LMFDB - Embedded newform 5780.2.a.q.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5780,2,Mod(1,5780)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5780, 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("5780.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5780 = 2^{2} \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5780.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:	$46.1535323683$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 24x^{10} + 206x^{8} - 16x^{7} - 776x^{6} + 152x^{5} + 1226x^{4} - 384x^{3} - 588x^{2} + 200x + 34 $ x^12 - 24x^10 + 206x^8 - 16x^7 - 776x^6 + 152x^5 + 1226x^4 - 384x^3 - 588x^2 + 200*x + 34
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 340)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$3.27990$ of defining polynomial
Character	$\chi$	$=$	5780.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.27990 q^{3} -1.00000 q^{5} -4.39608 q^{7} +7.75771 q^{9} +O(q^{10})$	$q+3.27990 q^{3} -1.00000 q^{5} -4.39608 q^{7} +7.75771 q^{9} +1.69753 q^{11} -1.18957 q^{13} -3.27990 q^{15} -5.70964 q^{19} -14.4187 q^{21} -2.36944 q^{23} +1.00000 q^{25} +15.6048 q^{27} -7.22678 q^{29} -5.03910 q^{31} +5.56772 q^{33} +4.39608 q^{35} -5.45850 q^{37} -3.90165 q^{39} +4.45431 q^{41} +3.94159 q^{43} -7.75771 q^{45} -7.43429 q^{47} +12.3255 q^{49} +6.57656 q^{53} -1.69753 q^{55} -18.7270 q^{57} +4.87751 q^{59} -3.62807 q^{61} -34.1035 q^{63} +1.18957 q^{65} -7.73044 q^{67} -7.77152 q^{69} -12.9021 q^{71} -9.83287 q^{73} +3.27990 q^{75} -7.46248 q^{77} +13.9301 q^{79} +27.9090 q^{81} -7.75068 q^{83} -23.7031 q^{87} -14.3594 q^{89} +5.22943 q^{91} -16.5277 q^{93} +5.70964 q^{95} +4.40439 q^{97} +13.1690 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{5} - 8 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^5 - 8 * q^7 + 12 * q^9	$ 12 q - 12 q^{5} - 8 q^{7} + 12 q^{9} + 8 q^{13} - 16 q^{21} - 8 q^{23} + 12 q^{25} - 16 q^{29} - 24 q^{31} + 8 q^{35} - 24 q^{37} + 8 q^{39} - 24 q^{41} + 8 q^{43} - 12 q^{45} + 8 q^{47} + 20 q^{49} + 16 q^{53} - 32 q^{57} + 8 q^{59} - 40 q^{61} - 24 q^{63} - 8 q^{65} + 16 q^{67} - 16 q^{69} - 16 q^{71} - 32 q^{73} + 24 q^{77} - 8 q^{79} + 4 q^{81} + 32 q^{83} + 16 q^{87} - 8 q^{89} - 8 q^{91} - 8 q^{93} - 32 q^{97} - 24 q^{99}+O(q^{100}) $ 12 * q - 12 * q^5 - 8 * q^7 + 12 * q^9 + 8 * q^13 - 16 * q^21 - 8 * q^23 + 12 * q^25 - 16 * q^29 - 24 * q^31 + 8 * q^35 - 24 * q^37 + 8 * q^39 - 24 * q^41 + 8 * q^43 - 12 * q^45 + 8 * q^47 + 20 * q^49 + 16 * q^53 - 32 * q^57 + 8 * q^59 - 40 * q^61 - 24 * q^63 - 8 * q^65 + 16 * q^67 - 16 * q^69 - 16 * q^71 - 32 * q^73 + 24 * q^77 - 8 * q^79 + 4 * q^81 + 32 * q^83 + 16 * q^87 - 8 * q^89 - 8 * q^91 - 8 * q^93 - 32 * q^97 - 24 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.27990	1.89365	0.946824	−	0.321751i	$-0.104271\pi$
$3$	3.27990	1.89365	0.946824	+	0.321751i	$0.104271\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.39608	−1.66156	−0.830781	−	0.556600i	$-0.812106\pi$
$7$	−4.39608	−1.66156	−0.830781	+	0.556600i	$0.812106\pi$
$8$	0	0
$9$	7.75771	2.58590
$10$	0	0
$11$	1.69753	0.511825	0.255912	−	0.966700i	$-0.417624\pi$
$11$	1.69753	0.511825	0.255912	+	0.966700i	$0.417624\pi$
$12$	0	0
$13$	−1.18957	−0.329926	−0.164963	−	0.986300i	$-0.552751\pi$
$13$	−1.18957	−0.329926	−0.164963	+	0.986300i	$0.552751\pi$
$14$	0	0
$15$	−3.27990	−0.846865
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−5.70964	−1.30988	−0.654941	−	0.755680i	$-0.727306\pi$
$19$	−5.70964	−1.30988	−0.654941	+	0.755680i	$0.727306\pi$
$20$	0	0
$21$	−14.4187	−3.14641
$22$	0	0
$23$	−2.36944	−0.494063	−0.247031	−	0.969007i	$-0.579455\pi$
$23$	−2.36944	−0.494063	−0.247031	+	0.969007i	$0.579455\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	15.6048	3.00315
$28$	0	0
$29$	−7.22678	−1.34198	−0.670990	−	0.741467i	$-0.734131\pi$
$29$	−7.22678	−1.34198	−0.670990	+	0.741467i	$0.734131\pi$
$30$	0	0
$31$	−5.03910	−0.905049	−0.452525	−	0.891752i	$-0.649477\pi$
$31$	−5.03910	−0.905049	−0.452525	+	0.891752i	$0.649477\pi$
$32$	0	0
$33$	5.56772	0.969216
$34$	0	0
$35$	4.39608	0.743073
$36$	0	0
$37$	−5.45850	−0.897371	−0.448686	−	0.893690i	$-0.648108\pi$
$37$	−5.45850	−0.897371	−0.448686	+	0.893690i	$0.648108\pi$
$38$	0	0
$39$	−3.90165	−0.624764
$40$	0	0
$41$	4.45431	0.695647	0.347823	−	0.937560i	$-0.386921\pi$
$41$	4.45431	0.695647	0.347823	+	0.937560i	$0.386921\pi$
$42$	0	0
$43$	3.94159	0.601087	0.300543	−	0.953768i	$-0.402832\pi$
$43$	3.94159	0.601087	0.300543	+	0.953768i	$0.402832\pi$
$44$	0	0
$45$	−7.75771	−1.15645
$46$	0	0
$47$	−7.43429	−1.08440	−0.542202	−	0.840248i	$-0.682409\pi$
$47$	−7.43429	−1.08440	−0.542202	+	0.840248i	$0.682409\pi$
$48$	0	0
$49$	12.3255	1.76079
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.57656	0.903360	0.451680	−	0.892180i	$-0.350825\pi$
$53$	6.57656	0.903360	0.451680	+	0.892180i	$0.350825\pi$
$54$	0	0
$55$	−1.69753	−0.228895
$56$	0	0
$57$	−18.7270	−2.48045
$58$	0	0
$59$	4.87751	0.634998	0.317499	−	0.948259i	$-0.397157\pi$
$59$	4.87751	0.634998	0.317499	+	0.948259i	$0.397157\pi$
$60$	0	0
$61$	−3.62807	−0.464527	−0.232264	−	0.972653i	$-0.574613\pi$
$61$	−3.62807	−0.464527	−0.232264	+	0.972653i	$0.574613\pi$
$62$	0	0
$63$	−34.1035	−4.29664
$64$	0	0
$65$	1.18957	0.147547
$66$	0	0
$67$	−7.73044	−0.944424	−0.472212	−	0.881485i	$-0.656544\pi$
$67$	−7.73044	−0.944424	−0.472212	+	0.881485i	$0.656544\pi$
$68$	0	0
$69$	−7.77152	−0.935581
$70$	0	0
$71$	−12.9021	−1.53119	−0.765597	−	0.643320i	$-0.777556\pi$
$71$	−12.9021	−1.53119	−0.765597	+	0.643320i	$0.777556\pi$
$72$	0	0
$73$	−9.83287	−1.15085	−0.575425	−	0.817854i	$-0.695164\pi$
$73$	−9.83287	−1.15085	−0.575425	+	0.817854i	$0.695164\pi$
$74$	0	0
$75$	3.27990	0.378730
$76$	0	0
$77$	−7.46248	−0.850428
$78$	0	0
$79$	13.9301	1.56726	0.783630	−	0.621227i	$-0.213366\pi$
$79$	13.9301	1.56726	0.783630	+	0.621227i	$0.213366\pi$
$80$	0	0
$81$	27.9090	3.10100
$82$	0	0
$83$	−7.75068	−0.850748	−0.425374	−	0.905018i	$-0.639857\pi$
$83$	−7.75068	−0.850748	−0.425374	+	0.905018i	$0.639857\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−23.7031	−2.54124
$88$	0	0
$89$	−14.3594	−1.52209	−0.761046	−	0.648698i	$-0.775314\pi$
$89$	−14.3594	−1.52209	−0.761046	+	0.648698i	$0.775314\pi$
$90$	0	0
$91$	5.22943	0.548193
$92$	0	0
$93$	−16.5277	−1.71385
$94$	0	0
$95$	5.70964	0.585797
$96$	0	0
$97$	4.40439	0.447198	0.223599	−	0.974681i	$-0.428219\pi$
$97$	4.40439	0.447198	0.223599	+	0.974681i	$0.428219\pi$
$98$	0	0
$99$	13.1690	1.32353
$100$	0	0
$101$	−17.1085	−1.70236	−0.851178	−	0.524877i	$-0.824111\pi$
$101$	−17.1085	−1.70236	−0.851178	+	0.524877i	$0.824111\pi$
$102$	0	0
$103$	19.0507	1.87713	0.938563	−	0.345109i	$-0.112158\pi$
$103$	19.0507	1.87713	0.938563	+	0.345109i	$0.112158\pi$
$104$	0	0
$105$	14.4187	1.40712
$106$	0	0
$107$	−5.14718	−0.497597	−0.248798	−	0.968555i	$-0.580036\pi$
$107$	−5.14718	−0.497597	−0.248798	+	0.968555i	$0.580036\pi$
$108$	0	0
$109$	−12.5417	−1.20128	−0.600638	−	0.799521i	$-0.705087\pi$
$109$	−12.5417	−1.20128	−0.600638	+	0.799521i	$0.705087\pi$
$110$	0	0
$111$	−17.9033	−1.69931
$112$	0	0
$113$	−7.86888	−0.740242	−0.370121	−	0.928984i	$-0.620684\pi$
$113$	−7.86888	−0.740242	−0.370121	+	0.928984i	$0.620684\pi$
$114$	0	0
$115$	2.36944	0.220952
$116$	0	0
$117$	−9.22831	−0.853158
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.11839	−0.738036
$122$	0	0
$123$	14.6097	1.31731
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0.869333	0.0771408	0.0385704	−	0.999256i	$-0.487720\pi$
$127$	0.869333	0.0771408	0.0385704	+	0.999256i	$0.487720\pi$
$128$	0	0
$129$	12.9280	1.13825
$130$	0	0
$131$	−3.22299	−0.281594	−0.140797	−	0.990039i	$-0.544966\pi$
$131$	−3.22299	−0.281594	−0.140797	+	0.990039i	$0.544966\pi$
$132$	0	0
$133$	25.1000	2.17645
$134$	0	0
$135$	−15.6048	−1.34305
$136$	0	0
$137$	11.5971	0.990807	0.495404	−	0.868663i	$-0.335020\pi$
$137$	11.5971	0.990807	0.495404	+	0.868663i	$0.335020\pi$
$138$	0	0
$139$	11.8858	1.00814	0.504071	−	0.863662i	$-0.331835\pi$
$139$	11.8858	1.00814	0.504071	+	0.863662i	$0.331835\pi$
$140$	0	0
$141$	−24.3837	−2.05348
$142$	0	0
$143$	−2.01932	−0.168864
$144$	0	0
$145$	7.22678	0.600151
$146$	0	0
$147$	40.4264	3.33431
$148$	0	0
$149$	1.64763	0.134979	0.0674895	−	0.997720i	$-0.478501\pi$
$149$	1.64763	0.134979	0.0674895	+	0.997720i	$0.478501\pi$
$150$	0	0
$151$	4.57789	0.372543	0.186272	−	0.982498i	$-0.440360\pi$
$151$	4.57789	0.372543	0.186272	+	0.982498i	$0.440360\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.03910	0.404750
$156$	0	0
$157$	4.92656	0.393182	0.196591	−	0.980486i	$-0.437013\pi$
$157$	4.92656	0.393182	0.196591	+	0.980486i	$0.437013\pi$
$158$	0	0
$159$	21.5704	1.71065
$160$	0	0
$161$	10.4163	0.820916
$162$	0	0
$163$	4.86520	0.381072	0.190536	−	0.981680i	$-0.438977\pi$
$163$	4.86520	0.381072	0.190536	+	0.981680i	$0.438977\pi$
$164$	0	0
$165$	−5.56772	−0.433447
$166$	0	0
$167$	−5.19442	−0.401957	−0.200978	−	0.979596i	$-0.564412\pi$
$167$	−5.19442	−0.401957	−0.200978	+	0.979596i	$0.564412\pi$
$168$	0	0
$169$	−11.5849	−0.891149
$170$	0	0
$171$	−44.2937	−3.38723
$172$	0	0
$173$	3.63412	0.276297	0.138149	−	0.990411i	$-0.455885\pi$
$173$	3.63412	0.276297	0.138149	+	0.990411i	$0.455885\pi$
$174$	0	0
$175$	−4.39608	−0.332312
$176$	0	0
$177$	15.9977	1.20246
$178$	0	0
$179$	0.680054	0.0508296	0.0254148	−	0.999677i	$-0.491909\pi$
$179$	0.680054	0.0508296	0.0254148	+	0.999677i	$0.491909\pi$
$180$	0	0
$181$	−0.159322	−0.0118423	−0.00592116	−	0.999982i	$-0.501885\pi$
$181$	−0.159322	−0.0118423	−0.00592116	+	0.999982i	$0.501885\pi$
$182$	0	0
$183$	−11.8997	−0.879651
$184$	0	0
$185$	5.45850	0.401317
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−68.5999	−4.98991
$190$	0	0
$191$	2.51179	0.181747	0.0908734	−	0.995862i	$-0.471034\pi$
$191$	2.51179	0.181747	0.0908734	+	0.995862i	$0.471034\pi$
$192$	0	0
$193$	−18.3611	−1.32166	−0.660829	−	0.750536i	$-0.729795\pi$
$193$	−18.3611	−1.32166	−0.660829	+	0.750536i	$0.729795\pi$
$194$	0	0
$195$	3.90165	0.279403
$196$	0	0
$197$	3.45752	0.246338	0.123169	−	0.992386i	$-0.460694\pi$
$197$	3.45752	0.246338	0.123169	+	0.992386i	$0.460694\pi$
$198$	0	0
$199$	19.8668	1.40832	0.704161	−	0.710040i	$-0.251323\pi$
$199$	19.8668	1.40832	0.704161	+	0.710040i	$0.251323\pi$
$200$	0	0
$201$	−25.3550	−1.78841
$202$	0	0
$203$	31.7695	2.22978
$204$	0	0
$205$	−4.45431	−0.311103
$206$	0	0
$207$	−18.3814	−1.27760
$208$	0	0
$209$	−9.69229	−0.670430
$210$	0	0
$211$	−12.6008	−0.867472	−0.433736	−	0.901040i	$-0.642805\pi$
$211$	−12.6008	−0.867472	−0.433736	+	0.901040i	$0.642805\pi$
$212$	0	0
$213$	−42.3174	−2.89954
$214$	0	0
$215$	−3.94159	−0.268814
$216$	0	0
$217$	22.1523	1.50380
$218$	0	0
$219$	−32.2508	−2.17931
$220$	0	0
$221$	0	0
$222$	0	0
$223$	20.7686	1.39077	0.695385	−	0.718637i	$-0.255234\pi$
$223$	20.7686	1.39077	0.695385	+	0.718637i	$0.255234\pi$
$224$	0	0
$225$	7.75771	0.517181
$226$	0	0
$227$	28.3700	1.88298	0.941492	−	0.337034i	$-0.109424\pi$
$227$	28.3700	1.88298	0.941492	+	0.337034i	$0.109424\pi$
$228$	0	0
$229$	11.9248	0.788010	0.394005	−	0.919108i	$-0.371089\pi$
$229$	11.9248	0.788010	0.394005	+	0.919108i	$0.371089\pi$
$230$	0	0
$231$	−24.4761	−1.61041
$232$	0	0
$233$	−14.3555	−0.940458	−0.470229	−	0.882544i	$-0.655829\pi$
$233$	−14.3555	−0.940458	−0.470229	+	0.882544i	$0.655829\pi$
$234$	0	0
$235$	7.43429	0.484960
$236$	0	0
$237$	45.6893	2.96784
$238$	0	0
$239$	2.01080	0.130068	0.0650339	−	0.997883i	$-0.479284\pi$
$239$	2.01080	0.130068	0.0650339	+	0.997883i	$0.479284\pi$
$240$	0	0
$241$	−25.0093	−1.61099	−0.805496	−	0.592601i	$-0.798101\pi$
$241$	−25.0093	−1.61099	−0.805496	+	0.592601i	$0.798101\pi$
$242$	0	0
$243$	44.7241	2.86905
$244$	0	0
$245$	−12.3255	−0.787448
$246$	0	0
$247$	6.79199	0.432164
$248$	0	0
$249$	−25.4214	−1.61102
$250$	0	0
$251$	−17.7910	−1.12296	−0.561480	−	0.827490i	$-0.689768\pi$
$251$	−17.7910	−1.12296	−0.561480	+	0.827490i	$0.689768\pi$
$252$	0	0
$253$	−4.02220	−0.252873
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.7548	0.670866	0.335433	−	0.942064i	$-0.391117\pi$
$257$	10.7548	0.670866	0.335433	+	0.942064i	$0.391117\pi$
$258$	0	0
$259$	23.9960	1.49104
$260$	0	0
$261$	−56.0633	−3.47023
$262$	0	0
$263$	2.48384	0.153160	0.0765800	−	0.997063i	$-0.475600\pi$
$263$	2.48384	0.153160	0.0765800	+	0.997063i	$0.475600\pi$
$264$	0	0
$265$	−6.57656	−0.403995
$266$	0	0
$267$	−47.0973	−2.88231
$268$	0	0
$269$	−6.72808	−0.410218	−0.205109	−	0.978739i	$-0.565755\pi$
$269$	−6.72808	−0.410218	−0.205109	+	0.978739i	$0.565755\pi$
$270$	0	0
$271$	−16.9724	−1.03100	−0.515500	−	0.856890i	$-0.672394\pi$
$271$	−16.9724	−1.03100	−0.515500	+	0.856890i	$0.672394\pi$
$272$	0	0
$273$	17.1520	1.03808
$274$	0	0
$275$	1.69753	0.102365
$276$	0	0
$277$	17.2009	1.03350	0.516751	−	0.856136i	$-0.327141\pi$
$277$	17.2009	1.03350	0.516751	+	0.856136i	$0.327141\pi$
$278$	0	0
$279$	−39.0919	−2.34037
$280$	0	0
$281$	−9.98782	−0.595823	−0.297912	−	0.954593i	$-0.596290\pi$
$281$	−9.98782	−0.595823	−0.297912	+	0.954593i	$0.596290\pi$
$282$	0	0
$283$	−8.10535	−0.481813	−0.240906	−	0.970548i	$-0.577445\pi$
$283$	−8.10535	−0.481813	−0.240906	+	0.970548i	$0.577445\pi$
$284$	0	0
$285$	18.7270	1.10929
$286$	0	0
$287$	−19.5815	−1.15586
$288$	0	0
$289$	0	0
$290$	0	0
$291$	14.4459	0.846836
$292$	0	0
$293$	12.3824	0.723386	0.361693	−	0.932297i	$-0.382199\pi$
$293$	12.3824	0.723386	0.361693	+	0.932297i	$0.382199\pi$
$294$	0	0
$295$	−4.87751	−0.283980
$296$	0	0
$297$	26.4896	1.53708
$298$	0	0
$299$	2.81861	0.163004
$300$	0	0
$301$	−17.3275	−0.998743
$302$	0	0
$303$	−56.1140	−3.22366
$304$	0	0
$305$	3.62807	0.207743
$306$	0	0
$307$	−4.47770	−0.255556	−0.127778	−	0.991803i	$-0.540784\pi$
$307$	−4.47770	−0.255556	−0.127778	+	0.991803i	$0.540784\pi$
$308$	0	0
$309$	62.4844	3.55462
$310$	0	0
$311$	−3.56095	−0.201923	−0.100961	−	0.994890i	$-0.532192\pi$
$311$	−3.56095	−0.201923	−0.100961	+	0.994890i	$0.532192\pi$
$312$	0	0
$313$	−33.9802	−1.92068	−0.960338	−	0.278840i	$-0.910050\pi$
$313$	−33.9802	−1.92068	−0.960338	+	0.278840i	$0.910050\pi$
$314$	0	0
$315$	34.1035	1.92152
$316$	0	0
$317$	10.3505	0.581343	0.290671	−	0.956823i	$-0.406121\pi$
$317$	10.3505	0.581343	0.290671	+	0.956823i	$0.406121\pi$
$318$	0	0
$319$	−12.2677	−0.686858
$320$	0	0
$321$	−16.8822	−0.942274
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−1.18957	−0.0659852
$326$	0	0
$327$	−41.1354	−2.27480
$328$	0	0
$329$	32.6817	1.80180
$330$	0	0
$331$	−9.06892	−0.498473	−0.249236	−	0.968443i	$-0.580180\pi$
$331$	−9.06892	−0.498473	−0.249236	+	0.968443i	$0.580180\pi$
$332$	0	0
$333$	−42.3454	−2.32052
$334$	0	0
$335$	7.73044	0.422359
$336$	0	0
$337$	−32.4687	−1.76868	−0.884342	−	0.466840i	$-0.845392\pi$
$337$	−32.4687	−1.76868	−0.884342	+	0.466840i	$0.845392\pi$
$338$	0	0
$339$	−25.8091	−1.40176
$340$	0	0
$341$	−8.55403	−0.463227
$342$	0	0
$343$	−23.4114	−1.26409
$344$	0	0
$345$	7.77152	0.418405
$346$	0	0
$347$	1.68449	0.0904282	0.0452141	−	0.998977i	$-0.485603\pi$
$347$	1.68449	0.0904282	0.0452141	+	0.998977i	$0.485603\pi$
$348$	0	0
$349$	4.00034	0.214133	0.107067	−	0.994252i	$-0.465854\pi$
$349$	4.00034	0.214133	0.107067	+	0.994252i	$0.465854\pi$
$350$	0	0
$351$	−18.5629	−0.990816
$352$	0	0
$353$	16.7412	0.891047	0.445523	−	0.895270i	$-0.353018\pi$
$353$	16.7412	0.891047	0.445523	+	0.895270i	$0.353018\pi$
$354$	0	0
$355$	12.9021	0.684771
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0.134324	0.00708933	0.00354466	−	0.999994i	$-0.498872\pi$
$359$	0.134324	0.00708933	0.00354466	+	0.999994i	$0.498872\pi$
$360$	0	0
$361$	13.6000	0.715789
$362$	0	0
$363$	−26.6275	−1.39758
$364$	0	0
$365$	9.83287	0.514676
$366$	0	0
$367$	−0.0179343	−0.000936161 0	−0.000468080	−	1.00000i	$-0.500149\pi$
$367$	−0.0179343	−0.000936161 0	−0.000468080		1.00000i	$0.500149\pi$
$368$	0	0
$369$	34.5553	1.79888
$370$	0	0
$371$	−28.9111	−1.50099
$372$	0	0
$373$	21.7079	1.12399	0.561997	−	0.827139i	$-0.310033\pi$
$373$	21.7079	1.12399	0.561997	+	0.827139i	$0.310033\pi$
$374$	0	0
$375$	−3.27990	−0.169373
$376$	0	0
$377$	8.59673	0.442754
$378$	0	0
$379$	25.0998	1.28929	0.644644	−	0.764483i	$-0.277005\pi$
$379$	25.0998	1.28929	0.644644	+	0.764483i	$0.277005\pi$
$380$	0	0
$381$	2.85132	0.146078
$382$	0	0
$383$	2.14578	0.109644	0.0548222	−	0.998496i	$-0.482541\pi$
$383$	2.14578	0.109644	0.0548222	+	0.998496i	$0.482541\pi$
$384$	0	0
$385$	7.46248	0.380323
$386$	0	0
$387$	30.5777	1.55435
$388$	0	0
$389$	27.7877	1.40889	0.704446	−	0.709757i	$-0.251195\pi$
$389$	27.7877	1.40889	0.704446	+	0.709757i	$0.251195\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−10.5711	−0.533240
$394$	0	0
$395$	−13.9301	−0.700900
$396$	0	0
$397$	−2.43399	−0.122158	−0.0610792	−	0.998133i	$-0.519454\pi$
$397$	−2.43399	−0.122158	−0.0610792	+	0.998133i	$0.519454\pi$
$398$	0	0
$399$	82.3255	4.12143
$400$	0	0
$401$	−1.74082	−0.0869322	−0.0434661	−	0.999055i	$-0.513840\pi$
$401$	−1.74082	−0.0869322	−0.0434661	+	0.999055i	$0.513840\pi$
$402$	0	0
$403$	5.99434	0.298599
$404$	0	0
$405$	−27.9090	−1.38681
$406$	0	0
$407$	−9.26596	−0.459297
$408$	0	0
$409$	−10.1645	−0.502601	−0.251300	−	0.967909i	$-0.580858\pi$
$409$	−10.1645	−0.502601	−0.251300	+	0.967909i	$0.580858\pi$
$410$	0	0
$411$	38.0373	1.87624
$412$	0	0
$413$	−21.4419	−1.05509
$414$	0	0
$415$	7.75068	0.380466
$416$	0	0
$417$	38.9843	1.90907
$418$	0	0
$419$	−21.7518	−1.06264	−0.531322	−	0.847170i	$-0.678305\pi$
$419$	−21.7518	−1.06264	−0.531322	+	0.847170i	$0.678305\pi$
$420$	0	0
$421$	11.9684	0.583306	0.291653	−	0.956524i	$-0.405795\pi$
$421$	11.9684	0.583306	0.291653	+	0.956524i	$0.405795\pi$
$422$	0	0
$423$	−57.6731	−2.80416
$424$	0	0
$425$	0	0
$426$	0	0
$427$	15.9493	0.771841
$428$	0	0
$429$	−6.62317	−0.319770
$430$	0	0
$431$	20.4363	0.984383	0.492192	−	0.870487i	$-0.336196\pi$
$431$	20.4363	0.984383	0.492192	+	0.870487i	$0.336196\pi$
$432$	0	0
$433$	10.3280	0.496331	0.248166	−	0.968718i	$-0.420172\pi$
$433$	10.3280	0.496331	0.248166	+	0.968718i	$0.420172\pi$
$434$	0	0
$435$	23.7031	1.13648
$436$	0	0
$437$	13.5287	0.647163
$438$	0	0
$439$	−3.05427	−0.145772	−0.0728862	−	0.997340i	$-0.523221\pi$
$439$	−3.05427	−0.145772	−0.0728862	+	0.997340i	$0.523221\pi$
$440$	0	0
$441$	95.6178	4.55323
$442$	0	0
$443$	−32.5311	−1.54560	−0.772799	−	0.634651i	$-0.781144\pi$
$443$	−32.5311	−1.54560	−0.772799	+	0.634651i	$0.781144\pi$
$444$	0	0
$445$	14.3594	0.680700
$446$	0	0
$447$	5.40405	0.255603
$448$	0	0
$449$	−4.74811	−0.224077	−0.112039	−	0.993704i	$-0.535738\pi$
$449$	−4.74811	−0.224077	−0.112039	+	0.993704i	$0.535738\pi$
$450$	0	0
$451$	7.56133	0.356049
$452$	0	0
$453$	15.0150	0.705466
$454$	0	0
$455$	−5.22943	−0.245159
$456$	0	0
$457$	33.3708	1.56102	0.780510	−	0.625143i	$-0.214959\pi$
$457$	33.3708	1.56102	0.780510	+	0.625143i	$0.214959\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	18.9996	0.884899	0.442450	−	0.896793i	$-0.354110\pi$
$461$	18.9996	0.884899	0.442450	+	0.896793i	$0.354110\pi$
$462$	0	0
$463$	25.5526	1.18753	0.593766	−	0.804638i	$-0.297641\pi$
$463$	25.5526	1.18753	0.593766	+	0.804638i	$0.297641\pi$
$464$	0	0
$465$	16.5277	0.766455
$466$	0	0
$467$	23.5691	1.09065	0.545324	−	0.838225i	$-0.316406\pi$
$467$	23.5691	1.09065	0.545324	+	0.838225i	$0.316406\pi$
$468$	0	0
$469$	33.9836	1.56922
$470$	0	0
$471$	16.1586	0.744549
$472$	0	0
$473$	6.69097	0.307651
$474$	0	0
$475$	−5.70964	−0.261976
$476$	0	0
$477$	51.0190	2.33600
$478$	0	0
$479$	9.00404	0.411405	0.205703	−	0.978615i	$-0.434052\pi$
$479$	9.00404	0.411405	0.205703	+	0.978615i	$0.434052\pi$
$480$	0	0
$481$	6.49324	0.296066
$482$	0	0
$483$	34.1642	1.55453
$484$	0	0
$485$	−4.40439	−0.199993
$486$	0	0
$487$	−11.8149	−0.535383	−0.267691	−	0.963505i	$-0.586261\pi$
$487$	−11.8149	−0.535383	−0.267691	+	0.963505i	$0.586261\pi$
$488$	0	0
$489$	15.9573	0.721616
$490$	0	0
$491$	−26.6121	−1.20099	−0.600494	−	0.799629i	$-0.705029\pi$
$491$	−26.6121	−1.20099	−0.600494	+	0.799629i	$0.705029\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−13.1690	−0.591901
$496$	0	0
$497$	56.7185	2.54417
$498$	0	0
$499$	−0.300796	−0.0134655	−0.00673275	−	0.999977i	$-0.502143\pi$
$499$	−0.300796	−0.0134655	−0.00673275	+	0.999977i	$0.502143\pi$
$500$	0	0
$501$	−17.0372	−0.761164
$502$	0	0
$503$	−26.9898	−1.20342	−0.601709	−	0.798715i	$-0.705513\pi$
$503$	−26.9898	−1.20342	−0.601709	+	0.798715i	$0.705513\pi$
$504$	0	0
$505$	17.1085	0.761317
$506$	0	0
$507$	−37.9974	−1.68752
$508$	0	0
$509$	−10.1068	−0.447975	−0.223988	−	0.974592i	$-0.571907\pi$
$509$	−10.1068	−0.447975	−0.223988	+	0.974592i	$0.571907\pi$
$510$	0	0
$511$	43.2261	1.91221
$512$	0	0
$513$	−89.0978	−3.93376
$514$	0	0
$515$	−19.0507	−0.839476
$516$	0	0
$517$	−12.6199	−0.555024
$518$	0	0
$519$	11.9195	0.523210
$520$	0	0
$521$	38.2819	1.67716	0.838580	−	0.544778i	$-0.183386\pi$
$521$	38.2819	1.67716	0.838580	+	0.544778i	$0.183386\pi$
$522$	0	0
$523$	44.1735	1.93157	0.965786	−	0.259342i	$-0.0835057\pi$
$523$	44.1735	1.93157	0.965786	+	0.259342i	$0.0835057\pi$
$524$	0	0
$525$	−14.4187	−0.629283
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−17.3857	−0.755902
$530$	0	0
$531$	37.8383	1.64204
$532$	0	0
$533$	−5.29870	−0.229512
$534$	0	0
$535$	5.14718	0.222532
$536$	0	0
$537$	2.23051	0.0962534
$538$	0	0
$539$	20.9229	0.901214
$540$	0	0
$541$	−2.24595	−0.0965609	−0.0482804	−	0.998834i	$-0.515374\pi$
$541$	−2.24595	−0.0965609	−0.0482804	+	0.998834i	$0.515374\pi$
$542$	0	0
$543$	−0.522560	−0.0224252
$544$	0	0
$545$	12.5417	0.537227
$546$	0	0
$547$	6.10422	0.260998	0.130499	−	0.991448i	$-0.458342\pi$
$547$	6.10422	0.260998	0.130499	+	0.991448i	$0.458342\pi$
$548$	0	0
$549$	−28.1456	−1.20122
$550$	0	0
$551$	41.2623	1.75783
$552$	0	0
$553$	−61.2379	−2.60410
$554$	0	0
$555$	17.9033	0.759952
$556$	0	0
$557$	−19.0616	−0.807665	−0.403833	−	0.914833i	$-0.632322\pi$
$557$	−19.0616	−0.807665	−0.403833	+	0.914833i	$0.632322\pi$
$558$	0	0
$559$	−4.68878	−0.198314
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−11.6509	−0.491028	−0.245514	−	0.969393i	$-0.578957\pi$
$563$	−11.6509	−0.491028	−0.245514	+	0.969393i	$0.578957\pi$
$564$	0	0
$565$	7.86888	0.331046
$566$	0	0
$567$	−122.690	−5.15250
$568$	0	0
$569$	−22.9973	−0.964098	−0.482049	−	0.876144i	$-0.660107\pi$
$569$	−22.9973	−0.964098	−0.482049	+	0.876144i	$0.660107\pi$
$570$	0	0
$571$	−6.44384	−0.269666	−0.134833	−	0.990868i	$-0.543050\pi$
$571$	−6.44384	−0.269666	−0.134833	+	0.990868i	$0.543050\pi$
$572$	0	0
$573$	8.23841	0.344164
$574$	0	0
$575$	−2.36944	−0.0988125
$576$	0	0
$577$	−12.7327	−0.530070	−0.265035	−	0.964239i	$-0.585383\pi$
$577$	−12.7327	−0.530070	−0.265035	+	0.964239i	$0.585383\pi$
$578$	0	0
$579$	−60.2224	−2.50276
$580$	0	0
$581$	34.0726	1.41357
$582$	0	0
$583$	11.1639	0.462362
$584$	0	0
$585$	9.22831	0.381544
$586$	0	0
$587$	−11.8614	−0.489571	−0.244785	−	0.969577i	$-0.578718\pi$
$587$	−11.8614	−0.489571	−0.244785	+	0.969577i	$0.578718\pi$
$588$	0	0
$589$	28.7715	1.18551
$590$	0	0
$591$	11.3403	0.466477
$592$	0	0
$593$	39.7191	1.63107	0.815534	−	0.578709i	$-0.196443\pi$
$593$	39.7191	1.63107	0.815534	+	0.578709i	$0.196443\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	65.1611	2.66687
$598$	0	0
$599$	29.6601	1.21188	0.605939	−	0.795511i	$-0.292798\pi$
$599$	29.6601	1.21188	0.605939	+	0.795511i	$0.292798\pi$
$600$	0	0
$601$	1.36790	0.0557978	0.0278989	−	0.999611i	$-0.491118\pi$
$601$	1.36790	0.0557978	0.0278989	+	0.999611i	$0.491118\pi$
$602$	0	0
$603$	−59.9706	−2.44219
$604$	0	0
$605$	8.11839	0.330060
$606$	0	0
$607$	−35.8146	−1.45367	−0.726834	−	0.686813i	$-0.759009\pi$
$607$	−35.8146	−1.45367	−0.726834	+	0.686813i	$0.759009\pi$
$608$	0	0
$609$	104.201	4.22242
$610$	0	0
$611$	8.84358	0.357773
$612$	0	0
$613$	−26.3513	−1.06432	−0.532159	−	0.846645i	$-0.678619\pi$
$613$	−26.3513	−1.06432	−0.532159	+	0.846645i	$0.678619\pi$
$614$	0	0
$615$	−14.6097	−0.589119
$616$	0	0
$617$	21.8178	0.878350	0.439175	−	0.898401i	$-0.355271\pi$
$617$	21.8178	0.878350	0.439175	+	0.898401i	$0.355271\pi$
$618$	0	0
$619$	−20.5969	−0.827858	−0.413929	−	0.910309i	$-0.635844\pi$
$619$	−20.5969	−0.827858	−0.413929	+	0.910309i	$0.635844\pi$
$620$	0	0
$621$	−36.9747	−1.48374
$622$	0	0
$623$	63.1250	2.52905
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−31.7897	−1.26956
$628$	0	0
$629$	0	0
$630$	0	0
$631$	47.8942	1.90664	0.953320	−	0.301963i	$-0.0976421\pi$
$631$	47.8942	1.90664	0.953320	+	0.301963i	$0.0976421\pi$
$632$	0	0
$633$	−41.3292	−1.64269
$634$	0	0
$635$	−0.869333	−0.0344984
$636$	0	0
$637$	−14.6620	−0.580930
$638$	0	0
$639$	−100.091	−3.95952
$640$	0	0
$641$	−17.9495	−0.708962	−0.354481	−	0.935063i	$-0.615342\pi$
$641$	−17.9495	−0.708962	−0.354481	+	0.935063i	$0.615342\pi$
$642$	0	0
$643$	14.7423	0.581379	0.290690	−	0.956817i	$-0.406115\pi$
$643$	14.7423	0.581379	0.290690	+	0.956817i	$0.406115\pi$
$644$	0	0
$645$	−12.9280	−0.509040
$646$	0	0
$647$	−12.1630	−0.478178	−0.239089	−	0.970998i	$-0.576849\pi$
$647$	−12.1630	−0.478178	−0.239089	+	0.970998i	$0.576849\pi$
$648$	0	0
$649$	8.27972	0.325007
$650$	0	0
$651$	72.6572	2.84766
$652$	0	0
$653$	23.1818	0.907173	0.453586	−	0.891212i	$-0.350144\pi$
$653$	23.1818	0.907173	0.453586	+	0.891212i	$0.350144\pi$
$654$	0	0
$655$	3.22299	0.125933
$656$	0	0
$657$	−76.2806	−2.97599
$658$	0	0
$659$	25.0197	0.974627	0.487314	−	0.873227i	$-0.337977\pi$
$659$	25.0197	0.974627	0.487314	+	0.873227i	$0.337977\pi$
$660$	0	0
$661$	−15.6369	−0.608205	−0.304103	−	0.952639i	$-0.598357\pi$
$661$	−15.6369	−0.608205	−0.304103	+	0.952639i	$0.598357\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−25.1000	−0.973337
$666$	0	0
$667$	17.1234	0.663022
$668$	0	0
$669$	68.1189	2.63363
$670$	0	0
$671$	−6.15877	−0.237757
$672$	0	0
$673$	−3.59920	−0.138739	−0.0693695	−	0.997591i	$-0.522099\pi$
$673$	−3.59920	−0.138739	−0.0693695	+	0.997591i	$0.522099\pi$
$674$	0	0
$675$	15.6048	0.600629
$676$	0	0
$677$	−9.56143	−0.367476	−0.183738	−	0.982975i	$-0.558820\pi$
$677$	−9.56143	−0.367476	−0.183738	+	0.982975i	$0.558820\pi$
$678$	0	0
$679$	−19.3620	−0.743047
$680$	0	0
$681$	93.0507	3.56571
$682$	0	0
$683$	16.0729	0.615014	0.307507	−	0.951546i	$-0.400505\pi$
$683$	16.0729	0.615014	0.307507	+	0.951546i	$0.400505\pi$
$684$	0	0
$685$	−11.5971	−0.443103
$686$	0	0
$687$	39.1119	1.49221
$688$	0	0
$689$	−7.82325	−0.298042
$690$	0	0
$691$	−4.43095	−0.168561	−0.0842806	−	0.996442i	$-0.526859\pi$
$691$	−4.43095	−0.168561	−0.0842806	+	0.996442i	$0.526859\pi$
$692$	0	0
$693$	−57.8918	−2.19913
$694$	0	0
$695$	−11.8858	−0.450855
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−47.0844	−1.78090
$700$	0	0
$701$	−4.78808	−0.180843	−0.0904216	−	0.995904i	$-0.528821\pi$
$701$	−4.78808	−0.180843	−0.0904216	+	0.995904i	$0.528821\pi$
$702$	0	0
$703$	31.1660	1.17545
$704$	0	0
$705$	24.3837	0.918343
$706$	0	0
$707$	75.2102	2.82857
$708$	0	0
$709$	−30.5105	−1.14585	−0.572924	−	0.819609i	$-0.694191\pi$
$709$	−30.5105	−1.14585	−0.572924	+	0.819609i	$0.694191\pi$
$710$	0	0
$711$	108.066	4.05279
$712$	0	0
$713$	11.9399	0.447151
$714$	0	0
$715$	2.01932	0.0755184
$716$	0	0
$717$	6.59522	0.246303
$718$	0	0
$719$	−45.8443	−1.70971	−0.854853	−	0.518871i	$-0.826353\pi$
$719$	−45.8443	−1.70971	−0.854853	+	0.518871i	$0.826353\pi$
$720$	0	0
$721$	−83.7486	−3.11896
$722$	0	0
$723$	−82.0280	−3.05065
$724$	0	0
$725$	−7.22678	−0.268396
$726$	0	0
$727$	14.7034	0.545318	0.272659	−	0.962111i	$-0.412097\pi$
$727$	14.7034	0.545318	0.272659	+	0.962111i	$0.412097\pi$
$728$	0	0
$729$	62.9635	2.33198
$730$	0	0
$731$	0	0
$732$	0	0
$733$	22.4183	0.828040	0.414020	−	0.910268i	$-0.364124\pi$
$733$	22.4183	0.828040	0.414020	+	0.910268i	$0.364124\pi$
$734$	0	0
$735$	−40.4264	−1.49115
$736$	0	0
$737$	−13.1227	−0.483379
$738$	0	0
$739$	8.02516	0.295210	0.147605	−	0.989046i	$-0.452844\pi$
$739$	8.02516	0.295210	0.147605	+	0.989046i	$0.452844\pi$
$740$	0	0
$741$	22.2770	0.818367
$742$	0	0
$743$	14.2933	0.524371	0.262186	−	0.965017i	$-0.415557\pi$
$743$	14.2933	0.524371	0.262186	+	0.965017i	$0.415557\pi$
$744$	0	0
$745$	−1.64763	−0.0603644
$746$	0	0
$747$	−60.1276	−2.19995
$748$	0	0
$749$	22.6274	0.826788
$750$	0	0
$751$	−44.3952	−1.62000	−0.810002	−	0.586427i	$-0.800534\pi$
$751$	−44.3952	−1.62000	−0.810002	+	0.586427i	$0.800534\pi$
$752$	0	0
$753$	−58.3527	−2.12649
$754$	0	0
$755$	−4.57789	−0.166606
$756$	0	0
$757$	−4.26298	−0.154940	−0.0774702	−	0.996995i	$-0.524684\pi$
$757$	−4.26298	−0.154940	−0.0774702	+	0.996995i	$0.524684\pi$
$758$	0	0
$759$	−13.1924	−0.478853
$760$	0	0
$761$	−38.7229	−1.40370	−0.701852	−	0.712323i	$-0.747643\pi$
$761$	−38.7229	−1.40370	−0.701852	+	0.712323i	$0.747643\pi$
$762$	0	0
$763$	55.1343	1.99600
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.80212	−0.209502
$768$	0	0
$769$	35.2510	1.27118	0.635592	−	0.772025i	$-0.280756\pi$
$769$	35.2510	1.27118	0.635592	+	0.772025i	$0.280756\pi$
$770$	0	0
$771$	35.2746	1.27038
$772$	0	0
$773$	39.5663	1.42310	0.711550	−	0.702635i	$-0.247993\pi$
$773$	39.5663	1.42310	0.711550	+	0.702635i	$0.247993\pi$
$774$	0	0
$775$	−5.03910	−0.181010
$776$	0	0
$777$	78.7043	2.82350
$778$	0	0
$779$	−25.4325	−0.911215
$780$	0	0
$781$	−21.9017	−0.783703
$782$	0	0
$783$	−112.772	−4.03016
$784$	0	0
$785$	−4.92656	−0.175836
$786$	0	0
$787$	−28.3880	−1.01192	−0.505961	−	0.862557i	$-0.668862\pi$
$787$	−28.3880	−1.01192	−0.505961	+	0.862557i	$0.668862\pi$
$788$	0	0
$789$	8.14673	0.290031
$790$	0	0
$791$	34.5922	1.22996
$792$	0	0
$793$	4.31583	0.153260
$794$	0	0
$795$	−21.5704	−0.765024
$796$	0	0
$797$	−40.3294	−1.42854	−0.714270	−	0.699870i	$-0.753241\pi$
$797$	−40.3294	−1.42854	−0.714270	+	0.699870i	$0.753241\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−111.396	−3.93598
$802$	0	0
$803$	−16.6916	−0.589034
$804$	0	0
$805$	−10.4163	−0.367125
$806$	0	0
$807$	−22.0674	−0.776809
$808$	0	0
$809$	3.11841	0.109637	0.0548187	−	0.998496i	$-0.482542\pi$
$809$	3.11841	0.109637	0.0548187	+	0.998496i	$0.482542\pi$
$810$	0	0
$811$	−42.1772	−1.48104	−0.740521	−	0.672033i	$-0.765421\pi$
$811$	−42.1772	−1.48104	−0.740521	+	0.672033i	$0.765421\pi$
$812$	0	0
$813$	−55.6677	−1.95235
$814$	0	0
$815$	−4.86520	−0.170421
$816$	0	0
$817$	−22.5051	−0.787352
$818$	0	0
$819$	40.5684	1.41757
$820$	0	0
$821$	−5.67311	−0.197993	−0.0989965	−	0.995088i	$-0.531563\pi$
$821$	−5.67311	−0.197993	−0.0989965	+	0.995088i	$0.531563\pi$
$822$	0	0
$823$	25.3303	0.882960	0.441480	−	0.897271i	$-0.354454\pi$
$823$	25.3303	0.882960	0.441480	+	0.897271i	$0.354454\pi$
$824$	0	0
$825$	5.56772	0.193843
$826$	0	0
$827$	−31.2464	−1.08654	−0.543272	−	0.839556i	$-0.682815\pi$
$827$	−31.2464	−1.08654	−0.543272	+	0.839556i	$0.682815\pi$
$828$	0	0
$829$	−25.5666	−0.887966	−0.443983	−	0.896035i	$-0.646435\pi$
$829$	−25.5666	−0.887966	−0.443983	+	0.896035i	$0.646435\pi$
$830$	0	0
$831$	56.4172	1.95709
$832$	0	0
$833$	0	0
$834$	0	0
$835$	5.19442	0.179760
$836$	0	0
$837$	−78.6342	−2.71800
$838$	0	0
$839$	40.5640	1.40043	0.700213	−	0.713934i	$-0.253088\pi$
$839$	40.5640	1.40043	0.700213	+	0.713934i	$0.253088\pi$
$840$	0	0
$841$	23.2263	0.800909
$842$	0	0
$843$	−32.7590	−1.12828
$844$	0	0
$845$	11.5849	0.398534
$846$	0	0
$847$	35.6891	1.22629
$848$	0	0
$849$	−26.5847	−0.912384
$850$	0	0
$851$	12.9336	0.443358
$852$	0	0
$853$	−17.1223	−0.586257	−0.293129	−	0.956073i	$-0.594696\pi$
$853$	−17.1223	−0.586257	−0.293129	+	0.956073i	$0.594696\pi$
$854$	0	0
$855$	44.2937	1.51481
$856$	0	0
$857$	2.27033	0.0775529	0.0387764	−	0.999248i	$-0.487654\pi$
$857$	2.27033	0.0775529	0.0387764	+	0.999248i	$0.487654\pi$
$858$	0	0
$859$	47.5024	1.62076	0.810381	−	0.585903i	$-0.199260\pi$
$859$	47.5024	1.62076	0.810381	+	0.585903i	$0.199260\pi$
$860$	0	0
$861$	−64.2253	−2.18879
$862$	0	0
$863$	−6.25896	−0.213058	−0.106529	−	0.994310i	$-0.533974\pi$
$863$	−6.25896	−0.213058	−0.106529	+	0.994310i	$0.533974\pi$
$864$	0	0
$865$	−3.63412	−0.123564
$866$	0	0
$867$	0	0
$868$	0	0
$869$	23.6468	0.802163
$870$	0	0
$871$	9.19587	0.311590
$872$	0	0
$873$	34.1680	1.15641
$874$	0	0
$875$	4.39608	0.148615
$876$	0	0
$877$	4.46831	0.150884	0.0754420	−	0.997150i	$-0.475963\pi$
$877$	4.46831	0.150884	0.0754420	+	0.997150i	$0.475963\pi$
$878$	0	0
$879$	40.6129	1.36984
$880$	0	0
$881$	43.5469	1.46713	0.733567	−	0.679617i	$-0.237854\pi$
$881$	43.5469	1.46713	0.733567	+	0.679617i	$0.237854\pi$
$882$	0	0
$883$	11.7503	0.395428	0.197714	−	0.980260i	$-0.436648\pi$
$883$	11.7503	0.395428	0.197714	+	0.980260i	$0.436648\pi$
$884$	0	0
$885$	−15.9977	−0.537757
$886$	0	0
$887$	26.3608	0.885108	0.442554	−	0.896742i	$-0.354072\pi$
$887$	26.3608	0.885108	0.442554	+	0.896742i	$0.354072\pi$
$888$	0	0
$889$	−3.82165	−0.128174
$890$	0	0
$891$	47.3763	1.58717
$892$	0	0
$893$	42.4471	1.42044
$894$	0	0
$895$	−0.680054	−0.0227317
$896$	0	0
$897$	9.24473	0.308673
$898$	0	0
$899$	36.4165	1.21456
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−56.8325	−1.89127
$904$	0	0
$905$	0.159322	0.00529604
$906$	0	0
$907$	24.5662	0.815709	0.407854	−	0.913047i	$-0.366277\pi$
$907$	24.5662	0.815709	0.407854	+	0.913047i	$0.366277\pi$
$908$	0	0
$909$	−132.723	−4.40213
$910$	0	0
$911$	4.74597	0.157241	0.0786205	−	0.996905i	$-0.474948\pi$
$911$	4.74597	0.157241	0.0786205	+	0.996905i	$0.474948\pi$
$912$	0	0
$913$	−13.1570	−0.435434
$914$	0	0
$915$	11.8997	0.393392
$916$	0	0
$917$	14.1685	0.467885
$918$	0	0
$919$	−9.47877	−0.312676	−0.156338	−	0.987704i	$-0.549969\pi$
$919$	−9.47877	−0.312676	−0.156338	+	0.987704i	$0.549969\pi$
$920$	0	0
$921$	−14.6864	−0.483932
$922$	0	0
$923$	15.3479	0.505181
$924$	0	0
$925$	−5.45850	−0.179474
$926$	0	0
$927$	147.790	4.85407
$928$	0	0
$929$	−15.2517	−0.500392	−0.250196	−	0.968195i	$-0.580495\pi$
$929$	−15.2517	−0.500392	−0.250196	+	0.968195i	$0.580495\pi$
$930$	0	0
$931$	−70.3742	−2.30642
$932$	0	0
$933$	−11.6795	−0.382371
$934$	0	0
$935$	0	0
$936$	0	0
$937$	51.9126	1.69591	0.847956	−	0.530067i	$-0.177833\pi$
$937$	51.9126	1.69591	0.847956	+	0.530067i	$0.177833\pi$
$938$	0	0
$939$	−111.452	−3.63708
$940$	0	0
$941$	0.643486	0.0209770	0.0104885	−	0.999945i	$-0.496661\pi$
$941$	0.643486	0.0209770	0.0104885	+	0.999945i	$0.496661\pi$
$942$	0	0
$943$	−10.5542	−0.343693
$944$	0	0
$945$	68.5999	2.23156
$946$	0	0
$947$	−38.8018	−1.26089	−0.630444	−	0.776234i	$-0.717127\pi$
$947$	−38.8018	−1.26089	−0.630444	+	0.776234i	$0.717127\pi$
$948$	0	0
$949$	11.6969	0.379696
$950$	0	0
$951$	33.9486	1.10086
$952$	0	0
$953$	52.2335	1.69201	0.846005	−	0.533175i	$-0.179001\pi$
$953$	52.2335	1.69201	0.846005	+	0.533175i	$0.179001\pi$
$954$	0	0
$955$	−2.51179	−0.0812796
$956$	0	0
$957$	−40.2367	−1.30067
$958$	0	0
$959$	−50.9818	−1.64629
$960$	0	0
$961$	−5.60746	−0.180886
$962$	0	0
$963$	−39.9304	−1.28674
$964$	0	0
$965$	18.3611	0.591064
$966$	0	0
$967$	−16.0124	−0.514924	−0.257462	−	0.966288i	$-0.582886\pi$
$967$	−16.0124	−0.514924	−0.257462	+	0.966288i	$0.582886\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	46.5708	1.49453	0.747265	−	0.664526i	$-0.231367\pi$
$971$	46.5708	1.49453	0.747265	+	0.664526i	$0.231367\pi$
$972$	0	0
$973$	−52.2510	−1.67509
$974$	0	0
$975$	−3.90165	−0.124953
$976$	0	0
$977$	−46.9970	−1.50357	−0.751784	−	0.659410i	$-0.770806\pi$
$977$	−46.9970	−1.50357	−0.751784	+	0.659410i	$0.770806\pi$
$978$	0	0
$979$	−24.3755	−0.779044
$980$	0	0
$981$	−97.2949	−3.10639
$982$	0	0
$983$	51.1844	1.63253	0.816264	−	0.577680i	$-0.196042\pi$
$983$	51.1844	1.63253	0.816264	+	0.577680i	$0.196042\pi$
$984$	0	0
$985$	−3.45752	−0.110166
$986$	0	0
$987$	107.193	3.41198
$988$	0	0
$989$	−9.33937	−0.296975
$990$	0	0
$991$	−5.26053	−0.167106	−0.0835531	−	0.996503i	$-0.526627\pi$
$991$	−5.26053	−0.167106	−0.0835531	+	0.996503i	$0.526627\pi$
$992$	0	0
$993$	−29.7451	−0.943932
$994$	0	0
$995$	−19.8668	−0.629821
$996$	0	0
$997$	40.7193	1.28959	0.644796	−	0.764355i	$-0.276942\pi$
$997$	40.7193	1.28959	0.644796	+	0.764355i	$0.276942\pi$
$998$	0	0
$999$	−85.1787	−2.69494

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5780.2.a.q.1.12		12
17.4	even	4		5780.2.c.j.5201.1		24
17.10	odd	16		340.2.u.a.321.6	yes	24
17.12	odd	16		340.2.u.a.161.6	✓	24
17.13	even	4		5780.2.c.j.5201.24		24
17.16	even	2		5780.2.a.r.1.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
340.2.u.a.161.6	✓	24	17.12	odd	16
340.2.u.a.321.6	yes	24	17.10	odd	16
5780.2.a.q.1.12		12	1.1	even	1	trivial
5780.2.a.r.1.1		12	17.16	even	2
5780.2.c.j.5201.1		24	17.4	even	4
5780.2.c.j.5201.24		24	17.13	even	4

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:	\( N \)	\(=\)	\( 5780 = 2^{2} \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5780.a (trivial)

\(f(q)\)	\(=\)	\(q+3.27990 q^{3} -1.00000 q^{5} -4.39608 q^{7} +7.75771 q^{9} +O(q^{10})\)	\(q+3.27990 q^{3} -1.00000 q^{5} -4.39608 q^{7} +7.75771 q^{9} +1.69753 q^{11} -1.18957 q^{13} -3.27990 q^{15} -5.70964 q^{19} -14.4187 q^{21} -2.36944 q^{23} +1.00000 q^{25} +15.6048 q^{27} -7.22678 q^{29} -5.03910 q^{31} +5.56772 q^{33} +4.39608 q^{35} -5.45850 q^{37} -3.90165 q^{39} +4.45431 q^{41} +3.94159 q^{43} -7.75771 q^{45} -7.43429 q^{47} +12.3255 q^{49} +6.57656 q^{53} -1.69753 q^{55} -18.7270 q^{57} +4.87751 q^{59} -3.62807 q^{61} -34.1035 q^{63} +1.18957 q^{65} -7.73044 q^{67} -7.77152 q^{69} -12.9021 q^{71} -9.83287 q^{73} +3.27990 q^{75} -7.46248 q^{77} +13.9301 q^{79} +27.9090 q^{81} -7.75068 q^{83} -23.7031 q^{87} -14.3594 q^{89} +5.22943 q^{91} -16.5277 q^{93} +5.70964 q^{95} +4.40439 q^{97} +13.1690 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{5} - 8 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^5 - 8 * q^7 + 12 * q^9	\( 12 q - 12 q^{5} - 8 q^{7} + 12 q^{9} + 8 q^{13} - 16 q^{21} - 8 q^{23} + 12 q^{25} - 16 q^{29} - 24 q^{31} + 8 q^{35} - 24 q^{37} + 8 q^{39} - 24 q^{41} + 8 q^{43} - 12 q^{45} + 8 q^{47} + 20 q^{49} + 16 q^{53} - 32 q^{57} + 8 q^{59} - 40 q^{61} - 24 q^{63} - 8 q^{65} + 16 q^{67} - 16 q^{69} - 16 q^{71} - 32 q^{73} + 24 q^{77} - 8 q^{79} + 4 q^{81} + 32 q^{83} + 16 q^{87} - 8 q^{89} - 8 q^{91} - 8 q^{93} - 32 q^{97} - 24 q^{99}+O(q^{100}) \) 12 * q - 12 * q^5 - 8 * q^7 + 12 * q^9 + 8 * q^13 - 16 * q^21 - 8 * q^23 + 12 * q^25 - 16 * q^29 - 24 * q^31 + 8 * q^35 - 24 * q^37 + 8 * q^39 - 24 * q^41 + 8 * q^43 - 12 * q^45 + 8 * q^47 + 20 * q^49 + 16 * q^53 - 32 * q^57 + 8 * q^59 - 40 * q^61 - 24 * q^63 - 8 * q^65 + 16 * q^67 - 16 * q^69 - 16 * q^71 - 32 * q^73 + 24 * q^77 - 8 * q^79 + 4 * q^81 + 32 * q^83 + 16 * q^87 - 8 * q^89 - 8 * q^91 - 8 * q^93 - 32 * q^97 - 24 * q^99

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