LMFDB - Embedded newform 5400.2.a.cg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5400,2,Mod(1,5400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5400, 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("5400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5400 = 2^{3} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5400.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:	$43.1192170915$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 11x^{2} + 16 $ x^4 - 11*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1080)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.04547$ of defining polynomial
Character	$\chi$	$=$	5400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.77753 q^{7} +O(q^{10})$	$q-4.77753 q^{7} +2.15068 q^{11} -6.09095 q^{13} -5.27492 q^{17} -3.27492 q^{19} +3.27492 q^{23} +6.09095 q^{29} +3.00000 q^{31} -2.62685 q^{37} -8.71780 q^{41} -10.3923 q^{43} -4.54983 q^{47} +15.8248 q^{49} +9.54983 q^{53} -3.46410 q^{59} +2.72508 q^{61} +2.62685 q^{67} +6.09095 q^{71} +2.15068 q^{73} -10.2749 q^{77} -1.27492 q^{79} +15.5498 q^{83} -10.3923 q^{89} +29.0997 q^{91} +16.1222 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 6 q^{17} + 2 q^{19} - 2 q^{23} + 12 q^{31} + 12 q^{47} + 18 q^{49} + 8 q^{53} + 26 q^{61} - 26 q^{77} + 10 q^{79} + 32 q^{83} + 56 q^{91}+O(q^{100}) $ 4 * q - 6 * q^17 + 2 * q^19 - 2 * q^23 + 12 * q^31 + 12 * q^47 + 18 * q^49 + 8 * q^53 + 26 * q^61 - 26 * q^77 + 10 * q^79 + 32 * q^83 + 56 * q^91

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.77753	−1.80573	−0.902867	−	0.429919i	$-0.858542\pi$
$7$	−4.77753	−1.80573	−0.902867	+	0.429919i	$0.858542\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.15068	0.648454	0.324227	−	0.945979i	$-0.394896\pi$
$11$	2.15068	0.648454	0.324227	+	0.945979i	$0.394896\pi$
$12$	0	0
$13$	−6.09095	−1.68933	−0.844663	−	0.535299i	$-0.820199\pi$
$13$	−6.09095	−1.68933	−0.844663	+	0.535299i	$0.820199\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.27492	−1.27936	−0.639678	−	0.768643i	$-0.720932\pi$
$17$	−5.27492	−1.27936	−0.639678	+	0.768643i	$0.720932\pi$
$18$	0	0
$19$	−3.27492	−0.751318	−0.375659	−	0.926758i	$-0.622584\pi$
$19$	−3.27492	−0.751318	−0.375659	+	0.926758i	$0.622584\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.27492	0.682867	0.341434	−	0.939906i	$-0.389088\pi$
$23$	3.27492	0.682867	0.341434	+	0.939906i	$0.389088\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.09095	1.13106	0.565530	−	0.824727i	$-0.308671\pi$
$29$	6.09095	1.13106	0.565530	+	0.824727i	$0.308671\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.62685	−0.431851	−0.215926	−	0.976410i	$-0.569277\pi$
$37$	−2.62685	−0.431851	−0.215926	+	0.976410i	$0.569277\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.71780	−1.36149	−0.680746	−	0.732520i	$-0.738344\pi$
$41$	−8.71780	−1.36149	−0.680746	+	0.732520i	$0.738344\pi$
$42$	0	0
$43$	−10.3923	−1.58481	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	−10.3923	−1.58481	−0.792406	+	0.609994i	$0.791172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.54983	−0.663662	−0.331831	−	0.943339i	$-0.607666\pi$
$47$	−4.54983	−0.663662	−0.331831	+	0.943339i	$0.607666\pi$
$48$	0	0
$49$	15.8248	2.26068
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.54983	1.31177	0.655885	−	0.754861i	$-0.272295\pi$
$53$	9.54983	1.31177	0.655885	+	0.754861i	$0.272295\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.46410	−0.450988	−0.225494	−	0.974245i	$-0.572400\pi$
$59$	−3.46410	−0.450988	−0.225494	+	0.974245i	$0.572400\pi$
$60$	0	0
$61$	2.72508	0.348911	0.174456	−	0.984665i	$-0.444183\pi$
$61$	2.72508	0.348911	0.174456	+	0.984665i	$0.444183\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.62685	0.320921	0.160460	−	0.987042i	$-0.448702\pi$
$67$	2.62685	0.320921	0.160460	+	0.987042i	$0.448702\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.09095	0.722863	0.361431	−	0.932399i	$-0.382288\pi$
$71$	6.09095	0.722863	0.361431	+	0.932399i	$0.382288\pi$
$72$	0	0
$73$	2.15068	0.251718	0.125859	−	0.992048i	$-0.459831\pi$
$73$	2.15068	0.251718	0.125859	+	0.992048i	$0.459831\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−10.2749	−1.17094
$78$	0	0
$79$	−1.27492	−0.143439	−0.0717197	−	0.997425i	$-0.522849\pi$
$79$	−1.27492	−0.143439	−0.0717197	+	0.997425i	$0.522849\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.5498	1.70682	0.853408	−	0.521243i	$-0.174532\pi$
$83$	15.5498	1.70682	0.853408	+	0.521243i	$0.174532\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.3923	−1.10158	−0.550791	−	0.834643i	$-0.685674\pi$
$89$	−10.3923	−1.10158	−0.550791	+	0.834643i	$0.685674\pi$
$90$	0	0
$91$	29.0997	3.05047
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.1222	1.63696	0.818479	−	0.574536i	$-0.194817\pi$
$97$	16.1222	1.63696	0.818479	+	0.574536i	$0.194817\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.98793	−0.297310	−0.148655	−	0.988889i	$-0.547494\pi$
$101$	−2.98793	−0.297310	−0.148655	+	0.988889i	$0.547494\pi$
$102$	0	0
$103$	16.4833	1.62414	0.812072	−	0.583558i	$-0.198340\pi$
$103$	16.4833	1.62414	0.812072	+	0.583558i	$0.198340\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.2749	0.993314	0.496657	−	0.867947i	$-0.334561\pi$
$107$	10.2749	0.993314	0.496657	+	0.867947i	$0.334561\pi$
$108$	0	0
$109$	−9.82475	−0.941041	−0.470520	−	0.882389i	$-0.655934\pi$
$109$	−9.82475	−0.941041	−0.470520	+	0.882389i	$0.655934\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	25.2011	2.31018
$120$	0	0
$121$	−6.37459	−0.579508
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.98793	0.265136	0.132568	−	0.991174i	$-0.457678\pi$
$127$	2.98793	0.265136	0.132568	+	0.991174i	$0.457678\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.98793	−0.261057	−0.130528	−	0.991445i	$-0.541667\pi$
$131$	−2.98793	−0.261057	−0.130528	+	0.991445i	$0.541667\pi$
$132$	0	0
$133$	15.6460	1.35668
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.82475	0.497642	0.248821	−	0.968549i	$-0.419957\pi$
$137$	5.82475	0.497642	0.248821	+	0.968549i	$0.419957\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−13.0997	−1.09545
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.40437	−0.606590	−0.303295	−	0.952897i	$-0.598087\pi$
$149$	−7.40437	−0.606590	−0.303295	+	0.952897i	$0.598087\pi$
$150$	0	0
$151$	18.2749	1.48719	0.743596	−	0.668629i	$-0.233119\pi$
$151$	18.2749	1.48719	0.743596	+	0.668629i	$0.233119\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−0.837253	−0.0668201	−0.0334101	−	0.999442i	$-0.510637\pi$
$157$	−0.837253	−0.0668201	−0.0334101	+	0.999442i	$0.510637\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−15.6460	−1.23308
$162$	0	0
$163$	17.4356	1.36566	0.682831	−	0.730577i	$-0.260749\pi$
$163$	17.4356	1.36566	0.682831	+	0.730577i	$0.260749\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.2749	0.872479	0.436240	−	0.899831i	$-0.356310\pi$
$167$	11.2749	0.872479	0.436240	+	0.899831i	$0.356310\pi$
$168$	0	0
$169$	24.0997	1.85382
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.09967	−0.463749	−0.231875	−	0.972746i	$-0.574486\pi$
$173$	−6.09967	−0.463749	−0.231875	+	0.972746i	$0.574486\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−23.0504	−1.72287	−0.861433	−	0.507871i	$-0.830433\pi$
$179$	−23.0504	−1.72287	−0.861433	+	0.507871i	$0.830433\pi$
$180$	0	0
$181$	19.8248	1.47356	0.736781	−	0.676131i	$-0.236345\pi$
$181$	19.8248	1.47356	0.736781	+	0.676131i	$0.236345\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−11.3446	−0.829603
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.8564	1.00261	0.501307	−	0.865269i	$-0.332853\pi$
$191$	13.8564	1.00261	0.501307	+	0.865269i	$0.332853\pi$
$192$	0	0
$193$	−19.5863	−1.40985	−0.704925	−	0.709281i	$-0.749020\pi$
$193$	−19.5863	−1.40985	−0.704925	+	0.709281i	$0.749020\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.00000	−0.641223	−0.320612	−	0.947211i	$-0.603888\pi$
$197$	−9.00000	−0.641223	−0.320612	+	0.947211i	$0.603888\pi$
$198$	0	0
$199$	−2.82475	−0.200241	−0.100121	−	0.994975i	$-0.531923\pi$
$199$	−2.82475	−0.200241	−0.100121	+	0.994975i	$0.531923\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−29.0997	−2.04240
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−7.04329	−0.487195
$210$	0	0
$211$	−5.82475	−0.400992	−0.200496	−	0.979694i	$-0.564255\pi$
$211$	−5.82475	−0.400992	−0.200496	+	0.979694i	$0.564255\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−14.3326	−0.972959
$218$	0	0
$219$	0	0
$220$	0	0
$221$	32.1293	2.16125
$222$	0	0
$223$	−0.952341	−0.0637735	−0.0318867	−	0.999491i	$-0.510152\pi$
$223$	−0.952341	−0.0637735	−0.0318867	+	0.999491i	$0.510152\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	21.2749	1.41207	0.706033	−	0.708179i	$-0.250483\pi$
$227$	21.2749	1.41207	0.706033	+	0.708179i	$0.250483\pi$
$228$	0	0
$229$	0.725083	0.0479148	0.0239574	−	0.999713i	$-0.492373\pi$
$229$	0.725083	0.0479148	0.0239574	+	0.999713i	$0.492373\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	9.55505	0.618065	0.309032	−	0.951051i	$-0.399995\pi$
$239$	9.55505	0.618065	0.309032	+	0.951051i	$0.399995\pi$
$240$	0	0
$241$	8.72508	0.562032	0.281016	−	0.959703i	$-0.409329\pi$
$241$	8.72508	0.562032	0.281016	+	0.959703i	$0.409329\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	19.9474	1.26922
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.55505	−0.603109	−0.301555	−	0.953449i	$-0.597506\pi$
$251$	−9.55505	−0.603109	−0.301555	+	0.953449i	$0.597506\pi$
$252$	0	0
$253$	7.04329	0.442808
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.7251	−1.16804	−0.584019	−	0.811740i	$-0.698521\pi$
$257$	−18.7251	−1.16804	−0.584019	+	0.811740i	$0.698521\pi$
$258$	0	0
$259$	12.5498	0.779809
$260$	0	0
$261$	0	0
$262$	0	0
$263$	28.5498	1.76046	0.880229	−	0.474549i	$-0.157389\pi$
$263$	28.5498	1.76046	0.880229	+	0.474549i	$0.157389\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	13.8564	0.844840	0.422420	−	0.906400i	$-0.361181\pi$
$269$	13.8564	0.844840	0.422420	+	0.906400i	$0.361181\pi$
$270$	0	0
$271$	23.0000	1.39715	0.698575	−	0.715537i	$-0.253818\pi$
$271$	23.0000	1.39715	0.698575	+	0.715537i	$0.253818\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−11.3446	−0.681634	−0.340817	−	0.940130i	$-0.610704\pi$
$277$	−11.3446	−0.681634	−0.340817	+	0.940130i	$0.610704\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	18.2728	1.09007	0.545033	−	0.838414i	$-0.316517\pi$
$281$	18.2728	1.09007	0.545033	+	0.838414i	$0.316517\pi$
$282$	0	0
$283$	−3.46410	−0.205919	−0.102960	−	0.994686i	$-0.532831\pi$
$283$	−3.46410	−0.205919	−0.102960	+	0.994686i	$0.532831\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	41.6495	2.45849
$288$	0	0
$289$	10.8248	0.636750
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−19.2749	−1.12605	−0.563026	−	0.826439i	$-0.690363\pi$
$293$	−19.2749	−1.12605	−0.563026	+	0.826439i	$0.690363\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−19.9474	−1.15359
$300$	0	0
$301$	49.6495	2.86175
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.67451	−0.0955692	−0.0477846	−	0.998858i	$-0.515216\pi$
$307$	−1.67451	−0.0955692	−0.0477846	+	0.998858i	$0.515216\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3.46410	−0.196431	−0.0982156	−	0.995165i	$-0.531313\pi$
$311$	−3.46410	−0.196431	−0.0982156	+	0.995165i	$0.531313\pi$
$312$	0	0
$313$	12.5430	0.708971	0.354486	−	0.935061i	$-0.384656\pi$
$313$	12.5430	0.708971	0.354486	+	0.935061i	$0.384656\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.0997	1.01658	0.508289	−	0.861186i	$-0.330278\pi$
$317$	18.0997	1.01658	0.508289	+	0.861186i	$0.330278\pi$
$318$	0	0
$319$	13.0997	0.733441
$320$	0	0
$321$	0	0
$322$	0	0
$323$	17.2749	0.961202
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	21.7370	1.19840
$330$	0	0
$331$	−0.549834	−0.0302216	−0.0151108	−	0.999886i	$-0.504810\pi$
$331$	−0.549834	−0.0302216	−0.0151108	+	0.999886i	$0.504810\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−14.6937	−0.800415	−0.400207	−	0.916425i	$-0.631062\pi$
$337$	−14.6937	−0.800415	−0.400207	+	0.916425i	$0.631062\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.45203	0.349397
$342$	0	0
$343$	−42.1605	−2.27645
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.27492	0.336855	0.168428	−	0.985714i	$-0.446131\pi$
$347$	6.27492	0.336855	0.168428	+	0.985714i	$0.446131\pi$
$348$	0	0
$349$	−22.9244	−1.22712	−0.613558	−	0.789650i	$-0.710262\pi$
$349$	−22.9244	−1.22712	−0.613558	+	0.789650i	$0.710262\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−15.0997	−0.803674	−0.401837	−	0.915711i	$-0.631628\pi$
$353$	−15.0997	−0.803674	−0.401837	+	0.915711i	$0.631628\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.1578	0.958330	0.479165	−	0.877725i	$-0.340940\pi$
$359$	18.1578	0.958330	0.479165	+	0.877725i	$0.340940\pi$
$360$	0	0
$361$	−8.27492	−0.435522
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5.72987	0.299097	0.149548	−	0.988754i	$-0.452218\pi$
$367$	5.72987	0.299097	0.149548	+	0.988754i	$0.452218\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−45.6246	−2.36871
$372$	0	0
$373$	−1.67451	−0.0867027	−0.0433513	−	0.999060i	$-0.513803\pi$
$373$	−1.67451	−0.0867027	−0.0433513	+	0.999060i	$0.513803\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−37.0997	−1.91073
$378$	0	0
$379$	2.72508	0.139978	0.0699891	−	0.997548i	$-0.477704\pi$
$379$	2.72508	0.139978	0.0699891	+	0.997548i	$0.477704\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8.17525	−0.417736	−0.208868	−	0.977944i	$-0.566978\pi$
$383$	−8.17525	−0.417736	−0.208868	+	0.977944i	$0.566978\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−20.3084	−1.02968	−0.514839	−	0.857287i	$-0.672148\pi$
$389$	−20.3084	−1.02968	−0.514839	+	0.857287i	$0.672148\pi$
$390$	0	0
$391$	−17.2749	−0.873630
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	32.1293	1.61252	0.806261	−	0.591561i	$-0.201488\pi$
$397$	32.1293	1.61252	0.806261	+	0.591561i	$0.201488\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.67451	−0.0836209	−0.0418104	−	0.999126i	$-0.513313\pi$
$401$	−1.67451	−0.0836209	−0.0418104	+	0.999126i	$0.513313\pi$
$402$	0	0
$403$	−18.2728	−0.910235
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.64950	−0.280035
$408$	0	0
$409$	−4.09967	−0.202716	−0.101358	−	0.994850i	$-0.532319\pi$
$409$	−4.09967	−0.202716	−0.101358	+	0.994850i	$0.532319\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	16.5498	0.814364
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−33.0816	−1.61614	−0.808071	−	0.589085i	$-0.799488\pi$
$419$	−33.0816	−1.61614	−0.808071	+	0.589085i	$0.799488\pi$
$420$	0	0
$421$	30.3746	1.48037	0.740183	−	0.672405i	$-0.234739\pi$
$421$	30.3746	1.48037	0.740183	+	0.672405i	$0.234739\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−13.0192	−0.630041
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.0668	−0.581238	−0.290619	−	0.956839i	$-0.593861\pi$
$431$	−12.0668	−0.581238	−0.290619	+	0.956839i	$0.593861\pi$
$432$	0	0
$433$	−21.2608	−1.02173	−0.510864	−	0.859662i	$-0.670674\pi$
$433$	−21.2608	−1.02173	−0.510864	+	0.859662i	$0.670674\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−10.7251	−0.513050
$438$	0	0
$439$	7.54983	0.360334	0.180167	−	0.983636i	$-0.442336\pi$
$439$	7.54983	0.360334	0.180167	+	0.983636i	$0.442336\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	29.2749	1.39089	0.695447	−	0.718578i	$-0.255207\pi$
$443$	29.2749	1.39089	0.695447	+	0.718578i	$0.255207\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	19.2252	0.907293	0.453646	−	0.891182i	$-0.350123\pi$
$449$	19.2252	0.907293	0.453646	+	0.891182i	$0.350123\pi$
$450$	0	0
$451$	−18.7492	−0.882864
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	23.7725	1.11203	0.556016	−	0.831171i	$-0.312329\pi$
$457$	23.7725	1.11203	0.556016	+	0.831171i	$0.312329\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10.8685	0.506195	0.253098	−	0.967441i	$-0.418551\pi$
$461$	10.8685	0.506195	0.253098	+	0.967441i	$0.418551\pi$
$462$	0	0
$463$	16.1222	0.749261	0.374630	−	0.927174i	$-0.377770\pi$
$463$	16.1222	0.749261	0.374630	+	0.927174i	$0.377770\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.6495	−0.492800	−0.246400	−	0.969168i	$-0.579248\pi$
$467$	−10.6495	−0.492800	−0.246400	+	0.969168i	$0.579248\pi$
$468$	0	0
$469$	−12.5498	−0.579498
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−22.3505	−1.02768
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−36.4306	−1.66456	−0.832279	−	0.554358i	$-0.812964\pi$
$479$	−36.4306	−1.66456	−0.832279	+	0.554358i	$0.812964\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−29.5024	−1.33688	−0.668441	−	0.743765i	$-0.733038\pi$
$487$	−29.5024	−1.33688	−0.668441	+	0.743765i	$0.733038\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12.5430	0.566057	0.283028	−	0.959112i	$-0.408661\pi$
$491$	12.5430	0.566057	0.283028	+	0.959112i	$0.408661\pi$
$492$	0	0
$493$	−32.1293	−1.44703
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−29.0997	−1.30530
$498$	0	0
$499$	17.8248	0.797945	0.398973	−	0.916963i	$-0.369367\pi$
$499$	17.8248	0.797945	0.398973	+	0.916963i	$0.369367\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.82475	−0.0813617	−0.0406808	−	0.999172i	$-0.512953\pi$
$503$	−1.82475	−0.0813617	−0.0406808	+	0.999172i	$0.512953\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16.1222	0.714603	0.357301	−	0.933989i	$-0.383697\pi$
$509$	16.1222	0.714603	0.357301	+	0.933989i	$0.383697\pi$
$510$	0	0
$511$	−10.2749	−0.454536
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−9.78523	−0.430354
$518$	0	0
$519$	0	0
$520$	0	0
$521$	37.3830	1.63778	0.818888	−	0.573953i	$-0.194591\pi$
$521$	37.3830	1.63778	0.818888	+	0.573953i	$0.194591\pi$
$522$	0	0
$523$	7.76546	0.339560	0.169780	−	0.985482i	$-0.445694\pi$
$523$	7.76546	0.339560	0.169780	+	0.985482i	$0.445694\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.8248	−0.689337
$528$	0	0
$529$	−12.2749	−0.533692
$530$	0	0
$531$	0	0
$532$	0	0
$533$	53.0997	2.30000
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	34.0339	1.46595
$540$	0	0
$541$	−21.4502	−0.922215	−0.461107	−	0.887344i	$-0.652548\pi$
$541$	−21.4502	−0.922215	−0.461107	+	0.887344i	$0.652548\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	35.7084	1.52678	0.763391	−	0.645936i	$-0.223533\pi$
$547$	35.7084	1.52678	0.763391	+	0.645936i	$0.223533\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−19.9474	−0.849786
$552$	0	0
$553$	6.09095	0.259014
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−17.3746	−0.736185	−0.368092	−	0.929789i	$-0.619989\pi$
$557$	−17.3746	−0.736185	−0.368092	+	0.929789i	$0.619989\pi$
$558$	0	0
$559$	63.2990	2.67726
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−30.8248	−1.29911	−0.649554	−	0.760315i	$-0.725044\pi$
$563$	−30.8248	−1.29911	−0.649554	+	0.760315i	$0.725044\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	16.3682	0.686189	0.343095	−	0.939301i	$-0.388525\pi$
$569$	16.3682	0.686189	0.343095	+	0.939301i	$0.388525\pi$
$570$	0	0
$571$	27.4743	1.14976	0.574881	−	0.818237i	$-0.305048\pi$
$571$	27.4743	1.14976	0.574881	+	0.818237i	$0.305048\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−14.6937	−0.611705	−0.305853	−	0.952079i	$-0.598941\pi$
$577$	−14.6937	−0.611705	−0.305853	+	0.952079i	$0.598941\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−74.2897	−3.08206
$582$	0	0
$583$	20.5386	0.850622
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.90033	0.326082	0.163041	−	0.986619i	$-0.447870\pi$
$587$	7.90033	0.326082	0.163041	+	0.986619i	$0.447870\pi$
$588$	0	0
$589$	−9.82475	−0.404822
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6.17525	−0.253587	−0.126794	−	0.991929i	$-0.540469\pi$
$593$	−6.17525	−0.253587	−0.126794	+	0.991929i	$0.540469\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	38.3353	1.56634	0.783169	−	0.621809i	$-0.213602\pi$
$599$	38.3353	1.56634	0.783169	+	0.621809i	$0.213602\pi$
$600$	0	0
$601$	−7.54983	−0.307964	−0.153982	−	0.988074i	$-0.549210\pi$
$601$	−7.54983	−0.307964	−0.153982	+	0.988074i	$0.549210\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−18.9950	−0.770984	−0.385492	−	0.922711i	$-0.625968\pi$
$607$	−18.9950	−0.770984	−0.385492	+	0.922711i	$0.625968\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	27.7128	1.12114
$612$	0	0
$613$	38.9424	1.57287	0.786434	−	0.617675i	$-0.211925\pi$
$613$	38.9424	1.57287	0.786434	+	0.617675i	$0.211925\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−41.8248	−1.68380	−0.841901	−	0.539633i	$-0.818563\pi$
$617$	−41.8248	−1.68380	−0.841901	+	0.539633i	$0.818563\pi$
$618$	0	0
$619$	−29.0997	−1.16961	−0.584807	−	0.811172i	$-0.698830\pi$
$619$	−29.0997	−1.16961	−0.584807	+	0.811172i	$0.698830\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	49.6495	1.98917
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	13.8564	0.552491
$630$	0	0
$631$	17.0000	0.676759	0.338380	−	0.941010i	$-0.390121\pi$
$631$	17.0000	0.676759	0.338380	+	0.941010i	$0.390121\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−96.3878	−3.81902
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−37.4980	−1.48108	−0.740542	−	0.672010i	$-0.765431\pi$
$641$	−37.4980	−1.48108	−0.740542	+	0.672010i	$0.765431\pi$
$642$	0	0
$643$	−40.9621	−1.61539	−0.807695	−	0.589601i	$-0.799285\pi$
$643$	−40.9621	−1.61539	−0.807695	+	0.589601i	$0.799285\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	44.9244	1.76616	0.883081	−	0.469221i	$-0.155465\pi$
$647$	44.9244	1.76616	0.883081	+	0.469221i	$0.155465\pi$
$648$	0	0
$649$	−7.45017	−0.292445
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.54983	0.0606497	0.0303249	−	0.999540i	$-0.490346\pi$
$653$	1.54983	0.0606497	0.0303249	+	0.999540i	$0.490346\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−0.476171	−0.0185490	−0.00927449	−	0.999957i	$-0.502952\pi$
$659$	−0.476171	−0.0185490	−0.00927449	+	0.999957i	$0.502952\pi$
$660$	0	0
$661$	23.0997	0.898473	0.449236	−	0.893413i	$-0.351696\pi$
$661$	23.0997	0.898473	0.449236	+	0.893413i	$0.351696\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	19.9474	0.772365
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.86077	0.226253
$672$	0	0
$673$	31.7682	1.22457	0.612287	−	0.790636i	$-0.290250\pi$
$673$	31.7682	1.22457	0.612287	+	0.790636i	$0.290250\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	−77.0241	−2.95591
$680$	0	0
$681$	0	0
$682$	0	0
$683$	39.4743	1.51044	0.755220	−	0.655471i	$-0.227530\pi$
$683$	39.4743	1.51044	0.755220	+	0.655471i	$0.227530\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−58.1676	−2.21601
$690$	0	0
$691$	−29.2749	−1.11367	−0.556835	−	0.830623i	$-0.687984\pi$
$691$	−29.2749	−1.11367	−0.556835	+	0.830623i	$0.687984\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	45.9857	1.74183
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6.56712	−0.248037	−0.124018	−	0.992280i	$-0.539578\pi$
$701$	−6.56712	−0.248037	−0.124018	+	0.992280i	$0.539578\pi$
$702$	0	0
$703$	8.60271	0.324457
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.2749	0.536864
$708$	0	0
$709$	14.5498	0.546431	0.273215	−	0.961953i	$-0.411913\pi$
$709$	14.5498	0.546431	0.273215	+	0.961953i	$0.411913\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	9.82475	0.367940
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−10.5074	−0.391860	−0.195930	−	0.980618i	$-0.562772\pi$
$719$	−10.5074	−0.391860	−0.195930	+	0.980618i	$0.562772\pi$
$720$	0	0
$721$	−78.7492	−2.93277
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.03559	0.0754958	0.0377479	−	0.999287i	$-0.487982\pi$
$727$	2.03559	0.0754958	0.0377479	+	0.999287i	$0.487982\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	54.8185	2.02754
$732$	0	0
$733$	51.3544	1.89682	0.948410	−	0.317047i	$-0.102691\pi$
$733$	51.3544	1.89682	0.948410	+	0.317047i	$0.102691\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.64950	0.208102
$738$	0	0
$739$	24.9244	0.916860	0.458430	−	0.888731i	$-0.348412\pi$
$739$	24.9244	0.916860	0.458430	+	0.888731i	$0.348412\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2.90033	−0.106403	−0.0532014	−	0.998584i	$-0.516943\pi$
$743$	−2.90033	−0.106403	−0.0532014	+	0.998584i	$0.516943\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−49.0887	−1.79366
$750$	0	0
$751$	−35.5498	−1.29723	−0.648616	−	0.761116i	$-0.724652\pi$
$751$	−35.5498	−1.29723	−0.648616	+	0.761116i	$0.724652\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	47.7753	1.73642	0.868211	−	0.496196i	$-0.165270\pi$
$757$	47.7753	1.73642	0.868211	+	0.496196i	$0.165270\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	37.3830	1.35513	0.677565	−	0.735462i	$-0.263035\pi$
$761$	37.3830	1.35513	0.677565	+	0.735462i	$0.263035\pi$
$762$	0	0
$763$	46.9380	1.69927
$764$	0	0
$765$	0	0
$766$	0	0
$767$	21.0997	0.761865
$768$	0	0
$769$	1.00000	0.0360609	0.0180305	−	0.999837i	$-0.494260\pi$
$769$	1.00000	0.0360609	0.0180305	+	0.999837i	$0.494260\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	18.1752	0.653718	0.326859	−	0.945073i	$-0.394010\pi$
$773$	18.1752	0.653718	0.326859	+	0.945073i	$0.394010\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	28.5501	1.02291
$780$	0	0
$781$	13.0997	0.468743
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−7.99563	−0.285014	−0.142507	−	0.989794i	$-0.545516\pi$
$787$	−7.99563	−0.285014	−0.142507	+	0.989794i	$0.545516\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	66.8854	2.37817
$792$	0	0
$793$	−16.5983	−0.589424
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−30.0997	−1.06618	−0.533092	−	0.846057i	$-0.678970\pi$
$797$	−30.0997	−1.06618	−0.533092	+	0.846057i	$0.678970\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.62541	0.163227
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−3.46410	−0.121791	−0.0608957	−	0.998144i	$-0.519396\pi$
$809$	−3.46410	−0.121791	−0.0608957	+	0.998144i	$0.519396\pi$
$810$	0	0
$811$	4.54983	0.159766	0.0798831	−	0.996804i	$-0.474545\pi$
$811$	4.54983	0.159766	0.0798831	+	0.996804i	$0.474545\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	34.0339	1.19070
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−36.5457	−1.27545	−0.637727	−	0.770263i	$-0.720125\pi$
$821$	−36.5457	−1.27545	−0.637727	+	0.770263i	$0.720125\pi$
$822$	0	0
$823$	13.6104	0.474429	0.237215	−	0.971457i	$-0.423766\pi$
$823$	13.6104	0.474429	0.237215	+	0.971457i	$0.423766\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−14.3746	−0.499853	−0.249927	−	0.968265i	$-0.580407\pi$
$827$	−14.3746	−0.499853	−0.249927	+	0.968265i	$0.580407\pi$
$828$	0	0
$829$	16.1993	0.562626	0.281313	−	0.959616i	$-0.409230\pi$
$829$	16.1993	0.562626	0.281313	+	0.959616i	$0.409230\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−83.4743	−2.89221
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	36.5457	1.26170	0.630849	−	0.775906i	$-0.282707\pi$
$839$	36.5457	1.26170	0.630849	+	0.775906i	$0.282707\pi$
$840$	0	0
$841$	8.09967	0.279299
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	30.4547	1.04644
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−8.60271	−0.294897
$852$	0	0
$853$	−51.1243	−1.75046	−0.875231	−	0.483705i	$-0.839291\pi$
$853$	−51.1243	−1.75046	−0.875231	+	0.483705i	$0.839291\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	40.9244	1.39795	0.698976	−	0.715145i	$-0.253640\pi$
$857$	40.9244	1.39795	0.698976	+	0.715145i	$0.253640\pi$
$858$	0	0
$859$	−31.4743	−1.07389	−0.536944	−	0.843618i	$-0.680421\pi$
$859$	−31.4743	−1.07389	−0.536944	+	0.843618i	$0.680421\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−35.4743	−1.20756	−0.603779	−	0.797152i	$-0.706339\pi$
$863$	−35.4743	−1.20756	−0.603779	+	0.797152i	$0.706339\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.74194	−0.0930138
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	29.7326	1.00400	0.501999	−	0.864868i	$-0.332598\pi$
$877$	29.7326	1.00400	0.501999	+	0.864868i	$0.332598\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−33.8038	−1.13888	−0.569439	−	0.822034i	$-0.692839\pi$
$881$	−33.8038	−1.13888	−0.569439	+	0.822034i	$0.692839\pi$
$882$	0	0
$883$	25.9232	0.872386	0.436193	−	0.899853i	$-0.356327\pi$
$883$	25.9232	0.872386	0.436193	+	0.899853i	$0.356327\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−14.3746	−0.482651	−0.241326	−	0.970444i	$-0.577582\pi$
$887$	−14.3746	−0.482651	−0.241326	+	0.970444i	$0.577582\pi$
$888$	0	0
$889$	−14.2749	−0.478765
$890$	0	0
$891$	0	0
$892$	0	0
$893$	14.9003	0.498621
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	18.2728	0.609434
$900$	0	0
$901$	−50.3746	−1.67822
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−2.62685	−0.0872231	−0.0436115	−	0.999049i	$-0.513886\pi$
$907$	−2.62685	−0.0872231	−0.0436115	+	0.999049i	$0.513886\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−10.3923	−0.344312	−0.172156	−	0.985070i	$-0.555073\pi$
$911$	−10.3923	−0.344312	−0.172156	+	0.985070i	$0.555073\pi$
$912$	0	0
$913$	33.4427	1.10679
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.2749	0.471399
$918$	0	0
$919$	−32.4743	−1.07123	−0.535613	−	0.844463i	$-0.679920\pi$
$919$	−32.4743	−1.07123	−0.535613	+	0.844463i	$0.679920\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−37.0997	−1.22115
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	28.7802	0.944249	0.472125	−	0.881532i	$-0.343487\pi$
$929$	28.7802	0.944249	0.472125	+	0.881532i	$0.343487\pi$
$930$	0	0
$931$	−51.8248	−1.69849
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−29.0262	−0.948246	−0.474123	−	0.880459i	$-0.657235\pi$
$937$	−29.0262	−0.948246	−0.474123	+	0.880459i	$0.657235\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−7.28929	−0.237624	−0.118812	−	0.992917i	$-0.537909\pi$
$941$	−7.28929	−0.237624	−0.118812	+	0.992917i	$0.537909\pi$
$942$	0	0
$943$	−28.5501	−0.929718
$944$	0	0
$945$	0	0
$946$	0	0
$947$	49.7492	1.61663	0.808315	−	0.588750i	$-0.200380\pi$
$947$	49.7492	1.61663	0.808315	+	0.588750i	$0.200380\pi$
$948$	0	0
$949$	−13.0997	−0.425233
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−55.6495	−1.80266	−0.901332	−	0.433129i	$-0.857410\pi$
$953$	−55.6495	−1.80266	−0.901332	+	0.433129i	$0.857410\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−27.8279	−0.898610
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−9.19397	−0.295658	−0.147829	−	0.989013i	$-0.547229\pi$
$967$	−9.19397	−0.295658	−0.147829	+	0.989013i	$0.547229\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−60.3182	−1.93570	−0.967852	−	0.251520i	$-0.919070\pi$
$971$	−60.3182	−1.93570	−0.967852	+	0.251520i	$0.919070\pi$
$972$	0	0
$973$	−19.1101	−0.612642
$974$	0	0
$975$	0	0
$976$	0	0
$977$	11.0997	0.355110	0.177555	−	0.984111i	$-0.443181\pi$
$977$	11.0997	0.355110	0.177555	+	0.984111i	$0.443181\pi$
$978$	0	0
$979$	−22.3505	−0.714325
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−36.0241	−1.14899	−0.574495	−	0.818508i	$-0.694802\pi$
$983$	−36.0241	−1.14899	−0.574495	+	0.818508i	$0.694802\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−34.0339	−1.08222
$990$	0	0
$991$	−24.6495	−0.783017	−0.391509	−	0.920174i	$-0.628047\pi$
$991$	−24.6495	−0.783017	−0.391509	+	0.920174i	$0.628047\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	25.2011	0.798125	0.399063	−	0.916924i	$-0.369336\pi$
$997$	25.2011	0.798125	0.399063	+	0.916924i	$0.369336\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5400.2.a.cg.1.1		4
3.2	odd	2		5400.2.a.ch.1.1		4
5.2	odd	4		1080.2.f.g.649.5	yes	8
5.3	odd	4		1080.2.f.g.649.6	yes	8
5.4	even	2		5400.2.a.ch.1.4		4
15.2	even	4		1080.2.f.g.649.4	yes	8
15.8	even	4		1080.2.f.g.649.3	✓	8
15.14	odd	2	inner	5400.2.a.cg.1.4		4
20.3	even	4		2160.2.f.o.1729.6		8
20.7	even	4		2160.2.f.o.1729.5		8
60.23	odd	4		2160.2.f.o.1729.3		8
60.47	odd	4		2160.2.f.o.1729.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.f.g.649.3	✓	8	15.8	even	4
1080.2.f.g.649.4	yes	8	15.2	even	4
1080.2.f.g.649.5	yes	8	5.2	odd	4
1080.2.f.g.649.6	yes	8	5.3	odd	4
2160.2.f.o.1729.3		8	60.23	odd	4
2160.2.f.o.1729.4		8	60.47	odd	4
2160.2.f.o.1729.5		8	20.7	even	4
2160.2.f.o.1729.6		8	20.3	even	4
5400.2.a.cg.1.1		4	1.1	even	1	trivial
5400.2.a.cg.1.4		4	15.14	odd	2	inner
5400.2.a.ch.1.1		4	3.2	odd	2
5400.2.a.ch.1.4		4	5.4	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:	\( N \)	\(=\)	\( 5400 = 2^{3} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5400.a (trivial)

\(f(q)\)	\(=\)	\(q-4.77753 q^{7} +O(q^{10})\)	\(q-4.77753 q^{7} +2.15068 q^{11} -6.09095 q^{13} -5.27492 q^{17} -3.27492 q^{19} +3.27492 q^{23} +6.09095 q^{29} +3.00000 q^{31} -2.62685 q^{37} -8.71780 q^{41} -10.3923 q^{43} -4.54983 q^{47} +15.8248 q^{49} +9.54983 q^{53} -3.46410 q^{59} +2.72508 q^{61} +2.62685 q^{67} +6.09095 q^{71} +2.15068 q^{73} -10.2749 q^{77} -1.27492 q^{79} +15.5498 q^{83} -10.3923 q^{89} +29.0997 q^{91} +16.1222 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 6 q^{17} + 2 q^{19} - 2 q^{23} + 12 q^{31} + 12 q^{47} + 18 q^{49} + 8 q^{53} + 26 q^{61} - 26 q^{77} + 10 q^{79} + 32 q^{83} + 56 q^{91}+O(q^{100}) \) 4 * q - 6 * q^17 + 2 * q^19 - 2 * q^23 + 12 * q^31 + 12 * q^47 + 18 * q^49 + 8 * q^53 + 26 * q^61 - 26 * q^77 + 10 * q^79 + 32 * q^83 + 56 * q^91

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