	0.806839	0.403419	+	0.915015i	$0.367822\pi$
$312$	0	0
$313$	16.7085	0.944420	0.472210	−	0.881486i	$-0.343456\pi$
$313$	16.7085	0.944420	0.472210	+	0.881486i	$0.343456\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.06275	0.115855	0.0579277	−	0.998321i	$-0.481551\pi$
$317$	2.06275	0.115855	0.0579277	+	0.998321i	$0.481551\pi$
$318$	0	0
$319$	9.29150	0.520224
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.2915	0.683918
$324$	0	0
$325$	3.64575	0.202230
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−8.70850	−0.480115
$330$	0	0
$331$	−3.29150	−0.180917	−0.0904587	−	0.995900i	$-0.528833\pi$
$331$	−3.29150	−0.180917	−0.0904587	+	0.995900i	$0.528833\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.00000	0.327815
$336$	0	0
$337$	10.2288	0.557196	0.278598	−	0.960408i	$-0.410130\pi$
$337$	10.2288	0.557196	0.278598	+	0.960408i	$0.410130\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4.35425	−0.235796
$342$	0	0
$343$	−18.5830	−1.00339
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	−8.87451	−0.475042	−0.237521	−	0.971382i	$-0.576335\pi$
$349$	−8.87451	−0.475042	−0.237521	+	0.971382i	$0.576335\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.29150	0.0687397	0.0343699	−	0.999409i	$-0.489058\pi$
$353$	1.29150	0.0687397	0.0343699	+	0.999409i	$0.489058\pi$
$354$	0	0
$355$	2.35425	0.124951
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.41699	−0.391454	−0.195727	−	0.980658i	$-0.562707\pi$
$359$	−7.41699	−0.391454	−0.195727	+	0.980658i	$0.562707\pi$
$360$	0	0
$361$	−12.0000	−0.631579
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−0.937254	−0.0490581
$366$	0	0
$367$	−33.6458	−1.75629	−0.878147	−	0.478391i	$-0.841220\pi$
$367$	−33.6458	−1.75629	−0.878147	+	0.478391i	$0.841220\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.47974	−0.336411
$372$	0	0
$373$	28.0000	1.44979	0.724893	−	0.688862i	$-0.241889\pi$
$373$	28.0000	1.44979	0.724893	+	0.688862i	$0.241889\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	20.5830	1.06008
$378$	0	0
$379$	8.77124	0.450549	0.225274	−	0.974295i	$-0.427672\pi$
$379$	8.77124	0.450549	0.225274	+	0.974295i	$0.427672\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	11.0000	0.562074	0.281037	−	0.959697i	$-0.409322\pi$
$383$	11.0000	0.562074	0.281037	+	0.959697i	$0.409322\pi$
$384$	0	0
$385$	2.70850	0.138038
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−14.2288	−0.721427	−0.360713	−	0.932677i	$-0.617467\pi$
$389$	−14.2288	−0.721427	−0.360713	+	0.932677i	$0.617467\pi$
$390$	0	0
$391$	19.9373	1.00827
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7.35425	−0.370032
$396$	0	0
$397$	24.1033	1.20971	0.604854	−	0.796336i	$-0.293231\pi$
$397$	24.1033	1.20971	0.604854	+	0.796336i	$0.293231\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.29150	0.0644946	0.0322473	−	0.999480i	$-0.489734\pi$
$401$	1.29150	0.0644946	0.0322473	+	0.999480i	$0.489734\pi$
$402$	0	0
$403$	−9.64575	−0.480489
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.70850	−0.431664
$408$	0	0
$409$	13.5830	0.671636	0.335818	−	0.941927i	$-0.390987\pi$
$409$	13.5830	0.671636	0.335818	+	0.941927i	$0.390987\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−6.00000	−0.295241
$414$	0	0
$415$	−5.00000	−0.245440
$416$	0	0
$417$	0	0
$418$	0	0
$419$	9.64575	0.471226	0.235613	−	0.971847i	$-0.424290\pi$
$419$	9.64575	0.471226	0.235613	+	0.971847i	$0.424290\pi$
$420$	0	0
$421$	−29.4575	−1.43567	−0.717836	−	0.696213i	$-0.754867\pi$
$421$	−29.4575	−1.43567	−0.717836	+	0.696213i	$0.754867\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.64575	0.225352
$426$	0	0
$427$	12.4797	0.603937
$428$	0	0
$429$	0	0
$430$	0	0
$431$	18.9373	0.912175	0.456088	−	0.889935i	$-0.349250\pi$
$431$	18.9373	0.912175	0.456088	+	0.889935i	$0.349250\pi$
$432$	0	0
$433$	30.9373	1.48675	0.743375	−	0.668875i	$-0.233224\pi$
$433$	30.9373	1.48675	0.743375	+	0.668875i	$0.233224\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	11.3542	0.543147
$438$	0	0
$439$	−35.1033	−1.67539	−0.837694	−	0.546140i	$-0.816097\pi$
$439$	−35.1033	−1.67539	−0.837694	+	0.546140i	$0.816097\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−36.0405	−1.71234	−0.856168	−	0.516697i	$-0.827161\pi$
$443$	−36.0405	−1.71234	−0.856168	+	0.516697i	$0.827161\pi$
$444$	0	0
$445$	8.35425	0.396029
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0.479741	0.0226404	0.0113202	−	0.999936i	$-0.496397\pi$
$449$	0.479741	0.0226404	0.0113202	+	0.999936i	$0.496397\pi$
$450$	0	0
$451$	0.583005	0.0274526
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.00000	0.281284
$456$	0	0
$457$	−14.5830	−0.682164	−0.341082	−	0.940034i	$-0.610793\pi$
$457$	−14.5830	−0.682164	−0.341082	+	0.940034i	$0.610793\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.58301	0.399750	0.199875	−	0.979821i	$-0.435946\pi$
$461$	8.58301	0.399750	0.199875	+	0.979821i	$0.435946\pi$
$462$	0	0
$463$	−9.29150	−0.431813	−0.215906	−	0.976414i	$-0.569271\pi$
$463$	−9.29150	−0.431813	−0.215906	+	0.976414i	$0.569271\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−15.4575	−0.715288	−0.357644	−	0.933858i	$-0.616420\pi$
$467$	−15.4575	−0.715288	−0.357644	+	0.933858i	$0.616420\pi$
$468$	0	0
$469$	9.87451	0.455962
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−11.4170	−0.524954
$474$	0	0
$475$	2.64575	0.121395
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−7.52026	−0.343609	−0.171805	−	0.985131i	$-0.554960\pi$
$479$	−7.52026	−0.343609	−0.171805	+	0.985131i	$0.554960\pi$
$480$	0	0
$481$	−19.2915	−0.879617
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.708497	−0.0321712
$486$	0	0
$487$	21.5203	0.975176	0.487588	−	0.873074i	$-0.337877\pi$
$487$	21.5203	0.975176	0.487588	+	0.873074i	$0.337877\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.58301	0.206828	0.103414	−	0.994638i	$-0.467023\pi$
$491$	4.58301	0.206828	0.103414	+	0.994638i	$0.467023\pi$
$492$	0	0
$493$	26.2288	1.18128
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.87451	0.173795
$498$	0	0
$499$	−3.81176	−0.170638	−0.0853189	−	0.996354i	$-0.527191\pi$
$499$	−3.81176	−0.170638	−0.0853189	+	0.996354i	$0.527191\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−19.4575	−0.867568	−0.433784	−	0.901017i	$-0.642822\pi$
$503$	−19.4575	−0.867568	−0.433784	+	0.901017i	$0.642822\pi$
$504$	0	0
$505$	11.2915	0.502465
$506$	0	0
$507$	0	0
$508$	0	0
$509$	31.7490	1.40725	0.703625	−	0.710571i	$-0.251563\pi$
$509$	31.7490	1.40725	0.703625	+	0.710571i	$0.251563\pi$
$510$	0	0
$511$	−1.54249	−0.0682356
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.708497	0.0312201
$516$	0	0
$517$	−8.70850	−0.382999
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.22876	−0.272887	−0.136443	−	0.990648i	$-0.543567\pi$
$521$	−6.22876	−0.272887	−0.136443	+	0.990648i	$0.543567\pi$
$522$	0	0
$523$	40.6863	1.77909	0.889543	−	0.456851i	$-0.151023\pi$
$523$	40.6863	1.77909	0.889543	+	0.456851i	$0.151023\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.2915	−0.535426
$528$	0	0
$529$	−4.58301	−0.199261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.29150	0.0559412
$534$	0	0
$535$	6.00000	0.259403
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−7.06275	−0.304214
$540$	0	0
$541$	−30.5830	−1.31487	−0.657433	−	0.753513i	$-0.728358\pi$
$541$	−30.5830	−1.31487	−0.657433	+	0.753513i	$0.728358\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.29150	0.0981572
$546$	0	0
$547$	−9.06275	−0.387495	−0.193748	−	0.981051i	$-0.562064\pi$
$547$	−9.06275	−0.387495	−0.193748	+	0.981051i	$0.562064\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	14.9373	0.636348
$552$	0	0
$553$	−12.1033	−0.514683
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−33.2915	−1.41061	−0.705303	−	0.708906i	$-0.749189\pi$
$557$	−33.2915	−1.41061	−0.705303	+	0.708906i	$0.749189\pi$
$558$	0	0
$559$	−25.2915	−1.06972
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6.00000	0.252870	0.126435	−	0.991975i	$-0.459647\pi$
$563$	6.00000	0.252870	0.126435	+	0.991975i	$0.459647\pi$
$564$	0	0
$565$	14.5830	0.613512
$566$	0	0
$567$	0	0
$568$	0	0
$569$	22.1033	0.926617	0.463309	−	0.886197i	$-0.346662\pi$
$569$	22.1033	0.926617	0.463309	+	0.886197i	$0.346662\pi$
$570$	0	0
$571$	35.9373	1.50393	0.751964	−	0.659205i	$-0.229107\pi$
$571$	35.9373	1.50393	0.751964	+	0.659205i	$0.229107\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.29150	0.178968
$576$	0	0
$577$	−37.3948	−1.55676	−0.778382	−	0.627791i	$-0.783959\pi$
$577$	−37.3948	−1.55676	−0.778382	+	0.627791i	$0.783959\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−8.22876	−0.341386
$582$	0	0
$583$	−6.47974	−0.268363
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.41699	0.0997600	0.0498800	−	0.998755i	$-0.484116\pi$
$587$	2.41699	0.0997600	0.0498800	+	0.998755i	$0.484116\pi$
$588$	0	0
$589$	−7.00000	−0.288430
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−26.3948	−1.08390	−0.541952	−	0.840410i	$-0.682314\pi$
$593$	−26.3948	−1.08390	−0.541952	+	0.840410i	$0.682314\pi$
$594$	0	0
$595$	7.64575	0.313445
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−20.8118	−0.850346	−0.425173	−	0.905112i	$-0.639787\pi$
$599$	−20.8118	−0.850346	−0.425173	+	0.905112i	$0.639787\pi$
$600$	0	0
$601$	−21.7085	−0.885508	−0.442754	−	0.896643i	$-0.645999\pi$
$601$	−21.7085	−0.885508	−0.442754	+	0.896643i	$0.645999\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−8.29150	−0.337098
$606$	0	0
$607$	10.2288	0.415172	0.207586	−	0.978217i	$-0.433439\pi$
$607$	10.2288	0.415172	0.207586	+	0.978217i	$0.433439\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−19.2915	−0.780451
$612$	0	0
$613$	20.0000	0.807792	0.403896	−	0.914805i	$-0.367656\pi$
$613$	20.0000	0.807792	0.403896	+	0.914805i	$0.367656\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−41.8118	−1.68328	−0.841639	−	0.540040i	$-0.818409\pi$
$617$	−41.8118	−1.68328	−0.841639	+	0.540040i	$0.818409\pi$
$618$	0	0
$619$	17.8745	0.718437	0.359219	−	0.933253i	$-0.383043\pi$
$619$	17.8745	0.718437	0.359219	+	0.933253i	$0.383043\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	13.7490	0.550843
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−24.5830	−0.980189
$630$	0	0
$631$	−16.5203	−0.657661	−0.328831	−	0.944389i	$-0.606655\pi$
$631$	−16.5203	−0.657661	−0.328831	+	0.944389i	$0.606655\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−20.5830	−0.816812
$636$	0	0
$637$	−15.6458	−0.619907
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−41.1660	−1.62596	−0.812980	−	0.582292i	$-0.802156\pi$
$641$	−41.1660	−1.62596	−0.812980	+	0.582292i	$0.802156\pi$
$642$	0	0
$643$	24.2288	0.955489	0.477744	−	0.878499i	$-0.341454\pi$
$643$	24.2288	0.955489	0.477744	+	0.878499i	$0.341454\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9.00000	−0.353827	−0.176913	−	0.984226i	$-0.556611\pi$
$647$	−9.00000	−0.353827	−0.176913	+	0.984226i	$0.556611\pi$
$648$	0	0
$649$	−6.00000	−0.235521
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4.06275	−0.158988	−0.0794938	−	0.996835i	$-0.525330\pi$
$653$	−4.06275	−0.158988	−0.0794938	+	0.996835i	$0.525330\pi$
$654$	0	0
$655$	−19.8745	−0.776561
$656$	0	0
$657$	0	0
$658$	0	0
$659$	43.6458	1.70020	0.850099	−	0.526624i	$-0.176542\pi$
$659$	43.6458	1.70020	0.850099	+	0.526624i	$0.176542\pi$
$660$	0	0
$661$	−2.83399	−0.110229	−0.0551147	−	0.998480i	$-0.517552\pi$
$661$	−2.83399	−0.110229	−0.0551147	+	0.998480i	$0.517552\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.35425	0.168851
$666$	0	0
$667$	24.2288	0.938141
$668$	0	0
$669$	0	0
$670$	0	0
$671$	12.4797	0.481775
$672$	0	0
$673$	46.2288	1.78199	0.890994	−	0.454015i	$-0.150009\pi$
$673$	46.2288	1.78199	0.890994	+	0.454015i	$0.150009\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−5.41699	−0.208192	−0.104096	−	0.994567i	$-0.533195\pi$
$677$	−5.41699	−0.208192	−0.104096	+	0.994567i	$0.533195\pi$
$678$	0	0
$679$	−1.16601	−0.0447474
$680$	0	0
$681$	0	0
$682$	0	0
$683$	27.5830	1.05543	0.527717	−	0.849420i	$-0.323048\pi$
$683$	27.5830	1.05543	0.527717	+	0.849420i	$0.323048\pi$
$684$	0	0
$685$	2.06275	0.0788135
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−14.3542	−0.546853
$690$	0	0
$691$	−39.2288	−1.49233	−0.746166	−	0.665760i	$-0.768108\pi$
$691$	−39.2288	−1.49233	−0.746166	+	0.665760i	$0.768108\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	22.5830	0.856622
$696$	0	0
$697$	1.64575	0.0623373
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−8.12549	−0.306896	−0.153448	−	0.988157i	$-0.549038\pi$
$701$	−8.12549	−0.306896	−0.153448	+	0.988157i	$0.549038\pi$
$702$	0	0
$703$	−14.0000	−0.528020
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18.5830	0.698886
$708$	0	0
$709$	15.7490	0.591467	0.295733	−	0.955271i	$-0.404436\pi$
$709$	15.7490	0.591467	0.295733	+	0.955271i	$0.404436\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−11.3542	−0.425220
$714$	0	0
$715$	6.00000	0.224387
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−52.5830	−1.96102	−0.980508	−	0.196481i	$-0.937049\pi$
$719$	−52.5830	−1.96102	−0.980508	+	0.196481i	$0.937049\pi$
$720$	0	0
$721$	1.16601	0.0434245
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.64575	0.209678
$726$	0	0
$727$	−17.8745	−0.662929	−0.331464	−	0.943468i	$-0.607543\pi$
$727$	−17.8745	−0.662929	−0.331464	+	0.943468i	$0.607543\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−32.2288	−1.19202
$732$	0	0
$733$	28.7085	1.06037	0.530186	−	0.847881i	$-0.322122\pi$
$733$	28.7085	1.06037	0.530186	+	0.847881i	$0.322122\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.87451	0.363732
$738$	0	0
$739$	−47.6863	−1.75417	−0.877084	−	0.480337i	$-0.840514\pi$
$739$	−47.6863	−1.75417	−0.877084	+	0.480337i	$0.840514\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−29.1660	−1.07000	−0.534999	−	0.844853i	$-0.679688\pi$
$743$	−29.1660	−1.07000	−0.534999	+	0.844853i	$0.679688\pi$
$744$	0	0
$745$	18.9373	0.693807
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.87451	0.360807
$750$	0	0
$751$	2.06275	0.0752707	0.0376353	−	0.999292i	$-0.488017\pi$
$751$	2.06275	0.0752707	0.0376353	+	0.999292i	$0.488017\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−17.2915	−0.629302
$756$	0	0
$757$	−28.5830	−1.03887	−0.519433	−	0.854511i	$-0.673857\pi$
$757$	−28.5830	−1.03887	−0.519433	+	0.854511i	$0.673857\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−37.6458	−1.36466	−0.682329	−	0.731046i	$-0.739033\pi$
$761$	−37.6458	−1.36466	−0.682329	+	0.731046i	$0.739033\pi$
$762$	0	0
$763$	3.77124	0.136528
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−13.2915	−0.479928
$768$	0	0
$769$	−26.1660	−0.943570	−0.471785	−	0.881713i	$-0.656390\pi$
$769$	−26.1660	−0.943570	−0.471785	+	0.881713i	$0.656390\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−49.2288	−1.77064	−0.885318	−	0.464987i	$-0.846059\pi$
$773$	−49.2288	−1.77064	−0.885318	+	0.464987i	$0.846059\pi$
$774$	0	0
$775$	−2.64575	−0.0950382
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.937254	0.0335806
$780$	0	0
$781$	3.87451	0.138641
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−12.9373	−0.461750
$786$	0	0
$787$	24.8118	0.884444	0.442222	−	0.896906i	$-0.354190\pi$
$787$	24.8118	0.884444	0.442222	+	0.896906i	$0.354190\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	24.0000	0.853342
$792$	0	0
$793$	27.6458	0.981729
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−23.8118	−0.843456	−0.421728	−	0.906722i	$-0.638576\pi$
$797$	−23.8118	−0.843456	−0.421728	+	0.906722i	$0.638576\pi$
$798$	0	0
$799$	−24.5830	−0.869684
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.54249	−0.0544332
$804$	0	0
$805$	7.06275	0.248929
$806$	0	0
$807$	0	0
$808$	0	0
$809$	10.9373	0.384533	0.192267	−	0.981343i	$-0.438416\pi$
$809$	10.9373	0.384533	0.192267	+	0.981343i	$0.438416\pi$
$810$	0	0
$811$	2.70850	0.0951082	0.0475541	−	0.998869i	$-0.484857\pi$
$811$	2.70850	0.0951082	0.0475541	+	0.998869i	$0.484857\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.29150	−0.0452394
$816$	0	0
$817$	−18.3542	−0.642134
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−24.5830	−0.857953	−0.428976	−	0.903316i	$-0.641126\pi$
$821$	−24.5830	−0.857953	−0.428976	+	0.903316i	$0.641126\pi$
$822$	0	0
$823$	52.1033	1.81621	0.908103	−	0.418747i	$-0.137531\pi$
$823$	52.1033	1.81621	0.908103	+	0.418747i	$0.137531\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	46.0405	1.60099	0.800493	−	0.599342i	$-0.204571\pi$
$827$	46.0405	1.60099	0.800493	+	0.599342i	$0.204571\pi$
$828$	0	0
$829$	−16.4575	−0.571593	−0.285797	−	0.958290i	$-0.592258\pi$
$829$	−16.4575	−0.571593	−0.285797	+	0.958290i	$0.592258\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−19.9373	−0.690785
$834$	0	0
$835$	−12.2915	−0.425365
$836$	0	0
$837$	0	0
$838$	0	0
$839$	48.4575	1.67294	0.836470	−	0.548013i	$-0.184616\pi$
$839$	48.4575	1.67294	0.836470	+	0.548013i	$0.184616\pi$
$840$	0	0
$841$	2.87451	0.0991210
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0.291503	0.0100280
$846$	0	0
$847$	−13.6458	−0.468874
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−22.7085	−0.778437
$852$	0	0
$853$	−3.29150	−0.112699	−0.0563495	−	0.998411i	$-0.517946\pi$
$853$	−3.29150	−0.112699	−0.0563495	+	0.998411i	$0.517946\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−35.8118	−1.22331	−0.611653	−	0.791126i	$-0.709495\pi$
$857$	−35.8118	−1.22331	−0.611653	+	0.791126i	$0.709495\pi$
$858$	0	0
$859$	36.0627	1.23045	0.615223	−	0.788354i	$-0.289066\pi$
$859$	36.0627	1.23045	0.615223	+	0.788354i	$0.289066\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	39.5830	1.34742	0.673711	−	0.738995i	$-0.264699\pi$
$863$	39.5830	1.34742	0.673711	+	0.738995i	$0.264699\pi$
$864$	0	0
$865$	11.3542	0.386056
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−12.1033	−0.410575
$870$	0	0
$871$	21.8745	0.741189
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.64575	0.0556365
$876$	0	0
$877$	−22.3542	−0.754849	−0.377425	−	0.926040i	$-0.623190\pi$
$877$	−22.3542	−0.754849	−0.377425	+	0.926040i	$0.623190\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	47.9778	1.61641	0.808206	−	0.588900i	$-0.200439\pi$
$881$	47.9778	1.61641	0.808206	+	0.588900i	$0.200439\pi$
$882$	0	0
$883$	−36.9373	−1.24304	−0.621519	−	0.783399i	$-0.713484\pi$
$883$	−36.9373	−1.24304	−0.621519	+	0.783399i	$0.713484\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.7490	0.830991	0.415495	−	0.909595i	$-0.363608\pi$
$887$	24.7490	0.830991	0.415495	+	0.909595i	$0.363608\pi$
$888$	0	0
$889$	−33.8745	−1.13611
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−14.0000	−0.468492
$894$	0	0
$895$	−7.29150	−0.243728
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−14.9373	−0.498185
$900$	0	0
$901$	−18.2915	−0.609378
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−16.2915	−0.541548
$906$	0	0
$907$	−37.8745	−1.25760	−0.628801	−	0.777566i	$-0.716454\pi$
$907$	−37.8745	−1.25760	−0.628801	+	0.777566i	$0.716454\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−24.1033	−0.798577	−0.399288	−	0.916825i	$-0.630743\pi$
$911$	−24.1033	−0.798577	−0.399288	+	0.916825i	$0.630743\pi$
$912$	0	0
$913$	−8.22876	−0.272332
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−32.7085	−1.08013
$918$	0	0
$919$	22.5830	0.744945	0.372472	−	0.928043i	$-0.378510\pi$
$919$	22.5830	0.744945	0.372472	+	0.928043i	$0.378510\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8.58301	0.282513
$924$	0	0
$925$	−5.29150	−0.173984
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−37.5203	−1.23100	−0.615500	−	0.788137i	$-0.711046\pi$
$929$	−37.5203	−1.23100	−0.615500	+	0.788137i	$0.711046\pi$
$930$	0	0
$931$	−11.3542	−0.372120
$932$	0	0
$933$	0	0
$934$	0	0
$935$	7.64575	0.250043
$936$	0	0
$937$	−17.4170	−0.568988	−0.284494	−	0.958678i	$-0.591826\pi$
$937$	−17.4170	−0.568988	−0.284494	+	0.958678i	$0.591826\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0.583005	0.0190054	0.00950271	−	0.999955i	$-0.496975\pi$
$941$	0.583005	0.0190054	0.00950271	+	0.999955i	$0.496975\pi$
$942$	0	0
$943$	1.52026	0.0495065
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−32.7490	−1.06420	−0.532100	−	0.846682i	$-0.678597\pi$
$947$	−32.7490	−1.06420	−0.532100	+	0.846682i	$0.678597\pi$
$948$	0	0
$949$	−3.41699	−0.110920
$950$	0	0
$951$	0	0
$952$	0	0
$953$	16.5830	0.537176	0.268588	−	0.963255i	$-0.413443\pi$
$953$	16.5830	0.537176	0.268588	+	0.963255i	$0.413443\pi$
$954$	0	0
$955$	2.70850	0.0876449
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.39477	0.109623
$960$	0	0
$961$	−24.0000	−0.774194
$962$	0	0
$963$	0	0
$964$	0	0
$965$	21.5203	0.692762
$966$	0	0
$967$	24.1255	0.775824	0.387912	−	0.921697i	$-0.373197\pi$
$967$	24.1255	0.775824	0.387912	+	0.921697i	$0.373197\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−30.4575	−0.977428	−0.488714	−	0.872444i	$-0.662534\pi$
$971$	−30.4575	−0.977428	−0.488714	+	0.872444i	$0.662534\pi$
$972$	0	0
$973$	37.1660	1.19149
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.54249	−0.177320	−0.0886599	−	0.996062i	$-0.528258\pi$
$977$	−5.54249	−0.177320	−0.0886599	+	0.996062i	$0.528258\pi$
$978$	0	0
$979$	13.7490	0.439420
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−17.8340	−0.568816	−0.284408	−	0.958703i	$-0.591797\pi$
$983$	−17.8340	−0.568816	−0.284408	+	0.958703i	$0.591797\pi$
$984$	0	0
$985$	21.8118	0.694981
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−29.7712	−0.946670
$990$	0	0
$991$	9.81176	0.311681	0.155840	−	0.987782i	$-0.450191\pi$
$991$	9.81176	0.311681	0.155840	+	0.987782i	$0.450191\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.87451	0.186234
$996$	0	0
$997$	4.93725	0.156364	0.0781822	−	0.996939i	$-0.475088\pi$
$997$	4.93725	0.156364	0.0781822	+	0.996939i	$0.475088\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.a.z.1.2	yes	2
3.2	odd	2		4320.2.a.p.1.2	✓	2
4.3	odd	2		4320.2.a.bd.1.1	yes	2
8.3	odd	2		8640.2.a.cs.1.1		2
8.5	even	2		8640.2.a.cl.1.2		2
12.11	even	2		4320.2.a.t.1.1	yes	2
24.5	odd	2		8640.2.a.cz.1.2		2
24.11	even	2		8640.2.a.dg.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4320.2.a.p.1.2	✓	2	3.2	odd	2
4320.2.a.t.1.1	yes	2	12.11	even	2
4320.2.a.z.1.2	yes	2	1.1	even	1	trivial
4320.2.a.bd.1.1	yes	2	4.3	odd	2
8640.2.a.cl.1.2		2	8.5	even	2
8640.2.a.cs.1.1		2	8.3	odd	2
8640.2.a.cz.1.2		2	24.5	odd	2
8640.2.a.dg.1.1		2	24.11	even	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:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +1.64575 q^{7} +1.64575 q^{11} +3.64575 q^{13} +4.64575 q^{17} +2.64575 q^{19} +4.29150 q^{23} +1.00000 q^{25} +5.64575 q^{29} -2.64575 q^{31} +1.64575 q^{35} -5.29150 q^{37} +0.354249 q^{41} -6.93725 q^{43} -5.29150 q^{47} -4.29150 q^{49} -3.93725 q^{53} +1.64575 q^{55} -3.64575 q^{59} +7.58301 q^{61} +3.64575 q^{65} +6.00000 q^{67} +2.35425 q^{71} -0.937254 q^{73} +2.70850 q^{77} -7.35425 q^{79} -5.00000 q^{83} +4.64575 q^{85} +8.35425 q^{89} +6.00000 q^{91} +2.64575 q^{95} -0.708497 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{7} - 2 q^{11} + 2 q^{13} + 4 q^{17} - 2 q^{23} + 2 q^{25} + 6 q^{29} - 2 q^{35} + 6 q^{41} + 2 q^{43} + 2 q^{49} + 8 q^{53} - 2 q^{55} - 2 q^{59} - 6 q^{61} + 2 q^{65} + 12 q^{67} + 10 q^{71}+ \cdots - 12 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^7 - 2 * q^11 + 2 * q^13 + 4 * q^17 - 2 * q^23 + 2 * q^25 + 6 * q^29 - 2 * q^35 + 6 * q^41 + 2 * q^43 + 2 * q^49 + 8 * q^53 - 2 * q^55 - 2 * q^59 - 6 * q^61 + 2 * q^65 + 12 * q^67 + 10 * q^71 + 14 * q^73 + 16 * q^77 - 20 * q^79 - 10 * q^83 + 4 * q^85 + 22 * q^89 + 12 * q^91 - 12 * q^97

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