LMFDB - Embedded newform 8037.2.a.n.1.6

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			1.6
Root			$-0.788915$ 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)

$f(q)$	$=$	$q-0.788915 q^{2} -1.37761 q^{4} +2.33649 q^{5} +4.65174 q^{7} +2.66465 q^{8} +O(q^{10})$	\(q-0.788915 q^{2} -1.37761 q^{4} +2.33649 q^{5} +4.65174 q^{7} +2.66465 q^{8} -1.84329 q^{10} -4.40108 q^{11} -3.88128 q^{13} -3.66983 q^{14} +0.653043 q^{16} +2.66524 q^{17} -1.00000 q^{19} -3.21878 q^{20} +3.47208 q^{22} +2.46241 q^{23} +0.459183 q^{25} +3.06200 q^{26} -6.40830 q^{28} -8.77607 q^{29} +0.297037 q^{31} -5.84450 q^{32} -2.10265 q^{34} +10.8687 q^{35} -5.49698 q^{37} +0.788915 q^{38} +6.22593 q^{40} +7.37730 q^{41} -1.43097 q^{43} +6.06299 q^{44} -1.94264 q^{46} +1.00000 q^{47} +14.6387 q^{49} -0.362256 q^{50} +5.34690 q^{52} +5.14098 q^{53} -10.2831 q^{55} +12.3953 q^{56} +6.92358 q^{58} -0.215782 q^{59} -13.7753 q^{61} -0.234337 q^{62} +3.30472 q^{64} -9.06856 q^{65} -13.5770 q^{67} -3.67168 q^{68} -8.57452 q^{70} -4.95815 q^{71} -0.329217 q^{73} +4.33665 q^{74} +1.37761 q^{76} -20.4727 q^{77} -9.12610 q^{79} +1.52583 q^{80} -5.82006 q^{82} +6.05664 q^{83} +6.22732 q^{85} +1.12891 q^{86} -11.7273 q^{88} -6.82110 q^{89} -18.0547 q^{91} -3.39225 q^{92} -0.788915 q^{94} -2.33649 q^{95} +2.14473 q^{97} -11.5487 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.788915	−0.557847	−0.278924	−	0.960313i	$-0.589978\pi$
$2$	−0.788915	−0.557847	−0.278924	+	0.960313i	$0.589978\pi$
$3$	0	0
$3$	0	0
$4$	−1.37761	−0.688806
$5$	2.33649	1.04491	0.522455	−	0.852667i	$-0.325016\pi$
$5$	2.33649	1.04491	0.522455	+	0.852667i	$0.325016\pi$
$6$	0	0
$7$	4.65174	1.75819	0.879096	−	0.476644i	$-0.158147\pi$
$7$	4.65174	1.75819	0.879096	+	0.476644i	$0.158147\pi$
$8$	2.66465	0.942096
$9$	0	0
$10$	−1.84329	−0.582900
$11$	−4.40108	−1.32698	−0.663488	−	0.748187i	$-0.730925\pi$
$11$	−4.40108	−1.32698	−0.663488	+	0.748187i	$0.730925\pi$
$12$	0	0
$13$	−3.88128	−1.07647	−0.538236	−	0.842794i	$-0.680909\pi$
$13$	−3.88128	−1.07647	−0.538236	+	0.842794i	$0.680909\pi$
$14$	−3.66983	−0.980803
$15$	0	0
$16$	0.653043	0.163261
$17$	2.66524	0.646417	0.323208	−	0.946328i	$-0.395239\pi$
$17$	2.66524	0.646417	0.323208	+	0.946328i	$0.395239\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−3.21878	−0.719741
$21$	0	0
$22$	3.47208	0.740250
$23$	2.46241	0.513449	0.256724	−	0.966485i	$-0.417357\pi$
$23$	2.46241	0.513449	0.256724	+	0.966485i	$0.417357\pi$
$24$	0	0
$25$	0.459183	0.0918366
$26$	3.06200	0.600507
$27$	0	0
$28$	−6.40830	−1.21105
$29$	−8.77607	−1.62968	−0.814838	−	0.579689i	$-0.803174\pi$
$29$	−8.77607	−1.62968	−0.814838	+	0.579689i	$0.803174\pi$
$30$	0	0
$31$	0.297037	0.0533494	0.0266747	−	0.999644i	$-0.491508\pi$
$31$	0.297037	0.0533494	0.0266747	+	0.999644i	$0.491508\pi$
$32$	−5.84450	−1.03317
$33$	0	0
$34$	−2.10265	−0.360602
$35$	10.8687	1.83715
$36$	0	0
$37$	−5.49698	−0.903697	−0.451849	−	0.892095i	$-0.649235\pi$
$37$	−5.49698	−0.903697	−0.451849	+	0.892095i	$0.649235\pi$
$38$	0.788915	0.127979
$39$	0	0
$40$	6.22593	0.984405
$41$	7.37730	1.15214	0.576070	−	0.817400i	$-0.304586\pi$
$41$	7.37730	1.15214	0.576070	+	0.817400i	$0.304586\pi$
$42$	0	0
$43$	−1.43097	−0.218221	−0.109110	−	0.994030i	$-0.534800\pi$
$43$	−1.43097	−0.218221	−0.109110	+	0.994030i	$0.534800\pi$
$44$	6.06299	0.914030
$45$	0	0
$46$	−1.94264	−0.286426
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	14.6387	2.09124
$50$	−0.362256	−0.0512308
$51$	0	0
$52$	5.34690	0.741481
$53$	5.14098	0.706168	0.353084	−	0.935592i	$-0.385133\pi$
$53$	5.14098	0.706168	0.353084	+	0.935592i	$0.385133\pi$
$54$	0	0
$55$	−10.2831	−1.38657
$56$	12.3953	1.65639
$57$	0	0
$58$	6.92358	0.909110
$59$	−0.215782	−0.0280925	−0.0140462	−	0.999901i	$-0.504471\pi$
$59$	−0.215782	−0.0280925	−0.0140462	+	0.999901i	$0.504471\pi$
$60$	0	0
$61$	−13.7753	−1.76375	−0.881876	−	0.471481i	$-0.843720\pi$
$61$	−13.7753	−1.76375	−0.881876	+	0.471481i	$0.843720\pi$
$62$	−0.234337	−0.0297608
$63$	0	0
$64$	3.30472	0.413090
$65$	−9.06856	−1.12482
$66$	0	0
$67$	−13.5770	−1.65870	−0.829348	−	0.558732i	$-0.811288\pi$
$67$	−13.5770	−1.65870	−0.829348	+	0.558732i	$0.811288\pi$
$68$	−3.67168	−0.445256
$69$	0	0
$70$	−8.57452	−1.02485
$71$	−4.95815	−0.588424	−0.294212	−	0.955740i	$-0.595057\pi$
$71$	−4.95815	−0.588424	−0.294212	+	0.955740i	$0.595057\pi$
$72$	0	0
$73$	−0.329217	−0.0385319	−0.0192660	−	0.999814i	$-0.506133\pi$
$73$	−0.329217	−0.0385319	−0.0192660	+	0.999814i	$0.506133\pi$
$74$	4.33665	0.504125
$75$	0	0
$76$	1.37761	0.158023
$77$	−20.4727	−2.33308
$78$	0	0
$79$	−9.12610	−1.02677	−0.513383	−	0.858159i	$-0.671608\pi$
$79$	−9.12610	−1.02677	−0.513383	+	0.858159i	$0.671608\pi$
$80$	1.52583	0.170593
$81$	0	0
$82$	−5.82006	−0.642718
$83$	6.05664	0.664802	0.332401	−	0.943138i	$-0.392141\pi$
$83$	6.05664	0.664802	0.332401	+	0.943138i	$0.392141\pi$
$84$	0	0
$85$	6.22732	0.675447
$86$	1.12891	0.121734
$87$	0	0
$88$	−11.7273	−1.25014
$89$	−6.82110	−0.723035	−0.361518	−	0.932365i	$-0.617741\pi$
$89$	−6.82110	−0.723035	−0.361518	+	0.932365i	$0.617741\pi$
$90$	0	0
$91$	−18.0547	−1.89265
$92$	−3.39225	−0.353667
$93$	0	0
$94$	−0.788915	−0.0813704
$95$	−2.33649	−0.239719
$96$	0	0
$97$	2.14473	0.217765	0.108882	−	0.994055i	$-0.465273\pi$
$97$	2.14473	0.217765	0.108882	+	0.994055i	$0.465273\pi$
$98$	−11.5487	−1.16659
$99$	0	0
$100$	−0.632577	−0.0632577
$101$	−15.2641	−1.51884	−0.759418	−	0.650603i	$-0.774516\pi$
$101$	−15.2641	−1.51884	−0.759418	+	0.650603i	$0.774516\pi$
$102$	0	0
$103$	−7.37116	−0.726302	−0.363151	−	0.931730i	$-0.618299\pi$
$103$	−7.37116	−0.726302	−0.363151	+	0.931730i	$0.618299\pi$
$104$	−10.3422	−1.01414
$105$	0	0
$106$	−4.05580	−0.393934
$107$	−0.950957	−0.0919325	−0.0459663	−	0.998943i	$-0.514637\pi$
$107$	−0.950957	−0.0919325	−0.0459663	+	0.998943i	$0.514637\pi$
$108$	0	0
$109$	−18.3372	−1.75638	−0.878191	−	0.478310i	$-0.841249\pi$
$109$	−18.3372	−1.75638	−0.878191	+	0.478310i	$0.841249\pi$
$110$	8.11248	0.773495
$111$	0	0
$112$	3.03779	0.287044
$113$	2.64058	0.248405	0.124202	−	0.992257i	$-0.460363\pi$
$113$	2.64058	0.248405	0.124202	+	0.992257i	$0.460363\pi$
$114$	0	0
$115$	5.75340	0.536508
$116$	12.0900	1.12253
$117$	0	0
$118$	0.170234	0.0156713
$119$	12.3980	1.13653
$120$	0	0
$121$	8.36954	0.760868
$122$	10.8676	0.983905
$123$	0	0
$124$	−0.409202	−0.0367474
$125$	−10.6096	−0.948949
$126$	0	0
$127$	3.79558	0.336803	0.168402	−	0.985718i	$-0.446139\pi$
$127$	3.79558	0.336803	0.168402	+	0.985718i	$0.446139\pi$
$128$	9.08184	0.802729
$129$	0	0
$130$	7.15433	0.627476
$131$	6.04453	0.528113	0.264056	−	0.964507i	$-0.414940\pi$
$131$	6.04453	0.528113	0.264056	+	0.964507i	$0.414940\pi$
$132$	0	0
$133$	−4.65174	−0.403357
$134$	10.7111	0.925299
$135$	0	0
$136$	7.10194	0.608987
$137$	−7.16604	−0.612236	−0.306118	−	0.951994i	$-0.599030\pi$
$137$	−7.16604	−0.612236	−0.306118	+	0.951994i	$0.599030\pi$
$138$	0	0
$139$	−2.97530	−0.252361	−0.126181	−	0.992007i	$-0.540272\pi$
$139$	−2.97530	−0.252361	−0.126181	+	0.992007i	$0.540272\pi$
$140$	−14.9729	−1.26544
$141$	0	0
$142$	3.91156	0.328250
$143$	17.0818	1.42845
$144$	0	0
$145$	−20.5052	−1.70286
$146$	0.259724	0.0214949
$147$	0	0
$148$	7.57271	0.622473
$149$	23.2818	1.90732	0.953661	−	0.300885i	$-0.0972819\pi$
$149$	23.2818	1.90732	0.953661	+	0.300885i	$0.0972819\pi$
$150$	0	0
$151$	−1.55011	−0.126146	−0.0630732	−	0.998009i	$-0.520090\pi$
$151$	−1.55011	−0.126146	−0.0630732	+	0.998009i	$0.520090\pi$
$152$	−2.66465	−0.216132
$153$	0	0
$154$	16.1512	1.30150
$155$	0.694023	0.0557453
$156$	0	0
$157$	9.85196	0.786272	0.393136	−	0.919480i	$-0.371390\pi$
$157$	9.85196	0.786272	0.393136	+	0.919480i	$0.371390\pi$
$158$	7.19972	0.572779
$159$	0	0
$160$	−13.6556	−1.07957
$161$	11.4545	0.902742
$162$	0	0
$163$	−19.4461	−1.52313	−0.761567	−	0.648086i	$-0.775570\pi$
$163$	−19.4461	−1.52313	−0.761567	+	0.648086i	$0.775570\pi$
$164$	−10.1631	−0.793602
$165$	0	0
$166$	−4.77817	−0.370858
$167$	4.56298	0.353094	0.176547	−	0.984292i	$-0.443507\pi$
$167$	4.56298	0.353094	0.176547	+	0.984292i	$0.443507\pi$
$168$	0	0
$169$	2.06431	0.158793
$170$	−4.91282	−0.376796
$171$	0	0
$172$	1.97132	0.150312
$173$	−3.09813	−0.235547	−0.117773	−	0.993040i	$-0.537576\pi$
$173$	−3.09813	−0.235547	−0.117773	+	0.993040i	$0.537576\pi$
$174$	0	0
$175$	2.13600	0.161466
$176$	−2.87410	−0.216643
$177$	0	0
$178$	5.38127	0.403343
$179$	7.05491	0.527308	0.263654	−	0.964617i	$-0.415072\pi$
$179$	7.05491	0.527308	0.263654	+	0.964617i	$0.415072\pi$
$180$	0	0
$181$	−3.25738	−0.242119	−0.121060	−	0.992645i	$-0.538629\pi$
$181$	−3.25738	−0.242119	−0.121060	+	0.992645i	$0.538629\pi$
$182$	14.2436	1.05581
$183$	0	0
$184$	6.56147	0.483718
$185$	−12.8436	−0.944282
$186$	0	0
$187$	−11.7300	−0.857780
$188$	−1.37761	−0.100473
$189$	0	0
$190$	1.84329	0.133726
$191$	26.9760	1.95191	0.975956	−	0.217967i	$-0.0699424\pi$
$191$	26.9760	1.95191	0.975956	+	0.217967i	$0.0699424\pi$
$192$	0	0
$193$	5.96369	0.429276	0.214638	−	0.976694i	$-0.431143\pi$
$193$	5.96369	0.429276	0.214638	+	0.976694i	$0.431143\pi$
$194$	−1.69201	−0.121480
$195$	0	0
$196$	−20.1665	−1.44046
$197$	3.04334	0.216829	0.108415	−	0.994106i	$-0.465423\pi$
$197$	3.04334	0.216829	0.108415	+	0.994106i	$0.465423\pi$
$198$	0	0
$199$	−11.4311	−0.810331	−0.405165	−	0.914243i	$-0.632786\pi$
$199$	−11.4311	−0.810331	−0.405165	+	0.914243i	$0.632786\pi$
$200$	1.22356	0.0865189
$201$	0	0
$202$	12.0421	0.847279
$203$	−40.8240	−2.86528
$204$	0	0
$205$	17.2370	1.20388
$206$	5.81522	0.405166
$207$	0	0
$208$	−2.53464	−0.175746
$209$	4.40108	0.304429
$210$	0	0
$211$	−9.05564	−0.623416	−0.311708	−	0.950178i	$-0.600901\pi$
$211$	−9.05564	−0.623416	−0.311708	+	0.950178i	$0.600901\pi$
$212$	−7.08228	−0.486413
$213$	0	0
$214$	0.750225	0.0512843
$215$	−3.34345	−0.228021
$216$	0	0
$217$	1.38174	0.0937985
$218$	14.4665	0.979793
$219$	0	0
$220$	14.1661	0.955079
$221$	−10.3446	−0.695850
$222$	0	0
$223$	8.56605	0.573625	0.286813	−	0.957987i	$-0.407404\pi$
$223$	8.56605	0.573625	0.286813	+	0.957987i	$0.407404\pi$
$224$	−27.1871	−1.81651
$225$	0	0
$226$	−2.08319	−0.138572
$227$	10.0388	0.666297	0.333149	−	0.942874i	$-0.391889\pi$
$227$	10.0388	0.666297	0.333149	+	0.942874i	$0.391889\pi$
$228$	0	0
$229$	23.8413	1.57548	0.787739	−	0.616010i	$-0.211252\pi$
$229$	23.8413	1.57548	0.787739	+	0.616010i	$0.211252\pi$
$230$	−4.53895	−0.299289
$231$	0	0
$232$	−23.3852	−1.53531
$233$	−18.1270	−1.18754	−0.593771	−	0.804634i	$-0.702362\pi$
$233$	−18.1270	−1.18754	−0.593771	+	0.804634i	$0.702362\pi$
$234$	0	0
$235$	2.33649	0.152416
$236$	0.297264	0.0193503
$237$	0	0
$238$	−9.78099	−0.634008
$239$	−9.47410	−0.612829	−0.306414	−	0.951898i	$-0.599129\pi$
$239$	−9.47410	−0.612829	−0.306414	+	0.951898i	$0.599129\pi$
$240$	0	0
$241$	−13.5864	−0.875179	−0.437590	−	0.899175i	$-0.644168\pi$
$241$	−13.5864	−0.875179	−0.437590	+	0.899175i	$0.644168\pi$
$242$	−6.60286	−0.424448
$243$	0	0
$244$	18.9771	1.21488
$245$	34.2032	2.18516
$246$	0	0
$247$	3.88128	0.246960
$248$	0.791499	0.0502602
$249$	0	0
$250$	8.37005	0.529368
$251$	−3.32273	−0.209729	−0.104864	−	0.994487i	$-0.533441\pi$
$251$	−3.32273	−0.209729	−0.104864	+	0.994487i	$0.533441\pi$
$252$	0	0
$253$	−10.8373	−0.681335
$254$	−2.99439	−0.187885
$255$	0	0
$256$	−13.7743	−0.860891
$257$	15.5246	0.968395	0.484198	−	0.874959i	$-0.339112\pi$
$257$	15.5246	0.968395	0.484198	+	0.874959i	$0.339112\pi$
$258$	0	0
$259$	−25.5705	−1.58887
$260$	12.4930	0.774781
$261$	0	0
$262$	−4.76862	−0.294606
$263$	14.6820	0.905329	0.452664	−	0.891681i	$-0.350474\pi$
$263$	14.6820	0.905329	0.452664	+	0.891681i	$0.350474\pi$
$264$	0	0
$265$	12.0118	0.737882
$266$	3.66983	0.225012
$267$	0	0
$268$	18.7039	1.14252
$269$	−2.91251	−0.177579	−0.0887894	−	0.996050i	$-0.528300\pi$
$269$	−2.91251	−0.177579	−0.0887894	+	0.996050i	$0.528300\pi$
$270$	0	0
$271$	−15.0791	−0.915991	−0.457996	−	0.888954i	$-0.651432\pi$
$271$	−15.0791	−0.915991	−0.457996	+	0.888954i	$0.651432\pi$
$272$	1.74052	0.105535
$273$	0	0
$274$	5.65340	0.341534
$275$	−2.02090	−0.121865
$276$	0	0
$277$	−16.0320	−0.963270	−0.481635	−	0.876372i	$-0.659957\pi$
$277$	−16.0320	−0.963270	−0.481635	+	0.876372i	$0.659957\pi$
$278$	2.34726	0.140779
$279$	0	0
$280$	28.9614	1.73077
$281$	−4.21388	−0.251379	−0.125690	−	0.992070i	$-0.540114\pi$
$281$	−4.21388	−0.251379	−0.125690	+	0.992070i	$0.540114\pi$
$282$	0	0
$283$	14.5854	0.867009	0.433505	−	0.901151i	$-0.357277\pi$
$283$	14.5854	0.867009	0.433505	+	0.901151i	$0.357277\pi$
$284$	6.83041	0.405310
$285$	0	0
$286$	−13.4761	−0.796859
$287$	34.3173	2.02568
$288$	0	0
$289$	−9.89647	−0.582145
$290$	16.1769	0.949938
$291$	0	0
$292$	0.453533	0.0265410
$293$	−22.8859	−1.33701	−0.668504	−	0.743708i	$-0.733065\pi$
$293$	−22.8859	−1.33701	−0.668504	+	0.743708i	$0.733065\pi$
$294$	0	0
$295$	−0.504173	−0.0293541
$296$	−14.6475	−0.851370
$297$	0	0
$298$	−18.3674	−1.06399
$299$	−9.55731	−0.552713
$300$	0	0
$301$	−6.65651	−0.383675
$302$	1.22291	0.0703704
$303$	0	0
$304$	−0.653043	−0.0374546
$305$	−32.1860	−1.84296
$306$	0	0
$307$	−3.64235	−0.207880	−0.103940	−	0.994584i	$-0.533145\pi$
$307$	−3.64235	−0.207880	−0.103940	+	0.994584i	$0.533145\pi$
$308$	28.2035	1.60704
$309$	0	0
$310$	−0.547525	−0.0310974
$311$	−3.05567	−0.173271	−0.0866354	−	0.996240i	$-0.527612\pi$
$311$	−3.05567	−0.173271	−0.0866354	+	0.996240i	$0.527612\pi$
$312$	0	0
$313$	3.86139	0.218259	0.109129	−	0.994028i	$-0.465194\pi$
$313$	3.86139	0.218259	0.109129	+	0.994028i	$0.465194\pi$
$314$	−7.77236	−0.438619
$315$	0	0
$316$	12.5722	0.707244
$317$	−10.0210	−0.562838	−0.281419	−	0.959585i	$-0.590805\pi$
$317$	−10.0210	−0.562838	−0.281419	+	0.959585i	$0.590805\pi$
$318$	0	0
$319$	38.6242	2.16254
$320$	7.72145	0.431642
$321$	0	0
$322$	−9.03664	−0.503592
$323$	−2.66524	−0.148298
$324$	0	0
$325$	−1.78222	−0.0988596
$326$	15.3413	0.849676
$327$	0	0
$328$	19.6579	1.08543
$329$	4.65174	0.256459
$330$	0	0
$331$	2.45479	0.134927	0.0674637	−	0.997722i	$-0.478509\pi$
$331$	2.45479	0.134927	0.0674637	+	0.997722i	$0.478509\pi$
$332$	−8.34370	−0.457920
$333$	0	0
$334$	−3.59981	−0.196973
$335$	−31.7225	−1.73319
$336$	0	0
$337$	4.75543	0.259045	0.129522	−	0.991577i	$-0.458656\pi$
$337$	4.75543	0.259045	0.129522	+	0.991577i	$0.458656\pi$
$338$	−1.62856	−0.0885822
$339$	0	0
$340$	−8.57883	−0.465253
$341$	−1.30728	−0.0707934
$342$	0	0
$343$	35.5332	1.91861
$344$	−3.81304	−0.205585
$345$	0	0
$346$	2.44416	0.131399
$347$	2.44667	0.131344	0.0656722	−	0.997841i	$-0.479081\pi$
$347$	2.44667	0.131344	0.0656722	+	0.997841i	$0.479081\pi$
$348$	0	0
$349$	36.2380	1.93978	0.969888	−	0.243553i	$-0.0783130\pi$
$349$	36.2380	1.93978	0.969888	+	0.243553i	$0.0783130\pi$
$350$	−1.68512	−0.0900736
$351$	0	0
$352$	25.7221	1.37099
$353$	27.7274	1.47578	0.737891	−	0.674920i	$-0.235822\pi$
$353$	27.7274	1.47578	0.737891	+	0.674920i	$0.235822\pi$
$354$	0	0
$355$	−11.5847	−0.614850
$356$	9.39683	0.498031
$357$	0	0
$358$	−5.56572	−0.294158
$359$	−10.7458	−0.567139	−0.283570	−	0.958952i	$-0.591519\pi$
$359$	−10.7458	−0.567139	−0.283570	+	0.958952i	$0.591519\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	2.56980	0.135066
$363$	0	0
$364$	24.8724	1.30367
$365$	−0.769212	−0.0402624
$366$	0	0
$367$	8.64940	0.451495	0.225748	−	0.974186i	$-0.427518\pi$
$367$	8.64940	0.451495	0.225748	+	0.974186i	$0.427518\pi$
$368$	1.60806	0.0838261
$369$	0	0
$370$	10.1325	0.526765
$371$	23.9145	1.24158
$372$	0	0
$373$	14.2937	0.740099	0.370049	−	0.929012i	$-0.379341\pi$
$373$	14.2937	0.740099	0.370049	+	0.929012i	$0.379341\pi$
$374$	9.25395	0.478510
$375$	0	0
$376$	2.66465	0.137419
$377$	34.0624	1.75430
$378$	0	0
$379$	3.14858	0.161732	0.0808658	−	0.996725i	$-0.474231\pi$
$379$	3.14858	0.161732	0.0808658	+	0.996725i	$0.474231\pi$
$380$	3.21878	0.165120
$381$	0	0
$382$	−21.2817	−1.08887
$383$	−24.2450	−1.23886	−0.619431	−	0.785051i	$-0.712636\pi$
$383$	−24.2450	−1.23886	−0.619431	+	0.785051i	$0.712636\pi$
$384$	0	0
$385$	−47.8343	−2.43786
$386$	−4.70484	−0.239470
$387$	0	0
$388$	−2.95461	−0.149998
$389$	−9.76065	−0.494885	−0.247442	−	0.968903i	$-0.579590\pi$
$389$	−9.76065	−0.494885	−0.247442	+	0.968903i	$0.579590\pi$
$390$	0	0
$391$	6.56294	0.331902
$392$	39.0070	1.97015
$393$	0	0
$394$	−2.40094	−0.120957
$395$	−21.3230	−1.07288
$396$	0	0
$397$	27.4624	1.37830	0.689149	−	0.724620i	$-0.257985\pi$
$397$	27.4624	1.37830	0.689149	+	0.724620i	$0.257985\pi$
$398$	9.01819	0.452041
$399$	0	0
$400$	0.299867	0.0149933
$401$	−11.1271	−0.555659	−0.277829	−	0.960630i	$-0.589615\pi$
$401$	−11.1271	−0.555659	−0.277829	+	0.960630i	$0.589615\pi$
$402$	0	0
$403$	−1.15288	−0.0574291
$404$	21.0280	1.04618
$405$	0	0
$406$	32.2067	1.59839
$407$	24.1927	1.19919
$408$	0	0
$409$	−11.0037	−0.544099	−0.272049	−	0.962283i	$-0.587701\pi$
$409$	−11.0037	−0.544099	−0.272049	+	0.962283i	$0.587701\pi$
$410$	−13.5985	−0.671583
$411$	0	0
$412$	10.1546	0.500282
$413$	−1.00376	−0.0493920
$414$	0	0
$415$	14.1513	0.694659
$416$	22.6841	1.11218
$417$	0	0
$418$	−3.47208	−0.169825
$419$	−32.6971	−1.59736	−0.798678	−	0.601758i	$-0.794467\pi$
$419$	−32.6971	−1.59736	−0.798678	+	0.601758i	$0.794467\pi$
$420$	0	0
$421$	−12.3141	−0.600155	−0.300077	−	0.953915i	$-0.597013\pi$
$421$	−12.3141	−0.600155	−0.300077	+	0.953915i	$0.597013\pi$
$422$	7.14413	0.347771
$423$	0	0
$424$	13.6989	0.665278
$425$	1.22384	0.0593647
$426$	0	0
$427$	−64.0794	−3.10102
$428$	1.31005	0.0633237
$429$	0	0
$430$	2.63770	0.127201
$431$	−15.1674	−0.730588	−0.365294	−	0.930892i	$-0.619031\pi$
$431$	−15.1674	−0.730588	−0.365294	+	0.930892i	$0.619031\pi$
$432$	0	0
$433$	−9.12022	−0.438290	−0.219145	−	0.975692i	$-0.570327\pi$
$433$	−9.12022	−0.438290	−0.219145	+	0.975692i	$0.570327\pi$
$434$	−1.09007	−0.0523252
$435$	0	0
$436$	25.2615	1.20981
$437$	−2.46241	−0.117793
$438$	0	0
$439$	26.1747	1.24925	0.624625	−	0.780925i	$-0.285252\pi$
$439$	26.1747	1.24925	0.624625	+	0.780925i	$0.285252\pi$
$440$	−27.4008	−1.30628
$441$	0	0
$442$	8.16097	0.388178
$443$	7.58343	0.360300	0.180150	−	0.983639i	$-0.442342\pi$
$443$	7.58343	0.360300	0.180150	+	0.983639i	$0.442342\pi$
$444$	0	0
$445$	−15.9374	−0.755506
$446$	−6.75789	−0.319995
$447$	0	0
$448$	15.3727	0.726293
$449$	−24.2298	−1.14347	−0.571737	−	0.820437i	$-0.693730\pi$
$449$	−24.2298	−1.14347	−0.571737	+	0.820437i	$0.693730\pi$
$450$	0	0
$451$	−32.4681	−1.52886
$452$	−3.63769	−0.171103
$453$	0	0
$454$	−7.91975	−0.371692
$455$	−42.1846	−1.97764
$456$	0	0
$457$	39.9049	1.86667	0.933336	−	0.359004i	$-0.116884\pi$
$457$	39.9049	1.86667	0.933336	+	0.359004i	$0.116884\pi$
$458$	−18.8088	−0.878876
$459$	0	0
$460$	−7.92596	−0.369550
$461$	−11.5560	−0.538219	−0.269109	−	0.963110i	$-0.586729\pi$
$461$	−11.5560	−0.538219	−0.269109	+	0.963110i	$0.586729\pi$
$462$	0	0
$463$	−36.0589	−1.67580	−0.837899	−	0.545826i	$-0.816216\pi$
$463$	−36.0589	−1.67580	−0.837899	+	0.545826i	$0.816216\pi$
$464$	−5.73116	−0.266062
$465$	0	0
$466$	14.3007	0.662467
$467$	−8.79092	−0.406795	−0.203398	−	0.979096i	$-0.565198\pi$
$467$	−8.79092	−0.406795	−0.203398	+	0.979096i	$0.565198\pi$
$468$	0	0
$469$	−63.1567	−2.91631
$470$	−1.84329	−0.0850247
$471$	0	0
$472$	−0.574984	−0.0264658
$473$	6.29782	0.289574
$474$	0	0
$475$	−0.459183	−0.0210688
$476$	−17.0797	−0.782846
$477$	0	0
$478$	7.47426	0.341865
$479$	32.2408	1.47312	0.736559	−	0.676373i	$-0.236449\pi$
$479$	32.2408	1.47312	0.736559	+	0.676373i	$0.236449\pi$
$480$	0	0
$481$	21.3353	0.972805
$482$	10.7185	0.488216
$483$	0	0
$484$	−11.5300	−0.524091
$485$	5.01115	0.227545
$486$	0	0
$487$	13.0245	0.590198	0.295099	−	0.955467i	$-0.404647\pi$
$487$	13.0245	0.590198	0.295099	+	0.955467i	$0.404647\pi$
$488$	−36.7065	−1.66162
$489$	0	0
$490$	−26.9834	−1.21899
$491$	−0.832162	−0.0375549	−0.0187775	−	0.999824i	$-0.505977\pi$
$491$	−0.832162	−0.0375549	−0.0187775	+	0.999824i	$0.505977\pi$
$492$	0	0
$493$	−23.3904	−1.05345
$494$	−3.06200	−0.137766
$495$	0	0
$496$	0.193978	0.00870987
$497$	−23.0640	−1.03456
$498$	0	0
$499$	−26.5619	−1.18907	−0.594537	−	0.804069i	$-0.702664\pi$
$499$	−26.5619	−1.18907	−0.594537	+	0.804069i	$0.702664\pi$
$500$	14.6159	0.653642
$501$	0	0
$502$	2.62135	0.116996
$503$	10.4607	0.466417	0.233209	−	0.972427i	$-0.425077\pi$
$503$	10.4607	0.466417	0.233209	+	0.972427i	$0.425077\pi$
$504$	0	0
$505$	−35.6644	−1.58705
$506$	8.54970	0.380081
$507$	0	0
$508$	−5.22884	−0.231992
$509$	15.8055	0.700566	0.350283	−	0.936644i	$-0.386085\pi$
$509$	15.8055	0.700566	0.350283	+	0.936644i	$0.386085\pi$
$510$	0	0
$511$	−1.53143	−0.0677466
$512$	−7.29697	−0.322484
$513$	0	0
$514$	−12.2476	−0.540217
$515$	−17.2226	−0.758920
$516$	0	0
$517$	−4.40108	−0.193559
$518$	20.1730	0.886349
$519$	0	0
$520$	−24.1645	−1.05969
$521$	−34.3572	−1.50522	−0.752608	−	0.658469i	$-0.771204\pi$
$521$	−34.3572	−1.50522	−0.752608	+	0.658469i	$0.771204\pi$
$522$	0	0
$523$	−18.5283	−0.810187	−0.405094	−	0.914275i	$-0.632761\pi$
$523$	−18.5283	−0.810187	−0.405094	+	0.914275i	$0.632761\pi$
$524$	−8.32702	−0.363767
$525$	0	0
$526$	−11.5828	−0.505035
$527$	0.791676	0.0344859
$528$	0	0
$529$	−16.9365	−0.736370
$530$	−9.47632	−0.411625
$531$	0	0
$532$	6.40830	0.277835
$533$	−28.6333	−1.24025
$534$	0	0
$535$	−2.22190	−0.0960612
$536$	−36.1780	−1.56265
$537$	0	0
$538$	2.29772	0.0990619
$539$	−64.4261	−2.77503
$540$	0	0
$541$	−11.7898	−0.506883	−0.253441	−	0.967351i	$-0.581562\pi$
$541$	−11.7898	−0.506883	−0.253441	+	0.967351i	$0.581562\pi$
$542$	11.8961	0.510983
$543$	0	0
$544$	−15.5770	−0.667859
$545$	−42.8446	−1.83526
$546$	0	0
$547$	32.5316	1.39095	0.695476	−	0.718549i	$-0.255193\pi$
$547$	32.5316	1.39095	0.695476	+	0.718549i	$0.255193\pi$
$548$	9.87203	0.421712
$549$	0	0
$550$	1.59432	0.0679821
$551$	8.77607	0.373873
$552$	0	0
$553$	−42.4523	−1.80525
$554$	12.6479	0.537358
$555$	0	0
$556$	4.09881	0.173828
$557$	0.146842	0.00622191	0.00311096	−	0.999995i	$-0.499010\pi$
$557$	0.146842	0.00622191	0.00311096	+	0.999995i	$0.499010\pi$
$558$	0	0
$559$	5.55399	0.234909
$560$	7.09776	0.299935
$561$	0	0
$562$	3.32439	0.140231
$563$	11.5980	0.488797	0.244399	−	0.969675i	$-0.421409\pi$
$563$	11.5980	0.488797	0.244399	+	0.969675i	$0.421409\pi$
$564$	0	0
$565$	6.16968	0.259561
$566$	−11.5066	−0.483659
$567$	0	0
$568$	−13.2117	−0.554351
$569$	41.0968	1.72287	0.861434	−	0.507869i	$-0.169567\pi$
$569$	41.0968	1.72287	0.861434	+	0.507869i	$0.169567\pi$
$570$	0	0
$571$	−16.0764	−0.672776	−0.336388	−	0.941724i	$-0.609205\pi$
$571$	−16.0764	−0.672776	−0.336388	+	0.941724i	$0.609205\pi$
$572$	−23.5321	−0.983928
$573$	0	0
$574$	−27.0734	−1.13002
$575$	1.13070	0.0471534
$576$	0	0
$577$	−1.01284	−0.0421652	−0.0210826	−	0.999778i	$-0.506711\pi$
$577$	−1.01284	−0.0421652	−0.0210826	+	0.999778i	$0.506711\pi$
$578$	7.80747	0.324748
$579$	0	0
$580$	28.2482	1.17294
$581$	28.1739	1.16885
$582$	0	0
$583$	−22.6259	−0.937068
$584$	−0.877248	−0.0363008
$585$	0	0
$586$	18.0550	0.745847
$587$	17.5502	0.724373	0.362187	−	0.932106i	$-0.382030\pi$
$587$	17.5502	0.724373	0.362187	+	0.932106i	$0.382030\pi$
$588$	0	0
$589$	−0.297037	−0.0122392
$590$	0.397750	0.0163751
$591$	0	0
$592$	−3.58976	−0.147538
$593$	−1.54459	−0.0634286	−0.0317143	−	0.999497i	$-0.510097\pi$
$593$	−1.54459	−0.0634286	−0.0317143	+	0.999497i	$0.510097\pi$
$594$	0	0
$595$	28.9679	1.18757
$596$	−32.0733	−1.31378
$597$	0	0
$598$	7.53990	0.308330
$599$	9.07124	0.370641	0.185320	−	0.982678i	$-0.440668\pi$
$599$	9.07124	0.370641	0.185320	+	0.982678i	$0.440668\pi$
$600$	0	0
$601$	−32.9625	−1.34457	−0.672285	−	0.740292i	$-0.734687\pi$
$601$	−32.9625	−1.34457	−0.672285	+	0.740292i	$0.734687\pi$
$602$	5.25142	0.214032
$603$	0	0
$604$	2.13545	0.0868904
$605$	19.5554	0.795038
$606$	0	0
$607$	24.1734	0.981166	0.490583	−	0.871394i	$-0.336784\pi$
$607$	24.1734	0.981166	0.490583	+	0.871394i	$0.336784\pi$
$608$	5.84450	0.237026
$609$	0	0
$610$	25.3920	1.02809
$611$	−3.88128	−0.157020
$612$	0	0
$613$	−28.8250	−1.16423	−0.582115	−	0.813107i	$-0.697775\pi$
$613$	−28.8250	−1.16423	−0.582115	+	0.813107i	$0.697775\pi$
$614$	2.87350	0.115965
$615$	0	0
$616$	−54.5526	−2.19799
$617$	11.1122	0.447361	0.223681	−	0.974662i	$-0.428193\pi$
$617$	11.1122	0.447361	0.223681	+	0.974662i	$0.428193\pi$
$618$	0	0
$619$	−22.0867	−0.887739	−0.443869	−	0.896092i	$-0.646395\pi$
$619$	−22.0867	−0.887739	−0.443869	+	0.896092i	$0.646395\pi$
$620$	−0.956096	−0.0383977
$621$	0	0
$622$	2.41066	0.0966587
$623$	−31.7300	−1.27124
$624$	0	0
$625$	−27.0851	−1.08340
$626$	−3.04631	−0.121755
$627$	0	0
$628$	−13.5722	−0.541589
$629$	−14.6508	−0.584165
$630$	0	0
$631$	−45.7340	−1.82064	−0.910320	−	0.413905i	$-0.864165\pi$
$631$	−45.7340	−1.82064	−0.910320	+	0.413905i	$0.864165\pi$
$632$	−24.3179	−0.967313
$633$	0	0
$634$	7.90575	0.313977
$635$	8.86834	0.351929
$636$	0	0
$637$	−56.8168	−2.25116
$638$	−30.4712	−1.20637
$639$	0	0
$640$	21.2196	0.838780
$641$	−11.0537	−0.436595	−0.218298	−	0.975882i	$-0.570050\pi$
$641$	−11.0537	−0.436595	−0.218298	+	0.975882i	$0.570050\pi$
$642$	0	0
$643$	0.288951	0.0113951	0.00569756	−	0.999984i	$-0.498186\pi$
$643$	0.288951	0.0113951	0.00569756	+	0.999984i	$0.498186\pi$
$644$	−15.7799	−0.621814
$645$	0	0
$646$	2.10265	0.0827277
$647$	25.0748	0.985791	0.492895	−	0.870089i	$-0.335939\pi$
$647$	25.0748	0.985791	0.492895	+	0.870089i	$0.335939\pi$
$648$	0	0
$649$	0.949676	0.0372780
$650$	1.40602	0.0551485
$651$	0	0
$652$	26.7892	1.04914
$653$	9.74553	0.381372	0.190686	−	0.981651i	$-0.438929\pi$
$653$	9.74553	0.381372	0.190686	+	0.981651i	$0.438929\pi$
$654$	0	0
$655$	14.1230	0.551830
$656$	4.81770	0.188099
$657$	0	0
$658$	−3.66983	−0.143065
$659$	−43.9605	−1.71246	−0.856228	−	0.516598i	$-0.827198\pi$
$659$	−43.9605	−1.71246	−0.856228	+	0.516598i	$0.827198\pi$
$660$	0	0
$661$	−37.1253	−1.44401	−0.722003	−	0.691890i	$-0.756778\pi$
$661$	−37.1253	−1.44401	−0.722003	+	0.691890i	$0.756778\pi$
$662$	−1.93662	−0.0752689
$663$	0	0
$664$	16.1388	0.626308
$665$	−10.8687	−0.421472
$666$	0	0
$667$	−21.6103	−0.836755
$668$	−6.28602	−0.243214
$669$	0	0
$670$	25.0264	0.966854
$671$	60.6265	2.34046
$672$	0	0
$673$	−45.2371	−1.74376	−0.871881	−	0.489718i	$-0.837100\pi$
$673$	−45.2371	−1.74376	−0.871881	+	0.489718i	$0.837100\pi$
$674$	−3.75163	−0.144507
$675$	0	0
$676$	−2.84382	−0.109378
$677$	46.6465	1.79277	0.896386	−	0.443274i	$-0.146183\pi$
$677$	46.6465	1.79277	0.896386	+	0.443274i	$0.146183\pi$
$678$	0	0
$679$	9.97675	0.382873
$680$	16.5936	0.636336
$681$	0	0
$682$	1.03134	0.0394919
$683$	20.4511	0.782541	0.391270	−	0.920276i	$-0.372036\pi$
$683$	20.4511	0.782541	0.391270	+	0.920276i	$0.372036\pi$
$684$	0	0
$685$	−16.7434	−0.639732
$686$	−28.0327	−1.07029
$687$	0	0
$688$	−0.934486	−0.0356269
$689$	−19.9536	−0.760170
$690$	0	0
$691$	−3.02845	−0.115208	−0.0576038	−	0.998340i	$-0.518346\pi$
$691$	−3.02845	−0.115208	−0.0576038	+	0.998340i	$0.518346\pi$
$692$	4.26803	0.162246
$693$	0	0
$694$	−1.93022	−0.0732701
$695$	−6.95175	−0.263695
$696$	0	0
$697$	19.6623	0.744763
$698$	−28.5887	−1.08210
$699$	0	0
$700$	−2.94258	−0.111219
$701$	−38.8723	−1.46819	−0.734093	−	0.679049i	$-0.762393\pi$
$701$	−38.8723	−1.46819	−0.734093	+	0.679049i	$0.762393\pi$
$702$	0	0
$703$	5.49698	0.207322
$704$	−14.5444	−0.548162
$705$	0	0
$706$	−21.8746	−0.823261
$707$	−71.0047	−2.67041
$708$	0	0
$709$	−15.5788	−0.585074	−0.292537	−	0.956254i	$-0.594499\pi$
$709$	−15.5788	−0.585074	−0.292537	+	0.956254i	$0.594499\pi$
$710$	9.13931	0.342992
$711$	0	0
$712$	−18.1758	−0.681168
$713$	0.731427	0.0273922
$714$	0	0
$715$	39.9115	1.49261
$716$	−9.71893	−0.363213
$717$	0	0
$718$	8.47749	0.316377
$719$	5.24493	0.195603	0.0978015	−	0.995206i	$-0.468819\pi$
$719$	5.24493	0.195603	0.0978015	+	0.995206i	$0.468819\pi$
$720$	0	0
$721$	−34.2887	−1.27698
$722$	−0.788915	−0.0293604
$723$	0	0
$724$	4.48741	0.166773
$725$	−4.02982	−0.149664
$726$	0	0
$727$	−22.1849	−0.822793	−0.411396	−	0.911457i	$-0.634959\pi$
$727$	−22.1849	−0.822793	−0.411396	+	0.911457i	$0.634959\pi$
$728$	−48.1094	−1.78305
$729$	0	0
$730$	0.606843	0.0224603
$731$	−3.81389	−0.141062
$732$	0	0
$733$	−25.0342	−0.924661	−0.462331	−	0.886708i	$-0.652987\pi$
$733$	−25.0342	−0.924661	−0.462331	+	0.886708i	$0.652987\pi$
$734$	−6.82364	−0.251865
$735$	0	0
$736$	−14.3916	−0.530480
$737$	59.7536	2.20105
$738$	0	0
$739$	37.8520	1.39241	0.696204	−	0.717844i	$-0.254871\pi$
$739$	37.8520	1.39241	0.696204	+	0.717844i	$0.254871\pi$
$740$	17.6935	0.650428
$741$	0	0
$742$	−18.8665	−0.692611
$743$	−46.4729	−1.70492	−0.852462	−	0.522788i	$-0.824892\pi$
$743$	−46.4729	−1.70492	−0.852462	+	0.522788i	$0.824892\pi$
$744$	0	0
$745$	54.3977	1.99298
$746$	−11.2765	−0.412862
$747$	0	0
$748$	16.1594	0.590845
$749$	−4.42361	−0.161635
$750$	0	0
$751$	22.7667	0.830767	0.415384	−	0.909646i	$-0.363647\pi$
$751$	22.7667	0.830767	0.415384	+	0.909646i	$0.363647\pi$
$752$	0.653043	0.0238140
$753$	0	0
$754$	−26.8723	−0.978632
$755$	−3.62182	−0.131812
$756$	0	0
$757$	34.7430	1.26276	0.631378	−	0.775475i	$-0.282490\pi$
$757$	34.7430	1.26276	0.631378	+	0.775475i	$0.282490\pi$
$758$	−2.48396	−0.0902215
$759$	0	0
$760$	−6.22593	−0.225838
$761$	−5.00392	−0.181392	−0.0906959	−	0.995879i	$-0.528909\pi$
$761$	−5.00392	−0.181392	−0.0906959	+	0.995879i	$0.528909\pi$
$762$	0	0
$763$	−85.2997	−3.08806
$764$	−37.1624	−1.34449
$765$	0	0
$766$	19.1272	0.691095
$767$	0.837511	0.0302408
$768$	0	0
$769$	12.1042	0.436489	0.218244	−	0.975894i	$-0.429967\pi$
$769$	12.1042	0.436489	0.218244	+	0.975894i	$0.429967\pi$
$770$	37.7372	1.35995
$771$	0	0
$772$	−8.21565	−0.295688
$773$	32.8937	1.18310	0.591552	−	0.806267i	$-0.298515\pi$
$773$	32.8937	1.18310	0.591552	+	0.806267i	$0.298515\pi$
$774$	0	0
$775$	0.136394	0.00489943
$776$	5.71497	0.205155
$777$	0	0
$778$	7.70032	0.276070
$779$	−7.37730	−0.264319
$780$	0	0
$781$	21.8212	0.780824
$782$	−5.17760	−0.185151
$783$	0	0
$784$	9.55970	0.341418
$785$	23.0190	0.821583
$786$	0	0
$787$	27.2431	0.971113	0.485557	−	0.874205i	$-0.338617\pi$
$787$	27.2431	0.971113	0.485557	+	0.874205i	$0.338617\pi$
$788$	−4.19255	−0.149353
$789$	0	0
$790$	16.8221	0.598502
$791$	12.2833	0.436743
$792$	0	0
$793$	53.4659	1.89863
$794$	−21.6655	−0.768880
$795$	0	0
$796$	15.7477	0.558161
$797$	7.60892	0.269522	0.134761	−	0.990878i	$-0.456973\pi$
$797$	7.60892	0.269522	0.134761	+	0.990878i	$0.456973\pi$
$798$	0	0
$799$	2.66524	0.0942896
$800$	−2.68369	−0.0948829
$801$	0	0
$802$	8.77830	0.309973
$803$	1.44891	0.0511310
$804$	0	0
$805$	26.7633	0.943284
$806$	0.909526	0.0320367
$807$	0	0
$808$	−40.6735	−1.43089
$809$	−16.1024	−0.566129	−0.283065	−	0.959101i	$-0.591351\pi$
$809$	−16.1024	−0.566129	−0.283065	+	0.959101i	$0.591351\pi$
$810$	0	0
$811$	−15.1000	−0.530233	−0.265116	−	0.964216i	$-0.585410\pi$
$811$	−15.1000	−0.530233	−0.265116	+	0.964216i	$0.585410\pi$
$812$	56.2397	1.97363
$813$	0	0
$814$	−19.0860	−0.668962
$815$	−45.4356	−1.59154
$816$	0	0
$817$	1.43097	0.0500633
$818$	8.68100	0.303524
$819$	0	0
$820$	−23.7459	−0.829242
$821$	−27.1298	−0.946836	−0.473418	−	0.880838i	$-0.656980\pi$
$821$	−27.1298	−0.946836	−0.473418	+	0.880838i	$0.656980\pi$
$822$	0	0
$823$	32.4650	1.13166	0.565829	−	0.824523i	$-0.308556\pi$
$823$	32.4650	1.13166	0.565829	+	0.824523i	$0.308556\pi$
$824$	−19.6416	−0.684246
$825$	0	0
$826$	0.791884	0.0275532
$827$	−9.28298	−0.322801	−0.161400	−	0.986889i	$-0.551601\pi$
$827$	−9.28298	−0.322801	−0.161400	+	0.986889i	$0.551601\pi$
$828$	0	0
$829$	3.47085	0.120548	0.0602738	−	0.998182i	$-0.480803\pi$
$829$	3.47085	0.120548	0.0602738	+	0.998182i	$0.480803\pi$
$830$	−11.1642	−0.387513
$831$	0	0
$832$	−12.8265	−0.444681
$833$	39.0157	1.35181
$834$	0	0
$835$	10.6614	0.368952
$836$	−6.06299	−0.209693
$837$	0	0
$838$	25.7952	0.891081
$839$	33.3573	1.15162	0.575811	−	0.817582i	$-0.304686\pi$
$839$	33.3573	1.15162	0.575811	+	0.817582i	$0.304686\pi$
$840$	0	0
$841$	48.0195	1.65584
$842$	9.71482	0.334795
$843$	0	0
$844$	12.4752	0.429413
$845$	4.82323	0.165924
$846$	0	0
$847$	38.9330	1.33775
$848$	3.35728	0.115290
$849$	0	0
$850$	−0.965502	−0.0331165
$851$	−13.5358	−0.464002
$852$	0	0
$853$	46.2074	1.58211	0.791055	−	0.611745i	$-0.209532\pi$
$853$	46.2074	1.58211	0.791055	+	0.611745i	$0.209532\pi$
$854$	50.5532	1.72989
$855$	0	0
$856$	−2.53397	−0.0866093
$857$	−6.16515	−0.210598	−0.105299	−	0.994441i	$-0.533580\pi$
$857$	−6.16515	−0.210598	−0.105299	+	0.994441i	$0.533580\pi$
$858$	0	0
$859$	8.85322	0.302068	0.151034	−	0.988529i	$-0.451740\pi$
$859$	8.85322	0.302068	0.151034	+	0.988529i	$0.451740\pi$
$860$	4.60598	0.157063
$861$	0	0
$862$	11.9658	0.407556
$863$	6.38420	0.217321	0.108660	−	0.994079i	$-0.465344\pi$
$863$	6.38420	0.217321	0.108660	+	0.994079i	$0.465344\pi$
$864$	0	0
$865$	−7.23876	−0.246125
$866$	7.19508	0.244499
$867$	0	0
$868$	−1.90350	−0.0646090
$869$	40.1647	1.36250
$870$	0	0
$871$	52.6961	1.78554
$872$	−48.8621	−1.65468
$873$	0	0
$874$	1.94264	0.0657106
$875$	−49.3530	−1.66844
$876$	0	0
$877$	20.1236	0.679527	0.339764	−	0.940511i	$-0.389653\pi$
$877$	20.1236	0.679527	0.339764	+	0.940511i	$0.389653\pi$
$878$	−20.6496	−0.696890
$879$	0	0
$880$	−6.71530	−0.226373
$881$	−18.6005	−0.626668	−0.313334	−	0.949643i	$-0.601446\pi$
$881$	−18.6005	−0.626668	−0.313334	+	0.949643i	$0.601446\pi$
$882$	0	0
$883$	−19.5773	−0.658829	−0.329415	−	0.944185i	$-0.606851\pi$
$883$	−19.5773	−0.658829	−0.329415	+	0.944185i	$0.606851\pi$
$884$	14.2508	0.479306
$885$	0	0
$886$	−5.98268	−0.200992
$887$	25.7083	0.863199	0.431599	−	0.902065i	$-0.357949\pi$
$887$	25.7083	0.863199	0.431599	+	0.902065i	$0.357949\pi$
$888$	0	0
$889$	17.6561	0.592165
$890$	12.5733	0.421457
$891$	0	0
$892$	−11.8007	−0.395117
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	16.4837	0.550990
$896$	42.2464	1.41135
$897$	0	0
$898$	19.1152	0.637884
$899$	−2.60682	−0.0869422
$900$	0	0
$901$	13.7020	0.456479
$902$	25.6146	0.852872
$903$	0	0
$904$	7.03622	0.234021
$905$	−7.61084	−0.252993
$906$	0	0
$907$	−26.6069	−0.883468	−0.441734	−	0.897146i	$-0.645636\pi$
$907$	−26.6069	−0.883468	−0.441734	+	0.897146i	$0.645636\pi$
$908$	−13.8296	−0.458950
$909$	0	0
$910$	33.2801	1.10322
$911$	−43.3674	−1.43683	−0.718413	−	0.695617i	$-0.755131\pi$
$911$	−43.3674	−1.43683	−0.718413	+	0.695617i	$0.755131\pi$
$912$	0	0
$913$	−26.6558	−0.882177
$914$	−31.4816	−1.04132
$915$	0	0
$916$	−32.8441	−1.08520
$917$	28.1176	0.928524
$918$	0	0
$919$	−34.2269	−1.12904	−0.564521	−	0.825419i	$-0.690939\pi$
$919$	−34.2269	−1.12904	−0.564521	+	0.825419i	$0.690939\pi$
$920$	15.3308	0.505442
$921$	0	0
$922$	9.11674	0.300244
$923$	19.2439	0.633422
$924$	0	0
$925$	−2.52412	−0.0829925
$926$	28.4474	0.934839
$927$	0	0
$928$	51.2917	1.68373
$929$	−32.6543	−1.07135	−0.535677	−	0.844423i	$-0.679944\pi$
$929$	−32.6543	−1.07135	−0.535677	+	0.844423i	$0.679944\pi$
$930$	0	0
$931$	−14.6387	−0.479764
$932$	24.9721	0.817987
$933$	0	0
$934$	6.93529	0.226930
$935$	−27.4069	−0.896303
$936$	0	0
$937$	21.9938	0.718506	0.359253	−	0.933240i	$-0.383031\pi$
$937$	21.9938	0.718506	0.359253	+	0.933240i	$0.383031\pi$
$938$	49.8253	1.62685
$939$	0	0
$940$	−3.21878	−0.104985
$941$	−49.2804	−1.60649	−0.803247	−	0.595646i	$-0.796896\pi$
$941$	−49.2804	−1.60649	−0.803247	+	0.595646i	$0.796896\pi$
$942$	0	0
$943$	18.1660	0.591565
$944$	−0.140915	−0.00458640
$945$	0	0
$946$	−4.96845	−0.161538
$947$	−11.1995	−0.363935	−0.181968	−	0.983305i	$-0.558247\pi$
$947$	−11.1995	−0.363935	−0.181968	+	0.983305i	$0.558247\pi$
$948$	0	0
$949$	1.27778	0.0414786
$950$	0.362256	0.0117532
$951$	0	0
$952$	33.0364	1.07072
$953$	24.9075	0.806833	0.403417	−	0.915016i	$-0.367823\pi$
$953$	24.9075	0.806833	0.403417	+	0.915016i	$0.367823\pi$
$954$	0	0
$955$	63.0291	2.03957
$956$	13.0516	0.422120
$957$	0	0
$958$	−25.4352	−0.821775
$959$	−33.3346	−1.07643
$960$	0	0
$961$	−30.9118	−0.997154
$962$	−16.8317	−0.542677
$963$	0	0
$964$	18.7169	0.602829
$965$	13.9341	0.448554
$966$	0	0
$967$	−35.3100	−1.13549	−0.567747	−	0.823203i	$-0.692185\pi$
$967$	−35.3100	−1.13549	−0.567747	+	0.823203i	$0.692185\pi$
$968$	22.3019	0.716810
$969$	0	0
$970$	−3.95337	−0.126935
$971$	−62.0781	−1.99218	−0.996091	−	0.0883366i	$-0.971845\pi$
$971$	−62.0781	−1.99218	−0.996091	+	0.0883366i	$0.971845\pi$
$972$	0	0
$973$	−13.8403	−0.443700
$974$	−10.2753	−0.329240
$975$	0	0
$976$	−8.99590	−0.287952
$977$	41.1779	1.31740	0.658699	−	0.752407i	$-0.271107\pi$
$977$	41.1779	1.31740	0.658699	+	0.752407i	$0.271107\pi$
$978$	0	0
$979$	30.0202	0.959451
$980$	−47.1187	−1.50515
$981$	0	0
$982$	0.656505	0.0209499
$983$	−46.0206	−1.46783	−0.733914	−	0.679243i	$-0.762308\pi$
$983$	−46.0206	−1.46783	−0.733914	+	0.679243i	$0.762308\pi$
$984$	0	0
$985$	7.11073	0.226567
$986$	18.4530	0.587664
$987$	0	0
$988$	−5.34690	−0.170107
$989$	−3.52364	−0.112045
$990$	0	0
$991$	−44.0674	−1.39985	−0.699923	−	0.714218i	$-0.746783\pi$
$991$	−44.0674	−1.39985	−0.699923	+	0.714218i	$0.746783\pi$
$992$	−1.73603	−0.0551190
$993$	0	0
$994$	18.1955	0.577128
$995$	−26.7087	−0.846723
$996$	0	0
$997$	34.8769	1.10456	0.552281	−	0.833658i	$-0.313758\pi$
$997$	34.8769	1.10456	0.552281	+	0.833658i	$0.313758\pi$
$998$	20.9551	0.663321
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.n.1.6		16
3.2	odd	2		893.2.a.b.1.11	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.b.1.11	✓	16	3.2	odd	2
8037.2.a.n.1.6		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)

\(f(q)\)	\(=\)	\(q-0.788915 q^{2} -1.37761 q^{4} +2.33649 q^{5} +4.65174 q^{7} +2.66465 q^{8} +O(q^{10})\)	\(q-0.788915 q^{2} -1.37761 q^{4} +2.33649 q^{5} +4.65174 q^{7} +2.66465 q^{8} -1.84329 q^{10} -4.40108 q^{11} -3.88128 q^{13} -3.66983 q^{14} +0.653043 q^{16} +2.66524 q^{17} -1.00000 q^{19} -3.21878 q^{20} +3.47208 q^{22} +2.46241 q^{23} +0.459183 q^{25} +3.06200 q^{26} -6.40830 q^{28} -8.77607 q^{29} +0.297037 q^{31} -5.84450 q^{32} -2.10265 q^{34} +10.8687 q^{35} -5.49698 q^{37} +0.788915 q^{38} +6.22593 q^{40} +7.37730 q^{41} -1.43097 q^{43} +6.06299 q^{44} -1.94264 q^{46} +1.00000 q^{47} +14.6387 q^{49} -0.362256 q^{50} +5.34690 q^{52} +5.14098 q^{53} -10.2831 q^{55} +12.3953 q^{56} +6.92358 q^{58} -0.215782 q^{59} -13.7753 q^{61} -0.234337 q^{62} +3.30472 q^{64} -9.06856 q^{65} -13.5770 q^{67} -3.67168 q^{68} -8.57452 q^{70} -4.95815 q^{71} -0.329217 q^{73} +4.33665 q^{74} +1.37761 q^{76} -20.4727 q^{77} -9.12610 q^{79} +1.52583 q^{80} -5.82006 q^{82} +6.05664 q^{83} +6.22732 q^{85} +1.12891 q^{86} -11.7273 q^{88} -6.82110 q^{89} -18.0547 q^{91} -3.39225 q^{92} -0.788915 q^{94} -2.33649 q^{95} +2.14473 q^{97} -11.5487 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8}+O(q^{10}) \) 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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Database

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