LMFDB - Embedded newform 5408.2.a.br.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5408,2,Mod(1,5408)] 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("5408.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5408, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5408 = 2^{5} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5408.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,0,7,0,-6,0,0,0,10,0,1,0,0,0,-18,0,-7,0,5,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	yes
Analytic conductor:	$43.1830974131$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 14x^{7} + 23x^{6} + 63x^{5} - 85x^{4} - 99x^{3} + 98x^{2} + 35x - 7 $ x^9 - 2x^8 - 14x^7 + 23x^6 + 63x^5 - 85x^4 - 99x^3 + 98x^2 + 35x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.148538$ of defining polynomial
Character	$\chi$	$=$	5408.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.851462 q^{3} -4.11908 q^{5} +4.22319 q^{7} -2.27501 q^{9} -4.95964 q^{11} -3.50724 q^{15} -4.42474 q^{17} -3.17673 q^{19} +3.59588 q^{21} +3.80838 q^{23} +11.9668 q^{25} -4.49147 q^{27} -1.10618 q^{29} -7.19114 q^{31} -4.22294 q^{33} -17.3956 q^{35} -0.218074 q^{37} +9.16464 q^{41} +11.1691 q^{43} +9.37095 q^{45} -4.49087 q^{47} +10.8353 q^{49} -3.76750 q^{51} +4.87022 q^{53} +20.4291 q^{55} -2.70486 q^{57} -0.982297 q^{59} -6.00168 q^{61} -9.60780 q^{63} -3.15809 q^{67} +3.24269 q^{69} -5.39071 q^{71} +5.36917 q^{73} +10.1893 q^{75} -20.9455 q^{77} +1.04391 q^{79} +3.00072 q^{81} +7.56101 q^{83} +18.2258 q^{85} -0.941874 q^{87} +6.65174 q^{89} -6.12298 q^{93} +13.0852 q^{95} -1.42288 q^{97} +11.2832 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 7 q^{3} - 6 q^{5} + 10 q^{9} + q^{11} - 18 q^{15} - 7 q^{17} + 5 q^{19} - 12 q^{21} + 12 q^{23} + 11 q^{25} + 34 q^{27} - 8 q^{29} - 20 q^{31} + 12 q^{33} + 6 q^{35} - 6 q^{37} + 31 q^{41} + 33 q^{43}+ \cdots + 47 q^{99}+O(q^{100}) $ 9 * q + 7 * q^3 - 6 * q^5 + 10 * q^9 + q^11 - 18 * q^15 - 7 * q^17 + 5 * q^19 - 12 * q^21 + 12 * q^23 + 11 * q^25 + 34 * q^27 - 8 * q^29 - 20 * q^31 + 12 * q^33 + 6 * q^35 - 6 * q^37 + 31 * q^41 + 33 * q^43 - 22 * q^45 + 14 * q^47 + 3 * q^49 + 12 * q^51 + 14 * q^53 + 38 * q^55 + 10 * q^57 - 9 * q^59 + 8 * q^61 - 20 * q^63 - 3 * q^67 + 26 * q^69 + 26 * q^71 + 5 * q^73 + 25 * q^75 - 6 * q^77 + 44 * q^79 + 45 * q^81 - 17 * q^83 - 6 * q^85 + 24 * q^87 + 41 * q^89 - 26 * q^93 + 4 * q^95 + 15 * q^97 + 47 * q^99

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
$3$	0.851462	0.491592	0.245796	−	0.969322i	$-0.420951\pi$
$3$	0.851462	0.491592	0.245796	+	0.969322i	$0.420951\pi$
$4$	0	0
$5$	−4.11908	−1.84211	−0.921054	−	0.389436i	$-0.872670\pi$
$5$	−4.11908	−1.84211	−0.921054	+	0.389436i	$0.872670\pi$
$6$	0	0
$7$	4.22319	1.59621	0.798107	−	0.602515i	$-0.205835\pi$
$7$	4.22319	1.59621	0.798107	+	0.602515i	$0.205835\pi$
$8$	0	0
$9$	−2.27501	−0.758337
$10$	0	0
$11$	−4.95964	−1.49539	−0.747693	−	0.664044i	$-0.768839\pi$
$11$	−4.95964	−1.49539	−0.747693	+	0.664044i	$0.768839\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−3.50724	−0.905565
$16$	0	0
$17$	−4.42474	−1.07316	−0.536578	−	0.843851i	$-0.680283\pi$
$17$	−4.42474	−1.07316	−0.536578	+	0.843851i	$0.680283\pi$
$18$	0	0
$19$	−3.17673	−0.728791	−0.364396	−	0.931244i	$-0.618724\pi$
$19$	−3.17673	−0.728791	−0.364396	+	0.931244i	$0.618724\pi$
$20$	0	0
$21$	3.59588	0.784686
$22$	0	0
$23$	3.80838	0.794102	0.397051	−	0.917797i	$-0.370034\pi$
$23$	3.80838	0.794102	0.397051	+	0.917797i	$0.370034\pi$
$24$	0	0
$25$	11.9668	2.39336
$26$	0	0
$27$	−4.49147	−0.864384
$28$	0	0
$29$	−1.10618	−0.205413	−0.102707	−	0.994712i	$-0.532750\pi$
$29$	−1.10618	−0.205413	−0.102707	+	0.994712i	$0.532750\pi$
$30$	0	0
$31$	−7.19114	−1.29157	−0.645784	−	0.763521i	$-0.723469\pi$
$31$	−7.19114	−1.29157	−0.645784	+	0.763521i	$0.723469\pi$
$32$	0	0
$33$	−4.22294	−0.735120
$34$	0	0
$35$	−17.3956	−2.94040
$36$	0	0
$37$	−0.218074	−0.0358512	−0.0179256	−	0.999839i	$-0.505706\pi$
$37$	−0.218074	−0.0358512	−0.0179256	+	0.999839i	$0.505706\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.16464	1.43128	0.715638	−	0.698472i	$-0.246136\pi$
$41$	9.16464	1.43128	0.715638	+	0.698472i	$0.246136\pi$
$42$	0	0
$43$	11.1691	1.70327	0.851636	−	0.524133i	$-0.175611\pi$
$43$	11.1691	1.70327	0.851636	+	0.524133i	$0.175611\pi$
$44$	0	0
$45$	9.37095	1.39694
$46$	0	0
$47$	−4.49087	−0.655061	−0.327530	−	0.944841i	$-0.606216\pi$
$47$	−4.49087	−0.655061	−0.327530	+	0.944841i	$0.606216\pi$
$48$	0	0
$49$	10.8353	1.54790
$50$	0	0
$51$	−3.76750	−0.527555
$52$	0	0
$53$	4.87022	0.668977	0.334488	−	0.942400i	$-0.391437\pi$
$53$	4.87022	0.668977	0.334488	+	0.942400i	$0.391437\pi$
$54$	0	0
$55$	20.4291	2.75466
$56$	0	0
$57$	−2.70486	−0.358268
$58$	0	0
$59$	−0.982297	−0.127884	−0.0639421	−	0.997954i	$-0.520367\pi$
$59$	−0.982297	−0.127884	−0.0639421	+	0.997954i	$0.520367\pi$
$60$	0	0
$61$	−6.00168	−0.768436	−0.384218	−	0.923242i	$-0.625529\pi$
$61$	−6.00168	−0.768436	−0.384218	+	0.923242i	$0.625529\pi$
$62$	0	0
$63$	−9.60780	−1.21047
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.15809	−0.385822	−0.192911	−	0.981216i	$-0.561793\pi$
$67$	−3.15809	−0.385822	−0.192911	+	0.981216i	$0.561793\pi$
$68$	0	0
$69$	3.24269	0.390374
$70$	0	0
$71$	−5.39071	−0.639760	−0.319880	−	0.947458i	$-0.603643\pi$
$71$	−5.39071	−0.639760	−0.319880	+	0.947458i	$0.603643\pi$
$72$	0	0
$73$	5.36917	0.628414	0.314207	−	0.949355i	$-0.398261\pi$
$73$	5.36917	0.628414	0.314207	+	0.949355i	$0.398261\pi$
$74$	0	0
$75$	10.1893	1.17656
$76$	0	0
$77$	−20.9455	−2.38696
$78$	0	0
$79$	1.04391	0.117449	0.0587243	−	0.998274i	$-0.481297\pi$
$79$	1.04391	0.117449	0.0587243	+	0.998274i	$0.481297\pi$
$80$	0	0
$81$	3.00072	0.333413
$82$	0	0
$83$	7.56101	0.829929	0.414964	−	0.909838i	$-0.363794\pi$
$83$	7.56101	0.829929	0.414964	+	0.909838i	$0.363794\pi$
$84$	0	0
$85$	18.2258	1.97687
$86$	0	0
$87$	−0.941874	−0.100979
$88$	0	0
$89$	6.65174	0.705083	0.352542	−	0.935796i	$-0.385318\pi$
$89$	6.65174	0.705083	0.352542	+	0.935796i	$0.385318\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−6.12298	−0.634924
$94$	0	0
$95$	13.0852	1.34251
$96$	0	0
$97$	−1.42288	−0.144472	−0.0722360	−	0.997388i	$-0.523013\pi$
$97$	−1.42288	−0.144472	−0.0722360	+	0.997388i	$0.523013\pi$
$98$	0	0
$99$	11.2832	1.13401
$100$	0	0
$101$	−0.877856	−0.0873499	−0.0436750	−	0.999046i	$-0.513907\pi$
$101$	−0.877856	−0.0873499	−0.0436750	+	0.999046i	$0.513907\pi$
$102$	0	0
$103$	16.4828	1.62410	0.812049	−	0.583589i	$-0.198352\pi$
$103$	16.4828	1.62410	0.812049	+	0.583589i	$0.198352\pi$
$104$	0	0
$105$	−14.8117	−1.44548
$106$	0	0
$107$	−2.55444	−0.246947	−0.123474	−	0.992348i	$-0.539403\pi$
$107$	−2.55444	−0.246947	−0.123474	+	0.992348i	$0.539403\pi$
$108$	0	0
$109$	1.99387	0.190978	0.0954890	−	0.995430i	$-0.469559\pi$
$109$	1.99387	0.190978	0.0954890	+	0.995430i	$0.469559\pi$
$110$	0	0
$111$	−0.185682	−0.0176241
$112$	0	0
$113$	17.5675	1.65261	0.826307	−	0.563220i	$-0.190438\pi$
$113$	17.5675	1.65261	0.826307	+	0.563220i	$0.190438\pi$
$114$	0	0
$115$	−15.6870	−1.46282
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−18.6865	−1.71299
$120$	0	0
$121$	13.5980	1.23618
$122$	0	0
$123$	7.80334	0.703603
$124$	0	0
$125$	−28.6967	−2.56672
$126$	0	0
$127$	1.70725	0.151494	0.0757469	−	0.997127i	$-0.475866\pi$
$127$	1.70725	0.151494	0.0757469	+	0.997127i	$0.475866\pi$
$128$	0	0
$129$	9.51007	0.837315
$130$	0	0
$131$	8.49064	0.741830	0.370915	−	0.928667i	$-0.379044\pi$
$131$	8.49064	0.741830	0.370915	+	0.928667i	$0.379044\pi$
$132$	0	0
$133$	−13.4159	−1.16331
$134$	0	0
$135$	18.5007	1.59229
$136$	0	0
$137$	−10.8252	−0.924863	−0.462432	−	0.886655i	$-0.653023\pi$
$137$	−10.8252	−0.924863	−0.462432	+	0.886655i	$0.653023\pi$
$138$	0	0
$139$	7.92594	0.672270	0.336135	−	0.941814i	$-0.390880\pi$
$139$	7.92594	0.672270	0.336135	+	0.941814i	$0.390880\pi$
$140$	0	0
$141$	−3.82381	−0.322023
$142$	0	0
$143$	0	0
$144$	0	0
$145$	4.55646	0.378393
$146$	0	0
$147$	9.22585	0.760935
$148$	0	0
$149$	−10.0480	−0.823165	−0.411582	−	0.911373i	$-0.635024\pi$
$149$	−10.0480	−0.823165	−0.411582	+	0.911373i	$0.635024\pi$
$150$	0	0
$151$	−2.75225	−0.223975	−0.111988	−	0.993710i	$-0.535722\pi$
$151$	−2.75225	−0.223975	−0.111988	+	0.993710i	$0.535722\pi$
$152$	0	0
$153$	10.0663	0.813815
$154$	0	0
$155$	29.6209	2.37920
$156$	0	0
$157$	12.3195	0.983199	0.491600	−	0.870821i	$-0.336412\pi$
$157$	12.3195	0.983199	0.491600	+	0.870821i	$0.336412\pi$
$158$	0	0
$159$	4.14681	0.328864
$160$	0	0
$161$	16.0835	1.26756
$162$	0	0
$163$	−1.05203	−0.0824015	−0.0412008	−	0.999151i	$-0.513118\pi$
$163$	−1.05203	−0.0824015	−0.0412008	+	0.999151i	$0.513118\pi$
$164$	0	0
$165$	17.3946	1.35417
$166$	0	0
$167$	8.16346	0.631708	0.315854	−	0.948808i	$-0.397709\pi$
$167$	8.16346	0.631708	0.315854	+	0.948808i	$0.397709\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	7.22709	0.552670
$172$	0	0
$173$	19.2395	1.46276	0.731378	−	0.681972i	$-0.238878\pi$
$173$	19.2395	1.46276	0.731378	+	0.681972i	$0.238878\pi$
$174$	0	0
$175$	50.5380	3.82031
$176$	0	0
$177$	−0.836388	−0.0628668
$178$	0	0
$179$	−18.7305	−1.39998	−0.699992	−	0.714151i	$-0.746813\pi$
$179$	−18.7305	−1.39998	−0.699992	+	0.714151i	$0.746813\pi$
$180$	0	0
$181$	−14.0447	−1.04394	−0.521968	−	0.852965i	$-0.674802\pi$
$181$	−14.0447	−1.04394	−0.521968	+	0.852965i	$0.674802\pi$
$182$	0	0
$183$	−5.11020	−0.377757
$184$	0	0
$185$	0.898264	0.0660417
$186$	0	0
$187$	21.9451	1.60478
$188$	0	0
$189$	−18.9683	−1.37974
$190$	0	0
$191$	−18.1132	−1.31062	−0.655312	−	0.755359i	$-0.727463\pi$
$191$	−18.1132	−1.31062	−0.655312	+	0.755359i	$0.727463\pi$
$192$	0	0
$193$	−2.63610	−0.189751	−0.0948753	−	0.995489i	$-0.530245\pi$
$193$	−2.63610	−0.189751	−0.0948753	+	0.995489i	$0.530245\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.2244	−1.72592	−0.862958	−	0.505276i	$-0.831391\pi$
$197$	−24.2244	−1.72592	−0.862958	+	0.505276i	$0.831391\pi$
$198$	0	0
$199$	18.7696	1.33054	0.665271	−	0.746602i	$-0.268316\pi$
$199$	18.7696	1.33054	0.665271	+	0.746602i	$0.268316\pi$
$200$	0	0
$201$	−2.68899	−0.189667
$202$	0	0
$203$	−4.67162	−0.327884
$204$	0	0
$205$	−37.7498	−2.63656
$206$	0	0
$207$	−8.66411	−0.602197
$208$	0	0
$209$	15.7554	1.08982
$210$	0	0
$211$	−1.03381	−0.0711702	−0.0355851	−	0.999367i	$-0.511329\pi$
$211$	−1.03381	−0.0711702	−0.0355851	+	0.999367i	$0.511329\pi$
$212$	0	0
$213$	−4.58999	−0.314501
$214$	0	0
$215$	−46.0064	−3.13761
$216$	0	0
$217$	−30.3695	−2.06162
$218$	0	0
$219$	4.57164	0.308923
$220$	0	0
$221$	0	0
$222$	0	0
$223$	5.34722	0.358076	0.179038	−	0.983842i	$-0.442701\pi$
$223$	5.34722	0.358076	0.179038	+	0.983842i	$0.442701\pi$
$224$	0	0
$225$	−27.2246	−1.81497
$226$	0	0
$227$	26.0767	1.73077	0.865386	−	0.501105i	$-0.167073\pi$
$227$	26.0767	1.73077	0.865386	+	0.501105i	$0.167073\pi$
$228$	0	0
$229$	−4.01313	−0.265195	−0.132597	−	0.991170i	$-0.542332\pi$
$229$	−4.01313	−0.265195	−0.132597	+	0.991170i	$0.542332\pi$
$230$	0	0
$231$	−17.8343	−1.17341
$232$	0	0
$233$	−2.43031	−0.159215	−0.0796075	−	0.996826i	$-0.525367\pi$
$233$	−2.43031	−0.159215	−0.0796075	+	0.996826i	$0.525367\pi$
$234$	0	0
$235$	18.4982	1.20669
$236$	0	0
$237$	0.888846	0.0577368
$238$	0	0
$239$	14.6547	0.947933	0.473966	−	0.880543i	$-0.342822\pi$
$239$	14.6547	0.947933	0.473966	+	0.880543i	$0.342822\pi$
$240$	0	0
$241$	24.3732	1.57002	0.785008	−	0.619486i	$-0.212659\pi$
$241$	24.3732	1.57002	0.785008	+	0.619486i	$0.212659\pi$
$242$	0	0
$243$	16.0294	1.02829
$244$	0	0
$245$	−44.6314	−2.85140
$246$	0	0
$247$	0	0
$248$	0	0
$249$	6.43791	0.407986
$250$	0	0
$251$	16.4253	1.03676	0.518378	−	0.855152i	$-0.326536\pi$
$251$	16.4253	1.03676	0.518378	+	0.855152i	$0.326536\pi$
$252$	0	0
$253$	−18.8882	−1.18749
$254$	0	0
$255$	15.5186	0.971813
$256$	0	0
$257$	−1.26309	−0.0787893	−0.0393946	−	0.999224i	$-0.512543\pi$
$257$	−1.26309	−0.0787893	−0.0393946	+	0.999224i	$0.512543\pi$
$258$	0	0
$259$	−0.920968	−0.0572261
$260$	0	0
$261$	2.51658	0.155773
$262$	0	0
$263$	−18.6363	−1.14916	−0.574580	−	0.818448i	$-0.694835\pi$
$263$	−18.6363	−1.14916	−0.574580	+	0.818448i	$0.694835\pi$
$264$	0	0
$265$	−20.0608	−1.23233
$266$	0	0
$267$	5.66370	0.346613
$268$	0	0
$269$	1.45156	0.0885034	0.0442517	−	0.999020i	$-0.485910\pi$
$269$	1.45156	0.0885034	0.0442517	+	0.999020i	$0.485910\pi$
$270$	0	0
$271$	−18.3946	−1.11739	−0.558695	−	0.829373i	$-0.688698\pi$
$271$	−18.3946	−1.11739	−0.558695	+	0.829373i	$0.688698\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−59.3509	−3.57900
$276$	0	0
$277$	−7.54671	−0.453438	−0.226719	−	0.973960i	$-0.572800\pi$
$277$	−7.54671	−0.453438	−0.226719	+	0.973960i	$0.572800\pi$
$278$	0	0
$279$	16.3599	0.979444
$280$	0	0
$281$	6.64428	0.396364	0.198182	−	0.980165i	$-0.436496\pi$
$281$	6.64428	0.396364	0.198182	+	0.980165i	$0.436496\pi$
$282$	0	0
$283$	2.07919	0.123595	0.0617976	−	0.998089i	$-0.480317\pi$
$283$	2.07919	0.123595	0.0617976	+	0.998089i	$0.480317\pi$
$284$	0	0
$285$	11.1415	0.659968
$286$	0	0
$287$	38.7040	2.28462
$288$	0	0
$289$	2.57830	0.151664
$290$	0	0
$291$	−1.21153	−0.0710212
$292$	0	0
$293$	−12.1943	−0.712396	−0.356198	−	0.934411i	$-0.615927\pi$
$293$	−12.1943	−0.712396	−0.356198	+	0.934411i	$0.615927\pi$
$294$	0	0
$295$	4.04616	0.235576
$296$	0	0
$297$	22.2761	1.29259
$298$	0	0
$299$	0	0
$300$	0	0
$301$	47.1692	2.71879
$302$	0	0
$303$	−0.747461	−0.0429405
$304$	0	0
$305$	24.7214	1.41554
$306$	0	0
$307$	19.9287	1.13739	0.568695	−	0.822549i	$-0.307449\pi$
$307$	19.9287	1.13739	0.568695	+	0.822549i	$0.307449\pi$
$308$	0	0
$309$	14.0345	0.798393
$310$	0	0
$311$	16.8046	0.952901	0.476450	−	0.879201i	$-0.341923\pi$
$311$	16.8046	0.952901	0.476450	+	0.879201i	$0.341923\pi$
$312$	0	0
$313$	−19.6091	−1.10837	−0.554186	−	0.832393i	$-0.686970\pi$
$313$	−19.6091	−1.10837	−0.554186	+	0.832393i	$0.686970\pi$
$314$	0	0
$315$	39.5753	2.22981
$316$	0	0
$317$	24.4330	1.37229	0.686147	−	0.727463i	$-0.259301\pi$
$317$	24.4330	1.37229	0.686147	+	0.727463i	$0.259301\pi$
$318$	0	0
$319$	5.48627	0.307172
$320$	0	0
$321$	−2.17501	−0.121397
$322$	0	0
$323$	14.0562	0.782107
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.69770	0.0938833
$328$	0	0
$329$	−18.9658	−1.04562
$330$	0	0
$331$	25.8773	1.42234	0.711171	−	0.703019i	$-0.248165\pi$
$331$	25.8773	1.42234	0.711171	+	0.703019i	$0.248165\pi$
$332$	0	0
$333$	0.496121	0.0271873
$334$	0	0
$335$	13.0084	0.710725
$336$	0	0
$337$	26.5515	1.44635	0.723176	−	0.690664i	$-0.242682\pi$
$337$	26.5515	1.44635	0.723176	+	0.690664i	$0.242682\pi$
$338$	0	0
$339$	14.9581	0.812412
$340$	0	0
$341$	35.6654	1.93139
$342$	0	0
$343$	16.1972	0.874567
$344$	0	0
$345$	−13.3569	−0.719111
$346$	0	0
$347$	−5.97049	−0.320513	−0.160256	−	0.987075i	$-0.551232\pi$
$347$	−5.97049	−0.320513	−0.160256	+	0.987075i	$0.551232\pi$
$348$	0	0
$349$	29.8008	1.59520	0.797600	−	0.603187i	$-0.206103\pi$
$349$	29.8008	1.59520	0.797600	+	0.603187i	$0.206103\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−28.5864	−1.52150	−0.760749	−	0.649046i	$-0.775168\pi$
$353$	−28.5864	−1.52150	−0.760749	+	0.649046i	$0.775168\pi$
$354$	0	0
$355$	22.2048	1.17851
$356$	0	0
$357$	−15.9108	−0.842091
$358$	0	0
$359$	7.17562	0.378715	0.189357	−	0.981908i	$-0.439360\pi$
$359$	7.17562	0.378715	0.189357	+	0.981908i	$0.439360\pi$
$360$	0	0
$361$	−8.90841	−0.468863
$362$	0	0
$363$	11.5782	0.607696
$364$	0	0
$365$	−22.1160	−1.15761
$366$	0	0
$367$	−8.42026	−0.439534	−0.219767	−	0.975552i	$-0.570530\pi$
$367$	−8.42026	−0.439534	−0.219767	+	0.975552i	$0.570530\pi$
$368$	0	0
$369$	−20.8497	−1.08539
$370$	0	0
$371$	20.5679	1.06783
$372$	0	0
$373$	29.6355	1.53447	0.767233	−	0.641368i	$-0.221633\pi$
$373$	29.6355	1.53447	0.767233	+	0.641368i	$0.221633\pi$
$374$	0	0
$375$	−24.4342	−1.26178
$376$	0	0
$377$	0	0
$378$	0	0
$379$	30.0698	1.54458	0.772291	−	0.635269i	$-0.219111\pi$
$379$	30.0698	1.54458	0.772291	+	0.635269i	$0.219111\pi$
$380$	0	0
$381$	1.45366	0.0744731
$382$	0	0
$383$	−20.9979	−1.07294	−0.536472	−	0.843918i	$-0.680243\pi$
$383$	−20.9979	−1.07294	−0.536472	+	0.843918i	$0.680243\pi$
$384$	0	0
$385$	86.2760	4.39703
$386$	0	0
$387$	−25.4099	−1.29166
$388$	0	0
$389$	4.11555	0.208667	0.104333	−	0.994542i	$-0.466729\pi$
$389$	4.11555	0.208667	0.104333	+	0.994542i	$0.466729\pi$
$390$	0	0
$391$	−16.8511	−0.852196
$392$	0	0
$393$	7.22945	0.364678
$394$	0	0
$395$	−4.29993	−0.216353
$396$	0	0
$397$	3.09264	0.155215	0.0776075	−	0.996984i	$-0.475272\pi$
$397$	3.09264	0.155215	0.0776075	+	0.996984i	$0.475272\pi$
$398$	0	0
$399$	−11.4231	−0.571872
$400$	0	0
$401$	21.7842	1.08785	0.543925	−	0.839134i	$-0.316937\pi$
$401$	21.7842	1.08785	0.543925	+	0.839134i	$0.316937\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−12.3602	−0.614182
$406$	0	0
$407$	1.08157	0.0536113
$408$	0	0
$409$	31.9544	1.58005	0.790023	−	0.613077i	$-0.210069\pi$
$409$	31.9544	1.58005	0.790023	+	0.613077i	$0.210069\pi$
$410$	0	0
$411$	−9.21729	−0.454655
$412$	0	0
$413$	−4.14842	−0.204130
$414$	0	0
$415$	−31.1444	−1.52882
$416$	0	0
$417$	6.74864	0.330482
$418$	0	0
$419$	3.25354	0.158946	0.0794730	−	0.996837i	$-0.474676\pi$
$419$	3.25354	0.158946	0.0794730	+	0.996837i	$0.474676\pi$
$420$	0	0
$421$	28.8187	1.40454	0.702270	−	0.711911i	$-0.252170\pi$
$421$	28.8187	1.40454	0.702270	+	0.711911i	$0.252170\pi$
$422$	0	0
$423$	10.2168	0.496757
$424$	0	0
$425$	−52.9499	−2.56845
$426$	0	0
$427$	−25.3462	−1.22659
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−33.3095	−1.60446	−0.802230	−	0.597015i	$-0.796353\pi$
$431$	−33.3095	−1.60446	−0.802230	+	0.597015i	$0.796353\pi$
$432$	0	0
$433$	3.61117	0.173542	0.0867709	−	0.996228i	$-0.472345\pi$
$433$	3.61117	0.173542	0.0867709	+	0.996228i	$0.472345\pi$
$434$	0	0
$435$	3.87965	0.186015
$436$	0	0
$437$	−12.0982	−0.578735
$438$	0	0
$439$	35.7325	1.70542	0.852710	−	0.522385i	$-0.174958\pi$
$439$	35.7325	1.70542	0.852710	+	0.522385i	$0.174958\pi$
$440$	0	0
$441$	−24.6504	−1.17383
$442$	0	0
$443$	−30.8088	−1.46377	−0.731886	−	0.681427i	$-0.761360\pi$
$443$	−30.8088	−1.46377	−0.731886	+	0.681427i	$0.761360\pi$
$444$	0	0
$445$	−27.3990	−1.29884
$446$	0	0
$447$	−8.55550	−0.404661
$448$	0	0
$449$	32.4518	1.53150	0.765748	−	0.643140i	$-0.222369\pi$
$449$	32.4518	1.53150	0.765748	+	0.643140i	$0.222369\pi$
$450$	0	0
$451$	−45.4533	−2.14031
$452$	0	0
$453$	−2.34344	−0.110104
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−21.9772	−1.02805	−0.514026	−	0.857775i	$-0.671847\pi$
$457$	−21.9772	−1.02805	−0.514026	+	0.857775i	$0.671847\pi$
$458$	0	0
$459$	19.8736	0.927620
$460$	0	0
$461$	13.7521	0.640497	0.320248	−	0.947334i	$-0.396234\pi$
$461$	13.7521	0.640497	0.320248	+	0.947334i	$0.396234\pi$
$462$	0	0
$463$	−7.23652	−0.336310	−0.168155	−	0.985761i	$-0.553781\pi$
$463$	−7.23652	−0.336310	−0.168155	+	0.985761i	$0.553781\pi$
$464$	0	0
$465$	25.2210	1.16960
$466$	0	0
$467$	31.6809	1.46602	0.733008	−	0.680220i	$-0.238116\pi$
$467$	31.6809	1.46602	0.733008	+	0.680220i	$0.238116\pi$
$468$	0	0
$469$	−13.3372	−0.615854
$470$	0	0
$471$	10.4895	0.483333
$472$	0	0
$473$	−55.3947	−2.54705
$474$	0	0
$475$	−38.0152	−1.74426
$476$	0	0
$477$	−11.0798	−0.507310
$478$	0	0
$479$	−0.386072	−0.0176401	−0.00882005	−	0.999961i	$-0.502808\pi$
$479$	−0.386072	−0.0176401	−0.00882005	+	0.999961i	$0.502808\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	13.6945	0.623121
$484$	0	0
$485$	5.86097	0.266133
$486$	0	0
$487$	−20.3910	−0.924005	−0.462002	−	0.886879i	$-0.652869\pi$
$487$	−20.3910	−0.924005	−0.462002	+	0.886879i	$0.652869\pi$
$488$	0	0
$489$	−0.895766	−0.0405079
$490$	0	0
$491$	5.35980	0.241885	0.120942	−	0.992660i	$-0.461408\pi$
$491$	5.35980	0.241885	0.120942	+	0.992660i	$0.461408\pi$
$492$	0	0
$493$	4.89457	0.220440
$494$	0	0
$495$	−46.4765	−2.08896
$496$	0	0
$497$	−22.7660	−1.02119
$498$	0	0
$499$	−22.5641	−1.01011	−0.505054	−	0.863088i	$-0.668528\pi$
$499$	−22.5641	−1.01011	−0.505054	+	0.863088i	$0.668528\pi$
$500$	0	0
$501$	6.95088	0.310542
$502$	0	0
$503$	−27.0739	−1.20717	−0.603583	−	0.797301i	$-0.706261\pi$
$503$	−27.0739	−1.20717	−0.603583	+	0.797301i	$0.706261\pi$
$504$	0	0
$505$	3.61596	0.160908
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−17.9132	−0.793991	−0.396995	−	0.917821i	$-0.629947\pi$
$509$	−17.9132	−0.793991	−0.396995	+	0.917821i	$0.629947\pi$
$510$	0	0
$511$	22.6750	1.00308
$512$	0	0
$513$	14.2682	0.629956
$514$	0	0
$515$	−67.8939	−2.99176
$516$	0	0
$517$	22.2731	0.979569
$518$	0	0
$519$	16.3817	0.719079
$520$	0	0
$521$	−15.5850	−0.682792	−0.341396	−	0.939920i	$-0.610900\pi$
$521$	−15.5850	−0.682792	−0.341396	+	0.939920i	$0.610900\pi$
$522$	0	0
$523$	11.1629	0.488118	0.244059	−	0.969760i	$-0.421521\pi$
$523$	11.1629	0.488118	0.244059	+	0.969760i	$0.421521\pi$
$524$	0	0
$525$	43.0312	1.87803
$526$	0	0
$527$	31.8189	1.38605
$528$	0	0
$529$	−8.49624	−0.369402
$530$	0	0
$531$	2.23474	0.0969793
$532$	0	0
$533$	0	0
$534$	0	0
$535$	10.5219	0.454903
$536$	0	0
$537$	−15.9483	−0.688221
$538$	0	0
$539$	−53.7392	−2.31471
$540$	0	0
$541$	−26.6285	−1.14485	−0.572424	−	0.819958i	$-0.693997\pi$
$541$	−26.6285	−1.14485	−0.572424	+	0.819958i	$0.693997\pi$
$542$	0	0
$543$	−11.9585	−0.513190
$544$	0	0
$545$	−8.21290	−0.351802
$546$	0	0
$547$	−14.4482	−0.617760	−0.308880	−	0.951101i	$-0.599954\pi$
$547$	−14.4482	−0.617760	−0.308880	+	0.951101i	$0.599954\pi$
$548$	0	0
$549$	13.6539	0.582734
$550$	0	0
$551$	3.51404	0.149703
$552$	0	0
$553$	4.40861	0.187473
$554$	0	0
$555$	0.764838	0.0324656
$556$	0	0
$557$	22.6385	0.959225	0.479613	−	0.877480i	$-0.340777\pi$
$557$	22.6385	0.959225	0.479613	+	0.877480i	$0.340777\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	18.6854	0.788899
$562$	0	0
$563$	−28.8754	−1.21695	−0.608477	−	0.793571i	$-0.708219\pi$
$563$	−28.8754	−1.21695	−0.608477	+	0.793571i	$0.708219\pi$
$564$	0	0
$565$	−72.3620	−3.04429
$566$	0	0
$567$	12.6726	0.532199
$568$	0	0
$569$	−6.59852	−0.276624	−0.138312	−	0.990389i	$-0.544168\pi$
$569$	−6.59852	−0.276624	−0.138312	+	0.990389i	$0.544168\pi$
$570$	0	0
$571$	−29.9931	−1.25517	−0.627587	−	0.778547i	$-0.715957\pi$
$571$	−29.9931	−1.25517	−0.627587	+	0.778547i	$0.715957\pi$
$572$	0	0
$573$	−15.4227	−0.644292
$574$	0	0
$575$	45.5741	1.90057
$576$	0	0
$577$	22.9455	0.955235	0.477617	−	0.878568i	$-0.341501\pi$
$577$	22.9455	0.955235	0.477617	+	0.878568i	$0.341501\pi$
$578$	0	0
$579$	−2.24454	−0.0932799
$580$	0	0
$581$	31.9316	1.32474
$582$	0	0
$583$	−24.1545	−1.00038
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−30.8418	−1.27298	−0.636488	−	0.771287i	$-0.719613\pi$
$587$	−30.8418	−1.27298	−0.636488	+	0.771287i	$0.719613\pi$
$588$	0	0
$589$	22.8443	0.941283
$590$	0	0
$591$	−20.6261	−0.848446
$592$	0	0
$593$	14.4145	0.591933	0.295967	−	0.955198i	$-0.404358\pi$
$593$	14.4145	0.591933	0.295967	+	0.955198i	$0.404358\pi$
$594$	0	0
$595$	76.9711	3.15551
$596$	0	0
$597$	15.9816	0.654084
$598$	0	0
$599$	40.4714	1.65362	0.826809	−	0.562483i	$-0.190154\pi$
$599$	40.4714	1.65362	0.826809	+	0.562483i	$0.190154\pi$
$600$	0	0
$601$	25.7782	1.05152	0.525758	−	0.850634i	$-0.323782\pi$
$601$	25.7782	1.05152	0.525758	+	0.850634i	$0.323782\pi$
$602$	0	0
$603$	7.18469	0.292583
$604$	0	0
$605$	−56.0112	−2.27718
$606$	0	0
$607$	34.2526	1.39027	0.695135	−	0.718879i	$-0.255344\pi$
$607$	34.2526	1.39027	0.695135	+	0.718879i	$0.255344\pi$
$608$	0	0
$609$	−3.97771	−0.161185
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−19.3912	−0.783203	−0.391601	−	0.920135i	$-0.628079\pi$
$613$	−19.3912	−0.783203	−0.391601	+	0.920135i	$0.628079\pi$
$614$	0	0
$615$	−32.1426	−1.29611
$616$	0	0
$617$	−28.7531	−1.15756	−0.578778	−	0.815485i	$-0.696470\pi$
$617$	−28.7531	−1.15756	−0.578778	+	0.815485i	$0.696470\pi$
$618$	0	0
$619$	32.8313	1.31960	0.659801	−	0.751441i	$-0.270641\pi$
$619$	32.8313	1.31960	0.659801	+	0.751441i	$0.270641\pi$
$620$	0	0
$621$	−17.1052	−0.686409
$622$	0	0
$623$	28.0915	1.12546
$624$	0	0
$625$	58.3701	2.33481
$626$	0	0
$627$	13.4151	0.535749
$628$	0	0
$629$	0.964921	0.0384739
$630$	0	0
$631$	38.2847	1.52409	0.762045	−	0.647524i	$-0.224195\pi$
$631$	38.2847	1.52409	0.762045	+	0.647524i	$0.224195\pi$
$632$	0	0
$633$	−0.880247	−0.0349867
$634$	0	0
$635$	−7.03229	−0.279068
$636$	0	0
$637$	0	0
$638$	0	0
$639$	12.2639	0.485154
$640$	0	0
$641$	−50.2675	−1.98545	−0.992724	−	0.120411i	$-0.961579\pi$
$641$	−50.2675	−1.98545	−0.992724	+	0.120411i	$0.961579\pi$
$642$	0	0
$643$	−15.4064	−0.607570	−0.303785	−	0.952741i	$-0.598251\pi$
$643$	−15.4064	−0.607570	−0.303785	+	0.952741i	$0.598251\pi$
$644$	0	0
$645$	−39.1727	−1.54242
$646$	0	0
$647$	22.4168	0.881295	0.440647	−	0.897680i	$-0.354749\pi$
$647$	22.4168	0.881295	0.440647	+	0.897680i	$0.354749\pi$
$648$	0	0
$649$	4.87183	0.191236
$650$	0	0
$651$	−25.8585	−1.01347
$652$	0	0
$653$	−24.1656	−0.945674	−0.472837	−	0.881150i	$-0.656770\pi$
$653$	−24.1656	−0.945674	−0.472837	+	0.881150i	$0.656770\pi$
$654$	0	0
$655$	−34.9736	−1.36653
$656$	0	0
$657$	−12.2149	−0.476550
$658$	0	0
$659$	0.393825	0.0153412	0.00767061	−	0.999971i	$-0.497558\pi$
$659$	0.393825	0.0153412	0.00767061	+	0.999971i	$0.497558\pi$
$660$	0	0
$661$	−40.6407	−1.58074	−0.790370	−	0.612629i	$-0.790112\pi$
$661$	−40.6407	−1.58074	−0.790370	+	0.612629i	$0.790112\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	55.2612	2.14294
$666$	0	0
$667$	−4.21277	−0.163119
$668$	0	0
$669$	4.55296	0.176027
$670$	0	0
$671$	29.7661	1.14911
$672$	0	0
$673$	−46.9874	−1.81123	−0.905616	−	0.424099i	$-0.860591\pi$
$673$	−46.9874	−1.81123	−0.905616	+	0.424099i	$0.860591\pi$
$674$	0	0
$675$	−53.7485	−2.06878
$676$	0	0
$677$	21.3124	0.819101	0.409551	−	0.912287i	$-0.365686\pi$
$677$	21.3124	0.819101	0.409551	+	0.912287i	$0.365686\pi$
$678$	0	0
$679$	−6.00910	−0.230608
$680$	0	0
$681$	22.2033	0.850834
$682$	0	0
$683$	12.2326	0.468066	0.234033	−	0.972229i	$-0.424808\pi$
$683$	12.2326	0.468066	0.234033	+	0.972229i	$0.424808\pi$
$684$	0	0
$685$	44.5900	1.70370
$686$	0	0
$687$	−3.41703	−0.130368
$688$	0	0
$689$	0	0
$690$	0	0
$691$	46.3780	1.76430	0.882152	−	0.470966i	$-0.156094\pi$
$691$	46.3780	1.76430	0.882152	+	0.470966i	$0.156094\pi$
$692$	0	0
$693$	47.6512	1.81012
$694$	0	0
$695$	−32.6476	−1.23839
$696$	0	0
$697$	−40.5511	−1.53598
$698$	0	0
$699$	−2.06932	−0.0782688
$700$	0	0
$701$	−34.0352	−1.28549	−0.642746	−	0.766079i	$-0.722205\pi$
$701$	−34.0352	−1.28549	−0.642746	+	0.766079i	$0.722205\pi$
$702$	0	0
$703$	0.692762	0.0261280
$704$	0	0
$705$	15.7505	0.593200
$706$	0	0
$707$	−3.70735	−0.139429
$708$	0	0
$709$	33.1776	1.24601	0.623005	−	0.782218i	$-0.285912\pi$
$709$	33.1776	1.24601	0.623005	+	0.782218i	$0.285912\pi$
$710$	0	0
$711$	−2.37490	−0.0890657
$712$	0	0
$713$	−27.3866	−1.02564
$714$	0	0
$715$	0	0
$716$	0	0
$717$	12.4779	0.465996
$718$	0	0
$719$	9.69809	0.361678	0.180839	−	0.983513i	$-0.442119\pi$
$719$	9.69809	0.361678	0.180839	+	0.983513i	$0.442119\pi$
$720$	0	0
$721$	69.6099	2.59241
$722$	0	0
$723$	20.7528	0.771807
$724$	0	0
$725$	−13.2375	−0.491627
$726$	0	0
$727$	18.8217	0.698059	0.349030	−	0.937112i	$-0.386511\pi$
$727$	18.8217	0.698059	0.349030	+	0.937112i	$0.386511\pi$
$728$	0	0
$729$	4.64629	0.172085
$730$	0	0
$731$	−49.4204	−1.82788
$732$	0	0
$733$	−43.3650	−1.60172	−0.800861	−	0.598850i	$-0.795624\pi$
$733$	−43.3650	−1.60172	−0.800861	+	0.598850i	$0.795624\pi$
$734$	0	0
$735$	−38.0020	−1.40172
$736$	0	0
$737$	15.6630	0.576953
$738$	0	0
$739$	6.36900	0.234288	0.117144	−	0.993115i	$-0.462626\pi$
$739$	6.36900	0.234288	0.117144	+	0.993115i	$0.462626\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−11.5334	−0.423119	−0.211560	−	0.977365i	$-0.567854\pi$
$743$	−11.5334	−0.423119	−0.211560	+	0.977365i	$0.567854\pi$
$744$	0	0
$745$	41.3885	1.51636
$746$	0	0
$747$	−17.2014	−0.629366
$748$	0	0
$749$	−10.7879	−0.394181
$750$	0	0
$751$	−48.1144	−1.75572	−0.877859	−	0.478919i	$-0.841029\pi$
$751$	−48.1144	−1.75572	−0.877859	+	0.478919i	$0.841029\pi$
$752$	0	0
$753$	13.9855	0.509661
$754$	0	0
$755$	11.3367	0.412586
$756$	0	0
$757$	−22.6438	−0.823003	−0.411502	−	0.911409i	$-0.634996\pi$
$757$	−22.6438	−0.823003	−0.411502	+	0.911409i	$0.634996\pi$
$758$	0	0
$759$	−16.0826	−0.583760
$760$	0	0
$761$	35.5528	1.28879	0.644395	−	0.764693i	$-0.277109\pi$
$761$	35.5528	1.28879	0.644395	+	0.764693i	$0.277109\pi$
$762$	0	0
$763$	8.42048	0.304842
$764$	0	0
$765$	−41.4640	−1.49913
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−37.5523	−1.35417	−0.677086	−	0.735904i	$-0.736757\pi$
$769$	−37.5523	−1.35417	−0.677086	+	0.735904i	$0.736757\pi$
$770$	0	0
$771$	−1.07547	−0.0387322
$772$	0	0
$773$	4.12049	0.148204	0.0741019	−	0.997251i	$-0.476391\pi$
$773$	4.12049	0.148204	0.0741019	+	0.997251i	$0.476391\pi$
$774$	0	0
$775$	−86.0549	−3.09118
$776$	0	0
$777$	−0.784169	−0.0281319
$778$	0	0
$779$	−29.1135	−1.04310
$780$	0	0
$781$	26.7360	0.956689
$782$	0	0
$783$	4.96840	0.177556
$784$	0	0
$785$	−50.7448	−1.81116
$786$	0	0
$787$	7.06875	0.251974	0.125987	−	0.992032i	$-0.459790\pi$
$787$	7.06875	0.251974	0.125987	+	0.992032i	$0.459790\pi$
$788$	0	0
$789$	−15.8681	−0.564918
$790$	0	0
$791$	74.1909	2.63793
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−17.0810	−0.605802
$796$	0	0
$797$	−2.36362	−0.0837237	−0.0418618	−	0.999123i	$-0.513329\pi$
$797$	−2.36362	−0.0837237	−0.0418618	+	0.999123i	$0.513329\pi$
$798$	0	0
$799$	19.8709	0.702982
$800$	0	0
$801$	−15.1328	−0.534691
$802$	0	0
$803$	−26.6291	−0.939721
$804$	0	0
$805$	−66.2492	−2.33498
$806$	0	0
$807$	1.23595	0.0435076
$808$	0	0
$809$	0.223889	0.00787153	0.00393576	−	0.999992i	$-0.498747\pi$
$809$	0.223889	0.00787153	0.00393576	+	0.999992i	$0.498747\pi$
$810$	0	0
$811$	−17.4773	−0.613710	−0.306855	−	0.951756i	$-0.599277\pi$
$811$	−17.4773	−0.613710	−0.306855	+	0.951756i	$0.599277\pi$
$812$	0	0
$813$	−15.6623	−0.549300
$814$	0	0
$815$	4.33340	0.151792
$816$	0	0
$817$	−35.4812	−1.24133
$818$	0	0
$819$	0	0
$820$	0	0
$821$	22.2561	0.776743	0.388372	−	0.921503i	$-0.373038\pi$
$821$	22.2561	0.776743	0.388372	+	0.921503i	$0.373038\pi$
$822$	0	0
$823$	35.6004	1.24095	0.620476	−	0.784225i	$-0.286939\pi$
$823$	35.6004	1.24095	0.620476	+	0.784225i	$0.286939\pi$
$824$	0	0
$825$	−50.5351	−1.75941
$826$	0	0
$827$	28.9263	1.00587	0.502933	−	0.864325i	$-0.332254\pi$
$827$	28.9263	1.00587	0.502933	+	0.864325i	$0.332254\pi$
$828$	0	0
$829$	−37.1439	−1.29006	−0.645031	−	0.764157i	$-0.723155\pi$
$829$	−37.1439	−1.29006	−0.645031	+	0.764157i	$0.723155\pi$
$830$	0	0
$831$	−6.42574	−0.222906
$832$	0	0
$833$	−47.9434	−1.66114
$834$	0	0
$835$	−33.6259	−1.16367
$836$	0	0
$837$	32.2988	1.11641
$838$	0	0
$839$	6.28498	0.216981	0.108491	−	0.994097i	$-0.465398\pi$
$839$	6.28498	0.216981	0.108491	+	0.994097i	$0.465398\pi$
$840$	0	0
$841$	−27.7764	−0.957805
$842$	0	0
$843$	5.65735	0.194850
$844$	0	0
$845$	0	0
$846$	0	0
$847$	57.4268	1.97321
$848$	0	0
$849$	1.77036	0.0607584
$850$	0	0
$851$	−0.830509	−0.0284695
$852$	0	0
$853$	−14.5503	−0.498194	−0.249097	−	0.968479i	$-0.580134\pi$
$853$	−14.5503	−0.498194	−0.249097	+	0.968479i	$0.580134\pi$
$854$	0	0
$855$	−29.7689	−1.01808
$856$	0	0
$857$	34.8702	1.19114	0.595572	−	0.803302i	$-0.296925\pi$
$857$	34.8702	1.19114	0.595572	+	0.803302i	$0.296925\pi$
$858$	0	0
$859$	26.0178	0.887717	0.443859	−	0.896097i	$-0.353609\pi$
$859$	26.0178	0.887717	0.443859	+	0.896097i	$0.353609\pi$
$860$	0	0
$861$	32.9550	1.12310
$862$	0	0
$863$	15.7106	0.534796	0.267398	−	0.963586i	$-0.413836\pi$
$863$	15.7106	0.534796	0.267398	+	0.963586i	$0.413836\pi$
$864$	0	0
$865$	−79.2492	−2.69455
$866$	0	0
$867$	2.19532	0.0745570
$868$	0	0
$869$	−5.17739	−0.175631
$870$	0	0
$871$	0	0
$872$	0	0
$873$	3.23708	0.109558
$874$	0	0
$875$	−121.192	−4.09703
$876$	0	0
$877$	10.0348	0.338852	0.169426	−	0.985543i	$-0.445809\pi$
$877$	10.0348	0.338852	0.169426	+	0.985543i	$0.445809\pi$
$878$	0	0
$879$	−10.3829	−0.350208
$880$	0	0
$881$	54.6749	1.84204	0.921022	−	0.389511i	$-0.127356\pi$
$881$	54.6749	1.84204	0.921022	+	0.389511i	$0.127356\pi$
$882$	0	0
$883$	16.6038	0.558762	0.279381	−	0.960180i	$-0.409871\pi$
$883$	16.6038	0.558762	0.279381	+	0.960180i	$0.409871\pi$
$884$	0	0
$885$	3.44515	0.115807
$886$	0	0
$887$	−55.3861	−1.85968	−0.929842	−	0.367959i	$-0.880057\pi$
$887$	−55.3861	−1.85968	−0.929842	+	0.367959i	$0.880057\pi$
$888$	0	0
$889$	7.21003	0.241817
$890$	0	0
$891$	−14.8825	−0.498581
$892$	0	0
$893$	14.2663	0.477402
$894$	0	0
$895$	77.1524	2.57892
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7.95472	0.265305
$900$	0	0
$901$	−21.5495	−0.717917
$902$	0	0
$903$	40.1628	1.33653
$904$	0	0
$905$	57.8513	1.92304
$906$	0	0
$907$	−20.9907	−0.696985	−0.348493	−	0.937311i	$-0.613306\pi$
$907$	−20.9907	−0.696985	−0.348493	+	0.937311i	$0.613306\pi$
$908$	0	0
$909$	1.99713	0.0662407
$910$	0	0
$911$	−37.8886	−1.25530	−0.627652	−	0.778494i	$-0.715984\pi$
$911$	−37.8886	−1.25530	−0.627652	+	0.778494i	$0.715984\pi$
$912$	0	0
$913$	−37.4999	−1.24106
$914$	0	0
$915$	21.0493	0.695868
$916$	0	0
$917$	35.8575	1.18412
$918$	0	0
$919$	27.2228	0.897997	0.448999	−	0.893532i	$-0.351781\pi$
$919$	27.2228	0.897997	0.448999	+	0.893532i	$0.351781\pi$
$920$	0	0
$921$	16.9685	0.559131
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−2.60965	−0.0858047
$926$	0	0
$927$	−37.4986	−1.23161
$928$	0	0
$929$	−26.5485	−0.871029	−0.435514	−	0.900182i	$-0.643434\pi$
$929$	−26.5485	−0.871029	−0.435514	+	0.900182i	$0.643434\pi$
$930$	0	0
$931$	−34.4208	−1.12810
$932$	0	0
$933$	14.3085	0.468438
$934$	0	0
$935$	−90.3935	−2.95618
$936$	0	0
$937$	−17.9293	−0.585724	−0.292862	−	0.956155i	$-0.594608\pi$
$937$	−17.9293	−0.585724	−0.292862	+	0.956155i	$0.594608\pi$
$938$	0	0
$939$	−16.6964	−0.544867
$940$	0	0
$941$	11.9807	0.390558	0.195279	−	0.980748i	$-0.437439\pi$
$941$	11.9807	0.390558	0.195279	+	0.980748i	$0.437439\pi$
$942$	0	0
$943$	34.9024	1.13658
$944$	0	0
$945$	78.1320	2.54163
$946$	0	0
$947$	−29.1251	−0.946438	−0.473219	−	0.880945i	$-0.656908\pi$
$947$	−29.1251	−0.946438	−0.473219	+	0.880945i	$0.656908\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	20.8038	0.674609
$952$	0	0
$953$	31.1838	1.01014	0.505072	−	0.863077i	$-0.331466\pi$
$953$	31.1838	1.01014	0.505072	+	0.863077i	$0.331466\pi$
$954$	0	0
$955$	74.6095	2.41431
$956$	0	0
$957$	4.67135	0.151003
$958$	0	0
$959$	−45.7170	−1.47628
$960$	0	0
$961$	20.7125	0.668145
$962$	0	0
$963$	5.81139	0.187269
$964$	0	0
$965$	10.8583	0.349541
$966$	0	0
$967$	−56.4990	−1.81688	−0.908442	−	0.418010i	$-0.862728\pi$
$967$	−56.4990	−1.81688	−0.908442	+	0.418010i	$0.862728\pi$
$968$	0	0
$969$	11.9683	0.384477
$970$	0	0
$971$	30.2789	0.971696	0.485848	−	0.874043i	$-0.338511\pi$
$971$	30.2789	0.971696	0.485848	+	0.874043i	$0.338511\pi$
$972$	0	0
$973$	33.4727	1.07309
$974$	0	0
$975$	0	0
$976$	0	0
$977$	37.4707	1.19879	0.599397	−	0.800452i	$-0.295407\pi$
$977$	37.4707	1.19879	0.599397	+	0.800452i	$0.295407\pi$
$978$	0	0
$979$	−32.9902	−1.05437
$980$	0	0
$981$	−4.53608	−0.144826
$982$	0	0
$983$	39.5666	1.26198	0.630989	−	0.775792i	$-0.282649\pi$
$983$	39.5666	1.26198	0.630989	+	0.775792i	$0.282649\pi$
$984$	0	0
$985$	99.7821	3.17932
$986$	0	0
$987$	−16.1486	−0.514017
$988$	0	0
$989$	42.5362	1.35257
$990$	0	0
$991$	−30.8316	−0.979398	−0.489699	−	0.871892i	$-0.662893\pi$
$991$	−30.8316	−0.979398	−0.489699	+	0.871892i	$0.662893\pi$
$992$	0	0
$993$	22.0335	0.699212
$994$	0	0
$995$	−77.3135	−2.45100
$996$	0	0
$997$	−54.0175	−1.71075	−0.855376	−	0.518007i	$-0.826674\pi$
$997$	−54.0175	−1.71075	−0.855376	+	0.518007i	$0.826674\pi$
$998$	0	0
$999$	0.979474	0.0309892

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5408.2.a.br.1.5	yes	9
4.3	odd	2		5408.2.a.bp.1.5	✓	9
13.12	even	2		5408.2.a.bs.1.5	yes	9
52.51	odd	2		5408.2.a.bq.1.5	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5408.2.a.bp.1.5	✓	9	4.3	odd	2
5408.2.a.bq.1.5	yes	9	52.51	odd	2
5408.2.a.br.1.5	yes	9	1.1	even	1	trivial
5408.2.a.bs.1.5	yes	9	13.12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5408 = 2^{5} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5408.a (trivial)

\(f(q)\)	\(=\)	\(q+0.851462 q^{3} -4.11908 q^{5} +4.22319 q^{7} -2.27501 q^{9} -4.95964 q^{11} -3.50724 q^{15} -4.42474 q^{17} -3.17673 q^{19} +3.59588 q^{21} +3.80838 q^{23} +11.9668 q^{25} -4.49147 q^{27} -1.10618 q^{29} -7.19114 q^{31} -4.22294 q^{33} -17.3956 q^{35} -0.218074 q^{37} +9.16464 q^{41} +11.1691 q^{43} +9.37095 q^{45} -4.49087 q^{47} +10.8353 q^{49} -3.76750 q^{51} +4.87022 q^{53} +20.4291 q^{55} -2.70486 q^{57} -0.982297 q^{59} -6.00168 q^{61} -9.60780 q^{63} -3.15809 q^{67} +3.24269 q^{69} -5.39071 q^{71} +5.36917 q^{73} +10.1893 q^{75} -20.9455 q^{77} +1.04391 q^{79} +3.00072 q^{81} +7.56101 q^{83} +18.2258 q^{85} -0.941874 q^{87} +6.65174 q^{89} -6.12298 q^{93} +13.0852 q^{95} -1.42288 q^{97} +11.2832 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 7 q^{3} - 6 q^{5} + 10 q^{9} + q^{11} - 18 q^{15} - 7 q^{17} + 5 q^{19} - 12 q^{21} + 12 q^{23} + 11 q^{25} + 34 q^{27} - 8 q^{29} - 20 q^{31} + 12 q^{33} + 6 q^{35} - 6 q^{37} + 31 q^{41} + 33 q^{43}+ \cdots + 47 q^{99}+O(q^{100}) \) 9 * q + 7 * q^3 - 6 * q^5 + 10 * q^9 + q^11 - 18 * q^15 - 7 * q^17 + 5 * q^19 - 12 * q^21 + 12 * q^23 + 11 * q^25 + 34 * q^27 - 8 * q^29 - 20 * q^31 + 12 * q^33 + 6 * q^35 - 6 * q^37 + 31 * q^41 + 33 * q^43 - 22 * q^45 + 14 * q^47 + 3 * q^49 + 12 * q^51 + 14 * q^53 + 38 * q^55 + 10 * q^57 - 9 * q^59 + 8 * q^61 - 20 * q^63 - 3 * q^67 + 26 * q^69 + 26 * q^71 + 5 * q^73 + 25 * q^75 - 6 * q^77 + 44 * q^79 + 45 * q^81 - 17 * q^83 - 6 * q^85 + 24 * q^87 + 41 * q^89 - 26 * q^93 + 4 * q^95 + 15 * q^97 + 47 * q^99

Introduction

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