LMFDB - Embedded newform 5328.2.a.bf.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5328,2,Mod(1,5328)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5328.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5328 = 2^{4} \cdot 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5328.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,1,0,-2,0,0,0,-1,0,-1,0,0,0,12,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$42.5442941969$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 74)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	5328.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30278 q^{5} +2.60555 q^{7} -2.30278 q^{11} +1.30278 q^{13} +6.00000 q^{17} -2.00000 q^{19} +3.90833 q^{23} +0.302776 q^{25} +3.90833 q^{29} +0.302776 q^{31} +6.00000 q^{35} +1.00000 q^{37} -9.90833 q^{41} -0.605551 q^{43} +4.60555 q^{47} -0.211103 q^{49} +6.00000 q^{53} -5.30278 q^{55} +10.6056 q^{59} +7.51388 q^{61} +3.00000 q^{65} +3.51388 q^{67} +6.00000 q^{71} -12.3028 q^{73} -6.00000 q^{77} -9.11943 q^{79} +2.78890 q^{83} +13.8167 q^{85} +9.21110 q^{89} +3.39445 q^{91} -4.60555 q^{95} -16.4222 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5} - 2 q^{7} - q^{11} - q^{13} + 12 q^{17} - 4 q^{19} - 3 q^{23} - 3 q^{25} - 3 q^{29} - 3 q^{31} + 12 q^{35} + 2 q^{37} - 9 q^{41} + 6 q^{43} + 2 q^{47} + 14 q^{49} + 12 q^{53} - 7 q^{55} + 14 q^{59}+ \cdots - 4 q^{97}+O(q^{100}) $ 2 * q + q^5 - 2 * q^7 - q^11 - q^13 + 12 * q^17 - 4 * q^19 - 3 * q^23 - 3 * q^25 - 3 * q^29 - 3 * q^31 + 12 * q^35 + 2 * q^37 - 9 * q^41 + 6 * q^43 + 2 * q^47 + 14 * q^49 + 12 * q^53 - 7 * q^55 + 14 * q^59 - 3 * q^61 + 6 * q^65 - 11 * q^67 + 12 * q^71 - 21 * q^73 - 12 * q^77 + 7 * q^79 + 20 * q^83 + 6 * q^85 + 4 * q^89 + 14 * q^91 - 2 * q^95 - 4 * 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
$5$	2.30278	1.02983	0.514916	−	0.857240i	$-0.327823\pi$
$5$	2.30278	1.02983	0.514916	+	0.857240i	$0.327823\pi$
$6$	0	0
$7$	2.60555	0.984806	0.492403	−	0.870367i	$-0.336119\pi$
$7$	2.60555	0.984806	0.492403	+	0.870367i	$0.336119\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.30278	−0.694313	−0.347156	−	0.937807i	$-0.612853\pi$
$11$	−2.30278	−0.694313	−0.347156	+	0.937807i	$0.612853\pi$
$12$	0	0
$13$	1.30278	0.361325	0.180662	−	0.983545i	$-0.442176\pi$
$13$	1.30278	0.361325	0.180662	+	0.983545i	$0.442176\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.90833	0.814942	0.407471	−	0.913218i	$-0.366411\pi$
$23$	3.90833	0.814942	0.407471	+	0.913218i	$0.366411\pi$
$24$	0	0
$25$	0.302776	0.0605551
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.90833	0.725758	0.362879	−	0.931836i	$-0.381794\pi$
$29$	3.90833	0.725758	0.362879	+	0.931836i	$0.381794\pi$
$30$	0	0
$31$	0.302776	0.0543801	0.0271901	−	0.999630i	$-0.491344\pi$
$31$	0.302776	0.0543801	0.0271901	+	0.999630i	$0.491344\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	6.00000	1.01419
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.90833	−1.54742	−0.773710	−	0.633540i	$-0.781601\pi$
$41$	−9.90833	−1.54742	−0.773710	+	0.633540i	$0.781601\pi$
$42$	0	0
$43$	−0.605551	−0.0923457	−0.0461729	−	0.998933i	$-0.514703\pi$
$43$	−0.605551	−0.0923457	−0.0461729	+	0.998933i	$0.514703\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.60555	0.671789	0.335894	−	0.941900i	$-0.390961\pi$
$47$	4.60555	0.671789	0.335894	+	0.941900i	$0.390961\pi$
$48$	0	0
$49$	−0.211103	−0.0301575
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−5.30278	−0.715026
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.6056	1.38073	0.690363	−	0.723464i	$-0.257451\pi$
$59$	10.6056	1.38073	0.690363	+	0.723464i	$0.257451\pi$
$60$	0	0
$61$	7.51388	0.962054	0.481027	−	0.876706i	$-0.340264\pi$
$61$	7.51388	0.962054	0.481027	+	0.876706i	$0.340264\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.00000	0.372104
$66$	0	0
$67$	3.51388	0.429289	0.214644	−	0.976692i	$-0.431141\pi$
$67$	3.51388	0.429289	0.214644	+	0.976692i	$0.431141\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−12.3028	−1.43993	−0.719965	−	0.694010i	$-0.755842\pi$
$73$	−12.3028	−1.43993	−0.719965	+	0.694010i	$0.755842\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.00000	−0.683763
$78$	0	0
$79$	−9.11943	−1.02602	−0.513008	−	0.858384i	$-0.671469\pi$
$79$	−9.11943	−1.02602	−0.513008	+	0.858384i	$0.671469\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.78890	0.306121	0.153061	−	0.988217i	$-0.451087\pi$
$83$	2.78890	0.306121	0.153061	+	0.988217i	$0.451087\pi$
$84$	0	0
$85$	13.8167	1.49863
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.21110	0.976375	0.488187	−	0.872739i	$-0.337658\pi$
$89$	9.21110	0.976375	0.488187	+	0.872739i	$0.337658\pi$
$90$	0	0
$91$	3.39445	0.355835
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.60555	−0.472520
$96$	0	0
$97$	−16.4222	−1.66742	−0.833711	−	0.552201i	$-0.813788\pi$
$97$	−16.4222	−1.66742	−0.833711	+	0.552201i	$0.813788\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12.4222	1.23606	0.618028	−	0.786156i	$-0.287932\pi$
$101$	12.4222	1.23606	0.618028	+	0.786156i	$0.287932\pi$
$102$	0	0
$103$	0.302776	0.0298334	0.0149167	−	0.999889i	$-0.495252\pi$
$103$	0.302776	0.0298334	0.0149167	+	0.999889i	$0.495252\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.697224	0.0674032	0.0337016	−	0.999432i	$-0.489270\pi$
$107$	0.697224	0.0674032	0.0337016	+	0.999432i	$0.489270\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.21110	0.302075	0.151038	−	0.988528i	$-0.451739\pi$
$113$	3.21110	0.302075	0.151038	+	0.988528i	$0.451739\pi$
$114$	0	0
$115$	9.00000	0.839254
$116$	0	0
$117$	0	0
$118$	0	0
$119$	15.6333	1.43310
$120$	0	0
$121$	−5.69722	−0.517929
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.8167	−0.967471
$126$	0	0
$127$	19.2111	1.70471	0.852355	−	0.522964i	$-0.175174\pi$
$127$	19.2111	1.70471	0.852355	+	0.522964i	$0.175174\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.6056	0.926611	0.463306	−	0.886199i	$-0.346663\pi$
$131$	10.6056	0.926611	0.463306	+	0.886199i	$0.346663\pi$
$132$	0	0
$133$	−5.21110	−0.451860
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−0.908327	−0.0776036	−0.0388018	−	0.999247i	$-0.512354\pi$
$137$	−0.908327	−0.0776036	−0.0388018	+	0.999247i	$0.512354\pi$
$138$	0	0
$139$	1.90833	0.161862	0.0809311	−	0.996720i	$-0.474211\pi$
$139$	1.90833	0.161862	0.0809311	+	0.996720i	$0.474211\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.00000	−0.250873
$144$	0	0
$145$	9.00000	0.747409
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−19.8167	−1.62344	−0.811722	−	0.584044i	$-0.801469\pi$
$149$	−19.8167	−1.62344	−0.811722	+	0.584044i	$0.801469\pi$
$150$	0	0
$151$	20.6056	1.67686	0.838428	−	0.545012i	$-0.183475\pi$
$151$	20.6056	1.67686	0.838428	+	0.545012i	$0.183475\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.697224	0.0560024
$156$	0	0
$157$	−7.21110	−0.575509	−0.287754	−	0.957704i	$-0.592909\pi$
$157$	−7.21110	−0.575509	−0.287754	+	0.957704i	$0.592909\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	10.1833	0.802560
$162$	0	0
$163$	−8.42221	−0.659678	−0.329839	−	0.944037i	$-0.606994\pi$
$163$	−8.42221	−0.659678	−0.329839	+	0.944037i	$0.606994\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.51388	−0.426677	−0.213338	−	0.976978i	$-0.568434\pi$
$167$	−5.51388	−0.426677	−0.213338	+	0.976978i	$0.568434\pi$
$168$	0	0
$169$	−11.3028	−0.869444
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.78890	0.668207	0.334104	−	0.942536i	$-0.391566\pi$
$173$	8.78890	0.668207	0.334104	+	0.942536i	$0.391566\pi$
$174$	0	0
$175$	0.788897	0.0596350
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−13.8167	−1.03271	−0.516353	−	0.856376i	$-0.672711\pi$
$179$	−13.8167	−1.03271	−0.516353	+	0.856376i	$0.672711\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.30278	0.169303
$186$	0	0
$187$	−13.8167	−1.01037
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.51388	−0.398970	−0.199485	−	0.979901i	$-0.563927\pi$
$191$	−5.51388	−0.398970	−0.199485	+	0.979901i	$0.563927\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	−26.4222	−1.87302	−0.936510	−	0.350640i	$-0.885964\pi$
$199$	−26.4222	−1.87302	−0.936510	+	0.350640i	$0.885964\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	10.1833	0.714731
$204$	0	0
$205$	−22.8167	−1.59358
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.60555	0.318573
$210$	0	0
$211$	−10.3028	−0.709272	−0.354636	−	0.935004i	$-0.615395\pi$
$211$	−10.3028	−0.709272	−0.354636	+	0.935004i	$0.615395\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.39445	−0.0951006
$216$	0	0
$217$	0.788897	0.0535538
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7.81665	0.525805
$222$	0	0
$223$	5.81665	0.389512	0.194756	−	0.980852i	$-0.437609\pi$
$223$	5.81665	0.389512	0.194756	+	0.980852i	$0.437609\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.8167	0.917044	0.458522	−	0.888683i	$-0.348379\pi$
$227$	13.8167	0.917044	0.458522	+	0.888683i	$0.348379\pi$
$228$	0	0
$229$	24.6056	1.62598	0.812990	−	0.582277i	$-0.197838\pi$
$229$	24.6056	1.62598	0.812990	+	0.582277i	$0.197838\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.51388	−0.557763	−0.278881	−	0.960326i	$-0.589964\pi$
$233$	−8.51388	−0.557763	−0.278881	+	0.960326i	$0.589964\pi$
$234$	0	0
$235$	10.6056	0.691830
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−17.5139	−1.13288	−0.566439	−	0.824103i	$-0.691679\pi$
$239$	−17.5139	−1.13288	−0.566439	+	0.824103i	$0.691679\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.486122	−0.0310572
$246$	0	0
$247$	−2.60555	−0.165787
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−21.2111	−1.33883	−0.669416	−	0.742887i	$-0.733456\pi$
$251$	−21.2111	−1.33883	−0.669416	+	0.742887i	$0.733456\pi$
$252$	0	0
$253$	−9.00000	−0.565825
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.21110	−0.200303	−0.100152	−	0.994972i	$-0.531933\pi$
$257$	−3.21110	−0.200303	−0.100152	+	0.994972i	$0.531933\pi$
$258$	0	0
$259$	2.60555	0.161901
$260$	0	0
$261$	0	0
$262$	0	0
$263$	13.8167	0.851971	0.425986	−	0.904730i	$-0.359927\pi$
$263$	13.8167	0.851971	0.425986	+	0.904730i	$0.359927\pi$
$264$	0	0
$265$	13.8167	0.848750
$266$	0	0
$267$	0	0
$268$	0	0
$269$	21.2111	1.29326	0.646632	−	0.762802i	$-0.276177\pi$
$269$	21.2111	1.29326	0.646632	+	0.762802i	$0.276177\pi$
$270$	0	0
$271$	22.4222	1.36205	0.681026	−	0.732259i	$-0.261534\pi$
$271$	22.4222	1.36205	0.681026	+	0.732259i	$0.261534\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.697224	−0.0420442
$276$	0	0
$277$	0.119429	0.00717582	0.00358791	−	0.999994i	$-0.498858\pi$
$277$	0.119429	0.00717582	0.00358791	+	0.999994i	$0.498858\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	0	0
$283$	−24.6056	−1.46265	−0.731324	−	0.682030i	$-0.761097\pi$
$283$	−24.6056	−1.46265	−0.731324	+	0.682030i	$0.761097\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−25.8167	−1.52391
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−11.0278	−0.644248	−0.322124	−	0.946697i	$-0.604397\pi$
$293$	−11.0278	−0.644248	−0.322124	+	0.946697i	$0.604397\pi$
$294$	0	0
$295$	24.4222	1.42192
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.09167	0.294459
$300$	0	0
$301$	−1.57779	−0.0909426
$302$	0	0
$303$	0	0
$304$	0	0
$305$	17.3028	0.990754
$306$	0	0
$307$	−17.9083	−1.02208	−0.511041	−	0.859556i	$-0.670740\pi$
$307$	−17.9083	−1.02208	−0.511041	+	0.859556i	$0.670740\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	15.9083	0.902078	0.451039	−	0.892504i	$-0.351053\pi$
$311$	15.9083	0.902078	0.451039	+	0.892504i	$0.351053\pi$
$312$	0	0
$313$	−9.02776	−0.510279	−0.255139	−	0.966904i	$-0.582121\pi$
$313$	−9.02776	−0.510279	−0.255139	+	0.966904i	$0.582121\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.21110	−0.517347	−0.258674	−	0.965965i	$-0.583285\pi$
$317$	−9.21110	−0.517347	−0.258674	+	0.965965i	$0.583285\pi$
$318$	0	0
$319$	−9.00000	−0.503903
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−12.0000	−0.667698
$324$	0	0
$325$	0.394449	0.0218801
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.0000	0.661581
$330$	0	0
$331$	13.2111	0.726148	0.363074	−	0.931760i	$-0.381727\pi$
$331$	13.2111	0.726148	0.363074	+	0.931760i	$0.381727\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.09167	0.442095
$336$	0	0
$337$	6.11943	0.333347	0.166673	−	0.986012i	$-0.446697\pi$
$337$	6.11943	0.333347	0.166673	+	0.986012i	$0.446697\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−0.697224	−0.0377568
$342$	0	0
$343$	−18.7889	−1.01451
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.1833	0.546671	0.273335	−	0.961919i	$-0.411873\pi$
$347$	10.1833	0.546671	0.273335	+	0.961919i	$0.411873\pi$
$348$	0	0
$349$	28.2389	1.51159	0.755796	−	0.654807i	$-0.227250\pi$
$349$	28.2389	1.51159	0.755796	+	0.654807i	$0.227250\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10.1833	−0.542005	−0.271002	−	0.962579i	$-0.587355\pi$
$353$	−10.1833	−0.542005	−0.271002	+	0.962579i	$0.587355\pi$
$354$	0	0
$355$	13.8167	0.733312
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.21110	0.169476	0.0847378	−	0.996403i	$-0.472995\pi$
$359$	3.21110	0.169476	0.0847378	+	0.996403i	$0.472995\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−28.3305	−1.48289
$366$	0	0
$367$	−3.81665	−0.199228	−0.0996139	−	0.995026i	$-0.531761\pi$
$367$	−3.81665	−0.199228	−0.0996139	+	0.995026i	$0.531761\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	15.6333	0.811641
$372$	0	0
$373$	−17.8167	−0.922511	−0.461256	−	0.887267i	$-0.652601\pi$
$373$	−17.8167	−0.922511	−0.461256	+	0.887267i	$0.652601\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.09167	0.262235
$378$	0	0
$379$	−24.3305	−1.24978	−0.624888	−	0.780715i	$-0.714855\pi$
$379$	−24.3305	−1.24978	−0.624888	+	0.780715i	$0.714855\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−36.8444	−1.88266	−0.941331	−	0.337486i	$-0.890424\pi$
$383$	−36.8444	−1.88266	−0.941331	+	0.337486i	$0.890424\pi$
$384$	0	0
$385$	−13.8167	−0.704162
$386$	0	0
$387$	0	0
$388$	0	0
$389$	37.1194	1.88203	0.941015	−	0.338365i	$-0.109874\pi$
$389$	37.1194	1.88203	0.941015	+	0.338365i	$0.109874\pi$
$390$	0	0
$391$	23.4500	1.18592
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−21.0000	−1.05662
$396$	0	0
$397$	6.18335	0.310333	0.155167	−	0.987888i	$-0.450409\pi$
$397$	6.18335	0.310333	0.155167	+	0.987888i	$0.450409\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7.81665	0.390345	0.195173	−	0.980769i	$-0.437473\pi$
$401$	7.81665	0.390345	0.195173	+	0.980769i	$0.437473\pi$
$402$	0	0
$403$	0.394449	0.0196489
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.30278	−0.114144
$408$	0	0
$409$	31.0278	1.53422	0.767112	−	0.641513i	$-0.221693\pi$
$409$	31.0278	1.53422	0.767112	+	0.641513i	$0.221693\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	27.6333	1.35975
$414$	0	0
$415$	6.42221	0.315254
$416$	0	0
$417$	0	0
$418$	0	0
$419$	36.1472	1.76591	0.882953	−	0.469462i	$-0.155552\pi$
$419$	36.1472	1.76591	0.882953	+	0.469462i	$0.155552\pi$
$420$	0	0
$421$	−3.72498	−0.181544	−0.0907722	−	0.995872i	$-0.528934\pi$
$421$	−3.72498	−0.181544	−0.0907722	+	0.995872i	$0.528934\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.81665	0.0881207
$426$	0	0
$427$	19.5778	0.947436
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.21110	0.443683	0.221842	−	0.975083i	$-0.428793\pi$
$431$	9.21110	0.443683	0.221842	+	0.975083i	$0.428793\pi$
$432$	0	0
$433$	34.9361	1.67892	0.839461	−	0.543421i	$-0.182871\pi$
$433$	34.9361	1.67892	0.839461	+	0.543421i	$0.182871\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.81665	−0.373921
$438$	0	0
$439$	−30.3305	−1.44760	−0.723799	−	0.690011i	$-0.757606\pi$
$439$	−30.3305	−1.44760	−0.723799	+	0.690011i	$0.757606\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	32.7250	1.55481	0.777405	−	0.629000i	$-0.216535\pi$
$443$	32.7250	1.55481	0.777405	+	0.629000i	$0.216535\pi$
$444$	0	0
$445$	21.2111	1.00550
$446$	0	0
$447$	0	0
$448$	0	0
$449$	15.2111	0.717856	0.358928	−	0.933365i	$-0.383142\pi$
$449$	15.2111	0.717856	0.358928	+	0.933365i	$0.383142\pi$
$450$	0	0
$451$	22.8167	1.07439
$452$	0	0
$453$	0	0
$454$	0	0
$455$	7.81665	0.366450
$456$	0	0
$457$	−2.60555	−0.121883	−0.0609413	−	0.998141i	$-0.519410\pi$
$457$	−2.60555	−0.121883	−0.0609413	+	0.998141i	$0.519410\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−12.4222	−0.578560	−0.289280	−	0.957245i	$-0.593416\pi$
$461$	−12.4222	−0.578560	−0.289280	+	0.957245i	$0.593416\pi$
$462$	0	0
$463$	−26.6972	−1.24073	−0.620363	−	0.784315i	$-0.713015\pi$
$463$	−26.6972	−1.24073	−0.620363	+	0.784315i	$0.713015\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	9.15559	0.422766
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.39445	0.0641168
$474$	0	0
$475$	−0.605551	−0.0277846
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−13.1194	−0.599442	−0.299721	−	0.954027i	$-0.596894\pi$
$479$	−13.1194	−0.599442	−0.299721	+	0.954027i	$0.596894\pi$
$480$	0	0
$481$	1.30278	0.0594015
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−37.8167	−1.71717
$486$	0	0
$487$	37.2111	1.68620	0.843098	−	0.537760i	$-0.180729\pi$
$487$	37.2111	1.68620	0.843098	+	0.537760i	$0.180729\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17.7250	0.799917	0.399959	−	0.916533i	$-0.369025\pi$
$491$	17.7250	0.799917	0.399959	+	0.916533i	$0.369025\pi$
$492$	0	0
$493$	23.4500	1.05613
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.6333	0.701250
$498$	0	0
$499$	42.2389	1.89087	0.945436	−	0.325809i	$-0.105637\pi$
$499$	42.2389	1.89087	0.945436	+	0.325809i	$0.105637\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−6.48612	−0.289202	−0.144601	−	0.989490i	$-0.546190\pi$
$503$	−6.48612	−0.289202	−0.144601	+	0.989490i	$0.546190\pi$
$504$	0	0
$505$	28.6056	1.27293
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.18335	0.185424	0.0927118	−	0.995693i	$-0.470446\pi$
$509$	4.18335	0.185424	0.0927118	+	0.995693i	$0.470446\pi$
$510$	0	0
$511$	−32.0555	−1.41805
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.697224	0.0307234
$516$	0	0
$517$	−10.6056	−0.466432
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−33.6333	−1.47350	−0.736751	−	0.676164i	$-0.763641\pi$
$521$	−33.6333	−1.47350	−0.736751	+	0.676164i	$0.763641\pi$
$522$	0	0
$523$	18.2389	0.797530	0.398765	−	0.917053i	$-0.369439\pi$
$523$	18.2389	0.797530	0.398765	+	0.917053i	$0.369439\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.81665	0.0791347
$528$	0	0
$529$	−7.72498	−0.335869
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−12.9083	−0.559122
$534$	0	0
$535$	1.60555	0.0694140
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.486122	0.0209387
$540$	0	0
$541$	25.9361	1.11508	0.557540	−	0.830150i	$-0.311745\pi$
$541$	25.9361	1.11508	0.557540	+	0.830150i	$0.311745\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4.60555	0.197280
$546$	0	0
$547$	20.6056	0.881030	0.440515	−	0.897745i	$-0.354796\pi$
$547$	20.6056	0.881030	0.440515	+	0.897745i	$0.354796\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7.81665	−0.333001
$552$	0	0
$553$	−23.7611	−1.01043
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−11.5139	−0.487859	−0.243929	−	0.969793i	$-0.578436\pi$
$557$	−11.5139	−0.487859	−0.243929	+	0.969793i	$0.578436\pi$
$558$	0	0
$559$	−0.788897	−0.0333668
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−28.0555	−1.18240	−0.591199	−	0.806525i	$-0.701345\pi$
$563$	−28.0555	−1.18240	−0.591199	+	0.806525i	$0.701345\pi$
$564$	0	0
$565$	7.39445	0.311087
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−18.4222	−0.772299	−0.386150	−	0.922436i	$-0.626195\pi$
$569$	−18.4222	−0.772299	−0.386150	+	0.922436i	$0.626195\pi$
$570$	0	0
$571$	16.6972	0.698757	0.349379	−	0.936982i	$-0.386393\pi$
$571$	16.6972	0.698757	0.349379	+	0.936982i	$0.386393\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.18335	0.0493489
$576$	0	0
$577$	22.2389	0.925816	0.462908	−	0.886406i	$-0.346806\pi$
$577$	22.2389	0.925816	0.462908	+	0.886406i	$0.346806\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.26662	0.301470
$582$	0	0
$583$	−13.8167	−0.572227
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−45.6333	−1.88349	−0.941744	−	0.336330i	$-0.890814\pi$
$587$	−45.6333	−1.88349	−0.941744	+	0.336330i	$0.890814\pi$
$588$	0	0
$589$	−0.605551	−0.0249513
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−18.4861	−0.759134	−0.379567	−	0.925164i	$-0.623927\pi$
$593$	−18.4861	−0.759134	−0.379567	+	0.925164i	$0.623927\pi$
$594$	0	0
$595$	36.0000	1.47586
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−20.7889	−0.849411	−0.424706	−	0.905331i	$-0.639622\pi$
$599$	−20.7889	−0.849411	−0.424706	+	0.905331i	$0.639622\pi$
$600$	0	0
$601$	−24.3028	−0.991331	−0.495665	−	0.868514i	$-0.665076\pi$
$601$	−24.3028	−0.991331	−0.495665	+	0.868514i	$0.665076\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−13.1194	−0.533381
$606$	0	0
$607$	13.4861	0.547385	0.273692	−	0.961817i	$-0.411755\pi$
$607$	13.4861	0.547385	0.273692	+	0.961817i	$0.411755\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.00000	0.242734
$612$	0	0
$613$	−29.8167	−1.20428	−0.602142	−	0.798389i	$-0.705686\pi$
$613$	−29.8167	−1.20428	−0.602142	+	0.798389i	$0.705686\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	42.5694	1.71378	0.856890	−	0.515500i	$-0.172394\pi$
$617$	42.5694	1.71378	0.856890	+	0.515500i	$0.172394\pi$
$618$	0	0
$619$	6.30278	0.253330	0.126665	−	0.991946i	$-0.459573\pi$
$619$	6.30278	0.253330	0.126665	+	0.991946i	$0.459573\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	24.0000	0.961540
$624$	0	0
$625$	−26.4222	−1.05689
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.00000	0.239236
$630$	0	0
$631$	−14.6972	−0.585087	−0.292544	−	0.956252i	$-0.594502\pi$
$631$	−14.6972	−0.585087	−0.292544	+	0.956252i	$0.594502\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	44.2389	1.75557
$636$	0	0
$637$	−0.275019	−0.0108967
$638$	0	0
$639$	0	0
$640$	0	0
$641$	20.5139	0.810249	0.405125	−	0.914261i	$-0.367228\pi$
$641$	20.5139	0.810249	0.405125	+	0.914261i	$0.367228\pi$
$642$	0	0
$643$	8.18335	0.322720	0.161360	−	0.986896i	$-0.448412\pi$
$643$	8.18335	0.322720	0.161360	+	0.986896i	$0.448412\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20.9361	0.823082	0.411541	−	0.911391i	$-0.364991\pi$
$647$	20.9361	0.823082	0.411541	+	0.911391i	$0.364991\pi$
$648$	0	0
$649$	−24.4222	−0.958655
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.90833	0.152945	0.0764723	−	0.997072i	$-0.475634\pi$
$653$	3.90833	0.152945	0.0764723	+	0.997072i	$0.475634\pi$
$654$	0	0
$655$	24.4222	0.954255
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−16.8806	−0.657574	−0.328787	−	0.944404i	$-0.606640\pi$
$659$	−16.8806	−0.657574	−0.328787	+	0.944404i	$0.606640\pi$
$660$	0	0
$661$	−30.5139	−1.18685	−0.593426	−	0.804888i	$-0.702225\pi$
$661$	−30.5139	−1.18685	−0.593426	+	0.804888i	$0.702225\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−12.0000	−0.465340
$666$	0	0
$667$	15.2750	0.591451
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−17.3028	−0.667966
$672$	0	0
$673$	20.6972	0.797819	0.398910	−	0.916990i	$-0.369389\pi$
$673$	20.6972	0.797819	0.398910	+	0.916990i	$0.369389\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−14.2389	−0.547244	−0.273622	−	0.961837i	$-0.588222\pi$
$677$	−14.2389	−0.547244	−0.273622	+	0.961837i	$0.588222\pi$
$678$	0	0
$679$	−42.7889	−1.64209
$680$	0	0
$681$	0	0
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	−2.09167	−0.0799187
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7.81665	0.297791
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.39445	0.166691
$696$	0	0
$697$	−59.4500	−2.25183
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−40.1194	−1.51529	−0.757645	−	0.652667i	$-0.773650\pi$
$701$	−40.1194	−1.51529	−0.757645	+	0.652667i	$0.773650\pi$
$702$	0	0
$703$	−2.00000	−0.0754314
$704$	0	0
$705$	0	0
$706$	0	0
$707$	32.3667	1.21727
$708$	0	0
$709$	−41.3305	−1.55220	−0.776100	−	0.630609i	$-0.782805\pi$
$709$	−41.3305	−1.55220	−0.776100	+	0.630609i	$0.782805\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.18335	0.0443167
$714$	0	0
$715$	−6.90833	−0.258357
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−51.6333	−1.92560	−0.962799	−	0.270220i	$-0.912904\pi$
$719$	−51.6333	−1.92560	−0.962799	+	0.270220i	$0.912904\pi$
$720$	0	0
$721$	0.788897	0.0293801
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.18335	0.0439484
$726$	0	0
$727$	−19.0917	−0.708071	−0.354035	−	0.935232i	$-0.615191\pi$
$727$	−19.0917	−0.708071	−0.354035	+	0.935232i	$0.615191\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−3.63331	−0.134383
$732$	0	0
$733$	−13.6333	−0.503558	−0.251779	−	0.967785i	$-0.581016\pi$
$733$	−13.6333	−0.503558	−0.251779	+	0.967785i	$0.581016\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.09167	−0.298061
$738$	0	0
$739$	2.66947	0.0981980	0.0490990	−	0.998794i	$-0.484365\pi$
$739$	2.66947	0.0981980	0.0490990	+	0.998794i	$0.484365\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−29.4500	−1.08041	−0.540207	−	0.841532i	$-0.681654\pi$
$743$	−29.4500	−1.08041	−0.540207	+	0.841532i	$0.681654\pi$
$744$	0	0
$745$	−45.6333	−1.67188
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.81665	0.0663791
$750$	0	0
$751$	−14.0000	−0.510867	−0.255434	−	0.966827i	$-0.582218\pi$
$751$	−14.0000	−0.510867	−0.255434	+	0.966827i	$0.582218\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	47.4500	1.72688
$756$	0	0
$757$	5.69722	0.207069	0.103535	−	0.994626i	$-0.466985\pi$
$757$	5.69722	0.207069	0.103535	+	0.994626i	$0.466985\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−16.8806	−0.611920	−0.305960	−	0.952044i	$-0.598977\pi$
$761$	−16.8806	−0.611920	−0.305960	+	0.952044i	$0.598977\pi$
$762$	0	0
$763$	5.21110	0.188655
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.8167	0.498890
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−22.0555	−0.793282	−0.396641	−	0.917974i	$-0.629824\pi$
$773$	−22.0555	−0.793282	−0.396641	+	0.917974i	$0.629824\pi$
$774$	0	0
$775$	0.0916731	0.00329299
$776$	0	0
$777$	0	0
$778$	0	0
$779$	19.8167	0.710005
$780$	0	0
$781$	−13.8167	−0.494399
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−16.6056	−0.592678
$786$	0	0
$787$	−10.7889	−0.384583	−0.192291	−	0.981338i	$-0.561592\pi$
$787$	−10.7889	−0.384583	−0.192291	+	0.981338i	$0.561592\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8.36669	0.297485
$792$	0	0
$793$	9.78890	0.347614
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−22.3305	−0.790988	−0.395494	−	0.918469i	$-0.629427\pi$
$797$	−22.3305	−0.790988	−0.395494	+	0.918469i	$0.629427\pi$
$798$	0	0
$799$	27.6333	0.977596
$800$	0	0
$801$	0	0
$802$	0	0
$803$	28.3305	0.999763
$804$	0	0
$805$	23.4500	0.826503
$806$	0	0
$807$	0	0
$808$	0	0
$809$	35.4500	1.24635	0.623177	−	0.782081i	$-0.285842\pi$
$809$	35.4500	1.24635	0.623177	+	0.782081i	$0.285842\pi$
$810$	0	0
$811$	7.14719	0.250972	0.125486	−	0.992095i	$-0.459951\pi$
$811$	7.14719	0.250972	0.125486	+	0.992095i	$0.459951\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−19.3944	−0.679358
$816$	0	0
$817$	1.21110	0.0423711
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.21110	0.112068	0.0560341	−	0.998429i	$-0.482154\pi$
$821$	3.21110	0.112068	0.0560341	+	0.998429i	$0.482154\pi$
$822$	0	0
$823$	−44.8444	−1.56318	−0.781589	−	0.623794i	$-0.785590\pi$
$823$	−44.8444	−1.56318	−0.781589	+	0.623794i	$0.785590\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34.6056	1.20335	0.601676	−	0.798740i	$-0.294500\pi$
$827$	34.6056	1.20335	0.601676	+	0.798740i	$0.294500\pi$
$828$	0	0
$829$	−27.7250	−0.962928	−0.481464	−	0.876466i	$-0.659895\pi$
$829$	−27.7250	−0.962928	−0.481464	+	0.876466i	$0.659895\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.26662	−0.0438856
$834$	0	0
$835$	−12.6972	−0.439406
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−12.9722	−0.447852	−0.223926	−	0.974606i	$-0.571887\pi$
$839$	−12.9722	−0.447852	−0.223926	+	0.974606i	$0.571887\pi$
$840$	0	0
$841$	−13.7250	−0.473275
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−26.0278	−0.895382
$846$	0	0
$847$	−14.8444	−0.510060
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.90833	0.133976
$852$	0	0
$853$	42.5416	1.45660	0.728299	−	0.685260i	$-0.240311\pi$
$853$	42.5416	1.45660	0.728299	+	0.685260i	$0.240311\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−42.8444	−1.46354	−0.731769	−	0.681553i	$-0.761305\pi$
$857$	−42.8444	−1.46354	−0.731769	+	0.681553i	$0.761305\pi$
$858$	0	0
$859$	−48.0555	−1.63963	−0.819816	−	0.572626i	$-0.805925\pi$
$859$	−48.0555	−1.63963	−0.819816	+	0.572626i	$0.805925\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	0	0
$865$	20.2389	0.688142
$866$	0	0
$867$	0	0
$868$	0	0
$869$	21.0000	0.712376
$870$	0	0
$871$	4.57779	0.155113
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−28.1833	−0.952771
$876$	0	0
$877$	−7.21110	−0.243502	−0.121751	−	0.992561i	$-0.538851\pi$
$877$	−7.21110	−0.243502	−0.121751	+	0.992561i	$0.538851\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	28.5416	0.961592	0.480796	−	0.876832i	$-0.340348\pi$
$881$	28.5416	0.961592	0.480796	+	0.876832i	$0.340348\pi$
$882$	0	0
$883$	−26.4222	−0.889178	−0.444589	−	0.895735i	$-0.646650\pi$
$883$	−26.4222	−0.889178	−0.444589	+	0.895735i	$0.646650\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−0.422205	−0.0141763	−0.00708813	−	0.999975i	$-0.502256\pi$
$887$	−0.422205	−0.0141763	−0.00708813	+	0.999975i	$0.502256\pi$
$888$	0	0
$889$	50.0555	1.67881
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9.21110	−0.308238
$894$	0	0
$895$	−31.8167	−1.06351
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.18335	0.0394668
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	0	0
$904$	0	0
$905$	46.0555	1.53094
$906$	0	0
$907$	−26.0000	−0.863316	−0.431658	−	0.902037i	$-0.642071\pi$
$907$	−26.0000	−0.863316	−0.431658	+	0.902037i	$0.642071\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−17.5778	−0.582378	−0.291189	−	0.956665i	$-0.594051\pi$
$911$	−17.5778	−0.582378	−0.291189	+	0.956665i	$0.594051\pi$
$912$	0	0
$913$	−6.42221	−0.212544
$914$	0	0
$915$	0	0
$916$	0	0
$917$	27.6333	0.912532
$918$	0	0
$919$	9.57779	0.315942	0.157971	−	0.987444i	$-0.449505\pi$
$919$	9.57779	0.315942	0.157971	+	0.987444i	$0.449505\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7.81665	0.257288
$924$	0	0
$925$	0.302776	0.00995520
$926$	0	0
$927$	0	0
$928$	0	0
$929$	18.4861	0.606510	0.303255	−	0.952909i	$-0.401927\pi$
$929$	18.4861	0.606510	0.303255	+	0.952909i	$0.401927\pi$
$930$	0	0
$931$	0.422205	0.0138372
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−31.8167	−1.04052
$936$	0	0
$937$	−18.0917	−0.591029	−0.295515	−	0.955338i	$-0.595491\pi$
$937$	−18.0917	−0.591029	−0.295515	+	0.955338i	$0.595491\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−13.8167	−0.450410	−0.225205	−	0.974311i	$-0.572305\pi$
$941$	−13.8167	−0.450410	−0.225205	+	0.974311i	$0.572305\pi$
$942$	0	0
$943$	−38.7250	−1.26106
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.63331	−0.118067	−0.0590333	−	0.998256i	$-0.518802\pi$
$947$	−3.63331	−0.118067	−0.0590333	+	0.998256i	$0.518802\pi$
$948$	0	0
$949$	−16.0278	−0.520283
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−49.7527	−1.61165	−0.805825	−	0.592154i	$-0.798278\pi$
$953$	−49.7527	−1.61165	−0.805825	+	0.592154i	$0.798278\pi$
$954$	0	0
$955$	−12.6972	−0.410873
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.36669	−0.0764245
$960$	0	0
$961$	−30.9083	−0.997043
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−9.21110	−0.296516
$966$	0	0
$967$	6.72498	0.216261	0.108130	−	0.994137i	$-0.465514\pi$
$967$	6.72498	0.216261	0.108130	+	0.994137i	$0.465514\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−22.5416	−0.723395	−0.361698	−	0.932295i	$-0.617803\pi$
$971$	−22.5416	−0.723395	−0.361698	+	0.932295i	$0.617803\pi$
$972$	0	0
$973$	4.97224	0.159403
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	−21.2111	−0.677910
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−12.0000	−0.382741	−0.191370	−	0.981518i	$-0.561293\pi$
$983$	−12.0000	−0.382741	−0.191370	+	0.981518i	$0.561293\pi$
$984$	0	0
$985$	13.8167	0.440235
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.36669	−0.0752564
$990$	0	0
$991$	−50.6972	−1.61045	−0.805225	−	0.592969i	$-0.797956\pi$
$991$	−50.6972	−1.61045	−0.805225	+	0.592969i	$0.797956\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−60.8444	−1.92890
$996$	0	0
$997$	−52.4222	−1.66023	−0.830114	−	0.557594i	$-0.811725\pi$
$997$	−52.4222	−1.66023	−0.830114	+	0.557594i	$0.811725\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5328.2.a.bf.1.2		2
3.2	odd	2		592.2.a.f.1.1		2
4.3	odd	2		666.2.a.j.1.2		2
12.11	even	2		74.2.a.a.1.2	✓	2
24.5	odd	2		2368.2.a.ba.1.2		2
24.11	even	2		2368.2.a.s.1.1		2
60.23	odd	4		1850.2.b.i.149.4		4
60.47	odd	4		1850.2.b.i.149.1		4
60.59	even	2		1850.2.a.u.1.1		2
84.83	odd	2		3626.2.a.a.1.1		2
132.131	odd	2		8954.2.a.p.1.2		2
444.443	even	2		2738.2.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.a.a.1.2	✓	2	12.11	even	2
592.2.a.f.1.1		2	3.2	odd	2
666.2.a.j.1.2		2	4.3	odd	2
1850.2.a.u.1.1		2	60.59	even	2
1850.2.b.i.149.1		4	60.47	odd	4
1850.2.b.i.149.4		4	60.23	odd	4
2368.2.a.s.1.1		2	24.11	even	2
2368.2.a.ba.1.2		2	24.5	odd	2
2738.2.a.l.1.2		2	444.443	even	2
3626.2.a.a.1.1		2	84.83	odd	2
5328.2.a.bf.1.2		2	1.1	even	1	trivial
8954.2.a.p.1.2		2	132.131	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5328 = 2^{4} \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5328.a (trivial)

\(f(q)\)	\(=\)	\(q+2.30278 q^{5} +2.60555 q^{7} -2.30278 q^{11} +1.30278 q^{13} +6.00000 q^{17} -2.00000 q^{19} +3.90833 q^{23} +0.302776 q^{25} +3.90833 q^{29} +0.302776 q^{31} +6.00000 q^{35} +1.00000 q^{37} -9.90833 q^{41} -0.605551 q^{43} +4.60555 q^{47} -0.211103 q^{49} +6.00000 q^{53} -5.30278 q^{55} +10.6056 q^{59} +7.51388 q^{61} +3.00000 q^{65} +3.51388 q^{67} +6.00000 q^{71} -12.3028 q^{73} -6.00000 q^{77} -9.11943 q^{79} +2.78890 q^{83} +13.8167 q^{85} +9.21110 q^{89} +3.39445 q^{91} -4.60555 q^{95} -16.4222 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} - 2 q^{7} - q^{11} - q^{13} + 12 q^{17} - 4 q^{19} - 3 q^{23} - 3 q^{25} - 3 q^{29} - 3 q^{31} + 12 q^{35} + 2 q^{37} - 9 q^{41} + 6 q^{43} + 2 q^{47} + 14 q^{49} + 12 q^{53} - 7 q^{55} + 14 q^{59}+ \cdots - 4 q^{97}+O(q^{100}) \) 2 * q + q^5 - 2 * q^7 - q^11 - q^13 + 12 * q^17 - 4 * q^19 - 3 * q^23 - 3 * q^25 - 3 * q^29 - 3 * q^31 + 12 * q^35 + 2 * q^37 - 9 * q^41 + 6 * q^43 + 2 * q^47 + 14 * q^49 + 12 * q^53 - 7 * q^55 + 14 * q^59 - 3 * q^61 + 6 * q^65 - 11 * q^67 + 12 * q^71 - 21 * q^73 - 12 * q^77 + 7 * q^79 + 20 * q^83 + 6 * q^85 + 4 * q^89 + 14 * q^91 - 2 * q^95 - 4 * q^97

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