LMFDB - Embedded newform 8649.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8649,2,Mod(1,8649)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8649, 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("8649.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8649 = 3^{2} \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8649.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:	$69.0626127082$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 31)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	8649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214 q^{2} -1.82843 q^{4} -1.00000 q^{5} -0.414214 q^{7} +1.58579 q^{8} +O(q^{10})$	\(q-0.414214 q^{2} -1.82843 q^{4} -1.00000 q^{5} -0.414214 q^{7} +1.58579 q^{8} +0.414214 q^{10} -3.24264 q^{11} -3.82843 q^{13} +0.171573 q^{14} +3.00000 q^{16} +5.82843 q^{17} +4.41421 q^{19} +1.82843 q^{20} +1.34315 q^{22} +4.00000 q^{23} -4.00000 q^{25} +1.58579 q^{26} +0.757359 q^{28} +6.82843 q^{29} -4.41421 q^{32} -2.41421 q^{34} +0.414214 q^{35} +1.00000 q^{37} -1.82843 q^{38} -1.58579 q^{40} +7.48528 q^{41} -10.8995 q^{43} +5.92893 q^{44} -1.65685 q^{46} -9.65685 q^{47} -6.82843 q^{49} +1.65685 q^{50} +7.00000 q^{52} -5.82843 q^{53} +3.24264 q^{55} -0.656854 q^{56} -2.82843 q^{58} -4.07107 q^{59} -2.82843 q^{61} -4.17157 q^{64} +3.82843 q^{65} +3.24264 q^{67} -10.6569 q^{68} -0.171573 q^{70} -0.0710678 q^{71} -1.82843 q^{73} -0.414214 q^{74} -8.07107 q^{76} +1.34315 q^{77} -6.75736 q^{79} -3.00000 q^{80} -3.10051 q^{82} +10.0711 q^{83} -5.82843 q^{85} +4.51472 q^{86} -5.14214 q^{88} -4.48528 q^{89} +1.58579 q^{91} -7.31371 q^{92} +4.00000 q^{94} -4.41421 q^{95} +5.17157 q^{97} +2.82843 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 6 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 6 q^{8} - 2 q^{10} + 2 q^{11} - 2 q^{13} + 6 q^{14} + 6 q^{16} + 6 q^{17} + 6 q^{19} - 2 q^{20} + 14 q^{22} + 8 q^{23} - 8 q^{25} + 6 q^{26} + 10 q^{28} + 8 q^{29} - 6 q^{32} - 2 q^{34} - 2 q^{35} + 2 q^{37} + 2 q^{38} - 6 q^{40} - 2 q^{41} - 2 q^{43} + 26 q^{44} + 8 q^{46} - 8 q^{47} - 8 q^{49} - 8 q^{50} + 14 q^{52} - 6 q^{53} - 2 q^{55} + 10 q^{56} + 6 q^{59} - 14 q^{64} + 2 q^{65} - 2 q^{67} - 10 q^{68} - 6 q^{70} + 14 q^{71} + 2 q^{73} + 2 q^{74} - 2 q^{76} + 14 q^{77} - 22 q^{79} - 6 q^{80} - 26 q^{82} + 6 q^{83} - 6 q^{85} + 26 q^{86} + 18 q^{88} + 8 q^{89} + 6 q^{91} + 8 q^{92} + 8 q^{94} - 6 q^{95} + 16 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 6 * q^8 - 2 * q^10 + 2 * q^11 - 2 * q^13 + 6 * q^14 + 6 * q^16 + 6 * q^17 + 6 * q^19 - 2 * q^20 + 14 * q^22 + 8 * q^23 - 8 * q^25 + 6 * q^26 + 10 * q^28 + 8 * q^29 - 6 * q^32 - 2 * q^34 - 2 * q^35 + 2 * q^37 + 2 * q^38 - 6 * q^40 - 2 * q^41 - 2 * q^43 + 26 * q^44 + 8 * q^46 - 8 * q^47 - 8 * q^49 - 8 * q^50 + 14 * q^52 - 6 * q^53 - 2 * q^55 + 10 * q^56 + 6 * q^59 - 14 * q^64 + 2 * q^65 - 2 * q^67 - 10 * q^68 - 6 * q^70 + 14 * q^71 + 2 * q^73 + 2 * q^74 - 2 * q^76 + 14 * q^77 - 22 * q^79 - 6 * q^80 - 26 * q^82 + 6 * q^83 - 6 * q^85 + 26 * q^86 + 18 * q^88 + 8 * q^89 + 6 * q^91 + 8 * q^92 + 8 * q^94 - 6 * q^95 + 16 * 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 $

$2$	−0.414214	−0.292893	−0.146447	−	0.989219i	$-0.546784\pi$
$2$	−0.414214	−0.292893	−0.146447	+	0.989219i	$0.546784\pi$
$3$	0	0
$3$	0	0
$4$	−1.82843	−0.914214
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−0.414214	−0.156558	−0.0782790	−	0.996931i	$-0.524942\pi$
$7$	−0.414214	−0.156558	−0.0782790	+	0.996931i	$0.524942\pi$
$8$	1.58579	0.560660
$9$	0	0
$10$	0.414214	0.130986
$11$	−3.24264	−0.977693	−0.488846	−	0.872370i	$-0.662582\pi$
$11$	−3.24264	−0.977693	−0.488846	+	0.872370i	$0.662582\pi$
$12$	0	0
$13$	−3.82843	−1.06181	−0.530907	−	0.847430i	$-0.678149\pi$
$13$	−3.82843	−1.06181	−0.530907	+	0.847430i	$0.678149\pi$
$14$	0.171573	0.0458548
$15$	0	0
$16$	3.00000	0.750000
$17$	5.82843	1.41360	0.706801	−	0.707413i	$-0.250138\pi$
$17$	5.82843	1.41360	0.706801	+	0.707413i	$0.250138\pi$
$18$	0	0
$19$	4.41421	1.01269	0.506345	−	0.862331i	$-0.330996\pi$
$19$	4.41421	1.01269	0.506345	+	0.862331i	$0.330996\pi$
$20$	1.82843	0.408849
$21$	0	0
$22$	1.34315	0.286360
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	1.58579	0.310998
$27$	0	0
$28$	0.757359	0.143127
$29$	6.82843	1.26801	0.634004	−	0.773330i	$-0.281410\pi$
$29$	6.82843	1.26801	0.634004	+	0.773330i	$0.281410\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	−4.41421	−0.780330
$33$	0	0
$34$	−2.41421	−0.414034
$35$	0.414214	0.0700149
$36$	0	0
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	−1.82843	−0.296610
$39$	0	0
$40$	−1.58579	−0.250735
$41$	7.48528	1.16900	0.584502	−	0.811392i	$-0.301290\pi$
$41$	7.48528	1.16900	0.584502	+	0.811392i	$0.301290\pi$
$42$	0	0
$43$	−10.8995	−1.66216	−0.831079	−	0.556155i	$-0.812276\pi$
$43$	−10.8995	−1.66216	−0.831079	+	0.556155i	$0.812276\pi$
$44$	5.92893	0.893820
$45$	0	0
$46$	−1.65685	−0.244290
$47$	−9.65685	−1.40860	−0.704298	−	0.709904i	$-0.748738\pi$
$47$	−9.65685	−1.40860	−0.704298	+	0.709904i	$0.748738\pi$
$48$	0	0
$49$	−6.82843	−0.975490
$50$	1.65685	0.234315
$51$	0	0
$52$	7.00000	0.970725
$53$	−5.82843	−0.800596	−0.400298	−	0.916385i	$-0.631093\pi$
$53$	−5.82843	−0.800596	−0.400298	+	0.916385i	$0.631093\pi$
$54$	0	0
$55$	3.24264	0.437238
$56$	−0.656854	−0.0877758
$57$	0	0
$58$	−2.82843	−0.371391
$59$	−4.07107	−0.530008	−0.265004	−	0.964247i	$-0.585373\pi$
$59$	−4.07107	−0.530008	−0.265004	+	0.964247i	$0.585373\pi$
$60$	0	0
$61$	−2.82843	−0.362143	−0.181071	−	0.983470i	$-0.557957\pi$
$61$	−2.82843	−0.362143	−0.181071	+	0.983470i	$0.557957\pi$
$62$	0	0
$63$	0	0
$64$	−4.17157	−0.521447
$65$	3.82843	0.474858
$66$	0	0
$67$	3.24264	0.396152	0.198076	−	0.980187i	$-0.436531\pi$
$67$	3.24264	0.396152	0.198076	+	0.980187i	$0.436531\pi$
$68$	−10.6569	−1.29233
$69$	0	0
$70$	−0.171573	−0.0205069
$71$	−0.0710678	−0.00843420	−0.00421710	−	0.999991i	$-0.501342\pi$
$71$	−0.0710678	−0.00843420	−0.00421710	+	0.999991i	$0.501342\pi$
$72$	0	0
$73$	−1.82843	−0.214001	−0.107001	−	0.994259i	$-0.534125\pi$
$73$	−1.82843	−0.214001	−0.107001	+	0.994259i	$0.534125\pi$
$74$	−0.414214	−0.0481513
$75$	0	0
$76$	−8.07107	−0.925815
$77$	1.34315	0.153066
$78$	0	0
$79$	−6.75736	−0.760262	−0.380131	−	0.924933i	$-0.624121\pi$
$79$	−6.75736	−0.760262	−0.380131	+	0.924933i	$0.624121\pi$
$80$	−3.00000	−0.335410
$81$	0	0
$82$	−3.10051	−0.342394
$83$	10.0711	1.10544	0.552722	−	0.833366i	$-0.313589\pi$
$83$	10.0711	1.10544	0.552722	+	0.833366i	$0.313589\pi$
$84$	0	0
$85$	−5.82843	−0.632182
$86$	4.51472	0.486835
$87$	0	0
$88$	−5.14214	−0.548153
$89$	−4.48528	−0.475439	−0.237719	−	0.971334i	$-0.576400\pi$
$89$	−4.48528	−0.475439	−0.237719	+	0.971334i	$0.576400\pi$
$90$	0	0
$91$	1.58579	0.166236
$92$	−7.31371	−0.762507
$93$	0	0
$94$	4.00000	0.412568
$95$	−4.41421	−0.452889
$96$	0	0
$97$	5.17157	0.525094	0.262547	−	0.964919i	$-0.415438\pi$
$97$	5.17157	0.525094	0.262547	+	0.964919i	$0.415438\pi$
$98$	2.82843	0.285714
$99$	0	0
$100$	7.31371	0.731371
$101$	8.48528	0.844317	0.422159	−	0.906522i	$-0.361273\pi$
$101$	8.48528	0.844317	0.422159	+	0.906522i	$0.361273\pi$
$102$	0	0
$103$	2.07107	0.204068	0.102034	−	0.994781i	$-0.467465\pi$
$103$	2.07107	0.204068	0.102034	+	0.994781i	$0.467465\pi$
$104$	−6.07107	−0.595317
$105$	0	0
$106$	2.41421	0.234489
$107$	−12.4142	−1.20013	−0.600064	−	0.799952i	$-0.704858\pi$
$107$	−12.4142	−1.20013	−0.600064	+	0.799952i	$0.704858\pi$
$108$	0	0
$109$	−10.8284	−1.03718	−0.518588	−	0.855024i	$-0.673542\pi$
$109$	−10.8284	−1.03718	−0.518588	+	0.855024i	$0.673542\pi$
$110$	−1.34315	−0.128064
$111$	0	0
$112$	−1.24264	−0.117419
$113$	−16.6569	−1.56695	−0.783473	−	0.621426i	$-0.786553\pi$
$113$	−16.6569	−1.56695	−0.783473	+	0.621426i	$0.786553\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	−12.4853	−1.15923
$117$	0	0
$118$	1.68629	0.155236
$119$	−2.41421	−0.221311
$120$	0	0
$121$	−0.485281	−0.0441165
$122$	1.17157	0.106069
$123$	0	0
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	8.89949	0.789702	0.394851	−	0.918745i	$-0.370796\pi$
$127$	8.89949	0.789702	0.394851	+	0.918745i	$0.370796\pi$
$128$	10.5563	0.933058
$129$	0	0
$130$	−1.58579	−0.139083
$131$	13.2426	1.15701	0.578507	−	0.815677i	$-0.303635\pi$
$131$	13.2426	1.15701	0.578507	+	0.815677i	$0.303635\pi$
$132$	0	0
$133$	−1.82843	−0.158545
$134$	−1.34315	−0.116030
$135$	0	0
$136$	9.24264	0.792550
$137$	−9.48528	−0.810382	−0.405191	−	0.914232i	$-0.632795\pi$
$137$	−9.48528	−0.810382	−0.405191	+	0.914232i	$0.632795\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	−0.757359	−0.0640085
$141$	0	0
$142$	0.0294373	0.00247032
$143$	12.4142	1.03813
$144$	0	0
$145$	−6.82843	−0.567070
$146$	0.757359	0.0626795
$147$	0	0
$148$	−1.82843	−0.150296
$149$	−1.00000	−0.0819232	−0.0409616	−	0.999161i	$-0.513042\pi$
$149$	−1.00000	−0.0819232	−0.0409616	+	0.999161i	$0.513042\pi$
$150$	0	0
$151$	5.31371	0.432423	0.216212	−	0.976346i	$-0.430630\pi$
$151$	5.31371	0.432423	0.216212	+	0.976346i	$0.430630\pi$
$152$	7.00000	0.567775
$153$	0	0
$154$	−0.556349	−0.0448319
$155$	0	0
$156$	0	0
$157$	9.17157	0.731971	0.365986	−	0.930621i	$-0.380732\pi$
$157$	9.17157	0.731971	0.365986	+	0.930621i	$0.380732\pi$
$158$	2.79899	0.222676
$159$	0	0
$160$	4.41421	0.348974
$161$	−1.65685	−0.130578
$162$	0	0
$163$	20.9706	1.64254	0.821271	−	0.570539i	$-0.193266\pi$
$163$	20.9706	1.64254	0.821271	+	0.570539i	$0.193266\pi$
$164$	−13.6863	−1.06872
$165$	0	0
$166$	−4.17157	−0.323777
$167$	−22.5563	−1.74546	−0.872731	−	0.488201i	$-0.837653\pi$
$167$	−22.5563	−1.74546	−0.872731	+	0.488201i	$0.837653\pi$
$168$	0	0
$169$	1.65685	0.127450
$170$	2.41421	0.185162
$171$	0	0
$172$	19.9289	1.51957
$173$	−8.31371	−0.632080	−0.316040	−	0.948746i	$-0.602353\pi$
$173$	−8.31371	−0.632080	−0.316040	+	0.948746i	$0.602353\pi$
$174$	0	0
$175$	1.65685	0.125246
$176$	−9.72792	−0.733270
$177$	0	0
$178$	1.85786	0.139253
$179$	15.2426	1.13929	0.569644	−	0.821891i	$-0.307081\pi$
$179$	15.2426	1.13929	0.569644	+	0.821891i	$0.307081\pi$
$180$	0	0
$181$	12.3137	0.915271	0.457635	−	0.889140i	$-0.348697\pi$
$181$	12.3137	0.915271	0.457635	+	0.889140i	$0.348697\pi$
$182$	−0.656854	−0.0486893
$183$	0	0
$184$	6.34315	0.467623
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	−18.8995	−1.38207
$188$	17.6569	1.28776
$189$	0	0
$190$	1.82843	0.132648
$191$	20.8995	1.51223	0.756117	−	0.654436i	$-0.227094\pi$
$191$	20.8995	1.51223	0.756117	+	0.654436i	$0.227094\pi$
$192$	0	0
$193$	7.14214	0.514102	0.257051	−	0.966398i	$-0.417249\pi$
$193$	7.14214	0.514102	0.257051	+	0.966398i	$0.417249\pi$
$194$	−2.14214	−0.153796
$195$	0	0
$196$	12.4853	0.891806
$197$	13.4853	0.960787	0.480393	−	0.877053i	$-0.340494\pi$
$197$	13.4853	0.960787	0.480393	+	0.877053i	$0.340494\pi$
$198$	0	0
$199$	18.4142	1.30535	0.652674	−	0.757638i	$-0.273647\pi$
$199$	18.4142	1.30535	0.652674	+	0.757638i	$0.273647\pi$
$200$	−6.34315	−0.448528
$201$	0	0
$202$	−3.51472	−0.247295
$203$	−2.82843	−0.198517
$204$	0	0
$205$	−7.48528	−0.522795
$206$	−0.857864	−0.0597702
$207$	0	0
$208$	−11.4853	−0.796361
$209$	−14.3137	−0.990100
$210$	0	0
$211$	10.4142	0.716944	0.358472	−	0.933540i	$-0.383298\pi$
$211$	10.4142	0.716944	0.358472	+	0.933540i	$0.383298\pi$
$212$	10.6569	0.731916
$213$	0	0
$214$	5.14214	0.351509
$215$	10.8995	0.743339
$216$	0	0
$217$	0	0
$218$	4.48528	0.303782
$219$	0	0
$220$	−5.92893	−0.399729
$221$	−22.3137	−1.50098
$222$	0	0
$223$	−23.7279	−1.58894	−0.794470	−	0.607304i	$-0.792251\pi$
$223$	−23.7279	−1.58894	−0.794470	+	0.607304i	$0.792251\pi$
$224$	1.82843	0.122167
$225$	0	0
$226$	6.89949	0.458948
$227$	18.4142	1.22219	0.611097	−	0.791556i	$-0.290728\pi$
$227$	18.4142	1.22219	0.611097	+	0.791556i	$0.290728\pi$
$228$	0	0
$229$	−5.48528	−0.362478	−0.181239	−	0.983439i	$-0.558011\pi$
$229$	−5.48528	−0.362478	−0.181239	+	0.983439i	$0.558011\pi$
$230$	1.65685	0.109250
$231$	0	0
$232$	10.8284	0.710921
$233$	9.17157	0.600850	0.300425	−	0.953805i	$-0.402872\pi$
$233$	9.17157	0.600850	0.300425	+	0.953805i	$0.402872\pi$
$234$	0	0
$235$	9.65685	0.629944
$236$	7.44365	0.484540
$237$	0	0
$238$	1.00000	0.0648204
$239$	21.2426	1.37407	0.687036	−	0.726623i	$-0.258911\pi$
$239$	21.2426	1.37407	0.687036	+	0.726623i	$0.258911\pi$
$240$	0	0
$241$	13.3431	0.859508	0.429754	−	0.902946i	$-0.358600\pi$
$241$	13.3431	0.859508	0.429754	+	0.902946i	$0.358600\pi$
$242$	0.201010	0.0129214
$243$	0	0
$244$	5.17157	0.331076
$245$	6.82843	0.436252
$246$	0	0
$247$	−16.8995	−1.07529
$248$	0	0
$249$	0	0
$250$	−3.72792	−0.235774
$251$	−6.41421	−0.404862	−0.202431	−	0.979297i	$-0.564884\pi$
$251$	−6.41421	−0.404862	−0.202431	+	0.979297i	$0.564884\pi$
$252$	0	0
$253$	−12.9706	−0.815452
$254$	−3.68629	−0.231299
$255$	0	0
$256$	3.97056	0.248160
$257$	−22.3137	−1.39189	−0.695945	−	0.718095i	$-0.745014\pi$
$257$	−22.3137	−1.39189	−0.695945	+	0.718095i	$0.745014\pi$
$258$	0	0
$259$	−0.414214	−0.0257380
$260$	−7.00000	−0.434122
$261$	0	0
$262$	−5.48528	−0.338882
$263$	23.3137	1.43758	0.718792	−	0.695225i	$-0.244695\pi$
$263$	23.3137	1.43758	0.718792	+	0.695225i	$0.244695\pi$
$264$	0	0
$265$	5.82843	0.358037
$266$	0.757359	0.0464367
$267$	0	0
$268$	−5.92893	−0.362167
$269$	26.1716	1.59571	0.797854	−	0.602850i	$-0.205968\pi$
$269$	26.1716	1.59571	0.797854	+	0.602850i	$0.205968\pi$
$270$	0	0
$271$	−0.686292	−0.0416892	−0.0208446	−	0.999783i	$-0.506636\pi$
$271$	−0.686292	−0.0416892	−0.0208446	+	0.999783i	$0.506636\pi$
$272$	17.4853	1.06020
$273$	0	0
$274$	3.92893	0.237355
$275$	12.9706	0.782154
$276$	0	0
$277$	14.1421	0.849719	0.424859	−	0.905259i	$-0.360324\pi$
$277$	14.1421	0.849719	0.424859	+	0.905259i	$0.360324\pi$
$278$	0	0
$279$	0	0
$280$	0.656854	0.0392545
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	−13.6569	−0.811816	−0.405908	−	0.913914i	$-0.633045\pi$
$283$	−13.6569	−0.811816	−0.405908	+	0.913914i	$0.633045\pi$
$284$	0.129942	0.00771066
$285$	0	0
$286$	−5.14214	−0.304061
$287$	−3.10051	−0.183017
$288$	0	0
$289$	16.9706	0.998268
$290$	2.82843	0.166091
$291$	0	0
$292$	3.34315	0.195643
$293$	−14.7990	−0.864566	−0.432283	−	0.901738i	$-0.642292\pi$
$293$	−14.7990	−0.864566	−0.432283	+	0.901738i	$0.642292\pi$
$294$	0	0
$295$	4.07107	0.237027
$296$	1.58579	0.0921720
$297$	0	0
$298$	0.414214	0.0239947
$299$	−15.3137	−0.885615
$300$	0	0
$301$	4.51472	0.260224
$302$	−2.20101	−0.126654
$303$	0	0
$304$	13.2426	0.759518
$305$	2.82843	0.161955
$306$	0	0
$307$	−11.2426	−0.641651	−0.320826	−	0.947138i	$-0.603960\pi$
$307$	−11.2426	−0.641651	−0.320826	+	0.947138i	$0.603960\pi$
$308$	−2.45584	−0.139935
$309$	0	0
$310$	0	0
$311$	−11.3137	−0.641542	−0.320771	−	0.947157i	$-0.603942\pi$
$311$	−11.3137	−0.641542	−0.320771	+	0.947157i	$0.603942\pi$
$312$	0	0
$313$	−1.82843	−0.103349	−0.0516744	−	0.998664i	$-0.516456\pi$
$313$	−1.82843	−0.103349	−0.0516744	+	0.998664i	$0.516456\pi$
$314$	−3.79899	−0.214389
$315$	0	0
$316$	12.3553	0.695042
$317$	7.82843	0.439688	0.219844	−	0.975535i	$-0.429445\pi$
$317$	7.82843	0.439688	0.219844	+	0.975535i	$0.429445\pi$
$318$	0	0
$319$	−22.1421	−1.23972
$320$	4.17157	0.233198
$321$	0	0
$322$	0.686292	0.0382455
$323$	25.7279	1.43154
$324$	0	0
$325$	15.3137	0.849452
$326$	−8.68629	−0.481089
$327$	0	0
$328$	11.8701	0.655414
$329$	4.00000	0.220527
$330$	0	0
$331$	9.24264	0.508021	0.254011	−	0.967201i	$-0.418250\pi$
$331$	9.24264	0.508021	0.254011	+	0.967201i	$0.418250\pi$
$332$	−18.4142	−1.01061
$333$	0	0
$334$	9.34315	0.511234
$335$	−3.24264	−0.177164
$336$	0	0
$337$	9.31371	0.507350	0.253675	−	0.967290i	$-0.418361\pi$
$337$	9.31371	0.507350	0.253675	+	0.967290i	$0.418361\pi$
$338$	−0.686292	−0.0373293
$339$	0	0
$340$	10.6569	0.577949
$341$	0	0
$342$	0	0
$343$	5.72792	0.309279
$344$	−17.2843	−0.931905
$345$	0	0
$346$	3.44365	0.185132
$347$	8.55635	0.459329	0.229664	−	0.973270i	$-0.426237\pi$
$347$	8.55635	0.459329	0.229664	+	0.973270i	$0.426237\pi$
$348$	0	0
$349$	−27.1127	−1.45131	−0.725655	−	0.688059i	$-0.758463\pi$
$349$	−27.1127	−1.45131	−0.725655	+	0.688059i	$0.758463\pi$
$350$	−0.686292	−0.0366838
$351$	0	0
$352$	14.3137	0.762923
$353$	−3.00000	−0.159674	−0.0798369	−	0.996808i	$-0.525440\pi$
$353$	−3.00000	−0.159674	−0.0798369	+	0.996808i	$0.525440\pi$
$354$	0	0
$355$	0.0710678	0.00377189
$356$	8.20101	0.434653
$357$	0	0
$358$	−6.31371	−0.333690
$359$	−7.10051	−0.374750	−0.187375	−	0.982288i	$-0.559998\pi$
$359$	−7.10051	−0.374750	−0.187375	+	0.982288i	$0.559998\pi$
$360$	0	0
$361$	0.485281	0.0255411
$362$	−5.10051	−0.268077
$363$	0	0
$364$	−2.89949	−0.151975
$365$	1.82843	0.0957042
$366$	0	0
$367$	−24.2132	−1.26392	−0.631959	−	0.775001i	$-0.717749\pi$
$367$	−24.2132	−1.26392	−0.631959	+	0.775001i	$0.717749\pi$
$368$	12.0000	0.625543
$369$	0	0
$370$	0.414214	0.0215339
$371$	2.41421	0.125340
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	7.82843	0.404798
$375$	0	0
$376$	−15.3137	−0.789744
$377$	−26.1421	−1.34639
$378$	0	0
$379$	7.38478	0.379330	0.189665	−	0.981849i	$-0.439260\pi$
$379$	7.38478	0.379330	0.189665	+	0.981849i	$0.439260\pi$
$380$	8.07107	0.414037
$381$	0	0
$382$	−8.65685	−0.442923
$383$	−5.10051	−0.260624	−0.130312	−	0.991473i	$-0.541598\pi$
$383$	−5.10051	−0.260624	−0.130312	+	0.991473i	$0.541598\pi$
$384$	0	0
$385$	−1.34315	−0.0684530
$386$	−2.95837	−0.150577
$387$	0	0
$388$	−9.45584	−0.480048
$389$	11.1421	0.564929	0.282464	−	0.959278i	$-0.408848\pi$
$389$	11.1421	0.564929	0.282464	+	0.959278i	$0.408848\pi$
$390$	0	0
$391$	23.3137	1.17902
$392$	−10.8284	−0.546918
$393$	0	0
$394$	−5.58579	−0.281408
$395$	6.75736	0.340000
$396$	0	0
$397$	−33.4853	−1.68058	−0.840289	−	0.542139i	$-0.817615\pi$
$397$	−33.4853	−1.68058	−0.840289	+	0.542139i	$0.817615\pi$
$398$	−7.62742	−0.382328
$399$	0	0
$400$	−12.0000	−0.600000
$401$	26.8284	1.33975	0.669874	−	0.742475i	$-0.266348\pi$
$401$	26.8284	1.33975	0.669874	+	0.742475i	$0.266348\pi$
$402$	0	0
$403$	0	0
$404$	−15.5147	−0.771886
$405$	0	0
$406$	1.17157	0.0581442
$407$	−3.24264	−0.160732
$408$	0	0
$409$	20.6569	1.02142	0.510708	−	0.859754i	$-0.329383\pi$
$409$	20.6569	1.02142	0.510708	+	0.859754i	$0.329383\pi$
$410$	3.10051	0.153123
$411$	0	0
$412$	−3.78680	−0.186562
$413$	1.68629	0.0829770
$414$	0	0
$415$	−10.0711	−0.494369
$416$	16.8995	0.828566
$417$	0	0
$418$	5.92893	0.289994
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	−31.1421	−1.51777	−0.758887	−	0.651222i	$-0.774257\pi$
$421$	−31.1421	−1.51777	−0.758887	+	0.651222i	$0.774257\pi$
$422$	−4.31371	−0.209988
$423$	0	0
$424$	−9.24264	−0.448862
$425$	−23.3137	−1.13088
$426$	0	0
$427$	1.17157	0.0566964
$428$	22.6985	1.09717
$429$	0	0
$430$	−4.51472	−0.217719
$431$	16.7574	0.807174	0.403587	−	0.914941i	$-0.367763\pi$
$431$	16.7574	0.807174	0.403587	+	0.914941i	$0.367763\pi$
$432$	0	0
$433$	27.1127	1.30295	0.651477	−	0.758669i	$-0.274150\pi$
$433$	27.1127	1.30295	0.651477	+	0.758669i	$0.274150\pi$
$434$	0	0
$435$	0	0
$436$	19.7990	0.948200
$437$	17.6569	0.844642
$438$	0	0
$439$	2.07107	0.0988467	0.0494233	−	0.998778i	$-0.484262\pi$
$439$	2.07107	0.0988467	0.0494233	+	0.998778i	$0.484262\pi$
$440$	5.14214	0.245142
$441$	0	0
$442$	9.24264	0.439628
$443$	4.75736	0.226029	0.113014	−	0.993593i	$-0.463949\pi$
$443$	4.75736	0.226029	0.113014	+	0.993593i	$0.463949\pi$
$444$	0	0
$445$	4.48528	0.212623
$446$	9.82843	0.465390
$447$	0	0
$448$	1.72792	0.0816366
$449$	40.6274	1.91733	0.958663	−	0.284543i	$-0.0918420\pi$
$449$	40.6274	1.91733	0.958663	+	0.284543i	$0.0918420\pi$
$450$	0	0
$451$	−24.2721	−1.14293
$452$	30.4558	1.43252
$453$	0	0
$454$	−7.62742	−0.357972
$455$	−1.58579	−0.0743428
$456$	0	0
$457$	−31.1127	−1.45539	−0.727695	−	0.685901i	$-0.759409\pi$
$457$	−31.1127	−1.45539	−0.727695	+	0.685901i	$0.759409\pi$
$458$	2.27208	0.106167
$459$	0	0
$460$	7.31371	0.341003
$461$	2.14214	0.0997692	0.0498846	−	0.998755i	$-0.484115\pi$
$461$	2.14214	0.0997692	0.0498846	+	0.998755i	$0.484115\pi$
$462$	0	0
$463$	−8.97056	−0.416897	−0.208449	−	0.978033i	$-0.566841\pi$
$463$	−8.97056	−0.416897	−0.208449	+	0.978033i	$0.566841\pi$
$464$	20.4853	0.951005
$465$	0	0
$466$	−3.79899	−0.175985
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	−1.34315	−0.0620207
$470$	−4.00000	−0.184506
$471$	0	0
$472$	−6.45584	−0.297154
$473$	35.3431	1.62508
$474$	0	0
$475$	−17.6569	−0.810152
$476$	4.41421	0.202325
$477$	0	0
$478$	−8.79899	−0.402456
$479$	15.7279	0.718627	0.359314	−	0.933217i	$-0.383011\pi$
$479$	15.7279	0.718627	0.359314	+	0.933217i	$0.383011\pi$
$480$	0	0
$481$	−3.82843	−0.174561
$482$	−5.52691	−0.251744
$483$	0	0
$484$	0.887302	0.0403319
$485$	−5.17157	−0.234829
$486$	0	0
$487$	19.3848	0.878408	0.439204	−	0.898387i	$-0.355261\pi$
$487$	19.3848	0.878408	0.439204	+	0.898387i	$0.355261\pi$
$488$	−4.48528	−0.203039
$489$	0	0
$490$	−2.82843	−0.127775
$491$	−1.58579	−0.0715655	−0.0357828	−	0.999360i	$-0.511392\pi$
$491$	−1.58579	−0.0715655	−0.0357828	+	0.999360i	$0.511392\pi$
$492$	0	0
$493$	39.7990	1.79246
$494$	7.00000	0.314945
$495$	0	0
$496$	0	0
$497$	0.0294373	0.00132044
$498$	0	0
$499$	−2.21320	−0.0990766	−0.0495383	−	0.998772i	$-0.515775\pi$
$499$	−2.21320	−0.0990766	−0.0495383	+	0.998772i	$0.515775\pi$
$500$	−16.4558	−0.735928
$501$	0	0
$502$	2.65685	0.118581
$503$	13.3848	0.596798	0.298399	−	0.954441i	$-0.403547\pi$
$503$	13.3848	0.596798	0.298399	+	0.954441i	$0.403547\pi$
$504$	0	0
$505$	−8.48528	−0.377590
$506$	5.37258	0.238840
$507$	0	0
$508$	−16.2721	−0.721957
$509$	32.7990	1.45379	0.726895	−	0.686749i	$-0.240963\pi$
$509$	32.7990	1.45379	0.726895	+	0.686749i	$0.240963\pi$
$510$	0	0
$511$	0.757359	0.0335036
$512$	−22.7574	−1.00574
$513$	0	0
$514$	9.24264	0.407675
$515$	−2.07107	−0.0912622
$516$	0	0
$517$	31.3137	1.37718
$518$	0.171573	0.00753848
$519$	0	0
$520$	6.07107	0.266234
$521$	20.4558	0.896187	0.448093	−	0.893987i	$-0.352103\pi$
$521$	20.4558	0.896187	0.448093	+	0.893987i	$0.352103\pi$
$522$	0	0
$523$	−8.00000	−0.349816	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000	−0.349816	−0.174908	+	0.984585i	$0.555963\pi$
$524$	−24.2132	−1.05776
$525$	0	0
$526$	−9.65685	−0.421059
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−2.41421	−0.104867
$531$	0	0
$532$	3.34315	0.144944
$533$	−28.6569	−1.24127
$534$	0	0
$535$	12.4142	0.536713
$536$	5.14214	0.222106
$537$	0	0
$538$	−10.8406	−0.467372
$539$	22.1421	0.953729
$540$	0	0
$541$	−31.2843	−1.34502	−0.672508	−	0.740090i	$-0.734783\pi$
$541$	−31.2843	−1.34502	−0.672508	+	0.740090i	$0.734783\pi$
$542$	0.284271	0.0122105
$543$	0	0
$544$	−25.7279	−1.10308
$545$	10.8284	0.463839
$546$	0	0
$547$	−19.7279	−0.843505	−0.421753	−	0.906711i	$-0.638585\pi$
$547$	−19.7279	−0.843505	−0.421753	+	0.906711i	$0.638585\pi$
$548$	17.3431	0.740862
$549$	0	0
$550$	−5.37258	−0.229088
$551$	30.1421	1.28410
$552$	0	0
$553$	2.79899	0.119025
$554$	−5.85786	−0.248877
$555$	0	0
$556$	0	0
$557$	−27.5147	−1.16584	−0.582918	−	0.812531i	$-0.698089\pi$
$557$	−27.5147	−1.16584	−0.582918	+	0.812531i	$0.698089\pi$
$558$	0	0
$559$	41.7279	1.76490
$560$	1.24264	0.0525112
$561$	0	0
$562$	0.828427	0.0349451
$563$	−13.2426	−0.558111	−0.279055	−	0.960275i	$-0.590021\pi$
$563$	−13.2426	−0.558111	−0.279055	+	0.960275i	$0.590021\pi$
$564$	0	0
$565$	16.6569	0.700759
$566$	5.65685	0.237775
$567$	0	0
$568$	−0.112698	−0.00472872
$569$	13.1421	0.550947	0.275473	−	0.961309i	$-0.411165\pi$
$569$	13.1421	0.550947	0.275473	+	0.961309i	$0.411165\pi$
$570$	0	0
$571$	−21.1005	−0.883029	−0.441514	−	0.897254i	$-0.645559\pi$
$571$	−21.1005	−0.883029	−0.441514	+	0.897254i	$0.645559\pi$
$572$	−22.6985	−0.949071
$573$	0	0
$574$	1.28427	0.0536044
$575$	−16.0000	−0.667246
$576$	0	0
$577$	−0.0294373	−0.00122549	−0.000612744	−	1.00000i	$-0.500195\pi$
$577$	−0.0294373	−0.00122549	−0.000612744		1.00000i	$0.500195\pi$
$578$	−7.02944	−0.292386
$579$	0	0
$580$	12.4853	0.518423
$581$	−4.17157	−0.173066
$582$	0	0
$583$	18.8995	0.782737
$584$	−2.89949	−0.119982
$585$	0	0
$586$	6.12994	0.253226
$587$	31.6569	1.30662	0.653309	−	0.757091i	$-0.273380\pi$
$587$	31.6569	1.30662	0.653309	+	0.757091i	$0.273380\pi$
$588$	0	0
$589$	0	0
$590$	−1.68629	−0.0694235
$591$	0	0
$592$	3.00000	0.123299
$593$	1.31371	0.0539475	0.0269738	−	0.999636i	$-0.491413\pi$
$593$	1.31371	0.0539475	0.0269738	+	0.999636i	$0.491413\pi$
$594$	0	0
$595$	2.41421	0.0989731
$596$	1.82843	0.0748953
$597$	0	0
$598$	6.34315	0.259391
$599$	−15.1005	−0.616990	−0.308495	−	0.951226i	$-0.599825\pi$
$599$	−15.1005	−0.616990	−0.308495	+	0.951226i	$0.599825\pi$
$600$	0	0
$601$	−6.51472	−0.265741	−0.132870	−	0.991133i	$-0.542419\pi$
$601$	−6.51472	−0.265741	−0.132870	+	0.991133i	$0.542419\pi$
$602$	−1.87006	−0.0762179
$603$	0	0
$604$	−9.71573	−0.395327
$605$	0.485281	0.0197295
$606$	0	0
$607$	−1.58579	−0.0643651	−0.0321825	−	0.999482i	$-0.510246\pi$
$607$	−1.58579	−0.0643651	−0.0321825	+	0.999482i	$0.510246\pi$
$608$	−19.4853	−0.790233
$609$	0	0
$610$	−1.17157	−0.0474356
$611$	36.9706	1.49567
$612$	0	0
$613$	12.3137	0.497346	0.248673	−	0.968587i	$-0.420006\pi$
$613$	12.3137	0.497346	0.248673	+	0.968587i	$0.420006\pi$
$614$	4.65685	0.187935
$615$	0	0
$616$	2.12994	0.0858178
$617$	33.2843	1.33997	0.669987	−	0.742373i	$-0.266300\pi$
$617$	33.2843	1.33997	0.669987	+	0.742373i	$0.266300\pi$
$618$	0	0
$619$	−20.3431	−0.817660	−0.408830	−	0.912611i	$-0.634063\pi$
$619$	−20.3431	−0.817660	−0.408830	+	0.912611i	$0.634063\pi$
$620$	0	0
$621$	0	0
$622$	4.68629	0.187903
$623$	1.85786	0.0744338
$624$	0	0
$625$	11.0000	0.440000
$626$	0.757359	0.0302702
$627$	0	0
$628$	−16.7696	−0.669178
$629$	5.82843	0.232395
$630$	0	0
$631$	50.0122	1.99095	0.995477	−	0.0950030i	$-0.0302861\pi$
$631$	50.0122	1.99095	0.995477	+	0.0950030i	$0.0302861\pi$
$632$	−10.7157	−0.426249
$633$	0	0
$634$	−3.24264	−0.128782
$635$	−8.89949	−0.353166
$636$	0	0
$637$	26.1421	1.03579
$638$	9.17157	0.363106
$639$	0	0
$640$	−10.5563	−0.417276
$641$	13.9706	0.551804	0.275902	−	0.961186i	$-0.411023\pi$
$641$	13.9706	0.551804	0.275902	+	0.961186i	$0.411023\pi$
$642$	0	0
$643$	35.3137	1.39264	0.696318	−	0.717733i	$-0.254820\pi$
$643$	35.3137	1.39264	0.696318	+	0.717733i	$0.254820\pi$
$644$	3.02944	0.119377
$645$	0	0
$646$	−10.6569	−0.419288
$647$	45.3137	1.78147	0.890733	−	0.454527i	$-0.150192\pi$
$647$	45.3137	1.78147	0.890733	+	0.454527i	$0.150192\pi$
$648$	0	0
$649$	13.2010	0.518185
$650$	−6.34315	−0.248799
$651$	0	0
$652$	−38.3431	−1.50163
$653$	6.14214	0.240360	0.120180	−	0.992752i	$-0.461653\pi$
$653$	6.14214	0.240360	0.120180	+	0.992752i	$0.461653\pi$
$654$	0	0
$655$	−13.2426	−0.517433
$656$	22.4558	0.876753
$657$	0	0
$658$	−1.65685	−0.0645909
$659$	1.65685	0.0645419	0.0322709	−	0.999479i	$-0.489726\pi$
$659$	1.65685	0.0645419	0.0322709	+	0.999479i	$0.489726\pi$
$660$	0	0
$661$	4.85786	0.188949	0.0944745	−	0.995527i	$-0.469883\pi$
$661$	4.85786	0.188949	0.0944745	+	0.995527i	$0.469883\pi$
$662$	−3.82843	−0.148796
$663$	0	0
$664$	15.9706	0.619778
$665$	1.82843	0.0709034
$666$	0	0
$667$	27.3137	1.05759
$668$	41.2426	1.59573
$669$	0	0
$670$	1.34315	0.0518902
$671$	9.17157	0.354065
$672$	0	0
$673$	9.34315	0.360152	0.180076	−	0.983653i	$-0.442366\pi$
$673$	9.34315	0.360152	0.180076	+	0.983653i	$0.442366\pi$
$674$	−3.85786	−0.148599
$675$	0	0
$676$	−3.02944	−0.116517
$677$	38.5980	1.48344	0.741720	−	0.670709i	$-0.234010\pi$
$677$	38.5980	1.48344	0.741720	+	0.670709i	$0.234010\pi$
$678$	0	0
$679$	−2.14214	−0.0822076
$680$	−9.24264	−0.354439
$681$	0	0
$682$	0	0
$683$	1.37258	0.0525204	0.0262602	−	0.999655i	$-0.491640\pi$
$683$	1.37258	0.0525204	0.0262602	+	0.999655i	$0.491640\pi$
$684$	0	0
$685$	9.48528	0.362414
$686$	−2.37258	−0.0905856
$687$	0	0
$688$	−32.6985	−1.24662
$689$	22.3137	0.850085
$690$	0	0
$691$	−0.0710678	−0.00270355	−0.00135177	−	0.999999i	$-0.500430\pi$
$691$	−0.0710678	−0.00270355	−0.00135177	+	0.999999i	$0.500430\pi$
$692$	15.2010	0.577856
$693$	0	0
$694$	−3.54416	−0.134534
$695$	0	0
$696$	0	0
$697$	43.6274	1.65251
$698$	11.2304	0.425079
$699$	0	0
$700$	−3.02944	−0.114502
$701$	13.4853	0.509332	0.254666	−	0.967029i	$-0.418034\pi$
$701$	13.4853	0.509332	0.254666	+	0.967029i	$0.418034\pi$
$702$	0	0
$703$	4.41421	0.166485
$704$	13.5269	0.509815
$705$	0	0
$706$	1.24264	0.0467674
$707$	−3.51472	−0.132185
$708$	0	0
$709$	17.3137	0.650230	0.325115	−	0.945674i	$-0.394597\pi$
$709$	17.3137	0.650230	0.325115	+	0.945674i	$0.394597\pi$
$710$	−0.0294373	−0.00110476
$711$	0	0
$712$	−7.11270	−0.266560
$713$	0	0
$714$	0	0
$715$	−12.4142	−0.464265
$716$	−27.8701	−1.04155
$717$	0	0
$718$	2.94113	0.109762
$719$	8.07107	0.301000	0.150500	−	0.988610i	$-0.451912\pi$
$719$	8.07107	0.301000	0.150500	+	0.988610i	$0.451912\pi$
$720$	0	0
$721$	−0.857864	−0.0319485
$722$	−0.201010	−0.00748082
$723$	0	0
$724$	−22.5147	−0.836753
$725$	−27.3137	−1.01441
$726$	0	0
$727$	40.8406	1.51469	0.757347	−	0.653012i	$-0.226495\pi$
$727$	40.8406	1.51469	0.757347	+	0.653012i	$0.226495\pi$
$728$	2.51472	0.0932017
$729$	0	0
$730$	−0.757359	−0.0280311
$731$	−63.5269	−2.34963
$732$	0	0
$733$	15.6274	0.577212	0.288606	−	0.957448i	$-0.406808\pi$
$733$	15.6274	0.577212	0.288606	+	0.957448i	$0.406808\pi$
$734$	10.0294	0.370193
$735$	0	0
$736$	−17.6569	−0.650840
$737$	−10.5147	−0.387315
$738$	0	0
$739$	45.8701	1.68736	0.843679	−	0.536848i	$-0.180385\pi$
$739$	45.8701	1.68736	0.843679	+	0.536848i	$0.180385\pi$
$740$	1.82843	0.0672143
$741$	0	0
$742$	−1.00000	−0.0367112
$743$	−5.65685	−0.207530	−0.103765	−	0.994602i	$-0.533089\pi$
$743$	−5.65685	−0.207530	−0.103765	+	0.994602i	$0.533089\pi$
$744$	0	0
$745$	1.00000	0.0366372
$746$	−4.14214	−0.151654
$747$	0	0
$748$	34.5563	1.26351
$749$	5.14214	0.187890
$750$	0	0
$751$	7.24264	0.264288	0.132144	−	0.991231i	$-0.457814\pi$
$751$	7.24264	0.264288	0.132144	+	0.991231i	$0.457814\pi$
$752$	−28.9706	−1.05645
$753$	0	0
$754$	10.8284	0.394348
$755$	−5.31371	−0.193386
$756$	0	0
$757$	23.3431	0.848421	0.424211	−	0.905564i	$-0.360552\pi$
$757$	23.3431	0.848421	0.424211	+	0.905564i	$0.360552\pi$
$758$	−3.05887	−0.111103
$759$	0	0
$760$	−7.00000	−0.253917
$761$	−30.4558	−1.10402	−0.552012	−	0.833836i	$-0.686140\pi$
$761$	−30.4558	−1.10402	−0.552012	+	0.833836i	$0.686140\pi$
$762$	0	0
$763$	4.48528	0.162378
$764$	−38.2132	−1.38251
$765$	0	0
$766$	2.11270	0.0763349
$767$	15.5858	0.562770
$768$	0	0
$769$	36.1127	1.30226	0.651129	−	0.758967i	$-0.274296\pi$
$769$	36.1127	1.30226	0.651129	+	0.758967i	$0.274296\pi$
$770$	0.556349	0.0200494
$771$	0	0
$772$	−13.0589	−0.469999
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	0	0
$775$	0	0
$776$	8.20101	0.294399
$777$	0	0
$778$	−4.61522	−0.165464
$779$	33.0416	1.18384
$780$	0	0
$781$	0.230447	0.00824606
$782$	−9.65685	−0.345328
$783$	0	0
$784$	−20.4853	−0.731617
$785$	−9.17157	−0.327347
$786$	0	0
$787$	42.4142	1.51190	0.755952	−	0.654627i	$-0.227174\pi$
$787$	42.4142	1.51190	0.755952	+	0.654627i	$0.227174\pi$
$788$	−24.6569	−0.878364
$789$	0	0
$790$	−2.79899	−0.0995836
$791$	6.89949	0.245318
$792$	0	0
$793$	10.8284	0.384529
$794$	13.8701	0.492230
$795$	0	0
$796$	−33.6690	−1.19337
$797$	−28.4558	−1.00796	−0.503979	−	0.863716i	$-0.668131\pi$
$797$	−28.4558	−1.00796	−0.503979	+	0.863716i	$0.668131\pi$
$798$	0	0
$799$	−56.2843	−1.99119
$800$	17.6569	0.624264
$801$	0	0
$802$	−11.1127	−0.392403
$803$	5.92893	0.209227
$804$	0	0
$805$	1.65685	0.0583964
$806$	0	0
$807$	0	0
$808$	13.4558	0.473375
$809$	12.0294	0.422932	0.211466	−	0.977385i	$-0.432176\pi$
$809$	12.0294	0.422932	0.211466	+	0.977385i	$0.432176\pi$
$810$	0	0
$811$	−13.7279	−0.482053	−0.241026	−	0.970519i	$-0.577484\pi$
$811$	−13.7279	−0.482053	−0.241026	+	0.970519i	$0.577484\pi$
$812$	5.17157	0.181487
$813$	0	0
$814$	1.34315	0.0470772
$815$	−20.9706	−0.734567
$816$	0	0
$817$	−48.1127	−1.68325
$818$	−8.55635	−0.299166
$819$	0	0
$820$	13.6863	0.477946
$821$	−8.48528	−0.296138	−0.148069	−	0.988977i	$-0.547306\pi$
$821$	−8.48528	−0.296138	−0.148069	+	0.988977i	$0.547306\pi$
$822$	0	0
$823$	36.2132	1.26231	0.631156	−	0.775656i	$-0.282581\pi$
$823$	36.2132	1.26231	0.631156	+	0.775656i	$0.282581\pi$
$824$	3.28427	0.114413
$825$	0	0
$826$	−0.698485	−0.0243034
$827$	−36.8995	−1.28312	−0.641561	−	0.767072i	$-0.721713\pi$
$827$	−36.8995	−1.28312	−0.641561	+	0.767072i	$0.721713\pi$
$828$	0	0
$829$	38.4264	1.33460	0.667302	−	0.744787i	$-0.267449\pi$
$829$	38.4264	1.33460	0.667302	+	0.744787i	$0.267449\pi$
$830$	4.17157	0.144797
$831$	0	0
$832$	15.9706	0.553680
$833$	−39.7990	−1.37895
$834$	0	0
$835$	22.5563	0.780595
$836$	26.1716	0.905163
$837$	0	0
$838$	11.5980	0.400646
$839$	14.6274	0.504995	0.252497	−	0.967598i	$-0.418748\pi$
$839$	14.6274	0.504995	0.252497	+	0.967598i	$0.418748\pi$
$840$	0	0
$841$	17.6274	0.607842
$842$	12.8995	0.444546
$843$	0	0
$844$	−19.0416	−0.655440
$845$	−1.65685	−0.0569975
$846$	0	0
$847$	0.201010	0.00690679
$848$	−17.4853	−0.600447
$849$	0	0
$850$	9.65685	0.331227
$851$	4.00000	0.137118
$852$	0	0
$853$	−15.5147	−0.531214	−0.265607	−	0.964081i	$-0.585572\pi$
$853$	−15.5147	−0.531214	−0.265607	+	0.964081i	$0.585572\pi$
$854$	−0.485281	−0.0166060
$855$	0	0
$856$	−19.6863	−0.672864
$857$	19.4853	0.665605	0.332802	−	0.942997i	$-0.392006\pi$
$857$	19.4853	0.665605	0.332802	+	0.942997i	$0.392006\pi$
$858$	0	0
$859$	−49.3848	−1.68499	−0.842493	−	0.538707i	$-0.818913\pi$
$859$	−49.3848	−1.68499	−0.842493	+	0.538707i	$0.818913\pi$
$860$	−19.9289	−0.679571
$861$	0	0
$862$	−6.94113	−0.236416
$863$	2.61522	0.0890232	0.0445116	−	0.999009i	$-0.485827\pi$
$863$	2.61522	0.0890232	0.0445116	+	0.999009i	$0.485827\pi$
$864$	0	0
$865$	8.31371	0.282675
$866$	−11.2304	−0.381626
$867$	0	0
$868$	0	0
$869$	21.9117	0.743303
$870$	0	0
$871$	−12.4142	−0.420640
$872$	−17.1716	−0.581503
$873$	0	0
$874$	−7.31371	−0.247390
$875$	−3.72792	−0.126027
$876$	0	0
$877$	53.8284	1.81766	0.908828	−	0.417170i	$-0.136978\pi$
$877$	53.8284	1.81766	0.908828	+	0.417170i	$0.136978\pi$
$878$	−0.857864	−0.0289515
$879$	0	0
$880$	9.72792	0.327928
$881$	−11.6863	−0.393721	−0.196861	−	0.980431i	$-0.563075\pi$
$881$	−11.6863	−0.393721	−0.196861	+	0.980431i	$0.563075\pi$
$882$	0	0
$883$	−30.2843	−1.01915	−0.509573	−	0.860427i	$-0.670197\pi$
$883$	−30.2843	−1.01915	−0.509573	+	0.860427i	$0.670197\pi$
$884$	40.7990	1.37222
$885$	0	0
$886$	−1.97056	−0.0662024
$887$	51.3259	1.72336	0.861678	−	0.507456i	$-0.169414\pi$
$887$	51.3259	1.72336	0.861678	+	0.507456i	$0.169414\pi$
$888$	0	0
$889$	−3.68629	−0.123634
$890$	−1.85786	−0.0622758
$891$	0	0
$892$	43.3848	1.45263
$893$	−42.6274	−1.42647
$894$	0	0
$895$	−15.2426	−0.509505
$896$	−4.37258	−0.146078
$897$	0	0
$898$	−16.8284	−0.561572
$899$	0	0
$900$	0	0
$901$	−33.9706	−1.13172
$902$	10.0538	0.334756
$903$	0	0
$904$	−26.4142	−0.878524
$905$	−12.3137	−0.409322
$906$	0	0
$907$	32.6863	1.08533	0.542665	−	0.839949i	$-0.317415\pi$
$907$	32.6863	1.08533	0.542665	+	0.839949i	$0.317415\pi$
$908$	−33.6690	−1.11735
$909$	0	0
$910$	0.656854	0.0217745
$911$	−0.958369	−0.0317522	−0.0158761	−	0.999874i	$-0.505054\pi$
$911$	−0.958369	−0.0317522	−0.0158761	+	0.999874i	$0.505054\pi$
$912$	0	0
$913$	−32.6569	−1.08078
$914$	12.8873	0.426274
$915$	0	0
$916$	10.0294	0.331382
$917$	−5.48528	−0.181140
$918$	0	0
$919$	−34.8995	−1.15123	−0.575614	−	0.817722i	$-0.695237\pi$
$919$	−34.8995	−1.15123	−0.575614	+	0.817722i	$0.695237\pi$
$920$	−6.34315	−0.209127
$921$	0	0
$922$	−0.887302	−0.0292217
$923$	0.272078	0.00895555
$924$	0	0
$925$	−4.00000	−0.131519
$926$	3.71573	0.122106
$927$	0	0
$928$	−30.1421	−0.989464
$929$	−7.51472	−0.246550	−0.123275	−	0.992373i	$-0.539340\pi$
$929$	−7.51472	−0.246550	−0.123275	+	0.992373i	$0.539340\pi$
$930$	0	0
$931$	−30.1421	−0.987869
$932$	−16.7696	−0.549305
$933$	0	0
$934$	−3.31371	−0.108428
$935$	18.8995	0.618080
$936$	0	0
$937$	15.6863	0.512449	0.256224	−	0.966617i	$-0.417521\pi$
$937$	15.6863	0.512449	0.256224	+	0.966617i	$0.417521\pi$
$938$	0.556349	0.0181654
$939$	0	0
$940$	−17.6569	−0.575903
$941$	35.0000	1.14097	0.570484	−	0.821309i	$-0.306756\pi$
$941$	35.0000	1.14097	0.570484	+	0.821309i	$0.306756\pi$
$942$	0	0
$943$	29.9411	0.975017
$944$	−12.2132	−0.397506
$945$	0	0
$946$	−14.6396	−0.475975
$947$	−19.1005	−0.620683	−0.310342	−	0.950625i	$-0.600443\pi$
$947$	−19.1005	−0.620683	−0.310342	+	0.950625i	$0.600443\pi$
$948$	0	0
$949$	7.00000	0.227230
$950$	7.31371	0.237288
$951$	0	0
$952$	−3.82843	−0.124080
$953$	−3.51472	−0.113853	−0.0569265	−	0.998378i	$-0.518130\pi$
$953$	−3.51472	−0.113853	−0.0569265	+	0.998378i	$0.518130\pi$
$954$	0	0
$955$	−20.8995	−0.676292
$956$	−38.8406	−1.25620
$957$	0	0
$958$	−6.51472	−0.210481
$959$	3.92893	0.126872
$960$	0	0
$961$	0	0
$962$	1.58579	0.0511278
$963$	0	0
$964$	−24.3970	−0.785773
$965$	−7.14214	−0.229913
$966$	0	0
$967$	−15.4437	−0.496634	−0.248317	−	0.968679i	$-0.579878\pi$
$967$	−15.4437	−0.496634	−0.248317	+	0.968679i	$0.579878\pi$
$968$	−0.769553	−0.0247344
$969$	0	0
$970$	2.14214	0.0687798
$971$	0.698485	0.0224154	0.0112077	−	0.999937i	$-0.496432\pi$
$971$	0.698485	0.0224154	0.0112077	+	0.999937i	$0.496432\pi$
$972$	0	0
$973$	0	0
$974$	−8.02944	−0.257280
$975$	0	0
$976$	−8.48528	−0.271607
$977$	0.485281	0.0155255	0.00776276	−	0.999970i	$-0.497529\pi$
$977$	0.485281	0.0155255	0.00776276	+	0.999970i	$0.497529\pi$
$978$	0	0
$979$	14.5442	0.464833
$980$	−12.4853	−0.398828
$981$	0	0
$982$	0.656854	0.0209611
$983$	38.8406	1.23882	0.619412	−	0.785066i	$-0.287371\pi$
$983$	38.8406	1.23882	0.619412	+	0.785066i	$0.287371\pi$
$984$	0	0
$985$	−13.4853	−0.429677
$986$	−16.4853	−0.524998
$987$	0	0
$988$	30.8995	0.983044
$989$	−43.5980	−1.38634
$990$	0	0
$991$	47.9411	1.52290	0.761450	−	0.648224i	$-0.224488\pi$
$991$	47.9411	1.52290	0.761450	+	0.648224i	$0.224488\pi$
$992$	0	0
$993$	0	0
$994$	−0.0121933	−0.000386748 0
$995$	−18.4142	−0.583770
$996$	0	0
$997$	32.5980	1.03239	0.516194	−	0.856472i	$-0.327348\pi$
$997$	32.5980	1.03239	0.516194	+	0.856472i	$0.327348\pi$
$998$	0.916739	0.0290189
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.l.1.1		2
3.2	odd	2		961.2.a.a.1.2		2
31.5	even	3		279.2.h.c.118.1		4
31.25	even	3		279.2.h.c.253.1		4
31.30	odd	2		8649.2.a.k.1.1		2
93.2	odd	10		961.2.d.l.531.2		8
93.5	odd	6		31.2.c.a.25.2	yes	4
93.8	odd	10		961.2.d.l.374.1		8
93.11	even	30		961.2.g.r.338.1		16
93.14	odd	30		961.2.g.o.816.1		16
93.17	even	30		961.2.g.r.816.1		16
93.20	odd	30		961.2.g.o.338.1		16
93.23	even	10		961.2.d.i.374.1		8
93.26	even	6		961.2.c.a.521.2		4
93.29	even	10		961.2.d.i.531.2		8
93.35	odd	10		961.2.d.l.388.1		8
93.38	odd	30		961.2.g.o.235.1		16
93.41	odd	30		961.2.g.o.844.2		16
93.44	even	30		961.2.g.r.448.2		16
93.47	odd	10		961.2.d.l.628.2		8
93.50	odd	30		961.2.g.o.547.2		16
93.53	even	30		961.2.g.r.732.1		16
93.56	odd	6		31.2.c.a.5.2	✓	4
93.59	odd	30		961.2.g.o.846.2		16
93.65	even	30		961.2.g.r.846.2		16
93.68	even	6		961.2.c.a.439.2		4
93.71	odd	30		961.2.g.o.732.1		16
93.74	even	30		961.2.g.r.547.2		16
93.77	even	10		961.2.d.i.628.2		8
93.80	odd	30		961.2.g.o.448.2		16
93.83	even	30		961.2.g.r.844.2		16
93.86	even	30		961.2.g.r.235.1		16
93.89	even	10		961.2.d.i.388.1		8
93.92	even	2		961.2.a.c.1.2		2
372.191	even	6		496.2.i.h.273.2		4
372.335	even	6		496.2.i.h.129.2		4
465.98	even	12		775.2.o.d.149.2		8
465.149	odd	6		775.2.e.e.501.1		4
465.242	even	12		775.2.o.d.749.3		8
465.284	odd	6		775.2.e.e.676.1		4
465.377	even	12		775.2.o.d.149.3		8
465.428	even	12		775.2.o.d.749.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.2.c.a.5.2	✓	4	93.56	odd	6
31.2.c.a.25.2	yes	4	93.5	odd	6
279.2.h.c.118.1		4	31.5	even	3
279.2.h.c.253.1		4	31.25	even	3
496.2.i.h.129.2		4	372.335	even	6
496.2.i.h.273.2		4	372.191	even	6
775.2.e.e.501.1		4	465.149	odd	6
775.2.e.e.676.1		4	465.284	odd	6
775.2.o.d.149.2		8	465.98	even	12
775.2.o.d.149.3		8	465.377	even	12
775.2.o.d.749.2		8	465.428	even	12
775.2.o.d.749.3		8	465.242	even	12
961.2.a.a.1.2		2	3.2	odd	2
961.2.a.c.1.2		2	93.92	even	2
961.2.c.a.439.2		4	93.68	even	6
961.2.c.a.521.2		4	93.26	even	6
961.2.d.i.374.1		8	93.23	even	10
961.2.d.i.388.1		8	93.89	even	10
961.2.d.i.531.2		8	93.29	even	10
961.2.d.i.628.2		8	93.77	even	10
961.2.d.l.374.1		8	93.8	odd	10
961.2.d.l.388.1		8	93.35	odd	10
961.2.d.l.531.2		8	93.2	odd	10
961.2.d.l.628.2		8	93.47	odd	10
961.2.g.o.235.1		16	93.38	odd	30
961.2.g.o.338.1		16	93.20	odd	30
961.2.g.o.448.2		16	93.80	odd	30
961.2.g.o.547.2		16	93.50	odd	30
961.2.g.o.732.1		16	93.71	odd	30
961.2.g.o.816.1		16	93.14	odd	30
961.2.g.o.844.2		16	93.41	odd	30
961.2.g.o.846.2		16	93.59	odd	30
961.2.g.r.235.1		16	93.86	even	30
961.2.g.r.338.1		16	93.11	even	30
961.2.g.r.448.2		16	93.44	even	30
961.2.g.r.547.2		16	93.74	even	30
961.2.g.r.732.1		16	93.53	even	30
961.2.g.r.816.1		16	93.17	even	30
961.2.g.r.844.2		16	93.83	even	30
961.2.g.r.846.2		16	93.65	even	30
8649.2.a.k.1.1		2	31.30	odd	2
8649.2.a.l.1.1		2	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:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q-0.414214 q^{2} -1.82843 q^{4} -1.00000 q^{5} -0.414214 q^{7} +1.58579 q^{8} +O(q^{10})\)	\(q-0.414214 q^{2} -1.82843 q^{4} -1.00000 q^{5} -0.414214 q^{7} +1.58579 q^{8} +0.414214 q^{10} -3.24264 q^{11} -3.82843 q^{13} +0.171573 q^{14} +3.00000 q^{16} +5.82843 q^{17} +4.41421 q^{19} +1.82843 q^{20} +1.34315 q^{22} +4.00000 q^{23} -4.00000 q^{25} +1.58579 q^{26} +0.757359 q^{28} +6.82843 q^{29} -4.41421 q^{32} -2.41421 q^{34} +0.414214 q^{35} +1.00000 q^{37} -1.82843 q^{38} -1.58579 q^{40} +7.48528 q^{41} -10.8995 q^{43} +5.92893 q^{44} -1.65685 q^{46} -9.65685 q^{47} -6.82843 q^{49} +1.65685 q^{50} +7.00000 q^{52} -5.82843 q^{53} +3.24264 q^{55} -0.656854 q^{56} -2.82843 q^{58} -4.07107 q^{59} -2.82843 q^{61} -4.17157 q^{64} +3.82843 q^{65} +3.24264 q^{67} -10.6569 q^{68} -0.171573 q^{70} -0.0710678 q^{71} -1.82843 q^{73} -0.414214 q^{74} -8.07107 q^{76} +1.34315 q^{77} -6.75736 q^{79} -3.00000 q^{80} -3.10051 q^{82} +10.0711 q^{83} -5.82843 q^{85} +4.51472 q^{86} -5.14214 q^{88} -4.48528 q^{89} +1.58579 q^{91} -7.31371 q^{92} +4.00000 q^{94} -4.41421 q^{95} +5.17157 q^{97} +2.82843 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 6 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 6 q^{8} - 2 q^{10} + 2 q^{11} - 2 q^{13} + 6 q^{14} + 6 q^{16} + 6 q^{17} + 6 q^{19} - 2 q^{20} + 14 q^{22} + 8 q^{23} - 8 q^{25} + 6 q^{26} + 10 q^{28} + 8 q^{29} - 6 q^{32} - 2 q^{34} - 2 q^{35} + 2 q^{37} + 2 q^{38} - 6 q^{40} - 2 q^{41} - 2 q^{43} + 26 q^{44} + 8 q^{46} - 8 q^{47} - 8 q^{49} - 8 q^{50} + 14 q^{52} - 6 q^{53} - 2 q^{55} + 10 q^{56} + 6 q^{59} - 14 q^{64} + 2 q^{65} - 2 q^{67} - 10 q^{68} - 6 q^{70} + 14 q^{71} + 2 q^{73} + 2 q^{74} - 2 q^{76} + 14 q^{77} - 22 q^{79} - 6 q^{80} - 26 q^{82} + 6 q^{83} - 6 q^{85} + 26 q^{86} + 18 q^{88} + 8 q^{89} + 6 q^{91} + 8 q^{92} + 8 q^{94} - 6 q^{95} + 16 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 6 * q^8 - 2 * q^10 + 2 * q^11 - 2 * q^13 + 6 * q^14 + 6 * q^16 + 6 * q^17 + 6 * q^19 - 2 * q^20 + 14 * q^22 + 8 * q^23 - 8 * q^25 + 6 * q^26 + 10 * q^28 + 8 * q^29 - 6 * q^32 - 2 * q^34 - 2 * q^35 + 2 * q^37 + 2 * q^38 - 6 * q^40 - 2 * q^41 - 2 * q^43 + 26 * q^44 + 8 * q^46 - 8 * q^47 - 8 * q^49 - 8 * q^50 + 14 * q^52 - 6 * q^53 - 2 * q^55 + 10 * q^56 + 6 * q^59 - 14 * q^64 + 2 * q^65 - 2 * q^67 - 10 * q^68 - 6 * q^70 + 14 * q^71 + 2 * q^73 + 2 * q^74 - 2 * q^76 + 14 * q^77 - 22 * q^79 - 6 * q^80 - 26 * q^82 + 6 * q^83 - 6 * q^85 + 26 * q^86 + 18 * q^88 + 8 * q^89 + 6 * q^91 + 8 * q^92 + 8 * q^94 - 6 * q^95 + 16 * q^97

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