LMFDB - Embedded newform 5400.2.a.ch.1.3

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			1.3
Root			$-1.31342$ 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)

$f(q)$	$=$	$q+0.418627 q^{7} +O(q^{10})$	$q+0.418627 q^{7} +6.50958 q^{11} -2.62685 q^{13} -2.27492 q^{17} +4.27492 q^{19} +4.27492 q^{23} -2.62685 q^{29} +3.00000 q^{31} -6.09095 q^{37} +8.71780 q^{41} +10.3923 q^{43} -10.5498 q^{47} -6.82475 q^{49} +5.54983 q^{53} -3.46410 q^{59} +10.2749 q^{61} +6.09095 q^{67} -2.62685 q^{71} -6.50958 q^{73} +2.72508 q^{77} +6.27492 q^{79} -0.450166 q^{83} -10.3923 q^{89} -1.09967 q^{91} +14.3901 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	0.418627	0.158226	0.0791130	−	0.996866i	$-0.474791\pi$
$7$	0.418627	0.158226	0.0791130	+	0.996866i	$0.474791\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.50958	1.96271	0.981356	−	0.192201i	$-0.0615626\pi$
$11$	6.50958	1.96271	0.981356	+	0.192201i	$0.0615626\pi$
$12$	0	0
$13$	−2.62685	−0.728557	−0.364278	−	0.931290i	$-0.618684\pi$
$13$	−2.62685	−0.728557	−0.364278	+	0.931290i	$0.618684\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.27492	−0.551748	−0.275874	−	0.961194i	$-0.588967\pi$
$17$	−2.27492	−0.551748	−0.275874	+	0.961194i	$0.588967\pi$
$18$	0	0
$19$	4.27492	0.980733	0.490367	−	0.871516i	$-0.336863\pi$
$19$	4.27492	0.980733	0.490367	+	0.871516i	$0.336863\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.27492	0.891382	0.445691	−	0.895187i	$-0.352958\pi$
$23$	4.27492	0.891382	0.445691	+	0.895187i	$0.352958\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.62685	−0.487793	−0.243897	−	0.969801i	$-0.578426\pi$
$29$	−2.62685	−0.487793	−0.243897	+	0.969801i	$0.578426\pi$
$30$	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
$37$	−6.09095	−1.00135	−0.500673	−	0.865637i	$-0.666914\pi$
$37$	−6.09095	−1.00135	−0.500673	+	0.865637i	$0.666914\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.71780	1.36149	0.680746	−	0.732520i	$-0.261656\pi$
$41$	8.71780	1.36149	0.680746	+	0.732520i	$0.261656\pi$
$42$	0	0
$43$	10.3923	1.58481	0.792406	−	0.609994i	$-0.208828\pi$
$43$	10.3923	1.58481	0.792406	+	0.609994i	$0.208828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.5498	−1.53885	−0.769426	−	0.638736i	$-0.779458\pi$
$47$	−10.5498	−1.53885	−0.769426	+	0.638736i	$0.779458\pi$
$48$	0	0
$49$	−6.82475	−0.974965
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.54983	0.762328	0.381164	−	0.924507i	$-0.375523\pi$
$53$	5.54983	0.762328	0.381164	+	0.924507i	$0.375523\pi$
$54$	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
$61$	10.2749	1.31557	0.657784	−	0.753206i	$-0.271494\pi$
$61$	10.2749	1.31557	0.657784	+	0.753206i	$0.271494\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.09095	0.744128	0.372064	−	0.928207i	$-0.378650\pi$
$67$	6.09095	0.744128	0.372064	+	0.928207i	$0.378650\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.62685	−0.311750	−0.155875	−	0.987777i	$-0.549820\pi$
$71$	−2.62685	−0.311750	−0.155875	+	0.987777i	$0.549820\pi$
$72$	0	0
$73$	−6.50958	−0.761888	−0.380944	−	0.924598i	$-0.624401\pi$
$73$	−6.50958	−0.761888	−0.380944	+	0.924598i	$0.624401\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.72508	0.310552
$78$	0	0
$79$	6.27492	0.705983	0.352992	−	0.935626i	$-0.385164\pi$
$79$	6.27492	0.705983	0.352992	+	0.935626i	$0.385164\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.450166	−0.0494121	−0.0247060	−	0.999695i	$-0.507865\pi$
$83$	−0.450166	−0.0494121	−0.0247060	+	0.999695i	$0.507865\pi$
$84$	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
$91$	−1.09967	−0.115277
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.3901	1.46110	0.730548	−	0.682862i	$-0.239265\pi$
$97$	14.3901	1.46110	0.730548	+	0.682862i	$0.239265\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−16.0646	−1.59849	−0.799245	−	0.601005i	$-0.794767\pi$
$101$	−16.0646	−1.59849	−0.799245	+	0.601005i	$0.794767\pi$
$102$	0	0
$103$	−7.76546	−0.765153	−0.382577	−	0.923924i	$-0.624963\pi$
$103$	−7.76546	−0.765153	−0.382577	+	0.923924i	$0.624963\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.72508	−0.263444	−0.131722	−	0.991287i	$-0.542051\pi$
$107$	−2.72508	−0.263444	−0.131722	+	0.991287i	$0.542051\pi$
$108$	0	0
$109$	12.8248	1.22839	0.614194	−	0.789155i	$-0.289481\pi$
$109$	12.8248	1.22839	0.614194	+	0.789155i	$0.289481\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.952341	−0.0873010
$120$	0	0
$121$	31.3746	2.85224
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−16.0646	−1.42551	−0.712753	−	0.701416i	$-0.752552\pi$
$127$	−16.0646	−1.42551	−0.712753	+	0.701416i	$0.752552\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−16.0646	−1.40357	−0.701787	−	0.712387i	$-0.747614\pi$
$131$	−16.0646	−1.40357	−0.701787	+	0.712387i	$0.747614\pi$
$132$	0	0
$133$	1.78959	0.155178
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.8248	1.43744	0.718718	−	0.695302i	$-0.244729\pi$
$137$	16.8248	1.43744	0.718718	+	0.695302i	$0.244729\pi$
$138$	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
$143$	−17.0997	−1.42995
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.67232	0.464695	0.232347	−	0.972633i	$-0.425359\pi$
$149$	5.67232	0.464695	0.232347	+	0.972633i	$0.425359\pi$
$150$	0	0
$151$	10.7251	0.872795	0.436397	−	0.899754i	$-0.356254\pi$
$151$	10.7251	0.872795	0.436397	+	0.899754i	$0.356254\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	9.55505	0.762576	0.381288	−	0.924456i	$-0.375481\pi$
$157$	9.55505	0.762576	0.381288	+	0.924456i	$0.375481\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.78959	0.141040
$162$	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
$167$	−3.72508	−0.288256	−0.144128	−	0.989559i	$-0.546038\pi$
$167$	−3.72508	−0.288256	−0.144128	+	0.989559i	$0.546038\pi$
$168$	0	0
$169$	−6.09967	−0.469205
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−24.0997	−1.83226	−0.916132	−	0.400877i	$-0.868706\pi$
$173$	−24.0997	−1.83226	−0.916132	+	0.400877i	$0.868706\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	7.46192	0.557730	0.278865	−	0.960330i	$-0.410042\pi$
$179$	7.46192	0.557730	0.278865	+	0.960330i	$0.410042\pi$
$180$	0	0
$181$	−2.82475	−0.209962	−0.104981	−	0.994474i	$-0.533478\pi$
$181$	−2.82475	−0.209962	−0.104981	+	0.994474i	$0.533478\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−14.8087	−1.08292
$188$	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
$193$	−10.9260	−0.786472	−0.393236	−	0.919438i	$-0.628644\pi$
$193$	−10.9260	−0.786472	−0.393236	+	0.919438i	$0.628644\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.00000	0.641223	0.320612	−	0.947211i	$-0.396112\pi$
$197$	9.00000	0.641223	0.320612	+	0.947211i	$0.396112\pi$
$198$	0	0
$199$	19.8248	1.40534	0.702670	−	0.711516i	$-0.251991\pi$
$199$	19.8248	1.40534	0.702670	+	0.711516i	$0.251991\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.09967	−0.0771816
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	27.8279	1.92490
$210$	0	0
$211$	16.8248	1.15826	0.579132	−	0.815234i	$-0.303392\pi$
$211$	16.8248	1.15826	0.579132	+	0.815234i	$0.303392\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.25588	0.0852547
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.97586	0.401980
$222$	0	0
$223$	−25.2011	−1.68759	−0.843794	−	0.536668i	$-0.819683\pi$
$223$	−25.2011	−1.68759	−0.843794	+	0.536668i	$0.819683\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.7251	−0.910966	−0.455483	−	0.890245i	$-0.650533\pi$
$227$	−13.7251	−0.910966	−0.455483	+	0.890245i	$0.650533\pi$
$228$	0	0
$229$	8.27492	0.546822	0.273411	−	0.961897i	$-0.411848\pi$
$229$	8.27492	0.546822	0.273411	+	0.961897i	$0.411848\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0.837253	0.0541574	0.0270787	−	0.999633i	$-0.491380\pi$
$239$	0.837253	0.0541574	0.0270787	+	0.999633i	$0.491380\pi$
$240$	0	0
$241$	16.2749	1.04836	0.524180	−	0.851608i	$-0.324372\pi$
$241$	16.2749	1.04836	0.524180	+	0.851608i	$0.324372\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−11.2296	−0.714520
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−0.837253	−0.0528470	−0.0264235	−	0.999651i	$-0.508412\pi$
$251$	−0.837253	−0.0528470	−0.0264235	+	0.999651i	$0.508412\pi$
$252$	0	0
$253$	27.8279	1.74953
$254$	0	0
$255$	0	0
$256$	0	0
$257$	26.2749	1.63898	0.819492	−	0.573090i	$-0.194256\pi$
$257$	26.2749	1.63898	0.819492	+	0.573090i	$0.194256\pi$
$258$	0	0
$259$	−2.54983	−0.158439
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−13.4502	−0.829373	−0.414686	−	0.909964i	$-0.636109\pi$
$263$	−13.4502	−0.829373	−0.414686	+	0.909964i	$0.636109\pi$
$264$	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
$277$	−14.8087	−0.889771	−0.444886	−	0.895587i	$-0.646756\pi$
$277$	−14.8087	−0.889771	−0.444886	+	0.895587i	$0.646756\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7.88054	−0.470114	−0.235057	−	0.971982i	$-0.575528\pi$
$281$	−7.88054	−0.470114	−0.235057	+	0.971982i	$0.575528\pi$
$282$	0	0
$283$	3.46410	0.205919	0.102960	−	0.994686i	$-0.467169\pi$
$283$	3.46410	0.205919	0.102960	+	0.994686i	$0.467169\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.64950	0.215423
$288$	0	0
$289$	−11.8248	−0.695574
$290$	0	0
$291$	0	0
$292$	0	0
$293$	11.7251	0.684987	0.342493	−	0.939520i	$-0.388729\pi$
$293$	11.7251	0.684987	0.342493	+	0.939520i	$0.388729\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−11.2296	−0.649422
$300$	0	0
$301$	4.35050	0.250758
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	19.1101	1.09067	0.545336	−	0.838218i	$-0.316402\pi$
$307$	19.1101	1.09067	0.545336	+	0.838218i	$0.316402\pi$
$308$	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
$313$	−16.9019	−0.955351	−0.477675	−	0.878536i	$-0.658520\pi$
$313$	−16.9019	−0.955351	−0.477675	+	0.878536i	$0.658520\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.0997	0.679585	0.339793	−	0.940500i	$-0.389643\pi$
$317$	12.0997	0.679585	0.339793	+	0.940500i	$0.389643\pi$
$318$	0	0
$319$	−17.0997	−0.957398
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−9.72508	−0.541118
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4.41644	−0.243486
$330$	0	0
$331$	14.5498	0.799731	0.399866	−	0.916574i	$-0.369057\pi$
$331$	14.5498	0.799731	0.399866	+	0.916574i	$0.369057\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	23.4115	1.27530	0.637652	−	0.770325i	$-0.279906\pi$
$337$	23.4115	1.27530	0.637652	+	0.770325i	$0.279906\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	19.5287	1.05754
$342$	0	0
$343$	−5.78741	−0.312491
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.27492	0.0684411	0.0342206	−	0.999414i	$-0.489105\pi$
$347$	1.27492	0.0684411	0.0342206	+	0.999414i	$0.489105\pi$
$348$	0	0
$349$	29.9244	1.60182	0.800909	−	0.598786i	$-0.204350\pi$
$349$	29.9244	1.60182	0.800909	+	0.598786i	$0.204350\pi$
$350$	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
$359$	26.8756	1.41844	0.709219	−	0.704988i	$-0.249048\pi$
$359$	26.8756	1.41844	0.709219	+	0.704988i	$0.249048\pi$
$360$	0	0
$361$	−0.725083	−0.0381623
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	24.7824	1.29363	0.646816	−	0.762646i	$-0.276100\pi$
$367$	24.7824	1.29363	0.646816	+	0.762646i	$0.276100\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.32331	0.120620
$372$	0	0
$373$	19.1101	0.989484	0.494742	−	0.869040i	$-0.335263\pi$
$373$	19.1101	0.989484	0.494742	+	0.869040i	$0.335263\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.90033	0.355385
$378$	0	0
$379$	10.2749	0.527787	0.263894	−	0.964552i	$-0.414993\pi$
$379$	10.2749	0.527787	0.263894	+	0.964552i	$0.414993\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	30.8248	1.57507	0.787536	−	0.616269i	$-0.211357\pi$
$383$	30.8248	1.57507	0.787536	+	0.616269i	$0.211357\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−33.3851	−1.69269	−0.846347	−	0.532632i	$-0.821203\pi$
$389$	−33.3851	−1.69269	−0.846347	+	0.532632i	$0.821203\pi$
$390$	0	0
$391$	−9.72508	−0.491819
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−5.97586	−0.299920	−0.149960	−	0.988692i	$-0.547914\pi$
$397$	−5.97586	−0.299920	−0.149960	+	0.988692i	$0.547914\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.1101	−0.954313	−0.477156	−	0.878818i	$-0.658333\pi$
$401$	−19.1101	−0.954313	−0.477156	+	0.878818i	$0.658333\pi$
$402$	0	0
$403$	−7.88054	−0.392558
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−39.6495	−1.96535
$408$	0	0
$409$	26.0997	1.29055	0.645273	−	0.763952i	$-0.276744\pi$
$409$	26.0997	1.29055	0.645273	+	0.763952i	$0.276744\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.45017	−0.0713580
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	19.2252	0.939212	0.469606	−	0.882876i	$-0.344396\pi$
$419$	19.2252	0.939212	0.469606	+	0.882876i	$0.344396\pi$
$420$	0	0
$421$	−7.37459	−0.359415	−0.179708	−	0.983720i	$-0.557515\pi$
$421$	−7.37459	−0.359415	−0.179708	+	0.983720i	$0.557515\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.30136	0.208157
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−29.5024	−1.42108	−0.710540	−	0.703656i	$-0.751549\pi$
$431$	−29.5024	−1.42108	−0.710540	+	0.703656i	$0.751549\pi$
$432$	0	0
$433$	8.18408	0.393302	0.196651	−	0.980474i	$-0.436993\pi$
$433$	8.18408	0.393302	0.196651	+	0.980474i	$0.436993\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18.2749	0.874208
$438$	0	0
$439$	−7.54983	−0.360334	−0.180167	−	0.983636i	$-0.557664\pi$
$439$	−7.54983	−0.360334	−0.180167	+	0.983636i	$0.557664\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−21.7251	−1.03219	−0.516095	−	0.856531i	$-0.672615\pi$
$443$	−21.7251	−1.03219	−0.516095	+	0.856531i	$0.672615\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−33.0816	−1.56122	−0.780609	−	0.625020i	$-0.785091\pi$
$449$	−33.0816	−1.56122	−0.780609	+	0.625020i	$0.785091\pi$
$450$	0	0
$451$	56.7492	2.67221
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−36.8492	−1.72373	−0.861867	−	0.507134i	$-0.830705\pi$
$457$	−36.8492	−1.72373	−0.861867	+	0.507134i	$0.830705\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.20822	−0.102847	−0.0514236	−	0.998677i	$-0.516376\pi$
$461$	−2.20822	−0.102847	−0.0514236	+	0.998677i	$0.516376\pi$
$462$	0	0
$463$	14.3901	0.668766	0.334383	−	0.942437i	$-0.391472\pi$
$463$	14.3901	0.668766	0.334383	+	0.942437i	$0.391472\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−34.6495	−1.60339	−0.801694	−	0.597735i	$-0.796068\pi$
$467$	−34.6495	−1.60339	−0.801694	+	0.597735i	$0.796068\pi$
$468$	0	0
$469$	2.54983	0.117740
$470$	0	0
$471$	0	0
$472$	0	0
$473$	67.6495	3.11053
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−18.9950	−0.867904	−0.433952	−	0.900936i	$-0.642881\pi$
$479$	−18.9950	−0.867904	−0.433952	+	0.900936i	$0.642881\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.0668	0.546799	0.273400	−	0.961901i	$-0.411852\pi$
$487$	12.0668	0.546799	0.273400	+	0.961901i	$0.411852\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	16.9019	0.762771	0.381386	−	0.924416i	$-0.375447\pi$
$491$	16.9019	0.762771	0.381386	+	0.924416i	$0.375447\pi$
$492$	0	0
$493$	5.97586	0.269139
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.09967	−0.0493269
$498$	0	0
$499$	−4.82475	−0.215986	−0.107993	−	0.994152i	$-0.534442\pi$
$499$	−4.82475	−0.215986	−0.107993	+	0.994152i	$0.534442\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−20.8248	−0.928530	−0.464265	−	0.885696i	$-0.653681\pi$
$503$	−20.8248	−0.928530	−0.464265	+	0.885696i	$0.653681\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−14.3901	−0.637831	−0.318915	−	0.947783i	$-0.603319\pi$
$509$	−14.3901	−0.637831	−0.318915	+	0.947783i	$0.603319\pi$
$510$	0	0
$511$	−2.72508	−0.120551
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−68.6750	−3.02032
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.20604	−0.271891	−0.135946	−	0.990716i	$-0.543407\pi$
$521$	−6.20604	−0.271891	−0.135946	+	0.990716i	$0.543407\pi$
$522$	0	0
$523$	−16.4833	−0.720762	−0.360381	−	0.932805i	$-0.617353\pi$
$523$	−16.4833	−0.720762	−0.360381	+	0.932805i	$0.617353\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.82475	−0.297291
$528$	0	0
$529$	−4.72508	−0.205438
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−22.9003	−0.991923
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−44.4262	−1.91357
$540$	0	0
$541$	−36.5498	−1.57140	−0.785700	−	0.618608i	$-0.787697\pi$
$541$	−36.5498	−1.57140	−0.785700	+	0.618608i	$0.787697\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	25.3161	1.08244	0.541220	−	0.840881i	$-0.317963\pi$
$547$	25.3161	1.08244	0.541220	+	0.840881i	$0.317963\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−11.2296	−0.478395
$552$	0	0
$553$	2.62685	0.111705
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−20.3746	−0.863299	−0.431649	−	0.902041i	$-0.642068\pi$
$557$	−20.3746	−0.863299	−0.431649	+	0.902041i	$0.642068\pi$
$558$	0	0
$559$	−27.2990	−1.15462
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.17525	0.344546	0.172273	−	0.985049i	$-0.444889\pi$
$563$	8.17525	0.344546	0.172273	+	0.985049i	$0.444889\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	42.5216	1.78260	0.891298	−	0.453417i	$-0.149795\pi$
$569$	42.5216	1.78260	0.891298	+	0.453417i	$0.149795\pi$
$570$	0	0
$571$	−40.4743	−1.69379	−0.846897	−	0.531756i	$-0.821532\pi$
$571$	−40.4743	−1.69379	−0.846897	+	0.531756i	$0.821532\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23.4115	0.974632	0.487316	−	0.873226i	$-0.337976\pi$
$577$	23.4115	0.974632	0.487316	+	0.873226i	$0.337976\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.188451	−0.00781828
$582$	0	0
$583$	36.1271	1.49623
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−38.0997	−1.57254	−0.786271	−	0.617882i	$-0.787991\pi$
$587$	−38.0997	−1.57254	−0.786271	+	0.617882i	$0.787991\pi$
$588$	0	0
$589$	12.8248	0.528435
$590$	0	0
$591$	0	0
$592$	0	0
$593$	28.8248	1.18369	0.591845	−	0.806052i	$-0.298400\pi$
$593$	28.8248	1.18369	0.591845	+	0.806052i	$0.298400\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−31.4071	−1.28326	−0.641629	−	0.767015i	$-0.721741\pi$
$599$	−31.4071	−1.28326	−0.641629	+	0.767015i	$0.721741\pi$
$600$	0	0
$601$	7.54983	0.307964	0.153982	−	0.988074i	$-0.450790\pi$
$601$	7.54983	0.307964	0.153982	+	0.988074i	$0.450790\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	36.4306	1.47867	0.739336	−	0.673336i	$-0.235139\pi$
$607$	36.4306	1.47867	0.739336	+	0.673336i	$0.235139\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	27.7128	1.12114
$612$	0	0
$613$	−47.6602	−1.92498	−0.962488	−	0.271324i	$-0.912538\pi$
$613$	−47.6602	−1.92498	−0.962488	+	0.271324i	$0.912538\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19.1752	0.771966	0.385983	−	0.922506i	$-0.373862\pi$
$617$	19.1752	0.771966	0.385983	+	0.922506i	$0.373862\pi$
$618$	0	0
$619$	1.09967	0.0441994	0.0220997	−	0.999756i	$-0.492965\pi$
$619$	1.09967	0.0441994	0.0220997	+	0.999756i	$0.492965\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.35050	−0.174299
$624$	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
$637$	17.9276	0.710317
$638$	0	0
$639$	0	0
$640$	0	0
$641$	40.9621	1.61791	0.808954	−	0.587872i	$-0.200034\pi$
$641$	40.9621	1.61791	0.808954	+	0.587872i	$0.200034\pi$
$642$	0	0
$643$	−37.4980	−1.47878	−0.739389	−	0.673278i	$-0.764886\pi$
$643$	−37.4980	−1.47878	−0.739389	+	0.673278i	$0.764886\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.92442	0.311541	0.155771	−	0.987793i	$-0.450214\pi$
$647$	7.92442	0.311541	0.155771	+	0.987793i	$0.450214\pi$
$648$	0	0
$649$	−22.5498	−0.885158
$650$	0	0
$651$	0	0
$652$	0	0
$653$	13.5498	0.530246	0.265123	−	0.964215i	$-0.414587\pi$
$653$	13.5498	0.530246	0.265123	+	0.964215i	$0.414587\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12.6005	0.490847	0.245423	−	0.969416i	$-0.421073\pi$
$659$	12.6005	0.490847	0.245423	+	0.969416i	$0.421073\pi$
$660$	0	0
$661$	−7.09967	−0.276145	−0.138073	−	0.990422i	$-0.544091\pi$
$661$	−7.09967	−0.276145	−0.138073	+	0.990422i	$0.544091\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−11.2296	−0.434810
$668$	0	0
$669$	0	0
$670$	0	0
$671$	66.8854	2.58208
$672$	0	0
$673$	16.1797	0.623682	0.311841	−	0.950134i	$-0.399054\pi$
$673$	16.1797	0.623682	0.311841	+	0.950134i	$0.399054\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	6.02409	0.231183
$680$	0	0
$681$	0	0
$682$	0	0
$683$	28.4743	1.08954	0.544769	−	0.838586i	$-0.316618\pi$
$683$	28.4743	1.08954	0.544769	+	0.838586i	$0.316618\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−14.5786	−0.555399
$690$	0	0
$691$	−21.7251	−0.826461	−0.413231	−	0.910626i	$-0.635600\pi$
$691$	−21.7251	−0.826461	−0.413231	+	0.910626i	$0.635600\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−19.8323	−0.751201
$698$	0	0
$699$	0	0
$700$	0	0
$701$	15.2274	0.575130	0.287565	−	0.957761i	$-0.407154\pi$
$701$	15.2274	0.575130	0.287565	+	0.957761i	$0.407154\pi$
$702$	0	0
$703$	−26.0383	−0.982053
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.72508	−0.252923
$708$	0	0
$709$	−0.549834	−0.0206495	−0.0103247	−	0.999947i	$-0.503287\pi$
$709$	−0.549834	−0.0206495	−0.0103247	+	0.999947i	$0.503287\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	12.8248	0.480291
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	24.3638	0.908616	0.454308	−	0.890845i	$-0.349886\pi$
$719$	24.3638	0.908616	0.454308	+	0.890845i	$0.349886\pi$
$720$	0	0
$721$	−3.25083	−0.121067
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−41.2657	−1.53046	−0.765230	−	0.643757i	$-0.777375\pi$
$727$	−41.2657	−1.53046	−0.765230	+	0.643757i	$0.777375\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−23.6416	−0.874417
$732$	0	0
$733$	27.1057	1.00117	0.500587	−	0.865686i	$-0.333118\pi$
$733$	27.1057	1.00117	0.500587	+	0.865686i	$0.333118\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	39.6495	1.46051
$738$	0	0
$739$	−27.9244	−1.02722	−0.513608	−	0.858025i	$-0.671692\pi$
$739$	−27.9244	−1.02722	−0.513608	+	0.858025i	$0.671692\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	33.0997	1.21431	0.607155	−	0.794584i	$-0.292311\pi$
$743$	33.0997	1.21431	0.607155	+	0.794584i	$0.292311\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.14079	−0.0416837
$750$	0	0
$751$	−20.4502	−0.746237	−0.373119	−	0.927784i	$-0.621712\pi$
$751$	−20.4502	−0.746237	−0.373119	+	0.927784i	$0.621712\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−4.18627	−0.152152	−0.0760762	−	0.997102i	$-0.524239\pi$
$757$	−4.18627	−0.152152	−0.0760762	+	0.997102i	$0.524239\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−6.20604	−0.224969	−0.112484	−	0.993653i	$-0.535881\pi$
$761$	−6.20604	−0.224969	−0.112484	+	0.993653i	$0.535881\pi$
$762$	0	0
$763$	5.36878	0.194363
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.09967	0.328570
$768$	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
$773$	−40.8248	−1.46836	−0.734182	−	0.678953i	$-0.762434\pi$
$773$	−40.8248	−1.46836	−0.734182	+	0.678953i	$0.762434\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	37.2679	1.33526
$780$	0	0
$781$	−17.0997	−0.611874
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−53.0290	−1.89028	−0.945139	−	0.326668i	$-0.894074\pi$
$787$	−53.0290	−1.89028	−0.945139	+	0.326668i	$0.894074\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.86077	0.208385
$792$	0	0
$793$	−26.9906	−0.958466
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−0.0996689	−0.00353045	−0.00176523	−	0.999998i	$-0.500562\pi$
$797$	−0.0996689	−0.00353045	−0.00176523	+	0.999998i	$0.500562\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−42.3746	−1.49537
$804$	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
$811$	−10.5498	−0.370455	−0.185227	−	0.982696i	$-0.559302\pi$
$811$	−10.5498	−0.370455	−0.185227	+	0.982696i	$0.559302\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	44.4262	1.55428
$818$	0	0
$819$	0	0
$820$	0	0
$821$	15.7611	0.550066	0.275033	−	0.961435i	$-0.411311\pi$
$821$	15.7611	0.550066	0.275033	+	0.961435i	$0.411311\pi$
$822$	0	0
$823$	43.0553	1.50081	0.750406	−	0.660977i	$-0.229858\pi$
$823$	43.0553	1.50081	0.750406	+	0.660977i	$0.229858\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−23.3746	−0.812814	−0.406407	−	0.913692i	$-0.633218\pi$
$827$	−23.3746	−0.812814	−0.406407	+	0.913692i	$0.633218\pi$
$828$	0	0
$829$	−44.1993	−1.53511	−0.767553	−	0.640985i	$-0.778526\pi$
$829$	−44.1993	−1.53511	−0.767553	+	0.640985i	$0.778526\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	15.5257	0.537935
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−15.7611	−0.544133	−0.272067	−	0.962278i	$-0.587707\pi$
$839$	−15.7611	−0.544133	−0.272067	+	0.962278i	$0.587707\pi$
$840$	0	0
$841$	−22.0997	−0.762058
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	13.1342	0.451298
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−26.0383	−0.892582
$852$	0	0
$853$	42.4065	1.45197	0.725985	−	0.687711i	$-0.241384\pi$
$853$	42.4065	1.45197	0.725985	+	0.687711i	$0.241384\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.9244	0.407330	0.203665	−	0.979041i	$-0.434715\pi$
$857$	11.9244	0.407330	0.203665	+	0.979041i	$0.434715\pi$
$858$	0	0
$859$	36.4743	1.24449	0.622243	−	0.782824i	$-0.286222\pi$
$859$	36.4743	1.24449	0.622243	+	0.782824i	$0.286222\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−32.4743	−1.10544	−0.552718	−	0.833368i	$-0.686409\pi$
$863$	−32.4743	−1.10544	−0.552718	+	0.833368i	$0.686409\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	40.8471	1.38564
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	57.4454	1.93979	0.969897	−	0.243517i	$-0.0783012\pi$
$877$	57.4454	1.93979	0.969897	+	0.243517i	$0.0783012\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−25.0860	−0.845168	−0.422584	−	0.906324i	$-0.638877\pi$
$881$	−25.0860	−0.845168	−0.422584	+	0.906324i	$0.638877\pi$
$882$	0	0
$883$	−43.3588	−1.45914	−0.729570	−	0.683906i	$-0.760280\pi$
$883$	−43.3588	−1.45914	−0.729570	+	0.683906i	$0.760280\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−23.3746	−0.784842	−0.392421	−	0.919786i	$-0.628362\pi$
$887$	−23.3746	−0.784842	−0.392421	+	0.919786i	$0.628362\pi$
$888$	0	0
$889$	−6.72508	−0.225552
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−45.0997	−1.50920
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7.88054	−0.262831
$900$	0	0
$901$	−12.6254	−0.420614
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−6.09095	−0.202247	−0.101123	−	0.994874i	$-0.532244\pi$
$907$	−6.09095	−0.202247	−0.101123	+	0.994874i	$0.532244\pi$
$908$	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
$913$	−2.93039	−0.0969817
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.72508	−0.222082
$918$	0	0
$919$	35.4743	1.17019	0.585094	−	0.810966i	$-0.301058\pi$
$919$	35.4743	1.17019	0.585094	+	0.810966i	$0.301058\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	6.90033	0.227127
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−32.2443	−1.05790	−0.528951	−	0.848652i	$-0.677415\pi$
$929$	−32.2443	−1.05790	−0.528951	+	0.848652i	$0.677415\pi$
$930$	0	0
$931$	−29.1752	−0.956180
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	24.6673	0.805847	0.402923	−	0.915234i	$-0.367994\pi$
$937$	24.6673	0.805847	0.402923	+	0.915234i	$0.367994\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−29.0838	−0.948104	−0.474052	−	0.880497i	$-0.657209\pi$
$941$	−29.0838	−0.948104	−0.474052	+	0.880497i	$0.657209\pi$
$942$	0	0
$943$	37.2679	1.21361
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25.7492	0.836736	0.418368	−	0.908278i	$-0.362602\pi$
$947$	25.7492	0.836736	0.418368	+	0.908278i	$0.362602\pi$
$948$	0	0
$949$	17.0997	0.555079
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10.3505	0.335285	0.167643	−	0.985848i	$-0.446384\pi$
$953$	10.3505	0.335285	0.167643	+	0.985848i	$0.446384\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7.04329	0.227440
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−21.3183	−0.685551	−0.342776	−	0.939417i	$-0.611367\pi$
$967$	−21.3183	−0.685551	−0.342776	+	0.939417i	$0.611367\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−21.0881	−0.676751	−0.338375	−	0.941011i	$-0.609877\pi$
$971$	−21.0881	−0.676751	−0.338375	+	0.941011i	$0.609877\pi$
$972$	0	0
$973$	1.67451	0.0536822
$974$	0	0
$975$	0	0
$976$	0	0
$977$	19.0997	0.611053	0.305526	−	0.952184i	$-0.401168\pi$
$977$	19.0997	0.611053	0.305526	+	0.952184i	$0.401168\pi$
$978$	0	0
$979$	−67.6495	−2.16209
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−47.0241	−1.49984	−0.749918	−	0.661531i	$-0.769907\pi$
$983$	−47.0241	−1.49984	−0.749918	+	0.661531i	$0.769907\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	44.4262	1.41267
$990$	0	0
$991$	20.6495	0.655953	0.327977	−	0.944686i	$-0.393633\pi$
$991$	20.6495	0.655953	0.327977	+	0.944686i	$0.393633\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0.952341	0.0301609	0.0150805	−	0.999886i	$-0.495200\pi$
$997$	0.952341	0.0301609	0.0150805	+	0.999886i	$0.495200\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5400.2.a.ch.1.3		4
3.2	odd	2		5400.2.a.cg.1.3		4
5.2	odd	4		1080.2.f.g.649.1	✓	8
5.3	odd	4		1080.2.f.g.649.2	yes	8
5.4	even	2		5400.2.a.cg.1.2		4
15.2	even	4		1080.2.f.g.649.8	yes	8
15.8	even	4		1080.2.f.g.649.7	yes	8
15.14	odd	2	inner	5400.2.a.ch.1.2		4
20.3	even	4		2160.2.f.o.1729.2		8
20.7	even	4		2160.2.f.o.1729.1		8
60.23	odd	4		2160.2.f.o.1729.7		8
60.47	odd	4		2160.2.f.o.1729.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.f.g.649.1	✓	8	5.2	odd	4
1080.2.f.g.649.2	yes	8	5.3	odd	4
1080.2.f.g.649.7	yes	8	15.8	even	4
1080.2.f.g.649.8	yes	8	15.2	even	4
2160.2.f.o.1729.1		8	20.7	even	4
2160.2.f.o.1729.2		8	20.3	even	4
2160.2.f.o.1729.7		8	60.23	odd	4
2160.2.f.o.1729.8		8	60.47	odd	4
5400.2.a.cg.1.2		4	5.4	even	2
5400.2.a.cg.1.3		4	3.2	odd	2
5400.2.a.ch.1.2		4	15.14	odd	2	inner
5400.2.a.ch.1.3		4	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+0.418627 q^{7} +O(q^{10})\)	\(q+0.418627 q^{7} +6.50958 q^{11} -2.62685 q^{13} -2.27492 q^{17} +4.27492 q^{19} +4.27492 q^{23} -2.62685 q^{29} +3.00000 q^{31} -6.09095 q^{37} +8.71780 q^{41} +10.3923 q^{43} -10.5498 q^{47} -6.82475 q^{49} +5.54983 q^{53} -3.46410 q^{59} +10.2749 q^{61} +6.09095 q^{67} -2.62685 q^{71} -6.50958 q^{73} +2.72508 q^{77} +6.27492 q^{79} -0.450166 q^{83} -10.3923 q^{89} -1.09967 q^{91} +14.3901 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more