LMFDB - Embedded newform 8037.2.a.n.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 13 x^{14} + 65 x^{13} + 47 x^{12} - 390 x^{11} + 4 x^{10} + 1115 x^{9} - 320 x^{8} + \cdots - 25 $ x^16 - 4x^15 - 13x^14 + 65x^13 + 47x^12 - 390x^11 + 4x^10 + 1115x^9 - 320x^8 - 1639x^7 + 618x^6 + 1250x^5 - 487x^4 - 456x^3 + 179x^2 + 62*x - 25
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 893)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.880178$ of defining polynomial
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.880178 q^{2} -1.22529 q^{4} -0.815813 q^{5} -4.34804 q^{7} +2.83883 q^{8} +O(q^{10})$	\(q-0.880178 q^{2} -1.22529 q^{4} -0.815813 q^{5} -4.34804 q^{7} +2.83883 q^{8} +0.718060 q^{10} -4.85877 q^{11} -3.63193 q^{13} +3.82704 q^{14} -0.0480977 q^{16} +1.61565 q^{17} -1.00000 q^{19} +0.999605 q^{20} +4.27658 q^{22} +4.93327 q^{23} -4.33445 q^{25} +3.19674 q^{26} +5.32759 q^{28} +3.39533 q^{29} +10.2386 q^{31} -5.63532 q^{32} -1.42206 q^{34} +3.54718 q^{35} -8.34698 q^{37} +0.880178 q^{38} -2.31595 q^{40} +10.1326 q^{41} -4.44071 q^{43} +5.95339 q^{44} -4.34216 q^{46} +1.00000 q^{47} +11.9054 q^{49} +3.81509 q^{50} +4.45016 q^{52} -10.8236 q^{53} +3.96385 q^{55} -12.3433 q^{56} -2.98850 q^{58} +1.63761 q^{59} +7.54565 q^{61} -9.01175 q^{62} +5.05628 q^{64} +2.96297 q^{65} +0.210448 q^{67} -1.97964 q^{68} -3.12215 q^{70} -0.376574 q^{71} +13.6995 q^{73} +7.34682 q^{74} +1.22529 q^{76} +21.1261 q^{77} +13.9332 q^{79} +0.0392387 q^{80} -8.91853 q^{82} +5.55821 q^{83} -1.31807 q^{85} +3.90861 q^{86} -13.7932 q^{88} +5.74118 q^{89} +15.7918 q^{91} -6.04467 q^{92} -0.880178 q^{94} +0.815813 q^{95} -15.3562 q^{97} -10.4789 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8}+O(q^{10}) $ 16 * q + 4 * q^2 + 10 * q^4 + q^5 - 9 * q^7 + 9 * q^8	$ 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8} - 15 q^{10} - 19 q^{13} + 6 q^{14} + 10 q^{16} + 8 q^{17} - 16 q^{19} + 11 q^{20} - 12 q^{22} + 5 q^{23} - 3 q^{25} - 9 q^{26} - 17 q^{28} + 2 q^{29} - 18 q^{31} - 3 q^{32} - 14 q^{34} + 11 q^{35} - 24 q^{37} - 4 q^{38} - 50 q^{40} + 6 q^{41} - 34 q^{43} + 4 q^{44} - 3 q^{46} + 16 q^{47} + 5 q^{49} - 26 q^{50} - 44 q^{52} + 23 q^{53} - 48 q^{55} + 3 q^{56} - 26 q^{58} + 32 q^{59} - 16 q^{61} - 32 q^{62} + 7 q^{64} + 18 q^{65} - 67 q^{67} + 19 q^{68} + 24 q^{70} - 19 q^{71} - 2 q^{73} + 29 q^{74} - 10 q^{76} - 14 q^{77} - 27 q^{79} - 15 q^{80} - 56 q^{82} + 17 q^{83} + 15 q^{85} + q^{86} - 13 q^{88} - 20 q^{89} - 42 q^{91} - 45 q^{92} + 4 q^{94} - q^{95} - 50 q^{97} - 13 q^{98}+O(q^{100}) $ 16 * q + 4 * q^2 + 10 * q^4 + q^5 - 9 * q^7 + 9 * q^8 - 15 * q^10 - 19 * q^13 + 6 * q^14 + 10 * q^16 + 8 * q^17 - 16 * q^19 + 11 * q^20 - 12 * q^22 + 5 * q^23 - 3 * q^25 - 9 * q^26 - 17 * q^28 + 2 * q^29 - 18 * q^31 - 3 * q^32 - 14 * q^34 + 11 * q^35 - 24 * q^37 - 4 * q^38 - 50 * q^40 + 6 * q^41 - 34 * q^43 + 4 * q^44 - 3 * q^46 + 16 * q^47 + 5 * q^49 - 26 * q^50 - 44 * q^52 + 23 * q^53 - 48 * q^55 + 3 * q^56 - 26 * q^58 + 32 * q^59 - 16 * q^61 - 32 * q^62 + 7 * q^64 + 18 * q^65 - 67 * q^67 + 19 * q^68 + 24 * q^70 - 19 * q^71 - 2 * q^73 + 29 * q^74 - 10 * q^76 - 14 * q^77 - 27 * q^79 - 15 * q^80 - 56 * q^82 + 17 * q^83 + 15 * q^85 + q^86 - 13 * q^88 - 20 * q^89 - 42 * q^91 - 45 * q^92 + 4 * q^94 - q^95 - 50 * q^97 - 13 * q^98

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 $

$2$	−0.880178	−0.622380	−0.311190	−	0.950348i	$-0.600728\pi$
$2$	−0.880178	−0.622380	−0.311190	+	0.950348i	$0.600728\pi$
$3$	0	0
$3$	0	0
$4$	−1.22529	−0.612644
$5$	−0.815813	−0.364843	−0.182421	−	0.983220i	$-0.558393\pi$
$5$	−0.815813	−0.364843	−0.182421	+	0.983220i	$0.558393\pi$
$6$	0	0
$7$	−4.34804	−1.64340	−0.821701	−	0.569918i	$-0.806975\pi$
$7$	−4.34804	−1.64340	−0.821701	+	0.569918i	$0.806975\pi$
$8$	2.83883	1.00368
$9$	0	0
$10$	0.718060	0.227071
$11$	−4.85877	−1.46497	−0.732487	−	0.680781i	$-0.761641\pi$
$11$	−4.85877	−1.46497	−0.732487	+	0.680781i	$0.761641\pi$
$12$	0	0
$13$	−3.63193	−1.00732	−0.503658	−	0.863903i	$-0.668013\pi$
$13$	−3.63193	−1.00732	−0.503658	+	0.863903i	$0.668013\pi$
$14$	3.82704	1.02282
$15$	0	0
$16$	−0.0480977	−0.0120244
$17$	1.61565	0.391853	0.195927	−	0.980619i	$-0.437229\pi$
$17$	1.61565	0.391853	0.195927	+	0.980619i	$0.437229\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0.999605	0.223518
$21$	0	0
$22$	4.27658	0.911770
$23$	4.93327	1.02866	0.514329	−	0.857593i	$-0.328041\pi$
$23$	4.93327	1.02866	0.514329	+	0.857593i	$0.328041\pi$
$24$	0	0
$25$	−4.33445	−0.866890
$26$	3.19674	0.626933
$27$	0	0
$28$	5.32759	1.00682
$29$	3.39533	0.630498	0.315249	−	0.949009i	$-0.397912\pi$
$29$	3.39533	0.630498	0.315249	+	0.949009i	$0.397912\pi$
$30$	0	0
$31$	10.2386	1.83890	0.919450	−	0.393207i	$-0.128634\pi$
$31$	10.2386	1.83890	0.919450	+	0.393207i	$0.128634\pi$
$32$	−5.63532	−0.996193
$33$	0	0
$34$	−1.42206	−0.243882
$35$	3.54718	0.599583
$36$	0	0
$37$	−8.34698	−1.37223	−0.686117	−	0.727491i	$-0.740686\pi$
$37$	−8.34698	−1.37223	−0.686117	+	0.727491i	$0.740686\pi$
$38$	0.880178	0.142784
$39$	0	0
$40$	−2.31595	−0.366184
$41$	10.1326	1.58245	0.791226	−	0.611524i	$-0.209443\pi$
$41$	10.1326	1.58245	0.791226	+	0.611524i	$0.209443\pi$
$42$	0	0
$43$	−4.44071	−0.677202	−0.338601	−	0.940930i	$-0.609954\pi$
$43$	−4.44071	−0.677202	−0.338601	+	0.940930i	$0.609954\pi$
$44$	5.95339	0.897507
$45$	0	0
$46$	−4.34216	−0.640216
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	11.9054	1.70077
$50$	3.81509	0.539535
$51$	0	0
$52$	4.45016	0.617126
$53$	−10.8236	−1.48674	−0.743368	−	0.668882i	$-0.766773\pi$
$53$	−10.8236	−1.48674	−0.743368	+	0.668882i	$0.766773\pi$
$54$	0	0
$55$	3.96385	0.534485
$56$	−12.3433	−1.64945
$57$	0	0
$58$	−2.98850	−0.392409
$59$	1.63761	0.213198	0.106599	−	0.994302i	$-0.466004\pi$
$59$	1.63761	0.213198	0.106599	+	0.994302i	$0.466004\pi$
$60$	0	0
$61$	7.54565	0.966122	0.483061	−	0.875587i	$-0.339525\pi$
$61$	7.54565	0.966122	0.483061	+	0.875587i	$0.339525\pi$
$62$	−9.01175	−1.14449
$63$	0	0
$64$	5.05628	0.632035
$65$	2.96297	0.367512
$66$	0	0
$67$	0.210448	0.0257103	0.0128552	−	0.999917i	$-0.495908\pi$
$67$	0.210448	0.0257103	0.0128552	+	0.999917i	$0.495908\pi$
$68$	−1.97964	−0.240066
$69$	0	0
$70$	−3.12215	−0.373168
$71$	−0.376574	−0.0446911	−0.0223456	−	0.999750i	$-0.507113\pi$
$71$	−0.376574	−0.0446911	−0.0223456	+	0.999750i	$0.507113\pi$
$72$	0	0
$73$	13.6995	1.60341	0.801704	−	0.597721i	$-0.203927\pi$
$73$	13.6995	1.60341	0.801704	+	0.597721i	$0.203927\pi$
$74$	7.34682	0.854051
$75$	0	0
$76$	1.22529	0.140550
$77$	21.1261	2.40754
$78$	0	0
$79$	13.9332	1.56761	0.783804	−	0.621009i	$-0.213277\pi$
$79$	13.9332	1.56761	0.783804	+	0.621009i	$0.213277\pi$
$80$	0.0392387	0.00438702
$81$	0	0
$82$	−8.91853	−0.984886
$83$	5.55821	0.610093	0.305046	−	0.952338i	$-0.401328\pi$
$83$	5.55821	0.610093	0.305046	+	0.952338i	$0.401328\pi$
$84$	0	0
$85$	−1.31807	−0.142965
$86$	3.90861	0.421477
$87$	0	0
$88$	−13.7932	−1.47036
$89$	5.74118	0.608564	0.304282	−	0.952582i	$-0.401583\pi$
$89$	5.74118	0.608564	0.304282	+	0.952582i	$0.401583\pi$
$90$	0	0
$91$	15.7918	1.65543
$92$	−6.04467	−0.630201
$93$	0	0
$94$	−0.880178	−0.0907834
$95$	0.815813	0.0837006
$96$	0	0
$97$	−15.3562	−1.55919	−0.779594	−	0.626286i	$-0.784574\pi$
$97$	−15.3562	−1.55919	−0.779594	+	0.626286i	$0.784574\pi$
$98$	−10.4789	−1.05853
$99$	0	0
$100$	5.31094	0.531094
$101$	−5.22378	−0.519786	−0.259893	−	0.965637i	$-0.583687\pi$
$101$	−5.22378	−0.519786	−0.259893	+	0.965637i	$0.583687\pi$
$102$	0	0
$103$	−2.05004	−0.201997	−0.100998	−	0.994887i	$-0.532204\pi$
$103$	−2.05004	−0.201997	−0.100998	+	0.994887i	$0.532204\pi$
$104$	−10.3104	−1.01102
$105$	0	0
$106$	9.52670	0.925314
$107$	0.189527	0.0183222	0.00916112	−	0.999958i	$-0.497084\pi$
$107$	0.189527	0.0183222	0.00916112	+	0.999958i	$0.497084\pi$
$108$	0	0
$109$	13.9567	1.33681	0.668406	−	0.743796i	$-0.266977\pi$
$109$	13.9567	1.33681	0.668406	+	0.743796i	$0.266977\pi$
$110$	−3.48889	−0.332653
$111$	0	0
$112$	0.209131	0.0197610
$113$	9.11161	0.857148	0.428574	−	0.903507i	$-0.359016\pi$
$113$	9.11161	0.857148	0.428574	+	0.903507i	$0.359016\pi$
$114$	0	0
$115$	−4.02463	−0.375298
$116$	−4.16026	−0.386270
$117$	0	0
$118$	−1.44139	−0.132690
$119$	−7.02492	−0.643973
$120$	0	0
$121$	12.6076	1.14615
$122$	−6.64151	−0.601294
$123$	0	0
$124$	−12.5452	−1.12659
$125$	7.61516	0.681121
$126$	0	0
$127$	−16.5310	−1.46689	−0.733444	−	0.679750i	$-0.762088\pi$
$127$	−16.5310	−1.46689	−0.733444	+	0.679750i	$0.762088\pi$
$128$	6.82021	0.602827
$129$	0	0
$130$	−2.60794	−0.228732
$131$	2.79358	0.244076	0.122038	−	0.992525i	$-0.461057\pi$
$131$	2.79358	0.244076	0.122038	+	0.992525i	$0.461057\pi$
$132$	0	0
$133$	4.34804	0.377023
$134$	−0.185232	−0.0160016
$135$	0	0
$136$	4.58656	0.393294
$137$	7.56218	0.646081	0.323040	−	0.946385i	$-0.395295\pi$
$137$	7.56218	0.646081	0.323040	+	0.946385i	$0.395295\pi$
$138$	0	0
$139$	−12.4358	−1.05479	−0.527396	−	0.849620i	$-0.676831\pi$
$139$	−12.4358	−1.05479	−0.527396	+	0.849620i	$0.676831\pi$
$140$	−4.34632	−0.367331
$141$	0	0
$142$	0.331452	0.0278149
$143$	17.6467	1.47569
$144$	0	0
$145$	−2.76996	−0.230032
$146$	−12.0580	−0.997929
$147$	0	0
$148$	10.2274	0.840691
$149$	−14.5109	−1.18878	−0.594388	−	0.804178i	$-0.702606\pi$
$149$	−14.5109	−1.18878	−0.594388	+	0.804178i	$0.702606\pi$
$150$	0	0
$151$	−3.82828	−0.311541	−0.155770	−	0.987793i	$-0.549786\pi$
$151$	−3.82828	−0.311541	−0.155770	+	0.987793i	$0.549786\pi$
$152$	−2.83883	−0.230259
$153$	0	0
$154$	−18.5947	−1.49841
$155$	−8.35275	−0.670909
$156$	0	0
$157$	7.73991	0.617712	0.308856	−	0.951109i	$-0.400054\pi$
$157$	7.73991	0.617712	0.308856	+	0.951109i	$0.400054\pi$
$158$	−12.2637	−0.975647
$159$	0	0
$160$	4.59736	0.363454
$161$	−21.4500	−1.69050
$162$	0	0
$163$	−21.9262	−1.71740	−0.858698	−	0.512482i	$-0.828726\pi$
$163$	−21.9262	−1.71740	−0.858698	+	0.512482i	$0.828726\pi$
$164$	−12.4154	−0.969479
$165$	0	0
$166$	−4.89221	−0.379709
$167$	−10.9343	−0.846121	−0.423061	−	0.906101i	$-0.639044\pi$
$167$	−10.9343	−0.846121	−0.423061	+	0.906101i	$0.639044\pi$
$168$	0	0
$169$	0.190914	0.0146857
$170$	1.16014	0.0889784
$171$	0	0
$172$	5.44114	0.414883
$173$	−10.7433	−0.816801	−0.408401	−	0.912803i	$-0.633913\pi$
$173$	−10.7433	−0.816801	−0.408401	+	0.912803i	$0.633913\pi$
$174$	0	0
$175$	18.8463	1.42465
$176$	0.233696	0.0176155
$177$	0	0
$178$	−5.05326	−0.378758
$179$	14.0472	1.04994	0.524969	−	0.851121i	$-0.324077\pi$
$179$	14.0472	1.04994	0.524969	+	0.851121i	$0.324077\pi$
$180$	0	0
$181$	11.0296	0.819822	0.409911	−	0.912125i	$-0.365560\pi$
$181$	11.0296	0.819822	0.409911	+	0.912125i	$0.365560\pi$
$182$	−13.8996	−1.03030
$183$	0	0
$184$	14.0047	1.03244
$185$	6.80957	0.500650
$186$	0	0
$187$	−7.85009	−0.574055
$188$	−1.22529	−0.0893632
$189$	0	0
$190$	−0.718060	−0.0520936
$191$	18.7394	1.35593	0.677967	−	0.735092i	$-0.262861\pi$
$191$	18.7394	1.35593	0.677967	+	0.735092i	$0.262861\pi$
$192$	0	0
$193$	0.0798209	0.00574564	0.00287282	−	0.999996i	$-0.499086\pi$
$193$	0.0798209	0.00574564	0.00287282	+	0.999996i	$0.499086\pi$
$194$	13.5162	0.970407
$195$	0	0
$196$	−14.5875	−1.04197
$197$	−7.09317	−0.505368	−0.252684	−	0.967549i	$-0.581313\pi$
$197$	−7.09317	−0.505368	−0.252684	+	0.967549i	$0.581313\pi$
$198$	0	0
$199$	−24.1281	−1.71040	−0.855200	−	0.518299i	$-0.826565\pi$
$199$	−24.1281	−1.71040	−0.855200	+	0.518299i	$0.826565\pi$
$200$	−12.3047	−0.870077
$201$	0	0
$202$	4.59786	0.323504
$203$	−14.7630	−1.03616
$204$	0	0
$205$	−8.26634	−0.577346
$206$	1.80440	0.125719
$207$	0	0
$208$	0.174687	0.0121124
$209$	4.85877	0.336088
$210$	0	0
$211$	14.8715	1.02380	0.511899	−	0.859045i	$-0.328942\pi$
$211$	14.8715	1.02380	0.511899	+	0.859045i	$0.328942\pi$
$212$	13.2620	0.910839
$213$	0	0
$214$	−0.166817	−0.0114034
$215$	3.62279	0.247072
$216$	0	0
$217$	−44.5176	−3.02205
$218$	−12.2844	−0.832005
$219$	0	0
$220$	−4.85685	−0.327449
$221$	−5.86794	−0.394720
$222$	0	0
$223$	−1.80447	−0.120837	−0.0604183	−	0.998173i	$-0.519243\pi$
$223$	−1.80447	−0.120837	−0.0604183	+	0.998173i	$0.519243\pi$
$224$	24.5026	1.63715
$225$	0	0
$226$	−8.01983	−0.533471
$227$	8.97335	0.595582	0.297791	−	0.954631i	$-0.403750\pi$
$227$	8.97335	0.595582	0.297791	+	0.954631i	$0.403750\pi$
$228$	0	0
$229$	1.86225	0.123061	0.0615304	−	0.998105i	$-0.480402\pi$
$229$	1.86225	0.123061	0.0615304	+	0.998105i	$0.480402\pi$
$230$	3.54239	0.233578
$231$	0	0
$232$	9.63876	0.632816
$233$	5.03087	0.329583	0.164792	−	0.986328i	$-0.447305\pi$
$233$	5.03087	0.329583	0.164792	+	0.986328i	$0.447305\pi$
$234$	0	0
$235$	−0.815813	−0.0532178
$236$	−2.00654	−0.130615
$237$	0	0
$238$	6.18318	0.400796
$239$	8.61365	0.557171	0.278585	−	0.960412i	$-0.410135\pi$
$239$	8.61365	0.557171	0.278585	+	0.960412i	$0.410135\pi$
$240$	0	0
$241$	6.96557	0.448691	0.224346	−	0.974510i	$-0.427976\pi$
$241$	6.96557	0.448691	0.224346	+	0.974510i	$0.427976\pi$
$242$	−11.0970	−0.713340
$243$	0	0
$244$	−9.24559	−0.591888
$245$	−9.71259	−0.620514
$246$	0	0
$247$	3.63193	0.231094
$248$	29.0655	1.84566
$249$	0	0
$250$	−6.70270	−0.423916
$251$	3.02652	0.191032	0.0955162	−	0.995428i	$-0.469550\pi$
$251$	3.02652	0.191032	0.0955162	+	0.995428i	$0.469550\pi$
$252$	0	0
$253$	−23.9696	−1.50696
$254$	14.5502	0.912961
$255$	0	0
$256$	−16.1156	−1.00722
$257$	−0.264314	−0.0164874	−0.00824372	−	0.999966i	$-0.502624\pi$
$257$	−0.264314	−0.0164874	−0.00824372	+	0.999966i	$0.502624\pi$
$258$	0	0
$259$	36.2930	2.25513
$260$	−3.63049	−0.225154
$261$	0	0
$262$	−2.45884	−0.151908
$263$	1.30114	0.0802320	0.0401160	−	0.999195i	$-0.487227\pi$
$263$	1.30114	0.0802320	0.0401160	+	0.999195i	$0.487227\pi$
$264$	0	0
$265$	8.83003	0.542425
$266$	−3.82704	−0.234651
$267$	0	0
$268$	−0.257859	−0.0157512
$269$	−10.5450	−0.642942	−0.321471	−	0.946919i	$-0.604177\pi$
$269$	−10.5450	−0.642942	−0.321471	+	0.946919i	$0.604177\pi$
$270$	0	0
$271$	20.2686	1.23123	0.615614	−	0.788048i	$-0.288908\pi$
$271$	20.2686	1.23123	0.615614	+	0.788048i	$0.288908\pi$
$272$	−0.0777092	−0.00471181
$273$	0	0
$274$	−6.65606	−0.402107
$275$	21.0601	1.26997
$276$	0	0
$277$	1.80880	0.108680	0.0543402	−	0.998522i	$-0.482694\pi$
$277$	1.80880	0.108680	0.0543402	+	0.998522i	$0.482694\pi$
$278$	10.9457	0.656481
$279$	0	0
$280$	10.0698	0.601788
$281$	−6.83976	−0.408026	−0.204013	−	0.978968i	$-0.565398\pi$
$281$	−6.83976	−0.408026	−0.204013	+	0.978968i	$0.565398\pi$
$282$	0	0
$283$	−9.45423	−0.561995	−0.280998	−	0.959708i	$-0.590665\pi$
$283$	−9.45423	−0.561995	−0.280998	+	0.959708i	$0.590665\pi$
$284$	0.461412	0.0273797
$285$	0	0
$286$	−15.5322	−0.918441
$287$	−44.0571	−2.60061
$288$	0	0
$289$	−14.3897	−0.846451
$290$	2.43805	0.143167
$291$	0	0
$292$	−16.7859	−0.982318
$293$	1.05340	0.0615404	0.0307702	−	0.999526i	$-0.490204\pi$
$293$	1.05340	0.0615404	0.0307702	+	0.999526i	$0.490204\pi$
$294$	0	0
$295$	−1.33598	−0.0777838
$296$	−23.6956	−1.37728
$297$	0	0
$298$	12.7721	0.739870
$299$	−17.9173	−1.03618
$300$	0	0
$301$	19.3084	1.11292
$302$	3.36957	0.193897
$303$	0	0
$304$	0.0480977	0.00275859
$305$	−6.15584	−0.352482
$306$	0	0
$307$	−14.8678	−0.848553	−0.424276	−	0.905533i	$-0.639471\pi$
$307$	−14.8678	−0.848553	−0.424276	+	0.905533i	$0.639471\pi$
$308$	−25.8855	−1.47497
$309$	0	0
$310$	7.35190	0.417560
$311$	19.9805	1.13299	0.566497	−	0.824064i	$-0.308298\pi$
$311$	19.9805	1.13299	0.566497	+	0.824064i	$0.308298\pi$
$312$	0	0
$313$	−29.8859	−1.68925	−0.844627	−	0.535356i	$-0.820178\pi$
$313$	−29.8859	−1.68925	−0.844627	+	0.535356i	$0.820178\pi$
$314$	−6.81250	−0.384452
$315$	0	0
$316$	−17.0722	−0.960384
$317$	2.02396	0.113677	0.0568386	−	0.998383i	$-0.481898\pi$
$317$	2.02396	0.113677	0.0568386	+	0.998383i	$0.481898\pi$
$318$	0	0
$319$	−16.4971	−0.923663
$320$	−4.12498	−0.230593
$321$	0	0
$322$	18.8799	1.05213
$323$	−1.61565	−0.0898974
$324$	0	0
$325$	15.7424	0.873232
$326$	19.2990	1.06887
$327$	0	0
$328$	28.7648	1.58827
$329$	−4.34804	−0.239715
$330$	0	0
$331$	−2.64046	−0.145133	−0.0725665	−	0.997364i	$-0.523119\pi$
$331$	−2.64046	−0.145133	−0.0725665	+	0.997364i	$0.523119\pi$
$332$	−6.81040	−0.373769
$333$	0	0
$334$	9.62413	0.526609
$335$	−0.171686	−0.00938021
$336$	0	0
$337$	−1.32646	−0.0722567	−0.0361283	−	0.999347i	$-0.511503\pi$
$337$	−1.32646	−0.0722567	−0.0361283	+	0.999347i	$0.511503\pi$
$338$	−0.168038	−0.00914007
$339$	0	0
$340$	1.61501	0.0875865
$341$	−49.7468	−2.69394
$342$	0	0
$343$	−21.3289	−1.15165
$344$	−12.6064	−0.679692
$345$	0	0
$346$	9.45605	0.508361
$347$	−30.2837	−1.62571	−0.812856	−	0.582464i	$-0.802089\pi$
$347$	−30.2837	−1.62571	−0.812856	+	0.582464i	$0.802089\pi$
$348$	0	0
$349$	−3.79790	−0.203297	−0.101648	−	0.994820i	$-0.532412\pi$
$349$	−3.79790	−0.203297	−0.101648	+	0.994820i	$0.532412\pi$
$350$	−16.5881	−0.886673
$351$	0	0
$352$	27.3807	1.45940
$353$	34.9208	1.85865	0.929324	−	0.369265i	$-0.120391\pi$
$353$	34.9208	1.85865	0.929324	+	0.369265i	$0.120391\pi$
$354$	0	0
$355$	0.307214	0.0163052
$356$	−7.03460	−0.372833
$357$	0	0
$358$	−12.3640	−0.653460
$359$	17.0379	0.899228	0.449614	−	0.893223i	$-0.351562\pi$
$359$	17.0379	0.899228	0.449614	+	0.893223i	$0.351562\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−9.70799	−0.510241
$363$	0	0
$364$	−19.3494	−1.01419
$365$	−11.1763	−0.584992
$366$	0	0
$367$	21.5304	1.12388	0.561939	−	0.827178i	$-0.310055\pi$
$367$	21.5304	1.12388	0.561939	+	0.827178i	$0.310055\pi$
$368$	−0.237279	−0.0123690
$369$	0	0
$370$	−5.99363	−0.311594
$371$	47.0614	2.44331
$372$	0	0
$373$	12.1014	0.626589	0.313294	−	0.949656i	$-0.398567\pi$
$373$	12.1014	0.626589	0.313294	+	0.949656i	$0.398567\pi$
$374$	6.90947	0.357280
$375$	0	0
$376$	2.83883	0.146401
$377$	−12.3316	−0.635110
$378$	0	0
$379$	−20.3598	−1.04581	−0.522907	−	0.852390i	$-0.675152\pi$
$379$	−20.3598	−1.04581	−0.522907	+	0.852390i	$0.675152\pi$
$380$	−0.999605	−0.0512786
$381$	0	0
$382$	−16.4940	−0.843906
$383$	32.4413	1.65767	0.828837	−	0.559490i	$-0.189003\pi$
$383$	32.4413	1.65767	0.828837	+	0.559490i	$0.189003\pi$
$384$	0	0
$385$	−17.2349	−0.878374
$386$	−0.0702566	−0.00357597
$387$	0	0
$388$	18.8158	0.955226
$389$	2.91435	0.147763	0.0738817	−	0.997267i	$-0.476461\pi$
$389$	2.91435	0.147763	0.0738817	+	0.997267i	$0.476461\pi$
$390$	0	0
$391$	7.97046	0.403083
$392$	33.7974	1.70703
$393$	0	0
$394$	6.24325	0.314530
$395$	−11.3669	−0.571930
$396$	0	0
$397$	−1.40508	−0.0705188	−0.0352594	−	0.999378i	$-0.511226\pi$
$397$	−1.40508	−0.0705188	−0.0352594	+	0.999378i	$0.511226\pi$
$398$	21.2371	1.06452
$399$	0	0
$400$	0.208477	0.0104239
$401$	36.5447	1.82496	0.912478	−	0.409127i	$-0.134167\pi$
$401$	36.5447	1.82496	0.912478	+	0.409127i	$0.134167\pi$
$402$	0	0
$403$	−37.1857	−1.85235
$404$	6.40064	0.318444
$405$	0	0
$406$	12.9941	0.644886
$407$	40.5560	2.01029
$408$	0	0
$409$	35.4334	1.75207	0.876033	−	0.482251i	$-0.160181\pi$
$409$	35.4334	1.75207	0.876033	+	0.482251i	$0.160181\pi$
$410$	7.27585	0.359328
$411$	0	0
$412$	2.51189	0.123752
$413$	−7.12038	−0.350371
$414$	0	0
$415$	−4.53446	−0.222588
$416$	20.4671	1.00348
$417$	0	0
$418$	−4.27658	−0.209174
$419$	−7.07769	−0.345768	−0.172884	−	0.984942i	$-0.555309\pi$
$419$	−7.07769	−0.345768	−0.172884	+	0.984942i	$0.555309\pi$
$420$	0	0
$421$	−3.97834	−0.193892	−0.0969462	−	0.995290i	$-0.530907\pi$
$421$	−3.97834	−0.193892	−0.0969462	+	0.995290i	$0.530907\pi$
$422$	−13.0896	−0.637192
$423$	0	0
$424$	−30.7263	−1.49220
$425$	−7.00297	−0.339694
$426$	0	0
$427$	−32.8088	−1.58773
$428$	−0.232225	−0.0112250
$429$	0	0
$430$	−3.18870	−0.153773
$431$	−26.1897	−1.26151	−0.630757	−	0.775981i	$-0.717255\pi$
$431$	−26.1897	−1.26151	−0.630757	+	0.775981i	$0.717255\pi$
$432$	0	0
$433$	16.9650	0.815287	0.407643	−	0.913141i	$-0.366351\pi$
$433$	16.9650	0.815287	0.407643	+	0.913141i	$0.366351\pi$
$434$	39.1834	1.88086
$435$	0	0
$436$	−17.1010	−0.818990
$437$	−4.93327	−0.235990
$438$	0	0
$439$	−29.0000	−1.38409	−0.692046	−	0.721853i	$-0.743291\pi$
$439$	−29.0000	−1.38409	−0.692046	+	0.721853i	$0.743291\pi$
$440$	11.2527	0.536450
$441$	0	0
$442$	5.16483	0.245666
$443$	−40.2450	−1.91210	−0.956050	−	0.293205i	$-0.905278\pi$
$443$	−40.2450	−1.91210	−0.956050	+	0.293205i	$0.905278\pi$
$444$	0	0
$445$	−4.68373	−0.222030
$446$	1.58826	0.0752062
$447$	0	0
$448$	−21.9849	−1.03869
$449$	0.0780147	0.00368174	0.00184087	−	0.999998i	$-0.499414\pi$
$449$	0.0780147	0.00368174	0.00184087	+	0.999998i	$0.499414\pi$
$450$	0	0
$451$	−49.2322	−2.31825
$452$	−11.1643	−0.525126
$453$	0	0
$454$	−7.89814	−0.370678
$455$	−12.8831	−0.603970
$456$	0	0
$457$	−37.3177	−1.74565	−0.872824	−	0.488034i	$-0.837714\pi$
$457$	−37.3177	−1.74565	−0.872824	+	0.488034i	$0.837714\pi$
$458$	−1.63911	−0.0765906
$459$	0	0
$460$	4.93132	0.229924
$461$	−8.61870	−0.401413	−0.200707	−	0.979651i	$-0.564324\pi$
$461$	−8.61870	−0.401413	−0.200707	+	0.979651i	$0.564324\pi$
$462$	0	0
$463$	4.72677	0.219672	0.109836	−	0.993950i	$-0.464967\pi$
$463$	4.72677	0.219672	0.109836	+	0.993950i	$0.464967\pi$
$464$	−0.163308	−0.00758137
$465$	0	0
$466$	−4.42806	−0.205126
$467$	−30.2837	−1.40136	−0.700680	−	0.713475i	$-0.747120\pi$
$467$	−30.2837	−1.40136	−0.700680	+	0.713475i	$0.747120\pi$
$468$	0	0
$469$	−0.915035	−0.0422524
$470$	0.718060	0.0331216
$471$	0	0
$472$	4.64888	0.213982
$473$	21.5764	0.992083
$474$	0	0
$475$	4.33445	0.198878
$476$	8.60754	0.394526
$477$	0	0
$478$	−7.58154	−0.346772
$479$	−11.0732	−0.505947	−0.252974	−	0.967473i	$-0.581409\pi$
$479$	−11.0732	−0.505947	−0.252974	+	0.967473i	$0.581409\pi$
$480$	0	0
$481$	30.3156	1.38227
$482$	−6.13094	−0.279256
$483$	0	0
$484$	−15.4480	−0.702181
$485$	12.5278	0.568858
$486$	0	0
$487$	−0.880924	−0.0399185	−0.0199592	−	0.999801i	$-0.506354\pi$
$487$	−0.880924	−0.0399185	−0.0199592	+	0.999801i	$0.506354\pi$
$488$	21.4208	0.969674
$489$	0	0
$490$	8.54880	0.386196
$491$	−7.52580	−0.339634	−0.169817	−	0.985476i	$-0.554318\pi$
$491$	−7.52580	−0.339634	−0.169817	+	0.985476i	$0.554318\pi$
$492$	0	0
$493$	5.48568	0.247063
$494$	−3.19674	−0.143828
$495$	0	0
$496$	−0.492451	−0.0221117
$497$	1.63736	0.0734456
$498$	0	0
$499$	−31.3254	−1.40232	−0.701159	−	0.713005i	$-0.747334\pi$
$499$	−31.3254	−1.40232	−0.701159	+	0.713005i	$0.747334\pi$
$500$	−9.33076	−0.417284
$501$	0	0
$502$	−2.66388	−0.118895
$503$	−0.732110	−0.0326432	−0.0163216	−	0.999867i	$-0.505196\pi$
$503$	−0.732110	−0.0326432	−0.0163216	+	0.999867i	$0.505196\pi$
$504$	0	0
$505$	4.26163	0.189640
$506$	21.0975	0.937900
$507$	0	0
$508$	20.2552	0.898679
$509$	38.0619	1.68706	0.843532	−	0.537079i	$-0.180472\pi$
$509$	38.0619	1.68706	0.843532	+	0.537079i	$0.180472\pi$
$510$	0	0
$511$	−59.5660	−2.63505
$512$	0.544128	0.0240473
$513$	0	0
$514$	0.232643	0.0102614
$515$	1.67245	0.0736970
$516$	0	0
$517$	−4.85877	−0.213688
$518$	−31.9443	−1.40355
$519$	0	0
$520$	8.41137	0.368863
$521$	7.86473	0.344560	0.172280	−	0.985048i	$-0.444887\pi$
$521$	7.86473	0.344560	0.172280	+	0.985048i	$0.444887\pi$
$522$	0	0
$523$	22.3512	0.977350	0.488675	−	0.872466i	$-0.337480\pi$
$523$	22.3512	0.977350	0.488675	+	0.872466i	$0.337480\pi$
$524$	−3.42293	−0.149532
$525$	0	0
$526$	−1.14524	−0.0499348
$527$	16.5420	0.720579
$528$	0	0
$529$	1.33717	0.0581380
$530$	−7.77200	−0.337594
$531$	0	0
$532$	−5.32759	−0.230980
$533$	−36.8010	−1.59403
$534$	0	0
$535$	−0.154618	−0.00668473
$536$	0.597425	0.0258048
$537$	0	0
$538$	9.28151	0.400154
$539$	−57.8457	−2.49159
$540$	0	0
$541$	−36.0667	−1.55063	−0.775314	−	0.631576i	$-0.782408\pi$
$541$	−36.0667	−1.55063	−0.775314	+	0.631576i	$0.782408\pi$
$542$	−17.8399	−0.766291
$543$	0	0
$544$	−9.10472	−0.390362
$545$	−11.3861	−0.487726
$546$	0	0
$547$	−11.3813	−0.486628	−0.243314	−	0.969948i	$-0.578235\pi$
$547$	−11.3813	−0.486628	−0.243314	+	0.969948i	$0.578235\pi$
$548$	−9.26584	−0.395817
$549$	0	0
$550$	−18.5366	−0.790404
$551$	−3.39533	−0.144646
$552$	0	0
$553$	−60.5820	−2.57621
$554$	−1.59207	−0.0676404
$555$	0	0
$556$	15.2374	0.646211
$557$	33.0905	1.40209	0.701045	−	0.713117i	$-0.252717\pi$
$557$	33.0905	1.40209	0.701045	+	0.713117i	$0.252717\pi$
$558$	0	0
$559$	16.1283	0.682156
$560$	−0.170611	−0.00720965
$561$	0	0
$562$	6.02021	0.253947
$563$	42.9397	1.80969	0.904846	−	0.425740i	$-0.139986\pi$
$563$	42.9397	1.80969	0.904846	+	0.425740i	$0.139986\pi$
$564$	0	0
$565$	−7.43336	−0.312724
$566$	8.32140	0.349775
$567$	0	0
$568$	−1.06903	−0.0448555
$569$	12.4835	0.523334	0.261667	−	0.965158i	$-0.415728\pi$
$569$	12.4835	0.523334	0.261667	+	0.965158i	$0.415728\pi$
$570$	0	0
$571$	−42.7357	−1.78843	−0.894217	−	0.447634i	$-0.852267\pi$
$571$	−42.7357	−1.78843	−0.894217	+	0.447634i	$0.852267\pi$
$572$	−21.6223	−0.904073
$573$	0	0
$574$	38.7781	1.61857
$575$	−21.3830	−0.891734
$576$	0	0
$577$	40.1286	1.67058	0.835289	−	0.549812i	$-0.185301\pi$
$577$	40.1286	1.67058	0.835289	+	0.549812i	$0.185301\pi$
$578$	12.6655	0.526814
$579$	0	0
$580$	3.39399	0.140928
$581$	−24.1673	−1.00263
$582$	0	0
$583$	52.5894	2.17803
$584$	38.8906	1.60930
$585$	0	0
$586$	−0.927181	−0.0383015
$587$	−6.70862	−0.276895	−0.138447	−	0.990370i	$-0.544211\pi$
$587$	−6.70862	−0.276895	−0.138447	+	0.990370i	$0.544211\pi$
$588$	0	0
$589$	−10.2386	−0.421873
$590$	1.17590	0.0484111
$591$	0	0
$592$	0.401470	0.0165003
$593$	−27.8616	−1.14414	−0.572070	−	0.820205i	$-0.693860\pi$
$593$	−27.8616	−1.14414	−0.572070	+	0.820205i	$0.693860\pi$
$594$	0	0
$595$	5.73102	0.234949
$596$	17.7800	0.728296
$597$	0	0
$598$	15.7704	0.644900
$599$	−21.7098	−0.887040	−0.443520	−	0.896264i	$-0.646271\pi$
$599$	−21.7098	−0.887040	−0.443520	+	0.896264i	$0.646271\pi$
$600$	0	0
$601$	−38.5144	−1.57104	−0.785518	−	0.618839i	$-0.787603\pi$
$601$	−38.5144	−1.57104	−0.785518	+	0.618839i	$0.787603\pi$
$602$	−16.9948	−0.692656
$603$	0	0
$604$	4.69074	0.190864
$605$	−10.2855	−0.418164
$606$	0	0
$607$	−24.0548	−0.976356	−0.488178	−	0.872744i	$-0.662338\pi$
$607$	−24.0548	−0.976356	−0.488178	+	0.872744i	$0.662338\pi$
$608$	5.63532	0.228542
$609$	0	0
$610$	5.41823	0.219378
$611$	−3.63193	−0.146932
$612$	0	0
$613$	25.8424	1.04377	0.521883	−	0.853017i	$-0.325230\pi$
$613$	25.8424	1.04377	0.521883	+	0.853017i	$0.325230\pi$
$614$	13.0863	0.528122
$615$	0	0
$616$	59.9733	2.41639
$617$	−30.9980	−1.24793	−0.623966	−	0.781452i	$-0.714479\pi$
$617$	−30.9980	−1.24793	−0.623966	+	0.781452i	$0.714479\pi$
$618$	0	0
$619$	−2.48476	−0.0998709	−0.0499355	−	0.998752i	$-0.515902\pi$
$619$	−2.48476	−0.0998709	−0.0499355	+	0.998752i	$0.515902\pi$
$620$	10.2345	0.411028
$621$	0	0
$622$	−17.5864	−0.705152
$623$	−24.9629	−1.00012
$624$	0	0
$625$	15.4597	0.618388
$626$	26.3049	1.05136
$627$	0	0
$628$	−9.48361	−0.378437
$629$	−13.4858	−0.537715
$630$	0	0
$631$	−47.2755	−1.88201	−0.941003	−	0.338398i	$-0.890115\pi$
$631$	−47.2755	−1.88201	−0.941003	+	0.338398i	$0.890115\pi$
$632$	39.5539	1.57337
$633$	0	0
$634$	−1.78145	−0.0707503
$635$	13.4862	0.535183
$636$	0	0
$637$	−43.2396	−1.71322
$638$	14.5204	0.574869
$639$	0	0
$640$	−5.56402	−0.219937
$641$	28.4778	1.12481	0.562403	−	0.826864i	$-0.309877\pi$
$641$	28.4778	1.12481	0.562403	+	0.826864i	$0.309877\pi$
$642$	0	0
$643$	30.8947	1.21837	0.609184	−	0.793029i	$-0.291497\pi$
$643$	30.8947	1.21837	0.609184	+	0.793029i	$0.291497\pi$
$644$	26.2825	1.03567
$645$	0	0
$646$	1.42206	0.0559503
$647$	11.2445	0.442068	0.221034	−	0.975266i	$-0.429057\pi$
$647$	11.2445	0.442068	0.221034	+	0.975266i	$0.429057\pi$
$648$	0	0
$649$	−7.95676	−0.312330
$650$	−13.8561	−0.543482
$651$	0	0
$652$	26.8659	1.05215
$653$	−37.6204	−1.47220	−0.736101	−	0.676872i	$-0.763335\pi$
$653$	−37.6204	−1.47220	−0.736101	+	0.676872i	$0.763335\pi$
$654$	0	0
$655$	−2.27904	−0.0890493
$656$	−0.487357	−0.0190281
$657$	0	0
$658$	3.82704	0.149194
$659$	−48.7461	−1.89888	−0.949439	−	0.313952i	$-0.898347\pi$
$659$	−48.7461	−1.89888	−0.949439	+	0.313952i	$0.898347\pi$
$660$	0	0
$661$	5.96247	0.231913	0.115957	−	0.993254i	$-0.463007\pi$
$661$	5.96247	0.231913	0.115957	+	0.993254i	$0.463007\pi$
$662$	2.32408	0.0903278
$663$	0	0
$664$	15.7788	0.612336
$665$	−3.54718	−0.137554
$666$	0	0
$667$	16.7501	0.648567
$668$	13.3977	0.518371
$669$	0	0
$670$	0.151114	0.00583805
$671$	−36.6626	−1.41534
$672$	0	0
$673$	37.9156	1.46154	0.730770	−	0.682624i	$-0.239161\pi$
$673$	37.9156	1.46154	0.730770	+	0.682624i	$0.239161\pi$
$674$	1.16752	0.0449711
$675$	0	0
$676$	−0.233924	−0.00899709
$677$	−12.7646	−0.490585	−0.245292	−	0.969449i	$-0.578884\pi$
$677$	−12.7646	−0.490585	−0.245292	+	0.969449i	$0.578884\pi$
$678$	0	0
$679$	66.7694	2.56237
$680$	−3.74177	−0.143490
$681$	0	0
$682$	43.7860	1.67665
$683$	−38.6880	−1.48035	−0.740177	−	0.672412i	$-0.765258\pi$
$683$	−38.6880	−1.48035	−0.740177	+	0.672412i	$0.765258\pi$
$684$	0	0
$685$	−6.16932	−0.235718
$686$	18.7732	0.716765
$687$	0	0
$688$	0.213588	0.00814296
$689$	39.3106	1.49761
$690$	0	0
$691$	37.1563	1.41349	0.706747	−	0.707467i	$-0.250162\pi$
$691$	37.1563	1.41349	0.706747	+	0.707467i	$0.250162\pi$
$692$	13.1637	0.500408
$693$	0	0
$694$	26.6550	1.01181
$695$	10.1453	0.384833
$696$	0	0
$697$	16.3708	0.620090
$698$	3.34283	0.126528
$699$	0	0
$700$	−23.0922	−0.872802
$701$	−39.7274	−1.50048	−0.750242	−	0.661163i	$-0.770063\pi$
$701$	−39.7274	−1.50048	−0.750242	+	0.661163i	$0.770063\pi$
$702$	0	0
$703$	8.34698	0.314812
$704$	−24.5673	−0.925914
$705$	0	0
$706$	−30.7365	−1.15678
$707$	22.7132	0.854218
$708$	0	0
$709$	−14.8254	−0.556781	−0.278391	−	0.960468i	$-0.589801\pi$
$709$	−14.8254	−0.556781	−0.278391	+	0.960468i	$0.589801\pi$
$710$	−0.270403	−0.0101480
$711$	0	0
$712$	16.2982	0.610802
$713$	50.5096	1.89160
$714$	0	0
$715$	−14.3964	−0.538395
$716$	−17.2119	−0.643238
$717$	0	0
$718$	−14.9964	−0.559661
$719$	−0.795114	−0.0296527	−0.0148264	−	0.999890i	$-0.504720\pi$
$719$	−0.795114	−0.0296527	−0.0148264	+	0.999890i	$0.504720\pi$
$720$	0	0
$721$	8.91366	0.331962
$722$	−0.880178	−0.0327568
$723$	0	0
$724$	−13.5144	−0.502259
$725$	−14.7169	−0.546572
$726$	0	0
$727$	12.7341	0.472282	0.236141	−	0.971719i	$-0.424117\pi$
$727$	12.7341	0.472282	0.236141	+	0.971719i	$0.424117\pi$
$728$	44.8301	1.66151
$729$	0	0
$730$	9.83709	0.364087
$731$	−7.17465	−0.265364
$732$	0	0
$733$	17.0346	0.629187	0.314593	−	0.949227i	$-0.398132\pi$
$733$	17.0346	0.629187	0.314593	+	0.949227i	$0.398132\pi$
$734$	−18.9506	−0.699479
$735$	0	0
$736$	−27.8006	−1.02474
$737$	−1.02252	−0.0376649
$738$	0	0
$739$	−14.3129	−0.526509	−0.263254	−	0.964726i	$-0.584796\pi$
$739$	−14.3129	−0.526509	−0.263254	+	0.964726i	$0.584796\pi$
$740$	−8.34368	−0.306720
$741$	0	0
$742$	−41.4224	−1.52066
$743$	7.69794	0.282410	0.141205	−	0.989980i	$-0.454902\pi$
$743$	7.69794	0.282410	0.141205	+	0.989980i	$0.454902\pi$
$744$	0	0
$745$	11.8381	0.433716
$746$	−10.6514	−0.389976
$747$	0	0
$748$	9.61861	0.351691
$749$	−0.824069	−0.0301108
$750$	0	0
$751$	−12.3805	−0.451771	−0.225886	−	0.974154i	$-0.572528\pi$
$751$	−12.3805	−0.451771	−0.225886	+	0.974154i	$0.572528\pi$
$752$	−0.0480977	−0.00175394
$753$	0	0
$754$	10.8540	0.395280
$755$	3.12316	0.113663
$756$	0	0
$757$	46.0844	1.67497	0.837483	−	0.546464i	$-0.184026\pi$
$757$	46.0844	1.67497	0.837483	+	0.546464i	$0.184026\pi$
$758$	17.9203	0.650893
$759$	0	0
$760$	2.31595	0.0840084
$761$	−51.6318	−1.87165	−0.935825	−	0.352464i	$-0.885344\pi$
$761$	−51.6318	−1.87165	−0.935825	+	0.352464i	$0.885344\pi$
$762$	0	0
$763$	−60.6844	−2.19692
$764$	−22.9611	−0.830704
$765$	0	0
$766$	−28.5541	−1.03170
$767$	−5.94768	−0.214758
$768$	0	0
$769$	7.35698	0.265300	0.132650	−	0.991163i	$-0.457651\pi$
$769$	7.35698	0.265300	0.132650	+	0.991163i	$0.457651\pi$
$770$	15.1698	0.546682
$771$	0	0
$772$	−0.0978035	−0.00352003
$773$	41.2224	1.48267	0.741333	−	0.671137i	$-0.234194\pi$
$773$	41.2224	1.48267	0.741333	+	0.671137i	$0.234194\pi$
$774$	0	0
$775$	−44.3785	−1.59412
$776$	−43.5936	−1.56492
$777$	0	0
$778$	−2.56515	−0.0919650
$779$	−10.1326	−0.363040
$780$	0	0
$781$	1.82969	0.0654714
$782$	−7.01542	−0.250871
$783$	0	0
$784$	−0.572623	−0.0204508
$785$	−6.31432	−0.225368
$786$	0	0
$787$	−16.0244	−0.571207	−0.285604	−	0.958348i	$-0.592194\pi$
$787$	−16.0244	−0.571207	−0.285604	+	0.958348i	$0.592194\pi$
$788$	8.69117	0.309610
$789$	0	0
$790$	10.0049	0.355957
$791$	−39.6176	−1.40864
$792$	0	0
$793$	−27.4053	−0.973190
$794$	1.23672	0.0438895
$795$	0	0
$796$	29.5639	1.04786
$797$	−6.54284	−0.231759	−0.115880	−	0.993263i	$-0.536969\pi$
$797$	−6.54284	−0.231759	−0.115880	+	0.993263i	$0.536969\pi$
$798$	0	0
$799$	1.61565	0.0571577
$800$	24.4260	0.863590
$801$	0	0
$802$	−32.1658	−1.13582
$803$	−66.5629	−2.34895
$804$	0	0
$805$	17.4992	0.616766
$806$	32.7301	1.15287
$807$	0	0
$808$	−14.8294	−0.521697
$809$	−41.9109	−1.47351	−0.736755	−	0.676160i	$-0.763643\pi$
$809$	−41.9109	−1.47351	−0.736755	+	0.676160i	$0.763643\pi$
$810$	0	0
$811$	−30.5564	−1.07298	−0.536490	−	0.843907i	$-0.680250\pi$
$811$	−30.5564	−1.07298	−0.536490	+	0.843907i	$0.680250\pi$
$812$	18.0889	0.634798
$813$	0	0
$814$	−35.6965	−1.25116
$815$	17.8877	0.626579
$816$	0	0
$817$	4.44071	0.155361
$818$	−31.1877	−1.09045
$819$	0	0
$820$	10.1286	0.353707
$821$	−12.6793	−0.442509	−0.221255	−	0.975216i	$-0.571015\pi$
$821$	−12.6793	−0.442509	−0.221255	+	0.975216i	$0.571015\pi$
$822$	0	0
$823$	−4.99309	−0.174048	−0.0870242	−	0.996206i	$-0.527736\pi$
$823$	−4.99309	−0.174048	−0.0870242	+	0.996206i	$0.527736\pi$
$824$	−5.81971	−0.202739
$825$	0	0
$826$	6.26720	0.218064
$827$	2.13423	0.0742144	0.0371072	−	0.999311i	$-0.488186\pi$
$827$	2.13423	0.0742144	0.0371072	+	0.999311i	$0.488186\pi$
$828$	0	0
$829$	33.2658	1.15537	0.577685	−	0.816260i	$-0.303956\pi$
$829$	33.2658	1.15537	0.577685	+	0.816260i	$0.303956\pi$
$830$	3.99113	0.138534
$831$	0	0
$832$	−18.3640	−0.636659
$833$	19.2350	0.666454
$834$	0	0
$835$	8.92034	0.308701
$836$	−5.95339	−0.205902
$837$	0	0
$838$	6.22962	0.215199
$839$	41.6102	1.43654	0.718272	−	0.695763i	$-0.244934\pi$
$839$	41.6102	1.43654	0.718272	+	0.695763i	$0.244934\pi$
$840$	0	0
$841$	−17.4717	−0.602473
$842$	3.50165	0.120675
$843$	0	0
$844$	−18.2219	−0.627224
$845$	−0.155750	−0.00535796
$846$	0	0
$847$	−54.8185	−1.88359
$848$	0.520590	0.0178772
$849$	0	0
$850$	6.16386	0.211419
$851$	−41.1779	−1.41156
$852$	0	0
$853$	−14.4784	−0.495732	−0.247866	−	0.968794i	$-0.579729\pi$
$853$	−14.4784	−0.495732	−0.247866	+	0.968794i	$0.579729\pi$
$854$	28.8775	0.988169
$855$	0	0
$856$	0.538033	0.0183896
$857$	17.8710	0.610460	0.305230	−	0.952279i	$-0.401267\pi$
$857$	17.8710	0.610460	0.305230	+	0.952279i	$0.401267\pi$
$858$	0	0
$859$	27.3938	0.934663	0.467332	−	0.884082i	$-0.345215\pi$
$859$	27.3938	0.934663	0.467332	+	0.884082i	$0.345215\pi$
$860$	−4.43895	−0.151367
$861$	0	0
$862$	23.0516	0.785140
$863$	19.7884	0.673606	0.336803	−	0.941575i	$-0.390654\pi$
$863$	19.7884	0.673606	0.336803	+	0.941575i	$0.390654\pi$
$864$	0	0
$865$	8.76456	0.298004
$866$	−14.9322	−0.507418
$867$	0	0
$868$	54.5469	1.85144
$869$	−67.6982	−2.29650
$870$	0	0
$871$	−0.764332	−0.0258984
$872$	39.6207	1.34173
$873$	0	0
$874$	4.34216	0.146876
$875$	−33.1110	−1.11936
$876$	0	0
$877$	19.2964	0.651593	0.325796	−	0.945440i	$-0.394368\pi$
$877$	19.2964	0.651593	0.325796	+	0.945440i	$0.394368\pi$
$878$	25.5251	0.861431
$879$	0	0
$880$	−0.190652	−0.00642687
$881$	21.9969	0.741094	0.370547	−	0.928814i	$-0.379170\pi$
$881$	21.9969	0.741094	0.370547	+	0.928814i	$0.379170\pi$
$882$	0	0
$883$	11.8238	0.397904	0.198952	−	0.980009i	$-0.436246\pi$
$883$	11.8238	0.397904	0.198952	+	0.980009i	$0.436246\pi$
$884$	7.18991	0.241823
$885$	0	0
$886$	35.4228	1.19005
$887$	13.3811	0.449293	0.224646	−	0.974440i	$-0.427877\pi$
$887$	13.3811	0.449293	0.224646	+	0.974440i	$0.427877\pi$
$888$	0	0
$889$	71.8773	2.41069
$890$	4.12252	0.138187
$891$	0	0
$892$	2.21100	0.0740297
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	−11.4599	−0.383062
$896$	−29.6545	−0.990688
$897$	0	0
$898$	−0.0686668	−0.00229144
$899$	34.7633	1.15942
$900$	0	0
$901$	−17.4872	−0.582583
$902$	43.3331	1.44283
$903$	0	0
$904$	25.8663	0.860299
$905$	−8.99807	−0.299106
$906$	0	0
$907$	−1.32116	−0.0438685	−0.0219342	−	0.999759i	$-0.506982\pi$
$907$	−1.32116	−0.0438685	−0.0219342	+	0.999759i	$0.506982\pi$
$908$	−10.9949	−0.364879
$909$	0	0
$910$	11.3394	0.375899
$911$	21.9003	0.725588	0.362794	−	0.931869i	$-0.381823\pi$
$911$	21.9003	0.725588	0.362794	+	0.931869i	$0.381823\pi$
$912$	0	0
$913$	−27.0061	−0.893770
$914$	32.8462	1.08646
$915$	0	0
$916$	−2.28179	−0.0753924
$917$	−12.1466	−0.401115
$918$	0	0
$919$	−5.65784	−0.186635	−0.0933174	−	0.995636i	$-0.529747\pi$
$919$	−5.65784	−0.186635	−0.0933174	+	0.995636i	$0.529747\pi$
$920$	−11.4252	−0.376678
$921$	0	0
$922$	7.58599	0.249831
$923$	1.36769	0.0450181
$924$	0	0
$925$	36.1796	1.18958
$926$	−4.16040	−0.136719
$927$	0	0
$928$	−19.1338	−0.628097
$929$	18.4882	0.606579	0.303289	−	0.952899i	$-0.401915\pi$
$929$	18.4882	0.606579	0.303289	+	0.952899i	$0.401915\pi$
$930$	0	0
$931$	−11.9054	−0.390184
$932$	−6.16426	−0.201917
$933$	0	0
$934$	26.6550	0.872179
$935$	6.40420	0.209440
$936$	0	0
$937$	51.9830	1.69821	0.849105	−	0.528224i	$-0.177142\pi$
$937$	51.9830	1.69821	0.849105	+	0.528224i	$0.177142\pi$
$938$	0.805393	0.0262970
$939$	0	0
$940$	0.999605	0.0326035
$941$	−9.98684	−0.325562	−0.162781	−	0.986662i	$-0.552046\pi$
$941$	−9.98684	−0.325562	−0.162781	+	0.986662i	$0.552046\pi$
$942$	0	0
$943$	49.9871	1.62780
$944$	−0.0787652	−0.00256359
$945$	0	0
$946$	−18.9911	−0.617452
$947$	−28.4962	−0.926002	−0.463001	−	0.886358i	$-0.653227\pi$
$947$	−28.4962	−0.926002	−0.463001	+	0.886358i	$0.653227\pi$
$948$	0	0
$949$	−49.7557	−1.61514
$950$	−3.81509	−0.123778
$951$	0	0
$952$	−19.9425	−0.646341
$953$	−53.3032	−1.72666	−0.863330	−	0.504640i	$-0.831625\pi$
$953$	−53.3032	−1.72666	−0.863330	+	0.504640i	$0.831625\pi$
$954$	0	0
$955$	−15.2878	−0.494703
$956$	−10.5542	−0.341347
$957$	0	0
$958$	9.74639	0.314891
$959$	−32.8806	−1.06177
$960$	0	0
$961$	73.8281	2.38155
$962$	−26.6831	−0.860299
$963$	0	0
$964$	−8.53482	−0.274888
$965$	−0.0651189	−0.00209625
$966$	0	0
$967$	−20.6041	−0.662582	−0.331291	−	0.943529i	$-0.607484\pi$
$967$	−20.6041	−0.662582	−0.331291	+	0.943529i	$0.607484\pi$
$968$	35.7909	1.15036
$969$	0	0
$970$	−11.0267	−0.354046
$971$	0.181435	0.00582253	0.00291127	−	0.999996i	$-0.499073\pi$
$971$	0.181435	0.00582253	0.00291127	+	0.999996i	$0.499073\pi$
$972$	0	0
$973$	54.0713	1.73345
$974$	0.775370	0.0248445
$975$	0	0
$976$	−0.362928	−0.0116171
$977$	−27.3635	−0.875437	−0.437718	−	0.899112i	$-0.644213\pi$
$977$	−27.3635	−0.875437	−0.437718	+	0.899112i	$0.644213\pi$
$978$	0	0
$979$	−27.8951	−0.891531
$980$	11.9007	0.380154
$981$	0	0
$982$	6.62404	0.211382
$983$	4.63273	0.147761	0.0738805	−	0.997267i	$-0.476462\pi$
$983$	4.63273	0.147761	0.0738805	+	0.997267i	$0.476462\pi$
$984$	0	0
$985$	5.78670	0.184380
$986$	−4.82838	−0.153767
$987$	0	0
$988$	−4.45016	−0.141578
$989$	−21.9072	−0.696609
$990$	0	0
$991$	58.4319	1.85615	0.928076	−	0.372392i	$-0.121462\pi$
$991$	58.4319	1.85615	0.928076	+	0.372392i	$0.121462\pi$
$992$	−57.6975	−1.83190
$993$	0	0
$994$	−1.44117	−0.0457110
$995$	19.6840	0.624026
$996$	0	0
$997$	−60.6282	−1.92011	−0.960057	−	0.279803i	$-0.909731\pi$
$997$	−60.6282	−1.92011	−0.960057	+	0.279803i	$0.909731\pi$
$998$	27.5719	0.872774
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.n.1.5		16
3.2	odd	2		893.2.a.b.1.12	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.b.1.12	✓	16	3.2	odd	2
8037.2.a.n.1.5		16	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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q-0.880178 q^{2} -1.22529 q^{4} -0.815813 q^{5} -4.34804 q^{7} +2.83883 q^{8} +O(q^{10})\)	\(q-0.880178 q^{2} -1.22529 q^{4} -0.815813 q^{5} -4.34804 q^{7} +2.83883 q^{8} +0.718060 q^{10} -4.85877 q^{11} -3.63193 q^{13} +3.82704 q^{14} -0.0480977 q^{16} +1.61565 q^{17} -1.00000 q^{19} +0.999605 q^{20} +4.27658 q^{22} +4.93327 q^{23} -4.33445 q^{25} +3.19674 q^{26} +5.32759 q^{28} +3.39533 q^{29} +10.2386 q^{31} -5.63532 q^{32} -1.42206 q^{34} +3.54718 q^{35} -8.34698 q^{37} +0.880178 q^{38} -2.31595 q^{40} +10.1326 q^{41} -4.44071 q^{43} +5.95339 q^{44} -4.34216 q^{46} +1.00000 q^{47} +11.9054 q^{49} +3.81509 q^{50} +4.45016 q^{52} -10.8236 q^{53} +3.96385 q^{55} -12.3433 q^{56} -2.98850 q^{58} +1.63761 q^{59} +7.54565 q^{61} -9.01175 q^{62} +5.05628 q^{64} +2.96297 q^{65} +0.210448 q^{67} -1.97964 q^{68} -3.12215 q^{70} -0.376574 q^{71} +13.6995 q^{73} +7.34682 q^{74} +1.22529 q^{76} +21.1261 q^{77} +13.9332 q^{79} +0.0392387 q^{80} -8.91853 q^{82} +5.55821 q^{83} -1.31807 q^{85} +3.90861 q^{86} -13.7932 q^{88} +5.74118 q^{89} +15.7918 q^{91} -6.04467 q^{92} -0.880178 q^{94} +0.815813 q^{95} -15.3562 q^{97} -10.4789 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8}+O(q^{10}) \) 16 * q + 4 * q^2 + 10 * q^4 + q^5 - 9 * q^7 + 9 * q^8	\( 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8} - 15 q^{10} - 19 q^{13} + 6 q^{14} + 10 q^{16} + 8 q^{17} - 16 q^{19} + 11 q^{20} - 12 q^{22} + 5 q^{23} - 3 q^{25} - 9 q^{26} - 17 q^{28} + 2 q^{29} - 18 q^{31} - 3 q^{32} - 14 q^{34} + 11 q^{35} - 24 q^{37} - 4 q^{38} - 50 q^{40} + 6 q^{41} - 34 q^{43} + 4 q^{44} - 3 q^{46} + 16 q^{47} + 5 q^{49} - 26 q^{50} - 44 q^{52} + 23 q^{53} - 48 q^{55} + 3 q^{56} - 26 q^{58} + 32 q^{59} - 16 q^{61} - 32 q^{62} + 7 q^{64} + 18 q^{65} - 67 q^{67} + 19 q^{68} + 24 q^{70} - 19 q^{71} - 2 q^{73} + 29 q^{74} - 10 q^{76} - 14 q^{77} - 27 q^{79} - 15 q^{80} - 56 q^{82} + 17 q^{83} + 15 q^{85} + q^{86} - 13 q^{88} - 20 q^{89} - 42 q^{91} - 45 q^{92} + 4 q^{94} - q^{95} - 50 q^{97} - 13 q^{98}+O(q^{100}) \) 16 * q + 4 * q^2 + 10 * q^4 + q^5 - 9 * q^7 + 9 * q^8 - 15 * q^10 - 19 * q^13 + 6 * q^14 + 10 * q^16 + 8 * q^17 - 16 * q^19 + 11 * q^20 - 12 * q^22 + 5 * q^23 - 3 * q^25 - 9 * q^26 - 17 * q^28 + 2 * q^29 - 18 * q^31 - 3 * q^32 - 14 * q^34 + 11 * q^35 - 24 * q^37 - 4 * q^38 - 50 * q^40 + 6 * q^41 - 34 * q^43 + 4 * q^44 - 3 * q^46 + 16 * q^47 + 5 * q^49 - 26 * q^50 - 44 * q^52 + 23 * q^53 - 48 * q^55 + 3 * q^56 - 26 * q^58 + 32 * q^59 - 16 * q^61 - 32 * q^62 + 7 * q^64 + 18 * q^65 - 67 * q^67 + 19 * q^68 + 24 * q^70 - 19 * q^71 - 2 * q^73 + 29 * q^74 - 10 * q^76 - 14 * q^77 - 27 * q^79 - 15 * q^80 - 56 * q^82 + 17 * q^83 + 15 * q^85 + q^86 - 13 * q^88 - 20 * q^89 - 42 * q^91 - 45 * q^92 + 4 * q^94 - q^95 - 50 * q^97 - 13 * q^98

Introduction

L-functions

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