LMFDB - Embedded newform 6032.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6032,2,Mod(1,6032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6032, 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("6032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6032 = 2^{4} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6032.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:	$48.1657624992$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3x^{9} - 17x^{8} + 47x^{7} + 104x^{6} - 235x^{5} - 283x^{4} + 364x^{3} + 330x^{2} + 12x - 8 $ x^10 - 3x^9 - 17x^8 + 47x^7 + 104x^6 - 235x^5 - 283x^4 + 364x^3 + 330x^2 + 12*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.63743$ of defining polynomial
Character	$\chi$	$=$	6032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.63743 q^{3} +3.03277 q^{5} +0.746946 q^{7} +3.95604 q^{9} +O(q^{10})$	$q-2.63743 q^{3} +3.03277 q^{5} +0.746946 q^{7} +3.95604 q^{9} +3.04261 q^{11} -1.00000 q^{13} -7.99871 q^{15} +5.09305 q^{17} -5.57244 q^{19} -1.97002 q^{21} -4.23685 q^{23} +4.19767 q^{25} -2.52149 q^{27} -1.00000 q^{29} -2.51122 q^{31} -8.02468 q^{33} +2.26531 q^{35} -1.54062 q^{37} +2.63743 q^{39} -8.37021 q^{41} +1.41166 q^{43} +11.9977 q^{45} -6.63347 q^{47} -6.44207 q^{49} -13.4326 q^{51} -3.27059 q^{53} +9.22753 q^{55} +14.6969 q^{57} -13.2033 q^{59} -8.00146 q^{61} +2.95495 q^{63} -3.03277 q^{65} -15.3195 q^{67} +11.1744 q^{69} +6.65576 q^{71} +8.08684 q^{73} -11.0711 q^{75} +2.27267 q^{77} -3.52952 q^{79} -5.21787 q^{81} +2.38741 q^{83} +15.4460 q^{85} +2.63743 q^{87} -7.71955 q^{89} -0.746946 q^{91} +6.62316 q^{93} -16.8999 q^{95} +11.6861 q^{97} +12.0367 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 13 q^{9}+O(q^{10}) $ 10 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 13 * q^9	$ 10 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 13 q^{9} - 14 q^{11} - 10 q^{13} - 7 q^{15} + 5 q^{17} - 11 q^{19} - 7 q^{23} + 10 q^{25} - 21 q^{27} - 10 q^{29} - 5 q^{31} + 5 q^{33} - 11 q^{35} + 8 q^{37} + 3 q^{39} + 14 q^{41} - 35 q^{43} + 7 q^{45} - 7 q^{49} - 20 q^{51} - 11 q^{53} - 8 q^{55} + 4 q^{57} - 23 q^{59} - 8 q^{61} - 43 q^{63} - 4 q^{65} - 27 q^{67} + 10 q^{69} - 3 q^{71} + 7 q^{73} - 23 q^{75} + 2 q^{77} - 9 q^{79} - 6 q^{81} - 48 q^{83} - 6 q^{85} + 3 q^{87} + 20 q^{89} - 3 q^{91} - 11 q^{93} - 11 q^{95} + q^{97} - 54 q^{99}+O(q^{100}) $ 10 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 13 * q^9 - 14 * q^11 - 10 * q^13 - 7 * q^15 + 5 * q^17 - 11 * q^19 - 7 * q^23 + 10 * q^25 - 21 * q^27 - 10 * q^29 - 5 * q^31 + 5 * q^33 - 11 * q^35 + 8 * q^37 + 3 * q^39 + 14 * q^41 - 35 * q^43 + 7 * q^45 - 7 * q^49 - 20 * q^51 - 11 * q^53 - 8 * q^55 + 4 * q^57 - 23 * q^59 - 8 * q^61 - 43 * q^63 - 4 * q^65 - 27 * q^67 + 10 * q^69 - 3 * q^71 + 7 * q^73 - 23 * q^75 + 2 * q^77 - 9 * q^79 - 6 * q^81 - 48 * q^83 - 6 * q^85 + 3 * q^87 + 20 * q^89 - 3 * q^91 - 11 * q^93 - 11 * q^95 + q^97 - 54 * q^99

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
$3$	−2.63743	−1.52272	−0.761361	−	0.648329i	$-0.775468\pi$
$3$	−2.63743	−1.52272	−0.761361	+	0.648329i	$0.775468\pi$
$4$	0	0
$5$	3.03277	1.35629	0.678147	−	0.734926i	$-0.262783\pi$
$5$	3.03277	1.35629	0.678147	+	0.734926i	$0.262783\pi$
$6$	0	0
$7$	0.746946	0.282319	0.141160	−	0.989987i	$-0.454917\pi$
$7$	0.746946	0.282319	0.141160	+	0.989987i	$0.454917\pi$
$8$	0	0
$9$	3.95604	1.31868
$10$	0	0
$11$	3.04261	0.917382	0.458691	−	0.888596i	$-0.348318\pi$
$11$	3.04261	0.917382	0.458691	+	0.888596i	$0.348318\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−7.99871	−2.06526
$16$	0	0
$17$	5.09305	1.23525	0.617623	−	0.786474i	$-0.288096\pi$
$17$	5.09305	1.23525	0.617623	+	0.786474i	$0.288096\pi$
$18$	0	0
$19$	−5.57244	−1.27841	−0.639203	−	0.769038i	$-0.720736\pi$
$19$	−5.57244	−1.27841	−0.639203	+	0.769038i	$0.720736\pi$
$20$	0	0
$21$	−1.97002	−0.429893
$22$	0	0
$23$	−4.23685	−0.883445	−0.441723	−	0.897152i	$-0.645632\pi$
$23$	−4.23685	−0.883445	−0.441723	+	0.897152i	$0.645632\pi$
$24$	0	0
$25$	4.19767	0.839535
$26$	0	0
$27$	−2.52149	−0.485261
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−2.51122	−0.451028	−0.225514	−	0.974240i	$-0.572406\pi$
$31$	−2.51122	−0.451028	−0.225514	+	0.974240i	$0.572406\pi$
$32$	0	0
$33$	−8.02468	−1.39692
$34$	0	0
$35$	2.26531	0.382908
$36$	0	0
$37$	−1.54062	−0.253277	−0.126638	−	0.991949i	$-0.540419\pi$
$37$	−1.54062	−0.253277	−0.126638	+	0.991949i	$0.540419\pi$
$38$	0	0
$39$	2.63743	0.422327
$40$	0	0
$41$	−8.37021	−1.30721	−0.653604	−	0.756837i	$-0.726744\pi$
$41$	−8.37021	−1.30721	−0.653604	+	0.756837i	$0.726744\pi$
$42$	0	0
$43$	1.41166	0.215276	0.107638	−	0.994190i	$-0.465671\pi$
$43$	1.41166	0.215276	0.107638	+	0.994190i	$0.465671\pi$
$44$	0	0
$45$	11.9977	1.78852
$46$	0	0
$47$	−6.63347	−0.967591	−0.483796	−	0.875181i	$-0.660742\pi$
$47$	−6.63347	−0.967591	−0.483796	+	0.875181i	$0.660742\pi$
$48$	0	0
$49$	−6.44207	−0.920296
$50$	0	0
$51$	−13.4326	−1.88094
$52$	0	0
$53$	−3.27059	−0.449250	−0.224625	−	0.974445i	$-0.572116\pi$
$53$	−3.27059	−0.449250	−0.224625	+	0.974445i	$0.572116\pi$
$54$	0	0
$55$	9.22753	1.24424
$56$	0	0
$57$	14.6969	1.94666
$58$	0	0
$59$	−13.2033	−1.71892	−0.859462	−	0.511199i	$-0.829201\pi$
$59$	−13.2033	−1.71892	−0.859462	+	0.511199i	$0.829201\pi$
$60$	0	0
$61$	−8.00146	−1.02448	−0.512241	−	0.858842i	$-0.671185\pi$
$61$	−8.00146	−1.02448	−0.512241	+	0.858842i	$0.671185\pi$
$62$	0	0
$63$	2.95495	0.372289
$64$	0	0
$65$	−3.03277	−0.376168
$66$	0	0
$67$	−15.3195	−1.87157	−0.935787	−	0.352567i	$-0.885309\pi$
$67$	−15.3195	−1.87157	−0.935787	+	0.352567i	$0.885309\pi$
$68$	0	0
$69$	11.1744	1.34524
$70$	0	0
$71$	6.65576	0.789893	0.394947	−	0.918704i	$-0.370763\pi$
$71$	6.65576	0.789893	0.394947	+	0.918704i	$0.370763\pi$
$72$	0	0
$73$	8.08684	0.946493	0.473247	−	0.880930i	$-0.343082\pi$
$73$	8.08684	0.946493	0.473247	+	0.880930i	$0.343082\pi$
$74$	0	0
$75$	−11.0711	−1.27838
$76$	0	0
$77$	2.27267	0.258995
$78$	0	0
$79$	−3.52952	−0.397102	−0.198551	−	0.980091i	$-0.563624\pi$
$79$	−3.52952	−0.397102	−0.198551	+	0.980091i	$0.563624\pi$
$80$	0	0
$81$	−5.21787	−0.579763
$82$	0	0
$83$	2.38741	0.262052	0.131026	−	0.991379i	$-0.458173\pi$
$83$	2.38741	0.262052	0.131026	+	0.991379i	$0.458173\pi$
$84$	0	0
$85$	15.4460	1.67536
$86$	0	0
$87$	2.63743	0.282762
$88$	0	0
$89$	−7.71955	−0.818271	−0.409136	−	0.912474i	$-0.634170\pi$
$89$	−7.71955	−0.818271	−0.409136	+	0.912474i	$0.634170\pi$
$90$	0	0
$91$	−0.746946	−0.0783012
$92$	0	0
$93$	6.62316	0.686790
$94$	0	0
$95$	−16.8999	−1.73390
$96$	0	0
$97$	11.6861	1.18655	0.593273	−	0.805001i	$-0.297836\pi$
$97$	11.6861	1.18655	0.593273	+	0.805001i	$0.297836\pi$
$98$	0	0
$99$	12.0367	1.20973
$100$	0	0
$101$	11.1746	1.11192	0.555959	−	0.831210i	$-0.312351\pi$
$101$	11.1746	1.11192	0.555959	+	0.831210i	$0.312351\pi$
$102$	0	0
$103$	0.212053	0.0208942	0.0104471	−	0.999945i	$-0.496675\pi$
$103$	0.212053	0.0208942	0.0104471	+	0.999945i	$0.496675\pi$
$104$	0	0
$105$	−5.97461	−0.583062
$106$	0	0
$107$	−2.55712	−0.247207	−0.123603	−	0.992332i	$-0.539445\pi$
$107$	−2.55712	−0.247207	−0.123603	+	0.992332i	$0.539445\pi$
$108$	0	0
$109$	−11.5349	−1.10484	−0.552422	−	0.833564i	$-0.686296\pi$
$109$	−11.5349	−1.10484	−0.552422	+	0.833564i	$0.686296\pi$
$110$	0	0
$111$	4.06328	0.385670
$112$	0	0
$113$	−10.9048	−1.02584	−0.512919	−	0.858437i	$-0.671436\pi$
$113$	−10.9048	−1.02584	−0.512919	+	0.858437i	$0.671436\pi$
$114$	0	0
$115$	−12.8494	−1.19821
$116$	0	0
$117$	−3.95604	−0.365736
$118$	0	0
$119$	3.80423	0.348734
$120$	0	0
$121$	−1.74251	−0.158410
$122$	0	0
$123$	22.0759	1.99051
$124$	0	0
$125$	−2.43326	−0.217638
$126$	0	0
$127$	6.60756	0.586326	0.293163	−	0.956062i	$-0.405292\pi$
$127$	6.60756	0.586326	0.293163	+	0.956062i	$0.405292\pi$
$128$	0	0
$129$	−3.72315	−0.327805
$130$	0	0
$131$	13.0066	1.13639	0.568196	−	0.822893i	$-0.307641\pi$
$131$	13.0066	1.13639	0.568196	+	0.822893i	$0.307641\pi$
$132$	0	0
$133$	−4.16232	−0.360919
$134$	0	0
$135$	−7.64709	−0.658157
$136$	0	0
$137$	−1.70065	−0.145296	−0.0726482	−	0.997358i	$-0.523145\pi$
$137$	−1.70065	−0.145296	−0.0726482	+	0.997358i	$0.523145\pi$
$138$	0	0
$139$	3.33796	0.283122	0.141561	−	0.989930i	$-0.454788\pi$
$139$	3.33796	0.283122	0.141561	+	0.989930i	$0.454788\pi$
$140$	0	0
$141$	17.4953	1.47337
$142$	0	0
$143$	−3.04261	−0.254436
$144$	0	0
$145$	−3.03277	−0.251858
$146$	0	0
$147$	16.9905	1.40135
$148$	0	0
$149$	11.5029	0.942354	0.471177	−	0.882039i	$-0.343829\pi$
$149$	11.5029	0.942354	0.471177	+	0.882039i	$0.343829\pi$
$150$	0	0
$151$	22.9200	1.86520	0.932601	−	0.360909i	$-0.117534\pi$
$151$	22.9200	1.86520	0.932601	+	0.360909i	$0.117534\pi$
$152$	0	0
$153$	20.1483	1.62889
$154$	0	0
$155$	−7.61594	−0.611727
$156$	0	0
$157$	−8.73917	−0.697462	−0.348731	−	0.937223i	$-0.613387\pi$
$157$	−8.73917	−0.697462	−0.348731	+	0.937223i	$0.613387\pi$
$158$	0	0
$159$	8.62594	0.684082
$160$	0	0
$161$	−3.16470	−0.249413
$162$	0	0
$163$	−0.338687	−0.0265280	−0.0132640	−	0.999912i	$-0.504222\pi$
$163$	−0.338687	−0.0265280	−0.0132640	+	0.999912i	$0.504222\pi$
$164$	0	0
$165$	−24.3370	−1.89463
$166$	0	0
$167$	3.57206	0.276415	0.138207	−	0.990403i	$-0.455866\pi$
$167$	3.57206	0.276415	0.138207	+	0.990403i	$0.455866\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−22.0448	−1.68581
$172$	0	0
$173$	−15.4472	−1.17443	−0.587213	−	0.809432i	$-0.699775\pi$
$173$	−15.4472	−1.17443	−0.587213	+	0.809432i	$0.699775\pi$
$174$	0	0
$175$	3.13544	0.237017
$176$	0	0
$177$	34.8228	2.61744
$178$	0	0
$179$	−16.3332	−1.22080	−0.610399	−	0.792094i	$-0.708991\pi$
$179$	−16.3332	−1.22080	−0.610399	+	0.792094i	$0.708991\pi$
$180$	0	0
$181$	5.53911	0.411719	0.205860	−	0.978582i	$-0.434001\pi$
$181$	5.53911	0.411719	0.205860	+	0.978582i	$0.434001\pi$
$182$	0	0
$183$	21.1033	1.56000
$184$	0	0
$185$	−4.67234	−0.343518
$186$	0	0
$187$	15.4962	1.13319
$188$	0	0
$189$	−1.88342	−0.136998
$190$	0	0
$191$	−6.68033	−0.483372	−0.241686	−	0.970355i	$-0.577700\pi$
$191$	−6.68033	−0.483372	−0.241686	+	0.970355i	$0.577700\pi$
$192$	0	0
$193$	14.5837	1.04976	0.524878	−	0.851177i	$-0.324111\pi$
$193$	14.5837	1.04976	0.524878	+	0.851177i	$0.324111\pi$
$194$	0	0
$195$	7.99871	0.572800
$196$	0	0
$197$	−15.4281	−1.09921	−0.549604	−	0.835426i	$-0.685221\pi$
$197$	−15.4281	−1.09921	−0.549604	+	0.835426i	$0.685221\pi$
$198$	0	0
$199$	8.04267	0.570130	0.285065	−	0.958508i	$-0.407985\pi$
$199$	8.04267	0.570130	0.285065	+	0.958508i	$0.407985\pi$
$200$	0	0
$201$	40.4041	2.84988
$202$	0	0
$203$	−0.746946	−0.0524253
$204$	0	0
$205$	−25.3849	−1.77296
$206$	0	0
$207$	−16.7612	−1.16498
$208$	0	0
$209$	−16.9548	−1.17279
$210$	0	0
$211$	8.30927	0.572033	0.286017	−	0.958225i	$-0.407669\pi$
$211$	8.30927	0.572033	0.286017	+	0.958225i	$0.407669\pi$
$212$	0	0
$213$	−17.5541	−1.20279
$214$	0	0
$215$	4.28123	0.291978
$216$	0	0
$217$	−1.87574	−0.127334
$218$	0	0
$219$	−21.3285	−1.44125
$220$	0	0
$221$	−5.09305	−0.342596
$222$	0	0
$223$	−23.2107	−1.55430	−0.777152	−	0.629312i	$-0.783337\pi$
$223$	−23.2107	−1.55430	−0.777152	+	0.629312i	$0.783337\pi$
$224$	0	0
$225$	16.6062	1.10708
$226$	0	0
$227$	−5.02157	−0.333293	−0.166647	−	0.986017i	$-0.553294\pi$
$227$	−5.02157	−0.333293	−0.166647	+	0.986017i	$0.553294\pi$
$228$	0	0
$229$	3.56761	0.235754	0.117877	−	0.993028i	$-0.462391\pi$
$229$	3.56761	0.235754	0.117877	+	0.993028i	$0.462391\pi$
$230$	0	0
$231$	−5.99400	−0.394376
$232$	0	0
$233$	−7.17726	−0.470198	−0.235099	−	0.971971i	$-0.575541\pi$
$233$	−7.17726	−0.470198	−0.235099	+	0.971971i	$0.575541\pi$
$234$	0	0
$235$	−20.1178	−1.31234
$236$	0	0
$237$	9.30886	0.604675
$238$	0	0
$239$	19.3421	1.25114	0.625568	−	0.780169i	$-0.284867\pi$
$239$	19.3421	1.25114	0.625568	+	0.780169i	$0.284867\pi$
$240$	0	0
$241$	21.3434	1.37485	0.687424	−	0.726256i	$-0.258741\pi$
$241$	21.3434	1.37485	0.687424	+	0.726256i	$0.258741\pi$
$242$	0	0
$243$	21.3262	1.36808
$244$	0	0
$245$	−19.5373	−1.24819
$246$	0	0
$247$	5.57244	0.354566
$248$	0	0
$249$	−6.29663	−0.399033
$250$	0	0
$251$	24.0299	1.51675	0.758376	−	0.651818i	$-0.225993\pi$
$251$	24.0299	1.51675	0.758376	+	0.651818i	$0.225993\pi$
$252$	0	0
$253$	−12.8911	−0.810457
$254$	0	0
$255$	−40.7378	−2.55110
$256$	0	0
$257$	0.348453	0.0217359	0.0108679	−	0.999941i	$-0.496541\pi$
$257$	0.348453	0.0217359	0.0108679	+	0.999941i	$0.496541\pi$
$258$	0	0
$259$	−1.15076	−0.0715048
$260$	0	0
$261$	−3.95604	−0.244873
$262$	0	0
$263$	−11.5734	−0.713647	−0.356823	−	0.934172i	$-0.616140\pi$
$263$	−11.5734	−0.713647	−0.356823	+	0.934172i	$0.616140\pi$
$264$	0	0
$265$	−9.91893	−0.609315
$266$	0	0
$267$	20.3598	1.24600
$268$	0	0
$269$	−17.3574	−1.05830	−0.529150	−	0.848528i	$-0.677489\pi$
$269$	−17.3574	−1.05830	−0.529150	+	0.848528i	$0.677489\pi$
$270$	0	0
$271$	25.8798	1.57209	0.786043	−	0.618171i	$-0.212126\pi$
$271$	25.8798	1.57209	0.786043	+	0.618171i	$0.212126\pi$
$272$	0	0
$273$	1.97002	0.119231
$274$	0	0
$275$	12.7719	0.770174
$276$	0	0
$277$	9.17686	0.551384	0.275692	−	0.961246i	$-0.411093\pi$
$277$	9.17686	0.551384	0.275692	+	0.961246i	$0.411093\pi$
$278$	0	0
$279$	−9.93448	−0.594762
$280$	0	0
$281$	3.87097	0.230923	0.115461	−	0.993312i	$-0.463165\pi$
$281$	3.87097	0.230923	0.115461	+	0.993312i	$0.463165\pi$
$282$	0	0
$283$	5.36071	0.318661	0.159331	−	0.987225i	$-0.449066\pi$
$283$	5.36071	0.318661	0.159331	+	0.987225i	$0.449066\pi$
$284$	0	0
$285$	44.5724	2.64024
$286$	0	0
$287$	−6.25210	−0.369050
$288$	0	0
$289$	8.93915	0.525833
$290$	0	0
$291$	−30.8213	−1.80678
$292$	0	0
$293$	1.00702	0.0588304	0.0294152	−	0.999567i	$-0.490635\pi$
$293$	1.00702	0.0588304	0.0294152	+	0.999567i	$0.490635\pi$
$294$	0	0
$295$	−40.0426	−2.33137
$296$	0	0
$297$	−7.67191	−0.445170
$298$	0	0
$299$	4.23685	0.245024
$300$	0	0
$301$	1.05443	0.0607765
$302$	0	0
$303$	−29.4723	−1.69314
$304$	0	0
$305$	−24.2666	−1.38950
$306$	0	0
$307$	−2.83674	−0.161901	−0.0809507	−	0.996718i	$-0.525796\pi$
$307$	−2.83674	−0.161901	−0.0809507	+	0.996718i	$0.525796\pi$
$308$	0	0
$309$	−0.559275	−0.0318161
$310$	0	0
$311$	−17.1047	−0.969919	−0.484960	−	0.874537i	$-0.661166\pi$
$311$	−17.1047	−0.969919	−0.484960	+	0.874537i	$0.661166\pi$
$312$	0	0
$313$	−16.8081	−0.950048	−0.475024	−	0.879973i	$-0.657561\pi$
$313$	−16.8081	−0.950048	−0.475024	+	0.879973i	$0.657561\pi$
$314$	0	0
$315$	8.96167	0.504933
$316$	0	0
$317$	−23.9295	−1.34402	−0.672008	−	0.740544i	$-0.734568\pi$
$317$	−23.9295	−1.34402	−0.672008	+	0.740544i	$0.734568\pi$
$318$	0	0
$319$	−3.04261	−0.170354
$320$	0	0
$321$	6.74424	0.376427
$322$	0	0
$323$	−28.3807	−1.57915
$324$	0	0
$325$	−4.19767	−0.232845
$326$	0	0
$327$	30.4225	1.68237
$328$	0	0
$329$	−4.95485	−0.273170
$330$	0	0
$331$	−23.5841	−1.29630	−0.648150	−	0.761513i	$-0.724457\pi$
$331$	−23.5841	−1.29630	−0.648150	+	0.761513i	$0.724457\pi$
$332$	0	0
$333$	−6.09476	−0.333991
$334$	0	0
$335$	−46.4604	−2.53840
$336$	0	0
$337$	−14.6765	−0.799478	−0.399739	−	0.916629i	$-0.630899\pi$
$337$	−14.6765	−0.799478	−0.399739	+	0.916629i	$0.630899\pi$
$338$	0	0
$339$	28.7607	1.56207
$340$	0	0
$341$	−7.64066	−0.413765
$342$	0	0
$343$	−10.0405	−0.542136
$344$	0	0
$345$	33.8894	1.82454
$346$	0	0
$347$	−28.8987	−1.55137	−0.775683	−	0.631123i	$-0.782594\pi$
$347$	−28.8987	−1.55137	−0.775683	+	0.631123i	$0.782594\pi$
$348$	0	0
$349$	−23.8281	−1.27549	−0.637744	−	0.770249i	$-0.720132\pi$
$349$	−23.8281	−1.27549	−0.637744	+	0.770249i	$0.720132\pi$
$350$	0	0
$351$	2.52149	0.134587
$352$	0	0
$353$	−7.70695	−0.410200	−0.205100	−	0.978741i	$-0.565752\pi$
$353$	−7.70695	−0.410200	−0.205100	+	0.978741i	$0.565752\pi$
$354$	0	0
$355$	20.1854	1.07133
$356$	0	0
$357$	−10.0334	−0.531024
$358$	0	0
$359$	24.0284	1.26817	0.634086	−	0.773263i	$-0.281377\pi$
$359$	24.0284	1.26817	0.634086	+	0.773263i	$0.281377\pi$
$360$	0	0
$361$	12.0521	0.634323
$362$	0	0
$363$	4.59575	0.241214
$364$	0	0
$365$	24.5255	1.28372
$366$	0	0
$367$	11.9867	0.625699	0.312849	−	0.949803i	$-0.398716\pi$
$367$	11.9867	0.625699	0.312849	+	0.949803i	$0.398716\pi$
$368$	0	0
$369$	−33.1129	−1.72379
$370$	0	0
$371$	−2.44295	−0.126832
$372$	0	0
$373$	−9.51362	−0.492596	−0.246298	−	0.969194i	$-0.579214\pi$
$373$	−9.51362	−0.492596	−0.246298	+	0.969194i	$0.579214\pi$
$374$	0	0
$375$	6.41757	0.331402
$376$	0	0
$377$	1.00000	0.0515026
$378$	0	0
$379$	3.90922	0.200803	0.100401	−	0.994947i	$-0.467987\pi$
$379$	3.90922	0.200803	0.100401	+	0.994947i	$0.467987\pi$
$380$	0	0
$381$	−17.4270	−0.892811
$382$	0	0
$383$	−22.9927	−1.17487	−0.587437	−	0.809270i	$-0.699863\pi$
$383$	−22.9927	−1.17487	−0.587437	+	0.809270i	$0.699863\pi$
$384$	0	0
$385$	6.89247	0.351273
$386$	0	0
$387$	5.58458	0.283880
$388$	0	0
$389$	−23.0737	−1.16988	−0.584941	−	0.811076i	$-0.698882\pi$
$389$	−23.0737	−1.16988	−0.584941	+	0.811076i	$0.698882\pi$
$390$	0	0
$391$	−21.5785	−1.09127
$392$	0	0
$393$	−34.3040	−1.73041
$394$	0	0
$395$	−10.7042	−0.538587
$396$	0	0
$397$	22.5388	1.13119	0.565594	−	0.824684i	$-0.308647\pi$
$397$	22.5388	1.13119	0.565594	+	0.824684i	$0.308647\pi$
$398$	0	0
$399$	10.9778	0.549578
$400$	0	0
$401$	−12.8582	−0.642107	−0.321053	−	0.947061i	$-0.604037\pi$
$401$	−12.8582	−0.642107	−0.321053	+	0.947061i	$0.604037\pi$
$402$	0	0
$403$	2.51122	0.125093
$404$	0	0
$405$	−15.8246	−0.786330
$406$	0	0
$407$	−4.68751	−0.232351
$408$	0	0
$409$	11.6157	0.574358	0.287179	−	0.957877i	$-0.407283\pi$
$409$	11.6157	0.574358	0.287179	+	0.957877i	$0.407283\pi$
$410$	0	0
$411$	4.48535	0.221246
$412$	0	0
$413$	−9.86216	−0.485285
$414$	0	0
$415$	7.24046	0.355420
$416$	0	0
$417$	−8.80364	−0.431116
$418$	0	0
$419$	19.4944	0.952362	0.476181	−	0.879347i	$-0.342021\pi$
$419$	19.4944	0.952362	0.476181	+	0.879347i	$0.342021\pi$
$420$	0	0
$421$	−35.6242	−1.73622	−0.868108	−	0.496376i	$-0.834664\pi$
$421$	−35.6242	−1.73622	−0.868108	+	0.496376i	$0.834664\pi$
$422$	0	0
$423$	−26.2423	−1.27594
$424$	0	0
$425$	21.3790	1.03703
$426$	0	0
$427$	−5.97666	−0.289231
$428$	0	0
$429$	8.02468	0.387435
$430$	0	0
$431$	34.1839	1.64658	0.823290	−	0.567621i	$-0.192136\pi$
$431$	34.1839	1.64658	0.823290	+	0.567621i	$0.192136\pi$
$432$	0	0
$433$	0.304977	0.0146563	0.00732813	−	0.999973i	$-0.497667\pi$
$433$	0.304977	0.0146563	0.00732813	+	0.999973i	$0.497667\pi$
$434$	0	0
$435$	7.99871	0.383509
$436$	0	0
$437$	23.6096	1.12940
$438$	0	0
$439$	−24.6917	−1.17847	−0.589235	−	0.807962i	$-0.700571\pi$
$439$	−24.6917	−1.17847	−0.589235	+	0.807962i	$0.700571\pi$
$440$	0	0
$441$	−25.4851	−1.21358
$442$	0	0
$443$	−6.53170	−0.310331	−0.155165	−	0.987889i	$-0.549591\pi$
$443$	−6.53170	−0.310331	−0.155165	+	0.987889i	$0.549591\pi$
$444$	0	0
$445$	−23.4116	−1.10982
$446$	0	0
$447$	−30.3381	−1.43494
$448$	0	0
$449$	22.9837	1.08467	0.542335	−	0.840162i	$-0.317540\pi$
$449$	22.9837	1.08467	0.542335	+	0.840162i	$0.317540\pi$
$450$	0	0
$451$	−25.4673	−1.19921
$452$	0	0
$453$	−60.4499	−2.84018
$454$	0	0
$455$	−2.26531	−0.106200
$456$	0	0
$457$	−34.9146	−1.63324	−0.816619	−	0.577178i	$-0.804154\pi$
$457$	−34.9146	−1.63324	−0.816619	+	0.577178i	$0.804154\pi$
$458$	0	0
$459$	−12.8421	−0.599416
$460$	0	0
$461$	28.1721	1.31211	0.656053	−	0.754715i	$-0.272225\pi$
$461$	28.1721	1.31211	0.656053	+	0.754715i	$0.272225\pi$
$462$	0	0
$463$	−32.7890	−1.52384	−0.761918	−	0.647674i	$-0.775742\pi$
$463$	−32.7890	−1.52384	−0.761918	+	0.647674i	$0.775742\pi$
$464$	0	0
$465$	20.0865	0.931490
$466$	0	0
$467$	−5.04586	−0.233495	−0.116747	−	0.993162i	$-0.537247\pi$
$467$	−5.04586	−0.233495	−0.116747	+	0.993162i	$0.537247\pi$
$468$	0	0
$469$	−11.4428	−0.528381
$470$	0	0
$471$	23.0490	1.06204
$472$	0	0
$473$	4.29513	0.197490
$474$	0	0
$475$	−23.3913	−1.07327
$476$	0	0
$477$	−12.9386	−0.592416
$478$	0	0
$479$	−36.7921	−1.68107	−0.840537	−	0.541754i	$-0.817760\pi$
$479$	−36.7921	−1.68107	−0.840537	+	0.541754i	$0.817760\pi$
$480$	0	0
$481$	1.54062	0.0702463
$482$	0	0
$483$	8.34668	0.379787
$484$	0	0
$485$	35.4413	1.60931
$486$	0	0
$487$	29.4307	1.33363	0.666816	−	0.745222i	$-0.267656\pi$
$487$	29.4307	1.33363	0.666816	+	0.745222i	$0.267656\pi$
$488$	0	0
$489$	0.893264	0.0403948
$490$	0	0
$491$	−34.2409	−1.54527	−0.772634	−	0.634852i	$-0.781061\pi$
$491$	−34.2409	−1.54527	−0.772634	+	0.634852i	$0.781061\pi$
$492$	0	0
$493$	−5.09305	−0.229379
$494$	0	0
$495$	36.5045	1.64075
$496$	0	0
$497$	4.97149	0.223002
$498$	0	0
$499$	39.0366	1.74752	0.873760	−	0.486358i	$-0.161675\pi$
$499$	39.0366	1.74752	0.873760	+	0.486358i	$0.161675\pi$
$500$	0	0
$501$	−9.42107	−0.420902
$502$	0	0
$503$	24.4495	1.09015	0.545074	−	0.838388i	$-0.316502\pi$
$503$	24.4495	1.09015	0.545074	+	0.838388i	$0.316502\pi$
$504$	0	0
$505$	33.8900	1.50809
$506$	0	0
$507$	−2.63743	−0.117132
$508$	0	0
$509$	−13.5587	−0.600980	−0.300490	−	0.953785i	$-0.597150\pi$
$509$	−13.5587	−0.600980	−0.300490	+	0.953785i	$0.597150\pi$
$510$	0	0
$511$	6.04043	0.267213
$512$	0	0
$513$	14.0509	0.620360
$514$	0	0
$515$	0.643108	0.0283387
$516$	0	0
$517$	−20.1831	−0.887651
$518$	0	0
$519$	40.7408	1.78832
$520$	0	0
$521$	42.8598	1.87772	0.938861	−	0.344298i	$-0.111883\pi$
$521$	42.8598	1.87772	0.938861	+	0.344298i	$0.111883\pi$
$522$	0	0
$523$	6.12317	0.267748	0.133874	−	0.990998i	$-0.457258\pi$
$523$	6.12317	0.267748	0.133874	+	0.990998i	$0.457258\pi$
$524$	0	0
$525$	−8.26950	−0.360910
$526$	0	0
$527$	−12.7898	−0.557131
$528$	0	0
$529$	−5.04907	−0.219525
$530$	0	0
$531$	−52.2328	−2.26671
$532$	0	0
$533$	8.37021	0.362554
$534$	0	0
$535$	−7.75516	−0.335285
$536$	0	0
$537$	43.0776	1.85894
$538$	0	0
$539$	−19.6007	−0.844263
$540$	0	0
$541$	32.0415	1.37757	0.688786	−	0.724965i	$-0.258144\pi$
$541$	32.0415	1.37757	0.688786	+	0.724965i	$0.258144\pi$
$542$	0	0
$543$	−14.6090	−0.626933
$544$	0	0
$545$	−34.9827	−1.49850
$546$	0	0
$547$	−36.4678	−1.55925	−0.779625	−	0.626247i	$-0.784590\pi$
$547$	−36.4678	−1.55925	−0.779625	+	0.626247i	$0.784590\pi$
$548$	0	0
$549$	−31.6541	−1.35096
$550$	0	0
$551$	5.57244	0.237394
$552$	0	0
$553$	−2.63636	−0.112109
$554$	0	0
$555$	12.3230	0.523081
$556$	0	0
$557$	2.26775	0.0960878	0.0480439	−	0.998845i	$-0.484701\pi$
$557$	2.26775	0.0960878	0.0480439	+	0.998845i	$0.484701\pi$
$558$	0	0
$559$	−1.41166	−0.0597068
$560$	0	0
$561$	−40.8701	−1.72554
$562$	0	0
$563$	18.8226	0.793278	0.396639	−	0.917975i	$-0.370176\pi$
$563$	18.8226	0.793278	0.396639	+	0.917975i	$0.370176\pi$
$564$	0	0
$565$	−33.0718	−1.39134
$566$	0	0
$567$	−3.89747	−0.163678
$568$	0	0
$569$	31.9273	1.33846	0.669231	−	0.743055i	$-0.266624\pi$
$569$	31.9273	1.33846	0.669231	+	0.743055i	$0.266624\pi$
$570$	0	0
$571$	39.4769	1.65206	0.826029	−	0.563628i	$-0.190595\pi$
$571$	39.4769	1.65206	0.826029	+	0.563628i	$0.190595\pi$
$572$	0	0
$573$	17.6189	0.736041
$574$	0	0
$575$	−17.7849	−0.741683
$576$	0	0
$577$	34.2643	1.42644	0.713220	−	0.700940i	$-0.247236\pi$
$577$	34.2643	1.42644	0.713220	+	0.700940i	$0.247236\pi$
$578$	0	0
$579$	−38.4634	−1.59849
$580$	0	0
$581$	1.78327	0.0739824
$582$	0	0
$583$	−9.95113	−0.412134
$584$	0	0
$585$	−11.9977	−0.496046
$586$	0	0
$587$	−45.9582	−1.89690	−0.948449	−	0.316931i	$-0.897348\pi$
$587$	−45.9582	−1.89690	−0.948449	+	0.316931i	$0.897348\pi$
$588$	0	0
$589$	13.9936	0.576597
$590$	0	0
$591$	40.6906	1.67379
$592$	0	0
$593$	−13.4979	−0.554291	−0.277145	−	0.960828i	$-0.589388\pi$
$593$	−13.4979	−0.554291	−0.277145	+	0.960828i	$0.589388\pi$
$594$	0	0
$595$	11.5374	0.472985
$596$	0	0
$597$	−21.2120	−0.868148
$598$	0	0
$599$	12.4176	0.507370	0.253685	−	0.967287i	$-0.418357\pi$
$599$	12.4176	0.507370	0.253685	+	0.967287i	$0.418357\pi$
$600$	0	0
$601$	40.0679	1.63440	0.817202	−	0.576351i	$-0.195524\pi$
$601$	40.0679	1.63440	0.817202	+	0.576351i	$0.195524\pi$
$602$	0	0
$603$	−60.6045	−2.46801
$604$	0	0
$605$	−5.28463	−0.214851
$606$	0	0
$607$	28.4026	1.15283	0.576413	−	0.817158i	$-0.304452\pi$
$607$	28.4026	1.15283	0.576413	+	0.817158i	$0.304452\pi$
$608$	0	0
$609$	1.97002	0.0798292
$610$	0	0
$611$	6.63347	0.268362
$612$	0	0
$613$	6.21411	0.250985	0.125493	−	0.992095i	$-0.459949\pi$
$613$	6.21411	0.250985	0.125493	+	0.992095i	$0.459949\pi$
$614$	0	0
$615$	66.9509	2.69972
$616$	0	0
$617$	6.31034	0.254045	0.127022	−	0.991900i	$-0.459458\pi$
$617$	6.31034	0.254045	0.127022	+	0.991900i	$0.459458\pi$
$618$	0	0
$619$	−3.72978	−0.149913	−0.0749563	−	0.997187i	$-0.523882\pi$
$619$	−3.72978	−0.149913	−0.0749563	+	0.997187i	$0.523882\pi$
$620$	0	0
$621$	10.6832	0.428701
$622$	0	0
$623$	−5.76609	−0.231014
$624$	0	0
$625$	−28.3679	−1.13472
$626$	0	0
$627$	44.7171	1.78583
$628$	0	0
$629$	−7.84646	−0.312859
$630$	0	0
$631$	6.98349	0.278008	0.139004	−	0.990292i	$-0.455610\pi$
$631$	6.98349	0.278008	0.139004	+	0.990292i	$0.455610\pi$
$632$	0	0
$633$	−21.9151	−0.871047
$634$	0	0
$635$	20.0392	0.795231
$636$	0	0
$637$	6.44207	0.255244
$638$	0	0
$639$	26.3304	1.04162
$640$	0	0
$641$	29.1089	1.14973	0.574866	−	0.818248i	$-0.305054\pi$
$641$	29.1089	1.14973	0.574866	+	0.818248i	$0.305054\pi$
$642$	0	0
$643$	−42.7427	−1.68561	−0.842803	−	0.538222i	$-0.819096\pi$
$643$	−42.7427	−1.68561	−0.842803	+	0.538222i	$0.819096\pi$
$644$	0	0
$645$	−11.2915	−0.444601
$646$	0	0
$647$	15.6262	0.614328	0.307164	−	0.951657i	$-0.400620\pi$
$647$	15.6262	0.614328	0.307164	+	0.951657i	$0.400620\pi$
$648$	0	0
$649$	−40.1726	−1.57691
$650$	0	0
$651$	4.94715	0.193894
$652$	0	0
$653$	−36.3003	−1.42054	−0.710270	−	0.703929i	$-0.751427\pi$
$653$	−36.3003	−1.42054	−0.710270	+	0.703929i	$0.751427\pi$
$654$	0	0
$655$	39.4460	1.54128
$656$	0	0
$657$	31.9919	1.24812
$658$	0	0
$659$	0.974686	0.0379684	0.0189842	−	0.999820i	$-0.493957\pi$
$659$	0.974686	0.0379684	0.0189842	+	0.999820i	$0.493957\pi$
$660$	0	0
$661$	−42.1645	−1.64001	−0.820004	−	0.572358i	$-0.806029\pi$
$661$	−42.1645	−1.64001	−0.820004	+	0.572358i	$0.806029\pi$
$662$	0	0
$663$	13.4326	0.521678
$664$	0	0
$665$	−12.6233	−0.489512
$666$	0	0
$667$	4.23685	0.164052
$668$	0	0
$669$	61.2167	2.36677
$670$	0	0
$671$	−24.3453	−0.939841
$672$	0	0
$673$	−15.5919	−0.601022	−0.300511	−	0.953778i	$-0.597157\pi$
$673$	−15.5919	−0.601022	−0.300511	+	0.953778i	$0.597157\pi$
$674$	0	0
$675$	−10.5844	−0.407393
$676$	0	0
$677$	−39.9722	−1.53625	−0.768127	−	0.640297i	$-0.778811\pi$
$677$	−39.9722	−1.53625	−0.768127	+	0.640297i	$0.778811\pi$
$678$	0	0
$679$	8.72890	0.334985
$680$	0	0
$681$	13.2440	0.507513
$682$	0	0
$683$	−9.20959	−0.352395	−0.176198	−	0.984355i	$-0.556380\pi$
$683$	−9.20959	−0.352395	−0.176198	+	0.984355i	$0.556380\pi$
$684$	0	0
$685$	−5.15768	−0.197065
$686$	0	0
$687$	−9.40931	−0.358988
$688$	0	0
$689$	3.27059	0.124599
$690$	0	0
$691$	−17.8286	−0.678233	−0.339116	−	0.940744i	$-0.610128\pi$
$691$	−17.8286	−0.678233	−0.339116	+	0.940744i	$0.610128\pi$
$692$	0	0
$693$	8.99076	0.341531
$694$	0	0
$695$	10.1233	0.383997
$696$	0	0
$697$	−42.6299	−1.61472
$698$	0	0
$699$	18.9295	0.715980
$700$	0	0
$701$	−7.35114	−0.277649	−0.138824	−	0.990317i	$-0.544332\pi$
$701$	−7.35114	−0.277649	−0.138824	+	0.990317i	$0.544332\pi$
$702$	0	0
$703$	8.58502	0.323790
$704$	0	0
$705$	53.0592	1.99833
$706$	0	0
$707$	8.34685	0.313915
$708$	0	0
$709$	−8.54988	−0.321098	−0.160549	−	0.987028i	$-0.551326\pi$
$709$	−8.54988	−0.321098	−0.160549	+	0.987028i	$0.551326\pi$
$710$	0	0
$711$	−13.9629	−0.523650
$712$	0	0
$713$	10.6397	0.398459
$714$	0	0
$715$	−9.22753	−0.345090
$716$	0	0
$717$	−51.0135	−1.90513
$718$	0	0
$719$	−5.14366	−0.191826	−0.0959131	−	0.995390i	$-0.530577\pi$
$719$	−5.14366	−0.191826	−0.0959131	+	0.995390i	$0.530577\pi$
$720$	0	0
$721$	0.158392	0.00589884
$722$	0	0
$723$	−56.2917	−2.09351
$724$	0	0
$725$	−4.19767	−0.155898
$726$	0	0
$727$	6.09700	0.226125	0.113063	−	0.993588i	$-0.463934\pi$
$727$	6.09700	0.226125	0.113063	+	0.993588i	$0.463934\pi$
$728$	0	0
$729$	−40.5928	−1.50344
$730$	0	0
$731$	7.18965	0.265919
$732$	0	0
$733$	−28.4338	−1.05023	−0.525113	−	0.851032i	$-0.675977\pi$
$733$	−28.4338	−1.05023	−0.525113	+	0.851032i	$0.675977\pi$
$734$	0	0
$735$	51.5283	1.90065
$736$	0	0
$737$	−46.6113	−1.71695
$738$	0	0
$739$	−1.85076	−0.0680813	−0.0340407	−	0.999420i	$-0.510838\pi$
$739$	−1.85076	−0.0680813	−0.0340407	+	0.999420i	$0.510838\pi$
$740$	0	0
$741$	−14.6969	−0.539905
$742$	0	0
$743$	34.5082	1.26598	0.632991	−	0.774159i	$-0.281827\pi$
$743$	34.5082	1.26598	0.632991	+	0.774159i	$0.281827\pi$
$744$	0	0
$745$	34.8856	1.27811
$746$	0	0
$747$	9.44469	0.345563
$748$	0	0
$749$	−1.91003	−0.0697911
$750$	0	0
$751$	5.23046	0.190862	0.0954311	−	0.995436i	$-0.469577\pi$
$751$	5.23046	0.190862	0.0954311	+	0.995436i	$0.469577\pi$
$752$	0	0
$753$	−63.3771	−2.30959
$754$	0	0
$755$	69.5110	2.52976
$756$	0	0
$757$	0.0985625	0.00358231	0.00179116	−	0.999998i	$-0.499430\pi$
$757$	0.0985625	0.00358231	0.00179116	+	0.999998i	$0.499430\pi$
$758$	0	0
$759$	33.9994	1.23410
$760$	0	0
$761$	1.41265	0.0512087	0.0256043	−	0.999672i	$-0.491849\pi$
$761$	1.41265	0.0512087	0.0256043	+	0.999672i	$0.491849\pi$
$762$	0	0
$763$	−8.61596	−0.311919
$764$	0	0
$765$	61.1051	2.20926
$766$	0	0
$767$	13.2033	0.476744
$768$	0	0
$769$	−12.1346	−0.437584	−0.218792	−	0.975771i	$-0.570212\pi$
$769$	−12.1346	−0.437584	−0.218792	+	0.975771i	$0.570212\pi$
$770$	0	0
$771$	−0.919020	−0.0330977
$772$	0	0
$773$	48.2628	1.73589	0.867946	−	0.496658i	$-0.165440\pi$
$773$	48.2628	1.73589	0.867946	+	0.496658i	$0.165440\pi$
$774$	0	0
$775$	−10.5413	−0.378654
$776$	0	0
$777$	3.03505	0.108882
$778$	0	0
$779$	46.6425	1.67114
$780$	0	0
$781$	20.2509	0.724634
$782$	0	0
$783$	2.52149	0.0901107
$784$	0	0
$785$	−26.5039	−0.945964
$786$	0	0
$787$	−9.03493	−0.322061	−0.161030	−	0.986949i	$-0.551482\pi$
$787$	−9.03493	−0.322061	−0.161030	+	0.986949i	$0.551482\pi$
$788$	0	0
$789$	30.5241	1.08668
$790$	0	0
$791$	−8.14531	−0.289614
$792$	0	0
$793$	8.00146	0.284140
$794$	0	0
$795$	26.1605	0.927817
$796$	0	0
$797$	26.7464	0.947405	0.473703	−	0.880685i	$-0.342917\pi$
$797$	26.7464	0.947405	0.473703	+	0.880685i	$0.342917\pi$
$798$	0	0
$799$	−33.7846	−1.19521
$800$	0	0
$801$	−30.5389	−1.07904
$802$	0	0
$803$	24.6051	0.868296
$804$	0	0
$805$	−9.59780	−0.338278
$806$	0	0
$807$	45.7790	1.61150
$808$	0	0
$809$	29.7920	1.04743	0.523716	−	0.851893i	$-0.324545\pi$
$809$	29.7920	1.04743	0.523716	+	0.851893i	$0.324545\pi$
$810$	0	0
$811$	1.41711	0.0497614	0.0248807	−	0.999690i	$-0.492079\pi$
$811$	1.41711	0.0497614	0.0248807	+	0.999690i	$0.492079\pi$
$812$	0	0
$813$	−68.2562	−2.39385
$814$	0	0
$815$	−1.02716	−0.0359798
$816$	0	0
$817$	−7.86639	−0.275210
$818$	0	0
$819$	−2.95495	−0.103254
$820$	0	0
$821$	41.0679	1.43328	0.716639	−	0.697444i	$-0.245680\pi$
$821$	41.0679	1.43328	0.716639	+	0.697444i	$0.245680\pi$
$822$	0	0
$823$	3.39057	0.118188	0.0590940	−	0.998252i	$-0.481179\pi$
$823$	3.39057	0.118188	0.0590940	+	0.998252i	$0.481179\pi$
$824$	0	0
$825$	−33.6850	−1.17276
$826$	0	0
$827$	−3.04776	−0.105981	−0.0529905	−	0.998595i	$-0.516875\pi$
$827$	−3.04776	−0.105981	−0.0529905	+	0.998595i	$0.516875\pi$
$828$	0	0
$829$	−34.3760	−1.19393	−0.596963	−	0.802268i	$-0.703626\pi$
$829$	−34.3760	−1.19393	−0.596963	+	0.802268i	$0.703626\pi$
$830$	0	0
$831$	−24.2033	−0.839604
$832$	0	0
$833$	−32.8098	−1.13679
$834$	0	0
$835$	10.8332	0.374900
$836$	0	0
$837$	6.33201	0.218866
$838$	0	0
$839$	−1.35948	−0.0469345	−0.0234672	−	0.999725i	$-0.507471\pi$
$839$	−1.35948	−0.0469345	−0.0234672	+	0.999725i	$0.507471\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−10.2094	−0.351631
$844$	0	0
$845$	3.03277	0.104330
$846$	0	0
$847$	−1.30156	−0.0447222
$848$	0	0
$849$	−14.1385	−0.485232
$850$	0	0
$851$	6.52739	0.223756
$852$	0	0
$853$	15.7069	0.537795	0.268897	−	0.963169i	$-0.413341\pi$
$853$	15.7069	0.537795	0.268897	+	0.963169i	$0.413341\pi$
$854$	0	0
$855$	−66.8568	−2.28645
$856$	0	0
$857$	−5.37358	−0.183558	−0.0917789	−	0.995779i	$-0.529255\pi$
$857$	−5.37358	−0.183558	−0.0917789	+	0.995779i	$0.529255\pi$
$858$	0	0
$859$	36.4514	1.24371	0.621853	−	0.783134i	$-0.286380\pi$
$859$	36.4514	1.24371	0.621853	+	0.783134i	$0.286380\pi$
$860$	0	0
$861$	16.4895	0.561960
$862$	0	0
$863$	−8.31067	−0.282898	−0.141449	−	0.989946i	$-0.545176\pi$
$863$	−8.31067	−0.282898	−0.141449	+	0.989946i	$0.545176\pi$
$864$	0	0
$865$	−46.8477	−1.59287
$866$	0	0
$867$	−23.5764	−0.800697
$868$	0	0
$869$	−10.7390	−0.364294
$870$	0	0
$871$	15.3195	0.519081
$872$	0	0
$873$	46.2308	1.56467
$874$	0	0
$875$	−1.81752	−0.0614433
$876$	0	0
$877$	14.7312	0.497437	0.248719	−	0.968576i	$-0.419991\pi$
$877$	14.7312	0.497437	0.248719	+	0.968576i	$0.419991\pi$
$878$	0	0
$879$	−2.65593	−0.0895824
$880$	0	0
$881$	−20.9084	−0.704423	−0.352211	−	0.935920i	$-0.614570\pi$
$881$	−20.9084	−0.704423	−0.352211	+	0.935920i	$0.614570\pi$
$882$	0	0
$883$	−12.4683	−0.419592	−0.209796	−	0.977745i	$-0.567280\pi$
$883$	−12.4683	−0.419592	−0.209796	+	0.977745i	$0.567280\pi$
$884$	0	0
$885$	105.609	3.55002
$886$	0	0
$887$	19.8058	0.665013	0.332507	−	0.943101i	$-0.392106\pi$
$887$	19.8058	0.665013	0.332507	+	0.943101i	$0.392106\pi$
$888$	0	0
$889$	4.93549	0.165531
$890$	0	0
$891$	−15.8759	−0.531864
$892$	0	0
$893$	36.9647	1.23698
$894$	0	0
$895$	−49.5347	−1.65576
$896$	0	0
$897$	−11.1744	−0.373103
$898$	0	0
$899$	2.51122	0.0837538
$900$	0	0
$901$	−16.6573	−0.554934
$902$	0	0
$903$	−2.78099	−0.0925457
$904$	0	0
$905$	16.7988	0.558412
$906$	0	0
$907$	−22.7348	−0.754898	−0.377449	−	0.926030i	$-0.623199\pi$
$907$	−22.7348	−0.754898	−0.377449	+	0.926030i	$0.623199\pi$
$908$	0	0
$909$	44.2073	1.46626
$910$	0	0
$911$	−32.7465	−1.08494	−0.542470	−	0.840075i	$-0.682511\pi$
$911$	−32.7465	−1.08494	−0.542470	+	0.840075i	$0.682511\pi$
$912$	0	0
$913$	7.26396	0.240402
$914$	0	0
$915$	64.0014	2.11582
$916$	0	0
$917$	9.71523	0.320825
$918$	0	0
$919$	1.78798	0.0589801	0.0294900	−	0.999565i	$-0.490612\pi$
$919$	1.78798	0.0589801	0.0294900	+	0.999565i	$0.490612\pi$
$920$	0	0
$921$	7.48171	0.246531
$922$	0	0
$923$	−6.65576	−0.219077
$924$	0	0
$925$	−6.46703	−0.212634
$926$	0	0
$927$	0.838891	0.0275528
$928$	0	0
$929$	31.5403	1.03480	0.517401	−	0.855743i	$-0.326899\pi$
$929$	31.5403	1.03480	0.517401	+	0.855743i	$0.326899\pi$
$930$	0	0
$931$	35.8981	1.17651
$932$	0	0
$933$	45.1125	1.47692
$934$	0	0
$935$	46.9963	1.53694
$936$	0	0
$937$	−7.26444	−0.237319	−0.118659	−	0.992935i	$-0.537860\pi$
$937$	−7.26444	−0.237319	−0.118659	+	0.992935i	$0.537860\pi$
$938$	0	0
$939$	44.3301	1.44666
$940$	0	0
$941$	5.89209	0.192077	0.0960383	−	0.995378i	$-0.469383\pi$
$941$	5.89209	0.192077	0.0960383	+	0.995378i	$0.469383\pi$
$942$	0	0
$943$	35.4634	1.15485
$944$	0	0
$945$	−5.71196	−0.185810
$946$	0	0
$947$	−35.6354	−1.15800	−0.578998	−	0.815329i	$-0.696556\pi$
$947$	−35.6354	−1.15800	−0.578998	+	0.815329i	$0.696556\pi$
$948$	0	0
$949$	−8.08684	−0.262510
$950$	0	0
$951$	63.1124	2.04656
$952$	0	0
$953$	−48.9694	−1.58628	−0.793138	−	0.609042i	$-0.791554\pi$
$953$	−48.9694	−1.58628	−0.793138	+	0.609042i	$0.791554\pi$
$954$	0	0
$955$	−20.2599	−0.655595
$956$	0	0
$957$	8.02468	0.259401
$958$	0	0
$959$	−1.27030	−0.0410200
$960$	0	0
$961$	−24.6938	−0.796574
$962$	0	0
$963$	−10.1161	−0.325986
$964$	0	0
$965$	44.2289	1.42378
$966$	0	0
$967$	51.6621	1.66134	0.830671	−	0.556764i	$-0.187957\pi$
$967$	51.6621	1.66134	0.830671	+	0.556764i	$0.187957\pi$
$968$	0	0
$969$	74.8522	2.40460
$970$	0	0
$971$	−55.5423	−1.78244	−0.891219	−	0.453573i	$-0.850149\pi$
$971$	−55.5423	−1.78244	−0.891219	+	0.453573i	$0.850149\pi$
$972$	0	0
$973$	2.49328	0.0799308
$974$	0	0
$975$	11.0711	0.354558
$976$	0	0
$977$	45.2676	1.44824	0.724119	−	0.689675i	$-0.242247\pi$
$977$	45.2676	1.44824	0.724119	+	0.689675i	$0.242247\pi$
$978$	0	0
$979$	−23.4876	−0.750667
$980$	0	0
$981$	−45.6326	−1.45694
$982$	0	0
$983$	−56.5119	−1.80245	−0.901225	−	0.433351i	$-0.857331\pi$
$983$	−56.5119	−1.80245	−0.901225	+	0.433351i	$0.857331\pi$
$984$	0	0
$985$	−46.7899	−1.49085
$986$	0	0
$987$	13.0681	0.415961
$988$	0	0
$989$	−5.98099	−0.190185
$990$	0	0
$991$	22.3229	0.709110	0.354555	−	0.935035i	$-0.384632\pi$
$991$	22.3229	0.709110	0.354555	+	0.935035i	$0.384632\pi$
$992$	0	0
$993$	62.2014	1.97390
$994$	0	0
$995$	24.3915	0.773264
$996$	0	0
$997$	−23.3514	−0.739545	−0.369773	−	0.929122i	$-0.620564\pi$
$997$	−23.3514	−0.739545	−0.369773	+	0.929122i	$0.620564\pi$
$998$	0	0
$999$	3.88466	0.122905

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.z.1.2		10
4.3	odd	2		3016.2.a.h.1.9	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.h.1.9	✓	10	4.3	odd	2
6032.2.a.z.1.2		10	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:	\( N \)	\(=\)	\( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6032.a (trivial)

\(f(q)\)	\(=\)	\(q-2.63743 q^{3} +3.03277 q^{5} +0.746946 q^{7} +3.95604 q^{9} +O(q^{10})\)	\(q-2.63743 q^{3} +3.03277 q^{5} +0.746946 q^{7} +3.95604 q^{9} +3.04261 q^{11} -1.00000 q^{13} -7.99871 q^{15} +5.09305 q^{17} -5.57244 q^{19} -1.97002 q^{21} -4.23685 q^{23} +4.19767 q^{25} -2.52149 q^{27} -1.00000 q^{29} -2.51122 q^{31} -8.02468 q^{33} +2.26531 q^{35} -1.54062 q^{37} +2.63743 q^{39} -8.37021 q^{41} +1.41166 q^{43} +11.9977 q^{45} -6.63347 q^{47} -6.44207 q^{49} -13.4326 q^{51} -3.27059 q^{53} +9.22753 q^{55} +14.6969 q^{57} -13.2033 q^{59} -8.00146 q^{61} +2.95495 q^{63} -3.03277 q^{65} -15.3195 q^{67} +11.1744 q^{69} +6.65576 q^{71} +8.08684 q^{73} -11.0711 q^{75} +2.27267 q^{77} -3.52952 q^{79} -5.21787 q^{81} +2.38741 q^{83} +15.4460 q^{85} +2.63743 q^{87} -7.71955 q^{89} -0.746946 q^{91} +6.62316 q^{93} -16.8999 q^{95} +11.6861 q^{97} +12.0367 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 13 q^{9}+O(q^{10}) \) 10 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 13 * q^9	\( 10 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 13 q^{9} - 14 q^{11} - 10 q^{13} - 7 q^{15} + 5 q^{17} - 11 q^{19} - 7 q^{23} + 10 q^{25} - 21 q^{27} - 10 q^{29} - 5 q^{31} + 5 q^{33} - 11 q^{35} + 8 q^{37} + 3 q^{39} + 14 q^{41} - 35 q^{43} + 7 q^{45} - 7 q^{49} - 20 q^{51} - 11 q^{53} - 8 q^{55} + 4 q^{57} - 23 q^{59} - 8 q^{61} - 43 q^{63} - 4 q^{65} - 27 q^{67} + 10 q^{69} - 3 q^{71} + 7 q^{73} - 23 q^{75} + 2 q^{77} - 9 q^{79} - 6 q^{81} - 48 q^{83} - 6 q^{85} + 3 q^{87} + 20 q^{89} - 3 q^{91} - 11 q^{93} - 11 q^{95} + q^{97} - 54 q^{99}+O(q^{100}) \) 10 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 13 * q^9 - 14 * q^11 - 10 * q^13 - 7 * q^15 + 5 * q^17 - 11 * q^19 - 7 * q^23 + 10 * q^25 - 21 * q^27 - 10 * q^29 - 5 * q^31 + 5 * q^33 - 11 * q^35 + 8 * q^37 + 3 * q^39 + 14 * q^41 - 35 * q^43 + 7 * q^45 - 7 * q^49 - 20 * q^51 - 11 * q^53 - 8 * q^55 + 4 * q^57 - 23 * q^59 - 8 * q^61 - 43 * q^63 - 4 * q^65 - 27 * q^67 + 10 * q^69 - 3 * q^71 + 7 * q^73 - 23 * q^75 + 2 * q^77 - 9 * q^79 - 6 * q^81 - 48 * q^83 - 6 * q^85 + 3 * q^87 + 20 * q^89 - 3 * q^91 - 11 * q^93 - 11 * q^95 + q^97 - 54 * q^99

Introduction

L-functions

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