LMFDB - Embedded newform 5571.2.a.g.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5571,2,Mod(1,5571)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5571, 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("5571.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5571 = 3^{2} \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5571.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:	$44.4846589661$
Analytic rank:	$1$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 619)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	5571.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.431096 q^{2} -1.81416 q^{4} -1.50079 q^{5} -4.83036 q^{7} +1.64427 q^{8} +O(q^{10})$	\(q-0.431096 q^{2} -1.81416 q^{4} -1.50079 q^{5} -4.83036 q^{7} +1.64427 q^{8} +0.646986 q^{10} -2.60102 q^{11} +5.36233 q^{13} +2.08235 q^{14} +2.91948 q^{16} -6.13234 q^{17} +1.28647 q^{19} +2.72267 q^{20} +1.12129 q^{22} +3.55264 q^{23} -2.74762 q^{25} -2.31168 q^{26} +8.76304 q^{28} +1.96205 q^{29} +7.06516 q^{31} -4.54711 q^{32} +2.64363 q^{34} +7.24937 q^{35} +3.47725 q^{37} -0.554591 q^{38} -2.46770 q^{40} -5.18803 q^{41} -8.24591 q^{43} +4.71865 q^{44} -1.53153 q^{46} -4.95150 q^{47} +16.3324 q^{49} +1.18449 q^{50} -9.72810 q^{52} +5.85664 q^{53} +3.90359 q^{55} -7.94241 q^{56} -0.845830 q^{58} +9.60280 q^{59} +6.86922 q^{61} -3.04576 q^{62} -3.87871 q^{64} -8.04774 q^{65} +14.9558 q^{67} +11.1250 q^{68} -3.12518 q^{70} +5.87145 q^{71} +4.41927 q^{73} -1.49903 q^{74} -2.33385 q^{76} +12.5639 q^{77} +8.82851 q^{79} -4.38153 q^{80} +2.23654 q^{82} -1.16250 q^{83} +9.20337 q^{85} +3.55478 q^{86} -4.27677 q^{88} -14.0013 q^{89} -25.9020 q^{91} -6.44504 q^{92} +2.13457 q^{94} -1.93072 q^{95} -5.05039 q^{97} -7.04084 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 9 q^{2} + 33 q^{4} - 21 q^{5} + 2 q^{7} - 27 q^{8}+O(q^{10}) $ 30 * q - 9 * q^2 + 33 * q^4 - 21 * q^5 + 2 * q^7 - 27 * q^8	$ 30 q - 9 q^{2} + 33 q^{4} - 21 q^{5} + 2 q^{7} - 27 q^{8} + 5 q^{10} - 23 q^{11} + 9 q^{13} - 7 q^{14} + 35 q^{16} - 4 q^{17} - q^{19} - 29 q^{20} - 4 q^{23} + 35 q^{25} - q^{26} - 13 q^{28} - 90 q^{29} + 2 q^{31} - 43 q^{32} - 9 q^{34} - 9 q^{35} + 19 q^{37} - 5 q^{38} - 12 q^{40} - 59 q^{41} - 4 q^{43} - 52 q^{44} - q^{46} - 4 q^{47} + 30 q^{49} - 31 q^{50} - 12 q^{52} - 34 q^{53} - 17 q^{55} - 2 q^{56} + 6 q^{58} - 13 q^{59} + 16 q^{61} - 28 q^{62} + 37 q^{64} - 31 q^{65} - 11 q^{67} + 52 q^{68} - 40 q^{70} - 42 q^{71} - 4 q^{73} - 16 q^{74} - 42 q^{76} - 29 q^{77} + 3 q^{79} - 21 q^{80} - 43 q^{82} + 11 q^{83} + 19 q^{85} + 11 q^{86} - 47 q^{88} - 58 q^{89} - 39 q^{91} + 7 q^{92} - 46 q^{94} - 23 q^{95} - 9 q^{97}+O(q^{100}) $ 30 * q - 9 * q^2 + 33 * q^4 - 21 * q^5 + 2 * q^7 - 27 * q^8 + 5 * q^10 - 23 * q^11 + 9 * q^13 - 7 * q^14 + 35 * q^16 - 4 * q^17 - q^19 - 29 * q^20 - 4 * q^23 + 35 * q^25 - q^26 - 13 * q^28 - 90 * q^29 + 2 * q^31 - 43 * q^32 - 9 * q^34 - 9 * q^35 + 19 * q^37 - 5 * q^38 - 12 * q^40 - 59 * q^41 - 4 * q^43 - 52 * q^44 - q^46 - 4 * q^47 + 30 * q^49 - 31 * q^50 - 12 * q^52 - 34 * q^53 - 17 * q^55 - 2 * q^56 + 6 * q^58 - 13 * q^59 + 16 * q^61 - 28 * q^62 + 37 * q^64 - 31 * q^65 - 11 * q^67 + 52 * q^68 - 40 * q^70 - 42 * q^71 - 4 * q^73 - 16 * q^74 - 42 * q^76 - 29 * q^77 + 3 * q^79 - 21 * q^80 - 43 * q^82 + 11 * q^83 + 19 * q^85 + 11 * q^86 - 47 * q^88 - 58 * q^89 - 39 * q^91 + 7 * q^92 - 46 * q^94 - 23 * q^95 - 9 * q^97

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 $

$2$	−0.431096	−0.304831	−0.152415	−	0.988317i	$-0.548705\pi$
$2$	−0.431096	−0.304831	−0.152415	+	0.988317i	$0.548705\pi$
$3$	0	0
$3$	0	0
$4$	−1.81416	−0.907078
$5$	−1.50079	−0.671175	−0.335587	−	0.942009i	$-0.608935\pi$
$5$	−1.50079	−0.671175	−0.335587	+	0.942009i	$0.608935\pi$
$6$	0	0
$7$	−4.83036	−1.82571	−0.912853	−	0.408288i	$-0.866126\pi$
$7$	−4.83036	−1.82571	−0.912853	+	0.408288i	$0.866126\pi$
$8$	1.64427	0.581336
$9$	0	0
$10$	0.646986	0.204595
$11$	−2.60102	−0.784237	−0.392118	−	0.919915i	$-0.628258\pi$
$11$	−2.60102	−0.784237	−0.392118	+	0.919915i	$0.628258\pi$
$12$	0	0
$13$	5.36233	1.48724	0.743621	−	0.668601i	$-0.233107\pi$
$13$	5.36233	1.48724	0.743621	+	0.668601i	$0.233107\pi$
$14$	2.08235	0.556532
$15$	0	0
$16$	2.91948	0.729869
$17$	−6.13234	−1.48731	−0.743656	−	0.668563i	$-0.766910\pi$
$17$	−6.13234	−1.48731	−0.743656	+	0.668563i	$0.766910\pi$
$18$	0	0
$19$	1.28647	0.295136	0.147568	−	0.989052i	$-0.452855\pi$
$19$	1.28647	0.295136	0.147568	+	0.989052i	$0.452855\pi$
$20$	2.72267	0.608808
$21$	0	0
$22$	1.12129	0.239060
$23$	3.55264	0.740776	0.370388	−	0.928877i	$-0.379225\pi$
$23$	3.55264	0.740776	0.370388	+	0.928877i	$0.379225\pi$
$24$	0	0
$25$	−2.74762	−0.549524
$26$	−2.31168	−0.453357
$27$	0	0
$28$	8.76304	1.65606
$29$	1.96205	0.364343	0.182171	−	0.983267i	$-0.441687\pi$
$29$	1.96205	0.364343	0.182171	+	0.983267i	$0.441687\pi$
$30$	0	0
$31$	7.06516	1.26894	0.634470	−	0.772947i	$-0.281218\pi$
$31$	7.06516	1.26894	0.634470	+	0.772947i	$0.281218\pi$
$32$	−4.54711	−0.803823
$33$	0	0
$34$	2.64363	0.453379
$35$	7.24937	1.22537
$36$	0	0
$37$	3.47725	0.571656	0.285828	−	0.958281i	$-0.407731\pi$
$37$	3.47725	0.571656	0.285828	+	0.958281i	$0.407731\pi$
$38$	−0.554591	−0.0899666
$39$	0	0
$40$	−2.46770	−0.390178
$41$	−5.18803	−0.810234	−0.405117	−	0.914265i	$-0.632769\pi$
$41$	−5.18803	−0.810234	−0.405117	+	0.914265i	$0.632769\pi$
$42$	0	0
$43$	−8.24591	−1.25749	−0.628745	−	0.777612i	$-0.716431\pi$
$43$	−8.24591	−1.25749	−0.628745	+	0.777612i	$0.716431\pi$
$44$	4.71865	0.711364
$45$	0	0
$46$	−1.53153	−0.225811
$47$	−4.95150	−0.722250	−0.361125	−	0.932517i	$-0.617607\pi$
$47$	−4.95150	−0.722250	−0.361125	+	0.932517i	$0.617607\pi$
$48$	0	0
$49$	16.3324	2.33320
$50$	1.18449	0.167512
$51$	0	0
$52$	−9.72810	−1.34904
$53$	5.85664	0.804472	0.402236	−	0.915536i	$-0.368233\pi$
$53$	5.85664	0.804472	0.402236	+	0.915536i	$0.368233\pi$
$54$	0	0
$55$	3.90359	0.526360
$56$	−7.94241	−1.06135
$57$	0	0
$58$	−0.845830	−0.111063
$59$	9.60280	1.25018	0.625089	−	0.780554i	$-0.285063\pi$
$59$	9.60280	1.25018	0.625089	+	0.780554i	$0.285063\pi$
$60$	0	0
$61$	6.86922	0.879514	0.439757	−	0.898117i	$-0.355065\pi$
$61$	6.86922	0.879514	0.439757	+	0.898117i	$0.355065\pi$
$62$	−3.04576	−0.386812
$63$	0	0
$64$	−3.87871	−0.484839
$65$	−8.04774	−0.998200
$66$	0	0
$67$	14.9558	1.82714	0.913569	−	0.406683i	$-0.133315\pi$
$67$	14.9558	1.82714	0.913569	+	0.406683i	$0.133315\pi$
$68$	11.1250	1.34911
$69$	0	0
$70$	−3.12518	−0.373530
$71$	5.87145	0.696813	0.348407	−	0.937344i	$-0.386723\pi$
$71$	5.87145	0.696813	0.348407	+	0.937344i	$0.386723\pi$
$72$	0	0
$73$	4.41927	0.517236	0.258618	−	0.965980i	$-0.416733\pi$
$73$	4.41927	0.517236	0.258618	+	0.965980i	$0.416733\pi$
$74$	−1.49903	−0.174258
$75$	0	0
$76$	−2.33385	−0.267711
$77$	12.5639	1.43179
$78$	0	0
$79$	8.82851	0.993285	0.496642	−	0.867955i	$-0.334566\pi$
$79$	8.82851	0.993285	0.496642	+	0.867955i	$0.334566\pi$
$80$	−4.38153	−0.489870
$81$	0	0
$82$	2.23654	0.246984
$83$	−1.16250	−0.127601	−0.0638003	−	0.997963i	$-0.520322\pi$
$83$	−1.16250	−0.127601	−0.0638003	+	0.997963i	$0.520322\pi$
$84$	0	0
$85$	9.20337	0.998246
$86$	3.55478	0.383322
$87$	0	0
$88$	−4.27677	−0.455905
$89$	−14.0013	−1.48414	−0.742068	−	0.670325i	$-0.766155\pi$
$89$	−14.0013	−1.48414	−0.742068	+	0.670325i	$0.766155\pi$
$90$	0	0
$91$	−25.9020	−2.71527
$92$	−6.44504	−0.671942
$93$	0	0
$94$	2.13457	0.220164
$95$	−1.93072	−0.198088
$96$	0	0
$97$	−5.05039	−0.512790	−0.256395	−	0.966572i	$-0.582535\pi$
$97$	−5.05039	−0.512790	−0.256395	+	0.966572i	$0.582535\pi$
$98$	−7.04084	−0.711232
$99$	0	0
$100$	4.98462	0.498462
$101$	−8.50227	−0.846007	−0.423004	−	0.906128i	$-0.639024\pi$
$101$	−8.50227	−0.846007	−0.423004	+	0.906128i	$0.639024\pi$
$102$	0	0
$103$	7.04198	0.693867	0.346934	−	0.937890i	$-0.387223\pi$
$103$	7.04198	0.693867	0.346934	+	0.937890i	$0.387223\pi$
$104$	8.81710	0.864588
$105$	0	0
$106$	−2.52477	−0.245228
$107$	14.6155	1.41293	0.706465	−	0.707748i	$-0.250289\pi$
$107$	14.6155	1.41293	0.706465	+	0.707748i	$0.250289\pi$
$108$	0	0
$109$	−3.64511	−0.349138	−0.174569	−	0.984645i	$-0.555853\pi$
$109$	−3.64511	−0.349138	−0.174569	+	0.984645i	$0.555853\pi$
$110$	−1.68282	−0.160451
$111$	0	0
$112$	−14.1021	−1.33253
$113$	−9.10477	−0.856505	−0.428252	−	0.903659i	$-0.640871\pi$
$113$	−9.10477	−0.856505	−0.428252	+	0.903659i	$0.640871\pi$
$114$	0	0
$115$	−5.33177	−0.497190
$116$	−3.55946	−0.330487
$117$	0	0
$118$	−4.13973	−0.381093
$119$	29.6215	2.71539
$120$	0	0
$121$	−4.23470	−0.384973
$122$	−2.96129	−0.268103
$123$	0	0
$124$	−12.8173	−1.15103
$125$	11.6276	1.04000
$126$	0	0
$127$	−13.4174	−1.19060	−0.595300	−	0.803504i	$-0.702967\pi$
$127$	−13.4174	−1.19060	−0.595300	+	0.803504i	$0.702967\pi$
$128$	10.7663	0.951617
$129$	0	0
$130$	3.46935	0.304282
$131$	−16.5240	−1.44371	−0.721855	−	0.692044i	$-0.756710\pi$
$131$	−16.5240	−1.44371	−0.721855	+	0.692044i	$0.756710\pi$
$132$	0	0
$133$	−6.21411	−0.538832
$134$	−6.44737	−0.556968
$135$	0	0
$136$	−10.0832	−0.864628
$137$	19.3142	1.65013	0.825063	−	0.565041i	$-0.191140\pi$
$137$	19.3142	1.65013	0.825063	+	0.565041i	$0.191140\pi$
$138$	0	0
$139$	−5.09316	−0.431997	−0.215998	−	0.976394i	$-0.569301\pi$
$139$	−5.09316	−0.431997	−0.215998	+	0.976394i	$0.569301\pi$
$140$	−13.1515	−1.11150
$141$	0	0
$142$	−2.53116	−0.212410
$143$	−13.9475	−1.16635
$144$	0	0
$145$	−2.94462	−0.244538
$146$	−1.90513	−0.157670
$147$	0	0
$148$	−6.30827	−0.518537
$149$	−15.6367	−1.28101	−0.640504	−	0.767955i	$-0.721274\pi$
$149$	−15.6367	−1.28101	−0.640504	+	0.767955i	$0.721274\pi$
$150$	0	0
$151$	14.9031	1.21279	0.606397	−	0.795162i	$-0.292614\pi$
$151$	14.9031	1.21279	0.606397	+	0.795162i	$0.292614\pi$
$152$	2.11530	0.171573
$153$	0	0
$154$	−5.41623	−0.436452
$155$	−10.6033	−0.851681
$156$	0	0
$157$	16.9495	1.35272	0.676359	−	0.736572i	$-0.263557\pi$
$157$	16.9495	1.35272	0.676359	+	0.736572i	$0.263557\pi$
$158$	−3.80593	−0.302784
$159$	0	0
$160$	6.82427	0.539506
$161$	−17.1605	−1.35244
$162$	0	0
$163$	−15.9532	−1.24955	−0.624775	−	0.780805i	$-0.714809\pi$
$163$	−15.9532	−1.24955	−0.624775	+	0.780805i	$0.714809\pi$
$164$	9.41189	0.734945
$165$	0	0
$166$	0.501147	0.0388966
$167$	−9.93294	−0.768634	−0.384317	−	0.923201i	$-0.625563\pi$
$167$	−9.93294	−0.768634	−0.384317	+	0.923201i	$0.625563\pi$
$168$	0	0
$169$	15.7546	1.21189
$170$	−3.96754	−0.304296
$171$	0	0
$172$	14.9594	1.14064
$173$	13.5963	1.03371	0.516855	−	0.856073i	$-0.327103\pi$
$173$	13.5963	1.03371	0.516855	+	0.856073i	$0.327103\pi$
$174$	0	0
$175$	13.2720	1.00327
$176$	−7.59361	−0.572390
$177$	0	0
$178$	6.03591	0.452410
$179$	−14.4590	−1.08072	−0.540358	−	0.841435i	$-0.681711\pi$
$179$	−14.4590	−1.08072	−0.540358	+	0.841435i	$0.681711\pi$
$180$	0	0
$181$	5.91625	0.439752	0.219876	−	0.975528i	$-0.429435\pi$
$181$	5.91625	0.439752	0.219876	+	0.975528i	$0.429435\pi$
$182$	11.1662	0.827697
$183$	0	0
$184$	5.84149	0.430640
$185$	−5.21863	−0.383681
$186$	0	0
$187$	15.9503	1.16640
$188$	8.98279	0.655138
$189$	0	0
$190$	0.832326	0.0603833
$191$	−22.2794	−1.61208	−0.806039	−	0.591862i	$-0.798393\pi$
$191$	−22.2794	−1.61208	−0.806039	+	0.591862i	$0.798393\pi$
$192$	0	0
$193$	−12.8221	−0.922955	−0.461478	−	0.887152i	$-0.652681\pi$
$193$	−12.8221	−0.922955	−0.461478	+	0.887152i	$0.652681\pi$
$194$	2.17720	0.156314
$195$	0	0
$196$	−29.6296	−2.11640
$197$	−1.62495	−0.115773	−0.0578866	−	0.998323i	$-0.518436\pi$
$197$	−1.62495	−0.115773	−0.0578866	+	0.998323i	$0.518436\pi$
$198$	0	0
$199$	12.0603	0.854933	0.427466	−	0.904031i	$-0.359406\pi$
$199$	12.0603	0.854933	0.427466	+	0.904031i	$0.359406\pi$
$200$	−4.51783	−0.319459
$201$	0	0
$202$	3.66529	0.257889
$203$	−9.47740	−0.665183
$204$	0	0
$205$	7.78615	0.543809
$206$	−3.03577	−0.211512
$207$	0	0
$208$	15.6552	1.08549
$209$	−3.34613	−0.231456
$210$	0	0
$211$	−21.0651	−1.45018	−0.725090	−	0.688654i	$-0.758202\pi$
$211$	−21.0651	−1.45018	−0.725090	+	0.688654i	$0.758202\pi$
$212$	−10.6249	−0.729719
$213$	0	0
$214$	−6.30067	−0.430705
$215$	12.3754	0.843995
$216$	0	0
$217$	−34.1273	−2.31671
$218$	1.57139	0.106428
$219$	0	0
$220$	−7.08172	−0.477450
$221$	−32.8836	−2.21199
$222$	0	0
$223$	−2.07074	−0.138667	−0.0693336	−	0.997594i	$-0.522087\pi$
$223$	−2.07074	−0.138667	−0.0693336	+	0.997594i	$0.522087\pi$
$224$	21.9642	1.46754
$225$	0	0
$226$	3.92503	0.261089
$227$	−1.64406	−0.109120	−0.0545599	−	0.998510i	$-0.517376\pi$
$227$	−1.64406	−0.109120	−0.0545599	+	0.998510i	$0.517376\pi$
$228$	0	0
$229$	−12.7092	−0.839848	−0.419924	−	0.907559i	$-0.637943\pi$
$229$	−12.7092	−0.839848	−0.419924	+	0.907559i	$0.637943\pi$
$230$	2.29851	0.151559
$231$	0	0
$232$	3.22613	0.211806
$233$	−25.1543	−1.64792	−0.823958	−	0.566651i	$-0.808239\pi$
$233$	−25.1543	−1.64792	−0.823958	+	0.566651i	$0.808239\pi$
$234$	0	0
$235$	7.43117	0.484756
$236$	−17.4210	−1.13401
$237$	0	0
$238$	−12.7697	−0.827736
$239$	−3.76267	−0.243387	−0.121693	−	0.992568i	$-0.538832\pi$
$239$	−3.76267	−0.243387	−0.121693	+	0.992568i	$0.538832\pi$
$240$	0	0
$241$	−5.67080	−0.365289	−0.182644	−	0.983179i	$-0.558466\pi$
$241$	−5.67080	−0.365289	−0.182644	+	0.983179i	$0.558466\pi$
$242$	1.82556	0.117352
$243$	0	0
$244$	−12.4618	−0.797788
$245$	−24.5116	−1.56599
$246$	0	0
$247$	6.89846	0.438939
$248$	11.6170	0.737682
$249$	0	0
$250$	−5.01260	−0.317025
$251$	−10.1988	−0.643741	−0.321871	−	0.946784i	$-0.604312\pi$
$251$	−10.1988	−0.643741	−0.321871	+	0.946784i	$0.604312\pi$
$252$	0	0
$253$	−9.24048	−0.580944
$254$	5.78418	0.362932
$255$	0	0
$256$	3.11611	0.194757
$257$	−0.739967	−0.0461579	−0.0230789	−	0.999734i	$-0.507347\pi$
$257$	−0.739967	−0.0461579	−0.0230789	+	0.999734i	$0.507347\pi$
$258$	0	0
$259$	−16.7964	−1.04368
$260$	14.5999	0.905445
$261$	0	0
$262$	7.12344	0.440087
$263$	−11.6292	−0.717090	−0.358545	−	0.933513i	$-0.616727\pi$
$263$	−11.6292	−0.717090	−0.358545	+	0.933513i	$0.616727\pi$
$264$	0	0
$265$	−8.78960	−0.539941
$266$	2.67888	0.164252
$267$	0	0
$268$	−27.1321	−1.65736
$269$	−20.9998	−1.28038	−0.640190	−	0.768217i	$-0.721144\pi$
$269$	−20.9998	−1.28038	−0.640190	+	0.768217i	$0.721144\pi$
$270$	0	0
$271$	−20.4684	−1.24337	−0.621684	−	0.783268i	$-0.713551\pi$
$271$	−20.4684	−1.24337	−0.621684	+	0.783268i	$0.713551\pi$
$272$	−17.9032	−1.08554
$273$	0	0
$274$	−8.32628	−0.503009
$275$	7.14662	0.430957
$276$	0	0
$277$	8.22513	0.494200	0.247100	−	0.968990i	$-0.420522\pi$
$277$	8.22513	0.494200	0.247100	+	0.968990i	$0.420522\pi$
$278$	2.19564	0.131686
$279$	0	0
$280$	11.9199	0.712351
$281$	−13.8741	−0.827658	−0.413829	−	0.910355i	$-0.635809\pi$
$281$	−13.8741	−0.827658	−0.413829	+	0.910355i	$0.635809\pi$
$282$	0	0
$283$	30.4615	1.81075	0.905373	−	0.424616i	$-0.139591\pi$
$283$	30.4615	1.81075	0.905373	+	0.424616i	$0.139591\pi$
$284$	−10.6517	−0.632064
$285$	0	0
$286$	6.01272	0.355540
$287$	25.0601	1.47925
$288$	0	0
$289$	20.6056	1.21210
$290$	1.26942	0.0745427
$291$	0	0
$292$	−8.01724	−0.469174
$293$	27.5968	1.61222	0.806111	−	0.591765i	$-0.201569\pi$
$293$	27.5968	1.61222	0.806111	+	0.591765i	$0.201569\pi$
$294$	0	0
$295$	−14.4118	−0.839088
$296$	5.71752	0.332324
$297$	0	0
$298$	6.74092	0.390491
$299$	19.0504	1.10171
$300$	0	0
$301$	39.8307	2.29581
$302$	−6.42465	−0.369697
$303$	0	0
$304$	3.75581	0.215411
$305$	−10.3093	−0.590307
$306$	0	0
$307$	−13.6959	−0.781667	−0.390834	−	0.920461i	$-0.627813\pi$
$307$	−13.6959	−0.781667	−0.390834	+	0.920461i	$0.627813\pi$
$308$	−22.7928	−1.29874
$309$	0	0
$310$	4.57106	0.259619
$311$	0.171615	0.00973139	0.00486569	−	0.999988i	$-0.498451\pi$
$311$	0.171615	0.00973139	0.00486569	+	0.999988i	$0.498451\pi$
$312$	0	0
$313$	−17.5584	−0.992461	−0.496230	−	0.868191i	$-0.665283\pi$
$313$	−17.5584	−0.992461	−0.496230	+	0.868191i	$0.665283\pi$
$314$	−7.30686	−0.412350
$315$	0	0
$316$	−16.0163	−0.900987
$317$	1.72271	0.0967569	0.0483784	−	0.998829i	$-0.484595\pi$
$317$	1.72271	0.0967569	0.0483784	+	0.998829i	$0.484595\pi$
$318$	0	0
$319$	−5.10332	−0.285731
$320$	5.82114	0.325412
$321$	0	0
$322$	7.39784	0.412265
$323$	−7.88906	−0.438959
$324$	0	0
$325$	−14.7337	−0.817276
$326$	6.87736	0.380902
$327$	0	0
$328$	−8.53051	−0.471018
$329$	23.9175	1.31862
$330$	0	0
$331$	23.5273	1.29318	0.646590	−	0.762838i	$-0.276195\pi$
$331$	23.5273	1.29318	0.646590	+	0.762838i	$0.276195\pi$
$332$	2.10895	0.115744
$333$	0	0
$334$	4.28205	0.234303
$335$	−22.4455	−1.22633
$336$	0	0
$337$	−8.02169	−0.436969	−0.218485	−	0.975840i	$-0.570111\pi$
$337$	−8.02169	−0.436969	−0.218485	+	0.975840i	$0.570111\pi$
$338$	−6.79173	−0.369421
$339$	0	0
$340$	−16.6964	−0.905487
$341$	−18.3766	−0.995150
$342$	0	0
$343$	−45.0790	−2.43404
$344$	−13.5585	−0.731024
$345$	0	0
$346$	−5.86133	−0.315107
$347$	11.7465	0.630584	0.315292	−	0.948995i	$-0.397898\pi$
$347$	11.7465	0.630584	0.315292	+	0.948995i	$0.397898\pi$
$348$	0	0
$349$	11.8697	0.635373	0.317686	−	0.948196i	$-0.397094\pi$
$349$	11.8697	0.635373	0.317686	+	0.948196i	$0.397094\pi$
$350$	−5.72151	−0.305828
$351$	0	0
$352$	11.8271	0.630387
$353$	−22.4752	−1.19624	−0.598118	−	0.801408i	$-0.704085\pi$
$353$	−22.4752	−1.19624	−0.598118	+	0.801408i	$0.704085\pi$
$354$	0	0
$355$	−8.81183	−0.467683
$356$	25.4006	1.34623
$357$	0	0
$358$	6.23322	0.329436
$359$	24.8782	1.31302	0.656510	−	0.754317i	$-0.272032\pi$
$359$	24.8782	1.31302	0.656510	+	0.754317i	$0.272032\pi$
$360$	0	0
$361$	−17.3450	−0.912895
$362$	−2.55047	−0.134050
$363$	0	0
$364$	46.9903	2.46296
$365$	−6.63240	−0.347156
$366$	0	0
$367$	32.9944	1.72230	0.861148	−	0.508355i	$-0.169746\pi$
$367$	32.9944	1.72230	0.861148	+	0.508355i	$0.169746\pi$
$368$	10.3718	0.540669
$369$	0	0
$370$	2.24973	0.116958
$371$	−28.2897	−1.46873
$372$	0	0
$373$	−9.20405	−0.476567	−0.238284	−	0.971196i	$-0.576585\pi$
$373$	−9.20405	−0.476567	−0.238284	+	0.971196i	$0.576585\pi$
$374$	−6.87613	−0.355556
$375$	0	0
$376$	−8.14159	−0.419870
$377$	10.5211	0.541866
$378$	0	0
$379$	5.20872	0.267554	0.133777	−	0.991011i	$-0.457289\pi$
$379$	5.20872	0.267554	0.133777	+	0.991011i	$0.457289\pi$
$380$	3.50263	0.179681
$381$	0	0
$382$	9.60455	0.491411
$383$	12.3327	0.630174	0.315087	−	0.949063i	$-0.397966\pi$
$383$	12.3327	0.630174	0.315087	+	0.949063i	$0.397966\pi$
$384$	0	0
$385$	−18.8558	−0.960978
$386$	5.52756	0.281345
$387$	0	0
$388$	9.16220	0.465140
$389$	16.7375	0.848626	0.424313	−	0.905516i	$-0.360516\pi$
$389$	16.7375	0.848626	0.424313	+	0.905516i	$0.360516\pi$
$390$	0	0
$391$	−21.7860	−1.10176
$392$	26.8549	1.35638
$393$	0	0
$394$	0.700511	0.0352912
$395$	−13.2498	−0.666668
$396$	0	0
$397$	23.7860	1.19378	0.596892	−	0.802322i	$-0.296402\pi$
$397$	23.7860	1.19378	0.596892	+	0.802322i	$0.296402\pi$
$398$	−5.19915	−0.260610
$399$	0	0
$400$	−8.02161	−0.401081
$401$	2.86075	0.142859	0.0714295	−	0.997446i	$-0.477244\pi$
$401$	2.86075	0.142859	0.0714295	+	0.997446i	$0.477244\pi$
$402$	0	0
$403$	37.8857	1.88722
$404$	15.4244	0.767395
$405$	0	0
$406$	4.08567	0.202768
$407$	−9.04439	−0.448314
$408$	0	0
$409$	−6.71964	−0.332265	−0.166132	−	0.986103i	$-0.553128\pi$
$409$	−6.71964	−0.332265	−0.166132	+	0.986103i	$0.553128\pi$
$410$	−3.35658	−0.165770
$411$	0	0
$412$	−12.7753	−0.629392
$413$	−46.3850	−2.28246
$414$	0	0
$415$	1.74467	0.0856423
$416$	−24.3831	−1.19548
$417$	0	0
$418$	1.44250	0.0705551
$419$	39.3202	1.92092	0.960459	−	0.278420i	$-0.0898107\pi$
$419$	39.3202	1.92092	0.960459	+	0.278420i	$0.0898107\pi$
$420$	0	0
$421$	5.48595	0.267369	0.133685	−	0.991024i	$-0.457319\pi$
$421$	5.48595	0.267369	0.133685	+	0.991024i	$0.457319\pi$
$422$	9.08108	0.442060
$423$	0	0
$424$	9.62988	0.467669
$425$	16.8494	0.817314
$426$	0	0
$427$	−33.1808	−1.60573
$428$	−26.5147	−1.28164
$429$	0	0
$430$	−5.33498	−0.257276
$431$	5.47777	0.263855	0.131927	−	0.991259i	$-0.457883\pi$
$431$	5.47777	0.263855	0.131927	+	0.991259i	$0.457883\pi$
$432$	0	0
$433$	−25.3491	−1.21820	−0.609099	−	0.793094i	$-0.708469\pi$
$433$	−25.3491	−1.21820	−0.609099	+	0.793094i	$0.708469\pi$
$434$	14.7121	0.706206
$435$	0	0
$436$	6.61280	0.316696
$437$	4.57035	0.218630
$438$	0	0
$439$	−8.92124	−0.425788	−0.212894	−	0.977075i	$-0.568289\pi$
$439$	−8.92124	−0.425788	−0.212894	+	0.977075i	$0.568289\pi$
$440$	6.41854	0.305992
$441$	0	0
$442$	14.1760	0.674284
$443$	39.4683	1.87520	0.937598	−	0.347720i	$-0.113044\pi$
$443$	39.4683	1.87520	0.937598	+	0.347720i	$0.113044\pi$
$444$	0	0
$445$	21.0131	0.996114
$446$	0.892689	0.0422700
$447$	0	0
$448$	18.7356	0.885173
$449$	7.56629	0.357075	0.178538	−	0.983933i	$-0.442863\pi$
$449$	7.56629	0.357075	0.178538	+	0.983933i	$0.442863\pi$
$450$	0	0
$451$	13.4942	0.635415
$452$	16.5175	0.776917
$453$	0	0
$454$	0.708746	0.0332631
$455$	38.8735	1.82242
$456$	0	0
$457$	−40.5680	−1.89769	−0.948846	−	0.315740i	$-0.897747\pi$
$457$	−40.5680	−1.89769	−0.948846	+	0.315740i	$0.897747\pi$
$458$	5.47888	0.256011
$459$	0	0
$460$	9.67267	0.450990
$461$	16.6102	0.773613	0.386807	−	0.922161i	$-0.373578\pi$
$461$	16.6102	0.773613	0.386807	+	0.922161i	$0.373578\pi$
$462$	0	0
$463$	−10.7052	−0.497514	−0.248757	−	0.968566i	$-0.580022\pi$
$463$	−10.7052	−0.497514	−0.248757	+	0.968566i	$0.580022\pi$
$464$	5.72815	0.265922
$465$	0	0
$466$	10.8439	0.502336
$467$	21.0194	0.972664	0.486332	−	0.873774i	$-0.338335\pi$
$467$	21.0194	0.972664	0.486332	+	0.873774i	$0.338335\pi$
$468$	0	0
$469$	−72.2418	−3.33582
$470$	−3.20355	−0.147769
$471$	0	0
$472$	15.7896	0.726774
$473$	21.4478	0.986169
$474$	0	0
$475$	−3.53473	−0.162184
$476$	−53.7379	−2.46307
$477$	0	0
$478$	1.62207	0.0741918
$479$	−11.8653	−0.542139	−0.271070	−	0.962560i	$-0.587377\pi$
$479$	−11.8653	−0.542139	−0.271070	+	0.962560i	$0.587377\pi$
$480$	0	0
$481$	18.6461	0.850191
$482$	2.44466	0.111351
$483$	0	0
$484$	7.68241	0.349200
$485$	7.57959	0.344171
$486$	0	0
$487$	0.526452	0.0238558	0.0119279	−	0.999929i	$-0.496203\pi$
$487$	0.526452	0.0238558	0.0119279	+	0.999929i	$0.496203\pi$
$488$	11.2948	0.511293
$489$	0	0
$490$	10.5668	0.477361
$491$	−26.2774	−1.18588	−0.592940	−	0.805246i	$-0.702033\pi$
$491$	−26.2774	−1.18588	−0.592940	+	0.805246i	$0.702033\pi$
$492$	0	0
$493$	−12.0319	−0.541891
$494$	−2.97390	−0.133802
$495$	0	0
$496$	20.6266	0.926160
$497$	−28.3612	−1.27218
$498$	0	0
$499$	19.1651	0.857950	0.428975	−	0.903316i	$-0.358875\pi$
$499$	19.1651	0.857950	0.428975	+	0.903316i	$0.358875\pi$
$500$	−21.0942	−0.943363
$501$	0	0
$502$	4.39665	0.196232
$503$	−22.4963	−1.00306	−0.501531	−	0.865140i	$-0.667230\pi$
$503$	−22.4963	−1.00306	−0.501531	+	0.865140i	$0.667230\pi$
$504$	0	0
$505$	12.7601	0.567819
$506$	3.98353	0.177090
$507$	0	0
$508$	24.3412	1.07997
$509$	−11.4434	−0.507222	−0.253611	−	0.967306i	$-0.581618\pi$
$509$	−11.4434	−0.507222	−0.253611	+	0.967306i	$0.581618\pi$
$510$	0	0
$511$	−21.3467	−0.944321
$512$	−22.8760	−1.01098
$513$	0	0
$514$	0.318997	0.0140703
$515$	−10.5686	−0.465706
$516$	0	0
$517$	12.8789	0.566415
$518$	7.24085	0.318145
$519$	0	0
$520$	−13.2326	−0.580290
$521$	6.38668	0.279805	0.139903	−	0.990165i	$-0.455321\pi$
$521$	6.38668	0.279805	0.139903	+	0.990165i	$0.455321\pi$
$522$	0	0
$523$	34.8404	1.52347	0.761733	−	0.647891i	$-0.224349\pi$
$523$	34.8404	1.52347	0.761733	+	0.647891i	$0.224349\pi$
$524$	29.9772	1.30956
$525$	0	0
$526$	5.01332	0.218591
$527$	−43.3260	−1.88731
$528$	0	0
$529$	−10.3788	−0.451251
$530$	3.78916	0.164591
$531$	0	0
$532$	11.2734	0.488762
$533$	−27.8199	−1.20501
$534$	0	0
$535$	−21.9348	−0.948324
$536$	24.5913	1.06218
$537$	0	0
$538$	9.05293	0.390299
$539$	−42.4809	−1.82978
$540$	0	0
$541$	−34.2589	−1.47291	−0.736453	−	0.676489i	$-0.763501\pi$
$541$	−34.2589	−1.47291	−0.736453	+	0.676489i	$0.763501\pi$
$542$	8.82386	0.379017
$543$	0	0
$544$	27.8844	1.19554
$545$	5.47055	0.234333
$546$	0	0
$547$	−21.6616	−0.926184	−0.463092	−	0.886310i	$-0.653260\pi$
$547$	−21.6616	−0.926184	−0.463092	+	0.886310i	$0.653260\pi$
$548$	−35.0390	−1.49679
$549$	0	0
$550$	−3.08088	−0.131369
$551$	2.52411	0.107531
$552$	0	0
$553$	−42.6449	−1.81345
$554$	−3.54582	−0.150647
$555$	0	0
$556$	9.23980	0.391855
$557$	25.7714	1.09197	0.545985	−	0.837795i	$-0.316156\pi$
$557$	25.7714	1.09197	0.545985	+	0.837795i	$0.316156\pi$
$558$	0	0
$559$	−44.2173	−1.87019
$560$	21.1644	0.894358
$561$	0	0
$562$	5.98106	0.252296
$563$	−20.5069	−0.864262	−0.432131	−	0.901811i	$-0.642238\pi$
$563$	−20.5069	−0.864262	−0.432131	+	0.901811i	$0.642238\pi$
$564$	0	0
$565$	13.6644	0.574864
$566$	−13.1318	−0.551972
$567$	0	0
$568$	9.65424	0.405083
$569$	44.5631	1.86818	0.934092	−	0.357032i	$-0.116211\pi$
$569$	44.5631	1.86818	0.934092	+	0.357032i	$0.116211\pi$
$570$	0	0
$571$	28.8727	1.20829	0.604143	−	0.796876i	$-0.293515\pi$
$571$	28.8727	1.20829	0.604143	+	0.796876i	$0.293515\pi$
$572$	25.3030	1.05797
$573$	0	0
$574$	−10.8033	−0.450921
$575$	−9.76130	−0.407075
$576$	0	0
$577$	−12.2296	−0.509124	−0.254562	−	0.967056i	$-0.581931\pi$
$577$	−12.2296	−0.509124	−0.254562	+	0.967056i	$0.581931\pi$
$578$	−8.88301	−0.369484
$579$	0	0
$580$	5.34201	0.221815
$581$	5.61528	0.232961
$582$	0	0
$583$	−15.2332	−0.630896
$584$	7.26646	0.300688
$585$	0	0
$586$	−11.8969	−0.491455
$587$	17.2501	0.711988	0.355994	−	0.934488i	$-0.384142\pi$
$587$	17.2501	0.711988	0.355994	+	0.934488i	$0.384142\pi$
$588$	0	0
$589$	9.08911	0.374510
$590$	6.21287	0.255780
$591$	0	0
$592$	10.1517	0.417234
$593$	−35.4901	−1.45740	−0.728702	−	0.684831i	$-0.759876\pi$
$593$	−35.4901	−1.45740	−0.728702	+	0.684831i	$0.759876\pi$
$594$	0	0
$595$	−44.4557	−1.82250
$596$	28.3674	1.16197
$597$	0	0
$598$	−8.21255	−0.335836
$599$	27.9759	1.14306	0.571532	−	0.820580i	$-0.306349\pi$
$599$	27.9759	1.14306	0.571532	+	0.820580i	$0.306349\pi$
$600$	0	0
$601$	3.04309	0.124130	0.0620652	−	0.998072i	$-0.480231\pi$
$601$	3.04309	0.124130	0.0620652	+	0.998072i	$0.480231\pi$
$602$	−17.1709	−0.699832
$603$	0	0
$604$	−27.0365	−1.10010
$605$	6.35541	0.258384
$606$	0	0
$607$	−18.9064	−0.767387	−0.383694	−	0.923460i	$-0.625348\pi$
$607$	−18.9064	−0.767387	−0.383694	+	0.923460i	$0.625348\pi$
$608$	−5.84971	−0.237237
$609$	0	0
$610$	4.44429	0.179944
$611$	−26.5516	−1.07416
$612$	0	0
$613$	44.2346	1.78662	0.893309	−	0.449442i	$-0.148377\pi$
$613$	44.2346	1.78662	0.893309	+	0.449442i	$0.148377\pi$
$614$	5.90426	0.238276
$615$	0	0
$616$	20.6584	0.832349
$617$	1.91143	0.0769511	0.0384756	−	0.999260i	$-0.487750\pi$
$617$	1.91143	0.0769511	0.0384756	+	0.999260i	$0.487750\pi$
$618$	0	0
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	19.2361	0.772541
$621$	0	0
$622$	−0.0739825	−0.00296643
$623$	67.6314	2.70960
$624$	0	0
$625$	−3.71246	−0.148499
$626$	7.56937	0.302533
$627$	0	0
$628$	−30.7491	−1.22702
$629$	−21.3237	−0.850231
$630$	0	0
$631$	−11.2642	−0.448419	−0.224210	−	0.974541i	$-0.571980\pi$
$631$	−11.2642	−0.448419	−0.224210	+	0.974541i	$0.571980\pi$
$632$	14.5164	0.577432
$633$	0	0
$634$	−0.742652	−0.0294945
$635$	20.1367	0.799100
$636$	0	0
$637$	87.5798	3.47004
$638$	2.20002	0.0870996
$639$	0	0
$640$	−16.1580	−0.638701
$641$	−48.5425	−1.91731	−0.958657	−	0.284565i	$-0.908151\pi$
$641$	−48.5425	−1.91731	−0.958657	+	0.284565i	$0.908151\pi$
$642$	0	0
$643$	−6.33981	−0.250018	−0.125009	−	0.992156i	$-0.539896\pi$
$643$	−6.33981	−0.250018	−0.125009	+	0.992156i	$0.539896\pi$
$644$	31.1319	1.22677
$645$	0	0
$646$	3.40094	0.133808
$647$	−5.36992	−0.211113	−0.105557	−	0.994413i	$-0.533662\pi$
$647$	−5.36992	−0.211113	−0.105557	+	0.994413i	$0.533662\pi$
$648$	0	0
$649$	−24.9771	−0.980435
$650$	6.35162	0.249131
$651$	0	0
$652$	28.9416	1.13344
$653$	−11.7733	−0.460724	−0.230362	−	0.973105i	$-0.573991\pi$
$653$	−11.7733	−0.460724	−0.230362	+	0.973105i	$0.573991\pi$
$654$	0	0
$655$	24.7991	0.968982
$656$	−15.1463	−0.591364
$657$	0	0
$658$	−10.3108	−0.401955
$659$	−0.123206	−0.00479943	−0.00239972	−	0.999997i	$-0.500764\pi$
$659$	−0.123206	−0.00479943	−0.00239972	+	0.999997i	$0.500764\pi$
$660$	0	0
$661$	9.04712	0.351892	0.175946	−	0.984400i	$-0.443702\pi$
$661$	9.04712	0.351892	0.175946	+	0.984400i	$0.443702\pi$
$662$	−10.1425	−0.394201
$663$	0	0
$664$	−1.91145	−0.0741788
$665$	9.32609	0.361650
$666$	0	0
$667$	6.97044	0.269896
$668$	18.0199	0.697211
$669$	0	0
$670$	9.67617	0.373823
$671$	−17.8670	−0.689747
$672$	0	0
$673$	−6.36878	−0.245498	−0.122749	−	0.992438i	$-0.539171\pi$
$673$	−6.36878	−0.245498	−0.122749	+	0.992438i	$0.539171\pi$
$674$	3.45812	0.133202
$675$	0	0
$676$	−28.5812	−1.09928
$677$	−13.5905	−0.522326	−0.261163	−	0.965295i	$-0.584106\pi$
$677$	−13.5905	−0.522326	−0.261163	+	0.965295i	$0.584106\pi$
$678$	0	0
$679$	24.3952	0.936203
$680$	15.1328	0.580317
$681$	0	0
$682$	7.92209	0.303352
$683$	33.0393	1.26422	0.632108	−	0.774881i	$-0.282190\pi$
$683$	33.0393	1.26422	0.632108	+	0.774881i	$0.282190\pi$
$684$	0	0
$685$	−28.9866	−1.10752
$686$	19.4334	0.741969
$687$	0	0
$688$	−24.0737	−0.917802
$689$	31.4052	1.19644
$690$	0	0
$691$	47.4035	1.80331	0.901657	−	0.432453i	$-0.142352\pi$
$691$	47.4035	1.80331	0.901657	+	0.432453i	$0.142352\pi$
$692$	−24.6659	−0.937656
$693$	0	0
$694$	−5.06386	−0.192221
$695$	7.64378	0.289945
$696$	0	0
$697$	31.8148	1.20507
$698$	−5.11700	−0.193681
$699$	0	0
$700$	−24.0775	−0.910044
$701$	2.90566	0.109745	0.0548726	−	0.998493i	$-0.482525\pi$
$701$	2.90566	0.109745	0.0548726	+	0.998493i	$0.482525\pi$
$702$	0	0
$703$	4.47337	0.168716
$704$	10.0886	0.380228
$705$	0	0
$706$	9.68898	0.364650
$707$	41.0691	1.54456
$708$	0	0
$709$	−27.9104	−1.04820	−0.524099	−	0.851657i	$-0.675598\pi$
$709$	−27.9104	−1.04820	−0.524099	+	0.851657i	$0.675598\pi$
$710$	3.79874	0.142564
$711$	0	0
$712$	−23.0219	−0.862782
$713$	25.1000	0.940001
$714$	0	0
$715$	20.9323	0.782825
$716$	26.2309	0.980294
$717$	0	0
$718$	−10.7249	−0.400249
$719$	−25.7321	−0.959646	−0.479823	−	0.877365i	$-0.659299\pi$
$719$	−25.7321	−0.959646	−0.479823	+	0.877365i	$0.659299\pi$
$720$	0	0
$721$	−34.0153	−1.26680
$722$	7.47736	0.278279
$723$	0	0
$724$	−10.7330	−0.398889
$725$	−5.39096	−0.200215
$726$	0	0
$727$	−46.8285	−1.73677	−0.868386	−	0.495889i	$-0.834842\pi$
$727$	−46.8285	−1.73677	−0.868386	+	0.495889i	$0.834842\pi$
$728$	−42.5898	−1.57848
$729$	0	0
$730$	2.85920	0.105824
$731$	50.5667	1.87028
$732$	0	0
$733$	−53.3687	−1.97122	−0.985609	−	0.169042i	$-0.945933\pi$
$733$	−53.3687	−1.97122	−0.985609	+	0.169042i	$0.945933\pi$
$734$	−14.2238	−0.525009
$735$	0	0
$736$	−16.1542	−0.595453
$737$	−38.9002	−1.43291
$738$	0	0
$739$	4.94427	0.181878	0.0909390	−	0.995856i	$-0.471013\pi$
$739$	4.94427	0.181878	0.0909390	+	0.995856i	$0.471013\pi$
$740$	9.46740	0.348029
$741$	0	0
$742$	12.1956	0.447714
$743$	−6.46500	−0.237178	−0.118589	−	0.992943i	$-0.537837\pi$
$743$	−6.46500	−0.237178	−0.118589	+	0.992943i	$0.537837\pi$
$744$	0	0
$745$	23.4674	0.859780
$746$	3.96783	0.145272
$747$	0	0
$748$	−28.9364	−1.05802
$749$	−70.5980	−2.57960
$750$	0	0
$751$	17.5196	0.639299	0.319649	−	0.947536i	$-0.396435\pi$
$751$	17.5196	0.639299	0.319649	+	0.947536i	$0.396435\pi$
$752$	−14.4558	−0.527148
$753$	0	0
$754$	−4.53562	−0.165178
$755$	−22.3664	−0.813997
$756$	0	0
$757$	22.6591	0.823560	0.411780	−	0.911283i	$-0.364907\pi$
$757$	22.6591	0.823560	0.411780	+	0.911283i	$0.364907\pi$
$758$	−2.24546	−0.0815587
$759$	0	0
$760$	−3.17462	−0.115156
$761$	−34.8860	−1.26462	−0.632308	−	0.774717i	$-0.717892\pi$
$761$	−34.8860	−1.26462	−0.632308	+	0.774717i	$0.717892\pi$
$762$	0	0
$763$	17.6072	0.637424
$764$	40.4183	1.46228
$765$	0	0
$766$	−5.31660	−0.192096
$767$	51.4934	1.85932
$768$	0	0
$769$	−9.05317	−0.326466	−0.163233	−	0.986588i	$-0.552192\pi$
$769$	−9.05317	−0.326466	−0.163233	+	0.986588i	$0.552192\pi$
$770$	8.12864	0.292936
$771$	0	0
$772$	23.2613	0.837192
$773$	−25.6348	−0.922020	−0.461010	−	0.887395i	$-0.652513\pi$
$773$	−25.6348	−0.922020	−0.461010	+	0.887395i	$0.652513\pi$
$774$	0	0
$775$	−19.4124	−0.697314
$776$	−8.30420	−0.298103
$777$	0	0
$778$	−7.21548	−0.258688
$779$	−6.67423	−0.239129
$780$	0	0
$781$	−15.2718	−0.546466
$782$	9.39185	0.335852
$783$	0	0
$784$	47.6821	1.70293
$785$	−25.4377	−0.907910
$786$	0	0
$787$	0.199605	0.00711515	0.00355758	−	0.999994i	$-0.498868\pi$
$787$	0.199605	0.00711515	0.00355758	+	0.999994i	$0.498868\pi$
$788$	2.94792	0.105015
$789$	0	0
$790$	5.71192	0.203221
$791$	43.9794	1.56373
$792$	0	0
$793$	36.8350	1.30805
$794$	−10.2540	−0.363902
$795$	0	0
$796$	−21.8793	−0.775491
$797$	22.5822	0.799904	0.399952	−	0.916536i	$-0.369027\pi$
$797$	22.5822	0.799904	0.399952	+	0.916536i	$0.369027\pi$
$798$	0	0
$799$	30.3643	1.07421
$800$	12.4937	0.441720
$801$	0	0
$802$	−1.23326	−0.0435478
$803$	−11.4946	−0.405636
$804$	0	0
$805$	25.7544	0.907723
$806$	−16.3324	−0.575284
$807$	0	0
$808$	−13.9800	−0.491815
$809$	2.59131	0.0911055	0.0455527	−	0.998962i	$-0.485495\pi$
$809$	2.59131	0.0911055	0.0455527	+	0.998962i	$0.485495\pi$
$810$	0	0
$811$	−36.4445	−1.27974	−0.639870	−	0.768483i	$-0.721012\pi$
$811$	−36.4445	−1.27974	−0.639870	+	0.768483i	$0.721012\pi$
$812$	17.1935	0.603373
$813$	0	0
$814$	3.89900	0.136660
$815$	23.9424	0.838667
$816$	0	0
$817$	−10.6081	−0.371130
$818$	2.89681	0.101285
$819$	0	0
$820$	−14.1253	−0.493277
$821$	−1.22959	−0.0429129	−0.0214565	−	0.999770i	$-0.506830\pi$
$821$	−1.22959	−0.0429129	−0.0214565	+	0.999770i	$0.506830\pi$
$822$	0	0
$823$	19.0447	0.663858	0.331929	−	0.943304i	$-0.392301\pi$
$823$	19.0447	0.663858	0.331929	+	0.943304i	$0.392301\pi$
$824$	11.5789	0.403370
$825$	0	0
$826$	19.9964	0.695763
$827$	1.35730	0.0471980	0.0235990	−	0.999722i	$-0.492488\pi$
$827$	1.35730	0.0471980	0.0235990	+	0.999722i	$0.492488\pi$
$828$	0	0
$829$	9.29337	0.322772	0.161386	−	0.986891i	$-0.448404\pi$
$829$	9.29337	0.322772	0.161386	+	0.986891i	$0.448404\pi$
$830$	−0.752118	−0.0261064
$831$	0	0
$832$	−20.7989	−0.721073
$833$	−100.156	−3.47020
$834$	0	0
$835$	14.9073	0.515888
$836$	6.07040	0.209949
$837$	0	0
$838$	−16.9508	−0.585555
$839$	−41.4301	−1.43033	−0.715164	−	0.698957i	$-0.753648\pi$
$839$	−41.4301	−1.43033	−0.715164	+	0.698957i	$0.753648\pi$
$840$	0	0
$841$	−25.1504	−0.867254
$842$	−2.36497	−0.0815023
$843$	0	0
$844$	38.2154	1.31543
$845$	−23.6443	−0.813390
$846$	0	0
$847$	20.4552	0.702847
$848$	17.0983	0.587159
$849$	0	0
$850$	−7.26369	−0.249143
$851$	12.3534	0.423469
$852$	0	0
$853$	−36.7516	−1.25835	−0.629176	−	0.777263i	$-0.716607\pi$
$853$	−36.7516	−1.25835	−0.629176	+	0.777263i	$0.716607\pi$
$854$	14.3041	0.489477
$855$	0	0
$856$	24.0317	0.821388
$857$	−28.9515	−0.988963	−0.494481	−	0.869188i	$-0.664642\pi$
$857$	−28.9515	−0.988963	−0.494481	+	0.869188i	$0.664642\pi$
$858$	0	0
$859$	41.6194	1.42004	0.710018	−	0.704184i	$-0.248687\pi$
$859$	41.6194	1.42004	0.710018	+	0.704184i	$0.248687\pi$
$860$	−22.4509	−0.765569
$861$	0	0
$862$	−2.36144	−0.0804311
$863$	28.4643	0.968937	0.484469	−	0.874809i	$-0.339013\pi$
$863$	28.4643	0.968937	0.484469	+	0.874809i	$0.339013\pi$
$864$	0	0
$865$	−20.4053	−0.693800
$866$	10.9279	0.371345
$867$	0	0
$868$	61.9123	2.10144
$869$	−22.9631	−0.778970
$870$	0	0
$871$	80.1978	2.71740
$872$	−5.99353	−0.202967
$873$	0	0
$874$	−1.97026	−0.0666451
$875$	−56.1654	−1.89874
$876$	0	0
$877$	4.10040	0.138461	0.0692303	−	0.997601i	$-0.477946\pi$
$877$	4.10040	0.138461	0.0692303	+	0.997601i	$0.477946\pi$
$878$	3.84591	0.129793
$879$	0	0
$880$	11.3964	0.384174
$881$	−23.8628	−0.803958	−0.401979	−	0.915649i	$-0.631678\pi$
$881$	−23.8628	−0.803958	−0.401979	+	0.915649i	$0.631678\pi$
$882$	0	0
$883$	−13.3019	−0.447645	−0.223822	−	0.974630i	$-0.571854\pi$
$883$	−13.3019	−0.447645	−0.223822	+	0.974630i	$0.571854\pi$
$884$	59.6561	2.00645
$885$	0	0
$886$	−17.0146	−0.571618
$887$	21.2442	0.713310	0.356655	−	0.934236i	$-0.383917\pi$
$887$	21.2442	0.713310	0.356655	+	0.934236i	$0.383917\pi$
$888$	0	0
$889$	64.8108	2.17369
$890$	−9.05864	−0.303646
$891$	0	0
$892$	3.75665	0.125782
$893$	−6.36995	−0.213162
$894$	0	0
$895$	21.7000	0.725350
$896$	−52.0052	−1.73737
$897$	0	0
$898$	−3.26180	−0.108848
$899$	13.8622	0.462330
$900$	0	0
$901$	−35.9149	−1.19650
$902$	−5.81728	−0.193694
$903$	0	0
$904$	−14.9707	−0.497917
$905$	−8.87907	−0.295150
$906$	0	0
$907$	−29.8311	−0.990525	−0.495262	−	0.868743i	$-0.664928\pi$
$907$	−29.8311	−0.990525	−0.495262	+	0.868743i	$0.664928\pi$
$908$	2.98257	0.0989802
$909$	0	0
$910$	−16.7582	−0.555530
$911$	−26.4575	−0.876578	−0.438289	−	0.898834i	$-0.644415\pi$
$911$	−26.4575	−0.876578	−0.438289	+	0.898834i	$0.644415\pi$
$912$	0	0
$913$	3.02367	0.100069
$914$	17.4887	0.578475
$915$	0	0
$916$	23.0565	0.761807
$917$	79.8170	2.63579
$918$	0	0
$919$	48.0450	1.58486	0.792429	−	0.609964i	$-0.208816\pi$
$919$	48.0450	1.58486	0.792429	+	0.609964i	$0.208816\pi$
$920$	−8.76686	−0.289035
$921$	0	0
$922$	−7.16058	−0.235821
$923$	31.4847	1.03633
$924$	0	0
$925$	−9.55416	−0.314139
$926$	4.61498	0.151658
$927$	0	0
$928$	−8.92164	−0.292867
$929$	17.1784	0.563605	0.281803	−	0.959472i	$-0.409068\pi$
$929$	17.1784	0.563605	0.281803	+	0.959472i	$0.409068\pi$
$930$	0	0
$931$	21.0111	0.688612
$932$	45.6339	1.49479
$933$	0	0
$934$	−9.06140	−0.296498
$935$	−23.9381	−0.782861
$936$	0	0
$937$	−51.5569	−1.68429	−0.842145	−	0.539252i	$-0.818707\pi$
$937$	−51.5569	−1.68429	−0.842145	+	0.539252i	$0.818707\pi$
$938$	31.1432	1.01686
$939$	0	0
$940$	−13.4813	−0.439712
$941$	−39.2864	−1.28070	−0.640349	−	0.768084i	$-0.721210\pi$
$941$	−39.2864	−1.28070	−0.640349	+	0.768084i	$0.721210\pi$
$942$	0	0
$943$	−18.4312	−0.600202
$944$	28.0351	0.912466
$945$	0	0
$946$	−9.24604	−0.300615
$947$	−15.5225	−0.504413	−0.252206	−	0.967673i	$-0.581156\pi$
$947$	−15.5225	−0.504413	−0.252206	+	0.967673i	$0.581156\pi$
$948$	0	0
$949$	23.6976	0.769255
$950$	1.52381	0.0494388
$951$	0	0
$952$	48.7056	1.57856
$953$	−22.1294	−0.716842	−0.358421	−	0.933560i	$-0.616685\pi$
$953$	−22.1294	−0.716842	−0.358421	+	0.933560i	$0.616685\pi$
$954$	0	0
$955$	33.4367	1.08199
$956$	6.82607	0.220771
$957$	0	0
$958$	5.11508	0.165261
$959$	−93.2947	−3.01264
$960$	0	0
$961$	18.9165	0.610211
$962$	−8.03828	−0.259164
$963$	0	0
$964$	10.2877	0.331345
$965$	19.2433	0.619464
$966$	0	0
$967$	−9.97343	−0.320724	−0.160362	−	0.987058i	$-0.551266\pi$
$967$	−9.97343	−0.320724	−0.160362	+	0.987058i	$0.551266\pi$
$968$	−6.96298	−0.223799
$969$	0	0
$970$	−3.26753	−0.104914
$971$	55.8372	1.79190	0.895951	−	0.444154i	$-0.146496\pi$
$971$	55.8372	1.79190	0.895951	+	0.444154i	$0.146496\pi$
$972$	0	0
$973$	24.6018	0.788699
$974$	−0.226952	−0.00727200
$975$	0	0
$976$	20.0545	0.641930
$977$	26.9677	0.862772	0.431386	−	0.902167i	$-0.358025\pi$
$977$	26.9677	0.862772	0.431386	+	0.902167i	$0.358025\pi$
$978$	0	0
$979$	36.4177	1.16391
$980$	44.4678	1.42047
$981$	0	0
$982$	11.3281	0.361493
$983$	−25.7677	−0.821862	−0.410931	−	0.911666i	$-0.634796\pi$
$983$	−25.7677	−0.821862	−0.410931	+	0.911666i	$0.634796\pi$
$984$	0	0
$985$	2.43872	0.0777040
$986$	5.18692	0.165185
$987$	0	0
$988$	−12.5149	−0.398152
$989$	−29.2947	−0.931518
$990$	0	0
$991$	−45.7646	−1.45376	−0.726881	−	0.686763i	$-0.759031\pi$
$991$	−45.7646	−1.45376	−0.726881	+	0.686763i	$0.759031\pi$
$992$	−32.1261	−1.02000
$993$	0	0
$994$	12.2264	0.387799
$995$	−18.1000	−0.573809
$996$	0	0
$997$	−19.9914	−0.633133	−0.316567	−	0.948570i	$-0.602530\pi$
$997$	−19.9914	−0.633133	−0.316567	+	0.948570i	$0.602530\pi$
$998$	−8.26202	−0.261530
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5571.2.a.g.1.14		30
3.2	odd	2		619.2.a.b.1.17	✓	30
12.11	even	2		9904.2.a.n.1.22		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
619.2.a.b.1.17	✓	30	3.2	odd	2
5571.2.a.g.1.14		30	1.1	even	1	trivial
9904.2.a.n.1.22		30	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5571 = 3^{2} \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5571.a (trivial)

\(f(q)\)	\(=\)	\(q-0.431096 q^{2} -1.81416 q^{4} -1.50079 q^{5} -4.83036 q^{7} +1.64427 q^{8} +O(q^{10})\)	\(q-0.431096 q^{2} -1.81416 q^{4} -1.50079 q^{5} -4.83036 q^{7} +1.64427 q^{8} +0.646986 q^{10} -2.60102 q^{11} +5.36233 q^{13} +2.08235 q^{14} +2.91948 q^{16} -6.13234 q^{17} +1.28647 q^{19} +2.72267 q^{20} +1.12129 q^{22} +3.55264 q^{23} -2.74762 q^{25} -2.31168 q^{26} +8.76304 q^{28} +1.96205 q^{29} +7.06516 q^{31} -4.54711 q^{32} +2.64363 q^{34} +7.24937 q^{35} +3.47725 q^{37} -0.554591 q^{38} -2.46770 q^{40} -5.18803 q^{41} -8.24591 q^{43} +4.71865 q^{44} -1.53153 q^{46} -4.95150 q^{47} +16.3324 q^{49} +1.18449 q^{50} -9.72810 q^{52} +5.85664 q^{53} +3.90359 q^{55} -7.94241 q^{56} -0.845830 q^{58} +9.60280 q^{59} +6.86922 q^{61} -3.04576 q^{62} -3.87871 q^{64} -8.04774 q^{65} +14.9558 q^{67} +11.1250 q^{68} -3.12518 q^{70} +5.87145 q^{71} +4.41927 q^{73} -1.49903 q^{74} -2.33385 q^{76} +12.5639 q^{77} +8.82851 q^{79} -4.38153 q^{80} +2.23654 q^{82} -1.16250 q^{83} +9.20337 q^{85} +3.55478 q^{86} -4.27677 q^{88} -14.0013 q^{89} -25.9020 q^{91} -6.44504 q^{92} +2.13457 q^{94} -1.93072 q^{95} -5.05039 q^{97} -7.04084 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 9 q^{2} + 33 q^{4} - 21 q^{5} + 2 q^{7} - 27 q^{8}+O(q^{10}) \) 30 * q - 9 * q^2 + 33 * q^4 - 21 * q^5 + 2 * q^7 - 27 * q^8	\( 30 q - 9 q^{2} + 33 q^{4} - 21 q^{5} + 2 q^{7} - 27 q^{8} + 5 q^{10} - 23 q^{11} + 9 q^{13} - 7 q^{14} + 35 q^{16} - 4 q^{17} - q^{19} - 29 q^{20} - 4 q^{23} + 35 q^{25} - q^{26} - 13 q^{28} - 90 q^{29} + 2 q^{31} - 43 q^{32} - 9 q^{34} - 9 q^{35} + 19 q^{37} - 5 q^{38} - 12 q^{40} - 59 q^{41} - 4 q^{43} - 52 q^{44} - q^{46} - 4 q^{47} + 30 q^{49} - 31 q^{50} - 12 q^{52} - 34 q^{53} - 17 q^{55} - 2 q^{56} + 6 q^{58} - 13 q^{59} + 16 q^{61} - 28 q^{62} + 37 q^{64} - 31 q^{65} - 11 q^{67} + 52 q^{68} - 40 q^{70} - 42 q^{71} - 4 q^{73} - 16 q^{74} - 42 q^{76} - 29 q^{77} + 3 q^{79} - 21 q^{80} - 43 q^{82} + 11 q^{83} + 19 q^{85} + 11 q^{86} - 47 q^{88} - 58 q^{89} - 39 q^{91} + 7 q^{92} - 46 q^{94} - 23 q^{95} - 9 q^{97}+O(q^{100}) \) 30 * q - 9 * q^2 + 33 * q^4 - 21 * q^5 + 2 * q^7 - 27 * q^8 + 5 * q^10 - 23 * q^11 + 9 * q^13 - 7 * q^14 + 35 * q^16 - 4 * q^17 - q^19 - 29 * q^20 - 4 * q^23 + 35 * q^25 - q^26 - 13 * q^28 - 90 * q^29 + 2 * q^31 - 43 * q^32 - 9 * q^34 - 9 * q^35 + 19 * q^37 - 5 * q^38 - 12 * q^40 - 59 * q^41 - 4 * q^43 - 52 * q^44 - q^46 - 4 * q^47 + 30 * q^49 - 31 * q^50 - 12 * q^52 - 34 * q^53 - 17 * q^55 - 2 * q^56 + 6 * q^58 - 13 * q^59 + 16 * q^61 - 28 * q^62 + 37 * q^64 - 31 * q^65 - 11 * q^67 + 52 * q^68 - 40 * q^70 - 42 * q^71 - 4 * q^73 - 16 * q^74 - 42 * q^76 - 29 * q^77 + 3 * q^79 - 21 * q^80 - 43 * q^82 + 11 * q^83 + 19 * q^85 + 11 * q^86 - 47 * q^88 - 58 * q^89 - 39 * q^91 + 7 * q^92 - 46 * q^94 - 23 * q^95 - 9 * q^97

Introduction

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