LMFDB - Embedded newform 6044.2.a.b.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6044,2,Mod(1,6044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6044, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6044 = 2^{2} \cdot 1511 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6044.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:	$48.2615829817$
Analytic rank:	$0$
Dimension:	$63$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	6044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.47185 q^{3} -3.39845 q^{5} +2.90753 q^{7} -0.833657 q^{9} +O(q^{10})$	$q-1.47185 q^{3} -3.39845 q^{5} +2.90753 q^{7} -0.833657 q^{9} -0.761837 q^{11} -4.45054 q^{13} +5.00201 q^{15} -7.63831 q^{17} -5.21134 q^{19} -4.27945 q^{21} -6.50102 q^{23} +6.54949 q^{25} +5.64257 q^{27} +1.47322 q^{29} -8.31067 q^{31} +1.12131 q^{33} -9.88111 q^{35} -0.635919 q^{37} +6.55053 q^{39} -4.80140 q^{41} +7.16927 q^{43} +2.83314 q^{45} -3.09705 q^{47} +1.45375 q^{49} +11.2425 q^{51} +2.65759 q^{53} +2.58907 q^{55} +7.67032 q^{57} -7.63988 q^{59} +4.38381 q^{61} -2.42388 q^{63} +15.1249 q^{65} -6.97663 q^{67} +9.56853 q^{69} -13.5680 q^{71} +11.1223 q^{73} -9.63986 q^{75} -2.21507 q^{77} +5.11617 q^{79} -5.80405 q^{81} -14.0685 q^{83} +25.9584 q^{85} -2.16836 q^{87} -13.8421 q^{89} -12.9401 q^{91} +12.2321 q^{93} +17.7105 q^{95} -8.18210 q^{97} +0.635111 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9}+O(q^{10}) $ 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9	$ 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9} + 21 q^{11} + 17 q^{13} + 26 q^{15} - 5 q^{17} + 57 q^{19} + 18 q^{21} + 16 q^{23} + 60 q^{25} + 14 q^{27} + 22 q^{29} + 36 q^{31} - q^{33} + 28 q^{35} + 21 q^{37} + 69 q^{39} - 3 q^{41} + 86 q^{43} + 39 q^{45} + 23 q^{47} + 63 q^{49} + 67 q^{51} + 18 q^{53} + 67 q^{55} + 27 q^{59} + 62 q^{61} + 58 q^{63} - 13 q^{65} + 62 q^{67} + 45 q^{69} + 29 q^{71} - q^{73} + 39 q^{75} + 19 q^{77} + 132 q^{79} + 51 q^{81} + 35 q^{83} + 60 q^{85} + 55 q^{87} - 2 q^{89} + 84 q^{91} + 29 q^{93} + 53 q^{95} - 10 q^{97} + 95 q^{99}+O(q^{100}) $ 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9 + 21 * q^11 + 17 * q^13 + 26 * q^15 - 5 * q^17 + 57 * q^19 + 18 * q^21 + 16 * q^23 + 60 * q^25 + 14 * q^27 + 22 * q^29 + 36 * q^31 - q^33 + 28 * q^35 + 21 * q^37 + 69 * q^39 - 3 * q^41 + 86 * q^43 + 39 * q^45 + 23 * q^47 + 63 * q^49 + 67 * q^51 + 18 * q^53 + 67 * q^55 + 27 * q^59 + 62 * q^61 + 58 * q^63 - 13 * q^65 + 62 * q^67 + 45 * q^69 + 29 * q^71 - q^73 + 39 * q^75 + 19 * q^77 + 132 * q^79 + 51 * q^81 + 35 * q^83 + 60 * q^85 + 55 * q^87 - 2 * q^89 + 84 * q^91 + 29 * q^93 + 53 * q^95 - 10 * q^97 + 95 * 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
$3$	−1.47185	−0.849773	−0.424887	−	0.905247i	$-0.639686\pi$
$3$	−1.47185	−0.849773	−0.424887	+	0.905247i	$0.639686\pi$
$4$	0	0
$5$	−3.39845	−1.51983	−0.759917	−	0.650020i	$-0.774761\pi$
$5$	−3.39845	−1.51983	−0.759917	+	0.650020i	$0.774761\pi$
$6$	0	0
$7$	2.90753	1.09894	0.549472	−	0.835512i	$-0.314829\pi$
$7$	2.90753	1.09894	0.549472	+	0.835512i	$0.314829\pi$
$8$	0	0
$9$	−0.833657	−0.277886
$10$	0	0
$11$	−0.761837	−0.229703	−0.114851	−	0.993383i	$-0.536639\pi$
$11$	−0.761837	−0.229703	−0.114851	+	0.993383i	$0.536639\pi$
$12$	0	0
$13$	−4.45054	−1.23436	−0.617179	−	0.786823i	$-0.711724\pi$
$13$	−4.45054	−1.23436	−0.617179	+	0.786823i	$0.711724\pi$
$14$	0	0
$15$	5.00201	1.29151
$16$	0	0
$17$	−7.63831	−1.85256	−0.926281	−	0.376833i	$-0.877013\pi$
$17$	−7.63831	−1.85256	−0.926281	+	0.376833i	$0.877013\pi$
$18$	0	0
$19$	−5.21134	−1.19556	−0.597782	−	0.801659i	$-0.703951\pi$
$19$	−5.21134	−1.19556	−0.597782	+	0.801659i	$0.703951\pi$
$20$	0	0
$21$	−4.27945	−0.933853
$22$	0	0
$23$	−6.50102	−1.35556	−0.677778	−	0.735267i	$-0.737057\pi$
$23$	−6.50102	−1.35556	−0.677778	+	0.735267i	$0.737057\pi$
$24$	0	0
$25$	6.54949	1.30990
$26$	0	0
$27$	5.64257	1.08591
$28$	0	0
$29$	1.47322	0.273570	0.136785	−	0.990601i	$-0.456323\pi$
$29$	1.47322	0.273570	0.136785	+	0.990601i	$0.456323\pi$
$30$	0	0
$31$	−8.31067	−1.49264	−0.746320	−	0.665587i	$-0.768181\pi$
$31$	−8.31067	−1.49264	−0.746320	+	0.665587i	$0.768181\pi$
$32$	0	0
$33$	1.12131	0.195195
$34$	0	0
$35$	−9.88111	−1.67021
$36$	0	0
$37$	−0.635919	−0.104544	−0.0522722	−	0.998633i	$-0.516646\pi$
$37$	−0.635919	−0.104544	−0.0522722	+	0.998633i	$0.516646\pi$
$38$	0	0
$39$	6.55053	1.04892
$40$	0	0
$41$	−4.80140	−0.749853	−0.374927	−	0.927054i	$-0.622332\pi$
$41$	−4.80140	−0.749853	−0.374927	+	0.927054i	$0.622332\pi$
$42$	0	0
$43$	7.16927	1.09330	0.546651	−	0.837360i	$-0.315902\pi$
$43$	7.16927	1.09330	0.546651	+	0.837360i	$0.315902\pi$
$44$	0	0
$45$	2.83314	0.422340
$46$	0	0
$47$	−3.09705	−0.451751	−0.225876	−	0.974156i	$-0.572524\pi$
$47$	−3.09705	−0.451751	−0.225876	+	0.974156i	$0.572524\pi$
$48$	0	0
$49$	1.45375	0.207678
$50$	0	0
$51$	11.2425	1.57426
$52$	0	0
$53$	2.65759	0.365048	0.182524	−	0.983201i	$-0.441573\pi$
$53$	2.65759	0.365048	0.182524	+	0.983201i	$0.441573\pi$
$54$	0	0
$55$	2.58907	0.349110
$56$	0	0
$57$	7.67032	1.01596
$58$	0	0
$59$	−7.63988	−0.994628	−0.497314	−	0.867571i	$-0.665680\pi$
$59$	−7.63988	−0.994628	−0.497314	+	0.867571i	$0.665680\pi$
$60$	0	0
$61$	4.38381	0.561289	0.280644	−	0.959812i	$-0.409452\pi$
$61$	4.38381	0.561289	0.280644	+	0.959812i	$0.409452\pi$
$62$	0	0
$63$	−2.42388	−0.305381
$64$	0	0
$65$	15.1249	1.87602
$66$	0	0
$67$	−6.97663	−0.852331	−0.426165	−	0.904645i	$-0.640136\pi$
$67$	−6.97663	−0.852331	−0.426165	+	0.904645i	$0.640136\pi$
$68$	0	0
$69$	9.56853	1.15192
$70$	0	0
$71$	−13.5680	−1.61022	−0.805112	−	0.593123i	$-0.797895\pi$
$71$	−13.5680	−1.61022	−0.805112	+	0.593123i	$0.797895\pi$
$72$	0	0
$73$	11.1223	1.30177	0.650884	−	0.759177i	$-0.274399\pi$
$73$	11.1223	1.30177	0.650884	+	0.759177i	$0.274399\pi$
$74$	0	0
$75$	−9.63986	−1.11312
$76$	0	0
$77$	−2.21507	−0.252430
$78$	0	0
$79$	5.11617	0.575614	0.287807	−	0.957688i	$-0.407074\pi$
$79$	5.11617	0.575614	0.287807	+	0.957688i	$0.407074\pi$
$80$	0	0
$81$	−5.80405	−0.644894
$82$	0	0
$83$	−14.0685	−1.54422	−0.772108	−	0.635491i	$-0.780798\pi$
$83$	−14.0685	−1.54422	−0.772108	+	0.635491i	$0.780798\pi$
$84$	0	0
$85$	25.9584	2.81559
$86$	0	0
$87$	−2.16836	−0.232472
$88$	0	0
$89$	−13.8421	−1.46725	−0.733627	−	0.679552i	$-0.762174\pi$
$89$	−13.8421	−1.46725	−0.733627	+	0.679552i	$0.762174\pi$
$90$	0	0
$91$	−12.9401	−1.35649
$92$	0	0
$93$	12.2321	1.26841
$94$	0	0
$95$	17.7105	1.81706
$96$	0	0
$97$	−8.18210	−0.830767	−0.415383	−	0.909646i	$-0.636353\pi$
$97$	−8.18210	−0.830767	−0.415383	+	0.909646i	$0.636353\pi$
$98$	0	0
$99$	0.635111	0.0638310
$100$	0	0
$101$	−9.14234	−0.909697	−0.454848	−	0.890569i	$-0.650307\pi$
$101$	−9.14234	−0.909697	−0.454848	+	0.890569i	$0.650307\pi$
$102$	0	0
$103$	−18.4788	−1.82077	−0.910387	−	0.413757i	$-0.864216\pi$
$103$	−18.4788	−1.82077	−0.910387	+	0.413757i	$0.864216\pi$
$104$	0	0
$105$	14.5435	1.41930
$106$	0	0
$107$	0.977964	0.0945433	0.0472717	−	0.998882i	$-0.484947\pi$
$107$	0.977964	0.0945433	0.0472717	+	0.998882i	$0.484947\pi$
$108$	0	0
$109$	10.6108	1.01633	0.508167	−	0.861259i	$-0.330323\pi$
$109$	10.6108	1.01633	0.508167	+	0.861259i	$0.330323\pi$
$110$	0	0
$111$	0.935978	0.0888391
$112$	0	0
$113$	14.5239	1.36629	0.683146	−	0.730282i	$-0.260611\pi$
$113$	14.5239	1.36629	0.683146	+	0.730282i	$0.260611\pi$
$114$	0	0
$115$	22.0934	2.06022
$116$	0	0
$117$	3.71022	0.343010
$118$	0	0
$119$	−22.2086	−2.03586
$120$	0	0
$121$	−10.4196	−0.947237
$122$	0	0
$123$	7.06695	0.637205
$124$	0	0
$125$	−5.26585	−0.470992
$126$	0	0
$127$	−20.7043	−1.83721	−0.918604	−	0.395180i	$-0.870682\pi$
$127$	−20.7043	−1.83721	−0.918604	+	0.395180i	$0.870682\pi$
$128$	0	0
$129$	−10.5521	−0.929059
$130$	0	0
$131$	14.8048	1.29351	0.646753	−	0.762700i	$-0.276127\pi$
$131$	14.8048	1.29351	0.646753	+	0.762700i	$0.276127\pi$
$132$	0	0
$133$	−15.1522	−1.31386
$134$	0	0
$135$	−19.1760	−1.65041
$136$	0	0
$137$	−9.67125	−0.826270	−0.413135	−	0.910670i	$-0.635566\pi$
$137$	−9.67125	−0.826270	−0.413135	+	0.910670i	$0.635566\pi$
$138$	0	0
$139$	−0.500225	−0.0424285	−0.0212143	−	0.999775i	$-0.506753\pi$
$139$	−0.500225	−0.0424285	−0.0212143	+	0.999775i	$0.506753\pi$
$140$	0	0
$141$	4.55840	0.383886
$142$	0	0
$143$	3.39059	0.283535
$144$	0	0
$145$	−5.00666	−0.415781
$146$	0	0
$147$	−2.13970	−0.176479
$148$	0	0
$149$	−10.9700	−0.898697	−0.449349	−	0.893356i	$-0.648344\pi$
$149$	−10.9700	−0.898697	−0.449349	+	0.893356i	$0.648344\pi$
$150$	0	0
$151$	22.3426	1.81821	0.909106	−	0.416564i	$-0.136766\pi$
$151$	22.3426	1.81821	0.909106	+	0.416564i	$0.136766\pi$
$152$	0	0
$153$	6.36773	0.514800
$154$	0	0
$155$	28.2434	2.26857
$156$	0	0
$157$	−5.12354	−0.408903	−0.204451	−	0.978877i	$-0.565541\pi$
$157$	−5.12354	−0.408903	−0.204451	+	0.978877i	$0.565541\pi$
$158$	0	0
$159$	−3.91158	−0.310208
$160$	0	0
$161$	−18.9019	−1.48968
$162$	0	0
$163$	22.7020	1.77816	0.889080	−	0.457752i	$-0.151345\pi$
$163$	22.7020	1.77816	0.889080	+	0.457752i	$0.151345\pi$
$164$	0	0
$165$	−3.81072	−0.296664
$166$	0	0
$167$	22.8434	1.76767	0.883837	−	0.467794i	$-0.154951\pi$
$167$	22.8434	1.76767	0.883837	+	0.467794i	$0.154951\pi$
$168$	0	0
$169$	6.80730	0.523638
$170$	0	0
$171$	4.34447	0.332230
$172$	0	0
$173$	14.8491	1.12895	0.564477	−	0.825449i	$-0.309078\pi$
$173$	14.8491	1.12895	0.564477	+	0.825449i	$0.309078\pi$
$174$	0	0
$175$	19.0428	1.43950
$176$	0	0
$177$	11.2448	0.845208
$178$	0	0
$179$	−15.2520	−1.13999	−0.569995	−	0.821648i	$-0.693055\pi$
$179$	−15.2520	−1.13999	−0.569995	+	0.821648i	$0.693055\pi$
$180$	0	0
$181$	−2.04497	−0.152001	−0.0760006	−	0.997108i	$-0.524215\pi$
$181$	−2.04497	−0.152001	−0.0760006	+	0.997108i	$0.524215\pi$
$182$	0	0
$183$	−6.45231	−0.476968
$184$	0	0
$185$	2.16114	0.158890
$186$	0	0
$187$	5.81915	0.425538
$188$	0	0
$189$	16.4060	1.19336
$190$	0	0
$191$	−21.8717	−1.58258	−0.791291	−	0.611440i	$-0.790591\pi$
$191$	−21.8717	−1.58258	−0.791291	+	0.611440i	$0.790591\pi$
$192$	0	0
$193$	−26.1538	−1.88259	−0.941295	−	0.337585i	$-0.890390\pi$
$193$	−26.1538	−1.88259	−0.941295	+	0.337585i	$0.890390\pi$
$194$	0	0
$195$	−22.2617	−1.59419
$196$	0	0
$197$	−6.84379	−0.487600	−0.243800	−	0.969826i	$-0.578394\pi$
$197$	−6.84379	−0.487600	−0.243800	+	0.969826i	$0.578394\pi$
$198$	0	0
$199$	7.65931	0.542954	0.271477	−	0.962445i	$-0.412488\pi$
$199$	7.65931	0.542954	0.271477	+	0.962445i	$0.412488\pi$
$200$	0	0
$201$	10.2685	0.724288
$202$	0	0
$203$	4.28343	0.300638
$204$	0	0
$205$	16.3173	1.13965
$206$	0	0
$207$	5.41962	0.376689
$208$	0	0
$209$	3.97020	0.274624
$210$	0	0
$211$	13.7250	0.944870	0.472435	−	0.881365i	$-0.343375\pi$
$211$	13.7250	0.944870	0.472435	+	0.881365i	$0.343375\pi$
$212$	0	0
$213$	19.9701	1.36833
$214$	0	0
$215$	−24.3644	−1.66164
$216$	0	0
$217$	−24.1635	−1.64033
$218$	0	0
$219$	−16.3704	−1.10621
$220$	0	0
$221$	33.9946	2.28672
$222$	0	0
$223$	−1.79789	−0.120396	−0.0601978	−	0.998186i	$-0.519173\pi$
$223$	−1.79789	−0.120396	−0.0601978	+	0.998186i	$0.519173\pi$
$224$	0	0
$225$	−5.46002	−0.364001
$226$	0	0
$227$	−29.8765	−1.98297	−0.991487	−	0.130203i	$-0.958437\pi$
$227$	−29.8765	−1.98297	−0.991487	+	0.130203i	$0.958437\pi$
$228$	0	0
$229$	−23.9401	−1.58200	−0.791002	−	0.611814i	$-0.790440\pi$
$229$	−23.9401	−1.58200	−0.791002	+	0.611814i	$0.790440\pi$
$230$	0	0
$231$	3.26025	0.214509
$232$	0	0
$233$	−3.09510	−0.202767	−0.101383	−	0.994847i	$-0.532327\pi$
$233$	−3.09510	−0.202767	−0.101383	+	0.994847i	$0.532327\pi$
$234$	0	0
$235$	10.5252	0.686588
$236$	0	0
$237$	−7.53024	−0.489142
$238$	0	0
$239$	2.53699	0.164104	0.0820521	−	0.996628i	$-0.473853\pi$
$239$	2.53699	0.164104	0.0820521	+	0.996628i	$0.473853\pi$
$240$	0	0
$241$	2.61486	0.168438	0.0842189	−	0.996447i	$-0.473161\pi$
$241$	2.61486	0.168438	0.0842189	+	0.996447i	$0.473161\pi$
$242$	0	0
$243$	−8.38502	−0.537899
$244$	0	0
$245$	−4.94049	−0.315636
$246$	0	0
$247$	23.1933	1.47575
$248$	0	0
$249$	20.7067	1.31223
$250$	0	0
$251$	22.0734	1.39326	0.696630	−	0.717431i	$-0.254682\pi$
$251$	22.0734	1.39326	0.696630	+	0.717431i	$0.254682\pi$
$252$	0	0
$253$	4.95272	0.311375
$254$	0	0
$255$	−38.2069	−2.39261
$256$	0	0
$257$	11.8356	0.738282	0.369141	−	0.929373i	$-0.379652\pi$
$257$	11.8356	0.738282	0.369141	+	0.929373i	$0.379652\pi$
$258$	0	0
$259$	−1.84896	−0.114889
$260$	0	0
$261$	−1.22816	−0.0760210
$262$	0	0
$263$	20.3820	1.25681	0.628403	−	0.777888i	$-0.283709\pi$
$263$	20.3820	1.25681	0.628403	+	0.777888i	$0.283709\pi$
$264$	0	0
$265$	−9.03170	−0.554813
$266$	0	0
$267$	20.3734	1.24683
$268$	0	0
$269$	31.4033	1.91469	0.957346	−	0.288945i	$-0.0933044\pi$
$269$	31.4033	1.91469	0.957346	+	0.288945i	$0.0933044\pi$
$270$	0	0
$271$	−6.34768	−0.385594	−0.192797	−	0.981239i	$-0.561756\pi$
$271$	−6.34768	−0.385594	−0.192797	+	0.981239i	$0.561756\pi$
$272$	0	0
$273$	19.0459	1.15271
$274$	0	0
$275$	−4.98964	−0.300887
$276$	0	0
$277$	30.7314	1.84647	0.923236	−	0.384232i	$-0.125534\pi$
$277$	30.7314	1.84647	0.923236	+	0.384232i	$0.125534\pi$
$278$	0	0
$279$	6.92825	0.414783
$280$	0	0
$281$	−7.35477	−0.438749	−0.219374	−	0.975641i	$-0.570402\pi$
$281$	−7.35477	−0.438749	−0.219374	+	0.975641i	$0.570402\pi$
$282$	0	0
$283$	3.95901	0.235339	0.117669	−	0.993053i	$-0.462458\pi$
$283$	3.95901	0.235339	0.117669	+	0.993053i	$0.462458\pi$
$284$	0	0
$285$	−26.0672	−1.54409
$286$	0	0
$287$	−13.9602	−0.824047
$288$	0	0
$289$	41.3438	2.43199
$290$	0	0
$291$	12.0428	0.705963
$292$	0	0
$293$	−0.585440	−0.0342018	−0.0171009	−	0.999854i	$-0.505444\pi$
$293$	−0.585440	−0.0342018	−0.0171009	+	0.999854i	$0.505444\pi$
$294$	0	0
$295$	25.9638	1.51167
$296$	0	0
$297$	−4.29872	−0.249437
$298$	0	0
$299$	28.9330	1.67324
$300$	0	0
$301$	20.8449	1.20148
$302$	0	0
$303$	13.4562	0.773036
$304$	0	0
$305$	−14.8982	−0.853066
$306$	0	0
$307$	−34.0202	−1.94164	−0.970818	−	0.239817i	$-0.922913\pi$
$307$	−34.0202	−1.94164	−0.970818	+	0.239817i	$0.922913\pi$
$308$	0	0
$309$	27.1981	1.54725
$310$	0	0
$311$	19.1516	1.08599	0.542994	−	0.839737i	$-0.317291\pi$
$311$	19.1516	1.08599	0.542994	+	0.839737i	$0.317291\pi$
$312$	0	0
$313$	−6.51824	−0.368433	−0.184216	−	0.982886i	$-0.558975\pi$
$313$	−6.51824	−0.368433	−0.184216	+	0.982886i	$0.558975\pi$
$314$	0	0
$315$	8.23746	0.464128
$316$	0	0
$317$	−17.0643	−0.958429	−0.479215	−	0.877698i	$-0.659078\pi$
$317$	−17.0643	−0.958429	−0.479215	+	0.877698i	$0.659078\pi$
$318$	0	0
$319$	−1.12235	−0.0628397
$320$	0	0
$321$	−1.43942	−0.0803404
$322$	0	0
$323$	39.8059	2.21486
$324$	0	0
$325$	−29.1487	−1.61688
$326$	0	0
$327$	−15.6176	−0.863653
$328$	0	0
$329$	−9.00478	−0.496450
$330$	0	0
$331$	24.2449	1.33262	0.666311	−	0.745674i	$-0.267873\pi$
$331$	24.2449	1.33262	0.666311	+	0.745674i	$0.267873\pi$
$332$	0	0
$333$	0.530138	0.0290514
$334$	0	0
$335$	23.7097	1.29540
$336$	0	0
$337$	16.3813	0.892349	0.446174	−	0.894946i	$-0.352786\pi$
$337$	16.3813	0.892349	0.446174	+	0.894946i	$0.352786\pi$
$338$	0	0
$339$	−21.3770	−1.16104
$340$	0	0
$341$	6.33138	0.342863
$342$	0	0
$343$	−16.1259	−0.870718
$344$	0	0
$345$	−32.5182	−1.75072
$346$	0	0
$347$	7.76351	0.416767	0.208384	−	0.978047i	$-0.433180\pi$
$347$	7.76351	0.416767	0.208384	+	0.978047i	$0.433180\pi$
$348$	0	0
$349$	1.76542	0.0945009	0.0472505	−	0.998883i	$-0.484954\pi$
$349$	1.76542	0.0945009	0.0472505	+	0.998883i	$0.484954\pi$
$350$	0	0
$351$	−25.1125	−1.34040
$352$	0	0
$353$	16.9062	0.899827	0.449914	−	0.893072i	$-0.351455\pi$
$353$	16.9062	0.899827	0.449914	+	0.893072i	$0.351455\pi$
$354$	0	0
$355$	46.1102	2.44727
$356$	0	0
$357$	32.6878	1.73002
$358$	0	0
$359$	7.39413	0.390247	0.195124	−	0.980779i	$-0.437489\pi$
$359$	7.39413	0.390247	0.195124	+	0.980779i	$0.437489\pi$
$360$	0	0
$361$	8.15811	0.429374
$362$	0	0
$363$	15.3361	0.804936
$364$	0	0
$365$	−37.7986	−1.97847
$366$	0	0
$367$	13.3070	0.694618	0.347309	−	0.937751i	$-0.387096\pi$
$367$	13.3070	0.694618	0.347309	+	0.937751i	$0.387096\pi$
$368$	0	0
$369$	4.00272	0.208373
$370$	0	0
$371$	7.72703	0.401168
$372$	0	0
$373$	4.51132	0.233587	0.116794	−	0.993156i	$-0.462738\pi$
$373$	4.51132	0.233587	0.116794	+	0.993156i	$0.462738\pi$
$374$	0	0
$375$	7.75055	0.400237
$376$	0	0
$377$	−6.55661	−0.337683
$378$	0	0
$379$	16.1107	0.827550	0.413775	−	0.910379i	$-0.364210\pi$
$379$	16.1107	0.827550	0.413775	+	0.910379i	$0.364210\pi$
$380$	0	0
$381$	30.4736	1.56121
$382$	0	0
$383$	−6.49960	−0.332114	−0.166057	−	0.986116i	$-0.553104\pi$
$383$	−6.49960	−0.332114	−0.166057	+	0.986116i	$0.553104\pi$
$384$	0	0
$385$	7.52780	0.383652
$386$	0	0
$387$	−5.97671	−0.303813
$388$	0	0
$389$	−3.42329	−0.173568	−0.0867840	−	0.996227i	$-0.527659\pi$
$389$	−3.42329	−0.173568	−0.0867840	+	0.996227i	$0.527659\pi$
$390$	0	0
$391$	49.6568	2.51125
$392$	0	0
$393$	−21.7905	−1.09919
$394$	0	0
$395$	−17.3871	−0.874839
$396$	0	0
$397$	−8.63350	−0.433303	−0.216651	−	0.976249i	$-0.569514\pi$
$397$	−8.63350	−0.433303	−0.216651	+	0.976249i	$0.569514\pi$
$398$	0	0
$399$	22.3017	1.11648
$400$	0	0
$401$	17.5006	0.873936	0.436968	−	0.899477i	$-0.356052\pi$
$401$	17.5006	0.873936	0.436968	+	0.899477i	$0.356052\pi$
$402$	0	0
$403$	36.9870	1.84245
$404$	0	0
$405$	19.7248	0.980132
$406$	0	0
$407$	0.484467	0.0240141
$408$	0	0
$409$	−3.67261	−0.181599	−0.0907993	−	0.995869i	$-0.528942\pi$
$409$	−3.67261	−0.181599	−0.0907993	+	0.995869i	$0.528942\pi$
$410$	0	0
$411$	14.2346	0.702143
$412$	0	0
$413$	−22.2132	−1.09304
$414$	0	0
$415$	47.8111	2.34695
$416$	0	0
$417$	0.736256	0.0360546
$418$	0	0
$419$	−12.7076	−0.620808	−0.310404	−	0.950605i	$-0.600464\pi$
$419$	−12.7076	−0.620808	−0.310404	+	0.950605i	$0.600464\pi$
$420$	0	0
$421$	−14.0878	−0.686597	−0.343298	−	0.939226i	$-0.611544\pi$
$421$	−14.0878	−0.686597	−0.343298	+	0.939226i	$0.611544\pi$
$422$	0	0
$423$	2.58188	0.125535
$424$	0	0
$425$	−50.0270	−2.42667
$426$	0	0
$427$	12.7461	0.616825
$428$	0	0
$429$	−4.99044	−0.240941
$430$	0	0
$431$	−30.2719	−1.45815	−0.729073	−	0.684436i	$-0.760049\pi$
$431$	−30.2719	−1.45815	−0.729073	+	0.684436i	$0.760049\pi$
$432$	0	0
$433$	−10.4177	−0.500645	−0.250322	−	0.968163i	$-0.580537\pi$
$433$	−10.4177	−0.500645	−0.250322	+	0.968163i	$0.580537\pi$
$434$	0	0
$435$	7.36906	0.353319
$436$	0	0
$437$	33.8790	1.62065
$438$	0	0
$439$	−34.7160	−1.65690	−0.828451	−	0.560062i	$-0.810777\pi$
$439$	−34.7160	−1.65690	−0.828451	+	0.560062i	$0.810777\pi$
$440$	0	0
$441$	−1.21192	−0.0577107
$442$	0	0
$443$	−21.5784	−1.02522	−0.512610	−	0.858621i	$-0.671321\pi$
$443$	−21.5784	−1.02522	−0.512610	+	0.858621i	$0.671321\pi$
$444$	0	0
$445$	47.0416	2.22998
$446$	0	0
$447$	16.1462	0.763689
$448$	0	0
$449$	−16.9342	−0.799176	−0.399588	−	0.916695i	$-0.630847\pi$
$449$	−16.9342	−0.799176	−0.399588	+	0.916695i	$0.630847\pi$
$450$	0	0
$451$	3.65789	0.172243
$452$	0	0
$453$	−32.8849	−1.54507
$454$	0	0
$455$	43.9763	2.06164
$456$	0	0
$457$	11.7666	0.550418	0.275209	−	0.961384i	$-0.411253\pi$
$457$	11.7666	0.550418	0.275209	+	0.961384i	$0.411253\pi$
$458$	0	0
$459$	−43.0997	−2.01172
$460$	0	0
$461$	−6.89670	−0.321211	−0.160606	−	0.987019i	$-0.551345\pi$
$461$	−6.89670	−0.321211	−0.160606	+	0.987019i	$0.551345\pi$
$462$	0	0
$463$	−7.05597	−0.327919	−0.163959	−	0.986467i	$-0.552427\pi$
$463$	−7.05597	−0.327919	−0.163959	+	0.986467i	$0.552427\pi$
$464$	0	0
$465$	−41.5701	−1.92777
$466$	0	0
$467$	4.48057	0.207336	0.103668	−	0.994612i	$-0.466942\pi$
$467$	4.48057	0.207336	0.103668	+	0.994612i	$0.466942\pi$
$468$	0	0
$469$	−20.2848	−0.936664
$470$	0	0
$471$	7.54108	0.347475
$472$	0	0
$473$	−5.46181	−0.251134
$474$	0	0
$475$	−34.1316	−1.56607
$476$	0	0
$477$	−2.21552	−0.101442
$478$	0	0
$479$	−26.1598	−1.19527	−0.597636	−	0.801768i	$-0.703893\pi$
$479$	−26.1598	−1.19527	−0.597636	+	0.801768i	$0.703893\pi$
$480$	0	0
$481$	2.83018	0.129045
$482$	0	0
$483$	27.8208	1.26589
$484$	0	0
$485$	27.8065	1.26263
$486$	0	0
$487$	17.6280	0.798803	0.399401	−	0.916776i	$-0.369218\pi$
$487$	17.6280	0.798803	0.399401	+	0.916776i	$0.369218\pi$
$488$	0	0
$489$	−33.4140	−1.51103
$490$	0	0
$491$	8.32219	0.375575	0.187788	−	0.982210i	$-0.439868\pi$
$491$	8.32219	0.375575	0.187788	+	0.982210i	$0.439868\pi$
$492$	0	0
$493$	−11.2529	−0.506805
$494$	0	0
$495$	−2.15839	−0.0970126
$496$	0	0
$497$	−39.4494	−1.76955
$498$	0	0
$499$	−14.7968	−0.662394	−0.331197	−	0.943562i	$-0.607452\pi$
$499$	−14.7968	−0.662394	−0.331197	+	0.943562i	$0.607452\pi$
$500$	0	0
$501$	−33.6221	−1.50212
$502$	0	0
$503$	14.7114	0.655949	0.327975	−	0.944687i	$-0.393634\pi$
$503$	14.7114	0.655949	0.327975	+	0.944687i	$0.393634\pi$
$504$	0	0
$505$	31.0698	1.38259
$506$	0	0
$507$	−10.0193	−0.444974
$508$	0	0
$509$	−32.9526	−1.46060	−0.730299	−	0.683128i	$-0.760619\pi$
$509$	−32.9526	−1.46060	−0.730299	+	0.683128i	$0.760619\pi$
$510$	0	0
$511$	32.3385	1.43057
$512$	0	0
$513$	−29.4054	−1.29828
$514$	0	0
$515$	62.7995	2.76728
$516$	0	0
$517$	2.35945	0.103768
$518$	0	0
$519$	−21.8556	−0.959354
$520$	0	0
$521$	39.3274	1.72297	0.861483	−	0.507787i	$-0.169536\pi$
$521$	39.3274	1.72297	0.861483	+	0.507787i	$0.169536\pi$
$522$	0	0
$523$	−15.7793	−0.689979	−0.344989	−	0.938607i	$-0.612117\pi$
$523$	−15.7793	−0.689979	−0.344989	+	0.938607i	$0.612117\pi$
$524$	0	0
$525$	−28.0282	−1.22325
$526$	0	0
$527$	63.4795	2.76521
$528$	0	0
$529$	19.2632	0.837532
$530$	0	0
$531$	6.36904	0.276393
$532$	0	0
$533$	21.3688	0.925587
$534$	0	0
$535$	−3.32356	−0.143690
$536$	0	0
$537$	22.4487	0.968733
$538$	0	0
$539$	−1.10752	−0.0477042
$540$	0	0
$541$	−24.2471	−1.04246	−0.521232	−	0.853415i	$-0.674527\pi$
$541$	−24.2471	−1.04246	−0.521232	+	0.853415i	$0.674527\pi$
$542$	0	0
$543$	3.00988	0.129166
$544$	0	0
$545$	−36.0604	−1.54466
$546$	0	0
$547$	6.17208	0.263899	0.131949	−	0.991256i	$-0.457876\pi$
$547$	6.17208	0.263899	0.131949	+	0.991256i	$0.457876\pi$
$548$	0	0
$549$	−3.65459	−0.155974
$550$	0	0
$551$	−7.67744	−0.327070
$552$	0	0
$553$	14.8754	0.632568
$554$	0	0
$555$	−3.18088	−0.135021
$556$	0	0
$557$	−0.612235	−0.0259412	−0.0129706	−	0.999916i	$-0.504129\pi$
$557$	−0.612235	−0.0259412	−0.0129706	+	0.999916i	$0.504129\pi$
$558$	0	0
$559$	−31.9071	−1.34953
$560$	0	0
$561$	−8.56492	−0.361611
$562$	0	0
$563$	32.7154	1.37879	0.689396	−	0.724385i	$-0.257876\pi$
$563$	32.7154	1.37879	0.689396	+	0.724385i	$0.257876\pi$
$564$	0	0
$565$	−49.3588	−2.07654
$566$	0	0
$567$	−16.8755	−0.708702
$568$	0	0
$569$	−7.36364	−0.308700	−0.154350	−	0.988016i	$-0.549328\pi$
$569$	−7.36364	−0.308700	−0.154350	+	0.988016i	$0.549328\pi$
$570$	0	0
$571$	24.5447	1.02716	0.513581	−	0.858041i	$-0.328319\pi$
$571$	24.5447	1.02716	0.513581	+	0.858041i	$0.328319\pi$
$572$	0	0
$573$	32.1919	1.34484
$574$	0	0
$575$	−42.5783	−1.77564
$576$	0	0
$577$	−19.9439	−0.830275	−0.415138	−	0.909759i	$-0.636267\pi$
$577$	−19.9439	−0.830275	−0.415138	+	0.909759i	$0.636267\pi$
$578$	0	0
$579$	38.4944	1.59977
$580$	0	0
$581$	−40.9046	−1.69701
$582$	0	0
$583$	−2.02465	−0.0838525
$584$	0	0
$585$	−12.6090	−0.521319
$586$	0	0
$587$	16.0650	0.663074	0.331537	−	0.943442i	$-0.392433\pi$
$587$	16.0650	0.663074	0.331537	+	0.943442i	$0.392433\pi$
$588$	0	0
$589$	43.3098	1.78455
$590$	0	0
$591$	10.0730	0.414349
$592$	0	0
$593$	−40.4436	−1.66082	−0.830409	−	0.557154i	$-0.811893\pi$
$593$	−40.4436	−1.66082	−0.830409	+	0.557154i	$0.811893\pi$
$594$	0	0
$595$	75.4750	3.09417
$596$	0	0
$597$	−11.2734	−0.461388
$598$	0	0
$599$	−14.3194	−0.585075	−0.292537	−	0.956254i	$-0.594500\pi$
$599$	−14.3194	−0.585075	−0.292537	+	0.956254i	$0.594500\pi$
$600$	0	0
$601$	9.22520	0.376304	0.188152	−	0.982140i	$-0.439750\pi$
$601$	9.22520	0.376304	0.188152	+	0.982140i	$0.439750\pi$
$602$	0	0
$603$	5.81611	0.236850
$604$	0	0
$605$	35.4105	1.43964
$606$	0	0
$607$	−39.6381	−1.60886	−0.804430	−	0.594048i	$-0.797529\pi$
$607$	−39.6381	−1.60886	−0.804430	+	0.594048i	$0.797529\pi$
$608$	0	0
$609$	−6.30456	−0.255474
$610$	0	0
$611$	13.7836	0.557623
$612$	0	0
$613$	−10.7193	−0.432947	−0.216474	−	0.976288i	$-0.569456\pi$
$613$	−10.7193	−0.432947	−0.216474	+	0.976288i	$0.569456\pi$
$614$	0	0
$615$	−24.0167	−0.968446
$616$	0	0
$617$	−18.3343	−0.738112	−0.369056	−	0.929407i	$-0.620319\pi$
$617$	−18.3343	−0.738112	−0.369056	+	0.929407i	$0.620319\pi$
$618$	0	0
$619$	35.3413	1.42049	0.710244	−	0.703956i	$-0.248585\pi$
$619$	35.3413	1.42049	0.710244	+	0.703956i	$0.248585\pi$
$620$	0	0
$621$	−36.6824	−1.47202
$622$	0	0
$623$	−40.2462	−1.61243
$624$	0	0
$625$	−14.8517	−0.594067
$626$	0	0
$627$	−5.84354	−0.233368
$628$	0	0
$629$	4.85735	0.193675
$630$	0	0
$631$	22.2313	0.885013	0.442506	−	0.896765i	$-0.354089\pi$
$631$	22.2313	0.885013	0.442506	+	0.896765i	$0.354089\pi$
$632$	0	0
$633$	−20.2012	−0.802925
$634$	0	0
$635$	70.3625	2.79225
$636$	0	0
$637$	−6.46995	−0.256349
$638$	0	0
$639$	11.3110	0.447458
$640$	0	0
$641$	−3.27011	−0.129162	−0.0645808	−	0.997912i	$-0.520571\pi$
$641$	−3.27011	−0.129162	−0.0645808	+	0.997912i	$0.520571\pi$
$642$	0	0
$643$	20.9310	0.825437	0.412719	−	0.910859i	$-0.364579\pi$
$643$	20.9310	0.825437	0.412719	+	0.910859i	$0.364579\pi$
$644$	0	0
$645$	35.8608	1.41202
$646$	0	0
$647$	−32.6152	−1.28224	−0.641118	−	0.767443i	$-0.721529\pi$
$647$	−32.6152	−1.28224	−0.641118	+	0.767443i	$0.721529\pi$
$648$	0	0
$649$	5.82035	0.228469
$650$	0	0
$651$	35.5651	1.39391
$652$	0	0
$653$	−5.71009	−0.223453	−0.111727	−	0.993739i	$-0.535638\pi$
$653$	−5.71009	−0.223453	−0.111727	+	0.993739i	$0.535638\pi$
$654$	0	0
$655$	−50.3136	−1.96591
$656$	0	0
$657$	−9.27218	−0.361742
$658$	0	0
$659$	27.2403	1.06113	0.530566	−	0.847644i	$-0.321979\pi$
$659$	27.2403	1.06113	0.530566	+	0.847644i	$0.321979\pi$
$660$	0	0
$661$	24.6191	0.957572	0.478786	−	0.877932i	$-0.341077\pi$
$661$	24.6191	0.957572	0.478786	+	0.877932i	$0.341077\pi$
$662$	0	0
$663$	−50.0350	−1.94320
$664$	0	0
$665$	51.4939	1.99685
$666$	0	0
$667$	−9.57741	−0.370839
$668$	0	0
$669$	2.64622	0.102309
$670$	0	0
$671$	−3.33975	−0.128929
$672$	0	0
$673$	−26.4593	−1.01993	−0.509966	−	0.860195i	$-0.670342\pi$
$673$	−26.4593	−1.01993	−0.509966	+	0.860195i	$0.670342\pi$
$674$	0	0
$675$	36.9559	1.42243
$676$	0	0
$677$	−48.2373	−1.85391	−0.926954	−	0.375174i	$-0.877583\pi$
$677$	−48.2373	−1.85391	−0.926954	+	0.375174i	$0.877583\pi$
$678$	0	0
$679$	−23.7897	−0.912966
$680$	0	0
$681$	43.9738	1.68508
$682$	0	0
$683$	−27.4743	−1.05127	−0.525637	−	0.850709i	$-0.676173\pi$
$683$	−27.4743	−1.05127	−0.525637	+	0.850709i	$0.676173\pi$
$684$	0	0
$685$	32.8673	1.25579
$686$	0	0
$687$	35.2362	1.34434
$688$	0	0
$689$	−11.8277	−0.450600
$690$	0	0
$691$	0.482348	0.0183494	0.00917470	−	0.999958i	$-0.497080\pi$
$691$	0.482348	0.0183494	0.00917470	+	0.999958i	$0.497080\pi$
$692$	0	0
$693$	1.84661	0.0701467
$694$	0	0
$695$	1.69999	0.0644843
$696$	0	0
$697$	36.6746	1.38915
$698$	0	0
$699$	4.55552	0.172306
$700$	0	0
$701$	2.92124	0.110334	0.0551669	−	0.998477i	$-0.482431\pi$
$701$	2.92124	0.110334	0.0551669	+	0.998477i	$0.482431\pi$
$702$	0	0
$703$	3.31399	0.124990
$704$	0	0
$705$	−15.4915	−0.583444
$706$	0	0
$707$	−26.5817	−0.999706
$708$	0	0
$709$	−35.6715	−1.33967	−0.669836	−	0.742509i	$-0.733636\pi$
$709$	−35.6715	−1.33967	−0.669836	+	0.742509i	$0.733636\pi$
$710$	0	0
$711$	−4.26513	−0.159955
$712$	0	0
$713$	54.0278	2.02336
$714$	0	0
$715$	−11.5228	−0.430926
$716$	0	0
$717$	−3.73407	−0.139451
$718$	0	0
$719$	−21.0347	−0.784461	−0.392231	−	0.919867i	$-0.628296\pi$
$719$	−21.0347	−0.784461	−0.392231	+	0.919867i	$0.628296\pi$
$720$	0	0
$721$	−53.7278	−2.00093
$722$	0	0
$723$	−3.84868	−0.143134
$724$	0	0
$725$	9.64882	0.358348
$726$	0	0
$727$	0.883799	0.0327783	0.0163891	−	0.999866i	$-0.494783\pi$
$727$	0.883799	0.0327783	0.0163891	+	0.999866i	$0.494783\pi$
$728$	0	0
$729$	29.7536	1.10199
$730$	0	0
$731$	−54.7611	−2.02541
$732$	0	0
$733$	27.9208	1.03128	0.515639	−	0.856806i	$-0.327554\pi$
$733$	27.9208	1.03128	0.515639	+	0.856806i	$0.327554\pi$
$734$	0	0
$735$	7.27166	0.268219
$736$	0	0
$737$	5.31505	0.195783
$738$	0	0
$739$	−8.53617	−0.314008	−0.157004	−	0.987598i	$-0.550184\pi$
$739$	−8.53617	−0.314008	−0.157004	+	0.987598i	$0.550184\pi$
$740$	0	0
$741$	−34.1371	−1.25406
$742$	0	0
$743$	−28.4527	−1.04383	−0.521914	−	0.852998i	$-0.674782\pi$
$743$	−28.4527	−1.04383	−0.521914	+	0.852998i	$0.674782\pi$
$744$	0	0
$745$	37.2810	1.36587
$746$	0	0
$747$	11.7283	0.429115
$748$	0	0
$749$	2.84346	0.103898
$750$	0	0
$751$	28.9595	1.05675	0.528373	−	0.849012i	$-0.322802\pi$
$751$	28.9595	1.05675	0.528373	+	0.849012i	$0.322802\pi$
$752$	0	0
$753$	−32.4887	−1.18395
$754$	0	0
$755$	−75.9302	−2.76338
$756$	0	0
$757$	31.0268	1.12769	0.563844	−	0.825882i	$-0.309322\pi$
$757$	31.0268	1.12769	0.563844	+	0.825882i	$0.309322\pi$
$758$	0	0
$759$	−7.28966	−0.264598
$760$	0	0
$761$	43.2340	1.56723	0.783616	−	0.621245i	$-0.213373\pi$
$761$	43.2340	1.56723	0.783616	+	0.621245i	$0.213373\pi$
$762$	0	0
$763$	30.8513	1.11689
$764$	0	0
$765$	−21.6404	−0.782411
$766$	0	0
$767$	34.0016	1.22773
$768$	0	0
$769$	3.14466	0.113399	0.0566997	−	0.998391i	$-0.481942\pi$
$769$	3.14466	0.113399	0.0566997	+	0.998391i	$0.481942\pi$
$770$	0	0
$771$	−17.4202	−0.627372
$772$	0	0
$773$	24.0370	0.864551	0.432275	−	0.901742i	$-0.357711\pi$
$773$	24.0370	0.864551	0.432275	+	0.901742i	$0.357711\pi$
$774$	0	0
$775$	−54.4306	−1.95521
$776$	0	0
$777$	2.72139	0.0976292
$778$	0	0
$779$	25.0218	0.896498
$780$	0	0
$781$	10.3366	0.369873
$782$	0	0
$783$	8.31273	0.297073
$784$	0	0
$785$	17.4121	0.621465
$786$	0	0
$787$	−29.6840	−1.05812	−0.529060	−	0.848584i	$-0.677455\pi$
$787$	−29.6840	−1.05812	−0.529060	+	0.848584i	$0.677455\pi$
$788$	0	0
$789$	−29.9992	−1.06800
$790$	0	0
$791$	42.2287	1.50148
$792$	0	0
$793$	−19.5103	−0.692831
$794$	0	0
$795$	13.2933	0.471465
$796$	0	0
$797$	−12.5328	−0.443934	−0.221967	−	0.975054i	$-0.571248\pi$
$797$	−12.5328	−0.443934	−0.221967	+	0.975054i	$0.571248\pi$
$798$	0	0
$799$	23.6562	0.836898
$800$	0	0
$801$	11.5395	0.407729
$802$	0	0
$803$	−8.47339	−0.299019
$804$	0	0
$805$	64.2373	2.26407
$806$	0	0
$807$	−46.2209	−1.62705
$808$	0	0
$809$	−51.1034	−1.79670	−0.898349	−	0.439282i	$-0.855233\pi$
$809$	−51.1034	−1.79670	−0.898349	+	0.439282i	$0.855233\pi$
$810$	0	0
$811$	−41.6877	−1.46385	−0.731927	−	0.681383i	$-0.761379\pi$
$811$	−41.6877	−1.46385	−0.731927	+	0.681383i	$0.761379\pi$
$812$	0	0
$813$	9.34283	0.327667
$814$	0	0
$815$	−77.1518	−2.70251
$816$	0	0
$817$	−37.3615	−1.30711
$818$	0	0
$819$	10.7876	0.376949
$820$	0	0
$821$	−7.41134	−0.258658	−0.129329	−	0.991602i	$-0.541282\pi$
$821$	−7.41134	−0.258658	−0.129329	+	0.991602i	$0.541282\pi$
$822$	0	0
$823$	−29.3910	−1.02451	−0.512253	−	0.858835i	$-0.671189\pi$
$823$	−29.3910	−1.02451	−0.512253	+	0.858835i	$0.671189\pi$
$824$	0	0
$825$	7.34401	0.255686
$826$	0	0
$827$	−13.7005	−0.476412	−0.238206	−	0.971215i	$-0.576559\pi$
$827$	−13.7005	−0.476412	−0.238206	+	0.971215i	$0.576559\pi$
$828$	0	0
$829$	25.4698	0.884602	0.442301	−	0.896867i	$-0.354162\pi$
$829$	25.4698	0.884602	0.442301	+	0.896867i	$0.354162\pi$
$830$	0	0
$831$	−45.2321	−1.56908
$832$	0	0
$833$	−11.1042	−0.384736
$834$	0	0
$835$	−77.6322	−2.68657
$836$	0	0
$837$	−46.8935	−1.62088
$838$	0	0
$839$	22.3302	0.770923	0.385462	−	0.922724i	$-0.374042\pi$
$839$	22.3302	0.770923	0.385462	+	0.922724i	$0.374042\pi$
$840$	0	0
$841$	−26.8296	−0.925160
$842$	0	0
$843$	10.8251	0.372837
$844$	0	0
$845$	−23.1343	−0.795843
$846$	0	0
$847$	−30.2953	−1.04096
$848$	0	0
$849$	−5.82707	−0.199985
$850$	0	0
$851$	4.13412	0.141716
$852$	0	0
$853$	−15.5334	−0.531853	−0.265927	−	0.963993i	$-0.585678\pi$
$853$	−15.5334	−0.531853	−0.265927	+	0.963993i	$0.585678\pi$
$854$	0	0
$855$	−14.7645	−0.504935
$856$	0	0
$857$	10.2045	0.348580	0.174290	−	0.984694i	$-0.444237\pi$
$857$	10.2045	0.348580	0.174290	+	0.984694i	$0.444237\pi$
$858$	0	0
$859$	26.2460	0.895501	0.447750	−	0.894159i	$-0.352225\pi$
$859$	26.2460	0.895501	0.447750	+	0.894159i	$0.352225\pi$
$860$	0	0
$861$	20.5474	0.700253
$862$	0	0
$863$	−57.0942	−1.94351	−0.971754	−	0.235995i	$-0.924165\pi$
$863$	−57.0942	−1.94351	−0.971754	+	0.235995i	$0.924165\pi$
$864$	0	0
$865$	−50.4639	−1.71582
$866$	0	0
$867$	−60.8519	−2.06664
$868$	0	0
$869$	−3.89769	−0.132220
$870$	0	0
$871$	31.0497	1.05208
$872$	0	0
$873$	6.82106	0.230858
$874$	0	0
$875$	−15.3106	−0.517594
$876$	0	0
$877$	53.9988	1.82341	0.911705	−	0.410846i	$-0.134767\pi$
$877$	53.9988	1.82341	0.911705	+	0.410846i	$0.134767\pi$
$878$	0	0
$879$	0.861680	0.0290637
$880$	0	0
$881$	−39.5009	−1.33082	−0.665410	−	0.746478i	$-0.731743\pi$
$881$	−39.5009	−1.33082	−0.665410	+	0.746478i	$0.731743\pi$
$882$	0	0
$883$	−12.5445	−0.422158	−0.211079	−	0.977469i	$-0.567698\pi$
$883$	−12.5445	−0.422158	−0.211079	+	0.977469i	$0.567698\pi$
$884$	0	0
$885$	−38.2148	−1.28458
$886$	0	0
$887$	−12.1803	−0.408975	−0.204487	−	0.978869i	$-0.565553\pi$
$887$	−12.1803	−0.408975	−0.204487	+	0.978869i	$0.565553\pi$
$888$	0	0
$889$	−60.1984	−2.01899
$890$	0	0
$891$	4.42174	0.148134
$892$	0	0
$893$	16.1398	0.540098
$894$	0	0
$895$	51.8333	1.73260
$896$	0	0
$897$	−42.5851	−1.42188
$898$	0	0
$899$	−12.2434	−0.408341
$900$	0	0
$901$	−20.2995	−0.676275
$902$	0	0
$903$	−30.6805	−1.02098
$904$	0	0
$905$	6.94972	0.231017
$906$	0	0
$907$	37.6756	1.25100	0.625499	−	0.780225i	$-0.284896\pi$
$907$	37.6756	1.25100	0.625499	+	0.780225i	$0.284896\pi$
$908$	0	0
$909$	7.62157	0.252792
$910$	0	0
$911$	−33.8504	−1.12151	−0.560757	−	0.827980i	$-0.689490\pi$
$911$	−33.8504	−1.12151	−0.560757	+	0.827980i	$0.689490\pi$
$912$	0	0
$913$	10.7179	0.354710
$914$	0	0
$915$	21.9279	0.724913
$916$	0	0
$917$	43.0456	1.42149
$918$	0	0
$919$	25.1675	0.830198	0.415099	−	0.909776i	$-0.363747\pi$
$919$	25.1675	0.830198	0.415099	+	0.909776i	$0.363747\pi$
$920$	0	0
$921$	50.0727	1.64995
$922$	0	0
$923$	60.3849	1.98759
$924$	0	0
$925$	−4.16494	−0.136943
$926$	0	0
$927$	15.4050	0.505967
$928$	0	0
$929$	−52.4965	−1.72235	−0.861177	−	0.508306i	$-0.830272\pi$
$929$	−52.4965	−1.72235	−0.861177	+	0.508306i	$0.830272\pi$
$930$	0	0
$931$	−7.57597	−0.248292
$932$	0	0
$933$	−28.1883	−0.922844
$934$	0	0
$935$	−19.7761	−0.646748
$936$	0	0
$937$	−45.6255	−1.49052	−0.745259	−	0.666775i	$-0.767674\pi$
$937$	−45.6255	−1.49052	−0.745259	+	0.666775i	$0.767674\pi$
$938$	0	0
$939$	9.59387	0.313084
$940$	0	0
$941$	−45.0402	−1.46827	−0.734134	−	0.679005i	$-0.762412\pi$
$941$	−45.0402	−1.46827	−0.734134	+	0.679005i	$0.762412\pi$
$942$	0	0
$943$	31.2140	1.01647
$944$	0	0
$945$	−55.7549	−1.81371
$946$	0	0
$947$	−27.1618	−0.882641	−0.441320	−	0.897350i	$-0.645490\pi$
$947$	−27.1618	−0.882641	−0.441320	+	0.897350i	$0.645490\pi$
$948$	0	0
$949$	−49.5003	−1.60685
$950$	0	0
$951$	25.1162	0.814448
$952$	0	0
$953$	−23.0784	−0.747583	−0.373792	−	0.927513i	$-0.621942\pi$
$953$	−23.0784	−0.747583	−0.373792	+	0.927513i	$0.621942\pi$
$954$	0	0
$955$	74.3300	2.40526
$956$	0	0
$957$	1.65193	0.0533995
$958$	0	0
$959$	−28.1195	−0.908025
$960$	0	0
$961$	38.0673	1.22798
$962$	0	0
$963$	−0.815286	−0.0262722
$964$	0	0
$965$	88.8824	2.86123
$966$	0	0
$967$	49.9896	1.60756	0.803778	−	0.594929i	$-0.202820\pi$
$967$	49.9896	1.60756	0.803778	+	0.594929i	$0.202820\pi$
$968$	0	0
$969$	−58.5883	−1.88213
$970$	0	0
$971$	−0.128492	−0.00412350	−0.00206175	−	0.999998i	$-0.500656\pi$
$971$	−0.128492	−0.00412350	−0.00206175	+	0.999998i	$0.500656\pi$
$972$	0	0
$973$	−1.45442	−0.0466265
$974$	0	0
$975$	42.9026	1.37398
$976$	0	0
$977$	29.0075	0.928033	0.464017	−	0.885827i	$-0.346408\pi$
$977$	29.0075	0.928033	0.464017	+	0.885827i	$0.346408\pi$
$978$	0	0
$979$	10.5454	0.337032
$980$	0	0
$981$	−8.84579	−0.282424
$982$	0	0
$983$	−39.2906	−1.25318	−0.626588	−	0.779351i	$-0.715549\pi$
$983$	−39.2906	−1.25318	−0.626588	+	0.779351i	$0.715549\pi$
$984$	0	0
$985$	23.2583	0.741071
$986$	0	0
$987$	13.2537	0.421870
$988$	0	0
$989$	−46.6075	−1.48203
$990$	0	0
$991$	39.3276	1.24928	0.624641	−	0.780912i	$-0.285245\pi$
$991$	39.3276	1.24928	0.624641	+	0.780912i	$0.285245\pi$
$992$	0	0
$993$	−35.6849	−1.13243
$994$	0	0
$995$	−26.0298	−0.825200
$996$	0	0
$997$	43.3054	1.37150	0.685748	−	0.727839i	$-0.259475\pi$
$997$	43.3054	1.37150	0.685748	+	0.727839i	$0.259475\pi$
$998$	0	0
$999$	−3.58822	−0.113526

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.b.1.18	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.b.1.18	✓	63	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:	\( N \)	\(=\)	\( 6044 = 2^{2} \cdot 1511 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6044.a (trivial)

\(f(q)\)	\(=\)	\(q-1.47185 q^{3} -3.39845 q^{5} +2.90753 q^{7} -0.833657 q^{9} +O(q^{10})\)	\(q-1.47185 q^{3} -3.39845 q^{5} +2.90753 q^{7} -0.833657 q^{9} -0.761837 q^{11} -4.45054 q^{13} +5.00201 q^{15} -7.63831 q^{17} -5.21134 q^{19} -4.27945 q^{21} -6.50102 q^{23} +6.54949 q^{25} +5.64257 q^{27} +1.47322 q^{29} -8.31067 q^{31} +1.12131 q^{33} -9.88111 q^{35} -0.635919 q^{37} +6.55053 q^{39} -4.80140 q^{41} +7.16927 q^{43} +2.83314 q^{45} -3.09705 q^{47} +1.45375 q^{49} +11.2425 q^{51} +2.65759 q^{53} +2.58907 q^{55} +7.67032 q^{57} -7.63988 q^{59} +4.38381 q^{61} -2.42388 q^{63} +15.1249 q^{65} -6.97663 q^{67} +9.56853 q^{69} -13.5680 q^{71} +11.1223 q^{73} -9.63986 q^{75} -2.21507 q^{77} +5.11617 q^{79} -5.80405 q^{81} -14.0685 q^{83} +25.9584 q^{85} -2.16836 q^{87} -13.8421 q^{89} -12.9401 q^{91} +12.2321 q^{93} +17.7105 q^{95} -8.18210 q^{97} +0.635111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9}+O(q^{10}) \) 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9	\( 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9} + 21 q^{11} + 17 q^{13} + 26 q^{15} - 5 q^{17} + 57 q^{19} + 18 q^{21} + 16 q^{23} + 60 q^{25} + 14 q^{27} + 22 q^{29} + 36 q^{31} - q^{33} + 28 q^{35} + 21 q^{37} + 69 q^{39} - 3 q^{41} + 86 q^{43} + 39 q^{45} + 23 q^{47} + 63 q^{49} + 67 q^{51} + 18 q^{53} + 67 q^{55} + 27 q^{59} + 62 q^{61} + 58 q^{63} - 13 q^{65} + 62 q^{67} + 45 q^{69} + 29 q^{71} - q^{73} + 39 q^{75} + 19 q^{77} + 132 q^{79} + 51 q^{81} + 35 q^{83} + 60 q^{85} + 55 q^{87} - 2 q^{89} + 84 q^{91} + 29 q^{93} + 53 q^{95} - 10 q^{97} + 95 q^{99}+O(q^{100}) \) 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9 + 21 * q^11 + 17 * q^13 + 26 * q^15 - 5 * q^17 + 57 * q^19 + 18 * q^21 + 16 * q^23 + 60 * q^25 + 14 * q^27 + 22 * q^29 + 36 * q^31 - q^33 + 28 * q^35 + 21 * q^37 + 69 * q^39 - 3 * q^41 + 86 * q^43 + 39 * q^45 + 23 * q^47 + 63 * q^49 + 67 * q^51 + 18 * q^53 + 67 * q^55 + 27 * q^59 + 62 * q^61 + 58 * q^63 - 13 * q^65 + 62 * q^67 + 45 * q^69 + 29 * q^71 - q^73 + 39 * q^75 + 19 * q^77 + 132 * q^79 + 51 * q^81 + 35 * q^83 + 60 * q^85 + 55 * q^87 - 2 * q^89 + 84 * q^91 + 29 * q^93 + 53 * q^95 - 10 * q^97 + 95 * q^99

Introduction

L-functions

Modular forms

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Database

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